FORMAL LOGIC
OR,
The Calculus of inference^
Neceflary and Probable.
BY
AUGUSTUS DE MORGAN
Of Trinity College Cambridge,
Fellow of the Cambridge Philofophical Society, Secretary of the Royal
Aftronomical Society, Profeflbr of Mathematics in
Univerfity College London.
ia nq arw
LONDON:
TAYLOR AND WALTON,
Bookiellers and Publifhers to Univerfity College,
28, Upper Gower Street.
M DCCC XLVII
PREFACE.
r I ^HE fyftem given in this work extends beyond that
-*- commonly received, in feveral directions. A brief
ftatement of what is now fubmitted for adoption into
the theory of inference will be the matter of this preface.
In the form of the proportion, the copula is made as
abftracl: as the terms : or is considered as obeying only
thofe conditions which are neceflary to inference.
Every name is treated in connection with its contrary
or contradictory name ; the diftinction between thefe words
not being made, and others fupplied in confequence.
Eight really feparable forms of predication are thus ob
tained, between any two names : the eight of the common
fyftem amounting only to fix, when, as throughout my
work, the two forms of a convertible proportion are
confidered as identical.
The complex proportion is introduced, confirming in
the coexiftence of two fimple ones. The theory of the
fyllogifm of complex propofitions is made to precede
that of the fimple or ordinary fyllogifm ; which laft is
deduced from it. I have only ufed the word complex,
/^ was already appropriated (fee page 85).
iv Preface.
By the introduction of contraries, the number of valid
fyllogiftic forms is increafed to thirty-two, connected to
gether by many rules of relation, but all fhewn to contain,
each with reference to its own difpofition of names and
contraries, only one form of inference.
The distinction of figure is avoided from the beginning
by introducing into every proportion an order of refer
ence to its terms.
A Simple notation, which includes the common one,
gives the means of reprefenting every fyllogifm by three
letters, each accented above or below. By inspection of
one of thefe fymbols it is feen immediately, i. What
fyllogifm is reprefented, 2. Whether it be valid or in
valid, 3. How it is at once to be written down, 4. What
axiom the inference contains, or what is the act of the
mind when it makes that inference (chapter XIV).
A fubordinate notation is ufed (page 60) in abbrevia
tion of the proposition at length.
Compound names are considered, both when the com-
poSItion is conjunctive, and when it is disjunctive. Diftinct
notation and rules of transformation are given, and the
compound fyllogifms are treated as reducible to ordinary
ones, by invention of compound names.
The theory of the numerical fyllogifm is investigated,
in which, upon the hypothesis of numerical quantity in
both terms of every proportion, a numerical inference
is made.
But, when the numerical relations of the feveral terms
are fully known, all that is unufual in the quantity of the
predicate is mown to be either fuperfluous, or elfe, as I
have called it, fpurious.
Preface. v
The old doctrine of modals is made to give place to
the numerical theory of probability. Many will object
to this theory as extralogical. But I cannot fee on what
definition, founded on real diftindtion, the exclufion of it
can be maintained. When I am told that logic confiders
the validity of the inference, independently of the truth
or falfehood of the matter, or fupplies the conditions
under which the hypothetical truth of the matter of the
premifes gives hypothetical truth to the matter of the
conclufion, I fee a real definition, which propounds for
consideration the forms and laws of inferential thought.
But when it is further added that the only hypothetical
truth mall be abfolute truth, certain knowledge, I begin
to fee arbitrary diftinction, wanting the reality of that
which preceded. Without pretending that logic can take
cognizance of the probability of any given matter, I
cannot underfland why the ftudy of the effect which
partial belief of the premifes produces with refpect to the
conclufion, fhould be feparated from that of the confe-
quences of fuppofing the former to be abfolutely true.
Not however to difpute upon names, I mean that I
fhould maintain, againft thofe who would exclude the
theory of probability from logic, that, call it by what
name they like, it fhould accompany logic as a fludy.
I have, of courfe, been obliged to exprefs, in my own
manner, my own convictions on points of mental philo-
fophy. But any one will fee that, in all which I have
propofed for adoption, it matters nothing whether my
views of the phenomena of thought, or others, be made
the bafis of the explanation. So far therefore, as I am
vi Preface.
confidered as propofing forms of fyllogifm, &c. to the
logician, and not giving inftruclion to the ftudent of the
fcience, the reader has nothing to do with my choice of
the terms in which mental operations are fpoken of.
In the appendix will be found fome remarks on the
perfonal controverfy between Sir W. Hamilton of Edin
burgh and myfelf, of which I fuppofe the celebrity of my
opponent, and the appearance of part of it in a journal
fo widely circulated as the Athenaum, has caufed many
ftudents of logic to hear or read fomething.
At the end of the contents of fome chapters in the
following table, are a few additions and corrections, to
which I requeft the reader s attention.
A. DE MORGAN
Univerfity College^ London,
O&ober 14, 1847.
TABLE OF CONTENTS.
%* The articles entered in Italic, are thofe, the contents of which belong
to the peculiar fyftem prefented in this work.
CHAPTER I. Flrft Notions (pages 125).
Firft notion of Logic, I ; Reduction of proportions to iimple
affirmation and negation, 2, 3 ; Diftinction between negation and
affirmation requiring a negative, 3 ; how two negatives make an
affirmative, 4 ; proportions, 4 ; their relations, contraries and contra
dictories, 5 ; Quantity of fubjedt and predicate, 6 ; Converfes, 7 ;
fundamental notion of inference, 8 ; Material reprefentation, 8, 9 ;
fyllogifm, 9 ; its elements, 9 ; fyllogifms of different kinds of conclu-
fion, 10, 11, 12 ; collection of refults, 12 ; rules of fyllogifm, 13, 14;
weakened conclulions and ftrengtbened premifes, 1 5 ; the figures, 1 6,
17, 1 8 ; collection ofeffentially different fyllogifms y 18, 19 ; examples,
19, 20; a fortiori fyllogifm, 20, 21, 22; hypothetical fyllogifm, 22,
23 ; demonftration, direct and indirect, 23, 24; converfion of a di
lemma, 25.
* # * This chapter may be omitted by thofe who have fome know
ledge of the ordinary definitions and phrafeology of logic. It is ftrictly
confined to the Ariftotelian forms and fyllogifms, and is the reprint of
a traft publifhed in 1839, under the title of Firft Notions of Logic
(preparatory to the ftudy of Geometry) 1 : the only alterations are ;
the change of phrafeology, as altering fome X is Y into fome Xs
are Ys, &c. ; the corre6lion of a faulty demonftration ; and a few
omiffions, particularly of fome infufficient remarks on the probability
of arguments.
CHAPTER 11. On Objettsjdeas.and Names (pages 26 46).
Definition of Logic, 26 ; our pofition with refpect to mind, 26,
27 ; Doubt on the uniformity of procefs in all minds, 27 ; exiftence of
things external to the perceiving mind, 28, 29 ; fubject and objecT:,
ideal and objective, 29, 30 ; idea the fole knowledge, 30 ; object,
why then introduced, 30 ; extent of its meaning, 30, 31 ; abftraction,
qualities, relations, 31, 32; innate ideas, 32 ; diftinction of neceffary,
viii Contents.
and not necefTary, 33, 34; names, 34; aflitmption of their correct
ufe a poftulate, 35 ; frequent vaguenefs of names, 35 ; the tendency
offcience to correct it, 35 ; definition, nominal and real, 36 ; the latter
purely objective, 36; reference of every name to every idea or object,
either as dire ft or contrary (i. e. contradictory], 37 ; the univerfe of a
proportion, limitation of the term univerfe, 37, 38 ; Notation for con
traries, 3 8 ; remarks on the manner in which language furnimes con
traries, 38, 39 ; converfion of particular into univerfal by invention of
fpecies, 39 ; the diftinttion of A, E, I, O, not more than an accident of
language in any particular cafe, 40; the introduction of a limit ed uni
verfe gives pojitive meaning to contraries originally defned by negation,
40, 41 ; inference 41, 42 ; qualities, how ufed in the forms of Logic,
42 ; formal Logic deals with names only, 42, 43 ; conclufion, ideal
and objective, remarks on the diftindlion of, 43, 44; AiTertions
fometimes made on the ftudy of neceffary confequences, 44, 45 ; vir
tual inclufion of the neceffary confequence in the premifes, remark
on, 45 ; Humble pofition of the logic treated in this work, 46.
CHAPTER III. On the abftratt Form of the Proportion
(pages 4654).
Separation of logic from metaphyfics, 46, 47 ; particularly necef
fary as to the import of the proportion, 47 ; Ufual mode of repre-
fenting abftradt terms, 47 ; the term may be nominal, ideal, or objec
tive, 47 ; objection to quantitative expreffions, as diflinguijhed from
quantuplicitative, 48 ; objection to the notion of cumulation as an ade
quate reprefentation of combination, 48, 49 ; Various meanings of the
copula is, 49 ; Abftr action of the logical characters of the word by
right of which all thofe meanings are proper for all inference, 50, 51 ;
meanings which only fatisfy fome charakteriftics may be adapted to
fome inferences, 51, 52; pojfibility of new meanings, 52; inadmif-
fibility of fome exifting meanings, 52, 53 ; fome cafes in which the
meanings may be Jhifted, 53, 54,
CHAPTER IV. On Proportions (pages 5476).
Formal ufe of names, 54; proportion defined, 54; Limited uni
verfe introduced, 5 5 ; ExpreJJed ftipulation that no name ufed Jills
this univerfe, 55 ; diftinRion of fimple and complex proportion, 56 ;
fign, affirmative and negative, 56; relative quantity, univerfal and
particular, 56 ; Only relative quantity or ratio, definite in univerfals,
57 ; fubject and predicate, 57 ; predicate always quantified by pofition,
57 ; Diftinction to be taken as to this quantification, 57 ; definite and
indefinite, ideal poffibility of perfect definitenefs throughout, 58 ; order,
58; convertibles and inconvertibles, 58, 59; remark on the alter
natives of logic, 59 ; ufual diflinction of contrary and contradictory,
not made in this work, 60 ; fubcontrary and fuper contrary proportions,
60 ; ftandard order of reference, which, as to clarification, renders
Contents. ix
figure unnecessary , 60; A, E, I, O, and their contranominals, 60;
thefe and their contranominals denoted by the fub-fy mbo Is and fuper-
fymbols A,, Ei, I., O 4 , A 1 , E f , I , O f , 60 ; Meanings of X)Y, X.Y,
XY, and X:Y, 60 ; The eight ft andard forms ; reduclion of all others
to them ; and reprefentation by i?iftances, 6j; new term, contranominal,
and exprejfion by means of it, 62; meaning of the new forms ofajjer-
tion, E 1 and I 1 , 62 ; reprefentations of the eight forms, 62; Quanti
ties of the dire ft and contrary terms, 6 3 ; Table of relations of inclufion,
&c., 63; Concomitants, 63 ; Reduction of the forms to one another,
by the orders of reference, XY, Xy, xy, xY, 63, 64; Inveftigation
of equivalences obtained by change of one or more of the four, fub-
jecJ, predicate, copula, and order, 64, 65 ; ftrengthened and weak
ened forms, 65 ; complex proportions, 65 ; P, the complex particular,
66; D, the identical, 66; D 4 , the fubidentica I, D f , the fuperidentica I,
C, the contrary, C|, the fubcontrary, C 1 , the fuper contrary, 67; fub
and fuper affirmation and negation, 68 ; Table of relations between
the fimple and complex, 69 ; Table of connexion of fimple and complex
propofitions by change of terms and orders, 70 ; Laws of this table,
70; Continuous interchange of complex relation, 70, 71, 72; its
laws, 72; necejfary,fufficient, actually pofjible, contingent, and their
contraries ; laws of connexion of thefe relations with the fimple and
complex forms, 72, 73, 74; nomenclature in conjuncJion with, or
amendment of, that of fub affirmative, & c., 75 ; fiatement of the evi
dent laws to which all fyllogifm might be reduced, 75, 76.
Additions and cor reel ions. Page 56, line 7, infer t except only
one which confifts of four fimple propofitions. Page 62, line 23;
Say X and Y are not complements (inftead of contraries] that is,
do not together either fill, or more than fill, the univerfe. Page
72, lines 4 and 3, from the bottom , The oppositions are incorrect.
It ought to be cannot do without and cannot fail with : muft precede,
and muft follow. The reader may eafily identify the eight forms of
predication as having X for fubjecl:, Y for predicate, with the copulae,
cannot be without, can be without, cannot be with, can be with,
cannot fail without, can fail without, cannot fail with, can fail with.
CHAPTER V.On the Syllogifm (pages 76106).
Definition of fyllogifm, premifes, middle term, concluding terms, 76 ;
Diftintlion of fimple and complex fyllogifm, 76 ; Reafons for beginning
with the latter, 76, 77 ; The common a fortiori fyllogifm is com
plex, 76; Diftinttion of fundamental and ftrengthened fyllogifm, 77 ;
Standard order of reference, the fubftitute for figure, 77; The forms
of the complex affirmatory and negatory fyllogifm, in fymbols and in
language, 78 ; its limiting forms, 79 ; its rules, 79 ; the demonftration
of the affirmatory forms, by help of a diagram, 79, 80; their a for
tiori char a tier, 81 ; the demonftration of the negatory forms, 81, 82 ;
reduclion of all the forms of each kind to any one, and rules, 82, 83,
84; Complex forms in which P enters, 84, 85 ; doubt on the goodnefs
x Contents.
of the terms fimple and complex, 85 ; Denial of the fanplicity of the
Jimple proportion, 85, 86; Are not disjunctive and conjunctive the
proper words ? 86 ; The denial of a conclufion, coupled with one of
the premifes, denies the other, 86 ; The fimple fyllogifm, 86 ; De-
monftration that a particular cannot lead to a univerfal, and that two
particulars are inconclulive, by help of the complex fyllogifm, 86, 87 ;
Opponent fyllogifms, 87, 88 ; Rules for the fymbols of opponent
fyllogifms, 87, 88 ; Of fundamental fyllogifms, there muft be
twice as many particular as univerfal, 88; Deduction of the fun
damental Jimple fyllogijms, eight univerfal, and fixteen particular,
from the eight affirmatory complex fyllogifms, 88, 89 ; Deduction of
the eight ftrengthened fyllogifms from the limiting forms of the affirm a-
tory complex ones, 90, 9 1 ; Connexion of the two modes offtrengthening a
premife, go, 91 ; The conclufion is never ftrengthened by ftrengthening
the middle term, nor only weakened by weakening it, 9 1 ; Table of
connexion of the ftrengthened fyllogifms with the reft, 91 ; deduction
of the ftrengthened fyllogifms from the negatory complex ones, and
difmiffal of the latter as of no more logical effect than the former, 92 ;
Direct rule of notation, applying to fyllogifms which begin and conclude
with like quantity, 92 ; Inverfe rule of notation, [N.B. the word
inverfe mould have been contrary^ applying to fyllogifms which begin
and conclude with unlike quantity, 93 ; Rules for all the retained fyllo
gifms, 93 ; Sub-rules for the particular fyllogifms [they would have
done as remarks, but are needlefs as rules] 94 ; Remarks, partly reca
pitulatory, 94, 95, 96; In all fundamental fyllogifms, the middle
term is univerfal in one premife, and particular in the other, 95 ; dif-
tinction thence arifing, 9 5 ; rule for connecting the fyllogifms which
are formed by interchanging the concluding terms, 96; converjion of a
particular into a univerfal, 96 ; diftinction of the particular quantity
in a conclufion into intrinfic and extrinfic, 97 ; the quantity of one
ter?n always intrinfic, and hence the fyllogifm can always be made uni
verfal, 97 ; Nominal mode of notation for, and reprefentation of, a
fundamental fyllogifm, 98 ; connexion of the nominal fy ft em with the
former (or proponent) fy ft em, 99 ; mode of deriving concomitants and
weakened forms, 100; more abftract mode of reprefentation derived
from the nominal, i oo ; nominal fyftem of ftrengthened fyllogifms,
101 ; mixed complex fyllogifm, 101 ; opponent forms, 102; verbal
defcription of the fimple fyllogif?n, 103 ; new view of the fyllogifm, in
which all is referred to the middle term, \ 04 ; rules thence derived,
105 ; compound names, and expulfion of quantity by reference of the
proportion to poffibiliiy or impojfibility of a compound name, 105 ;
fyftem of fyllogifm thence arifing, 106.
Additions and corrections. Page 79, in the firft diagram, for
DjD 4 D, read D 4 D|D| ; page 88, line 23, inftead of has the other two
for its opponents, read has its opponents in the fet ; page 90, line 4,
from the bottom, for premifs read premife : the firft fpelling has been
common enough, but it feems ftrange that the cognate words promife,
furmife, demife, &c. mould not have dictated the fecond. Page 96 ;
Contents. xi
The inverted forms of the llrengthened fyllogifms are omitted : of
thefe, four are their own inverfions, namely, A,A ! I , A A I 4 , E E lj,
and EjEjI : of the remainder, A^ O and E A Oi are inverfions ;
and alfo A EjOj and EjA,O ? . Page 100, line 12, from the bottom;
for on read on), the firft time it occurs. Page 101 : Read
the fymbols of the ftrengthened fyllogifms fo as to begin from the
middle in both premifes : thus, Xyz! is y)X+y)z=Xz. Page 101.
I might have faid a word or two on the cafe in which a complex
particular is combined with a univerfal ; to form the refults will be an
eafy exercife for the reader. Page 102, line 7, from the bottom, for
IiA I, read IjAJj.
CHAPTER VI. On the Syllogifm (pages 107126).
Remarks connected with the exiftence of the terms, 107, 108, 109,
1 10, in, 112, 113. The conclujion not feparable from the premifes
except as to truth, 107, 108; conditions, and conditional fyllogifm,
109; incompletenefs of reduction of conditional to categorical, 109,
no; univerfe of proportions, no; exiftence of the terms of a pro-
pofition, in; its ajumption in fyllogifm, particularly as to the middle
term, 112, 113; poftulate more extenfive than the dictum de omni et
nullo, involved as well in the formation of premifes as in fyllogifm,
114, 115; Invention of names, 115; notation for conjunctive and
disjunctive names, 115, 1 16 ; exprejjion of complex relations and their
contraries, 1 1 6 ; copulative and disjunctive fyllogifms and dilemma,
117; Conjunctive poftulate, 117; deduction of other evident propoji-
tions from it, 118, 119; The collective and, as conjunctive, oppofed
to the disjunctives and and or diftributively ufed in univerfal s, and or
disjunctive (in the common fenfe) in particulars, 119; Disjunctives
may be rejected from univerfals, and conjunctives from particulars, 119;
Tranfpojition, introduction of, and rules for, 120 ; Table of the tranf-
pofed forms of A and E with compound names, 121 ; Examples of dif-
junctive fyllogifms, dilemmas, &c. treated by the above method, 122,
123, 124; Sorites, 124; Extended rules for the formation of the
various claffes of Sorites, 125, 126.
Additions and corrections. Page 121, line 8, from the bottom.
For [x,y][p,q])u read [X,Y][p,q])u.
CHAPTER VII. On the Arijlotelian Syllogifm (127141).
Limitations impofed either by Ariftotle or his followers, 127;
Dictum de omni et nullo, 127 ; defefi of this, 128 ; exclufion of
contraries, 128 ; Standard forms, 129; Major and minor terms, and
diftinftion of figure, 129 ; Selection of the Ariflotelian fyllogifms from
among thofe of this work, 130 131 ; Symbolic words, and meaning of
their letters, 131; Reduction to the firft figure, 131,132; Old form of
the fourth figure, 132, 133; Suggejlion as to two fgures fubdivided,
133; Poflible ufe of the diftindtion of figure, 133, 134; Collection of
xii Contents.
the figures in detail, 134, 135, 136; Aldrich s verfes on the rules,
1 36; Explanation of thefe rules, and fubftitutesfor thefyjtem in which
contraries are allowed, 137, 138, 139; Method of determining what
terms are taken dire ft from thepremifes, and what contrariwije. 140 ;
Reafonfor the duplication of thefyjiem of chapter P., 140, 141.
CHAPTER VIII. On the numerically definite Syllogifm (pages
141170).
Reafonfor its introduction, 141 ; definition of numerical definitenefs,
141, 142; difiinftion between it and perfect definitenefs, 142, 143 ;
Notation for thefimple numerical proportion, 144; Forms of inference
when only the dire ft middle term is numerically definite, 145, 146;
Canon of the middle term, 145 ; Double inference in the cafe of one
premife negative, 145, 146 ; This double inference is true in the Arif-
totelian fyllogifm Bokardo, 146; Application of the phrafeology of
complex names to the relations of propofitions, 146, 147, 148, 149;
Identical propofitions, 146, 147 ; Nece/ary confequence, 147 ; Reafons
for rejecting the ufual diftinftion of Contrary and Contradiftory, and
for introducing fubcontrary and f up er contrary, 148 ; Remarks on a uni-
verfe of proportions, 149; Abolition of the numerical quantification of
the predicate, 1 50, 1 5 I ; The cafes in which it appears either identical
with thofe in which it does not appear, orfpurious, 150, 151; numeri
cal forms of the ufual propofitions, 151; Modes of contradicting the
numerical forms, 152; Definition of fpurious propofitions : reafonsfor
refufing their introduction, and excluding them when they appear, 153,
154; Note in defence of the word fpurious, 153; Spurious conclufions
may refult from premifes not fpurious, 153, 154; Law of inference,
154; Contranominal forms of numerical propofitions, partial, (which
are fpurious) and complete, 155, 156 ; When one is impojfible, the
other is fpurious, 157; Fundamental form of inference, 157; Of two
contranominals, one is always partially fpurious, 158; deduction of the
remaining forms from the fundamental one, 1 5 8, 1 59 ; Equations of
connexion between the numerical quantities, 159; Enmneration of the
ufefulfubdivifionsofthe numerical hypothefis, 1 60; Exhibition of the
fixteen varieties of numerical fyllogiftic inference, 161 ; Deduction of
all the ordinary fyllogifms from them, 161, 162; Cafes in which de
finite particulars allow of inference by defcription with refpeft to the
middle term, 163 ; Double choice in the mode of exprejfing thefe fyllo
gifms, 163; Exceptional fyllogifms, averted to be what are mofl
frequently meant when univerfals are ufed, 1 64 ; Formation of ab
infirmiori fyllogifms, their connexion with the ordinary ones, 165 ; For
mation of fyllogifms oftranfpofed quantity, 166 ; Enumeration of them,
1 66, 167; Rules for their formation, 167, 168; Example of their
occurrence, 168 ; Example of the formation of an opponent numerical
fyllogifm, 1 68 ; Remark on what becomes of the fecond inference in a
partially definite fyft em, 169 ; Nonexiftence of definite numerical com
plex fyllogifms, 169, 170.
Contents. xiii
Additions and corrections. Page 143, line 12 : Supply the propo-
fitions X)M,P and Y)N,Q, as deducible from the numbers of in-
flances in the feveral names. Page 148, line 10, from the bottom:
for propofitions read prepofitions. Page 152, line 4: for m read
m. Page 153, line 22 : for will prefently mow us, read have mown
us in page 145. Page *54 ^ ne 2 from the bottom, for ys read zs.
Page 155, //# & from the bottom, for mXY read #7XY. P^g^ 162,
line 2, tf/hr ^ table : for lail chapter read chapter V. Page 166,
line ij,for m*xy read m xy. Page 167, line 24 : for 62 read 92.
CHAPTER IX. On Probability (pages 170191).
Remark on old and new views of knowledge, 1 70 ; Neceflary
truths not always identities, inftance, two and two are four, 171 ;
degrees of belief or knowledge, 171; Degree of knowledge treated as
a magnitude, 172; Diftinftion of ideal and objective probability,
172, 173 ; Rejection of the latter, 173 ; Definition of probability as
referring to degree of belief, 173 ; Illuftration of degree of belief as a
magnitude, 1 74 ; What is perception of magnitude, 1 74 ; Meafure-
ment of magnitude, 175; Illuftration of various degrees of belief,
1 76 ; Difference of certain and probable, not that of magnitudes
of various kinds, but that of finite and infinite of the fame kind,
176, 177 ; the real diftinftion not thereby abrogated, 178 ; Poftulate
on the acceptance of which the theory of probabilities depends, 1 79 ;
the aflumption of this poftulate, in other cafes, not always fo well
founded as is fuppofed, 1 79, 1 80, 1 8 1 ; the difficulties of this poftu
late intentionally introduced and infifted on, 181, 182 ; Meafure of
probability or credibility, and alfo of authority, 182, 183 ; Rule for
the formation of this meafure, 1 84 ; Objective verification of a re
mote conclufion of this rule, 1 84, 185; Probability of the joint hap
pening of independent events, 1 86; Confequences of this rule, 187;
Problem in which the primary cafes are unequally probable, 187,
1 88; Rule of inverfe probabilities, 1 8 8, 189, 190; this rule alfo
holds in calculating the probabilities of reftridled cafes from the unre-
ftrifted ones, 190, 191.
CHAPTER X. On probable Inference (pages 191 210).
Argument and teftitnony, 191, 192 ; argument never the only vehi
cle of information except when demonftrative, 192 ; truth or falfehood
not the fimple iffue in argument, 192, 193 ; difficulty thereby introduced
into the judgment of truth or falfehood, 193; entrance of teftimony,
194; remark on the precept to negleft authority, 194; Compofition
of independent teftimonies, 195 ; on the majority of witnefles, 196 ;
the fame problem, when the event aflerted has an antecedent proba
bility, 197; queftion of collufion, 198, 199 ; extenflon of the laft
problem to more complicated events, 200 ; Compofition of indepen-
xiv Contents.
dent arguments on the fame fide, 201 ; manner in which the weak-
nefs of an argument may become an argument or a teftimony, 202, 208 ;
Compaction of arguments on contrary fides, 203 ; the fame on fubcon-
trary fides, 204 ; Compofition of argument and teftimony in a queftion
of contrary fides, 205 ; More weight due to argument than to teftimony
of the fame probability, 206 ; Utter rejection of authority, what it
amounts to, 207, 208 ; Effefts of the fame arguments on different
minds, 209 \Efeft of probable confequence upon an ajfertion, 209, 210;
Old fuicidal ajfertion, explained by probability, 210.
Additions and corrections. Page 199, line 4, from the bottom:
for (i-\) read (\ \} m . Page 201, line 14, from the bottom :
for 1 read T V
CHAPTER XL On Inclusion (pages 211226).
Explanation of induction, 211; Reduction of the procefs to a fyl-
logifm, 2ii ; Induction by connexion, and inftance, 212 ; Ordinary
induction not a demonftrative procefs, 212, 213 ; Pure induction,
incomplete, probability of it, 213, 214; Ordinary miftakes on this
fubject, 215 ; Examination of Mr. T. B. Macaulay s enumeration of
initances in which fcientific analyfis is ufelefs, 216, 217, 218, 219,
220, 221, 222, 223, 224; probability of fyllogifms with particular
premifes, 224, 225, 226; Circumftantial evidence, 226.
CHAPTER XII. 0;z old logical Terms (pages 227237).
Dialectics, 227; fimple and complex terms, 227; apprehenfion,
judgment, difcourfe, 227; Univerfal and fingular, 228; Individuals,
228 ; categories, predicaments, 228 ; fubftance, 228 ; firft and fecond
fubftance, 229 ; quantity, continuous and difcrete, 229 ; Quality,
habit, difpolition, pamon, 229 ; Relation, 229 ; Action, paffion, imma
nent, tranfient, univocal, equivocal, 229, 230; Remaining categories,
230; predicables, genus, fpecies, 230 ; difference, property, acci
dent, 231 ; caufe, material, formal, efficient, final, 231 ; form, mo
tion, fubject, object, 231 ; Subjective, objective, adjunct, 232;
modals, fubftitution of the theory of probabilities for them, 232 ;
Their ufe in the old philofophy, 232, 233 ; Notions of old logicians
on quantity, 234; Intenfion or comprehenfion, and extenfion, ob
jections to their oppofition as quantities, and references to places in
this work where the diftinction has occurred, 234, 235, 236; In-
ftance, 236 ; Enthymeme, Ariftotle s, and modern, 236, 237.
Additions and corrections. Page 230, lines 16 and 15, from the
bottom; tranfpofe the words former and latter. Page 234 line ^ from
bottom, for after read before. Page 237, note y I find that etymolo-
gifts are decidedly of opinion that prjo-ic, fpeech, and /OEM, flow, have
different roots, and that the former is fpeech in its primitive meaning.
The reader muft make the alteration, which however does not affeft
my fuggeftion.
Contents. xv
CHAPTER XIIL On Fallacies (pages 237286).
No claffification of fallacies, 237 ; Amufcment derived from, 238 ;
fallacy, fophifm, paradox, paralogifm, 238; Ariftotle s claffification,
240 ; Pofition of ancients and moderns as to fallacies, 240 ;
Confequences of the neglect of logic, 241 ; Ariflotle s fpecies of
fallacies enumerated, 241; Equivocation, 241, 242; Change of
meanings with time, 243 ; Importance once attached to fuccefsful
equivocation, 244 ; Government fallacies, 244 ; Qualifications of
meaning, 244, 245 ; Phrafes interpreted by their component words,
245 ; AiTumption of right over words, 246, 247 ; Equivocating forms
of predication, 247 ; Amphibology, 247 ; Defects in the ftructure
of language, 247 ; Compofition and divifion, 248 ; Accent, 248,
249; Fallacy of alteration of emphafis, 249, 250; diction, 250;
Accident and a ditto fecundum quid, &c., 250, 251, 252; Examina
tion of fome cafes of legal ftridtnefs, 252, 253, 254; Petitio prin-
cipii, 254; often wrongly imputed, 255; Ariftotle s meaning of it,
256 ; Meaning of the old logicians, 256 ; derivation from the fyllo-
gifm of principle and example, 257; Charge of petit "io principii
againft all fyllogifms, 257, 258, 259; Syllogifm fometimes only re
quired for diminution of comprehenfion, 259; Imperfect dilemma,
fophifm of Diodorus Cronus, 259, 260; Ignoratio elenchi, 260;
proof of negative, and negative proof, 261, 262 ; aflertions of difpu-
tants in their own favour, 262 ; Fallacy of tendencies and necefTary
confequences, 263 ; Fallacy of attributing refults of teftimony to ar
gument, 264 ; Argumentum ad hominem, 265 ; Parallel cafes, 265,
266 : Fallacies of illuflration, 266, 267 ; Fallacia confequentis, 267 ;
Incorrect logical forms, 267, 268 ; Non caufa pro caufa, 268, 269 ;
Fallacia plurimum interroga tionum, 269, 270; Practices of barrifters,
270; Incorrect ufe of univerfal form, 270, 271 ; Fallacy of the
extreme cafe, 271 ; Ufe of the extreme cafe, 271, 272 ; Carriage of
principles, 272 ; Ufe of the word general, 272 ; Confufion of logic
and perfpedtive, 272, 273 ; General truths, 273 ; Implied univerfals
not fairly flated, 273 ; Fallacies of quantity, 274; Proverbs, 275 ;
Fallacies of probability, 275, 276; Fallacy of analogy, 276; Fallacy
of judging by refults, 276, 277; Equivocations of ftyle, 277; Fal
lacy of fynonymes, 277, 278 ; Fallacies arifing out of connection of
principles and rules, 279, 280, 281; Want of rule nifi va. common
language, 280; Fallacy of importation of premifes, 281 ; Fallacy of
retaining conclufions after abandoning premifes, 282 ; Fallacies of
citation and quotation, 282, 283, 284, 285, 286.
Additions and corrections. Page 250, lines 3 and 5 y for mil-
lenium read millennium, and for Newtonion read Newtonian.
CHAPTER XIV. On the verbal Defcription of the Syllogifm
(pages 286 296).
Conditions to be fatisfied, 287 ; Double mode of defcription and
xvi Contents.
reference of one to the other, 287, 288; Language propofed, 288 ;
Defcription of the cafes of fyllogifm in that language, 289, 290 ;
Connexion of the univerfal and concomitant fyllogifm with the complex
one, 291, 292; Quantitative formation of the fyllogifm, 293, 294, 295 ;
Rules for the formation of the numerical fyllogifm, 295, 296.
APPENDIX I. Account of^ a Controverfy between the Author
of this Work and Sir William Hamilton of Edinburgh ; and
final reply to the latter (pages 297 323).
APPENDIX II. On fome Forms of Inference differing from
thofe of the Ariflotelians (pages 323 336).
ELEMENTS OF LOGIC.
CHAPTER I.
Firft Notions.
THE firft notion which a reader can form of Logic is by
viewing it as the examination of that part of reafoning
which depends upon the manner in which inferences are formed,
and the investigation of general maxims and rules for conftru6t-
ing arguments, fo that the conclufion may contain no inaccuracy
which was not previoufly aflerted in the premifes. It has fo far
nothing to do with the truth of the facts, opinions, or prefump-
tions, from which an inference is derived ; but fimply takes care
that the inference (hall certainly be true, if the premifes be true.
Thus, when we fay that all men will die, and that all men are
rational beings, and thence infer that fome rational beings will
die, the logical truth of this fentence is the fame whether it be
true or falfe that men are mortal and rational. This logical truth
depends upon ihejtruflure of the fentence^ andjiotjjupon the par
ticular matters fpoken of. Thus,
Inftead of Write,
All men will die. Every Y is X.
All men are rational beings. Every Y is Z.
Therefore fome rational beings Therefore fome Zs are Xs.
will die.
The fecond of thefe is the fame propofition, logically confidered,
as the firft ; the confequence in both is virtually contained in,
and rightly inferred from, the premifes. Whether the premifes
be true or falfe, is not a queftion of logic, but of morals, philofo-
phy, hiftory, or any other knowledge to which their fubjecT:-
2 Firji Notions of Logic.
matter belongs : the queftion of logic is, does the conclufion
certainly follow if the premifes be true ?
Every act of reafoning muft mainly confift in comparing to
gether different things, and either finding out, or recalling from
previous knowledge, the points in which they refemble or differ
from each other. That particular part of reafoning which is
called inference^ confifts in the comparifon of feveral and different
things with one and the fame other thing ; and afcertaining the
refemblances, or differences, of the feveral things, by means of
the points in which they refemble, or differ from, the thing with
which all are compared.
There muft then be fome proportions already obtained before
any inference can be drawn. All propofitions are either affer-
tions or denials, and are thus divided into affirmative and negative.
Thus, X is Y, and X is not Y, are the two forms to which
all propofitions may be reduced. Thefe are, for our prefent
purpofe, the moil fimple forms ; though it will frequently hap
pen that much circumlocution is needed to reduce propofitions
to them. Thus, fuppofe the following affertion, If he fhould
come to-morrow, he will probably ftay till Monday; how is
this to be reduced to the form X is Y ? There is evidently
fomething fpoken of, fomething faid of it, and an affirmative
connection between them. Something, if it happen, that is, the
happening of fomething, makes the happening of another fome
thing probable ; or is one of the things which render the hap
pening of the fecond thing probable.
X is Y
r~ u u ru- 1 fan event from which it may be
The happening of his . _ _ _
}* is 1 inferred as probable that he
arrival to-morrow j | w m ftay till Monday.
The forms of language will allow the manner of afferting to
be varied in a great number of ways ; but the reduction to the
preceding form is always poffible. Thus, fo he faid is an affir
mation, reducible as follows :
What you have juft 1 f the thing which
faid (or whatever 1S
, r . r , r .
elfe c fo refers to)
Firjl Notions of Logic. 3
By changing 4 is into * is not, we make a negative propofi-
tion j but care muft always be taken to afcertain whether a
proportion which appears negative be really fo. The principal
danger is that of confounding a propofition which is negative
with another which is affirmative of fomething requiring a nega
tive to defcribe it. Thus, c he refembles the man who was not
in the room, is affirmative, and muft not be confounded with
4 he does not refemble the man who was in the room. Again,
4 if he mould come to-morrow, it is probable he will not ftay till
Monday, does not mean the fimple denial of the preceding pro
pofition, but the affirmation of a directly oppofite propofition.
It is,
X is Y
_,. 1 f an event from which it may be
1 he happening or his . . r 11- i_ i i i_
J- is J inferred to be /^probable that
arrival to-morrow, J ^ he will ftay till Monday :
whereas the following,
,, , . f ,. "I fan event from which it may be
1 he happening or his . . .. . ... . .
. , is not inferred as probable that he
arrival to-morrow, j { ^ fay
would be exprelTed thus : c If he mould come to-morrow, that is
no reafon why he mould ftay till Monday.
Moreover, the negative words not, no, &c., have two kinds of
meaning which muft be carefully diftinguifhed. Sometimes they
deny, and nothing more : fometimes they are ufed to affirm the
direct: contrary. In cafes which offer but two alternatives, one
of which is necefTary, thefe amount to the fame thing, fince the
denial of one, and the affirmation of the other, are obvioufly
equivalent propofitions. In many idioms of converfation, the
negative implies affirmation of the contrary in cafes which offer
not only alternatives, but degrees of alternatives. Thus, to the
queftion, 4 Is he tall ? the fimple anfwer, No, moft frequently
means that he is the contrary of tall, or confiderably under the
average. But it muft be remembered, that, in all logical reafon-
ing, the negation is fimply negation, and nothing more, never
implying affirmation of the contrary.
The common propofition that two negatives make an affirm
ative, is true only upon the fuppofition that there are but two
4 Firft Notions of Logic.
poffible things, one of which is denied. Grant that a man muft
be either able or unable to do a particular thing, and then not
unable and able are the fame things. But if we fuppofe various
degrees of performance, and therefore degrees of ability, it is
falfe, in the common fenfe of the words, that two negatives make
an affirmative. Thus, it would be erroneous to fay, John is
able to tranflate Virgil, and Thomas is not unable ; therefore,
what John can do Thomas can do, for it is evident that the
premifes mean that John is fo near to the beft fort of tranflation
that an affirmation of his ability may be made, while Thomas is
confiderably lower than John, but not fo near to abfolute defi
ciency that his ability may be altogether denied. It will generally
be found that two negatives imply an affirmative of a weaker
degree than the pofitive affirmation.
Each of the propofitions, c X is Y, and X is not Y, may
be fubdivided into two fpecies : the univerfal, in which every
poffible cafe is included ; and the particular, in which it is not
meant to be afTerted that the affirmation or negation is univerfal.
The four fpecies of propofition are then as follows, each being
marked with the letter by which writers on logic have always
diftinguifhed it.
A Univerfal Affirmative Every X is Y
E Univerfal Negative No X is Y
I Particular Affirmative Some Xs are Ys
O Particular Negative Some Xs are not Ys
In common converfation the affirmation of a part is meant to
imply the denial of the remainder. Thus, by c fome of the apples
are ripe, it is always intended to fignify that fome are not ripe.
This is not the cafe in logical language, but every propofition is
intended to make its amount of affirmation or denial, and no
more. When we fay, Some X is Y, or, more grammatically,
Some Xs are Ys, we do not mean to imply that fome are not :
this may or may not be. Again, the word fome means, one or
more, poffibly all. The following table will mew the bearing
of each propofition on the reft.
Every Xis l"affirms Some Xs are Ts and denies \
(.some Xs are not is
Firji Notions of Logic. 5
No Xis 7~affirms Some Xs are not Ts and denies] Ver ^ ^ ~~
(.some As are Is
Some Xs are Ts does not contradift< . > but denies No X is T
[Some Xs are not Ts )
Some Xs are not Ts does not oontndim g v v- ( but denies Every XtsT
Contradictory propofitions are thofe in which one denies any
thing that the other affirms ; contrary propofitions are thofe in
which one denies every thing which the other affirms, or affirms
every thing which the other denies. The following pair are
contraries,
Every X is Y and No X is Y
and the following are contradictories,
Every X is Y to Some Xs are not Ys
No X is Y to Some Xs are Ys
A contrary, therefore, is a complete and total contradictory;
and a little confideration will make it appear, that the decifive
diftinction between contraries and contradictories lies in this,
that contraries may both be falfe, but of contradictories, one
muft be true and the other falfe. We may fay, Either P is true,
or fomethlng in contradiction of it is true ; but we cannot fay,
Either P is true, or every thing in contradiction of it is true.
It is a very common miftake to imagine that the denial of a
proportion gives a right to affirm the contrary; whereas it (hould
be, that the affirmation of a propofition gives a right to deny the
contrary. Thus, if we deny that Every X is Y, we do not affirm
that No X is Y, but only that Some Xs are not Ys ; while, if we
affirm that Every X is Y, we deny No X is Y, and alfo Some
Xs are not Ys.
But, as to contradictories, affirmation of one is denial of the
other, and denial of one is affirmation of the other. Thus,
either Every X is Y, or Some Xs are not Ys : affirmation of either
is denial of the other, and vice verfa.
Let the ftudent now endeavour to fatisfy himfelf of the fol
lowing. Taking the four preceding propofitions, A, E, I, O,
let the fimple letter fignify the affirmation, the fame letter in pa-
renthefes the denial, and the abfence of the letter, that there is
neither affirmation nor denial.
Fir/I Notions of Logic.
From A follow (E), I, (O)
From E (A), (I), O
From I (E)
From O .... (A)
From (A) follow. O
From (E) . . . . I
From (I) (A),E,O
From (O) ... A, (E), I
Thefe may be thus fummed up : The affirmation of a univerfal
proportion, and the denial of a particular one, enable us to affirm
or deny all the other three ; but the denial of a univerfal propo-
fition, and the affirmation of a particular one, leave us unable to
affirm or deny two of the others.
In fuch propofitions as Every X is Y, l Some Xs are not Ys,
&c., X is called the /*>#, and Y the predicate, while the verb
c is or c is not/ is called the copula. It is obvious ^that the
words of the proportion point out whether the fubjecl: is fpoken
of univerfally or partially, but not fo of the predicate, which it is
therefore important to examine. Logical writers generally give
the name of diflnbuted fubjefts or predicates to thofe which are
fpoken of univerfally ; but as this word is rather technical, I fhall
fay that a fubjecl: or predicate enters wholly or partially, accord
ing as it is univerfally or particularly fpoken of.
1. In A, or Every X is Y, the fubjecl: enters wholly, but
the predicate only partially. For it obvioufly fays, c Among the
Ys are all the Xs, c Every X is part of the colleaion of Ys, fo
that all the Xs make a part of the Ys, the wjtole it may be.
Thus, Every horfe is an animal, does not fp^Pof all animals,
but ftates that all the horfes make up a portionfff the animals.
2. In E, or c No X is Y, both fubjecl: and predicate enter
wholly. No X whatfoever is any one out of all the Ys ;
< fearch the whole collection of Ys, and every Y (hall be found
to be fomething which is not X.
3. In I, or c Some Xs are Ys, both fubjecl: and predicate enter
partially. c Some of the Xs are found among the Ys, or make
up a part (the whole poffibly, but not known from the preceding)
of the Ys.
4. In O, or Some Xs are not Ys, the fubjecl: enters partially,
and the predicate wholly. c Some Xs are none of them any
whatfoever of the Ys ; every Y will be found to be no one out
of a certain portion of the Xs.
It appears then that,
In affirmatives, the predicate enters partially.
Firji Notions of Logic. /
In negatives, the predicate enters wholly.
In contradictory proportions, both fubjecl: and predicate enter
differently in the two.
The converfe of a propofition is that which is made by inter
changing the fubjecl: and predicate, as follows :
The propofition. Its converfe.
A Every X is Y Every Y is X
E No X is Y No Y is X
I Some Xs are Ys Some Ys are Xs
O Some Xs are not Ys Some Ys are not Xs
Now, it is a fundamental and felf-evident propofition, that no
confequence muft be allowed to aflert more widely than its pre-
mifes ; fo that, for inftance, an aflertion which is only of fome
Ys can never lead to a refult which is true of all Ys. But if a
propofition aflert agreement or difagreement, any other propofi
tion which aflerts the fame, to the fame extent and no further,
muft be a legitimate confequence ; or, if you pleafe, muft
amount to the whole, or part, of the original aflertion in another
form. Thus, the converfe of A is not true : for, in Every X
is Y, the predicate enters partially ; while in Every Y is X,
the fubjecl: enters wholly. All the Xs make up a part of the
Ys, then a part of the Ys are among the Xs, or fome Ys are Xs/
Hence, the only legitimate converfe of c Every X is Y is, c Some
Ys are Xs. But in No X is Y, J both fubjecT: and predicate enter
wholly, and c No Y is X is, in fact, the fame propofition as
No X is Y. And Some Xs are Ys is alfo the fame as its con
verfe c Some Ys are Xs : here both terms enter partially. But
Some Xs are not Ys admits of no converfe whatever ; it is per
fectly confident with all aflertions upon Y and X in which Y is
the fubjecl:. Thus neither of the four following lines is incon-
fiftent with itfelf.
Some Xs are not Ys and Every Y is X
Some Xs are not Ys and No Y is X
Some Xs are not Ys and Some Ys are Xs
Some Xs are not Ys and Some Ys are not Xs.
Having thus difcufled the principal points connected with the
fimple aflertion, I pafs to the manner of making two aflertions
8 Firft Notions of Logic.
give a third. Every inftance of this is called ^fylloglfm^ the two
affertions which form the bafis of the third are called premlfes^
and the third itfelf the conclufion.
If two things both agree with a third in any particular, they
agree with each other in the fame ; as, if X be of the fame colour as
Y, and Z of the fame colour as Y, then X is of the fame colour as
Z. Again, if X differ from Y in any particular in which Z
agrees with Y, then X and Z differ in that particular. If X be
not of the fame colour as Y, and Z be of the fame colour as Y,
then X is not of the colour of Z. But if X and Z both differ
from Y in any particular, nothing can be inferred; they may
either differ in the fame way and to the fame extent, or not.
Thus, if X and Z be both of different colours from Y, it neither
follows that they agree, nor differ, in their own colours.
The paragraph preceding contains the effential parts of all in
ference, which confifts in comparing two things with a third, and
finding from their agreement or difference with that third, their
agreement or difference with one another. Thus, Every X is
Y, every Z is Y, allows us to infer that X and Z have all thofe
qualities in common which are neceffary to Y. Again, from
every X is Y, and No Z is Y, we infer that X and Z differ
from one another in all particulars which are effential to Y. The
preceding forms, however, though they reprefent common reafon-
ing better than the ordinary fyllogifm, to which we are now com
ing, do not conftitute the ultimate forms of inference. Simple iden
tity or non-Identity is the ultimate ftate to which every affertion
may be reduced j and we mail, therefore, firft afk, from what
identities, &c., can other identities, &c., be produced ? Again,
fmce we name objects in fpecies, each fpecies confifting of a
number of individuals, and fmce our affertion may include all or
only part of a fpecies, it is further neceffary to afk, in every in
ftance, to what extent the conclufion drawn is true, whether of
all, or only of part ?
Let us take the fimple affertion, c Every living man refpires ;
or every living man is one of the things (however varied they
may be) which refpire. If we were to enclofe all living men in
a large triangle, and all refpiring objects in a large circle, the pre
ceding affertion, if true, would require that the whole of the tri
angle mould be contained in the circle. And in the fame way we
Firjt Notions of Logic. 9
may reduce any aflertion to the expreflion of a coincidence, total
or partial, between two figures. Thus, a point in a circle may
reprefent an individual of one fpecies, and a point in a triangle
an individual of another fpecies : and we may exprefs that the
whole of one fpecies is aflerted to be contained or not contained
in the other by fuch forms as, All the A is in the O > c None
of the A is in the O -
Any two afTertions about X and Z, each exprefling agreement
or difagreement, total or partial, with or from Y, and leading to a
conclufion with refpecT: to X or Z, is called a fyllogifm, of which
Y is called the middle term. The plaineft fyllogifm is the folio w-
Every X is Y
Every Y is Z
Therefore Every X is Z
All the A is in the Q
All the O is in the D
Therefore All the A is in the n
In order to find all the poflible forms of fyllogifm, we muft
make a table of all the elements of which they can confift ;
namely
X and Y Z and Y
Every X is Y A Every Z is Y
No XisY E No ZisY
Some Xs are Ys I Some Zs are Ys
Some Xs are not Ys O Some Zs are not Ys
Every Y is X A Every Y is Z
Some Ys are not Xs O Some Ys are not Zs
Or their rynonymes,
A and O D and Q
All the A is in the O A All the D is in the Q
None of the A is in the Q E None of the D is in the Q
Some of the A is in the Q I Some of the D is in the Q
Some of the A is not in the Q O Some of the D is not in the O
All the O is in the A A All the Q is in the n
Some of the Q is not in the A O Some of the Q is not in the n
Now, taking any one of the fix relations between X and Y,
and combining it with either of thofe between Z and Y, we
have fix pairs of premifes, and the fame number repeated for
every different relation of X to Y. We have then thirty-fix
io Firft Notions of Logic.
forms to confider : but, thirty of thefe (namely, all but (A, A)
(E, E), &c.,) are half of them repetitions of the other half. Thus,
< Every X is Y, no Z is Y, and Every Z is Y, no X is Y,
are of the fame form, and only differ by changing X into Z and
Z into X. There are then only 15+6, or 21 diftinft forms,
fome of which give a neceffary conclufion, while others do not.
We (hall felea the former of thefe, claffifying them by their
conclufions ; that is, according as the inference is of the form
A, E, I, or O.
I. In what manner can a univerfal affirmative conclufion be
drawn ; namely, that one figure is entirely contained in the other ?
This we can only affert when we know that one figure is entirely
contained in the circle, which itfelf is entirely contained in the
other figure. Thus,
Every X is Y
Every Y is Z
Every X is Z
All the A is in the Q A
All the O is in tne D A
All the A is in the D A
is the only way in which a univerfal affirmative conclufion can
be drawn.
II. In what manner can a univerfal negative conclufion be
drawn ; namely, that one figure is entirely exterior to the other ?
Only when we are able to affert that one figure is entirely within,
and the other entirely without, the circle. Thus,
Every X is Y
No Z is Y
No X is Z
All the A is in the O A
None of the n is in the O E
None of the A is in the D E
is the only way in which a univerfal negative conclufion can be
drawn.
III. In what manner can a particular affirmative conclufion be
drawn ; namely, that part or all of one figure is contained in the
other ? Only when we are able to affert that the whole circle is
part of one of the figures, and that the whole, or part of the cir
cle, is part of the other figure. We have then two forms.
Every Y is X
Every Y is Z
Some Xs are Zs
All the O is m tne A A
All the O is in the D A
Some of the A is in the a I
Firft Notions of Logic. \ i
Every Y is X All the Q is in the A A
Some Ys are Zs Some of the Q is in the D I
Some Xs are Zs j Some of the A is in the n I
The fecond of thefe contains all that is ftriftly neceflary to the
conclufion, and the firft may be omitted. That which follows
when an aflertion can be made as to fome, muft follow when the
fame aflertion can be made of all.
IV. How can a particular negative propofition be inferred ;
namely, that part, or all of one figure, is not contained in the
other ? It would feem at firft fight, whenever we are able to
aflert that part or all of one figure is in the circle, and that part
or all of the other figure is not. The weakeft fyllogifm from which
fuch an inference can be drawn would then feem to be as follows.
Some Xs are Ys
Some Zs are not Ys
.Some Zs are not Xs
Some of the A is in the Q
Some of the D is not in the Q
. Some of the A is not in the n
But here it will appear, on a little confederation, that the con
clufion is only thus far true ; that thofe Xs which are Ys cannot
be tbofe Zs which are not Ys ; but they may be other Zs, about
which nothing is aflerted when we fay that fome Zs are not Ys.
And further confideration will make it evident, that a conclufion
of this form can only be arrived at when one of the figures is
entirely within the circle, and the whole, or part of the other
without ; or elfe when the whole of one of the figures is without
the circle, and the whole or part of the other within ; or laftly,
when the circle lies entirely within one of the figures, and not
entirely within the other. That is, the following are the diftind
forms which allow of a particular negative conclufion, in which
it fhould be remembered that a particular propofition in the pre-
mifes may always be changed into a univerfal one, without affect
ing the conclufion. For that which necefTarily follows from
" fome," follows from " all."
Every X is Y
Some Zs are not Ys
Some Zs are not Xs
All the A is in the Q A
Some of the D is not in the Q O
Some of the n is not in the A O
12
No X is Y
Some Zs are Ys
/.Some Zs are not Xs
Firji Notions of Logic.
None of the A is in the O
Some of the D is in the O
Some of the n is not in the
Every Y is X
Some Ys are not Zs
Some Xs are not Zs
All the O is in tne A
Some of the O is not m tne D
Some of the A is not in the D
E
I
O
A
O
O
It appears, then, that there are but fix diftint fyllogifms. All
others are made from them by ftrengthening one of the premifes,
or converting one or both of the premifes, where fuch converfion
is allowable ; or elfe by firft making the converfion, and then
ftrengthening one of the premifes. And the following arrange
ment will ftiow that two of them are univerfal, three of the others
being derived from them by weakening one of the premifes in a
manner which does not deftroy, but only weakens, the conclu-
fion.
i. Every X is Y 3. Every X is Y
Every Y is Z No Z is Y
Every X is Z No X is Z
6. Every Y is X
Some Ys are not Zs
Some Xs are not Zs
2. Some Xs are Ys 4. Some Xs are Ys 5. Every X is Y
Every Y is Z No Z is Y Some Zs are not Ys
Some Xs are Zs Some Xs are not Zs Some Zs are not Xs
We may fee how it arifes that one of the partial fyllogifms is
not immediately derived, like the others, from a univerfal one.
In the preceding, A E E may be confidered as derived from
A A A, by changing the term in which Y enters univerfally into
a univerfal negative. If this be done with the other term inftead,
we have
No X is Y) from which univerfal premifes we cannot deduce a
Every Y is Z) univerfal conclufion, but only fome Zs are not Xs.
If we weaken one and the other of thefe premifes, as they
ftand, we obtain
Some Xs are not Ys No X is Y
Every Y is Z and Some Ys are Zs
No conclufion
Some Zs are not Xs
Firji Notions of Logic. i 3
equivalent to the fourth of the preceding : but if we convert the
firft premife, and proceed in the fame manner,
From No Y is X we obtain Some Ys are not Xs
Every Y is Z Every Y is Z
Some Zs are not Xs Some Zs are not Xs
which is legitimate, and is the fame as the laft of the preceding
lift, with X and Z interchanged.
Before proceeding to fhow that all the ufual forms are con
tained in the preceding, let the reader remark the following rules,
which may be proved either by collecting them from the preceding
cafes, or by independent reafoning.
1. The middle term muft enter univerfally into one or the other
premife. If it were not fo, then one premife might fpeak of one
part of the middle term, and the other of another ; fo that there
would, in fact, be no middle term. Thus, Every X is Y, Every
Z is Y, J gives no conclufion : it may be thus ftated ;
All the Xs make up a part of the Ys
All the Zs make up a part of the Ys
And, before we can know that there is any common term of
comparifon at all, we muft have fome means of fhowing that the
two parts are to fome extent the fame ; or the preceding premifes
by themfelves are inconclufive.
2. No term muft enter the conclufion more generally than it
is found in the premifes ; thus, if X be fpoken of partially in the
premifes, it muft enter partially into the conclufion. This is ob
vious, fmce the conclufion muft aflert no more than the premifes
imply.
3. From premifes both negative no conclufion can be drawn.
For it is obvious, that the mere aflertion of difagreement between
each of two things and a third, can be no reafon for inferring
either agreement or difagreement between thefe two things. It
will not be difficult to reduce any cafe which falls under this rule
to a breach of the firft rule : thus, No X is Y, No Z is Y, gives
Every X is (fomething which is not Y)
Every Z is (fomething which is not Y)
1 4 Firft Notions of Logic.
in which the middle term is not fpoken of univerfally in either.
Again, No Y is X, fome Ys are not Zs, may be converted into
Every X is (a thing which is not Y)
Some (things which are not Zs) are Ys
in which there is no middle term.
4. From premifes both particular no conclufion can be drawn.
This is fufficiently obvious when the firft or fecond rule is broken,
as in c Some Xs are Ys, Some Zs are Ys. But it is not immediately
obvious when the middle term enters one of the premifes uni
verfally. The following reafoning will ferve for exercife in the
preceding refults. Since both premifes are particular in form,
the middle term can only enter one of them univerfally by being
the predicate of a negative proportion ; confequently (Rule 3)
the other premife muft be affirmative, and, being particular, nei
ther of its terms is univerfal. Confequently both the terms as to
which the conclufion is to be drawn enter partially, and the con
clufion (Rule 2) can only be a particular affirmative proportion.
But if one of the premifes be negative, the conclufion muft be
negative (as we mall immediately fee). This contradiction (hows
that the fuppofition of particular premifes producing a legitimate
refult is inadmiffible.
5. If one premife be negative, the conclufion, if any, muft be
negative. If one term agree with a fecond and difagree with a
third, no agreement can be inferred between the fecond and
third.
6. If one premife be particular, the conclufion muft be par
ticular. This may be fhown as follows. If two propofitions
P and Q_, together prove a third, R, it is plain that P and the
denial of R, prove the denial of Q. For P and Q^cannot be true
together without R. Now if poffible, let P (a particular) and Q
(a univerfal) prove R (a univerfal). Then P (particular) and
the denial of R (particular) prove the denial of Q.. But two
particulars can prove nothing.
In the preceding fet of fyllogifms we obferve one form only
which produces A, or E, or I, but three which produce O.
Let an affertion be faid to be weakened when it is reduced
from univerfal to particular, and ftrengthened in the contrary cafe.
Thus, Every X is Z* is called ftronger than Some Xs are Zs.
Firji Notions of Logic. 1 5
Every ufual form of fyllogifm which can give a legitimate re-
fult is either one of the preceding fix, or another formed from
one of the fix, either by changing one of the aflertions into its
converfe, if that be allowable, or by ftrengthening one of the
premifes, without altering the conclufion, or both. Thus,
Some Xs are Ys 7 f Some Ys are Xs
Every Y is Z { may be written { Every y is Z
What follows will ftill follow from { ""? C ! s
1 Every Y is Z
for all which is true when Some Ys are Xs, is not lefs true when
c Every Y is X.
It would be poflible alfo to form a legitimate fyllogifm by
weakening the conclufion, when it is univerfal, fmce that which
is true of all is true of fome. Thus, c Every X is Y, Every Y
is Z, which yields < Every X is Z, alfo yields c Some Xs are Zs.
But writers on logic have always confidered thefe fyllogifms as
ufelefs, conceiving it better to draw from any premifes their
ftrongeft conclufion. In this they were undoubtedly right ; and
the only queftion is, whether it would not have been advifable
to make the premifes as weak as poffible, and not to admit any
fyllogifms in which more appeared than was abfolutely neceflary
to the conclufion. If fuch had been the practice, then
Every Y is X, Every Y is Z, therefore Some Xs are Zs
would have been confidered as formed by a fpurious and unne-
ceflary excefs of afTertion. The minimum of aflertion would be
contained in either of the following,
Every Y is X, Some Ys are Zs, therefore Some Xs are Zs
Some Ys are Xs, Every Y is Z, therefore Some Xs are Zs
In this chapter, fyllogifms have been divided into two clafTes :
firft, thofe which prove a univerfal conclufion ; fecondly, thofe
which prove a partial conclufion, and which are (all but one)
derived from the firft by weakening one of the premifes, in fuch
manner as to produce a legitimate but weakened conclufion.
Thofe of the firft clafs are placed in the firft column, and of the
other in the fecond.
i6
Firft Notions of Logic.
Univerfal. Particular.
A Every X is Y Some Xs are Ys
A Every Y is Z Every Y is Z
A Every X is Z
A Every X is Y
E No Y is Z
Some Xs are Zs
Some Xs are Ys
No Y is Z
J
A
I
I
E
E No X is Z
Some Xs are not Zs O
Every X is Y
Some Zs are not Ys
A
O
Some Zs are not Xs O
Every Y is X A
Some Ys are not Zs O
Some Xs are not Zs O
In all works on logic, it is cuftomary to write that premife
firft which contains the predicate of the conclufion. Thus,
Every Y is Z Every X is Y
Every X is Y would be written, and not Every Y is Z
Every X is Z Every X is Z
The premifes thus arranged are called major and minor ; the pre
dicate of the conclufion being called the major term, and its fub-
jecl: the minor. Again, in the preceding cafe we fee the various
fubjecls coming in the order Y, Z ; X, Y ; X, Z : and the num
ber of different orders which can appear is four, namely -
YZ ZY YZ ZY
XY XY YX YX
XZ XZ XZ XZ
which are called the four figures, and every kind of fyllogifm in
each figure is called a mood. I now put down the various moods
of each figure, the letters of which will be a guide to find out
thofe of the preceding lift from which they are derived. Co
means that a premife of the preceding lift has been converted ;
-f- that it has been ftrengthened ; Co-f, that both changes have
taken place. Thus^
Firji Notions of Logic. 17
A Every Y is Z A Every Y is Z
I Some Xs are Ys becomes A Every Y is X : (Co -f )
I Some Xs are Zs I Some Xs are Zs
And Co -{- points out the following : If fome Xs be Ys, then
fome Ys are Xs (Co) ; and all that is true when Some Ys are Xs,
is true when Every Y is X (-{-) ; therefore the fecond fyllogifm
is legitimate, if the firft be fo.
Firft Figure.
A Every Y is Z A Every Y is Z
A Every X is Y I Some Xs are Ys
A Every X is Z I Some Xs are Zs
E No Y is Z E No Y is Z
A Every X is Y I Some Xs are Ys
E No X is Z O Some Xs are not Zs
Second Figure.
E No Z is Y (Co) E No Z is Y (Co)
A Every X is Y I Some Xs are Ys
E No X is Z O Some Xs are not Zs
A Every Z is Y A Every Z is Y
E No X is Y (Co) O Some Xs are not Ys
E No X is Z O Some Xs are not Zs
Third Figure.
A Every Y is Z E No Y is Z
A Every Y is X (Co +) A Every Y is X (Co 4.)
I Some Xs are Zs O Some Xs are not Zs
I Some Ys are Zs (Co) O Some Ys are not Zs
A Every Y is X A Every Y is X
I Some Xs are Zs O Some Xs are not Zs
A Every Y is Z E No Y is Z
I Some Ys are Xs (Co) I Some Ys are Xs (Co)
I Some Xs are Zs O Some Xs are not Zs
i8
Firji Notions of Logic.
Fourth Figure.
A Every Z is Y (+)
A Every Y is X
I Some Xs are Zs
A Every Z is Y
E No Y is X
E No X is Z
I Some Zs are Ys
A Every Y is X
I Some Zs are Xs
E No Z is Y (Co)
A Every Y is X (Co +)
O Some Xs are not Zs
E No Z is Y (Co)
I Some Ys are Xs (Co)
O Some Xs are not Zs
The above is the ancient method of dividing fyllogifms ; but,
for the prefent purpofe, it will be fufficient to confider the fix
from which the reft can be obtained. And fmce fome of the
fix have X in the predicate of the conclufion, and not Z, I fhall
join to them the fix other fyllogifms which are found by tranf-
pofmg Z and X. The complete lift, therefore, of fyllogifms with
the weakeft premifes and the ftrongeft conclufions, in which a
comparifon of X and Z is obtained by comparifon of both with
Y, is as follows :
Every X is Y
Every Y is Z
Eveiy Z is Y
Every Y is X
Some Xs are Ys
No Y is Z
Some Zs are Ys
No Y is X
Every X is Z
Every X is Y
No Y is Z
Every Z is X
Every Z is Y
No Y is X
Some Xs are not Zs
Every X is Y
Some Zs are not Ys
Some Zs are not Xs
Every Z is Y
Some XsarenotYs
No X is Z
Some Xs are Ys
Every Y is Z
No Z is X
Some Zs are Ys
Every Y is X
Some Zs are not Xs
Every Y is X
Some Ys are not Zs
Some Xs are not Zs
Every Y is Z
Some Ys are not Xs
Some Xs are Zs
Some Zs are Xs
Some Xs are not Zs
Some Zs are not Xs
In the lift of page 12, there was nothing but recapitulation of
forms, each form admitting a variation by interchanging X and
Z. This interchange having been made, and the refults col
lected as above, if we take every cafe in which Z is the predi
cate, or can be made the predicate by allowable converfion, we
Flrjl Notions of Logic. 1 9
have a collection of all poflible weakeft forms in which the refult
is one of the four c Every X is Z, < No X is Z, Some Xs are Zs,
c Some Xs are not Zs ; as follows. The premifes are written
in what appeared the moft natural order, without diftincSHon of
major or minor.
Every X is Y
Every Y is Z
Every X is Z
Some Xs are Ys Some Zs are Ys
Every Y is Z Every Y is X
Some Xs are Zs Some Xs are Zs
Every X is Y Every Z is Y
No ZisY No XisY
No X is Z No X is Z
Some Xs are Ys Every Z is Y Every Y is X
No Z is Y Some Xs are not Ys Some Ys are not Zs
Some Xs are not Zs Some Xs are not Zs Some Xs are not Zs
Every affertion which can be made upon two things by com-
parifon with any third, that is, every fimple inference, can be
reduced to one of the preceding forms. Generally fpeaking, one
of the premifes is omitted, as obvious from the conclufion ; that
is, one premife being named and the conclufion, that premife is
implied which is neceflary to make the conclufion good. Thus,
if I fay, " That race muft have poflefTed fome of the arts of life,
for they came from Afia," it is obvioufly meant to be aflerted,
that all races coming from Afia muft have pofTefled fome of the
arts of life. The preceding is then a fyllogifm, as follows :
That race is c a race of Afiatic origin :
Every c race of Afiatic origin* is a race which muft
have pofleffed fome of the arts of life :
Therefore, That race is a race which muft have pofleffed
fome of the arts of life.
A perfon who makes the preceding aiTertion either means to
imply, antecedently to the conclufion, that all Afiatic races muft
have poflefTed arts, or he talks nonfenfe if he aflert the conclu-
2O Firft Notions of Logic.
fion pofitively. C X muft be Z,for it isY, can only be an inference
when c Every Y is Z. This latter propofition may be called
the fupprefled premife ; and it is in fuch fupprefled propofitions
that the greateft danger of error lies. It is alfo in fuch propofi
tions that men convey opinions which they would not willingly
exprefs. Thus, the honeft witnefs who faid, I always thought
him a refpe&able man he kept his gig, would probably not
have admitted in direct terms, Every man who keeps a gig muft
be refpectable.
I mall now give a few detached illuftrations of what precedes.
" His imbecility of character might have been inferred from
his pronenefs to favourites ; for all weak princes have this fail
ing." The preceding would ftand very well in a hiftory, and
many would pafs it over as containing very good inference.
Written, however, in the form of a fyllogifm, it is,
All weak princes are prone to favourites
He was prone to favourites
Therefore He was a weak prince
which is palpably wrong. (Rule I.) The writer of fuch a fen-
tence as the preceding might have meant to fay, for all who
have this failing are weak princes ; in which cafe he would have
inferred rightly. Every one mould be aware that there is much
falfe form of inference arifing out of badnefs of ftyle, which is
juft as injurious to the habits of the untrained reader as if the
errors were miftakes of logic in the mind of the writer.
* X is lefs than Y ; Y is lefs than Z : therefore X is lefs than
Z.* This, at firft fight, appears to be a fyllogifm ; but, on re
ducing it to the ufual form, we find it to be,
X is (a magnitude lefs than Y)
Y is (a magnitude lefs than Z)
Therefore X is (a magnitude lefs than Z)
which is not a fyllogifm, fmce there is no middle term. Evident
as the preceding is, the following additional propofition muft be
formed before it can be made explicitly logical. l If Y be a mag
nitude lefs than Z, then every magnitude lefs than Y is alfo lefs
than Z. There is, then, before the preceding can be reduced
to a fyllogiftic form, the neceffity of a deduction from the fecond
Firfl Notions of Logic. 2 r
premife, and the fubftitution of the refult inftead of that premife.
Thus,
X is lefs than Y
Lefs than Y is lefs than Z : following from Y is lefs than Z.
Therefore X is lefs than Z
But, if the additional argument be examined namely, if Y be
lefs than Z, then that which is lefs than Y is lefs than Z it will
be found to require precifely the fame confiderations repeated ;
for the original inference was nothing more. In fact, it may
eafily be feen as follows, that the proportion before us involves
more than any fimple fyllogifm can exprefs. When we fay that
X is lefs than Y, we fay that if X were applied to Y, every part
of X would match a part of Y, and there would be parts of Y
remaining over. But when we fay, Every X is Y, meaning
the premife of a common fyllogifm, we fay that every inftance of
X is an inftance of Y, without faying any thing as to whether
there are or are not inftances of Y ftill left,, after thofe which
are alfo X are taken away. If, then, we wifh to write an ordi
nary fyllogifm in a manner which mall correfpond with c X is lefs
than Y, Y is lefs than Z, therefore X is lefs than Z, we muft
introduce a more definite amount of aflertion than was made in
the preceding forms. Thus,
Every X is Y, and there are Ys which are not Xs
Every Y is Z, and there are Zs which are not Ys
Therefore Every X is Z, and there are Zs which are not Xs
Or thus :
The Ys contain all the Xs, and more
The Zs contain all the Ys, and more
The Zs contain all the Xs, and more
The moft technical form, however, is,
From Every X is Y ; [Some Ys are not Xs]
Every Y is Z ; [Some Zs are not Ys]
Follows Every X is Z ; [Some Zs are not Xs]
This fort of argument is called a fortiori argument, becaufe the
premifes are more than fufficient to prove the conclufion, and the
extent of the conclufion is thereby greater than its mere form
would indicate. Thus, X is lefs than Y, Y is lefs than Z,
22 Firft Notions of Logic.
therefore, a fortiori^ X is lefs than Z, means that the extent to
which X is lefs than Z muft be greater than that to which X is
lefs than Y, or Y than Z. In the fyllogifm laft written, either
of the bracketted premifes might be ftruck out without deftroying
the conclusion ; which laft would, however, be weakened. As
it ftands, then, the part of the conclufion, Some Zs are not
Xs, follows a fortiori.
The argument a fortiori may then be defined as a univerfally
affirmative fyllogifm, in which both of the premifes are fhewn to
be lefs than the whole truth, or greater. Thus, in c Every X is
Y, Every Y is Z, therefore Every X is Z, we do not certainly
imply that there are more Ys than Xs, or more Zs than Ys, fo
that we do not know that there are more Zs than Xs. But if
we be at liberty to ftate the fyllogifm as follows,
All the Xs make up part (and part only) of the Ys
Every Y is Z ;
then we are certain that
All the Xs make up part (and part only) of the Zs.
But if we be at liberty further to fay that
All the Xs make up part (and part only) of the Ys
All the Ys make up part (and part only) of the Zs
then we conclude that
All the Xs make up part of part (only) of the Zs
and the words in Italics mark that quality of the conclufion from
which the argument is called a fortiori.
Moft fyllogifms which give an affirmative conclufion are gene
rally meant to imply a fortiori arguments, except only in mathe
matics. It is feldom, except in the exacl: fciences, that we meet
with a propofition, Every X is Z, which we cannot immediately
couple with c fome Zs are not Xs.
When an argument is completely eftabliftied, with the excep
tion of one aiTertion only, fo that the inference may be drawn as
foon as that one aflertion is eftablifhed, the refult is ftated in a
form which bears the name of an hypothetical fyllogifm. The
word hypothefis means nothing but fuppofition ; and the fpecies
of fyllogifm juft mentioned firft lays down the aflertion that a
confequence will be true if a certain condition be fulfilled, and
Firft Notions of Logic. 23
then either aflerts the fulfilment of the condition, and thence the
confequence, or elfe denies the confequence, and thence denies
the fulfilment of the condition. Thus, if we know that
When X is Z, it follows that P is Q ;
then, as foon as we can afcertain that X is Z, we can conclude
that P is Q ; or, if we can fhew that P is not Q, we know that
X is not Z. But if we find that X is not Z, we can infer no
thing ; for the preceding does not aflert that P is Q^onfy when
X is Z. And if we find out that P is Qjwe can infer nothing.
This conditional fyllogifm may be converted into an ordinary
fyllogifm, as follows. Let K be any c cafe in which X is Z/ and
V, a cafe in which P is Q ; then the preceding afTertion amounts
to Every K is V. Let L be a particular inftance, the X of
which may or may not be Z. If X be Z in the inftance under
difcuflion, or if X be not Z, we have, in the one cafe and the
other,
Every K is V Every K is V
L is a K L is not a K
Therefore L is a V No conclufion
Similarly, according as a particular cafe (M) is or is not V, we
have
Every K is V Every K is V
M is a V M is not a V
No conclufion M is not a K
That is to fay : the aflertion of an hypothefis is the afTertion of
its neceflary confequence, and the denial of the necefTary confe
quence is the denial of the hypothefis : but the aflertion of the
neceflary confequence gives no right to aflert the hypothefis, nor
does the denial of the hypothefis give any right to deny the truth
of that which would (were the hypothefis true) be its neceflary
confequence.
Demonftration is of two kinds : which arifes from this, that
every propofition has a contradictory ; and of thefe two, one
muft be true and the other muft be falfe. We may then either
prove a propofition to be true, or its contradictory to be falfe.
x It is true that every X is Z, and * it is falfe that there are fome
Xs which are not Zs,* are the fame propofition ; and the proof
of either is called the indirect proof of the other.
24 Firft Notions of Logic.
But how is any propofition to be proved falfe, except by prov
ing a contradiction to be true ? By proving a necefTary confe-
quence of the propofition to be falfe. But this is not a complete
anfwer, fmce it involves the neceflity of doing the fame thing ;
or, fo far as this anfwer goes, one propofition cannot be proved
falfe unlefs by proving another to be falfe. But it may happen,
that a neceffary confequence can be obtained which is obvioufly
and felf-evidently falfe, in which cafe no further proof of the
falfehood of the hypothecs is neceflary. Thus the proof which
Euclid gives that all equiangular triangles are equilateral is of the
following ftructure, logically confidered.
(i.) If there be an equiangular triangle not equilateral, it fol
lows that a whole can be found which is not greater than its
part.*
(2.) It is falfe that there can be any whole which is not greater
than its part (felf evident).
(3.) Therefore it is falfe that there is any equiangular triangle
which is not equilateral ; or all equiangular triangles are equila
teral.
When a propofition is eftabliflied by proving the truth of the
matters it contains, the demonftration is called direfl ; when by
proving the falfehood of every contradictory propofition, it is
called indireft. The latter fpecies of demonftration is as logical
as the former, but not of fo fimple a kind ; whence it is defira-
ble to ufe the former whenever it can be obtained.
The ufe of indirect demonftration in the Elements of Euclid
is almoft entirely confined to thofe propofitions in which the con-
verfes of fimple propofitions are proved. It frequently happens
that an eftabliflied aflertion of the form
Every X is Z (i)
may be eafily made the means of deducing,
Every (thing not X) is not Z . . (2)
which laft gives
Every Z is X (3)
* This is the propofition in proof of which nearly the whole of the de
monftration of Euclid is fpent.
Fir/I Notions of Logic. 25
The converfion of the fecond propofition into the third is
ufually made by an indirect demonftration, in the following manner :
If poflible, let there be one Z, which is not X, (2) being true.
Then there is one thing which is not X and is Z ; but every
thing not X is not Z ; therefore there is one thing which is Z
and is not Z : which is abfurd. It is then abfurd that there
fhould be one fingle Z which is not X ; or, Every Z is X.
The following propofition contains a method which is of fre
quent ufe.
HYPOTHESIS. Let there be any number of propofitions or
afTertions, three for inftance, X, Y, and Z, of which it is the
property that one or the other muft be true, and one only. Let
there be three other propofitions, P, Q, and R, of which it is
alfo the property that one, and one only, muft be true. Let it
alfo be a connexion of thofe afTertions, that
When X is true, P is true
When Y is true, QJs true
When Z is true, R is true
CONSEQUENCE : then it follows that
When P is true, X is true
When QJs true, Y is true
When R is true, Z is true
For, when P is true, then Q,and R muft be falfe ; confequently,
neither Y nor Z can be true, for then Q_ or R would be true.
But either X, Y, or Z muft be true, therefore X muft be true ;
or, when P is true, X is true. In a fimilar way the remaining
afTertions may be proved.
Cafe i. If When P is Q, X is Z
When P is not Q, X is not Z
It follows that When X is Z, P is Q^
When X is not Z, P is not Q_
rWhen X is greater than Z, P is greater than Q
Cafe 2. If < When X is equal to Z, P is equal to Q
C When X is lefs than Z, P is lefs than Q
f When P is greater than Q, X is greater than Z
It follows that < When P is equal to Q, X is equal to Z
I When P is lefs than Q, X is lefs than Z
26
CHAPTER II.
On Objefts, Ideas, and Names.
LOGIC is derived from a Greek word (to yof) which fignifies
communication of thought, ufually by fpeech. It is the
name which is generally given to the branch of inquiry (be it called
fcience or art), in which the act of the mind in reafoning is con-
fidered, particularly with reference to the connection of thought
and language. But no definition yet given in few words has
been found fatisfactory to any confiderable number of thinking
perfons.
All exifting things upon this earth, which have knowledge of
their own exiftence, poffefs, fome in one degree and fome in
another, the power of thought, accompanied by perception, which
is the awakening of thought by the effect of external objects
upon the fenfes. By thought I here mean, all mental action, not
only that comparatively high ftate of it which is peculiar to man,
but alfo that lower degree of the fame thing which appears to be
poffeffed by brutes.
With refpect to the mind, confidered as a complicated ap
paratus which is to be ftudied, we are not even fo well off as
thofe would be who had to examine and decide upon the me-
chanifm of a watch, merely by obfervation of the functions of
the hands, without being allowed to fee the infide. A mechani
cian, to whom a watch was prefented for the firft time, would be
able to give a good guefs as to its ftructure, from his knowledge of
other pieces of contrivance. As foon as he had examined the law of
the motion of the hands, he might conceivably invent an inftru-
ment with fimilar properties, in fifty different ways. But in the
cafe of the mind, we have manifeftations only, without the
fmalleft power of reference to other fimilar things, or the leaft
knowledge of ftructure or procefs, other than what may be
derived from thofe manifeftations. It is the problem of the watch
to thofe who have never feen any mechanifm at all.
On ObjeEls, Ideas, and Names. 27
We have nothing more to do with the fcience of mind,
ufually called metaphyfics,* than to draw a very few neceflary
diftinctions, which, whatever names we ufe to denote them, are
matters of fact connected with our fubjedT:. Some modes of
expreffing them favor one fyflem of metaphyfics, and fome
another; but flill they are matters of obferved fad!:. Our words
muft be very imperfect fymbols, drawn from comparifon of the
manifeftations of thought with thofe of things in corporeal ex-
iftence. For inftance, I juft now fpoke of the mind as an
apparatus, or piece of mechanifm. It is a ftructure of fome fort,
which has the means of fulfilling various purpofes ; and fo far it
refembles the hand, which by the difpofition of bone and mufcle,
can be made to perform an immenfe variety of different motions
and grafps. Where the refemblance begins to be imperfect, and
why, is what we cannot know. In all probability we fhould
need new modes of perception, other fenfes befides fight, hear
ing, and touch, in order to know thought as we know colour,
lize, or motion. But the purpofe of the prefent treatife is only
the examination of fome of the manifeftations of thinking power
in their relation to the language in which they are expreiTed.
Knowledge of thought and knowledge of the refults of thought,
* All fyftems make an affumption of the uniformity of procefs in all
minds, carried to an extent the propriety of which ought to be a matter of
fpecial difcuflion. There are no writers who give us fo much muft with fo
little ivfiy, as the metaphyficians. If perfons who had only feen the outfide
of the timepiece, were to invent machines to anfwer its purpofe, they might
arrive at their objeft in very different ways. One might ufe the pendulum
and weight, another the fprings and the balance : one might difcover the
combination of toothed wheels, another a more complicated a6lion of lever
upon lever. Are we fare that there are not differences in our minds, fuch
as the preceding inftance may fuggeft by analogy ; if fo, ho f w are we fure ?
Again, if our minds be as tables with many legs, do we know that a weight
put upon different tables will be fupported in the fame manner in all. May
not the fame leg fupport much or all of a certain weight in one mind, and
little or nothing in another ? I have feen ftriking inftances of fomething like
this, among thofe who have examined for themfelves the grounds of the
mathematical fciences.
I would not diffuade a ftudent from metaphyfical inquiry j on the con
trary, I would rather endeavour to promote the defire of entering upon fuch
fubjefts : but I would warn him, when he tries to look down his own throat
with a candle in his hand, to take care that he does not fet his head on fire.
28 On Objetts, Ideas, and Names.
are very different things. The watch abovementioned might
have the functions of its hands difcovered, might be ufed in find
ing longitude (and even latitude) all over the world, without the
parties ufmg it having the fmalleft idea of its interior ftrudture.
That our minds, fouls, or thinking powers (ufe what name
we may) exift, is the thing of all others of which we are moft
certain, each for himfelf. Next to this, nothing can be more
certain to us, each for himfelf, than that other things alfo exift ;
other minds, our own bodies, the whole world of matter. But
between the character of thefe two certainties there is a vaft dif
ference. Any one who mould deny his own exiftence would,
if ferious, be held beneath argument : he does not know the
meaning of his words, or he is falfe or mad. But if the fame
man mould deny that any thing exifts except himfelf, that is, if
he mould affirm the whole creation to be a dream of his own
mind, he would be abfolutely unanfwerable. If I (who know he
is wrong, for 7 am certain of my own exiftence) argue with him,
and reduce him to filence, it is no more than might* happen in
his dream. A celebrated metaphyfician, Berkeley, maintained
that with regard to matter, the above is the ftate of the cafe :
that our impreffions of matter are only impreffions, communi
cated by the Creator without any intervening caufe of communi
cation.
Our moft convincing communicable proof of the exiftence of
other things, is, not the appearance of objects, but the neceffity
of admitting that there are other minds befides our own. The
external inanimate objects might be creations of our own
thought, or thinking and perceptive fun&ion : they are fo fome-
times, as in the cafe of infanity, in which the mind has frequently
the appearance of making the whole or part of its own external
world. But when we fee other beings, performing fimilar func
tions to thofe which we ourfelves perform, we come fo irrefiftibly
to the conclufion that there muft be other fentients like ourfelves,
that we mould rather compare a perfon who doubted it to one who
denied his own exiftence, than to one who fimply denied the real
external exiftence of the material world.
* It is not impoflible that in a real dream of deep, fome one may have
created an antagonift who beat him in an argument to prove that he was
awake.
On Objetts, Ideas , and Names. 29
When once we have admitted different and independent
minds, the reality of external objects (external to all thofe minds)
follows as of courfe. For different minds receive impreflions at
the fame time, which their power of communication enables
them to know are fimilar, fo far as any impreffions, one in each
of two different minds, can be known to be fimilar. There muft
be zfomewhat independent of thofe minds, which thus acts upon
them all at once, and without any choice of their own. This
fomewhat is what we call an external object : and whether it
arife in Berkeley s mode, or in any other, matters nothing to us
here.
We mall then, take it for granted that external objetts actually
exift, independently of the mind which perceives them. And this
brings us to an important diftinction, which we muft carry with
us throughout the whole of this work. Befides the actual exter
nal object, there is alfo the mind which perceives it, and what
(for want of better words or rather for want of knowing whether
they be good words or not) we muft call the image of that objett
in the mind, or the idea which it communicates. The termfub-
jeft is applied by metaphyficians to the perceiving mind : and
thus it is faid that a thing may be confidered fubjetfivety (with re
ference to what it is in the mind) or objectively (with reference
to what it is independently of any particular mind). But logicians
ufe the word fubject in another fenfe. In a proportion fuch as
bread is wholefome , the thing fpoken of, c bread , is called the
fubject of the propofition : and in fact the wordfubjetf is in com
mon language fo frequently confounded with objeft, that it is al-
moft hopelefs to fpeak clearly to beginners about themfelves as
filbjtft$. I mall therefore adopt the words ideal and objective,
idea and objeft^ as being, under explanations, as good as any
others : and better than fubjeft and objeft for a work on logic.
The word idea> as here ufed, does not enter in that vague fenfe
in which it is generally ufed, as if it were an opinion that might be
right or wrong. It is that which the object: gives to the mind,
or the ftate of the mind produced by the object. Thus the idea
of a horfe is the horfe in the mind : and we know no other horfe.
We admit that there is an external objeft^ a horfe, which may
give a horfe in the mind to twenty different perfons : but no one
of thefe twenty knows the object ; each one only knows his idea.
30 On Objefts, Ideas, and Names.
There is an object, becaufe each of the twenty perfons receives
an idea without communicating with the others : fo that there is
fomething external to give it them. But when they talk about
it, under the name of a horfe, they talk about their ideas. They
all refer to the object, as being the thing they are talking about,
until the moment they begin to differ: and then they begin to fpeak,
not of external horfes, but of impreflions on their minds ; at leaft
this is the cafe with thofe who know what knowledge is ; the pofi-
tive and the unthinking part of them ft ill talk of the horfe. And
the latter have a great advantage* over the former with thofe
who are like themfelves.
Why then do we introduce the term object at all, fince all our
knowledge lies in ideas ? For the fame reafon as we introduce the
term matter into natural philofophy, when all we know is form,
fize, colour, weight, &c., no one of which is matter, nor even all
together. It is convenient to have a word for that external
fource from which fenfible ideas are produced : and it is juft as
convenient to have a word for the external fource, material or
not, from which any idea is produced. Again, why do we fpeak
of our power of confidering things either ideally or objectively,
when as we can know nothing but ideas, we can have no right
to fpeak of any thing elfe ? The anfwer is that, juft as in other
things, when we fpeak of an object, we fpeak of the idea of an
objetf. We learn to fpeak of the external world, becaufe there
are others like ourfelves who evidently draw ideas from the fame
fources as ourfelves : hence we come to have the idea of thofe
fources, the idea of external objects, as we call them. But we do
not know thofe fources ; we know only our ideas of them.
We can even ufe the terms ideal and objective in what may
appear a metaphorical fenfe. When we fpeak of ourfelves in the
manner of this chapter, we put ourfelves, as it were, in the pofi-
tion of fpectators of our own minds : we fpeak and think of our
* One man aflerts a faEl on his own knowledge, another aflerts his full
con<vition of the contrary fa6l. Both ufe the evidence of their fenfes : but
the fecond knows that full conviftion is all that man can have. The firft
will carry it hollow in a court of juftice, in which perfons are conftantly
compelled to fwear, not only that they have an impreflion, but that the im-
preffion is correft j that is to fay, is the impreffion which mankind in general
would have, and muft have, and ought to have.
On Objeffs, Ideas, and Names. 3 1
own minds objectively. And it muft be remembered that by the
word object, we do not mean material object only. The mind
of another, any one of its thoughts or feelings, any relation of
minds to one another, a treaty of peace, a battle, a difcuffion
upon a controverted queftion, the right of conveying a freehold,
are all objects, independently of the perfons or things engaged
in them. They are things external to our minds, of which we
have ideas.
An object communicates an idea : but it does not follow that
every idea is communicated by an object. The mind can create
ideas in various ways ; or at leaft can derive, by combinations
which are not found in external exiftence, new collections of
ideas. We have a perfectly diftinct idea of a unicorn, or a flying
dragon : when we fay there are no fuch things, we fpeak ob
jectively only : ideally, they have as much exiftence as a horfe or
a fheep ; to a herald, more. Add to this, that the mind can
feparate ideas into parts, in fuch manner that the parts alone are
not ideas of any exifting feparate material objects, any more than
the letters of a word are conftituent parts of the meaning of the
whole. Hence we get what are called qualities and relations. A
ball may be hard and round, or may have hardnefs and round-
nefs : but we can not fay that hardnefs and roundnefs are feparate
external material objects, though they are objects the ideas of
which neceflarily accompany our perception of certain objects.
Thefe ideas are called abftratt as being removed or abftracted from
the complex idea which gives them : the abftraction is made by
comparifon or obfervation of refemblances. If a perfon had never
feen any thing round except an apple, he would perhaps never
think of roundnefs as a diftinct object of thought. When he faw
another round body, which was evidently not an apple, he would
immediately, by perception of the refemblance, acquire a feparate
idea of the thing in which they refemble one another.
Abftraction is not performed upon the ideas of material objects
only. For inftance, from conduct of one kind, running through
a number of actions, performed by a number of perfons, we get
the ideas of goodnefs, wickednefs, talent, courage. But we muft
not imagine that we can make ideally external reprefentation of
thefe words. They are objefls^ that is to fay, the mind confiders
them as external to itfelf : but they are not material objects.
32 On Objetts, Ideas, and Names.
Some people deny their exiftence, and look upon them as only
abftracl: words, or words under which we fpeak of minds or
bodies without fpecifying any more than one of the ideas pro
duced by thefe minds or bodies. For inftance, they aflert that
when we fay knowledge gives power it is really that perfons
with knowledge are therefore able, or have power, to produce,
or to do, what perfons without it cannot. This is a queftion
which it does not concern me here to difcufs.
Seeing that the mind poflefles a power of originating new
combinations of ideas, and alfo of abftracling from complex ideas
the more fimple ones of which, it feems natural to fay, they are
compofed, it has long been a queftion among metaphyficians
whether the mind has any ideas of its own which it poflefles in
dependently of all fuggeftion from external objects. It is not
neceflary that I mould attempt to lead the ftudent to any con-
clufion* on this fubject: for our purpofe, the diftinction between
ideas and objects, though it were falfe, is of more importance
than that between innate and acquired ideas, though it be true.
But one of thefe two things muft be true : either we have ideas
which we do not acquire from or by means of communication
with the external world (experience, trial of our fenfes) or there
is a power in the mind of acquiring a certainty and a generality
which experience alone could not properly give. For inftance,
we are fatisfied as of our own exiftence that feven and three col
lected are the fame as five and five, whatever the oljeRs may be
* It has always appeared to me much fuch a queftion as the following.
There are hooks which certainly catch fifh if put into the water ; and moft
certainly they have been put into the water. There are then fifh upon them.
But thefe fifh might have been on fome of them when they were put into the
water. It is to no purpofe to inquire whether it was fo or not, unlefs there
be fome diftin&ion between the fifties which may make it a queftion whether
fome of them could have been bred in the river into which the hooks were
put. The mind has certainly a power of acquiring and retaining ideas,
which power, when put into communication with the external world, it muft
exercife. There is no mind to experiment upon, except thofe which have
had fuch communication. Are there found any ideas which we have reafon
to think could not have been acquired by this communication ? any fifhes
which could not have come out of the river ? Metaphyficians feem to admit
that if any ideas be innate, they are thofe of fpace, time, and of caufe and
effeft : they feem alfo to admit, that if there be any ideas, which, not being
innate, are fure to be acquired, they are thefe very ones.
On Objects, Ideas, and Names. 3 3
which are counted: the thing is true of fingers, pebbles, counters,
fheep, trees, &c. &c. &c. We cannot have allured ourfelves of
this by experience : for example, we know it to be true of peb
bles at the North Pole, though we have never been there ; we
are as fure of it as of our own exiftence. I do not mean that we
have a rational conviction only, fit to act upon, that it is fo at the
North Pole, becaufe it is fo in every place in which it has been
tried : if we had nothing elfe, we ftiould have this ; but we feel
that this lefler conviction is fwallowed up by a greater. We have
the lefler conviction that the pebbles at the pole fall to the ground
when they are let go : we are very fure of this, without afferting
that it cannot be otherwife : we fee no impoffibility in thofe peb
bles being fuch as always to remain in air wherever they are
placed.* But that feven and three are no other than five and
five is a matter which we are prepared to affirm as pofitively of
the pebbles at the North Pole as of our own fingers, both that it
is fo, and that it muft be fo. Whence arifes this actual difference
in point of fact, between our mode of viewing and knowing
* Metaphyficians, in their fyftems, have often taken this diftinftion to be
one of fyftem only, treating it as a thing to be accepted or rejefted with
the fyftem, inftead of an aftual and Jndifputable phenomenon which re
quires explanation under any fyftem. Dr. Whewell, of all Englifh writers
on natural fcience I know, is the one who has made the faft, as a faft, per
vade his writings, fometimes attached to a fyftem, fometimes not. The
following remarks on the general fubjeft are worth confideration : " It is
indeed, extremely difficult to find, in fpeaking of this fubjeft, expreflions
which are fatisfaftory. The reality of the objeils which we perceive is a
profound, apparently an infoluble problem. We cannot but fuppofe that
exiftence is fomething different from our knowledge of exiftence : that what
exifts, does not exift merely in our knowing that it does : truth is truth
whether we know it or not. Yet how can we conceive truth, otherwife than
as fomething known ? How can we conceive things as exifting, without
conceiving them as objefts of perception ? Ideas and Things are conftantly
oppofed, yet necefTarily coexiftent. How they are thus oppofite and yet iden
tical, is the ultimate problem of all philofophy. The fucceffive phafes of
philofophy have confifted in feparating and again uniting thefe two oppofite
elements ; in dwelling fometimes upon the one and fometimes upon the other,
as the principal or original or only element ; and then in difcovering that
fuch an account of the ftate of the cafe was infufficient. Knowledge requires
ideas. Reality requires things. Ideas and things coexift. Truth is, and is
known. But the complete explanation of thefe points appears to be beyond
our reach."
34 On Objects, Ideas, and Names.
different fpecies of affertions ? the truth of the laft named affer-
tion is not born with us, for children are without it, and learn it
by experience, as we know. The mufl be fa cannot be acquired
from experience in the common way, for that fame experience
on which we rely tells us that however often a thing may have
been found true, whatever rule may have been eftablifhed by re
peated inftances, an exception may at laft occur. There feems
then to be in the mind a power of developing, from the ideas
which experience gives, a real and true diftinction of necefTary
and not neceffary, poffible and impoffible. The things which
are without us always confirm our neceffary propofitions : but
how we derive that complete aflurance that they will do fo as
faithfully as hitherto they have done fo, is not within our power
to fay.
Connected with ideas are the names we give them ; the fpoken
or written founds by which we think of them, and communicate
with others about them. To have an idea, and to make it the
fubjedt of thought as an idea, are two perfectly diftinct things :
the idea of an idea is not the idea itfelf. I doubt whether we
could have made thought itfelf the fubjedt of thought without
language. As it is, we give names to our ideas, meaning by a
name not merely a fingle word, but any collection of words which
conveys to one mind the idea in another. Thus a-man-in-a-
I lack-coat-riding-along-the high-road-on-a-bay-horfe is as much
the name of an idea as man, black, or horfe. We can coin
words at pleafure ; and, were it worth while, might invent a
fingle word to ftand for the preceding phrafe.
Names are ufed indifferently, both for the objects which pro
duce ideas, and for the ideas produced by them. This is a dif-
advantage, and it will frequently be neceffary to fpecify whether
we fpeak ideally or objectively. In common converfation we
fpeak ideally and think we fpeak objectively : we take for granted
that our own ideas are fit to pafs to others, and will convey to
them the fame ideas as the objects themfelves would have done.
That this may be the cafe, it is neceffary firft, that the object
fhould really give us the fame ideas as to others ; fecondly, that
our words fhould carry from us to our correfpondents the fame
ideas as thofe which we intended to exprefs by them. How,
and in what cafes, the firft or the fecond condition is not ful-
On Objects, Ideas, and Names. 35
filled, it is impoflible to know or to enumerate. But we have
nothing to do here except to obferve* that we are only incidentally
concerned with this queftion in a work of logic. We prefume
fixed and, if objective, objectively true ideas, with certain names
attached : fo that it is never in doubt whether a name be or be not
properly attached to any idea. This method muft be followed in
all works of fcience : a conceivably attainable end is firft pre-
fumed to be attained, and the confequences of its attainment are
ftudied. Then, afterwards, comes the queftion whether this end
is always attained, and if not, why. The way to mend bad roads
muft come at the end, not at the beginning, of a treatife on the
art of making good ones.
Every name has a reference to every idea, either affirmative or
negative. The term horfe applies to every thing, either pofitively
or negatively. This (no matter what I am fpeaking of) either is
or is not a horfe. If there be any doubt about it, either the idea
is not precife, or the term horfe is ill underftood. A name ought
to be like a boundary, which clearly and undeniably either {huts
in, or fhuts out, every idea that can be fuggefted. It is the im
perfection of our minds, our language, and our knowledge of
external things, that this clear and undeniable inclufion or exclu-
fion is feldom attainable, except as to ideas which are well within
the boundary : at and near the boundary itfelf all is vague. There
are decided greens and decided blues : but between the two
colours there are (hades of which it muft be unfettled by uni-
verfal agreement to which of the two colours they belong. To
the eye, green pafles into blue by imperceptible gradations : our
fenfes will fuggeft no place on which all agree, at which one is
to end and the other to begin.
But the advance of knowledge has a tendency to fupply means
of precife definition. Thus, in the inftance above cited, Wol-
lafton and Fraunhofer have difcovered the black lines which al
ways exift in the fpe&rum of folar colours given by a glafs prifm,
in the fame relative places. There are definite places in the fpec-
trum,by the help of which the place of any {hade of colour therein
exifting may* be afcertained, and means of definition given.
When a name is complex, it frequently admits of definition,
* It is quite within the portabilities of the application of fcience to the
36 On Objefts, Ideas y and Names.
nominal or real. A name may be faid to be defined nominally
when we can of right fubftitute for it other terms. In fuch a
cafe, a perfon may be made to know the meaning of the word
without accefs to the object of which it is to give the idea. Thus,
an ijland is completely defined in c land furrounded by water.
In definition, we do not mean that we are necefTarily to have
very precife terms in which to explain the name defined : but, as
the terms of the definition fo is the name which is defined ; ac
cording as the firft are precife or vague, clear or obfcure, fo is the
fecond. Thus there may be a queftion as to the meaning of
land: is a marfh flicking up out of the water an ifland ? Some
will fay that, as oppofed to water, a marfh is land, others may
confider marfh as intermediate between what is commonly called
[dry] land and water. If there be any vaguenefs, the term ifland
muft partake of it : for ifland is but fhort for ( land furrounded by
water, whether this phrafe be vague or precife. This fort of de
finition is nominal^ being the fubftitution of names for names. It
is complete, for it gives all that the name is to mean. An ifland,
as fuch, can have nothing neceflarily belonging to it except what
neceflarily belongs to c land furrounded by water/ By real de
finition, I mean fuch an explanation of the word, be it the whole
of the meaning or only part, as will be fufficient to feparate the
things contained under that word from all others. Thus the
following, I believe, is a complete definition of an elephant-, c an
animal which naturally drinks by drawing the water into its nofe,
and then fpirting it into its mouth. As it happens, the animal
which does this is the elephant only, of all which are known upon
the earth : fo long as this is the cafe, fo long the above definition
anfwers every purpofe ; but it is far from involving all the ideas
which arife from the word. Neither fagacity, nor utility, nor the
production of ivory, are neceflarily connected with drinking by
help of the nofe. And this definition is purely objective ; we do
not mean that every idea we could form of an animal fo drinking
is to be called an elephant. If a new animal were to be difco-
vered, having the fame mode of drinking, it would be a matter
of pure choice whether it fhould be called elephant or not. It
arts that the time fhould come when the fpe6lrum, and the lines in it, will
be ufed for matching colours in every linen-draper s mop.
On Objefts, Ideas, and Names* 37
muft then be fettled whether it fhall be called an elephant, and
that race of animals fhall be divided into two fpecies, with diftinc-
tive definitions ; or whether it fhall have another name, and the
definition above given fhall be incomplete, as not ferving to draw
an entire diftinc~tion between the elephant and all other things.
It will be obferved that the nominal definition includes the real,
as foon as the terms of fubftitution are really defined : while the
real definition may fall fhort of the nominal.
When a name is clearly underftood, by which we mean when
of every objecl: of thought we can diftinclly fay, this name does
or does not, contain that objecl: we have faid that the name ap
plies to everything, in one way or the other. The word man
has an application both to Alexander and Bucephalus : the firft
was a man, the fecond was not. In the formation of language,
a great many names are, as to their original fignification, of a
purely negative character : thus y parallels are only lines which do
not meet, aliens are men who are not Britons (that is, in our
country). If language were as perfecT: and as copious as we
could imagine it to be, we ftiould have, for every name which has
a pofitive fignification, another which merely implies all other
things : thus, as we have a name for a tree, we fhould have an
other to fignify every thing that is not a tree. As it is, we have
fometimes a name for the pofitive, and none for the negative, as
in tree : fometimes for the negative and none for the pofitive, as
in -parallels : fometimes for both, as in a frequent ufe of perfon
and thing. In logic, it is defirable to confider names of inclufion
with the correfponding names of exclufion : and this I intend to
do to a much greater extent than is ufual : inventing names of
exclufion by the prefix not, as in tree and not-tree, man and not-
man. Let thefe be called contrary ,* or contradictory^ names.
Let us take a pair of contrary names, as man and not-man.
It is plain that between them they reprefent everything imaginable
or real, in the univerfe. But the contraries of common language
ufually embrace, not the whole univerfe, but fome one gene
ral idea. Thus, of men, Briton and alien are contraries : every
man muft be one of the two, no man can be both. Not-Briton
and alien are identical names, and fo are not-alien and Briton.
* I intend to draw no diftin&ion between thefe words.
38 On ObjeEts, Ideas , and Names.
The fame may be faid of integer and fraction among numbers,
peer and commoner among fubje&s of the realm, male and fe
male among animals, and fo on. In order to exprefs this, let us
fay that the whole idea under confideration is the univerfe (mean
ing merely the whole of which we are confidering parts) and let
names which have nothing in common, but which between them
contain the whole idea under confideration, be called contraries
/, or with refpeft to, that univerfe. Thus, the univerfe being
mankind, Briton and alien are contraries, as are foldier and civi
lian, male and female, &c. : the univerfe being animal, man and
brute are contraries, &c.
Names maybe reprefented by the letters of the alphabet: thus
A, B, &c., may ftand for any names we are confidering, fimple or
complex. The contraries may be reprefented by not- A, not-B,
&c., but I fliall ufually prefer to denote them by the fmall letters
#, , &c. Thus, everything in the univerfe (whatever that uni
verfe may embrace) is either A or not- A, either A or a, either
B or , &c. Nothing can be both B and b -, every not-B is ,
and every not-/ is B : and fo on.
No language, as may well be fuppofed, has been conftructed
beforehand with any intention of providing for the wants of any
metaphyfical fyftem. In moft, it is feen that the neceflity of
providing for the formation of contrary terms has been obeyed.
Our own language has borrowed from the Latin as well as from
its parent : thus we have imperfeft, dif agree able, as well as un
formed and witlefs. There is a choice of contraries without very
well fettled modes of appropriation : ftanding for different de
grees of contrariety. Thus we have not perfeft which is not fo
ftrong a term as imperfett ; and not imperfetf, the contrary of a
contrary, which is not fo ftrong as perfefl. The wants of com
mon converfation have fometimes retained a term and allowed
the contrary to fink into difufe ; fometimes retained the contrary
and neglected the original term ; fometimes have even introduced
the contrary without introducing any term for the original no
tion, and allowed no means of expreiling the original notion
except as the contrary of a contrary. If we could imagine a
perfect language, we mould fuppofe it would contain a mode of
Signifying the contrary of every name : this indeed our own lan
guage may be faid to have, though fometimes in an awkward and
On Objects, Ideas y and Names. 39
unidiomatic manner. One inflexion, or one additional word,
may ferve to fignify a contrary of any kind : thus not man is
effective to denote all that is other than man. But there is a
wider want, which can only be partially fupplied, for its complete
fatisfaction would require words almoft beyond the power of
arithmetic to count : and all that has been done to make it lefs
confifts, in our language and in every other, moftly in the forma
tion of compound terms, be they fubftantive and adjective, dou
ble fubftantives, or any others. A clafs of objects has a fub-clafs
contained within it, the individuals of which are diftinguifhed
from all others of the clafs by fomething common to them and
them only. If the diftinguifhing characteriftic have been fepa-
rated, and a word formed to fignify the abftract idea, that word,
or an adjective formed from it (if it be not an adjective) is joined
with the general name of the clafs. Thus we have ftrong men,
white horfes, &c. Or it may happen that the individuals of the
fub-clafs take, in right of the diftinguiming characteriftic, a per
fectly new name, and by the moft varied rules. A corn-grinding
man is called from the implement he ufes, a miller ; a meat-
killing man from the organ which he fupplies, a butcher^ (if the
firft idea of the etymology of this word be correct). Other men
ufe mills and other trades feed the mouth : ftill cuftom has fet
tled thefe terms, though the firft is only connected with its origin
by the fpelling, and the fecond by a derivation which muft be
fought in another language. But again, it will more often hap
pen that a diftinctive characteriftic, belonging to fome only, gives
no distinctive name to thofefome, which ftill remain an unnamed
feme out of the whole, to be feparated by the defcription of their
characteriftic when wanted, inftead of being the all of a name
invented to exprefs them, and them alone of their clafs. In fuch
a predicament, for inftance, are men who have never feen the
fea, as diftinguifhed from thofe who have feen it. Hence it ap
pears that particular propofitions are not fo diftinct from uni-
verfal ones in real character as they are generally made to be.
If I fay c fome As are Bs the reader may well fuppofe that it is
not often neceflary to advert to this fact : had it been fo, a name
would have been invented fpecially to fignify ( As which are Bs.
If this name had been C, the proportion would have been c every
C is B.
40 On Objetts, Ideas, and Names.
The fame convenience which dilates the formation of a name
for one fub-clafs and not for another, rules in the formation of
contrary terms, as already noted. And thefe caprices of language
for logically confidered they are nothing elfe, though their for
mation is far from lawlefs make it defirable to include in a for
mal treatife the moft complete confideration of all propofitions,
with reference not only to their terms, but alfo to the contraries
of thofe terms. Every negative proportion is affirmative, and
every affirmative is negative. Whatever completely does one of
the two, include or exclude, alfo does the other. If I fay that
c no A is B, then, b being the name of every thing not B in the
univerfe of the propofition, I fay that c every A is b : and if I
fay that every A is B, I fay that c no A is b. Whether a lan
guage will happen to poflefs the name B, or , or both, depends
on circumftances of which logical preference is never one, ex
cept in treatifes of fcience. The Englifh may poflefs a term for
B, the French only for b : fo that the fame idea muft be prefented
in an affirmative form to an Englimman, as in every A is B,
and in a negative one to a Frenchman, as c no A is b. 9 From
all this it follows that it is an accident of language whether a pro
pofition is univerfal or particular, pofitive or negative. We,
having the names A and B, may be able to fay c every A is B :
another language, which only names the contrary of B, muft fay
< no A is b. 3 A third language, in which As have not a feparate
name, but are only individuals of the clafs C, muft fay ( fome Cs
are Bs ; while a fourth, which is in the further predicament of
naming only , muft have it c fome Cs are not s. When we
come to confider the fyllogifm, we ftiall have full confirmation of
the correctnefs and completenefs of this view.
It may be objected that the introduction of terms which are
merely negations of the politive ideas contained in other terms
is a fpecies of fiction. I anfwer, that, firft, the fiction, if it be a
fiction, exifts in language, and produces its effects : nor will it
eafily be proved more fictitious than the invention of founds to
ftand for things. But, fecondly, there is a much more effective
anfwer, which will require a little development.
When writers on logic, up to the prefent time, ufe fuch con
traries as man and not-man, they mean by the alternative, man
and everything elfe. There can be little effective meaning, and
On Objects, Ideas, and Names. 41
no ufe, in a claflification which, becaufe they are not men, in
cludes in one word, not-man, a planet and a pin, a rock and a
featherbed, bodies and ideas, wifhes and things wimed for. But
if we remember that in many, perhaps moil, proportions, the
range of thought is much lefs extenfive than the whole univerfe,
commonly fo called, we begin to find that the whole extent of a
fubje6t of difcuflion is, for the purpofe of difcuflion, what I have
called a univerfe, that is to fay, a range of ideas which is either
exprefTed or underilood as containing the whole matter under
confideration. In fuch univerfes, contraries are very common :
that is, terms each of which excludes every cafe of the other,
while both together contain the whole. And, it muft be ob-
ferved that the contraries of a limited univerfe, though it be a
fufficient real definition of either that it is not the other, are fre
quently both of them the objects from which pofitive ideas are
obtained. Thus, in the univerfe of property, perfonal and real
are contraries, and a definition of either is a definition of the
other. But though each be a negative term as compared with
the other, no one will fay that the idea conveyed by either is that
of a mere negation. Money is not land, but it is fomething. And
even when the contrary term is originally invented merely as a
negation, it may and does acquire pofitive properties. Thus
alien is ftriclly not-Briton : but fuppofe a man taken in arms
againft the crown on fome fpot within its dominions, and claim
ing to be a prifoner of war. The anfwer that he is a Britiih fub-
jecT: is a negation : to eftabliih his pofitive claim he firft muft
prove himfelf an alien, and moreover that he is in another pofitive
predicament, namely, that he is the fubjecl: of a power at war
with Great Britain. Accordingly, of two contraries, neither
muft be confidered as only the negation of the other : except when
the univerfe in queftion is fo wide, and the pofitive term fo li
mited, that the things contained under the contrary name have
nothing but the negative quality in common.
Perception of agreements and difagreements is the foundation
of all affertion : the acquirement of fuch perception with refpecl:
to any two ideas by the comparifon of both with a third, is the
procefs of all Inference. To infer, by comparifon of abftract ideas,
is the peculiar privilege of man j to need inference is his imper
fection. To what point man would carry inference if he wanted
42 On Objetts, Ideas, and Names.
language, how much further the lower animals could carry what
they have of it if they had language, are queftions on which it is
vain to fpeculate. The words is and is not, which imply the
agreement or difagreement of two ideas, muft exift, explicitly or
implicitly, in every afTertion. And what we call agreement or
difagreement, may be reduced to identity or non-identity. When
we fay John is a man, we have the firft and moft objective form of
afTertion. Looked at in the moft objective point of view it is
only this, John is one of the individual objects who are called
man. Looked at ideally, the proportion is more general. The
idea of man, gathered from inftances, prefents itfelf as a collective
mafs of ideas, of which we can figure to ourfelves an inftance
without neceflarily calling up the idea of any man that ever ex-
ifted. In the ideal conception of man, Achilles is a man as
much as the Duke of Wellington, whether the former ever ex-
ifted objectively or no : of all the ideas of man which the mind
can imagine, the former is one as well as the latter.
The feparation of ideas, or formation of abftract ideas, and
afTertion by means of them, prefents nothing, for our purpofe,
which differs from the former cafe. If we fay this picture is
beautiful, the mere phrafe is incomplete, for c beautiful is only an
attribute, a purely ideal reference to a claffification which the
mind makes, dictated by its own judgment. The picture being
a material object, cannot be anything but an object, cannot be
long to any clafs of notions, unlefs that clafs contain objects.
What the propofition may mean is to a certain extent dependent
upon the implied fubftantive to which beautiful belongs : that is>
to the clafs of objects which the propofition implies the mind to
have feparated into beautiful and not beautiful. It may be that
the picture is a beautiful picture : or a beautiful work of art, tak
ing its place in that divifion by which not only pictures, but fta-
tues, buildings, reliefs, &c. are feparated into beautiful and other-
wife : or a beautiful creation of human thought, placed among
works of art, imagination, or fcience, &c. in the fubdivifion beau
tiful : or finally, it may be a beautiful thing, placed with all ob
jects of perception in a fimilar fubdivifion.
In all aiTertions, however, it is to be noted, once for all, that
formal logic, the object of this treatife, deals with names and not
with either the ideas or things to which thefe names belong. We
On Objetts, Ideas, and Names. 43
are concerned with the properties of < A is B and c A is not B
fo far as they prefent an idea independently of any fpecification
of what A and B mean : with fuch ideas upon propofitions as
are prefented hy their forms, and are common to all forms of the
fame kind. The reality of logic is the examination of the ufe of
is and is not : the tracing of the confequences of the application
of thefe words. The argument when the fun mines it is day :
but it is not day, therefore the fun does not mine, contains a
theory and two facts, the latter of which is made to follow from
the former by the theory. That inference is made is feen in the
word therefore : and the fentence is capable of being put upon its
trial for truth or falfehood by logical examination. But this exa-
mination rejects the meaning of fun and day, the truth of the
theory and of the facts ; and only inquires into the right which
the fentence, of its own ftructure, gives us to introduce the word
therefore. It merely enters upon c when A is, B is ; but B is
not ; therefore A is not: and decides that this is a correct junc
tion of precedents and confequent, an exhibition of necefTary con
nexion between what goes before and after therefore, and a de
velopment, in the latter, of what is virtually, though not actually,
exprefled in the former. What A and B may mean is of no
confequence to the inference, or right to bring In A is not.
Thus A and B, diverted of all fpecific meaning, are really
names as names, independently of things : or at leaft may be fo
confidered. For the truth of the propofition, under all mean
ings, gives us a right to fuppofe, if we like, that names are the
meanings- that is to fay, that we may put it thus, c When the
name A is, the name B is: but the name B is not; therefore the
name A is not.
It is not therefore the object of logic to determine whether
conclufions be true or falfe ; but whether what are aflerted to
be conclufions are conclufions. By a conclufion is meant that which
is and muft be Jhut in with certain other preceding things put in
firft : it is that which muft have been put into a fentence becaufe
certain other things were put in. To Infer a conclufion is to bring
in, as it were, the direft ftatement of that which has been virtu
ally ftated already has been Jhut in. When we fay A is B,
B is C ; we conclude A is C ; it would be more correct to fay
( A is B, B is C ; we have concluded A is C . We mould never
44 On Objects y Ideas y and Names.
think of faying we have put into a box a man s upper drefs of
the colour of the trees ; therefore we muft put In a green coat ;
we fhould fay c we have put in To infer the conclufion then
is to bring in a ftatement that we have concluded.
Inference does not give us more than there was before : but it
may make us fee more than we faw before : ideally fpeaking,
then, it does give us (in the mind) more than there was before.
But the homely truth that no more can come out than was in,
though accepted as to all material objects even by metaphyficians
who are generally well pleafed to find the key of a box which
contains what they want, though fure that it will put in no more
than was there already has been applied to logic, and even to
mathematics, in depreciation of their rank as branches of know
ledge. Thofe who have made this ftrangeft of human errors
muft have aflumed an ideal omnifcience, and looked at human
imperfection objectively. Omnifcience need neither compare
ideas, nor draw inferences : the conclufion which we deduce
from premifes, is always prefent with them ; truths are concomi
tants^ not conferences. When we fay that one affertion follows
from another, we fpeak purely ideally, and defcribe an imperfec
tion of our own minds : it is not that the confequence follows
from the premifes, but that our perception of the confequence fol
lows our perception of the premifes : the confequence, objectively
fpeaking, is in, and with, and of, the premifes. We fpeak wrongly
if we fpeak ideally, when we fay that A is C, is in < A is B and
B is C : in fact, it is only by giving an objective view to the
argument, that we can even aflert that it will be feen. To un
cultivated minds, this fimple conclufion is never concomitant
with the premifes, and only with fome difficulty a confequence.
From the certainty that a confequence may be made to come
out, which is an allegorical ufe of the word o/, we afliime a right
to declare, by the fame fort of allegory,* that it was in. The
premifes therefore contain the conclufion : and hence fome have
fpoken as if in ftudying how to draw the conclufion, we were
ftudying to know what we knew before. All the propofitions
of pure geometry, which multiply fo faft that it is only a fmall
* I am of opinion that it is more confiftent with analogy to fay that the
hypothecs is contained in its neceflary confequence, than to fay that the for
mer contains the latter. My reafon will appear in the courfe of the work.
On Objetfsy Ideas, and Names. 45
and ifolated clafs even among mathematicians who know all that
has been done in that fcience, are certainly contained in, that is
neceflarily deducible from, a very few fimple notions. But to be
known from thefe premifes is very different from being known with
them.
Another form of the aflertion is that confequences are virtually
contained in the premifes, or (I fuppofe) as good as contained in
the premifes. Perfons not fpoiled by fophiftry will fmile when
they are told that knowing two ftraight lines cannot enclofe a
fpace, the whole is greater than its part, &c. they as good as
knew that the three interfe&ions of oppofite fides of a hexagon
infcribed in a circle muft be in the fame flraight line. Many
of my readers will learn this now for the firft time : it will com
fort them much to be aflured, on many high authorities, that they
virtually knew it ever fmce their childhood. They can now pon
der upon the diftinclion, as to the ftate of their own minds, be
tween virtual knowledge and abfolute ignorance.
There muft always be fome contention as to the relative value
of their knowledge between the ftudents of the things which we
can fee muft have been, and of the things which, for what
we can fee, might have been otherwife. How much of the dif-
tinclion is due to our ignorance, no one can tell. In the mean
time, it is of more ufe to point out the advantage, as things are,
of ftudying both kinds of knowledge, than to attempt to inftitute
a rivalry between them. Thofe who have undervalued the ftudy
of neceflary confequences, have allowed themfelves, in illuftrat-
ing their argument, phrafes * which taken literally, mean more
perhaps than they intended.
* We might fometimes take them to mean that the ftudy of necefTary
connexion in logic, mathematics, &c., is at leaft ufelefs, if not pernicious.
Now we mould fuppofe, if this be what they mean, that clofe connexion,
fhort of abfolute necefllty, muft partake fomewhat of the fame charaaer. If
the abfolute mathematical neceflity that three angles of a triangle are equal
to two right angles is therefore to be avoided, the ftudy of phyfics, in which
there are the neceffities which we exprefs by the term laws of nature, muft
do fome harm. Hiftory, in which we may fo often count upon the aftions
which motives will produce, cannot be quite faultlefs : and there are laws of
formation in language which might as well be kept out of fight, for they
aft almoft with the uniformity of laws of nature. True knowledge muft
confift in the ftudy of the aftions of madmen : that a certain man imagined
46 On Objects, Ideas, and Names.
The ftudy of logic, then, confidered relatively to human know
ledge, {lands in as low a place as that of the humble rules of
arithmetic, with reference to the vaft extent of mathematics and
their phyfical applications. Neither is the lefs important for its
lowlinefs : but it is not every one who can fee that. Writers on
the fubject frequently take a fcope which entitles them to claim
for logic one of the higheft places : they do not confine them-
felves to the connexion of premifes and conclufion, but enter
upon the periculum et commodum of the formation of the premifes
themfelves. In the hands of Mr. Mill, for example (and to fome
extent in thofe of Dr. Whateley) logic is the fcience of diftin-
guifhing truth from falfehood, fo as both to judge the premifes
and draw the conclufion, to compare name with name, not only
as to identity or difference, but in all the varied affociations of
thought which arife out of this comparifon.
CHAPTER III.
On the Abflraft Form of the Proportion.
IN the preceding chapter, I have endeavoured to put together
fuch notions on the actual fources of our knowledge as may
give the reader the means of thinking upon points which any
fyftem of logic, however reftricted, muft neceffarily fuggeft.
We cannot attempt to connect our ufe of words with our notions
of things, without the occurrence of a great many difficulties, a
great many fources of adverfe theories, and of never-ending dif-
putes. We cannot even reprefent phenomena, as phaenomena,
except in the language of fome fyftem, and it may be of a wrong
one. The confidence which the favourers of thefe feveral the
ories place in their corredtnefs is a fufficient reafon for keeping
the account of the procefs of the underftanding, fo far as it can
be made an exact fcience, as diftinct as poflible from all of them :
for they differ widely, and if they agree in anything which can
himfelf to be Csefar, when he might juft as well have been Newton or Ne
buchadnezzar, muft be a real bit of knowledge, not virtually contained in
anything elfe, wholly or partially.
On the Abftratt Form of the Proposition. 47
be diftincSHy apprehended, it is only in having names of great
authority enrolled among the partifans of every one.
In order to examine the laws of inference, of the way of dif-
tin&ly perceiving the right to fay * therefore, fo that, ( whence
it muft be, &c., &c., in a manner which may be admitted, fo
far as we go, by all, we mufl make this feparation very complete.
All admit propofitions, as man is animal, ( no man is faultlefs;
all are, after a little thought, agreed upon the modes of inference :
but upon the import of a fimple proportion, there is every kind
of difference. How much we mean, when we fay c man is ani
mal, and how we arrive at our meaning, is matter for volumes
on different fides of unfettled queftions.
In order properly to examine the laws of inference, or of any
thing elfe, we muft firft endeavour to arrive at a diftincl: abftrac-
tion of fo much of the idea we are concerned with, as is itfelf the
precedent reafon, if it be right fo to fpeak, of the law in queftion.
This is an eafy procefs upon familiar things. We do not give
the carriers of goods much credit for profundity, in feeing that,
on a given road, there is only the difference of weight by which
they are concerned to know how one parcel differs from another ;
and further that, as long as they have to carry a pound, it matters
nothing whether it be of fugar or iron. It is this procefs which
we want to perform to the utmoft, upon the fimple propofition.
Writers on logic, from Ariftotle downwards, have made a large
and important ftep in fubftituting for fpecific names, with all their
fuggeftions about them, the mere letters of the alphabet, A, B,
C, &c. Thefe letters arefymbols, and general iymbols : each of
them ftands for any one we pleafe of its clafs. But what are
they iymbols of, names, ideas,* or the objects which give thofe
ideas ? The anfwer is, that this is precifely one of thofe confi-
derations which we may leave behind, in abftra&ing what is
neceffary to an examination of the laws of inference. The only
condition is, that we are to confine ourfelves to one or the other.
When we fay man is animal, it may be that the name man is
contained in the name animal, that the idea of man is contained
in that of animal, or that the object man is in the object animal.
Or if there were twenty more different appropriations of the
Meaning of courfe (page 30) ideas of ideas, and ideas of objefts.
48 On the AbJlraSl Form
fymbols, the fame thing might be faid of each. This is, I believe,
the firft ufe of the general fymbol in order of time ; the algebrai
cal ufe of letter or other fymbol, to defignate number, being both
fubfequent and derived.
When therefore we fay c Every X is Y , we underftand that
X is a fymbol which reprefents an inftance of a name, idea, ob
ject, &c., as the cafe may be. There may be more or fewer of
fuch inftances ; they may be numerable or innumerable. And the
fame of Y. The language of logicians has generally been unfa
vourable to the diftincl: perception of their terms being diftribu-
tively applicable to clafles of inftances. They have rather been
quantitative than quantvpllcitative : expreffing themfelves as if, in
faying that animal is a larger or wider term than man, they would
rather draw their language from the idea of two areas, one of
which is larger than the other, than from two collections of indi-
vifible units, one of which is in number more than the other.
They have even carried this fo far as to make it doubtful, except
from context, whether their diftincT:ion between univerfal and par
ticular is that of all and fome, or of the whole and part. If their
inftances had been white fquares ^ their c all A is B and c fome A
is B might have applied as well to c All the fquare is white and
c Some of the fquare is white as to All the fquares are white and
c Some of the fquares are white. I fhall take particular care
to ufe numerical language, as diftinguifhed from magnitudinal,
throughout this work, introducing of courfe, the plurals Xs, Ys,
Zs, &c.
I may mention here another mode of fpeaking, which will, I
think, appear objectionable to all who are much ufed to confi-
deration of quantity. When a compound idea contains two or
more fimpler ones, fome logicians have fpoken as if the com
bination were legitimately reprefented by arithmetical addition.
Thus the combination of the ideas of animal and rational muft
give the idea of man : for the two notions co-exift in nothing
elfe that we know of. Accordingly, fome write animal + ratio
nal = man. If this be intended as an abftraclion of the notation of
arithmetic, for the purpofe of fitting to it entirely different mean
ing, there is of courfe no objection which I need confider here :
but it feems to me that more is meant, and that thofe who have
ufed this notation imagine a great refemblance between combining
of the Proportion. 49
ideas, and cumulating them. What the difference is, I cannot
pretend to fay, any more than I can pretend to fay what the dif
ference is between chemically combining volumes of oxygen and
hydrogen, fo as to produce water, and fimple cumulation of them
in the fame veflel, fo as to produce a mixed gas : every beginner
knows that the electric fpark, or fome other inexplicable agency,
is neceffary to turn the mixed gas into a new chemical combina
tion. But that the difference exifts in the former cafe alfo, feems
to me as clear as any thing I can imagine. Even in chemiftry the
cumulative notation, which was once thought an all-fufficient mode
of expreffing the refults of the atomic theory, has failed with the
progrefs of knowledge. To a confiderable extent, the introduc
tion of modes of cumulation as yet anfwers the purpofe : but
there ftill remain ifomeric compounds, differing in properties, but
of the fame compofition. For example, the tartaric and racemic
acids: of which Profeffor Graham fays {Elements of Chemiftry p.
158), "A nearer approach to identity could fcarcely be con
ceived than is exhibited by thefe bodies, which are, indeed, the
fame both in form and compofition But by no treat
ment can the one acid be tranfmuted into the other." If the
above mode of confounding cumulation and combination be ad-
miffible, I fuppofe we might eafily give ourfelves a right to fay that
2 -f- 2 + addition = 4
an equation at which the mathematician would flare.
So much for the characleriftics of the terms of a proportion,
as wanted for the abftracl: forms of inference. It remains to
confider thofe of the connecting copulae is and is not.
The complete attempt to deal with the term is would go to
the form and matter of every thing in exiftence^ at leaft, if not to
the poflible form and matter of all that does not exift, but might.
As far as it could be done, it would give the grand Cyclopaedia,
and its yearly fupplement would be the hiftory of the human race
for the time. That logic exifts as a treated fcience, arifes from
the characlieriftics of the word, requifite to be abftra6ted in ftudy-
ing inference, being few and eafily apprehended. It may be ufed
in many fenfes, all having a common property. Names, ideas,
and objects, require it in three different fenfes. Speak of names^
and fay c man is animal : the is is here an is of applicability ; to
50 On the Abftratt Form
whatfoever (idea, object, &c.) man is a name to be applied, to
that fame (idea, object, &c.) animal is a name to be applied. As
to ideas, the is is an is of pofTeffion of all efTential characteriftics ;
man is an idea which poflefies, contains, prefents, all that is con-
ftitutive of the idea animal. As to abfolute external objects, the
is is an is of identity, the moft common and pofitive ufe of the
word. Every man is one of the animals ; touch him, you touch
an animal, deftroy him, you deftroy an animal.
Thefe fenfes are not all interchangeable. Take the is of iden
tity, and the name man is not, as a name, the name ani?nal : the
idea man is not, as an idea, the idea animal. Now we muft afk,
what common property is poiTeiTed by each of thefe three notions
of is y on which the common laws of inference depend. Common
laws of inference there certainly are. If the applicability of the
name A be always accompanied by that of B, and that of B by
that of C, then that of A is always accompanied by that of C. If
the idea A contain all that is eiTential to the idea B, and B all that
is eflential to C, then A contains all that is effential to C. If the
object A be actually the object B, and if B be actually C, then
A is actually C.
The following are the characteriftics of the word is which, ex-
ifting in any propofed meaning of it, make that meaning fatisfy
the requirements of logicians when they lay down the proportion
* A is B/ To make the ftatement diftinct, let the proportion
be doubly fingular, or refer to one inftance of each, one A and
one B : let it be c this one A is this one B.
Firft, the double fingular proportion above mentioned, and
every fuch double-fingular, muft be indifferent to converfion : the
A is B, and the c B is A* muft have the fame meaning, and be
both true or both falfe.
Secondly, the connexion /j, exifting between one term and
each of two others, muft therefore exift between thofe two
others ; fo A is B and A is C muft give c B is C.
Thirdly, the effential diftinction of the term is not is merely
that is and is not are contradictory alternatives, one muft, both
cannot, be true.
Every connexion which can be invented and fignified by the
terms is and is not, fo as to fatisfy thefe three conditions, makes
all the rules of logic true. No doubt abfolute identity was the fug-
of the Propojition. 51
getting connexion from which all the others arofe : juft as arith
metic was the medium in which the forms and laws of algebra
were fuggefted. But, as now we Invent algebras by abftract-
ing the forms and laws of operation, and fitting new meanings to
them, fo we have power to invent new meanings for all the
forms of inference, in every way in which we have power to
make meanings of is and is not which fatisfy the above condi
tions. For inftance, let X, Y, Z, each be the fymbol attached
to every inftance of a clafs of material objects, let is placed be
tween two, as in < X is Y mean that the two are tied together,
fay by a cord, and let X be confidered as tied to Z when it is
tied to Y which is tied to Z, &c. There is no fyllogifm but
what remains true under thefe meanings. Thus
The fyllogifm Is true in the fenfe
Every X is Y Every X is tied to a Y
Some Zs are not Ys Some Zs are not tied to Ys
/. Some Zs are not Xs . .Some Zs are not tied to Xs
This laft inftance might be confidered as a material reprefen-
tation of attachment together of ideas in the mind.
We muft diftinctly obferve that it is not every cafe of infe
rence which demands all the characteriftics to be fatisfied. Thus
in the moft common cafe of all, c Every A is B, every B is C,
therefore every A is C, of all the three conditions only the fe-
cond is wanted to fecure the validity of this cafe. Though it be fel-
dom thought worth while to make this obfervation, yet it is uni-
verfal practice to act upon it, and fo as to introduce into formal
logic apparent contradictions of its own rules. For example,th e
following are allowed to pafs for fyllogifms, in the ordinary defi
nition of that word.
1 Every A is greater than fome one B j every B is greater than
fome one C, therefore every A is greater than fome one C.
And the fame when inftead of greater than is read equal to or
lefs than. The form which moft commonly appears is the
pair of doubly fingular propofitions, A (one thing) is greater
than B ; B is greater than C ; therefore A is greater than C.
Here c greater than greater is greater, the fecond rule is fatisfied,
and no other is wanted. But this meaning for is (or this fubfti-
tute for it, if the reader like it better) will not fatisfy all the con-
52 On the Abjlratt Form
ditions, and therefore will not apply to all the forms of inference.
But is in the fenfe is equal to does fatisfy all the conditions.
This fenfe of / j, namely agreement in magnitude, is the copula of
the mathematician s fyllogifm, when he is reafoning on quantity
only.
It will probably be affirmed that the generalization thus made,
or mown to be poffible, in the conception of the word is for
purpofes of inference, amounts only to a very frequent, if not
moft ufual, ufe of the word, namely, as fignifying a certain mode,
not of identity, but of agreement in quality. As when we fay
thefe two things are the fame in colour or c the one thing is
the other in colour : that the name man is the name animal,
in a certain refpect, namely, in what the latter can be applied to :
that the idea man is animal, in both pofleffing certain charac-
teriftics : that every object man is an object animal, in actual
fubftance : that A is B in magnitude, when we fay A equals
B ; and fo on. But I admit only the converfe, namely, that all
thefe ufes fatisfy the conditions. It would hardly be for any one
to fay, that every poffible ufe of is which fatisfies three fuch fim-
ple requirements, has been or can be exhausted. Even the
material example which was juft now given, cannot be iden
tified with any common ufe, or eafily imaginable one, of the
common verb. But if no invented meaning, proper to fatisfy
the conditions,, can be found, other than already exifts in more
or lefs of ufe, ftill, thefe conditions are the laws to which the
word muft fubmit in its logical acceptation.
There are common ufes of the word which are not admitted
in logic : and among them, one of the moft common, connection
of an object with its quality, and of an idea with one of its con-
ftituent or aflbciated ideas. As when we fay, the rofe is red,
prudence is defirable. Here the logical conditions are not fatis-
fied. For example, c red is the rofe, though a poetical inverfion
of the firft afTertion, is not logically true. It is ufual to confider
fuch propofitions, in logic, as elliptical ; thus c the rofe is red is
confidered as c the rofe is a red object, or an object of red colour ;
in which the is now takes one of the fenfes which allows of con-
verfion. Similarly, in all other cafes, the fubject and predicate
are made to take the fame character ; both names, both ideas, or
both objects. This reduction renders unneceiTary both the ftudy
of the Proportion. 53
of the varieties of meaning of the word is (meaning varieties out
of the pale of the conditions above enumerated), and alfo that of
the tranfitions of meaning within the circle of which the infe
rence remains good.
The moft common ufes of the verb are ; firft abfolute iden
tity, as in c the thing he fold you is the one I fold him : fecondly,
agreement in a certain particular or particulars underftood, as
in He is a negro* faid of a European in reference to his colour :
thirdly, pofTeflion of a quality, as in c the rofe is red : fourthly,
reference of a fpecies to its genus, as in c man is an animal. All
thefe ufes are independent of the ufe of the verb alone, denoting
exiftence, as in c man is [i. e. exifts]. In all thefe fenfes, and in
all which might be added confidently with the conditions in page
50, fome proportions fometimes admit of having the fenfe of is
fhifted, and fome do not. Thus, in negative propofitions, the is
of agreement in particulars may be lawfully converted into that
of identity : if c No A is B in colour/ then abfolutely No A is
B. But c Every A is B in colour, does not prove Every A
is B. But the firft pair might be connected by a fyllogifm.
The is of agreement in particulars may always be reduced to
the is of identity, by alteration of the predicate ; thus Every A
is B in colour is Every A is a thing having the colour of one
of the Bs. J When a fyllogifm has a negative conclufion, and
the middle term is, or can be made, the predicate of both pre-
mifes, then the whole fyllogifm can be transformed from one in
which there is only the is of agreement to one in which there is
no is but that of identity. For example, fuppofe the premifes to
be No X is Y (in colour) ; every Z is Y (in colour), not
meaning neceflarily that all the Ys are of one colour, but reading
it as c No X is of the colour of any one of the Ys ; every Z is
of the colour of one of the Ys. The conclufion is that no Z
is X (in colour), or c no Z is of the colour of any one of the
Xs. But from this it follows that no Z is X, for if any one Z
were abfolutely X, it would have * the colour of that X. This
* The reader muft not paint any of the letters during the procefs. The
fenfe in which we fay a door is the fame door as before, after it has been
painted of a different colour, is not the fenfe of logical identity : it is the
fame in all but colour and colouring matter j and the is is one of agreement.
Except as a joke in fufficient anfvver to a captious objeftion or a trap, no
54 On Proportions.
laft conclufion can be brought directly from altered premifes :
thus, is being that of identity, we have No X is [a thing hav
ing the colour of one of the Ys] ; every Z is [a thing having
the colour of one of the Ys] ; therefore no Z is X. But fup-
pofe we take the following premifes, Some Ys are not Xs (in
colour) ; every Y is Z (in colour). From this it follows that
fome Zs are not Xs (in colour), and thence that fome Zs are
not Xs. But we cannot now alter the premifes, fo as to produce
the laft conclufion from X, Z, and a middle term.
CHAPTER IV.
On Proportions.
A NAME is a fymbol which is attached to one or more
objects of thought, on account of fome refemblance, or
community of properties. Or elfe it is a fymbol attached to
fome one or more objects of thought, to diftinguifh them from
others having the fame properties. Objects of the fame name
are, fo far as that name is concerned, undiftinguimable. And
one object may have many names, as being one in each of many
clafles of objects of thought.
Names, as explained in chapter II, are exclufively the ob
jects of formal logic. The identity and difference of things is
defcribed by aflerting the right to aflert, or the right to deny, the
application of names. And names, whether fimple or complex,
will be reprefented by letters of the alphabet, as X, Y, Z.
A proportion is the aflertion of agreement, more or lefs, or
difagreement, more or lefs, between two names. It exprefTes
that of the objects of thought called Xs, there are fome which
are, or are not, found among the objects of thought called Ys :
change whatever muft take place in the terms of conclufion, during infe
rence. The American calculating boy, Zerah Colburn, was afked how
many black beans it would take to make ten white ones ; to which he very
properly anfwered Ten, if you fkin em : but the ten fkinned beans would
not be the fame beans as before : except, indeed, to thofe to whom black is
white.
On Propojitions. 55
that there are objects which have both names, or which have
one but not the other, or which have neither.
For the moft part, the objects of thought which enter into a
proportion are fuppofed to be taken, not from the whole univerfe
of poffible objects, but from fome more definite collection of
them. Thus when we fay " All animals require air," or that
the name requiring air belongs to every thing to which the name
animal belongs, we fhould underftand that we are fpeaking of
things on this earth : the planets, &c., of which we know no
thing, not being included. By the univerfe of a propofition, I
mean the whole range of names in which it is exprefTed or un-
derftood that the names in the propofition are found. If there
be no fuch expreilion nor underftanding, then the univerfe of the
propofition is the whole range of poffible names. If, the uni
verfe being the name U, we have a right to fay every X is Y,
then we can only extend the univerfe fo as to make it include all
poffible names, by faying Every X which is U is one of the Ys
which are Us, or fomething equivalent.
Contrary names, with reference to any one univerfe, are thofe
which cannot both apply at once, but one or other of which al
ways applies. Thus, the univerfe being man, Briton and alien
are contraries ; the univerfe being property, real and perfonal are
contraries. Names which are contraries in one univerfe, are
not necefTarily fo in a larger one. Thus in geometry, when the
univerfe is one plane, pairs of ftraight lines are either parallels or
interfectors, and never both : parallels and interfectors are then
contraries. But when the ftudent comes to folid geometry, in
which all fpace is the univerfe, there are lines which are neither
parallels nor interfectors ; and thefe words are then not contra
ries. But names which are contraries in the larger and contain
ing univerfe, are neceflarily contraries in the fmaller and contained,
unlefs the fmaller univerfe abfolutely exclude one name, and then
the other name is the univerfe.
In future, I always underftand fome one univerfe as being that
in which all names ufed are wholly contained : and alfo (which
it is very important to bear in mind) that no one name mentioned
in a propofition fills this univerfe, or applies to everything in it.
Nothing is more eafy than to treat the fuppofition of a name
being the univerfe as an extreme cafe. And I (hall denote con-
56 On Proportions.
traries by large and fmall letters : thus, X being a name, x is the
contrary name. And everything (in the univerfe underftood) is
either X or x : and nothing is both.
A propofition may be either Jlmple and incomplete, or complex
and complete. The fimple propofition only afTerts that Xs are
Ys, or are not Ys : the complex propofition, which always con-
fifts of two fimple ones, difpofes in one manner or the other of
every X and every Y. Thus c Every X is Y is a fimple pro
pofition : but it forms a part of two complex propofitions. It
may belong either to * every X is Y and every Y is X/ or to
c Every X is Y and fome Ys are not Xs.
The propofitions advanced in common life are ufually com
plex, with one fimple propofition expreffed and one underflood :
but books of logic have hitherto confidered only the fimple pro
pofition. And this laft fhould be confidered before the complex
form.
The fimple propofition mufl be confidered with refpect to
ftgn, relative quantity, and order.
Simple propofitions are oftwofigns: affirmative and negative.
It is either Xs are Ys, or Xs are not Ys. The phrafes are
and are not, or is and is not, which mark the diftinftion, are
called copultz.
The relative quantity of a propofition has reference to the
numbers of inftances of the different names which enter it. The
diftincliions of quantity ufually recognized are all and fome* :
leading to the diftincliion of universal and particular. Thus
Every X is Y and c Every X is not Y are the univerfal affir
mative and negative propofitions : the latter is ufually ftated as
c No X is Y. And ( fome Xs are Ys and fome Xs are not
Ys are the particular affirmative and negative propofitions. And
when the proportions are reduced frri&ly to thefe four forms,
* Some, in logic, means one or more, it may be all. He who fays that
fome are , is not to be held to mean that the reft are not. f Some men
breathe, fome horfes are diftinguifhable by fhape from their riders would
be held falfe in common language. The reafon is, as above noted, that
common language ufually adopts the complex particular propofition, and
implies that fome are not in faying that fome are. The ftudent cannot be
too careful to remember this diftinclion. A particular propofition is only a
may be particular.
On Proportions . 57
the firft named, X, is called the fubjeft, and the fecond named,
Y, the predicate.
It has been propofed to confider the univerfal proportions as
definite with refpect to quantity : but this is not quite correct.
The phrafe all Xs are Ys does not tell us how many Xs
there are, but that, be the unknown number of Xs in exiftence
what it may, the unknown number mentioned in the proportion
is the fame. That which is definite is the ratio of the number
of Xs of the proportion to the Xs of the univerfe. So under-
ftood, however, the definite quantity, as an abbreviation, may be
faid to belong to univerfals. And the indefinitenefs of the parti
cular proportion is only hypothetical. It is in our power to fup-
pofe the feme to be one half of the whole, or two-thirds, or any
other fraction.
The quantity of the fubject is expreiTed ; that of the predicate,
though not exprefTed, is neceflarily implied by the meaning of
language. The predicate of an affirmative is particular : the
predicate of a negative is univerfal. If I fay Xs are Ys/ even
though I fpeak of all the Xs, I only really fpeak of fo many Ys
as are compared with Xs and found to agree : and thefe need
not be all the Ys. c Every horfe is an animal, declares that fo
many horfes as there are to fpeak of, fo many animals are fpoken
of : and leaves it wholly unfettled whether there be or be not
more animals left. But if I fhould fay c Xs are not Ys, though
it fhould be only one X, as in this X is not a Y, yet I fpeak of
every Y which exifts. The aflertion is c this X is not any one
whatfoever of all the Ys in exiftence. A perfon who fhould
wifh to verify by actual infpection, thefe 20 Xs are Ys might,
perchance, be enabled to affirm the refult upon the examination
of only 20 Ys, if he came rrft upon the right ones. But he
could not verify this one X is not a Y until he had examined
every Y in exiftence. This is the common doctrine, but though
admitting of courfe that the affirmative propofition only enables
us to infer of fome inftances of the predicate, yet I think it more
correct to fay that the predicate itfelf is fpoken of univerfally, but
indivifibly^ and that in the negative propofition the predicate is
fpoken of univerfally and divifibly. c Some Xs are Ys tells
us that each X mentioned is either the rrft Y, or the fecond Y,
or the third Y, &c., no Y being excluded from comparifon. But
58 On Proportions.
Some Xs are not Ys tells us that each X mentioned is abfo-
lutely not the firft Y, nor the fecond, nor the third, &c ; is not,
in fact, any one of all the Ys. Still, however, the predicate of
an affirmative yields no more than it would do if the Ys finally
accepted as Xs were fpecially feparated, and confidered as the
only Ys fpoken of.
The relation of the univerfal quantity to the whole quantity of
inftances in exiftence is definite^ being that whole quantity itfelf.
But the particular quantity is wholly indefinite : Some Xs are
Ys gives no clue to the fraction of all the Xs fpoken of, nor to
the fraction which they make of all the Ys. Common language
makes a certain conventional approach to definitenefs, which has
been thrown away in works of logic. Some, ufually means a
rather fmall fraction of the whole ; a larger fraction would be
exprefTed by c a good many ; and fomewhat more than half by
moft ; while a ftill larger proportion would be a great majo
rity or nearly all . A perfectly definite particular, as to quan
tity, would exprefs how many Xs are in exiftence, how many
Ys, and how many of the Xs are or are not Ys : as in 70 out
of the 100 Xs are among the 200 Ys . In this chapter I mail
treat only the Indefinite particular^ leaving the definite particular
for future confideration.
The order of a propofition has relation to the choice of fub-
ject and predicate. Thus Every X is Y and every Y is X
though both eftablifh a univerfal affirmative relation between X
O
and Y, yet are in fact two different propofitions. They are called
converfe forms. When the fubjedt and predicate are of the fame
fort of quantity, both univerfal or both particular, the converfe
forms give the fame propofition. Thus No X is Y and c No
Y is X are the fame ; neither has any meaning, except perhaps
of emphafis, which the other has not. And Some Xs are Ys
is the fame as Some Ys are Xs . The univerfal negative, then,
in which both terms are univerfal, and the particular affirmative,
in which both are particular are neceflarify convertible propofi
tions. But the univerfal affirmative, in which the fubjedt is uni
verfal and the predicate particular, and the particular negative, in
which the fubject is particular, and the predicate univerfal ^are
not neceffarily convertible, and are generally called inconvertible.
They may be convertible, in one cafe, and inconvertible in an-
On Propo/itions. 59
other. But the term inconvertible is not incorrect, for the fol
lowing reafon.
The agreements and difagreements which are treated in logic
are of this character ; there can only be agreement with one, but
there may be difagreement with all. If this X be a Y it is
one Y only : it is this X is either the firft Y, or the fecond
Y, or the third Y, &c. If there be 100 Ys, there is, to thofe
who can know it, 99 times as much negation as affirmation in
the proportion : and yet moft afluredly it is properly called affir
mative. But if it be this X is not a Y, we have this X is
not the firft Y, and it is not the fecond Y, and it is not the third
Y, &c. The affirmation is what is commonly called disjunctive,
the negation conjunctive. A disjunctive negation would be no
propofition at all, except that one and the fame thing cannot
be two different things : any X is either not the firft Y or not
the fecond Y. And in like manner a conjunctive affirmation
would be an impoffibility : it would ftate that one thing is two
or more different things.
We muft be prepared, then, to confider cafes of oppofition in
which on the one fide there is fixed neceffity, and on the other
fide poffibility of alternatives : and we muft be prepared to de
note thefe by oppofite terms, which, looking to etymology only,
denote fixed neceffities of oppofite characters. This happens in
the cafe above : convertible means abfolutely and neceflarily con
vertible, inconvertible means convertible or inconvertible as the
cafe may be. Taking the four forms of one order, we find that
each of the univerfals cannot exift with either propofition of op
pofite form. Thus Every X is Y cannot be true if either
No X is Y or Some Xs are not Ys : while No X is Y
cannot be true if either Every X is Y or Some Xs are Ys.
But each of the particulars is neceffarily inconfiftent with nothing
but the univerfal of oppofite form. That Some Xs are Ys
cannot be true if No X is Y but it may be true if Some Xs
are not Ys. And Some Xs are not Ys cannot be true if
Every X is Y, but it may be true though Some Xs are
Ys.
The pair Every X is Y and fome Xs are not Ys are called
contradictory : and fo are the pair No X is Y and Some Xs
are Ys. Of each pair of contradictories, one muft be true and
60 On Proportions.
one muft be falfe : fo that the affirmation of either is the denial
of the other, and the denial of either is the affirmation of the
other. The pair Every X is Y and c No X is Y are ufually
called contraries; contrariety implying the utmoil extreme of
contradiction. Contraries may both be falfe, but cannot both be
true. The pair Some Xs are Ys, and c Some Xs are not Ys,
which may both be true, but cannot both be falfe, are ufually
called fub contraries. But, for reafons hereafter to be given, I
intend to abandon the diftin6lion between the words contrary
and contradiftory^ and to treat them as fynonymous. And the
propofitions ufually called contraries , c Every X is Y and c No
X is Y I fhall czllfubcontraries : while thofe ufually called fub-
contrarles ( Some Xs are Ys and Some Xs are not Ys I fhall
call fupercontraries
I fhall now proceed to an enlarged view of the propofition,
and to the ftruclure of a notation proper to reprefent its different
cafes.
As ufual, let the univerfal affirmative be denoted by A, the par
ticular affirmative by I, the univerfal negative by E, and the par
ticular negative by O. This is the extent of the common fym-
bolic expreffion of propofitions : I propofe to make the following
additions for this work. Let one particular choice of order, as
to fubject and predicate, be fuppofed eftablifhed as a flandard of
reference. As to the letters X, Y, Z, let the order be always
that of the alphabet, XY, YZ, XZ. Let x, y, z, be the con
trary names of X, Y, Z ; and let the fame order be adopted
in the ftandard of reference. Let the four forms, when choice
is made out of X, Y, Z, be denoted by A 4 , E, L, Oi ; but when
the choice is made from the contraries, let them be denoted by
A , E 1 , I 1 , O 1 . Thus, with reference to Y and Z, c Every Y is
Z is the Ai of that pair and order : while Every y is z is the
A T . I mould recommend AI and A 1 to be called the fub-A and
the fuper-A of the pair and order in queftion : the helps which
this will give the memory will prefently be very apparent. And
the fame of L and I f , &c.
Let the following abbreviations be employed ;
X) Y means <E very X is Y
X:Y < Some Xs are not Ys
X. Y means No X is Y
XY Some Xs are Ys
On Proportions. 6 r
There are eight diftinft modes, independent of contraries, in
which a fimple propofition may be made by means of X and Y.
Thefe eight modes are X)Y and Y)X, X:Y and Y:X, X.Y and
Y. X, and XY and YX. But the eight are equivalent only to
fix.: for X.Y and Y. X are the fame, and fo are XY and YX.
Again, there are fix fimple proportions between x and y, fix be
tween X and y, fix between x and Y. Taking in contraries,
there are then twenty-four apparent modes of forming a fimple
propofition from X and Y : but thefe are not all diftincl:. Eight
of them contain all the reft : thefe eight being the A 4 , E,, L, O t ,
A 1 , E ! , I 1 , O 1 , above defcribed. This is feen in the following
table, the ftudy of which fhould be carefully made,
A, X)Y = X.y = y)x
O, X:Y = Xy =y:x
E, X.Y = X)y = Y)x
I. XY =X:y = Y:x
A T x)y =x.Y = Y)X
! x:y = xY = Y:X
E ! x.y = x)Y = y)X
I f xy = x:Y = y:X
I fuppofe moft readers will readily fee the truth of the identities
here affirmed : if not, the following mode of illuftration (which
will be very ufeful when I come to treat of the fyllogifm) may be
tried. Let U be the name which is the univerfe of the propo
fition : and write down in a line as many Us as there are diftin6t
objects to which this name applies. A dozen will do as well for
illuftration as a million. Under every U which is an X write
down X : and x, of courfe, under all the reft. Follow the fame
plan with Y. The occurrence of letters in the fame column
mows that they are names of the fame object. The following
are fpecimens of the eight ftandard varieties of affertion, to which
all the reft may be referred.
A, UUUUUUUUUUUU
XXXXX x x x x x x x
YYYYYYYYyyyy
UUUUUUUUUUUU
XXXXXXX xxxxx
yyyy YYYYYYyy
UUUUUUUUUUUU
XXXX xxxxxxxx
y y y y y y y YYYYY
A 1 UUUUUUUUUUUU
XXXXXXXX xxxx
YYYYYyyyyyyy
UUUUUUUUUUUU
XXXXX xxxxxxx
YYyy yyyy YYYY
UUUUUUUUUUUU
XXXXXXXXxxxx
yyyyyYY YYYYY
6 2 On Propo/itions.
In the firft fcheme, Ai, there exift twelve Us, the firft five of
which are both Xs and Ys, the next three Ys but not Xs, the laft
four neither Xs nor Ys. This cafe, fo conftru&ed that X)Y is
true, mows X.y and y)x.
The proportions AI and A 1 , X)Y and x)y, may be called con-
tranominal, as having each names contrary of thofe in the other.
It appears, then, that as to inconvertibles, contranominal and con-
verfe are terms of the fame meaning, for X)Y and y)x are the
fame, and x:y and Y:X. And fmce it is more natural to fpeak
of direft names than of their contraries, it will be beft to attach
to A 1 and O ? the ideas of Y)X and Y:X ; but not fo as to forget
their derivation from x)y and x:y. Obferve alfo that each uni-
verfal propofition has converted contranominals for its affirmative
forms. Thus X)Y = y)x : and though X.Y is not y.x, yet if
we make X.Y take the affirmative form X)y, it is equivalent to
Y)x. In particular propofitions, the negative forms have the
fame property. The contranominals of the convertible propo
fitions Ei and L are of totally different meaning. They have
never till now been introduced into logic, and a few words of
explanation are wanted.
Firft as to I 1 or xy. We here exprefs that fome not-Xs are
not-Ys, or that there are things in the univerfe which are neither
Xs nor Ys. That is, X and Y are not contraries. Next as to
E f or x.y. We here exprefs that no not-X is not-Y, or that
everything in the univerfe is either X or Y, or both. Thefe laft
words are important : by omitting them, we mould imagine that
x.y fignifies that X and Y are contraries ; which is not necef-
farily true.
Accordingly, the eight ftandard forms of expreffion, with re
ference to the order X Y, and exhibited in the form in which it
will be moft convenient to think and fpeak of them, are as
follows,
A 4 or X) Y Every X is Y
O, or X: Y Some Xs are not Ys
E, or X.Y No X is Y
I, or XY Some Xs are Ys
A T orY)X Every Y is X
O or Y:X Some Ys are not Xs
E f or x.y Everything is either X or Y
I or xy Some things are neither Xs norYs.
Returning to the table, we now fee the following general laws.
I. Each triad of equivalents contains two inconvertibles and one
convertible. 2. Of the four, X, Y, x, y, each of the eight forms
On Proportions. 63
fpeaks univerfally of two, and particularly of two. 3. A pro-
pofition fpeaks in different ways of each name and its contrary ;
univerfally of one and particularly of the other. 4. The propo-
fitions called contradictory, from the common meaning of this
word, may be fo called in another fenfe : for they fpeak in the
fame manner of contraries. Thus X) Y fpeaks univerfally of X,
and particularly of Y : its denial, X: Y or y:x, fpeaks univerfally
of x, and particularly of y.
Any two of the eight forms being taken, it is clear either that
they cannot exift together, or that one muft exift when the other
exifts, or that one may exift either with or without the other.
The alternatives of each cafe are prefented in the following table.
Con- Is indif- Is con - Is indif-
Denies tains ferent to Denies tained in ferent to
A,
OiE.E 1
LF
A ! f
Oi
A,
E.E 1
A O LF
A 1
E f E,
FL
A 4 O,
O 1
A 1
E E,
AiOJ L
E,
LA,A
0,0
ET
I,
E,
A,A f
E 1
FA A,
cm
EJ,
F
E f
A Ai
EJ 4 o o
Let the concomitants of a proportion be thofe to which it is
wholly indifferent. Then it appears that each univerfal has for
concomitants its contranominal and the contradictory of the laft :
but each particular has all for concomitants except only its own
contradictory. Each univerfal denies, befides its own contra
dictory, the two univerfals of oppofite name ; and contains the
two particulars of the fame name. The two concomitants of a
univerfal may be defcribed as its univerfal and its particular con
comitant.
There is a certain fort of repetition in our choice of the four
forms, combined with the four felections XY, Xy, xy, xY. If
any one of the four forms A, EI A 1 E 1 be applied to all the above,
it will give the four forms derived from XY. Thus the A 4 of
XY, Xy, xy, xY, are feverally the A,, E,, A T and E f of XY ;
and the E of X Y, Xy, xy, and xY are feverally the E ! , A T , E,,
and AI of XY : and fo on. It will ferve for exercife to verify
the above, and ftill more the cafes contained in the following.
There are four things in a proportion, each of which may be
changed into its contrary : fubject, predicate, order, and copula.
Let S be the direction to change the fubject into its contrary : P
64 On Proportions.
the fame for the predicate : let T be the direction to transform
the order : and F the direction to change the form, from affirma
tive to negative, or from negative to affirmative. When T enters,
let it be done laft, to avoid confufion. Thus SPT performed
upon X)Y gives x)Y from S, x)y, from P, and y)x from T j
which is X)Y, fo that in this cafe alteration of fubject, predicate,
and order, is no alteration at all. Let L be the reprefentation of
no alteration at all. To inveftigate equivalent alterations, ob-
ferve, firft, that F and P, fingly, are identical : thus F performed
on X.Y gives X)Y, and P on X.Y gives X.y. And X)Y =
X.y. This perfect identity of F and P in effect, remains in
all combinations into which T does not enter. But when T
enters, it is S and F which are identical. Thus ST performed
on Y)X gives X)y or X .Y : and FT performed on Y)X gives
X.Y. The reafon is, that T interchanges fubjecl: and predicate ;
fo that F, after T, makes a change which is counterbalanced by
a change in what was the fubjecl:. Accordingly, remembering
that each operation performed twice is no operation at all (thus
PP is L, and TT is L), we have in all cafes
P = F, SP = SF, PF=L, SPF=S
ST=FT, SPT = FPT, SFT^T, SPFT=PT
all which fhould be tried for exercife. Again, in a convertible
propofition, transformation is no alteration or T = L : in an incon
vertible one, transformation changes it into its contranominal ;
or T = SP. Now fet out as follows; L, in convertible propo-
fitions is T ; which in inconvertible*, is SP ; which, in convertibles
again, is SPT ; which, in inconvertibles again, is TT, or L.
Put thefe down as follows, writing under them the operations
which are always equivalent to them, as fhewn above,
L
PF
T [SP
SFT SF
SPT
PFT
L
PF
The combinations written under one another are always the
fame in effect : thofe feparated by double lines have the fame
effect on convertibles : thofe feparated by fmgle lines, have the
fame effecl: on inconvertibles. Again P, for convertibles, is the
fame as PT ; which, for inconvertibles is the fame as PSP, or
S ; which, for convertibles again, is the fame as ST ; which, for
inconvertibles, is SSP or P. Thefe treated as before, give the
table
On Proportions. 65
PT
SPFT
S
SPF
! ST
FT
In thefe two cycles there are L and all the fifteen feleclions
which can be made out of S, P, F, T. And every poflible cafe
of equivalent changes is contained in thefe two tables. Thus
PT is in all cafes equivalent to SPFT ; in convertible cafes, to
P and to F ; in inconvertible ones, to S and to SPF. And no
other combination is in any cafe equivalent to PT. In verifica
tion of thefe tables, obferve that the operation F always occurs
in the lower line, and never in the upper ; and that this opera
tion changes convertibles into inconvertibles, and vice verfa.
We ought then to expect, that the equivalences which, con
taining F, apply to inconvertibles, will be thofe which when F is
ftruck out, apply to convertibles ; and vice verfa. And fo we (hall
find it : for inftance, SPFT and SPF are equivalent when per
formed on inconvertibles ; ftrike out F and we have SPT and
SP, which are equivalent when performed on convertibles.
It appears, then, that any change which can be made on a
propofition, amounts in effecl to L, P, S, or PS. This is another
verification of the preceding table : for all our forms may be de
rived from applying thofe which relate to XY in the cafes of
Xy, xY, and xy.
We have seen that A t and A f both contain L and I f ; and that
E and E 1 both contain O and O ! . Hence each of the untverfals
may be faid to be the Jlrengtkened form of either of its particulars
of the fame fign : and each of the particulars the weakened form
of its univerfals of the fame fign. The only diftinction which
appears between the two forms of the convertible particulars,
XY and YX, xy and yx, is that the ftrengthened forms derived
from extending the fubje6ls are different. Thus xy gives x)y or
Y)X ; but yx gives y)x or X)Y.
A complex propofition is one which involves within itfelf the
afTertion or denial of each and all of the eight fimple propofitions.
If thefe eight propofitions were all concomitants, or if any num
ber of them might be true, and the reft falfe, there would be
256 poflible cafes of the complex propofition. As it is, owing
to the connexion eftablifhed in the table of page 63, there are
butfeven.
66 On Proportions.
Firft, let the names X and Y be fo related that neither of the
four univerfals are true. Then all the four particulars are true :
and this is the firft cafe. Let it be called a complex particular^
and denoted by P. Then, denoting coexiftence of fimple pro-
pofitions by writing + between their feveral letters, we have
p=o f +o,+r+i,
This cafe is of the leaft frequent mention in the theory of the
fyllogifm.
Next, let one of the univerfal propofitions be true. Then five
of the other propofitions are fettled, either by affirmation or de
nial. There remain the two concomitants, which are contra
dictory ; fo that only one is true. Accordingly, with the excep
tion of the complex particular juft defcribed, every complex pro-
pofition muft confift of the coexiftence of a univerfal and one of
its concomitants. But there are not therefore eight more fuch
propofitions : for A 1 + Ai and A 4 + A 1 are the fame, and fo are
Ei + E f and E f + Ei. The remaining number is then reduced
to fix, which are
Ai + 1 , A t + A f ,
E. + I f , Ei + E , E + L,
Thefe muft be feparately examined.
Firft, take Ai-f A ? (the order XY always underftood). We
have then X)Y and Y)X. That is, there is no object whatfo-
ever which has one of thefe names, but what alfo has the other.
The names X and Y are then identical^ not as names, but as
fubjects of application. Where either can be applied, there can
the other alfo. Thus, in geometry (the univerfe being plane
rectilinear figure) equilateral and equiangular are identical names.
Not that they agree in etymology nor in meaning : more than
this, a few words would explain the firft to many who could not
comprehend the fecond without difficulty. But they agree in
that what figure foever has a right to either name, it has the fame
right to the other. It will tend to uniformity of language, if we
call X, in this cafe, an identical of Y, and Y an identical of X.
Let the fymbol of an identical be D : then we have
On Propofitiom. 67
Next, take A, + O f . We have then X)Y and Y:X. Every X
is Y, and fo far there is a character of identity. But fome Ys
are not Xs ; there are more Ys than Xs, and X ftops fhort of a
complete claim of identity with Y. Let X be called zfubiden-
tical of Y (thus man is a fubidentical of animal], and let Di de
note this cafe. Then
Let A 1 + O 4 exift. We have then Y)X and X: Y. Every Y
is X, and fo far there is identity. But fome Xs are not Ys, there
are more Xs than Ys, or X goes beyond a claim of identity with
Y. Let X be now called a f up er identical of Y, and let it be
denoted by D f . Then
The terms fuperidentical and fubidentical are obvioufly correla
tive. If X be either of Y, Y is the other of X. Now let us
confider E + E f . We have then X.Y and x.y. There is no
thing which is both X and Y, there is nothing which is neither.
Confequently X and Y are contraries, or juft fill up the univerfe.
Let C be the mark of this relation. Then
Next, take E, + I f . We have then X . Y and xy. Nothing is
both X and Y, but there are things which are neither. X and
Y are clear of one another, but do not amount to contraries,
for they do not fill up the univerfe. Let them be called fubcon-
trarles, (thus in the univerfe metal, gold and filver are fubcontra-
ries, and let C denote the relation. Then
c,=E,+r
Laftly, take E f + L. We have x.y and XY. The names fill
the univerfe ; for there is nothing but what is either X or Y.
But they overfill it ; for fome things are both Xs and Ys. There
is then all the completenefs of a contrary and more. Let X and
Y be called fuper contraries,* and let C ! denote the relation. Then
we have
* The fupercontrary relation, though eflential to a complete fyftem of
fyllogifm, is not frequently met with. The other extreme of the fupercon-
68 On Proportions.
To complete our language, let Ai or X)Y, with reference to the
order XY, be called fub-qffirmathe ; and A 1 or Y)X,fuper affirma
tive. Let EI or X.Y be called fubnegatfoe } and E or x.j,fuperneg-
ative. Let the particulars L, I 1 , and O|, O 1 , have alfo thefe feveral
names. This extenfion of our language will require a little ex
planation.
When I fay that X is a fubidentical of Y, I mean that the
etymological fuggeftions are actually fatisfied. The whole name
X, and more, is contained in Y. But when I fay that X is a
univerfal ^affirmative of Y, or X)Y, I mean no more than that
we have the proportion whofe form is not fuperaffirmative, ac
cording to the etymology of that word. An algebraift would
well underftand the diftinclion at a glance. He has often to
diftinguifh the cafe in which a is lefs than b from that in which
a is lefs than or equal to b : the cafe in which the extreme limit
of the afTertion is not included from that in which it is included.
Again, the word negative had better be viewed as not fo much
prefenting exclufion for its firft idea, as indufion in the contrary.
Thus a fubnegative, when univerfal, is to fuggeft complete in-
clufion in the contrary, meaning the extreme cafe, poffibly ;
namely, that the fubnegative names may be contraries. Again,
fupernegative is to fuggeft the idea of fupercontrary, with the
loweft extreme, the relation of contrary, poffibly included.
For exercife in this language, and in the ideas which it is
meant to prefent, I now ftate the following refults.
Univerfal affirmation, though as a general term, it is to include
fuper and fub affirmation, yet looked at as one of the three, and
diftinguifhed from the reft, it means identity. The fame of ne
gation and contrariety. Subidentity requires univerfal fubaffir-
mation and particular fupernegation. Identity is univerfal fub
and fuper affirmation, both. Superidentity requires univerfal
fuperaffirmation and particular fubnegation. Subcontrariety re
quires univerfal fubnegation and particular fuperaffirmation. Con
trary, or the fubidentical, is fo much the eafieft of all our complex relations,
that the latter rarely allows the former to appear. The firft inftance that
fuggefted itfelf to me was man and irrational (as defcriptive of the quality of
the individual and not of the fpecies) in the univerfe animal. Thefe more
than fill that univerfe, idiot being common to both. But it is more natural
to fay that rational (in this fenfe) is fubidentical of man.
On Proportions. 69
trariety is univerfal fub and fuper negation, both. Supercontra-
riety requires univerfal fupernegation and particular fubaffirma
tion. Again, univerfal fubaffirmation is either fubidentity or iden
tity : particular fubaffirmation is a denial of contrariety and fub-
contrariety. Univerfal fuperaffirmation is either fuperidentity or
identity : particular fuperaffirmation denies contrariety and fuper-
contrariety. Univerfal fubnegation is either fubcontrariety or
contrariety : particular fubnegation denies fubidentity and iden
tity. Univerfal fupernegation is either fupercontrariety or con
trariety : particular fupernegation denies fuperidentity and iden
tity. All this is exprefled in the following table,
DI affirms
A,
and
O f
A,
affirms Di or
D
D
Ai
and
A 1
A
Di or
D
orD
D 1
A 1
and
Oi
A 1
D 1 or
D
Ci
Ei
and
P
Ei
C, or
C
C
Ei
and
E f
E
Ci or
C
orC f
C 1
E
and
Ii
E 1
C 1 or
C
Dl
A 1
or
Oi
Oi
denies DI and D
D
O f
or
Oi
Di and
D,
or D ? and
D
D 1
Ai
or
T
D and
D
Ci
E 1
or
II
I,
Ci and
C
C
r
or
II
I
Ci and
C
or C 1 and
C
C 1
Ei
or
P
P
C and
C
Denial of D
C
c
Every fubidentical of a name is the fubcontrary of its contrary ;
every fubcontrary is the fubidentical of the contrary. Treat the
word contrary as negative, the word identical as pofitive ;. and
the two as of different figns. Then the algebraical rule like
figns give a pofitive, unlike figns a negative, holds in every cafe :
including the variety of it fo well known as c two negatives make
an affirmative/ When the modifying prepofition comes firft it
muft be retained ; when it comes fecond, it muft be changed.
Thus the fubcontrary of a contrary is a fubidentical : but the con
trary of a fubcontrary is a fuperidentical. In putting two rela
tions together, however, we have got into iyllogifm, as we mall
prefently fee.
The following tables will mow a connexion between the ex-
preffions, for different orders and felections, which it may be ufeful
to verify.
7 o
On Proportions.
XY
A,0 Di
YX
A OiD
xY
E L C 1
Yx
E I, C 1
Xy
Ei r Ci
yX
r a
xy
A ! OiD !
yx
AiO Di
A O,D
AiO Di
Ei I 1 C,
Ei r Ci
E 1 L C 1
E 1 li C 1
AiO Di
A OiD
Ei I Ci
Ei I 1 Ci
A f O.D f
AiO Di
AiO D,
A ? OiD
E I, C
E L C
E 1 1, C 1
E 1 1. C
A,O Di
A O.D 1
A OiD
AiO Di
Ei I 1 Ci
. I Ci
This table only contains fome of the rules already laid down
in pp. 64, 65. It exprefles that, for inftance, the AI, O 1 , and Di
of XY, are feverally the fame as the Ei, I , and d of yX. This
table may be exhibited thus, the identicals counting as inconvert-
ibles, the contraries as convertibles.
Change of
Subjeft
Predicate
Subjeft and Predicate
Order
Subject and Order
Predicate and Order
Subject, Predicate, and Order
In all cafes, change of fubjecl: is change both of fign and pre-
pofition ; change of predicate is change of fign ; change of fub-
jedT: and predicate is change of prepofition. Thefe three cafes
are of great importance in the iyllogifm : and the reader would
do well to connect in his mind
In Convertibles,
changes
In Inconvertibles,
changes
Sign and Prepofition
Sign
Prepofition
Neither
Sign
Sign and Prepofition
Prepofition
Sign and Prepofition
Sign
Prepofition
Prepofition
Sign and Prepofition
Sign
Neither
Subjeft with
Subjefl and Predicate
Predicate
Sign and prepofitlon
Prepofition
Sign
It is defirable to confider the feveral complex relations as to
the continuous tranfition from one into another : the growth of
names concerns not only the etymologift, but the logician alfo.
With the analogies and affinities by which the dominion of
one name is extended to inftance after inftance, and clafs after
clafs and fometimes, in fcientific language at leaft, deprived of
a part of what it has held I have here nothing to do. It is
enough that the phenomena exift which may be defcribed as the
gradual transformation of one relation into another. The words
butt and bottle, for example, are now fubcontraries in the uni-
verfe receptacle : but the etymology of the fecond word fhows
On Proportions. 71
that it was a fubidentical of the firft, being a diminutive. And
if we were to take the whole clafs butt, bufs, boot, bufhel, box,
boat, bottle, pottle, &c, which are all of one origin, the number
of tranfitions would be found to be very large.
I afliime that all the inftances of a name are counted and
arranged in its univerfe : a conceivable, though not attainable,
fuppofition. Alfo, that the inftances of the name are arranged
contiguoufly, as in page 61. Whatever the reafon may be
which dictates the particular arrangement chofen, it will generally
happen that the inftances near to the boundary poflefs the cha-
ra&eriftics of the name in a fmaller degree than thofe nearer the
middle. Let the contiguous arrangement be made of all the in
ftances of the name Y, the univerfe being U. Let another name
X begin to grow, commencing with one inftance, that is, being
applied to one of the objects in the univerfe U, be it a Y or not;
then to another contiguous, and fo on. We are to enumerate the
ways in which fuch changes, whether of increafe or diminu
tion, may caufe one name to change its relation to another.
According as the change is made by acceffion or retrenchment,
it may be denoted by ( + ) or ( ).
Let the name X begin within the limits of the name Y : its
initial relation to Y is then Di . And the poflibility of the
following continuous changes is obvious :
Hence D 4 may become D f through either D or P, but C or C 1
only through P. Next, let X begin without the limits of Y :
the initial relation is Ci. We may have then
Let X begin both within and without Y : its initial relation is
then P. And we have
But when ( ) follows Di or D, d or C, we have nothing
except
J2 On Proportions.
If we begin at the other extreme, with the name U, we have
U (-) D f U (-) C 1
Beginning from D 1 and C 1 we have
D (-)D(-)Di D f (-)P(-)C,
D f (-)P(-)D, D (-)P( + )C f
C f (-) C (-) d C ? (-) P (-) Di
But when ( + ) follows D f or D, C 1 or C, we have only
C
From the above lift it appears that the tranfition which is ac
companied by a change of prepofition only can be made either
through the letter without prepofition or through P : and in all
cafes with one continued mode of alteration. But when the tranf
ition involves change of letter, it can only be made through P :
with continuation of the mode of alteration when the prepofi-
tions are different, and change in the mode when they are the
fame. The following fuccefiions contain the arrangement of the
refults.
With one altera- With one altera- With two altera
tion (-J-) tion ( ) tions (-J )
Di D D 1 D 1 D Di Di P Ci
Di P D f D 1 P Di Ci P Di
Ci C C 1 C 1 C Ci
C|PC ! C ! PCi (-+)
D 1 P C f
Di PC 1 D f P Ci C 1 P D f
Ci P D T C 1 P D.
The following confiderations will further ferve to illuftrate the
want of the extenfion of the doctrine of proportions made in
this chapter, and alfo the completenefs of it. Among our moft
fundamental diftincl:ions is that of necejjlty and fufficlency ; of
what we cannot do without^ and what we can do with ; of that
which muft precede^ and that which can follow. The contraries
of thefe are non-necejjtty and non-fufficiency. In thefe four words,
applied to both Y and y, we have the defcription of the eight re-
On Proportions. 73
lations of X to Y. For inftance A, or X)Y tells us that to
have an X, we muft take a Y, or to be X, it is neceffary to be
Treating all in the fame way, we have
Y.
A.
A f
E,
E 1
L
r
O,
O
X ) Y To take an X it is neceffary to take a Y
Y)X
X.Y
x.y
X Y
xy
X:Y
Y:X
X
. . fufficient
X
. . neceffary
X
fufficient
X
. not neceffary
X
. not fufficient
X
. not neceffary
X
. not fufficient
Y
y
y
y
y
Y
Y
And the convertibility of the ordinary mode of defcription with
this new one may be eafily mown in any cafe. For example,
what can we mean by faying that to take a X, it is not fufficient
to take what is not Y ? Clearly that by taking not Y, or y, we
may at the fame time take a x, or that there are xs which are ys.
And fo on for the reft.
Of the four pairs XY, Xy, xy, xY, we know that each
propofition may be exprefled by three, and refufes to be exprefled
by one. If we now admit the two words impojjible and contingent^
meaning by the latter that which, as the cafe may be, is poffible
or impoffible, we mail eafily fee the following table for the uni-
verfals :
XY Xy xy xY
A. X)Y
E, X.Y
A 1 Y)X
N
I
S
C
I
N
c
S
S
C
N
I
c
S
I
N
The letters N, I, S, C, are the initials of neceffary, &c. And
we read in the firft line, that if X ) Y, then to be X it is necef
fary to be Y ; to be X, it is impoffible to be y ; to be x it is fuf
ficient to be y ; and to be x, it is contingently poffible or impoffible
to be Y. Again, if by n and s we mean not neceffary and not
fufficient , by P, aftually poffible ; and by C, as before (C being
its own contrary), we have the following table for the parti
culars :
74
On Proportions.
XY Xy xy xY
0, X:Y
1, X Y
O ! Y:X
I 1 xy
n
P
s
C
P
n
C
s
s
C
n
P
C
s
P
n
Of the four contrary pairs, n, P, s, C, are related to the par
ticulars precifely as N, I, S, C, are to the univerfals. The inter
change of Y and y is always accompanied by the interchange of
N and I, S and C, n and P, s and C ; the interchange of X
and x is that of N and C, S and I, n and C, s and P ; of both
X and x, Y and y, is that of N and S, C and I, n and s, C
and P.
The complex relations may be thus defcribed. According as
X is fubidentical, identical, or fuperidentical of Y, to be X it is
neceflary and not fufficient, necefTary and fufficient, or not ne
ceflary and fufficient, to be Y : according as X is fubcontrary,
contrary, or fupercontrary of Y, to be X it is neceflary and not
fufficient, neceflary and fufficient, or not neceflary and fufficient,
to be y. Or, as in the following table :
XY Xy xy xY
D,
Ns
I
Sn
P
C,
I
Ns
P
Sn
D !
Sn
P
Ns
I
C
P
Sn
I
Ns
D
NS
I
NS
I
C
I
NS
I
NS
P
nsP
nsP
nsP
nsP
Inftead of 1C and PC, write I and C : for " impoffible, and
poflible or impoffible as the cafe may be " is " impoffible " &c.
The names of the complex relations, fubidentity, identity, &c
I fuppofe will be held tolerably fatisfa&ory : thofe of the fimple
relations fuggefted in page 68, fubaffirmative &c. have nothing in
their favor except analogy with the former, and clofe connexion
with the notation. A little practice in their ufe might ren
der thefe laft names available : but it will be advifable to con-
On Propofitions. 75
necl them with names more defcriptive of the meaning, and to
adopt thefe laft, whether we reject or maintain their fynonymes.
When X ) Y, the relation of X to Y is well underftood as
that of the fpecies to the genus. We may adopt thefe words,
with the understanding that the word fpecies includes the
extreme cafe in which the fpecies is as extenfive as the genus.
When X : Y, we may call X a non-fpecies of Y, and Y a non-genus
of X. When X . Y we may call X an exdufive or excludent of
Y, or elfe a non-participant-, and alfo Y of X. When XY, we
may fay that each is participant, or non-ex clufive^ of the other.
When x . y, which means that X and Y together fill up, or more
than fill up, the univere, we may fay that they are complement al
names. When x y, which only means that X and Y do not be
tween them contain the univerfe, we may call them non-comple-
mental. We have then
Inconvertibles. Name of X with refpeft to Y.
AI X)Y fpecies, or fubaffirmative.
d X:Y non-fpecies, or particular fubnegative.
A f Y)X genus, or fuperaffirmative.
O f Y:X non-genus, or particular fupernegative.
Convertibles. Name of X and Y with refpeft to each other.
Ei X.Y Exclufives, or non-participants, or fubnegatives.
L X Y Non-exclufives, or participants, or particular fubaffir-
E f x.y Complements, or fupernegatives. [matives.
I 1 xy Non-complements, or particular fuperaffirmatives.
The following exercifes in thefe terms, really contain the de-
fcription of all the fyllogifms in the next chapter.
Inclufion in the fpecies is inclufion in the genus ; and inclufion
of the genus is inclufion of its parts (fpecies or not).
Exclufion from the genus is exclufion from the fpecies ; and ex-
clufion of the genus is exclufion of its parts (fpecies or not).
Inclufion or exclufion of the fpecies is part inclufion or exclu
fion of the genus.
When the fpecies is complemental, fo is the genus : and when
the genus is not complemental, neither is the fpecies.
Exclufion from one complement is inclufion in the other.
Complements of the fame are participants.
76 On the Syllogifm.
Two fpecies of one genus, are not complements ; neither are
two exclufions from the fame.
The complement of a genus is a non-fpecies ; and the com
plement is a non-fpecies of the non-complement.
CHAPTER V.
On the Syllogifm.
A SYLLOGISM is the inference of the relation between
_\_ two names from the relation of each of thofe names to a
third. Three names therefore are involved, the two which ap
pear in the conclufion, and the third or middle term, with which
the names, or terms, of the conclufion are feverally compared.
The ftatements expreffing the relations of the two concluding
terms to the middle term, are the two premifes. In this chapter,
no ratio of quantities is confidered except the definite all and the
indefinite feme.
A fyllogifm may be either Jimple or complex. A fyllogifm is
fimple when in it two fimple propofitions produce the affirmation
or denial of a third : or the affirmation of a third, we may fay,
fmce every denial of one fimple propofition is the affirmation
of another. A complex fyllogifm is one in which two complex
propofitions produce the affirmation or denial of a third complex
propofition.
It might be fuppofed that we ought to begin with the fimple
fyllogifm, and from thence proceed to the complex. On this
point I have fome remarks to offer, in j unification of following
precifely the reverfe plan.
Hitherto the complex fyllogifm has never made its appearance
in a work on logic, except in one particular cafe, in which it is
allowed to be treated as a fimple fyllogifm, though moft obvioufly
it is not fo. I allude to the common a fortiori argument, as in
c A is greater than B, B is greater than C, therefore A is greater
than C. There is no middle term here : the predicate of the
firft propofition is a thing greater than B, J the fubjecT: of the
fecond propofition is B.
Admitting fully that the quality of the premifes, that which
On the Syllogifm. 77
entitles the conclufion to be made, as it is faid, a fortiori marks
this argument out as, if anything, ftronger, clearer, and (could
fuch a thing be) truer, than a fimple fyllogifm ; yet it is plain
that the very additional circumflance on which this additional
clearnefs depends, takes the argument out of a fyllogifm, as de
fined by all writers. By beginning with the complex fyllogifm,
and thence defcending to the fimple one, it will be feen that we
begin with cafes which prefent this a fortiori and clearer charac
ter. I think I mall (hew that the complex fyllogifm is eafier
than the fimple one.
Next, the fyllogifm hitherto confidered has never involved any
contrary terms ; the confequence of which has been that various
legitimate modes of inference have been neglected. Moreover,
feveral of the ufual fyllogifms are more ftrong than need be in
the premifes, in order to produce the conclufion. Thus Y)X
and Y)Z being admitted as premifes, the neceffary conclufion is
XZ. But if Y)X be weakened into YX, the fame conclufion
follows. If we call a fyllogifm fundamental^ when neither of its
premifes are ftronger than is necefTary to produce the conclufion,
it is obvious that every fundamental fyllogifm which has a parti
cular premife, gives at leaft as ftrong a conclufion when that
particular is ftrengthened into a univerfal. But, except when
ftrengthening the premife alfo enables us to ftrengthen the con
clufion, in which cafe we have a new and different fyllogifm, it
feems hardly fyftematic to mix with fundamental arguments fyl
logifms which have quality or quantity more than is necefTary for
the conclufion.
The ufe of the complex fyllogifm will, as we fhall fee, give
an independent and fyftematic derivation to thefe ftrengthened
fyllogifms, as well as to the reft.
Let X and Z be the terms of the conclufion ; and let Y be
the middle term. Let the premife in which X and Y are com
pared come firft of the two. Let the order of reference in each
cafe be that of the alphabet
XY YZ XZ
So that by ftating what X is with refpect to Y, and what Y is
with refpedt to Z, our fyllogifm involves the ftatement of what
X therefore muft be, or therefore cannot be, with refpecl: to Z.
We can, in every cafe, exprefs the refult in fimple words. Thus,
78 On the Syllogjfm.
one of our fyllogifms being what I fhall reprefent by DDjDi is
as follows. If X be a fubidentical of Y, and Y a fubidentical of
Z, then X is a fubidentical of Z. But all this merely amounts
to the following c A fubidentical of a fubidentical is a fubidentical.
We have then to examine every way in which D t or D 1 or C
or C f can be combined with Di or D 1 or Ci or C 1 , giving fixteen
cafes in all, and all conclufive in one way or the other. Inftead
of taking an accidental order, and afterwards claffifying the re-
fults, it will be better to predial the order which will give clafli-
fication. That order will be to take I. a D followed by another
of the fame prepofition 2. a C followed by another of different
prepofition 3. a D followed by another of a different prepofition.
4. a C followed by another of a like prepofition. This arrange
ment gives us
1. D.D. D D f DiC, D f C f
2. C,D T C D, CiC C C,
3. DiD 1 D D, D 4 C ? D Ci
4. C,D, C D C,C, C C 1
Each of thefe cafes will be examined by a method fimilar to that
propofed in page 61. But a clear perception of the meaning of
the words will at once dictate the fixteen refults, which are as
follows, preceded by the mode in which the fyllogifms are to
be exprefTed.
DiDiDi Subidentical of fubidentical is fubidentical.
D D D 1 Superidentical of fuperidentical is fuperidentical.
DiCiCi Subidentical of fubcontrary is fubcontrary.
D C C 1 Superidentical of fupercontrary is fupercontrary.
CD Ci Subcontrary of fuperidentical is fubcontrary.
C Di C Supercontrary of fubidentical is fupercontrary.
CC Di Subcontrary of fupercontrary is fubidentical.
C CiD Supercontrary of fubcontrary is fuperidentical.
DiD :C Subidentical of fuperidentical is not fupercontrary.
D Dr. Ci Superidentical of fubidentical is not fubcontrary.
DiC :D ! Subidentical of fupercontrary is not fuperidentical.
D Ci:Di Superidentical of fubcontrary is not fubidentical.
CiDr.D Subcontrary of fubidentical is not fuperidentical.
C D iDj Supercontrary of fuperidentical is not fubidentical.
CiCr.C Subcontrary of fubcontrary is not fupercontrary.
C C 1 : Ci Supercontrary of fupercontrary is not fubcontrary.
On the Sylloglfm. 79
In the denials, the extreme limit is included : in the affirma
tions it is not. Thus not fuperidentical and not fubidentical*
both include not identical ; J and the fame of contraries. In the
affirmations, extreme limitation of one premife does not alter the
conclufion : but that of both reduces the conclufion to its extreme
limit. Thus
Subcontrary of identical is fubcontrary.
Contrary of fuperidentical is fubcontrary.
Contrary of identical is contrary.
and fo on. The rules of this fpecies of fyllogifm are as follows.
For affirmatory conclufions ; (i.) Like names in the premifes give
D in the conclufion, and unlike names C. (2.) D in the firft
premife requires premifes of the fame prepofition ; C in the firft
premife, of different prepofitions. (3.) The prepofition of the
conclufion agrees with that of the firft premife. For negatory
conclufionS) the preceding rules are reverfed. Thefe rules will do
for the prefent, but they afterwards merge in others.
The fixteen forms of complex conclufion above given are of
the clearnefs of axioms, as foon as the terms are diftincSHy appre
hended. The following diagrams will affift, and fhould be ufed
until the propofitions fuggeft their own meaning. Though there
be four, yet thefe four are really but one, as will be mown.
X
If D D D
X
Y
r 7,
} \
f
X
C.D Ci Y
Y D C C
Y C D.C
X
C.C D.Y
Z
} ? ?
Y C f C.D
8o On the Syllogifm.
In each diagram are three lines, partly thick and partly open :
thefe are meant to be laid over one another, but are kept feparate
for diftin&nefs. A point on the firft line fignifies a X or a x ;
and one on the fecond or third, a Y or a y, and a Z or a z.
The univerfe of the propofitions is fuppofed to be the whole
breadth. Points which come under one another are fuppofed to
reprefent the fame object of thought, varioufly named. Thus
in the firft diagram, when the thick lines contain the points
named X, Y, and Z, it is fhown that we mean to fay there are
objects to which all the three names apply : for there are points
under one another in the thick part of all the three lines.
When we read by the letters on the left, the thick lines are
meant to reprefent the parts in which the Xs, Ys, and Zs muft
be placed : and when by thofe on the right, the open lines.
Accordingly, looking at the third diagram, and at the left, we fee
Ci D Ci : while in the diagram, it is clear that X is a fubcon-
trary of Y, or that X . Y and x y ; and that Y is a fuperidentical
of Z, or that Z ) Y and Y : Z. And the conclufion is equally
manifeft, namely, that X is a fubcontrary of Z. But, looking at
the left, and feeing C 1 Di C , we take the open parts to reprefent
the fpaces in which Xs, Ys, and Zs are found, and the thick
parts for thofe in which xs, ys, and zs are found. Here then we
fee that X is a fupercontrary of Y, that Y is a fubidentical of Z,
and that, consequently^ X is a fupercontrary of Z.
Some attempts at laying down the premifes fo as to evade the
conclufions, will be inftructive to any one who does not imme
diately fee the latter. And formal demonstration is always prac
ticable. Thus if X be a fubcontrary of Y, that is, if X and Y
do not fill the univerfe, and have nothing in common ; and if Y
be a fuperidentical of Z, or entirely contain Z, without being
filled by it : then it is clear that X muft be more a fubcontrary
of Z than of Y, by all the inftances which there are of a Y not
being a Z. The diagram, however, is fo much clearer than this
fort of demonftration, that the reader, until he has great com
mand of the language, may as well look to the former to fee that
he is right in the latter.
It may be convenient, as a matter of language, to fpeak of a
name as a kind of collective whole, confiding of inftances. And
thus we may talk of one name being entirely in another, or
partly in and partly out &c, as in fact: we have already done.
On the Syllogifm. 8 1
All the complex fyllogifms which conclude by affirmation are
obvioufly of the a fortiori character : I fhould rather fay, thofe
of the firft three diagrams properly and obvioufly, thofe of the
fourth by an eafy extenfion of language. The marks I 2 3 in the
middle of the diagrams fhow how this is. In the firft, on the
left, X is more of a fubidentical of Z than it is of Y : the in
ftances in which its ^-identity appears confift of all thofe which
prove the fubidentity of X to Y, together with all thofe which
prove the fubidentity of Y to Z. In the third, read from the
right, X is more fupercontrary to Z than it is to Y, by all the
inftances which {how the fubidentity of Y to Z. In the fourth
diagram (from the left) we cannot fay that X is more fubidentical
of Z than of fomething elfe, fimply becaufe there is no previous
fubidentity among the relations. But flill the diftinguifhing
chara&eriftic of the conclufion takes its quantity from the addi
tion of thofe of both the premifes.
If either of the premifes be brought to the limit which fepa-
rates it from the relation of an oppofite prepofition ; that is, if
C ? or Ci be changed into C, or elfe D or D 4 into D : the nature
of the conclufion is not altered, except by the lofs of the a for
tiori character. One of the quantities which have hitherto con
tributed to the quantity of the conclufion, now difappears. Thus
Ci D gives Ci as well as C D ; and C D 1 gives C as well as
Ci D ; C t C gives Di as well as C C 1 .
Let one of the premifes pafs over the limit, and take the oppo
fite prepofition. Choofe Ci D 1 , which gives Ci, and continues
to give it, though weakened, when the firft Ci becomes C. Then
let Ci become C : fo that our premifes are C f D 1 . The dia
gram is then as follows
X
C D f y
Z
? ] ?
The quantity of the conclufion now depends upon the differ
ence between the number of inftances in (12) and (23) and its
quality upon whether (12) has fewer inftances than (23), or the
fame number, or more. As I have drawn it, C t is the conclufion,
ftill : ftrengthen the firft premife ftill more, and the conclufion
82
On the Syllogifm
will pafs through C into C or elfe into P, and in the fecond
cafe may pafs into D 1 , as in the following diagram
X
CT) T Y
Z
Nothing is impoffible except DI or D. Hence C 1 D 1 enables
us only to deny DI and its limit D. Treat the other cafes in the
fame manner, and, remembering that denial is to include denial
up to the limit (while affirmation only affirms to any thing fhort
of the limit) we have
DI D 1 denies C f
DiC 1 . . D 1
C, D, . . D
Ci C, C 1
D D denies C
D C, . . DI
C D 1 . . D,
C C 1 Ci
The rules given above in page 79 may be collected from the
inftances.
As long as we keep contraries out of view, the ultimate ele
ment of inference is of a twofold character. It is either X and
Z are both Y ; therefore X is Z or elfe c X is Y and Z is not
Y ; therefore X is not Z : X, Y, Z, being fingle inftances of
three names ; and Y the fame inftance in both premifes. But the
ufe of contraries enables us to give an affirmative form to the latter
cafe. It is X is Y, and not-Z is Y ; therefore c X is not-Z .
Connected with this change of expreffion is the following
theorem : that all the eight affirmatory complex fyllogifms are
reducible to any one among them : and the fame of the negatory
ones. The reader may trace this theorem to the order of the
figures i, 2, 3, being the fame in all the four diagrams. Taking
DiDiDj as the moft fimple and natural form, and looking at the
diagram of CiD d, we fee the laft as DiDiDi in c X is fubi
dentical of y ; y is fubidentical of z ; therefore X is fubidentical
of z. If we write the terms of the fyllogifm after its defcriptive
letters, as in DiDiDi (XYZ) we have the following refults ;
DiD.Di (XYZ) = DiD,D, (XYZ)
C!D C!(XYZ) = D,D,D, (Xyz)
C.C D, (XYZ) = D,DiDi (XyZ)
D D D (XYZ) = D,DiD. (xyz)
D C C 1 (XYZ) = D.D.D, (xyZ)
C D,C ! (XYZ) = D 4 DiD, (xYZ)
C f CiD (XYZ) = D,DiD, (xYz)
On the Syllogifm. 83
Thinking of the firft defcription only as to relations, and of the
fecond only as to terms, we fee the following rules of connexion.
In the firft and fecond premifes and terms, there are X and Y
in the terms, or their contraries, according as there are fub-
accents or fuperaccents in the relations. But in the conclufion,
the term is Z for D and C f , z for D f and C. And we may
thus reduce any fyllogifm involving any one of the eight varieties
of relation combined with any one of the varieties of terms,
either to DiD t D t or to XYZ. Thus C,D C, (XyZ) is D.DiD,
(XYz), or DiCiC, (XYZ). Not to load the fubjeft with de-
monftration of forms, I will give at once the general rules by
which changes of accent and letter are governed : remarking
that they apply throughout the whole of my fyftem.
The varieties in queftion are eight :
XYZ, xyz ; xYZ, Xyz ; XyZ, xYz ; XYz, xyZ.
in which (thinking of XYZ) all are kept; or all changed; or
one only kept ; or one only changed. Learn to connect each
letter with the proportions in which it occurs ; marking the pro-
pofitions, premifes and conclufion, as I, 2, 3. Connect X with
J >3 5 Y witn J ?2 ; Z with 2,3. Keeping all, or changing all,
makes no alteration of letters : keeping only one, or changing
only one, alters the letters in the premifes in which that one
occurs. Thus, be the accents what they may, if in DDD we
change only the firft letter into its contrary, the fyllogifm becomes
CDC ; and the fame if we keep only the firft letter unchanged.
As to accents, remember that change of Z produces no effecT: :
look then only at X and Y. When either letter is changed into
its contrary, change the accents belonging to the premifes in
which that letter comes firft ; 13 for X, 2 for Y, 123 for XY.
For example, what is CiC ! D, (Xyz). Here, as to letters, X
alone (1,3) is unchanged: then CCD becomes DCC. As to
accents, Y is changed, which comes firft only in 2 : change C f
into C. Hence CiC D, (Xyz) = DiCC (XYZ). Here we
have parted from a fyllogifm in Xyz to the correfponding equi
valent in XYZ : the rules equally hold for the inverfe procefs,
and for all combinations of letters. For the change of XYZ
into Xyz, and that of Xyz into XYZ, have only one defcription :
the firft only left unchanged. Now fuppofe it required to know
84 On the Syllogifm.
what fyllogifm in xYz anfwers to DiCCi(Xyz). The key words
are, the third only unchanged. Alter then DCC into DDD by the
firft rule, and change all the accents. Thus DiCid(Xyz) =
D D ! D T (xYz). The independent rules are that change of fub-
jec~t only, changes both letter and accent ; predicate only, letter ;
fubjecl: and predicate, accent. Thus to find what D C C (xYz)
is, exprefled in XYz, the changes are, in the three premifes S,
neither, S, and D C ! C T (xYz) = C,CT>i(XYz). The following
table may be verified for exercife : it fhows the efFecl of all
changes except that of the middle term.
XYZ xYZ XYz xYz
DDiDi C DiC 1 D,CiC C C,D
C DiC D t D 4 D t COD 1 D,C,C,
D,CiCi C C,D f DiD.Di C D,C !
C CiD 1 aC.C, C DiC 1 DiDiD,
Similarly, D D D would have QD Ci D T C ? C f &c. When
the middle term only is changed, the table may ftand thus ;
XYZ DiD.D, C ^C DiCiC, C dD 1
XyZ CiC Di D C C CiD C, D f D D f
It will of courfe have been obferved that the eight fyllogifms
go in pairs, each one of a pair differing from the other in accen
tuation, and nothing elfe. When we take fets of four, the ones
put together mould be thofe in which the firft premife, or the
fecond, or the conclufion (whichever we take for a ftandard)
has Di and C ! , or elfe has D f and Ci*
The fame rules of transformation apply to negatory complex
fyllogifms ; thus D ? D:Ci(XYZ) is C D f :Di(Xyz). In fed* thefe
rules do not depend upon the character of the inference, nor even
upon its validity, but merely on the efFe&s produced in the fingle
propofitions by changes of term. Thus the flatement D DiCi
(XYZ), an invalid inference, is the fame flatement (equally in
valid of courfe) as is exprefled in DjC D (xyZ).
An examination of the complex particular relation P = L + I f
+ Oj + O f , whether by the diagram or by unaflifted thought, will
mow that when this relation exifts between X and Y, it alfo exifts
between x and Y, X and y, x and y. Hence PC, CP, PD, DP,
On the Sylloglfm. 85
give P. Moreover, two complex particulars give no poffibility
of any conclufion, all being equally poflible. Thus PP may give
Ci or C or C 1 , or D 4 or D or D .
Now combine one of the others, as Di, with P : examine PD
and DiP. It will be found that the complex particular of a fub-
identical may be either complex particular, fubidentical, or fuper-
contrary ; or that PD may be either P, D or C 1 . Examine all
the cafes, and the rules will be found in
(D,C,)P P(D,C ! )
(D C )P P(CiD )
thus interpreted. Either premife from between the parenthefes,
with P, in order as written, may have either, and muft have one,
of the three for its conclufion. That D t P muft give either Di
Ci or P, and fo muft C t P : but PC muft have either P, Ci, or D 1 .
Before proceeding to the fimple fyllogifm, as I have called it,
I will ftate that I much doubt the propriety of the terms fimple
and complex. Undoubtedly the phrafes are hiftorically juft, for
each of the fyllogifms which I propofe to call complex is, as
we mail fee, neceflarily compofed of three of thofe which are
always called fimple. But in another point of view, the phrafe-
ology ought to be reverfed ; the fimple fyllogifm is the affirma
tion of the exiftence of one out of feveral of the complex ones.
Thus X)Y+Y)Z=X)Z, or A.AiAi, is really (D 4 or D, not
known which) (D or D, not known which) (Di or D, not
known which) and aflerts that there is either DiDiDi or D 4 DD,
or DDiDi or DDD.
But it will be faid, furely the complex propofition requires the
conjunctive exiftence of two fimple ones : Di=A 4 +O ? ; and is
therefore compound at leaft. I anfwer that, on the other hand,
the fimple propofition requires the disjunctive exiftence of two
complex ones : as A t =Di or D. Which is moft fimple, both, or
one or the other ? to me, I think, the firft. Certainly the fyllo
gifm DiDiDi is one which I more readily apprehend than AiAA.
Indeed, to moft minds, the latter is the former, if they are left
to themfelves : and the cafes DiDDi, &c. are only admitted when
produced and infifted on.
But further, is the fimple propofition properly called fimple ?
Is there in it but one afTertion to deny or admit ? Is but one
86 On the Syllogifm.
queftion anfwered ? When I affirm Every X is Y, I affirm
i. Comparifon of X and Y. 2. Coincidences. 3. The greateft
poffible amount of them. 4. That every X has been ufed in ob
taining them. In c Some Xs are Ys the firft two of the preced
ing are employed. In No X is Y, we have, I. Comparifon
of Xs and Ys. 2. Exclufions. 3. The greateft amount. 4. The
comparifon of every X with every Y. And Some Xs are not
Ys omits the third, and fubftitutes Xs for every X in the fourth.
Now the fubidentical, for inftance, only contains, befides what
is in the fubaffirmative, the notion that there are more Ys than
Xs in exiftence. The fubcontrary confifts, over and above what
is in the fubnegative, in that Xs and Ys are not every thing that
the propofition might have applied to : and fo on. On thefe
confiderations, I think it may be allowed to treat the words fim-
ple and complex as only of hiftorical reference, and to confider
the firft as disjunctively connected with the fecond, the fecond
as conjunctively connected with the firft, in the manner above
noted. I think I fhall make it clear enough, that the paflage
from the conjunctions to the disjunctions is better fuited to a
demonstrative fyftem than the converfe. If the plan which I
propofe fhould gain any reception, I fhould imagine that disjunc
tive and conjunctive would be the names given to the claffes
which I have called fimple and complex : the conjunctive com-
pofed of feveral of the disjunctive, the disjunctive confifting of
one or the other out of feveral of the conjunctive.
When a propofition R, is the neceflary confequence of two
others, P and Q, it neceflarily follows that the denial of R, muft
be the denial of one at leaft of P and Q. For every propofition
admits but of affirmation or denial : and he who affirms both P
and Q^muft affirm R. If then P be affirmed and R denied, the
denial of Q_ muft follow : if Q be affirmed and R denied, the
denial of P muft follow.
PL fimple fyllogifm is one, the two premifes and conclufion of
which are to be found among the fimple propofitions A, Ei, L,
O 4 , A T , E , I , O 1 . Thus we have AjEiEi or X)Y + Y.Z =
X.Z, as an inftance. The order of reference is always XY,
YZ, XZ.
The following theorems will beneceflary; I. A particular
premife cannot be followed by a univerfal conclufion.
On the Syllogifm. 87
If poffible, let AJi for example, have a univerfal conclufion.
Take the complex premifes D 4 P or (Ai + O f )(Ii + r + Oi + O 1 ).
All that can be inferred is that one of three conclufions (page 85)
is valid, and neither D nor C : either D t or P or C. But if a
univerfal be true, one of two conclufions muft be valid (page 69)
and one of them D or C. If then Ai and L alone yielded a
univerfal conclufion, quite as much muft DiP : or a form which
is indifferent to three conclufions, and not having D nor C, is ne-
ceflarily productive of one of two conclufions, one of which is
D or C. This contradiction cannot exift : or AJi cannot yield
a univerfal conclufion.
2. From two particular premifes no condujion can follow.
If poffible, let IJi yield a conclufion ; which by the laft the
orem, muft be only particular. Now PP or (Ii + I f H-Oi-fO 1 )
(Ii + 1 1 + Oi + O 1 ) is indifferent to all complex conclufions : quite
as much is Ui. But if thefe premifes yield a particular conclu
fion, two complex conclufions are denied (page 69). This con
tradiction cannot exift : or particular premifes can yield no
conclufion.
Let a fimple fyllogifm with premifes and conclufion all univer
fal, be called univerfal: and with either premife (and therefore
the conclufion) particular, be called particular. Then every
univerfal fyllogifm has two particular fyllogifms deducible from it.
Thus if AtEiEi be valid, then AI joined with the denial of EI
gives the denial of EI : or AHiIifeems to be valid. But the altera
tion of the places of the propofitions requires us to fay that it is
A Ui which is valid : and this point requires clofe attention.
Take AiEEi or X)Y + Y.Z = X.Z. Then X)Y with the
denial of X.Z(or XZ) gives the denial of Y.Z(or YZ) ; and
we have
This is valid, if the firft be (as it is) valid : but its fymbol is not
AJiL. For the middle term is, in our notation, made middle in
the order of reference, which is therefore YX, XZ, YZ : and
the fyllogifm is A Ui. Similarly we have
XZ + Y.Z=X:Y
produced by coupling the denial of X.Z with Y.Z. But this is
LEOj : for the order of reference is now XZ, ZY, XY, and
88 On the Syllogifm.
Ei is not changed by change of order. The rule is as follows.
When the denials of the conclufion and of a premife are made to
take the places of that premife and the conclufion, the order of
reference remains undifturbed as to the tranfpofed terms, and is
changed as to the ftanding term. This laft muft therefore have
the prepofition of the inconvertible propofition changed ; but
not that of the convertible propofition.
Thus E AiE , if valid, gives ETOi and I 1 AT. Again, in a
fimilar way it may be fhown that from each particular fyllogifm
follows a univerfal : thus LE O 1 , if valid, {hows that denial of
O f , and E f , give denial of L or A E Ei. In this cafe neither is
valid. And ETOi, befides E AiE , alfo gives A.IT.
Such clarification of thefe opponent forms as is ufeful, will pre-
fently be given.
Since there are eight forms of afTertion, with reference to each
of the orders X Y YZ, it follows that there are fixty-four com
binations of a pair of premifes each. But of thefe the only ones
which have a chance of yielding a conclufion are, I. fixteen
with premifes both univerfal ; 2. thirty-two with one univerfal
and one particular. If, for a moment, U ftand for univerfal and
P for particular, the form of a fyllogifm is either UUU, PUP,
UPP, or UUP. Of thefe, the firft, fecond, and third are fo
related that each form has the other two for its opponents : but
the fourth has its own form in each of its opponents.
Now examine one of the complex affirmative fyllogifms, fay
DiDiDi, by the diagram in page 79. The premifes are Ai + O 1
and AI + O 1 , giving the four combinations AiAi, A|O , O Ai and
O O 1 . The conclufion is Ai -f- O : but it is not merely twofold,
but threefold : for the a fortiori character explained in page 81,
(hows that O 1 is obtainable on two different grounds, and is the
fum, as it were, of two different and neceffary parts of the con
clufion. That every X is Z, follows from X)Y and Y)Z, or
we have the fyllogifm.
AiAiAi X)Y + Y)Z=X)Z
But as far as the Zs which are below (12) are concerned, it
follows that they are not Xs becaufe they are the Ys which are
not Xs : or we have
O AO
On the Syllogifm. 89
and as to the Zs below (23) they are not Xs becaufe they are
not Ys, among which are all the Xs. Accordingly we have
AiO O X)Y + Z:Y = Z:X
or DiDiDi requires the coexiftence of AiAiAi, O AiO , AiO O .
Apply this reafoning to the contraries x, y, z, or elfe examine
D D D in the fame way, and we find that D D D requires the
coexiftence of A A A , OiA O,, A f OiOi.
By applying the preceding refults to x, Y, Z, &c. as in page
82, or, as is better at firft, by examining all the cafes of the dia
gram in page 79, we get the following table of derivations from
the eight affirmatory complex fyllogifms. The firft column
mews the terms which muft be ufed, to deduce all from DiD 4 Di
rAiAiAi X)Y + Y)Z=X)Z
XYZ DiDiD, . A A,0 Y:X + Y)Z=Z:X ( I2 )
lA.O O X)Y + Z:Y=Z:X (23)
f-A A A Y)X + Z)Y = Z)X
xyz D D D . . -f OiA Oi X:Y + Z)Y = X:Z (12)
UVOiOi Y)X + Y:Z = X:Z (23)
rE A,E ! x.y +Y)Z= x.z
x YZ C DiC . . \ LA.L XY +Y)Z=XZ (12)
LE O L x.y+Z:Y = XZ (23)
rTTA TT Y V L V ^ V Y V
JLi/\ IL| -A.. I -\- Z-j ) I =^V.Zy
Xyz C.D Ci . . \ TAT xy +Z)Y=xz (12)
lEiOJ X.Y + Y:Z=xz (23)
(A TT IT Y\V L V V Y V
zi.iJlrlll/i j\. ] I -p I ./.j zzz. yv.^L/
O EJ 1 Y:X + Y.Z=xz (12)
lA,IT X)Y+yz =xz (23)
rA E E Y)X+y.z =x.z
xyZ D C C . . \ OiE L X:Y+y.z =XZ (12)
LA IiI, Y)X + YZ =XZ (23)
(-E E.A 1 x.y +Y.Z=Z)X
x Yz C CiD . . <( LEiOi XY +Y.Z=X:Z (12)
lE I Oi x.y +yz =X:Z (23)
rE.E Ai X.Y+y.z =X)Z
XyZ CiC Di . . \ I E O 1 xy +y.z=Z:X (12)
LE.1,0 1 X.Y + YZ=Z:X (23)
Before forming any rule, or making any remark, I proceed to
90 On the Syllogifm.
collet the refults of the remaining cafes. And firft, let a pre-
mife be brought to its limit, D or C : fay that DiDiDi becomes
DDiDi. In the diagram it immediately appears that one of the
particular conclufions is loft ; not contradicted, but nullified : for
(12) difappears, becaufe X and Y are identical names. That is,
AiAiAi remains, and AiO O 1 : but the conclufion of O AiO 1 is
nullified. But this very circumftance creates, not a new conclu
fion, for it is only a part of one already exifting, but a new form
of deduction. The premifes are now Ai + A 1 and Ai + O , and
the conclufion is A t + O . The fyllogifms A,AAi and AiO O f
are as before, and for the fame reafons : but there is now the
combination A Ai among the premifes, which produces the con
clufion L, and we have
A AJi Y)X + Y)Z=XZ
This fyllogifm, though new as far as D 4 DiDi is concerned, is
only a ftrengthened form of LAJi, a concomitant of E AjE 1 .
For (page 65) L is true whenever A is true, fo that A Ai in
cludes LAi and its necefTary confequence L. But if L had been
ftrengthened into A* inftead of A 1 , we fhould have had AiAJj
which though perfectly valid, yet admits of a ftronger conclu
fion, as feen in AiAiAj.
Of the two modes of ftrengthening a particular propofition
(as L into AI or A 1 ) there is one which ftrengthens the quantity
of the firft form of the propofition, and another that of the fecond.
Thus XY or L becomes X)Y or AI when the firft form, and
Y)X or A 1 , when the fecond form, is ftrengthened. Similarly
Oi or X:Y becomes X.Y or E, and y.x or E 1 , according as
the form ftrengthened is X:Y or y:x. The prepofition remains
the fame, or changes, according as the firft or fecond form is
ftrengthened. If the firft form of the fecond premife of a fyllo
gifm, or the fecond form of the firft premife, be ftrengthened, no
ftrength is added to the conclufion. Thus, as far as the fyllo
gifms in this chapter are concerned, LAi gives as much as A Ai,
and E t Oi as EtE. But if the firft form of the firft premifs, or
the fecond form of the fecond, be ftrengthened, the conclufion
has its firft form ftrengthened.
A very fimple and obvious theorem contains all thefe refults.
The concluding terms are, in our order of reference, the firft
On the Syllogifm. 91
term of the firft premife and the fecond term of the fecond. The
conclufion is never ftrengthened by augmenting the quantity of
the middle term, nor only weakened (it may be altogether de-
ftroyed) by weakening the middle term. A wider field of com-
parifon does not by itfelf give more comparifons : nor can more
comparifons arife except by augmenting the number of things
compared in that field. Since the conclufion can obvioufly
fpeak of no more than was in the premifes, no term of that con
clufion can be augmented in quantity, until the fame thing has
taken place in its premife. But no ftrengthening of a propofi-
tion ftrengthens both terms : confequently, to make fuch a thing
effective, it muft be the concluding, and not the middle, term
which is ftrengthened.
The following table is only worth inferting as a collection of
exercifes. The fourth column (hows the eightjtrengthened par
ticular fyllogifms^ as I will call them, having univerfal premifes but
only a particular conclufion, not ftronger than might have been
inferred from the particular fyllogifm itfelf.
Alteration
and fub-
ftrengthened
occurring
of
into
removes
ftitutes
from
in
DiD.Di
DD.D,
O AiO )
A AiL
LAili )
C DiC
D D D 1
D DD
A OiOi}
A LL )
D C C 1
DiDiDi
D,DDi
AiO O )
AiAT
I A I )
CiD G
D D D
DD D
s,
AilT )
DiCiCi
D,CiCi
DCiCi
O Eil )
A EiO,
LEiOi I
C CiD
D C C
D CC 1
A LL )
A OiOij
D D D
DiC,Ci
DiCCi
AilT )
AiE O
I E O )
CiC Di
D C C
DC C
\
AiO O )
DiDiD,
GD Ci
CD C,
I AT ^
E A Oi
OiA Oi)
D D D
C DiC
C DC
E O L )
E I O, }
C C,D
C.D Ci
CiDC,
EiOJ I
EiAiO
O AiO )
DiD,D,
C DiC
CDiC
LA.Ii 5
EiLO ]
CiC D,
CiC Di
CC Di
I E O )
E E L
OiE L |
D C C
C C,D
C CD 1
E I Oi )
E O L J
C DiC
CiC Di
C,CDi
EiLO J
E.Eil
O Eil )
DiCiCi
C CiD
CCiD
LEiOi J
EiOil j
CiD C,
I will now examine the negatory complex fyllogifms, premifing
however than we cannot get any new conclufions from them.
92 On the Syllogifm.
For we have now got all the fixteen cafes in which both pre-
mifes are univerfal : and we know that there can be no fyllogifm
with a particular premife, except it have one of thofe with uni
verfal premifes for its opponents.
Take DD f :C or A, + O and A f + Oi together deny E + L,
that is, deny the coexiftence of E 1 and L, that is, deny either E 1
or L, that is, affert either I 1 or E t . This fyllogifm then may be
written thus,
(Ai + O 1 ) (A + Oi) (either Ei or I )
Now the fact is that this disjunction is fuperfluous ; it is I
which is always afferted, and E 4 is never a neceffary confequence
of DiD 1 . For AjA gives I 1 as already mown, and "Aid and
O A are inconclufive (and O d of courfe). And the rationale
of the inference is as follows : fince X is a fubidentical of Y, and
Y a fuperidentical of Z, it follows that Y is fuperidentical both of
X and Z ; confequently, Y not rilling the univerfe (our fuppo-
fition throughout) it follows that there are things which are nei
ther Xs nor Zs, namely, all which are not Ys. Again, in
CiCiiC 1 , which the fame reafoning mows to be only CiCiI ,
none either of X or of Z is in Y, therefore every inftance in Y
is both x and z. And thus it will appear that in every negatory
complex conclufion the whole middle term, or the whole of its
contrary, makes the fubject matter of the ftrengthened particular
fyllogifm which is all that can be collected.
Our conclufion is that no negatory complex fyllogifm is of
any more logical effect than the ftrengthened particular derived
from it. Thus we may fay that, fo far as the extent and cha
racter of the inference is concerned, the former is the latter.
I will now pafs to the general rules of the complete fyftem of
fyllogifms ;
The reader muft take pains to remember two rules of forma
tion, perfect contraries of each other, for the dependence of the
accents (or prepofitions] on the fign (affirmative or negative cha
racter) of the firft premife. I exprefs them in the briefeft way
poffible.
Direfl Rule. Affirmation (in the firft premife) makes the fecond
premife agree with both the other propofitions, or ifolates no
thing : negation makes the fecond premife differ from both the
On the Syllogifm. 93
others, or ifolates the fecond premife. Inverfe rule. Affirmation
Ifolates the firft premife, makes the firft premife differ from both
the others in prepofition : negation ifolates the conclufion, makes
the conclufion differ from both the others. Thefe rules might
be expreffed fo as to make their contrariety more complete.
Thus in the . n " r rule, affirmative commencement mows ,.,
mverie unlike
prepofitions in the two premifes, and the conclufion ,9
Y 1 the firft premife in prepofition : but negative commence
ment (hows , prepofitions in the two premifes, and the con
clufion j" g the firft premife in prepofition.
The fubjecls of the following rules are,
1. The eight affirmatory complex fyllogifms.
2. The eight univerfal fimple fyllogifms.
3. The eight ftrengthened particular fimple fyllogifms.
4 The fixteen particular fimple fyllogifms.
Omit the negatory complex fyllogifms, as fully contained in
the third of this enumeration, and the complex fyllogifms which
contain the unaccented D or C, as carrying a momentary accent
for the rule, to be expunged when the formation is completed.
Confider DI, D, D f , Ai, A f , L, I 1 , as of the affirmative figns, and
Ci, C, C 1 , Ei, E 1 , Oi, O f , as negative.
Rule i. In the complex fyllogifm all parts are complex; in
the univerfal fimple fyllogifm all parts are univerfal ; in the
ftrengthened particular only the conclufion is particular ; in the
particular only a premife is univerfal.
Rule 2. Premifes of like fign have an affirmative conclufion ;
of unlike fign, a negative.
Rule 3. The complex, the univerfal, the particulars which
begin with a particular, follow the direcl: rule ; the ftrengthened
particulars, and the particulars which begin with a univerfal (all
that commence with a univerfal, and conclude with a particular)
follow the inverfe rule. [Or thus ; all which begin and end
alike, follow the direcl: rule ; all which begin and end differently,
the inverfe.]
The complex fyllogifms and univerfals are eafily remembered
94 On the Syllogifm.
by rule : the particulars almoft as eafily. The following Tub-
rules may be noted, as far as thefe laft are concerned.
Sub-rule I. Firft and fecond premifes. A and O in the firft
premife demand unlike prepofitions in the two premifes : E and
I demand like prepofitions. Thus AiOi muft be inconclufive :
A 4 O muft be conclufive. But E t Oi muft be conclufive : and
EiO 1 muft be inconclufive.
Sub-rule 2. Firft premife and conclufion. A univerfal in the firft
premife demands an unlike prepofition in the conclufion : a par
ticular firft premife, a like prepofition in the conclufion.
Sub-rule 3. Second premife and conclufion. Every fecond pre
mife demands its own prepofition in a conclufion of like fign :
and the other prepofition in a conclufion of unlike fign.
As far as the four fpecies are concerned, every fyllogifm
formed according to the three rules is valid ; and every one
not fo formed is invalid. The following remarks are partly
recapitulatory, partly new.
Remark I. Every complex fyllogifm gives one univerfal fyllo
gifm * and two particular ones, its concomitants : and the con
comitants are formed by changing one of the premifes of the
univerfal and the conclufion, into their particular concomitant
propofitions (page 63.)
Remark 2. Every fyllogifm has its contranominal, which afTerts
of the contraries in the fame manner as the firft does of the di-
recl: terms : and contranominals have all their accents different,
as in O AiO 1 and OiA d (page 62.)
Remark 3. Every fyllogifm has two opponents, made by inter
changing the contradictories of one premife and of the conclufion,
and altering the accent of the remaining premife, if inconverti
ble (A or O) (page 88.)
Remark 4. Every complex fyllogifm has two fuch opponents
formed in the fame way, the Ds being the inconvertibles, the Cs
the convertibles. Thus (:) meaning denial of, the opponents of
CiD C. are C.iCiiD and :C,Di:Ci. The firft of thefe is
(E, + r)(WE )(0 ! or A.)
containing the valid fyllogifms EE Ai, EJiO 1 , I E O ; being
* Syllogifnty not preceded by complex, means fimple fyllogifm.
On the Syllogifm. 95
EiE Ai and its concomitants. And :CDr.C gives E AiE (the
contranominal of EiA Ei) and its concomitants. And the fame
of the reft.
Remark 5. Each univerfal fyllogifm has two weakened forms,
made by weakening one premife and the conclufion. When the
firft premife is weakened, it is without change of prepofition :
but when the fecond, with change. Thus the weakened forms
of EiA Ei are O,A ! Oi and EJiO 1 .
Remark 6. Each particular fyllogifm has two ftrengthened
forms, one of which is a univerfaj, the other only a ftrengthened
particular. Thus the ftrengthened forms of OiA Oi are EiA Ei
and E f A Oi.
Remark 7. In every fyllogifm except the ftrengthened particu
lar, the middle term is univerfal in one premife, and particular in
the other : and its contrary is therefore the fame. But in the
ftrengthened particular, the middle term is univerfal in both
premifes, or particular in both. This affords a complete crite
rion of fyllogifm, as will be noticed hereafter : in facl:, the com-
pletenefs of this fyftem crowds us with relations, from many of
which general rules might be deduced, though they need only
appear here by cafual remark.
In O A.O 1 , A 0,0,, LA.L, E.O.I 1 , O EJ , A LL, I.EiO,,
EJiO , the middle term enters univerfally in the univerfal, and
particularly in the particular. In all the others it enters particu
larly in the univerfal, and univerfally in the particular. In the
firft fet, the convertible premifes are all fubs, the inconvertibles
are fubs in the fecond premife, and fupers in the firft. In the
fecond fet, thefe rules are inverted.
Remark 8. Of the twelve poffible pairs of premifes AA, AE,
AI, AO, EA, EE, El, EO, IA, IE, OA, OE, which can give
a conclufion, each one wlll^ in two ways, which two ways are
inverted in their accents. Thus EO appears in E O L and
EtOJ . The two premife-letters and one accent dictate all the
reft : thus I A can belong to nothing but I 1 AT. When the
fyftem is well learnt, it will be found unneceffary to write more
than I A, for the fymbol of I 1 AT. I now fpeak only of funda
mental fyllogifms : the ftrengthened fyllogifm AAT might be
fignified by A 1 A 1 .
Remark 9. The fyllogifms of the three firft clafles are all really
96 On the Syllogifm.
fpecimens of one, thofe of the fourth of two, among them, with
the eight variations XYZ, xYZ, XYz, xYz, XyZ, xyZ, Xyz,
xyz. The rules for conducing thefe changes are
Change of fubjecl: is change of both accent and letter.
Change of predicate is change of letter.
Change of both is change of accent.
thus to pafs from E EjA to AiEEi we note in XY change of
fubjecl:, in YZ change of neither, in XZ change of fubjecT: :
therefore xYZ is the fet of terms into which XYZ muft be
changed : and the E EjA 1 fyllogifm of either fet is the AiE t Ei
fyllogifm of the other.
The 24 fyllogifms, which are 24 with reference to the order
XY, YZ, XZ, are only 12 if the order ZY, YX, ZX, be
allowed. Thus AJT of the firft is the TAT of the fecond.
Thefe fyllogifms are eflentially the fame in the mode of inference
they afford. To change a fyllogifm into another of the fame mode
of inference, invert the premifes and change the prepofition of
all the inconvertibles. Thus A OiOi and O AiO 1 are of the fame
inference. The pairs which in this point of view are identical are
A.AiA, =A I A ! A !
O A.O^A OiOi
A,0 T =OiA f O,
IiAJ, =A I. L
E ? L=0 1 E f L
E,A Ei=A 1 E 4 E 1
PAT =AJT
E,0.r=0 ! Eir
E ! EiA f =EE f A.
LEiOi =EJ,O
ETOi =rE ! O ?
The ninth remark admits of confiderable extenfion. The fame
of a logical proportion may have a much more definite character
in fome cafes than in others. It may be a felecled, or at leaft a
diftinguimabley^?, which want nothing but a nominal diftin&ion
to make the particular proportion eafily and ufefully univerfal.
Whether it can be done more or lefs eafily, and more or lefs ufe
fully, is no queftion of formal logic. If it be fuppofed done, the
particular is converted into a univerfal. In c fome Xs are Ys,
if we make a name for every X which is Y, fay M, we have
then Every M is Y . This proportion may be purely identi
cal, or it may not. If we call every X which is Y by the name
M merely becaufe it is Y, then our univerfal is only Every X
which is Tis Y . But if the name M be conferred from any
other circumftance, which diftinguifhes the Xs that are Ys from
other Xs, then the change from the particular to the univerfal by
On the Syllogifm. 97
means of the new reftri&ion impofed by the new name, is the
expreffion of new knowledge.
The quantities in the conclufion are of two kinds. There are
thofe which are brought in with the terms, and which continue
in the conclufion fuch as they were introduced in the premifes :
and there are thofe which depend on the union of the premifes,
and which are what they are only in virtue of the joint exiftence
of the premifes. For example, in LAJi we have 4 fome Xs are
Ys, but every Y is Z, therefore fome Xs are Zs : if we afk,
what Xs are Zs, the anfwer is, thofe which are Ys, and no others,
fo far as this conclufion affirms. But when we look at O AiO 1
or * fome Ys are not Xs, and every Y is Z ; therefore fome Zs
are not Xs : and if we then afk what Zs are not Xs ; the anfwer
is, that this quantity does not enter with Z, but depends upon the
other premife, namely, upon the number of Ys which are not Xs.
In a particular fyllogifm, let us call the quantity of the fubjet
in the conclufion Intrinfic or extrlnfic according as it is that of
the premife which introduces that fubjecl:, or of the other premife.
Examination will fhow that in every particular fyllogifm which
concludes in L or I 1 , in which both terms are particular, the
quantities of the terms are, of the one intrinfic, of the other ex-
trinfic : but that where the conclufion is in O 4 or O 1 , either the
quantity of the fubjecl: is intrinfic and that of the contrary of the
predicate extrinfic, or vice verfa.
When the quantity of a particular term in the conclufion is
intrinfic, the invention of a name will convert the syllogifm into
a univerfal. Thus LA,A, or XY + Y)Z = XZ, if M be taken
to reprefent all thofe Xs which are Ys, and nothing elfe, becomes
M)Y + Y)Z = M)Y, of the form AiA.A,. Again, O A.O or
Y:X-f Y)Z = Z:X, thrown into the form x:y + z)y=x:z, be
comes m.y-fz)y=m.z, of the form EiAiEi, when the xs which
are ys are diftinguifhed from the reft of the univerfe by the name
m. There is nothing either illegitimate or uncommon in diftin-
guifhing by a peculiar name certain fome (or even uncertain fome,
if certainly always the fame fome] of another name. Again, fince
we know that every univerfal fyllogifm is reducible to the form
AiAtAj by ufe of contraries, we have now reafon to know that
there is no fundamental inference, of the kind treated in this chap
ter, which is any other than that in AiAiAi, or, the contained
H
98 On the Syllogifm.
of the contained is contained. And there is no better exercife than
learning to read off each of the fyllogifms, univerfal and particu
lar, into this one form, by perception, and without ufe of rules.
Take as an inftance X:Y+y.z=XZ : what is the container,
what is the contained, and what is the middle container of one
and contained of the other. It is a parcel of Xs which are con
tained in y, all y in Z, and therefore that parcel of Xs in Z.
This general principle fuggefts a notation for all the complex,
univerfal, and fundamental particular, fyllogifms. If we abbre
viate X)Y + Y)Z = X)Z into XYZ), and if we denote by
XYZ, without ), that it is only a parcel of Xs (all or fome,
defined or undefined, but always the fame), we have the fol
lowing,
For A.AiA, read XYZ) or zyx)
_ Q A.O 1
_ A,0
xYZ
Zyx
For E A.E 1 read xYZ) or zyX)
LAiIi XYZ
E O L ZyX
For A,EE read X Yz) or Zyx)
_ O EJ 1 xYz
_ AJT zyx
For E E,A ! read xYz) or ZyX)
LEiO, XYz
ETOi zyX
For A A A read xyz) or ZYX)
__ O,A ! Oi Xyz
A OiOi zYX
For EA E read Xyz) or ZYx)
TAT xyz
EiOJ zYx
For A E E read xyZ) or zYX)
_ QiE L XyZ
A lJi ZYX
For EiE A, read XyZ) or zYx)
_I E T O ? xyZ
EiLO 1 ZYx
Here, ufmg P,Q,R, as general terms, PQR) denotes that all
Ps are Qs, and all Qs are Rs, whence all Ps are Rs : while
PQR only denotes that there is a parcel of Ps among the Qs,
and all Qs are among the Rs, whence that parcel of Ps is among
the Rs.
The rules for the connection of thefe fyftems are not compli
cated, confidering the extent of the cafes they are to include.
Let the letters A,E, &c. be called proponents ; X,Y,Z, nominals:
and by the order of the nominals we always mean that X is firfl^
&c. both in XYZ, and ZYX. The nominals being direft
(X,Y,Z) and contrary (x,y,z), remember that,/r/?,
On the Syllogifm. 99
t firft {fi r ft an d Second
An affirmative?/^?^/ proponent denotes that \L\\t\fecond and third
( third third and firft
nominals agree (are both direct or both contrary).
{firft (fi r ft an d Second
A nega,tive\fecond proponent denotes that t\\Q\ fecond and third
(third third and firft
nominals differ (are one direct, one contrary).
Thus EIO muft give Xyz or xYZ or zyX or ZYx
IEO muft give XYz or xyZ or zYX or Zyx
Secondly^ whether the middle term be Y or y depends only on
the accent of the middle proponent : a fub-accent gives Y, a
fuper-accent gives y. In the univerfal lyllogifm however, either
gives either.
Thirdly^ the XYZ fyllogifms are the particulars which begin
with a particular : and the ZYX fyllogifms are the particulars
which begin with a univerfal.
For example, required OiE L . Seeing the particular Oi, at the
beginning, take the order XYZ, feeing the fuperaccent in E 1
make it XyZ. Seeing the negative Oi , let the exifting difagree-
ment of the firft and fecond nominals continue : and the fame of
the fecond and third from the negative E. Confequently XyZ
is the fyllogifm exprefTed in nominals. Or the rationale of
the inference in OiE L is that a parcel of Xs are among the Zs
becaufe among the ys which are all among the Zs.
Again, required the nominal mode of expreffing ET Oi . See
ing the univerfal E at the beginning, write down ZYX ; for the
fuperaccent in I 1 , write down ZyX ; for the negative in E 1 ,
continue yX ; for the affirmative in I 1 , write zy : hence zyX is
the nominal form of ETOi.
Required the proponent mode of exprefling xYz. Here xY,
Yz, (how us that the premifes are negative forms, and the direc
tion of the order x, Y, z, that the firft premife is particular.
Then OE are the premifes, and I the conclufion. And Y tells
us that the middle proponent has a fubaccent. Whence OEJ
is, fo far as it goes, the proponent expreffion. And, by the laws
of form, the other accents muft be as in O f EJ ! , fince the fyllo
gifm follows the direcl: rule (page 93).
ioo On the Syllogifm.
Required the proponent mode of expreffing ZYx. Here we
note in fucceffion univerfal commencement firft premife ne
gative fecond, affirmative middle accent fub. This gives ELO
of the inverfe rule, or EJjO T .
Required the proponent notation for the univerfal xYZ) or
zyX). We fee at once EAiE, or E AiE .
The concomitants of a univerfal are found by changing the
firft nominal into the contrary, in each of the forms, and throw
ing away the fign of univerfality [ ) ] . Thus the concomitants
of XyZ) or zYx) are xyZ and ZYx.
The weakened forms of a univerfal are found by merely
throwing away the fymbol of univerfality [ ) ] from the two
forms of the univerfal. Thus the weakened forms of XYZ)
which is alfo zyx) are XYZ and zyx.
But we have not yet reached the climax of fymbolic fimplicity
in the mere reprefentation of fyllogifms. An algebraift would
fay that the ftrucSture of the inference, as now confidered, does
not depend upon the names ; but only upon their reference to
the names in the fundamental form XYZ). He would there
fore propofe a fimple iymbol to reprefent letting alone, and
another to reprefent changing Into the contrary. Thefe, with a
fign of complete univerfality, and another of inverfion of order,
are all that he would find necefTary. Let o and I fignify letting
alone and changing into the contrary : let the terminal parenthefis
denote complete univerfality, as before, and let inverfion of order
be denoted by a negative fign prefixed. Thus XYZ or LAJi,
would be denoted by ooo ; Zyx or AiO O by on ; AiE 4 Ei
or XYz) by ooi) or its equivalent on. Thus on tells us
that fome of the Zs are ys, all the ys are xs, whence fome of the
Zs are xs. To write its proponent form, obferve that inftrucl:s
us to write a univerfal firft ; 1 1 to make it affirmative ; I in the
middle to fuperaccent the middle propofition ; 01 to make the
fecond premife negative. We have then AiO O 1 or X)Y +
Z : Y = Z:X which is Zy + y)x = Zx, as aflerted.
All that relates to univerfals in the preceding, applies to the
complex fyllogifms. Let a couple of parenthefes imply a complex
fyllogifm : thus DiDiDi may be (XYZ) or (ooo). Then in (oio)
or (XyZ), we are to fee that X is a fubidentical of y, and y of Z,
whence X is the fame of Z. But Xy and yZ warn us to write
On the Syllogifm. 101
contraries for the firft and fecond premifes and y to fuperaccent
the middle letter : whence CjC Di is the fyllogifm expreffed by
the names XYZ. The equivalent forms (101) and (zYx) ex-
prefs it by faying that z is a fubidentical of Y and Y of x, whence
z is a fubidentical of x.
I now look at the ftrengthened particular fyllogifms. All in
ference which is fundamental, that is, which will come from
nothing weaker than the premifes given, has been reduced to the
one eafy cafe of the contained of the contained is contained.
The ftrengthened particular, the type of which is A AJi, obeying
the inverfe rule of formation, and written at more length in Y)X
+ Y)Z = XZ, may be ftated thus all names are common as to
what they contain in common. If we denote this ftrengthened
fyllogifm by XYZI, a fymbol intended to imply fomething be
tween XYZ and XYZ) in the amounts of quantity introduced,
we (hall find that the eight ftrengthened fyllogifms muft be re-
prefented by
A AJi =XYZI AiAT = xyzl
A ? E,Oi=XYzI AiE O ! = xyZI
E A ! O, = Xyzl E,A t O f = xYZJ
E E L = XyZI EiEJ T = xYzl
The rules of connexion are precifely thofe for the particular
fyllogifms : and inverfion is abfolutely ineffective. Thus XYZI
=ZYXI.
A few words will ferve to difpofe of the mixed complex fyllo
gifms in which a complex premife is combined with a fimple one,
univerfal or particular. Firft, when a complex and a univerfal
are premifed, and figns and accents are as in the dire ft rule (page
92), the conclufion is as it would be if the A were heightened
into D, or E into C. Thus EiD 1 gives Ci, the fame as CiD f .
For Ei is C or C 4 , and both CD f and CiD 1 give Ci, but with
different quantities. But if the premifes be conftru&ed on the
inverfe rule, there is no more inference than can be obtained
when the complex premife is lowered into a univerfal : or we
have only a ftrengthened particular. Thus in DiE or (A 4 -f-
O )E f , AiE gives the ftrengthened particular A,E ? O T , and O E 1
is inconclufive. And when the complex premife is combined
with a particular, we have only what would follow if the com
plex premife were lowered into a univerfal. Thus D t l f , or
IO2 On the Sylloglfm.
(A, + O )I f can only give AJT ; and DT or (A 1 + Oi)I ! gives no
conclufion, for AT is inconclufive.
The claflification of opponent forms may be thus treated.
We know that opponent forms of AEE, for inftance, be it A
EiEi or A E E 1 , muft be IEO and All. Now whether AE 4 E.
mail have LEiOi or EJiO 1 , whether A lJi orLAJi, depends upon
the introdu&ion of a new and arbitrary notion of the order to be
adopted. Our firft fyllogifm being defcribed by XY, YZ, XZ,
the opponent which ends in the contradiction of the firft premife
is in XZ, YZ, XY j which, keeping Z middle, is either to be
defcribed with reference to XZ, ZY, XY, or to YZ, ZX, YX.
Now in adopting the firft of thefe three orders, there is nothing
which compels us therefore to prefer the fecond to the third, or
vice verfa.
The effect of the change of order which confifts in the inter
change of Z and X is as follows. The premifes change places ;
A and O with altered accents, altered alfo in the conclufion, E
and I with unaltered accents. Thus AJT becomes FAT ;
E O L becomes OiE L . Accordingly, it is matter of new ar
rangement whether for inftance, LEiOi or EJiO 1 {hall be called
the opponent of AiEiEj ; and I prefer to give the name to both.
The confequence is, the following diftribution of opponents ;
AT> T- A AT TA TT
EE EO OE AE EA AI IA EI
The three fets reprefent letters combined in reprefentation of pre
mifes : the firft two containing fix fyllogifms each, the third
twelve. The third muft be divided into two fets of fix each, in
one of which the fubaccents are in greater number, in the other
the fuperaccents. There are then four fets in all. Pick any
two out of a fet, which only differ in change of order : thefe two
have the fame opponent forms, namely, the other four of the
fet. For inftance, A f IJ and LA L, in which fubaccents predo
minate. Take AE, EA, EI, IE, and complete fyllogifms in
fuch manner as to make fubaccents predominate : giving AiEiEi,
EiA Ei, EJiO LEiOi. The laft four are the opponents of the
firft two.
In the fet of ftrengthened particulars the opponent forms will
be found to be univerfals weakened in the conclufion without
On the Syllogifm. 103
being weakened in the premifes. Thus AiA 1 ! 1 has A E O 1 for
one of its opponents : but A E 1 may produce the univerfal con-
clufion E 1 as well as its weaker form O 1 .
Some readers, particularly thofe who have a tincture of algebra,
are more helped by fymbolic notation than by language : with
others it is the converfe. To fuit the latter, obferve that the
language of page 78 may eafily be adapted to fimple fyllogifms.
Thus Ai being fubaffirmation, L may be fome fubaffirmation, O
may be fome fupernegation ; and fo on. Thus inftead of ETOi we
may fay that fupernegation of fome fuperaffirmation gives fome
fubnegation. Practice in this language would make the phrafe
fuggeft fomething more than the notation it is derived from.
The phrafe refers to Z : there is a term partially fuperaffirmed
of Z, namely Y ; and a complete fubnegative of Y, namely X.
The partial fubaffirmation declares fome things neither Y nor Z ;
the complete fupernegation declares that whatever is not Y is X.
Confequently there are fome Xs which are not Zs : or X is a
partial fubnegative of Z. This fubject will be refumed.
In what precedes are two views of the deduction of all the
varieties of fyllogifm. The firft, taking the complex fyllogifm as
the fource, connects the ftrengthened fyllogifms and the parti
cular ones with the univerfals, and thus in fact reduces every
thing to the conftituents of DiDiDi or DDiDi. The fecond pro
ceeds from AiAjAj, A AJi, AJT, and LAJ, and forms the clafTes
of univerfal, ftrengthened, and particular, fyllogifms by fubfti-
tuting contraries in every way in which it can be done. Thefe
two fyftems have clofe connexion, but not fo clofe as might
perhaps be thought : for LAJi is not one of thofe which are
connected with AjAjAi in the formation of a complex fyllo
gifm.
The two new views which I now proceed to give are alfo
clofely connected, and different from the former ones, in which
we held it equally admiffible to refer one of the concluding terms
to the middle, as in X)Y, or the middle to one of the concluding
terms, as in Y)X. But now I afk whether it be not poflible fo
104 On the Syllogifm.
to conftruct the fyftem, that we may firft lay down the middle
term and its contrary, as conftituting the univerfe of the fyllo-
gifm, and then complete the premifes and their conclufion, by
properly laying down the concluding terms in their places. We
may fucceed, if, in the firft inftance, we confider none but con
vertible propofitions. And this we can do; for univerfal ex-
clufion and particular inclufion comprehend all aflertion. Thus
univerfal inclufion is only univerfal exclufion from the contrary,
and particular exclufion is only particular inclufion in the con
trary.
Setting out then with the middle term and its contrary, and
reftri&ing ourfelves to E and I, let E fignify (univerfal) exclu
fion from the middle term, and e from its contrary ; let I fignify
(particular) inclufion in the middle term, and i in its contrary.
Choofmg a pair of concluding terms, we rejecT: II, li, and ii on
grounds already demonftrated, and very eafily feen in this view,
and proceed to confider Ee, EE and ee, El and ei, Ei and el.
Ee. From this a univerfal conclufion muft follow. If one
term be completely excluded from the middle and the other from
its contrary, the terms are completely excluded each from the
other. The fundamental forms are,
E,A E,,X.Y + Z.y=X.Z ; A 4 EiEi, X.y + Z.Y = X.Z
and by ufe of XZ, Xz, xZ, xz, we thus bring out the eight uni
verfal fyllogifms.
EE and ee. From thefe a particular inclufion muft follow.
Exclufion of both terms from a third, gives partial inclufion of
their contraries in each other : for all that third term belongs
to the contraries of the other two. The fundamental forms
are,
EiEJ 1 , X.Y + Z.Y=xz ; A^TX.y + Z.y^xz
from which, as before, the eight ftrengthened fyllogifms are de
duced.
EI and ei. From thefe a particular inclufion muft follow.
The exclufion of one term from a third, and the inclufion of
part of a fecond term in that third, tell us that part of the par
ticularized term is in the contrary of the univerfalized term.
The fundamental forms are,
On the Sylloglfm. 105
E.I.O 1 , X.Y-f ZY =Zx ; A.O O , X.y + Zy =Zx
LEiOi, XY +Z.Y = Xz ; O.A O,, Xy + Z.y=Xz
from which the fixteen particular fyllogifms are deduced.
Ei and el. From thefe no conclufion can be drawn. All that
is fignified is that one concluding term is wholly excluded from
a third, and the fecond partially excluded (or included in the
contrary).
It thus appears that a fyllogifm with one particular premife is
valid when the premifes reduced to convertible forms, fhow the
middle term in both or the contrary of it in both ; otherwife,
invalid. Alfo, that the conclufion in its convertible form, takes
directly from the particular premife and contrariwife from the
univerfal.
It alfo appears that a fyllogifm with both premifes univerfal is
always valid ; with a univerfal conclufion when the premifes
(made convertible) mow one the middle term and the other its
contrary ; with a particular conclufion when both mow the mid
dle term or both its contrary. And the convertible form of the
conclufion takes directly from both in the firft cafe, and contra
riwife from both in the fecond.
The other view which I here propofe is really a different
mode of looking at that juft given. By the time we have made
every name carry its contrary, as a matter of courfe, we become
prepared to take the following view of the nature of a propofi-
tion. A name by itfelf is a found or a fymbol : its relation to
things (be they objects or ideas) is twofold. There may be in
rerum natura that to which the name applies, or there may not.
I do not here fpeak of how many things there may be to which
a name applies : it is not effential to know whether they be more
or fewer, either abfolutely or relatively. The introduction of
contraries may be made the expulfion of quantity. With refer
ence to application, then, let a name be called pojjible or impof-
fible according as the thing to which it applies can be found or
not.
A name may be compounded of others ; the compound name
being that of everything to which all the components apply.
Thus wild animal is the name of all things to which both the
names wild and animal apply. To call this compound name
io6 On the Syllogifm.
impoffible is to fay that there is not fuch a thing as a wild animal :
to call it poffible is to fay that there is fuch a thing.
X and Y being two names, the compound name may be re-
prefented by XY when poffible, and by XY) when impoffible.
This does not alter the meaning of our fymbol XY, as hitherto
ufed : as yet it has been there are Xs which are Ys and now
it is XY, the name of that which is both X and Y, is the
name of fome thing or things ; and thefe two are the fame in
meaning, fo far as their ufe in inference is concerned. Nor need
XY), as juft defined, be treated as a departure from, otherwife
than as an extenfiori of, the ufe of X)Y. In X)Y, we aflert
that X is fomething, namely Y : in X) we aflert that X is nothing
whatever. The proper notation, however, for indicating that
the name X has no application, is X)u, u being the contrary of
U, which laft includes everything in the univerfe fpoken of; fo
that u may denote nonexiftence.
The proportion c Every X is Y aflerts that Xy is the name
of nothing, or X)Y = Xy). Similarly No X is Y aflerts that
XY is the name of nothing, or X.Y = XY). But c Some Xs
are Ys and * Some Xs are not Ys merely aflert the poffibility
of the names X Y and Xy.
A fyllogifm, then, is the aflertion that from the poffibility or
impoffibility of the names produced by compounding X or x,
Z or z, each with Y or y, may be inferred the poffibility or im
poffibility of a name compounded of X or x with Z or z. The
rules of the laft fyftem are now fo eafily changed into the lan
guage of the prefent one, that it is hardly worth while to ftate
more than one for example. Thus, if X compounded with Y,
and Z compounded with y, both give impoffible names, then X
compounded with Z gives an impoffible name. This is XY) +
Zy)=ZX) or X.Y + Z.y=Z.X, or EiA E.
The view here taken of compound names will be extended
in the next chapter.
On the Syllogifm. 107
CHAPTER VI.
On the Syllogifm.
WHEN the premifes of a fyllogifm are true, the conclufion
is alfo true, and when the conclufion is falfe, one or
both of the premifes are falfe. There are two kinds of modifica
tions which it may be ufeful to confider : thofe which concern
the entrance of the proportion into the argument ; and thofe
which affect the connexion of the fubject and predicate.
As to the proportion itfelf, it may be true or falfe abfolutely,
or it may have any degree of truth, credibility, or probability.
This relation will be hereafter confidered ; and, according to the
principles of Chapter IX. fo far as the proportion is probable
it is credible, and fo far as it is credible, it is true. But as to
other modes of looking at the fyllogifm, are we entitled to fay
that every thing which can be announced as to the premifes may
be announced in the fame fenfe as to the conclufion ? The an-
fwer is, that we cannot make fuch announcement abfolutely ; but
of the premifes as derived from that conclufion we can make it.
In what manner foever two premifes are applicable, their conclu
fion as from thofe premifes is alfo applicable : becaufe the conclu
fion is in the premifes. For inftance, in the fyllogifm all men are
trees, all trees are rational, therefore all men are rational, the
premifes are abfurd and falfe, and the conclufion taken indepen
dently is rational and true : but that conclufion, as from thofe
premifes, is as abfurd as the premifes themfelves. Again, in c all
pirates are convicted, all convicts are punifhed, therefore all
pirates are punifhed, the premifes are deferable, and fo is the
conclufion with thofe premifes. But the conclufion is not de-
firable in itfelf: as that pirates fhould be punifhed with or with
out trial. Neither may we fay c X ought to be Y and Y ought
to be Z, therefore X ought to be Z except in this manner, that
we affirm X ought to be Z in a particular way. We may not
even fay that when c X ought to be Y, and Y is Z it follows
that X ought to be Z, for it may be that Y ought not to be Z.
Thus a royalift, in 1655, would fay that the hundred excluded
lo8 On the Syllogifm.
members of Cromwell s parliament ought to be allowed to take
their feats, and alfo that all who took any feats in that parliament
were rebels ; but he would not infer that the hundred members
ought to be rebels. There is nothing which, being the property
of the premifes, is necefTarily the independent property of the
conclufion, except abfolute truth. It mould be noted that in
common language and writing, the ufual meaning of conclufions
is that they are ftated as of their premifes and to ftand or fall
with them, even as to truth. Though a conclufion may be true
when its premifes are falfe, the proponent does not mean, for the
moft part, to claim more than his premifes will give, nor that
any thing mould ftand longer than the premifes ftand.
Next, we are not to argue from what we may fay of a propo-
fition to what we may fay of the inftances it contains, except as
to what concerns the truth of thofe inftances, or elfe to what
concerns the inftances as parts of a whole. If I fay Every X
is Y I afTert, no doubt, of each X independently of the reft :
that is, the truth of Every X is Y involves the truth of c this
X is Y. But if, to take fomething elfe, I maintain c Every X
is Y to be a defirable rule, I do not therefore aflert this X is
Y to be a defirable cafe, except upon an implied neceffity that
there mould be a rule. And if I fay that every X is Y is
unintelligible, I do not fay that this X is Y is unintelligible;
and fo on. Thus, where there muft be a rule, as in law, every
man s houfe is his caftle is defirable, becaufe there is but one
alternative c no man s houfe, &c. But the proportion, by itfelf,
may not be defirable as to the inftance of a generally reputed
thief or receiver.
There is one cafe, however, in which a term cannot be ap
plied to the general propofition, unlefs it can be applied in a
higher degree to the inftances. The propofition c Every X is Y
cannot be announced as of any degree of probability, unlefs each
inftance has a much higher degree of probability. If ^, :/, ^, &c.
be the probabilities of the feveral inftances, fuppofed independent,
that of the propofition (Chapter IX.) is/n^> ... which product muft
be lefs than that of any one of the fraHons of which it is formed.
I now come to the confideration of circumftances which mo
dify the internal ftruclure of the premifes themfelves. And firft
of conditions.
On the Syllogifm. 109
A conditional propofition is only a grammatical variation of
the ordinary one ; as in If it be X, then it is Y. The common
form of this, Every X is Y, is called categorical^ or predicative.
Of the two forms, categorical and conditional, either may always
be reduced to the other ; as follows,
Every X is Y or If X, then it is Y 9
4 No X is Y or If X, then it is not Y
The particular propofitions might be given conditionally in
various ways, but the transformation is not fo common. Thus
4 fome Xs are Ys might be if X, then it may be Y or c if X,
then Y muft not therefore be denied of it, &c.
Of the two common fubject-matters of names, ideas and
propofitions, it is moft common to apply the categorical form to
the firft, and the conditional form to the fecond : in truth we
might call the conditional form a grammatical convenience for
the expreflion of dependence of propofitions on one another,
and of names which require complicated forms of expreflion.
Thus in pages 2 and 3, the conditional forms, containing //*, are
more fimple than the correfponding categorical forms.
A condition may be either necejjary^ o>\ fufficient, or both. A
neceflary condition is that without which the thing cannot be ; a
fufficient condition is one with which the thing muft be. In
pages 73, 74, I have fufficiently pointed out the completenefs of
the connexion between the conditional and the categorical forms.
In any one cafe the fufficient muft contain all that is neceflary,
and may contain more.
After what is faid in page 23, it is not neceflary to dwell on
the reduction of a conditional* fyllogifm to a categorical one.
The premifes contain the conclufion : whatever gives us the
premifes, gives us the conclufion. But I think that the reduc
tion of conditional to categorical forms, though juft, and, for in
ference, complete, is not the reprefentation of the whole of what
pafles in our minds.
As an example of what I mean, look forward to the nume
rical fyftem of Chapter VIII. Precedent to all propofitions,
* Wallis, as far as I know, was the firft who aflerted that all fyllogifms
are, or can be made, categorical. He did this in the fecond thefis attached
to his logic, headed Syllogifmi Hypot hetici, aliique Compojiti, referendi funt
omnes ad Ariftotclicos Categoricorum Modos.
I io On the Syllogifm.
there are the numerical conditions which prefcribe the limits of
the univerfe under confideration. Say there are 250 inftances in
that univerfe : this is the firft condition. Of thefe 100 are Xs and
200 are Ys ; giving a fecond and third condition. If we take a
proportion, as 2oXY, and afk whether it be fpurious or not, we
have reference to the three conditions underftood. But this is not
neceflary : for it would be poffible categorically to exprefs thefe
conditions by c 2oXs out of 100 in a univerfe of 250 inftances
containing 200 Ys are to be found among thofe 200 Ys ? It is
of courfe the rule of brevity not to drag about thefe conditions
with every proportion which is employed, but rather to ftate
them once for all. There is however fomething more. The
conditions are a reftriclion upon the arguments intended to be
introduced, and a reftri&ion throughout. The attachment of
them to each individual propofition does not exprefs this : if they
be feen in twenty confecutive proportions, there is no more than
a prefumption that they are to be feen in the twenty-firft. It is
better that the limits allowed fhould be marked out by one boun
dary than that the feveral arguments fhould each have a defcrip-
tion of the boundary to itfelf.
Juft as a univerfe of names is defined by fpecifying one or
more names to conftitute collectively thefammum genus ^ or uni-
verfe, fo one of proportions may be defined by ftating propofi-
tions which are to be true, or which are not to be contradicted, as
the cafe may be. Thefe propofitions may be conditions preced
ing all, or fome only, of the premifes which are ufed in argu
ment ; or fome may precede fome, and others others. In
analyfing arguments, it would be found that many propofitions
which enter as premifes, enter each with a condition underftood,
and well underftood, to be granted. Whatever the conditions
may be, fo long as the confequent propofitions acl logically toge
ther to produce the final refult, then that fame refult depends at
laft only on the conditions, and muft be affirmed when the con
ditions, and their connexion with their confequents, are affirmed.
But then it muft be underftood that the refult alfo ftands upon
the conditions, and may fall with them. Let us now examine
the common fyllogifm, and fee whether there be any preceding
conditions, on which the refult depends.
On looking into any writer on logic, we fhall fee that existence
On the Syllogifm. 1 1 1
is claimed for the fignifications of all the names. Never, in the
ftatement of a proportion, do we find any room left for the alter
native, fuppofe there Jhould be no fuck things. Exiftence as ob
jects, or exiftence as ideas, is tacitly claimed for the terms of
every fyllogifm. The exiftence of an idea we muft grant when
ever it is diftinclly apprehended, and (therefore) not felf-contra-
diclory : we cannot for inftance admit the notion of a lamp
which is both metal and not metal ; but, as an idea, we are at
liberty to figure to ourfelves fuch a lamp as that with which
Aladdin made his fortune. An attempt at a felf-contradi&ing
idea is no idea ; we have not that apprehenfion of it in which an
idea confifts : but in no other way can we fay that the attempt
to produce an idea fails. It may then be more convenient here
to dwell on objective definition of terms, as more eafily con
ceived with relation to exiftence and non-exiftence. Accordingly,
let us take the propofitions X)Y and X.Y, of the character of
which the particulars muft partake, as to the point before us.
By the meaning of y, in relation to Y, it follows that every thing
is either Y or y : if we fay that Y does not exift, then every thing
is y. If then X exift, and Y do not, the propofition X)Y, or
X.y is falfe, and X)y, or X.Y is true. If neither X nor Y
exift, I will not fo far imitate fome of the queftions of the fchools
as to attempt to fettle what nonexifting things agree or difagree.
If Y exift, but not X, then y)x is certainly true, but not thence
X)Y, for when x is, as here, the whole univerfe, the proof of
y)x = X)Y fails to prefent intelligible ideas, that is, fails to be a
proof. But Y)x or Y. X is true.
If all my readers were mathematicians, I might purfue thefe
extreme cafes, as having intereft on account of their analogy
with the extreme cafes which the entrance of zero and of infinite
magnitude oblige him to confider. But as thofe who are not
mathematicians would not be interefted in the analogy, and thofe
who are can purfue the fubjecl: for themfelves, I will go on to
fay that the preceding order is not the natural one. We cannot,
to ufeful purpofe, laying down the truth of the propofition, firfl,
then proceed to enquire how the non-exiftence of one or both
terms afFe&s the propofition. The exiftence of the terms muft
be firft fettled, and then the truth or falfehood of the propofition.
The affirmative propofition requires the exiftence of both terms :
ii2 On the Syllogifm.
the negative propofition, of one ; being necefTarily true if the
other term do not exift, and depending upon the matter, as
ufual, if it do exift.
Let us make the exiftence of the terms to be preceding con
ditions of the propofitions. The fyllogifm AiAjAi is then as
follows,
If X and Y both exift, Every X is Y
If Z alfo exift Every Y is Z
Therefore If X, Y, Z all exift Every X is Z.
As to the concluding terms, X and Z, they remain, as it were,
to tell their own ftory. Whatever conditions accompany their
introduction unto the premifes, thefe fame conditions may be
conceived to accompany them in the conclufion. But the middle
term difappears : and, not fhowing itfelf in the conclufion, the
conditions which accompany it muft be exprefsly preferved.
The conclufion then is every X is Z, if Y exift which may be
thrown into theform of a dilemma, Either every X is Z, or Y
does not exift .
But taking X and Z to exift, let us confider the following fyl
logifm, as it appears to be^
Every X is (Y, if Y exift)
Every (Y, if Y exift) is Z
Therefore Every X is Z.
If this be not a valid fyllogifm, what expreffed law of the ordi
nary treatifes does it break ? The middle term, a curious one, is
ftriclly middle : but there is no rule for excluding middle terms
of a certain degree of fingularity. That it does break, and very
obvioufly, an implied rule, I grant. And as to this work, the
rule laid down in Chapter III. is broken in its fecond condition
(page 50). The two ufes of the word is do not amount to one
fuch ufe as is made in the conclufion. That X is (conditionally)
Y which is (on the fame condition) Z, gives that X is (on the
fame condition) Z. Accordingly, the abfolute conclufion is only
true upon fuch conditions as give the middle term abfolute ex
iftence.
But it muft be particularly noted that it is enough if this ex-
On the Syllogifm. I i 3
iftence be given to the middle term by the fulfilment of the
conditions which precede the entrance of one of the concluding
terms. The condition of the act of inference is, that the com-
parifon muft be really made, if the terms to be compared with
the middle term really exift, or, which is the fame, if the condi
tions under which they are to enter be fatisfied. The other terms
being ready, there muft then be a real middle term : and there
will be, if the mere entrance of one of the concluding terms be
proof of the exiftence of a middle term ; while, if the other terms
cannot be brought in, from nonexiftence, there is no occafion to
inquire about a middle term, for it is otherwife known that the
comparifon cannot be completed. I will take two concrete
inftances, in the firft of which one of the concluding terms, if
exifting, is held to furnifh a middle term as real as itfelf, and in
the fecond of which no fuch fuppofition occurs. Of courfe I have
nothing here to do with the truth of the premifes.
Philip Francis, (if the author of Junius), was an accufer whofe
filence was fimultaneous with a government appointment : an
accufer &c. reflects difgrace upon the government (if they knew
that their nominee was the accufer): therefore Francis (if &c.)
reflects difgrace upon government (if &c.).
Homer (if there were fuch a perfon) was a perfect poet (if
ever there were one) : a perfect poet (if &c.) is faultlefs in
morals : therefore Homer (if &c.) was faultlefs in morals.
The firft inference is good, even though we grant that our
only poffible mode of knowing of the exiftence of an accufer Sec.
is by eftabliftiing that Francis was Junius : it is even good againft
one who mould aflert that the accufer &c. is a contradiction in
terms in every actual and imaginable cafe except that of Junius.
In the fecond cafe, we put it that the man Homer (if he ever
exifted ; fome critics having contended for the contrary) was a
perfect poet, if ever there were one. There may never have
been one ; and then Homer (exiftent or nonexiftent) was not a
perfect poet. There is no condition here, which being fulfilled,
is held to amount to an aflertion that the middle term muft have
exifted : but the condition of the exiftence of the middle term is
independent. Accordingly, the fecond inference is not good : it
mould be Homer (if &c.) was a perfect poet, if ever there were
one : that is, or elfe there never was a perfect poet.
H4 On the Syllogifm.
Thefe points refer to the matter of a fyllogifm, and not to the
form ; or rather, perhaps, hold a kind of intermediate relation.
There is another procefs which is often necefTary, in the
formation of the premifes of a fyllogifm, involving a transforma
tion which is neither done by fyllogifm, nor immediately reducible
to it. It is the fubftitution, in a compound phrafe, of the name
of the genus for that of the fpecies, when the ufe of the name
is particular. For example, man is animal, therefore the head
of a man is the head of an animal is inference, but not fyllo
gifm. And it is not mere fubftitution of identity, as would be c the
head of a man is the head of a rational animal* but a fubftitution
of a larger term in a particular fenfe.
Perhaps fome readers may think they can reduce the above to
a fyllogifm. If man and bead were connected in a manner which
could be made fubjecl: and predicate, fomething of the fort might
be done, but in appearance only. For example, Every man is
an animal, therefore he who kills a man kills an animal. It
may be faid that this is equivalent to a ftatement that in Every
man is an animal ; fome one kills a man ; therefore fome one
kills an animal, the firft premife, and the fecond premife condi
tionally^ involve the conclufion as conditionally. This I admit :
but the laft is not a fyllogifm : and involves the very difficulty in
queftion. c Every man is an animal ; fome one is the killer of
a man : here is no middle term. To bring the firft premife
into Every killer of a man is the killer of an animal is juft the
thing wanted. By the principles of chapter III, undoubtedly
the copula is might in certain inferences be combined with the
copula kills^ or with any verb. But fo fimple a cafe as the pre
ceding is not the whole difficulty. If any one mould think he
can fyllogize as to the inftances I have yet given, let him try the
following. c Certain men, upon the report of certain other men
to a third fet of men, put a fourth fet of men at variance with a
fifth fet of men. Now every man is an animal : and therefore
1 Certain animals, upon the report of certain other animals, &c.
Let the firft defcription be turned into the fecond, by any num
ber of fyllogifms, and by help of c Every man is an animal.
The truth is, that in the formation of premifes, as well as in
their ufe, there is a poftulate which is conftantly applied, and there
fore of courfe conftantly demanded. And it mould be demanded
openly. It contains the dictum de omni ct nu/Io (fee the next chap-
On the Syllogifm. 1 1 5
ter), and it is as follows. For every term ufed univerfally lefs
may be fubftituted, and for every term ufed particularly, more.
The fpecies may take the place of the genus, when all the genus
is fpoken of: the genus may take the place of the fpecies when
fome of the fpecies is mentioned, or the genus, ufed particularly,
may take the place of the fpecies ufed univerfally. Not only in
fyllogifms, but in all the ramifications of the defcription of a com
plex term. Thus for men who are not Europeans may be
fubftituted c animals who are not Englifh. If this poftulate be
applied to the unftrengthened forms of the Ariftotelian Syllogifm,
(page 17) it will be feen that all which contain A are immediate
applications of it, and all the others eafily derived.
I now pafs to the confideration of the invention of names, and
of the distinctions which are made to exift for the want of it.
Any one may invent a name, that is, may choofe a found or
fymbol which is to apply to any clafs of ideas or of objects. The
clafs mould, no doubt, be well defined : but fmall caution is here
necefTary, for invented words are generally much more definite
than thofe which have undergone public ufage. They come from
the coiner s hand as fharp at the edge as a new halfpenny : and in
procefs of time we look in vain for any edge at all. The right
of invention being unlimited, and the actual ftock having been
got together without any uniform rule of formation, there can
be no reafon why we jhould admit any diftinttion which can be ab
rogated by the invention of a name^ fo far as inference is con
cerned. I do not difpute that the modes of fupplying the want
of names may be of importance in many points of view : what
I deny is, that they create any peculiar modes of inference.
The invention of names muft either be by actually pointing
out objects named, or by defcription in terms of other names.
With the former mode of invention, as let this, that, &c (mow
ing them) be called X we can have nothing to do. As to the
latter, we may make a fymbolic defcription of the procefs by join
ing together the names to be ufed, with a fymbol indicative of
the mode of ufing them, in extenfion of the fyftem in page 106.
Thus, P, Q, R, being certain names, if we wifh to give a name
to everything which is all three, we may join them thus, PQR :
if we wifh to give a name to every thing which is either of the
three (one or more of them) we may write P,Q,R : if we want
to fignify any thing that is either both P and Q, or R, we have
1 1 6 On the Syllogifm.
PO,R. The contrary of PQR is p,q,r; that of P,Q,R is pqr ;
that of PQ,R is (p,q)r : in contraries, conjunction and disjunc
tion change places. This notation would enable us to exprefs
any complication of the preceding conditions : thus, to name that
which is one and one only of the three, we have Pqr, Qrp,
Rpq ; for that which is two and two only, PQr, QRp, RPq.
Thus, XY includes the inftances common to X and Y ; but
X,Y includes all X and all Y : accordingly X,Y is a wider term
than XY, except when X and Y are identical. As in page 106,
XY, the term, fuppofed to exift, is XY, the propofition of chapter
IV ; if we wifh to diftinguim, we may make X-Y the term, and
XY the propofition, the hyphen having its common grammatical
ufe. Thus, X-Y P-Q tells us the fame as XYP-Q, both mean
ing, for Inference, no more than that there exift objects or ideas
to which the four names are applicable. But the firft tells it
thus, fome XYs are POs ; and the fecond thus, fome things are
Xs, Ys, and POs.
With refpecl: to this and other cafes of notation, repulfive
as they may appear, the reader who refufes them is in one of two
circumftances. Either he wants to give his aflent or diflent to
what is faid of the form by means of the matter, which is eafmg
the difficulty by avoiding it, and ftepping out of logic : or elfe he
defires to have it in a fhape in which he may get that moft futile
of all acquifitions, called a general idea* which is truly, to ufe
the contrary adjective term as colloquially, nothing particular^ a
whole without parts.
If the difficulty of abftract afTertion be to be got over, the
eafieft way is by firft conquering that of abftracl: expreffion, to
the extent of becoming able to make a little ufe of it.
Suppofe we afk for the alternative of the following fuppofition,
c Both X, and either P, or Q and one of the two R or S. This
is no impoffible complication : for inftance, He was rich, and
if not abfolutely mad, was weaknefs itfelf fubje&ed either to bad
advice or to moft unfavourable circumftances. The reprefen-
tation of the complex term is X {P, O(R,S)} ; of the contrary,
* " Je vous avoue, dit . . . ., que j ai cru en deviner quelque chofe, et que
je n ai pas entendu le refte. I/abbe de . . . . a ce difcours, fit reflexion que
c etait ainfi que lui-meme avait toujours lu, et que la plupart des homines
ne lifaient omere autrement."
On the Syllogjfm. 1 1 7
x, p(q,rs) or x,pq,prs. If not the above, he was either not rich,
or both not mad and not very weak, or neither mad nor badly
advifed, nor unfavourably circumftanced.
When a name thus formed, whether conjunctively or disjunc
tively, enters a fimple inference, it gives rife to what have been
called the copulative fyllogifm, the disjunctive fyllogifm, and the
dilemma. The two laft are not well diftinguimed by their defi
nitions as given : the disjunctive fyllogifm feems to be that in
which names are confidered disjunctively, the dilemma that in
which proportions are fo ufed. But a propofition entering as part
of a propofition, enters merely as a name, the predicates being
ufually only true or fa Ij *, or fome equivalent terms. A propofi
tion may only enter for its matter, or it may enter in fuch a way
that its truth is the matter : in this laft cafe it is only as a name
that it is the fubject of inference. Thus, It is true that he was
fired at is the aflertion (that he was fired at) is a true aflertion.
I believe the beft way would be to apply the term disjunctive
argument fo as to include the dilemma, marking by the latter
word (as a term rather of rhetoric than of logic) every argument
in which the disjunctive propofition is meant to be a difficulty
for the opponent on every cafe, or horn^ of it.
Whatever has right to the name P, and alfo to the name Q,
has right to the compound name PQ^ This is an abfolute
identity, for by the name PQ_ we fignify nothing but what has
right to both names. According X)P + X)Q==X)PQ> not a
fyllogifm, nor even an inference, but only the aflertion of our
right to ufe at our pleafure either one of two ways of faying the
fame thing inftead of the other. But can we not effect the re
duction fyllogiftically ? Let Y be identical with PO ; we have
then PQ)Y and Y)PQ, and alfo Y)P and Y)Q. Add to thefe
X)P and X)Q, and we have all the propofitions aflerted. But
we cannot deduce from them alone X)Y, the refult wanted, by
any fyllogiftic combination of the fix. Nor muft it be thought
furprifing that we cannot, by a train of argument, arrive at de-
monftration of it being allowable to give to anything which has
right to two names, a third name invented exprefsly to fignify
that which has fuch right. We might as well attempt to fyllo-
gize into the refult, that a perfon who fells the meat he has killed
is a butcher.
n8 On the Syllogifm.
I lay ftrefs upon this, to an extent which may for a moment
appear like diligently grinding nothing in a mill which might be
better employed, for two reafons. Firft, the young mathema
tician is very apt to try, in algebra, to make one principle deduce
another by mere force of fymbols : and the above attempt may
{how him what he is liable to. Secondly, I am inclined to fup-
pofe that the diftinction drawn between the clafles of fyllogifms
to which I prefently come, and the ordinary categorical ones, is
due to what muft be defcribed in my language as a want of per
ception of the abfolute, lefs than inferential (fo to fpeak) identity
ofX)P + X)Q_and XJPQi But all other proportions of the
kind, however fimple, may be made deductions. For inftance,
if X be both P and Q, and if P be R, and Q^bc S, then X is
both Q_and S is thus deduced: X)P + P)R = X)R, and X)Q_
+ Q)S=X)S, and X)R + X)S is X)RS. Even P)R + Q)R =
P,O)R is deducible; being P)R + Q)R=r)p+r)q=r)pq=
P,Q)R. Thus it is feen that, as foon as the conjunctive poftu-
late is laid down, the identity of the correfponding disjunctive
poftulate with it may be mown. Next, if X muft be either P
or O, or X)P,O, and if P be always R, and Q_be always S, then
X)R,S may be deduced from the preceding.
Firft, that X)P and Y)Q_give XY )PQ_can be deduced ; evi
dent as it may be, it is a fucceflion of applications. XY)X-f
X)P gives XY)P,and XY)Y + Y)Q_gives XY)Q, and XY)P
+ XY)Q_is XY)PQ>y the poftulate. Next, X)P,Q^ is pq)x,
and P)R is r)p, and O)S is s)q, whence, as juft proved rs)pq.
Now, rs)pq + pq)x = rs)x, which is X)R,S. It will be a good
exercife for the reader to tranflate this proof into ordinary lan
guage.
I may now proceed to extend this idea and notation relative
to proportions of complex terms. The complexity confifts in
the terms being conjunctively or disjunctively formed from other
terms, as in PQ, that to which both the names P and Q_belong
conjunctively; and as in P,Q_that to which one (or both) of
the names P and Q_ belong disjunctively. The contrary of PQ_
is p,q; that of P,O is pq. Not both is either not one or not the
other, or not either. Not either P nor j^(which we might denote
by :P,Q_or .P,Q) is logically c not P and not Qj or pq : and
this is then the contrary of P,Q.
On the Syllogifm. 1 1 9
The disjunctive name is of two very different characters, ac
cording as it appears in the univerfal or particular form : fo very
different that it has really different names in the two cafes,
copulative and disjunctive. This diftinction I here throw away :
oppofmg disjunctive, (having one or more of the names) to
conjunctive, (having all the names). The disjunctive particle or
has the fame meaning with the diftributive copulative and, when
ufed in a univerfal. Thus, * Every thing which is P or Q is
R or S means Every P and every Q is R or S. But PQ_ is
always both P and QJn one. Accordingly
Conjunctive
Disjunctive
P QR ufes and collectively.
P,O,R in a univerfal ufes and diftributively,
P,O,R in a particular ufes or disjunctively, in
the common fenfe of that word.
( Either P or Q^ is true, is nn ambiguous phrafe, which is
P,O)T or T)P,O according to the context.
The manner in which the component of a name enters, whe
ther conjunctively or disjunctively, is to pafs as it were for a part
of the quality of the name itfelf. Thus the contrary of P (con
junctive, as indicated by the abfence of the comma) is ,p (dis
junctive, as indicated by the comma). To teft this affertion about
the mode of making contraries, let us afk what is that of one
only of the two P or QJ* We know it of courfe to be both or
neither. The name propofed is Pq, Qp and its contrary is
(p,Q)(q,P), that is, one of the two p,O, and one of the two q,P.
It is then either pq, pP, qQ, or PQj the fecond and third can
not exift, therefore it is pq, PQ, as already feen. I need hardly
have remarked that (P,O)(R,S) is PR, PS, OR, OS.
Obferve that though X)PO gives X)P, and that XPO gives
XP, we may not fay that XY)P gives X)P, nor that X)P,Q_
gives X)P. But any disjunctive element may be rejected from
a univerfal term, and any conjunctive element from a par
ticular one. Thus P)QR gives P)Q_and P,O)R gives P)R.
Alfo P.O,R gives P.Q_and PO:R,S gives P:R. All thefe rules
are really one, namely that PO is of the fame extent at leaft as
POR. This will appear from our rules of tranfpofition prefently
given.
]2o On the Syllogifm.
Let change from one member of the propofition to the other
be called tranfpofition. I proceed to inquire how many tranfpo-
fitions the various forms will bear, and what they are. It will
however be necefTary to complete our forms by the recognition, as
a propofition, of the fimple afTertion of exiftence or non-exiftence.
By XU we mean that there are in the univerfe things to which
the name X applies, and we fpeak only of fuch things under the
name. Accordingly X)U and XU do not differ in meaning.
By u, the contrary of U, we can only denote non-exiftence ;
thus X.U or X)u throws the name X out of confideration.
Thus Y)X = U)X,y; Y.X = YX)u, &c. To fignify, for in-
ftance, that X and Y are complements (contraries or fubcon-
traries, page 75) we have U)X,Y, which our rules will tranf-
pofe into xy)u, or x.y.
Having to confider fubject and predicate, conjunctive and dif-
junctive, affirmative and negative, univerfal and particular, we
muft think of fixteen different forms. Thus the four forms of
the univerfal affirmative are
XY)PQ ; X,Y)PQ_; XY)P,Q 5 X,Y)?,Q_
It will be beft here to neglect the contranominal converfes of
A and O equally with the fimple converfes of E and I : thus
XY)PQ may be read as identical with p,q)x,y. There is alfo
one obvious tranfpofition which we muft not merely neglecl: but
throw out ; fmce it does not give a refult identical with its prede-
ceflbr. I mean the tranfpofition of M)PQJnto MP)Q;. the
fecond follows from the firft but not the firft from the fecond.
Alfo the correfponding change of M.P,Qjnto Mp.O, for the
fame reafon.
This being premifed, the following are the rules ;
Dlreft tranfpofition is the change from one member to the
other without alteration of name or junction : contrary, with
alteration of both.
The convertibles (E,I) allow direct tranfpofition of conjunctive
elements either way, from fubject to predicate, or from predicate
to fubjecl: : and thefe are the only direct tranfpofitions. Thus
X.YZ = XY.Z, and X-YZ = XY-Z.
The inconvertibles (A,O) allow contrary tranfpofition of con
junctive elements from fubject to predicate, and of disjunctive
On the Syllogijm.
121
elements from predicate to fubjecl : beft remembered by allow
ing SP to ftand for conjunctive and PS for disjunctive. And thefe
are the only contrary tranfpofitions. Thus XY)M = X)M,y
and M)X,Y=My)X.
An element that can be rejected cannot be tranfpofed, and
vice verfa. Thus X,Y)M gives X)M, and Y cannot be tranf
pofed.
The following table exhibits the varieties of the forms A and
E, equivalents being written under one another, and converfions,
contranominal or fimple, oppofite.
XY)P,Q
Xp)Q,y
Xq)P,y
X)P,Q,y
Y)P,Q,x
P)Q,x,y
q)P,x,y
XYpq)u
pq) x >y
Yq)P,x
Y P )Q,x
pqY)x
pqX)y
XYq)P
XYp)Q_
U)P,Q,x,y
XY.PQ_
XP.QY
XQ.PY
X.PQY
Y.PQX
P.QXY
Q.PXY
XYPQ.U
PQ.XY
QY.XP
PY.XQ.
PQY.X
PQX.Y
QXY.P
PXY.Q.
U.XYPQ_
XY)PO
X)PQ,7
Y)PQ,x
XY[p,q])u
p>q) x >y
[p,q]Y)x
[p,q]X)y
U)[x,y],PQ_
XY.P,Q_
X.[P,QJY
Y.[P,Q]X
XY[P,Q].U
P,Q-XY
[P,Q]Y X
[P,Q]X.Y
U.XY[P,0]
X,Y)P,Q_
[X,Y] P )Q_
[X,Y]q)P
[X,Y]pq)u
pq)xy
q)*y,P
p) x y>0.
U)xy,P,Q,
X,Y.PQ_
[X,Y]P.Q.
[X,Y]Q.P
[X,Y]PQ.U
P6-X,Y
Q^X,Y]P
P.[X,Y]Q.
U.[X,Y]PQ.
X,Y.P,Q. P,Q.X,Y
[X,Y][P,Q].UU.[X,Y][P,Q]
X.YJPQ. P ,q)xy
[x,y][p,q])u U)xy,PQ
If for ) we write (:) in the left hand divifions, and erafe the(.)
and ufe the hyphens of page 1 15, on the right, we have the tranf
pofitions of O and I. And if we write p and q for P and Q_on
the left, and change the form X)Y into X.y, we thereby change
the forms of A into thofe of E. If more than two elements
were ufed, the tranfpofitions would now be perfectly eafy.
It appears that there are no lefs than fixteen A forms into
122 On the Syllogifm.
which XY)P,Q_may be varied : the reafon is that both fubjeft
and predicate are tranfpofibly conftru&ed. But XY)PQ fhows
only a tranfpofible fubjecl: 5 X,Y)P,Q only a tranfpofible predi
cate : and thefe have only four forms each. Laftly, X,Y)PQ,
having neither tranfpofible, has only two forms. By tranfpofi
bly conftrufted, I mean capable of having the elements feparated
by tranfpofition. The whole term is always tranfpofible : that is,
the complete fubjeft, or the complete predicate, may be looked
on as conjunctive or disjunctive, at pleafure. Thus in X)Y, if
we confider this as XU)Y,u, we may make this yU)x,u or y)x.
So that the ordinary contranominal converfion may be confidered
as a cafe of the more general rule. Juft as, in arithmetic, a num
ber, 5, may be made to obey the laws of a + b as + 5, or of ab
as I x 5.
Syllogifms of complex terms might be widely varied, even if we
chofe to confider only each firft cafe of the preceding table as
fundamental. Thus
XY)P,Q + VW)P,Q=(x,y)-(v,w) A, A 1 ! 1
would give fixty-four varieties of premifes. I now proceed to
{how that the ordinary disjunftive and dilemmatic forms are
really common fyllogifms with complex terms, reducible to ordi
nary fyllogifms by invention of names.
Example I. Every S is either P, Q, R 5 no P is S ; no Q>
S ; therefore every S is R. Let S reprefent the true propofi-
tiorT (fmgular), and let P, O, R be names of proportions, and
this then reprefents a very common form, which would be ex-
preffed thus either A is B, or C is D, or E is F ; but A is not
B, C is not D ; therefore E is F. I fay that, where the necef-
fary names exift, the final ftep of this could not be diftinguifhed
from a common fyllogifm ; which accordingly it becomes by in
vention of names.
We have S)P,O,R, whence Spq)R. But S.P and S.Q_or S)p
and S)q give S)po~with which S)S combined gives S)pqS. And
S)pqS + pqS)R = S)R. Let M be the name of what is S and not P
and not O, and the thing required is done. Here then is a fyllogifm
of the ordinary kind, to one premife of which we are led by a
ufe of the conjunctive populate (page 1 16) : the neceffity for which
is the diftinaion between the clafs we are confidering and others.
It happens here that two of the terms of our final fyllogifm are
On the Syllogifm. 123
identical : for Spq is of no greater extent than S. But the ufe
made of S)S is perfe&ly legitimate.
Example 2. < If A be B, E is F ; and if C be D, E is F ; but
either A is B or C is D ; therefore E is F. This can be re
duced to
P)R + O)R + S)P,Q=S)R
which is immediately made a common iyllogifm by changing
P)R + O)RintoP,O)R.
Example 3. From P follows Q_; and from R follows S ; but
Q_and S cannot both be true ; therefore P and R cannot both
be true. This may be reduced to
P)Q+R)S + T.QS=T.PR
orPR)QS + T.QS=T.PR
Example 4. Every X is either P, O, or R ; but every P is
M, every O is M, every R is M ; therefore every X is M.
This is a common form of the dilemma ; it is obvioufly reduci
ble to P,Q,R)M + X)P,O,R = X)M.
Example 5. Every X is either P or Q, and every O is X.
This is wholly inconclufive, and leads to an identical refult, as
follows ; X)P,Q gives Xp)O, which with Q)X gives Xp)X,
a neceflary proportion.
Example 6. If we throw X)R into the form X)R,R, we have
Xr)R, or Every X which is not R is R, a contradiction in
terms. But it evidently implies that there can be no Xs which
are not Rs ; and thus alfo we return to X)R. Take c every X
is either P, Q, or R ; every P is M ; every Q is M ; and every
M is R. Here X)P,Q,R = Xr)P,O, whicrTwith P,Q)M gives
Xr)M, which with M)R gives Xr)R or X)R.
Example 7. Every X is either P or Q, and only one. This
gives two proportions, X)P,Q + X.PQ. Now X)XP,XO is
identical with X)P,Q, and this may be looked on as an extreme
cafe of
X)P,Q + X)Y=X)PY,QY
but X.PQ gives XP)q and XQ)p, from which we can obtain
Hence X)P,Q_+X)p,q = X)[P,Q,][p,q.]
=X)P P ,P q) Q P ,g q =X)P q ,Q P
fincc Pp and Qq arc fubjcil to X.Pp and X.Qq. All this being
124 On the Syllogifm.
worked out in fyllogiftic detail, fhows us that the tranfition from
c Every X is P or Q, and no X is both to c Every X is either
P and not Q, or Q and not P J is capable of being made fyllo-
giftically. The ftudent of logic may thus acquire the idea,
which fo foon becomes familiar to the ftudent of mathematics,
of perfectly felf-evident propofitions which are deducible from
one another, as diftinguifhed from thofe which are not.
Example 8. Every X is one only of the two, P or Q ;
every Y is both P and Q, except when P is M, and then it is
neither ; therefore no X is Y. Here is a cafe in which it is the
fact of the exception and not its nature which determines the
inference : M may be anything. This ought to appear in our
reduction : and it does appear in this way. From X)P,Q it is
obvious that X)P,Q,R,S, and fyllogifUcally demonftrable from
X)P,O, and Xrs)X. Now in the fecond premife we have
Y)PQm,pqM, or [p,q,M][P,Q,m])y
r pQ,Pq,PM,QM,pm,qm)y
from which, by rejection, follows pO,Pq)y. And the firft pre
mife is X)Pq,Qp. Whence X)y oTx.Y.
It is not neceflary to multiply examples : I will conclude this
part of the fubjecl: by pointing out that the ordinary propofitions
X)Y, &c. are, with reference to their inftances, disjunctively
compofed : the difference between the univerfal and particular
lying in the latter being indefinite in the number of its inftances.
Thus, if there be three Xs and four Ys, the four propofitions
are, applying the name to each inftance, as feen written at length in
X,X,X)Y,Y,Y,Y; X,X,X.Y,Y,Y,Y; (X,X,X)(Y,Y,Y,Y);
and (X,X,X):Y,Y,Y,Y.
The propofition in page 25, is a cafe of the preceding method.
I leave the reader to mow it, and alfo that the hypothefis is
{lightly overrated.
I now come to the forties, the heap or chain of fyllogifms, in
which the conclufion of the firft is a premife of the fecond, and
fo on. Take a fet of terms, P, Q, R, S, &c. and let the order
of reference be PQ, QR, RS, &c. Then A t AiAAi &c. is a
forites, and the only one ufually confidered : thus,
R)S + S)T=P)T
On the Syllogifm. 125
The firft two links give P)R, which with the third gives P)S,
which with the fourth gives P)T. Thus we have links, inter
mediate conclufions^ and a final conclufion.
A great number of different forites may be formed, under the
following conditions,
The firft particular propofition which occurs, be it link or
conclufion, prevents any future link from being particular : for
all the conclufions thence become particular.
Examine the cafes of fyllogifm which proceed by the firft
rule of accentuation (page 92), that is, which have beginning and
ending both univerfal, or both particular : thefe only can occur
in a forites, except at the end, or in the place where a particular
propofition firft enters. It will be found that the conclufion,
when the argument goes on, muft come after fomething con
nected with that which comes after it by the firft rule of ac
centuation : except at the place where a particular conclufion
comes in for the firft time. For inftance, EiE 1 gives Aj, which,
ftill keeping conclufions univerfal, muft be followed by A t or EI,
which follow E 1 by the firft rule. Again, take O t E T , which gives
L ; this muft be followed either by A 4 or Ei, which follow E 1 by
the fame rule : and fo on. Accordingly,
Any chain of univerfals, in which affirmation is followed by a
like prepofition, and negation by a different one, as A t AiEiA !
E AiEiE , &c. may be part of the chain of a forites. And the
chain muft be either of this kind wholly, or once only broken in
one of two ways : either by the direct entrance of a particular
propofition, or by a breach of the rule. In a chain of this kind,
unbroken, the conclufions are affirmative or negative, according
as an even or odd number of negatives goes to the formation of
them. All the conclufions have the fame accent as the firft link.
Let a particular premife be introduced, as in AiEiET &c.
The accent of the particular introduced muft be the fame as
or contrary to that of the firft link, according as the preceding
number of negatives is odd or even. For the accent of the firft
link remains as long as the conclufion is univerfal, and a fyllo
gifm with the fecond premife particular follows the fecond
rule. Thus, inferting the intermediate conclufions, the above is
A^^E.jE^Ai)!^! 1 ). And after (I 1 ) muft come A 1 or E 1 , fo
that the firft rule ftill continues. But the accent of the conclu
fions changes.
126 On the Syllogifm.
Now let the rule of accentuation be broken. The accent of
the conclufion ftill requires the firft rule to be refumed. Thus,
EiE (rule unbroken) gives Ai, and EiEi (rule broken) gives I ,
and Ai requires Ai or EI to follow E , while I requires A 1 or
E ! to follow Et. This one breach of rule only changes the con
clufion from univerfal to particular. The accent of the conclu
fion changes as before.
The links of a forites, then, are either a chain of univerfals
following the firft rule of accentuation, or fuch a chain with one
breach of the rule, or fuch a chain with one particular inferted,
of the fame or contrary accent to the firft link, according as the
preceding negatives are odd or even, and made the commence
ment of the refumption of the rule (if broken). In all the cafes
the conclufion is affirmative or negative according as the preced
ing negatives are even or odd in number : the unbroken chain
has a univerfal conclufion with the accent of the firft link, and
the broken one a particular with the contrary accent.
A E EiA E 1
E A A E 1
E.A AiEiE A,
EiOTO O 1
EiEJTO 1
Here are examples of the three kinds. The chain is in the
firft row, the intermediate and final conclufions in the fecond.
Thus the fecond example prefents the fyllogifms EiA f Ei, EiAtO 1 ,
O EJ 1 , FE O , O AiO 5 and at length is
The forites ufually confidered are only AiA t A. . . . and
A A A* ..... To thefe might be added without abandoning
the Ariftotelian fyllogifm, fuch as AiEiA A A 1 ---- , AiEjA AiAi
.... But it would not be very eafy to follow the chain in thought
without introducing the intermediate conclufions, and thus de-
ftroying the fpecific character of the procefs.
And juft as the ordinary univerfal fyllogifm can be reduced to
AAjAi, fo the univerfal forites can always be reduced to a chain
of A,. Thus A E E.A E 1 or
is u)T +T)S + S)r +r)0
127
CHAPTER VII.
On the Ariftotelian Syllogifm.
FROM the time of Ariftotle until now, the formal inference
has been a matter of ftudy. In the writings of the great
philofopher, and in a fomewhat fcattered manner, are found the
materials out of which was conftructed the fyftem of fyllogifm
now and always prevalent : and two diftinct principles of exclu-
fion appear to be acted on. Perhaps it would be more correct
to fay that the followers collected two diftinct principles of ex-
clufion from the writings of the mafter, by help of the afltimption
that everything not ufed by the teacher was forbidden to the
learner. I cannot find that Ariftotle either limits his reader in
this manner, or that he anywhere implies that he has exhaufted
all poffible modes of fyllogizing. But whether thefe exclufions
are to be attributed to the followers alone, or whether thofe who
have more knowledge of his writings than myfelf can fix them
upon the leader, this much is certain, that they were adopted,
and have in all time dictated the limits of the fyllogifm. Of all
men, Ariftotle is the one of whom his followers have worfhipped
his defects as well as his excellencies : which is what he himfelf
never did to any man living or dead ; indeed, he has been accufed
of the contrary fault.
The firft of thefe exclufions is connected with the celebrated
diftum de omni et nullo, namely, that what is diftributively affirmed
or denied of all, is diftributively affirmed or denied of every fome
which that all contains. It is there faid that in every fyllogifm
the middle term muft be univerfal in one of the premifes, in order
that we may be fure that the affirmation or denial in the other
premife may be made of fome or all of the things about which
affirmation or denial has been made in the firft. This law, as
we mail fee, is only a particular cafe of the truth : it is enough
that the two premifes together affirm or deny of more than all
the inftances of the middle term. If there be a hundred boxes,
into which a hundred and one articles of two different kinds are
128 On the Arlftotelian Syllogifm.
to be put, not more than one of each kind into any one box,
fome one box, if not more, will have two articles, one of each
kind, put into it. The common doctrine has it, that an article
of one particular kind mud be put into every box, and then fome
one or more of another kind into one or more of the boxes, be
fore it may be affirmed that one or more of different kinds are
found together. This exclufion is a fimple miftake, the mere
fubftitution of the aflertion that none but a certain law of infe
rence can exift, for the determination that no other Jhall exift.
Any one is at liberty to limit the inferences he will ufe, in any
manner he pleafes : but he may err if he declare his own arbi
trary boundary to be a natural limit impofed by the laws of
thought.
The other exclufion may involve, on the fame terms, an error
of the fame kind ; or may equally be the expreffion of arbitrary
will: but there is what is more reafonably matter of opinion about
it. Ariftotle will have no contrary terms : not-man, he fays, is not
the name of anything. He afterwards calls it an indefinite or
aorlft name, becaufe, as he aflerts, it is both the name of exifting
and non-exifting things. If he had here made the diftinction
between ideal and objective, he would have feen that man and
not-man equally belong to both (objectively) exifting and non-
exifting things : man, for example, belongs as a name to Achilles
and the feven champions of Chriftendom, whether they ever ex-
ifted in objective reality or not : and not-man belongs, in either
cafe, to their horfes. I think, however, that the exclufion was
probably dictated by the want of a definite notion of the extent
of the field of argument, which I have called the univerfe of the
propofitions. Adopt fuch a definite notion, and, as fufficiently
fliown, there is no more reafon to attach the mere idea of ne
gation to the contrary, than to the direct term.
The exclufion of contraries throws out the propofitions E ?
and I , or x.y and xy, which cannot be exprefled without either
contraries, as in x.y=x)Y = y)X, and xy=x:Y = y:X, or refe
rence to things not named by X and Y, as in Every thing is
either X or Y* and c Some things are neither Xs nor Ys/ the
moft natural readings of No not-Xs are not-Ys, and c Some not-
Xs are not-Ys. There remain then fix modes of connexion of
X and Y, namely X)Y and Y)X, X:Y and Y:X, and XY( =
On the Arijlotelian Syllogifm. 129
YX) and X.Y( = Y.X). Thefe fix are made eight ; for in the
common fyftem, XY and YX are confidered as diftincl in form,
and alfo X.Y and Y.X. But thefe eight are only treated as
four : for reference to order is not made in the fimple propofi-
tion. Thus X)Y and Y)X are both denoted by A, XY and
YX by I, X.Y and Y.X by E, and X:Y and Y:X by O. But
the ftandard of order which is neglected as to the proportion by
itfelf, is adopted in the fyllogifm in the following manner.
The predicate of the conclufion is called the major term, and
the fubjecT: of the conclufion the minor term. This language is
fafhioned upon the idea of an affirmative propofition, in which
major and minor have reference to magnitude. In every X is Z
Z is a name which entirely contains X and is therefore at leajl
as great as X, greater than or equal to X. Here is, before it
was introduced into mathematics, the idea now fo familiar to the
mathematician, of allowing his language to include the extreme
limit of its meaning. When the fame terms are applied to
negative proportions, the notion of magnitudinal inclufion is
loft ; and major and minor, being ftill retained, muft be pre-
fumed to refer to real or fuppofed importance. The premifes
are called major and minor, according as they contain the major
or minor term of the conclufion : and the major premife is always
written firft. Accordingly, Z and X being the major and minor
terms, there are four poffible arrangements, which are called the
four figures. Ariftotle gives three, and tradition has it that
Galen fupplied the fourth in number and order.
i. YZ 2. ZY 3. YZ 4. ZY
XY XY YX YX
XZ XZ XZ XZ
To me, the moft fimple arrangement is that which takes up
what was left off" with, as in the fourth figure : and X is in Y,
Y is in Z, therefore X is in Z is more natural than < Y is in Z,
X is in Y, therefore X is in Z.
It is now plain, that whenever one only of the three propofi-
tions is convertible, there are two diftincl: ways in which the
fyllogifm may be written : when two only, four : and when all
three (if there were fuch a thing), eight.
K
130 On the Ariftotelian Syllogifm.
The fyftem rejeas all conclufions which may be made
ftronger : thus when X . Z follows, it does not allow X : Z to
make a diftina form. But when X)Z is the conclufion, it does
not rejea ZX, for, not confidering ZX as identical with XZ, it
does not confider X)Z as a ftrengthened form of ZX. But it
does not rejea fyllogifms in which as ftrong a conclufion can be
deduced from a weaker premife : accordingly, we muft fearch
for Ariftotelian forms among the ftrengthened fyllogifms of
chapter V, as well as among the fundamental ones. Now,
taking all the forms which ftiow neither E or I f , let us write
down the fymbols of them, and the number of cafes we may
expea from each. Moreover, fince transformation of order
makes no difference here, I put the fyllogifms together as in
page 96, into twelve pairs.
Fundamental AiAiAi, A 1 A 1 A 1 , i ; O T A,O T , A OiOi, i ;
AiO O ,OiA O t , i ; E AiE , A E E , rejeaed; LAJi, A LL, 4;
E O L, OiE L, rejeaed ; EiA E., AiE 4 Ei, 4; I AT, AJT, re
jeaed; EiOJ , O EJ , rejeaed; E EiA , E t E Ai, rejeaed;
LEiOi, EJiO , 4; ETO,, I E O , rejeaed.
Weakened A 4 A Ji, I .
Strengthened A AJ, I ; A t AT, rejeaed ; A EiO,, EiAiO ,
2 ; AiE O , E A Oi, rejeaed ; E E L, rejeaed ; EiEJ 1 , rejeaed.
There are then fifteen fundamental, one weakened, and three
ftrengthened, forms of fyllogifm in the received fyftem. I now
put them down, with their derivations, forms of expreffion in
full, ordinary fymbols, figures into which they fall, and the magic
words by which they have been denoted for many centuries,
words which I take to be more full of meaning than any that
ever were made.
Fundamental.
AiAiA, A A A Y)Z+X)Y=X)Z AAA I Barbara
O AiO A OiOi Y:Z + Y)X = X:Z OAO III Bokardo
AiO O O.A Oi Z)Y + X:Y = X:Z AGO II Banks
LAJ A LL Y)Z + XY =XZ All I Darn
Y)Z + YX =XZ All III Datlfi
ZY + Y)X=XZ IAI IV Dimaris
YZ +Y)X = XZ IAI III Difamls
E.A E, A.EE
E.1,0 LE,0
On the Ariftotelian Syllogifm.
Fundamental.
X)Y = X.Z
X)Y = X.Z
Y.X = X.Z
Z)Y + X.Y = X.Z
Y.Z + XY=X:Z
Z.Y + XY=X:Z
Y.Z + YX=X:Z
EAE
I
Ce la rent
EAE
AEE
II
IV
Cefare
Camenes
AEE
EIO
II
I
Cameftres
Ferlo
EIO
EIO
EIO
II
III
IV
Feftino
Ferifon
Frefifon
A,A,I,
A AJ,
A A L
A AJi
Weakened.
Y)X=XZ
AAI IV Bramantip
Strengthened.
Y)Z + Y)X = XZ AAI III Darapti
:X:Z EAO III Felapton
;X:Z EAO IV Fefapo
The words which reprefent the different moods (as they are
called) are ufually collected under their figures in the following
lines.
Barbara, Celarent, Darii, Ferioque prioris.
Cefare, Cameftres, Feftino, Baroko, fecundse.
Tertia Darapti, Difamis, Datifi, Felapton,
Bokardo, Ferifon habet. Q^uarta infuper addit
Bramantip, Camenes, Dimaris, Fefapo, Frefifon.
The vowels of the different words give the fymbol of the
fyllogifm ; thus A,A,A, are feen in Barbara. The confonants
in the firft figure have no fpecial meaning : but in the other
figures every confonant except T and N (which are only eu
phonic) has its meaning as follows ; every mood of every figure
can (with two exceptions) in one way or another, be reduced to
a mood of the firft figure : and the letters mow the way of doing
it. The initial tells to which mood the reduction brings us :
thus Cefare is reduced to Celarent, and alfo Cameftres ; Feftino
is reduced to Ferio, and fo on. The two exceptions are denoted
by the letter K (as in Baroko and Bokardo) ; we (hall prefently
notice them further. And S means that the preceding premife is to
be fimply converted. P, that what was called converfion per acci-
132 On the Ariftotelian Syllogifm.
dens is to be made, ZX for X)Z, or X)Z for ZX : accordingly,
P only occurs in the weakened or ftrengthened fyllogifms. M
means that the premifes are to be tranfpofed. Thus the meaning of
the word D if amis is nothing lefs than what follows. ( There is a
fyllogifm in which the middle term is the fubject of both pre
mifes, and when reduced to the firft figure it becomes Darn :
the major premife, which muft be converted in reduction, is a
particular affirmative : the minor premife, which muft become
the major one in reduction, is a univerfal affirmative : and the
conclusion, which muft be converted in reduction, is a particular
affirmative. Thus,
YZ + Y)X = XZ Difamis
becomes Y)X + ZY =ZX Darii
The moods Baroko and Bokardo do not admit of reduction to
the firft figure, by any fair ufe of the phrafe : but the logicians
were determined they fhould do fo, and they accordingly hit
upon the following plan, which they called reduction per impoffi-
bile. AOO and OAO being the opponent forms (pages 88,
and 102) of AAA, the two moods in queftion were connected
with Barbara (whence their letter B) by fhowing that the latter
would make the denial of their conclufions force one premife to
contradict the other. Thus, Baroko, or if Z)Y and X : Y then
X:Z was proved in the firft figure as follows. If under thefe
premifes, X:Z be not true, then X)Z is true ; but Z)Y is true :
and Z)Y + X)Z, by Barbara, gives X)Y. But X:Y: there
fore, if Baroko be not a legitimate form, X)Y and X:Y are both
true at once, which is abfurd. Had contraries been ufed,
Z) Y + X : Y = X : Z would have been thrown into the firft figure
as y)z + Xy=:Xz, "Darn, or y.Z + Xy = X :Z, Ferio. And
Y:Z + Y)X = X:Z, Bokardo, is feen reduced to the firft figure
in Y)X + zY = zX, Darn.
Ariftotle did not ufe the fourth figure, confidering it, as is
faid, to be only an inverfion of the firft. The introduction of it
among the figures is attributed to Galen, and it does not often
appear in ordinary works of logic before the beginning of the laft
century. If the order of the premifes be inverted, fo as to make
the firft figure appear, the major and minor terms will appear
wrongly placed in the conclusion. The words ufed for thefe
On the Arijhtelian Syllogifm. 133
indirect moods of the firft figure were ufually the fifth and fol
lowing
ones in
Barbara^ Celarent, Dari i, Ferio, Baralip-/0
Celantes, Dabitis, Fapefmo, Frifefom-orum
the final fyllables in Italics being only euphonic (Frifefmo-orum
would have been more correct). Some ufed the words Farefmo
and Firefmo.
In calling the moods of the fourth figure by the name of in
direct moods of the firft figure, notice was taken of the circum-
ftance that a tranfpofition of the premifes would give the ar
rangement of the firft figure, in every thing but the proper
arrangement of major and minor terms, which is inverted. A
little confideration will mow the reader that the earlier Ariftote-
lians were wifer than the later ones in this matter. Confider the
fourth and firft figures as coincident, and the arbitrary notion of
arrangement by major and minor vanimes. It was not till this
mere matter of difcipline was made an article of faith that the
fourth figure had any ground of feceffion from the firft.
It might feem as if the union of the firft and fourth figures
would demand that of the fecond and third : the firft pair con
taining all the moods in which the middle term occupies different
places in the two premifes, the fecond pair thofe in which it has
the fame place in both. If this were done, each of the two main
fubdivifions muft be itfelf fubdivided into two. And this would
perhaps have been the more fkilful mode of divifion.
The diftinction of figures has been condemned by many, and
particularly by Kant. Whether attacked or defended, it is eflen-
tial that the true grounds of the fide taken fhould be more ex
plicitly ftated than is often done. The root of the diftinction of
figure is undoubtedly the diftinction between the two forms XY
and YX, X . Y and Y. X. It would be equally abfurd, either to
deny the identity of XY and YX, confidered as material of
inference, or to deny their difference in many other points of
view. In this work I am concerned only with what can be
inferred, and to what extent of quantity, and accordingly the dif-
tindtion is to me immaterial. But if I had not merely to ftudy
the way of ufing premifes, but alfo that of arriving at them, it
might very well happen that the afpects under which the fame
134 On the Arlftotelian Syllogifm.
inference is feen in different figures would give it very different
(hades of character. A fimple inftance will mow that though
the comparifon, and its extent, are all that can be attended to in
forming the conclufion, thefe points of meaning are not the only
ones. A perfon who wifhed to conteft the old ufe of the word
green y as applied to unripe fruit, would fay that fome green
fruits are ripe/ if he wanted fpecially to {how the mifapplication
of the word. But if he rather wanted to mow the badnefs of
the method of denying ripenefs, he would fay fome ripe fruits
are green. The proportions are endlefs in which, X and Y
being the terms, it is at one time X which is brought to Y for
comparifon, and at another Y to X. The fubjecl: of a propofi-
tion is always the objecl: of examination ; whether the form be
X)Y, X.Y, XY, or X:Y, we examine and report upon the Xs.
If we arrange the four figures feparately, we mail better fee
their feveral peculiarities.
Flrft Figure.
Barbara Y)Z + X)Y = X)Z
Darll Y)Z + XY =XZ
Celarent Y.Z + X)Y=X.Z
Ferlo Y.Z + XY=X:Z
What is here declared, is in every cafe the ditfum de omni et
nullo in its fimplefr. form, in a manner which juftifies the prefe
rence given to this figure. The middle term being completely
contained in, or completely excluded from, the major term \ fuch
inclufion or exclufion then follows of all fuch part of the minor
term as is declared in the fecond premife to be in the middle
term. The inference then is in this fentence c What is true of
the whole middle term, is true of its part/ And it is obvious
that in this figure the major premife mutt be univerfal, the minor
premife affirmative. The four forms are all found among the
conclufions. I think that the inverfion of the premifes which
the lyftem of chapter V. employs will be found to give the forms
which are mod eafily tranflated into language independent of the
middle term. The fentence All (or fome) of the Xs are what
muft be Zs, therefore all (or fome) of the Xs are Zs* includes
Barbara and Darn: and All (or fome) of the Xs are what can
not be Zs, and therefore cannot be Zs, contains Celarent and
Ferio.
On the Ariftotelian Syllogifm. 135
Second Figure.
Cefare Z.Y + X)Y = X.Z Came/Ires Z)Y + X.Y = X.Z
FtftifM Z.Y + XY =X:Z
Baroko Z)Y + X:Y = X:Z
In this figure (in which only negatives can be proved) the ap
pearance of the dictum is not fo direct. The terms of the
conclufion are both objects of examination, and one is wholly
included, and the whole or part of the other excluded (Cefare,
Cameftres, and Baroko) or one is wholly excluded, and the whole
or part of the other included (Cefare, Cameftres, and Feftino).
Or rather, to juftify the diftinction, we fhould fay that the whole
of the major term is mc Ul and the whole of the minor
which S ives Cfare in which the wh e f the
is therefore excluded from the major ; or elfe the whole of the
5 a y. in which that part of the minor is excluded from the
Feftino
major. And it is evident enough why the premifes muft be of
different figns.
In the firft figure, though all the forms be efTentially one,
(page 98,) the reduction of either to the form Barbara requires
either the explicit ufe of contraries, or invention of a name fub-
identical to X. Accordingly, no mood of that figure is reducible
to any other by the ufually admitted reductions. But this cannot
be faid of any of the other figures. In the one before us, Cefare
and Cameftres are identical, even without changing the figure.
That which is Cefare when X is major and Z minor, is Camef
tres when X is minor and Z major. In the firft figure, the
fame attempt made on Celarent or Darll^ removes them into
another figure.
Third Figure.
Daraptl Y)Z + Y)X = XZ
Difamis YZ + Y)X=XZ
Datifi Y)Z + YX =XZ
Felapton Y.Z-f Y)X = X:Z
Bokardo Y:Z + Y)X=X:Z
Ferlfon Y.Z + YX =X:Z
The firft and fecond figures contain a pair of univerfals each,
136 On the Arijlotelian Syllogifm.
with one particular derived from each, by a legitimate weakening of
one premife and the conclufion at the fame time : but in no in-
ftance is the quantity of the middle term weakened. And all
the fyllogifms in thefe two figures are fundamental (page 77).
In the cafe now before us, both the leading fyllogifms are not
fundamental, but ftrengthened, and capable of being weakened in
two different ways. The middle term is here examined in both
premifes : if it be wholly included in, or excluded from, one of
the concluding terms, and wholly or partly included in, or ex
cluded from, the other (but not fo that there ftiall be exclu-
fion from both) we have it that the whole or part mentioned in
one cafe is included in, or excluded from, that which the whole
is included in, or excluded from, in the other. There can be
none but particular conclufions.
Bramantip
Dlmarls
Z)Y
ZY
Fourth
+ Y)X = XZ
+ Y1X = XZ
Figure.
Camenes
Z)Y-
hY.
X = X
.z
Fefapo Z.Y + Y)X = X:Z
Frefifon Z.Y + YX =X:Z
We have now one univerfal fyllogifm in a form which does not
admit of being weakened in this figure, and two ftrengthened
fyllogifms, each of which has one weakened form, one of them,
Bramantip) admitting a ftronger conclufion in another figure.
Every conclufion except A appears. The mode of inference of
the three firft fyllogifms has been defcribed in the other figures.
In Fefapo and Frefifon, the perfect exclufion of the major term
from the middle, accompanied by the total or partial inclufion of
the middle in the minor, fecures the exclufion from the major,
of as much of the minor as it has in common with the middle.
I mall now proceed to the rules ufually given, and to fome
remarks on the degree in which they apply to the more general
fyftem in chapter V. Aldrich gives them as follows
Diftribuas medium : nee quartus terminus adfit :
Utraque nee praemifTa negans, nee particularis :
Se&etur partem conclufio deteriorem j
Et non diftribuat, nifi cum praemifTa, negetve.
On the Ariftotelian Syllogifm. 137
Thefe rules, I need hardly fay, are perfectly correct, when the
contraries of the terms are excluded, and alfo all notion of quan
tity except all, or the indefinite fome. Taking them in the natu
ral order, which verification has a little difturbed, we have ,
1. There are to be but three terms, of which it is underftood
two only appear in the conclufion, the excluded or middle term
appearing in both of the premiies. This is true in my fyftem,
when by terms are underftood alfo contraries of terms. I fhould
fuppofe that there can be no objection to the admiffion of con
traries, unlefs there be one to the conception of a contrary. Any
one may, with Ariftotle, object to the word not-man, as not the
name of anything : on the grounds which immediately induced
him to call it an aorift, or indefinite, name. But it can hardly
be affirmed that any one admitting not-man as a name, mould
thereupon refufe to recognife the identity of horfe is not man,
with c horfe is not-man.
2. The middle term is to be dijlributed in one or the other of
the premifes. By diftributed is here meant univerfally fpoken of.
I do not ufe this term in the prefent work, becaufe I do not fee
why, in any deducible meaning of the word diftributed, it can be
applied to univerfal as diftinguifhed from particular. In ufing a
name, it feems to me that we always diftribute : that is, fcatter
as it were, the general name over the inftances to which it is to
apply. When I fay fome horfes are animals, I diftribute certain
horfes among the animals ; and when all, all. Leaving the word,
the principle is one which clearly muft be true whenever we are
reftricted in quantity to all or fome (indefinite), and when con
traries are not admitted. In the former cafe we have, in one
form or another, to make m-\-n greater than y (chapter VIII.)
when we cannot know what relation either m or n has to 17, unlefs
one of them, or both, be equal to . We have no alternative
then, but to require that m or n fhall be u. The cafes in which
there is apparently no dependence on y\ will be difcufled in
the next chapter.
But when contraries are introduced, this rule is not univerfally
true. The exception is feen in
AATorX)Y + Z)Y=xz.
If all the Xs be Ys, and alfo all the Zs, it follows that there are
things which are neither Xs nor Zs, namely, all which are not
138 On the Ariftotelian Syllogifm.
Ys. It is here, as elfewhere, implied that the middle term is
not the univerfe of the proportion.
When we come, then, to ufe contraries, the fimple rule of the
middle term is no longer univerfally true. What other rule are
we to put in its place ? We know, of courfe, that every fyllo-
gifm can be reduced to an Ariftotelian fyllogifm, and even to one
or other of two among them, AiA 4 Ai or LAJi, or to the firft of
thefe, if we be at liberty to ufe invention of names (page 97).
Again, each term, or its contrary, is mentioned univerfally in
every proportion : fo that there is certainly one way in which
every pair of premifes may be made to exhibit a middle term
univerfally ufed in one of them. The rule to be fubftituted for
the diflribuas medium is, that all pairs of univerfals are con-
clufive, but a univerfal and a particular require that the middle
term mould alfo be a univerfal and a particular, that is, univerfal
in one and particular in the other. Thus, in X)Y-f Z)Y, as it
ftands, the middle is particular in both ; tranfpofe into y)x + y)z
and the middle is now univerfal in both, by which we fee the
Ariftotelian concluflon. Again, in X)Y + ZY, which is of the
fame kind, the tranfpofition gives y)x + Z : y, which is faulty,
becaufe, though there be a particular premife, there is not any
where a particular middle term. The cafes in which the middle
is of the fame name in both places (univerfal in four, particular
in four), are the ftrengthened fyllogifms only. There is nothing
to be furprifed at in its thus appearing that the particularity of
the middle term is juft as much a teft of a good fyllogifm as its
univerfality : of every name and its contrary, one enters univer
fally, and one particularly, in every proportion which contains
it ; and the fyftem in chapter V. is as much concerned with con
trary as with direct terms. It is thence vifible beforehand, to the
mathematician at leaft, that any teft muft be defective, unlefs
univerfal and particular enter into it in the fame manner.
The above contains a complete canon of validity, as foon as
the law of the three terms is underftood, which is only a law of
definition. We may ftate it as follows : Two premifes conclude
when both are univerfal, always ; when one only is univerfal,
fo often as it happens that the middle term (be it Y or y) is once
only univerfal ; when neither is univerfal, never. By this rule
alone the thirty-two conclufive cafes can be diftinguimed from
the thirty-two inconclufive ones.
On the Ariftotelian Syllogifm. 139
3. When both premifes are negative, there is no Ariftotelian
fyllogifm. In the fyftem completed by contraries, there are eight
fuch fyllogifms, as many in fa<5t, as there are with premifes both
affirmative. But a pair of negative premifes never conclude with
both terms of the premifes, but with the contrary of one or
both : and this muft be fubftituted, as a rule of conclufion, for
the one juft named.
4. Both premifes muft not be particular. This rule, which
relates wholly to quantity, muft be preferved in every fyftem
which admits no definite ratio, except that of one to one, or
all (pages 56, 57). I cannot learn that any writer on logic
ever propounded even the very fimple cafe of c Moft Ys are
Xs, moft Ys are Zs, therefore fome Xs are Zs, as a legiti
mate inference. And this, though it is certain that the quanti
tative prefix moft (plurimi) has before now excited difcuffion as
to whether it belonged to a univerfal or a particular.
5. By fettetur part em conclufio deteriorem it is underftood that
the negative is called weaker or lower (deter lor) than the affir
mative, and the particular than the univerfal ; and that the con
clufion is to be as weak as negative, or as particular, if there be
a premife which is negative or particular. This rule muft be
preferved, when contraries are introduced, fo far as relates to
particulars. But fo far as negatives are concerned, the rule muft
be that one negative premife gives a negative conclufion, and two
an affirmative one.
7. The laft line, et non dlflrlbuat^ nifi cum premljfa^ negetve^
fpoils the fymmetry to procure a verfe. The conclufion is not
to be negative without a negative premife : that is, affirmative
premifes give an affirmative conclufion. Alfo, no term is to be
diftributively, (/. e. univerfally) taken in the conclufion, unlefs it
were fo taken in its premife. A breach of this rule would be
equivalent to drawing a conclufion about what was not (or about
more than was) introduced into the premifes.
When contraries are introduced, the diftinclrion between pofi-
tive and negative is made to appear, what it really is, one of
language, or rather one of choice of names. But the diftin&ion
of form is not abolifhed, but is exactly what it was before. We
cannot lay down any rules for the formation of the conclufion
unlefs, in our eight ftandard forms, we preferve the mode of
140 On the Ariftotelian Syllogifm.
writing which belongs to the fundamental derivation of the
forms (page 61). Thus, the order being XY, A is x)y and not
Y)X, and O ! is x:y and not Y:X. This method of writing
being reftored, when neceflary, in pages 89 and 91, it follows
immediately that the rule of accentuation in the notation gives
the rule by which we determine whether the conclufion takes
the terms from the premifes, or prefers contraries. According as
the prepofition of the conclufion agrees with or differs from that of
a premife, fo does the conclufion take a term from that premife,
or its contrary. Thus, AiAiAi takes both terms from the pre
mifes, but AjA 1 ! 1 takes a contrary from the firft premife only.
This laft we fee if we write the fyllogifm as X)Y+y)z=xz.
Accordingly, we have
Syllogifms taking both concluding terms direct from the premifes.
Univerfals which begin with A ; particulars which begin with I :
eight in number ; being all which ifolate no accent.
Taking the fir/} term only from the premife. Univerfals begin
ning with E ; particulars beginning with O : eight in number ;
being all which ifolate the middle accent.
Taking the fecond term only from the premife. Strengthened
forms and particulars which begin with A : eight in number,
being all which ifolate the firft accent.
D
Taking neither term from the premifes. Strengthened forms
and particulars which begin with E : eight in number, being all
which ifolate the third accent.
This is a new mode of ftating the law of accentuation (pages
92-3) which I have preferred to place here, for fear of overload
ing chapter V. with rules. I have not ftated one half of thofe
which fuggefted themfelves. This multiplicity of relations is a
prefumption of the completenefs of the fyftem.
In the Ariftotelian fyftem, there is multiplication of the fame
modes of inference, under the varieties of figure. In that which
I propofe, there is a reduplication of moft of the effential cafes ;
for whatever cafe is found, the fame is alfo found with X and Z
interchanged, and alfo the order of the premifes. Again, whatever
cafe is found, it is found contranominally ; or with all the accents
(or prepofitions) altered. There are other ways (and many of
them) in which the fyftem is only in one half a duplicate of what
it is in the other. If all thefe modes of dividing the fyftem into
On the Ariftotelian Syllogifm. 1 4 1
two correlative parts divided it into the fame two parts, there can
be no queftion that one alone of thofe parts fhould have been
prefented as the object of confideration. But this does not hap
pen in any inftance : fo that it is impoflible to difpenfe with the
whole of the thirty-two cafes. The Ariftotelian cafes do not
form or include any half whatever of this fyftem.
CHAPTER VIII.
On the numerically definite Syllogifm.
IN the laft chapter I confidered no other quantity in names
except all and fome : the latter meaning one or more, it
may be all. To this extent of quantity we are limited in moft
kinds of reafoning, by want of knowledge of the definite extent
of our propofitions : and the few phrafes (page 58), as moft,
a good many, &c. by which we endeavour to eftablifh differ
ences of extent in ordinary converfation, have been hitherto held
inadmiffible into logic. In this fcience it feems to have been
always intended that the bafes on which its forms are conftructed
fhall be nothing but the fuppofition of the moft imperfect and
inaccurate knowledge. Though in geometry we are permitted
to aflume as the object of reafoning the ideal ftraight line, the
4 length without breadth of Euclid, which has no objective pro
totype, and though we fee the advantage of reafoning upon ideas,
and allowing the efTential inaccuracies of material application to
produce no effect except in material application, yet in the con
fideration of the pure forms of thought, the learner has always
been denied the advantage of ftudying the more perfect fyftem of
which his inferences are the imperfect imitation.
The ordinary univerfal propofitions are of a certain approach
to definite character, both of them with refpedt to their fubjects,
and the negative one with refpect to its predicate alfo. In X) Y
for example, what is known is as much known of any one X as
of any other. Perfect definitenefs would confift in a more exact
degree of defcription, and would require a higher degree of know
ledge. But in this chapter I fpeak only of numerical definite-
142 On the numerically
nefs, of the fuppofition that we know bow many things we are
talking about. We may be well content to examine what we
fhould do if we were a ftep or two higher in the fcale of creation,
if by fo doing we can manage to add fomething to our methods
of inference in the higheft to which we have as yet attained.
A numerically definite propofition is of this kind. Suppofe
the whole number of Xs and Ys to be known : fay there are
100 Xs and 200 Ys in exiftence. Then an affirmative propo
fition of the fort in queftion is feen in 45 Xs (or more*) are
each of them one of 70 Ys : and a negative propofition in
* 45 Xs (or more) are no one of them to be found among
70 Ys.
But it muft be particularly noticed that in fpeaking of a num
ber of Xs, as 45 Xs, I do not mean certain 45 Xs which can
be diftinguifhed from all the reft, fo that of any X it is poffible
to be known whether it belong to the 45 of the propofition, or to
the remaining 55. This degree of definitenefs is one ftep higher
than that which I here propofe to confider, and which is defcribed
by c there are 45 Xs which are contained among 70 Ys, it not
being known which Xs are the 45 Xs, nor which Ys are the
70 Ys : or elfe by c there are 45 Xs which are not any of them
identical with any one of 70 Ys, the precife Xs and Ys in
queftion being unknown.
It cannot of courfe be difputed that if any thing fhould necef-
farily follow from any 45 Xs being found among any 70 Ys, it
will not the lefs follow from our knowing which are the Xs and
which are the Ys. But this laft fuppofition only brings us to
really univerfal propofitions. If, there being 100 Xs, 45 of
them can be fpecifically feparated from the reft, fo as to be
known, the procefs of feparation is equivalent to putting them
* Thefe words (or more) mow that the word definite has reference only
to the lower boundary. Of courfe nothing can be fhown in right of " 45
or more, perhaps " except what is true in right of the 45. It is defirable
that as the premifes, fo fhould be the conclufion, of a fyllogifm : this would
not be the cafe if weufed premifes definite both ways. For example, there
being 100 Ys in exiftence, it will prefently appear that f Exaftly 55 Ys are
Xs and exaflly 60 Ys are Zs, though it enable us to fay that 15 Xs are
Zs does not allow us to fay Exaftly 1 5 Xs are Zs, but only 1 5 Xs (or
more) are Zs.
definite Syllogifm. 143
under a feparate name, fubidentical to X, and the reft, which are
equally diftinguifhable, under another name, alfo fubidentical to
X, and contrary of the firft name, when the univerfe is X.
Whether the name be long or fhort, does not matter, nor
whether it carry the feparating diftinclion in its etymology or
not. To feparate in any way inftance from inftance by lan
guage, is to name.
If then 45 definite Xs were known to be contained among
70 definite Ys, and if thefe Xs were each named M, and thofe
Ys each N, and if the reft of the Xs and Ys were named P
and Q, we fhould have the following propofitions,
M)X, P)X, N)Y, Q)Y, M)N, M.P, N.Q,
and all inferences. Moreover, in each cafe, we fhould have the
total number of inftances which are contained under each name ;
the numbers carrying with them evidence that every X is either
M or P, and every Y either N or Q. Subftitute M.N for
M)N and we have the correfponding negative propofition.
But if 45 unfeparated and infeparable Xs be fuppofed known
each to be among 70 fimilarly fituated Ys, there is no immediate
method of making any other propofition out of the terms X and
Y except its converfe, that 45 of thefe 70 Ys are 45 Xs, and (if
the whole number of Ys be known, fay 200) that there are 45
Xs which are not any one among 200 70, or 130 Ys. This is
then a fimple propofition, which becomes of a highly complex
chara&er, when the Xs and Ys named in it are taken as defi
nitely feparable from the reft. I fhall call it the/tmple numerical
propofition.
The diftin&ion may be eafily illuftrated by example. " All
the planets but one is a particular propofition ; it is c fome
planets : there is no one planet of right included in it. But
all the planets except Neptune is a univerfal propofition : c a-
planet-not-Neptune is a name of Mercury, of Venus, &c. ;
and of every planet it can be ftated whether it be in the name
or not. That which is true inferentially of all the planets but
one left particular, is true of c all the planets but Neptune:
but that which is true of the latter is not neceflarily true of the
former.
Taking X, Y, Z as the terms of the fyllogifm, | the number
1 44 On the numerically
of Xs in exiftence, >j the number of Ys, and the number of
Zs, and v the number of inftances in the univerfe, there are of
courfe fixteen poffible cafes of knowledge, more or lefs, of thefe
primary quantities, from all unknown to all known. Of thefe
fixteen cafes, it will be requifite to confider two only. Firft,
when the extent of the middle term is known, and all the
reft unknown ; fecondly, when all are known. The algebraical
formulae of the latter cafe will enable us to point out how the
fuppofition of any lefs degree of knowledge would affect our
power of inference.
I propofe the following notation. Let mXY denote either of
the equivalent propofitions, that m Xs are to be found among the
Ys, or that m Ys are to be found among the Xs. Let mK :Y
denote either of the equivalent propofitions, that there are m Xs
which are not any one among Ys, or n Ys which are not any
one among m Xs.
The fymbol loX is the algebraical fymbol for ten equal Xs
added together, X being a magnitude : it is then a collective
fymbol. In this work, X being a name, it implies every one
out of ten inftances of that name, diftributively^ but not collec
tively. This diftin&ion is very material, not only in this chap
ter, but throughout every part of logic. c Every X is Y is
diftributively true, when, by c Every X* we mean each one X :
fo that the propofition is c The firft X is Y, and the fecond X
is Y, and the third X is Y, &c. In this cafe the fubjecl: is X,
and the word every belongs to the quantity of the propofition.
But c every X is Y is collectively true, when we do not mean
that any one X is a Y, nor that any number of Xs are Ys, but
that all the Xs make a Y. In this cafe the propofition is fin-
gular : there is but one inftance of the fubjecl: mentioned, that
fubjecl: being, not X, but the collection c all the Xs. Thus s the
ten men are members of a committee* is distributive : c the ten
men are a committee is collective.
If, in fuch a propofition as loXY, we were to fuppofe the
I o Xs fpecifically feparated from the reft, being certain affignable
ten individuals from among all the Xs, then loX becomes a name
for each of the ten, as much as X, and may be confidered as a
univerfal term. And now loXY and {ioX})Ymean the fame
things.
definite Syllogifm. 145
Let >j be known, and only of the four, y, |, ?, . The only
collections of premifes which it is neceflary to confider are
mXY+nYZ
Without fome knowledge of the number of ys, of which by
fuppofition we have none, it would be ufelefs to attempt to draw
an inference from a pair in which Y and y enter together, par
tially quantified, as in mX Y + nL : ry. And nZy merely amounts
to nZ:*Y.
The above three are all we need confider : and even of thefe
the third is incapable of inference, fmce both premifes are nega
tive, and moreover, not reducible to a pofitive form by ufe of
contraries, the only way in which negative premifes really acquire
a conclufion in chapter V.
Let us firft confider the premifes mXY -\-nYZ. They tell
us that among the Ys we find m Xs and n Zs : accordingly,
neither m nor n exceeds >?. If m and n together fall fhort of 17,
nothing can be inferred : Y is extenfive enough (that is, there
are inflances enough of Y) to hold the m Xs and the n Zs with
out any coincidence of an X with a Z. As to other Xs or Zs,
we do not know whether they exift ; or, if they exift, we do not
know that any one of them is a Y. But if m and n together
exceed w, it is impoffible that m Xs and n Zs can find place
among Ys, except by putting either two Xs or two Zs, or an
X and a Z, with one of the Ys. Now as by the nature of the
fuppofitions, there cannot be two Xs, nor two Zs, to one Y, we
muft have the inference iXZ as often as there are units in the
excefs of m-}- n over rj. That is,
u XZ
Next, let us take mXY + nZ : sY. There may be two in
ferences, perfectly diftincl: from each other, the connexion of
which can only be explained in the more general fyftem to which
we fhall prefently come. Firft, let m and s together exceed .
Then m + s of the Ys have the common property of being
Xs, and of being clear of the n Zs. Accordingly, we have
mX Y + nZ : s Y = (m + s *)X : nZ
L
146 On the numerically
Next, let n -f- s be greater than *j. Take the s Ys among which
no one of the n Zs is found. Becaufe n + s is greater than >i, w
is greater than j, the number of Ys left. Accordingly,
n (-n s} of the n Zs cannot be any Ys, and therefore cannot
be any of the wXs which are Ys. Hence we have
In the appendix to this chapter (at the end of the work) will
be feen the manner in which all the Ariftotelian fyllogifms can
be brought under the firft cafe, and the firft* inference of the
fecond cafe. No Ariftotelian fyllogifm can be deduced from the
fecond inference except when S = YI, in which cafe it agrees with
the firft. For, when s is not >?, we muft, to make fuch a fyllo
gifm, have ^2=>?, and then, to make Z:iY Ariftotelian, s not
being , we muft have all the Zs in w, or = . We thus get
:Z, the premifes of Bokardo. But the conclufion is
u)Z, that of Bokardo being jX:Z. And this will
be found to be the only Ariftotelian fyllogifm which has this
fecond and numerically quantified inference, depending upon the
number of Zs exceeding the number of Ys unnamed in the par
ticular premife.
I now proceed to fuppofe that all the quantities are taken into
account. Some preliminary confiderations will be ufeful, as
follows.
Let two propofitions be called identical, when, either of them
being true, the other muft be true alfo : fo that nothing can be
inferred from the one, which does not equally follow from the
other. Such propofitions are X.Y and Y.X, fuch are X)Y
and y)x, and fo on. Again, two propofitions may be identical
relatively to a third : thus, P being true, Q_ and R may either
follow from the other ; accordingly, as long as it is underftood
that P is true, Q_ and R may, relatively to that fuppofition, be
treated as identical.
The word identical^ as applied to propofitions^ is here made to
mean more than ufual, but not with more licenfe than when the
word is applied to names. Thus, man and rational animal are
* I was not in pofleflion of the fecond inference till I had written what is
in page 157.
definite Sylloglfm. 147
not identical names, qua names, for they neither fpell nor found
alike : the identity understood is that of meaning ; where one
applies, there (hall the other apply alfo. Similarly, as to propo-
fitions (of which fubjecT:, predicate, and copula are the material
parts, juft as fpelling and found are thofe of names), identity does
not confift in famenefs of parts, nor in reducibility to famenefs, but
in fimultaneous truth or falfehood, fo that what either is, be it true
or falfe, the other is alfo, in every cafe. Thus two propofitions,
one of which fignifies that an end has been gained, and the other
that the fole and fufficient means of gaining it have been ufed,
are identical.
All the theory of names, their application or non-application,
may be applied to propofitions , their truth or falfehood. To fay
that a propofition is true in a certain cafe, is to fay that a certain
name applies to a certain cafe : to fay that it is falfe, is to fay
that a certain name does not apply, but that its contrary does.
That contrary is what logicians ufually call contradictory : and
the name is not fimply true or falfe, but the adjective attached to
the propofition. The conditions under which we are to fpeak
limit us to a number of cafes which conftitute what we may now
call, not the univerfe of the names in the propofitions, but the
univerfe of the truth or falfehood of the propofitions. Thus we
fhall fuppofe ourfelves now to be fpeaking, not of all inftances to
which the name U applies, but of all in which the propofition U
is true, or in which the name true U applies. A cafe in
which a propofition P is true may be marked P, one in which it
is falfe, p. We may now apply the names fubidentical, &c. and
the fymbols, together with all the iyllogifms, complex and fimple ;
but on each a remark may be neceflary.
Subidentical, identical, and fuper identical. If P be a propo
fition fubidentical of Q, that is, if every cafe in which P is true be
one in which Q is true, but fo that Q is fometimes true when P
is not, the propofition Q is ufually mentioned as ejfential to P,
and as a necejfary confequence of it. Whenever P is true, O is
true ; Q necefTarily follows from P ; if Q be falfe, P cannot be
true ; Q is eflential to P ; are all mere fynonymes. Accord
ingly necejfary confequent" 1 and *- f up er identical or identical 1 are
fynonymous terms : that is (page 68), necejfary confequent and
fuper affirmative. Identity of courfe confifts in each propofition
1 48 On the numerically
being true when the other is true. I think that, according to
general notions, it would be held more juft to fay that a propo-
fition contains its neceflary confequence than that it is contained:
but a moment s confideration will mow that the latter analogy is
at leaft as found. If the fecond be true whenever the firft is
true, it may be true in other cafes alfo : fo that we only fay the
fecond contains the firft, and it may be more.
Subcontrary^ contrary^ and fuper contrary. It is ufual to call
No X is Y and c Every X is Y by the name of contraries,
and to fay that c contraries may be both falfe, but cannot be both
true. This is a technical ufe of the word : in common lan
guage we mould fay that either a propofition or its contrary muft
be true ; c have you any thing to fay to the contrary generally
means what a logician would exprefs by putting the word con
tradictory in the place of contrary. I am compelled to ufe the
words contrary and contradictory as fynonymous : at which com-
pulfion I am well pleafed, never having feen any good reafon
why, in the fcience which confiders the relations of difta^ the
contrarla mould be any thing but the contra difta. The proper
word for contrary, commonly ufed to exprefs the relation of X) Y
and X. Y, is fubcontrary. Here are two propofitions P and Q
which cannot both be true, but may both be falfe : here is a pair
which can never be atferted of the fame inftance, and of which,
in many inftances, neither can apply. In the fame manner, the
propofitions X Y and X : Y, ufually called fubcontrary (for no
reafon that I can find except that they are written under the fo
called contraries in a fcheme or diagram very common in books
of logic) mould be called fupercontrary : they are never both
falfe, and may be both true. This is a complete inverfion of
the ufual propofitions : an inverfion which feems to me to be
imperatively required, if only my ufe offub and fuper in Chapter
IV. be allowed.
In applying thefe names to propofitions, it muft be remem
bered that we make the fame fort of afcent which we make in
pafling from fpecific to univerfal arithmetic, in ufmg a fymbol
to ftand for any number at pleafure. For inftance ; Perhaps
it may be thought that XY and X:Y may fometimes be only
contraries, and not fupercontraries, becaufe there may be names
which make one only true and not both. But this is not correct :
definite Syllogifm. 149
for we are considering the proportion itfelf as an in/lance among
propofitions^ not the propofition as fubdivifible into inftances, in
which name is compared with name. In fpeaking of propo
fitions, it is change from ufe of one name to ufe of another, or
from ufe of one number to ufe of another, which is change of
inftance : not change from one inftance of name to another.
And juft as in a univerfe of names, every name introduced is
fuppofed to belong, or not to belong, to every inftance in that
univerfe : fo in a univerfe of propofitions, I fuppofe every propo
fition, or its contrary, to apply (whether it be or be not known
which applies) in every inftance. We have never confidered
fuch a thing as the univerfe U, in which there are cafes in which
neither X nor x applies : we fuppofe there is always a power of
declaring that the name X muft either belong or not belong
to each inftance. In like manner, all the propofitions in each
univerfe now confidered, are fuppofed to be connected with all
the names in queftion : fo that X, Y, being two of them in their
order of reference, AI or Oi is true in each cafe, and A f or O 1 ,
EI or L, and E f or I 1 . We might, if we pleafed, enter upon a
wider ryftem. For though we cannot imagine of any object of
thought, but that it is either X or not X, be X what name it
may, yet we can imagine of propofitions that they may be wholly
inapplicable, as being neither true nor falfe. The firft aflertion
is all the more true, that it could hardly be exemplified without
exciting laughter : as I fhould do if I reminded the reader that a
book is either a cornfield or not a cornfield. We have never
confidered names under more predicaments than two ; never,
for inftance, as if we were to fuppofe three names X 15 X 2 , X 3 , of
which everything muft be one or the other, and nothing can be
more than one. But we fhould be led to extend our fyftem if
we confidered propofitions under three points of view, as true,
falfe, or inapplicable. We may confine ourfelves to fingle alter
natives either by introducing not-true (including both falfe and
inapplicable) as the recognized contrary of true : or elfe by con
fining our refults to univerfes in which there is always applica
bility, fo that true or falfe holds in every cafe. The latter hypo-
thefis will beft fuit my prefent purpofe.
This digreflion is fomewhat out of place here, but I have
preferred to retain the matter of it until I had occafion to ufe it.
1 50 On the numerically
I now proceed to aflert that the fimple numerical proportion
has no occafion for a numerically definite predicate. Let us
confider firft an affirmative proportion, fay c Of 10 Xs, each
is to be found among fome 15 Ys/ Of courfe it is fuppofed
there are 15 or more Ys in exiftence. With this let us compare
10 Xs are to be found among the Ys/ Thefe two proportions
are identical : if 10 Xs be among 15 Ys, there are 10 Xs among
the Ys : and if 10 Xs be among the Ys they are certainly 10
Ys ; put on 5 more Ys at pleafure, and they can be faid to be
among 15 Ys in juft as many ways as we can choofe 5 more Ys
to make up the 15. Note, that if the 10 Xs were among certain
fpecified 15 Ys, then, though the firft propofition would give the
fecond, the fecond would not neceflarily give the firft. But we
are now fuppofing that numerical fele6tion is only numerically
definite : definite as to the number, not as to the inftances which
make up that number. When therefore we fay 10 Xs are
among 15 Ys we fay neither more nor lefs than when we fay
10 Xs are among the Ys. It is in fact 10 of the Xs are 10
of the Ys and the converfe 10 of the Ys are 10 of the Xs is
the fame proportion.
Now let us take a negative propofition, 10 of the Xs are not
to be found, any one of them^ among fome 15 Ys, abbreviated
into c 10 Xs are not in 15 Ys. If there be 25 Ys in exiftence
this propofition muft be true ; mean X and Y what they may.
It is as true as that the X which is one Y is not any other Y.
Say there are 25 or more Ys : take any 10 Xs you choofe, and
put them down on any 10 Ys you choofe. Then certainly
there are 15 Ys left, no one of which is any of thofe 10 Xs.
Again, if there be 25 Xs in exiftence, ftill the propofition muft
be true. For if the 15 Ys were all there are, and they were all
Xs, there ftill remain 10 Xs which are not any one in the 15 Ys.
Accordingly, the propofition c m Xs are all clear of Ys, when
ever either the whole number of Xs, or the whole number of
Ys, exceeds m + n^ fays no more than is conveyed in our perma
nent underftanding that no object of thought can be more than
one X or one Y. But let it be otherwife ; let neither Xs nor
Ys be as many as m + n in number. Say there are 20 Xs and
23 Ys and let 10 Xs be clear of 15 Ys. There muft now be
at leaft 15 + 10 20, or 5 Ys which are no Xs at all^ and at
leaft 15 + 10 23, or 2 Xs which are no Ys at all. Firft, it is
definite Syllogifm. 151
plain that there are no 10 Xs among thofe Ys which are clear of
15 Ys : for there are but 23 Ys in all. Therefore, 2 at leaft
of thefe 10 Xs muft be Xs which are not Ys : which with 8
Xs that may be Ys, will be clear of the remaining 15 Ys.
Therefore 2 Xs at leaft are not Ys. Again, there are no 15
Ys among thofe Xs which are clear of 10 Xs, for there are but
20 Xs in all. Five Ys which are not Xs muft exift, which
with 10 that may be Xs, will be clear of the remaining IO Xs.
Accordingly, if the whole number of Xs be , and the whole
number of Ys be >?, the proportion there are m Xs which are
no one to be found among n Ys is eflentially true of every cafe of
that univerfe, whenever m + n is lefs than either or u. But
when m + n is greater than both and >?, there are two propo-
fitions, neceflarily involved, which are not eflentials of all cafes
of that univerfe : namely, that there are m + n Ys which are
not any Xs, and m + n u Xs which are not any Ys.
But, it may be afked, if y fhould be lefs than , and m + n
greater than 17, but ftill lefs than , may we not affirm that
m + n y Xs are not Ys ? Undoubtedly we may, but then we
d^ "ot affirm fo much as already belongs to every cafe of the
univerfe. For if be greater than u, no more than y Xs can be
Ys, and there are left >j Xs which cannot be Ys : and n
is, in the cafe fuppofed, more than m + n y.
Let v be the number of inftances in the univerfe, and being
the number of Xs and of Ys. The following ufes of the notation
will be readily feen to exprefs preceding refults, or others imme
diately deducible.
greater than u ( >j)X :Y or ( )Xy
greater than ( g)Y:|X or (u-)Yx
m + n greater than and than n gives
A.
O.
A
O 1
E,
L
E f
Y)X=,XY =(i/-
=my:(u
Y.X=Xy
XY =
x.y =( y -
I 1 xy =mxy
152 On the numerically
I now examine the modes of contradicting wXY and mX : wY.
As to the firft, it is obvious that (m always meaning that m are,
but that more may be) either m or more Xs are Ys, or elfe
m-fi or more Xs are not Ys. The contradiction then is
either of the equivalents
It will be fatisfactory to evolve the contradiction of mX : n Y
by a method which will again demonftrate the cafes in which no
contradiction exifts ; or in which the proportion is always true.
Let us put the two names in the leaft favourable pofition for
making mX:nY true. Let/> then be the number of Xs which
are not Ys, all the reft being Ys. Take the p Xs which are not
Ys (p muft not be fo great as w, for then the proportion is
made good by the Xs which are not any Ys) and m p from
thofe which are Ys. All the m Xs thus obtained are clear of
y (m p) or YJ m+p Ys. Let this juft be n : that is, let
p=zm + n-n. Then />, the number of Xs which are Ys, is
(m-\-n ) or + n m n. Let but one more X be Y,
and the proportion begins to be contradicted : for now m-\-n
v I Xs are not Ys, we muft take up y + I n of thofe which are
Ys to make m Xs, and there only remain u (+ I n) or n I
Ys clear of the m Xs. And it is plain that if we cannot do it by
ufmg firft all the Xs which are not Ys at all, ftill lefs can it be
done by ufmg thofe which are. Accordingly the contradiction of
Y is
Then, in order to have a proportion which can be contra
dicted, m + n muft be greater than , or equal to + I at leaft,
for otherwife + M m n + i would be greater than u, or more
Ys than u muft be Xs, which is abfurd : and fimilarly m + n
muft be greater than u. Otherwife, all contradiction is abfurd,
or 77zX:Y is always true.
AfTuming thefe laft conditions, however, the contradiction of
mX : n Y is made earer. To be capable of contradiction, it muft
amount to (m-\-n j)X : Y\ Y. Now when m-\-n u Xs are not
Ys, and no more, -M -w n Xs are Ys. One or more
definite Syllogifm. 153
above this, or let ( + >j -m n-\- i)XY, and mX:nY cannot be
true.
Thus much for contradictory or contrary propofitions. I
fhall prefently confider the contranominal proportions .
We muft guard ourfelves from prefcribing the ufe of any premife
which neceflarily belongs to all cafes in the univerfe (of propo
fitions). Let P be a proportion which may or may not be true,
laid down as a premife, and Q a propofition which is true in
every cafe. Let R be their neceflary confequence, or legitimate
inference : then it is not whenever P and Q are true, R is
true/ but c whenever P is true, R is true. So far as R is a con
fequence of Q, fo far it is a confequence of every thing which
neceflarily gives O ; and thus it is a confequence of the fuppofed
conftitution of the univerfe from which the propofitions are
taken. Now this conftitution is always underftood ; it may be
a convenience that R mould be deduced by firft deducing Q, but
it cannot be a necefiity. And R is a confequence of P and this
conftitution, not of P and Q.
For example, let the univerfe of propofitions be all that can
be formed out of the fuppofitions of the exiftence of 20 Xs, and
30 \ s, and 40 Zs, in one univerfe of names. Let us join to
gether I5XY and loZ : 2oY. Our rules of inference will pre
fently mow us, that 5X : loZ is the neceflary confequence of
thefe premifes : but this refult is not only true when I5XY is
true, without anything elfe, but even without that; becaufe
5 + 10 falls fhort of 40.
Again, we muft guard ourfelves from adopting the conclufion
which follows from premifes, when that conclufion is true in all
cafes by the conftitution of the univerfe : it is then a fort of
jpitrious* conclufion, legitimate enough as an inference, but of a
perfectly diftincl: character from inferences which would bear
* To this word, as here ufed, I have heard much objeftion ; and when I firft
took it, it was unwillingly, and for want of a better. But on further confi-
deration I am well fatisfied with it. The objection arifes from the idea of
falfe or worthlefs being generally attached to the word. But, though it may
be ufual for fpurious things to be worthlefs, it is not neceflary. If a London
maker of razors mould put the name of a great Sheffield houfe upon them,
thofe razors would be fpurious. Suppofe them as good as thofe of the Shef
field maker, or better, they are ftill fpurious : though it may be true enough
1 54 On the numerically
doubt but for the premifes, or would bear contradiction under
other premifes. Say that in the above univerfe we join the pro-
pofitions I5XY and 3oZ : 20 Y. Both thefe proportions are
capable of contradiction : the fecond is 2oZ : uY ( means 30,
but the fymbol reminds the reader that 30 is all) or loY : Z
( being 40). Now, by laws of inference, I5XY + 3OZ : 2oY
yields 5X : 3oZ, which is always true in that univerfe.
Here is a cafe in which premifes capable of contradiction give
a conclufion which is not.
The rule of inference is obvioufly as follows. We cannot
fhow that Xs are Zs by comparifon of both with a third name,
unlefs we can affign a number of inftances of that third name,
more than filed up by Xs and Zs : that is to fay, fuch that the
very leaft number of Xs and Zs which it can contain are together
more in number than there are feparate places to put them in.
If our premifes, for example, feparate fome 30 Ys, and dictate
that among thofe 30 Ys there muft be 20 Xs and 15 Zs, it is
clear that there muft be at leaft 5 Zs which are Xs. For if we
put down the 20 Xs which are to go in, and try to put the
Zs into feparate places, we are flopped as foon as we have filled
up the 10 remaining out of the 30 Ys, and mult put the otrier
5 Zs among the Ys which have been made Xs. Accordingly,
fo many Xs at leaft muft be Zs as there are units in the number
by which the Xs and Zs to be placed, together exceed the num
ber of places for them. All the other rules of inference are
modifications of this. For example, to prove that 10 Xs are not
Zs, we muft fhow fome number of inftances (be they Ys or ys,
or part one and part the other) overfull (in the above fenfe) of
Xs and zs, to the amount of 10 at leaft ; fo that 10 Xs are zs,
or are not Zs. To prove that fome xs are ys, we muft fhow a
number of inftances in which the leaft numbers of xs and zs
that the chances are rather in favour of their refembling the ware of Peter
Pindar s hero. In this work, a fpurious inference is that which paffes for
the confequence of certain premifes, but does not in reality follow from
thofe premifes any more than from an infinity of others : being true by the
conftitution of the univerfe. It is made to have the mark of thofe premifes,
when in truth we cannot know whether thofe premifes be poffible or not,
until we have firft examined a conftitution which virtually contains our con
clufion.
definite Sylloglfm. 155
which it can contain, overfill it, or in which the greatefl number
of Xs and Zs which it can contain underfill it, or do not fill it,
though made completely feparate.
In examining the fundamental laws of fyllogiftic inference, it
is not neceflary to confider any thing but the pofitive forms.
For wXiwY, when not fpurious (and we fhall fee that the
fpurious cafes may be reje&ed) is (z + >j)X:>jY, which is
(m + n >?)Xy or (m + n |)xY. There are, then, but two
fundamental cafes : one in which the predicates are the fame,
one in which they are contraries. We fhall accordingly have to
confider
X Y + nZ Y and mX Y + wZy :
m
and it will prefently appear that not more than one, even of
thefe, is abfolutely neceflary. In each cafe we muft afk, what
collective inftances of Y or of y, or partly of one and partly of
the other, receive any dilation as to how they are to be filled
with Xs, with xs, with Zs, or with zs : and what is the leaft
number of each which can be allowed to every fuch collection.
But there is yet fomething to do, fuggefted by the preceding
remarks. Let us take one proportion, a type of all we fhall
have to confider, fay mXY. This means that XY is true to at
leaft m inftances. Now, this propofition may involve Xy, or
xY, or xy. Firft, as to Xy. To get the leaft number of Xs
among the ys, we muft put the greateft number among the Ys.
If all the Xs will go among the Ys (or if be greater than or
equal to |) there need be no Xs among the ys : but if not (or if
n be lefs than |) then ! Xs muft be among the ys, in every
cafe. Accordingly
m
XY gives (t-
where by | underftand o, not only when | is equal to >,, but
when it is lefs. This refult is fpurious, fince it is true or falfe,
by the mere conftitution of the univerfe, independently of mX Y.
Secondly, as to xY. Since mXY is equally mYX, the fame
reafoning fhows that
mXY gives (u-|)xY
where jj | is to be underftood in the fame way. This refult is
alfo fpurious for a like reafon.
156 On the numerically
Thirdly, as to xy. Since there muft be m Xs among the Ys,
the greateft poflible number of xs is y m. If this be as great
as v |, the whole number of xs, there need be no xs among
the ys : but if m be lefs than u , there muft then be at leaft
u |(>j m) xs among the ys, or u + m f . Confequently
Y[ xy.
I here put the fign = becaufe thefe proportions are really equi
valents. Treat the fecond in the fame way as that which de
duced it from the firft, and we have
(v-\-m u |)xy=(y + u + m u | v >j v |)XY
If y + m be not greater than rj + |, the equivalent does not exift.
We are already well acquainted with one cafe of this proportion.
Let m = %: then mXY is X)Y and the equivalent becomes
(u )xy, which, as v v is the whole number of ys, is y)x.
The rule is, if two names have a certain number of inftances
at leaft in common, to the whole number in the univerfe add
that number of inftances, and fee if the fum exceed the whole
number of inftances of both names together. If it do fo, the
excefs fhows the leaft number of inftances which the contraries
of thefe two names muft have in common. Follow this rule,
and we have
n |)xy
>j)Xy
mXy =(y + m |)xY
mxy =(% + y + m y)XY
Here are exhibited the equivalent contranominal forms. The
following refults may now be deduced.
Firft, thefe contranominals being formed in the fame way,
each from the other, in any one pair, whatever we prove of the
firft from the fecond, we alfo prove of the fecond from the firft.
The mathematician would call them conjugate pairs. Next, fmce
all the four pairs are but verfions of the firft, with difference of
names, whatever we prove univerfally of the firft pair, we prove
of all. Now, taking the firft of any pair and making it poflible,
which is done by allowing m not to exceed the number of either
of the names mentioned, the fecond may be poflible or impofli-
definite Syllogifm. 157
ble, according as the fubtraction indicated can be done or not.
But whenever the fecond Is impojpble^ the fir/} is fpurious. Take
raXY, and let (u + m | n)xy be impoflible, or u + m (and ftill
more u] lefs than f + . Now as all the | Xs and >j Ys muft find
place in the v inftances of the univerfe, and | + >j exceeds y, we
muft, in every cafe of the univerfe of propofitions, have at leaft
(| + >j v)XY. But I/ + TW is lefs than | + or % + Y\ U greater
than m : confequently, wXY is fpurious, a larger propofition
being always true.
As we are not to admit fpurious propofitions among our pre-
mifes, we had better write all premifes double, putting down each
of the forms, and making double forms of inference. The pre-
fence of the fymbols of all necefTary fubtra&ions will remind the
reader of the fuppofitions which muft be made, to infure a legiti
mate fyllogifm. I now take the feveral forms.
m XY n ZY =
(u + m | >,) xy ^(u + n Zv) zy
The law of inference here tells us (page 154,) that m + n being
greater than >,, (m + n u)XZ, be it fpurious or not, follows from
the upper premifes. The lower premifes alfo give their inference
if
(u + m | >,) + (:,-{-__>,) be greater than u n
v v being the number of the ys. This laft is equivalent to fay
ing that u + m + n is greater than ! + >? + . Firft, remark that
one fpurious premife necefTarily gives a fpurious conclufion. Say
that u + m is lefs than + , or that mXY is fpurious. Then,
fmce u + m is lefs than | + u, and n does not exceed , it follows
that u + m + n is lefs than ! + >! + ; whence the contranominal
of the conclufion does not exift, or the conclufion is fpurious, as
afTerted.
Next, obferve that the conclufion may be fpurious, though
neither of the premifes be fo. For though v + m be greater than
+ *, and u + n than + , and therefore zu + m + n greater than
l + ^+2>j, or v + m + n + (uYi) greater than n4-l + ^ it by no
means follows that u + m + n alone is greater than *j + | + . It
is alfo vifible in the mode of formation of the fecond inference,
that to fay u + m exceeds | + j, and v + n exceeds + , only gives
158 On the numerically
exiftence to the premifes : to give them conclufion, the fum of
the two excefles muft itfelf exceed v n, the whole number
of ys.
Thirdly, we muft not omit to examine the poffible cafe in
which a premife is partially fpurious. For example, there are 10
Xs and 20 Ys in a univerfe of 25 inftances ; accordingly, 10 +
20 25, or 5, of the Xs muji be Ys. Let one of the premifes be
8XY : this is not then all contingent, and capable of contra
diction ; we only learn fomething about 3 out of the 8 Xs. And
I call this propofition partially fpurious. But it will give no
trouble : for we muft deal with the premifes and their contra-
nominal equivalents before we can pronounce for a conclufion ;
and of two proportions which are contranominal equivalents of
each other, one muft be partially fpurious. To fhow this, obferve
that if mXY be not partially fpurious, it is becaufe v is greater
than| + ; or 2u than | + j + i/ ; or (v |) + (i/ ->j) than v. But
then the numbers of xs and ys together exceed the whole num
ber of inftances in the univerfe ; whence fome xs muft be ys, or
the contranominal equivalent of wXY is partially fpurious.
Now, to write down the various forms of inference. There
are fixteen ways of trying for an inference : we may combine a
propofition in X Y, or xy, or xY, or Xy, with one in XZ, or xz,
or xZ, or Xz. But thefe fixteen cafes really combine four and
four into only four diftincl: cafes. Thus the one we have been
confidering, really contains the combinations of XY and YZ,
XY and yz, xy and YZ, and xy and yz. It is in our power to
make either pair the principal pair, and to give the other pair as
contranominals of the firft pair.
Thus, we may write the cafe of inference we have been con
fidering, as in the firft of the following lift, the others being ob
tained from the firft, by changing X into x, or Z into z, or both.
The fign + placed in the middle implies the coexiftence of the
four propofitions : and independent numeral letters are introduced
as feen, which will prefently be connected with the others by
equations, inftead of being exprefled in terms of them.
I - J>XZ
=
m xy ~^Vyz j "" \p*
xz
The equations prefently given for
this cafe apply with certain changes
to the other cafes.
mxY nYZ} CpxZ
(/> Xz
+
m Xyn yz, /
definite Syllogifm. 159
Here X and x are made to change
their former places : in the equa
tions, | and % muft change places.
mXY , Yz) fpXz
m xy -r f ==
mxY .nYz \ r/> x z
m Xy+n yZf ~ j/XZ
Here Z and z change places : as
muft and f in the equations.
Here X and x, and alfo Z and z,
change places ; as muft ! and | ,
and and <f, in the equations.
In the new manner of writing the form we have already con-
fidered, being the firft of the four, we have juft written
m for
rf for
p for m+ n
/> f for
Let us write | ? , >, , (\ for w |, v >j, y ^, the numbers of xs,
ys, and zs : and then, | + | f , >, + >, , + , being all the fame, (for
each is u) we may write y $ for | >, ? , ^ f | for | , and fo on.
That is, in the difference of two, one of which is accented, we
may interchange the letters if we pleafe. The equations of con
nection for the firft or ftandard cafe, are then
n =
or
or m + n + t? | W J
For the fecond cafe we muft write w =w + | ^=772 + V | f ,
and fo on. I now proceed to the feveral divifions into which
our ufual modes of thinking make it convenient to feparate the
cafes of this moft general form.
160 On the numerically
Firft, when every thing is numerically definite. In this cafe,
as feen, every form requires an examination of the premifes and
conclufion, as to whether they are or are not fpurious.
Secondly, when u, the number of inftances in the whole uni-
verfe of names, is wholly unknown. In this cafe | f is indefinite
when | is definite, and vice verfa ; and fimilarly one at leaft of
each two, n or >/, or , is indefinite. There are then no fpu
rious conclufions ; or, which is the fame thing, none which are
known to be fuch : for the fpurioufnefs of a premife or conclu
fion confifts in our knowing that it muft be true of its two terms,
independently of all comparifon of thofe terms with a third.
Thirdly, when |, >?, , are all indefinite, as well as u. In this
cafe, as here ftated, there is no poffibility of inference. We can
not tell whether m + n be or be not greater than , if we do not
know what u is, in any manner, or to any extent.
But here we introduce that degree of definitenefs by which
we diftinguifh the univerfal from the particular (or pojjlble parti
cular, fee page 56) propofition. If we can know that either of
the two, m and , is the fame as >j (greater neither can be) then
we know that m + n is greater than u. And at the fame time
we make Y univerfal, in one or the other of the premifes. And
the fame if we can know that either m* or w ? is y\
The following are the forms which may all be derived from
the firft, by ufing all the varieties of contrary names and contra-
nominal equivalents. If we want, for inftance, to fhow the con
nection of the fourteenth with the firft, we throw the firft into
the form
We then change x into X, and Z into z, changing at the
fame time | into | ! and into : and thus we get
Now, for m + w write m\ that is, for m write m 1 n 1 + 4
and we have
which is one of the forms of the fourteenth. And (n -\-m* |)xz
is only the contranominal of (m 1 + n )XZ.
definite Syllogifm.
161
mXY
2. m xy
3. mXY + n yz =
4. m xy
5. mxY
6. ai Xy
7-
8.
9-
J 4 <f)xz
-Oxz
|)xz
11. mXY
12. w xy
13. mxY
14. m Xy
15. TTZXY
1 6. w Xy
The fyllogifms of chapter V are all particular cafes of the
above lift, obtained as follows :
m =y
A 1 !,!,
9-
m =v
A O.Oi
m =>j, =^
A f A A
m =-n , n = !
A E E 1
W =)}
I.A.I,
w =rj
LE.O,
=>j, w =|
A.A.Ai
=>j, w =|
AiEiEi
m = >j , w = >}
A AJi
m=.y,n =n
A EiOi
/ =|, =^
A,AT
m=% n =?
A.E O 1
^ = |
A LI,
10.
*f=e
A ! 0,0i
n ={
PAT
n =
FE O 1
m = ?,n =t
A f A ! A ?
m = ?,n =?
A E E
m=
AJT
ii.
m=Z
A.0
n\=t
LA.I.
n ={
LEiO,
m=$, f =^
AiAjAi
m= ^ n i = r
AiE.E,
^ f = >, !
AJT
12.
m =j
AO O
w = , , =^
AiA 4 Ai
n? = *\n =t
AiE.E,
f =.
I AT
tf=J
I E O 1
== , , = ?
A A A
^W,J f =f
A E E
m f = /? ! , f =yi f
AiAT
! := , =,
AiE O
= ? f =?
A AJ,
W = ? = f
A E.0,
3- m =
4. 772 =
M
i6a
On the numerically
5-
m = v
EJ.0 1
J 3-
w =>i
EiOJ 1
m =u, n =
E t A ! E
^2 = u , n =t?
EiE Ai
n = v
O AiO 1
n =n
O EJ
n =YI m = |
E AiE
n =y m =|
E EiA 1
m =u , w =
EAiO
m =YI n =YI
EiEJ 1
E A Oi
m =f n =?
E E Ii
6.
;;z = |
EJiO
14.
m = %
E.OJ 1
w =
OiA ! Oi
n =?
OiE L
w _| _
EA ! Ei
w i_| 5 n __^
E.E Ai
7-
>i
ETOi
15.
m =S
E O Ii
=<t f
O A.O 1
d =
O EJ 1
m _.| 5 _^
E AiE
m __| 5 w =< ^
E EiA 1
8.
w ! = >i f
ETOi
16
w f = >i
E O L
m = v =^ f
E f A,E
rr?=-n\ n 1 =
E EiA 1
w f =>i !
OiA f Oi
n =i f
O.E Ii
w i jji w f = |
EiA ! E,
M 1 =jj f W ? = |
EiE Ai
m i J n =>,
E A ! Oi
77Z f = )j f , W ? =H f
E E I,
w = | =^
E.^0 1
m ! =| =C
E,E,I f
We have thus another mode of eftablifhing the completenefs
or the fyftem of fyllogifm, laid down in the laft chapter : that is,
of the fyftem in which there is only the common univerfal and
particular quantity. Thefe fyllogifms of numerical quantity, in
which conditions of inference belonging to every imaginable cafe
are reprefented by the general forms which numerical fymbols
take in algebra, muft of neceflity be the moft general of their
kind. And examination makes it clear that, except the preced
ing, there can be no fyllogifm exifting between X, Y, Z, and their
contraries. Many fubordinate laws of connexion might be no
ticed between the general forms and their particular cafes. Thus,
each univerfal occurs three times, each fundamental particular
twice, and each ftrengthened particular twice. The firft form
in pages 158, 159, gives only affirmative, the fourth only negative,
premifes : the fecond and third one of each kind, commencing with
a negative in the fecond, and with an affirmative in the third.
There are two remarkable fpecies of fyllogifm (or rather,
which ought to have been remarkable) : which I (hall now pro
ceed to notice.
The diftindion of larger and fmaller part, when divifion into
definite Syllogifm, 163
two parts is made, is as much received into the common idiom
of language as the diftinction of whole and part itfelf. Moft of
the Xs are Ys, is nearly as common as All the Xs are Ys :
though feweft of the Xs are Ys, is only feen as moft of the
Xs are not Ys/ The fyllogifms which can be made legitimate
by the ufe of this language will do equally well for any fraction,
provided we couple with it the fraction complemental to unity
(which in the cafe of one half is one half itfelf). Let a and /3
ftand for two fractions which have unity for their fum, as f and
. Let a XY and a X:Y indicate that lefs than the fraction a
of the Xs are or are not Ys. Let *XY and "X:Y indicate
that more than the fraction a of the Xs are or are not Ys.
Then the following fyllogifms arife from the cafes with the
numbers prefixed.
i. YX
4. y:X +< 3 y:Z =xz
5. "YtX+eYZ = Z:X
8, X + ?:Z =X:Z
9.
YX
12. y:X -f^yZ =Z:X
13. Y : X+?Y:Z=xz
1 6. X + ?Z =XZ
It will be feen that here are but three really diftinct forms ; of
which the fimpleft examples are as follows,
Moft Ys are Xs ; Moft Ys are Zs ; therefore fome Xs are Zs.
Moft Ys are Xs ; Moft Ys are not z; therefore fome Xs are
not Zs.
Moft Ys are not Xs ; Moft Ys are not Zs ; therefore fome
things are neither Zs nor Xs.
It is hardly neceflary to obferve that in one of the premifes
c more than may be reduced to as much as : but not in both.
Thus, if two-fevenths exactly of the Ys be Xs, and more than
five-fevenths of the Ys be Zs, it follows that fome Xs are Zs.
The above fyllogifms admit a change of premife, as follows :
If we fay that more than ths of the Ys are Xs, we thereby fay
that lefs than f ths of the Ys are xs : or YX and ^Y : X are the
fame proportions. Thus, moft are is equivalent to a minority
(none included) are not. Hence we have
and fo on. Or we may combine the two forms, x as in
164 On the numerically
The above are the only fyllogifms in which indefinite particu
lars give conclufions, by reafon of that approach to definitenefs
which confifts in defcribing what fractions of the middle term are
fpoken of, at leaft, or at moft. But they are not the only fyllo
gifms of the fame general fpecies. In every cafe inference follows
when there is a certain preponderance ; and the largenefs of the
inference depends upon the extent of that preponderance. Thus
in (12) there is an Xz inference when T/Z + W + H exceeds + :
fo many units as there are in this excefs, fo many Xs (at leaft)
are zs. Now in every cafe, a pair of univerfal premifes give in
ference : and in every cafe there muft be a degree of approach
to univerfality at which inference begins. The ordinary fyllo
gifms, I fufpedl:, are, and are meant to be, not fuch as c Every
X is Y, every Y is Z, therefore every X is Z, but c generally
fpeaking X is Y, and generally fpeaking Y is Z, therefore gene
rally fpeaking X is Z. And by c generally fpeaking is meant
the aflertion that an enormous majority of inftances make the
affertion true. A fyllogifm of this fort is the oppofite of the a
fortiori fyllogifm ; and might be faid to be true ab infirmiori. If
we have X)Y with p exceptions, and Y)Z with q exceptions ;
then, in form (i.) we have m=l t p^n = n q^ and m + n y) =
^p q. As long, then, as the number of exceptions altogether
fall fliort of the number of Xs, there is inference : if the total
number of exceptions be very fmall, compared with the number
of Xs, there is the c generally fpeaking kind of inference. Ex
amine all the univerfal cafes, and it will be found that the fame
law prevails ; namely, that there is inference when the numbers
of exceptional inftances in both premifes together do not amount
to the number of inftances in the univerfal term of the conclu-
fion ; and that there is exceptional univerfality (as we may call it)
in the conclufion, whenever the whole amount of exception is
very fmall, compared with that number of inftances.
This leads us to what I will call the theory of exceptional
particular fyllogifms. We have feen that the eight complex
affirmatory fyllogifms, which are all a fortiori in their conclu
fions, afford each two particular fyllogifms. We have denoted
coexiftence by + ; and the coexiftence of two proportions gives
more than either. Let us denote exceptive coexiftence by :
thus, P Q means that the propofition P is true except in the
definite Syllogifm. 165
inftances contained in O. Thus, X)Y X: Y means that every
X (with fome exceptions) is Y. This is, of courfe, A 4 O, and
only differs from L in the mode of expreffion not being fome
more than none at all but c fome lefs than all. In the expref
fion
(A, 0,)(Ar- 0.)(A 00
we have the fymbol of the ab Infirmlorl fyllogifm ftated above,
fubjecT: to the poffibility of nonexiftence if the number of excep
tions in the two premifes mould exceed the number of inftances
in the univerfal term of the conclufion. If we look at Aid, as
a fymbol defcriptive of premifes, we fee one of the inconclufive
forms ; that is, a form from which we cannot draw an inference.
But this is only becaufe our inferences are all pofitive, and imply
aflertion of fufficitncy in the premifes. There is no ufe (except
to mow the manner in which the parts of a fyftem hang together)
in declarations of Infufficlency : for we know that all collections
of premifes, whatever they may be fufficient for, will be infuf-
ficient for an infinite number of different things. And it is
important to remember that while fufficiency is accompanied by
muft be, infufficiency only allows may be. From AiAi the con
clufion AI muft be true : from AOj (and as far as thefe are con
cerned) it may be falfe. Accordingly AidOj and OiAiOi may
ferve to exprefs the two defects of (Ai Oi)(Ai O)(Ai O t )
from AiAiAi, exifting in the ab Infirmlorl fyllogifm, and poflibly
preventing conclufion altogether : juft as AiO O f and O AiO
mow the additional conditions by the fulfilment of which AiAi AI
is elevated into the a fortiori iyllogifm DiDiDi. It is worth
while to dwell upon the varieties of this cafe. The ab Infirmlorl
fyllogifms of the ftrengthened particulars were previoufly confi-
dered.
In all the cafes yet treated, we have had, more or lefs, the
power of giving inftances in common language, without recourfe
to numerical relation expreffed in unufual terms. This of courfe,
is always the cafe in the fyllogifms of chapter V. ; and we
have given one common Injlance (though never met with in books
of logic) from each fet of ab Infirmlorl fyllogifms. But there are
ftill cafes of the fame fort to be confidered. Though in our de
finite relation (page 56) of all, we ufually (in books of logic at
1 66 On the numerically
leaft) make the relation exift, for each propofition, between the
terms of the propofition itfelf, yet it may be afked whether we
cannot fometimes infer fuch a fpecies of univerfal as this, for
every Z there is an X which is Y ; Z being one of the names
of the fecond premife. If we examine the firft two cafes, which
will be guide enough, we mail find the following refults from the
new fuppofitions now made.
1. m = ^n=^ gives XY + YZ = >/xz : or if for every Z
there be an X which is Y, and for every X a Z which is Y,
then, fo many ys as there are, fo many things which are neither
X nor Z. This fyllogifm has little new meaning, and no new
application : it requires =, and therefore X)Y and Z)Y.
2. ;?Z T =, gives xy-f-w YZ = wxz, or if for every Z there be
that which is neither X nor Y, and if fome Ys be Zs, there are
as many inftances which are neither X nor Z. This is a new and
effective form.
2. = , gives 77zVy4-J ! YZ=w XZ, a new form.
Thefe two cafes will be prefently further confidered. Now,
obferve that if m + n in the firft form, or m + n in the fecond,
be i/, that is, if the pair m and n be and , or n and >/, or and
f , we have inference of the kind required. The firft form gives
no new fyllogifm : fince v is more than u, Ys which are Xs, and Ys
which are Zs, to the number of i/, give the form (i.) by the main
law of inference (page 154). In the fecond form, if m + n = U)
we diftribute among the Ys and ys, Zs and xs to the full number
of both, fo that wherever there are not xs (that is, wherever
there are Xs) there are Zs : or X)Z as obtained from the form.
But everyway of conftructing m\y + YZ = (w + ;z | f )XZ
which gives rr? + n-=u^ is only a cafe of A^iAi. For m 1 can not
exceed y\ and n cannot exceed u : and w ! + n being v or >j + 1 ,
we muft have n? =n f and u = y ; whence the afTertion made. The
forms we are now in fearch of, fo far as quite new, are all con
tained in the two new ones above noted ; and of thefe, the fecond
is but a transformation of the firft. The eight varieties derived
from ufe of contraries, or from the forms in page 161, beginning
with the fimpleft, are
definite Syllogifm . 167
=Z:|X
Thefe are fyllogifms, which exhibit a curious kind of antago-
nifm to the particular fyllogifms. Take the fyllogifm AiO O 1 , the
terms being M,Y,Z ; we have then M)Y + Z: Y = Z:M. Of
courfe the conclufion M : Z is not legitimate from thefe premifes
alone : but if M have as many inftances as Z, then M : Z is
legitimate. For if Ms, as many as there are Zs, be among the
Ys, and fome of the Zs be not among the Ys, though all the
reft were, there would not be enough to match all the Ms, or
fome Ms are not Zs. Now, let M be a name given to an X
which is Y, and let fuch Xs have as many inftances as Z, and
the above becomes the firft of the fyllogifms in the laft lift.
Thus, LO Oi is legitimate, if the quantity of the fubjecl: men
tioned in Ii be taken from the Zs. The fecond fyllogifm is EiLO 1 ,
altered into OJiOi in the fame manner.
The reader may find all the refults of the above cafe in the
following rule, in which it is underftood that all the fuper-propo-
fitions are to be written either way : thus, A 1 is written x)y,
or Y)X, and O T is 7zx:>?y, or wY: |X (page 62). Write down
any pair of particulars, followed by I if the pair be of the fame
fign, and O if the pair be of different figns : as in OOI or IOO.
Accent the pair in contradiction to either the direct rule (page
62) as far as the words affirmative and negative are concerned :
that is, let a negative beginning ifolate nothing, and an affir
mative beginning ifolate the middle propofition : or elfe, ac
cent the pair according to the inverfe rule. Thus, OiOJi
and O OT contradift the direcl rule, and O O L and OiOJ 1
preferve the inverfe rule. To make thefe fyllogifms good (in the
particular way in queftion) proceed thus : When the dlrett rule
is contradicted, take the quantity of the firft concluding term
from the total of thefecond^ if the fecond premife be affirmative,
and from its contrary, if negative. When the inverfe rule is
preferved, take the quantity of the fecond from the total of
the firft. Thus, in O OT the direft rule is contradicted :
and it ftands m 1 x:Jy + n 1 y:Z 1 z,=p l xz,. The fecond premife is
negative, the total of its predicate f inftances, that of the con
trary . Accordingly, x:n f y + l y: f z= l xz, or Y:|X-f
168 On the numerically
w Z :Y = w xz, which is one of the forms already obtained. Again,
O O L preferves the inverfe rule, and is f/z T x:v) f y + Z:^Y
pXZ. The total of the firft term is inftances. Hence,
is derived from one of the forms given, by interchanging X
and Z.
This clafs of fyllogifms with tranfpofed quantity naturally leads
to the queftion, Is it ufed ? Do fuch fyllogifms occur in ordi
nary or in literary life ? If not, there is no reafon for fele&ing
them from the infinite number of cafes which the numeri
cally definite fyftem affords. To try this, fuppofe a perfon, on
reviewing his purchafes for the day, finds, by his countercheques,
that he has certainly drawn as many cheques on his banker (and
may be more) as he has made purchafes. But he knows that he
paid fome of his purchafes in money, or otherwife than by
cheques. He infers then that he has drawn cheques for fome-
thing elfe except that day s purchafes. He infers rightly enough ;
but his inference cannot be reduced to a common fyllogifm, with
the names in queftion for terms. It is really a fyllogifm of tranf
pofed quantity, as follows :
For every memorandum of a purchafe a countercheque
is a tranfa&ion involving the drawing of a cheque.
Some purchafes are not c tranfaclions involving, &c.
Therefore fome countercheques are not memoranda of pur
chafes.
It may be worth while to give one inftance of the verification
of the contradictory form. By page 152 it appears that the con
tradiction of wXY is (| z+i)Xy, or (vj m + i)xY, and that
of TTz Xy is (!-w ! + i)XY, or (vf-m 1 + i)xy.
To wXY join the contrary of (m + n t^)XZ, or (1 + ^ m
+i)Xz: we have then
^ m n+ i)zX;
the inference of which is (m + Z + y m n+ I |)Yz, that is,
(^ w + i)Yz, the contrary of wYZ.
Returning to the forms in page 161, it will be obferved that we
have no double inferences. In every cafe we have made ufe of
one form of inference : if u be known, the other is a real equi
valent ; or elfe it is impoffible, and as we have feen, then the
definite Syllogifm. 169
firft is fpurious. If v be not known, then the fecond is either
perfectly indefinite, or elfe identical with the one chofen. Ex
amination will fhow that in every one of the cafes cited in page
161, the neglected form of inference is only faved from perfect
indefinitenefs when we are able to apply the word all to one or other
of the terms : the number being as indefinite as before ; the rela
tion thus obtained being definite. Take the firft form, and make
= v) ; by the firft inference we then get the fyllogifm LAJi : by
the fecond, we get (m + v f ) xz > indefinite both in number
and relation. We do not know what y, |, and are. If we
knew as much as that m + u is lefs than ! + , we fhould know
our inference to be fpurious,* it being not the lefs an inference.
Now, add the condition m f : the firft inference gives the fyl
logifm AiAiAi, the fecond inference now becomes (u )xz :
definite relation enters, and we have z)x, or X)Z, or AI, as
before. And the fame of the other forms.
The reader may perhaps fuppofe that I ought to have com
menced this chapter with the complex numerical fyllogifm, in
imitation of the method which I followed in treating the ordinary
fyllogifm. But in truth there is no fyftem of complex fyllogifm
of per feel: numerical definitenefs both in premifes and conclusion.
To fhow this, let w,XY with the comma, mean that there are
exactly m Xs which are Ys, neither more nor fewer. Accord
ingly w,XY is a fynonyme for ;/zXY + (^ m)xY. Now com
bine #z,XY and ,ZY, or
We then have mXY + nZY =
(r, #z)xY-f(>i #)zY = (r; m W
OTXY + (n )zY=(z )Xz
(n w)xY + ZY =(n
* I muft again remind the reader, of the diftinftion between fpurious and
illegitimate, which exifts in my language. The fpurious inference follows
from the premifes, and is perfe&ly good and true : but from the conftitution
of the univerfe, it will always be true, whatever premifes in that univerfe are
taken. The illegitimate inference is that which does not follow from the
premifes. A conclufion not known to be fpurious, that is, there not being
the means of knowledge, is not fpurious : but an illegitimate conclufion can
not be made legitimate, that is, following from the premifes, by any further
knowledge.
170 On the numerically definite Syllogifm.
Two only of thefe have meaning : let them be the two upper
ones. We can affign then Z or z to (m + n y) + (m ), or
to 2m- v\ of the Xs. But there are not all of the Xs here : for
m is lefs than vj, and than f , whence 2m is lefs than ^ + 1, or
2m -vi lefs than |. The reft of the Xs, | + ij 2i in number,
may, for aught thefe premifes declare, be either Zs or zs.
CHAPTER IX.
On Probability.
THE moft difficult inquiry which any one can propofe to
himfelf is to find out what any thing is : in all probability
we do not know what we are talking about when we afk fuch a
queftion. The philofophers of the middle ages were much con
cerned with the is, or effence, of things : they argued to their own
minds, with great juftice, that if they could only find out what a
thing is, they mould find out all about it : they tried, and failed.
Their fucceflbrs, taking warning by their example, have inverted
the propofition ; and have fatisfied themfelves that the only way
of finding what a thing is, lies in finding what we can about it ;
that modes of relation and connexion are all we can know of the
effence of any thing ; in fhort, that the proverb tell me who you
are with, and I will tell you what you are, applies as much to
the nature of things as to the characters of men. We are apt
to think that we know more of the effence of objects than of
ideas ; or rather, of ideas which have an objeftive fource, than
of thofe which are the confequence of the mind s adion upon
them. I doubt whether the reverfe be not the cafe : at any rate,
when we content ourfelves with inquiry into properties and rela
tions, we have certain knowledge upon our moft abftract ideas.
The object of this chapter is the confideration of the degrees of
knowledge itfelf. That which we know, of which we are cer
tain, of which we are well affured nothing could perfuade us to
the contrary, is the exiftence of our own minds, thoughts, and
perceptions, the two laft when a&ually prefent. This higheft
knowledge, this abfolute certainty, admits of no imagination of
the poflibility of falfehood. We cannot, by flopping to confider,
On Probability. 171
make ourfelves more fure than we are already, that we exift,
think, fee, &c. Next to this, come the things of which we can
not but fay at laft we are as certain of them as of our own exif-
tence ; but of which, neverthelefs, we are obliged to fay that we
arrive at them by procefs, by reflection. Thefe we call neceffary
truths (page 33). The neceffity of admitting thefe things caufes
fome to imagine that they are merely identities, that they amount
to faying that when a thing is, it is : but this is not correct. To
fay that two and two make four (which muft be), and that a
certain man wears a black coat (when he does fo) both involve
the pure identity that whatever is, is ; and not one more than
the other. Nor is two and two identically four, though necef-
farily fo. Our definitions of number arife in the procefs of fim-
ple counting. Throw a pebble into a bafket, and we fay one :
throw in another, and we fay two ; yet one more, and we fay
three^ and fo on. The full definitions of the fuccefiive numbers
are feen in
That three and one are four is definition : it is our pleafure to
give the name four to 3+1. But that 3+1 is 2 + 2 is neither
definition nor pure identity. It is not even true that two and
two is four; that
s
It is true, no doubt, that two and two is four, in amount,
value, &c. but not in form, conftruction, definition, &c.
There is no further ufe in drawing diftinction between the
knowledge which we have of our own exiftence, and that of two
and two amounting to four. This abfolute and inaflailable feel
ing we fhall call certainty. We have lower grades of knowledge,
which we ufually call degrees of belief^ but they are really degrees
of knowledge. A man knows at this moment that two and two
make four : did he know it yefterday ? He feels perfectly certain
that he knew it yefterday. But he may have been feized with a
fit yefterday, which kept him in unconfcioufnefs all day : and
thofe about him may have been warned by the medical man not
to give him the leaft hint of what has taken place. He could
fwear, as oaths are ufually underftood, that it was not fo : if he
172 On Probability.
could not fwear to this, no man could fwear to anything except
neceflary truths. But he could not regard the aflertion that it
was not fo, as incapable of contradiction : he knows it well, but,
as long as it may poflibly be contradicted, he cannot but fay that
he might know it better.
It may feem a ftrange thing to treat knowledge as a magnitude,
in the fame manner as length, or weight, or furface. This is
what all writers do who treat of probability, and what all their
readers have done, long before they ever faw a book on the fubject .
But it is not cuftomary to make the ftatement fo openly as I
now do : and I confider that fome juftification of it is neceflary.
By degree of probability we really mean, or ought to mean,
degree of belief. It is true that we may, if we like, divide pro
bability into ideal and objective, and that we muft do fo, in order
to reprefent common language. It is perfectly correct to fay It
is much more likely than not, whether you know It or not^ that
rain will foon follow the fall of the barometer. We mean that
rain does foon follow much more often than not, and that there
do exift the means of arriving at this knowledge. The thing is fo,
every one will fay, and can be known. It is not remembered,
perhaps, that there is an ideal probability , a pure ftate of the mind,
involved in this aflertion : namely, that the things which have been
are correct reprefentatives of the things which are to be. That
up to this 2 1 ft of June, 1847, tne above ftatement has been true,
ever fmce the barometer was ufed as a weather-glafs, is not de
nied by any who have examined it : that the connexion of
natural phenomena will, for fome time to come, be what it has
been, cannot be fettled by examination : we all have ftrong rea-
fon to believe it, but our knowledge is ideal^ as diftinguifhed
from objective. And it will be found that, frame what circum-
ftances we may, we cannot invent a cafe of purely objective pro
bability. I put ten white balls and ten black ones into an urn,
and lock the door of the room. I may feel well aflured that,
when I unlock the room again, and draw a ball, I am juftified
in faying it is an even chance that it will be a white one. If all
the metaphysicians who ever wrote on probability were to witnefs
the trial, they would, each in his own fenfe and manner, hold me
right in my aflertion. But how many things there are to be
taken for granted ! Do my eyes ftill diftinguifh colours as be-
On Probability. 173
fore ? Some perfons never do, and eyes alter with age. Has
the black paint melted, and blackened the white balls ? Has any
one elfe pofTefTed a key of the room, or got in at the window,
and changed the balls ? We may be very fure^ as thofe words
are commonly ufed, that none of thefe things have happened, and
it may turn out (and I have no doubt will do fo, if the reader try
the circumftances) that the ten white and ten black balls will be
found, as diftinguifhable as ever, and unchanged. But for all
that, there is much to be afTumed in reckoning upon fuch a
refult, which is not fo objective (in the fenfe in which I have
ufed the word) as the knowledge of what the balls were when
they were put into the urn. We have to aflume all that is re-
quifite to make our experience of the paft the means of judging
the future.
Having made this illuftration to draw a diftinction, I now pre-
mife that I throw away objective probability altogether, and con-
fider the word as meaning the ftate of the mind with refpeft to
an aflertion, a coming event, or any other matter on which ab-
folute knowledge does not exift. c It is more probable than im
probable means in this chapter I believe that it will happen
more than I believe that it will not happen. 5 Or rather c I ought
to believe, &c. : for it may happen that the ftate of mind which
/j, is not the ftate of mind which fhould be. D Alembert be
lieved that it was two to one that the firft head which the throw
of a halfpenny was to give would occur before the third throw :
a jufter view of the mode of applying the theory would have
taught him it was three to one. But he believed it, and thought
he could (how reafon for his belief: to him the probability was
two to one. But I (hall fay, for all that, that the probability is
three to one : meaning, that in the univerfal opinion of thofe who
examine the fubjecl:, the ftate of mind to which a perfon ought
to be able to bring himfelf is to look three times as confidently
upon the arrival as upon the non-arrival.
Probability then, refers to and implies belief, more or lefs, and
belief is but another name for imperfect knowledge, or it may be,
exprefles the mind in a ftate of imperfect knowledge. There is
accurate meaning in the phrafe c to the beft of his knowledge and
belief j the firft word applying to the ftate of his circumftances
with refpect to external objects, the fecond to the ftate of his
1/4 On Probability.
mind with refpect to the circumftances. But we cannot make
any ufe of the diftinction here : what we know is to regulate
what we believe ; nor can we make any effective ufe of what
we know, except in obtaining and defcribing what we believe, or
ought to believe. According to common idiom, belief is often
a lower degree of knowledge : but it is imperative upon us to
drop all the quantitative diftinctions of common life, or rather
to remodel them, when we come to the construction of a
fcience of quantity.
I have faid that we treat knowledge and belief as magnitudes :
I will now put a broad illuftration of what I mean. We know,
(fuppofe it known] that an urn contains nothing but two balls,
one white and one black, undiftinguimable by feeling : and we
know (fuppofe this alfo) that a ball is to be drawn. Disjunctively
then we know white will be drawn : black will be drawn, one
or the other muft be. How do we ftand as to c white will be
drawn, and c black will be drawn, feparately ? Clearly in no
preponderance with refpect to either. May we then properly
and reafonably fay that we divide our knowledge and belief of
the event one or the other into two halves, and give one half
to each. I can conceive much objection to this fuppofition :
but, whether they formally make it or not, I am fure writers on
probability act upon it, and are accepted by their readers.
Let us confider what magnitude is, that is to fay, how we
know we are talking about a magnitude. We know that when
ever we can attach a diftinct conception of more and lefs to dif
ferent inftances, fo as to fay this has more than that, we are
talking of comparable magnitudes. We fpeak of a quantity of
talent, or of prudence : we fay one man has more talent than
another, and one man more prudence than another : but we never
fay that one man has more talent than another has prudence. If
we occafionally fay he (the fame one man) has more talent than
prudence, it is only as an abbreviation : we mean that he has not
prudence enough to guide his talent. Juft as we might fay (though
we do not) that there is more cart than horfe, when the horfe
cannot draw the cart : juft as, fpeaking very loofely, we do fay,
the prejfure of the atmofphere is not fifty Inches ; meaning that it
is not enough to balance the prefTure of fifty inches of mercury
in the barometer. And thus, both up to, and beyond our means
On Probability. 175
of meafurement, we form to ourfelves diftinct notions of com
parable magnitudes, and incomparable magnitudes, as well as of
the meaning of the fomewhat incorrect, but eafily amended,
figures of fpeech by which we fometimes talk of comparing the
latter.
But the object of all quantitative fcience is not merely magni
tude, but the meafurement of magnitude. And when are we en
titled to fay that we can meafure magnitude ? As foon as we
know how, from the greater, to take off a part equal to the lefs :
a procefs which neceflarily involves the teft of which is the
greater, and which is the lefs, and, in certain cafes, as it may
happen, of neither being the greater nor the lefs. As to fome
magnitudes, the clear idea of meafurement comes foon : in the
cafe of length, for example. But let us take a more difficult
one, and trace the fteps by which we acquire and fix the idea :
fay weight. What weight is, we need not know : the Newto
nian, who makes it depend on the earth s attraction, and the
Ariftotelian, who referred it to an impulfe which all bodies pof-
fefs to feek their natural places, are quite at one on their notions
of the meafurable magnitude which their feveral philofophies dif-
cufs. We know it as a magnitude before we give it a name :
any child can difcover the more that there is in a bullet, and the
lefs that there is in a cork of twice its fize. Had it not been for
the fimple contrivance of the balance, which we are well aflured
(how, it matters not here) enables us to poife equal weights
againft one another, that is, to detect equality and inequality,
and thence to afcertain how many times the greater contains the
lefs, we might not to this day have had much clearer ideas on the
fubject of weight, as a magnitude, than we have on thofe of
talent, prudence, or felf-denial, looked at in the fame light. All
who are ever fo little of geometers will remember the time when
their notions of an angle, as a magnitude, were as vague as, per
haps more fo than, thofe of a moral quality : and they will alfo
remember the fteps by which this vaguenefs became clearnefs
and precifion.
Now a very little confideration will mow us that, the moment
we begin to talk of our belief (the mind s meafure of our know
ledge) of propofitions fet before us, we recognize the relations
called more and lefs. Does the child feel that the bullet has
176 On Probability.
more fomething than the cork one bit better than an educated
man feels that his belief in the ftory of the death of Caefar is
more than his belief in that of the death of Remus. Let any
one try whether he have not in his mind the means of arranging
the following fet in order of magnitude of belief, including within
that term all the range which comes between certain knowledge
of the falfehood, and certain knowledge of the truth, of an affer-
tion. Let them be I. Caefar invaded Britain with the fole view
of benefiting the natives. 2. Two and two make five. 3. Two
and two make four. 4. Caefar invaded Britain. 5. Romulus
founded Rome. He will probably difcover the gradations of
neceffary truth, moral certainty, reafonable prefumption, utter
incredibility, and neceflary falfehood. Thefe are but names given
to different ftates of the mind with refpect to knowledge of pro-
pofitions afferted ; and I fay they exprefs different ftates of
quantity.
The only difficulty, and a ferious one it can be made, may be
ftated in the following queftion ; Are we to confider the fort
of belief which we have of a neceffary propofition (as two and
two make four), that is, abfolute knowledge, to which contra
diction is glaring abfurdity as only a ftrengthened or augmented
fpecimen of the fort of knowledge which we have of any con
tingent propofition (fuch as Caefar invaded Britain) which may
have been, or might have been, falfe, and can be contradicted
without abfurdity ? I anfwer, we can eafily (how that the dif
ference of the two cafes is connected with the difference be
tween finite and infinite, not between two magnitudes of dif
ferent kinds. The mathematician will eafily apprehend this,
and will look upon the various difficulties which furround even
the explanation as upon things to which he is well accuftomed,
and which he underftands by many parallel inftances. We can
invent circumftances under which a contingent propofition mall
make any degree of approach to neceffity which we pleafe, but
fo that no actual attainment mail be arrived at. If an urn con
tain balls, and if one ball muft be drawn, then, the balls being
all white, it is neceffary that a white ball muft be drawn, as
neceffary as that two and two being in any place, there are four
in that place : for there are no degrees of neceffity. But let it
be that there are black balls alfo, at the rate of one to a thoufand
On Probability. 177
white ones : the drawing of a white ball is no longer neceflary ;
but there is ftill a ftrong degree of aflurance that a white ball
will be drawn. We do not readily fee how much : becaufe the
urn has no vifible relation to our ufual cafes of judgment. But
let it be made to reprefent the life of a youth of twenty : and let
the drawing of a white ball reprefent his living to come of age,
and of a black one his death in the interval. There ought to be
feven black balls to the thoufand white ones to make the cafes
parallel. And yet we know that our aflurance of his furvival is
generally very ftrong : be it wife aflurance or not, it exifts, and
we acl: upon it. Now fuppofe the rate to be one black to a mil
lion of white : the aflurance is much increafed, but ftill there is
no neceflity ; the black ball may be drawn. Take one black to
a million of million of white, or a million of million of million,
&c. : long before we have arrived at fuch a point, we have loft
all conception of the quantitative difference between our belief in
drawing a white ball, and our belief that two and two are four.
We fay it is almofl impojfftble that one trial mould give a black
ball : and this very phrafe is a recognition of the famenefs for
which I am contending. Except on the fuppofition of fuch
famenefs, there is no almofl impojjtble^ nor nearly certain. Be
tween the impoflible and the poflible, the certain and the not
certain, there muft be every imaginable difference, if we do not
admit unlimited approach. For it will clearly not be contended
that, reprefenting certainty, fay by 100, we can make an ap
proach to it by an uncertainty counting as, fay 90, but nothing
higher. Reprefenting the ftate of abfolute knowledge by 100, any
one, with a little confideration, will fay that the laws of thought
fix no numerical limit to our approach towards this ftate : but
that things mort of certainty are capable of being brought within
any degree of nearnefs to certainty. On fuch confiderations, I
mail aflume that neceflity on the one hand, a certainty for, and
impoflibility on the other, a certainty againft, are extreme limits,
which being reprefented by quantities, may allow our knowledge
of all contingent propofitions to be reprefented by intermediate
quantities.
It muft be fully allowed, nay, imperatively infifted on, that
nothing in the numerical view, tending to connect neceflary and
contingent propofitions, can at all leflen the diftinction between
N
178 On Probability.
them : nor give the latter any refemblance to the former, except
only in the quantities by which they are indicated. Though
there be only one black ball to as many white ones as would fill
the vifible univerfe, yet between that cafe and the one of no
black balls muft always exift the eflential difference, that in the
former a black ball may be drawn, and in the latter it cannot.
But this very great diftinction between the necefTarily certain
and the contingent, is it compatible with their being reprefented
by numerical quantities as near to one another as we pleafe ? I
anfwer that all who are acquainted with the relations of quantity
are aware that nearnefs of value is no bar to any amount of dif
ference of properties. A common fraction, for inftance, may
be made as near as we pleafe in value to an integer : but there
do not exift, even among propofitions, more eflential, or more
ftriking, differences, than thofe which exift between the properties
of integers and of fractions. There are crowds of theorems (I
fhould rather fay unlimited crowds of clafles of theorems) which
are always true when integers are ufed, and never true when
fractions are ufed. Let any quantities be named, integer or frac
tional, and it is eafy to make clafles of theorems which are true
for thofe quantities, and not for any others, however near to them.
The reader who is not a mathematician muft rely upon the know
ledge of the one who is, that the difference between two quan
tities, no matter how nearly equal, may be connected with other
differences as complete, and by practice as eafily recognized, as
the difference between neceflary and contingent truth.
I will take it then that all the grades of knowledge, from
knowledge of impoflibility to knowledge of neceffity, are capable
of being quantitatively conceived. The next queftion is, are
thefe quantities capable, in any cafe^ of meafurement, or of com-
parifon with one another. At prefent, we ftand as the child
ftands with refpect to the bullet and the cork : perceptive of
more and lefs, but without a balance by which to make compa-
rifons. To mow the poftulate on which our balance depends,
let us fuppofe an urn, which, to our knowledge, contains white,
black, red, green, and blue balls, one of each colour. It is within
our knowledge that a ball muft be drawn : accordingly we have
full knowledge (and of courfe entire belief] that the refult c no
balP is impoflible, and that c white, or black, or red, or green, or
On Probability. 179
blue is neceffary. To the refult c white we accord a certain
probability, that is, a certain amount of belief. If a man tell us
that white will be drawn, we may hold him ram, but we do not
pronounce his communication incredible : let another tell us that
c black, or red, or green, or blue will be drawn, and we hold him
not fo rafh, and his communication more credible. We may
hold with either, if he will defcribe his knowledge and belief as
partial, and give them their proper amounts. Now, whether we
mall proceed, or flop fhort at this point, depends upon our ac
ceptance or non-acceptance of the following POSTULATE :
When any number of events are disjunctively poffible, fo that
one of them may happen, but not more than one, the meafure
of our belief that one out of any fome of them will happen,
ought to be the amount of the meafures of our feparate beliefs
in each one of thofe fome.
I mean that any one mould fay, A, B, C, being things of
which not more than one can happen, c my belief that one of the
three will happen is the fum of my feparate beliefs in A, and in
B, and in C. This is the poftulate on which the balance de
pends ; and there is a fimilar poftulate before we can ufe the
phyfical balance. The only difference (and that but apparent)
is that we are to fpeak of weights collectively, and of events dif-
junclively. The weight of the (conjunftive) mafs is the fum of the
weights of its parts : the credibility of the (disjunctive) event is the
fum of the credibilities of its components. There are feveral
may-bes, any one of which may become a has-been : when we
fpeak dlsjunftwelj) it is of the will-be, which cannot be faid of
more than one : the may-be of an event defcribed as contained in
c A, B, C, is to be reprefented as in quantity the fum of thofe
in < A, in B/ and in < C.
Is it matter of mere necefiity that, talking of phyfical weight,
the weight of the whole is equal to the fum of the weights of the
parts ? We have learnt to admit this poftulate, of which no
man ever doubted : but no one can fay that it was neceffary.
The laws of matter and mind being both what they are, the con
nexion between phyfical colleftion and mental fummation is, I
grant, neceffary : the fimpleft of manual, and the fimpleft of
mental, operations, are and, with us, muft be, concomitants.
But, in the firft place, it is not true that the weight of the
i So On Probability.
whole is equal to the fum of the weights of the parts, in the
manner in which the reader probably imagines it to be true. Let
the firft part we hang on the balance be the weight which is
correctly meafured by W. Then if we hang under it another
weight, as correctly reprefented by V, we think we are quite fure
when we fay that the collective mafs muft have a weight W + V
becaufe its parts have the weights W and V. But its parts have
not the weights W and V. The
weight of V is diminimed by the
upward attraction of W, and is,
fay 5 V M : the weight of W is
as much increafed by the down
ward attraction of V, and is W
+ M. And though V M and
W + M added together do give V + W, yet it was not in this
way that the reader made out his neceflary truth. The univer-
fal equality of action and reaction did not exift in the thoughts of
the firft perfon who formed a diftinct conception of the weight
of the whole as compofed of the fum of that of the parts : and he
was only right by the (fo far as he was concerned) accidental
circumftance, that two things of which he knew nothing, coun
terbalanced each other s effects. Nor do we know at this mo
ment, as of neceffity, that the propofition is correct. We have
much reafon to think that the law of equality of action and reac
tion is mathematically true : but, let it fail to the amount of only
one grain in a thoufand million of tons, and the propofition is
not true, but only nearly true.
Again, the co-exiftence of thofe laws of mind and matter
which beft, fo to fpeak, fit each other, and which make the phe
nomena of the external world, after due confideration, appear to
be almoft what they muft have been, is not, to our apprehenfions,
a neceflary coexiftence. We can imagine the following refult,
though we cannot trace what the full confequences of it would
be on the expreflion of the laws of thought. Conceive fentient
beings, to whom the fimpleft mode of arithmetical fucceffion is
not o, i, 2, 3, &c. but i, 10, 100, 1000, &c. their powers of
numeration being fo constructed that the fecond of thefe fuccef-
fions has that character of fundamental fimplicity which we
attach to the firft. Of courfe, their primary fymbols would be
On Probability. 1 8 1
fignificative of I, 10, 100, &c. It would be impoflible for us to
conceive any mode by which ten or any other number could be
thus fundamentally attached to unity, in a manner fhared by no
third number : but, I am not faying, Imagine how this could
be, but, Imagine that it is. There is no contradiction in the
fuppofition, either to itfelf, or, till we know much more of the
mind than we now do, to anything elfe. Beings fo conftituted
would have logarithmic brains ; and if, thus conftituted, they
were placed among our material laws of exiftence, the manner
in which the weight of the whole is to be inferred from thofe of
the parts, would be a profound myftery for ages, only to be folved
in an advanced ftage of mathematical fcience. A recent mode
of conftructing mathematical tables, which generally carries with
it the name of its eminent inventor, Gaufs, would conftitute
one of their principal neceffities : they would have to ufe it as
their only mode (except a6r.ua! experiment) of finding out that
what we reprefent by 156 and 200, together make (and thi-s
making would be a complicated procefs) 356.
Inftead, then, of trying to eftablifh it as perfectly natural and
neceflary to fay that our belief of * one of the two A or B, when
both cannot happen, is, quantitatively fpeaking, the fum of our
belief in A, and our belief in B, I have rather endeavoured to
fhow that the analogous cafes with which we firft think of com
paring this propofition, other kinds of compofition, are not fo
natural and neceflary as is fuppofed. There are two ways of
levelling ; by bringing up the lower, or bringing down the higher.
And I particularly wifh in this chapter to prevent the reader
from accepting the arithmetical doctrine of probability quite fo
rapidly as is ufiially done. In furtherance of this object, I pro
ceed to the following poflible objection.
It may be faid, you have, by thus formally identifying proba
bility with belief, and ftating a poftulate which, in exprefs terms,
has not the moft axiomatic degree of evidence, rendered fome-
what difficult that which in the ordinary view of fimple chances,
is very eafy. This charge, I hope, is true : fuch was my inten
tion, at leaft. And my reafon is, that in the ordinary view of
the fubject, one of two things occurs : either probability is fepa-
rated by definition from ftate of belief, though it be known that
the two words will afterwards be confounded without any per-
i 8 2 On Probability
miffion ; or elfe the poftulate is tacitly affumed, and the difficulty
which I fuppofe myfelf charged with introducing, is flurred over.
Take a common queftion ; An urn has two white balls and
five black ones : there are feven equally likely drawings, two
white ; therefore the chance or probability of drawing a white
ball is called two-fevenths. But the chance of either particular
white ball is one-feventh. Now firft, if any one mould fay
that this is mere definition, I can, of courfe, allow it : but it then
remains to fhow what connexion this defined probability has with
any ordinary acceptation of the word. But if, probability mean
ing belief, or fentiment of probability actually exifting in the
mind, or index of the proper degree of belief, &c. &c. the
above ftatement be made as fundamentally evident, I mould then
afk how it is known that the probability of one or the other
white ball being drawn is properly fet down as the fum of the
probabilities of the feparate white balls. And I cannot conceive
any anfwer except that it is by an afTumption of the poftulate.
That fuch aflumption will finally be knowingly made, on the
fulleft conviction, by every one who ftudies the theory, I have
no doubt whatever : nor that it has been made, no matter in
what words, nor with what clearnefs of avowal, by every one
who has ftudied that theory. And therefore I hold it defirable
that the beginner mould know what I have here told him.
It is indifferent, as far as the theory is concerned, what nu
merical fcale of belief we take. We might, if we pleafed, copy
Fahrenheit s thermometer, fet down knowledge of impoffi-
bility as 32, perfect certainty as 212, and other ftates of mind
accordingly. Thus, 122 would reprefent perfect indecifion,
belief inclining neither way, an even chance. But this would
complicate our formulae : the ufual and preferable plan is to af-
fume o as the index of knowledge of impoffibility, I as that of
certainty, and intermediate fractions for the intermediate ftates.
This mode of eftimation makes formulae and procefles fo much
more eafy than any other, that it muft be adopted ; but there is
a ftrong objection to it in one point of view : as follows.
When we fpeak of belief in common life, we always mean
that we confider the object of belief more likely than not : the
ftate of mind in which we rather reject than admit, we call
wwbelief. When the mind is quite unbalanced either way, we
On Probability. 183
have no word to exprefs it, becaufe the ftate is not a popular*
one. The quantitative theory calls by the name of belief every
admiflion of poflibility. When there is only one black ball to a
million of white ones, there is fome belief that a black ball will
be drawn ; a much larger belief in a white one. It would be
advantageous in fome refpech that o fhould reprefent the ftate
of indifference, + i, that of knowledge of certainty, and I,
that of knowledge of impoflibility. But this would complicate
formulae too much. I confider it therefore defirable to ufe the
common meafures and formulae, but to aflbciate them with the
one juft propofed, in the following manner.
When a perfon tells us that his belief in an afTertion is, fay W,
meaning that he confiders it 3 for and 7 againft, or 7 to 3 againft,
we fhould fay in common talk that he difbelieves, but not very
ftrongly. In the language of this theory, we fay that he both
believes and difbelieves, the latter more ftrongly than the former.
Let us add that his authority is againft the conclufion. If he fay
that it is in his mind an even chance, or that he has no opinion
one way or the other, let us fay that he gives no authority either
way. If we adapt this definition to the fuppofition that -j- I and
i reprefent the extremes of authority for and againft, we have
the following rules. The meafure of authority is twice the mea-
fure of belief diminifhed by unity, for, when pofitive, againft,
when negative : the meafure of belief is half of unity increafed
(algebraically) by the meafure of authority. If a reprefent the
meafure of belief, and A that of authority, then
It is alfo advifable to have a term to reprefent what are ufually
called the odds. Some might think it defirable to rid the fubjecl:
as much as poflible of words derived from gambling : aftrono-
mers have done the fame thing with the phrafes of aftrology, and
chemifts with thofe of alchemy. When it is 7 for and 3 againft,
* Many minds, and almoft all uneducated ones, can hardly retain an
intermediate ftate. Put it to the firft comer, what he thinks on the queftion
whether there be volcanoes on the unfeen fide of the moon larger than
thofe on our fide. The odds are, that though he has never thought of the
queftion, he has a pretty ftiff opinion in three feconds.
184 On Probability.
it might be faid that the relative tefttmony for, is J, and that
againft, \. But the brevity of the firft phrafe will infure its con
tinuance, let who will try to change it.
The ordinary rule is a confequence of the notions hereinbefore
laid down, and of the particular mode of meafurement adopted.
It is as follows ; When all the things that can happen can be
refolved into a number of equally probable (or credible) cafes,
fome favourable and fome unfavourable to the event under con-
fideration, then the fraction which the favourable cafes are of all
the cafes, meafures the probability (or credibility) of the arrival
of the event : and the fraction which the unfavourable cafes are
of all the cafes, meafures the probability (or credibility) of the
non-arrival. There are, for inftance, in an urn, 5 white, 4 black,
and 3 red balls, 12 in all. It is aflumed that we know them to
be equally likely to be drawn ; which here means no more than
that we know nothing to the contrary. That one ball muft be
drawn, is fuppofed certainly known. Accordingly, our belief in
c one or another is reprefented by I : which is, by the poftulate,
the fum of the feveral credibilities of the balls ; which laft are all
equal. Therefore each ball has T V : and by the fame poftulate,
the event c one or other of the white balls or the drawing of a
white ball, has T V ; of a black ball -? ; of a red ball, T \.
Inftances like the above, in which we invent all the cafes and
have arbitrary power over their number, are the only ones on
which we can employ a priori numerical reafoning. They are
alfo the only ones on which we can try experiments. It is im
portant to know whether, as a matter of fact, our belief, nume
rically formed, will be approximately juftified by the refults of
trial. And this juftification is found to exift, in the following
way. It is a remote, but certain, conclufion from the theory,
requiring mathematical reafoning too complicated to introduce
here, that events will, in the long run, happen in numbers pro
portional to the objective probabilities under which the trials are
made. For inftance ; if a die be correctly formed, fo that no
one face has more tendency than another to fall upwards, the
probability of throwing an ace is -- ; that of not throwing an ace
is |.. The theory tells us its own worthleflhefs, if in the long
run, not-ace do not occur five times as often as ace. If 60,000
trials were made, the theory would tell us to expect about 10,000
On Probability. 1 8 5
aces and about 50,000 not-aces. Practice confirms the theory :
not, that I know of, in the actual cafe juft cited, but in fimilar
ones. I will ftate an inftance.
Throw a half-penny up, and if it give tall, repeat the throw,
and fo on, till head arrives : and let this fucceffion be called zfet.
The probability that a fet fhall confift of one throw, is {hewn
by the theory to be ^ ; that it fhall have two throws, i ; three
throws, |.; and fo on. If a very large number of fets be tried,
we are to expect that about half will be of one throw, about a
quarter of two throws, about an eighth of three throws ; and fo
on, as long as the number is large enough to give any profpect
of fomething like an average. This experiment has been tried
twice : once by the celebrated Buffon, and once by a young
pupil of mine, for his own fatisfaction ; both in 2,048 fets. The
refults were as follows ; the third column fhowing the number
of each kind which the theory aflerts to be moil probable.
B H
Head at the firft throw
1061
1048
1024
No head till the 2nd throw
494
507
512
3rd
232
248
256
4th
137
99
128
5th
56
71
64
6th
29
38
32
7 th
25
17
16
8th
8
9
8
gth
6
5
4
loth
o
3
2
nth
i
I
1 2th
o
""1
1 3th
o
i ^th
o
i
i5th
o
o
^ X
1 6th
o
I
&c
o
o
J
2048 2048 2048
In BufFon s trials, there were altogether 1992 tails to 2048
heads, .and in Mr. H s there were 2044 tails to 2048 heads.
Inftances in which we can command all the cafes are to the
1 86 On Probability.
mind, in this theory, what acceflible lengths are to the eye. We
can meafure the latter by a rule, and fo train the organ to judge
of lengths which cannot be approached, or cafes in which the
rule is not at hand.
I fhall now refer the reader to other works on the fubjecT:, for
further details on the operative part, and proceed to juft as much
as is neceflary for the particular purpofe of the next chapter,
namely, the application of the hypothecs of meafure of belief to
queftions of argument and teftimony. Two theorems will be
enough : the firft relating to independent events, the fecond to
the probability of events which are neither wholly independent,
nor wholly confequent, either upon the other. The word event
is ufed in the wideft poffible fenfe : it does not even neceflarily
mean future event. Unlefs our knowledge, either of the cir-
cumftances, or of the event itfelf, thereby undergo fome altera
tion, it is nothing to us now whether it has happened, or is to
happen.
Let there be two events, P and Q, of which the probabilities
are the fractions a and b ; and let them be wholly independent of
one another, the arrival or non-arrival of either being perfectly
independent of that of the other. The probability that both
fhall happen is the product of a and b : and fimilarly for more
events than two. Suppofe, to take an inftance, that a is f and b
is |. We muft then confider P as an event which has 3 ways
of failing to 4 of happening : if we would have an urn from
which the credibility of drawing a white ball fhould be that of
the happening of P, we muft put in 4 white balls and 3 not
white (fay black) balls. Similarly to reprefent Q, we muft have
an urn of 3 white and 2 black balls. Now to afcertain the
profpect of drawing white from both urns, we muft count all
the cafes. A ball from the urn of 7 may be combined with one
from the urn of 5, in 7 x 5 or 35 ways. But a white ball from
the firft urn may be combined with a white ball from the fecond,
in 4x 3 or 12 different ways. There are then 35 cafes in all,
12 of which are favourable : hence the probability in favour of
white from both (which is that of the two events both happen
ing) is
H or i^ or x $- or ab.
35 7X5 75
On Probability. 187
Similar reafoning may be applied to more events than two.
This theorem has a large number of confequences, fome of
which we may notice.
When a is the probability for, I a is the probability againft.
This I (hall always denote by # : fimilarly will (land for I b;
and fo on.
Required the probability that of a number of independent
events, P,Q,R, &c one or more fhall happen. Let #,,<:, & c - be
the feveral probabilities, then that of their all failing is the pro-
duel: # />V .... and that of their not all failing (or of one or
more happening) is I a tfc* .... Accordingly, if there be
only two events, for one or both we have I (i a)(i b)
which is a + b ab. If the number of events be , and all
equally probable (fo that a = b = c, &c.) for c one or more we
have i lw or I (i a} n .
It is a confequence of this laft that, however unlikely an event
may be, it is fure (in the common fenfe of the word) to happen,
if the trial can be repeated as often as we pleafe. However
fmall a may be, or however near to unity i a, n may be taken
fo great that ( i a] n fhall be as fmall as we pleafe, or i (i a} n
as near to unity as we pleafe, or the probability that the unlikely
event will happen once or more in n times, as great as we pleafe.
Let a =!:(&+ i), which means that the odds are k to i againft
the event on any one trial : the following rough deductions will
(how what kind of refults the formula gives, true within an in-
ftance or two when k is confiderable. In ^k inftances it is an
even chance that the event happens once or more ; in 2 3/f, it is
9 to i ; in 4 6, 99 to i ; 6 9/, 999 to i ; 9*2/, 9999 to i :
and in 23^, it is ten thoufand millions to i. Thus, fuppofe at
each trial it is a hundred to one againft fuccefs. Then of thofe
who try 70 efforts, as many will fucceed once or more as will
altogether fail, in the long run. Of thofe who try 6900 times,
only one of a thoufand will always fail. A perfon who will not
examine an afTertion that comes to him with ten to one againft
it, muft count it an even chance that he throws away one or
more truths, if he follow his plan feven times.
Let us now fuppofe that there are reafons why the feveral in
ftances which can arrive are not equally credible. Suppofe the
urn to contain a white, a black, and a red ball, and ourfelves to
1 88 On Probability.
have reafons to think the balls not equally probable or credible,
but that 6, 5, and 2 are the proportions of the degrees of belief
we fhould accord to them feverally. If then 6x reprefent the
probability of a black ball, 5* and 2x will reprefent thofe of the
other two feverally. By the poftulate, 1 3* reprefents that of one
or the other. But this is certainty ; whence x muft be T y, and
T 6 _, T 5 _, and T 2 T are the probabilities of the white, black, and red
balls. That is to fay, when the feveral inftances are unequally
probable, we muft count each inftance as though it occurred a
number of times proportioned to its probability, and then proceed
as in the cafe of equally probable inftances. Thus, in the above,
inftead of faying (as we mould do if the balls were equally pro
bable) that the probability of the white ball is
I 6 6m
- we fay it is 7 - or r
i + i + i, 6 + 5 + 2; 6m
would do, m being any number or fraction whatfoever.
Now fuppofe two urns, one of all white balls, and the other
of all black ones. If we actually draw a ball, and find it white,
we know that the urn chofen to draw from muft have been the
firft : the fecond could not have given that drawing. But fup
pofe the firft urn to have 99 white balls to one black, and the
fecond one white to 1000 black. If we now draw again, and
draw a white one, not knowing from which we drew, we feel
almoft certain, from the drawing, that we have chofen the firft
urn. We ftill feel almoft certain that the fecond urn would have
given a black ball. This inverfion of circumftances, this conclu-
fion that the circumftances under which the event did happen,
are moft probably thofe which would have been moft likely to
bring about the event, is of the utmoft evidence to our minds :
but the queftion now before us is, are we to call it a fecond poftu
late, or is it deducible from the other one ? It is . fo deducible,
and is not a fecond poftulate ; but it has not been ufual to give a
very diftincl: account of the deduction.* If it could not be made,
* So well eftablifhed is this fpecies of inverfion in the mind, that both
Laplace and Poiflbn, the two moft eminent mathematical writers on the fub-
jecl, of the prefent century, have in a certain cafe affumed that an equation
which gives the moft probable value of x in terms of j, is therefore the one
which gives the moft probable value of y in terms of x. This is carrying
the principle too far.
On Probability. 189
the following procefs would, no doubt, be fufficient : it has often
been held fo. Let the urns have 6 white balls to i black, and 2
white balls to 9 black. Then the probabilities of drawing a
white ball from the two are 7- and ^, which are in the propor
tion of 33 to 7. If, becaufe when we choofe the firft urn, we
have nearly five times as much chance of a white ball as the
fecond one would give, we conclude that a known white ball
from an unknown urn is in that proportion more likely to have
come from the firft urn ; we (hall have J4 and * f r the proper
degrees of belief in the two urns. For if 33* be that for the
firft urn, then 7* muft, by the aflumption, be that for the
fecond : and for one or the other, we have 40*. But this is
certainty ; whence x muft be ^.
To reduce this refult to dependence upon the firft poftulate,
proceed as follows. The probability that two events are con-
netted, our belief, that is, in the connexion, muft be the fame
whether the two events, or either of them, have happened, or
whether they be yet to happen : unlefs there be fomething
in the happening which alters our knowledge, and puts us in
a different ftate for forming a judgment. Suppofe I make up
my mind, rightly or wrongly, as to how far I will believe that
a white ball, If drawn, will have been drawn from the firft urn.
An inftant after, I am told that the trial I anticipated has been
made, and the contingency which I fuppofed has occurred ; a
white ball has been drawn. I know no more than I took myfelf
to know in my hypothefis ; and cannot therefore have any
means of altering my opinion. Now, without altering the pro
portions in the urns, change the numbers of the balls, fo that
there may be the fame total number in each : let them be
{66 white, n black} (14 white, 63 black)
Now put each ball in an urn by itfelf, 154 urns in all. This
gives T ^- T to any one ball, if I choofe an urn at hazard. But it
was fo before : as to the firft of the two urns for inftance, \ was
the probability of choofing that urn, and T X T that of choofing one
particular ball from it : and ^-x 7 V is j^. If we then remove
all the urns with black balls, fo that a white ball muft be drawn,
the chance of its being one of the 66 is -f| or |. If without
removing the black balls, we think of the probability of a white
1 90 On Probability.
ball, if drawn, being of the 66, or of the 14, the credibilities of
thofe fuppofitions are as 66 to 14. If, having chofen an urn, we
find it contains a white ball, the fame probabilities are ftill in
that proportion.
The rules derived from fimilar reafoning, whether for judging
of the probabilities of precedents from an obferved confequent,
or for judging of the probabilities of events which reftricT: each
other, are precifely the fame, as follows. If the probability of
the obferved event, fuppofed ftill future, from the feveral poffible
precedents, feverally fuppofed actually to exift, be #,,, &c : then,
when the event is known to have happened, the probabilities
that it happened from the feveral precedents are
a for the firft, - - - for the fecond, &c.
a+b+c+. ..
Again, if there be feveral events, which are not all that could
have happened ; and if, by a new arrangement (or by additional
knowledge of old ones) we find that thefe feveral events are now
made all that can happen, without alteration of their relative cre
dibilities : their probabilities are found by the fame rule. If a, ,
r, &c. be the probabilities of the feveral events, when not reftricted
to be the only ones : then, after the reftri&ion, the probability
of the firft is a-r(a + b + ...), of the fecond, b+(a + b + ...)
and fo on.
We may obtain a very diftincl: notion of this laft theorem, as
follows. Suppofe two events, which are among thofe that can
happen, and let one, fay, be twice as probable as the other. This
means, that among all the independent, and equally likely, cafes,
there are twice as many favourable to the firft as to the fecond.
Now, fuppofe by fome alteration of fuppofitions, the introduction
of new knowledge, for inftance, it is found, all the cafes remain
ing as before, that all are prevented from happening except thefe
two events. This new ftate of things does not alter the cafes in
number : accordingly, the proportion of the probabilities of the
two events is as before, two to one. But now one of them muft
happen : or the fum of thefe probabilities muft be unity. It
follows then that one of them is f , and the other f. The fame
reafoning may be applied to more complicated cafes.
It frequently happens, when different problems are folved by
On Probability. 191
the fame formula, that they may be confidered as the fame pro
blem in two different points of view : and alfo that one and the
fame problem may be confidered as belonging to either clafs.
For inftance ; Let there be two witnefles, whofe credibilities
(or the probabilities that in any given inftance they are correct)
are a and b. As long as we do not know that they are talking
about the fame thing, the probability that both will tell truth is
ab. But the moment we know that they both aflert the fame
thing, the problem is changed : they muft now be either both
right or both wrong ; before, one might have been right and
the other wrong. To take the firft view of the problem, we
have now an obferved event, both ftate that the circumftance
did happen. There are two precedents ; the event did, or did
not, happen. If it did, the probability of the obferved event
(which is then that both are right) would be ab ; if it did not, it
would then be (ia)(i b}. Accordingly, the probability that
the obferved event did happen, will be, by the rule above, ab
divided by ab + (i a}(i b).
If we take the fecond view, we have, before the reftriction,
four poflible cafes, the probabilities of which are ah, #(1 />),
b(i a] and (i a)(i b}. After the reftri&ion, only the firft
and fourth are poflible : whence the conclufion is as juft given.
Full exemplifications of thefe methods will be found in the next
chapter.
CHAPTER X.
On probable Inference.
THERE are two fources of conviction, argument and tefti-
mony, reafon why the thing fhould be, ftatement that the
thing is. When the argument is neceffarily good, we call it
demonftratwn : when the ftatement can be abfolutely relied on,
we call it authority. Both words are ufed in lower than their
abfolute fenfes ; thus, very cogent arguments are often called
demonftration, and very good evidence, authority.
I fhall fuppofe all the arguments I fpeak of to be logically
192 On Probability.
valid ; that is, having conclufions which certainly follow from
the premifes. If then the premifes be all true, the conclufion is
certainly true. If a, />, c, &c. be the probabilities of the indepen
dent premifes, or the independent proportions from which pre
mifes are deduced, then the product abc. . . is the probability that
the argument is every way good.
Argument being offer of proof, its failure is only failure of
proof: and the conclufion may yet be true. But teftimony is an
affertion of the truth of the conclufion ; and its failure can only
be failure of truth. If a proportion of Euclid turn out to be
badly demonftrated, the enunciation need not therefore be falfe.
An argument may prove, difprove, or neither prove nor dif-
prove : a teftimony cannot be true, falfe, or neither true nor
falfe. This diftinction generally gains no more than a one-fided
admiffion : perfons begin to fee it when fome over-zealous bro
ther writes weakly on their own fide of a queftion ; but they are
very apt to think, with refpect to the other fide, that anfwering
the arguments is difproving the conclufion.
Teftimony is, for the above reafon, more eafily underftood than
argument. It is the moft effective mode of conveying know
ledge to the uneducated. But it muft not be fuppofed that, in
any ftage of reafon, argument can be the only vehicle of infor
mation, even on fubjects called argumentative. This point is
one of great importance.
When argument is demonftration, it eftablifhes its conclufion
againft all teftimony. The idea of an infallible witnefs bearing
evidence againft a demonftrated conclufion, is a contradiction.
That n confecutive numbers have a fum which is divifible by w,
whenever n is odd, is demonftrated. If a thoufand of the beft
qualified witneffes that ever lived, both for honefty and arith
metic, were to fwear that they had difcovered 101 very high
confecutive numbers, the fum of which is not divifible by 101,
any mere beginner in mathematics would be more fure that a
thoufand good witnefles had loft their wits or their characters,
than any one elfe can be of anything not admitting of demon
ftration.
But when argument does not amount to demonftration, not
only is the truth or falfehood of the conclufion matter of credi
bility, but the iffue of the argument is not that mere truth or
On Probability. 193
telfehood. It does not ftand thus : c According as this argument
is good or bad, fo is the conclufion true or falfe, but c According
as this argument is good or bad, fo is the conclufion true in this
way, or not true in this way, (that is, either falfe, or true in fome
other way). If we were to fay c men are trees, and trees have
reafon, therefore men have reafon, we have a perfectly logical
argument, falfe in the matter of both premifes : but we cannot
deny the conclufion.
Suppofe now that an argument is prefented to us of which we
are fatisfied that the like will prove their conclufions to be true
in the particular modes averted, in nine cafes out of ten. What
are we to fay of the truth or falfehood of the conclufion ? We
have T% of belief to its being true in one particular way : how
much {hall we add for other poffible ways? Are we to reft in
the conclufion as having 9 to i for it, or are we to allow more?
We cannot fay, let us confine ourfelves to the grounds we have
got, and believe or difbelieve, not in the conclufion, but in the
conclufion as obtained in that one way.
I take it for granted that the mind muft have a {rate with
refpecl: to every aflertion prefented to it, with reafon, or without
reafon. Every propofition, the terms of which convey any mean
ing, at once, when brought forward, puts the hearer into fome
degree of belief, or, if we ufe the common phrafe, of belief or
unbelief: including, of courfe, the intermediate ftate, which is as
clearly marked upon our fcale as any other. Men who are
accuftomed to fufpend their opinion, as it is called, that is, to
throw themfelves into the intermediate ftate when they have
no definite reafon to think either way, are interefted in this
queftion as much as any others. If there be fome ftate, though
not numerically appreciable, in which their belief muft be, there
is fome ftate, which they would rather know numerically than
not, in which it ought to be. In the preceding cafe, fuppofe it
known that 9 to i, or T % is granted to the conclufion from the
argument alone, and any one wifhes to fufpend his opinion as to
the remaining T V Is he to grant half of that T ^, and fay that
i 9 <?-|-^r or g is what he would wifh to make the meafure of his
belief, if he knew how ? The confideration of this queftion will
enter among others.
The manner in which he deals with the refult of the argument
1 94 On Probability.
muft depend upon teftimony^ ufmg the word in its wideft fenfe.
Firft, every man has, as juft noticed, a teftimony in his own
mind as to every proportion. He may fet out with the inter
mediate ftate : he may have no reafon to lean either way, and
may know it ; that is to fay, he may have to apply an argument
of .^ to an exifting probability of 1. Or he may have previous
good reafon, or bad reafon, which makes him lean to the affer-
tion or denial ; and the meafure of this leaning muft then be
combined with T 9 o. Or he may have other teftimony to combine
with that of his own previous ftate. Any way, he cannot have
a definite opinion on the bare truth or falfehood of the conclufion
of the argument, without appeal to the previous ftate of his own
mind at leaft, if not to that of others.
It is generally faid that we are to throw away authority, and
judge by argument alone ; that our reafon is to be convinced,
and not biafled by the opinion of others ; that no conclufions are
worth anything, except thofe which a man forms for himfelf. All
the forms in which this frequent caution is exprefled, I take to
be diftortions of the very needful warning not to allow authority
more weight than is properly due to it : a warning, by the way,
which is juft as much wanted with refpecl: to argument as to
authority. For every miftake which has been made by taking
authorities on trufl (that is, taking bad witneiTes to prove the
goodnefs of afTerted good ones), one miftake at leaft has been
made by taking arguments on preponderance: that is, treating
them as proving their conclufion, as foon as they (how it to be
more likely than its contradiction.
To form the habit of allowing authority no more weight than
is due to it, and the fame of argument, is undoubtedly one great
object of mental cultivation : but it ought not to be forgotten
that it is another and juft as great an object to form the habit of
allowing them no hfs. Suppofe an argument of value T 9 ^ is pre-
fented, and that at the fame time we have the teftimony of a
witnefs againft the conclufion, of whom we know that he leads
us right IOOO times for each once that he mifleads us. Is there
any fenfe in reducing this witnefs to one of no authority, or of an
even chance, upon the principle of depending on argument only?
Except the argument be demonftration, we muft be prepared to
admit that a witnefs may be as good as an argument, or better.
On Probability. 195
I fhall now proceed to the feveral problems which this fubjecl:
requires, confidering firft teftimony alone, next argument alone,
and then the two in combination.
Problem I. There are independent teftimonies to the truth of
an aflertion, of the value u, i/, f , &c. (one of them being the
initial teftimony of the mind itfelf which is to form the judg
ment) : required the value of the united teftimony.
Let J be I ^, &c. as in page 187. Here is a problem of
the fame clafs as in page 190 ; the reflections are, that all the
teftimonies are right, or all wrong, the independent chances of
which are ^. . . and /Ay. . . Hence the probabilities are
/Ay.
Obferve, firft, that any numbers proportional to /*, // &c. will
do as well : and if the produfts have a common denominator,
(as generally they have) the numerators only need be ufed. Se
condly, the eafieft way of exprefling the refult is by faying that it
is pv?. to /Ay. . . for, or fjf. . . to /^. . . againft.
For inftance, let it be in my mind 99 to one againft an afler
tion, that is, I bear only the teftimony ^ In favour of it. Let
four witnefles, for whofe accuracy it is 2 to i, 3 to i, 4 to i, 5
to i, depofe in favour of it : I want to know how it ought to
ftand in my mind. The teftimonies for and againft, are
JL 2 3 4 5. nH 99 * i i i
100 3 4 5* 6 l ^ ? ;> ? 6 J
Hence, neglecling the common denominator, it ought to be
1x2x3x4x5 to 99x1x1x1x1, or 120 to 99, or 40 to 33,
for the aflertion.
Obferve that in faying the witnefs gives teftimony, fay f , it is
of no confequence whether it be a queftion of judgment, or of
veracity, or of both together. I mean that, come how it may, I
am fatisfied that when he fays anything, it is 2 to i he fays what
is correcl.
An eafy rule for the more common modes of expreflion pre-
fents itfelf thus. The combined relative teftimony is the product
of the feparate relative teftimonies. Thus, two witnefles of 6
truths to one error, and of 7 truths to one error, are equivalent
196 On Probability.
to one witnefs of 42 (or 6x7) truths to one error. Three wit-
nefles of 8, 6, 5 truths to 7, 3, 1 1 errors are equivalent to one
witnefs of 80 truths to 77 errors.
A jury of twelve equally truftworthy perfons, after conferring
together, agree to an affertion on which previoufly I had no
leaning. Suppofmg me fully fatisfied that fuch agreement gives
100 to i for their refult, what am I to think of the deliberate
opinion of any one among them, that is, of his opinion after he
has had the advantage of difcuffion with others.
Let p be the value of fuch teftimony from any one ; then by
the queftion
^12 : (i_ /yt ) 12 : : 100 : i, or ^ : i ^ :: v loo : i
fay as 1-468 to i. That is, I think inconfiftently if I rely on
the united verdift as upon 100 to I, unlefs I am prepared to
think it 1468 to 1000, or about 3 to 2, for each juror alone.
Of m + n equally truftworthy jurors, a majority m are for, and
n againft, a conclufion. If> be the value of the teftimony of
each, then the odds are to be taken as being /* OT (i /*)" for, and
^"(x ^ againft. But
which are exactly as if the majority m n had been all, and
unanimous. From the original formula it will appear that two
equally good teftimonies on oppofite fides produce no effect on
the refult.
If then, the unanimity of the jury box in this country could
be confidered as that of deliberate conviction, we might fay that
a larger jury, with the condition that the majority mould exceed
the minority by 12 at leaft, would be always as good, and often
better. But there are various confiderations which prevent the
above refult from being applicable. The neceffity of being unan
imous, as our law ftands, may lower the value of the verdict. On
the other hand, a jury of 30, required to find by a majority of 12,
would generally proceed to a vote before they had put the matter
to each other with the real defire to gain opinion which the pre-
fent practice produces : confequently, the value of their verdicl:
would perhaps be lower than that of the majority only, required
to be unanimous.
On Probability. i 97
The theory thus appears to confirm the notion on which we
often aft, that a given excefs of majority over minority, is of the
fame value whatever the numbers in the two may be. And this
might be the cafe, if the thing called deliberation in a large body,
were as well adapted to the difcovery of truth as the fame thing
in a fmaller one. The reader muft remember that this teft does
not compare the one witnefs on his own judgment with a num
ber after common deliberation ; but the firft, after common deli
beration with others, is compared with the whole.
But in this, and all the problems of this chapter, the diftinftion
muft be carefully drawn between the credibility of a circumftance
at one time and at another. For example, a witnefs enters with
i o to i in his favour, and owing to combination with others, the
refult comes out that it is 100 to i he is in error in the par
ticular matter on which he gives evidence. We cannot believe
both that it is 10 to I he is right, and 100 to I that he is wrong.
What we believe is the latter, for the cafe in queftion.
As another inftance, fuppofe m independent witnefles of equal
goodnefs O) unite in affirming that a certain ball was drawn from
a lottery of n balls : collufion being fuppofed impoffible. My
knowledge of the circumftances of the affirmation here alters the
problem. If n be confiderable, it is almoft impoffible that the
witnefles, by independent falfehood or error, fhould all pitch on
the fame wrong ball. To find the bias this ought to give me to
the conclufion that they have told the truth, I muft obferve that
there being n I balls not drawn, whichever of thefe any one
choofes, by error, the chance of any one of the reft choofmg the
fame is i~(w i), the probability that all the m i {hall choofe
the fame is i-i-(_i)-i. Hence, the odds are as p m to (i /*) "
multiplied by the laft-named expreffion, or as ( n i) m " l p m to
( i -v} m . If n be very great, the odds may be enormous for the
aflertion, even though //, the credibility of each witnefs, may be
fmall. In cafes of ordinary evidence, the thing aflerted is ufually
one out of almoft an infinite number of equally poffible aflertions,
and the agreement of even two witnefles (for when m is two or
upwards, n appears in the formula) is certain conviftion, if, as
aflumed, we know the two witnefles to be not in collufion. If
^=i-i- 5 which is as much as to fay that the evidence of each
witnefs makes a ball no more likely to have been the one drawn,
198 On Probability.
becaufe he fays it, that it was on our mere knowledge that a ball
bad been drawn, it turns out I to n- I for the truth of the afler-
tion, juft as it was before the evidence. But let ^ = (1 +)-r>
a being any fraftion, however fmall, that is, let each witnefs
make the aflertion more probable than at firft, however little :
then the odds for its truth become
which odds may be made as great as we pleafe, by fufficiently m-
creafmg m. That is to fay, however little each witnefs may be
good for, in real fupport of the aflertion, or in making it more
probable than it is of itfelf, a fufficient number of witnefles, cer
tainly independent, will give it any degree of credibility what
ever.
The ftudent of this fubjeft is always ftruck by the frequency
of the problems in which the fcience confirms an ordinary notion
of common life, or is confirmed by it, according to his ftate of
mind with refpedt to the whole doctrine. It is impoffible to fay
that we have a theory made to explain common phenomena, and
hence affording no reafon for furprife that it does explain them.
The firft principles are too few and two fimple, the train of
deduction ends in conclufions far too remote. I believe hundreds
of cafes might be cited in which the refults of this theory are
found already eftablifhed by the common fenfe of mankind : in
many of them, the mathematical fciences were not powerful
enough to give the modes of calculation, when the principles of
the theory were firft digefted.
There are problems, however, in which we cannot eafily
come into pofleffion of data on which many will agree. The
fimple queftion of independent witnefles is not one of them :
but the queftion of collufion is. One of the difficulties is as
follows. We cannot inftitute independent hypothefes upon the
goodnefs of the witnefles and the probability of their having con
ferred upon their evidence. They declare, expreffly or by im
plication, that they have not done fo : if they have, there is falfe-
hood in one part of their evidence ; or, which makes the difficulty
ftill greater, there may have been general, but (as they aflert or
imply) not particular conference : they may have been biafled by
On Probability. 199
each other, without knowing how or to what extent. The firft
ftep in one view of the problem is eafily made, as follows.
Let n be the value of the evidence of each witnefs, m their
number, n the number of afTertions they have power to choofe
from, all as before. Let A be the probability that there has been
particular conference between them. There are then four cafes
to which the problem is reftricted : (i) they have conferred and
agreed to fpeak truth ; (2) they have not conferred and all fpeak
truth ; (3) they have conferred and agreed on a falfehood ; (4)
they have not conferred and have all lighted upon the fame falfe
hood. The a priori probabilities of thefe four cafes are
and the odds that they fpeak the truth (fuppofing n fo great that
we may reject the fourth cafe) are ^ m to A( I //) ". Now
comes the practical difficulty of this queftion ; How are A and
I* to be connected ? Every cafe which is worth examining fup-
pofes that the greater the chance of there having been particular
conference, the lefs is the witnefs worth from that very circum-
ftance. For it is to be remembered that we are not generally
able to give the witnefs a character wholly independent of his
evidence in the cafe before us ; in hiftorical queftions, for in-
ftance, it frequently happens that we have nothing but the wit
nefles to try* the cafe by, and nothing but the cafe to try the
witnefles by. A very common occurrence is this ; that a cafe
is one in which no one would throw any doubt upon the wit
nefles, except for fufpicion of conference, and juft as much doubt
as there is fufpicion of conference. This makes ^=1 A, and
gives ( i A) : A"^ 1 for the odds in favour of the aflertion. On this
fuppofition, it follows that whenever the chances are againft all
the witnefles having conferred particularly, their number, if great
enough, ought to give any degree of credibility to the aflertion.
* This gives rife to two great tendencies, which very nearly divide the
world among them. Some fettle the cafe in their own minds, and then try
the witnefles : fome fettle the witnefles and then try the cafe : not a few
bring their fecond refult back again to juftify their firft aflumption. When
there are two unknown quantities with only one equation, it is eafy for thofe
who will aflume either to find the other. But the difficulty is to find the
moft probable value of both.
200 On Probability.
Problem 2. Let there be any number of different aflertions, of
which one muft be true, and only one : or of which one may be
true, and not more than one : or of which any given number
may be true, but not more : required the probability of any one
poflible cafe.
The folution of all thefe varieties depends on one principle,
explained in page 190 ; requiring the previous probabilities of all
the confident cafes to be compared. As an inftance, fuppofe
four aflertions, A,B,C,D, and fuppofe /^v,^, to be the probabi
lities from teftimony, for each of them. If either of them have
feveral testimonies, their united force muft be afcertained by the
laft problem. Firft, let it be that one of them muft be true,
and one only. The probabilities in favour of A,B,C,D, are in
the proportion of ^v /jV, v/u^V, ^//Vcr*, and oyAV. Either of
thefe, divided by the fum of all, reprefents the probability of its
cafe. Secondly, let it be that one of them only can be true, and
all may be falfe. Put on the fifth quantity /A^V, for the cafe
in which all are falfe. For example, there are four diftincl:
aflertions, not more than one of which can be true. The fepa-
rate evidences for thefe four aflertions give them the probabili
ties 7, T 3 T , ^. and . There is a certain aflertion which is true if
either of the firft three be true : required the probability of that
aflertion. Here, neglecting the common denominator, which
is 7x11x8x5 in every cafe, the probabilities of the feveral
aflertions, and that of all being falfe, are as 2.8.7.1, 3.5.7.1,
1.5.8.1, 4.5.8.7, and 5.8.7.1, or as 112, 105, 40, 1120, and
280. The odds for one of the firft three cafes againft one of
the other two are 112 + 105+40 to 1120 + 280 or as 257 to
1400 ; or it is 1400 to 257 againft the truth of the aflertion.
Suppofe the condition were that two of the aflertions, but not
more, may be true, and that one muft be true. Then the pof
fible cafes, meaning by an accent that the aflertion is not true,
are AB C D 1 , BA C D 1 , CA B D , DA B C 1 , ABC D , ACB D T ,
ADB C , BCA D 1 , BDA C 1 , CDA B . Confequently, the pro
babilities of thefe cafes are in the proportion of //y ^V, v/^ pV,
p/AV, &c. And the odds in favour of, fay A, being true, are as
the fum of all the terms which contain //, to the fum of thofe
which contain /A
When we wifh to fignify that no evidence is offered either for
On Probability. 201
or againft one of the afTertions, we muft put it down as having
the teftimony . To put down o in the place of ^ would be to
make an infallible witnefs declare that it is not true. Suppofe
there are four affertions, one of which muft be true and one
only : evidence of goodnefs is offered for the firft, and none
either way for the others. Required the probability of the firft.
The probabilities of the four affertions are in the proportion of
4.1.1.1, 1.3.1.1, 1.3.1.1, and 1.3.1.1, and it is 4 to 9 for the
firft, or 9 to 4 againft it.
Problem 3. Arguments being fuppofed logically good, and the
probabilities of their proving their conclusions (that is, of all
their premifes being true) being called their validities, let there
be a conclufion for which a number of arguments are prefented,
of validities a, , r, &c. Required the probability that the con
clufion is proved.
This problem differs from thofe which precede in a material
point. Teftimonies are all true together or all falfe together :
but one of the arguments may be perfectly found, though all the
reft be prepofterous. The queftion then is, what is the chance
that one or more of the arguments proves its conclufion. That
all mail fail, the probability is #W that all mall not fail, the
probability is i *W Accordingly, if we fuppofe n equal
arguments, each of validity a, the probability that the conclufion
is proved is I ( i a) n . And, as in page 1 8 7, if the odds againft
each argument be k to i, then, the number of fuch arguments
being as much as k, the conclufion is rendered as likely as not.
But are we really to believe, having arguments againft the
validity of each of which it is 10 to i, that feven fuch arguments
make the conclufion about as likely to be true as not. If fuch
be the cafe, the theory, ufually fo accordant with common notions,
is ftrangely at variance with them. This point will require fome
further confideration.
In this problem I confider only argument, and not teftimony,
which, neverthelefs, cannot be finally excluded (fee page 194).
If the conclufion be one on which our minds are whollv un-
biaffed to begin with, it may feem that we have no efcape from
the preceding refult. And to it we muft oppofe, for confidera
tion at leaft, the common opinion of mankind that ftrong argu
ments are the prefumption of truth, weak arguments of falfehood.
2O2 On Probability.
If a controverfialift were to bring forward a hundred arguments,
and if his opponent were fo far to anfwer them as to make it ten
to one againft each, there can be no doubt that the latter would
be confidered as having fairly contradicted the former.
We muft not forget that argument, in a great many cafes, in
volves and produces the effect of teftimony, and this in an eafily
explicable and perfectly juftifmble manner. If I were to pick up
a bit of paper in the ftreets, on which an argument is written,
for a conclufion on which I have no previous opinion, and by an
unknown writer, and if I could fay that that argument left on
my mind the impreilion of ten to one againft its validity, I might
be prepared to allow it to ftand as giving T T of probability, and
upon that fuppofition to combine it with my previous opinion, | ,
as in the next problem. But fuppofe it is on a queftion of
phyfics, and Newton is the propofer of it, and that it is his only
argument, and therefore, I conclude, his beft. The cafe is now
entirely altered : poffibly the conclufion is one on which the
following argument would have great probability : If this con
clufion were true, it could be proved ; if it could be proved,
Newton could have proved it ; therefore if it were true,
Newton could have proved it : but Newton cannot prove it ;
therefore it is not true. If the cafe be fuch that the two pre-
mifes of this laft argument have each 9 to i for it, or 7 %- ; then,
though the original argument give T V for the conclufion, the mere
circumftance of Newton bringing this argument as his beft is T 8 o x o
againft it. If Newton at the fame time declare his belief in the
conclufion, we have on one fide his argument and his authority,
on the other fide the argument arifmg from his being reduced
to fuch an argument.
That fuch confiderations have weight, we know : and that
they ought to have weight, v/e may eafily fee. It is of courfe,
dependent upon the particular conclufion what weight fhall be
attached to the afTertion, if this conclufion were true it could
be proved. The courts of law conftantly act: upon this princi
ple. They confider (very juftly I think) that evidence, however
good it may be, is much lowered by not being the beft evidence
that could be brought forward. If a man be alive, and capable
of being produced with fufficient eafe, they will not take any
number of good witnefles to the fact of his having been very
On Probability. 203
recently alive. In enumerating the arguments, then, for or
againft a proportion, thofe muft be included, if any, which arife
out of the nature, mode of production, or producers, of any
among them. And until this has been properly done, we are
not in a condition to apply the methods of the prefent chapter.
Problem 4. A conclufion and its contradiction being produced,
one or the other of which muft be true^ and arguments being
produced on both fides, required the probability that the conclu
fion is proved, difproved (/. e. the contradiction proved), or left
neither proved nor difproved.
Collect all the arguments for the conclufion, as in the laft
problem, and let a be the probability that one or more of them
prove the conclufion. Similarly, let b be the probability that one
or more of the oppofite arguments prove the contradiction. Both
thefe cafes cannot be true, though both may be falfe. The pro
babilities of the different cafes are thus derived. Either the
conclufion is proved, and the contradiction not proved, or the
conclufion not proved and the contradiction proved, or both are
left unproved. The probabilities for thefe cafes are as a(i b},
b(ia] and (i a}(i ), and the probability that the conclufion
is proved is a(ib] divided by the fum of the three, and fo on.
The fraction (ia}(ib] divided by this fum may be called the
incondufivenefs of the combined arguments. The manner in
which this incondufivenefs is to be diftributed between the hypo-
thefis of the truth and falfehood of the conclufion muft depend
upon teftirnony, in the complete fenfe of the word.
The predominance of one fide or the other, as far as argu
ments only are concerned, depends on which is the greateft,
a(ib) or b(ia\ or fimply on which is the greateft, a or b.
If the arguments on both fides be very ftrong, or a and b both very
near to unity, then, though a(i b) and b(ia) are both fmall,
yet (i #)(! b) is very fmall compared with either. The ratio
ofa(i-b) to b(ia] on which the degree of predominance de
pends, may, confiftently with this fuppofition, be anything what
ever. But we cannot pretend that, when oppofite fides are thus
both nearly demonftrated, the mind can take cognizance of the
predominance which depends upon the ratio of the fmall and
imperceptible defects from abfolute certainty. The neceflary
confequence is, that the arguments are evenly balanced, and are
204 On Probability.
as if they were equal : there is no fenfible notion of predominance.
This is the ftate to which moft well conducte d oppofitions of
argument bring a good many of their followers. They are fairly
outwitted by both fides, and unable to anfwer either, and the
conclufion to which they come is determined by their own pre
vious impreffions, and by the authorities to which they attach
moft weight ; and thefe are, of courfe, thofe which favour their
own previoufly adopted fide of the queftion.
When no argument is produced on one fide of the queftion,
the cafe is very different from the cafe of the preceding problems,
in which no teftimony is produced. Here the queftion is, * Has
the conclufion been proved or not proved - 3 and when no argu
ment is produced, we are certain it has not been proved. Ac
cordingly, if no argument were urged for the contradiction, we
mould have I =i, or b = o.
If, in the preceding problem, the two fides of the queftion be
not contradictions, but fubcontradi&ions, of which neither need
be true, but both cannot be, the problem is folved in the fame
way, for the cafes are juft the fame. But we may introduce a
diftinclion which the former cafe would not admit. When one
muft be true, every argument againft one is of equal force for
the other ; which is not the cafe when neither need be true.
Let there, then, be arguments for the firft conclufion and againft
it, and let a and p be the probabilities that one or more of the
arguments for, prove it, or againft, difprove it. Let b and q be
the fimilar probabilities for the fecond conclufion. Then, there
are thefe cafes : I. The arguments (or fome of them) for the
firft are valid, againft it invalid, and thofe for the fecond are
invalid (it matters nothing whether thofe againft the fecond be
valid or invalid). 2. The arguments for the firft are invalid,
thofe for the fecond valid, and againft it invalid. 3. The argu
ments againft the firft are valid, and thofe for it invalid. 4. The
arguments againft the fecond are valid, and thofe for it invalid.
5. All the arguments are invalid. Accordingly, the probabilities
that the firft is proved, that it is difproved, that the fecond is
proved, that it is difproved, and that neither of the two is proved
nor difproved, are in the proportion of a(ip}(ib}^ ( ! #)/>,
*(i-?)(i-*)> ( -%> and (i- Xi-JJO-JX 1 -?)-
Problem 5. Given both teftimony and argument to both fides
On Probability. 205
of a con tradition, one fide of which muft be true, required the
probability of the truth of each fide.
This is the moft important of our cafes, as reprefcnting all
ordinary controverfy. Collect all the teftimonies, and let their
united force for the firft fide be ^ and, from the nature of this
cafe, i (A for the other fide. Let a and b be the probabilities
that the firft fide and the fecond fide are proved by one or more
of the arguments in their favour. Now, obferve that, for the
truth of either fide, it is not efTential that the argument for it
fhould be valid, but only that the argument againft it fhould be
invalid. Accordingly, the probabilities of the two fides are in
the proportion of ^(i b] and (i ^(i #), and the probabili
ties of the two fides are reprefented by
Firft, let there be no teftimony either way : we muft then have
A*=-a-=i At; confequently, thefe probabilities are as I b to
I a. Let no argument have been offered for the fecond fide,
or let b = o. Then we have I to Ia, for the odds, or
i-7-(2 a) for the probability of the firft fide being true. It has
been ufual to fay that if an argument be prefented of which the
probability is *, the truth of the conclufion has alfo the probabi
lity a. Probably the above was the cafe intended as to teftimony,
&c., and the probability fhould then have been
or
2 a
which is always greater than a. Or, as we might expeft, the
poffibility of the conclufion being true, though the argument
fhould be invalid, always adds fomething to the probability of its
being true. Moreover, 1-^(2 0) is always greater than : or
any argument, however weak, adds fomething to the force of
the previous probability. The fame thing is true in every cafe.
Suppofe a new argument to be produced for the firft fide, of the
force k. The effeft upon the formula is to change Ia into
(i_rf)(i_^ anc l t he odds in favour of the conclufion are in-
creafed in the proportion of i to ik. But this is to be under-
206 On Probability.
flood ftri&ly in the fenfe defcribed in page 202, namely, we are
to fuppofe that the newly produced argument is Tingle, that is,
does not by the circumftances of its production caufe itfelf to be
accompanied by an argument for the fecond fide, or againft the
firft. If this laft fhould happen, and the argument thus created
for the fecond fide have the force /, the odds are altered in the
proportion of I /to i k.
From the above it appears that oppofite arguments of the
force a and b are exactly equivalent to a teftimony the odds for
the truth of which are as I b to I #. Thus, fuppofe we have
for a conclufion witnefTes whofe teftimonies are worth , *, ,
T 9 ^; arguments for of the feveral forces, |, ii, i ; and arguments
againft of the forces f, T 2 T , . Writing numerators only, we put
down
For, 2, 2, 4, 9 ; 7, 9, I :
Againft, I, I, 3, I ; 4, I, 3.
Hence it is, 2. 2. 4. 9. 7. 9. i to i. i. 3. i. 4. I. 3,
252 to i for the conclufion.
or
An argument, we fhould infer beforehand, is better than a
teftimony of the fame force ; for the failure of the argument is
nothing againft the conclufion, but the failure of the teftimony is
its overthrow. So fays the formula alfo : the introduction of a
teftimony of the value /, not before received, alters the exifting
odds in the proportion of k to I k : but the introduction of an
argument of the fame force alters them in the greater proportion
of i to i k. Thus, the introduction of the teftimony of a
perfon who is as often wrong as right (-} alters the odds in the
proportion of i to I, or does not alter them at all : but the intro
duction of an argument which is as likely as not to prove the
conclufion, alters them in the proportion of I to I -, or of
2 to i.
Are we not in the habit, unconfcioufly, of recognizing fome
fuch diftin&ion ? Do we not give much more weight to argu
ment than to teftimony ? I fufpect the anfwer fhould be in the
affirmative : that an argument of 3 to i does convince us much
more than a teftimony of 3 to i. I fufpecl: we fhow it, not in
numerical appreciation, of courfe, but in liftening to and allow-
On Probability. 207
ing weight to arguments, when we fhould refufe teftimony of
the fame character.
It may be doubted, however, whether we have much fcope
for experiment on the lower degrees either of teftimony or argu
ment. Perhaps it is not often we meet a witnefs, whether as
bearing teftimony of veracity to a fact, or of judgment to a con-
clufion, whofe evidence is as low as 4 ; and the fame perhaps of
an argument.
I have fpoken, in the previous part of this chapter, of the
rejection of authority, that is, of teftimony, authority being only
high teftimony. Let us now examine by the formula and fee
what it amounts to. Let a be the probability that the argument
proves its conclufion : and let us therefore perfift in faying that
a is the probability for the truth of the conclufion. In the for
mula, b being=o, let ^ be made a+(i -f *), it will be found that
the probability for the conclufion, //. divided by /n + (i / a)(i tf),
comes out a, as required. Confequently, in the cafe of a fingle
argument, the total rejection, as it would be thought, of all tefti
mony, is really equivalent to accompanying every argument by a
teftimony lefs than , depending upon its own force. It is to
declare that, by the laws of thought, an argument of T 7 o- is of its
own nature accompanied by a witnefs of vV, one of f. by a wit
nefs of f , and fo on ; this is clearly not what was meant. Nor,
I fuppofe, can it be meant that we are arbitrarily to ftart with the
teftimony , and to reduce our own evidence, and that of all
others, to the fame. If there be any fenfe in which the rejection
of authority is defenfible, it muft be when we are required to
proceed as if we were in perfect ignorance what the value of the
authority is. We cannot fuppofe it to be as likely to have one value
as another. Suppofe, for inftance, that the arguments have un
known propofers : we cannot treat their authorities as if they were
juft as likely to be exceffively high or low as to be very near
to none at all. 7 he more rational fuppofition is that the autho
rity fhould be more likely to be fmall than great, as likely to be
againft as for, and very unlikely to be exceffively great either for
or againft. I cannot here enter into the mode in which fuch an
hypothefis can be exprefTed or ufed : but the refult of the fimpleft
formula which fatisfies the above conditions, is as follows :
Let r=( i />)"H *~ *)> ^ an d a meaning as above; then the
208 On Probability.
probability that the conclufion is true, which has a for the
validity of its argument, &c. is
r(r 3 6r 2 + 3r + 6rlogr + 2)-f-(r i) 4
where logr means the Napertan logarithm (99-43^5 of the
common logarithm will be near enough for the prefent purpofe).
If, for inftance, r=2, which, on the fuppofition of no previous
balance of teftimony, would give 2 to i for the conclufion, the
formula juft written gives -636, or 636 to 364, fomething lefs
than 2 to i.
In the cafe firft difcuiTed in page 202, it may be thought that
the weaknefs of a propofed argument, from one who fhould have
brought a better, if there had been one, may be confidered as a
teftimony againft the conclufion rather than an argument. Sup-
pofe his argument, for inftance, to have only the probability T V-
He tells us then, that after he has done his beft, it is 9 to i
againft the propofition being proved. If we are very confident
that it could be proved, if true, and that he could do it, if any
one, he comes before us as a teftimony of 9 to i againft the
truth of the conclufion, or very nearly fo. If we take, then, all
that his argument wants of demonftration, as fo much evidence
from him againft the conclufion, this amounts to fuppofmg that,
a being the validity of his argument, a is alfo his teftimony for
the conclufion (and I a that againft it). If there be only argu
ment for, and none againft, and if our minds be previoufly unbi-
afled, we reprefent this cafe by putting a for //, in the formula,
and the odds for the conclufion are then as a to (i a}*. On
this fuppofition, which I incline to think well worthy of attention,
we mould not confider an unoppofed argument from an acute
reafoner as giving the conclufion to be as likely as not, unlefs
a = (i cTf- or # = 382, a little more than |. Were it not for
our peculiar introduction of teftimony, then, the conclufion being
as likely as not to begin with, an argument which has any pro
bability of proving it, would have made it more likely than not,
as before feen.
But that the introduced teftimony fhould be exactly as above,
is a mere fuppofition. If it were a mathematical propofition,
for inftance, and Euler were to declare himfelf unable to give
more than a probability of proof, I, for one, fhould confider him
On Probability. 209
as giving a much higher rate of teftimony againft the truth of
the affertion than is fuppofed in the preceding. But all this has
reference to the queftion how to meafure teftimonies and va
lidities in particular cafes, which is quite a diftincl: thing from
the inveftigation of the way to ufe them when meafured.
In cafes in which the number of arguments is multiplied, it
generally happens that they ftand or fall together, in parcels :
namely, that the fame failure which makes one invalid, neceiTarily
makes others invalid. In this cafe, independent arguments muft
be felec~red, and the probabilities for them alone employed.
We fee in this problem an illuftration of the commonly ob-
(erved refult, that the fame argument produces very different
final conclufions in two different minds ; and this when, fo far as
can be judged, both are difpofed to give the fame probabilities to
the feveral premifes of the argument. The initial odds, come
how they may, or p to I ^, fhould be altered by the arguments
in the proportion of I b to I a. Accordingly, b and a being
the fame to both parties, their belief in the conclufion may have
any kind of difference, if/* be not the fame thing to both.
Problem 6. Given an affertion, A, which has the probability
a ; what does that probability become, when it is made known
that there is the probability m that B is a neceffary confequence
of A, B having the probability b ? And what does the probabi
lity of B then become ?
Firft, let A and B not be inconfiftent. The cafes are now as
follows, with refpecl: to A. Either A is true, and it is not true
that both the connexion exifts and B is falfe : or A is falfe. This
is much too concife a ftatement for the beginner, except when it
is fuppofed left to him to verify it by collecting all the cafes. The
odds for the truth of A, either as above or by the collection,
are a{i m(i b)} to ia. As to B, either B is true, or B is
falfe and it is not true that A and the connexion are both true.
Accordingly, the odds for B are as b to (ib)(ima).
The reader muft remember that when B neceffarily follows
from A, B muft be true when A is true, but may be true when
A is falfe ; while A muft be falfe when B is falfe. And now we
fee that a proportion is not neceffarily unlikely, becaufe it is very
likely to lead to an incredibility, or even to an absolute impoffi-
bility. Let = o, or let B be impoffible : then the odds for A
2i o On Probability.
are as a(im] to I a. Say that it is 9 to I that the connec
tion exifts ; then thefe odds are as a to 10(1 a). If a be
greater than -J-, ftill A remains more likely than not, even when
it is 9 to I that it leads to the abfurdity B.
Secondly, let A and B be inconfiftent, fo that both cannot be
true. Either then A is true, B falfe, and the connexion does
not exift ; or A is falfe. The odds for A are then as a(ib)
(im) to i a. With refpect to B, either B is true and A is
falfe, or B is falfe, and A and the connexion are not both true.
The odds for B are then as b(ia] to (ib)(ima).
Among the early fophifms with which the Greeks tried the
power of logic, as a formal mode of detecting fallacies, was the
conftruction of what we may cz\\ fuiddal proportions, aflertions
the truth of which would be their own falfehood. If a man
fhould fay c I lie, he fpeaks neither truth nor falfehood ; for if
he fay true, he lies, and if he lie, he fpeaks truth. Such a fpeech
cannot be interpreted. Again, the Cretan, Epimenides, faid that
all the Cretans were incredible liars ; is he to be believed or not ?
If we believe him, we muft, he being a Cretan, difbelieve him.
Some ftated it thus ; c If we believe him, then the Cretans are
liars, and we fhould not believe him ; then there is no evidence
againft the Cretans, or we may believe him, fo that the evidence
againft the Cretans revives, &c. &c. &c. Refer fuch a propo-
fition to the theory of probabilities, and the difficulty immediately
difappears. Whatever the credit of Epimenides as a witnefs
may be, that is, whatever, upon his word, the odds may be for his
propofition, the fame odds are there againft him from the propo-
fition itfelf. Thefe equal conflicting teftimonies balance one
another (problem i) and leave the effect of other teftimonies to
the fame point unaltered. The fophifm of Epimenides, as ftated,
is but an extreme cafe of the fecond of the problems before us.
The propofition B is inconfiftent with A, and the connexion is
certain (m=i): the odds for B muft then be as b(ia) to
(!_)(! #) 5 or as b to I , exactly what they are independ
ently of the previous aflertion.
On Induftion. 21 1
CHAPTER XL
On Induction.
THE theory of what is now called induttlon muft occupy a
large fpace in every work which profefles to treat of the
matter of arguments ; but there is not much to fay upon the gen
uine meaning of the word, in any fyftem of formal logic. And that
little would be lefs, if it were not for the miftaken oppofition
which it has long been cuftomary to confider as exifting between
the inductive procefs and the reft of our fubject.
By induction (hrayurn) is meant the inference of a univerfal
proportion by the feparate inference of all the particulars of
which it is compofed : whether thefe particulars defcend fo low
as fingle inftances or not. Thus if X be a name which includes
P>2>R> fo that every thing which is X muft be one of the
three : then if it be mown feparately that every P is Y, and that
every Q is Y, and that every R is Y ; it follows that every X
is Y. And this laft is faid to be proved by induftion. Thus
(Chapter VI).
X)P,Q,R + P) Y + Q) Y + R)Y=X) Y
is an inductive procefs. In form, it may be reduced as in page
123, to one ordinary fyllogifm.
Complete induction is demonftration, and ftrictly fyllogiftic in
its character. In the preceding procefs we have y)p, y)q, y)r,
which give y)pqr : and X)P,Q,R is pqr)x ; whence y)x, or X) Y.
It is a queftion of names, that is, it depends upon the exiftence
or nonexiftence of names, whether a complete induction mall
preferve that form, or lofe it in the appearance of a Barbara fyllo
gifm, formed by help of the conjunctive poftulate of Chapter VI.
But when the number of fpecies or inftances contained under
a name X is above enumeration, and it is therefore practically
impoflible to collect and examine all the cafes, the final induc
tion, that is, the ftatement of a univerfal from its particulars,
becomes impoflible, except as a probable ftatement: unlefs it
mould happen that we can detect fome law connecting the fpe
cies or inftances, by which the refult, when obtained as to a
certain number, may be inferred as to the reft.
212 On Induction.
This laft named kind of induftion by connexion^ is common
enough in mathematics, but can hardly occur in any other kind
of knowledge. In an innumerable feries of proportions, repre-
fented by P 15 P 2 ,P 3 ,P 4 , &c, it may and does happen that means
will exift of fhowing that when any confecutive number, fuppofe
three, of them are true, the next muft be true. When this
happens, a formal induction may be made, as foon as the three
firft are eftablifhed. For by the law of connexion, Pi,P 2 , and P 3 ,
eftablim P 4 ; but P 2 ,P 3 , and P 4 , eftablim P 5 ; and then P 3 ,P 4 ,
and P 5 , eftablim P 6 ; and fo on ad infinitum. It is to be obferved
that this is really induftion : there is no way, in this procefs, of
compelling an opponent to admit the truth of P 100 without
forcing him, if he decline to admit it otherwife, through all the
previous cafes.
As an eafy inftance, obferve the proof that the fquare of any
number is equal to the fum of as many confecutive odd numbers,
beginning with unity, as there are units in that number : as feen
in
Take any number, n ; and write n ns (reprefenting a unit by
a dot) in rank and file. To enlarge this figure into ( n + I ) ( n + I )s,
we muft place n more dots at each of two adjacent fides, and
one more at the corner. So that the fquare of n is turned into
the fquare of + 1 by adding ^n + I, which is the (n + i)th odd
number. Thus loox 100 is turned into lOix 101 by adding
the lOift odd number, or 201. If then the theorem alleged be
true of n x , it is therefore true of (n + i ) x (n + i). But it is
true of the firft number, I x i being i ; therefore it is true of the
fecond, or 2x2=1+3; therefore it is true of the third, or
3x3 = 1+3 + 5; and fo on.
But when we can neither examine every cafe, nor frame a
method of connecting one cafe with another, no abfolutely de-
monftrative induction can exift. That which is ufually called by
the name is the declaration of a univerfal truth from the enumer
ation of fome particulars, being the aflumption that the unex-
amined particulars will agree with thofe which have been ex
amined, in every point in which thofe which have been examined
agree with one another. The refult thus obtained is one of
On Induction. 213
probability ; and though a moral certainty, or an unimpeachably
high degree of probability, can eafily be obtained, and actually is
obtained, and though moft of our conclufions with refpect to the
external world are really thus obtained, yet it is an error to put
the refultof fuch an induction in the fame clafs with that of a de-
monftration. There is no objection whatever to any one faying
that the former refults are to his mind more certain than thofe of
the latter : the fact may be that they are fo. The difference
between neceffary and contingent propofitions lies in the quali
ties from which they receive thofe adjectives, more than in
difference of credibility. I know that a ftone will fall to the
ground, when let go : and I know that a fquare number muft be
equal to the fum of the odd numbers, as above : and though,
when I flop to think, I do become fenfible of more affurance for
the fecond than for the firft, yet it is only on reflection that I
can diftinguim the certainty from that which is fo near to it.
The rule of probability of a pure induttlon is eafily given.
Suppofmg the fimple queftion to be whether X is or is not Y,
there being no previous circumftances whatfoever to make us
think that any one X is more likely than not to be Y, or lefs
likely than not. Thefe are the circumftances of what I call a
pure induction. To begin with, it is I to I that the firft X ex
amined mall be a Y : if this be done, and Xj be a Y, then it is
2 to i that X 2 mail be a Y ; mould it fo happen, then it is 3 to
I that X 3 mall be a Y. Generally, when the firft m Xs have
all been examined, and all turn out to be Ys, it is m + I to I
that the (m + i)th X mail be a Y.
The fimplicity of this rule muft not lead the ftudent to fuppofe
he can find a fimple reafon for it. Let 10 Xs have been exam
ined and found to be Ys : what do we affert when we fay it is
I 1 to i that the I ith X mail be a Y ? We affert that if an in
finite number of urns were collected, each having white balls and
black balls in infinite number but in a definite ratio, and fo that
every poffible ratio of white balls to black ones occurs once ; and
if every poffible way of drawing eleven balls, the firft ten of
which are white, were felected and put afide : then, of thofe put
afide, there are eleven in which the eleventh ball is white, for
one in which the eleventh ball is black. The reader will find
fome difficulty in forming a diftinct conception of this, and of
214 n Induction.
courfe will find it impoflible to have any axiomatic perception of
the truth or falfehood of the refult.
It may be worth while to {how that a fuppofition making fome
degree of approach to the preceding circumftances will give fome
approach to the refult. Firft, in lieu of an infinite number of
balls in each box, which is fuppofed only that withdrawal of a
definite number may not alter the ratio, let each ball drawn be
put back again, which will anfwer the fame purpofe. Let there
be only ten urns with ten balls in each, of which let the firft
have one white, the fecond two white, &c. and the laft all white.
The number of ways of drawing eleven white balls fucceilively
out of any one urn is the eleventh power of the number of white
balls in the urn : that of drawing ten white balls followed by one
black one is the tenth power of the number of white balls mul
tiplied by the number of black ones. If we were to put together
all the firft, and then all the fecond, we fhould find about 21
times as many ways of arriving at the firft refult (ten white, fol
lowed by a white) as the fecond (ten white followed by a black).
But if we now increafed the number of urns, and took a hundred,
having one, two, Sec. white balls, we fhould find inftead of 21,
a number much nearer to 1 1 ; and fo on.
Accordingly, when without any previouily formed bias, we
find that m Xs, fucceffively examined, are each of them a Y, we
ought then to believe it to be m + I to I that the next, or
(m-\- i)th X, will be a Y. And further, a being a fraction lefs
than unity, we have a right to fay there is the probability I ^"H- 1
that the Xs make up the fraction a or more, of the Ys. Or
thus ; if the fraction a be, fay , and if m be 10 : then if the
10 firft Xs be all Ys, the probability that or more of the Xs
are Ys is juft that of drawing one or more black balls in n
drawings, from an urn in which of the balls are always white.
If, for example, the firft 100 Xs were all Ys, it would be
found to be 1000 to I that 93^ per cent, at leaft, of all the Xs
are Ys.
If as before, the firft m Xs obferved have all been Ys, and we
afk what probability thence, and thence only, arifes that the next
n Xs examined fhall all be Ys, the anfwer is that the odds in fa
vour of it are m-}- 1 to w, and againft it n to m+ i. No induc
tion then, however extenfive, can by itfelf, afford much probability
On Induction. 2 1 5
to a univerfal conclufion, if the number of in fiances to be exam
ined be very great compared with thofe which have been exam
ined. If 100 inftances have been examined, and 1000 remain, it
is 1000 to 101 againft all the thoufand being as the hundred.
This refult is at variance with all our notions ; and yet it is
demonftrably as rational as any other refult of the theory. The
truth is, that our notions are not wholly formed on what I have
called the pure induction. In this it is fuppofed that we know no
reafon to judge, except the mere mode of occurrence of the in
duced inftances. Accordingly, the probabilities fhown by the
above rules are merely minima, which may be augmented by
other fources of knowledge. For inftance, the ftrong belief,
founded upon the moft extenfive previous induction, that pheno
mena are regulated by uniform laws, makes the firft inftance of a
new cafe, by itfelf, furnifh as ftrong a prefumption as many in
ftances would do, independently of fuch belief and reafon for it.
With this however I have nothing farther to do, except to
obferve that, in the language of many, induction is ufed in a fenfe
very different from its original and logical one. It is made to
mean, not the collection of a univerfal from particulars, but the
mode of arrival at a common caufe for varied, but fimilar, phe
nomena. A great part of what is thus called induction confifts
in difcovery of differences, not refemblances. Under this confufed
ufe of language, the ufual theory is introduced, namely, that
Ariftotle was oppofed to all induction, that Bacon was oppofed
to every thing elfe, that the whole world up to the time of Bacon
followed Ariftotle, that the former was the firft who fhowed the
way to oppofe the latter, that each had a logic of his own, &c.
&c. The whole of this account abounds with miftatements.
The admitted, and fufficiently ftriking difference between the
philofophy of modern and ancient times, in all natural and mate
rial branches of inquiry, is not fo eafily explained as by choofmg
two men, one to bear all the blame, the other all the credit : nor
are Copernicus, Gilbert, Tycho Brahe, Galileo, and the other
predeceffbrs of the Novum Organum, deftined to be always de
prived of their proper rank.
What is now called induction, meaning the difcovery of laws
from inftances, and higher laws from lower ones, is beyond the
province of formal logic. Its inftruments are induction properly
2 1 6 On Induction.
fo called, feparation of apparently related, but really diftin& par
ticulars (the neglect of which was far more hurtful to the old
philofophy than a neglect of indu&ion proper would have been,
even had it exifted) mathematical deduction, ordinary logic, &c.
&c. &c. It is the ufe of the whole box of tools : and it would
be as abfurd to attempt it here, as to append a chapter on car
pentry to a defcription of the mode of cutting the teeth of a faw.
The procefTes of Ariftotle and of Bacon are equally thofe which
we are in the habit of performing every day of our lives. But
fome perform them well, and fome ill. It is extraordinary that
there mould be fuch divifion of opinion on the queftion whether
a careful analyfis of them, and ftudy of the parts into which they
decompofe, is of any ufe towards performing them well. On
this point, and on the character of Bacon s office in philofophy,
a living writer, to whom I mould think it likely that many yet
unborn would owe their firft notions of Bacon s writings, ex-
preiTes himfelf in a manner which I quote, and comment on at
length, as the beft expofition I can find, of a clafs of opinions
which is very prevalent, and, I fully believe, to the prejudice of
fober thought and accurate knowledge.
The vulgar notion about Bacon we take to be this, that he invented a
new method of arriving at truth, which method is called Induction, and that
he detefted Tome fallacy in the fyllogiftic reafoning which had been in vogue
before his time. This notion is about as well founded as that of the people
who, in the middle ages, imagined that Virgil was a great conjuror. Many
who are far too well informed to talk fuch extravagant nonfenfe, entertain
what we think incorrect notions as to what Bacon really effefted in this
matter.
The induftive method has been pra<SHfed ever fmce the beginning of the
world, by every human being. It is conftantly pra&ifed by the moil igno
rant clown, by the moft thoughtlefs fchoolboy, by the very child at the
breaft. That method leads the clown to the conclufion that if he fows
barley, he mall not reap wheat. By that method a fchoolboy learns that a
cloudy day is the beft for catching trout. The very infant, we imagine, is
led by inclusion to expeft milk from his mother or nurfe, and none from
his father.
Not only is it not true that Bacon invented the indu6Hve method ; but
it is not true that he was the firft perfon who correftly analyfed that method
and explained its ufes. Ariftotle had long before pointed out the abfurdity
of fuppofmg that fyllogiftic reafoning could ever condu61 men to the difco-
very of any new principle, had fhown that fuch difcoveries muft be made by
On Induction. 217
inclusion, and by Inclusion alone, and had given the hiftory of the induaive
procefs, concifely indeed, but with great perfpicuity and precifion.
Again, we are not inclined to afcribe much praftical value to that analy-
fis of the induaive method which Bacon has given in the fecond book of
the Novum Organum. It is indeed an elaborate and correft analyfis. But
it is an analyfis of that which we are all doing from morning to night, and
which we continue to do even in our dreams. A plain man finds his fto-
mach out of order. He never heard Lord Bacon s name. But he proceeds
in the ftriaeft conformity with the rules laid down in the fecond book of
the Nojvum Organum, and fatisfies himfelf that minced pies have done the
mifchief. " I eat minced pies on Monday and Wednefday, and I was kept
awake by indigeftion all night." This is the comparentia ad intellettum in-
Jlantiarum convenientium. I did not eat any on Tuefday and Friday, and
I was quite well." This is the comparentia injlantiarum in proximo qu<?
natura dataprivantur. " I ate very fparingly of them on Sunday, and was
very nightly indifpofed in the evening. But on Chriftmas-day I almoft
dined on them, and was fo ill that I was in great danger." This is the
comparentia injlantiarum fecundum magis et minus. " It cannot have been
the brandy which I took with them ; for I have drunk brandy daily for
years without being the worfe for it." This is the reje&io naturarum. Our
invalid then proceeds to what is termed by Bacon the Vindemiatio, and pro
nounces that minced pies do not agree with him.
We repeat that we difpute neither the ingenuity nor the accuracy of the
theory contained in the fecond book of the Novum Organum $ but we think
that Bacon greatly overrated its utility. We conceive that the induaive
procefs, like many other procefles, is not likely to be better performed
merely becaufe men know how they perform it. William Tell would not
have been one whit more likely to cleave the apple if he had known that
his arrow would defcribe a parabola under the influence of the attraftion of
the earth. Captain Barclay would not have been more likely to walk a
thoufand miles in a thoufand hours, if he had known the place and name of
every mufcle in his legs. Monfieur Jourdain probably did not pronounce
D and F more correftly after he had been apprifed that D is pronounced
by touching the teeth with the end of the tongue, and F by putting the
upper teeth on the lower lip. We cannot perceive that the ftudy of g ram-
mar makes the fmalleft difference in the fpeech of people who have always
lived in good fociety. Not one Londoner in ten thoufand can lay down
the proper rules for the ufe of will and flail. Yet not one Londoner in a
million ever mifplaces his w/// anclyM. Dr. Robertfon could, undoubtedly,
have written a luminous differtation on the ufe of thefe words. Yet, even in
his lateft work, he fometimes mifplaced them ludicroufly. No man ufes
figures of fpeech with more propriety becaufe he knows that one figure of
fpeech is called a metonymy, and another a fynecdoche. A drayman in a
paflion calls out You are a pretty fellow/ without fufpefting that he is
uttering irony, and that irony is one of the four primary tropes. The old
fyftems of rhetoric were never regarded by the moft experienced and dif-
cernmg judges as of any ufe for the purpofe of forming an orator. " E<r O
2i 8 On Induction.
hanc vim intelligo" faid Cicero " effe in praeceptis omnibus, non uteafecuti
oratores eloquentiae laudem Tint adepti, fed quae fua fponte homines eloquen-
tes facerent, ea quofdam obfervafle, atque id egiffe } fie efle non eloquentiam
ex artificio, fed artificium ex eloquentia natum." We muft own that we
entertain the fame opinion concerning the ftudy of Logic, which Cicero
entertained concerning the ftudy of Rhetoric. A man of fenfe fyllogizes in
celarent and cefare all day long without fufpefting it : and though he may
not know what an ignoratio elenchl is, has no difficulty in expofmg it when
ever he falls in with h.( Lord Bacon; in Critical and Hiftorical Efays
contributed to the Edinburgh Review. By Thomas Babington Macaulay.)
This brilliant paflage has, I have no doubt, appeared to many
completely decifive of the queilion which it affirms : and, as fo
often happens in like cafes, there is a certain exaggeration againft
which it is of truth. It is good againft thofe who confound
analyfis and recombination of exifting materials with introduction
of them : and who might profefs to fee in agriculture fomething
which would have benefited mankind, though plants and animals
had not been natural produces of the foil. But I now proceed
to examine it, againft thofe who affirm that Ariftotle and Bacon
are of no ufe, and who very frequently fall into the common
logical fallacy of fuppofmg that their cafe is proved, as foon as it
is made out that they are not of all the ufe : which Mr. Macau-
lay himfelf has done, except as againft the exaggerators aforefaid.
We reafon inductively from morning till night, and even in
our dreams. True : and how badly we often do it, particularly
in fleep. A plain man is then produced, to reafon on Bacon s
principles : and Mr. Macaulay has imitated a plain man better
than he intended, by making him do it wrongly. Look over
the indu&ion, and it will appear that the cafe is not made out ;
an exclufion is wanting : it may have been the mixture of minced
pies and brandy which did the mifchief. The plain man fhould
have tried minced pies without brandy ; but he had drunk the
latter daily for years, and it never ftruck him. This is precifely
one of the points in which we are moft apt to deceive ourfelves,
and for which we moft need to have recourfe to the complete-
nefs of a fyftem of rules ; fomething is left taken for granted.
The things of courfe, our daily habits, are neglected in the
confideration of anything of a lefs ufual character : the plain man
left oft the minced pies upon trial ; but not the brandy : Chrift-
On Induction. 219
mas mifchief muft be referred, he thinks, entirely to Chriflmas
fare, if at all.
But even if this omiffion had been fupplied, and the refult found
to confirm the conclufion, yet the plain man has flopped where
the plain man frequently does flop, at what Bacon calls the
Vindemiatio prima, the rudiments of interpretation. Complete-
nefs is feldom anything but fludy and fyflem. Philofophy ought
to bring him to the refult that daily brandy has made that fpirit
ceafe to give the flimulus which, were its ufe only occafional,
would enable his flomach to bear an unufually rich diet for a
fhort time. Our plain friend is precifely in the pofition of a
bankrupt who curfes the times, on reafoning flrictly Baconian as
far as it goes, and forgets that a cafual tightnefs in the money
market would never have upfet him, if it had not been for the
previous years of extravagant living and ram fpeculation.
But there are many procefTes which are not better performed
becaufe men know " how they perform them." Mr. Macaulay
here means " becaufe men know the laws of that part of the
procefs which nature does for them." That men mould not
know better how to perform for knowing how they perform
is almofl a contradiction in terms. William Tell knew how to
moot all the better for knowing which end of the arrow he was
accuflomed to fit to the firing : had he wanted this knowledge,
his chance of cleaving the apple would have been much dimi-
nifhed. But he would not have been improved by knowing
that his arrow defcribed a parabola. True, becaufe it did not do
fo. The centre of gravity of the arrow would defcribe a para
bola, if it were not for the refiflance of the air ; or fomething fo
near it as to be undiflinguifhable. But, taking the defcription
as roughly correct, William Tell did know, inductively, that the
arrow defcribes a curve, concave to the earth : and had made
thoufands of experiments in connexion of the two ends of that
curve, which were all that he was concerned with. It is no ar
gument againfl the fludy, as a fludy, of mduftion^ that the amount
of ufeful refult which it had recorded in the mind of William
Tell in the fhape of habit, would not have been augmented by
deduttlve knowledge of an intermediate flatus with which he had
nothing to do. But let knowledge advance, under both modes
of progrcfs, and Tell becomes an artillery officer, the rude arrow
22O On Induction.
a truly fhaped and balanced ball, means of meafurement are ap
plied, the true curve is more correctly reprefented than by the
parabola, and thirty pounds of iron are thrown to four times the
diftance which an arrow ever reached, and with a certainty al-
mofr. equal to that of the legend.
But if Captain Barclay had known the places and names of
the mufcles, he would not have been more likely to walk a thou-
fand miles in a thoufand hours. The inftance is far fetched :
becaufe the feat confided in the exhibition of power of endurance
acquired by practice. If my denial feem as far fetched, it is the
fault of the propofer. Captain Barclay muft, by habit, by in
duction, have acquired facility in varying his pace and gefture
fo as to eafe the mufcles. Had he been well acquainted with the
difpojition and ufes of thefe organs to begin with (towards which
knowledge of their places and names would have contributed) he
would have learnt this art more eafily. Though not altogether
ad elenchum^ yet I may fay that in this cafe the effect of fuch
knowledge would have been that he would have been lefs likely
to have performed the feat. Had he directed his attention to
fome fcience of obfervation, he would not have needed to have
fought fame, or exhauftion of remarkable energy, in fuch a tri
fling purfuit. And further, in a very common cafe, mechanics
has taught what few ever learn by induction, though they have
conftant opportunities of doing it : namely, that in walking, the
ordinary practice of fwinging the arms is injurious and tiring
that a very trifling amount of it tells ferioufly in a long journey.
Here is one ufeful refult, which natural induction does not com
monly teach, and there may be many more of the fame kind :
the queftion between it and regular ftudy requires the confidera-
tion, not only of what is done, and whether it might be done
better, but of what is not done.
Next, M. Jourdain did not pronounce D and F more cor
rectly after his attention had been called to the details of the act
of pronunciation. None but Moliere ever knew whether he did
or not : but all who have watched the progrefs of inflruction
know that the bad habits or natural imperfections of children are
removed or alleviated by making them practice mechanical pro
nunciation, with perceptive adoption of rules. In every one of
a few detached initances in which I have feen children at their
On In duff ion. 221
reading lefTons in France, I have noticed that a return upon the
habits of pronunciation is always a part of the exercife: and that
the letters are pronounced with that diftincl: effort which makes
the pupil fenfible of the action required. I have always attri
buted to this practice the more uniform flandard of pronunciation
which prevails among the educated French, as compared with
ourfelves.
But the f^idy of grammar makes no difference in the fpeech
of people who have always lived in good fociety. If Mr. Macau-
lay mean merely as to the ufe of yfttf // and will, and the like, it
may certainly be faid that the perpetual ufe of fpeech (which is
not reafoning) does enable every one to form the habits of thofe
about him. But that grammar, as a whole, produces no effect
upon the fpeech of good fociety, is one fide of a balanced matter
of opinion. Many contend that it has produced, in our gene
ration and the one above it, a very unfortunate effecl: : they aver
that the purity and character of our Englifh has been deteriorated
by Lindley Murray and his fchool, and that we much want better
grammar teaching. On the fubject of Jhall and will, it is re
markable that Mr. Macaulay, whom a vigorous faculty of illuf-
tration, combined with immenfe reading, enables to ftrew his
path with inftances, has to invent his cafe, and to refer to a
treatife which Robertfon could have written. But it is not
enough : if we grant that fuch a treatife would have been lumi
nous, we may be fafe ; but would it have been corrett ? And
further, knowledge muft abdicate at once, if we pronounce ufe-
lefs all that has been clearly explained by thofe who have not
rightly practifed. Bacon himfelf might have taken exfors ipfa
fecandl for his motto.
Next, it is faid that no man ufes figures of fpeech more cor
rectly becaufe he knows that one is metonymy and another fynec-
docbe. True ; and in like manner no man confults his books
more eafily becaufe he has a bookcafe. But, having the book-
cafe, he arranges his books in it, and then he knows where to
find them. Mr. Macaulay dwells throughout upon nomencla
ture. I might infift upon its fuperftructure : but even mere
naming is ufeful, when the meaning of the name is clearly under-
ftood. A mind well flocked with underftood names cannot
keep itfelf from being conftantly in the act of claflification,
222 On Induction.
which contains induction. The mere involuntary reference of
inftance number two to inftance number one, which is made
when we remember that the fecond muft have the fame name as
the firft, is comparifon and induction, leads to reflection, culti
vates tafte, and gives power. The drayman, who calls out in a
paffion, " You are a pretty fellow !" without knowing that he
is uttering irony, is an incomplete picture : there is omitted a
wifh relative to the eyes of his opponent, and an adjective which
is (in fuch quarrels) fometimes prophetically, but feldom defcrip-
tively, true. The value of the difference between this favage
irony and the more elegant form of it which is fo pleafmg in the
defcription of the plain man s induction quoted above, is not
within the comprehenfion of the drayman : the foundation of a
better mode of expreilion than undifciplined rhetoric furnimes, fo
far as its adoption is matter of tafte, was laid by thofe who placed
irony among the primary tropes. Good tafte is a refult of com-
parifons, which could not have been made without nomenclature.
Did Cicero declare that fyftems of rhetoric are not of any ufe ?
The very quotation appears to mean that thefe fyftems, prcecepta,
have their power ; that men get them by obfervation, and put
them into practice. The ea fecuti oratores refers to what was
done in the firft inftance, by the firft eloquent men,fud fponte.
Moft truly does he fay that the art of rhetoric is derived from
eloquence, and not vice verfa : moft falfely, as far as can be
judged, does he feem to infmuate that it was all done at one
ftep ; firft, fome one or more confummate orators, fecondly, a
finifhed fyftem, drawn from obfervation of their methods. Per
haps he intended a particular reference to a certain orator then
namelefs : the fentence, thus conftrued, contains nothing but
matter which Tully is likely enough to have whifpered to
Cicero.
A fyftem is a tool, and it muft be employed upon materials
which different men furnifh from their different means. But
the coat muft be cut according to the cloth, both in fize and
quality : no reproach to the fciffors, nor prejudice to their fupe-
riority over the fharpened wood of the favage, even though prac
tice will enable him to ufe the latter better than any civilized
man who is not a tailor can ufe the former. The formation of
tools, mental or material, is a cyclical procefs. The firft iron
On Inclusion. 223
was obtained by help of wood ; one of the firft ufes of it was to
make better tools, to get more iron, with which better tools ftill
were made, and fo on. And in this way we may trace back any
art to natural tools, and to materials which are to be had for the
gathering. The aflertion made by Mr. Macaulay, and many
others, that in logic only, of all the abftract fciences, our natural
means are as good as thofe which refult from diligent analyfis, is
one which terminates in an iflue of fact. The inftances given
are contained in the aflertion that a man offenfe fyllogizes in cefare
and celarent all day long without fufpefting it, and though he
does not know what an ignoratio elencbi is, can always detect
it when he meets with it.
Mr. Macaulay begins with an indefinite term, a man offenfe :
and the claufe is deficient in logical perfpicuity. Firft, what is a
man offenfe ? I grant that I mould doubt the fenfe of a man who
could not make the inferences defcribed by cefare and celarent.
But do men become men of fenfe by nature, without education ?
if yes, I deny the aflertion that men of fenfe reafon (correctly) in
cefare, &c. The man of fenfe who is not educated is as likely
to aflert that cefaro is all that can be obtained, or to invent the
form fefape, as the plain man to forget to try the mince pies
without brandy before he concludes. If no, then the aflertion
is itfelf ignoratio elencbi : for the very queftion is how to make
men of fenfe ; can they not be, ceteris paribus, formed better and
fafter with ftudy of logic than without : it being agreed on all
hands that this man of fenfe is always a practical logician.
Next, a man of fenfe reafons, &c. without fufpecting it.
Sufpecting what ? that he is reafoning, or that he is reafoning in
cefare ? I fuppofe the latter : that is to fay, I take it to be meant
that a man of fenfe may (not muft, for fome Ariftotelians are men
of fenfe) not know that the logicians call the form of reafoning
he ufes cefare. This is eafily granted : but what is it but the
celebrated Ignoratio elenchl of Locke, who fancied that he raifed
an objection againft the pretenfions of the logicians, when he
declared he never could believe that God had made men only
two-legged, and left it to Ariftotle to make them rational. No
one ever denied that men reafoned before Ariftotle, and would
have reafoned ftill if he had never lived.
Mr, Macaulay, probably without fo much as a new application
224 n Induction.
to the inkftand, after falling into the ignoratio elenchi, fmgles out
this very fallacy as the one which a man of fenfe is fure to detect.
But if there be a fallacy which is the ftaple of paralogifm, it is
this one. Dele ft at domi, for ordinary difcuffion (efpecially after
dinner) is little elfe ; tmpedit forts, for three fourths of public
debate, from the Houfes of Parliament downwards, is made up
of it. A man who expofes it in converfation is confidered a
tirefome, and if he do it often, an uncourteous perfon : he " has
no converfation," he " harps upon one fubjea," he " won t let
you fpeak."
I have made the above comments upon a very marked paflage
of an eminent writer, in preference to introducing their fub-
ftance as a diflertation of my own, that I might have the advan
tage of the reader feeing that I meet real arguments, inftead of
my own verfion or fele&ion. It would probably be difficult to
find a better concentration of the fubftance of the antagonift
views, with refpect to the formal ftudy of reafoning, than is
contained in my quotation from Mr. Macaulay : and I may fafely
take his adoption of them as proof that thefe views yet require
the notice of a writer on logic.
There is one refult of the theory of probabilities, clofely con
nected with induction proper, which it will be advifable to notice
here.
When the fyllogifm is declared illegitimate, on account of
both premifes being particular, a probable conclufion of great
ftrength may be admitted in many cafes. This muft be the
more^infifted on, becaufe it is too common to attend to nothing
but the demonftrative fyllogifm, leaving all of which the con-
clufions are only probable, however probable, entirely out of
view.
I take as the inftance the fyllogifm, or imperfect fyllogifm,
c Some Xs are Ys, fome Zs are Ys, therefore there is fome pro
bability that fome Xs are Zs. If the number of Xs and Zs
together exceed the number of Ys (as in Chapter VIII) there is
a certainty that fome Xs are Zs. Let us then fuppofe this is
not the cafe.
Let the whole number of Ys in exiftence be u, and let m and
On Induction. 225
n be the numbers of Xs and Zs which are among them. I
(hall confider two diftincl: cafes : Firft, when the diftribution
of the Xs and Zs among the Ys is utterly unknown ; fecondly,
when their diftribution is that of contiguity, that is, when the Ys
being for fome reafon arranged in a particular order, the Xs
which are Ys are fucceffive Ys, and the fame of the Zs which
are Ys.
For the firft cafe a very rough notion will do, confined to the
fuppofition that few Xs and Zs are mentioned, compared with
the whole number of Ys. When the Xs and Zs together make
a large proportion of the Ys in number, then, if we have no
reafon for making them contiguous, or otherwife limiting the
equally probable arrangements, it may be faid to be a moral cer
tainty that fome Xs are Zs.
In the firft cafe, if we divide 43 times the product of m and n
by TOO times u, it gives us a fufficient notion (not large enough)
of the common logarithm of /, the odds in favour of fome
Xs being Zs being k to I. Say there are rooo Ys, and that
100 Xs are Ys and 100 Zs are Ys. Then 43 x loox 100
divided by loox iooo is 4-3, which is the logarithm of 20,000-
It is then more than 20,000 to I that, in this cafe, one or
more Xs are Zs. A more exact rule is as follows. To ^mn
divided by 100*7 add its hundredth part, and to the refult add
fuch a fraction of itfelf as m + n is of 2j. Thus 43^2^-7- 100^
being 4-3, which, with its hundredth part is 4*343, and m + n
(200) being the tenth part of 2 (or 2000), we add to 4*343 its
tenth part, giving 4*777, which is about the logarithm of 60,000,
ftill under the mark. It is more than 60,000 to I that fome Xs
are Zs. When the fractions are very fmall, this rule is accurate
enough, if be confiderable. Its refult is, that if u be very con-
fiderable, and if a perceptible fraction of the Ys be Xs, and a
perceptible fraction Zs, and if we really have no reafon to make
the limitation of contiguity or the like, then we are juftified in
treating it as a moral certainty that fome Xs are Zs. But I fuf-
pect the relation of contiguity, to which I now proceed, better
reprefents the actual ftate of the cafe in ordinary argument.
When the Xs which are Ys are contiguous, and alfo the Zs
which are Ys, the probability that no Xs are Zs is the fraction
having the product of y m n -f i and u m n + 2 for nu-
226 On Induction.
merator, and the product of u m+ i and Y\ -f i for denomi
nator. Thus in the example above propofed, 1000 Ys containing
among them 100 Xs and 100 Zs (each fet contiguous) we have
80 1 x 802 for numerator and 901 X 901 for denominator. This
fraction is about 8-tenths ; fo that it is now 8 to 2, or 4 to i,
againft any Xs being Zs.
In order to find the probability againft the number of Xs
which are Zs exceeding /, add k to both the multipliers in the
numerator, which then become v m n + k+i and >j m n
+ / + 2. For example, there are 100 Ys, containing 30 Xs and
60 Zs (each fet contiguoufly) : what is the chance againft the
number of Xs which are Zs exceeding 10? The numerator is
2iX22 : the denominator is 71x41. This fraction is 462 by
291 1 ; whence it is 462 to 2449 againft, or 2449 to 462 (more
than 5 to i] for, the number of Xs which are Zs exceeding 10.
The chances, it is to be remembered, are all minima : ex
cept when we mean that m Xs, and not more, are Ys, &c. Thefe
queftions may ferve to give fome notion of the manner in which
arguments not logically conclufive, may be morally fo.
What is called circumftantial evidence is a fpecies of induction
by probability. The thing required to be found has the marks
P,O,R,S, &c. : this Y has the marks P,Q,R,S, &c. : there is then
a certain amount of circumftantial evidence that this Y is the
thing we want to find. If it can be fhown that there is but one
thing which has all thefe marks, then the circumftantial evidence
is demonftrative. But if there were, fay 100 Ys, of which 5
have the mark P, 5 the mark Q, &c., then having afcertained
one Y which has all the marks, the queftion is, what chance is
there againft another Y having them all : the fame chance, at
leaft, is there that the Y found is the one fought. Inftead how
ever, of attempting the problem in this way, which is never
reforted to for want of data (I mean that the refemblance which
the rough procefles of our minds bear to thofe of the theory of
probabilities does not here exift) I take it as follows. If the
pofleflion of the mark P give a certain probability to the Y
found being that fought, it is as a witnefs whofe teftimony has a
certain credibility. Similarly for Q,R,S, &c. Compound thefe
teftimonies, when known, by the rule in page 195, and the refult
is the value of the circumftantial evidence.
227
CHAPTER XII.
On old Logical Terms.
IN this chapter I propofe to fay fomething on a few terms of
the old Logic, which though they keep their places in works
on the fubject, and have fome of them parted into common lan
guage, are very little ufed. They relate generally to the fimple
notion, and the name by which it is exprefTed : and have little of
fpecial reference, either to the propofition or fyllogifm. They
are moftly derived from Ariftotle, whofe incidental expreffions
became or give rife to technical terms, and whofe fingle fentences
were amplified into chapters. And here, as in other places, I
have nothing to do with the degree of correctnefs with which
Ariftotle s meaning was apprehended, nor even with how much
was drawn from Ariftotle and how much added to him, but only
with the actual phrafes and their ufual meaning.
The words logic and dialectics* are now ufually taken as
meaning the fame thing : the old diftinction is that dialectics is
the part of logic in which common and probable, but not necef-
fary, principles, are ufed. But the diftinction is neither clearly
laid down, nor faithfully adhered to, even by Ariftotle himfelf.
The term (in this work always called name] was divided into
fimple and complex : the fimple term was the mere name, the
complex term was what all moderns call the affirmative propofi
tion. Thus man and run were fimple terms : man runs^ a com
plex term. Later writers rejected this confufion : and divided
the acts of the mind confidered in logic into apprehenfion, judg
ment^ and difcourfe^ taking cognizance of notions, propofitions,
and arguments. The common meaning of the word difcourfe,
* Our language is capricious with regard to the ufe of angular and plural
of words in ic : thus we have logic and dialectic/, arithmetic and mathema
tics, phyfic and phyficj for medicine and natural philosophy. Some modern
writers are beginning to adhere uniformly to the angular, in which I cannot
follow them, for I am afraid an Englifh ear would not bear with mat/ie-
mat ic as a fubftantive. Would it not better confift with the genius of our
language if the plurals were to be always ufed, and the fingulars made ad-
jeilives without the termination /?
228 On old Logical Therms.
(which now generally applies to fomething fpoken) is derived
from its place in this divifion. The word argument, which is
now equivalent to reafoning againft opposition expreffed or implied,
was originally nothing but the middle term of a fyllogifm.
The fimple term was univerfal or fingular : univerfal, when
of more inflances than one, as man, horfe, ftar ; fingular, when
of one inftance only, as the fun, the firft man, the pole-ftar, this
book. Singular names were called individuals, from the etymo
logy of the word, as belonging to objects not divifible into
inftances to each of which the name could be applied. I have
not dwelt upon the diftinction between fingular and univerfal,
becaufe it is ineffective in inference. And moreover, a fingular
propofition is only objectively fingular, but ideally plural. Julius
Caefar was a Roman * : in point of fact, there was but one Caefar.
But take any imaginary repetition of the circumftances of Caefar s
life ; fuch, for inftance as occurs to thofe who have thought of
the poflibility of the fame courfe of events returning into exift-
ence after a certain cycle : and then the term Caefar becomes
plural. Or, even without fo forced a fuppofition, we may fay
that, if we defcribe Caefar, we muft defcribe a Roman : that our
definition of Caefar is fo clofe as to fit only one man that ever
lived, makes no effential difference in the character of the pro
pofition.
But a further diftinction which was made divided fingular
terms into fubjects of univerfal, and fubjects of particular, propo-
fitions. A determinate (or definite) individual, as Caefar, this
man, was the former : a vague (or indefinite) individual, as a
certain man, the firft comer, was the latter. The diftinction is
that of c fome man and c this one man.
Certain notions of effence or relation, accompanying the ap-
prehenfion of a name, were called categories, or predicaments,
meaning c modes of affertion with refpedt to the object named.
Ariftotle gave ten categories, and might have given ten hundred.
In their ufual Latin form they were fubflantia, quantitas, quali-
tas, relatio, aElio, pajjio, ubi, quando, fetus, habitus.
The word tranflated by fubftance, xo-ia, means mode of being:
and its literal Latin is ejfentia, effence. It is called fubftance (that
which ftands under) as fupporting accidents, prefently explained.
It is far too metaphyfical a term to come into common life with-
On old Logical Terms. 229
out fome degradation : and accordingly it there means that of
which a thing is compofed, whether material or not. Accordingly
we have the material fubftance of a coat, the intellectual fub
ftance of an argument. But, as we ufe the word, its meaning
belongs to the other predicaments. In fact, the fubftance of the
old logicians ftands, as to exiftence, in the fame fituation as mat
ter (page 30) with refpect to our fenfible perceptions, or objeft
with refpecl: to our ideas. The fubftance, it was faid, is per fe
fulfijlens, while the accident could not be faid ejje, but inejfe.
The diftin&ion between the fubftance (mode of being) and the
material fubftance (in the modern fenfe) may be helped by the
diftinction between fubftantia prlma and fubjlantla fccunda, the
firft referring to the individual, the fecond to the general term.
Thus the fubftance of]ohn,asjohn,wasfub/}antiaprima; as
man, fubftantia fecunda. All thefe very metaphyfical notions
were the ftudent s firft introduction to logic, and were confidered
as of the utmoft importance.
The predicament of quantity, derived from the notion of whole
and part, was conceived as either continuous or difcrete. In con
tinuous quantity, the unit was divifible, in difcrete, indivifible.
Thus ten feet is continuous, ten men difcrete. The diftinction
is precifely that of magnitudinal and numerical.
Duality was fubdivided into I. Habit and difpofition, the latter
term being ufed for the imperfect ftate of the former 2. Power
and want of it 3. Patibilis qualitas and pajfio, applied to the
ideas of that which is undergone, the firft permanently, the fecond
for a time. 4. Form and figure.
Relation then, as now, referred to the fuggeftions derived from
comparifon of two things or ideas. It was divided into verbal
and real (fecundum did andfecundum ejfe]. Thus the relation of
profit to profitable was verbal : that of father to fon, or of above
to below, real. The two things related, or correlatives, were called
frbjefi and term : fo that of two correlatives, giving two oppofite
relations, the fubjecl: of either was the term of the other. The
fundamentum of the relation was that in which it took its rife,
when it had a beginning.
Attion and pajfion, the production and reception of an effect,
requiring the producing agent, and the receiving patient, were
divided into immanent, or enduring in the agent, and tranfient,
230 On old Logical Terms.
or pafling out to another. Actions were univocal^ or
according as their effects were of the fame or different fpecies.
A few years before the publication of Newton s Principia, it was
taught in a work imported into Cambridge that when mice bred
mice, the action was univocal, but when the sun bred mice (the
writer muft have been thinking of Ariftotle and fome of the
fchoolmen) aequivocal. There was alfo the terminus a quo and the
terminus ad quern to reprefent the (late before and the ftate after
the action. Thus, when all this nonfenfe was fent to Coventry,
the terminus a quo was an immenfe quantity of univocally bred
learning of the preceding kind ; the terminus ad quern was the
rooting up of the wheat of logic with the tares.
The where (as to abfolute pofition), the when^ and the fite
(relative pofition) gave no peculiar terms of fubdivifion. The
habitus (s%eiv) referring to poffejjion generally in the firft inftance,
was materialized by fome of the old logicians till it related to
drefs only, or habit in the thence acquired meaning.
The word predicament (and category as well) has been intro-
cuced into common language to fignify a fet of circumftances
under which any thing takes place. It is then no longer con
fined to the above predicaments, nor is there any occafion that it
mould be.
The predicables (xotTwopxpeva) are diflinguifhed from predica
ments (xaryyopiai) in that the former belong to any fimple notion
or name, and may be predicated of it : the latter belong to the
connexion (when affirmative) between two names. They are
laid to be five in number, genus, fpecies^ differentia^ proprium^
and accidens.
The words genus and fpecies have preferved their old meaning.
If there be a number of names of which each is fubidentical of
the one which follows, fay V, W, X, Y, Z : then of any two, fay
W and X, X is a genus containing the fpecies W. Here Z is
the fummum genus^ and V the infima fpecies : X is the genus
proximum of W, Y the genus remotum. In what I have called a
univerfe^ which is a fummum genus^ having for its infima fpecies
the individual inftance of any name in it, the fuperidentical is the
genus, the fubidentical the fpecies. Subcontraries (and contraries)
are oppofite fpecies ; fupercontraries and complex particulars have
no ancient name.
On old Logical Terms. 23 1
The differentia is that by which one clafs (be it fpecies or
genus, the difference being accordingly termed fpecific or generic)
is diftinguifhed from another. Thus the difference (or one differ
ence) feparating the fpecies man from the other fpecies of the
genus animal, is the epithet rational.
The proprium (or property) is that which belongs to the fpecies
only, whether it be to all or only to fome : thus to ftudy, and to
fpeak, are equally proprla of man. But the old commentators
give definitions of the property as follows. There are four
kinds. I. That which belongs to the fpecies alone, but not to
all. 2. To all the fpecies, but not to that alone. 3. To the
fpecies only, and to all of it, but not at all times. 4. To the
fpecies alone, to all, and always.
The accident (or accident) is that which may fometimes be
long to the individual of a fpecies, but not neceffarily, nor to that
fpecies alone. In modern language, the term is limited to what
is unufual and unexpected.
The word caufe was ufed by the ancients in a wider fenfe
than by us : more nearly in the fenfe of the Latin caufa, or the
Italian cofa. Caufes were diftinguifhed into material, formal,
efficient, and final. The material caufe was the very matter of
a thing, confidered as a kind of giver of exiftence ; the formal
caufe was its form, in the fame light ; the efficient caufe (our
common Englifh word) the agent or precedent ; and the final
caufe, the ultimate end or objet, confidered as a reafon for the
exiftence of the thing. Sometimes writers ftill talk of final
caufes, and are as unintelligible to moft readers as if they had
talked of final beginnings.
The word form was ufed in a wider fenfe than that of figure
or fhape, to mean, as it were, law of exiftence, mode, difpofition,
arrangement. Mere figure or fhape was only one of the acci
dental forms, as diftinguifhed from fubjlantlal forms, belonging to
the fubftance. And motion was as widely ufed as form : it meant
any alteration. Thus, corruption was one of the motions of mat-
ter. Change from place to place, to which the modern word is
confined, was local motion.
The original ufe of the terms fubjeft and objeft is to denote a
thing confidered as that which may have fomething inherent in
it, or attached to it, or fpoken of it, &c. ; and as that which may
232 On old Logical Terms.
be objetted to the mind or reafon, or made to come in its way.
Thus it was faid that matter is the fubjeft of thofe properties
which are the objefts of the mind in natural philofophy. The
tranfition to the modern fenfe of objeff, namely, end propofed, is
natural enough. In modern times, fubjeft and object are ufed*
with refpet to knowledge : the fubjecl: being the mind in which
it is, the object being the external fource from which it comes.
For [ubjeflive and objeSfive I have in this work ufed ideal and ob-
jeftive (page 29). Adjunct was the technical term for that
which is in the fubjecl:.
A modal proportion was one in which the affirmation or nega
tion was exprefled as more or lefs probable : including all that is
technically under probability (Chapter IX) from neceffity to
impoffibility. The theory of probabilities I take to be the un
known God which the fchoolmen ignorantly worfhipped when
they fo dealt with this fpecies of enunciation, that it was faid to be
beyond human determination whether they moft tortured the
modals, or the modals them. Their gradations were neceffary^
contingent^ pojffible^ impojjible ; contingent meaning more likely
than not, poffible lefs likely than not. Thefe they connected
with the four modes of enunciation, A, I, O, E, and when by
fame is meant more than half, the connexion is good. The con-
troverfy about modal forms continues up to this day among
logicians who are not mathematicians : I mould fuppofe that the
latter would never give it a thought, except as a branch of the
theory of probabilities, and except as to the confideration how
the terms by which the non-mathematical logician indicates his
degrees of belief are to be placed upon the numerical fcale. In
like manner he reads the thermometer by graduation, and though
he admits the freezing and boiling point, which have an origin
in nature, he leaves temperate, fummer heat, blood heat, &c. to
the fancy of thofe who choofe to employ them.
At the fame time it is clear that thefe modal forms were con-
fidered not merely as ufeful in expreflion of the nature and amount
of belief, but as fuggeftive of real branches of inquiry, fubfervient
to that great a priori inquiry into the nature of things to which
* See a full account of thefe words in Sir William Hamilton s notes to
Reid, p. 806, &c.
On old Logical Terms. 233
mediaeval logic was applied. We are not fit to judge of the in-
ftrumental part of this philofophy, unlefs we confider alfo the
materials on which it was founded. In an age in which much
more faith was demanded of the ftudent than now ; when he was
much more frequently required to decide in one way or the
other upon a fmgle teftimony j when, in addition to the non-
mythic wonders recorded in ancient writers, which there was no
mode of contradicting, all that was known of immenfe regions
and countries refted upon very few accounts, and thofe filled
with ftories quite as ftrange : the abfence of other means of
diftinguifhing truth from falfehood obliged thofe who thought to
lay much ftrefs upon a priori confiderations. It matters little to
us whether we infer the necejfity of man being a walking animal
from the non-arrival of exceptions, and thence the univerfality
of the rule, or the univerfality from the fuppofed perfect induction
of inftances, and thence the neceflity. But it was of much more
confequence to the old logician : of more real confequence. He
did not know but that any day of the week might bring from
Cathay or Tartary an account of men who ran on four wheels
of flefh and blood, or grew planted in the ground like Polydorus
in the ./Eneid, as well evidenced as a great many nearly as mar
vellous ftories. As he could not pretend to inductive and demon-
ftrative univerfality, even upon the queftion of the form of his
own race, he was obliged to combine with his argument the an
tecedent teftimony of his own and other minds, in the manner
which the real doctrine of modals (page 205) ftiows to be necef-
fary in all non-demonftrated conclufions. It is true that he fre
quently confounded the predifpofition of minds with the confti-
tution of objects ; the teftimony with the thing teftified about.
We fhall never have true knowledge of the fchools of the
middle ages, until thofe who have ftudied both their philofophy,
their phyfics, and their ftate of tradition, will look at their
weapons of controverfy as both ofFenfive and defenfive, and give
a fair account of the amount of protection afforded by the firft,
in the exifting ftate of the fecond and third. It would alfo be
advifable to confider whether, looking at the power of communi
cation by land and fea, and all the circumftances of literary inter-
courfe, it would have been practicable to place the knowledge of
the earth and its details upon any better footing of evidence.
234 On old Logical Terms.
One leading feature of the fchoolmen, acute as they were, and
as to reprefentation of notions, inventive, and which is fhared by
many more modern writers who have not difciplined themfelves
mathematically, is feen in their employment of quantity : there
are inftances of the ftrange ufe, the wrong ufe, and the no-ufe.
Moft of them arife from indiftincl: apprehenfion of continuity,
which obliges them to accept fuch ftages of quantity as are ex-
prefled by exifling terms, without any effort to fill up gaps.
There is alfo a flovenlinefs of definition in what relates to quan
tity. Thus dozens of inftances might be given in which the
fame of the particular propofition is fo defined that we might
fuppofe it is c fome, not all, inftead of fome, it may be all, and
the former is the exprefs definition of fome writers : and it is only
when we find in rules that X Y does not allow us to infer X : Y,
nor to contradict X)Y, that we afcertain the real intended
meaning. " Logicians," fays Sir William Hamilton, " have
referred the quantifying predefignations plurimi^ and the like, to
the moft oppofite heads ; fome making them univerfal, fome
particular, and fome between both." They muft have had curi
ous ideas of quantity who made the propofition c moft Xs are
Ys either univerfal, or between the univerfal and particular : I
fhould fuppofe that thofe who did the latter muft have imagined
fome to refer to a minority.
There is a ftrange notion of quantity revived in modern times,
which confifts in making plurality of attributes a part of the quan
tity of a notion. It is called its intenfive quantity, or its intenfion, or
comprebenfion. It is oppofed to extenfive quantity, or extenfion,
which is the more common notion of quantity, referring to the
number of fpecies or of individuals (it may be either, the individual
is the real inf ma fpecies) contained under the name. Thus man is
not fo extenfive as animal, but more intenfive ; the attribute ratio-
nal gives greater comprehenfion. But man refiding in Europe is
lefs extenfive and more comprehenfive than either. It is faid that
the greater the intenfive quantity the lefs the extenfive, but this is
not true, unlefs no two of the figns of intenfion be properties of
the fame fpecies. Thus, according to fuch ftatements as I have
feen, c man, refiding in Europe, drawing breath north of the
equator, feeing the fun rife after thofe in America, would be a
more intenfively quantified notion than man refiding in Europe ;
On old Logical Terms. 235
but certainly not more extenfive^ for the third and fourth elements
of the notion muft belong to thofe men to whom the firft and fe-
cond belong. Thus, in the Port-Royal Logic, one of the earlieft
modern works (according to Sir W. Hamilton), in which the dif-
tin&ion is drawn, it is faid that the comprehenfion of the idea of a
triangle includes fpace, figure, three fides, three angles, and the
equality of the angles to two right angles. But the idea of recti
linear three-fided figure has juft as much extenfion.
The relation between comprehenfion and extenfion exifts, and
is ufeful : but not, I think, as that of different kinds of quantity.
In page 148, where I hold that the propofition is contained in its
neceflary confequence, the view is one of extenfion : the ordinary
view is one of comprehenfion. c Every cafe in which P is true,
is a cafe in which Q is true, tells us that all the P-cafes are con
tained, as to extent (number and location of inftances), among
the Q-cafes. But, as to comprehenfion, every P-cafe contains
all that diftinguifhes a O-cafe from other things. When, in
page 47, it is faid that the idea of man is contained in that of
animal, I fpeak of extenfion : all the inftances to which the firft
idea applies are among thofe to which the fecond applies. But,
as to comprehenfion, the idea of animal is contained in that of
man : all that defines animal goes to the definition of man, and
other things befides. In page 50, the " is of pofTeffion of all
efTential characteriftics," refers to comprehenfion ; the " is of
identity" to extenfion : both pofleffing equally the characters
under which the verb may occur in logic. There is no diftinction
which affecls inference : for X)Y has exactly the fame proper
ties whether we interpret it as expreffing that Y has all the ex
tenfion of X, and may be more j or that X has all that Y has in
comprehenfion, and may be more.
In pages 115, &c. we have the mode of reprefenting names
of more or lefs comprehenfion. Thus, P, Q, R, &c. being cha-
ra&eriftics, the obvious propofition PQ)P, illuftrates the theorem
that where the comprehenfion of one name has all that of a fe
cond (as PQ_has that of P) the extent of the fecond is at leaft
as great as that of the firft. And the felf-evident poftulate in page
115, by which we may diminifh the extent of a term univerfally
ufed, or increafe that of one particularly ufed, may be exprefled in
language of comprehenfion. That is, we may augment the com-
236 On old Logical Terms.
prehenfion of a univerfal, or diminifh that of a particular. Thus,
X)Y gives XP)Y, and X.Y gives XP.Y : but X)YP gives
X)Y.
It will be eafily feen that comprehenfion has the firft attribute
of quantity (page 174) : there is more and lefs about it. But it is
not of the meafurable kind (page 175). As to extent, 200 in-
flances bear a definite ratio to 100, which we can ufe, becaufe
our inftances are homogeneous. But different qualities or defcrip-
tions can never be numerically fummed as attributes, to any pur-
pofe arifing out of their number. Does the idea of rational ani-
mal, two defcriptive terms, fuggeft any ufeful idea of duplication^
when compared with that of animal alone. When we fay that
a chair and a table are more furniture than a chair, which is true,
we never can cumulate them to any purpofe, except by abftracT:-
ing fome homogeneous idea, as of bulk, price, weight, &c. To
give equal quantitative weight to attributes, as attributes, feems
to me abfurd : to ufe them numerically otherwife, is at prefent
impoffible.
The reader will have feen the origin of feveral very common
terms, which are ufed in a fenfe coinciding with, or at leaft much
refembling, that put upon them by the fchoolmen. But there is
one which has diametrically changed its meaning ; it is the word
Inflance. The word inftantla (and alfo eWrau^) implied a cafe
againft, not for ; the latter was exemplum : fo that inftance to the
contrary would have been tautology.
I have referred the word enthymeme to this chapter, though it
is always regularly explained in connexion with the fyllogifm.
According to Arifrotle, Ev^/^/xa scrn cruhhoyio-fAoi; drex^ If ELXOTCUV
tea} cr^E/wv, an enthymeme is an imperfect fyllogifm from probables
and figns : the modern critics reject the word arc?^, imperfeft^ as
interpolated. The word fegn feems to mean indication, fymptom,
or effect, which makes the caufe almoft necefTary or highly
probable. But the fchools took the word enthymeme to mean a
fyllogifm with a fuppreffed and implied premife, fuch as c He muft
be mortal, being a man. I cannot help fufpecting that Ariftotle*
* He fays all that is communicated (\sysreu) of the predicate, will be
aflerted in words (jnQn<maC) of the fubjeft. Thefe two different tenfes of
two different verbs are often both translated by dlcitur. Why did they
On Fallacies. 237
made no difference between a fupprefTed premife, clearly intended
and diftinctly received, and one formally given. It feems to me
that we might as well diftinguifh a written from a fpoken fyllo-
gifm, as to the logical character of the two.
CHAPTER XIII.
On Fallacies.
THERE is no fuch thing as a clarification of the ways in
which men may arrive at an error : it is much to be
doubted whether there ever can be. As to mere inference, the
main object of this work, it is reducible to rules : thefe rules
being all obeyed, an inference, as an inference, is good ; confe-
quently a bad inference is a breach of one or more of thefe rules.
Except, then, by the production of examples to exercife a be
ginner in the detection of breaches of rule, there is nothing to do
in a chapter on fallacies, fo far as thofe of inference are con
cerned. Neverthelefs, there are many points connected with the
matter of premifes, to which it is very defirable to draw a reader s
attention : and above all to queftions in which it is not at firft
obvious whether the miftake be in the matter or in the form ; or
in which it may be the one or the other, according to the fenfe
put upon the words.
If there be anything ri dent em dicere verum quod vetat^ writers
on logic have in all ages moft grievoufly neglected the prohibition
in treating this fubject, and have given the ftudent a prefcriptive
right to fome amufement. One reafon of this was, that the
occur ? For various reafons, I allow myfelf to fufpeft, though not fcholar
enough to maintain, that Xo yo? generally meant communication, paflage
from one mind to another by any means, as much at leaft with reference to
the receiving, as to the imparting, mind : and that it is here oppofed to p?<n ? ,
fpeech, in that fenfe. Throw the verbs back to their primary meanings,
and it will be That which is picked up of the predicate, mall /ow out about
the fubjeft." If my conjecture be correft, the modern enthymeme is here
put on the fame footing as the fully exprefled fyllogifm.
238 On Fallacies.
Greeks endeavoured to try the new art by inventing inferences
the falfehood of which could not be detected by its rules. Thefe,
as may be fuppofed, were whimfical efforts of reafoning : never-
thelefs, they have been handed down from book to book, unfur-
paffed in their way. Another reafon is, that jefts, puns, &c. are
for the moil part only fallacies fo obvious that they excite laugh
ter ; and the greater number of them can be mown to break one
or another of the rules of logic. Accordingly, they furnim
ftriking examples of thefe rules ; the application of which, in fe-
rious terms, has itfelf a tafte of the ludicrous. Boccacio has, by
his inimitable mode of narration, made a good ftory the jeft of
which could be defcribed as confifling in nothing more than the
aflumption that what can be predicated of ftorks* in general
can be predicated of roafted ftorks : which is what logicians
would call the fallacia accidentis^ or arguing a ditto Jimpltciter^
ad dictum fecundum quid.
The terms fallacy^ fophifm, paradox^ and paralogifm^ are ap
plied to offences againft logic ; but not with equal propriety,
Fallacy and fophifm may technically have been firft applied to
arguments in which there is a failure of logic : but it is now very
common to apply them alfo to arguments in which there is a
falfehood of fa<5t, or error of principle, though logically treated ;
and if this laft ufe be not correct, writers on logic have fanc-
tioned it in their examples. Many perfons go further, and call
the erroneous ftatement itfelf a fallacy : that men are in the
habit of walking on their heads, they would fay is a very obvious
fallacy. A paradox is properly fomething which is contrary to
general opinion : but it is frequently ufed to fignify fomething
felf-contradi6lory : thus the newfpaper which recently avowed
* A fervant who was roafting a ftork for his mafter was prevailed upon
by his fweetheart to cut off a leg for her to eat. When the bird came upon
table, the mafter defired to know what was become of the other leg. The
man anfwered that ftorks had never more than one leg. The mafter, very
angry, but determined to ftrike his fervant dumb before he punifhed him,
took him next day into the fields where they faw ftorks, ftanding each on
one leg, as ftorks do. The fervant turned triumphantly to his mafter : on
which the latter fhouted, and the birds put down their other legs and flew
away. " Ah, Sir," faid the fervant, " you did not fhout to the ftork at din
ner yefterday : if you had done fo, he would have mown his other leg too."
On Fallacies. 239
its opinion that the repeal of the corn laws would make food
both cheap and dear is faid to have maintained a paradox. The
modern ufe of the word implies difrefpect, but it was not fo for
merly. Thus in the fixteenth century the opinion of the earth s
motion was ftyled the paradox of Copernicus by writers who
meant neither praife nor blame, but only reference to the opinion
of Copernicus as an unufual one. The more precife writers of
our day ufe the word paradox for an opinion fo very fingular and
improbable, that the holder of it is chargeable with an undue
bias in favor of fingularity or improbability for its own fake.
Paralogifm^ by its etymology, is beft fitted to fignify an offence
againft the formal rules of inference. It has been frequently
abufed by mathematical writers, who have fignified by it errors
of ftatement, and undue affumptions : but it is not completely
fpoiled for the purpofe, and I (hall therefore ufe it to denote a
formal error in inference, as a particular clafs of fallacy or fo-
phifm, words which it would now be difficult to diftinguifh in
meaning. Some have defined paraloglfm to be that by which a
man deceives himfelf, and fophifm that by which he tries to de
ceive others : on what grounds I do not know.
The queftion of a premife being right or wrong in fact or
principle, unlefs indeed it contradict itfelf, does not belong to
logic : nor could it fo belong unlefs logic were made, in the wideft
fenfe, that attempt at the attainment of the cognltto vert which
fome have defined it to be. All that relates to the collection of
true premifes with refpect to the vegetable world belongs to
botany; with refpecl: to the heavenly bodies, to aftronomy; with
refpect to the relation of man to his Creator, to theology. Even
were it within the province of logic, it would be impoffible, in
lefs fpace than an encyclopaedia, to enter upon queftions con
nected with the matter of fyllogifms. With regard to paralogifms,
or logical fallacies, (fo called, as an error about the meafure of
fpace is called a geometrical error) the clarification under breach
of rules would be good in form, but would afford no bafis for
the treatment of the fubject. Thofe who bring them forward
feldom proceed in direct defiance of rule, but in various modes
of evafion. Thefe it would be almoft impoffible to arrange in
fatis factory order.
Ariftotle made a claffification of fallacies, which was of courfe
240 On Fallacies.
adhered to by the writers of the middle ages. In this, as in
every other place, when I fpeak of Ariftotle and his fyftem, I
fpeak of it as underftood by thofe writers. How far they dif-
tinctly comprehended their mafter is a queftion into which I
could not enter here, even if I were competent to write on the
fubject. It is, however, fufficiently apparent that the logic of
Ariftotle is not of the purely formal character which marked the
dialectics of the middle ages : there is a much more decided
introduction of the attempt to write on the matter of fyllogifm
than many perfons think there is. The clarification of fallacies
feems to be one proof of this : and the interpretation of that claf-
fification by the middle writers feems to add their teftimony to
the after tion : in this part of the fubject they abandon techni
calities almoft entirely.
It ought to be efpecially remembered that we are very diffe-
ently fituated from thofe writers, not as to what is fallacy, but
as to what the fpecimens of it produced are likely to be. Out
of a world of general principles declared by authority, or declared
to be felf-evident by authority, they had to produce logical de
ductions ; and, of courfe, the pure fyllogifm and its rules were
to them as familiar as the alphabet. The idea of an abfolute and
glaring offence againft the ftructure of the fyllogifm being fup-
ported one moment after it was challenged, would no more
fuggeft itfelf to the mind of a writer on logic than it would now
occur to a writer on aftronomy that the accidental errer (which
might happen to any one) of affixing four ciphers inftead of five
in multiplying by a hundred thoufand would be maintained after
expofure. Accordingly, their formal chapters on fallacies would
naturally relate, if not entirely to fallacies of matter, at leaft to
thofe in which the fallacy of matter very clofely hinges upon that
of form. And fo it is in all the old fyftems which I have exam
ined. The Ariftotelian divifion (or rather feledtion, for it is far
from including everything) lends itfelf eafily to this adaptation.
We, on the contrary, live in an age in which formal logic has
long been nearly banimed from education : entirely, we may fay,
from the education of the habits. The ftudents of all our uni-
verfities (Cambridge excepted) may have heard lectures and
learnt the forms of fyllogifm to this day : but the practice has
been fmall : and out of the univerfities (and too often in them)
the very name of logic is a bye-word.
On Fallacies. 241
The philofophers who made the difcovery (or what has been
allowed to pafs for one) that Bacon invented a new fpecies
of logic which was to fuperfede that of Ariftotle, and their fol
lowers, have fucceeded by falfe hiftory and falfer theory, in driv
ing out from our fyftem all ftudy of the connexion between
thought and language. The growth of inaccurate expreflion
which this has produced, gives us fwarms of legislators, preachers,
and teachers of all kinds, who can only deal with their own
meaning as bad fpellers deal with a hard word, put together
letters which give a certain refemblance, more or lefs as the cafe
may be. Hence, what have been aptly called " the flipfhod judg
ments and crippled arguments which every-day talkers are content
to ufe." Offences againft the laws of fyllogifm (which are all laws
of common fenfe) are as common as any fpecies of fallacy : not
that they are always offences in the fpeaker s or writer s mind,
but that they frequently originate in his attempt to fpeak his
mind. And the excufe is, that he meant differently from what
he faid : which is received becaufe no one can throw the firft ftone
at it, but which in the middle ages would have been regarded
as a plea of guilty. The current notions about what logic is, are
beautiful and wonderful. I have heard a difputant, an educated
man, a graduate, efcape from allowing himfelf to be convinced
that he was arguing with a middle term particular in both pre-
mifes by declaring that faffs were better than fyllogifms : the form
of his argument would have proved that men are plants, becaufe
both require air. " I" he faid, " produce you faff s, like Bacon :
you quibble about their combination, like Ariftotle."
The Ariftotelian fyftem of fallacies contains two fubdivifions.
In the firft, which are in diffione, or in voce, the miftake is faid
to confift in the ufe of words : in the fecond, which are extra
diffionem, or in re, it is faid to be in the matter.
Of the firft fet fix kinds were diftinguifhed, as follows :
i. Mquivocatio or Homonymia^ in which a word is ufed in two
different fenfes ; giving really no middle term (if the middle term
be in queftion) or a term in the conclufion which is not the fame
name as that ufed in the premifes. For example, All criminal
actions ought to be punifhed by law : profecutions for theft are
criminal actions ; therefore, profecutions for theft ought to be
puniftied by law. Here the middle term is doubly ambiguous,
242 On Fallacies.
both criminal and atiion having different fenfes in the two pre-
mifes. But here, as in many other cafes, the choice lies with
the fophift to bring the fallacy under the head to which we refer
it or not. It may pleafe him to affert that he means the fame
thing by criminal attion in both premifes ; in which cafe, the in
ference is logical, but one or the other premife muft be denied
as to the matter. Again, Finis rei eft illius perfeftio ; mors eft
finis vfce 5 ergo mors eft vite perfettio. Here the ambiguity
may be thrown either on finis or on perfeZto. The following
example can be traced through books for three centuries. Every
dog runs on four legs ; Sirius (the dog-ftar) is a dog ; therefore
Sirius runs on four legs. It has been the defea of many old
works on logic that all their examples have been of that obvious
abfurdity, which is well enough in one or two inftances. Such
as < Nothing is better than wifdom and virtue ; dry bread is bet
ter than nothing 5 therefore, dry bread is better than wifdom and
virtue/ Some of the old examples are < A moufe eats cheefe ;
a moufe is one fyllable ; therefore one fyllable eats cheefe/ And
again, Ifte pannus eft de Anglia ; Anglia eft terra ; ergo, li
pannus eft de terra/
Where the fyllogifm is formally put, equivocation of t
die term is generally feen with great eafe. The moft difficult
exception is, I think, the old fallacy, in which giving the name
of the genus is confounded with giving the name of the fpecies,
and thereby, of courfe, giving the name of the genus. As in <
call you an animal is to fpeak truth ; to call you an afs is to call
you an animal ; therefore, to call you an afs is to fpeak truth.
This equivocation will puzzle a beginner as to its form, and the
more fo from the evident falfehood of the matter. The middle
term is " He who fays that you are one among all animals,
fpeaks truth ; and the one who calls you an afs or a goofe, cer
tainly fays that you are one among all animals. The equivocation
is in the two different ufes of the word one ; in the firft premife,
it is an entirely indefinite one ; in the fecond it is a lefs indefinite
one This one is not attached to the quantity of the middle term,
which is univerfal in the firft premife, and particular in the fe
cond but is part of the middle term itfelf.
The manner in which the ferious fallacy of equivocation moit
frequently appears, is in the conneaion of the old affociations of
On Fallacies. 243
a word which has fliifted its meaning with the altered meaning of
the fame. The word loyal, for inftance, originally meaning no
more (and no lefs) than lawful, which, as applied to a man, meant
one who refpefted the laws, and had not forfeited any right by
mifbehaviour, now means attached to the Crown and to the title
of the holder of it. In contefts for fucceffion, the winner would,
of courfe, affiime that lawful men were on his fide. In more
recent times, the term was always felf-applied, at elections, by
thofe who fupported the party which had the confidence of the
Crown for the time being : but on fuch occafions, abftinence
from the fallacy which the French call the vote du fait is the
utmoft which can be expeded of human nature.
The word publication has gradually changed its meaning, ex
cept in the courts of law. It flood for communication to others,
without reference to the mode of communication, or the number
of recipients. Gradually, as printing became the eafieft and moft
ufual mode of publication, and confequently the one moft fre
quently reforted to, the word acquired its modern meaning : if
we fay a man publifhes his travels, we mean that he writes and
prints a book defcriptive of them. I fufpecl: that many perfons
have come within .the danger of the law, by not knowing that to
write a letter which contains defamation,, and to fend it to another
perfon to read, is ptiblijhing a libel-, that is, by imagining that
they were fafe from the confequences of publifhing, as long as
they did not print. In the fame manner, the well-eftablimed
rule that the firft publifher of a difcovery is to be held the difco-
verer, unlefs the contrary can be proved, is mifunderftood by
many, who put the word printer in the place of publifher. I
could almoft fancy that fome perfons think rules ought to travel
in meaning, with the words in which they are exprefled.
A fimilar change has taken place in the meaning of the word
to utter, the fenfe of which is to give out, but which now means
ufually to give out of the mouth in words. As yet, I am not
aware that any perfon charged with the utterance of counterfeit
coin has pleaded that no one ever uttered coin except the prin-
cefs in the fairy tale : but there is no faying to what we may
come, with good example, and under high authority.
It may almoft be a queftion whether, in the time of Ariftotle,
fuccefsful equivocation, that is, undeteded at the moment, would
244 On Fallacies.
not have been held binding on the difputant who had failed to
dete6t it. The genius of uncultivated nations leads them to
place undue force in the verbal meaning of engagements and
admiflions, independently of the underftanding with which they
are made. Jacob kept the blefiing which he obtained by a trick,
though it was intended for Efau : Lycurgus feems to have fairly
bound the Spartans to follow his laws till he returned, though he
only intimated a fhort abfence, and made it eternal : and the
Hindoo god who begged for three fteps of land in the fhape of
a dwarf, and took earth, fea and Iky in that of a giant, feems to
have been held as claiming no more than was granted. The
great ftrefs laid by Ariftotle on fo many different forms of verbal
deception, may have arifen from a remaining tendency among
difputants to be very ferious about what we fhould now call play
upon words.
Governments permit what would otherwife be equivocation to
take a ftrong air of truth, by legiflating in detail againft the prin
ciples of their own meafures. The window-tax is a fpecial in-
ftance. A newfpaper calls it a tax upon the light which God s
beneficence has given to all. The anfwer would be plain enough,
namely, that it is an income tax levied upon a ufe of that light
which (how truly matters not here) is afferted to be a fair criterion
of income. But this anfwer is deftroyed by the permiffion to
block up windows, and thereby evade the tax : which is thus
made to fall upon the light ufed, and not upon the means of
ufmg it which the fize of the houfe affords. According to the
principle of this import, the blocked window is as fair a crite
rion of the income of the occupant as the open one, and fhould
have been fo confidered.
Among the forms which the fallacy of equivocation frequently
affumes, is that of the fophift altering or qualifying the known
meaning of a word in his own mind, without giving the other
party any notice : fo that there may be, if not two meanings in
one mind, yet different meanings in the two minds concerned.
A perfon afferts that Nobody denies, &c. &c/ Should this go
down, the point is gained ; what nobody denies muft be undeni
able. But fhould it be contefted (and it will generally be found
that the things which nobody denies are matters of fome diffe
rence of opinion, while thofe which nobody can deny are quite
On Fallacies. 245
fure to be points of conftant controverfy) the evafion is ready.
It is no fenfible perfon, or nobody that underftands the fubject,
nobody that is anybody, in fhort : while perhaps it cannot be
fettled who does, or who does not, underftand the fubjedr., until,
among other things, the very point in difpute is determined.
There is a wide range of equivocations arifmg out of mean
ings which are fometimes implied and fometimes not. A large
clafs of them is made by the ufual, but not univerfal, practice,
of giving to the thing the name of that which it is intended
to be, whether the attempt be fuccefsful or not. This is now
abbreviation or courtefy ; but it was the rule. According to old
definitions, bad reafoning is reafoning, fyllogifmus fophifticus is a
fyllogifm, and in an old book now before me, the fruits and effects
of demonftration are fcience, opinion, and ignorance^ the latter
containing belief of falfehood derived from bad demonftration,
which we mould now call no demonftration.
One fallacy of our time, and a very favourite one, is the fet-
tlement of the merit of a perfon, or an opinion, not by arguing
the place of that perfon or opinion in its fpecies, but by arbitrary
alteration of the boundary of the fpecies, with the intent of ex
cluding the individual in queftion altogether.
It is fomewhat analogous to the proceeding of the landlord
who unroofs the houfe to get rid of a tenant. Thus we have
had the controverfy whether Pope was a poet^ not whether he
was a good poet or a bad one, but whether he was a poet at all.
The difputants, or fome of them, claimed a right to define a poet,
and decided that none but verfe-makers of a certain goodnefs (to
be fettled by themfelves) were poets. They might juft as well
have decided, on their own authority, that none but men of a
certain amount of reafoning power were men. Had they done
this laft, as long as they fixed the amount at a figure which in
cluded themfelves under the name, nobody would have thought
they materially altered the extent of the term : it is not eafy to
fee why they have rights fo arbitrary, over words the objective
definitions of which are nearly as well fixed as that of man.
Another form of the fallacy of equivocation is the afluming,
without exprefs ftatement, that the meaning of a phrafe can be
determined by joining the meanings of its feveral words : which
is not always true in any language. When two words come to-
On Fallacies.
zether, it often happens that their diaionary meanings would
never enable us to arrive at their known and ufual (and therefc
proper) compound meaning : though they might help us in e:
plaining how that laft meaning arofe. A perfon undertakes to
crofs a bridge in an incredibly fhort time : and redeems his pledge
by croffing the bridge as one would crofs a ftreet, that is, by
traverfmg the breadth. Now, though it be true that, in general,
to crofs is to go over the breadth, or fhorter dimenfion, yet i
the cafe before us, the phrafe is elliptical, and figmfies croffing
the river upon the bridge. Nor can it be faid that this common
meaning is incorreft : that which is common and well known is,
in language, always correft. No reafonable perfon would fay
that a French newfpaper is wrong in reporting an army to be a
chevalfur la riviere, becaufe a river is not a horfe. This literal
(or rather unlettered) mode of interpretation is adopted among
gamblers in fettling bets : and is of itfelf enough to raife a ftrong
preemption that their occupation is not that of well-educated
men.
It is common enough in controverfy, for one fide or the other
to have fixed meanings of words in his own mind, on which he
proceeds without any inquiry as to whether thofe meanings will
be conveyed by the words to the other fide, or to the reader. _
is very difficult to avoid this form of the fallacy, without giving
the meanings of the moft effential terms, on the firft occafions
of their occurrence. It is not uncommon to meet with a write
who appears to believe, at leaft who certainly aas upon the
notion that the right over words refides in him, and that other
are wrong fo far as they differ from him. I do not only mean
that there are many who have an undue belief in their own
judgments, both as to words and things : but I fpeak of thofi
who, though mowing a proper modefty in refpeft to their own
conclufions, feem to be unable to do the fame with refped to
their definitions of words. If all mankind had fpoken one lan
guage we cannot doubt that there would have been a powerful,
perhaps a univerfal, fchool of philofophers who would have 1
lieved in the inherent connexion between names and things ;
who would have taken the found man to be the mode of agita-
tincr the air which is effentially communicative of the ideas of
rcafon, cookery, bipedality, &c. The writers of whom I fpeak,
On Fallacies. 247
are more or lefs of this fchool ; they treat words as abfolute
images of things by right of the letters which fpell them. " The
French," faid the failor, "call a cabbage njhoe; the fools ! why
can t they call it a cabbage, when they muft know it is one ?"
Equivocation may be ufed in the form of a propofition ; as
for inftance, in throwing what ought to be an affirmative into
the form of a qualified negative, with the view of making the
negative form produce an impreffion. Thus a controverfial
writer will afTert that his opponent has not attempted to touch a
certain point, except by the abfurd aflertion, &c. &c. &c. To
which the other party might juftly reply, " Your own words
mow that I have made the attempt, though your phrafe has a
tendency, perhaps intended, to make your reader think that there
is none, or at leaft to blind him to the difference between none
and none that you approve of"
2. Thefallacia amphibolic , or amphibologies ^ differs in nothing
from trte laft, except in the equivocation being in the conftruc-
tion of a phrafe, and not in a fingle term : as in confounding that
which is Plato s (property) with that which is Plato s (writing).
Or, as in c Qui funt domini fui funt fui juris ; fervi funt do-
mini fui ; ergo fervi funt fui juris. The ambiguities of con-
ftru&ion in our language, arifmg from want of inflexions and
genders are tolerably (and intolerably) numerous. The dif
ficulty of determining the emphatic word often gives a doubt as
to the meaning. But very often indeed there is a want of the
diftin&ion which the algebraift makes when he writes three-and-
four tens as diftinguimed from three and four-tens: (3 + 4).io
and 34-4.10. It cannot, for inftance, be faid whether c I intend
to do it and to go there to-morrow means that it will be done
to-morrow or not. It may be either (I intend to do it and to go
there) to-morrow, or I intend to do it and (to go there to-mor
row). The prefumption may be for the firft conftruftion : but
it is only a prefumption, not a rule of the language. In an inftance
cited by Dr. Whateley If this day happen to be Sunday,
this form of prayer {hall be ufed and the faft kept the next day
following," the conftru&ion is ambiguous, and the intended mean
ing probably againft the prefumption. There is a book of the
laft century, written by a " teacher of mathematics, and writing
mafter to Eton College." Were mathematics taught at Eton*
248 On Fallacies.
or not ? Punctuation may be an aSHStance ; but it fo often hap
pens that the author leaves that point to the printer, that it is
hardly fafe to rely upon it. Printers punctuate correctly when
the meaning is clear : but when it is ambiguous, they may be
as apt to take the wrong meaning as any other readers.
3, 4, The fallacia compofitlonts, and fallacia divifionis, confiSt
in joining or Separating thofe things which ought not to be joined
or feparated. If we may fay that A is X and B is Y, fo that A
and B is X and Y, we have no right to infer that we may form
the compound and collective names A and B, and C X and Y,
and fay that A and B is C .X and Y. Thus two and three are
even and odd : but five is not even and odd. Again, two and
five are four and three ; but neither is two four, nor five three.
It muft be remembered that the word all, in a proposition, is not
necefTarily fignificative of a univerfal propofition : it may be a
part of the defcription of the fubject. Thus in c all the peers are
a houfe of Parliament, we do not ufe the words all the peers in
the fame fenfe as when we fay all the peers derive their titles
from the Crown. In the fecond cafe the fubjecl: of the propo
fition is peer ; and the term all is distributive, fynonymous with
each and every. In the firSt cafe the fubjecl: is all the peers, and
the term all is collective, no more diftinguifhing one peer from
another than one of John s fingers is distinguished from another
in the phrafe, c John is a man. The fame remarks may be made
on the word fame; as in fome peers are dukes, and fome peers
are the committee of privileges. The all and feme of the quan
tity of the propofition are distributive terms j the all and fome of
the fubjecl: are collective. Again, all men are a fpecies (of ani
mals) which no number of men are, wanting the reft. All men
here make the one individual object of thought of a fmgular pro
pofition. This amounts to an ambiguity of construction, an
amphibologia, as do moft fources of fallacy falling under this head,
which can therefore hardly be confidered as anything more than
a cafe of the laft. We want another idiom or the algebraical
distinction, as in c All (peers) hold of the Crown ; (all peers) are
a houfe of Parliament.
5. The fallacia profodite or accentus was an ambiguity arifing
from pronunciation, and its introduction feems to lead to very
minute fubdivifion of the fubjecl, and to enfure the entrance of
On Fallacies. 249
none but ludicrous examples. Burgerfdicius does not think it
unworthy of himfelf to defcend to the following, c Omnis equus
eft beftia ; omnis juftus eft aequus, ergo omnis juftus eft beftia.
An older writer has c Tu es qui es ; quies eft requies ; ergo, tu
es requies. Thefe are mere puns ; and the makers of them
were fairly beaten by the contriver of Two men eat oyfters for
a wager, one eat ninety-nine, the other eat two more, for he eat
a hundred and won. But more ferious fallacies may be referred
to this head. A very forced emphafis upon one word may, ac
cording to ufual notions, fuggeft falfe meanings. Thus, c thou
(halt not bear falfe witnefs againft thy neighbour, is frequently
read from the pulpit either fo as to convey the oppofite of a pro
hibition, or to fuggeft that fubornation is not forbidden, or that
anything falfe except evidence is permitted, or that it may be
given for him, or that it is only againft neighbours that falfe wit
nefs may not be borne.
A ftatement of what was faid, with the fuppreilion of fuch
tone as was meant to accompany it, is thefallacia accentus. Gef-
ture and manner often make the difference between irony or
farcafm, and ordinary aflertion. A perfon who quotes another,
omitting anything which ferves to mow the animus of the meaning;
or one who without notice puts any word of the author he cites
in italics, fo as to alter its emphafis ; or one who attempts to
heighten his own afTertions, fo as to make them imply more than
he would openly fay, by italics, or notes of exclamation, or
otherwife, is guilty of the fa Ha da accentus.
To this fallacy I mould refer one of very common occurrence,
the alteration of an opponent s proportion fo as to prefent it in
a manner which is logically equivalent, but which alters the em
phafis, either as noticed in page 134, or in any other manner. It
is generally not reafoning, but retort, which is the object of the
alteration : for inference cannot be altered by changing a propo-
fition into a logical equivalent, but a fmart repartee may be very
effective againft Some Xs are Ys, but flat enough againft c fome
Ys are Xs. And even when the proponent miftakes his own
meaning, and mifcalculates his own emphafis, ftill, if the miftake
be obvious, there is fallacy in taking advantage of it ; for he who
communicates in fuch incorrect terms as mow what the correct
ones are, does, in fact, communicate in correct terms, to all who
250 On Fallacies.
fee the {bowing. Of courfe, refpect for logic never flood in the
way of a fuccefsful retort from the time of Ariftotle till now, nor
will on this fide of the millenium. A fpeculator once wrote to
a fcientific fociety, to challenge them to an (on his part) anti-
Newtonion controverfy, relying on it that he could contend in
mechanics, though avowedly ignorant of geometry. He was
anfwered by a recommendation to ftudy mathematics and dyna
mics. His rejoinder was an angry pamphlet, in which, indignant
at the unfairnefs, as he took it to be, of the recommendation, he
exclaimed, I did not confefs my ignorance of dynamics. Had
he been worth the anfwering, it would have been impoffible to
refift the reply No, but you fhowed it. Had he written, as he
meant It was not dynamics of which I confeffed ignorance,
and had an opponent written, as many would have done, You
fay, fir, that you did not confefs your ignorance of dynamics :
indeed you did not, you contented yourfelf with an ample difplay
of it, he would have ufed thefal/acia accentus. Nor would he, in
my opinion, have been clear of it though he had only taken advan
tage of a wrong, but evidently wrong, placement of emphafis on
the part of the afTailant. The ufe of fuch a weapon, as to its
legitimacy, depends entirely upon the manner in which the quef-
tion mall be fettled how far irony is allowable. Where the anfwer
is in the affirmative, a very obvious fallacy, as a farcafm, may be
permitted. But I may here obferve, that irony itfelf is generally
accompanied by \he fallacla accentus \ perhaps cannot be afTumed
without it. A writer difclaims attempting a certain tafk as above
his powers, or doubts about deciding a proportion as beyond his
knowledge. A felf-fufficient opponent is very effective in aflur-
ing him that his diffidence is highly commendable, and fully jufti-
fied by the circumftances.
6. The fa Had a figurte dittionls^ as explained, means literally
a miftake in grammar and nothing elfe ; as that becaufe Jluvius
is aqua it is humid A, or that becaufe aqua is feminine, fo is poeta.
All thefe fallacies in diflione come under the head of ambiguous
language, and amount to nothing but giving the iyllogifm four
terms, two of them under the fame name. The fallacies extra
dittionem are fet down as follows.
I. Thefallada accident is ; and 2. That a ditto fecundum quid
ad diftum fwipllciter. The firft of thefe ought to be called that
On Fallacies. 251
of a ditto fimpliciter ad dittum fecundum quid, for the two are
correlative in the manner defcribed in the two phrafes. The firft
confifts in inferring of the fubjeft with an accident that which
was premifed of the fubjeft only : the fecond in inferring of the
fubjea only that which was premifed of the fubjeft with an acci
dent. The firft example of the fecond muft needs be What you
bought yefterday, you eat to-day ; you bought raw meat yefter-
day ; therefore, you eat raw meat to-day/ This piece of meat
has remained uncooked, as frefh as ever, a prodigious time. It
was raw when Reifch mentioned it in the Margarita Pkilofo-
pblca in 1496 : and Dr. Whateley found it in juft the fame ftate
in 1826. Of the firft, we may give the inftance Wine is per
nicious ; therefore, it ought to be forbidden. The expreffed
premife refers to wine ufed immoderately : the conclufion is
meant to refer to wine however ufed. This fpecies of fallacy
occurs whenever more or lefs ftrefs is laid upon an accident,
or upon any view of the fubjec~t, in the conclufion, than was
done in the premifes. As in the following : All that leads to
fuch philofophy as that of the fchoolmen, with their logic, muft
be unworthy to be ftudied, except hiftorically/ The intent of
fuch a fentence is not formally to propofe the falfe fyllogifm,
The fchoolmen had that which led them to a falfe philofophy ;
the fchoolmen had logic ; therefore, logic led them to a falfe phi
lofophy, but only to take the chance of the ftrefs thus laid upon
logic producing a difpofition to fuppofe that the logic was in fault.
The premifes are really :
The philofophy of the fchool-1 f
men (who paid particular atten- | is | a falfe philofophy.
tion to logic) J
*] fthat the guides to which
Every falfe philofophy I is \ fhould be neglefted, except
J [as hiftory.
whence it is rightly inferred that the guides to fuch a philofophy
as that of the fchoolmen (who ftudied logic) are only of hiftorical
ufe. And the fame thing might equally be inferred of the fchool
men who ate mutton, a practice to which moft of them were as
much addidted, no doubt, as to making fyllogifms. The art of
252 On Fallacies.
the fophift confifts in making the accident which is either un
fairly introduced, or withdrawn, or fubftituted, have an apparently
relevant relation to the fubjecl itfelf. Undoubtedly, the fchool-
men s logic has a connexion with their philofophy which the
mutton they ate has not : but as long as it is not the connexion
which permits the inference, it is abfolutely irrelevant.
All the fallacies which attempt the fubftitution of a thing in
one form for the fame thing (as it is called) in another, belong to
this head : fuch as that of the man who claimed to have had one
knife twenty years, giving it fometimes a new handle, and fome-
times a new blade. The anfwer given by the calculating boy
(page 54, note) was, relatively to the queftion, a worthy anfwer,
and took advantage of the common notion that a bean, after
being fkinned, is ftill a bean, as before. More ferious difficulties
have arifen from the attempt to feparate the ejfcntial from the
accidental^ particularly with regard to material objects. The
Cartefians denied weight, hardnefs, &c. to be eflential to mat
ter, until at laft they made it nothing but fpace, and contended
that a cubic foot of iron contained no more matter than a cubic
foot of air.
The law, in criminal cafes, demands a degree of accuracy in
the ftatement of the fecundum quid which many people think is
abfurd : and it appears to me that the lawyers often help the
popular mifapprehenfion, and give it excufe, by confounding
errors of things with errors of words, after the example of the
world at large. Any error of any kind, provided it be fmall in
amount, pafTes for a miftake in words only, by virtue of its fmall-
nefs. By a miftake in words, I mean the addition or omiffion
of words which, whatever they might do under another ftate of
things, do not, as matters ftand, affect the meaning.
Take two inftances, as follows ; Some years ago, a man was
tried for ftealing a ham, and was acquitted upon the ground that
what was proved againft him was that he had ftolen a portion of
a ham. Very recently, a man was convicted of perjury, in the
year 1846, and an objection (which the judge thought of impor
tance enough to referve) was taken, on the ground that it ought
to have been in the year of our Lord 1846. There may, of
courfe, be acknowledged rules, which, as long as they are rules,
muft be obeyed, and which may make the fecond miftake as ne-
On Fallacies. 253
ceffarily vitiate an indictment as the firft. But, in difcufling the
policy of the rules, it would feem to me that the two cafes are
entirely different. In both, no doubt, the reft of the indictment
might, by implication, make good the meaning required : but
there feems a great difference between allowing the remainder to
correct an error, and allowing it to make good an infufficiency
(fuppofing the date, in the fecond cafe, to be really inefficient).
In the fecond cafe, the accufed may fee the omiffion as well as
another, and may confider of his defence againft every alterna
tive : in the firft, he may be actually led to appear in court
with a defence not relevant to what will be brought againft him.
The fecond may be a hardfhip, the firft is an injuftice. And this,
even on the fuppofition that the reft of the indictment is to be
allowed in explanation : for we have no more right to fuppofe
that the true parts will correct the erroneous ones, than that the
erroneous parts will affect the conftrudtion of the true ones. But
there is good reafon to think that the fufficient defcription of one
fentence may fupply what is wanted in the inefficient defcription
of another, when infufficiency is all.
But, perhaps, it will be held to be the better rule, that the re
mainder of the indictment mould not be allowed in explanation.
It will then be admitted by all that a material error, or a material
infufficiency, mould be allowed to nullify the charge. The dif
ference between the law and common opinion entirely relates
to what conftitutes a material amount of one or the other. And
here it is impoffible to bring the two together : for the law muft
judge fpecies, while the common opinion will never rife above
the cafe before it. In the two inftances, which by many will be
held equally abfurd, a great difference will be feen by any who
will imagine the two defcriptions, in each cafe, to be put before
two different perfons. One is told that a man has ftolen a ham ;
another that he has ftolen a part of a ham. The firft will think
he has robbed a provifion warehoufe, and is a deliberate thief:
the fecond may fuppofe that he has pilfered from a cook-mop,
pofiibly from hunger. As things ftand, the two defcriptions
may fuggeft different amounts of criminality, and different mo
tives. But put the fecond pair of defcriptions in the fame way.
One perfon is told that a man perjured himfelf in the year 1846 ;
and another, that he perjured himfelf in the year of our Lord
254 On Fallacies.
1846. As things ftand, there is no imaginable difference : for
there is only one era from which we reckon. The two defcrip-
tions mean the fame thing : nor can it even be faid that one is
complete and the other incomplete ; but only that one is lefs
incomplete than the other. The next queftion might have been,
what lord was meant, our Lord Jefus Chrift, or our Lord the
King ? both being phrafes of law. The anfwer will be, that the
number 1846 leaves no doubt which was meant. A very good
anfwer, certainly ; but equally conclufive as to the fimple phrafe
* in the year 1846. The firft cafe is one in which the two de-
fcriptions have a real difference of meaning : it is not fo in the
fecond.
3. The petitio principle is one of the logical terms which has
almoft found its way into ordinary life. It is tranflated by the
phrafe begging the queftion, that is, afluming the thing which is
to be proved. This is alfo called reafoning in a circle, coming
round, in the way of conclufion, to what has been already for
mally affumed, in a manner exprefled or implied. I (hall referve
what I have to fay on the juftice of this tranflation, and take it
for the prefent as good.
Every colle&ive fet of premifes contains all its valid conclu-
fions ; and we may fairly fay that, fpeaking objectively of the
premifes, the affumption of them is the aflumption of the con
clufion ; though, ideally fpeaking, the prefence of the premifes
in the mind is not neceflarily the prefence of the conclufion. But
by this fallacy is meant the abfolute aflumption of the fmgle con
clufion, or a mere equivalent to it, as a fmgle premife. If the
conclufion be c Every X is Z and if it be formally known that
A and X are identical names, and alfo B and Z, then to aflume
Every A is B as a premife in proving Every X is Z would
be a manifeft petitio principii, or begging of the queftion. But
even this muft be faid hypothetically ; it is fuppofed fully agreed
between the difputants that the two identities are granted. Let
it be otherwife, and there is no petitio principii : it is then fair to
propound A)B, which, if difputed, is to be proved, and afterwards
to reafon as in A)B + B)Z = A)Z, X)A + A)Z = X)Z. Striftly
fpeaking, there is no formal petitio principii except when the very
proportion to be proved, and not a mere fynonyme of it, is
aflumed. This of courfe, rarely occurs : fo that the fallacy to
On Fallacies. 255
be guarded againft is the aflumption of that which is too nearly
the fame as the conclufion required. And then the fallacy is
nothing diftinct in itfelf : but merely amounts to putting forward
and claiming to have granted that which ftiould not be granted.
When this is done, it matters little as to the character of the
fallacy, whether the undue claim be made for a propofition which
is nearer to, or further from, the conclufion to be proved. When
proof is offered, the advancement of the conclufion in other words
is of courfe not petitio principii : when proof is not offered, the
aflumption of that which (with other things proved) would prove
the conclufion, is a fallacy of the fame character in all cafes.
There is an opponent fallacy to the petitlo principii which, I fuf-
pecl, is of the more frequent occurrence : it is the habit of many
to treat an advanced propofition as a begging of the queftion the
moment they fee that, if eftablifhed, it would eftablifti the quef
tion. Before the advancer has more than ftated his thefis, and
before he has time to add that he propofes to prove it, he is
treated as a fophifl on his opponent s perception of the relevancy
(if proved) of his firft ftep. Are there not perfons who think
that to prove any previous propofition, which neceflarily leads to
the conclufion adverfe to them, is taking an unfair advantage ?
There is another cafe in which begging the queftion may be
unjuftly imputed. It fhould be remembered that demonstrative
inference is not the only kind of inference : there is elucidatory
inference, recapitulatory inference, &c. A propofition may have
its aflerted explanation prefented as a fyllogifm, the inference of
which, as demonftration, might well be called a refult of petitio
principii. Say it never could have been doubted that men would
apply fcience to the production of food/ If there fhould be any
hefitation about this, the explanation of man under the phrafe
which is exclufively characleriftic of him, rational animal, would
remove it : the animal muft have food, the rational being will
have fcience. But it would be begging the queftion to aflert that
the fyllogifm of elucidation c A rational animal is, &c. ; man is,
&c ; therefore man is, &c. is a demonftration. And out of this
arifes the fallacy of prefuming that an author meant demonstration,
when he can only be fairly conftrued to have attempted elucida
tion of what he fuppofed would, upon that elucidation, be granted.
The forms of language are much the fame in the two cafes.
256 On Fallacies.
It has been obferved that Ariftotle hardly ever ufes the phrafe
f M v amicr&z/, prlnclpium peter e : it is TO if f %ij? and TO iv a/?%>i5
that which is (ought to come) out of, or is in, the principle. By
the word prlnclpium he diftin&ly means that which can be known
ofltfelf. He lays down five ways of ajjuming that which ought
to come out of a felf-known principle, of which begging the quef-
tion is the firft. The others are affuming the univerfal to prove
the particular ; affirming a particular to help to prove the uni
verfal ; affuming all the particulars of which the univerfal may
be compofed ; and afTuming fomething which obvioufly demon-
ftrates the conclufion.
Among the earlier modern writers, as far as I have feen them,
there is fome diverfity in their defcription of the petltlo prlnclpll.
That the prlnclpium was meant to be the thing known of itfelf,
the f%>7 of Ariftotle, as far as the introduction of the word is
concerned, feems clear enough. Was it not then by a mere cor
ruption that it was frequently confounded with the conclufion,
the quod in principle quaefitum fuit ? Did not the fame in
accuracy, * which confounds the TO ev agxy of Ariftotle with the
a^XYi itfelf, govern the change of the word ? Moft writers take
the fallacy of the petltlo prlnclpll as meaning that in which the
conclufion is deduced either from itfelf, or from fomething which
requires proof more, or at leaft as much, Ignatius aut aque Igno-
turn. But fome, in their definitions, and ftill more in their ex
amples, fupport the following meaning, which I ftrongly fufpecl:
to be the true derivation of the phrafe, however the prlnclpium
and quod In prlnclplo might afterwards have been confounded with
one another. The philofophy of the time confifted in a large
variety of general propofitions (principles) deduced from autho
rity, and fuppofed to be ultimately derived from intrinfic evidence,
felf-known, or elfe by logical derivation from fuch principles.
Thefe were at the command of the difputant, his opponent could
not but admit each and all of them : the laws of difputation de
manded f the aflent which the geometer requires for his poftu-
* Sir W. Hamilton of Edinburgh (notes on Reid, p. 761,) fays that
prlnclpium is always ufed for that on which fomething elfe depends.
f Does a traditional remnant of this convention ftill linger in the not un-
frequent notion that a difputant is entitled to the conceflion of his principia ?
We ufed to hear You muft grant me my firft principles, elfe I cannot
On Fallacies. 257
lates. Except when, now and then, literary fociety was fhaken
to its very foundations by a difpute which affected any of them,
as a nominalift controverfy or the like moral earthquake. The
mofl frequent fyllogifm was one which, having the form Barbara^
had a principium for its major, and an exemplum for its minor :
as in All men are mortal (principium] ; Socrates is a man (ex
emplum} ; therefore Socrates is mortal. The petitio principii,
then, occurred, when any one, to prove his cafe, made it an ex
ample of a principle which was not among thofe received, with
out offering to bring the former under the logical empire of the
latter. And fome writers define the fallacy as occurring ft
contingent in fyllogifmo principium petere ; where by prlnclplum
they mean the principle which generally occurs in the major pre-
mife, and by their inftances they clearly fhow that they mean to
include nothing but the fimple fyllogifm of principle and example.
They would leave us to infer that if any one fhould happen to
conftruct a fyllogifm in which both premifes are principles, one
or both not received, the inference, though denied by fimple
denial of one or both premifes, would not be confidered as tech
nically the petitio principii, which with them was, as it were,
petitio principii exemplum continentis.
It has often been afferted that all fyllogifm is a begging of the
queflion, or a petitio principii in the modern fenfe, an affumption
of the conclufion. That all premifes do, when the argument is
objectively confidered, contain their conclufion, is beyond a doubt :
and a writer on logic does but little who does not make his reader
fully alive to this. But the phrafe, as applied to a good fyllo
gifm, is a mifapprehenfion of meaning : for its definition refers
it to what is affumed in one premife. The moft fallacious pair
of premifes, though expreffly constructed to form a certain con
clufion, without the leaft reference to their truth, would not be
affuming the queftion, or an equivalent. But a further charge
has been made againft the fyllogifm, namely that very often the
conclufion, fo far from being deduced from the principle, is
actually required to deduce it : that for inftance, in All men are
argue. Cardinal Richelieu s anfwer to his applicant s ilfaut <vi<vre, namely,
Je nen <vois pas la mcejfite, had fomething of inhumanity in it : but, as
applied to the Mais, Monfieur, il faut fe difputer oi the preceding aflumption,
it would generally be quite the reverfe.
S
258 On Fallacies.
mortal; Plato is a man; therefore Plato is mortal we do not
know that Plato is mortal becaufe all men are mortal, but that
we need to know that Plato is mortal, in order to know that it
is really true that all men are mortal. There is much ingenuity
in this argument : but I think a little confideration, not of the
fyllogifm, but of how we ftand with refpect to the fyllogifm, will
anfwer it.
When we fay that A is B, we do not merely mean that the
thing called A is the thing called B : if we fpoke of objects as
objects, it would not matter under what name, and A is B would
be no other than c B is B and the very proportion itfelf would
be of its own nature a mere identity, an affertion that what is, is.
It feems to me that between objects, thus viewed, there can nei
ther be proportions nor fyllogifms. A may remind us of a thing
as fuggefting one idea to our minds ; B of the fame thing as
fuggefting another : and the proportion c A is B then aflerts that
the two ftates of our mind are from the fame external fource.
Our logic, in wholly feparating names from objects, and dealing
only with the former, makes a fort of fymbolic reprefentation of
the diftinction between ideas and objects.
Now the objection above ftated to the fyllogifm appears to me
to be founded upon thinking of the object, as if it had no names.
Suppofe all things marked, each with every name which can be
applied to it. Undoubtedly then, each one marked man will
have the mark mortal upon him, and fome the mark Plato, it
may be : and by the time all the marks are put on, and to a per-
fon who is fuppofed to be immediately cognizant of the fimul-
taneous exiftence of two or more marks on the fame thing, it
would be an abfurdity to attempt any fyllogifm at all. What
coexiftence of marks could there be which he muft not be fup
pofed to have noted in making the induction necefTary for a uni-
verfal propofition. When he collected the elements of All men
are mortal he faw among the reft and fet it down. But
man o
man
fuppofe that his knowledge is not acquired, as to different marks,
all at once : but that each coincidence of marks is to be a fepa-
rate acquifition to his mind. Then he does not know, by the
time he has found out that All men are mortal whether Plato
be mortal or not. Plato may be a ftatue, a dog, or a book written
On Fallacies. 259
by a man of that name. Plato does not carry man with it : his
major tells him nothing about Plato, until he has the minor,
c Plato is a man and then, no doubt, he has abfolutely acquired
the conclufion Plato is mortal. The whole objection tacitly
aflumes the fuperfluity of the minor ; that is, tacitly aflumes we
know Plato to be a man, as foon as we know him to be Plato.
Grant the minor to be fuperfluous, and no doubt we grant the
neceffity of connecting the major and the conclufion to be fuper
fluous alfo. Grant any degree of neceffity, or of want of necef
fity, to the minor, and the fame is granted to the connection of
the major and conclufion.
In the preceding cafe, the fyllogifm is looked upon as one of
communication, by the authors of the objection ; while at the
fame time it is tacitly aflumed that the minor does not commu
nicate : Plato, by virtue of our acquaintance with the name, is
taken to be a man.
Moreover, it is to be noted that the proportion ufed in argu
ment, whether to ourfelves or to others, is very frequently not
fo much the mere attribution of one idea to another, as a decla
ration that pro hac vice the idea contained in the more extenfive
term is all that is wanted, and that the differences which con-
ftitute the fpecies are not to the purpofe. Or (page 234) it is
the diminution of the comprehenfion which is neceflary, and the
increafe of extenfion is only contingent. It is ftripping the com
plex idea of the unneceflary parts, to prevent only what is requi-
fite. Thus any one who will aflert that, in the Mofaic account,
no animal life whatever was deftroyed by {laughter before the
deluge, muft be convinced by being reminded that an antedilu
vian (Cain) killed Abel who was a man and therefore an animal.
With the petitio principle may be clafTed (for it might alfo be
referred to other fallacies) cafes of the imperfect dilemma. Sup-
pofe we fay Either M or N muft be true : if M be true, Z is
impoffible ; if N be true, Z is impoffible ; therefore Z is impof
fible. Now if the disjunctive premife ought to have been ei
ther M or N or Z is true, here would have been almoft an ex-
prefs petitio principii. For example, fay c A body muft either be
in the ftate A or the ftate B ; it cannot change in the ftate A ;
it cannot change in the ftate B ; therefore, it cannot change at
all. Now, if the alternative A or B be neceflary, the correct
260 On Fallacies.
ftatement may be c A body muft either be in the ftate A, or in
the ftate B, or in the ftate of tranfition from one to the other.
Of this kind is the celebrated fophifm of Diodorus Cronus, that
motion is impoflible, for all that a body does, it does either in
the place in which it is, or in the place in which it is not, and it
cannot move in the place in which it is, and certainly not in the
place in which it is not. Now, motion is merely the name of
the tranfition from the place in which it is (but will not be) to
that in which it is not (but will be). It is reported that the in
ventor of this fophifm fent for a furgeon to fet his diflocated
fhoulder, and was anfwered that his fhoulder could not have been
put out either in the place in which it was, or in the place in
which it was not ; and therefore, that it was not hurt at all.
4. The ignoratio elenchi^ or ignorance of the refutation^ is what
we fhould now call anfwering to the wrong point : or proving
fomething which is not contradictory of the thing afferted. It
may be confidered either as an error of form or of matter ; and
it is, of all the fallacies, that which has the wideft range. Such,
for inftance, as the cafe of a writer I have read, who admits that
certain evidence, if given at all, would prove a certain point ; and
admits that fuch evidence has been given : but refufes to admit
the point as proved, becaufe the evidence was given in anfwer to
objections, and in a fecond pamphlet. The pleadings in our courts
of law, previous to trial, are intended to produce, out of the varieties
of ftatement which are made by parties, the real points at iflue ;
fo that the defence may not be ignoratio elenchi^ nor the cafe the
counter-fallacy, which has no correlative name, but might be
called ignoratio conclufionls. If a man were to fue another for
debt, for goods fold and delivered, and if defendant were to reply
that he had paid for the goods furnifhed, and plaintiff were to
rejoin that he could find no record of that payment in his books ;
the fallacy would be palpably committed. The rejoinder, fup-
pofed true, (hows that either defendant has not paid, or plaintiff
keeps negligent accounts ; and is a dilemma, one horn of which
only contradicts the defence. It is plaintiff s bufmefs to prove
the fale, from what is in his books, not the abfence of payment
from what is not ; and it is then defendant s bufmefs to prove
the payment by his vouchers.
It is commonly faid that no one can be required to prove a
On Fallacies. 261
negative, and often that no one can prove a negative. There is
much confufion about this : for any one who proves a pofitive,
proves an infinite number of negatives. Every thing that can be
proved to be in St. Paul s Cathedral at any one moment is fairly
proved not to be in more places than I can undertake to enume
rate. What is meant is, that it is difficult, and may be impoflible,
to prove a negative without proving a pofitive. Accordingly,
when the two fides of the queftion confift of a pofitive and nega
tive, the burden of proof is generally confidered to lie upon the
perfon whofe intereft it is to eftablifh the pofitive. This being
underftood, it is ignoratio elenchi to attempt to transfer the charge
of proving the negative to the other party. But this rule is by
no means without exception : there are many departures from it
in the law, for example, though not under the moft logical
phrafes. For inftance, a homicide, as fuch, is confidered by the
law a murderer, unlefs, failing juftification, he can prove that he
had no malice. Here, in the language of the law, the homicide,
fuppofed unjuftifiable, is in itfelf a preemption of malice, which
the accufed is to rebut. It is not true, in point of fact, that fuch
prefumption exifts on the mere cafe of homicide, independent of
the manner of it : if the law will confult its own records, it will
find that, for one homicide with malice of which it has had to
take cognizance, there are dozens at leaft, done in heat of blood,
and called manflaughters. But the cafe ftands thus ; the alter
natives are few, fo that proving the negative of one, which the
accufed is called on to do, can be done by proving the affirmative
one out of a fmall number. There are but malice, heat of blood,
mifadventure, infanity, &c. to which the action can be referred.
Of thefe few things, it is eafier for the accufed to eftablifh fome
one out of feveral, above all when motive is in queftion (of which
only himfelf can be in pofTeffion of the moft perfect knowledge)
than it is for the profecutor to eftablifh a particular one. And
the principle on which he is called on to eftablifh a negative (or
rather another pofitive) is that the burden of proof fairly lies on
the one to whom it will be by much the eafieft. The proof of
a negative, then, being as eafy as, in fact identical with, the
proof of one of the pofitive alternatives, fuch proof may, from
the circumftances, lie upon a difputant, particularly when the
number of the alternatives is few. But the negative proof^ a
262 On Fallacies.
very different thing, is of its own nature hardly attainable, and
therefore hardly to be required. A book has been miflaid ; is it
in one room or the other ? If found in the fecond room, there
is proof of the negative as to the firft : and almoft any one who
can read can be trufted to fay, on his own knowledge, that in a
certain room there is a certain book. But to give negative proof
as to the firft room, it muft be made certain, firft, that every
book in the room has been found and examined, fecondly, that
it has been correctly examined. No one, in fact, can prove
more than that he cannot find the book : whether the book be
there or not, is another queftion, to be fettled by our opinion of
the vigilance and competency of the fearcher. Controverfialifts
conftamly lay too much ftrefs on their own negative proofs, on
their / cannot find, even as to cafes in which it is palpably not
their intereft to find.
Somewhat akin to the preceding is the conftant fallacy of con-
troverfialifts, conveyed in their ftrong aflertion of the refults of
their own arguments. Few can bear to admit that there is a
queftion for others to decide ; and after fumming up both fides,
to feparate the points which the reader is to pronounce upon.
They muft decide for him, and thus act both counfel and judge :
probably becaufe their arguments are not fo convincing to their
own minds as they wifh them to be to the reader s. They prove,
at the utmoft, their own conviction that they have the right fide :
but the thing to be proved is that fuch conviction is well founded.
They know the maxim Si vis me flere, dolendum eft primum ipfe
tibi, and think it will hold good of the reafon, as well as of the
feelings : as it will, to fome. The confequence is, that the deli
berate reader fufpects them, and feels inclined rather to differ
than agree : he will not dance to a writer who pipes too much.
Juft as " I ll tell you a capital thing," fets the hearer upon avoid
ing laughter, and gives him notice to try ; fo * I intend to give
moft unimpeachable proof, puts the judicious reader upon look
ing for inadmiffible aflumptions, and he is feldom allowed by
fuch writers to look in vain. But, if the difputant who begins by
declaring his intention to be irrefiftible, be fufpicious, the one
who ends by announcing that he is fo, is abfolutely felf-convicted.
If it be very clear, why ftiould he fay it ? Does he tell his reader
that he muft remember to diftinguifh the black letters from the
On Fallacies. 263
white paper, or does he print at the lop of the book c keep this
fide uppermoft ? Thefe things (eflential as they are) he really
does leave to the reader : but he dares not truft the latter to find
out (though he fays it is as clear as black and white) that his ar
guments are fo ftrong and fo good, that nothing but wilful dif-
honefty, or hopelefs prejudice, can refift their force.
Another common form of the ignoratio elenchi^ lies in attri
buting to the conclufion afTerted fome ultimate end or tendency.
Thus, an argument in favour of checking the power of the
Crown is called Jacobinifm ; of an increafe of that power, abfo-
lutifm : though the argument propofed may be found, indepen
dently of its propofer s wifhes. This is a cafe in which the refult
of the method is juftifiable, though the method is wrong. Many
readers will remember the advice given by an old judge to a
young one, Give your judgments without reafons ; moft likely
your decifions will be right ; and it is juft as likely that your
reafons will be wrong. This advice mould be followed by many
of thofe who judge or decide arguments. The propofer is of a
known opinion, which gives him a ftrong bias towards the con
clufion of the argument. He is a witnefs (page 205), and the
effecl: upon the mind of the receiver is to be that of the united
argument and teftimony. The teftimony is, in the receiver s
mind, of a low order ; the propofer is a radical, and the receiver
is of opinion that a radical would pick a pocket : or elfe, perhaps,
the propofer is a tory, and the receiver is of the belief that a tory
muft have picked a pocket. Thefe opinions may be right or
wrong ; but they exift : and there is certainly no formal fallacy
in admitting them, as affecting the teftimony, to fubtraft from
the probability of the truth of the conclufion. But there is a
formal fallacy, a decided ignoratio elencbi^ in throwing all the in-
difpofition to receive upon the invalidity of the argument.
There is a much more culpable form of the fame fpecies.
If fuch a conclufion were admitted, it would lead to fuch and
fuch another conclufion, which is not to be admitted. In quef-
tions of abfolute demonftration, this procefs is found : if B be
certainly falfe, and if it be the neceflary confequence of A, then
A muft alfo be falfe. But it is unfound when it takes the form,
* I believe B to be falfe ; I believe it to follow from A j there
fore I afTuine a right to difbelieve A whatever evidence may be
264 On Fallacies.
offered for it ? This fallacy is fufficiently expofed in page 209.
There is a tradition of a Cambridge profeflbr who was once
afked in a mathematical difcuflion I fuppofe you will admit that
the whole is greater than its part/ and who anfwered, Not I,
until I fee what ufe you are going to make of it. This was no
doubt the extreme cafe ; the more ordinary one arifes in a great
meafure from the great fallacy of all, the determination to have
a particular conclufion, and to find arguments for it. Obferve a
certain perfon who is led on by a wily opponent in converfation :
nothing is prefented to him except what his reafon fully concurs
in, and no inference except what is indifputable. At a fudden
turn of the argument, he fees a favourite conclufion, which he
cares more for than for all the reafonings that ever were put
together, upfet and broken to pieces. He confiders himfelf an
ill-ufed man, entrapped, fwindled out of his lawful goods ; and
he therefore returns upon his fteps, and finds out that fome of
the things which he admitted when he did not fee their con-
fequences, are no longer admiffible. Neither he nor the oppo
nent has the leaft idea of the nature of probable arguments, and
of their oppofition : both proceed as if the train of reafoning were
either demonftration or nothing. The conclufion, formed perhaps
upon teftimony, which is more likely to be a guide to truth for
the mind in queftion than any appreciation of argument which
that mind could make, muft, according to the maxims of the age,
be referred to argument, and argument only. The perpetual and
wilful fallacy of that mind is the determination that all argument
fhall fupport, and no argument mall make, the conclufion. If
there were only a diftincl: perception of another fource of con
viction, fo ftrong that ordinary argument can neither materially
weaken, nor materially confirm it, there would be fenfe in the
conclufion ; fenfe, becaufe there is truth. Right or wrong, fuch
is the fource of moft convictions in, perhaps, moft minds : fuch
fource ought therefore to be acknowledged. It would be an ex
cellent thing, if, in any difputed matter, thofe who are better
fatisfied by authority of the truth of one fide of the conclufion
than of the validity of argument in general, would avow it, keep
their own fide, and let others do the fame. But here is the diffi
culty : the perfons who mould avow fuch a ftate of mind are as
much difpofed to make converts as others : they do not like to
On Fallacies. 265
debar themfelves from diflemination of their opinions. Accor
dingly they propound their beft arguments, be they what they
may, as what ought to produce all the conviction which them
felves feel. On this point fee page 194.
The whole clafs of argumenta ad kominem^ having fome refe
rence to the particular perfon to whom the argument is addrefTed,
will generally be found to partake of the fallacy in queftion. Such
are recrimination and charge of inconfiftency, as, You cannot ufe
this afTertion, becaufe in fuch another cafe you oppofe it. But
if the original argument itfelf fhould be a perfonal attack, then
fuch a retort as the preceding may be a valid defence.
In many fuch argumenta ad hominem^ it is not abfolutely the
fame argument which is turned againft the propofer, but one
which is aflerted to be like to it, or parallel to it. But parallel
cafes are dangerous things, liable to be parallel in immaterial
points, and divergent in material ones. A celebrated writer on
logic afTerts, that no one who eats meat ought to object to the
occupation of a fportfman on the ground of cruelty. The parallel
will not exift until, for the perfon who eats meat, we fubftitute
one who turns butcher for amufement. There is, or was, a vul
gar notion that butchers cannot fit on a jury. Suppofe that fuch
a law were propofed, on the ground of the habits arifing from
continual infliction of death. Would it really be a counter-argu
ment that men who eat meat have the fame animus and are liable
to acquire the fame habits. It is contended (juftly or not) that
a defire to take life for fport is a cruel defire ; to anfwer that
thofe who eat flefh from which life has been taken by others have
therefore alfo cruel defires, ought to be called arguing a ditto
fecundum quid ad diffum fecundum alterum quid. The matter is
clear enough. Cruelty of intention (the thing in queftion) muft
be fettled by our judgment of the circumftance in which the
fport confifts. A perfon who feeks bodily exercife and the ex
citement of the chafe, and who can acknowledge to himfelf that
his object is gained on the birds which he mifles, as well as upon
thofe which he hits, even if thoughtlefs, cannot be faid to act
with cruelty of intention. But the fportfman, as he calls himfelf,
who collects his game in one place, merely that he may kill,
without exercife, or feeling of fkill, is either culpably thoughtlefs,
oj elfe a favage, who delights in the infliction of death. Let any
266 On Fallacies.
man afk himfelf, whether in the event of his being called upon to
vote for a perfectly abfolute fovereign, he would feel much con
cerned to inquire whether the candidate was or was not a fportf-
man of the firft kind : and then let him afk himfelf the fame
queftion with refpedt to the fecond.
The moft amufing, and perhaps the moft common, example
of the ignoratio elenchl, is the taking exception to fome part of
an illuftration which has nothing to do with the parallel. The
word illuftration (though it mean throwing light upon a thing)
is ufually confined to that fort of light which is derived from
mowing a procefs of difficulty employed upon an eafier cafe.
The firft fallacy may be committed by the illuftrator. He has
before him the fubject matter of the premifes, their connexion in
the procefs of inference, and the refult produced. Either may
be illuftrated ; thus, if it be doubtful whether fuch premifes
may be employed, the illuftrator may throw away his mode of
connexion, and choofe another : if the procefs of inference be
doubtful, he may choofe other premifes : and fo on. But he may
illuftrate the wrong point : and this is a fallacy very common to
teachers and lecturers. The greateft difficulty in the way of
learners is not knowing exactly in what* their difficulty confifts ;
and they are apt to think that when fomething is made clear, it
muft be the fomething. I am of opinion that the examples
given of fyllogifms in works of logic are examples of wrong
illuftration. The point in queftion is the form, the object is to
produce conviction of the form, of its necefTary validity. If the
ftudent receive help from an example ftated both in matter and
form, the odds are that the help is derived from the plainnefs
of the matter, and from his conviction of the matter of the con-
clufion. If this be the cafe, he has not got over his difficulty.
Many learners are puzzled to fee that Every Y is X is not a
neceflary confequence of* Every X is Y. If the want of con-
* Every learner, in eveiy fubjeft, fhould accuftom himfelf to endeavour
to ftate the point of difficulty in writing, whether he want to mow the re
fult to another or not. I wifh I had kept a record of the number of times
which I have infifted on this being done, previoufly to undertaking the ex
planation, and of the proportion of them in which the writer has acknow
ledged that he faw his way as foon as he attempted to aflc the road in precife
written language. That proportion is much more than one half. Truly
faid Bacon, that writing makes an exaft man.
On Fallacies. 267
nexion be eftablimed by an inftance, as by appealing to their
knowledge that every bird is not a goofe, though every goofe be
a bird, their knowledge of the proportion is not logical. The
right perception may, no doubt, be acquired by reflection on in-
ftances : but the minds which are beft fatisfied by material in-
ftances, are alfo thofe which give themfelves no further trouble.
The illuftration being fuppofed correct, there is more than one
fallacious mode of oppofing it. Some perfons will difpute the very
method of illuftration of form, in which the fame mode of infer
ence is applied toeafier matter ; but thefe are mere beginners, hardly
even entitled to a name which fuppofes the poffibility of progrefs.
Others will deny the analogy of the matter, and thefe there is no
means of meeting : for illuftration is ad hominem^ and the per
ception of it cannot be made purely and formally inferential : a
denier of the force of an illuftration is inexpugnable as long as
he only denies. But when he attempts more, when he indicates
the point in which the illuftration fails, he very often falls into the
error of attacking an immaterial point. If any one were to con
tend (as fome do) that it is unlawful to take the life of any ani
mal, he might be afked what he would fay if Guy Faux had
trained a pigeon to carry the match to the vault, would it have
been lawful to moot the bird on its way or not ? There are not
a few who would think it an anfwer to fay that he could not
have trained the pigeon, or that pigeons were not then trained
to carry.
5. ^\\Q fallacla confequentis (now very often called a non fe-
quitur] is the fimple affirmation of a conclufion which does not
follow from the premifes. If the fchoolmen had lived in our
day, they would have joined with this the affirmation of logical
form applied to that which wants it, a very common thing among
us. A little time ago, either the editor or a large-type correfpon-
dent (I forget which) of a newfpaper imputed to the clergy the
maintenance of the c logic of the following as 4 confecutive and
without flaw. This was hard on the clergy (particularly the
Oxonians) for there was no middle term, neither of the conclud
ing terms was in the premifes, and one negative premife gave a
pofitive conclufion. It ran thus,
Epifcopacy is of Scripture origin.
268 On Fallacies.
The church of England is the only epifcopal church in Eng
land,
Ergo, the church eftablifhed is the church that fhould be fup-
ported.
Many cafes offend fo (lightly that the offence is not perceived.
For inftance c knowledge gives power, power is defirable, there
fore knowledge is defirable is not a fyllogifm ; there is no mid
dle term. It is a forites, as follows, knowledge is a giver of
power, the giver* of power is the giver of a defirable thing, the
giver of a defirable thing is defirable, therefore knowledge is de
firable.
It mould be noted, however, that the copula c gives refem-
bles is greater than (page 5) and is an admiffible copula in in
ferences with no converfion, provided that c A gives B and B gives
C/ implies c A gives C. The fame may be faid of the verbs to
bring, to make, to lift, &c. And many of thefe verbs are, by
the unfeen operation of their having the effect of is in inference,
often fupplanted by the latter verb in phrafeology. Thus we
fay * murder is death to the perpetrator where the copula is
brings ; c two and two are four the copula being c have the
value of &c. But this practice may lead to fallacies, as above
mown : which muft be avoided by attention to the clafs of verbs
which communicate their action or ftate, fuch as make, give,
bring, lift, draw, rule, hold, &c. &c. All thefe verbs are applied
to denote the caufe of the feveral actions : fo, to give that which
gives, or to bring that which brings, is to give or to bring. The
boy who was faid to rule the Greeks becaufe he ruled his mo
ther, who ruled Alcibiades, who ruled the Athenians, who ruled
the Greeks, would have been corredMy faid fo to do, if the mat
ters of rule had been the fame throughout.
6. The non caufa pro caufa. This is the miftake of imagin
ing neceffary connexion where there is none, in the way of caufe,
confidered in the wideft fenfe of the word. The idioms of lan
guage abound in it, that is, make their mere expreffions of phe
nomena attribute them to apparent caufes, without intent to
afTert real connexion. Thus we fay that a tree throws a madow,
* Becauie power is defirable. See page 115, as to this llcp.
On Fallacies. 269
to dcfcribe that it hinders the light. When the level of a billiard
table is not good, the favoured pocket is faid to draw the balls.
A particular cafe of this fallacy, which is often illuftrated by
the words poft hoc^ ergo propter hoc, is the conclufion that what
follows in time follows as a confequence. When things are feen
together, there is frequently an aflumption of necefTary connexion.
There is, of courfe, a prefumption of connexion : if A and B have
never been feen apart, there is probability (the amount of which
depends upon the number of inftances obferved) that the removal
of one would be the removal of the other. It is when there is
only one inftance to proceed upon that the ailumption falls under
this fallacy ; were there but two, induclive probability might be
faid to begin. The fallacy could then confift only in eftimating
the probability too high.
As may be fuppofed, the non caufa pro caufa arifes more often
from mere ignorance than any other fallacy. To take the two
inftances that I happened to meet with neareft to the time of
writing this page ; Walpole, remarking on the uniform practice
among the old writing-mafters of putting their portraits at the
beginning of their works, remarks that thefe men feem to think
their profeffion gives pofterity a particular intereft in their fea
tures. Probably they did not think about it : the ufage of the
day prevented any man from being chargeable with undue va
nity who exhibited his phyfiognomy, and moft of the writing
mafters were tbemfelves engravers^ and either did their own por
traits, or more probably made ufe of their acquaintance with the
more celebrated engravers for whom they did the under drudgery,
to get themfelves done on eafy terms. Again, Noble (in his con
tinuation of Granger) remarks that Saunderfon had fuch a pro
found knowledge of mufic, that he could diftinguim the fifth
part of a note. The author did not know, firft, that any perfon
who cannot diftinguifh lefs than the fifth part of a note to begin
with, mould be bound over to keep the peace if he exhibit the
leaft intention of learning any mufical inftrument in which in
tonation depends upon the ear ; and fecondly, that if Saunderfon
were not fo gifted by nature, knowledge of mufic would no more
have fupplied the defect, than knowledge of optics would give
him fight.
Thefallacia plurium interrogationum confifts in trying to get
270 On Fallacies.
one anfwer to feveral queftions in one. It is fometimes ufed by
barrifters in the examination of witnefTes, who endeavour to get
yes or no to a complex queftion which ought to be partly anfwered
in each way, meaning to ufe the anfwer obtained, as for the whole,
when they have got it for a part. An advocate is fometimes
guilty of the argument a ditto fecundum quid ad diftum Jimpli-
citer : it is his bufinefs to do for his client all that his client might
honeftly do for himfelf. Is not the word in Italics frequently
omitted ? Might any man honeftly try to do for himfelf all that
counfel frequently try to do for him ? We are often reminded of
the two men who ftole the leg of mutton ; one could fwear he
had not got it, the other that he had not taken it. The counfel
is doing his duty by his client ; the client has left the matter to
his counfel. Between the unexecuted intention of the client,
and the unintended execution of the counfel, there may be a
wrong done, and, if we are to believe the ufual maxims, no
wrong doer. The anfwer of the owner of the leg of mutton is
fometimes to the point, c Well, gentlemen, all I can fay is, there
is a rogue between you. That a barrifter is able to put off his
forenfic principles with his wig, nay more, that he becomes an
upright and impartial judge in another wig, is curious, but cer
tainly true.
The above were the forms of fallacy laid down as moft effen-
tial to be ftudied by thofe who were in the habit of appealing to
principles fuppofed to be univerfally admitted, and of throwing
all deduction into fyllogiftic form. Modern difcuffions, more
favourable, in feveral points, to the difcovery of truth, are con
ducted without any conventional authority which can compel
precifion of ftatement : and the neglect of formal logic occa-
fions the frequent occurrence of thefe offences againft mere rules
which the old enumeration of fallacies feems to have confidered
as fufficiently guarded againft by the rules themfelves, and fuf-
ficiently defcribed under one head, thefallacia confequentis. For
example, it would have been a childifh miftake, under the old
fyftem, to have afferted the univerfal propofition, meaning the
particular one, becaufe the thing is true in moft cafes. The rule
was imperative : not all muft be fome^ and even #//, when not
known to be #//, was fame. But in our day nothing is more
common than to hear and read affertions made in all the form,
On Fallacies. 271
and intended to have all the power, of univerfals, of which no
thing can be faid except that moft of the cafes are true. If a
contradiction be aflerted and proved by an inftance, the anfwer
is Oh ! that is an extreme cafe. But the aflertion had been
made of all cafes. It turns out that it was meant only for ordi
nary cafes ; why it was not fo flated muft be referred to one of
three caufes ; a mind which wants the habit of precifion which
formal logic has a tendency to fofter, a defire to give more
ftrength to a conclufion than honeftly belongs to it, or a fallacy
intended to have its chance of reception.
The application of the extreme cafe is very often the only teft
by which an ambiguous aflumption can be dealt with : no won
der that the aflumer fhould dread and proteft againft a procefs
which is as powerful as the fign of the crofs was once believed
to be againft evil fpirits. Where anything is aflerted which is
true with exceptions, there is often great difficulty in forcing the
aflertor to attempt to lay down a canon by which to diftinguifh
the rule from the exception. Every thing depends upon it : for
the queftion will always be whether the example belongs to the
rule or the exception. When one cafe is brought forward which
is certainly exception, the aflertor will, in nine cafes out of ten,
refufe to fee why it is brought forward. He will treat it as a
fallacious argument againft the rule, inftead of admitting that it
is a good reafon why he fhould define the method of diftinguifh-
ing the exceptions : he will virtually, and perhaps abfolutely, de
mand that all which is certainly exception mail be kept back,
fimply that he may be able to aflume that there is no occafion to
acknowledge the difficulty of the uncertain cafes.
The ufe of the extreme cafe, its decifive effecT: in matters of
demonftration, may furnifh prefumption as to what it is likely to
be in matters of aflerted near approach. As in the following in
ftance. It feems almoft matter of courfe, when ftated, to thofe
who have not ftudied the fubject of life contingencies, that the
proper value of a life annuity is that of the annuity made certain
during the average exiftence of fuch lives as that of the annuitant.
That if, for example, perfons aged 22 live, one with another,
40 years, an office which receives from every fuch perfon the
prefent value of forty payments certain, will, without gain or lofs,
in the long run, be able to pay the annuities. If this be (as was
272 On Fallacies.
ftoutly contended by fome writers of the laft century) a univerfal
truth, it will hold in this extreme cafe. Let there be two per
fons, one of whom is certain to die within a year from the grant
(and therefore never claims anything) and the other of whom is
certain to live for ever. It is clear that the value of an annuity
to both is o + the value of a perpetual annuity. But the average
life of both is eternal : one perpetual duration makes the average
of any fet in which it is, perpetual. Hence by the falfe rule the
value is two perpetual annuities, or juft double of the truth.
We might fuppofe that moft perfons have no idea of a uni
verfal propofition : but ufe the language, never intending all to
fignify more than moft. And in the fame manner principles are
ftated broadly and generally, which the aflertor is afterwards at
liberty to deny under the phrafe that he does not carry them fo
far as the inftance named. It would not do to avow that the
principle is not always true : fo it is ftated to be always true^ but
not capable of being carried more than a certain length. Are
not many perfons under fome confufion about the meaning of
the word general? In fcience it always has the meaning of uni
verfal : and the fame in old Englifh. Thus the catechifm of
the church of England aflerts that there are two facraments
which are generally neceflary to falvation : meaning neceflary for
all of the genus in queftion, be it man, Chriftian, member of the
church, or any other. But in modern and vernacular Englifh,
general means only ufual^ and generally means ufually.
A great deal of what is called evafion belongs to this head,
or to that of the Ignoratlo elenchi, as the fophift anfwers. The
advocates, for inftance, of the abfolute unlawfulnefs of war never
tell, unlefs prefled, what they think of the cafe of refiftance to
invafion. Is the country to be given up to the firft foreigner
who choofes to come for it ? Sometimes the extreme cafe comes
into play : fometimes the aflertion that no one will come ; which
is irrelevant as to the queftion what would be right if he did
come.
Among amufmg modern evafions are There is no occafion to
confider that and C I do nt confider it in that point of view.
Any one who watches the manner in which men defend their
opinions will frequently fee * A is B and B is C, therefore A is
C anfwered, not by denial of either premife, but by that is not
On Fallacies. 273
the proper point of view or c I don t fee it in that light. This
fhould be called the confufion between logic and perfpective.
The denial of one univerfal is often made to amount to, or to
pafs into, the aflertion of the oppofite, or fubcontrary, univerfal.
This craving after general truths, the moft manifeft fault of the
old logicians in their choice of premifes, did not expire with them.
Bacon fays c the mind delights in fpringing up to the moft gene
ral axioms, that It may find reft. Many perfons are defirous of
fettled opinions, which is well ; unlefs by fettled opinions they
mean univerfal, as is often the cafe. That fome are and fome
are not is no fettlement : it makes every cafe require examina
tion, to fee under which it falls. And with the above we may
couple the tendency to believe that refutation of an argument is
proof of the falfehood of its conclufion, and that a falfe confe-
quence muft be a falfe proportion. Hence it arifes that fo many
perfons dare not give up any argument in favour of a proportion
which they fully believe : they think they abandon the propo-
fition.
It fometimes happens that an aflertion is made, which it is
difficult to fuppofe can be anything but a cafe of a univerfal pro-
pofition : and yet the afTertor takes care not to make his pro-
pofition univerfal, but perfifts in the particular cafe. A logician
in our day has aflerted that when Calvin fays that all officers of
the church mould be elected by the people, he muft be under-
ftood as fpeaking in reference to deacons only, becaufe the
aflertion is made in the chapter on deacons. If it had been
roundly ftated that all univerfal propofitions are to have their
univerfes limited by the headings of the works or chapters in
which they occur for inftance, that the aflertion that all men
are mortal, occurring in a hiftory of England, is to be taken as
made of Englifhmen only there would have been at leaft no
ambiguity. But as it is, we are left to furmife whether this be
meant, or whether the proportion be to apply to Calvin only, or
to Reformers only, or to men whofe names begin with C, &c.
The odds are that the application of a univerfal propofition will
be dictated by the heading of a chapter : but the extent to v/hich
a premife is aj/erted as true is not to be judged of by that to which
it is wanted for ufe : and the lefs, the nearer we go to the day of
the old logicians.
274 On Fallacies.
Wrong views of the quantity of a proportion are as frequent
as any fallacies. Some, meaning moft, and feme , meaning few,
are frequently confounded. This is the neceflary confequence
of the nature of human knowledge, in which we can but rarely
form a definite idea of the proportion which the extent fpoken of
bears to the whole. It is part of the value of the mathematical
theory of probabilities, that the mind is accuftomed to the view
of refults drawn from perfectly definite fuppofed cafes -, as ufelefs,
it may be, in themfelves, as many of the queftions in a book of
arithmetic, but neverthelefs good for exercife. It is not furprif-
ing that fallacies about quantity mould be capable of moft ftrik-
ing expofure in queftions concerning meafurable quantity, that
is, in queftions of mathematics : nor that there mould be clafles
of fallacy of which it is difficult to illuftrate the detection by any
other inftances. What can be more clear, for example, to ordi
nary apprehenfions than the broad ftatement that c of things of
the fame kind, that which is fometimes right muft be better than
that which is always wrong. But a little confideration will
fuggeft that what is always wrong may be as good as that which
is fometimes right, if we do not know how to diftinguifh the
cafes in which the latter is right : and alfo that what is not
much wrong, generally, may be more ufeful than that which is
moftly very wrong, when it is not abfolutely right. A watch
which does not go is right twice a day : but it is not fo ufeful as
one which does go, though very badly.
To give an account of all the fallacies which depend upon
wrong notions of quantity would require much fpace, and more
affumption of mathematical knowledge in my reader than is con-
fiftent with my plan. But I may mention the miftaken ufe of
abfolute terms and notions in queftions of degree. There can
be, a difputant will fay, but a right and a wrong ; and if this be
not right, it is wrong. Many perfons will announce that their
watches are quite right, abfolutely at the true time, to a fecond :
and will end by giving the time which was mown when they
looked, as being accurately that of the inftant at which they
announce it. The proverb Fruftra fit per plura, quod fieri po-
tefl per pandora contains an inaccuracy of degree : a bargain
which cofts twenty millings and is worth fifteen, is not twenty
(hillings loft, but only five, though the vexation of the party
overreached will feldom fufFer him to fee this.
On Fallacies. 275
Proverbs in general are liable to this miftalce. They are often
ufed in exactly the fame manner as the firft principles of the old
logicians. In fact, remembering that thefe firft principles were
bandied from mouth to mouth till they were perfectly proverbial,
as we now call it, among the learned ; and obferving the appli
cation of our modern proverbs, as made by the mafs of thofe
who have not profited by mental difcipline, we may fee that the
faults of the fchoolmen are only thofe of the ordinary human
mind. It is hard indeed if there be a purpofe which a proverb
cannot be found to ferve : it is a univerfal propofition of no very
definite meaning, fanclioned by ufage, having the appearance of
authority, and capable of ftretching or contracting like Prince
Ahmed s pavilion. One only is allowable In generalibus latet
error : this deftroys all the reft, and then, when clofely looked
at commits fuicide.
All miftakes of probability are eflentially miftakes of quantity,
the fubftitution of one amount of knowledge and belief for an
other. It is often difficult to convey a proper notion of the
degree of force which is meant to be given ; and ftill more fo
to retain it throughout the whole of a difcuffion. A perfon be
gins by ftating an explanation as poffible, or probable enough to
require confideration, as the cafe may be. The forms of language
by which we endeavour to exprefs different degrees of probability
are eafily interchanged ; fo that, without intentional difhonefty
(but not always) the propofition may be made to flide out of one
degree into another. I am fatisfied that many writers would
ftirink from fetting down, in the margin, each time they make a
certain afTertion, the numerical degree of probability with which
they think they are juftified in prefenting it. Very often it hap
pens that a conclufion produced from a balance of arguments,
andyfr/? prefented with the appearance of confidence which might
be reprefented by a claim of fuch odds as four to one in its favour,
is afterwards ufed as if it were a moral certainty. The writer
who thus proceeds, would not do fo if he were required to write
% in the margin every time he ufes that conclufion. This would
prevent his falling into the error in which his partifan readers are
generally fure to be more than ready to go with him, namely,
turning all balances for, into demonftration, and all balances
againft, into evidences of impoflibility.
One of the great fallacies of evidence is the difpofition to dwell
276 On Fallacies.
on the actual poffibility of its being falfe : a poffibility which muft
exift when it is not demon ft rative. Counfel can bewilder juries
in this way till they almoft doubt their own fenfes. A man is
fhot, and another man, with a recently difcharged piftol in his
hand, is found hiding within fifty yards of the fpot, and ten mi
nutes of the time. It does not follow that the man fo found
committed the murder : and cafes have happened, in which it has
turned out that a perfon convicted upon evidence as ftrong as the
above, has been afterwards found to be innocent. An aftute
defender makes thefe cafes his prominent ones : he omits to men
tion that it is not one in a thoufand againft whom fuch evidence
exifts, except when guilty.
All the makers of fyftems who arrange the univerfe, fquare the
circle, and fo forth, not only comfort themfelves by thinking of
the neglect which Copernicus and other real difcoverers met
with for a time, but fometimes fucceed in making followers.
Thefe laft forget that for every true improvement which has
been for fome time unregarded, a thoufand abfurdities have met
that fate permanently. It is not wife to tofs up for a chance of
being in advance of the age, by taking up at hazard one of the
things which the age pafles over. As little will it do to defpife
the ufual track for attaining an object, becaufe (as always hap
pens) there are fome who are gifted with energies to make a road
for themfelves. Dr. Johnfon tells a ftory of a lady who ferioufly
meditated leaving out the claffics in her fon s education, becaufe
(he had heard Shakfpeare knew little of them. Telford is a
{landing proof (it is fuppofed by fome) that fpecial training is
not eflential for an engineer.
The difpofition to judge the prudence of an action by its refult,
contains a fallacy when it is applied to fmgle inftances only, or
to few in number. That which, under the circumftances, is the
prudent rule of conduct, may, neverthelefs end in fomething as bad
as could have refulted from want of circumfpection. But upon
dozens of inftances, fuch a balance would appear in favour of pru
dence as would leave no doubt in favourofthe rule of conduct, even
in the inftances in which it failed. The fallacy confifts in judging
from the refult about the conduct of one who had only the previous
circumftances to guide him. c You acted unwifely, as is proved
by the refult, is a paralogifm, except when it implies c You did,
On Fallacies. 277
as it happens in this inftance, take a courfe which did not lead to
the defined refult. Take a ftrong cafe, and the abfurdity will be
feen. A chemifr. makes up a prefcription wrongly, and his cuf-
tomer leaves him for another : this other, fo it may happen, makes
it up frill more wrongly, and poifons the patient. Who would
venture to fay that he acted unwifely, as is proved by the refult,
in leaving the tradefman whom he knew to be carelefs, for another
of whom he knew no harm. The only way in which blame can
be imputed, is when it can be faid You acted unwifely, in not
finding out, as you might have done, that the refult which has
happened is the one which was likely to happen. One refult
proves very little as to the fuperior wifdom of the courfe which
produced it ; feveral may give a prefumption of it, and the greater
the number, the greater the prefumption.
So little is this thought of, that the common phrafe, c I acted
for the beft, meaning originally I acted in the manner which
under the circumftances, appeared likely to lead to the beft re-
fults, very often lofes its proper meaning, and is ufed as fynony-
mous with c I acted with good intentions/
Thefe, and many other points, I can only flightly touch on :
I will proceed to notice a few other caufes of error.
And firft, of equivocations of ftyle. I have before referred to
fuch a phenomenon as the alteration of a good ryllogifm into a
bad one, to make the fentence read better. But nothing ever
reads well (for a continuance) except the natural current of a
writer s thought. I fhould like it to be the law of letters, that
every book fhould have inferted in it the printer s affidavit, fetting
forth the number of verbal erafures in the manufcript, fair copies
being illegal. It would be worth at leait one review.
There is a wilful and deliberate equivocation, which it is
fuppofed the age demands. It is the ufe of fynonymes, or fup-
pofed fynonymes, to prevent the fame word from occurring twice
in the fame paflage. So far is the neceffity of this practice recog
nized, that there are few printing-offices in London, the readers
of which do not query the fecond introduction of any word which
prominently appears twice. And then the author obeys the hint,
frrikes out one of the offenders, fticks in a dictionary equivalent,
and would have been content if the printer s reader had done it
for him. And fo he writes a good ftyle. To be fure, he does not
278 On Fallacies.
fay what he meant, exactly ; for fynonymes are feldom or never
logical equivalents : but what is that to elegance of expreflion ?
The demand for non-recurrence of words arifes from the pub
lic (I beg its pardon) not knowing how to read. If, when a
word occurs twice, the proper emphafes were looked for, and
obferved, there would be nothing ofFenfive about the repetition.
It is the reader who makes one and one into two, by giving both
units equal value. Take this fentence from Johnfon, (the firft
I happened to light on, in the preface to Shakfpeare), and read
it firft as follows : " He therefore indulged his natural difpofition :
and his difpofition^ as Rymer has remarked, led him to comedy :"
and then as follows " He therefore indulged his natural difpofi
tion ; and his difpofition, as Rymer has remarked, led him to
comedy." This reading is what the context requires, and the ill
effect of the repetition is next to nothing. Take the next fen
tence : " In tragedy he often writes, with great appearance of
toil and ftudy, what is written at lajl with little felicity : but in
his comic fcenes he feems to produce, without labour^ what no
labour can improve." Thefe were the firft inftances I found,
from a chance opening of the Elegant Extracts, purpofely chofen
as a mifcellany. The laws of thought generally dictate this rule,
that the firft occurrence of a word is the more emphatic of the
two : the leflbn of experience is, that a writer who prevents re
currence by the ufe of the dictionary of fynonymes, is a good
ftyle-maker for none but a bad reader, and may very poflibly be
a good arguer for none but a bad logician. Of courfe, I fhould
not deny that recurrence of both word and emphafis is a defect,
if it be frequent.
The confufion between the means and the end, and putting
one in the place of the other, is well enough known in morals :
but there is a correfponding tendency to forget the diftinction
between the principle which is to be acted on, and the rule of
action by which adherence to that principle is fecured. A refe
rence to the derived rule is in all refpedts as good as one to the
firft principle, between parties who underftand both, and the
connexion between them. But thofe who underftand the rule
only, are apt to forget that a rule may or may not be the true
expreflion of a principle, according to the circumftances in which
it is propofed to apply it. If, indeed, it were of univerfal appli-
On Fallacies. 279
cation, thofe who do and thofe who do not underftand the prin
ciple might be on the fame footing as to fecurity : but there are
few fuch rules.
The preceding caution may be applied in all departments of
thought, in law and in logic, in morals and in arithmetic. It is
impoflible, for inftance, to ftate the rule of three in fuch a man
ner as eafily to include the cafes in which it mail apply, and ex
clude thofe to which it does not. To fay that it muft be ufed
where the fourth quantity, the one fought, is to be a fourth pro
portional to the three which are given, though correct, ftill leaves
it open to inquiry what are the cafes in which this condition is to
be fatisfied : and many cafes might be, and are propofed, in which
the inquiry is not eafy to a beginner. In law, there are not only
rules, but rules for their application. To an unlearned fpectator,
particularly in the courts of equity, in which the advocate addrefles
a judge, and not a jury, the argument takes that technical form
which makes many perfons think that the whole law is, at beft,
only arbitrary rule. It may be that fome of thofe who there
addrefs the court can make nothing better of it : and juft as there
are arithmeticians, and good ones too, who are but the flaves, and
never the mafters, of their procefles, fo there may be advocates,
and even judges, who have not one element of the legiflator in
them. But there are enough of a higher fpecies.
The great art of ufmg rules is to apply them in aid, and not
in contravention, of the principles which they are intended to
embody. A rule may have exceptions, it is faid ; but this is
hardly a correct ftatement. A rule with exceptions is no rule,
unlefs the exceptions be definite and determinable : in which cafe
the exceptions are exclufions by another rule. The parallel is
perfect between rules and proportions (page 143). Thus, c All
Europe, except Spain and Portugal is a univerfal propofition ;
but All the ftates of Europe except two 7 is a particular one. A
rule which applies to all ftates except Spain and Portugal is a
rule : but a rule which applies to all except two (unknown) is no
rule. When it is ftated, in ordinary language, that every rule is
fubject to exception, it is meant, for the moft part, that the cir-
cumftances under which adherence to the rule gains the object,
are thofe which moft frequently occur, and that the circumftances
under which adherence to the rule would defeat the object are
280 On Fallacies.
rare. If this were remembered, much confufion would often be
faved. We want a word which (hall fo far exprefs rule, that it
fhall imply that which will generally fucceed, without the notion
of obligation which accompanies that of rule, and which perpetu
ally mifleads. We want, in fact, the rule nifi of the courts, which
is to be a rule unlefs caufe be ftiown againft it : and which will,
in moft cafes, be ultimately made abfolute, but is not abfolute
from the beginning.
The common miftake is, that the rule nifi is an abfolute rule,
and that therefore it may be fubftituted for its leading object or
firft principle, and that even the very words which exprefs that
objecl: gained, may be taken as equally expreffive of fatisfaction
of the rule, and vice verfd. For inftance, it is commonly ftated
that the rule by which a difcoverer is determined, is publication ;
that he who firft publifhes the difcovery, is to he held the difco
verer ; one lapfe more, and it is faid that he is the difcoverer ;
yet one more, and it will be faid that the publication is the difco
very. The very remarkable circumftances attending the recent
difcovery of the planet Neptune, involving points of peculiar in-
tereft and delicacy, have caufed this rule to be much difcufTed, and
have brought out every variety of ftatement of it. The thing to
be determined is the aftual truth of the queftion, the real hiftory
of the human mind with regard to it. No one has a right under
any rule, no matter what its authority, nor by whom impofed, to
fubftitute the thing which is not, for the thing which is, or the
lefs probable for the more probable. If philofophers were to at
tempt, by a law of their own framing, to fubftitute the conven
tional refult for the real one, the common fenfe of mankind would
difpute their authority, and reverfe their decifion. The firft rule
(nifi) is undoubtedly that the firft printer is the firft publifher, the
fecond, that the firft publiftier is the difcoverer. Thefe will, un
lefs caufe be fhown againft them, be made abfolute in every cafe.
A notion which is very prevalent, namely, that the firft publifher
has therefore the rights of the difcoverer, is as incorrect as that
the firft printer is therefore the firft publifher. To take the cur
rent language, one would fuppofe that printing one hundred
copies would be held better than circulating one thoufand in
manufcript, and that even though the firft publifher could be
proved to have plngiarifed, he has ftill the rights of difcovery.
On Fallacies. 281
Juft as (page 244) early notions make laws of literal interpre
tation fuperfede thofe of intended meaning, fo, in the earlier
ftages of law, rules are often made to over-ride the principles on
which they profefs to be founded, and to defeat truth and common
fenfe. There is more excufe here than there would be in a
queftion of fcience, for peace and convenience are main objects
of law, and it may be that rigid adherence to a rule, as a rule, at
a certain avowed facrifice of truth and juftice, may be the only
practicable means of preventing a larger facrifice of both. In old
times, the rule of affiliation, Pater eft quern nupticz demonftrant,
was held fo abfolutely, that the hufband of the mother would be
the legal father, though the two had been confined in two diffe
rent jails a hundred miles apart for twelve months preceding the
birth of the child. The modern law has made this rule to be no
more than it ought to be, namely, one which muft hold unlefs
the contrary be proved.
It is not uncommon, in difputation, to fall into the fallacy of
making out conclufions for others by fupplying premifes. One
fays that A is B ; another will take for granted that he muft be
lieve B is C, and will therefore confider him as maintaining that
A is C. But it may be that the other party, maintaining that
A is B, may, by denying that A is C, really intend to deny that
B is C. In religious controverfy, nothing is more common than
to reprefent feels and individuals as avowing all that is efteemed
by thofe who make the reprefentation to be what, upon their pre
mifes, they ought to avow. All parties feem more or lefs afraid
of allowing their opponents to fpeak for themfelves. Again, as
to fubjecls in which men go in parties, it is not very uncommon
to take one premife from fome individuals of a party, another
from others, and to fix the logical conclufion of the two upon
the whole party : when perhaps the conclufion is denied by all,
fome of whom deny the firft premife by affirming the fecond,
while the reft deny the fecond by affirming the firft. Any feel:
of Chriftians might be made atheifts by logical confequence, if it
were permitted to join together the premifes of different fections
among them into one argument. This is a fallacy which, how
ever common, could eafily be avoided, and would be, if thofe
who ufe it cared for anything but victory. But there is another
form of the fame, which every one is fubjecT: to, and which it is
282 On Fallacies.
not fo eafy to perceive. It is that of drawing upon our former
felves for the premifes which are to guide us for the time being.
Conclufions remain in our minds long after the grounds on which
they were formed are abandoned : and it may happen that one
premife of an argument will ftill have force, when the very rea-
fons on which the fecond premife is now admitted are contra
dictory of thofe which once induced us to admit the firft. Thus
many who have learnt to advocate the legal toleration of opinions
which they ftill believe, by force of education, to be abfolute
crimes againft fociety, are logically the advocates of toleration of
crime ; whereas, the arguments which they have learned to think
valid for the firft premife, ought, if worth anything, to teach
them to deny the fecond. I have myfelf heard from one mouth
in one converfation (of courfe not in one part of it) that all fins
againft the Creator are fins againft fociety, that all fins againft
fociety ought to be punifhed by fociety, that certain opinions then
named are fins againft the Creator, and that it is the height of
injuftice to punifh any one for his opinions.
In printed controverfy, the ftatement of the oppofite opinion
or aflertion may be made by defcription without citation (by
chapter or page), by defcription with citation, or by quotation
with or without defcription. The firft is not allowable. The
prefumption is ftrong that a perfon who oppofes an opinion, im
putes an error, or makes a charge, upon the writings of another,
is bound at leaft to cite, in a manner which cannot be miftaken,
the part of thofe writings to which he refers. There are writers
who refer defcriptively and even commentatively, putting the
reference of citation, and thus (as Bayle fays Moreri conftantly
does) lead the reader to fuppofe that the words of their para-
phrafe and comment are thofe of the paflage itfelf. I do not fee
that quotation is obligatory, though highly defirable : but the
reader muft remember, when there is only citation, that it is not
the author cited who fpeaks, but the perfon who brings him for
ward. It is a man s own account of his own witnefs : with the
advantage of an apparent offer of enabling the reader to go and
verify the ftatement for himfelf. If the citer be honeft, the paf-
fage in queftion exifts : if judicious, it is to the effecT: ftated.
Confequently, whenever the citer s honefty or judgment is ex-
prefsly in queftion, no mere citation is admiflible.
On Fallacies. 283
When citations are few they ought perhaps to be quotations :
when they are many, it may be impracticable to make them fo.
But extenfive citation ought to be encouraged. Lazy readers
do not like it : they are not pleafed to have a power of verifica
tion offered of which they do not mean to avail themfelves ; and
they would rather, in cafe of being mifled, have to throw the
blame upon the author than upon their own non-acceptance of
the offered means of verification. Accordingly, they exprefs
their difguft at " pages loaded with references." But the more
diligent readers confider every citation as a boon. At the fame
time it is to be remembered that there are writers who, relying
on the common difinclination to verify, add a large number of
citations, and give the appearance of a ftrong body of authorities,
which are often nothing to the purpofe, and fometimes not taken
from actual examination, but copied from other writers.
Perhaps the greateft and moft dangerous vice of the day, in
the matter of reference, is the practice of citing citations, and
quoting quotations, as if they came from the original fources,
inftead of being only copies. It is in truth the reader s own
fault if he be taken in by this, or by the falfe appearance of au
thority juft alluded to ; for it is in his own power to certify him-
felf of the truth : though there may be difficulty when the cita
tions are many, or when fome of them are from very rare books.
Honefty and policy both demand the exprefs ftatement of every
citation and quotation which is made through another fource.
If a perfon quote what he finds of Cicero in Bacon, it mould be
4 Cicero (cited by Bacon) fays, &c. It has happened often
enough that a quoter has been convicted of altering his author,
and has had no anfwer to make except that he took the pafTage
from fome previous quoter.
Quotations are frequently made with intentional omiffion and
alteration. But no rule ought to be more inflexible than that
all which is within the marks of quotation ought to be a literal
tranfcript of the book quoted. Sometimes the omiffion is made
becaufe part of the fentence is unnecefTary, as the quoter thinks.
But this is juft the point which he has no bufinefs to decide
without letting his reader know that he has decided it, which is
eafily done by the recognized mark of omiffion ( ) If
a perfon would quote the /Eneid for the antiquity of Carthage,
284 On Fallacies.
he has no bufmefs to write down, as from Virgil, c Urbs antiqua
fuit Carthago : it fhould be Urbs antiqua fuit Carthago,
if he decide upon omitting c Tyrii tenuere coloni. In this cafe,
not only may the omiflion make the proportion appear more
categorical than it is in the original, turning it from There was
an old city, Carthage, rather towards c Carthage was an old city ;
but a reader may choofe to think that the omitted words qualify
the epithet, or even offer proof deftru&ive of it. What if he
fhould deny the antiquity of Tyre ? The omijjion may (or may
not) be right, but the omiflion without notice, or fupprejfion^ is
certainly wrong.
Moreover, it is dangerous to truth to fhorten without notice,
inafmuch as thofe who quote the quotation will be apt to do the
fame thing; that is, thinking they have the whole pafTage, to
fhorten it further. What this may end in, no one can predict :
but miflakes have been brought about in this way quite as abfurd
as any that ever were made. It may reafonably be fuppofed
that many very ludicrous errors arife thus. A good many years
ago, I fucceeded, by means of a fhortened quotation, put away
until it was wanted, in arriving at, and publifhing, the conclufion
that Archimedes was once fuppofed to have been an anceftor of
Henry IV. of France. The real purport of the fentence was that
he was fuppofed to have been an anceftor of the Sicilian martyr
St. Lucia, on whofe day Henry IV. was born. It has happened
that A has been faid to have aflerted in a fecond book, that B
related the death of C, when the truth is that A faid in the firft
book that B died many years before C (See the Companion to the
Almanack for 1846, page 27). I do not fpeak of omiflions made
becaufe the part omitted would prove more than the quoter
likes : this of courfe is fraud.
Unjuftifiable as unnoted omiffions may be, ftill more fo are
additions and alterations. Writers have fometimes inferted glofles
of their own, into the text which they quote, either as addition
or alteration. Explanatory additions may eafily be made within
brackets [ ], which are underftood marks of fuch a thing : but
alterations are intolerable. But why, the reader may afk, are
fuch things infifted on ? Is not the fimple rule, Be bone ft,
enough to include thefe and hundreds of things like them, with
out detail ? To this I reply that within a twelvemonth before
On Fallacies. 285
the time I write this, a clergyman, a man of high education and
character both, publifhed a fermon in which he gave a verfe
from the Bible within marks of quotation, in which he wilfully
ftruck out one word, and inferted another, without notice : and
his fermon went through feveral editions, either without detec
tion, or without that detection leading to fuccefsful remonftrance.
I do not fuppofe there was difhonefty here ; but rather the fol
lowing reafoning ; I am fure it was meant; therefore I may
ftate that it was faid. Such reafoning is one of the curfes of our
literature.
There is one alteration within the marks of quotation which
may at firft feem reafonable : it is alteration of grammar to bring
the quoted phrafes into connected Englifh with the quoter s
context. As when a man fays " I know" and another perfon,
quoting him, fays " He knows." But it is furely juft as eafy
to put down He fays " I know." There is often an alteration
of emphafis in this adaptation of grammar, and generally an in
troduction of irony : and it is the premier pas to fomething worfe.
As far as I have feen, thofe who do it as a matter of courfe, are
apt fometimes to put their own paraphrafes under marks of quo
tation. A writer fhould fuit his own grammar to that of his
quotation, and not the converfe.
Omiffion of context, preceding or following the quotation,
may alter its character entirely : and this is one of the moft fre
quent of the fallacies of reference, both intentional and uninten
tional. The only way to infure full confidence is to give the egg
in its fhell : that is, to begin at a point which clearly precedes
the immediate fubject of quotation, and to continue until the
matter is as clearly paft : to give a fentence preceding and a
fentence following the matter quoted for its own fake, diflin-
guiming the latter. This is not always conclufive : becaufe the
fubject may be refumed in a fentence or two, or in another part
of the book. But it will inform the reader, in moft cafes, whe
ther he is or is not likely to differ from the quoter as to the
meaning of the part quoted. And this refers particularly to quo
tations of opinion : thofe of fact may often be more briefly
treated with fafety.
In quoting ancient authors, in cafes where the text is not no
torious, the various readings mould be given, efpecially when it
286 On Fallacies.
is an author whofe text has an indifferent reputation for accuracy.
Or if this cannot be done, the edition fhould be cited. Shameful
things have occurred in controverfy, by omiffion of a part of the
ordinary text, which the quoter chofe to confider as an interpola
tion, without choofing to confider that the reader ought to have
liberty to judge for himfelf on that point.
Among the cafes of indirect citation, fhould be included that
in which a book is mentioned as exifting, not on the authority
of the writer s own eyes, but on that of a catalogue. The num
ber of nonexifting books which are entered in catalogues and
copied, as to their titles, into other works, is greater than any
one who has not examined for himfelf would fuppofe poflible.
In thofe who know this, confidence is deftroyed ; and this fome-
times affects queftions of opinion. I am told that Dugald Stewart,
who had a ftrong notion of the practical impofftbility of pre-
fenting Euclid in a fyllogiftic form, never would believe that it
had been done by Herlinus and Dafypodius* Such a work is
entered in catalogues : but I mutt fay that the ftate of catalogues
is fuch that Stewart or any one elfe had full right to doubt of
any work, upon no other than catalogue evidence. The work
does exift, and I have a copy of it. But, feeing how matters
ftand, no one has a right to declare that an old book ever was
written, without informing his reader on what fort of evidence
he relies.
CHAPTER XIV.
On the Verbal Defcription of the Syllogifm.
IN page 75, I have made a firft attempt to exprefs the rela
tions of propofitions in language which will make fyllo-
gifms capable of verbal defcription, and the inference of their
conclufions matter of felf-evidence. It is defirable that this fhould
be more fully done, and I accordingly renew the attempt, with
the beft words of defcription which I can find or make. Any
one who can fuggeft words which better convey the meaning to
himfelf, will find it eafy to fubftitute them for thofe which I
have ufed.
Defcription of the Syllogifm. 287
The conditions to be fatisfied are, that the words fhould have
as much imported meaning as poflible, that every word and its
contrary fhould have the connexion of contrariety well marked,
and that the verbal defcriptions mould be capable of being eafily
formed from the fymbolic notation. As may be fuppofed, thefe
conditions are to fome extent contradictory of each other : the
facrifice of either to the others is then to be made to the moft
advantageous effect.
There are two ways in which it may be neceflary to defcribe
* the fyllogifm. Firft, the one hitherto uied throughout this work,
in which one concluding term is referred to the other by the in
tervention of the middle term : what X is of Y, and what Y is of
Z, determine what X is of Z. Secondly, that in which the two
terms are referred to one another by comparifon of both with the
middle term : what X and Z feverally are of Y determine what
X is of Z.
In the firft mode, the middle term is mentioned, and its de-
fcription is middle in the fentence ; while the reference term is
understood in the predicate of each defcription. Thus when we
fay c a fubcontrary of a fupercontrary is a fubidentical, it is that
a fubcontrary of a fupercontrary (of Z] is a fubidentical (of Z) ;
and the fupercontrary of Z is the middle term.
In the fecond mode, the middle term is underftood in the fub-
ject, and the concluding terms in the predicate, of the defcrip
tion of the fyllogifm. Thus when we fay genus and fpecies are
genus and fpecies, 1 it means that two terms which are feverally
genus and fpecies of the middle term (one entirely containing,
the other entirely contained in, the middle term) are genus and
fpecies to one another (the firft genus, the fecond fpecies).
Now it will be very eafily feen, that the way to change the
firft defcription into the fecond is as follows. Say the defcrip
tion runs thus, c P of Q is R. If Q be its own correlative, as
happens when Y and Z are convertibly connected, then c P of
Q merely becomes P and Q : but if Q^have another, O, for
its correlative, then c P of O becomes c P and QV Again, if
R be its own correlative, its plural takes its place : but if R have
R for its correlative, it becomes c R and RV Thus fubcon
trary of fupercontrary is fubidentical of the firft mode, becomes
fubcontrary and fupercontrary are fubidentical and fuperidenti-
288 On the Verbal Description
cal meaning that Ci and C 1 of the middle term are Di and D
of each other. But fubcontrary of fuperidentical is fubcontrary
becomes c fubcontrary and fubidentical are fubcontraries.
I need hardly fay that c P of Q is R with refpecl: to X in
terms of Z, muft be read Q of P is R with refpecT: to Z in
terms of X. This rule we have already ufed.
It is thus mown that it is only neceflary to dwell on the firfr.
mode ; and now arifes the queftion what words are to be em
ployed in defcribing the eight ftandard propofitions. After a
good deal of confideration, I prefer to denote the univerfal rela
tions by pofitive terms, and their contrary particulars by the cor-
refponding negative ones : not without full perception of the
facrifice which enfues of the firft condition above mentioned to
the third.
The words genus and fpecles immediately fuggeft themfelves to
denote the relation of Y to X and X to Y in X)Y. Thefe are
to be underftood as employed up to their limit ; or the genus and
fpecies may be coextenfive. For two names which have no
thing in common, as in X . Y, I propofe to fay that they are ex-
ternals of each other. And for two names which have nothing
out of one or the other, as in x .y, that they are complements of
each other. Remember that complemental does not mean only
jufl complemental (which is contrary), but may be contrary or
fupercontrary.
In X :Y, I call X a non-fpecies of Y, and Y a non-genus of X.
Thefe words have not as much as I could wim of imported
meaning, nor are there any pofitive terms which I can propofe
to fupply their places. They appear as fynonymous with not
entirely contained in and not containing the whole. In XY, let
X and Y be non-externals ; and in xy, let X and Y be non-com
plements. Accordingly, in defcribing what X is with refpect to
Y, we have as follows, mowing the fubftitutions which occur in
reading the fyllogiftic fymbols into this language.
Ai, fpecies
A f , genus
Ei, external
E f , complement
Oi, non-fpecies.
O f , non-genus.
Ii, non-external.
I 1 , non-complement,
If we consider genus and complement as larger terms, and fpecies
of the Syllogifm. 289
and external as fmaller ones^ and if we put down each univerfal
followed by its two weakened particulars, writing firft that which
is of the fame accent, we have
Univerfal.
A 1 Genus
Ai Species
E f Complement
E External
Firft weakened form.
I 1 non-complement
L non-external
O f non-genus
Oi non-fpecies
Second weakened form.
L non-external.
I 1 non-complement.
Oi non-fpecies.
O 1 non-genus.
Thus it appears that the primary weakened form of a larger
name contains a larger name, and of a fmaller a fmaller : and
the contrary for the fecondary forms. The words primary and
fecondary do not refer to importance, but only to order of deri
vation : thus AI was in our table X)Y, weakened into XY,
before it became y)x, weakened into yx or xy.
The rules for forming particular fyllogifms by weakening uni
verfal premifes may now be repeated. In a univerfal fyllogifm,
fubftitute for they?r/? premife and for the conclufion their primary
weakened forms, or for thefecond premife and for the conclufion
their fecondary weakened forms. In a ftrengthened fyllogifm,
fubftitute for the_/?r/? premife its fecondary form, or for thefecond
premife its primary form.
I now write down the whole body of fyllogifms, that the rea
der may exercife himfelf in the independent comprehenfion of
their meaning, and in aflent to their inferences ; deducing the
particular ryllogifms from the univerfals only.
Univerfal and particular Sylloglfms.
Symbol. Defcription of X with refpeft to Z.
I Ai AI AI Species of fpecies is fpecies.
} LAJi Non-external of fpecies is non-external.
[AiIT Species of non-complement is non-complement,
f A 1 A 1 A 1 Genus of genus is genus.
<j I A I 1 Non-complement of genus is non-complement.
j^A IiIi Genus of non-external is non-external,
f AiEiEi Species of external is external.
<j LEiOi Non-external of external is non-fpecies.
AjO O Species of non-genus is non-genus.
On the verbal Defcription
E E 1 Genus of complement is complement.
E O 1 Non-complement of complement is non-genus.
OOi Genus of non-fpecies is non-fpecies.
TEiA Ei External of genus is external.
J OA Oi Non-fpecies of genus is non-fpecies.
[EiliO 1 External of non-external is non-genus.
[E AiE 1 Complement of fpecies is complement.
J O AiO 1 Non-genus of fpecies is non-genus.
JETOi Complement of non-complement is non-fpecies.
f EiE ! A External of complement is fpecies.
J OiE Ii Non-fpecies of complement is non-external.
[EiOJ 1 External of non-fpecies is non-complement.
fE EiA Complement of external is genus.
J O EJ 1 Non-genus of external is non-complement.
[ E O L Complement of non-genus is non-external.
Strengthened Syllogifms.
Ai AT Species of genus is non-complement.
A AiL Genus of fpecies is non-external.
AiE O Species of complement is non-genus.
A EiOi Genus of external is non-fpecies.
EiAiO 1 External of fpecies is non-genus.
E A Oi Complement of genus is non-fpecies.
EiEJ ? External of external is non-complement.
E E L Complement of complement is non-external.
No perfon could propofe to himfelf a better exercife in the
acquifition of command over language, than practifmg the de-
monftrations of thefe relations, or more properly their reduction
into fpecific ftiowing, as to the matter of the inference, in what its
extent confifts. For inftance, the complement of a non-com
plement is a non-fpecies : How, and by how much ? The non-
complement leaves fomething which is neither in the term un-
derftood, nor in that non-complement. This, the complement
of that non-complement muft fill up : and by this then, at leaft,
the complement of the non-complement is not in the term under-
ftood, of which it is therefore fo far non-fpecies.
of the Syllogifm . 291
In the preceding view, I have particularly confidered the con
nexion between contrary forms, and the adaptation of language
to that connexion. But in the firft derivation of the fimple fyl-
logifms (page 88) the univerfals were related, not to their con
traries, but to their particular concomitants. I now proceed to
the confideration of this view, and to the j unification, on felf-
evident principles, of the afTertion that there is a real and ftriking
affinity between the univerfal fyllogifm and its concomitants, as
AiAiAi and O ! AO ! , E f E 4 A f and ETOi, &c.
The complex proportions Di, D f , and C contain each a uni
verfal which, in common language, is generally confounded with
it, and a particular, the exiftence of which is therefore for the
moft part fuppofed in thought to accompany the univerfal. The
remaining univerfal, E 1 , is differently circumftanced : if we fay
that X and Y complete the univerfe, we fhould generally mean
that they only juft complete it, and fhould not think of the fuper-
contrary relation, or of their overcompleting it. To be contained
but not to fill ; to contain with room to fpare, or to overfill ; to
exclude and be excluded without completion ; and to exclude and
be excluded with completion (or to complete and be completed
without inclufion); are our moft ufual ideas of the relations of
the extent of names.
The reduction of the complex propofition to the fimple uni
verfal, when done by removal of the concomitant particular^ is
in all cafes a lowering of the quantity, by the removal of an ex-
cefs, as follows :
Di means that X is contained in Y, and more is contained.
D 1 means that X contains Y, and contains more.
Ci means that X excludes Y, and excludes more.
C 1 means that X completes Y, and *more than completes.
Drop the fecond claufes, and DI, &c. are reduced to AI, &c.
Drop the firft claufes, and it would feem as if we had ftill the
* The alteration of grammar here feen is in deference to the word com
plete , the beft I can get. In this propofition, the verb refers to the uni verfe,
and it is X(joins in completing the univerfe) Y and joins in completing more
(than the univerfe).
292 On the verbal Defer iption
complex propofitions ; for more will contain its tacit reference to
that which it is more than. Let this tacit reference be dropped,
and then we have, inftead of the whole complex propofition, only
its particular. And this abandonment is actually made in com
mon language, by what would be called perhaps a lax, but is a
very logical, ufe of the word more. There are more than fifti on
the dry land, would be perfectly intelligible, and not as implying
that there were any fifh : c he was actuated by more than the
motive, &c. very often means other than the motive* &c.
Now, in the complex fyllogifm, as we have feen (page 81), the
exceffive part of the conclufion (whence comes its fecond claufe,
its additive more] is the fum of the exceffive parts of the premifes.
If one of the complex premifes be deprived of its aflertion of
excefs, or lowered into a fimple univerfal, the conclufion ftill
remains, though not a fortiori, neceffarily. This being done, the
valid excefs of the conclufion depends upon the excefs of the remain
ing premife ; and the concomitant particular fyllogifm, confidered
as part of the mixed complex fyllogifm, is the expreffion of this,
without the reft. Finally, the excefs may be ufed in the lax, or
non-correlative, fenfe, and then the concomitant fyllogifm ftands
by itfelf.
For example, OiA T Oi may be read thus : Confider Oi as
concomitant of A in D 1 . c X contains more than [fomething
that is not in] Y ; Z contains X ; therefore, Z contains more
than [fomething that is not in] Y. If more than Y mean c Y
and more, this would be D A D 1 . Again, O EJ is more than
X [fomething not X] is contained in Y ; Y excludes Z ; there
fore, X excludes more than Z [fomething not in Z]. If c more
than X were c X and more, &c. : this would be DiEiCi. And
fo on for other cafes.
I now proceed to what I may call the quantitative defcription
of the fyllogifm : by which I mean the expreffion of its cafes in
terms of the quantities only of its names and propofitions, leaving
the alternative of affirmation and negation to be fettled by the
law of thefe quantities. My reafon for the prefentation of the
fyftem in fo many different points of view will be obvious enough :
that which claims to be complete, muft mow itfelf to contain juft
the fame, and no more, as to refults, whatever may be the prin
ciple which is chofen as the bafis of conftruftion.
of the Syllogifm. 293
Every propofition, in fpeaking of two names, fpeaks of their
contraries, and (page 63) of the four terms, two direct and two
contrary, two are univerfal and two are particular. Since univer-
fal and particular are themfelves properly contraries, (for Every
X is c Xs, known to be all and Some Xs are <Xs, not known to
be all ) let us fignify the univerfal and particular forms of the
propofition by V and v. Again, fpeaking of a name, let its mode
of entry, univerfal and particular, be denoted by T and t. Writ
ing down V( or v) applied to T( or t), T( or t) we can make eight
varieties, which give us the eight ftandard forms applied to one
order, fay XY ; as follows :
A, = V(Tt) A 1 = V(tT) E, = V(TT)
O ! =v(Tt)
0. =v (tT) I 1 = v (TT) I, = v (tt)
E f =V(tt)
Thus P or xy, may be defcribed as the particular in which
both terms are univerfal : for X and Y are both univerfal in xy,
or x: Y, or y:X. And v(TT) defcribes it thus.
If, underftanding the order to be XY, YZ, XZ, we write
down any three propofitions, we make an attempt at a fyllogifm,
valid or not, as the cafe may be : as in
V(Tt).v(tt).V(tT) or VvV(Tt,tt,tT)
which muft be AJiA 1 . It will affift the memory to obferve that
fub-fymbols have VT or vt at the beginning, fuper-fymbols vT or
Vt. Alfo, that affirmatives have an even number of capitals
(none* or two) and negatives an odd number (one or three). A
univerfal and its particular concomitant have the fame entries of
T and t, and contranominals have inverted modes of entry of
thefe letters. The convertibles have T in both places, or t : the
inconvertibles have T and t.
Firft, it is unneceflary to write down the term-letters of the
conclufion, for they muft be taken from the premifes, in every
cafe in which the conclufion is the ftrongeft that can be drawn
from the premifes ; and our fyftem has no others (nor, indeed,
* The reader muft here follow the mathematician in confidering o as an
even number.
294 On the verbal Defer ipt ion
has the Ariftotelian any other except Bramantip). Thus, TT,tt
being the term letters of the premifes, ftrike out the fecond T
and the firft t, which refer to the middle term, and Tt muft be
long to the conclufion. To prove this, obferve that we know
that t in the premife cannot give T in the conclufion : therefore
T cannot give t ; for if, the term being Z, T gave t, then, put
ting z properly in its place, t would give T, which it cannot.
Again, we know that the valid forms, as to propofitions, are
VVV, VVv, vVv, Vvv ; fo that v occurring once only, muft
come third, and V muft come in the firft pair. Further, in the
four term letters of the premifes, VVV, vVv, Vvv, require Tt,
or tT, to come in the middle, while VVv alone requires TT,
or tt. Obferve thefe laws, and every formation which can take
place under them leads to a valid fyllogifm. Putting dots to re-
prefent a blank place, we form the eight univerfal fyllogifms by
filling up the blanks in VVV(. . t,T . .) and VVV(. . T,t . .) ;
the eight ftrengthened fyllogifms from VVv(..T,T..) and
VVv(. . t,t . .) ; the eight particulars which begin with a univer
fal from Vvv(. . t,T . .) and Vvv(. . T,t . .) ; and the eight par
ticulars which begin with a particular from vVv(. . t,T . .) and
vVv(..T,t. .). And, under the rules juft given, we have no
other cafes.
Taking the preceding as a bafis, we might make the rules of
accentuation follow from it. For, fince the firft blank in our
fymbol, and the firft concluding term, muft agree, and fince ac
cents depend only on the firft two letters in the fymbol of a pro-
pofition, we may proceed as follows. Let K and L, each of
them, mean T or t, as the cafe may be, but with the provifo that
what it means in either place it (hall mean in the other. Then,
in VVV(KT,tL,KL) and in vVv(KT,tL,KL), in which fym-
bols of conclufion are introduced, we fee that the firft and third
accents muft agree, which is part of the direct rule. As to the
firft and fecond accents, they agree in the firft inftance above, if
K be t, which puts an even number of capitals in the firft fym
bol VKT, or an affirmative propofition at the commencement :
they difter if K be T, which puts a negative propofition firft.
In the fecond inftance, they agree if K be T, which puts an
affirmative firft, &c. I leave it to the reader to deduce the other
cafes of this rule, the inverfe rule, and alfo that premifes give an
of the Syllogifm. 295
affirmative, or a negative, conclufion, according as they have like
or unlike figns. And thus it will appear, that the fymbolic rules
given in chapter V, are really expreflions of the general rules of
quantity.
It will be obferved that the concomitant fyllogifms of a univer-
fal have the fame term letters as that univerfal, and only change
VVV into Vvv, or vVv. Alfo, that the inverted fyllogifms of
page 96 only invert the order of all the term-letters, and the
letters of the premifes, when different.
Thus, E.A Ei being VVV(TT,tT), its concomitants TAT
and EiOJ 1 , are vVv(TT,tT) and Vvv(TT,tT). But the in
verted form AEiEi is VVV(Tt,TT). Contranominals have
different quantities in all the term-letters. The weakened forms
of a univerfal change the firft premife letter and the firft term
letter, or the fecond of both. Thus, E t E A, being V V V(TT,tt),
its weakened forms, OiE L and EiOJ , are vVv(tT,tt) and
Vvv(TT,tT).
The forms of the numerical fyllogifm (page l6r) may be re
covered by few and eafy rules, in which the premifes as they ftand
determine the conclufion, as follows : Let | be defignated as
the number of X, and | as that of x ; and fo on. Let a term
of the conclufion be called direff when it is in the premife, and
inverfe when its contrary is in the premife. Then,
1. In every cafe, the conclufion has the fum of the quantities
mentioned in the premifes, as part of the exprefiion of its quan
tity.
2. For every inverfe term in the conclufion, the number of
its direct term appears in the quantity of the conclufion, fub-
tracted. Thus, x in a premife, with X in the conclufion, muft
have I 1 in the concluding quantity. But the direct terms of
the conclufion never introduce anything into the concluding
number.
3. When the entrances of the middle term are fimilar (YY,
or yy), the terms of the two forms of conclufion are both direct
and both inverfe, with fubtraction of the number of the middle
term in the former, addition of the number of its contrary in the
latter. Thus, yy gives n 1 in the direct, +xj in the inverfe
form.
4. When the entrances of the middle term are diflimilar (Yy,
296 On the verbal Defer ipt ion, &c.
or yY), each form of conclufion has one direct and one inverfe
term ; and no number from the middle term enters the conclud
ing quantity.
Thus, the conclufions from wxY + wYZ are immediately
written down as
v)xZ and (m + n + J ! )Xz :
while thofe from mx Y + n yz, are at once
T | f )Xz and (
There are relations exifting between the forms of the fyllo-
gifm which I have not confidered. For inftance, the defcription
of X with refpecl: to Z being that it is a fpecies, (Ai), the de
fcription of its contrary, x, is that it is a fupercontrary, (E 1 ). If
then we give the name of contradefcriptives to AI and E f we find
that A 1 and EA, L and O 1 , I 1 and Oi, are alfo contradefcriptives.
The arrangement of iyllogifms by contradefcriptives, and the laws
of connexion thence refulting, will be an eafy exercife for the
ftudent.
APPENDIX.
I.
Account of a Controverfy between the Author of this Work and
Sir William Hamilton of Edinburgh ; and
final reply to the latter.
THIS appendix contains an account of a controverfy in which fome
c- ori? th( U ma " ers treated in the preceding work involved me with
bir William Hamilton, ProfbiTor of Logic and Metaphyfics in the Uni-
verfity of Edinburgh. It has produced four publications (to which I
mail refer as I, II, III, IV) namely:
I. Statement in anfwer to an affertion made by Sir William Hamil-
% i b j A uguftus De Morgan, .... (London, oftavo, R. and
E. Taylor, pp. 16, publifhed April 30, 1847.)
_ II. A letter to Auguftus De Morgan, Efq on his claim to an
independent redifcovery of a new principle in the theory of fyllogifm.
From Sir William Hamilton, Bart. Subjoined, the whole previous cor-
refpondence, and a poftfcript in anfwer to Profeffor De Morgan s State
ment (London and Edinburgh, oftavo, Longman and Co., Maclachlan
and Co. pp. 44, exclufive of Profpeftus hereinafter mentioned: re
ceived by me May 22, 1847.)
III. Letter from me to Sir W. Hamilton, dated Mav 24, publifhed
in the Atbenaum Journal of May 29.
I V. Letter from Sir W. Hamilton to me, dated June 2, publifhed in
the lame Journal of June 5.
There are two queftions involved, one concerning my character, the
other purely literary. The former ftands thus. March 13, Sir W
Hamilton informed me by letter that (the Italics are his own words) /,
him // is manifeft that for a certain principle I was wholly indebted to his
information, and that ifljhouldgive Vi forth as a /peculation of my own
(which I had done to himfelf, and meant to do, as he knew, and have
imce done, m print) I mould, even though recognizing always his pri-
r chr $ y f an in J urious breaf h of confidence towards him and
offalfe dealing towards the public. This hypothetical charge, and dero
gatory fuppofition of which he may formerly have furmifed the poJRMitj
(iuch are his fubfequent qualifications of it) is unrefervedly retraced at
the beginning and end of II: but it is frequently infmuated in the mid
dle, by propofmg things as difficult to be explained otherwife, by hint
that others may believe it, by hopes that they will not, by charges of
alfenood, &c. &c. For the formal charge is fubftituted imputation of
298 Appendix.
lapfe of memory, intellectual confufion, &c. The following is the pro
gramme of the firft intended argument, (II. p. 4.)
I confefs, that, for a time, I regarded your pretenfion, as an attempt
at plagiarifm, cool as it was contemptible.
From this view, feeling, information, reflection turned me ; and I
now, Sir, tender you my fmcere apology, for admitting, though founded
on your own ftatements, an opinion fo derogatory of one, otherwife fo
well entitled to refpect.
In itfelf, this view was, to me, painful and revolting. The cha-
rafter, too, which you bear among your friends, I found to be wholly
incompatible with a fuppofition fo odious. You are reprefented as an
active and able man, profound in Mathematics, curious in Logic, wholly
incapable of intentional deceit, but not incapable of chronological mif-
takes. Your habitual confufion of times is, indeed, remarkable, even
from our correfpondence. Your dates are there, not unfrequently of
the wrong month, and not always, even of the right year. With much
acutenefs, your works mow you deficient in architectonic power, the
concomitant of lucid thinking; and, that you are not guiltlefs of intel-
lectual rafhnefs is fufficiently manifeft, from your pretention to advance
Logic, without having even maflered its principles.
With regard to the fubfequent infmuation of a retracted charge, my
explanation (believing as I do, that Sir W. Hamilton always fpeaks fub-
jective truth) is that his mind infenfibly fell back to its old bias as he
felt that the fubftitute for his charge wanted ftrength : my conclufion is,
that it is unneceflary henceforward to notice any thing he may fay or
write on my character : and my determination is to act accordingly.
SirW. Hamilton s pamphlet contains about a fcore and a half of quo
tations, on which hang fundry jokes and fneers, fome of them at mathe
maticians in general, and myfelf as one of the body. On thefe I mail
only fay that my notions of the common fenfe of controverfy, and my
determination to perfift, generally, in the tone of refpect to my oppo
nent s learning and character which I have hitherto preferved, would,
were there nothing elfe, prevent my adopting the habit of which they
are fpecimens. But as no man willingly Hands an unreturned fire of
facetiae without defiring to prove that his forbearance does not arife from
want of ammunition, I will permit myfelf (declaiming the animus under
which fuch things are ufually written) juft to mow that quotation, ap
plication, allufion, fneer, joke, and fling at an opponent s ftudies, are all
among the weapons which I could have employed, if I had thought
them worthy of my antagonift, or of thofe whom I want to convince.
I might, for inftance, have written fomething like the following ;
Among the aflets of the old logicians, difcovered when the fchools
were fwept out, there was found, as is well known, the queftion \Jtrum
cbimtera bombinans in vaeuo pojfet comedere fecundas intentions s : a very
good title, as Curll would have faid, wanting nothing but a treatife
written to it. Now whether it be comedere, or whether the fchoolmen
invented comedere, Sir W. Hamilton, on whom their mantle has fallen,
has written the treatife, and fuccefsfully maintained the affirmative. His
Appendix. 299
notion that his communication could give any hint, is clearly and aptly
defcribed by chimera, his ftyle by bombinans, his proof by vacuum; and
the fecond intentions, above noticed, chewed up and given forth with
his firft ones, are a practical example of the poffibility of the Q.E.I. He,
or rather the bombinating chimaera which has perfonified itfelf in his
form, as the sAof ovsipog did in that of Neftor, is thus both retraftor
and detractor. But though the tranfition from flops to folids generally
indicates convalefcence, yet, as here made manifeft, the paffage from
liquid to dental may be only the growing weaknefs, the perifcence, of the
cafe.
I afTert the following documents to be all that are relevant with
refpecl: to the literary part of the controverfy. They are given at the
end of this appendix.
A is an extracl from a communication of mine to the Cambridge
Philofophical Society, made before I received any communication what-
foever from Sir W. Hamilton. I affert it to contain a diftincT: an
nouncement and ufe of the principle of quantification of the middle term,
be that middle term fubjecl or predicate. On this point the reader is to
judge.
B is a communication from Sir W. Hamilton to me. The reader
is to judge firft, whether it contain anything which is intelligible with
refpecl: to any fyftem of fyllogifm ; fecondly, whether, if it fhould fo con
tain anything, that fomething would have been information to me who
had written A, on fome matter afterwards found in C.
C is the relevant part of an addition made by me to A, when the
latter came before me in proof. The reader is to judge firft, whether C
contain anything more than an application of A; fecondly, iffo, whe
ther that fomething more is derived from anything intelligibly hinted at
in B.
The only bare faft on which Sir W. Hamilton and myfelf are at
ifTue is this. I affert and maintain that the matter of C was written in
my poffeffion before I received B : Sir W. Hamilton holds me mif-
taken, and thinks he can prove from the correfpondence that in this
point my memory has failed. This I continue to treat as irrelevant :
for we are both agreed that the corpus delifti, if deliclum there be, lies
in C containing fomething not fubftantially contained in A, but furH-
ciently hinted at in B. Any reader who thinks that C does contain
fomething fuggefted by B which is not in A, may declare againft the
correftnefs of my memory; any one who thinks the contrary, will hold
it of no confequence whether my memory on the difputed fa6l be good
or bad. With the firft reader I have no cafe : with the fecond I have
all I think worth caring about.
Sir. W. Hamilton maintains my letters to be effential parts of the
cafe. They may become fo, as foon as it is pointed out what C contains
which is hinted at in B, and not contained in fubftance or principle, in
A. When Sir W. Hamilton points out, by citation from C, what he
alleges to have been taken, and by citation from B, what he thinks it
has been taken from, and when I thereupon fail to produce equivalent
: 3oo Appendix.
knowledge from A or elfe to expofe the irrelevance of his citation from
B then thofe letters may become of importance. This he has not done,
though fpecially challenged to do fo : and when I come to difcufs III
and IV, it mall appear that he admits he has not done it.
I now give the beft account I can of the origin of the difpute, pre-
mifmg, that up to this 3d of September, 1847, 1 do not abfolutely know
what the fyftem is which I am charged with appropriating. There is a
fyftem which I think is moil probably the thing in queftion : but a fyf
tem containing a defe6t of fo glaring a character, that I will not attribute
it to Sir W. Hamilton, who defcribes his own as "adequately tefted
and matured" until he expreffly claims it, or until I have the moil indu
bitable proof.
In the common, or Ariftotelian propofition, the quantities of the fub-
jecl: and predicate are determined, the firft by exprejfion or implication,
the fecond by the nature of the copula (fee page 57 of this work). And
the only quantities confidered are all and fome ; the latter meaning any
thing that not none may mean, fome, it may be all but not known to be
all, perhaps not more than one. The matter contained in A fuggefted
itfelf to me in the fummer of 1846, and was forwarded to Cambridge
with the reft of the memoir on the 4th of October.
I will now introduce Sir W. Hamilton s defcription of the various
kinds of quantity (II p. 31, 32).
Your " Statement" is chiefly plaufible from a wretched confufion
of diftinft things. This confufion, with which you delude yourfelf,
and many of your readers, is of two independent fchemes of logical
quantification ; the one, affertingtf# increafe in the expreffly quantified
terms, the other, a minuter divifion of the forms of quantification itfelf.
To difmtricate this entanglement, we have fimply to confider, in their
contrails, the three following fchemes of quantification :
The firft fcheme is that which logically confines all exprefled
quantity to the Subjett, prefuming the Predicate to be taken in ne-
gative propofitions, always determinately in its greateft and leaft ex-
tenfion (univerfally and fingularly), in affirmative propofitions, always
indeterminately in fome part of its extenfion (particularly).
The fecond fcheme is that which logically extends the expreffion
of quantity to both the propoiitional terms, and allows the Predicate to
be of any quantity, in propofitions of either quality. This not only
fupplies a capital defect, but affords a principle on which Logic ob-
tains a new and general development.
The third fcheme is that which logically admits more exprejjed
* quantities than a determinately leail or greateft extenfion (quantity fin-
gular and univerfal), and an indeterminately partial extenfion (quantity
particular.) This, though it corrects, perhaps, an omiffion, yields no
principle for a general logical development.
The firft doclrine is the common or Ariftotelic ; the fecond is mine ;
and in the third in fo far as you have gone, and apart from the con-
* fideration of right or wrong I do not queftion your originality.
Now, the fecond and third fchemes are both oppofed to the firft,
but in different refpeds ; coniequently the fecond and third may, each
Appendix. 301
of them, combine with itfelf, either the whole other, or that part of
the firft to which it is not itfelf oppofed. More is impoffible.
Let the following be noted:* Tour OLD view (that in the body of
the Cambridge Memoir} is a combination of the THIRD fcheme of quan-
< tification with the FIRST/ your NEW view (that in its Addition) is a
combination of the THIRD f^eme of quantification with the SECOND: and
the confufion, of which you are NOW guilty, is the recent and uniform,
and perverfe identification, in your PRESENT " Statement," of the SECOND
fcheme with the THIRD.
Before, however, proceeding to comment on your confufion of the
fecond and third fchemes, I may alfo relieve a confufion in the term
definite and its reverfe, indefinite, as applied to logical quantification.
| In the/ry?, common, or Ariftotelic meaning, definite, or more pre-
cifely predejinite (ttOptTOf, VtpOffapirrtf,) is equivalent to exprejed,
overt, or, more proximately, to defignate and pre-defignate ; in this
* fenfe, definite quantity denotes expre/ed, in oppofition to merely under-
flood, quantity.
< In the fecond meaning, that which I have always ufed, (and certain
< ancients, I find, were before me,) definite is equivalent to determinately
marked out ; a fenfe in which definite quantity is extenfion undivided
< or indivifibk, univerfal or jtngular (this including any collecled plu-
rality of individuals) as oppofed to particular quantity.
In the third meaning, which you have ufurped, definite is equivalent
to numerically fpecified; and in this fenfe, a definite is an arithmetically
< articulate quantity, as oppofed to one arithmetically inarticulate.
This your meaning of the word I did not, before the appearance of
< your " Statement," apprehend ; for of courfe I prefumed you to ufe it
in its firft or common meaning, from which you never hint that you
confciouily intend to deviate.
Three fchemes of quantity are here mentioned.
Firft, the ordinary one.
^Secpndly, that in which the ordinary quantities, allm&fome, are ap
plied in every way to both fubjeft and predicate.
Thirdly, that in which numerically definite quantity is applied to
fubjeft or predicate or both : the effential diftinftion of this cafe is nume
rical definitenefs : it really contains the fecond fyftem, when numerical
quantity is algebraically exprefled. Of thefe, it appears, Sir W. Ha
milton claims the fecond, or rather, the application of fuch a fcheme to
the fyllogifm. What then is it ? I fuppofe it to be the following. My
order of reference is X Y.
* Let the following alfo be noted : My old view (that in the body of the Cam
bridge paper) is entirely on the/r/? fcheme, except in one digrefR-ve fedion and one
iubfequent paragraph (from both of which A is quoted) in which the frond and third
are combined : my next view (that in the addition) is alfo a combination of the fecond
and third khemes: and my "Statement" contained alfo a uniform, but not recent,
identification of the fame fecond and third fchemes, which I never feparated in thought
until 1 law this paragraph. Any one who can form an opinion of the way in which
the iubjeft would prefent itfelf to the mind of a mathematician, will fee that the fecond
jCheme would prefent itfelf concomitantly with, and as an effential part of, the alge
braical form of the third. A. De M.
302 Appendix.
All X is all Y means that X and Y are identical : it is my D. All
X is fame Y is A t . Some X is all T is A f . Some X is fome Y is I|.
As to negative propofitions, All X is not all T is E 4 . Some X is not
all T is O A . All X is not fome T is O f . Some X is not fome T is true
of all pairs of terms one of which is plural. In its indefinite form, it is
what I have in Chapter VIII. called fpurious.
The propofitions of this fyftem are then the complex D, or Ai-f-A f ,
the fix Ariftotelian forms A 4 , A 1 , Ej, O, O 1 , I|, and the fpurious form,
which may be called U. In looking over (Sept. 5) Sir W. Hamilton s
pamphlet, I happened to light on the affertion (incidentally made) that
his iyftem gives thirty-fix valid moods in each figure. On examining
the preceding fyftem, I find this to be the cafe. I mould not have pub-
limed the refults, had not Sir W. Hamilton made it necefTary for me
to comment on them. I mall denote the propofition U, or Some Xs
are not fome Ys by X : : Y ; and I mall, fuppofmg each cafe to be formed
in the firft figure, then tranfpofe it into my own notation.
1. There are ff "teen forms in which D enters. Whenever D is either
of the premifes, the other premife and conclufion agree. Thus we have
AjDAj, DUU, &c. &c.
2. Fifteen Ariftotelian forms AiAjA,, A f A ! A ! ; AjEjEj, ^A Ej ;
A.0 1 , O t A 0; A OA, O A t O f ; A 1,1,, 1^,1,; E^O 1 , I|EA;
A AJj ; A EA, EjA.O .
3. Six more U fyllogifms A O U, O^U ; A UU, UA A U; I t O U,
0,1,17.
The two things to be confidered are ; the introduction of the iden
tical propofition ; and that of the fpurious one, as I call it.
It is, I fuppofe, a fundamental rule of all formal logic, that every pro
pofition muft have its denial, its contradiction. Now D has no fimple
contradiction in this fyftem : that O T and O, both contradict it (and alfo
E,) is true : but the mere contradiction is the disjunction O 1 or O, .
A perfon who can mow that one or the other of thefe is true, has de-
monftratively contradicted D, even though it could be proved impoffible
to determine which of the two it is.
The propofition U is ufually fpurious. But if we introduce it, we
muft introduce its contradictory alfo. Now if either X or Y be plural
names, it muft be true : confequently, the contradiction of U is * X and
Y are fmgular names, and X is Y. When a fyllogifm having the pre
mife U is introduced, either that premife may be contradicted, or it may
not. If it may, there is no form to do it in : if it may not, then it is a
fpurious propofition, and cannot, by combination with others, prove
anything but a like fpurious conclufion.
Let X : : Y denote Some Xs are not fome Ys, and X,Y, denote there
is but one X and one Y, and X is Y. Then either X : : Y or X,Y, muft
be true, and one only. A logical iyftem which admits one and not the
other, which contains an aflertion incapable of contradiction without
going out of the fyftem, can hardly be faid to be " adequately tefted and
matured," and is not felf- complete. The propofition X,Y, includes in
itfelfthe conditions of D, and is a kind of fingnlar form of D.
Appendix. 303
I prefume, from the number of Sir W. Hamilton s moods, thirty-fix,
as above obtained, that the contradidlion neither of D nor of U finds a
place. Admit them, and the contradiction of U alone (call it V) de
mands fixteen new moods in each figure. I will now proceed.
In my publication, fpeaking now of (A) what was fent to Cambridge
before I communicated with Sir W. Hamilton, I had no quantification
intermediate between the ordinary one, and the numerical one applied
to either fubjecl or predicate, as wanted in the canon of the middle term
there given. Look at the laft of the feven fyllogifms in tiizfecond extra ft,
where lotb the predicates, being of the middle term are quantified, and the
condition of validity is quantitatively ftated. But for * Y,-|-Y 2 lefs than
I mould be read yi-j-y 2 greater than i. The equivalence of this
to Y,-j-Y 2 lefs than I is a miftake. In theyfr/? extra ft, the general
canon is given which is afterwards ufed in C.
Up to the time when Sir W. Hamilton publifhed his letter in reply
to my ftatement, (II), I never had feparated the idea of his fecond fcheme
of quantification from that of the third.
Thus then we flood on Oclober 3, when I fent my paper to Cam
bridge. Sir W. Hamilton had been teaching the application of the
ordinary quantities to both fubjecl: and predicate : I had arrived at the
algebraical reprefentation of the numerical quantification of terms, whe
ther fubjecl: or predicate matters not, as long as they were middle terms.
1846, Oftober 6. My communication (containing A) was in the
hands of Dr. Whewell (as he informs me) for tranfmiffion to the Cam
bridge Society : I never faw it again till the next February. Oftober
7, Sir W. Hamilton wrote to me, in anfwer to an application of mine
on the biftory of the fyllogifm, further informing me that he taught an
extenfion and fimplification of its theory, which he offered to commu
nicate. November 2, (the offer having been accepted) Sir W. Ha
milton forwarded the communication B, which I give entire ; coniifting
of a letter, and the Requifites which he had furnifhed to his fludents,
for a prize EfTay. December 28, he wrote again, forwarding a printed
Profpeftus of his intended work on logic. This is not material ; for,
on receiving it, I thought certain, what from the previous communica
tion I had thought poflible, that Sir W. Hamilton was in pofleffion of
the theory of numerically definite fyllogifms (but this was a miftake of
mine, as will prefently appear). I accordingly, to preferve my own
rights, immediately forwarded (as will prefently be ftated more in de
tail) an identifying defcription of the meets of paper on which my nu
merical theory was written, and an account of both my fyftems (in
letters dated December 31, 1846, and January I, 1847). Of this, Sir
W. Hamilton (who has publifhed both letters) is my witnefs. 1 847,
February 27, I dated the addition to the proof fheet of my Cambridge
paper, which was defpatched to Cambridge the next day. This addi
tion contains C, which itfelf contains (in fubftance) all that part of my
letter of January I which refers to the difputed point. March 13, Sir
W. Hamilton wrote the letter containing the charge of plagiarifm ; hav
ing been for two months prevented by illnefs from refuming the fubjecl.
304 Appendix.
All fubfequent correfpondence referred to proceedings, and not to the
fubject matter of the charge.
Many days before the middle of October, I had applied the fyftem of
quantification in the manner fhewn in C. Sir W. Hamilton thinks my
memory has failed here : I know better. My memory does not depend
upon a date, but upon the opening of the Univerfity College Seffion,
which takes place in the middle of October. But it matters nothing,
for the notion of the complete quantification of a predicate, when wanted
becaufe it is the middle term, will prove the pofTeflion of that procefs as
well as quantification in all cafes whether wanted or not. On receiving
B, I looked with curiofity at 2, on which, in fact, Sir W. Hamilton
grounds his declaration of having made a communication. He demands
of his pupils,
The reafons why common language makes an ellipjts of the exprejjed
quantity, frequently of the fab j eft, and more frequently of the predicate,
though both have always their quantities in thought.
On looking at this, and feeing mention of the quantities which the
terms have in thought, in common language, I took it for granted that
the common quantities were fpoken of: namely, that of the fubject from
the tenor of the proportion, that of the predicate from the nature of the
copula. I never mould have imagined that in the common language of
common people, there were any other quantities, even if, in their minds,
the predicate have thefe. Had this been all, I mould have paffed it over,
as referring to common quantities, and making common people a little
more of logicians, as to the predicate, than I have found them to be.
That this common language meant the language of any fcientific fyftem, I
had not the leaft idea : ftill lefs that it referred to the language of the
writer s own unprinted fyftem, current only between himfelf and his
hearers. And, though I gained a fufpicion that Sir W. Hamilton might
have (which he had not) adopted numerical quantification, it was not
from this pafTage, which by itfelf was nothing, but from what is now
coming, which made this paffage ambiguous.
On looking further into B, (which fee) I found that Sir William s
fyftem, whatever it might be, noted defects in the converjion of propoji-
tions, and a general canon of fyllogifm. Now I had two fyftems, each
of which had its own way of adding to the converfions, and each its
own canon of fyllogifm. In my firlt fyftem (which has now grown into
Chapter V) the permanent introduction of the contranominals is a com
pletion of converfion : and the reduction, by the remarks in pages 96,
&c. of all fyllogifms to univerfal affirmative premifes, was the canon of
fyllogifm. In the fecond, feen in A and C, which has grown into
Chapter VIII, there is the univerfality of fimple converfion, and the
canon of the middle term. Sir W. Hamilton may deny (I believe he
does) that thefe are canons : let it be fo ; but I took them for canons,
and thought of them when I faw the word canon in his fummary. And
then the queftion was, had Sir W. Hamilton one of thefe fyftems, or a
third one ? I had been throughout our correfpondence well pleafed with
the idea that I had hit upon fomething in common with Sir W. Ha
milton ; and in my anfwer to communication B I faid,
Appendix. 305
* I am not at all clear that I (hall mt have to claim only fecondary
originality on feveral points. When I fee " defects of the common
doctrine of converfion " and a " fupreme canon " of categorical fyllo-
gifm, I muft wait for further information I think I may yet be
able to flatter myfelf that I have followed you in foine points unknow-
ingly.
The reader will obferve that this inftructive communication is fup-
pofed to tell me, that in my thoughts the predicate has all kinds of
quantity : though in truth both have their quantities is not Englifh for
either may have any one of two fpecies of quantity. Sir W, Hamilton
has exprefled (perhaps) the dictum which is to have taught me new
quantification, in terms of that new quantification unknown. By both
have quantities he feems to aflert that he meant both have all quantities,
That both have their quantities, is true in the common fyftem : thefe
words, which exprefs a truth of the common fyftem, Sir W. Hamilton
declares to be a fure mode of communicating the difference between his
fyftem and the common one. This may do in his own lecture room,
in which he has the arbitrium et jus et norma loquendi in his own
hands. A diftinctively unmeaning phrafe may, in virtue of his expla
nations, pafs current between him and his pupils : and a private bank,
of courfe, muft receive its own notes. But they are not lawful tender
anywhere : nor good tender out of the neighbourhood.
I mail now proceed to the letters in the Athenaum (III and IV).
Thefe contain the iflues raifed by the pamphlets : my fhort letter con
tains the ftrength of my cafe : I am to prefume that my opponent s
letter contains the ftrength of his anfwer, and I think it does fo. At
leaft I can fee nothing ftronger in his pamphlet.
MR. DE MORGAN. SIR W. HAMILTON.
I take this mode of acknowledg- In reply to your letter in the laft
ing the receipt of your printed number of the Athemeum; you
letters to me. I promifed you an were not wrong to abandon your
anfwer, if you would bring for- promife "of trying the ftrength of
ward the grounds of your afTertion my polition ;" for never was there
that I had acted with breach of a weaker pretenfion than that, by
confidence and falfe dealing. But you, fo fuicidally maintained. You
you now admit that your grounds would, indeed, have been quite
are no grounds ; you declare your right had you never hazarded a
conviction that (though chargeable fecond word ; for every additional
with confufion, want of memory, fentence you have written is ano-
&c. &c.) I have acted with good ther mif-ftatement, calling, fome-
faith ; and you offer a proper re- times, for another correction,
traction and apology. You ftate in
various places and manners, that
though you are fatisfied of my in
tegrity, all may not be fo ; and,
thereupon, you call for an anfwer.
But I think that others will be
I
306 Appendix.
quite fatisfied with your own an-
fwer to your own charge.
There is nothing left which I
care to difcufs with you. Our
views of logic, their coincidences,
their differences, their firft dates,
my memory, &c. I am content to
leave to thofe who will read my
ftatement and your letters, with
two remarks.
There is no ftrength in an abandoned pofition. My pamphlet was
publifhed in defence of my own character: when Sir Wm. Hamilton
retraced his charge of breach of confidence and falfe dealing, there was
nothing to which I flood engaged, nothing I cared to write feparate
pamphlets on, efpecially when the approach of this prefent publication
was confidered. Any one who reads page 9 of my pamphlet, in which
the promife was made, will fee that it has reference to what I there call
" the infamy which would attach to any one who had deferved the
terms he ufed for the conduct he defcribed." I certainly forgot to fay
" unlefs you retraft : " but as he had already refufed to retracl (though
he had propofed to fufpend the charge, provided I would then undergo
an examination) it did not enter into my head to provide for fuch a con
tingency. The affertions about weaknefs, misftatement, &c. are for the
reader s judgment. I did not, in this letter, allude expreffly to Sir W.
Hamilton s various infmuations that the old charge might be true : both
becaufe, at the firft hurried reading, I did not become aware of their
extent ; and alfo, becaufe I wifhed to take time before I made up my
mind as to the way of treating what I faw of them.
MR. DE MORGAN. SIR W. HAMILTON.
i . As foon as the queftion of You do not deny, that your cor-
charafter was difpofed of, it was refpondence afferts a claim to the
your bufmefs to mow that my Ad- principle communicated to you by
dition* written after I communi- me ; but you complain that I have
cated with you, contained fome not mown that your Addition in-
principle not contained in my Me- volves a new doctrine, uncontained
moir,\ written before I communi- in tbat part! [from the overt con-
cated with | you. This you do traditions of its other parts I had]
not do. You affert, and you de- of your Memoir which you de-
fcribe, and you fum up ; but you do clared to contain the principles
not quote, except a few words, ufed in your Addition. And this
which are not in that part of my you can fay, when I explicitly
Memoir which I declared to con- Hated that " throughout the whole
tain the principles ufed in my paper (the Memoir) not only is
A ti dition. there much in contradiction there
* Here given in C. f Here given in A, fo far as relevant.
J Sir W. Hamilton s part of this is B.
Appendix. 307
is abfolutely nothing in (more then
fortuitous) conformity with the
theory of a quantified predicate"
(L. p. 34). This, too, you can
fay whilit before your eyes, unan-
fwered, there was lying " my for
mal requeft, that you would point
out any pa/Jage of your previous
writings in which this doctrine
(that afferted in your * Statement,
of a quantification of the middle
term, be it fubject or predicate) is
contained" (Ibid) for 1 could find
none j and none has by you been
indicated.
I do deny, in one fenfe, that my correfpondence afferts a claim to the
principle communicated by Sir W. Hamilton : for I deny that he com
municated any principle. I prefume of courfe that the Profpedtus and
tter fent on the 28th of December are out of the queftion : fmce I
gave the fyftem on which the charge was made by return of port. Sir
W. Hamilton has very properly confined himfelf, in his pamphlet, to
his communication (B) of November 2, as containing the communica
tion which he afferts me to have ufed. Let the reader look through it
and afk himfelf what new principle is communicated, and where.
Sir W. Hamilton afferts that he has mown my Addition to contain
a new doctrine, not contained in one definite part of my memoir, by the
contradictions of its other parts. Let P, Q^ R, be parts of a memoir ;
and S an addition. By mowing that P and (^contradict one another,
Sir W. Hamilton thinks he mows that S contains a doctrine not in
volved in R. The fact is, that all my memoir except Seftion iii. On
the quantity of proportions and one other paragraph (from both which
A is taken) belongs to the fyftem of Chapter V. in this work : while
Seftion in., the other paragraph, and the addition, belong to Chapter
VIII. Let the reader take notice that Sir W. Hamilton (who, by the
way, feems to confider I explicitly ftated as a fufficient anfwer to
you have not mown ) does fa&fomething in my memoir in conformity
Wttb the theory of a quantified predicate. He fays it is fortuitous:
but it did not feem to him requifite to bring it forward, and point out
itefortuitoufnefs. This point is for the reader to judge of. "How
dare you," he fays, " rob me of my quantified predicate." " Good Sir,"
I anfwer, " I had it before I knew you." " What if you had," he
replies, " it is enough if I inform you that it was only by accident."
Sir W. Hamilton cannot find either in the memoir or the addition
(he fays here only in the previous writings, but in his pamphlet (p. 34)
he ftates it of both memoir and addition], any thing about the doctrine
of quantification of the middle term, whether it be fubject or predicate,
which doarine he fays // repugnant to all that is there taught. It is
308 Appendix.
true that in the next fentence he refers to previous writings, as cited. ^ I
will therefore conclude that Sir William included the addition by mif-
take, and meant the memoir only. Whether my Seftion iii. (A) is or
is not full of quantification of the middle term, without reference to
whether that middle term be fubjeft or predicate, I am quite content to
leave to the reader. Sir W. Hamilton fays he cannot find it. This I
believe, and wonder at : but it does not follow that it is not there. Let
the reader look.
Again, when Sir W r . Hamilton averted that C contains fomethmg
which I got from him, and which is therefore not in A, I repeat that
he ought to have pointed out what it is. His affertion that he cannot
find it in A neither proves that it is not in A, nor that it is in C.
This is the pinch which obliged him to write forty-four pages of ac-
cufation in anfwer to fixteen of defence : and this is the point on which
the queftion will finally turn, I am tedioufly often obliged to bring
the whole matter to its ABC; but what elfe can I do with an oppo
nent who writes an ignoratio elencbi of forty-four pages long.
Sir W. Hamilton is not good at finding. Immediately after what
he has quoted from himfelfas above, comes the following paffage;
< In regard to your third affertion, that perfettly definite qxantifiea-
< tion dejtroys the nece/tty of diflinguijhing fuljeft and predicate; this
* is altogether a miftake. It is not " definite quantification," (in what-
ever fenfe the word definite bt employed), but the quantification of both
< tbe terms which deftroys the neceffity of diftinguiming fubjeft and
predicate ;" and this by mowing, that proportions are merely equations,
and enabling us to convert them allJimplyS
I now quote from myfelf. Of the two fentences now coming, Sir
W. Hamilton quotes the firft, omits tbe fecond, which mows that my
phrafe perfeBly definite means definite in botb terms, and then makes
the preceding remark.
< In faft, perfeftly definite quantification deftroys the neceffity of dii-
< tinguifhing fubjeft and predicate. To fay that fome 20 Xs out of 50,
are all to be found among 70 Ys, or that 20 out of 50 Xs are 20 out of
70 Ys, is precifely the fame thing as faying that 20 out of 70 Ys are
20 out of 50 Xs.
In a writer of whom difhoneft intention might be concluded, we
mould know how to explain the omiflion of the fecond fentence. But
there is no difhonefty in Sir W. Hamilton : the omiffion _ muft be
referred to the fame difpofition which prevents him from feeing quan
tification of the middle term in A. What I take that difpofition to be,
matters nothing to my reader. Perhaps this fentence alone will enable
fome to deteft that I had not any idea of the fecond fyftem ofquantifi-
tion independently of the third.
MR. DE MORGAN. SIR W. HAMILTON.
2. All the alleged inconfiften- You fay, that my expofure of
cies which you find in my letters, your inconfiftencies is unavailing,
&c will not help you till you have except " I mow that my commu-
Appendix. 309
nication was intelligible." You
forget that it is for you to explain
how, having "fubfcribed to" as
having " rightly underftood" twen
ty-two fentences of my profpeftus
(L. pp. 19, 1 6), you could fubfe-
quently declare that communica
tion to be unintelligible ! ! (L. p.
59). I have now no doubt, how
ever, that you then " fubfcribed
to " more fentences than, by you,
were " rightly underftood." In
deed, had you only betimes avowed
that all you had " fubfcribed to, as
rightly underftood," was to you
really unintelligible, and that the
repetition of my do&rine was in
your mouth mere empty found,
two pamphlets might have eafily
been fpared.
Firft, the profpefius is not the " communication. * The communica
tion is that of November 2 (B). Let the reader look at it, and fee
whether it be intelligible communication of new principle.
In my pamphlet I have feveral times fpoken of the communication,
though there were two. This was natural enough, inafmuch as there
was one communication (that of Nov. 2), on which the charge was made
againft which that pamphlet was a defence. Sir Wm. Hamilton has
never ventured to maintain that I derived anything from the communi
cation of Dec. 28, containing the profpeclus, to which I replied on the
evening I received it, as prefently mentioned. But he makes, in various
places of which the above is one, a mixture of the two communica-
done this : and even then, you
will have to mow that your com
munication was intelligible.
In glancing over my letters and
the mafs of notes which you have
written on them, I fee that I have
feveral times ufed inaccurate lan
guage, as people do in hurried let
ters. Still more often you have
mifunderftood me. If my occa-
fional inaccuracy and your occa-
fional mifunderftanding mould be
held to furnifh fome excufe for you
when you precipitately charged me
with diflionourable conducl, I mail
be better pleafed than not.
tions.
Secondly, I have looked carefully at pages 19 and 16 of Sir Wm.
Hamilton s letter, and at all the reft of our correfpondence, without find
ing that I have ever admitted that I fubfcribed to any part of the prof-
peftus as by me " rightly underftood." Page 59 is no doubt a mifprint
for 39. I have neither found, nor have I the flighteft remembrance of,
any fubfcription of mine to any thing Sir Wm. Hamilton ever wrote as
" rightly underftood."
I repeat the account given in my pamphlet of the manner in which I
fubfcribed to this profpeftus ;
The next communication is dated Dec. 28, and confifted of I. A
letter. 2. A printed profpeftus of Sir William Hamilton s intended
work on logic. Nothing turns on this, for the fimple reafon that my
anfwer contained the moft exprefs and formal proof that, come by it
how I might, I was then in the moft complete written poffeffion of all
I have fmce publifhed. . . . The profpedtus which accompanied this letter
3 1 o Appendix.
is very full on the refults which Sir William Hamilton can ^ produce
* from his principles ; but gives nothing, I think, certainly nothing intel-
* ligible to me, on thofe principles themfelves.
As foon as I faw thefe refults, I inftantly faw that many of them
agreed with my own. I had then no doubt that we poffeffed fomething
in common ; and I faid fo very diftinftly in my reply. As the reader
will prefently fee, this firft impreflion has not been confirmed. Feeling
it now time to fecure whatever of independent difcovery might belong
* to me, I anfwered Sir William Hamilton in two letters, dated Decem-
her 31 and January I. In thefe letters
I.I returned the printed profpeftus with the refults underlined
which my fyftem would produce.
2. I ftated that I had a fyftem written on certain meets of paper,
which I defcribed as to number, fize, &c., adding the head words of
each page. I felt inclined to get the fignature of fome good witnefs put
upon thefe papers ; but at the fame time I felt reluftant that Sir Wil-
liam Hamilton mould fee, if it ever became neceffary to produce thefe
papers, that I had been taking precautions againft him. I therefore de-
< termined to make himfelf my witnefs.
3. I ftated diftindtly the firft principles of both my fyftems, and the
fyllogiftic formulae to which they lead.
Thirdly, I fubftantiate the above, fo far as the fubfcription is con
cerned, by quoting two paflages from Sir W. Hamilton s publication
of my letter of December 3 1 .
I received your obliging communication this morning and am now
fully fatisfied that I have, in one of my views of fyllogifm, arrived at
your views in fubftance, or fomething fo like them, that I could fub-
fcribe in my own fenfe to a great part of your paper This
chapter [meaning the one on the meets of paper above referred to]
I might exprefs in your words wherever they are underlined in the
profpedtus which I return, hoping you will fend another. _
Where are thofe words " rightly underftood " which Sir W. Ha
milton attributes to me three times in one paragraph ?
He muft have been quoting from memory. Seeing bis refults, I
found they were alfo my refults; fo I told him that I could " fubfcribe"
(and I cannot find I have ufed this word more than once, and it is in
page 19 referred to by Sir William) " in my own fenfe to a great part
of" his "paper." If words can fpeak meaning, I here tell him that I
fubfcribe in my own fenfe, leaving it to the future to mow whether I
fubfcribe in bis, that is, whether I under ft and him rightly.
[I was reading this for the prefs, when I found out the words which,
applied in one fenfe hypothetically to one of his refults, Sir W. Hamilton
has transferred in a different fenfe to all. One of his refults, fpeaking
of the moods, is the eftablifhment of Their numerical equality under
all the figures, the Italics being his. I could not make out the Englifh
of this. The others I underftood in the grammatical fenfe. For ex
ample, The abrogation of the fpecial laws of fyllogifm is intelligible :
I did not know whether my fenfe of thefe words, that is to fay, my
Appendix. 3 1 1
abrogation of thofe laws, was the fame as Sir W. Hamilton s ; ftill that
he did abrogate certain laws was clear. But numerical equality of moods
I could only underftand as referring to the numerical quantities which
I fuppofed (the reader will remember that I fent back the profpedus by
the next poft, and had little time to look at it) Sir W. Hamilton s fyftem
to contain. It means, I find, that there are the fame number of moods
in all figures : but to attribute numerical equality to different things is a
mode of faying that there is the fame number of them in different fets to
which I was unaccuftomed. Having however, as I thought, divined
what the Englim of this might mean, I underlined it, adding (as Sir W.
Hamilton ftates in one of the foot-notes, which I never remarked till
now) thefe words, " If I underftand this rightly I may underline it I
think." I meant, " If I can make out the words" This underftand
rightly, Sir W. Hamilton actually takes from this fentence, joins it to
my " fubfcription " mentioned in another document , and reprefents me
as declaring that I have "fubfctibed to as rightly underftood" twenty-two
fentences, &c., and himfelf as quoting from one paffage.]
But, had I betimes avowed my non-underftanding, two pamphlets
might have been fpared. Where are we now ? I did avow my not un-
derftanding the firft communication, and my fubfcribing to the fecond
in my own fenfe. To which Sir W. Hamilton fubfequently anfwered
to the effecT: that I fpoke falfe, that I did underftand the firft, for that
I had fent him, in letters written immediately after the fecond was
received, his " fundamental doctrine " and " many of its moft important
confequences." What have I been contending for all along, except
that the doctrine of Mis firft communication was to me mere empty found,
and that all I was able to produce when I received the fecond, was my
own ? But Sir. W. Hamilton actually gives me a right to fay, with
reference to the fecond, the more developed and more intelligible com
munication, that I did not underftand it, infifts upon my faying it, and
reproaches me for not faying it. Well then, to ufe a Scottifli phrafe,
the lefs I lie when I fay I did not underftand the firft, which is the
point at iffue. So that, as to the matter of our controverfy, Sir W. Ham
ilton admits that there was (fortuitous he calls it) entrance of the theory
of the quantified predicate in my writings prior to his communications ;
and as to the conduct of it, he admits that I did not underftand his
communication ; and in the face of fact, reproaches me with maintain
ing that I did till after the pamphlets were written : when it was of the
effence of my ftatement, firft, that I did not underftand, fecondly that
neither I nor any one elfe could have underftood, fave only the pupils
to whom the requifites were addrefled.
MR. DE MORGAN. SIR W. HAMILTON.
Your copious and flaming cri- I difregard your mifreprefenta-
ticifms on my intellect (by which tion that " I avenge myfelf for the
you avenge yourfelf for the retrac- retraction of my afperfion on your
tion of your afperfion on my integ- integrity by my copious and ilafh-
rity), I will profit by fo far as I ing criticifms on your intellect."
3 1 2 Appendix.
difcover them to be true : the reft When your (excufable) irritation
{hall amufe me; and the whole has fubfided, you will fee that I
will be good for the printer. Take could only fecure you from a ver-
one retort from me on the fame did of plagiarifm by bringing you
terms. You have much fkill in in as fuffering under an illufion.
forming new words ; and, as is What, however, is all in all ; my
fair, you put your own image and criticifms will not, I think, be
fuperfcription on your own coin- found untrue,
age. I think you have got into If guilty of lefe majefty by re-
the habit of afTuming the fame ference to the Queen s Englim,
authority over that already exifting have I not my accufer as abettor ?
portion of our language which is For you not only paffed my min-
commonly faid to belong to the tages (quantify and quantification)
Queen and that you need an in- as current coin ; but, in borrow-
terpreter. If I can arrive at your ing, actually " thanked me for the
meaning by the time I write the words" (L. p. 22). However,
preface to my work on logic, I my verbal innovations are, at leaft,
will ftate your claim, accompanied not elementary blunders, I do
by your own words ; if not, I can not, for example, confound a term
flill ftate your own words. Till with a prepojttion, the middle with
then, I have nothing more to fay. the conclusion of a fyllogifm.
Sir W. Hamilton unconfcioufly adapts his language to a very true
fuppofition, namely, that he has, in his pamphlet, made himfelf the jury
in this cafe. He is unfortunate about the mintage. I fay to him You
make new words well, but I am afraid you alter the old ones. To
which he replies Why, you thanked me for my new words. So I did,
and fo I do again : but what has that to do with the lefe majefty part
of my infmuation.
Sir W. Hamilton fays that I have fomewhere (where he does not
fay) ufed term for proportion, middle for conclujion, collectively for dif-
tributively. This may be ; fuch flips of the pen are common enough.
He fets them down as blunders of ignorance. I am not afraid the
reader will follow him. He ought to have faid where they occur, that
is, when he firft mentioned them, in his pamphlet. Till I put thefe
letters together, I was fatisfied, on Sir Wm. Hamilton s ftatement, that
I had done all thefe enormities : but now, after the cafe of " rightly
underftood" which I have juft had to difcufs, I do not feel fo well fatif-
fied,
SIR W. HAMILTON,
Finally, I beg leave to remind you. There is now evidence in your
pofTeffion that for feven years, at leaft, the doctrine of a quantified pre
dicate has been puclickly taught by me ; whilft, on your part, there is
a counter aflertion or innuendo, which, as you cannot prove, it concerns
your character formally to annul.
I never denied that Sir W. Hamilton had taught a doftrine of the
quantified predicate. By the time I wrote my pamphlet, I was pretty
Appendix. 3 1 3
fure that it was not the fame as mine. Sir W. Hamilton s anfwer
confirmed me in this, as appears in page 300.
I now come to mention a part of the difcuffion which I mould per
haps have omitted, if I had not pledged myfelf in my pamphlet to give
an account of a certain offer which I there made to Sir William Ham
ilton, in the event of that offer not being accepted. It is a curious
inftance of that difpofition to hold a correfpondent or an opponent
capable of folving enigmas, and bound to do it, which appears in his
prefuming that (fee B, paragraph 2) an obfcure reference to what is
done in common language would enable me to guefs at the uncommon
language of his fyftem and his lectures. I infert it, alfo, as a fpecimen
of the various mifunderftandings and mifapprehenfions which Sir W.
Hamilton imputes to me, referring to a matter which readers will fepa-
rately comprehend. Had I fpace or inclination to deal with them all,
I believe I could ferve them all in the fame way.
Oft. 7, 1846, I learnt from Sir Wm. Hamilton that his doctrine
had obtained confiderable publicity through the notes and effays of his
fludents. In my reply, referring to this fyflem, and to his offer of
communicating it, I afked if he had a pupil whom he could truft with
the communication ; the anfwer was B, prefently given. But, Dec. zS, in
fending the profpectus, Sir W. Hamilton informed me that, before
forwarding it (the firfl communication in which that he had other than
Ariftotelian quantification was intelligibly announced) he had waited for
a reply from Mr. . That gentleman continues Sir W.
Hamilton, in words fome of which I place in Italics * was a pupil of
mine fix years ago, and obtained one of the higheft honours of the clafs ;
he was therefore fully competent to afford you information, which I
f begged him to do, in regard to my logical doctrines as they were taught
fo far back. I knew him to be a graduate of your College, and he tells
me that he was for three years a pupil of your own. If you are ftill
* interefted in the matter, you can therefore obtain from him as an
acquaintance, what information you wifh, more agreeably than from a
ftranger. When he attended me, befides the twofold wholes in which
the fyllogifm proceeds, the quantification of the predicate, and the effect
of that on the doctrine of converfion, on the doctrine of fyllogirtic
moods, on the fpecial fyllogiftic rules, &c., were topics difcuj/ed, and
partly given out for exercifes. They were, in fa ft, then mere common-
< place.
Jan.l$, 1847, Mr. called on me at Univerfity College, after
an evening lecture of mine, put his notes into my hands, and has fmce
dated (in which I have no doubt he is correct, though I do not remem
ber it) that he informed me he was doubtful whether they contained
exactly what I wanted, and that he would gladly furnifh any additional
"information. Now I conceived, as I thought it was intended by Sir
W. Hamilton I mould do, that the notes of one of the belt fludents,
even if not exactly what I wanted, were fure to contain fomething of
the mere commonplace (by which I took to be meant the ordinary matter
of the lectures) which was difcu/ed, and given out as exercifes to thofe
3 1 4 Appendix.
who attended. But in thefe notes I found nothing on quantification (I
had now this key word, which did not appear in the main communica
tion B) differing from what is ufual ; and after expreffmg this in my
pamphlet, I proceeded as follows :
But if there really be anything in which Sir William Hamilton has
preceded me, I mail be, of all men except himfelf, moft interefted in
his having his full rights. And I make him this offer, and will take his
acceptance of it as reparation in full for his fufpicions and aiTertions.
With the confent of the gentleman to whom thefe notes belong, which
I am fure will not be refufed to our joint application, I will forward to
him a copy of their table of contents, having more than a hundred and
< fifty headings. From thefe Sir William Hamilton mail felect thofe
which are, in his opinion, fure to contain proof of his priority on any
point which I have inveftigated. Of thefe I will have copies made and
fent to him : and will print in the work on Logic which I am preparing
(and in fome one part of it) the parts which he {hall felecl: as fit to
prove (or to mow that he could prove, let him call it as he likes) his
< cafe, or the germs of his cafe (as he pleafes, again). Provided always,
that the matter mall not run beyond fome eight or a dozen octavo pages
< of fmall print. And I on my part propofe that I {hall be allowed to
print, to one-half the amount felected by Sir William Hamilton, of ad-
ditional extract : but if this be refufed I will not infift on it. With this
* I will put a heading fully defcriptive of the reafon and meaning of the
infertion, and fuch diftinct reference and account at the beginning of
the preface as mall be fure to call the reader s attention to it. So that
my book mall eftablifh the claim, if it can be eftablifhed from the notes
* of one of the beft fludents. If this offer be not accepted, an account of
it will take the place of any other refult. If Sir William Hamilton, or
any one elfe, can propofe anything to make this offer fairer, I mall pro-
bably not be found indifpofed to accept the addition. And though, I
< will frankly fay, my prefent conviction is that the acceptance of the
offer would alone caufe my work to knock Sir William Hamilton s
* affertions to atoms, yet I will pledge myfelf, in any cafe, to abide by it.
Had our places in this difcuffion been changed, I mould have taken
care that no reader of my anfwer mould have been left in ignorance of
fo fair an offer on the part of my opponent : more efpecially if that
opponent had been accufed by me of fraud and falfehood, in a manner
which I felt obliged formally to retract. But Sir Wm. Hamilton does
not notice the offer, even by an allufion : and refers to the notes in the
following way :
( In regard to Mr and his Notes, I beg leave to fay, that in
my relative letters, neither to that gentleman nor to you, did I ever
refer to his Notes of my lectures, but exclufively to his perfonal infor-
mation in regard to them. And for a fufficient reafon. The Paragraphs
on Logic dictated to, and taken down by, my fludents, on which I after-
wards prelect, were written fo far back as the year 1837, and prior to
many of my new views, and to the whole doctrine of a quantified predi-
* cate. Thefe views, as developed, were, and are, introduced in a great
Appendix. 315
meafure as corrections of the common doclrine ; in the older Notes
efpecially, they may, therefore, not appear in the dictated and numbered
* Paragraphs at all j whilft, frequently, (particularly at firft,) they were
given out as data, on which, previous to farther comment, the ftudents
were called on or excited to write expofitory EiTays. I diftindlly recol-
left, that in the Seffion during which Mr. attended my courfe of
Logic (1840-1) it was required, on the hypothecs of a quantified pre-
dicate, to Hate in detail, the valid moods of each fyllogiftic figure ; and
I, further, diftinftly recoiled, that Mr. was one of thofe who
effayed this problem. If wrong on this point, I mail admit that my
memory is as treacherous as yours. It was, indeed, quite natural, that
Mr. mould give, and that you ihould receive, his Notes ; but,
of courfe, you could have fought or obtained no perfonal information
from him, in reference to the point in queftion, without mentioning the
fad Were it, however, requilite to give proof from Notes of fo
manifeft a fad, I doubt not that fcores of ftudents would be willing to
place theirs at my difpofal.
On the appearance of Sir W. Hamilton s pamphlet, Mr.
wrote him a very ftraightforward letter, of which he fent me a copy,
with permiffion to both of us to ufe it. The general tenor is that Sir
W. Hamilton is corred in his ftatements of what he had taught (which
ftatements I never impugned as to fad ; I did not know what they
meant). On the point in queftion Mr. fays (the Italics are
mine) ;
During the Seffion in which I attended your ledures (1840 and
* 1841) your new fyftem, bafed on the thorough going quantification of
the predicate (the fecond of the three fyftems mentioned in page 3 1 of
your publilhed letter) and its confequences in making all proportions
limply convertible &c. was not developed by you in your ordinary feries
f of Leftures. I believe it was not touched upon in them, but it was partly
( explained to the clafs verbally* and then given out as a fubjeflfor Ef-
4 fays. When the Effays were given in they were read aloud in the clafs,
and commented upon by you, and in fo doing you fully explained the
fyftem as " a full extension and thereby a complete fimplification of the
fyllogiftic theory."
Thefe fads which were ftrongly fixed in my memory, becaufe I
* believe on that occafion I happened to be the only EfTayift who had
rightly apprehended and worked out the thefis, will account for the
circumftance that my notes, which were originally taken in Ihorthand,
although containing a full Report of all your ordinary Lectures, are
completely filent on the fubjed."
The reader may find out, if he can, where Sir W. Hamilton re
ferred to perfonal information as diftinguiihed from notes, or to his
teaching of his new fyftem, as a matter diftind from that of his ordinary
ledlures : and muft judge what his fuccefs is in faying what he means
* I think this fhould be extempore-, meaning that Sir W. Hamilton ufually reads his
le&ures.
3 1 6 Appendix.
to fay. And he may find out further, how I was to guefs that the
mere commonplace of the topics difcujfed in Sir William s teaching was to
come, after an interval of fix years, from his old pupil s perfonal infor
mation, and not from the full and (as I found them) excellent notes which
he made at the time.
I mould add that Mr. , fubfequently to the printed contro-
verfy, anfwered every query which I put to him on Sir W. Hamilton s
fyftem, but did not feel juftified (as in a like cafe I mould not either) in
anfwering pofitively as to the minute details of it, after laying it by for
years.
^ I have mentioned one or two inftances in which, as feems to me,
Sir W. Hamilton has a ftrange idea of the fenfe of his own words : I
will now take one of the cafes in which he has dealt as ftrangely with
mine. The way in which we ufe language, is one of the means which
the reader has, for forming his judgment on the whole of this difpute :
and he muft decide which of us is incapable of giving to the phrafes of
the other their proper fignification.
When I returned to Sir W. Hamilton his profpeftus, with thofe
parts underlined which I could interpret in my own fenfe, the more
important parts relating to logical mood and figure were not thus un
derlined. In the accompanying letter, I ufed thefe words, To mood
and figure, I have attended but little ; what I get on thefe points will
be from your hint, or from your book. The whole letter was on what
I had done in the way of inveftigation, not of elementary reading : and
I may fafely fay that it is clear I meant that I had not made mood and
figure, as conftituent parts of a theory of fyllogifm, fubjefts of inveftiga
tion, with a view to new properties. But Sir W. Hamilton, in two
places, makes me avow ignorance of the ordinary fyftem of mood and
figure. In a foot-note to the above, he fays, " And yet, though con-
feffedly to feek in the very alphabet of the fcience, Mr. De Morgan
would be a logical inventor ! What is here acknowledged in terms, is
< fufficiently manifefted from miftakes."* And in his pamphlet (II. p.
9), he reprefents me as no proficient no thorough ftudent, in the
fcience ; and refers to this paragraph of mine as the ground of the
afTertion. It would have been ftrange, if, avowing ignorance of the
ordinary doftrine of mood and figure, I had faid that what I mould get
on thefe points muft be from Sir W. Hamilton s hint or unpublifhed
book, when any ordinary treatife would have given it : fo ftrange, that
this claufe ought, I think, to have fuggefted the obvious meaning. Is
Sir W. Hamilton s interpretation a fair one ? I do not doubt that he
meant it to be fair. What I afk is, has he the power to read fairly as
well as the will ?
The two preceding cafes (that of the notes and that of the avowed
ignorance] are fpecimens of Sir W. Hamilton s give and take, of the
* Sir W. Hamilton fhould have cited a few : but when he declares I have made
elementary blunders, he does not give fo much as a reference. The plan is a fafe one.
Appendix. 3 1 7
manner in which he expefts to be underftood, and of that in which he
claims" a right to underftand. They are alfo, of courfe, fpecimens of
my own.
In (A), the fymbols A, E, I, O, are the A, E ( , I|, O 4 , of this work :
and a, e, i, o, are the A 1 , E , I 1 , O f .
(A) From the paper as fent to Cambridge before I bad any communica
tion whatfoever from Sir William Hamilton (without any corrections).
SECTION III. On the quantity of proportions .
" The logical ufe of the word feme, as merely more than none,
needs no further explanation. ExacT: knowledge of the extent of a pro-
pofition would confift in knowing, for inftance in fome Xs are not Ys
both what proportion of the Xs are fpoken of, and what proportion
exifts between the whole number of Xs and of Ys. The want of this
information compels us to divide the exponents of our proportion into
o, more than o not neceffarily I, and I. An algebraifl learns to con-
fider the diftin&ion between o and quantity as identical, for many
purpofes, with that between one quantity and another : the logician
mull (all writers imply) keep the diftinftion between o and a, however
fmall a may be, as facred as that between o and I a ; there being but
the fame form for the two cafes. We mail now fee that this matter
has not been fully examined.
" Inference muft confift in bringing each two things which are to be
compared into comparifon with a third. Many comparifons may be
made at once, but there muft be this procefs in every one. When the
comparifon is that of identity, of is or is not, it can only be in its ulti
mate or individual cafe, one of the two following : This X is a Y,
this Z is the very fame Y, therefore this X is this Z ; or elfe This X
is a Y, this Z is not the very fame Y, therefore this X is not this Z.
And colleftively, it muft be either Each of thefe Xs is a Y ; each of
thefe Ys is a Z ; therefore each of thefe Xs is a Z ; or elfe Each of
thefe Xs is a Y, no one of thefe Ys is a Z, therefore no one of thefe
Xs is a Z.
" All that is efTential then to a fyllogifm is that its premifes fhall
mention a number of Ys, of each of which they fhall affirm either that
it is both X and Z, or that it is one and is not the other. The pre
mifes may mention more : but it is enough that this much can be picked
out ; and it is in this laft procefs that inference confifts.
" Ariftotle noticed but one way of being fure that the fame Ys are
fpoken of in both premifes ; namely, by fpeaking of all of them in one
at leaft. But this is only a cafe of the rule : for all that is neceffary is
that more Ys in number than there exift feparate YsJbaH be fpoken of in
both premifes together. Having to make m-\-n greater than unity, when
neither m nor n is fo, he admitted only that cafe in which one of the
two m or n, is unity and the other is anything except o. Here then
are two fyllogifms which ought to have appeared, but do not,
3 i 8 Appendix.
Moft of the Ys are Xs Moft of the Ys are Xs
Molt of the Ys are Zs Moft of the Ys are not Zs
.*. Some Xs are Zs .. Some of the Xs are not Zs
And inftead of moft, or ^ -\-a, of the Ys, may be fubftituted any two
fractions which have a fum greater than unity. If thefe fractions be m
and n, then the middle term is at leaft the fraction m-{- n \ of the Ys.
It is not really even neceffary that all the Ys mould enter in one pre-
mifs or the other: for more than the fraction m-\-n I of the whole
may be repeated twice.
And in truth it is this mode of fyllogifing that we are frequently
obliged to have recourfe to ; perhaps more often than not in our uni-
verfal fyllogifms. All men are capable of fome inftruction ; all who
are capable of any inftru&ion can learn to diftinguifh their right and left
hands by name ; therefore all men can learn to do fo. J Let the word
all in thefe two cafes mean only all but one, and the books on logic tell
us with one voice that the fyllogifm has particular premifes, and no con-
clufion can be drawn. But in fact idiots are capable of no inftruction,
many are deaf and dumb, fome are without hands : and yet a conclufion
is admiflible. Here m and n are each very near to unity, and m-\-n I
is therefore near to unity. Some will fay that this is a probable con-
clufio h : that in the cafe of any one perfon it means there is the chance
m that he can receive inftruction, and n that one fo gifted can be made
to name his right and left hand : therefore m X n (very near unity) is
the chance that this man can learn fo much.
" But I cannot fee how in this inftance the probability is anything
but another fort of inference from the demonftrable conclufion of the
fyllogifm, which muft exift under the premifes given. Befides which,
even if we admit the fyllogifm as only probable with regard to any
one man, it is abfolute and demonftrative in regard to the propofition
with which it concludes.
" But this is not the only cafe in which the middle term need not
enter univerfally : this however is matter for the next Section. I now
go on to another point."
Extratt II.
" I now take the two cafes in which particular premifes may give a
conclulion : namely
I 7/ XY+XY=XZ XY+Y:Zz=X:Z O ro
on the fuppofition that the Ys mentioned in both premifes are in num
ber more than all the Ys. If Y x and Y 2 {land for the fractions of the
whole number of Ys mentioned or implied in the two premifes, and y r
2 for the fractions of the ys implied or mentioned, we mail by a
Appendix. 3 1 9
repetition of the procefs on YX-}-YZ=XZ (the other being obtained
in the courfe of the procefs) arrive at the following refults or their
counterparts : remembering that Y T 4-Yj, is greater or lefs than i, ac
cording as^-j-^ is lefs or greater.
Dcfignation. Syllogifm. Condition of its exiftence.
I 7/ YX + YZ = XZ Y, +Y 2 greater than I
O fo YX+Y:Z=rXZ ........................
O oi X:Y + yz =X:Z Y,+Y 2 lefs than I
O oi
loo X:Y + Z:Y= XZ .....................
(B) Communication received on the ^tb or th of November from Sir
William Hamilton, being the pretext for his charge that I have, with
injurious breach of confidence towards himfelf, and falfe dealing to
wards the public, appropriated his " Fundamental Doctrine of Syllo
gifm" privately communicated to me : and, after the retraction of that
charge, noticed in pages 297, S,for the aj/ertion that I have done the
fame thing unconfcioujly.
" 1 6, Great King Street,
November 2nd, 1846.
" DEAR SIR, I have been longer than I anticipated in anfwering
your laft letter. I now fend you a copy of the requifites for the prize
Elfay, which I gave out to my ftudents at the clofe of laft feffion. It
will mow you the nature of my doctrine of fyllogifm, in one of its
halves. The other, which is not there touched on, regards the two
wholes, or quantities in which a fyllogifm is caft. I had intended fend
ing you a copy of a more articulate ftatement which I meant, at any
rate, to have drawn up ; but I have not as yet been able to write this.
I will fend it when it is done. From what you ftate of your fyftem
having * little in common with the old one, and from the contents of
your Firft Notions, we mall not, I find, at all interfere, for my doctrine
is limply that of Ariftotle, fully developed.
It will give me great pleafure if I can be of any ufe, in your invefti-
gations concerning the hiftory of Logical doctrines. I have paid great
attention to this fubject, on which I found, that I could obtain little or
no information from the profelfed hiftorians of Logic ; and my collec
tion of Logical books is probably the moil complete in this country.
But, as I mentioned to you in my former letter, it is only in fubordinate
matters that in abftratt Logic there has been any progrefs.
" I remain, dear Sir, very truly yours,
"W. HAMILTON."
320 Appendix.
Effay on the new Analytic of Logical Forms.
Without wifhing to prefcribe any definite order, it is required that
there fhould be rtated in the Effay,
1. What Logic poftulates as a condition of its applicability.
2. The reafons why common language makes an ellipjts of the ex-
preffed quantity frequently of the fubjecl, and more frequently of the
predicate , though both have always their quantities in thought. \This
paragraph is the one on which Sir W. Hamilton principally relies],
3. Converfion of proportions on the common doctrine.
4. Defeds of this.
5. Figure and Mood of Categorical fyllogifm, and Reduction, on
common doctrine (General ftatement).
6. Defects of this (General ftatement).
7. The onefupreme Canon of Categorical Syllogifms.
8. The evolution, from this canon, of all thefpecies of Syllogifm.
9. The evolution, from this canon, of all the general laws of cate
gorical Syllogifms.
10. The error of \h.z fpecial laws for the feveral Figures of Catego
rical Syllogifm.
11. How many Figures are there.
12. What are the Canons of the feveral Figures.
13. How many moods are there in all the Figures : mowing in con
crete examples, through all the Moods, the uneffential variation which
Figure makes in a fyllogifm.
(Thofe which follow 13 were wrong numbered.)
15. What relation do the Figures hold to extenfion and comprehen-
Jton.
1 6. Why have the fecond and third Figures no determinate major
and minor premifes and two indifferent conclufions : while the firft Fi
gure has a determinate major and minor premife, and a fmgle proximate
conclufion.
17. What relation do the Figures hold to Deduction and Induction.
N.B. This EiTay open for competition to all ftudents of the clafs of
Logic and Metaphyfics during the laft or during the enfuing feflion.
April I5th, 1846.
(C) Extract from the Addition to my Paper, taken, as can be Jhown,
from the papers which I gave the means of identifying in January
laft, and which papers (though I hold it immaterial) I ajjert to have
been written before I received any logical communication from Sir
William Hamilton. (To be compared with the extracts given in A).
" Since this paper was written, I found that the whole theory of the
fyllogifm might be deduced from the confideration of propofitions in a
form in which definite quantity of afTertion is given both to the fubject
and the predicate of a propofition. I had committed this view to
paper, when I learned from Sir William Hamilton of Edinburgh, that
Appendix. 321
he had for fome time part publicly taught a theory of the fyllogifm
differing in detail and extent from that of Ariftotle. From the pro-
fpeaus of an intended work on logic, which Sir William Hamilton has
recently iffued, at the end of his edition of Reid, as well as from infor
mation conveyed to me by himfelf in general terms, I mould fuppofe it
will be found that I have been more or lefs anticipated in the view juft
alluded to. To what extent this has been the cafe, I cannot now
afcertam ; but the book of which the profpeaus juft named is an
announcement, will fettle that queflion. From the extraordinary extent
of its author s learning in the hiftory of philofophy, and the acutenefs of
his written articles on the fubjea, all who are interefted in logic will
look for its appearance with more than common intereft.
" The footing upon which we mould be glad to put proportions, if
our knowledge were minute enough, is the following. We mould ftate
how many individuals there are under the names which are the fubjea
and predicate, and of how many of each we mean to fpeak. Thus
mftead of Some Xs are Ys, it would be, Every one of a fpecified Xs
is one or other of b fpecified Ys. And the negative form would be as
m No one of a fpecified Xs is any one of b fpecified Ys. If propofi-
tions be ftated in this way, the conditions of inference are as follows.
Let the effeaive number of a propofition be the number of mentioned
cafes of the fubjel, if it be an affirmative propofition, or of the middle
term, if it be a negative propofition. Thus, in < Each one of 50 Xs is
one or other of 70 Ys, is a propofition, the effeaive number of which
is always 50. But No one of 50 Xs is any one of 70 Ys is a propo
fition, the effeaive number of which is 50 or 70, according as X or Y
is the middle term of the fyllogifm in which it is to be ufed. Then
two proportions, each of two terms, and having one term in common,
admit an inference when i. They are not both negative. 2. The
fum of the effeaive numbers of the two premifes is greater than the
whole number of exifting cafes of the middle term. And the excefs of
that fum above the number of cafes of the middle term is the number
of the cafes in the affirmative premifs which are the fubjeas of inference.
Thus, if there be 100 Ys, and we can fay that each of 50 Xs is one or
other of 80 Ys, and that no one of 20 Zs is any one of 60 Ys ; the
effeaive numbers are 50 and 60. And 50+60 exceeding 100 by 10,
there are i o Xs, of which we may affirm that no one of them is any
one of 20 Zs mentioned.
The following brief fummary will enable the reader to obferve the
complete deduaion of all the Ariftotelian forms, and the various modes
of inference from fpecifc particulars, of which a fliort account has
already been given.
" Let a be the whole number of Xs ; and / the number fpecified in
the premifs. Let c be the whole number of Zs ; and w the number
fpecified in the premifs. Let b be the whole number of Ys ; and u and
v the numbers fpecified in the premifes of x and z. Let X,Y M denote
that each of/ Xs is affirmed to be one out of u Ys : and X, : Y M that
each of/ Xs is denied to be any one out of u Ys. Let X,,, n fignify m
322 Appendix.
Xs taken out of a larger fpecified number n ; and fo on. Then the five
poffible fyllogifms, on the condition that no contraries are to enter either
premifes or conclufion, are as follows :
X,
2. X.Y.+Y.Z. =X l + 1 ^ f< Z w -Z t + v _ b>w X t .
3 . Y B X,+ Y.Z., = X. 4- v -i, t Z W =Z U + v _ b , w X,
4. X / Y M -|-Z W : Y X t + v -b, t Z w .
5. y u X t +Z w :Y v =:X u+v ^, t :^
" The condition of inference exprefles itfelf; in the X m>t of the con
clufion, m muft neither be o nor negative. The firft cafe gives no
Ariftotelian fyllogifm ; the middle term never entering univerfally (of
neceffity) into any of its forms, under any degree of fpecification which
the ufual modes of fpeaking allow. The other cafes divide the old fyl
logifms among themfelves in the following manner : they are written fo
as to mow that there is fometimes a little difference of amount of fpeci
fication between the refults of different figures, which changes in the
reduction from one figure to another. The Roman numerals mark the
figures.
?.
/ ~ tf , vi^b
t </z, v b
i < a , z> in ^
*/ < b, v b
ui^.by v < b
T* ~~~ * ~~~ J ""~
Y . Z+XJY^X . Z
X)Y,,+Z . Y=Z . X
X)Y ? ,+Y . ZzrZ . X
Y . Z=X,Y M = X,: Z
Z.Y-f-X^-X^Z
=X :Z
Y.
Z . Y+Y 7( X<=X 7 ,
Y v : Z+Y)X^X,, ,
:Z
:Z
Barbara I.
Bramantip IV
D^r I.
Dimaris IV.
Darapti III.
D if amis III.
>*/// III.
Celarent I.
C<?>^ II.
Cameftres II.
Camenes IV.
Fm<? I.
/V/?/#0 II.
Baroko II.
Felapton III.
III.
Frefifon IV.
Bokardo III.
I conclude by fubmitdng to the reader what I began with, namely,
that until Sir William Hamilton produces fomething from C, intelligi
bly hinted at in B, and neither fubftantially contained in the matter, nor
Appendix. 323
immediately deducible from the principles, of A, he has no right what
ever to aflert that I have borrowed from him confcioufly or unconfci-
oufly. I have not found any perfon who thinks that fuch a thing can
be produced : and I leave every reader to form his own opinion whether
it can he done or not.
APPENDIX II.
On fame forms of inference differing from thofe of the Arlftotellans.
I THINK it deiirable to ftate all I know of any attempt to deal with
the forms of inference othenvife than in the Ariftotelian method.
Since the time of Wallis, three well known mathematicians have written
on the fubjecl, Euler, Lambert, and Gergonne : there may have been
others, but I have not met with them.
Euler s Lettres a une PrincefTe d Allemagne fur quelques fujets de
Phyfiqueet de Philofophie (3 vols. 8vo. Peter/burg 1768-1772, accord
ing to Fufs) contain the reprefentation of the fyllogifm by fenfible terms,
namely, areas. There was a Paris edition by Condorcet and Lacroix,
in 1787, as is ftated by Dr Henry Hunter, who publifhed an Englifh
tranflation from it and from the original edition, London, 1795, 2 vols.
8vo. Euler makes ufe of circles to reprefent the terms. In a tradl
publifhed (or completed) in 1831, in the Library of Ufeful Knowledge,
under the name of the Study and Difficulties of Mathematics I fell
upon this method before I knew what Euler had done, ufing, for dif-
tinftion, fquares, circles, and triangles, as in Chapter I. of this work.
The author of the " Outlines" prefently mentioned, has what I con-
fider a very happy improvement on Euler. The propofition fome X
is Y, is reprefented by the latter as the circle of X, partly infide and
partly outfide the Y. The author of the " Outlines" puts a broken
fegment of the circle of X infide the circle of Y, leaving it unfettled
whether the reft of the circle is united to the broken piece, or tranf-
ferred elfe where.*
But Euler had been preceded in the publication of this idea by Lam
bert, in his Neues Organon, &c. Leipzig, 1764, 2 vols. 8vo. In
this work, the terms are reprefented by lines, and identical extents by
parts of the lines vertically under one another, as in page 79. The
whole notion is reprefented by continuous line, the part left indefinite
in particular proportions by dotted line. Some of the contranominal
forms are more diftinftly mentioned than is ufual, but there is no intro
duction that I can find of any form of inference which is not Ariftote
lian.
* I fhould fay that Euler does not ufe the numerical, but the magnitu.iinal notion,
(fee page 48 of this work).
324 Appendix.
In the feventh volume of the Annales de Matbematiques (Nifmes,
1816 and 1817, 4to.) there is a paper by the editor, M. Gcrgonne,
entitled E/ai de dialettique rationale. I did not fee this ^ paper, nor
Lambert s work, until after my memoir in the Tranfactions of the
Cambridge Society had been publimed. The fecond would have given
me no hint : the firft might have done fo. There is the idea, and fome
formal ufe, of a complex proportion : but the divifion is erroneous.
The fubidentical, identical, and fuperidentical forms are there; thefe
are not eafily miffed : the others which Gergonne ufes are, the complete
exclufion (the contrary or fub contrary of my fyftem, which, disjunctively,
are only the common univerfal negative) and partial inclufion with par
tial ex clufwn (the complex particular, or fupercontrary, of mine). The
ufe of contraries is expreffly* forbidden, the old converfion by contra-
pofition formally declared/^, and the particular proportion afferted
to be incapable of being made univerfal. But M. Gergonne s complex
proportions, fuch as they are, are ufed in a manner refembling that in
chapter V, of this work, though requiring a feparate tatonnement for
many things the analogues of which appear as connected refults of my
fyftem. Accordingly, I am bound to attribute to M. Gergonne the firft
publication of the idea of a complex fyllogifm, and of the comparifon of
the fimple one with it. But numerical ftatement is not hinted at.
Sir William Hamilton s fyftem dates, as to its publication in lectures,
from 1841, as far as has yet appeared. What I have to fay of it will
be found in another appendix.
In 1842, there was publimed anonymoufly Outline of the laws of
thought ; London and Oxford (Pickering, and Graham) ottavo in twos
(fmall). The author is the Rev. Wm. Thomfon, tutor of Queen s
College, Oxford. It is a very acute work, and learned. The fyftem of
proportions is extendedby the introduction of both the common quanti-
rcations of the predicate into the affirmatives only, which introduces the
proportions U and Y, as the author calls them, or "All Xs are all Ys,"
and " Some Xs are all Ys."
The memoir in the Cambridge Tranfactions in which I gave the firft
account of what has fmce grown into Chapters IV, V, VIII, and X, of
this work, is defcribed as to date in the preceding appendix. With re
ference to the fubject of chapter V, I may note the following defects
of that memoir : I . That only one arrangement of X and Z as pre-
mifes being taken, only half the fyftem is given, and many correlative
arrangements are not obtained (fee page 140). 2. That owing to my
not feeing diftindtly that each univerfal propofition has two weakened
forms, the fyllogifms AjA I and E E I t are confidered as a clafs apart.
3. That much of the power of forming eafy rules is not gained, by the
order of reference being made XY, ZY, XZ, inftead of XY, YZ, XZ.
The former appears at firft the more natural order, and is certainly
I am told that fome works on logic ufed in the Irifh colleges formally announce
"I law
any
that the truth of the [ordinary] laws of fyllogifm depends upon the exclufion of contra
ries : but I have not met with any of them.
Appendix. 325
more eafily defcribed ; namely, to refer each of the concluding terms to
the middle term, with which both are compared. I obfcrvc, fmce,
that M. Gergonne adopts this laft order of" reference : but the other is
by an immenfe deal more convenient in its refults, as I think I have
mown.
With refpecl: to the numerical quantification, what I did in the Me
moir and Addition is given in full in the preceding appendix. Sir
William Hamilton, who diftinclly renounces all claim to the " arithme
tically articulate" fyftem, and doubts whether it afford any bafis for a
logical developement, ftates that he had formerly obtained the " ultra-
total quantification" (page 317) and thrown it away as a cumbrous and
ufelefs fubtlety, without publifhing it, as I underftand, in any way. To
his reply, he appends a note which I think it defirable to republifh at
length, as a document in the hiftory of this fpeculation, and that I may
make that hiftory complete (II. p. 41).
I have avoided, in the previous letter and poflfcript, all details in
regard to the third fcheme of quantification (p. 32) ; becaufe that fcheme
except in fo far as it is confounded with the jfcrwfc/, has no bearing in
the controverfy; and I admit that whatever Mr. De Morgan has
therein accomplifhed, he has accomplished independently of me. Fur-
ther, I mail not deny him any claim of priority to whatever he may
have ftated in our correfpondence, in reference to this third fcheme.
Finally, I mail acknowledge, for I think it not improbable, that his
fyllogifm (p. 19) fuggefted a reconfideration, on my fickbed, of a cer-
tain former fpeculation, in regard to the ultratotal quantification of the
middle term in both premifes together ; a fpeculation determined by
the vacillation of the logicians, touching the predefignations more, moft,
&c. but which I had laid, afide, as a ufelefs and cumbrous fubtlety.
Arirtotle, followed by the logicians, did not introduce into his doc-
trine of fyllogifm, any quantification between the abfolutely univerfal
and the merely particular predefignations, for valid reafons. 1, Such
quantifications were of no value or application in the one whole (the
univerfal, potential, logical), or, as I would amplify it, in the two cor-
relative and counter wholes (the logical, and the formal, aftual,
metaphyfical,) with which Logic is converfant. For all that is out of
clarification, all that has no reference to genus and fpecies, is out of
Logic, indeed out of Philofophy ; for Philofophy tends always to the
univerfal and neceflary. Thus the higheft canons of dedu&ive reafon-
ing, the difla de Omni et de Nullo, were founded on, and for, the
procedure from the univerfal whole to the fubjeft parts ; whilft, con-
verfely, the principle of indudive reafoning was eftablifhed on, and for,
the (real or prefumed) collection of all the fubjeft parts as conilituting
the univerfal whole. 2, The integrate or mathematical whole, on
the contrary, (whether continuous or difcrete) the philofophers con-
temned. For whilft, as Ariftotle obferves, in mathematics genus and
fpecies are of no account ; it is, almoft exclufively, in the mathemati-
cal whole, that quantities are compared together, through a middle
term, in neither premife, equal to the whole. But this reafoning, in
326 Appendix.
which the middle term is never univerfal, and the conclufion always
* particular, is, as vague, partial, and contingent, of little or no value
in philofophy. It was accordingly ignored in Logic ; and die prede-
* fignations more moft, &c., as I have faid, referred, to univerfal, or,
(as was moft common) to particular, or to neither, quantity. This
* difcrepancy among Logicians long ago attracted my attention ; and I
faw, at once, that the poffibility of inference confidered abfolutely,
depended, exclufively on the quantifications of the middle term, in both
* premifes, being, together, more than its poffible totality its diftribution,
* in any one. At the fame time I was impreffed I*, with the almoft
utter inutility of fuch reafoning, in a philofophical relation : ^and 2,
alarmed with the load of valid moods which its recognition in Logic
would introduce. The mere quantification of the predicate, under the
* two pure quantities of definite and indefinite, and the two qualities of
affirmative and negative, gives (abflraftly) in each figure, thirty fix
valid moods ; which, (if my prefent calculation be correft,) would be
multiplied, by the introduction of the two hybrid or ambiguous quan-
< tifications of a majority and a half, to the fearful amount of four bun-
dred and eighty valid moods for each figure. Though not, at the
time, fully aware of the ftrength of thefe objeftions, they however
prevented me from breaking down the old limitation ; but as my fu-
preme canon of Syllogifm proceeds on the mere formal poffibility of
* reafoning, it of courfe comprehends all the legitimate forms of quanti-
* fication. It is ; What worjl relation of fubjett and predicate, fubfifts
between either of two terms and a common third term, with which one,
at lea ft, is pojitively related ; that relation fubfifts between the two
< terms themfehes : in other words ; In as far as two notions both
agree, or one agreeing, the other dif agrees, with a common third notion:
< in fo far, thofe notions agree or dif agree with each other. This canon
applies, and proximately, to all categorical fyllogifms, in extenfion
and comprehenfion, affirmative and negative, and of any figure. It
determines all the varieties of fuch fyllogifms ; is developed into all
their general, and fuperfedes all their fpecial, laws. In fliort, without
* violating this canon, no categorical reafoning can, formally, be wrong.
Now, this canon fuppofes that the two extremes are compared together,
* through the fame common middle ; and this cannot but be, if the
middle, whether, fubjeft or predicate, in both its quantifications to-
gether, exceed its totality, though not taken in that totality in either
premife.
But, as I have ftated, I was moved to the reconfideration of this
< whole matter ; and it may have been Mr. De Morgan s fyllogifm in
our correfpondence (p. 19), which gave the fuggeilion. The remit
was the opinion, that thefe two quantifications mould be taken into
account by Logic, as authentic forms, but then relegated, as of little
* ufe in pra&ice, and cumbering the fcience with a fuperfluous mafs of
moods. As to Mr. De Morgan s ftatement in our correfpondence (p.
2 1) of the principle on which (by his later fyftem) fuch fyllogifms
proceed, this, to ufe his own exprcffion, " I did not comprehend at
Appendix. 327
all ;" nor do I now,* having, to fpeak with the Rabbis, " referved it
* for the advent of Elias." I faw however, that, be it what it might,
* it had no analogy with mine ; indeed, even from the fuller expofition
of his dodlrines, contained in the body of the Cambridge Memoir and
its Addition, which I afterwards received, I can find no indication
* of his having generalifed either, I the comprebetijive principle of all
inference, that the two quantifications of the middle term, Jbould, to-
get her, exceed it as a Jingle whole y or, 2, under a non-diftributed
( middle, the TWO exclujive forms of its quantification. On receipt,
* however, of Mr. De Morgan s Cambridge Memoir, I faw, or thought
I faw, in the body of the paper, on his old view, fome manifeftation of
f a lefs erroneous doctrine upon this point, than that afterwards contained
in his Letters and Addition, upon his new. Accordingly, to obviate
all mifconftruftion, I wrote immediately the following letter,f of which
* an account has been previoufly given (p. 26, note).
EDINBURGH, 30^ March, 1847.
Your paper read to the Society I have curforily perufed ; but though
oppofed to many of its doctrines, I admire the ingenuity which charac-
* The paflages which Sir William Hamilton does notunderftand, are the following,
and alfo that relating to the effective terms, in C of the preceding appendix.
" Now fuppofe propofitions in which the quantitative part of the preceding is made
more definite. Say that
X t Y u | and X t : Y u |
mean
Every one of t Xs No one of t Xs
is one or other of u Ys is any one of u Ys
Let the effe&ive number of cafes in a propofition be the number which it makes ef
fective in inference. Then the effective number in a pofitive propofition is the num
ber of cafes of thefubjefl.
The effective number in a negative propofition is the number of cafes of the middle
term.
And the criterion of inference being poflible, is that the fum of the effective num
bers of the two premifes (not both negative) is greater than the whole number of cafes
of the middle term.
And the excefs is the number of cafes involved in the inference, of all which are
mentioned in the conclufion-term (or terms) of the pofitive premifs (or premifes).
For inftance, let b be the whole number of Ys in exiftence : I afk whether we can
infer anything from
X t Y u effective number t
Z w : Y v .... v
Anfwer, if t -J- v be greater than b, we can infer
Xt-fv b : Zw
Or, if each of t Xs be one or other of u Ys, and no one of w Zs be any one of v Ys,
then if t and v together are more in number than there are Ys, we may infer that no
one of t -|- v b Xs is any one of the w Zs juft fpoken of."
f This letter (the firft paragraph of which is omitted, as not relevant to this appen
dix,) was addrefied to me, and was fent open to my friend Dr. Sharpey, to be deli
vered to me. Dr. Sharpey refufed to deliver (and, as it happened, I was as much
prepared to refufe to receive) any thing on the literary fubject matter of the controverfy
which did not contain a retraction of Sir W. Hamilton s then fubfifting charge againft
me. Accordingly, I never faw it till it appeared in print.
328 Appendix.
* terifes it throughout. On one point, I find we coincide, in principle,
at leaft, againft logicians in general. They have referred the quantify-
ing predefignations plurimi, and the like, to the moll oppofite heads ;
* fome making them univerfal, fome, particular, and fbme between
both ; (for you are not correft in faying, (p. 6), that logicians are
unanimous in regarding them as particular, [though molt do]). This
conflidlion attracted my attention ; and a little confideration mowed
me, that befides the quantification of the pure quantities, univerfal vn&
particular, (which I call definite and indefinite,} there are two others of
* thefe, mixed and half developed, which ought to be taken into account
by the logician, as affording valid inference ; but which, without fcien-
tific error, cannot be referred either to univerfal, (definite,) or to par-
* ticular, (indefinite) quantity, far lefs left to vacillate ambiguoufly be-
tween thefe. I accordingly introduced them into my modification, in
Englifh doggerel, of " AJJ erit A" &c. f which [in the original caft] I
formerly faid was at your fervice ; and as it affords a brief view of my
doclrine on this point, I may now quote it.
A, it affirms of this, that, all,*
Whilft E denies of any,
I, it affirms, whilft O denies,
Of fome (or few or many).
1 Thus A affirms, as E denies,
And definitely either ;
Thus I affirms, as O denies,
And definitely neither.
* A half, left femi-definite,
Is worthy of its fcore ;
U, then, affirms, as Y denies,
This, neither lefs nor more.
Indefinito-definites,
To UI, YO, laft we come }
And that affirms, and this denies,
Of more, moft, (half plus fome),
" The rule of the logicians, that the middle term mould be once at
" leaft diftributed [or indiftributable,] (i.e. taken univerfally or fmgu-
"larly, = definitely,) is untrue. For it is fufficient, if, in both the
" premifes together, its quantification be more than its quantity as a
"definite whole. (Ultratotal)" - "It is enough for a
" valid fyllogifm, that the two extreme notions mould (or mould not),
"ofneceffity, partially coincide in the third or middle notion; and
" this is neceffarily mown to be the cafe, if the one extreme coincide
* Better : A, it affirms of this, tbefe, all:
Appendix. 329
" with the middle, to the extent of a half, (dimidiate quantification) ;
"and the other, to the extent of aught more than a half, (ultradimi-
"diate quantification).
" The firft and higheft quantification of the middle term (. .) is
"fufficient not only in combination with itfelf, but with any of all the
" three inferior. The fecond (. ,) fuffices, in combination with the
* higheft,^ with itfelf, and with the third, but not with the loweft.
The third (.) fuffices, in combination with either of the higher, but
" not with itfelf, far Ids with the loweft. The fourth and loweft (,)
" fuffices only in combination with the higheft." [i. Definite;
"2. Indefinite-definite; 3. Semi-definite; 4. Indefinite.]"
Of the efFeft of this new fyftem of quantification in amplifying the
fyllogiftic moods, (which in all the figures remain the fame,) I fay no-
thing. It mould be noted, however, that the letters A, E, &c. do not
* mark the quantification [and qualification] of proportions, (as of old)
but of proportional terms. The fentences within inverted commas are
taken from notes for the " Effay towards," &c.
Before concluding, I ought to apologife, in the circumftances, for
the details, that have infenfibly lengthened out, of a part of my doc-
trine, which I have found, to a certain extent, coincident with what
appears in your paper. I was anxious, however, that you and others
mould have no grounds for furmifmg, that I borrowed any thing from
my predeceflbrs without due acknowledgment. On fecond thoughts,
however, I deem it more proper to make this communication through
a third party.
The difcuffion between Sir William Hamilton and myfelf called a
very able third party into the field, who addrefled the following letter
to the editor of the Atbenaum, in which journal it was publimed, June
19* 1.847-
* Sir, As two great logical innovations the one due to Sir William
Hamilton, the other due to Mr. De Morgan ufed in conjunction, have
led me to the fimpleft and moft general formulae of fyllogifm that ever
have been given (formulae which correct: a ferious miftake into which
both Sir William Hamilton and Mr. De Morgan have fallen), I think
it will gratify thofe interefted in logical fcience if you would give them
publicity through your columns.
n l , ", 7/ m , &c. are any numbers. When placed before a term, as
n"xs> n u marks the total number of the clafs x ; placed before a pro-
pofition, it marks the number of things of which we mean to fpeak.
Thus, , of n n xs are of n m ys, means that a number of things n l are
alleged to have both the characleriftics x and y ; and are to the whole
clafs of xs as to IF , and to the whole clafs ofys as n l to n ul : fimi-
larly with the negative propofition of n n xs are not of n in ys, n l
things being here faid to have the charafteriitic x, and to want the
charafteriftic y. It is clear, from the nature of a propofition, that in
< affirmatives, n { can never be greater than the lead extenfive of the
terms, and in negatives never greater than the number of the clafs
whofe chara&eriftic it is faid to have. But within thefe limits the pro-
330 Appendix.
( portion n l to ?i u may be wholly undetermined ; we then mark it with
the word fome t we call this, with Sir William Hamilton, indefinite
* quantity. It may be perfectly determined; as of equality when we mark
it with all, every, or, following Mr. De Morgan, any other arithmeti-
* cal proportion as a half. (Sir William Hamilton has erred in calling
* a half, femi-definite ; it is thoroughly definite). All this we call defi-
* nite quantity. Lailly, the indefinitude may be reduced within limits
* indefinite-definite, as moft, &c.
The firft formula contains all fyllogifms with an affirmative conclu-
fion, without any exception.
I. n l of n u xs are of Iir jv
IV of v zj are of n m ys
(fli+fliv __ ) O f n \\ xs ar e o f n v zs
* As Sir William Hamilton s principle takes away all diftinftion of
* fubjeft and predicate in affirmative proportions, it will be feen that, by
* varying the proportions of the fymbols, n l t &c., every poffible affirma-
* tive logical inference, in whatever mood or figure, emerges.
The fyllogifms with negative queftions or conclufions, are not fo
fimple. They fall into two divifions, according as, in the negative
* premifs, the things fpoken of have the characleriftic of the extreme, or
* of the middle; and from each of thefe, two conclufions, not one, are
* drawn, according as the things to be fpoken of in the conclufion have
the charafteriftic of the extreme in the affirmative premifs, or of that
6 in the negative premifs.
II. n l of n xs are of n lll y s
1 n lv of n v zs are not of nys concludes ;
* doubly i (n l -f- n ^ v ) of n u xs are not ofn v zs
2 n l ^ IV " ^" ri* zs are not ^" n n xs.
It is to this formula I referred as correcting a ferious error into which
e Sir William Hamilton and Mr. De Morgan have fallen of holding, as
a general principle of all inference, that the two quantifications of the
middle term mould exceed it as a whole ; for this fyllogifm proceeds
wholly irrefpeftive of the total quantity of the middle, which is excluded
from our fymbolic conclufion.
III. n l of n v xs are of n m ys
n lv of nys are not of n v zs concludes ; alfo,
doubly i (n lv + a 1 m ) of n"xs are not of n v zs
< 2 (n -f- a 1 -j- n v n m #") of n v zs are not of x u xs.
Such are the three fymbolical formulae of every poffible logical infe-
* rence. I have the demonftrations that thefe are in all their extent valid,
and are the only poffible forms ; but it is fufficient to give here the re-
fults.
It will furprife no one who confiders that the negative proportion is
not converted in the fame fenfe as the affirmative, that the negative
* fvllogiftic formulae are not reducible to one. For the rule of negative
Appendix. 331
converfion changes the things fpoken of, and is as follows : of ji n xs
f are not of n m ys; converts ( m -|- n l //") of ?i w ys are not of ti u xs. The
confequence of a form univerfally true, (7/ m //") of ?i ui ys are not of n"xs.
* As to the two conclufions, they are but the converfe of each other.
* It will not be difficult to interpret thefe, by /?" as every or n l :
n u indefinite fome, &c. The ufual Ariftotclic forms will be feen to
be derived from them. Thus the mood Cefare, and the corresponding
indirect mood (or, if you will, the mood of the fourth figure, call it at
4 another time Celantes or Cadere at will, but let it be Celantcs for the
* nonce), come forth from the third formula.
# IV = 1IX gives no y is z . . . # IV : # v indefinite
n " every z is y . . . : n m indefinite.
* Hence in Cefare, no x is z from our firft,
* and in Celantes, no z is x from our fecond conclufion, and fo of all
* the others.
I owe it to Sir William Hamilton and Mr. DC Morgan to fay that
* without their improvements I could not have advanced one ftep. Mr. De
* Morgan has even attempted a like reduction to general formulae, and has
* failed, chiefly through a mifapprehenfion of Sir William Hamilton s prin-
ciple of quantified predicate. He has introduced a fuperfluous quantity,
one logically ufelefs, or worfe than ufelefs, as the refult has mown.
* This confufion explains his errors. Had it not- been for this circum-
* fiance, I mould not have had the honour of prefenting thefe formula:
* to logicians.
Permit me to add what I think alfo of fome value. I am not of thofe
who think with Sir William Hamilton that the fyllogifm always pro-
ceeds in the two counter wholes of intenfion and extenfion that it
mufl always be an involution or evolution in refpect of claffification.
* This is, no doubt, true in the moil important reafonings of fcience ; but
it is not fcientifically accurate to afTert this univerfally.
Quality, which is the comprehenfive element, is of three kinds not
two, as heretofore affirmed ; for fmce Kant, the divifion of affirmatives
* into analytic and fynthetic, or (as Sir William Hamilton wifhes) expli-
cative and ampliative, has been eftablifhed. James Bernouilli has puz-
* zled himfelf to reduce thefe two to the fame form, but without fuccefs ;
* for that contains an immediate relation of part to whole, and only a re-
mote one of part to part, while this contains an immediate relation of
part to part, and remote of part to whole. Thefe, as diflindt kinds of
* quality, are erroneoufly elided in language. As the words ampliative
and reftfifiive are generally oppofed in logic, perhaps we might replace
the old divifion of proportions, according to quality, into affirmative and
* negative by one into Explicative, Ampliative y and Reftrittive.
Where, then, both premifes are ampliative, the fyllogifm proceeds
purely by force of extenfion. There is neither involution nor evolution
* neither induction nor deduction but a paffage or tranfition from one
* mark to another, or from clafs to clafs. Of this kind are all fingular,
*or, as Ramus calls them, proper fyllogifms. Let us call this new
33 2 Appendix.
clafs of fyllogifms tradudtive, to contrail it with the inductive and de-
dudtive.
The ufe of thefe in philofophy as independent modes of inference will
eafily appear. When we collect the fcattered fragments of our know-
ledge into unity of fcience, we ufe induction and induftive fyllogifm y
when we apply the principles of fcience to fpecial events of things, we
ufe deduction and deduftive fyllogifm; but when, abandoning one fcheme
of clarification, we transfer our knowledge direttly to another, we ufe
traduftion and traduttive fyllogifm. Thus, in political fcience, what
has been predicated by hiftorians of men claffed geographically is tranf-
( ferred to men clailed according to conftitutions of government by tra-
duction. This lafl efcapes Sir William Hamilton s rule, and never
concludes through a comprehenfive containing and contained.
I mail not add, at prefent, any attempt to prove a priori the exclufive
validity of fyllogiftic inference.
I admit that I ought not, without good ground, to diffent from a ma-
* cured opinion of Sir William Hamilton in any part of philofophy, flill
more in logic ; but I obey the force of demonilration, and, as Ludo-
vicus Vives faid in refpect to Ariflotle, Verecunde diffentio.
f Yours, &c.
JAMES BROUN.
Temple, June 9, 1847.
My reply to this confifled in forwarding, on the fame Ipth of June,
to the editor of the Athenaeum, a fummary of the refults of chapter VIII,
then written. This fummary appeared on the 26th : I do not infert it,
becaufe the chapter in queftion is a better anfwer ; and though the pub
lication faved my rights, the republication is unnecefTary. Mr. Broun s
three forms are the firft (without the contranominal), the ninth, and the
eleventh, of page 161. Mr. Broun was wrong in deducing from the
two latter forms that the principle of the middle term was erroneous :
for in thefe very forms the two quantifications exceed the whole : being
the whole (in premife one) plus fome (in the other). As to the fuper-
fluous quantity, it only becomes fuperfluous when fuch quantifications
are introduced as diftinguifh fpurious from admiffible proportions : fee
pages 145, 146, in which it is mown that the forms are correct.
Nothing but clofe comparifon, and that after practice, would detect
the accordance of the two fymbolic modes of expreilion in pages 145
and 161. I am not therefore furprifed that Mr. Broun mould, having
obtained cafes of that in page 161, pronounce that in page 145 erro
neous.
In the anfwer which I made, I promifed to ftate diftinctly how much
of the chapter was written before Mr. Broun s letter appeared. This
I now do. With the exception of pages 145, 146, the matter of which
is moftly from my Cambridge Memoir, the whole of it was then written,
excepting fuch verbal alterations and occafional introduction of fentences,
as take place at the prefs, or at the lafl reading of the manufcript. I had
Appendix. 333
thought that there would be no neceflity to introduce thofe pages, ex
cept (lightly, and in anfvver to certain objections which feemed likely to
occur. The examination which the affertion that they are erroneous
made me give my previous forms, pointed out the defirablenefs of intro
ducing them as they now Hand.
September 17, 1847. I had finifhed the preceding appendix, when I
became aware of the exiftence of the < Commentationes PhilofophiczE
Seleftiores of Godfrey Ploucquet, of Tubingen, Utrecht, 1781, quarto.
The laft title (p. 561) is De Arte Charadleriftica. Subjicitur Methodus
calculandi in logicis, ab auftore inventa. 1763. I find by a catalogue*
that this methodus calculandi had been previoufly publifhed in 1773, at
Tubingen, at the end of a work entitled Principia de Subftantiis et
Phaenomenis : alfo that the * Methodus demonftrandi direcle omnes
fyllogifmorum fpecies of the fame author (which is probably the thing
I am going to defcribe) was publifhed at Tubingen in 1763. From the
title of a work which, I am informed, exifts, namely, Sammlung der
Schriften welche von logifchen Calcul des Prof. Ploucquet betreffen
Tubingen, 1773, one would fuppofe that this fyflem had obtained great
local currency. I give a fliort account of it : premifing that Ploucquet
appears to have been a well informed mathematician, much given to
pure fpeculation on mental fubjecls.
The calculus (a term which Ploucquet ufes in as wide a fenfe as I do
when I call the contents of Chapter V. a part of the calculus of infe
rence) confifts in the invention of a fimple notation, and the mechanical
fubftitution, in one premife, of an identical equivalent to the middle term
therein contained, taken from the other premife (this laft being one in
which the middle term is univerfal). There is neither ufe of contraries,
nor numerical definition : but there is every variety of quantity of the
predicate which can be produced by fimple converfion of the ordinary
forms. A term ufed univerfally is denoted by the capital letter ; par
ticularly, by the fmall letter : affirmation by juxtapofition ; negation, by
interpofmg < Thus X)Y is Xy ; X.Y is X> Y ; XY is xy ; X:Y is
x> Y. The following is a complete fpecimen :
Sint prse- Pm
miffae s > M
Calculo : s > mP quoddam s non eft P
Omnis ducatus eft aureus
Quaedam moneta non eft aurea.
Da
m> A
Calc. mj> aD. feu mj> D, quasdam moneta non eft ducatus.
As Ploucquet feems to think that this a6lual application of the calculus
to concrete inftances, by aid of their initial letters, is a material part of
* The fecond edition of Mr. Blakey s * Eflay on Logic recently publi&ed, contains
a catalogue of upwards of a thoufand works on logic, briefly titled.
334 Appendix.
his fyftem, I have inferred the cafe entire. The rationale of the fyftem
confifts in that fubftitution of identicals for each other, which I under-
ftand Sir William Hamilton (with perfect truth) to employ in every
cafe. Thus we have in the above Some of the Ss are not any Ms,
are rot thofe Ms which make up all the Ps, are not therefore any Ps.
This demand for identical fubftitutes requires both kinds of quantity for
every predicate, and Ploucquet ufes them accordingly, as far as wanted
to eilabliih the Ariftotelian fyllogifms. Sir W. Hamilton goes further,
and invents fyllogifms for all the kinds of quantity. Thus Ploucquet
ufes mP or * fome Ms are all the Ps and P > m or all Ps are not fome
of the Ms ; but not MP or p > m.
At the fame time with the knowledge of Ploucquet I obtained that
of the work of a follower and extender, M. W. Drobitfch, author of
Neue Darftellung der Logik . . . Nebft einen logifch-mathematifchen
Anhange, Leipzig, 1836, oftavo. As far as the fymbolic part is con
cerned, Mr. Drobitfch begins by a convention which would reconcile
any one to the found, not merely of Barbara and Celarent, but even of
Baroko and Frejtfon. He makes S and P the fubjeft and predicate of
the conclulion and M the middle term ; and puts the Ariftotelian vowel
between them : thus S)P is SAP, and P:S is POS. Hence his pre-
mifes may be map fam or mop fam ; and one of his fyllogifms is mep-
famfep. In the algebraical part, he ufes large and fmall letters for the
univerfal and particular, or for the whole and part extent of a term.
He alfo introduces the figns and <J to fignify identity and (what I
call) fubidentity. This ufe of the mathematical figns involves an ex-
tenfion, which is made by all thofe who fignify the identity of X and
Y by X=Y. The mathematician thinks of extent as quantity only :
the logician includes both quantity and pofition. Thus when the for
mer fays that five feet are lefs than {even feet, he means any five feet,
be they part of the feven feet or not : the latter, when he fays that X
is a name of lefs extent than Y, means not only that the former can be
contained in the latter, but that it is. To make negative propofitions,
Mr. Drobitfch takes a limited univerfe (call it U, as I have done) an
extent greater than the utmoft extent of all the names, otherwife inde
finite. And here he falls into fome confulion : X and Y being the
names, he fays U muft be of greater extent than X+Y: now if we had
X)Y, U need only be of greater extent than Y. If from the genus Y
be taken all the fpecies X, the remainder is denoted by Y X. Ac
cordingly, the contrary of X is U X.
Mr. Drobitfch then lays down eight forms of predication, of which,
however, he only ufes the ordinary ones. And I cannot find out that
the limited univerfe, or the contrary, has any ufe except to furnifh means
of notation. The eight forms are ; firft, X~ y, or my X)Y ; fecondly,
XzrY, or X)Y+Y)X ; thirdly, x=y, or XY; fourthly, u=Y, or
Y)X; fifthly, X <U Y, which tells us that X is all contained in
what is left of the univerfe after Y is removed, or is X.Y ; fixthly,
X=z <JZ <JU Y, a very roundabout way of faying that X is /^con
trary of Y, or X.Y-j-xy; feventhly, xzzU Y or X:Y ; eighthly,
Appendix. 335
x=X Y, which tells us that Y is a fubidentical of X, or Y)X+ X:Y.
This is in faft a mixture of two fyflems, both in principle and nota
tion. The forms are A,, A 1 , O| (and O ! ), Ej, I 4 , D, D (and D 1 ), and
C 4 . Allb C is virtually given : but E f , l f , C 1 , do not appear. The
ordinary rules under which the mathematicians ufe m and <!, remain
true in this logical ufe of them : and thus there is an elegant mode or
exhibiting the inference in fyllogifms. For initance, in Cameftres we
havePirm, S <U M.-.<U m .-. <U P; orS<U P.
It would have been more confident to have made zr, <J, and > , (in
troducing this laft) ferve all purpofes. But it has happened very often
that a fyilem of notation, already exhibited, has been extended by a better
one, and mended only, inilead of being reconilrudled. Ploucquet had
ufed the large and fmall letters, and > for denial : the latter fymbol a
ilrange one, if mathematical analogy were intended. Mr. Drobitfch
has ingenioufly contrived that <^ mould reprefent denial, and has been
led to what might have ufefully amended all he had to begin with. Tak
ing little x to reprefent a part of the extent of X, &c. and U for the ex
tent of the univerfe, the following notation might have been adopted :
Firft when < and > both include their limit, . We fhould have
A! X<Y or Y>X
O, x<U Y or U Y>x
E, X<U Y or U Y>X
I 4 x<Y orY>x
A Y<X or X>Y
O 1 y<U X or U X>y
E X>U Y or U Y<X
I" is inexpreffible.
To exprefs I 1 , we muft invent a fymbol for a part of U X.
Next, when < and > do not include their limits, we have
D t X <Y or Y> X
D X=Y or Y=X
D X>Y or Y<X
C, X<U Y orU Y>X
C XzrU Y or U Y=X
C ! X>U Y or U Y<X
is inexpreffible.
I am inclined to think that the reprefcntation of quantity and location
both under one fymbol is objectionable, if that fymbol be one already
appropriated in mathematics to quantity only. I would on no account
accuftom myfelf to read A <!B as A is lefs than (becaufe a part of) B.
Mr. Drobitfch is much more complete than his predeceflbrs in his enu
meration of the various kinds of forites.
Qttober 29, 1 847. While this flieet was paffing through the prefs,
I became acquainted with " A fyllabus of logic, in which the views of
Kant are generally adopted, and the laws of fyllogifm fymbolically ex-
prefled. By Thomas Solly, Efq." Cambridge, 1839, 8vo. The
fymbolical expreffion here given is of a peculiar character : the algebraic
figns are adopted in a fenfe which preferves the rules of fign, while the
fymbols reprefent the terms of the fyllogifm, or elfe the notions of par
ticular and univerfal. Thus, if p Hand for particular, u for univerfal,
and m for one of the terms of a fyllogifm, mi=.u or m uQ implies
336 Appendix.
that m is a univerfal term, and (mu}(np)o implies the alternative
that either m is univerfal, or n is particular. By means of fuch alter
native relations, the conditions of validity of the various figures are
expreffed. Mr. Solly contends for fix forms in each figure, by intro
ducing all forms which have weakened conclufions, and proves a priori,
from his equations, that fix and no more are poflible in each figure. If
I had admitted weakened forms, there would have been fixteen more
fyllogifms, which might be deduced, either from the eight univerfals, or
from the fixteen particulars.
THE END.
C. WHITTINGHAM, CHISWICK.
28, Upper Gower Street,
November 1, 1847.
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TURNER S ELEMENTS OF CHEMISTRY. Eighth Edition.
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