:CO
A NEW
LAW OF THOUGHT
AND ITS
LOGICAL BEARINGS
E. E. CONSTANCE JONES
GIRTON COLLEGE STUDIES No. IV
GIRTON COLLEGE STUDIES
EDITED BY LILIAN KNOWLES, LITT.D., HEADER IN ECONOMIC HISTORY
IN THE UNIVERSITY OF LONDON
No. 4.
A NEW
LAW OF THOUGHT
AND ITS LOGICAL BEARINGS
CAMBRIDGE UNIVERSITY PRESS
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A NEW
LAW OF THOUGHT
AND ITS LOGICAL BEARINGS
BY
E. E. CONSTANCE JONES
Author of A Primer of Logic
WITH A PREFACE BY
PROFESSOR STOUT
"One of the greatest pains of human nature
is the pain of a new idea." BAGEHOT.
Cambridge :
at the University Press
191 1
Cambridge :
PRINTED BY JOHN CLAY, M.A..
AT THE UNIVERSITY PRESS
PREFACE
of the three fundamental Laws of Thought, which
are traditionally regarded as the cardinal principles
of Formal Logic, are concerned with the relation of pro
positions to each other. According to the Law of
Contradiction, two propositions of the form " A is B "
and " A is not B" cannot both be true. According to
the Law of Excluded Middle, they cannot both be false.
Now it is clear that if there is another principle which
expresses the fundamental condition of the possibility of
any proposition taken by itself, without reference to
others, this also must be regarded as a fundamental Law
of Thought, and as being logically prior to the Laws of
Contradiction and Excluded Middle. It is the aim of
Miss Jones in the following pages to show that there is
such a Law, and to exhibit in detail its vital importance
in the treatment of the whole range of topics with which
Formal Logic deals. This Law of " Significant Assertion "
is formulated as follows : Every Subject of Predication is
an identity (of denotation) in diversity (of intension). In
other words, every affirmative proposition asserts, and
every negative proposition denies, the union of different
attributes within the unity of the same thing. In every
affirmative proposition, the subject-term designates some
thing as characterised in one way, and the predicate
vi PREFACE
designates the same thing as characterised in another
way. This Law of Significant Assertion is substituted
by Miss Jones for the traditional Law of Identity, as
expressed in the formula " A is A." "A is A," if it has
any significance at all, must, she holds, be taken as an
attempt to express the essential nature of all predication ;
but so regarded it is plainly untenable; for to say
" A is A " is merely to say " A " twice, and not to assert
anything about "A." There is no proposition, unless
what is characterised as " A " in the subject-term is also
characterised as "B" in the predicate-term.
The service which Miss Jones has rendered to Logic
in this little volume lies not so much in the mere enun
ciation of the " Law of Significant Assertion " as in her
thorough and systematic application of it, so as to clear
up special logical problems. By way of illustration, I may
refer to her discussion of the doctrine of " the fourfold
implication of propositions in Connotation and Denota
tion," and to her account of immediate inferences, and of
the syllogism. As regards syllogistic inference, it may be
worth while to refer to a point which Miss Jones has not
expressly noticed. The Law of Significant Assertion
supplies the most direct, simple and general vindication
of the syllogism against the charge of petitio principii.
The charge is based on the fact that the conclusion asserts
of the same thing the same predicate which has already
been ascribed to it in the major premiss. The straight
forward reply is, that in the conclusion this predicate is
brought into connexion with an attribute with which it
has not been connected in either of the premisses. A
PREFACE Vll
remarks to B, " That woman in the corner is a scare-crow."
B replies, "Sir, that woman is my wife." For A, it is a
startling novelty, and no mere repetition, to discover that
he has called B s wife a scare-crow. The novelty is plainly
due to a new synthesis of attributes with the same deno
tation, the combination of the attribute of being B s wife
with that of being the woman whom A has just called a
scare-crow.
Miss Jones seems to have made out a good case for
regarding the Law of Significant Assertion as a funda
mental Law of Thought. But its claim to be the only
justifiable rendering of the Law of Identity is not so
clear. The best writers on Logic tend to interpret this
law as expressing the immutability of truth. According
to them, it means that the truth of a proposition is
unaffected by variation of time, place and circumstances,
or of the minds which apprehend it. Either this prin
ciple, or, if the pragmatists be right, its contradictory,
seems to demand recognition as a fundamental law of
thought, and it is certainly a principle of Identity. But
it is of course no substitute for the Law of Significant
Assertion. The question which of the two is the most
appropriate interpretation of the cryptic formula " A is A "
is of quite subordinate interest.
G. F. STOUT.
ST ANDREWS.
March, 1911.
CONTENTS
PAGE
INTRODUCTORY SUMMARY . . . . 1
STATEMENT OF THE CASE .... 4
FALLACIES . . .. . . , .68
DEFINITIONS OF CERTAIN TERMS 70
INTRODUCTORY SUMMARY
MY object in the following brief essay is to propound a
certain analysis of Categorical Propositions of the forms
$ is P, S is not P, to show that this is the only general
it. is nofifilU^ t.n }.rrp-nt, anrl t.r inrlinafci ifa
ERRATA.
(1) p. 9 last line, read (2) the intensional, connotational, or
implicational ;
(2) p. 20 top, read In A = B there is between A and B equality
of quantity or value in intensional diversity thus there is not only
denotational identity between A and equal to B, but also qualitative
sameness in qualitative diversity between A and B.
(3) p. 23, line 3 from foot. For E is F read A is E.
(4) p. 54, lines 16 and 17, delete and these may conflict.
(5) p. 66, line 6 from foot, delete three.
(6) p. 71, line 15. For name read names.
TCM . A is related to B implies A is
not
00
j.
INTRODUCTORY SUMMARY
MY object in the following brief essay is to propound a
certain analysis of Categorical Propositions of the forms
8 is P, S is not P, to show that this is the only general
analysis which it is possible to accept, and to indicate its
bearing upon logical science. According to the analysis
in question, S is P asserts Identity of Denotation in
Diversity of Intension, and S is not P denies this. The
example given by Professor Frege (whose analysis of 8
is P I understand to agree roughly with mine) is
" The morning star is the evening star "
terms " morning star " and " evening star " apply to one
thing, but the meaning, intension, or qualitative implica
tion of " morning star " is not the same as that of
" evening star." " The largest city in the world is the
Metropolis of England" is another illustration, where
again it is clear that the two names or terms, the Subject
and Predicate of the Assertion, apply to one place but
have different meanings or definitions. S is not P asserts
Difference of Denotation (Otherness) in Difference of
Intension (Diversity) e.g. " Cambridge is not Oxford,"
/^"""x
( j . A is related to B implies A is not B,
00- . / ;
j. 1
2 INTRODUCTORY SUMMARY
We need propositions of the form S is P, S is not P,
for significant assertion, and without them no satisfactory
statement can be given of the " three fundamental Laws
of Thought," which are put forward as the basis of logical
science. The first two of these Laws are commonly
formulated as: (1) A is A, (2) A is not non-A } and the
third sometimes as A is either A or non-A (3). Desperate
efforts have been made by logicians to give a valuable
meaning to A is A ; but if A is A, interpreted as A is A,
is retained as the first fundamental Law, there is no
possible passage from it to A is B, and A is A or A is B
(S is P) must be given up. This is fully recognised by
Lotze, who gives up (theoretically) S is P. A is A tells
us no more than A is A, and if we begin with it, we must
also end with it, if we are to be consistent. I maintain
that we must not begin with it, but must begin instead
with a Law of significant assertion assertion of the forms
S is P, S is not P, forms which provide the only straight
forward and effective statement of the second and third
Laws of Thought, thus :
S is P [cannot both be true (L. of Contradiction)
S is not P [cannot both be false (L. of Excluded Middle).
It follows from these two Laws that of any Subject of
Predication (S) either P or not-P can be affirmed. Thus
from them, and S is P, S is not P, analysed as above, we
obtain the principle that :
Every Subject of Predication is an Identity-in-
Diversity.
It follows further that every Predicate (P) is neces
sarily incompatible with not-P (absence of intension P)
and necessarily compatible with not-not-P. (This suggests
a formal principle of necessary connection of attributes.)
INTRODUCTORY SUMMARY 3
I contend that if we start, not with A is A, but with
the principle that Every Subject of Predication is an
identity (of denotation) in diversity (of intension), this
Law (1), and the Laws of (2) Contradiction and (3) Ex
cluded Middle (of which (1) for the first time makes
logically possible the formulation given above) do furnish
a real and adequate and obvious basis and starting-point
of " Formal " Logic. Granted propositions of the form
S is P, with the identity-in-diversity analysis and the
corresponding analysis of 8 is not P, together with the
traditional Laws of Contradiction and Excluded Middle,
the whole scheme of Immediate and Mediate Inference
can be built up systematically and explicitly, as I hope to
show. The possibility of Conversion, e.g. implies that the
Predicate, as well as the Subject, of any Proposition has
Denotation, and a Denotation that is implicitly quan
tified; the one indispensable condition of Mediate In
ference is identity of Denotation of the Middle Term in
both premisses. Without propositions of the forms S is P,
8 is not P, thought cannot live or move ; but the
disastrous acceptance of A is A, with its baffling am
biguities, has stood in the way of their being rightly
analysed by logicians and explicitly recognised by them as
fundamental forms of significant assertion, without which
not even the Laws of Contradiction and Excluded Middle
can receive satisfactory expression 1 .
1 In the following pages I have occasionally borrowed from writings
of my own in cases where I have not felt able to improve upon the
statement already printed.
12
A NEW "LAW OF THOUGHT" AND
ITS LOGICAL BEAEINGS
"I am the pillars of the house,
The keystone of the arch am I ;
Take me away, and roof and wall
Would fall to ruin utterly."
K. TYNAN.
STATEMENT OF THE CASE.
IT will be admitted that up to the present time no
adequate and unquestionable basis of the Science of Logic
has been found that the Method of Logic, itself the
Science of Method, is not wholly satisfactory. Logic is
often defined as the Science of the Laws of Thought
the Laws, that is, of Identity, Contradiction, and Excluded
Middle ; but on the one hand the statement of these Laws
is not uniform, and the interpretation of at least the first
of them, the Law of Identity (A is A, whatever is is,
Everything is what it is), is matter of perpetual dispute ;
on the other hand no one of these Laws alone, nor all of
them together, can or do take account of, or can explain
and justify, the common indispensable form of Categorical
Assertion S is P e.g. Trees are green, All Men are
mortal, George V is the present King of England, Per
severance is admirable, Honesty is the best policy, The
quality of Mercy is twice bless d. On the contrary, A is A
appears to exclude it, and there is no passage from A is A
to A is B. And if anyone who accepts A is A, and the
corresponding expression of the Law of Contradiction, A
is not not- A, is driven into giving A is B or not-B as the
Law of Excluded Middle, it is for him to show what
logical connexion there is between the last " Law " and
the two previous ones. Logic undoubtedly, like all other
Sciences, like literature, like common thought and common
speech, uses the forms 8 is P, S is not P uses them at
every step. It must use them, of course; it has no choice;
without them, it would be impossible to affirm or deny;
but it adopts them in the same fashion as Bentham
adopted the Greatest Happiness of the Greatest number as
his ultimate ethical principle that is to say, without
any reasoned justification. No " plain man " certainly,
would be expected to give any reason why he should use
propositions of the form A is B rather than of the form
A is A; lout a, logician who declares that A is A is the
first Law of Thought, and (if he is consistent), that
A is not not- A and A is either A or not A 1 are the other
two, may fairly be called upon to explain the fact that he
habitually says that Roses are red and Violets are blue,
rather than Roses are roses, Red is red, Violets are violets,
and so on. For logicians to find fault with a so-called
" Law " which is a pure tautology, which is expressed in
a form which may indeed have important uses, and may
be employed epigrammatically or rhetorically, but in
which no ordinary sensible person would think of trying
to convey straightforward information, or matter of fact
much less a fundamental principle is no new thing.
1 This, however, is generally stated A is either B or not B and
sometimes the A is B form is slipped into even in stating the Law of
Contradiction, by upholders of the A is A Law of Identity.
6 A NEW
To lay it down (1) that we can never legitimately affirm
of any subject a predicate different from itself, while at
the same time (2) it has to be allowed that this rule
cannot be even stated without being broken, without
using assertions of the form S is P, was we know, a state
of mind possible in the time of Plato; it was possible
because those who asserted (1) thought it self-evident
that the Predicate ought always to be the same as its
Subject, "that to apply many Predicates to one and the
same Subject is to make one thing into many things."
And as for (2), they could not deny it; while to give
up (1) seemed to be a denial of self-evident truth, to give
up (2) was sheerly impossible. The situation is rather
intolerable.
That there is a difficulty about S is P we need not
question, that logicians who accept A is A are impera
tively called upon to show how this " Law " can be adapted
to propositions of form A is B (S is P) is too obvious
to need pointing out. Some writers have tried to give a
meaning to A is A which does not seem to prohibit
diversity of Predicate from Subject a meaning which
is itself expressed in the A is B, not in the tautological
A is A, form ; Mr Bradley e.g. interprets the Law of
Identity to mean that "if what I say is really true, it
stands for ever." A is A thus expounded into A is B
does not of course exclude propositions of A is B form.
Dr Bosanquet frankly admits that, while he would not
accept either A is B or A is A as a schematic ex
pression of the Law of Identity, he would prefer A is B
to A is A 1 .
1 " If I were asked " he says, " how I should represent a true
Identity, such as a judgment must express, in a schematic form with
AND ITS LOGICAL BEARINGS 7
The only logician, as far as I know, who, while re
taining A is A in its purity has made a determined effort
to reconcile it with propositions of the A is B (8 is P)
form, is Lotze. He holds (Logic, Bk I. ch. n.) that "our
thought is subject to a limitation, has to conform to a
law... in the categorical judgment each constituent can
only be conceived as self-same [= ?]. This primary law of
thought, the principle of identity, we express positively
in the formula A = A." He states the conclusion to
symbolic letters, I should say the problem was insoluble. Every A is B
would be much better than Every A is A; but as the letters are not parts
in any whole of meaning, they are things cut asunder with an axe ."
(Dr Bosanquet in Mind, 1888, p. 357.) (The objection that in A is B
"the letters are not parts in any whole of meaning" seems either
(1) inaccurate, for there is a symbolic whole, viz. (A, B, I which has
a meaning and an important one, or (2) irrelevant, if what is meant is
a concrete special " whole of meaning.")
It is clear from other passages in the same article that for
Dr Bosanquet, individual identity is not distinguished from qualitative
one-ness of two things e.g., he speaks of some "present impression"
as being identical with a former impression " (p. 360), and says that
" the element of identity between two outlines can be accurately pointed
out and limited, but the moment they cease to be two, it ceases to be an
identity" (p. 359). He objects to drawing "a sharp line between the
unity of the individual human being... and the unity of human beings
in identical sentiments, ideas, purposes or habits" (p. 362), and says
that a number of persons may have "a really identical purpose and
endeavour and consciousness of certain facts" (p. 364). Again (p. 365),
he says "Any indiscernible resemblance [ = ?] between two different
contents, in specified respects, will do whatever identity will do, because
it is identity under another name" (if so, what need is there of a
Distributed Middle in Syllogism?); and on p. 366 speaks of "indis
cernible likeness [ = ?] or identity." With this meaning of identity it
certainly is not clear how "a true identity" could be satisfactorily
expressed as A is B. Connotationally, qualitatively, A is not B.
8 A NEW "LAW OF THOUGHT"
which he is driven, thus : " This absolute connexion of
two concepts 8 and P, in which the one is unconditionally
the other and yet both stand over against each other as
different, is a relation quite impracticable in thought :
by means of this copula, the simple is of the cate
gorical judgment, two different contents cannot be con
nected at all ; they must either fall entirely within one
another, or they must remain entirely separate, and the
impossible judgment ( S is P resolves itself into the
three others, ( S is 8, P is P, 1 S is not P ." (Engl.
transl. p. 59.)
Whether A is A is understood as A-ness is A-ness, or
in any other possible way in which A is A is honestly
interpreted as A is A (not as A is B), the acceptance of
it as a first and fundamental Law is absolutely suicidal
for Logic from a theoretical point of view. But it must
be confessed that its nominal acceptance does not appear
to have seriously affected the construction of the Science.
A is A cannot justify or support this, it even seems in
consistent with it, but the restrictions logically imposed
by A is A have (almost universally) been not only not
respected, they have not even been borne in mind, and
A is A itself has received a variety of interpretations
(generally of the form A is B) which it was natural to
ignore as they mostly did not interfere with either theory
or practice, and it was thus easy for logicians to go on
systematising and constructing in complete independence
of the " First Law of Thought."
No doubt the speculative incompatibility between it
and ordinary assertion has been for the most part a
"contradiction that was not seen." When it has been
seen, common sense has had no hesitation in driving a coach-
AND ITS LOGICAL BEARINGS 9
and-six through the venerable but insubstantial obstacle.
Lotze, keenly aware of the contradiction and loyal to
tradition, but oblivious for the moment of the needs and
actualities of living thought, imagined that he must,
and could, give up S is P. The actual starting-point
of Logic has been not A is A, but the Law of Contra
diction and the Law of Excluded Middle, and the effort
to analyse S is P (not-P) , and in Conversion, Mediate
Inference etc., it is propositions of those forms that have
been dealt with. But those forms were accepted un
critically, and together with A is A. Logic has lacked a
First Law which could furnish a legitimate and logical
starting-point and be capable of development and general
application, have a real and important difference from,
and connexion with, the Law of Contradiction and the
Law of Excluded Middle, be effective throughout the
Science of Logic, and justify, explain and support logical
procedure. Though A is A may be sometimes a con
venient mode of expression, we cannot start from it as
the fundamental prepositional form and we do not see
how to get from it to A is B. A is B is the inevitable
point of departure, and this has, as the limit on one side
(the side of tautology) A is A (which excludes diversity
of intension), and on the other (the side of Contra
diction), A is not- A (which excludes identity of denota
tion). A is A, f A, Aj 5 is of course quite different from
I think that every name or term has two aspects :
(1) the denotational, extensional, or applicational ;
(2) the intensional or connotational ;
10 A NEW
corresponding to the two aspects of the things of which
they are names i.e. the aspects of (1) Thatness and
(2) Whatness, to use Mr Bradley s terms. Everything
of which we can think or speak is (1) Something and
(2) some definite sort of something. Everything must be
thought as having (1) existence (in the widest sense
mere thing-hood) and (2) some fixed definite nature and
constitution. For the sake of clearness, I propose in what
follows to confine the term identity to denotational one
ness, as distinct from one-ness in the intensional sense,
which makes possible general names, classing, and classi
fication. Without both (1) and (2) no assertion is possible,
nothing can be Subject or Predicate of a proposition,
The Law of Identity may have been an attempt to
express the qualitative fixity of nature of anything in
brief and self-evident form ; if so, the expression A is A
is unfortunately incapable of expressing what was in
tended. If it does express a meaning, that meaning is
clearly not self-evident, for there is nothing about which
there has been more dispute than the meaning of A is A.
It seems to me that until we have A is B (S is P) there
is nothing to accept or reject, nothing to doubt or dispute,
and that the true significance of contradiction is to deny
of something some predicate which has already been
affirmed of it. It might seem that for conceptualists the
problem of A is A was simplified, as their whole interest
was in Quality, Intension, as distinct from Extension
or Thatness; but it is demonstrable that no significant
affirmation can be purely qualitative.
If we genuinely accept A is A as the expression of a
fundamental and primary logical principle, the difficulty
is, how theoretically to get beyond it. If we reject it,
AND ITS LOGICAL BEARINGS 11
what we need, and what we find, to put in its place, is a
principle of significant Assertion Assertion of the form
S is P. The laws of Contradiction and Excluded Middle
are laws of the relations of assertions, and they cannot
be expressed in satisfactory and unambiguous form with
out the use of S is P, S is not P, propositions. So even
for them we require a prior principle, explaining and
justifying the S is P proposition itself. Such a logical
principle, based on a new analysis of S is P, I think I can
provide.
I call the analysis in question " new " because although
I put it forward in print in 1890, and although Dr Keynes
in his Formal Logic has practically adopted it as appro
priate to "logical equations" (loc. cit. 4th edit. pp. 189,
190), it has not received much attention no doubt be
cause many other accounts of the Categorical Proposition
have looked so like it (and in fact sometimes came so
near it) that the fundamental difference has not been
recognised.
"Oh, the little more and how much it is,
And the little less and what worlds away."
And although my own conviction has remained un
shaken because the doctrine has seemed to me to stand
all the tests that I could apply in a thoroughly satisfactory
manner, I should not have taken up the question again at
this time but for two circumstances. One is that I have
rather suddenly become aware that my analysis furnishes
a law of Categorical Assertion which together with the
Laws of Contradiction and Excluded Middle stated in
S is P, S is not P, form does provide Formal Logic with
an adequate foundation, and gives a systematising prin
ciple, in complete accord both with common thought and
12 A NEW "LAW OF THOUGHT"
common usage, and with the accepted structure of logical
science, and is perhaps further of direct philosophical im
portance.
The other circumstance to which I refer is, that I have
recently had my attention drawn to the fact that Professor
Frege s analysis of Categoricals (published in 1892) was
apparently the same as my own, and that a similar view
was adopted by Mr B. Russell (1903) in his Principles
of Mathematics, where Frege s theory of the import of
propositions is expounded with sympathetic approbation.
Recognising in Terms the two aspects of Extension
(or Denotation) and Intension (as Jevons and most other
modern writers on Logic do), I approach from that point
of view the question : How are the propositions of the
forms 8 is P, S is not P, to be analysed ? I hold that one
or other of these two symbolic expressions may be applied
to every Categorical Proposition. Further, that in Pro
positions of which the Term-names are Class-names e.g.
All Lions are carnivora conversion, involving Quanti
fication of the Predicate, is possible and legitimate.
By the Extension or Denotation of a Term I mean
the things to which it applies, by its Intension I mean those
properties or qualities of the things which it signifies. As
Dr Keynes says : " The extension of a name consists of
objects of which the name can be predicated ; its in
tension consists of properties which can be predicated of
it " (Formal Logic, 4th ed. p. 22). " Intension may be
used to indicate in the most general way what may be
called the implicational aspect of names " (loc. cit. p. 26).
E.g. (1) Quadruped in extension denotes lions, tigers,
horses, dogs, kittens, etc., etc., in intension it means
having four feet ; gold in extension applies to this cup,
AND ITS LOGICAL BEARINGS 13
that ring, those sovereigns, etc., in intension it means
yellow, heavy, malleable, insoluble in aqua regia ; man
in extension denotes Henri Bergson, Josiah Royce, J. J.
Sargent, Mary Findlater, Jane Barlow, Madame Curie,
etc., in intension it signifies having rationality and
anirnality.
It is to be observed that we may know the applica
tion or extension of a name and not know its intension
(definition or signification) and vice versa. E.g. I know
that metal in extension denotes gold, silver, copper, iron,
lead, tin, mercury, aluminium, etc., and I know these
when I see them, but I am not able to give a satis
factory statement of the intension which they have in
common.
Or again I know, or I may know, all the inhabitants
of a country parish and be able to greet them correctly
by name when I meet them, but may be entirely unable
to give a recognisable description of any of them. Or
I may know real diamonds from paste, or one disease
from another, and always apply the names rightly, and yet
be unable to set out even to myself the connotation or
intension.
On the other hand I may have full descriptive know
ledge of a person or plant or precious stone, and yet not be
able to recognise the person or plant or jewel though it may
much concern me to do so. I may even know much more
about a person than his ordinary acquaintances, or even
than his dearest friend, and be able to give a much more
accurate description of his appearance and manner, and
yet not know him when I meet him. Or I may recognise,
though I cannot define, Justice; and define, though I
might not recognise, a chiliagon.
14 A NEW " LAW OF THOUGHT"
Extension and Intension or both may be imaginary.
I may put together elements universally recognised as
charming, and draw a fancy portrait or a fictitious character;
or I may attribute to an actual person impossible or in
compatible perfections.
What I insist on is that all the names we use have both
Extension and Intension ; and either of these may be a
guide to the other. I may have the things to which
a name applies put before me (Extensive definition) and
from examination of them reach the intension : or have
intension given, and go out and by means of it determine
extension.
It may occasionally be possible and be convenient to
apply the terms Extension, Intension, to things as well as
to their names, but I suggest as generally appropriate to
things and not names, the terms Quiddity and Quality
for the aspects of Thatness and Whatness in things, and
Entity for Quiddity + Quality, and for that which is asserted
in a proposition as distinct from the assertion, I would
suggest the term Assertum.
According to my analysis, propositions of the form
8 is P assert identity of denotation (extension) in diversity
of intension (s~r) ; while correspondingly S is not P
asserts difference (or otherness) of denotation in intensional
diversity (IT) C}T) i. e. it denies identity of denotation.
In S is not P the intension of P is asserted to be absent
from what is denoted by S. The purpose of S is not P is
not to assert that the intension of S is diverse from the
intension of P that goes without saying, and is essential
to S is P. The speaker who asserts S is P starts from a
AND ITS LOGICAL BEARINGS 15
whole (si?) ; the hearer or learner hears first S, then P,
and puts the two together into the whole, (s P) , from
which the speaker started. The is of the S is P can
not mean sameness of intension (exact similarity) for S
and P ex vi terminorum are diverse, have different inten
sions as Lotze avows (cp. ante], two different concepts
or contents cannot be connected at all by the simple
" is" of the categorical judgment ; if S and P were taken
in intension only, we could say of S nothing but that
S is not P , and if 8 is P did not indicate one-ness of
denotation, then S and P would not refer to an identical
object, and we should again have to say S is not P. If
terms were taken in denotation only, we should not know
what to do with more than one Term in affirmation.
An intension S neither (1) excludes from the denota
tion of S every other intension P, nor (2) does the
addition of P to S change the intension S. If (1), no
significant affirmation would be possible ; if (2), we should
never be sure what we were affirming. The thing which
is S is of course modified by the addition to it of the
intension P, but not so the intension S.
" Suppose I assert that all fronds of the Mountain
Buckler are erect. The meaning of the assertion is fixed
and definite, and, if true, it is true once for all. If I
go on to say that the fronds are also lance-shaped in
form, pinnately divided, that the pinnae stand opposite
(generally), that they are narrow and tapering and are
pinnatifidly divided do any of these affirmations, or all
of them, in the least alter or modify the meaning of my
original statement that all the fronds are erect ? It must
16 A NEW "LAW OF THOUGHT"
be admitted that no one of them alters the meaning of
any other; but what is very importantly modified is my
knowledge of, and power of identifying, the thing itself,
the actual object in time and space, the Subject of Attri
butes to which all these successive characteristics are
assigned. All the successive predicates are related as
joint characteristics of the whole which they qualify ; they
are related not as modifying each other, but as modifying it.
That the interior angles of an isosceles triangle are
equal to two right-angles, is a general truth, the meaning
of which is not affected by the further general truth that
the angles at the base of an isosceles triangle are equal to
each other, or that any triangle is half of a parallelogram.
The meaning of the assertion: This is an engraving of
a picture by Gainsborough, is not modified or changed by
the further assertions : The picture is a portrait of Lady
Mulgrave, it is one of the artist s masterpieces, it is
supposed to be now in America. But though any one of
these statements does not alter or modify the meaning
of the others, each one does modify the hearer s knowledge
concerning the object which is spoken of." (Mind, 1908,
p. 391, etc.)
Propositions of forms S is P, S is not P having been
admitted, and analysed as above (pp. 14, 15), we are of course
justified in formulating the commonly accepted logical
Laws of Contradiction and Excluded Middle as follows :
S is P ) cannot both be true (Law of Contradiction.
S is not P ) cannot both be false (Law of Excluded Middle).
Both these Laws appear to be self-evident, and it is
perhaps partly because A is A has been supposed to be
self-evident tear e^o^rfv, that it has so long held its
AND ITS LOGICAL BEARINGS 17
ground, and that such persevering and desperate efforts
have been made to give it an interpretation which would
fit it for the post of the First Law of Thought. If Logic
were provided with laws really self-evident, laws also on
which the Science could be based, and according to which
its structure could be built up, then indeed the founda
tion and method of Logic would be adequate and satis
factory. The old Law of Identity, however, affords only
a simulacrum of self- evidence. The Law of Identity in
Diversity which I propose to put in its place is simply a
law of significant assertion a law which while it is arrived
at through analysis of propositions of the form 8 is P
(8 is not P) implies conditions which make such assertions
possible, conditions without which knowledge itself seems
impossible without assertions of this form, knowledge
could not be communicated, nor even formulated. I hold
that there is no alternative and really primary analysis
which will bear investigation; that all valid interpreta
tions of categoricals which go beyond this must be based
upon it.
That 8 is P asserts an Identity of Extension in
Diversity of Intension seems to me to be on reflection
self-evident. I should at any rate claim that its denial
must be admitted to be inconceivable, and as to the form
8 is P, as the Eleatic Stranger in the Sophistes observed,
men " admit it implicitly and involuntarily in their common
forms of speech, they cannot carry on a conversation with
out it."
Taking together the three Laws of Identity in
Diversity, of Contradiction, and of Excluded Middle, we
may say that of any Subject (8), P must be affirmed or
denied, but not both, i.e. of any subject (S), P, or not-P,
j. 2
18 A NEW "LAW OF THOUGHT"
but not both, can be predicated. Thus we reach the
principle that
Any Subject of Predication is an identity of denota
tion in diversity of intension.
Such subject is a whole to which the two terms S and
P (or not-P) are applicable as names. For every 8 is P,
(s p) , or not-P, (snot-PJ ; to the thing spoken of, in both
cases, diverse intensions, S and P, or S and not-P, are
assigned, and S and P (or not-P) are equally names of the
denotational whole (s P) (or ( Snot - p )), and are therefore
interchangeable, and can be substituted one for the other.
The Law of Identity just formulated implies, I believe, that
Everything is an identity of extension in diversity
of intension.
At any rate we could only disprove this by showing
that there is something which is not a Subject of Predica
tion. But to show this, we must expressly treat it as a
Subject of Predication. Thus the proof that it is not,
involves that it is. Everything is a possible Subject of
Predication, and, directly the question is raised, it becomes
an actual Subject of Predication.
In the Law of Contradiction what is asserted is, that
if the diverse intensions signified by S and P are identical
in denotation, (SP) , then the diverse intensions S, and
not-P (absence of intension P), are not identical in de
notation ( S ) fnot-Pj . What the Law of Excluded
AND ITS LOGICAL BEARINGS
19
Middle asserts is that if the intensions signified by S and
P (or not-P) have not identical denotation, then the in
tensions S and not-P (or P) have identical denotation.
P, and not-P (= intension P absent), are mutually exclu
sive, and together exhaustive of all possibilities.
Though $ is P is not a self-evident and universally
applicable proposition, S is P or not-P is both, but we
should not have been in a position to assert it, unless we
had first established the forms S is P, S is not P. If it
were inevitable to analyse S is P as Lotze feels forced to
do into :
S is S,
P is P,
S is not P,
no such principle could be formulated.
>S is P, S is not P, express the two kinds or qualities
of assertion, affirmative and negative, in the most general
and abstract way, as a = b may stand for any and every
equation.
There are large and important groups of assertions
which though they can be expressed as S is P are more
appropriately exhibited in skeleton form and diagram-
matically as A is not B, (&j (&j , since what they
affirm is a relation between two things which have not
identical extension, however vital the relation between
them may be.
E.g., A is equal to B, G is father of D, E is to the
left of F:
22
20 A NEW "LAW OF THOUGHT"
In A = B there is between A and B equality of
quantity or value in intensional diversity i.e., there
is not only denotational identity, but also qualitative
sameness in qualitative diversity.
In all the above instances two non-identical things are
considered in their relation to each other, in the "system"
of related things (numbers or quantities, family con
nexions, positions in space) to which they respectively
belong. But the matter of fact in each case can be
perfectly well expressed (in any given case) as an identity
in diversity thus:
A is equal-to-B, E is to-the-left-of-F, and so on.
A is not B and E is not F, and it cannot be predicated
of A that it is B, nor of E that it is F ; but we can predi
cate of A that it is something equal to B, of E that it is
something to the left of F.
The copula in : A is equal-to-B,
E is to-the-left-of-F
(as always in S is P propositions) signifies identity of
denotation, and the special kind of relation between A
and B, E and F is here expressed in the Predicate.
In dealing with any " Relative Propositions," a know
ledge of the special system to which they belong is
required. The symbolic forms 8 is P, 8 is not P, are the
only ones that can be applied everywhere, and as they are
of the extremest generality, they are also of the extremest
abstractness and simplicity. The Subject and Predicate
in any 8 is P give the whole (s P) from which it is
inferrible that P is S, not-P is not-S, not-S is not-P.
S is not P gives the " system " MM MM ; from which it
AND ITS LOGICAL BEARINGS 21
can be inferred that P is not S, not-S is P, not-P is S. Of
course every variety of systematic relation between S and
P is possible, as has just been indicated.
A fortiori arguments are simply a special case of
arguments which turn upon Relativity of Terms.
A Proposition which has a relative term for S or P
or both, besides the ordinary Immediate Inferences (Educ
tions) which can be drawn from it in just the same way
as from non-relative Propositions, furnishes other imme
diate inferences to anyone acquainted with the system to
which it refers. These inferences cannot be educed except
by a person knowing the " system " ; on the other hand,
no knowledge is needed of the objects referred to,
except a knowledge of their place in the system, and
this knowledge is in many cases co-extensive with
ordinary intelligence ; consider, e.g., the relations of mag
nitude of objects in space, of the successive parts of time,
of family connexions, of number. From such a Pro
position as : C is a grandfather of D, in addition to such
inferences as could be drawn from a non-relative Propo
sition (a-grandfather-of-D is C, not-a-grandfather-of-D
is not-C, etc.), it is, of course, possible for anyone having
the most elementary knowledge of family relationship to
infer further that :
D is a grandchild of C,
A parent of D is a child of C,
A child of D is a great-grandchild of C,
The father of C is a great-grandfather of D, etc.
22 A NEW " LAW OF THOUGHT"
From C is equal to D (besides Something equal to D is
C, No not-equal to D is C, etc.), it can be inferred that
D is equal to C,
C is not less than D, / (Home thm g) \ (^\ (^\
D is not greater than C, \ equal to / \^S V_y
C is not greater than D,
Whatever is greater than C is
greater than D,
and so on (compare C is an inference from D).
In each of the above examples we are not dealing with
one object or group in the same way as in non-relative
Propositions, e.g.,
All men are
mortal,
All men \ / Byzantium
I I a lark
Constantinople,^ / \ r
This bird is a lark ;
but we are considering, besides the identity of appli
cation of S and P, two objects denotationally distinct,
namely C and D. (See my General Logic, p. 34.)
If, making an advance in complication, and passing to
Mediate Inference, it is asserted that :
M is P
and S is M,
the hearer is entitled to the construction /s M P) , where
three diverse intensions are referred to one denotation;
and each of the terms M, P, S, applies to, is a name of,
the whole (s.MJM , and each one maybe asserted of either
of the others. Thus this construction entitles the hearer to
AND ITS LOGICAL BEARINGS
the assertion S is P, which had not been actually asserted.
The speaker who asserted first M is P, and then S is M,
must have had before him at starting the whole fs M
which his hearer reached as a result of putting together
M is P, S is M.
Thus the conditions of affirmative Mediate Inference
with S, M, and P as terms may be formulated as follows :
If the denotation of any two terms, M and P, is identical,
any third Term S, which is identical in denotation with
either of them, is also identical in denotation with the
other. If in accordance with this pronouncement I reach
a whole of denotation which has intensions M, P, and S,
thus (s M P) , it is obvious that I am as much entitled to
assert identity of denotation between S and P as between
M and P or S and M.
For negative Mediate Inference with Terms, S, M, P
the following canon may be suggested :
If of two terms S, P, one is, and one is not, identical
in denotation with a third Term M\ then 8 and P are
not identical in denotation.
Hypotheticals are all either (1) Immediate Inferences
(e.g., If M is P, P is M), or Mediate Inferences which are
(2) fully expressed (e.g., If M is P, and S is M, then
S is P) which like (1) may be called Self-contained; or
(3) Mediate Inferences which are elliptical and enthy-
mematic e.g. If M is P, S is P (v S is M), If A is B,
E is F (v B is C, and C is D, and D is E); If S is M,
S is P (v M is P); If S is M, S is not P (v M is not P).
Conditionals (as distinguished by Dr Keynes and
24 A NEW "LAW OF THOUGHT"
Mr Johnson from Hypotheticals) are merely Categoricals
with a Subject-Term which is resolvable into Genus
4- Differentia, expressed in Hypothetical form because the
Predicate is limited to that part of the Class or Genus
which is qualified or conditioned by the Differentia.
E.g. If any triangle is equilateral, it is equiangular
equiangularity in a triangle follows from, depends upon,
is inseparable from, its equilaterality. A Hypothetical,
like an Inference, must start from something given (a
proposition or propositions). Inferences can be put in
hypothetical form, and Hypotheticals in inferential form.
I will examine concrete examples later, but may
observe here that there are some propositions Categorical,
Hypothetical, and Alternative which seem to be used as
a rhetorical device; e.g., If Newton was not a greater
mathematician than Kepler, the whole is not greater than
the part, We are the victims of misunderstanding, or the
truth is not true. These only amount to a very strong
asseveration (1) that Newton was greater than Kepler;
(2) that we are the victims of misunderstanding. We
can, no doubt, and do, use prepositional forms in cases
where instead of the/orm (S is P, etc.) resulting naturally
from the content, the only connection of content is that
imposed by the form (S is P, If A is B, C is D, etc.) upon
an indifferent or even recalcitrant content : E.g., we may
give the form S is P to an A not- A (A not- A) content, as
in the above examples (the whole is not greater than the
part, the truth is not true).
In the above brief and simple statement is contained,
I believe, the essential framework of the theory of Import
AND ITS LOGICAL BEARINGS 25
of Categorical assertions of forms 8 is P, S is not P, of
Hypothetical assertion, of Immediate Inference, and
Mediate Inference. The expansion and complication
introduced by its application to Class-Propositions will
be considered forthwith. Here I will only remark that
the twofold relation (affirmative and negative) of Subject
and Predicate in assertion (which must be distinguished
from the relation between Subject and Attribute) is quite
disparate from the fivefold relation possible between two
classes taken in extension, and no theory of the one can
be made perfectly symmetrical with any theory of the
other.
It is, I think, to the prominence given to Class-
Propositions and the predominant use of such Propositions
as examples (whether symbolic or significant) in books of
Logic, that the blurring of the clear and simple outlines
of Assertion (affirmative and negative) is largely due.
Such propositions as R is Q, Tully is Cicero, Courage
is Valour, Generosity is not Justice, London is the largest
city in the world, convert quite simply into : The largest
city in the world is London, Justice is not Generosity,
etc.
In dealing with ordinary class-propositions with
quantified Subject and unquantified Predicate, the matter
becomes more elaborate, and mistake more possible,
because in conversion the unexpressed but implied
Quantification (All, Some) of the old Predicate-name has
to be expressed, since that name is now the Subject-Term;
and on the other hand the expressed quantification (all,
some) of the old Subject-name sinks into mere implicitness,
that name being the new Predicate-name. Further, when
a class-name occurs as Term without quantification, a
26 A NEW "LAW OF THOUGHT"
different quantification is understood, when it is a Subject-
Term, from what is understood when it is a Predicate-
Term. E.g., in Trees are plants, Trees would be quantified
by all. In Cedars are trees, trees would be quantified
by some. And if we converted this last proposition, it
would be to: Some Trees are Cedars.
Propositions of the A, E, I, O form commonly have
some sign of quantity attached to the Subject and not to
the Predicate, and are said to have a quantified Subject
and an unquantified Predicate. It has been held by
certain reformers in Logic that all Predicates are natu
rally quantified in thought, and ought to be explicitly
quantified in speech. This view does not seem to be borne
out by reflection; but careful reflection does appear to
show that Quantification is an indispensable instrument
of Conversion.
The place of Quantification in Logic is very curious,
its function being often as completely hidden from those
whose processes of Conversion involve it, as the subterranean
course of a train in one of the loop-tunnels of the Swiss
Alps would be to an observer who only saw it rush into
one opening, and emerge again in a few minutes from
another, just above or just below. My meaning will be
best elucidated by taking an ordinary proposition and
tracing the changes which it undergoes in Conversion.
Let the proposition be
All human beings are rational (1)
The ordinary converse of this is /[Ail] human
Some rational creatures are human [[gome] rational]
beings (2), \ [creatures]
or
Some rational creatures are human (3).
AND ITS LOGICAL BEAEINGS 27
If I merely alter the relative position of S and P in (1)
as it stands, and say
Rational are all human beings,
it is clear that Conversion in the logical sense has not
taken place; for Rational is still the Predicate, and all
human beings is still the Subject. The proposition has
been merely turned round. But it may be put into the
form
All human beings are rational creatures (4),
and with this we can deal. It is not, however, any more
than the adjectival (1), directly convertible. If altered
into
Rational creatures are all human beings,
the proposition thus obtained, besides being awkward,
is ambiguous it is by no means clear which term is to
be taken as Subject, and the all might even be understood
to qualify (or quantify) Rational creatures.
The first step towards real Conversion is taken when
we pass from (4) to the quantificated proposition
All human beings are some rational creatures (5).
From this we go on to the quantificated converse
Some rational creatures are all human beings (6) ;
and from (6) to the unquantificated converse of (5)
Some rational creatures are human beings (7).
From (7) we can pass to the corresponding adjectival
Proposition
Some rational creatures are human (8).
It is to be observed that in going from (4) to (7), we have
not only inserted a sign of quantity before the new Subject-
name (rational creatures) which, as the old Predicate, had
not any to start with : we have also dropped the sign of
quantity which the new Predicate (human beings) had
28 A NEW "LAW OF THOUGHT"
when it was the old Subject-name. Thus, as we began
with an unquantificated proposition, so we end with an
unquantificated proposition. The propositions which
logicians (on the whole) have recognised and dealt with
are unquantificated propositions ; it is for enabling us to
pass (by an elliptical procedure) from unquantificated to
unquantificated propositions that the ordinary rules of
Conversion and Reduction of Class-Propositions and
Syllogisms are framed; it is of unquantificated propositions
that the "nineteen valid moods" of the traditional
Categorical Syllogism are composed.
In converting an E proposition, we should proceed
as follows : Let the proposition to be converted be,
No R is Q (1). (1) = (2) Any R is not Q (by
grammatical equivalence). Quantificating (2)
we get, Any R is not any Q (3). (3) converts to, Any Q
is not any R (4). By disquantificating
(4) we reach (5), Any Q is not R. And
(5) = No Q is R (by grammatical equi-
valence).
My view then is that the usage of Logic and of ordi
nary speech is on the whole to be justified, and yet that
Quantification is possible and valid in a subordinate office,
as a necessary transformation stage of propositions. This
can be made clear by reference to the Import of Categorical
Propositions. What a Categorical proposition affirms or
denies is, identity of denotation of the S and the P in
diversity of intension. Denotation of S and of P in an
affirmative Categorical Proposition are the same; intension
of the S and P being, of course, always diverse in
propositions of the form 8 is P. And denotation is
sufficiently indicated by the S ; identity or otherness is
AND ITS LOGICAL BEARINGS 29
indicated by the copula (is or is not) while diversity of
intension comes into view only when the Predicate is
enunciated. In regard to any assertion, we want to know
in the first place ivhat it is of which something is affirmed
or denied; this knowledge is given with the enunciation
of the Subject, which indicates the thing or things spoken
of. We want, in the second place, to know what it is that
is affirmed or denied of the thing or things indicated by
the Subject. This information is supplied by the Pre
dicate that is, by its signification or intension, since it
is evident that in affirmative propositions the application
of the Predicate is identical with, in negative propositions
is altogether distinct from, that of the Subject. Hence it
seems clear that in the Predicate of any proposition, it is
intension, and not denotation, which is naturally and
generally prominent. This is confirmed by the considera
tion that we commonly use Adjectival Predicates, if
appropriate Adjectival Terms are available; and that
such terms cannot in English (though they can in many
languages) take the sign of the plural, while the Substantive
Terms which they qualify can, and no one doubts that the
application of an Adjectival Term is the same as that of
the Substantive Term which it qualifies. Now if it is the
primary function of the S in any Categorical Proposition
to indicate denotation, while it is the primary function
of the P to indicate intension, it seems obvious that
quantifying is appropriate, and may be necessary, in the
case of S, but not in the case of P, under ordinary
circumstances. And a further reason against admitting
Quantification (except as a transformation stage) in most
propositions, is deducible from the consideration that what
propositions affirm or deny is the identity of denotation
30 A NEW "LAW OF THOUGHT"
(in diversity of intension) of S and P; for in a quantificated
affirmative, though indeed identity of the terms is still
asserted (as it is bound to be), the fact that the denotation
of both terms is made prominent tends to blur this
especially where difference of extent of the classes referred
to is suggested. It might indeed be maintained that
where both terms of our propositions are taken purely in
denotation, quantincated propositions are most appropriate,
being the form of proposition which makes the denotation
of both S and P most prominent. But both terms cannot
be taken purely in denotation. If, e.g., in S is P, both
S and P were taken in denotation only, then to say 8 is P
would be exactly equivalent to saying S is S, for the
denotation of P is the very same as that of S. On the
other hand, the view here advocated of the Import of
Categorical Propositions justifies the recognition of
Quantification as a phase of propositions. For the
Predicates of propositions have denotation as well as the
Subjects, and (in affirmative propositions) a denotation
which is identical with that of the Subjects. It is
therefore possible, and under certain conditions allowable
and necessary, to make this prominent by quantification.
And the Subjects of propositions have intension ; and this
may be allowed to come into prominence by dropping the
sign of quantity which inevitably fixes attention rather
upon the denotation than the intension of a term. What
Sir Wm Hamilton hoped for from the doctrine of Quanti
fication was, that by its help the relations of classes,
as well as the relation of Subject and Predicate, could
have been exactly expressed by the form of Assertion.
But Quantification is entirely and for ever unequal to the
accomplishment of such a task.
AND ITS LOGICAL BEARINGS 31
The above may be further confirmed and illustrated
by a consideration of the traditional logical treatment of
O Propositions. Of the four Class Propositions A, E, I, 0,
the first three have always been regarded as capable, the
fourth as incapable, of Conversion.
We have seen that propositions on their way to
Conversion have to undergo the process of Quantification.
But the reason why O (Some R is not Q) is pronounced
inconvertible is not because there is any more difficulty
in quantifying its Predicate than in quantificating the
other propositions, but because, when the quantiftcated
converse of (any Q is not some R) has been reached, the
quantification cannot be dropped without an illegitimate
alteration of signification. For the commonly accepted
signification of the disquantificated converse of O (Any
Q is not R) implies a quantification different from that
which has been dropped the dropped quantification being
some, the quantification understood as involved in the
unquantificated Proposition (Any Q is not R) reached
by dropping it, being any. And as, at the same time,
ordinary thought and speech will not admit the explicitly
quantificated form, it is inevitable that a Logic which
deals with the forms of ordinary thought and speech
should regard as inconvertible. Let us take as a con
crete instance the Proposition, Some trees are not oaks (1).
This becomes by quantification (2) Some trees are not
any oaks, which converts to (3) Any oaks are not some
trees. Dropping the quantification of (3), we get (4)
Any oaks are not trees, and this would be understood
to mean (5) Any oaks are not any trees (= No oaks are
trees). (General Logic, p. 58 &c.)
32 A NEW "LAW OF THOUGHT"
(1) All lilies (8) are beautiful (P)
converts to :
(2) Some beautiful things (P) are lilies (S),
and this again converts to :
(3) Some lilies (S) are beautiful things (P).
Obviously the quantification some in (2) must have been
implicit, though unexpressed, in (1); and the explicit
quantification some in (3), must have been implicit in (2).
It is clear that it is the quantified Subject and
Predicate in Class-Propositions which correspond to the
S and P in S is P. E.g., in (3) some lilies is S, and [some]
beautiful things is P. Similarly with (1) and with (2). If
in (1) e.g., the denotation of " beautiful " were not limited
to the denotation of "all lilies" if, that is, All lilies (S),
were not denotationally identical with only [some] beautiful
things (P), \^Jj, then what the proposition asserts would
be identity (of denotation) between lilies and all beautiful
things an interpretation which is neither intended nor
admissible.
(1) No men (S) are angels (P)
converts to :
(2) No angels are men.
Angels was implicitly quantified universally in (1),
otherwise the explicit universal quantification of that
term-name in (2) would not be possible. By implicitly
quantified I mean that there is no explicit quantification
but that explicit quantification is justified.
In Conversion, as we have been seeing, the Subject-name
of the converse is supplied with a sign of quantity which
it had not at first, and the Predicate-name of the converse
is deprived of the sign of quantity which it originally
AND ITS LOGICAL BEARINGS 33
had. To sum up : The explanation of this change intro
duced into Categorical Propositions, when they undergo
conversion, is that "the natural way of thinking a
Categorical Proposition is to emphasise the extension -
aspect in the Subject and the intension-aspect in the
Predicate; where an adjective of quantity is expressed,
it is inevitable that the aspect of extension should
have attention drawn to it. Further, the quantification
of the new Subject-name makes clear that this name
has had, throughout, an extensive aspect, though that
aspect was not emphasised or explicitly brought into
notice as long as it was a Predicate-name. The mere
transposition of the Predicate into the place of the Subject
could not suffice to give it extension unless it had had
extension from the beginning, since Conversion is not a
legitimate process if it does more than infer something
which is true supposing the inferend is true! (Primer
of Logic, p. 34.)
S is P, 8 is not P, are susceptible of Obversion, and
there is no difficulty in applying this process to Class-
Propositions in accordance with the simple procedure
applicable to the former. S is P, (S P) , obverts to S is not
not-P ; S is not P, obverts to 8 is not-P.
/All robins\ y^SX
All robins are insect-eaters, (gomeTnsect-) (>-< ) b verts
to No robins (S) are [any] not-insect- eaters (P) (=A11
robins are-not not-insect-eaters). Robins being in
cluded in the group of insect-eaters, are (by Law of
j. 3
34 A NEW "LAW OF THOUGHT"
Excluded Middle) excluded from all those things, whatever
they may be, that do not eat insects : fcobinsj
No painters are mathematicians -/P )(M\- obverts to :
x f ^_s
All Pcmifers (S) are not [any] mathematicians (P).
In inference by Added Determinants of the form: If
R is Q, then XR is Q, ^3/, it is because the relation
of identity in extension between R and Q remains un
affected by the intension added to the Subject, that we
can add this intension.
E.g. If all ices are unwholesome, then strawberry ices
are unwholesome.
Inference by added Determinants of the form : If
R = Q then ZR = ZQ, as applied to quantity or number,
depends on the principle : If equals be added to equals
the wholes are equals.
E.g. If 2 + 2 = 4 ........................ (1)
then 2 + 2 + 3 = 4 + 3 .................. (2).
Here there are in (1) two related objects or groups :
These two are now transformed, by the addition to
each of another object, exactly similar, into objects of
which both denotation and intension have been modified,
but in exactly the same way in both, so that the relation
of equality is maintained.
So, if 40 shillings = 2 pounds,
then 40 shillings x 4 = 2 pounds x 4.
AND ITS LOGICAL BEARINGS 35
Again, if 500 will buy one motor,
then 1000 will buy two motors.
Or, if two Northerners can tackle three Southerners,
four Northerners could tackle six Southerners.
I will venture at this point to quote and consider a
passage from a little Logic book 1 which has been reprinted
many times since it was first published in 1870, and is
still largely used in schools and colleges, and recommended
for examinations. The author says: "There are modes
in which all persons do uniformly think and reason, and
must think and reason. Thus if two things are identical
with a third common thing they are identical with each
other. This is a law of thought of a very simple and
obvious character, and we may observe concerning it:
1. That all people think in accordance with it, and
agree that they do so as soon as they under
stand its meaning.
2. That they think in accordance with it whatever
may be the subject about which they are thinking.
Thus if the things 2 considered are
London,
The Metropolis,
The most populous city in Great Britain,
since the Metropolis is identical with London, and
London is identical with the most populous city in
Great Britain, it follows necessarily in all minds that
the Metropolis is identical with the most populous city
in Great Britain.
Again, if we compare the three following things 2
Iron,
The most useful metal,
The cheapest metal,
1 Jevons Elementary Lessons in Logic. 2 Italics mine.
32
36 A NEW " LAW OF THOUGHT"
and it be allowed that The most useful metal is Iron,
and Iron is the cheapest metal/ it follows necessarily in
all minds that the most useful metal is the cheapest.
We here have two examples of the general truth that
things identical with the same thing are identical with each
other 1 , and this we may say is a general or necessary form
of thought and reasoning.
Compare again the following three things^
The earth,
Planets,
Bodies revolving in elliptic orbits."
As far as I know I am the first person to question
this " simple and obvious law of thought," that " if two
things^ are identical with a third common thing 1 they are
identical with each other." And yet it is not a law
either of thought or of things, and it is not simple and
obvious, on the contrary it is untrue and impossible.
No thing can be identical with any other
thing : London, The Metropolis, The most XtJndonX
populous city in Great Britain, are not three /metropolis A
r f J most I
things, but three names of one thing. If not, V populous /
we could not say : The Metropolis is iden- \^crty^/
tical with the most populous city in Great
Britain.
The explanation of this passage from Jevons, so chaotic
when we come to examine it, is, I suppose, that like so
many other thinkers, Jevons, in spite of all his ability
and originality, was not clear about the different sorts
of oneness and difference, and (as in his "great rule of
inference " the " Substitution of Similars ") persistently
confused together Identity of Denotation or Extension,
1 Italics mine. Compare Hansel s interpretation of the Law of
Identity.
AND ITS LOGICAL BEARINGS 37
and Sameness of Intension, denotative one-ness, and
qualitative one-ness. We can no more substitute
" similars " in inference than we can " identify " one
thing with another thing. " Interchangeability of de-
notational identicals" would be a much better name for
what Jevons means.
Similarity is the category of classing, not of affirma
tion this pencil or this stamp may be similar in the
highest degree to that, but this is not that. On the
other hand, this is " the man who was," but how tragically
different. This girl is incredibly like what her grand
mother was at 17, but I do not therefore take her for
her grandmother, who at 17 was fair and fresh and active,
but is now faded and infirm. I have so far learnt to
discriminate between cases in which exact similarity is, and
those in which it is not, evidence of individual identity.
Similar confusion occurs in a curious form in Mill
(Logic, I. 116, 9th ed.) when he gives as examples of
propositions in which simple Resemblance is asserted the
following :
" The colour I saw yesterday was a white colour,"
" The sensation I feel is one of tightness."
Here there seems to be confusion between assertion
(S is P, identity in diversity) and classing (grouping of
this instance with other instances, in virtue of resemblance
or similarity), and complete oblivion of anything like a
general view of import. We find a like want of clearness
in a passage in Jevons Elementary Lessons, p. 65, when
he says : " The proposition Gold is a yellow substance
states stick an agreement of gold with other yellow sub
stances that we know it to have the colour yellow," etc.
38 A NEW "LAW OF THOUGHT"
"Mill tends to drop out of account in his treatment
of names and propositions not only all surplusage of in
tension beyond connotation, but also all explicit reference
to the extension aspect. But this the application of
names is in the very forefront of importance. For Mill,
connotation swells and grows till it almost fills the
picture, whether we are dealing with Terms or with
Import of Propositions. Connotation (where there is
Connotation) may determine application. But without
application somehow determined, all use of names and
terms is impossible. Mill himself seems to admit this
when he says of Hobbes s definition of Categorical affirma
tive Propositions ( In every proposition what is signi
fied is the belief of the speaker that the predicate is a
name of the same thing of which the subject is a name ),
that it is true of all propositions and is the only account
of import which is rigorously true of all propositions
without exception. It is odd that Mill, while setting
aside and belittling Hobbes s analysis, should have been
content to furnish finally as his own contribution to the
theory, nothing better than an analysis (and an ex
ceedingly unsystematic one) of the imports of different
classes of propositions.
Hobbes, as we have seen, lays all the stress on appli
cation of names on the denotation, not the connotation,
aspect and this carried on into Syllogism would justify
the fundamental importance of identity of application of
the Middle Term (however this identity may be deter
mined). (Cp. the requirement that the Middle Term in
a class syllogism must be distributed. ) The same
would hold of Immediate Inference. And it may be
observed that Jevons doctrine of Substitution of Similars
AND ITS LOGICAL BEARINGS 39
does really lay like emphasis on the supreme part played
by application for the substitution referred to by the
great rule of inference which Jevons gives, is Sub
stitution not of Similars but of terms having identical
application. The rule runs as follows :
The one supreme rule of inference consists... in the
direction to affirm of anything whatever is known of its
like, equal or equivalent. The Substitution of Similars
is a phrase which seems aptly to express the capacity of
mutual replacement existing in any two objects which are
like or equivalent [= ?] to a sufficient [= ?] degree (Prin
ciples of Science, p. 17, 3rd edit.).
That the substitution here referred to is in fact
substitution of terms having identical application is
obvious on the most cursory examination, and is ap
parent at first sight from Jevons own examples in
illustration, e.g.
(a) Snowdon (1)
Highest mountain in England or Wales (2)
(Something) 3590 feet in height (3).
(6) The Lord Chancellor (1)
The Speaker of the House of Lords (2).
(c) God s image (1)
Man (2)
Some reasonable creature (3).
It hardly needs pointing out that in (a) and (c) (1),
(2) and (3), and in (6) (1) and (2), respectively, are not
qualitative similars, but numerical, historical, or ex-
tensional, identicals intension is in each case different,
but extension (and therefore application) the same. On
the other hand, taking things that are so similar as
to be intrinsically indistinguishable, we see at once that
40 A NEW "LAW OF THOUGHT"
they cannot be thus substituted the one for the other.
That house/ e.g., may be similar in the highest degree
to another standing next it, but the one is not the other,
and in inference could not be substituted for it. This
copy of Giorgione s Richiesta may be an exact copy,
yet could not be substituted for it as The highest
mountain in England or Wales could be substituted
for Snowdon. One of a pair of twins may be so like
the other as to be commonly mistaken for him yet
owing to the one having come into the world a brief
space of time before the other, he may be the heir to
a dukedom and inheritor of an immense fortune, while
the other is neither the one nor the other, and to c sub
stitute the one for the other would be inadmissible and
even felonious" (Mind, 1908, p. 531, etc.).
" When Jevons (Principles of Science, ch. in.) discusses
the Import, etc., of Categorical Propositions, expressing
them as Equations (A = B, etc.), and speaking of them as
Identities, I find that some of his examples and some of
his explanations are quite in accordance with my analysis.
E.g., when he takes the Proposition, Tower Hill/is/the
place where Raleigh was executed, and says that it ex
presses an identity of place ; and whatever is true of the
one spot is true of the spot otherwise defined, but in
reality the same. But when he goes on to say that the
same analysis can be applied to e.g., the Proposition
(1) Colour of Pacific Ocean = Colour of Atlantic
Ocean, finding no distinction between this and e.g.
(2) Deal Landing-place of Caesar, except that in
(1) we assert identity of single qualities while in (2) we
express identity of groups of qualities, it is clear that
there is confusion between extensional and intensional
AND ITS LOGICAL BEARINGS
41
same-ness. The colour of the Pacific Ocean may be
exactly similar to that of the Atlantic, but we certainly
cannot say that the one is the other in the sense in which
we can say that Deal is the place where Caesar landed.
This confusion ruins Jevons whole account of inference,
and is even betrayed by the very name-Substitution of
Similars which he has chosen to characterise his theory "
(Mind, 1893, pp. 450, 451).
It would hardly be worth while to take warning
examples from Jevons and Mill, if thinkers generally
had outgrown this confusion between the different kinds
of Same-ness or One-ness which has had such a devas
tating effect upon theories of import in particular; but
there are indications that this is not the case.
For instance, Mrs Ladd Franklin, in discussing the
Import of Categoricals, says : " The reason that so many
different views are possible is a very simple one. Every
term is a double-edged machine it effects the separating
out of a group of objects, and it epitomises a certain
complex of marks. From this double nature of the term,
it follows... that a proposition which contains two terms
must have a fourfold implication.... Whoever says, for in
stance, that All politicians are statesmen must be prepared
to maintain that the objects politicians are the same as
some of the objects statesmen ; and also that the quality-
complex politician entails the quality-complex statesman,
and is indicative of the presence of some of the objects
statesmen.... In other words to say that a is b is to affirm
that both from the objects a and from the qualities a are
inferrible both objects b and qualities 6. [But if a is b,
objects a actually are objects b, and from qualities a,
qualities b need not be inferrible e.g. a man may be
42 A NEW "LAW OF THOUGHT"
a politician (a) without being a statesman (b)]. Now it
is open to the logician to say that any one of these four
implications is the most important or the most prominent
implication of the proposition, but it is not open to him to
say that less than all four of them is the complete impli
cation 1 . Any one of the four is a sufficient groundwork
on which to work out the entire system of reasoning"
(Mind 1890, p. 561).
By this I believe is meant that we may understand
both S and P in denotation, or both in intension (con
notation), or S in denotation and P in intension, or S in
intension and P in denotation. But when it is said that
to affirm a is b is to affirm that both from the objects a
and from the qualities a, both objects b and qualities 6 are
inferrible, I reply that such " inference " is only possible
provided we have already understood a is b to assert
identity of denotation of a and b (in diversity of inten
sion). As regards the concluding assertion, I proceed
shortly to examine " the four " and to show that not one
of them is even possible.
Mrs Franklin s view of the " fourfold implication of Pro
positions in Connotation and Denotation " is approved by
Dr Keynes who (in his Formal Logic, 3rd ed. p. 147, etc.),
expounds the matter as follows :
"(i) If we read the subject of a proposition in de
notation and the predicate in connotation, we have what is
sometimes called the predicative mode of interpreting the
proposition. This way of regarding propositions un
doubtedly corresponds in the great majority of cases
with the course of ordinary thought ; that is to say, we
1 Of course the important question is: Exactly how are "all four"
implicated ?
AND ITS LOGICAL BEARINGS 43
naturally contemplate the subject as a class of objects of
which a certain attribute or complex of attributes is pre
dicated "(P- 149).
" (ii) Subject in denotation, predicate in denotation.
If we read both the subject and the predicate of a
proposition in denotation, we have a relation between two
classes, and hence this is called the class mode of inter
preting the proposition. It must be particularly observed
that the relation between the subject and the predicate is
now one of inclusion in or exclusion from, not one of pos
session. It may at once be admitted that the class mode
of interpreting the categorical proposition is neither the
most ultimate, nor generally speaking that which we
naturally or spontaneously adopt. It is, however, extremely
convenient for manipulative purposes, and hence is the
mode of interpretation usually selected, either explicitly
or implicitly, by the formal logician" (p. 151).
" (iii) Subject in connotation, predicate in connotation.
If we read both the subject and the predicate of a
proposition in connotation, we have what may be called
the connotative mode of interpreting the proposition. In
the proposition All S is P, the relation expressed between
the attributes connoted by S and those connoted by P is
one of concomitance the attributes which constitute the
connotation of S are always found accompanied by those
which constitute the connotation of P " (p. 154).
" (iv) Subject in connotation, predicate in denotation.
Taking the proposition All S is P, and reading the
subject in connotation and the predicate in denotation, we
have The attributes connoted by S are an indication of
the presence of an individual belonging to the class P.
This mode of interpretation is always a possible one, but
44 A NEW "LAW OF THOUGHT"
it must be granted that only rarely does the import of a
proposition naturally present itself to our minds in this
form " (p. 146).
I proceed to examine the four readings here recom
mended to us. (See my article on Logical Judgment in
Mind, 1893, pp. 452, etc.)
Since, it is said, terms may have Denotation (Exten
sion) or Connotation, or both, any Proposition of the form
S is P may be read wholly in Denotation, or wholly in
Connotation, or S in Denotation and P in Connotation, or
S in Connotation and P in Denotation ; thus giving four
possibilities. If there can be four valid formal theories of
Assertion, since each differs considerably from the others
it ought no doubt to be possible, as Mrs Ladd Franklin
affirms, to have four systems of Logic corresponding to
those four theories respectively. It would indeed be
interesting to have even the most meagre outline of even
one of these four possible theories. But leaving this point,
let us look at the alternative readings of S is P pro
positions which are here formulated. That these four
alternatives are possible, or indeed that any of them
is so, I most emphatically dispute. If the assertion
expressed by S is P is to be read wholly in Extension,
(1) then since the Application of S is (by the force
of the copula) identical with the Application of P,
if we ignore the element of Connotation or Intension
(in which alone there is difference) we must express
the assertion as S is S. S is not P is clearly not capable
of being even supposably read in Extension only, since
diversity of Signification in Subject and Predicate is
rendered indispensable by the negative copula.
(2) If S is P is to be read in Connotation (or Inten-
AND ITS LOGICAL BEARINGS 45
sion, or Comprehension) only, again the affirmative S is P
must melt (cp. Lotze) into 8 is 8; for how can any
connotation be any other connotation ? If it is said that
S is P expresses a combination of the connotations of S
and P, it seems sufficient to point out that the only way
in which connotations can be combined is by co-existing
in one extension.
Again, if in S is P (3) S is taken in Denotation only,
and P in Connotation only, or (4) S in Connotation and
P in Denotation, what is the force of is ? Between what
is Identity supposed to be asserted ? We can no more
say that Denotation is Intension than we can say that
This kitten is Animality. Is it not plain that, for is to
have any assertive force, there must be denotational
Identity between S and P (in S is P), and that for any
significance to attach to the assertion, there must be a
diversity of Connotation or Intension ?
In order that e.g., All R (= S) is [some] Q (= P) may
be interpreted (" in Extension ") to mean :
Class R/is /included in class Q ; /ciass>\
/ [some] in- \
or (" in Connotation ") to mean : I cluded in /
\Class Q/
Attributes R I are I accompanied by Attributes Q;
not only must both aspects have been taken account of in
both Subject and Predicate of the original Proposition but
the interpreting Propositions are unintelligible without a
similar analysis having been applied to them as they
stand, and the interpretations into " in Extension " and
"in Connotation" are seen to be entirely founded not on the
form of the propositions, but on the intensions of the sub
ject and predicate. Granted that All R is [some] Q (1),
may mean Class R is included in Class Q (2); this
46 A NEW "LAW OF THOUGHT"
(2) again has got to be analysed as denotational identity
in intensional diversity. Unless so understood, is is not
admissible, for in intension Class R and included in Class
Q are diverse, we could not say that in intension the one
is the other. So it is the denotation of Class R that is
identical with the denotation of the intensionally diverse
predicate, [some] included in Class Q.
The Identity-in-Diversity analysis starts simply and
solely from what is asserted, the whole
And the analysis (identity-in-diversity) being abso
lutely general and highly abstract, a mere skeleton
analysis, admits of further determination of various kinds ;
the only proviso is, that these further determinations start
from and presuppose the skeleton analysis.
E.g. in : All Isosceles Triangles have the angles at the
base equal
(= All Isosceles Triangles are having the angles
at the base equal),
the identity-in-diversity analysis offers no obstacle to the
view that the intension of the Predicate is inseparable
from that of the Subject. It allows indeed of this being
recognised in the fullest way. In fact the inseparability
of the intension of P from that of S quite inevitably
involves identity of denotation (in diversity of inten
sion).
Mr Russell in Mind, 1905, proposes to substitute for
Frege s analysis of Categoricals (of which in 1903 he
approved) a very complicated statement e.g. instead of
understanding "The father of Charles II was executed" to
express identity of denotation (Bedeutung) in diversity of
intension (Sinn), he would interpret it as follows : " It is
AND ITS LOGICAL BEARINGS 47
not always false of x that x begat Charles II, and that
x was executed, and that If y begat Charles II, y is
identical with x is always true of y" (p. 482).
As regards this I would point out that in my view
(1) the speaker who asserts that The Father of Charles II
was executed starts from the subject-matter of assertion,
/Father of\
the complex whole : Charles n 1 and in order to deal in
V Executed /
any way with this, it has to be first of all analysed on the
identity in diversity plan, so that " Father of Charles II "
and " executed " are referred to the denotation of the
subject as its intension. (2) The statement : " It is not
always false of x, etc." involves several repetitions of
identity in diversity :
(a) It is not always false of x t (b) that x begat Charles
II and that (c) x was executed, and that (d) if y begat
Charles II, (e) y is identical -with x t (f) is always true
of y.
Unless these clauses are to be understood as identities-
in-diversity, what can be made of them, what is the
connexion between their elements ? I understand that
Mr Russell s object in giving up Frege s view and putting
forward this complicated substitute, is to eliminate " de
noting phrases" and so get rid of inconvenient implications
of " existence." But he does not seem to have escaped
identity of denotation in diversity of intension (in the
ordinary sense of denotation and intension) and I cannot
avoid the conviction that any form of proposition is in
capable of determining questions of " existence," just as
much as forms of proposition (8 is P, etc.) are incapable
48 A NEW "LAW OF THOUGHT"
of completely and determinately expressing relations of
classes.
In: All Cavicornia are Ruminants,
All Antelopes are Cavicornia,
All Antelopes are Ruminants,
the relation of Terms may be diagrammatically represented
thus:
The true Middle Term is the some Cavicornia of the
Minor Premiss ; for it is only that part of the denotation
of Cavicornia which is common to both Antelopes and
Ruminants, that is the bond of connexion between them.
The Ruminants that Antelopes are, are the Ruminants
whose denotation is identical with that of those Cavi
cornia that are identical with Antelopes. Of those
Ruminants whose denotation does not coincide with that
of any Cavicornia, and of those Cavicornia whose denotation
does not coincide with that of any Antelopes, it must be
said that they are not Antelopes, and that Antelopes are
not they. It is the indefiniteness of the some by which
Ruminants in the Major Premiss, and Cavicornia in the
Minor Premiss are implicitly quantified, that makes it
necessary to sweep in the whole extension of Cavicornia,
so as to make sure that those Cavicornia with which (as
being Antelopes) we are concerned, are Ruminants.
Sameness of Denotation (identity) of Middle Term in
Mediate Inference is that which connexion between Major
and Minor Terms must depend upon, for it cannot depend
on sameness of intension or exact similarity (cp. Jevons
"Substitution of Similars")] intensional sameness, the
AND ITS LOGICAL BEARINGS
closest similarity, would not justify substitution if it
would, there would be no reason why the Middle Term in
a Syllogism should be distributed the intension M would
be all that could be required as a link,
and (as in all S is M, all P is M), S
might be identified with one part of the
Class M, and P with another part, and as a
result S identified with P, which is absurd,
In:
No diamonds are red
This stone is red. ( D
/. This stone is not a diamond.
No diamonds = P
[any] and [some] red = M.
This stone = S
Some red is, denotationally, part of any red.
In:
All sapphires are blue
This stone is not blue.
. . This stone is not a sapphire.
A II sapphires = P
[any] and [some] blue = M.
This stone = S
Some blue is identical with part of any blue.
It is some red and some blue which are the true middle
terms. Compare the account of the "Antelope" syllogism
above.
It is because Class relations as expressed in the A, I, O
forms are indeterminate, that in Mediate Inference we
cannot make the Terms correspond exactly with the clear
and perfectly definite forms of the 8 is P, S is not P type
when we are dealing with unquantified class-syllogisms.
J. 4
50
A NEW "LAW OF THOUGHT"
London :
Capital of
England:
Largest
city
This may be done however in the case of what has been
called Traduction, where all the subjects are singular and
have identical denotation, e.g.
London is the largest city in the
world,
London is the capital of England,
The capital of England is the
largest city in the world.
It is done exactly in every Mediate Inference (Traduc-
tional or other) in which the denotations of all the Terms
are determinate, e.g.
The Syndics and Night Watch are
two of Rembrandt s masterpieces;
The Syndics and Night Watch are
two of the pictures in the New
Museum at Amsterdam;
Two of the pictures in the New
Museum are two of Rembrandt s
masterpieces.
It seems unnecessary here to consider the differences
of Syllogistic Figure, and of Mood in as far as variation
of Mood in Class- Propositions goes beyond the three cases
possible when we use the S is P, S is not P forms only
i.e. (1) M is P
SisM
(M P S]
ight Watch
Two of R. s
masterpieces
Two of the
pictures in
the New
Museum
(3)
SisP
P is not M
SisM
S is not P
PisM
S is not M
S is not P
00
AND ITS LOGICAL BEARINGS 51
Such differences of Mood and Figure may result from
the indeterminateness of A, I, O, and further variations
of determination due to the fact that the some of ex
plicit quantification is itself indeterminate. If we allow
conversion of Class-Propositions to be possible, we must
admit that in every case the Terms are either explicitly
or implicitly quantified; owing to the conventions of
customary speech, the quantification is generally im
plicit ; when made explicit it is mostly indeterminate ;
but its possibility is incontestable proof of the denotation
of Predicates. If in affirmative Categoricals it were
possible (which it is not) to simply add the intension of
the Predicate to the denotation of the Subject, we should
avoid all difficulties due to the implicit some and the in
determinateness of denotation of Predicate ; but then the
Propositions would be incapable of Conversion.
An examination of concrete Hypothetical, Conditional,
and Disjunctive (Alternative) Propositions shows that
here too the analysis of Categorical Affirmation as identity
of denotation in diversity of intension is
applicable. Take the following Conditionals :
If any child is spoilt, he is troublesome,
asserts the identity of denotation of spoilt
child with troublesome child.
If any rose is blue, it is a curiosity, asserts denota-
tional identity of Blue Rose with a Curiosity.
Take the following Hypotheticals, of
which (1) is Self-contained, i.e. the conse
quent is a necessary consequence of the
antecedent taken alone :
(1) If all men are fallible and the Archbishop is a
man, the Archbishop is fallible.
42
52
A NEW "LAW OF THOUGHT
What is asserted is, that granting that the denotation
of man is part of the denotation of
fallible, and that the denotation of the
Archbishop is part of the denotation of
man, then it follows that the denotation
of Archbishop is part of the denotation
of fallible.
(2) If Charles I had not deserted Strafford, he would
be deserving of sympathy.
This asserts that supposing denotation of Charles I
to be identical with denotation of one
who did not desert Straff ord, then (be
cause not to have deserted Stafford
would have been to deserve sympathy) the
denotation of Charles I would have been
the denotation of one deserving of sym
pathy. In this example it is not from the expressed
antecedent alone that the consequence follows, but from
that antecedent taken in conjunction with another (un
expressed) proposition.
(3) If the building goes on, he will not recover.
This may be expanded into :
If the work goes on, great noise will be made ;
If great noise is made, he will be disturbed by it ;
If he is disturbed, he will not sleep ;
If he does not sleep, he will die.
The conclusion he will die results from a series of
suppositions in which building going on (1) is identified
(denotationally) with making noise (2), making noise with
disturbing him (3), disturbing him with preventing his
sleeping (4), preventing his sleeping with preventing his
recovery (5). What holds the argument together deno-
AND ITS LOGICAL BEARINGS
53
tationally is just as much of the denotations of (2), (3),
(4) and (5) as are identical with
the denotation of (1). (This is not
affected by the circumstance that
here denotational identities follow
from intensional connexions).
The efficacy of the identity-
in-diversity analysis is I think
nowhere more strikingly seen than
in its application to Hypothetical,
especially Hypotheticals of the elliptical and often com
plicated sort which we so commonly employ, and of which
the illustrations (2) and (3) examined above are instances.
I will here take as one more illustration, Lewis
Carroll s " Logical Paradox," the discussion of which has
at intervals amused the readers of Mind since 1894 1 .
The case presented by Lewis Carroll is, that in a certain
barber s shop there are three attendants, Allen, Brown
and Carr, and at no time are they all out together,
i.e. Allen or Brown or Carr is always in (1). According
to this we may have A, B and C all in, only A and B in,
only A and C in, only B and C in (a), only A in, only B
in (b), or only C in ; and
(1) all times are times at
which one man is in. But
(2) if Allen is out Brown
is out (because Allen
has been ill and cannot
go out without Brown).
So (a) B and C are in and Allen is out, and
(b) B is in and Allen and Carr are out, are in-
1 See Mind for 1894, 1895, 1905.
Times
when A is
out
Times when
A and B are
out
Times when
A and C are
out
Times when
A and B and
C are out
54 A NEW
admissible cases (a) is barred by (2) because A is out
implies B is out (=A11 times
that Allen is out are times
that Brown is out). So Brown
and Carr cannot be in when ( ^^wh^ ) ( > ( c )
Allen is out. And Allen is
out = Allen and Brown are
both out (by (2)).
So in (b) Carr and Allen are out = Carr and Allen
and Brown are out (c), and by (1) All times are times
when A or B or C is in. So (b) as it stands is barred by
(2), and as amended to (c), is barred by (1).
The interest of this analysis of Lewis Carroll s instance
is that the whole case is subject to two conditions :
(1) That A or B or C must always be in;
(2) That A cannot be out without B ; and these may
conflict, and it is not easy at first sight to see exactly
how to combine the fulfilment of both conditions, and
exactly what denotational identities are justified by the
combination. As in all elliptical Hypothetical, when the
argument is expanded to a full statement the whole con
ditions need to be explicitly taken account of; and as has
been indicated, the whole argument in any case may be
completely set out in a series of propositions asserting
identity-in-diversity 1 .
In any concrete case in which it is possible to assert that :
If A is true, the truth of C follows,
If A is true, the truth of C does not follow,
it will be found on examination that either each Hypo
thetical is elliptical or A is itself contradictory.
1 An interesting solution of Lewis Carroll s Paradox " is offered by
Mr Bertrand Eussell in Mind for 1905 (pp. 400, 401). He says that he
AND ITS LOGICAL BEARINGS 55
E.g. in :
If this is Inference, the conclusion is contained in
the premisses ;
If this is Inference, the conclusion goes beyond the
premisses,
it is clear that the Hypotheticals are elliptical.
Disjunctive (or Alternative) Propositions are equi
valent to Hypotheticals or Conditionals, and may be
analysed in the same way.
E.g. Any topaz is pink or yellow
= Any topaz is pink or (if not pink) is yellow
= If any topaz is not pink, it is yellow.
They must come some other time than Saturday after
noon or I cannot receive them, may mean :
Saturday afternoon is a time when I shall be away
from home ;
considers the paradox to be " a good illustration of the principle that a
false proposition implies every proposition. Putting p for Carr is
out, q for Allen is out, and r for Brown is out, Lewis Carroll s
two Hypotheticals are :
(1) q implies r.
(2) p implies that q implies not-r.
Lewis Carroll supposes that q implies r and q implies not-r
are inconsistent, and hence infers that p must be false. But as a matter
of fact q implies r and q implies not-r must both be true if q is
false, and are by no means inconsistent. Thus the only inference from
Lewis Carroll s premisses (1) and (2) is that if p is true, q is false,
i.e. that if Carr is out, Allen is in. This is the complete solution of the
paradox."
But (i) if q implies r and q implies not-r are not inconsistent,
how do we know (on the above reasoning) that q is false ? (ii) We seem
to admit here both that the truth of q implies r (1), and also that the
falsity of q implies r. (iii) In (1) q implies r unconditionally, in (2) the
implication is conditional on the truth of p.
56 A NEW "LAW OF THOUGHT"
A time when I shall be away from home is a time
when I cannot receive visitors;
. . If they come on Saturday after
noon they come at a time
when I cannot receive visitors.
In regard to the interpretation of
Alternative Propositions, the one ques
tion in dispute regarding the alter
nants of the proposition is : Are they exclusive or un-
exclusive ? Though there has been great division of
opinion among logicians on this point, and though there
are Alternative Propositions such as : " He came in either
second or third," " We start either Wednesday or Thurs
day/ in which it is quite clear that while we cannot deny
both alternatives, neither can we assert both, yet there
can be no doubt that in such cases the exclusiveness of
the alternatives is due not to the form of proposition,
but to the nature of the cases in question. " It thus
seems that the only account which we can give of the
general or formal import of Alternatives that is to say
of the import which is common to every one of them
is that if we deny one alternative, we must affirm the
other. It should be observed that although terms used
as alternatives are not necessarily exclusive in extension
or denotation, they are exclusive in intension, in as far as
they are not tautologous (in which case the alternation
seems to vanish). Thus, in All his parishioners are
criminals or paupers, the alternatives, though not de-
notationally exclusive since the same parishioner may be
both criminal and pauper are necessarily exclusive in-
tensionally, since we cannot say that they are synonymous."
(Primer of Logic, p. 26.)
AND ITS LOGICAL BEARINGS 57
As regards the doctrine of Opposition, it is of par
ticular interest from my point of view, because on the
identity-in-diversity analysis it presents in a clear com
pact form the equation of the fivefold class-relation to
the traditional fourfold schedule of class-propositions.
Any two classes indicated by intension or by symbols
may have one of five extensional relations to each other.
Let us take R and Q to symbolise two classes. The
scheme may then be set out as follows :
/r^
(C
(1) (2) (3) (4) (5)
A. All R is Q = (l)or (2).
E. No R is Q = (5).
I. Some R is Q : (1) or (2) or (3) or (4).
O. Some R is not Q = (3) or (4) or (5).
Of these propositions A and E may both be false, but
they cannot both be true ; I and O may both be true,
but they cannot both be false ; of A and 0, and of E
and I, one is true and the other false. If A is true, I is
true, if I is false, A is false ; if E is true, is true ; if
is false, E is false. Reference to the diagrams makes
the whole scheme at once self-evident, and the diagrams
exhibit the identity or non-identity of Subject and Pre
dicate in every case. To take concrete examples of Contra
dictories (A and 0, E and I), we may say:
Either all beliefs are true ((1) or (2)), or some are
not true ((3) or (4) or (5)).
Either no men are perfectly happy ((5)), or some are
so ((1) or (2) or (3) or (4)).
Conditionals come under the same rule as Cate-
58
A NEW "LAW OF THOUGHT"
goricals in respect of Opposition, and any proposition of
the form : If A , then C, contraposits to // not-C, then not- A,
and may be contradicted by : the
truth of C does not follow from
the truth of A i.e. though A is
true C is doubtful (which exactly
expresses the relation between I
and A, O and E). To take a
concrete case :
(1) If money go before,
all ways lie open,
may be contradicted
by:
(2) Though money go be
fore, it does not follow that all ways lie
open. (Primer of Logic, p. 37.)
The Identity of Denotation in Diversity of Intension
analysis applies whether we are considering what is
asserted ; or the assertor the speaker or teacher who
starts from the whole (si>) ; or the hearer or learner, who
receives the assertion piecemeal, and finishes with the
whole.
The two different attitudes afford some explanation of
different theories of import of propositions, etc. It is
plain, e.g., that the account of judging according to which
it consists in putting two ideas together, and the Canons
of Syllogism : (1) Two terms agreeing with one and the
same third term agree with each other, (2) two terms
of which one agrees and the other does not agree with
one and the same third term, do not agree with each
other: are adapted to the point of view of hearer or
AND ITS LOGICAL BEARINGS 59
learner ; while the view of Brentano and Hillebrand, that
in any judgment S is P, one object fay is present to
the mind, is evidently appropriate to the point of view of
teacher or speaker the hearer has to build up the whole
he reaches it in the end, he does not start from it.
The speaker who has before him a whole composed of
parts of denotation e.g. a division or classification of
Triangle, thus :
Triangle
Equilateral Isosceles Seal
ene
or a clockmaker with a clock, or a schoolboy with a knife,
or an astronomer contemplating the planetary system, or
a General in a campaign with a plan of operations sketched
out in his mind all these can communicate to others
piecemeal as much as they wish of that which is cognised
by them, by means of propositions of the forms 8 is P,
S is not P. No doubt if they can set before their audience
the actual table of classification, the actual piece of
mechanism, the actual knife, a working model of a
planetary system and so on, the exposition is immensely
helped, or may even be rendered unnecessary. Of course
such helps are used in teaching wherever possible
blackboard-drawings, models, lantern slides, etc.
The difficulties of (1) impersonal and (2) elliptical
propositions, such as (1) It rains, (2) Fire! Wolf! are
very much mitigated if it is remembered that in every
case the speaker must start, not from the words of his
pronouncement, but from the matter of fact, not from the
expressed assertion, but from what is asserted. The corre-
60 A NEW "LAW OF THOUGHT"
spondence of the verbal assertion to that which it asserts
is often regarded as artificial, and the verbal expression is
called a " verbal device," or by some name that has an
equally opprobrious implication. Of course it cannot be
denied that if I have before me a red rose and assert :
This rose is red, my spoken assertion consists entirely of
words, and in particular contains a copula to which the
red rose seems to present nothing even remotely corre
spondent. Accordingly some logicians wish to reject the
copula, and some think it a verbal device in the very
worst sense, a useless, embarrassing and unjustified re
dundancy of expression. From my point of view, how
ever, all this is mistaken. If what is asserted in any
S is P is identity of denotation in diversity of intension,
then in asserting it we want not only the diverse terms
with their intensions and denotations, but also something
which indicates and conveys to the hearer the identity
of denotation between the terms, and this function the
copula is admirably fitted to perform. And the negative
copula is just as well suited to its particular task. In
fact the copula seems to me a very economical and
effective means of carrying out a delicate and indis
pensable part of the whole function of communicating by
means of speech. It is one of the many instances in
which men " have builded better than they knew."
In such propositions as : The round-square is non
existent, we cannot dispense with a one-ness of denotation
(extension) in the subject, because, without this, round
and square would have simply their intensional diversity
there would be no even hypothetical joining together of
round and square, no problem, no difficulty, no reason to
assert "non-existence," to raise any question. Since in
AND ITS LOGICAL BEARINGS 61
space, as known to us, roundness cannot be square, and
squareness cannot be round, the denotation to which the
two qualifications are assigned can " exist " only in the
universe (or region) of hypothesis or supposition. This
hypothetical combination is denied a place in the "uni
verse " of actual space.
Where intensions, attributes, are (1) incompatible, or
(2) inseparable, then the attempt (1) to combine them in
one subject, one denotation, as round-square, or (2) to
separate them, as equiangularity from equilaterality in a
triangle, is an attempt which can never be realised. We
may "suppose" the conjunction (or separation), we can
assert it, and trace its consequences, but that is all, as I
might suppose that I could fly like an eagle, swim like
a fish, and be stronger than an elephant, and deduce
various things that I could do on these suppositions.
In using impossible combinations as Subjects (or
Predicates) of Propositions, or a Subject which has a
Predicate which cannot co-inhere with it in one denota
tion, we are perhaps sometimes simply extending forms
and processes of language, appropriate in some cases,
to cases to which they are not primarily and directly
applicable.
Suppose I say:
No roses are blue rBJ KB j ,
this may be expressed also as :
There are no blue roses, or Blue roses = 0, or
Blue roses are non-existent.
All these seem admissible ways of expressing the matter
of fact indicated by the diagram MM Mn .
62 A NEW "LAW OF THOUGHT"
Apply this to the round-square case :
(1) No squares are round Cs j M-M .
There are no round-squares.
Round-squares = 0.
Round-squares are non-existent.
Even on this view, however, we have to postulate the
conjunction of round and square in a suppositional de
notation.
In speaking of Squares and Rounds in (1), we are
naturally understood to be referring to the region or
universe of space as known to as, by Rounds and Squares
we mean plane figures of a definite familiar shape.
But when we say Round-squares do not exist we
assign only our Predicate to that same
extended universe, and the Subject
which is round and square belongs to
a region of the merest, and we may even say wildest,
hypothesis a region entirely separate from the region in
which squares that are merely square, and rounds that
are simply round, have their " existence."
The round-squares are declared to be non
existent, they are identified (in denotation)
with something that is non-existent.
But that non-existence does not signify complete and
unmitigated non-existence, but only the absence of spatial
existence in talking about round-squares we are talking
about something, although it is an incoherent and un-
realisable something.
Whatever is thought of as having denotation, is thereby
thought of as having " being," existence of some sort, of
^uhat sort has to be fixed by intensional determination.
AND ITS LOGICAL BEARINGS 63
All the wheels that go to Croyland are shod with
silver, was a picturesque way of saying that
no ordinary work-a-day wheels did ever go to
Croyland. The wheels that went there were
shod with silver, that is to say, they belonged
to the same region as silver-shod wheels, viz.,
the region of imagination shod with india-rubber would
probably have been an even more far-fetched idea, at the
time when the saying was framed but fens have been
drained and roads constructed, no doubt the rubber tyres
of motors have found their way to the ruins of the ancient
abbey, and it might some day occur to a cranky millionaire
going in that direction to have the tyres of his wheels of
silver metal now so much less precious than formerly,
if only to illustrate the legend.
" Existence " of some sort we must attribute to every
thing of which we speak. But no particular kind of
existence can be implied by forms (such as S is P) which
propositions that deal both with the " real " world and
with mere fancy or fiction, have in common. The kind of
existence anything has is shown by the predicates we
can give it. Any proposition S is P that I assert, is an
entity, has some sort of existence. But the important
question is, What sort ? Is it true, for instance ? Well, this
must be tested by criteria. I cannot doubt (1) what is
self-evident, as that a whole is greater than its part ;
or (2) what is to me matter of direct experience, as
that that flash of lightning was followed by a clap of
thunder ; or (3) what is logically deduced from that which
is accepted as true, e.g. if twenty shillings are equal to
1, forty shillings are equal to 2 ; or (4) what is in
harmony with all which I accept, as that parallel lines do
64 A NEW "LAW OF THOUGHT"
not enclose a space ; or (5) that which is implied in what
is accepted as true e.g. that propositions of the form
S is P (8 is not P) (by the help of which alone the
Laws of Contradiction and Excluded Middle can be
asserted, and in which, in fact, most of our assertions,
whether self-evident or disputable, must be affirmed or
denied, supported or called in question) that proposi
tions of these forms are possible and valid.
As already insisted on, what S is P asserts, is that the
denotation of S, whatever it is, is the denotation of P.
If we start with an S which has not any denotation to
begin with, nothing can ever bestow that which is lacking.
But of what sort the denotation of S is, is settled by its
intension, and the intension of its predicate, and by context,
as in the old-fashioned school-room game in which one
person thinks of a thing, and another person has to try
and find out what it is by asking questions, to which the
answer must be Yes or No. The thing questioned about
is thought of by the questioner as being something, as
having some existence, but of what sort it is, in what
region it is, is revealed to him only when he knows what
predicates, what intension, can be assigned to it. As to
"Real" Existence, it is subject to as much ambiguity as
Identity is, and the ambiguity in this case is far more
difficult to clear up. How are we to define or describe
Reality ? What about the future, what about the past ?
The roses that have faded and fallen this year, and those
that will blossom next year? What about ideas of the
non-existent, which become operative in the world of
Time and Matter?
What are we to say of the ideal of an architect, painter,
poet, novelist, reformer, which guides the action of the man,
AND ITS LOGICAL BEARINGS 65
and leads to physically embodied results which may be
widely influential ? Or even of the delusions of a madman,
which are intensely " real " to him and may lead him to
realise the most disastrous actions in the everyday world
of space and time ? I remember reading a tragic story of
the Captain of a ship who on a voyage went out of his
mind. He fell under the delusion that various members
of the crew were conspiring to mutiny, and with marvellous
caution and cunning, induced first one and then another
of the officers and men to share his suspicions of some
of their number, and help him to secure them. He
succeeded so well that most of the crew (it was not a large
one) were overpowered one by one, and bound and made
helpless. Then, having laid all his plans with superhuman
ingenuity, with the strength and fury of a madman and
armed with weapons which he had secreted, he fell upon
the unfortunate victims, and the ship arrived in port
with the Captain a raging lunatic and most of the crew
murdered. What view are we to take of "reality" in
such a case ?
Or again of the perverted judgment of a dipsomaniac,
or of such a mother as the one in The Green Graves of
Balgowrie, which leads to cruel ill-treatment of the children
of the person so afflicted ; or the " fixed idea " of an old-
fashioned miser who leaves his unfortunate sons and
daughters half-starved and uneducated, to save a lawyer s
fee, draws his own will, with the result that it does not
carry out his intentions, and himself dies of starvation.
The finding of the North Pole by Dr Cook, and the
near approach to the South Pole by Lieutenant Shackleton
were, some months ago, on the same level of " reality " as
far as the general public knew, and neither achievement
J. 5
66 A NEW "LAW OF THOUGHT"
could have been even discussed or questioned, unless it
had been provisionally credited with "denotation,"
"existence," or "reality," in the region at least of
supposition. We identify the denotation of P with the
denotation of S just the same whether we merely suppose
$ is P, or question it, or affirm it, or consciously suspend
our judgment.
The reproach of unreality is, it would seem, only
pertinent when one kind of reality is mistakenly iden
tified with another kind.
I hope that I have in the foregoing pages made good
my undertaking, and shown that the substitution for the
old Law of Identity, A is A, of the principle that Every
Subject of Predication is an Identity (of Denotation) in
Diversity (of Intension), does provide the explicit recog
nition and justification of S is P, S is not P propositions
which Logic has hitherto needed but not had, and does
furnish Formal Logic with a real and obvious basis, and
an adequate constructive principle.
My scheme, I hold, elucidates (among other things)
the relations of Denotation (Extension) and Intension ;
the general Import of Categorical Propositions and their
relation to Conditionals, Hypotheticals and Alternatives ;
Immediate and Mediate Inference ; the relations to each
other, and to logical science, of the three Laws of Thought ;
the meaning and place of Quantification; the general
relation between Relative and Non-relative Propositions ;
the fundamental difference between the relation of Subject
and Predicate in Assertion and other relations which have
been sometimes confounded with it e.g. the relations of
AND ITS LOGICAL BEARINGS 67
(1) Subject (Substance) and Attribute, and (2) relations
of Classes; the difference between extensional one-ness,
and qualitative one-ness.
On my principles, as I think, the whole of Formal
Logic becomes a systematised and harmonious whole, with
a sound basis, an obvious and all-pervading principle, and
a simple and coherent structure.
52
68 A NEW "LAW OF THOUGHT"
FALLACIES.
It remains to say a word about Fallacies Fallacies
may be brought into a simple connexion with the Identity
in Diversity analysis of Categoricals by the consideration
that all fallacy consists in either identifying what is
distinct or distinguishing what is identical, so that we
get a primary division of Fallacies into (a) those of mis
taken distinction, which are Fallacies of Tautology, and
(b) those of mistaken identification, which are Fallacies
in which there is failure of continuity, and may be called
Fallacies of Discontinuity. The classification possible on
these lines is summed up in the following Table. (See
Primer of Logic.}
AND ITS LOGICAL BEARINGS
69
1
l : a
ill
1 t
H
91
HH
M C
-38.
<H 3
H.S
JS c rOS
J 8^2
g S " S, c
15 I ^1
eg g
HH >H g
5.3
If d
11
fil Ml
1" Jils j
l
flO
<1 9
3
* be
scfl
II
II
J
70 A NEW "LAW OF THOUGHT"
DEFINITIONS OF CERTAIN TERMS.
Proposition] _
A ,. > I use in the same sense, as denoting the
Assertion }
statements of Categorical, Inferential (Hypothetical), or
Alternative form, e.g. S is P ; If A is B, C is D ; C is D
or A is not B. In as far as the words in an assertion can
be considered apart from what is asserted I would use the
name verbal expression or, if more convenient, sentence.
By ivhat is asserted (or the assertion) I mean that
matter of fact, or belief, or state of the case, which the
assertion or proposition sets forth in words. The speaker
apprehends, or is conscious of, something which he conveys
by means of words to his audience ; that something is what
is asserted, e.g. he sees the door open and conveys the
fact in the assertion: "The door is open"; or he is aware
of feeling very chilly, and conveys the fact by asserting
the proposition : " I am chilled to the bone," or he believes
that "Twice two is four" and asserts it. The assertion
or proposition is of course a "verbal device" (though
a necessary and indeed an indispensable one) and in
particular the copula is sometimes accused of being a
device in a specially bad sense and of having nothing
corresponding to it in what is asserted. Such objections
seem beside the mark no one attempts to deny the
difference between what is asserted and the assertion of
it the point is : Does the assertion made by the speaker
convey to the hearer a knowledge of what the speaker
asserts ? If so, it answers its purpose fully and perfectly.
The copula in particular seems to me one of the most
admirable of human devices briefly and simply and
AND ITS LOGICAL BEARINGS 71
modestly helping to fulfil the function of conveying to
every hearer the information, the matter of fact, the
somewhat asserted, which any speaker desires to com
municate.
The distinction which I draw between the attitude
of speaker or teacher on the one hand, and hearer or
learner or seeker on the other, which is of great interest
and importance in Logic, seems to be specially enlightening
here.
Denotation (Extension) of a term means the sphere
of its application the things of which the term is the
name, the things to which the term applies.
Intension of a term means the properties of the things
to which the term applies it " may be used to indicate
in the most general way the implicational aspect of name "
(Keynes).
Connotation is that part of the Intension of a Term
which is set out in the Definition, and on account of
which the name is applicable.
Sameness (1) one-ness of denotation, identity;
(2) similarity, resemblance, likeness, qualitative one-ness
(Same-ness might conveniently be restricted to qualitative
one-ness).
Identity denotational one-ness, existential or exten-
sional unity antithetic to Distinctness, Otherness.
Compare " mistaken identity."
One-ness = antithetic to Difference =
(1) denotational one-ness, identity ;
(2) qualitative one-ness, same-ness.
Unity = (1) Identity, (2) Same-ness, (3) any system,
or whole made up of parts, and (4) the relation between
such parts.
72 A NEW "LAW OF THOUGHT"
Difference = (1) Distinctness or Otherness, such as the
difference between this new shilling and that new shilling
of the same minting ; (2) Diversity e.g. such a difference
as there is between justice and generosity, humanity and
mortality, or between an egg and the robin into which
it hatches; (3) (Differentia) the characteristics by which
any sub-Class (or species) is distinguished (differenced)
from the rest of its wider containing class (or Genus).
Diversity see Difference.
Distinctness see Difference.
Otherness see Difference. (Compare: give me another,
give me a different one.)
Similarity. There is similarity between two things
when they resemble each other produce impressions
which we call like ; and there is similarity between the
different phases of one thing in so far as it remains
unaltered. Similarity may be slight and partial, or so
great as to amount to what has been called indistinguishable
resemblance (= qualitative one-ness). Similarity (Resem
blance) is antithetic to Diversity.
The phrase exact similarity as equivalent to qualitative
one-ness is sometimes objected to on the ground that,
e.g., squareness or snow-whiteness or mortality have
extensional as well as intensional one-ness that the
mortality of Socrates is identical with the mortality of
Newton or even that extensional and intensional one-ness
in such cases coalesce or are indistinguishable This
appears to involve a monadistic existence of qualitative
or conceptual entities. I think that granting such entities
if we could assert of any one of them, S, that it is P, we
could not do this without postulating or implying that
it is an identity-in-diversity. For S is given as intension-
AND ITS LOGICAL BEARINGS 73
ally diverse from P that is, intensionally S is not P.
If, therefore, S is P, the one-ness indicated must be
something different from intensional one-ness. It must
be a one-ness of being to which the diverse intensions
are referred that is an extensional one-ness. Under
no other condition can one-ness of intensional diversity
be asserted. Even in the case of, e.g., obtuse-angled
triangularity, we cannot say that Triangularity is Obtuse-
angled-ness, but only that a triangle may be obtuse-
angled.
The relation of Identity-in-Diversity of Subject and
Predicate in Predication (1) must be distinguished from
(2) relation of Subject and Attribute (the subject
of Predication may be the attribute of a Subject
(Substance) e.g. Triangularity is a property of plane
figures).
(3) From relations of classes. Relation of S and P
in Predication is twofold only either (a) a relation of
identity or coincidence (of denotation or extension) or
(b) a relation of denotational exclusion, while the relations
possible between two classes are five.
(4) From the relation of successive similar percepts
to a conception or general notion which is implicated in
every general name. When Mill says that the import
of such propositions as : The colour I saw yesterday was
a white colour, The sensation I feel is one of tightness,
is to assert resemblance, he seems to confuse (1) with (4).
If his account of the import of these two propositions is
correct, then every proposition which has a general name
for Predicate is a proposition " asserting" Resemblance,
e.g. Rosa is fair-haired, This orange is ripe, Arsenic is
a cause of death.
74 A NEW " LAW OF THOUGHT"
(5) From the relation of members of a class (a) to
each other or (b) to the class.
Identical. Used to mean denotationally the same,
the same individual or thing.
System. By System is meant a group of two or more
related objects or items.
Logical Inference. When we can say that
If a proposition (or pair of propositions) A is true,
another proposition C is true ;
then C is a logical inference (eductive or deduc
tive) from A that is, the truth of C is implied in the
truth of A, we cannot affirm C and deny A, C follows
from A.
Logical Inference has to be distinguished from
Instinctive or " Psychological " Inference, and from what
may be called Tentative Inference, which may be (1) a
sudden apergu, a revelation, an intuition, or (2) an
Hypothesis or guess, deliberately framed for purposes of
investigation. " If we take the simplest possible case
of mediate inference or syllogism (Deduction) we have
S is M, M is P, entitling us to the inference S is P.
Here we have, no doubt, as the conclusion, an assertion
or proposition, which (qua assertion or proposition) differs
in some way from either of the premisses ; or from both
taken together. At the same time the content of the
assertion S is P is certainly in some way contained in
and justified by M is P and S is M. The exact connexion
seems to me to be as follows : When as audience or seeker
or pupil, we have learnt that M is P and S is M, and
grasped the contents of the two assertions and their
connexion, we find that we have really produced a con
struction in which the connotations or intensions S and M
AND ITS LOGICAL BEARINGS 75
and P are referred to one denotation (s,M,PJ. Having this
whole before us as an object of thought, or imagination,
it is apparent that it entitles us to say not only that
S is M and M is P, but also that S is P, and even further,
if we wish, that P is M and M is S and P is S, Not-P
is not M, not-M is not S, etc. The same might be shown
in a similar though even simpler way, of any case of
Immediate Inference (Eduction).
We may, of course, syllogise and otherwise infer in
an unintelligent mechanical way, using the accepted laws
of Formal Inference and of the Systems concerned as
mere rules of thumb/ guides blindly obeyed. But if
the acceptance of these rules can be justified, it must be
seen that they are valid. It is, e.g., the vision actual or
possible of a constructed whole, S that is M, M that is P,
fs.M.Pj that justifies to our mind the assertion of (among
other statements) S is P as an inference from 8 is M and
M is P.
I believe that a clear distinction between the contrasted
attitudes of hearer, reader, or learner on the one hand, and
speaker or teacher on the other hand, is very important
indeed for the theory of Logical Inference." (See Mind,
1908, pp. 533, 534.)
CAMBRIDGE : PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
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