A syllabus of logic : in which the views of Kant are generally adopted, and the laws of syllogism symbolically expressed

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SYLLABUS OF LOGIC,

IN WHICH

THE VIEWS OF KANT ARE GENERALLY ADOPTED, AND

THE LAWS OF SYLLOGISM SYMBOLICALLY

EXPRESSED.

BY

THOMAS SOLLY, ESQ.

LATE OP CAIUS COLLEGE, CAMBRIDGE.

J. & J. J. DEIGHTON;
J. W. PARKER, LONDON; & J. H. PARKER, OXFORD.

1839.

CAMBRIDGE:

PRINTED BY METCALFE AND PALMER, TRINITY STREET.

BC7/

S7

PREFACE.

THE object which I have proposed to myself
in writing this Treatise on Logic, is the com-
bination of a brief but complete account of
the Aristotelian system, with some of Kant's
philosophical views of the nature and divi-
sions of the science. With respect to the
first-mentioned part of my task, I have en-
deavoured to give a strictly h priori charac-
ter to the derivation of the fundamental laws
of syllogism, and the results of their combi-
nation in the various forms of reasoning.
This is attempted, partly by employing a
method in the derivation of these laws of a
rather more exhaustive character than that
which has usually been adopted by logicians,

11 PREFACE.

and partly by introducing mathematical ana-
lysis, for the exhibition of the symmetry in
their forms. Symbolical representation is also
employed in the second section of the Intro-
duction, for the purpose of explaining the
nature of abstract conceptions, and the method
of thinking some of them purely, or inde-
pendently of sense. .

I have derived from the deductio ad absurdum
the grounds for a division of the twenty-four
categorical syllogisms into eight systems of
three each. By means of this arrangement,
which to the best of my knowledge is en-
tirely new, the equality of the number of sound
moods in the first three figures, and several
other properties of categorical syllogism, may
be demonstrated a priori without the assist-
ance of symbolical reasoning. And although
these results admit of no immediate application
to practice, yet are they useful in giving the
student such firm hold upon the fundamental
principles of the science, that they will never
afterwards desert him. Besides, it should be
considered a sufficient merit that they add a
theoretical completeness to the science, which
it can never obtain when the symmetry of

PREFACE. Ill

its results is arrived at by empirical methods
alone.

There is another subject which has been
very much neglected in all the works on Logic
with which I am acquainted. Aristotle's Ana-
lytics is the only book in which I have dis-
covered any attempt at a theory of the
modality of syllogism, and on this particular
point he appears to me to have failed. I have
accordingly devoted an entire section to the
consideration of this subject, at the end of
the second book, in which I have endeavoured
to expose the fallacious nature of his reason-
ing. It must, moreover, be remembered, that
this subject is one of the most important
connected with Logic, for any misconception
respecting it may give birth to fallacies of a
very complicated nature, and extremely diffi-
cult to detect. Perhaps not a few of the
errors on the nature of the will might be
ultimately traced to this source.

I will now mention as briefly as may be
the extent to which I am indebted to Kant,
or rather his translators, for it is by their
means alone that I have had any access to
his works.

IV PREFACE.

The division of the science into Transcen-
dental and Universal, (for the latter of which
terms I have substituted ' Formal ' as being
more generally intelligible) is adopted from
the Criticism, and cannot perhaps be entirely
comprehended without a reference to that
work. The first part of the second section
in the Introduction, and a great part of the
third, are little more than an abridged para-
phrase of some portion of his Logic. There
are also several other places throughout the
Introduction which have at least originated
in some idea derived from his works, though
it is impossible individually to specify them
here.

In the first book, which contains the merely
formal analysis, I have adopted the opinion
of Kant respecting the distinct nature of
Categorical, Hypothetical, and Disjunctive
propositions. But as the latter part of the
first book is entirely opposed to his opinion
on the philosophical correctness of the dis-
tinction of figures, it will be necessary to
account for my having thought proper to
retain the scholastic theory on that subject.
Many other reasons might probably be urged

PREFACE. V

in favour of this view, but one alone is amply
convincing to myself. The deductio ad absur-
dum can only be applied to some syllogisms in
the first figure, by the introduction of those
moods of other figures which cannot be re-
duced to the first by conversion ; and the
indirect proof of these syllogisms would only
restore the original syllogisms in the first
figure. There are one or two other points
in which I have not followed Kant, and which
are immediately consequent upon the adoption
of the system;' of figures. These it is not
necessary to specify here ; for if the reader is
acquainted with the works of that philosopher,
he will readily detect them for himself; and if
he is not, he would not understand my expla-
nation.

Material fallacies, ambiguous terms, and
many other similar subjects which are usually
considered in works on this science, have found
no place in the following pages. All these
subjects have been already discussed by Dr.
Whately in such a very able and lucid manner,
that nothing more remains to be said about
them. And even had this not been the case,
my silence respecting them could not have

VI PREFACE.

been considered an omission, as they never
entered into the plan of my work.

I have attempted little more than an ana-
lysis of the formal laws of reasoning, and how
far this attempt has been attended with suc-
cess, the reader can now determine for himself.

Cambridge, May 13, 1839.

TABLE OF CONTENTS.

INTRODUCTION.

SECTION I — NATURE AND DIVISIONS OF THE SCIENCE.

Art.

Legality of the Understanding .. ... ... 1

Legality of the Reason ... ... ... 2

A priori Character of their Laws ... ... 3

Division of Logic into Transcendental and Uni-
versal or Formal ... ... ... ... 4

Proper Province of Logic ... 5

Logic not an Organum ... ... ... 6

Notice on Whately's Logic ... ... 7—9

SECTION II. — COGNITION.

Definition of Cognition ... ... ... 10

Distinct and Indistinct Cognition ... .. 11

Conceptions, how obtained ... ... ... 12

Their Matter and Sphere 13

Their Rank ... .14

Symbolical expression of the Nature of Abstract
Conceptions ... ... ... ... 15 — 17

Of certain pure Conceptions ... 18 — 20

Vlll CONTENTS.

SECTION III. — JUDGMENTS.

Art.

Judgments distinguished as to their ... ... 21

Quantity, into Universal, Particular, Singular 22

Quality, into Affirmative, Negative, Indefinite ... 23
Relation, into Categorical, Hypothetical, Disjunctive 24

Modality, into Problematical, Assertive, Necessary 25

Propositions ... ... ... ... .. 26

Synthetical and Analytical Judgments ... 27

How confounded ... 28

Definition 29

Several applicable to the same thing 30

BOOK I.

SECTION I. — CATEGORICAL PROPOSITIONS.

Their Form ... ... 31

Their Number 32

Law for Quantity of Predicates ... ... 33

SECTION II. — MUTUAL RELATION OF CATEGORICAL

PROPOSITIONS.

Opposition of three kinds — Contradictory, Con-
trary, Sub-contrary .,,., ... ... 34 — 36

Subalternation .. ... ... ... 37

Conversion ... .. ... ... ... 38

SECTION III. — HYPOTHETICAL AND DISJUNCTIVE PRO-
POSITIONS.

Form of Hypothetical Propositions 39

Admits no Variations ... •. ;,, _ ... .-. 40

Their Contradictory Categorical - - •kt't{ -••; 44
Their Nature distinct from that of Categorical

Propositions ... ... .. . v .'.; .', 5 ...f - 4-2

Disjunctive Propositions ... ... .... 4*3

Have no Contradictory ... ... .. 44

CONTENTS. IX

SECTION IV. — SYLLOGISM. Art.

Syllogism defined 45

Its Divisions ... .. ^< ... ... .. 46

Categorical Syllogism .. ... ... 47

Method to be adopted in determining its Laws .. 48

Its Elements ... 49

Division of the Enquiry ... .. -'^0 50

Law for the Middle Term ... l'c. j&i 51

Quality of the Premises ... ... 52

Forms of Premises from which a Conclusion for

the Reason is possible ... .. ... 53

Conclusions inexpressible in the legitimate Cate-
gorical forms ... ... ... .. 54

Illicit Processes ... .. ... ... 55

Quality of Conclusion... ... 56

Recapitulation of the Laws of Categorical Syllo-
gism 57

Secondary Laws ... ... ... ... 58

Table of Sound Moods 59

Division into Figures and complete Table of Cate-
gorical Syllogism ... ... ... ... 60

Laws peculiar to different Figures ... 61

Transformation of Figures and Table 62

Rejection of certain forms of Syllogism considered.

Table of Moods for each Figure ... ... 63

Hypothetical Syllogism .. 64

Disjunctive Syllogism ... ... ... 65

Dilemma 66

Other forms of Syllogism ... .. ... 67

Enthymeme ... ... ... ... ... 68

Sorites ^U^I 69

SECTION V. — THE DEDUCTIO AD ABSURDUM, ITS

NATURE 70

Affords the grounds for a Symmetrical Division •

of the 24 Syllogisms into eight Systems ... 71

X CONTENTS.

SECTION VI. — SYMBOLICAL EXPRESSION FOR THE

SYLLOGISTIC LAWS. Art>

Fundamental Equations ... ... ... 72

Symmetry of the Equations to the first three

Figures ... 73

Number of possible Solutions 74

Derivation of Secondary Laws 75

Truth of Premises 5>W, 76

BOOK II.

SECTION I. — LIMITATIONS OF THE FORM OF JUDG-
MENTS.

Conception of Substance never a Predicate ... 77

Law between Predicate and Copula ... ... 78

No formal proposition of Identity ... ... 79

Relative merits of the four Figures ... ... 80

True Conclusions from false Premises ... 81

Conclusions of the Reason ... ... ... 82

SECTION II. — MODALITY OF SYLLOGISM.

Does not affect the form of Conclusion ... 83

Proper and Consequential Modality distinguished 84

Proper and Derived Modality distinguished ... 85

No conclusion from Problematical Premises ... 86

The law of Derived Modality 87

Fallacies in Aristotle's Analytics arising from the
breach of this law 88 — 90

APPENDIX.

Examples for practice 155

Mathematical Note 160

Index to the principal Technical Terms ... 161

ERRATA.

Page 4, lines 5 and 7, from the bottom, for "represented," read

" imaged."

Page 40, line 1, for " cognitions," read " representation."

Page 89, lines 5, 6, in the table, dele "Subject Premiss."

Page 108, line 10 from bottom, insert "not" between "B" and

" being."

SYLLABUS OF LOGIC

INTRODUCTION.
SECTION I.

ON THE NATURE AND DIVISIONS OF THE SCIENCE.

(1.) THE subjection of the Understanding
to certain invariable laws is the first indis-
pensable condition to all knowledge. Kant
has shewn, in his Criticism on the pure Reason,
that the mind legislates for matter, or in
other words, that the laws we discover in
the external world derive their very possi-
bility from the laws of Mind. But let us for
an instant imagine the possibility of nature
following fixed laws as an object of our senses,
quite independently of any laws in our under-
standing, which for argument's sake we will
suppose to be without them. It is evident
that upon this hypothesis the Understand-
ing could never take cognizance of these

u

2 A SYLLABUS OF LOGIC.

laws of nature, nor even of the existence
of the objects they concerned. For these
objects can only become known to us by
means of certain laws, according to which
we can severally distinguish them from each
other : and how could we even distinguish
between ourselves and nature, or between the
'me* and the ' not me/ without some law in
the faculty by which objects are known? —
Every change in our representations might
either arise from a change in nature, or from
a change in our own indeterminate state.
But even could we separate ourselves from
nature, there would still remain the question,
' How can we conjoin any phenomena in a
synthesis for the purposes of knowledge, unless
we are conscious of something fixed and
determinate to which we can refer all the
phenomena to be conjoined, and also of some
law for the mode of their conjunction?' With-
out these requisites the cessation of the phe-
nomena and of their conjunction in the mind
must be simultaneous. And, moreover, even
while the phenomena lasted, no act of con-
joining them in any one instant of time could
ever be considered the same as, or be united
with a similar act of conjoining them in any
succcessive instant, unless we were to grant
something fixed in our consciousness, and

INTRODUCTION. O

quite independent of time, as a common
ground for the unity of these successive
syntheses. Hence, upon every hypothesis, the
legality of the knowing faculty is a necessary
condition to its use; and it now only remains to
extend this remark to the other logical faculty
— the Constructive Reason.

(2.) The Understanding has been defined
by Kant, as ' the faculty of rules ;' by Coleridge,
as ' the faculty of judging according to sense/
Its operation may be considered as two-fold,
accordingly as it dissects a representation by
analysis, _or conjoins several representations
in a synthesis. But there can be no act of
analysis without the consciousness of a prior
synthesis ; and hence it follows, that the latter
is an indispensable condition to every act of
the faculty. Now the Reason, considered as
to its logical use, (its transcendental use has
nothing to do with our present purpose, and
need not be considered here,) differs from
the Understanding in this : Whereas the
Understanding merely conjoins the diversity
of representations in synthesis, or dissects
them by analysis, in either case referring this
diversity to the unity of consciousness; the
Constructive Reason on the other hand con-
joins the very unities of these syntheses,
which unities are acts in the consciousness

B2

A SYLLABUS OF LOGIC.

or logical functions of the Understanding, and
entirely distinct from the diverse representa-
tions contained in them.* But as this kind
of reasoning is rather difficult to comprehend
in the abstract, we shall endeavour to ex-
plain our meaning by an example: — If any
person were to state the two propositions,
' all tyrants are unhappy/ and ' Nero was a
tyrant/ my Understanding alone could never
have enabled me to discover that ' Nero was
unhappy/ For this faculty unaided by the
Reason could only conjoin certain concep-
tions according to a rule. For instance, 1

* If we might be permitted to offer an illustration of so abstract
a subject, we would represent the synthesis of representations in
the Understanding by the arc of a circle connecting two points,
whose centre or unity of the syntheses may be of course at a
finite distance. But if another point be taken in the same right
line with the first two, we can only join all these points, and make
the centres of the two arcs correspond, by placing the common
centre at an infinite distance, where it will represent the Reason
conjoining the unities of the syntheses of the Understanding. It
must also be observed, that as on the one hand the centre of this
circle is infinitely distant, but its arc, which is a straight line joining
the three points, is finite and determinate ; so on the other hand
the rational act of conjoining the unities of the syntheses of the
Understanding transcends all possible experience, and cannot be
represented, but can only be thought, whereas the synthesis,
which results from the act of reason, is quite as easily repre-
sented as those from which it was originally derived. We are
perfectly aware of the manifold objections which might be urged
against this illustration, but we are inclined to think it may, in
some measure, assist the reader in comprehending the idea we
wish to convey.

INTRODUCTION.

may severally conjoin the conceptions of un-
happiness and Nero with the conception of
tyrant, as in each case belonging to the same
subject : but in order to conjoin the two
extreme parts of these judgments, namely,
Nero and unhappiness, I must simultaneously
reflect on my previous conjunction of each
of them with the conception of tyrants, and
therefore on the two acts of the Understand-
ing by which these judgments take place.
Before then I can arrive at any conclusion
from the two given propositions, I must pos-
sess some faculty by which I can conjoin the
very acts of the Understanding that are con-
tained in them, and this is what is meant by
the Constructive Reason.

The ease with which the Reason arrives at
a conclusion from judgments that have been
reduced to their most simple logical form,
and the perfect similarity of such a conclu-
sion when obtained to any immediate act of
the Understanding, are circumstances which
have rendered the peculiar function of the
Reason, in its logical use, extremely liable to
be overlooked. But a little attention to the
subject will make it evident, that the con-
junction of the acts of the Understanding
requires quite another faculty than that of the
Understanding itself, which merely conjoins

6 A SYLLABUS OF LOGIC.

the parts of possible experience according to
rules.*

Having given this short account of the
logical use of the Reason, we shall now pro-
ceed to shew that its functions are entirely
determined by laws. For as the conclusions
of the Reason are constructed with materials
derived from the use of the Understanding,
they can only concern the objects of the latter
faculty, and must in their intrinsic nature be
possible as its immediate acts. But this har-
mony of the two faculties can only be secured
by the subjection of the Reason to laws. Here
then at length we are justified in assuming,
that the whole use of our intellectual and ra-
tional faculties is based upon certain universal
laws, and that the Science which treats of them
is not a mere chimera of the imagination, but
founded on a reality which is the primary con-
dition to all other knowledge whatever.

* After the author had written his account of the logical use of
the Reason, he met with the following passage from Kant. It
refers, however, to the whole use of the faculty : —

" The Understanding may be a faculty of the unity of pheno-
mena.by means of rules ; Reason is thus the faculty of the unity
of the rules of the Understanding under principles. Reason,
therefore, never refers directly to experience, or to an object, but
to the Understanding, in order to give to the diverse cognitions
of this, unity a priori by means of conceptions, which may be
termed unity of Reason, and which is of quite another kind to
that which can be derived from the Understanding." — Anonymous
Translation of Kant's Criticism.

INTRODUCTION. 7

(3.) The laws of the use of the Understand-
ing must be a priori, and cannot be derived by
the method of induction from any number of
mental phenomena. For it has been shewn
that they are necessary to all use of the Un-
derstanding, (as it cannot even determine an
object without them,) and consequently to
experience itself; and we should therefore be
guilty of a glaring circle in our reasoning, if
we endeavoured to derive from experience
those laws which have previously been made
the very grounds of its possibility.*

But if the laws of the Understanding are

* The commonest example of this most fallacious attempt to
deduce the pure conceptions of the Understanding from expe-
rience, is to be found in the grounds that are sometimes given for
the causal relation of phenomena. For instance : Q. Why do I
expect a spark from the concussion of flint and steel? A. Because
in all past time such a concussion has been immediately succeeded
by a spark — Q. But why does this succession having taken
place in all past time, give me any right to expect that it will take
place again? A. Because any succession that has taken place
very often, has been afterwards observed to take place again. And
if we were to ask for a reason, why the uniformity of nature up
to the present moment should be considered as any guide for the
future, we should get precisely the same answer, or in other
words, the fact would be stated as the ground for itself. And
we might thus continue the question and answer to infinity, with-
out ever getting any nearer the point. The difficulties of this
subject were first brought to light by the subtilty of Hume. And
although he never succeeded in offering any satisfactory solution
of them, yet his " sceptical doubts" aroused the attention of Kant,
and became the occasion of the Criticism on Pure Reason, in which
the difficulty is satisfactorily explained.

8 A SYLLABUS OF LOGIC.

independent of all empirical matter, a fortiori
must this be the case with the formal laws of
the Reason, which simply regard the conjunc-
tion of the acts of the former faculty, and
are therefore removed one step farther from
their empirical contents. Hence it follows
that both Reason and Understanding are en-
tirely self-regulated, or subject to a priori
laws. And the determination of these metho-
dically, and their arrangement in a system,
is the business of Logic in the widest accep-
tation of the word.

Division of Logic.

(4.) Logic is divided by Kant into Trans-
cendental, and Universal or Formal.*

Transcendental Logic agrees with Formal
in excluding all consideration of particular
objects, but differs from it in admitting
that of the pure conception of an object in
general.

Formal Logic entirely excludes all consider-
ation of the objects thought, and merely re-

* The reader who has no acquaintance with the Kantian
system, must not be surprised if he does not clearly under-
stand the distinction between these branches of the science.
As, however, Formal Logic is exclusively the subject of the first
book, and a few considerations from Transcendental are intro-
duced in the second — the two books together may throw some
light upon the nature of the division in question.

INTRODUCTION. 9

gards the form of our judgments, and their
relations to each other.

It will be seen from the above definitions
that Transcendental Logic embraces a very
wide field, and includes within its limits much
that is metaphysical, and entirely foreign to
the other branch of the science.*

As Formal Logic alone is the proposed sub-
ject of this treatise, we should have been jus-
tified in taking leave of Transcendental Logic
here: but certain limitations are imposed upon
the very forms of the judgments of the Un-
derstanding by conceptions which are peculiar
to the last-mentioned science. And although
they may be justly assumed in every treatise
on Formal Logic, yet, as they are the origin of
several very peculiar results,f we thought the
notice of them in our second book might be
deemed not wholly superfluous.

(5.) This apology has been offered for an
intended reference to what properly lies out
of the sphere of the subject, on account of
the great injury which the sciences must in-

• For instance, the conceptions of substance, causality, &c.

•f Some of the limitations are alluded to in the commencement
of the first book. The absence of a formal proposition of
identity is one of them. For if we say A is B, we do not know
but that other things may be B also. Qne of the peculiar results is
the " possibility of a true conclusion from false premises in every
form of reasoning."

10 A SYLLABUS OF LOGIC.

variably incur whenever their boundaries are
not strictly recognised. Formal Logic has,
perhaps, suffered more on this score than
any other science : for, to say nothing of many
encroachments on the side of Transcendental
Logic, it has been a common custom with
logicians to introduce into their treatises a
great deal of matter derived entirely from
empirical psychology, which is a distinct
branch of knowledge. Thus, the choice and
arrangement of arguments, the best applica-
tion of particular syllogistic forms, and other
similar considerations, have frequently been
made the subject of rules which can only
be derived from practice, and should never
be mixed up with Logic, which is an a priori
science. Observations of this kind can never
form part of a system, but are merely an
aggregate of information, which is very useful
no doubt, but belongs more particularly to the
Art of Rhetoric. Knowledge of this kind may
properly be termed an art, and so far the old
logicians were at least consistent. But this
term can never be applied to Logic without an
absurdity as manifest as if we were to speak of
the Art of the Differential Calculus, or Conic
Sections.*

* We have thought it better to put a few observations on Dr.
Whately's Definitions of Logic in a notice by themselves at the
end of this section.

INTRODUCTION. 11

(6.) Logic is not an organum of the sciences;
for it does not contain a single reference to
any one branch of knowledge, and therefore
has no resting-place or fulcrum from which
to commence its investigations : it is how-
ever of the greatest use in testing the work
of another organum, by exposing all its results.
When a new law has been arrived at by Induc-
tion, Logic will determine all its consequences;
and should one of these prove at variance with
truth, it is certain that the induction is errone-
ous, and that new observations are necessary.

Notice on Whately's Logic.

(7.) As the distinction between the theory
of Logic and its application to practice has
been well explained in the very valuable trea-
tise by the Archbishop of Dublin, we cannot
help feeling in some measure surprised that
he has united these very different subjects in
the same definition. In the introduction to
his work, Dr. Whately defines Logic as the
"Science, and also the Art of Reasoning;"
and afterwards explains in a note, " that as
a science is conversant about knowledge only,
an art is the application of knowledge to
practice." Now it follows from the latter
unobjectionable definition, that the rules for
the art must be merely practical, and derived

12 A SYLLABUS OF LOGIC.

from the experience of " how we reason best/'
or in other words from empirical psychology ;
for if they were a priori, they would ipso
facto become laws of the science. Hence
we see that the rules of the art, which are
all empirical and depend on observation, dif-
fer from the laws of the science, which are
a priori and derived from the reason, not
only in their use and nature, but also in their
origin. The question however is not one of
fact, or even of theory, as Dr. Whately has
himself introduced the very distinction for
which we are contending : but the question
is, whether that arrangement can be consi-
dered conducive to the interests of science,
which combines under the same name two
branches of knowledge whose nature and origin
are equally distinct from each other.

(8.) There are two other passages in Dr.
Whately's work, which appear to the author
to contradict each other, and to convey equally
false notions of the nature of the science.
Soon after the definition upon which we have
just commented, Dr. Whately adds, " Its most
appropriate office however is, that of instituting
an analysis of the process of the mind in reason-
ing." Now it is not the process of the mind in
reasoning, but the principles with which that
process must accord, that is the proper object of

INTRODUCTION. 13

Logic.* We hardly ever reason to ourselves in
syllogism, but only in a manner which agrees
with its laws. For instance, if I think of the
future death of a hale and hearty man, in whom
there do not appear the slightest symptoms of
decay, I do not consider it a merely probable
circumstance, but morally certain. Now this
conclusion could only be logically deduced by
my virtually assuming the proposition that ' All
men must die/ and therefore that John being
a man must die also. But although the actual
law which connects humanity and mortality
must have been thought in my mind, when
I first considered it as morally certain that
John must die, yet the proposition that ' all
men must die' (which is, as it were, the ex-
ponent of the law) had, in all probability, never
occurred to me : perhaps, indeed, so far from
thinking of all mankind, I had not thought
of any other person than John. We see then
that the object of Logic is not the actual pro-
cess , of our reasoning, but rather the princi-
ples to which that process can always be re-
ferred.f And it must be remembered that

* Dr. Whately admits that Logic " investigates the principles
on which augmentation is conducted ; " and had his definition
terminated here, it would not have been liable to any objections.

f The simple state of the case is this : In the actual process
of our minds we generally connect the mere conceptions in a law,
without thinking of all that is contained in their sphere. But

14 A SYLLABUS OF LOGIC.

this is no unimportant point in the defini-
tion of Logic, as an erroneous conception
on this head is extremely calculated to bring
the science into disrepute. For, if any one
discovers that the process of reasoning in his
own mind is rarely, if ever, in the form of
a complete syllogism, he immediately leaves
the science with the conviction that it cer-
tainly does not contain the principles of all
reasoning, and probably not of any.

(8.) The other passage to which we would
allude, is to be found in a note near the com-
mencement of the second book, in which Dr.
Whately states, that " Logic is entirely conver-
sant about language."

This appears to us to be at variance with
the passage we have just quoted, in which the
most appropriate office of Logic is said to be
the analysis of the process of the mind in
reasoning. Now, that language follows cer-
tain laws, is unquestionably true. But whence
do these laws arise, unless from the necessity

in the logical development we consider the aggregate of indi-
viduals that come under the conception contained in the subject
of an universal proposition. As, for instance, in the above ex-
ample, if I look at John and think of his future death, I merely
connect the conception of death with that of humanity, to the
conditions of which I .see that h'e corresponds. But if I am
asked to express logically my reasons for expecting his death, I
immediately commence with ' All men die,' &c.

INTRODUCTION. 15

of language conforming itself to the mental
laws of which it is the exponent ? If language
did not receive its stamp from mind (the laws
of which we assume to be universally the same
in all countries and ages), we could never be
certain that we might not at some future
time meet with a people whose language re-
quired quite a new logic, and in that case
the science would lose its a priori character,
and rest on probability alone.

16

A SYLLABUS OF LOGIC.

SECTION IT.

ON COGNITION.

(10.) COGNITION is the generic name for all
representations that are sufficiently completed
for the logical use, and must therefore contain
a reference to our consciousness on the one
hand, and to an object on the other.

If a cognition contain sensation, it is em-
pirical ; if it does not, it is pure.

Cognition may be divided into the two spe-
cies— Intuition and Conception, each of which
may be either pure or empirical.

Intuition is a merely sensual* cognition of
an object.

Conception is an intellectual cognition, and
refers to many objects.

Perhaps we can exhibit these co-divisions
of cognition in a more intelligible manner by
means of the following scheme : —

COGNITION TABLE.

.

INTUITION.

CONCEPTION.

Pure

A straight line.

Substance.

Empirical..

A horse that is seen.

f Abstract conception
\ of a horse.

* Pure Intuitions concern the mere forms of the Sensitivity,
Space, and Time, and are therefore sensual without containing
sensation.

INTRODUCTION. 17

To investigate the ultimate original of cog-
nition, and shew by what process a merely
sensual representation obtains objective vali-
dity,* and thus becomes an intuition, is entirely
foreign to our present purpose, as it belongs to
the science of Metaphysics. We shall there-
fore only explain that formal property of our
cognition upon which its logical perfection
mainly depends, and conclude the section with
investigating those logical acts of the under-
standing by which it arrives at conceptions.

(11.) Cognition may be divided into distinct
and indistinct.

Indistinct cognition is that in which a diver-
sity is thought, that is not exhibited objectively
to the consciousness.

Distinct cognition, on the contrary, is that
in which no diversity is thought, which is not
exhibited objectively to the consciousness.

Let us take an example of an indistinct
intuition.

While walking on the sea-shore, I perceive
a large object in the offing, which, from its
general appearance, I immediately know to be
a ship. Now, from my previous knowledge of

* We give objective validity to a representation, when we
conceive it to arise from something independent of the peculiar
state or nature of our own mind, and therefore believe that other
people would view it as we do.

C

18 A SYLLABUS OF LOGIC.

the construction of a ship, I know that it must
have a rudder, shrouds, and a great variety of
tackle, which are not to be found in the image
as it appears to me, although I must really see
them : I therefore say that my intuition of the
ship is indistinct. If, on the other hand, I had
never seen a ship before, and had no conception
of the nature of the object presented to me,
I should say that my intuition of the object
was distinct; and this would arise from my
ignorance of what there was in it to distin-
guish. But if any person then asked me ' if
I could see what the object was ?' the question
' what ?' would superadd in my mind a con-
ception of this object, having some nature
peculiar to itself, or having been made for
some particular purpose, which was not ex-
hibited to me in my intuition, and I should
therefore answer that I could not see distinctly
what it was.

Hence we see that an intuition is indistinct,
when it suggests a conception of the diverse,
which it does not exhibit objectively to the
consciousness. And it is the relation of our
conception to the representation, and not the
latter alone, upon which distinctness must
depend. For if distinctness were defined as
the consciousness of the diversity of the repre-
sentation, without introducing any limitation

INTRODUCTION. 19

to this diversity, distinctness would be impos-
sible, as the diversity in any object is infinite,
and can never be exhausted.*

In the same manner we. say that a conception
is indistinct, when we attach to it the concep-
tion that an analysis of it is possible, or in
other words that it contains a diversity, and
yet cannot dissect, and severally represent to
our consciousnessf the parts of this diversity.
For instance, we may be said to have an in-
distinct conception of the nature of the human
mind, when we attach to it the conception
that some analysis of its different powers is
possible, and yet cannot exhibit these powers
separately to our consciousness.

Conceptions.

(12.) The logical acts by which the un-
derstanding arrives at conceptions are three,
— Comparison, Reflection, and Abstraction.

Comparison is that act of the understanding
by which several representations are referred
to the consciousness simultaneously.

* It is evident from this that distinctness is relative, not posi-
tive. And we have dwelt the more upon this, as it does not seem
to be very clearly laid down in Kant's Logic.

f If the remarks in the text are founded on truth, it will follow
that an increase of knowledge may sometimes render a previously
distinct conception indistinct, by destroying the equality be-
tween that which we know to be in it, and that of which we are
conscious.

C2

20 A SYLLABUS OF LOGIC.

Reflection is that act of the understanding
by which we determine what is common to
several representations, and consequently how
we may embrace them in one conception.

Abstraction is that act of the understanding
by which we separate all that is not common
to several representations.

For example, if I compare several men,
I reflect on their resemblance as bipeds that
walk erect, and abstract their differences as to
height, complexion, &c., and thus arrive at
a conception that answers to all of them.

Matter and Sphere of Conceptions.

Conceptions may be considered either as to
their matter, or as to their sphere.

(13.) The matter of a conception consists of
the various representations contained in it.

The sphere of a conception consists of the
things that come under it, or answer its condi-
tions.

As any addition to the matter of a con-
ception is a new condition to be answered,
and as a part of the original sphere of the
conception probably does not answer this new
condition ; it follows that the conception thus
altered by an addition to its matter will have
a less sphere than that of the original concep-
tion. Hence it is evident that the sphere and

INTRODUCTION. 21

matter of a conception vary inversely, and that
the more matter there is in a conception, the
less is its sphere. Let us take an example.
The conception of a horse has more matter
than the conception of an animal : for the
former conception must contain all that is
contained in the latter conception, and some-
thing more besides. But the sphere of the
conception of a horse is less than the sphere of
the conception of an animal ; as there are
many more things that answer the conditions
of the latter than the conditions of the former
conception.

Rank of Conceptions.

(l^,) Superior and inferior conceptions are
merely relative terms. The inferior conception
contains all the matter of the superior concep-
tion, and something more besides, but is itself
contained under the superior conception. For
example, the conception ' horse' is inferior to
that of ' animal/

A superior conception is also termed a genus,
and an inferior conception a species. There
can be a highest genus, but there cannot be
a lowest species ; for we may continue to
abstract matter from a conception till we have
left so little in it, that the next step would
take the conception away entirely. This con-

22 A SYLLABUS OF LOGIC.

ception must then be the highest, for it is
impossible to think of another superior to it,
under which it might rank. But we can never
arrive at a lowest species ; for though we con-
tinue to increase the matter of the conception,
and thereby lessen its sphere, we can never be
certain that we have exhausted all the possible
partial representations, which may make dis-
tinct species under this conception, and thus
render it a genus : for instance, I may gradually
abstract from my conception of* horse* through
the steps of ' quadruped/ 'animal/ and 'organic
being/ till at last I arrive at mere being,
from which I cannot abstract any thing more.
This conception must therefore be considered
as a highest genus. But if I take the con-
ception of 'horse/ and continue gradually to
increase its contents, I can never be certain
that there may not be other unknown dis-
tinctions in horses, which might constitute
the grounds of a division into still lower
species.

But although we can never arrive at a lowest
species (which must of course be a conception),
we can very easily complete the determination
of a cognition by fixing its individuality in
Time and Space.

INTRODUCTION. 23

Symbolical expression of the nature of Abstract
Conceptions.

(15.) A conception may be symbolically
represented as the common measure of the
representations from which it is derived. For
if we express the various parts of a repre-
sentation by the letters a, b, c, &c., and the
whole representation as their product,* we
may consider the following quantities as the
expressions for three different representations,

abode ft b c df, abode,

where it is evident that the conception ob-
tained in the manner already explained, will
be symbolically represented by the common
measure bed.

It frequently happens that we cannot image
to our minds the conceptions we obtain by
reflection and abstraction; for sometimes the
representations from which we wish to obtain
a conception, have not a single sensible partial
representation in common, but only a law con-
necting their parts. And although this law
may always be thought, yet it cannot be
imaged to the mind, as the image could only

* In the following symbolical exposition of the nature of ab-
stract conceptions, we have thought it better to retain the names
for the operations that are suggested by arithmetical algebra. The
terms, ' product,' « multiplication, '&c. must therefore be understood
as merely referring to the corresponding symbolical operations.

24 A SYLLABUS OF LOGIC.

represent a particular case. If, for instance,
I witness a great many cases of the rebound
of elastic bodies from a smooth surface, per-
haps the only thing these cases have in
common is the equality of the angles of in-
cidence and reflection, which cannot be imaged
to the mind, but can only be thought. For
an image could only give us the represen-
tation of the equality of two particular angles,
but the conception of the equality generally
could only be thought in the understanding.
This may be symbolically expressed as follows.
Let the representations be

b . 0 (ft), e . 0 (e), h . 0 (A).

Now, in arithmetical algebra these quantities
would have no common measure, but in sym-
bolical algebra we may separate the symbol of
affection 0, and consider that as the symbolical
representation of the conception of the law.
And it must be observed, that as a general law
may be thought and employed in reasoning,
but cannot be imaged (which is only possible
for a particular case), in like manner the func-
tion 0, which is a symbol of affection, may be
employed in analysis as a medium of reasoning,
but can never be itself interpreted in all its
generality in any subordinate science, as its
interpretation is only possible by its union
with some particular symbol of quantity.

INTRODUCTION. 25

(16.) It is impossible to obtain a correct
abstract conception of the organic productions
of nature. For let us take as an example the
conception of man. If we were to proceed
in accordance with the simple method already
proposed, and abstract every thing that is not
common to all men, we should obtain a con-
ception.in which no one could ever recognise
the least semblance of humanity : it must
possess neither arms, legs, eyes, ears, nose,
teeth, hair, or the power of speech, — for men
exist who are separately deprived of each of
these things. Our abstract conception of the
human form divine would therefore contain
a trunk without a single limb, and a scull
without a single feature.

Again, if we determine in our own minds
the greater part of a horse (all but the tail
for instance), in conformity with our general
conception of that animal, we may allow
variations in that one part between very
wide limits, and yet consider the whole
result as coming under our conception of
a horse. In the same manner we might
allow any other part of the horse to vary
between very wide limits, provided the rest
resembled the corresponding parts of horses
generally, or of any well made horse in par-
ticular : but were we to introduce all these

26 A SYLLABUS OF LOGIC.

variations simultaneously, the result would
be a monster, to which our conception of
' horse ' would be no longer applicable. Hence
we see that the several parts of our conception
must be considered as functions of other parts,
and all mutually dependent on one another.
And retaining the symbolical language we
have already employed, where u, v, w, &c.
express the partial variable representations,
we might write

abstract conception = F(n,v, w, x, y, *)

where 0 (u, vf w, x9 y, &) = 0 ;
and every set of values that answers the con-
ditions of the equation will correspond to an
individual of the species.

(17.) A friend once suggested to the author
that there was a great analogy between an
abstract conception and an enveloping curve;
and after a little consideration he discovered
that the symbolical expression for the envelope
is not a mere illustration, but a strictly correct
representation of the nature of an abstract
conception.

Instead of representing each individual of a
species by a function of a particular set of values
for u, v, w, &c., let us consider the whole locus*

* As the original (i.e. geometrical) signification of the word
'locus' is necessarily confined to functions containing not more
than three variables, it may be as well to remark, that we have
adopted it here in the strictly analogous sense of ' interpretation
of an equation.' - .

INTRODUCTION. 27

of an equation as but one individual. If then
this equation contains a constant (a)* to which
we give continuously successive values, we
shall obtain a family of these equations to
individuals whose loci will intersect each
other (in a locus of one dimension less than
themselves), and the envelope or locus of
these intersections will be a correct symboli-
cal representation of the conception abstracted
from them all. For we may consider an ab-
stract conception as one which contains all
the points of resemblance between any in-
dividual, and those immediately preceding and
succeeding it.

We therefore discover a perfect analogy
between abstract conceptions and enveloping
surfaces or curves : and the only difference
in their respective symbolical representations
consists in the number of variables which we
are justified in introducing in either case.
For the equations whose interpretations relate
to space cannot of course have more variables

* It may be as well to remind the reader, that though the con-
stant (a) is a symbol of quantity in the function considered simply
symbolically, yet it becomes a symbol of affection (in fact apart of
the function) in the interpretation : the symbol (a) is therefore
to be considered as a part of the law between the members of the
individual ; just in the same manner as a constant in the equation
to a surface or curve is only interpreted through its affecting the
variables. •

23 A SYLLABUS OF LOGIC.

than space has dimensions, and are therefore
confined to three : whereas the variables intro-
duced in our abstract conceptions concern
many other things besides space, and are
therefore unlimited in number. We subjoin
the symbolical expression for the enveloping
or abstract conception.

Let F(w, v, w, x, y, *,- a) represent the locus
of an individual : then the locus of the en-
veloping conception will be expressed by the
equations

F(u, v,w, x, y, z,a) = 0 ... (1)

da

And if a is eliminated from (1) and (2) we
shall have an equation of the form

$ (u, v, w, x, y, z) = 0,

which represents the locus of the abstract
conception.

(18.) Before we conclude this section, it may
be as well to add a few remarks on the method
of thinking certain conceptions purely or inde-
pendently of sense.

The idea of a limit, which is the basis of the
differential calculus, supplies the only possible
means of thinking the conceptions Substance
and Causality purely, i. e. independently of all
empirical matter. For in order to think the

INTRODUCTION. 29

conception of Substance at all, I must imagine
certain phenomena, and then conceive the ob-
jective existence of a substratum for their
support. But this alone is insufficient, for
although the conception of Substance is ob-
tained, yet it is not pure as long as it con-
tains any sensation or phenomenon. I must,
therefore, diminish my sensation till it approx-
imate to zero. This, however, must not be
effected by lessening its extension in space,
for in that case the object itself would vanish,
as its identity and objective existence can
only be determined in that form of sensation.
Neither, on the other hand, may the sensation
be diminished as to its intensity : for although
this operation would only affect the conception
of its being an object for me, and not of its
being an object at all, yet this mode of dimi-
nution would introduce the conception of suc-
cessive states, and therefore other matter
besides the bare conception of Substance.

The only way that remains is that of dimi-
nishing the time in which the phenomenon is
viewed, till we arrive at the limit. If -then
we consider time as the independent variable,
and the phenomenon as a function of it, the
pure conception of Substance will be properly
represented as the limiting ratio between the
phenomenon and the time in which it is viewed.

30 A SYLLABUS OF LOGIC.

Let S = the pure conception of Substance,
P = phenomenon, and T = time in which the
phenomenon is viewed, we shall then have
s dP
'- dT

And as the conception of Substance contains
permanence, it follows that S is constant. We
may therefore consider

P = S T.

as the equation to a phenomenon to which
the pure conception of Substance alone is
applicable.

(19.) Very nearly the same reasoning will
apply to the conception of Causality. This
can only be imaged to the mind in a suc-
cession of phenomena occupying finite por-
tions of time. But while our conception con-
tains sensible phenomena, it is not pure : we
therefore diminish the conceived duration of
the times in which they exist, till they approx-
imate to zero. When they arrive at this limit
the conception becomes pure, for the pheno-
mena will only occupy successive instants of
time; and must therefore vanish. Hence we
see that the two cases resemble each other, in-
asmuch as they are each represented by the
limiting ratio of the phenomenon to the variable
time. They differ however in this : whereas
the phenomenon from which the conception

INTRODUCTION. 31

of Substance alone was derived did not vary
in time, but bore a constant relation to it ;
in the latter case the phenomenon will vary,
and consequently be some unknown function
of time. If then the pure conception of
Causality = C, and the other symbols retain
their former signification, we shall have
P = $ (T, S),

(20.) If the reader will examine his con-

ception of -^ and also of one of these cate-
dx

gories, the analogy between them will become

immediately evident. The ratio -^ cannot

dx

be imaged to our minds (because we cannot
image the ratio between two nothings), but it
can be thought. For we can very well under-
stand that there may be such a relation between
two quantities, as that the nearer they approx-
imate to zero, the nearer does their ratio ap-
proximate to equality with some given ratio,
and that the two approximations are com-
pleted simultaneously. Exactly in the same
manner the pure conception of Substance can-
not be imaged;* for we cannot image that,

* " The invisible was assumed as the supporter of the apparent,
rtav (j>aivo/j.£V(av — as their substance, a term which in any other
interpretation expresses only the striving of the imaginative power
under conditions that involve the necessity of its frustration."—
Coleridge's Friend.

32 A SYLLABUS OF LOGIC.

which, considered as to space and time, and
the sensation contained in them, is equal to
zero. But it may be thought : for we can
very well understand that there should be in
our minds a law or a priori relation between
phenomena and the time in which they are
viewed, and that this law should retain its
signification even when the phenomena and
their time both = 0. It is, however, quite
beyond the powers of symbolical representa-
tion to give the nature of these conceptions
themselves. Nothing more is here intended,
than to express, by an analogy to the mathe-
matical idea of a limit, the only method by
which these conceptions can be thought pure,
and independent of the possible experience to
which they are necessary.

33

SECTION III.

JUDGMENTS.

(21.) A judgment is that act of the under-
standing by which it determines how certain
representations may be conjoined in the con-
sciousness according to some rule of its own.

Kant has divided the consideration of judg-
ments as to their form into the four points
of Quantity, Quality, Relation, and Modality;
and each of these again into three subdivisions
or moments,* from which he derives the twelve
categories, or pure conceptions of the under-
standing.

Quantity.

(22.) The three moments of Quantity are
the Universal, the Particular, and the Singular.
These determine the quantity of the subject of
a judgment, and refer respectively to the whole
sphere of a conception, to a part of the sphere

* Our choice of the word ' moment,' was entirely determined by
its having been employed in this signification by the anonymous
Translator of the Criticism.

D

34 A SYLLABUS OF LOGIC.

of a conception, or to an individual ; e. g. All
men are mortal ; some men are mortal ; John is
mortal.

In Transcendental Logic we distinguish be-
tween the Universal and Singular, because the
subject of the former is in respect of its formal
quantity unlimited, whereas the subject of the
latter is completely determined. If, for in-
stance, I speak of ' all men/ there is nothing
in my bare conception of man which can put
any limit to the number of individuals that
may come under it, or answer its conditions;
and the quantum of the sphere of the con-
ception is therefore formally indeterminate :
whereas, if I speak of an individual, John, my
cognition contains all that is found in the
conception of man, and also the possibility of
his complete determination in space and time,
for on that his identity depends.* But For-

* That our conception of identity requires us to think the pos-
sibility of determining the object in a particular space at a parti-
cular time, is evident from the consideration that there is no limit
to the number of individuals who may exactly resemble each other
in all other respects. Now the particularity of space and time is
determined by a reference to ourselves ; for we can only think of
a particular time as being at such a period before or after the pre-
sent time, or that in which we are thinking, and of a particular
space as being at a certain distance from the present space, or that
in which we are now sensible : hence, the identity of all external
objects ultimately depends on a reference to the identity of the
thinking subject. But if I endeavour to represent my own iden-
tity, I must have recourse to the identity of external objects;

INTRODUCTION. 35

mal Logic takes no cognizance of the object
thought, and consequently does not recognise
the distinction : for in speaking of an indi-
vidual man, I speak of all that answers to my
cognition, and that is all that Formal Logic
can require for an Universal. This science
therefore acknowledges only the two moments,
Universal and Particular.

Quality.

(23.) The three moments of Quality are — the
Affirmative, the Negative, and the Indefinite.

The first, as its name implies, affirms the
predicate of the subject ; the second denies
it ; but the third affects the matter of a con-
ception which it limits by entirely excluding it
from some particular conception. Thus, ' not
A' is indefinite, as it applies to any thing that
lies out of the sphere of A. As every thing

e. g. I may represent myself to myself as the person who was
in a particular room at a certain time : hence it follows, that I
think of the identity of other objects by means of a reference to
myself, and of myself by means of a reference to other objects.
There is, then, between the 'me* and the 'not me' a kind of
polarity, which existing in space and time according to some un-
discoverable law of the consciousness, gives rise to the conception
of identity. This may be illustrated by representing space and time
as the axes of co-ordinates, and our own mind as the origin from
whence any values of x and y may be measured for the determi-
nation of an object. Or if we think of our own identity by means
of that of another object, we must place the origin at the object,
and consider our own mind as the point to be determined.

D 2

36 A SYLLABUS OF LOGIC.

must be either A or ' not A,' it is the same
thing whether ' not A* be affirmed or A be
denied of any subject; and for this reason
the Indefinite is not considered in Formal
Logic, as the Negative moment can always
take its place.

Relation.

(24.) The three moments of Relation are —
the Categorical, the Hypothetical, and the
Disjunctive.

The first of these considers the relation of
cognitions to the same substratum or sub-
stance. The second considers the dependence
of one cognition as consequent upon another
as antecedent. The third considers a recipro-
city of relation between cognitions in such a
manner that any one of them can be known
by the determination of all the rest. The
most general form of a Categorical judgment
would be ' A is B,' or ' No A is B,' in which
the conception B is asserted to belong, or not
to belong, to the subject A. An Hypotheti-
cal is of the form t If A is B, C is I)/ in which
the judgment that f C is D ' is made to depend
as consequence upon the judgment that ' A is
B' as antecedent. Either ' A is B, or ' C is D,'
is a Disjunctive judgment, and (the determina-
tion of either part would determine the other.

INTRODUCTION. 37

On these three points of a judgment we
touch but lightly here, because they affect
the laws of Formal Logic, and are therefore
more fully considered in another place.

Modality.

(25). Modality concerns the manner in
which we think a judgment with regard to its
truth, and is divided into the three moments
of Problematical, Assertive, and Necessary.
The first degree of holding true is that of
problematical, which merely signifies the pos-
sibility of the judgment, inasmuch as it does
not violate any of the universal laws of think-
ing.

The second degree accords to a judgment
the agreement with the matter of the senses,
as well as with their necessary forms. The third
degree makes some a priori law of thinking the
matter of the judgment.

A problematical judgment merely implies a
formal, but not a material possibility. For
instance, the conception of an elderly lady
riding through the air on a broomstick, is
formally possible, as it involves no contradic-
tion, but is not generally considered materially
possible.

A judgment may be assertive without its
contents having come immediately under the

38 A SYLLABUS OF LOGIC.

cognizance of the person who forms it. No-
thing more is meant by the above definition,
than that assertive judgments must have
sufficient and empirical grounds for their
truth.

The third moment, Necessity, is applicable to
any a priori law, whether rational, intellectual,
or intuitive. For instance, that A is C, if A is
B, and B is C, is a necessary truth of reason.
That no change in phenomena can be self-
originated, or that every effect must have its
cause, is a necessary truth of the understand-
ing. That the straight line is the shortest
between two given points, is a necessary truth
of intuition.

(26.) Propositions are judgments whose mo-
dality is either assertive or necessary: for a
problematical judgment, so far as its modality
alone is concerned,* merely implies that a cer-
tain judgment is not necessarily false, but bears
no positive testimony to its truth. And we
could not therefore say with propriety that
any thing is proposed in a problematical judg-

* It was necessary to introduce this restrictive clause ; for
problematical judgments, considered independently of their moda-
lity, are generally intended to imply some anticipation of a truth,
and their main use is as a step to the assertive. For instance,
there may be such a person as Mr. Stiggins in New York, but no
man would ever make such a judgment unless he had some
grounds for believing it true.

INTRODUCTION. 39

ment, as that term has evidently a positive
signification.

Propositions alone can enter into the consi-
deration of Formal Logic ; for the laws of
Reason require something determinate for the
matter of the judgments whose combinations
they regulate, and are not conversant about
mere possibilities which have only a negative
value in relation to truth. This science more-
over regards all propositions as assertive : for it
cannot admit the distinction between assertive
and necessary modality, as this is determined
by the matter of judgments, and is quite inde-
pendent of their form.

Synthetical and Analytical Judgments.

(27.) Judgments may be divided, as to their
matter, into Synthetical and Analytical.

Synthetical judgments are those whose
predicates are not contained in the conception
of their subjects.

Analytical judgments, on the other hand,
predicate of their subjects something that is
already contained in them, or, in other words,
their predicate is a superior conception to
their subject.

Hence we see that synthesis, or the principle
of synthetical judgments is the conjunction of

40 A SYLLABUS OF LOGIC.

two different cognitions, neither of which is
contained in the other.

Analysis, or the principle of analytical judg-
ments, is the dissection of a conception into its
partial representations.

As the matter contained in the conceptions
of the subject and predicate determine the
nature of the judgment in respect of the above
division, it follows that the same proposition
may express an analytical judgment to one
person, and a synthetical judgment to another.
The proposition ' all members of the Univer-
sity are members of some particular college/ is
an analytical judgment to a person who is well
acquainted with the constitution of the Univer-
sity, and whose conception of a member of the
University already contains the conception of
his belonging to some particular college. But
the same proposition is synthetical to a person
who is entirely ignorant of that constitution,
and is therefore unacquainted with the neces-
sary conditions to being a member of the
University.

(28.) But the most remarkable instance of
confusion between synthesis and analysis arises
from the circumstance, that what is analysis
considered objectively, is very frequently a
synthesis if considered subjectively. Thus
the process by which a chemist examines any

INTRODUCTION. 41

substance for the purpose of discovering its
qualities, is rightly named analysis when re-
ferred to the object examined, but is synthesis
when considered in relation to himself. For
instance, if a person, who is perfectly ignorant
of the constituent elements of water, separates
the two gases by means of some chemical
process, the investigation is correctly styled
analysis, if considered in reference to the ob-
ject ( water/ as he has disjoined its component
parts : but the result in the operator's own
mind is a synthesis ; for his previous concep-
tion of water contained nothing more than
fluidity, the absence of all colour and taste,
and perhaps the property of dissolving a great
many salts. But by the analysis of the object
he has increased this conception, by adding
to it, that water is composed of oxygen and
hydrogen ; and consequently his mental act is
a synthesis.

Definition.

(29.) Definition is a judgment which deter-
mines all the partial representations contained
in any conception, and is therefore equivalent
to a completed analysis.

There is a specious resemblance of definition
which is grounded on a synthesis, and has
really no right to the name.

42 A SYLLABUS OF LOGIC.

Definition, in the strict acceptation of the
word, regards only the matter of a conception,
and not its sphere. But sometimes the term is
improperly applied to the exposition of any
conception whose sphere is the same with
that of the conception to be defined. Let us
suppose that a conception a b, or one which
contains the partial representations a and by
is connected by a certain law with the con-
ception c d, and in such a manner that the
sphere of a b is identical with the sphere of
c d. In this case it would not be a correct
definition of the conception a b, to state that it
contained the partial representations c and d ;
though such a judgment would lead us to
the same individuals as if the correct defi-
nition had been given. Such a judgment
would be a true synthetical judgment, but
not a definition.

(30.) It follows from this, that two persons
may give different definitions of the same
thing, and yet both of them be equally correct :
for though the things be the same in each
case, yet the conceptions by which each indivi-
dual may recognise these things may be very
different, and perhaps have hardly a single
point in common. Let us take as an example
the conception of water. The common con-
ception contains little more than that water

INTRODUCTION. 43

is a perfectly tasteless and colourless liquid.
Now these properties are connected by a law
of nature with a combination of oxygen and
hydrogen, in certain proportions and under
certain circumstances; and a definition which
contained either account of water would equally
refer to the same thing, but to very different
conceptions. The rustic would probably not
have much faith in the chemist's proof that
a certain liquid was water by his exhibiting
the gases in a separate state ; neither would the
chemist have much faith in the rustic's defini-
tion : and yet the definition would be correct
in each case, as it would express that concep-
tion by which each was in the habit of recog-
nising the thing itself. The example in Mr.
Newman's Logic is a very good one. He
observes that it would be extremely incorrect
to define man as a cooking animal, although it
is highly probable that man, and man only,
answers to that description. But as we do not
recognise men by this peculiarity in their
nature — in short, as it does not form part of
our conception of humanity, such a definition
would only mark out the sphere, and by no
means determine the matter of our conception
of man. When two conceptions are united by
some law which is immediately and univer-
sally recognised, it becomes a matter of indif-

44 A SYLLABUS OF LOGIC.

ference from which of these conceptions we
derive our definition. For instance, the com-
mon definition of a triangle is ' a figure of
three sides/ whereas in strictness it should be
'a figure of three angles/ The usual defini-
tion is in reality a pure synthetical judgment
of intuition.

FORMAL LOGIC.

BOOK I.
SECTION I.

PROPOSITION S.

(30.) Propositions are of three kinds —
Categorical, Hypothetical, and Disjunctive.

Form of Categorical Propositions.

Categorical propositions consist of two terms
and a copula, of which the terms designate the
matter, and the copula their relation to each
other. Their most general form may be re-
presented by the proposition ' A is B,' in which
the terms A and B are connected by the copula
* is/ Particular examples are, ' horses are
animals/ 'men are not monkies/

Many categorical propositions do not come
immediately under this simple form, but are
easily reduced to it by a periphrasis ; e. g.
the proposition ' I took a ride this morning*

46 A SYLLABUS OF LOGIC.

maj be put under the form, ' I am a person
who took a ride this morning/

The terms are susceptible of no other* for-
mal variations than the moments of quantity,
which determine to how much of the sphere
of a conception the proposition refers.

There can be only two such formal varia-
tions— the whole, and less than the whole, or
part. For we evidently cannot speak of more
than the whole, as the idea involves an absur-
dity : neither can we make any distinction
between different quantities that are less than
the whole, without introducing the considera-
tion of the matter of a conception as well as
of its form. It follows, therefore, that ' the
whole' and 'less than the whole' are the only
variations of quantity which Formal Logic can
recognise : e. g. we may speak of ' all albinos/
i. e. the whole sphere of the conception ' albino,'

* It is sometimes convenient to consider the matter of a pro-
position with respect to its quality as well as quantity, and to
speak of ' not A/ ' not B,' &c. : in this case, the * not A' is
called the external sphere of A. The attribute of quality is,
however, but rarely accorded to the terms : for if the predicate
is an external sphere, the form of the proposition is precisely
the same as if the negation had been applied to the copula.
Thus, A is a t not B ' is exactly equivalent to A 'is not ' B.
And an external sphere is but rarely introduced in the subject
for other reasons, as well as on account of its extreme awkward-
ness. For instance, we do not say, ' not men' are ' not English-
men,' but 'all Englishmen are men,' though the meaning of the
two propositions is precisely the same.

PROPOSITIONS. 47

or of ' some albinos/ i. e. a part of the sphere
of that conception.

When the whole of either term is compared
with the other, it is said to be distributed ;
when a part only is so compared, it is said to
be undistributed. For instance, in the pro-
position ( All A is B,' the term A is distri-
buted ; but in the proposition ( Some A is B,'
it is undistributed.

The only formal variations of which the
copula is susceptible, are two moments of
quality, affirmation, and negation. For if we
compare any two terms A and B in a cate-
gorical proposition, we can only affirm or deny
the one of the other : e. g. A is B, A is not
B. In the first example the quality is said
to be affirmative, in the latter negative.

Hence, the two terms and the copula, which
constitute the three elements of a categorical
proposition, severally admit of two gradations
or variations in form : and if there were no
law for their limitation, the number of possible
combinations of these elements would = 23 = 8.
These eight combinations would be,

* All A is Some B. All A is All B.

* Some A is Some B. Some A is All B.

* All A is not All B. All A is not Some B.

* Some A is not All B. Some A is not Some B.

48 A SYLLABUS OF LOGIC.

(32.) But we find by experience, that of
these eight forms of categorical propositions,
only four are ever introduced in practice :
for the quantity of the term that is placed
last in the general categorical form (Le. A is B)
is entirely determined by the quality of the
copula,* The variations of the copula and of
this latter term must therefore be taken toge-
ther, and the whole number of combinations
for categorical propositions will — 22 = 4.

As the first and lastf terms of a categorical
proposition do not bear precisely the same
relation to the copula, independently of their
mere position, they are distinguished respec-
tively by the names of subject and predicate:

* The law for the dependence of the quantity of th
upon the quality of the copula, can only be assumed in Formal
Logic : for the a priori grounds upon which it rests are to be
found in the consideration of one of the categories or conceptions
of the understanding, and therefore belong to the Transcendental
branch of the science. This subject will be considered in a
future section, but a cursory view of it may not be wholly out of
place here. The category of substance can never be introduced
in the predicate of any proposition, and this term must therefore
be a conception. But the quantum of the sphere of a conception
is entirely indeterminate, as there is no formal limit to the number
of individuals that may answer its conditions. It is therefore
impossible to compare the limits of the sphere of the predicate
with those of the subject, as those of the former are unknown.
The subject then can only be placed wholly out of, or wholly in,
the sphere of the predicate, and the latter will be distributed or
undistributed accordingly.

f i. e. the first and last in the simplest form of a categorical
proposition, e.g. ' All A is B.'

PROPOSITIONS. 49

and the propriety of introducing such a dis-
tinction into Formal Logic depends entirely
on this formal difference of relation, and not
at all on the real nature of the distinction,
which is a subject of Transcendental Logic,
and cannot therefore be considered here.
Neither does it at all depend on the order in
which the terms are arranged ; * for this is
frequently reversed in some languages, and
occasionally even in English.

Of the eight combinations given in the last
page, those marked by an asterisk are the four
legitimate Categorical propositions. As, how-
ever, the quantity of the predicate is a known
function of the quality of the copula, it is
never expressed, but always understood ; the
four propositions will therefore assume the
following form :

All A is B. No A is B.

Some A is B. Some A is not B ;

where it must be observed that the proposition
' No A is B ' is equivalent to the proposition

* This order is seldom introduced in English, except in poetry,
emphatic diction, or where the subject is a sentence. The fol-
lowing are examples of these three cases :

" Oh, many are the Poets that are sown

By Nature \-Wordsworth.

i. e. the poets are many that, &c. "Brave indeed is that man
who," &c. 2. e. that man who, &c. is brave indeed. " It is very
easy to say a great deal in a letter that cannot be hinted at in a
personal interview," i. e. to say a great deal in a letter, &c. is very
easy. E

oO

A SYLLABUS OF LOGIC.

' All A is not B/ or the entire exclusion of
A from B.

As the difference in form of these four pro-
positions arises from variations in the subject
and copula, they will admit of two correspond-
ing co-divisions. One of these is into Univer-
sal and Particular, and respects the quantity of
the subject ; the other is into Affirmative and
Negative, and respects the quantity of the
copula. They are thus divided into Universal
Affirmative represented by A ; Universal Nega-
tive by E ; Particular Affirmative by I ; and
Particular Negative by O :

AFFIRMATIVE.

NEGATIVE.

Universal.

A.
All A is B.

E.

No A is B.

Particular.

I.

Some A is B.

0.

Some A is not B.

•(33.) The law by which the number of legi-
timate combinations is reduced to four, may
be stated as follows :

Affirmative copulas have undistributed pre-
dicates; Negative copulas have distributed
predicates. For if B is predicated of A, A is
subsumed under the conception B as being a
part of its sphere ; for instance, in the pro-

PROPOSITIONS. 51

position ' all horses are animals : ' the subject
' horses ' is subsumed under the conception of
the predicate € animals/ as being a part of
the sphere of that conception. But it is evi-
dent that the proposition does not refer to
the whole sphere of the conception e animals/
for in that case there could be no other ani-
mals than horses.

The annexed diagram is a general illustra-
tion of the law :

All A is B. Some A is B.
In each of these diagrams we see that A is
compared with only a part of B ; that is to
say, B is undistributed.

It frequently happens that the sphere of the
predicate is not any larger than the sphere of
the subject : this however is incidental, and
can only arise from the peculiar nature of the
matter of the proposition, and never from its
logical form : for in the example s all carnivor-
ous animals have teeth of a certain form/ it
may be equally true that ' all animals with
teeth of this form are carnivorous/ But the
latter proposition requires additional knowledge
of the subject matter, and cannot be deduced
from the mere form of the original statement.

E2

52 A SYLLABUS OF LOGIC. .

If, on the other hand, the predicate B is
denied of the subject A, the sphere of A is
entirely excluded from the whole sphere of B :
for instance, in the proposition ' no durable
friendship can be based on a participation in
crime,' friendship is entirely excluded from all
those things which can be based on a partici-
pation in crime : or in the particular propo-
sition ' some of the most talented men do not
possess the best private characters; these ' some
men 'are entirely excluded from all who pos-
sess the best private characters. In each case
then the predicate is distributed, as the pro-
position refers to the whole of its sphere. The
diagrams will take the following forms :

No A is B Some A is not B

in which it is evident that the whole of B lies
out of as much of A as is introduced in the
subject.

The accompanying table contains the results
of these remarks :

SUBJECTS. PREDICATES.

Universal Affirmative A ... Distributed — Undistributed.
Particular Affirmative I ... Undistributed — Undistributed.
Universal Negative E ... Distributed — Distributed.
Particular Negative O ...Undistributed — Distributed.

PROPOSITIONS. 53

SECTION II.

MUTUAL RELATIONS OF CATEGORICAL
PROPOSITIONS.

(34.) The mutual relations of the four propo-
sitions A, E, I, O, have been usually classed by
logicians under the generic term of Opposition.
But as there is no opposition whatever in some
of these relations, it may be more correct to
restrict the acceptation of the word to those
cases which contain its meaning, and apply
that of Subalternation to the rest.

Opposition.

Opposition is of three kinds — Contradictory,
Contrary, and Sub-contrary.

Contradictory opposition is the relation which
exists between two propositions that simply
contradict each other. One of them therefore
must be false, and the other true. For each
of them must be either true or false, and if
one of them is true, the other which asserts
that it is false, must be itself false ; and if
one of them is false, the other which asserts
that it is false, must be itself true.

Contradictories differ from each other in
the quality of their copulas ; for as one denies

54? A SYLLABUS OF LOGIC.

what the other asserts, the former must be
affirmative, and the latter negative.

They differ also in the quantity of their sub-
jects. For a contradictory contains nothing
more than the falsity of that proposition to which
it is opposed, and must therefore express the
change that takes place in the relation of the
terms, when its opposite first ceases to be true.
But the first formal change affecting the truth
of an universal proposition, affects only a part
of its subject ; as that cannot be true of the
whole, which is false of a part. And the first
formal change affecting the truth of a parti-
cular proposition, must necessarily affect the
whole of the subject; as that cannot be false
of every part, which is not at the same time
false of the whole.

These remarks will be better understood by
the assistance of the accompanying diagrams :

B

m

Fig. 1. a. Fig. 1. b. Fig. 2. a. Fig. 2. b.

Fig. 1. 0, represents the proposition, All A
is B ; and Fig. 2. «, represents the proposition,
No A is B. Now it is very evident that the
first changes which will render these proposi-
tions no longer true, must take place when
the sphere of A first begins to emerge from

PROPOSITIONS. 55

that of B, as in Fig. 1. b, or when it first begins
to impinge on it, as in Fig. 2. b. And the pro-
positions which express these changes must
necessarily be the respective contradictories of
the original propositions. c Some A is not B '
will therefore contradict the proposition ' All
A is B/ and ' Some A is B/ the proposition
' No A is B.' Contradictories are therefore op-
posed to each other both in quality and quan-
tity : the universal affirmative A is opposed to
the particular negative O; and the universal
negative E, to the particular affirmative I.
Example : ' Some Englishmen are as light-
hearted as Frenchmen ;' e No Englishmen are
as light-hearted as Frenchmen.'

(35.) Contrary opposition exists between
two propositions which contain each other's
contradictory, and something more besides.
Hence it follows that they cannot be both true,
but may be both false : for if one is false, that
part of its contrary which merely contradicts
it, must be itself true ; but the other part, or
surplus statement, may be either true or false.

As contraries state more than each other's
contradictories, the quantity of their subjects
must be greater, and therefore distributed;
whence it follows that A and E are the only
propositions between which this species of
opposition exists. Examples of this opposi-
tion are — 'All smuggling is dishonest'; 'No

56 A SYLLABUS OF LOGIC.

smuggling is dishonest.' 'All Englishmen are
haughty'; 'No Englishmen are haughty/ &c.
(36.) Subcontrary opposition, as its name
in some measure indicates, is the opposition
of propositions contained under contraries.
The subjects of the subcontraries must of
course be less than the subjects of the con-
traries which contain them, and are there-
fore undistributed. As I and O the sub-con-
traries are contradictories of the two contra-
ries E and A, and as the contraries cannot
be both true, but may be both false, it fol-
lows that the subcontraries may be both true^
but cannot be both false. Examples are, ( Some
men are liars'; ' Some men are not liars': both
of which are true. ( Some men are perfect' ;
' Some men are not perfect': of which propo-
sitions one is false.

Subalternation .

(37.) Subalternation is the relation which
exists between an universal proposition, and
the particular that is contained in it. The
former is usually called the Subalternant, the
latter the Subalternate.

The truth of the subalternant necessarily
involves the truth of its subalternate, as what
is true of the whole, must be also true of a
part ; and the falsity of the subalternate in-
volves the falsity of the subalternant, as what

PROPOSITIONS. 57

is false of a part cannot be true of the whole.
But the converse of these propositions does not
hold ; for what is true of a part, is not neces-
sarily true of the whole, and what is false of
the whole, is not necessarily false of a part.
Hence, we cannot infer the truth of the uni-
versal from that of the particular subalternate
to it, nor the falsity of the particular from
that of its subalternant universal. Examples
are — ' All dogs are animals/ whence it follows,
' Some dogs are animals.' ' Some politicians are
no better than they should be/ whence it cannot
be logically inferred that ' All politicians are
no better than they should be/ The annexed
scheme is usually employed to elucidate the
relations of categorical propositions : —

A f Contraries f E

f f

c c

s > *

? X **'o 2

5 0^ X I

t * *

/ /

/ c Subcontraries c O

f means false ; t means true ; and c contingent.

The letters f, f, c, that are placed against
the Universal Affirmative A, are intended to

58

A SYLLABUS OF LOGIC.

represent that A is false when E or O are true,
and contingent when I is true, and similarly of
the rest of the table.

Perhaps the following table is more imme-
diately intelligible —

If one

The other

' s

O

'43

^

r

c

a

fc

O

i.

H

§

* i

0 -

a I

^'

O

G <

t-

O

'

2

3

. OT

HH

i

1"

1

i

pq

PH x

s

d

«sj

«

eq

ts
Ol

w

G «

<u

i

i

. M

Between A and

•a Between E and I..

Between A and E.

Between I and O.

(True False.

\ False.

-True.

'f True False.

\ False Contingent.

f False.. True.

\ True Contingent.

PROPOSITIONS. 59

Conversion.

(38.) Conversion is the transposition of the
terms of a proposition, in such a manner that
the subject becomes the predicate, and the
predicate the subject. Conversion is of two
kinds, simple and limited. The former is the
simple transposition of the terms, retaining
the previous quantities ; the latter requires
the limitation of the subject.

In all conversion, no term must be distri-
buted in the converted proposition, that is not
distributed in the original proposition : hence
E and I are simply convertible, as in E
both terms are distributed, and in I, neither.
No term then is distributed in the converted
proposition, that was not previously distri-
buted in the original proposition. But A is
not simply convertible, as its predicate is
undistributed, and would become distributed
by simple conversion. It admits however of
limited conversion, and then becomes I ; for
in I, no terms are distributed, and therefore
none can be distributed in it, which are un-
distributed in the original proposition. E may
also be converted by limitation, as well as
simply, and then becomes O. O can never be
converted, as its subject is undistributed, and
therefore can never become the predicate of

60 A SYLLABUS OF LOGIC.

a negative proposition, which is always dis-
tributed.

CONVERSION.

Simple. Limited.

E to E A to I

I to I E to O

Examples : —

No cats are cows. All dogs are animals.

No cows are cats. Some animals are dogs.

PROPOSITIONS. 61

SECTION III.

HYPOTHETICAL AND DISJUNCTIVE PROPOSITIONS.

(39.) An hypothetical proposition consists
of two judgments and a copula, r.-y

The judgments constitute the matter of the
proposition, and the copula expresses their
relation, or that function of the understanding
by which they are conjoined in one con-
sciousness.

The whole proposition represents that one
of these judgments which is called the ante-
cedent, contains all the necessary grounds for
the truth of the other, which is accordingly
called the consequent.

The most general form of these propo-
sitions is the following. f If A is B, C is D/ in
which the judgment ' A is B' is the antecedent,
( C is D' the consequent, and the word e if the
copula.

(40.) As an hypothetical proposition merely
asserts such a connexion between two judg-
ments, that the truth of the one may always

62 «A SYLLABUS OF LOGIC.

be inferred from the truth of the other,* it
follows that both the form and the matter of
these judgments constitute the matter alone of
the hypothetical, and cannot be the means
of introducing any variations into its form.
Neither can such variations arise .from the
form of the copula; for the conception of
a law connecting a consequent with its grounds,
may be thought, or not thought, with regard
to any judgments, but will neither admit of
a negative quality, or of a quantum or degree.
Hence there is but one form for hypothetical
propositions, and all the variations that take
place in them must be considered as affecting
their matter alone. For instance, the propo-
sition ' if some A is B, some C is D/ does not
differ in form from the proposition ' if all
A is B, all C is D.' It is true that the cate-
gorical judgments in these examples have
formal differences, as in the former they are
both particular, and in the latter both univer-
sal. But in either case, such a relation is
asserted to exist between them, that the truth
of the one must be the invariable consequence

* The understanding merely asserts a relation of such a nature
between the judgments that the truth of the one may be inferred
from that of the other. But in order that this result should be
actually inferred, another judgment is necessary which shall
assert the truth of the antecedent, and then the reason will
deduce the truth of the consequent.

PROPOSITIONS. 63

of the truth of the other ; and it is this
universality alone, and not that of the judg-
ments themselves, with which the form of
an hypothetical proposition is concerned.

(41.) From these observations it will readily
be seen that an hypothetical proposition can-
not have an hypothetical contradictory : for
a contradictory states nothing more than the
falsity of that proposition to which it is op-
posed, and, if this latter is an hypothetical,
merely denies that there is any law by which
the truth of the consequent can be rightly
inferred from that of the antecedent. But
this does not establish any similar relation
between any other two judgments, and is not
therefore hypothetical in its nature. The con-
tradictory must accordingly be a categorical
proposition, in which I predicate of my consci-
ousness, that it does not give objective validity
to such a relation between two judgments, that
one contains the grounds for the other.

(42.) Many logicians have considered the
hypothetical proposition as merely another
form of a categorical ; and this error has been
rather favoured by the seeming ease with
which an hypothetical may be put under such
a categorical form as shall answer all the
logical ends of the original proposition. But
the two propositions have been shewn by

64 A SYLLABUS OF LOGIC.

Kant to be based on fundamentally different
acts of the understanding, which ought never
to be confounded with each other. For in-
stance, the hypothetical proposition < if A is,
B is/ appears to be fully expressed in the
categorical form, ' all the cases of A being, are
cases of B being ;' and so far as any logical
deductions from either one or the other are
concerned, the propositions are equivalent.
But there is this fundamental difference be-
tween them : the former represents the depen-
dence of one judgment upon grounds contained
in the other, and therefore enunciates a law
to which the understanding has accorded its
assent as being universally valid ; whereas the
latter form either merely asserts the fact ' that
all the cases of A being are cases of B being,'
without superadding the conception that they
are so universally by a law ; or else, if it means
that 'all possible cases of A being are ne-
cessarily cases of B being/ it has only placed
what constituted the form in the hypothetical
proposition in the matter of the categorical,
but has by no means transfused the virtue of
the form of the one into the form of the other.
In fact, this method is no more a formal reduc-
tion of an hypothetical to a categorical proposi-
tion, than if we merely said, " the hypothetical
proposition ' if A is, B is/ is true."

PROPOSITIONS. 65

Disjunctive Propositions.
(43.) Disjunctive propositions consist of any
number of judgments, which they disjoin in the
relation of reciprocal dependence. Hence the
whole conception of their possibility is divided
between them, and the truth of either may be
inferred from the falsity of the others, or in-
versely, the falsity of all the others from the
truth of one. Their general form is ' either A
is B, or C is D.' All the observations that have
been made on hypothetical propositions, in
order to prove the impossibility of any vari-
ations in their form, apply with equal force
to the disjunctive : and as the disjunctive
expresses a distinct function of thinking,
any attempt to bring it into a categorical
form will meet with no better success than in
the case of hypotheticals. It is true that we
can express the meaning of a disjunctive in
a categorical form ; but then, what constitutes
the form in the one proposition must be in-
troduced in the matter of the other. For in-
stance, the disjunctive proposition ' A is either
B or C ' may be expressed thus : ' the cases
of A being B are identical with the cases of
A being not C/ But here we see that the com-
pleteness of the division, which is the real
principle of disjunctive propositions, is ex-
pressed in the conception of the identity of

66 A SYLLABUS OF LOGIC.

one judgment with the negation of another,
which constitutes the matter of the catego-
rical proposition. And hence, as in the case of
the hypothetical, we have no more succeeded
in the transformation of the disjunctive to
the categorical form than, if we had said, " the
proposition ' A is either B or C ' is true,"

(44.) Disjunctives resemble hypothetical
propositions in not admitting a contradictory
of the same form with themselves : for the
contradictory of a disjunctive must merely
contain the falsity of that proposition, and
therefore denies that a certain cognition is
exactly divided out among the members of
the disjunction, but by no means gives a new
and correct division. Hence the contradicto-
ries to both these forms of propositions are
simply categorical in their nature.

SYLLOGISM. 67

SECTION IV.

SYLLOGISM.

(45.) Kant has defined syllogism as that
function of thinking by which we derive one
judgment from another, and has in this
manner included all those judgments which
may be formally derived from another single
judgment by the methods of opposition, sub-
alternation, &c.

These syllogisms, styled immediate from
their wanting a middle term, he arranges
under the title of syllogisms of the under-
standing, and the propriety of such a clas-
sification, as far as the meaning alone is
concerned, is evident from the nature of the
faculty employed. But as it is rather ques-
tionable whether the derivation of the word
will bear out this general acceptation, we
shall follow the usual practice of logicians,
and restrict its signification to those syllo-
gisms which Kant defines as the syllogisms of
reason.

Syllogism, .then, is that function of the
reason by which a third judgment is derived

F 2

68 A SYLLABUS OF LOGIC.

from the union of two others, neither of which
contain it when taken separately.

(46.) The first division of which syllogism
is susceptible, is determined by the nature of
the judgments in respect of their moments of
relation. These have already been shewn to
be three — the categorical, hypothetical, and
disjunctive ; and as there are a great many
combinations of judgments of these several
moments, from all of which other judgments
may be deduced, it follows that there must
also be a great many different syllogistic forms
of ratiocination. But as the principles in-
volved in all of them are the same as those in
the three simplest forms (which are usually
named after the three moments), it will be
sufficient to investigate the laws which regu-
late these alone, and, with one or two excep-
tions, leave the consideration of the rest to
the ingenuity of the reader. In these three
species of syllogism, the categorical, the hypo-
thetical, and the disjunctive, a categorical
conclusion is deduced from one judgment of
categorical form, and another of that form
from which the syllogism takes its name.
We shall now proceed to the separate consi-
deration of each of these three forms of rea-
soning.

SYLLOGISM. 69

Categorical Syllogism.

(47.) In this syllogism the two given judg-
ments, which are also called premises, and
the derived judgment or conclusion, are all
categorical propositions. The nature of this
species of argument may be popularly stated
as follows : ' If two cognitions are severally
compared with a third cognition, (that is, ob-
jectively the same in each case,*) they may
afterwards be compared f with one another. J

* Perhaps an objection may be brought against this exposition,
on the grounds of its admitting negative premises. But it must
be observed that when both premises are of this quality, the real
middle is an external sphere, and consequently undistributed, or
virtually two middles. Arid this fallacy is quite excluded by the
clause in parentheses.

f The affirming or denying one cognition of another is what is
here intended by comparison. For if two cognitions are referred
to the consciousness simultaneously, they must either be thought
as belonging to the same subject or substratum, in which case
one may be affirmed of the other ; or as not belonging to the
same, in which case one may be denied of the other.

J As it is our present object to discover the universal laws of
all categorical syllogism, we have preferred this simple though not
very elegant definition to the celebrated ' dictum de omni out
nullo' of Aristotle, which is merely a particular case of it, and
may be stated thus : ' What is affirmed or denied of all, is affirmed
or denied of each ;' but this would only have given a particular
class of categorical syllogisms, and is therefore insufficient for
our present purpose. Were we however to admit either the em-
pirical considerations of the use of the reason, or the metaphy-
sical considerations of the conceptions of the understanding, we
should then be justified in confining our attention to the dictum
alone. For in the actual use of the reason we discover the fact,
that this form of syllogistic ratiocination is at once more natural

70 A SYLLABUS OF LOGIC.

This is the principle of all categorical syllo-
gism ; and whenever the conditions are really
answered, and the comparison made according
to certain laws which we are about to deter-
mine, the conclusion is necessary, and the
reasoning incontrovertible. Let us take as an
example, ' Animals of the same species are sup-
posed to have originated from the same pair ;
all dogs are animals of the same species,
therefore all dogs are supposed to have ori-
ginated from the same pair/ Here, dogs and
animals supposed to have originated from the
same pair, are severally compared with animals
of the same species, and are afterwards com-
pared with one another.

(48.) We must now determine those laws,
in conformity with which the various forms
of categorical propositions may be so com-
bined as to answer the required conditions of
syllogism.

For this purpose we must not give a mere
aggregate of rules, even though those rules
should be in themselves sufficient. This me-
thod, which has been adopted in most (if not

and intelligible than the rest, and in the nature of one of the con-
ceptions of the understanding, we can discover the reason of the
fact. But we must not regard what lies out of the field of Formal
Logic, and have therefore given a definition which will include all
the forms of categorical syllogism that are possible in theory,
however awkward some of them may be in practice.

SYLLOGISM. 71

all) works on Logic, is exceedingly unscien-
tific, as it only shews that certain rules are
requisite and sufficient to provide against cer-
tain forms of paralogism, but by no means
proves that these are the only forms to which
a syllogism is exposed. Now the method of
exhaustion will secure our reasoning against
all fallacy, so far at least as mere Formal Logic
is concerned.* We shall therefore dissect cate-
gorical syllogism into its ultimate elements,
and discover what rules are necessary and
sufficient for the prevention of fallacy in each
of them.

(49.) Every categorical syllogism contains
three terms — the major term, the minor term,
and the middle term ; and three propositions —
the major premiss, the minor premiss, and the
conclusion. In the major premiss, the major
term is compared with the middle term. In

* It may be as well to notice here, the very prevalent custom
of introducing into treatises on Logic, the consideration of
material as well as formal fallacies. In all the works on this
science that have come under our observation, (with the excep-
tion at least of Kant's,) nearly the first rule for syllogism is to the
effect that the middle term must not have different meanings in
the two premises. Now Logic merely considers the formal laws
of reasoning, but has nothing whatever to do with its matter ; and
the introduction of such a rule as this in a work on that science,
is something like beginning a treatise on Geometry with an
injunction to the student to draw his circles correctly: as if
mathematical reasoning could be at all affected by the perfection
of the diagram.

72 A SYLLABUS OF LOGIC.

the minor premiss, the minor term is com-
pared with the middle term. And in the con-
clusion, the major and minor are compared
together. The major term is always the pre-
dicate, and the minor the subject of the con-
clusion. Hence may be deduced the following
rules for distinguishing the different terms and
premises in any given syllogism : —

1st. The term that is common to the two

premises is the middle term.
2nd. The term that is the predicate of the
conclusion is the major term ; and the
premiss that contains it, the major pre-
miss.

3rd. The term that is the subject of the
conclusion is the minor term ; and the
premiss that contains it, the minor pre-
miss.

The three propositions are usually placed in
the following order : the major premiss, the
minor premiss, and the conclusion. And the
particular form of a syllogism, as far as it
depends on the particular categorical forms
of its three propositions, is termed its mood :
though this name is also given to any ternary
combination of the symbols A, E, I, O, with-
out reference to its conformity to the syllo-
gistic laws.

(50.) For the simplification of the subject

SYLfeOGlSM. 73

we shall divide it into the three following

questions : —

1st. What forms of premiss are sufficient for
a conclusion of the reason,* without con-
sidering its capability of being expressed
in any of the four categorical forms, (i.e.

2nd. When can the conclusion be expressed
in any of the legitimate forms, and when
can it not ?

3rd. What laws are sufficient and necessary
to secure the legitimacy of the con-
clusion ?

As the first of these questions excludes all
reference to the categorical form of the con-
clusion, and only seeks for premises that give
a conclusion valid for the reason, it does away
with the distinction of subject and predicate,
so far as the conclusion is concerned ; and
therefore with the distinction of major and
minor premiss. Hence, in this part of our
investigation, both premises are on exactly the
same footing.

* As it is impossible that the reader should understand the
following pages, unless he has a clear conception of what is meant
by a conclusion ' possible for the reason, but not expressible in
any of the four legitimate categorical forms,' we subjoin the fol-
lowing example. If the premises are 'some B is A, some C is not
B,' the reason may logically deduce that some C is not some A.
But this conclusion is not in one of the four legitimate forms, and
is therefore styled a conclusion only for the reason.

74- A SYLLABUS OF LOGIC.

We may also discard all considerations of
the extremes. For any peculiarities in the
form of these terms, can merely require cor-
responding peculiarities when they recur in
the conclusion, as they constitute the matter
only of that judgment. They may therefore
affect the nature of the conclusion, but cannot
affect the possibility of its existence.

All that remains then for our consideration,
is the middle term, arid the copula in both the
premises.

(51.) Now the middle term admits of no
variations but those of quantity. Every law
therefore regarding it must respect this, and
this alone. And we find accordingly that the
middle term must be distributed in at least one
of the premises.

For if the middle term is undistributed,
a part of it only is compared with each
premiss. And as it cannot be formally known
that the major and minor are compared with
the same parts, the middle term becomes
virtually two terms, and the major and minor
terms cannot be considered as formally com-
pared with the same. But it is not necessary
that the middle term should be distributed in
more than one premiss. For if the whole
middle term is compared with one of the
extremes in one premiss, and only a part of it

SYLLOGISM. 75

with the other extreme in the other premiss,
that part must be compared with both ex-
tremes, and in this case therefore the middle
term cannot be considered as two terms, or
ambiguous.

(52.) The first law for the quality of the
premises may be derived from the law for the
distribution of the middle term. This could
not be the case if every possible comparison
of the spheres of two terms, both internal and
external, positive and negative, found a cor-
responding expression among the categorical
forms : for in that case, a law regarding
quantity could not affect quality. This how-
ever has been shewn not to be the case, as
external spheres are never made subjects ; and
hence arises the possibility of deducing from
the law for the distribution of the middle
term the following law for the quality of the
premises.

No conclusion is possible from two negative
premises. For in premises of this form, the
extremes are each placed in the external
sphere of as much of the middle term as is
compared with them, and consequently the
external sphere becomes the real term with
which both extremes are compared. But the
external sphere is not distributed in any
categorical form, and consequently the virtual

76 A SYLLABUS OF LOGIC.

middle term would be undistributed, and no
conclusion possible.

No. B is A. No. C is B.

Fig.l. ^\ C* J Fig. 2.

In the accompanying diagram we see that
A and C are each placed in the external sphere
of B ; but as the external sphere is undistri-
buted in both propositions, we do not know
that A and C are compared with the same
parts of it, and cannot therefore compare them
with one another. In Fig. 1. we find that ' no
C is A/ and in Fig. 2. the contradictory ' some

c is A;

As the external spheres of conceptions do
not admit the variations of quantity, or bear
the same mutual relations in any categorical
propositions as the internal spheres, they are
never understood unless they are expressly
mentioned. It is for this reason that the law
for the distribution of the middle term is
generally understood to relate to the internal
spheres only, and that the law against two
negative premises cannot be subsumed under
it as a particular case. It is therefore given

SYLLOGISM. 77

as a separate law, that at least one premiss
must be affirmative.

The extreme term contained in the affirma-
tive premiss will of course agree and coincide
with the middle term, and will therefore
agree or disagree with all with which the
middle term agrees or disagrees in the other
premiss. Hence the latter premise may be
either affirmative or negative, and there is no
other law for the quality of the premises, than
that given above.

(53.) We have now exhausted the elements
of the premises, (for we have shewn that our
present inquiry does not involve the distinction
of the premises, or the consideration of the
major and minor terms,) and have arrived at
the following laws for the middle term, and
the quality of the two copulas.

In order that a conclusion for the reason,
though not necessarily categorically expressible,
be possible from any particular forms of
premises, it is only necessary —

1st. That these categorical forms contain at
least one distributed term.

2nd. That one of these forms be affirmative.

As at present there is no distinction between
the premises, we are not considering permuta-
tions but combinations ; and as each proposition
may be combined with itself, the number will

78

A SYLLABUS OF LOGIC.

be

5.4
1 .2

= 10, of which we shall find by the

following table, that 6 are unobjectionable,
that 3 are excluded by their negative premises,
and 1 by the want of a distributed term.

-§

?orms of premises from
which no conclusion

is possible.
.^

Forms of premi-
ses from which
a conclusion at

to

3

No

No

least .for the

'cj

O

distributed

affirmative

reason, is pos-
sible

O

term.

term.

r

A

_

A A

A combined with /

E

T

A E
A T

J

A O

f

E

_

EE

E combined with -/

T

.

.

E I

I combined with 4

O
I
O

I I

E O

I 0

O combined with...

0

00

The above table is only intended to state,
that some conclusion, though perhaps not
categorically expressible, is sometimes possible
in each of the six forms in the last column,
but always impossible in the four forms in
the other two columns. And it must be ob-
served that we can only say sometimes pos-
sible ; for though all these six forms contain
one or more distributed terms, yet if the
middle is not one of those terms, it will be
undistributed, and no conclusion possible. For

SYLLOGISM. 79

instance, let B be the middle term, and A I the
form of the premises, if we say ' All B is A/
' Some C is B,' the conclusion that ' Some C
is A/ is strictly deducible. But if we say ' All
A is B; ' Some C is B,' therefore ' Some C is A/
our conclusion would not be logically correct,
as the middle term B would be undistributed
in each premiss. We repeat, therefore, that
the last table merely indicates the possibility
of deducing a conclusion for the reason from
some premises of certain forms, and the im-
possibility of deducing any conclusion what-
ever from premises of certain other forms.

(54.) We may now dismiss the first part of
our present investigation, and consider the
second question. When can the conclusion
be expressed in any of the legitimate forms,
and when can it not ?

As the variations of quality have not been
allowed to affect the terms * in the categorical
forms of the admissible premises, the conclu-
sion from them that is possible for the reason,
but not categorically expressible, cannot be so
restricted, on account of its being valid for

* Perhaps the following explanation will be more easily under-
stood than the text. As external spheres have not been admitted
as terms in the forms of premises given in the above table, it is
impossible that they should appear in the conclusion ; the imprac-
ticability therefore of these conclusions cannot arise from the
appearance of an external sphere in either of their terms.

80 A SYLLABUS OF LOGIC.

only the external sphere of a conception. It
must therefore belong to one of the four
rejected forms (mentioned in Art. 32,) whose
predicates do not as functions of the copula fol-
low the same law with the accepted forms. In
other words, the conclusion must belong to
one of these four forms, in which the predicate
of an affirmative is distributed, and of a ne-
gative undistributed. But it cannot belong
to a proposition of the former class, as in that
case one of the legitimate categorical proposi-
tions, in which the predicate is undistributed,
would state less than this sound conclusion,
and would therefore be contained in it, and be
itself true ; but this is contrary to the hypothe-
sis. Hence the only case that remains in
which a conclusion can be valid for the reason,
but not categorically expressible, is when a
conclusion whose quality is negative has
a predicate whose quantity ought to be un-
distributed. And we shall accordingly find
that the rule for the quantity of the predicate
of the conclusion will exclude all those pre-
mises which give conclusions only possible for
the reason.

(55.) Our next object is to determine what
laws are sufficient and necessary to secure the
legitimacy of the conclusion.

For this, let us examine the three elements

SYLLOGISM. 81

of the conclusion — the subject, the predicate,
and the copula.

As the only variations of which the terms
admit are those of quantity, the laws concern-
ing them must respect that, and that only.

As, moreover, logic can only consider the
formal quantity of a term, either extreme may
contain in the conclusion the same quantity as
in its premiss, but not more. Hence the only
rule for the terms is this :

If the major or minor terms are undistri-
buted in the premises, the predicate and
subject must be respectively undistributed
in the conclusion.

When this rule is violated in a syllogism
whose major term is undistributed in the
major premiss, but whose predicate is distri-
buted in the conclusion, the resulting fallacy
is called an illicit process of the major.

When the minor term is undistributed in
the minor premiss, and the subject is distri-
buted in the conclusion, the resulting fallacy
is called an illicit process of the minor.

(56.) With regard to the quality of the
copula, the only law for the conclusion is this :

If either premiss be negative, the conclusion
must also be negative ; but if both premises are
affirmative, the conclusion must also be affir-
mative.

G

82 A SYLLABUS OF LOGIC.

For as one premiss must be affirmative, the
extreme term which it contains must agree
with the middle term in that premiss, and
therefore agrees or disagrees in the conclu-
sion, with whatever the middle term agrees or
disagrees with in the other premiss. What-
ever therefore is the quality of this latter pre-
miss must also be the quality of the conclusion.

(57.) The elements of a categorical syllo-
gism have now been completely exhausted,
and it is absolutely certain that if all the given
rules are preserved inviolate, a formally incor-
rect conclusion can never be obtained.

All then that is necessary to ensure the
legitimacy of a syllogism, is comprised in the
five following rules : —

1. There cannot be more than one negative
premiss,

2. If there is one negative premiss, the con-
clusion is negative ; if there is no nega-
tive premiss, the conclusion is affirmative.

3. The middle term must be distributed in
at least one of the premises.

4. If the predicate of the conclusion is
distributed, the major term must be dis-
tributed in the major premise.

5. If the subject of the conclusion is dis-
tributed, the minor term must be distri-
buted in the minor premiss.

SYLLOGISM. 83

The rules that have been given for the
premises will only ensure a conclusion pos-
sible for the reason, and hence it will follow
that the rules for the conclusion will in certain
cases affect the premises also. For if the
conclusion is negative, its predicate must be
distributed ; and therefore, by Rule 4, the major
term must also be distributed in the major
premiss.

(58.) Although the five given rules are quite
sufficient in themselves, yet two others are
derived from them, with which the student
should be well acquainted, as they are of very
easy application. They are —

1. No conclusion can be drawn from par-
ticular premises.

2. Only a particular conclusion can be drawn
where one of the premises is particular.

As the middle term must always be distri-
buted in one premiss at least, and as no term
can be distributed in the conclusion that is
not distributed in the premises, it follows
that there must be at least one more distri-
bution in the premises than in the conclusion.
But there can be only one distribution in the
predicates of the premises, as there can be
only one negative premiss, and in that case
there must be a distribution in the predicate
of the conclusion, as that must also be nega^-

02

84 A SYLLABUS OF LOGIC.

tive. Hence it follows that the number of
distributions in the subjects of the premises
must exceed by at least unity, the number
of distributions in the subject of the con-
clusion. If then both premises are particular,
there will be no distributions in the subjects of
the premises ; and as the above condition will
not be answered, there can be no conclusion.
And if one premiss is particular, there will
be only one distribution in the subjects of the
premises, and therefore there can be none
in the subject of the conclusion, i. e. the con-
clusion must be particular.

(59.) We shall now be able to extend our
table for the premises only, and determine the
moods in which a syllogism, perfect in all its
parts, is possible. It must however be remem-
bered, that as the table of sound premises paid
no respect to the distinction of major and
minor premiss, the order of the premises was
there indifferent. But as the conclusion is
here considered in its legitimate categorical
forms, the distinction holds, and we must
therefore examine the two permutations of
each of those forms of premises.

SYLLOGISM.

85

Premises.

Conclu-
sion.

Moods in which Paralogism is
unavoidable.

Moods in which
Syllogism is
possible.

A AA
A A I

Illicit Process
of Minor.

Undistributed
Middle.

N

Illicit Process
of Major.

A A with 1

A
I

_.

_.

A I with |

A
I

A I A

.

A I I

I A with |

A
I

I A A

or I A A

-_

I A I

A E with |

E
O

AE E
AEO

E A with |

E
O

_

E A E
E A O

A O with |

E
0

AOE

or A O E

or AOE

AOO

O A with |

E
O

O A E

or 0 AE

or O A E

O AO

E I with/

E
O

EIE

E I O

I E with /

E
O

IEE
IE O

I O with /

E
0

IO E

or I O E

I OE
100

0 I with |

E
O

0 I E

O I E
0 I O

or OIE
or O I O

SUBJECT. PREDICATE.

A Distributed Undistributed.

E Distributed Distributed.

I Undistributed Undistributed.

O Undistributed Distributed.

86 A SYLLABUS OF LOGIC.

The following example will shew how the
above table should be studied.

Let us take the mood O A E. In this mood
the predicate is distributed in the major pre-
miss, and the subject in the minor premiss,
and both subject and predicate in the con-
clusion. Hence it follows that either some
term must be distributed in the conclusion
which was not distributed in the premises, or
else the middle term cannot be distributed in
either premiss. We cannot therefore determine
at once which form the fallacy will take, but
may be quite certain that there must be either
an illicit process of major or minor, or else an
undistributed middle. Again, in the mood
O I E, both subject and predicate are distri-
buted in the conclusion, whereas no term is
distributed in the minor premiss, and it there-
fore follows that there must be an illicit process
of the minor. It is also evident that the
middle^ term cannot be distributed in the
minor premiss, and that if it is distributed in
the major premiss, the major term must be
undistributed, and consequently there must be
a fallacy either of undistributed middle or
illicit major.

(60.) It will be seen from the last column in
the table, that there are eleven moods in which
syllogism is possible, or in which the syllogistic

SYLLOGISM. 87

rules may be observed. But it does not follow
that the rules of syllogism are necessarily
observed in these eleven moods, as paralogism
is possible in all but two of them ;* for in the
premises of some of the moods we have but
one distributed term, and in others two or
more ; and unless the position of the terms is
such as to obey the laws for the distribution
of the middle term, and prevention of illicit
process, fallacy will still be the inevitable
result.

The position of the terms will accord-
ingly give rise to a further classification of
categorical syllogisms. And as each premise
admits of two permutations, in one of which
the middle term is predicate, and in the other
subject, there will be four combinations deter-
mined by its position, which are usually called
the four syllogistic figures.

In the first figure, the middle term is the
subject of the major premiss, and predicate of
the minor.

In the second figure, the middle term is the
predicate of both premises.

In the third figure, the middle term is the
subject of both premises.

In the fourth figure, the middle term is the

* These two moods are E A O, and E I 0, which are true in
all the four figures.

88

A SYLLABUS OF LOGIC.

predicate of the major, and subject of the
minor premiss.

The following scheme will enable the reader
to understand the distinction at a glance.

Let S represent the minor term (or subject
of conclusion), M the middle term, and P the
major term, (or predicate of conclusion).

1st Fig.

2nd Fig.

3rd Fig.

4th Fig.

Major Premiss

M — P

P — M

M — P

P — M

Minor Premiss

S — M

S — M

M— S

M— S

Conclusion

S — P

S— P

S — P

S— P

In each of these figures several moods are
sound, and others unsound. For instance, the
mood A A A, which is sound in the first
figure, in the second figure would have its
middle undistributed in each premiss, while
the mood A E E, which is sound in the second
figure, would contain an illicit process of the
major in the first figure. The accompanying
table will shew what moods are sound, and
what unsound, in each figure, and in the latter
case, the peculiar form of the fallacy.

SYLLOGISM.

TABLE OF CATEGORICAL SYLLOGISM.

Same of
Mood.

?orm of
Mood.

1st
Figure.

2nd
Figure.

3rd

Figure.

4th
Figure.

A A A<
A A U

Subject
Premiss

o-in

D-ul
D-UJ

D-U^j
D-ul

u-u/

M — P

S — M
S — P

P — M

S — M
S — P

M — P
M — S
S — P

P — M

M — S
S — P

Sound

Undistributed
Middle

Illicit
Minor

Illicit
Minor

Sound

undistributed
Middle

Sound

Sound

„,{

D-U>|

u-ul
u-uj

Sound

Undistributed
Middle

Sound

Undistributed
Middle.

I A I I

U-U)
D-UV

u-uj

Undistributed
Middle

Jndistributed
Middle

Sound

Sound

AEE^

D-in

D-DV
D-D/

Illicit
Major

Sound

Illicit
Major

Sound

AEO<j

D-U>|
D-Dl
U-Dj

Illicit
Major

Sound

Illicit

Major

Sound

AOOJ

D— U^
U-D)
U-Dj

Illicit
Major

Sound

Illicit
Major

Undistributec
Middle

EAE<[

D — D^j
D — U V
D — Dj

Sound

Sound

Illicit
Minor

Illicit
Minor

BACK

D — D^k
D-UV
U— Dj

Sound

Sound

Sound

Sound

E I O<j

U— D^
D — U
U— Dj

Sound

Sound

Sound

Sound

OACK

U— D^
D— U1
U— Dj

Undistributec
Middle.

Illicit
Major

Sound

Illicit
Major

90 A SYLLABUS OF LOGIC.

In the first column of this table, the
name of the mood is given. In the second, its
form with regard to the distribution of its
terms, where D signifies distributed, and U
signifies undistributed. In the other four
columns, which are headed by the forms of the
four figures, the nature of each mood in each
figure is expressed by the word e sound,' if the
reasoning is unobjectionable, and if not so, by
the name of its particular fallacy.

By comparing the form of one of the
moods with the general form for any one of
the figures given in the upper line of the table,
we shall see if any D in the one corresponds in
position with an M in the other, in which case
the middle will be distributed. We must also
observe if either S and P in the conclusion
correspond with a D in the form of the mood,
and if so, they must correspond respectively
with a D in the premises, or else there will be
an illicit process. With this explanation there
will be no difficulty in understanding the
manner in which the table is formed. For
instance, let us take the mood A A A. We
find in the form of the mood that the subject
in each premiss is distributed, and upon turning
to the form of the first figure, we find that the
subject of the major premiss is the middle
term, and are therefore justified in concluding

SYLLOGISM. 91

that the mood A A A in the first figure has
its middle distributed. Again, we find that
the only term in the conclusion corresponding
to a D, is S, which also corresponds to a D in
the premises. We are therefore certain that
no term is distributed in the conclusion that is
not also distributed in the premises, or, in other
words, that there is no illicit process; and as
the middle is distributed, it follows that the
syllogism A A A in the first figure is sound.

If however we compare the form of the
mood A A A, with the form of the third figure,
we find that S the subject of the conclusion
corresponds to D, and is therefore distributed ;
but that S the predicate of the minor premiss
corresponds to U, and is therefore undistri-
buted : whence it follows that A A A in the
third figure has an illicit process of the minor.

(61.) Several laws may be obtained for each
figure, by an examination of their peculiar
forms.

In the first figure, the minor premiss must
be affirmative : for if it were negative, the
major premiss must be affirmative, and there-
fore have its predicate, which is the major
term, undistributed. But the conclusion must
be negative, and therefore its predicate would
be distributed, which would accordingly give
rise to an illicit process of the major.

92 A SYLLABUS OF LOGIC.

As the minor premiss must be affirmative,
the middle term, which is its predicate, must
be undistributed in that premiss, and therefore
distributed in the other premiss, in which it
holds the place of subject. Hence it follows
that the major premiss must always be uni-
versal in the first figure.

In the second figure, the middle term is the
predicate in each premiss ; and as it must be
distributed in one of them, one premiss must
be negative, and therefore the conclusion also.
In the third figure the minor premiss must
always be affirmative for the same reasons as
in the first figure ; and therefore its predicate
(which is the minor term), being undistributed,
the conclusion must be particular, for were it
universal, there would be an illicit process of
the minor. There are several other rules
which may be derived from the primary laws,
but as all of them may be more simply evolved
by means of algebraical symbols, we shall post-
pone their consideration for the present.

Transformation of the figures of Syllogism.

(62.) It has been already observed, that
Aristotle's dictum (or the first figure) is the
form in which our reasoning appears the most
natural, and is most easily comprehended.
This is the reason why the other three

SYLLOGISM. 93

figures have been considered as unnatural
deviations from it, and that laws have been
laid down for the reduction of all syllogisms
to the original form of the first. But as this
peculiarity of the dictum is based on the laws
of an understanding-conception, and as Formal
Logic can pay no attention to a distinction
which originates in grounds that lie out of its
field, it will be more correct to consider the
laws for transforming a syllogism from any
one figure into another, especially as they will
include the laws for reduction (or transforma-
tion to the first) as a particular case. As the
distinction of figure depends entirely on the
position of the terms, conversion, by which
alone this can be altered, is the only method
of transformation. But this operation is not
always possible, as in many of the moods the
propositions are not of a convertible form :
e. g. it is impossible to transform the mood
A E O from the second figure into the first ;
for the major premiss is not simply convertible,
and limited conversion would give the mood
I E O, which contains an illicit process of the
major. The following table can therefore only
give those conditions which must be answered
by any syllogism in each figure for its trans-
formation into any other; but in order to
know if a syllogism is capable of answering

A SYLLABUS OF LOGIC.

these conditions, it will be necessary to ex-
amine its particular mood.

TRANSFORMATION TABLE.

Figure to be
transformed.

Propositions to be converted for transformation into the

First figure

Second figure

Third figure

— -s

Fourth figure

First

Major

Minor

Conclusion *
or
Major & Minor

Second

Major

Major and

Minor

Minor or
Major & Con-
clusion

Third

Minor

Major and
Minor

Major

Fourth

Conclusion
or

Major & Minor

Minor

Major

(63.) It appears from the table at Art. 60,
that there are six moods in each figure which
answer all the required conditions for a sound
categorical syllogism. Five of these have
been very generally rejected by logicians, on
the grounds of their having particular conclu-
sions, although universal are warranted by the
premises.f But this regard to a more prac-

* " It is hardly necessary to observe, that whenever the conclu-
sion is converted, the major and minor terms, which are respec-
tively its predicate and subject, must be interchanged, and
therefore the premises that contain them.

f These five are the moods A A I in the first figure, E A O
in the first and second, and A E O in the second and fourth.

SYLLOGISM, 95

tical utility is always very unphilosophical in
an a priori science, and especially if it be at
all destructive of the symmetry of our results.
The mathematician will readily acknowledge
his obligations to symmetry for the light it
throws upon truths already known, as well as
for its efficiency as an organum for the dis-
covery of new ones : and the same regard
should be paid to it in every pure science, but
more particularly in one which is intended to
increase the accuracy and rigid strictness of
our thought. In the present case there are
six syllogisms in each of the four figures, and
this reduction would have only nineteen, four
in the first and second, six in the third, and
five in the fourth. Some of the old schoolmen
carried their veneration for Aristotle so high,
as to reject the fourth or Galenic * figure
entirely, and thus reduced the whole number
to fourteen. But this reduction is nearly as
unphilosophical as the other, for the only
rational ground upon which it can rest will
apply to the second and third figures also,
though not perhaps to the same extent. We
shall therefore submit to the reader the whole

* The fourth figure was first recognised by Galen, the great
medical philosopher, who flourished in the second century.

96 A SYLLABUS OP LOGIC.

twenty-four syllogisms, collected under their
respective fig u res . *

Fig. 1. A A A, A A I, All, E A E, E A O, E I O.

Fig. 2. A E E, A E O, A O O, E A E, E A O, E I O.

Fig. 3. I A I, A A I, A I I, O A O, E A O, E I O.

Fig. 4. A E E, A A I, I A I, A E O, E A O, E I O.

Hypothetical Syllogism.
(64.) One premiss of a hypothetical syllogism
is a hypothetical proposition ; the other premiss
is a categorical proposition, and either asserts

* There is a barbarous practice of naming the various forms of
a categorical syllogism by certain words which constitute mnemo-
nic hexameters. A subject which should always be treated in
a rational point of view (»'. e. as appertaining to the reason), is in
this manner degraded to a mere historical record of the deduc-
tions of others, and draws upon the memory alone. Having
entered my protest against these lines, I still think it proper to
subjoin them, in order that the student may understand the allu-
sion when he hears such phrases as ' a syllogism in barbara,' &c.
The vowels contained in the words of these lines give the names
of the moods, and the consonants refer to their other peculiari-
ties,— such, for instance, as the methods of reducing them to the
first figure. The two first lines give the nature of the four cate-
gorical propositions.

Asserit A, negat E, verum generaliter ambae.
Asserit I, negat O, sed particulariter amba?.

Barbara, Celarent, Darii, Ferioque, prioris
Cesare, Camestres, Festino, Baroko, secundse.
Tertia, Darapti, Disamis, Datisi, Felapton,
Bokardo, Ferison habet : Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison :
Quinque Subalterni, totidem Generalibus orti,
Nomen habent nullum, nee, si bene collegis, usum.

SYLLOGISM. 97

the antecedent, or denies the consequent. In
the former case, which is called the modus
ponens, the conclusion infers the truth of the
consequent ; in the latter case, which is called
the modus tollens, the conclusion infers the
falsity of the antecedent. The general forms
of these two cases are, ' If A is, B is ; but A is ;
therefore B is ;' and ' If A is, B is ; but B is
not ; therefore A is not/ Example, ' If what
we learn from the bible is true, we ought not
to do evil that good may come ; but what we
learn from the bible is true ; therefore we ought
not to do evil that good may come.'

These are the only two forms which a
hypothetical syllogism can assume. For no
variation can enter on the side of the hypo-
thetical (which is usually styled the major)
premiss, as there is but one form of such
propositions ; neither can there be any other
form for the categorical (which is usually
styled the minor) premiss than those already
mentioned, for nothing can be inferred by
denying the antecedent or asserting the con-
sequent. As, moreover, the latter premiss
only concerns the truth or falsity of the
members of the hypothetical, all variations in
its form, when considered merely as a cate-
gorical proposition, must affect its matter and
not its form when considered as the minor
premiss. H

98 A SYLLABUS OF LOGIC.

Disjunctive Syllogism.

(65.) In this syllogism, we commence with
a disjunctive judgment, and proceed either by
asserting the truth of one member of the
division, and thence inferring the falsity of
all the rest, which is called the ' modus ponens,'
or else by asserting the falsity of all the
members but one, and hence inferring the
truth of that one, which latter method is called
the "modus tollens/ The general form of
these two cases will be, ' Either A is, or B is,
or C is ; but A is ; therefore neither B is, nor
C is/ And ' Either A is, or B is, or C is ; but
neither B is, nor C is; therefore A is.' We
may take as an example ' Either the Pope is
infallible, or there is at least one great error
in the Romish church ; but the Pope is not
infallible ; therefore there is at least one great
error in the Romish church/

These may be shewn to be the only forms
of a disjunctive syllogism by reasoning very
similar to that employed in the case of the

hypothetical.

Dilemma, fyc.

(66.) Besides the three simple forms of syl-
logism already mentioned, there are several
others,of which perhaps the Dilemma is the most
important. This is an hypothetical syllogism,
whose consequent is divided into members by

SYLLOGISM. 99

a disjunctive judgment. Thus, 'if A is B,
either C is D, or E is F ' is the general form
of the major premiss of a dilemma ; and as it
is hardly ever used except in the modus tol-
lens, the minor premiss and conclusion will
be 'But neither C is D, nor E is F, there-
fore A is not B.'

(67.) The following are examples of com-
binations of premises which differ from those
already given in respect of the moments of
relation. Thus, two hypotheticals will give —

' If A is B, C is D ; but if C is D, E is F ;
therefore if A is B, E is F.'

Or, ' If A is B, C is D ; but if E is F, C is
not D ; therefore if E is F, A is not B/

' If A is B, C is D ; and if A is not B, E is F;
therefore either C is D, or E is F.'

In the same manner two disjunctives will
give—

' A is either B or C ; but B is either D or E ;
therefore A is either C or D or E.J

Or a categorical and disjunctive —

« A is either B or C ; D is A ; therefore D
is either B or C.9

Enthymeme.

(68.) The Enthymeme is a syllogism abridged
by the suppression of one of its premises, which
is nevertheless understood, as the argument

H 2

100 A SYLLABUS OF LOGIC.

would not be valid without it. For, as the
conclusion and either premiss are sufficient to
indicate what the other premiss must be, we
rarely express both premises in practice, but
generally leave one of them to the hearer or
reader to supply. The following is an example
of a categorical enthymeme :

f The science of Logic is very useful, as it
enables us to detect the formal fallacies in
the arguments of our adversaries'. Here the
major premiss is suppressed. The completed
syllogism will stand thus —

* Whatever enables us to detect the formal
fallacies in the arguments of our adversaries
is very useful ; the science of Logic enables
us, &c.; therefore the science of Logic is
very useful.' Had the minor premiss been
suppressed, the enthymeme would have been—

' The science of Logic is very useful, for
any thing is useful that enables us to de-
tect the formal fallacies in the arguments of
our adversaries.'

The following is an example of an hypo-
thetical enthymeme : —

' It will certainly rain, for the sky looks
very black/

In this case the major premiss is suppressed.
The syllogism when completed would stand
thus —

SYLLOGISM. 101

' If the sky looks black, it will certainly rain.

* The sky does look black.

c Therefore it will certainly rain.'
Had the minor premiss been suppressed, the
enthymeme would have been of the following
form : —

' It will certainly rain, for it always rains
if the sky looks black.'

The following examples are two disjunc-
tive enthymemes in which the major and minor
premises of the same syllogism are respec-
tively suppressed : —

' He must be in York, for he is not in
London.'

The suppressed premiss is, ' he must be
either in London or York.'

If the minor premiss is suppressed the en-
thymeme will become —

' He must be in York, for he must be either
in London or York.'

Sorites.

(69.) In a Sorites the conclusion of a syllogism
is not expressed, but made the suppressed
premiss of an enthymeme whose conclusion
may be made the suppressed premiss of ano-
ther, and similarly for any number of enthy-
memes. Thus, ' B is A, C is B, D is C, E is D ;

102 A SYLLABUS OF LOGIC.

therefore E is A* is a categorical Sorites.
Again, ' If A is, B is ; if B is, C is ; if C is,
D is ; but D is not ; therefore A is not/ is a
specimen of an hypothetical Sorites.

( 103 )

SECTION V.

THE DEDUCTIO AD ABSURDUM, OR INDIRECT
PROOF.

(70.) THIS name is given to a circuitous method
of proving one proposition from two or more
others, by means of at least three syllogisms.
As, however, the principle of the proof is quite
independent of the number of given proposi-
tions, our present object will be fully answered
by an investigation of that case, in which they
are limited to two. And, moreover, as the
forms of syllogism contain all the principles
or functions of the reason, by which one pro-
position can be thought as necessarily con-
nected with several others, it follows, that
any conclusion at which we can arrive by the
deductio ad absurdum, might also have been
obtained by the direct application of one of
the regular syllogistic forms of ratiocination.
We may therefore consider the given proposi-
tions, and the one to be deduced from them,
as the premises and conclusion of a syllo-
gism, and proceed to shew how this conclu-

104 A SYLLABUS OF LOGIC.

sion may be obtained by a different chain of
reasoning.

The method, then, consists first, in assum-
ing the falsity of the conclusion and truth of
one premiss, and deducing from these propo-
sitions as premises, the falsity of the other
premiss, as a conclusion ; secondly, in taking
this conclusion as the consequent, and the
premises of the last syllogism as the antece-
dent in a hypothetical syllogism, and inferring
the falsity of the antecedent by the modus
tollens ; thirdly, in dividing this falsity in a
disjunctive syllogism into its three members,
viz. the falsity of each proposition separately,
or of both together, and inferring the falsity
of that member of the division which is the
contradictory of the original conclusion by
the modus ponens. But the reader will under-
stand this method more easily by examining
the following general form :

Let A and B represent the premises, and
C the conclusion of any syllogism. In order
to prove C by the indirect method, we com-
mence with assuming that C is not true.
The three syllogisms may be then stated as
follows :

First syllogism : ' A is ; C is not ; therefore
B is not.

Second syllogism : ' If A is, and C is not,

INDIRECT PROOF. 105

it follows that B is not ; but B is ; there-
fore it is false that "A is and C is not/"

Third syllogism : ' Either both propositions
" A is" and " C is not" are false, or else one
of them is false ; but that " A is " is not false ;
therefore that "C is not" is false, (i.e. C

is').

The hypothetical syllogism is rarely if ever
expressed in practice ; the disjunctive, perhaps,
never. But, in an analysis of the indirect
proof, it would be just as unreasonable to
neglect the consideration of these syllogisms
on the ground of their being rarely expressed,
as it would be in an analysis of syllogism
generally to neglect the consideration of one
of the premises on the ground of our usually
reasoning in enthymemes.

Although the dcductio ad alsurdum requires
premises from which the conclusion might
have been deduced by the direct method, yet
is it frequently very useful when these premises
are of such a nature as not to admit of a very
convenient syllogistic form. It is on this account
not unfrequently used in the propositions of
geometry, though Euclid has occasionally intro-
duced it when the direct proof would have been
equally simple. An instance of this will be
found in the fourth proposition of the third book,
in which it is required to prove that ' If in a

106 A SYLLABUS OF LOGIC.

circle two straight lines cut one another which
do not both pass through the centre, they do
not bisect each the other/ The indirect proof
assumes that they bisect each other, and then
shews that they must both be perpendicular
to a line joining the centre and the point of
their intersection, which is absurd ; therefore,
&c. Whereas the direct proof states that all
chords which are bisected by the line drawn
from the centre to the point of intersection
must be perpendicular to that line ; but both
of these chords are not perpendicular to that
line; therefore both of these chords are not
bisected by the line drawn from the centre
to the point of their intersection, which can-
not therefore be the point of their bisection.*
But every proposition which admits the deduc-
tio ad absurdum, will also admit of a direct
proof from the same data, though in a great
many cases the direct proof would be exceed-
ingly clumsy, and not nearly as simple as the
indirect. There is a very striking instance
of this in the usual proof of the falsity of
a hypothetical or disjunctive proposition from
two categorical premises. The reasoning is
rather abstruse, but it bears too completely
upon the present subject to be entirely

* This syllogism is the mood A O O in the second figure.

INDIRECT PROOF. 107

overlooked, and is therefore subjoined in a
note.*

* Hypothetical and disjunctive differ from categorical proposi-
tions in not possessing the quality of negation objectively. For
if the cognition upon which any one of them is grounded (t. e. the
dependence of one particular cognition upon another, or its divi-
sion into members) does not hold in nature, there is no other
hypothetical or disjunctive proposition which can simply assert
this want of objectivity. This remark has been made already
(Art. 42 — 44) where it was observed that a proposition which
simply denies the dependence of a particular cognition on another,
or the completeness of the division of a cognition, in neither case
gives any new dependence of cognitions, or any new exhausting
division, and consequently cannot be either hypothetical or dis-
junctive, but must be simply categorical. If then two categorical
propositions are granted as premises, from which we are to deduce
as conclusion the falsity of an hypothetical or disjunctive proposi-
tion, these two premises must be referred to the thinking subject as
the only middle term by which they can be united in an act of
reason, for all common objective grounds are denied them by the
very nattfre of the case. This necessity for a subjective reference
renders the reasoning so much more abstruse, that the mind
naturally chooses the other method of deductio ad absurdum,
which, from its objective nature, is much easier of comprehension.
In this method we assume the falsity of our desired conclusion
(which conclusion is, in this case, the contradictory of an hypo-
thetical or disjunctive proposition) and therefore assume the truth
of the hypothetical or disjunctive, and conjoining it with one of
our given premises, deduce the falsity of our other premiss ; in
this manner we entirely avoid the necessity of any reference to
the thinking subject. We will exemplify this reasoning as
follows : —

Let the given premises be, * A is ; B is not,' from which we
are to deduce the falsity of the hypothetical proposition,

' If A is, B is.'

If I would deduce this falsity (which being the contradictory of
a hypothetical must be contained in a categorical proposition)
by the direct method, I must refer the two premises to the
thinking subject ' I' in some such manner as the following : —

108 A SYLLABUS OF LOGIC.

(71.) The deductio ad dbsurdum supplies the
principle upon which may be founded a rather
pretty and symmetrical arrangement of the
twenty-four categorical syllogisms. It also
suggests a method of proving the necessary
equality of the moods true in the first three
figures without the aid of mathematical
analysis, and of shewing the reason why the
number of negative syllogisms is exactly double
the number of the affirmative. And although
we do not propose to derive any particular
practical advantage from its consideration, yet
anything that tends to give- additional order
and theoretical completeness must always have
a sufficient value in a pure science to warrant
its insertion.

As in every categorical syllogism we may

' Whatever is in ray present consciousness is conjoined in it
with my assent to the conception of B not being.

4 My assent to the conception of A being is in my present con-
sciousness.

' Therefore my assent to the conception of A being is conjoined
in my present consciousness with my assent to the conception of
B%eing.'

This conclusion is the categorical statement of the falsity of the
hypothetical which can only be shewn directly by this or some
other equally clumsy method. The indirect proof however, is
simple enough, and may be stated thus :

Let us assume that the hypothetical is true, or that ' if A is,
B is ; but A is; therefore B is ; but B is not, therefore,' &c.

As we have already stated, the simplicity of the proof in this
case arises from every premiss contained in it having an objective
reference, which the contradictory of a hypothetical has not.

INDIRECT PROOF.

109

either employ the major premiss and contra-
dictory of the conclusion to disprove the minor
premiss, or the minor premiss and contradic-
tory of the conclusion to disprove the major
premiss, it follows that there must be two dis-
tinct syllogisms, with either of which we may
commence an indirect proof. The forms which
they respectively assume will readily appear
from the following table : —

Major Premiss and
Contradictory of
Conclusion being
Premises.

Minor Premiss and
Contradictory of
Conclusion being
Premises.

First Figure.
(M-P)
< S — M > becomes

U - pj

Second Figure.
(M- P)

\ s ~ p f

U -Mj

Third Figure.

f S - P )
{ S -MV

\M-PJ

Second Figure.
(P-M)
< S — M > becomes
\S-Pj

First Figure.
(P- M)

1S ~ P f
(S -M/

Third Figure.
(S - M)

1S ~ P^
U — M)

Third Figure.
I'M- P)
< M — S V becomes
\S- PJ

Second Figure.
( S - P)
1 M - P \

(M-SJ

First Figure.
(S -P)

{M - si

\M-PJ

Fourth Figure.
( P -M)
< M — S V becomes

Is-pJ

Fourth ligure.
(S- P)
fP-Ml

(M- s

Fourth Figure.

rM- s S

\ s~p t
IP-MJ

From this table it is immediately evident
that for every sound mood in the first figure,

110

A SYLLABUS OF LOGIC.

there must also be a sound mood in the second
and third ; that for every sound mood in the
second figure, there must also be one in the
first and third ; and for every sound mood in
the third figure, one in the first and second.
Hence it follows, that the number of moods
that are sound in each of the first three figures
must be the same. But the form of the fourth
figure is such, that it only admits of an indirect
proof by syllogisms in the same figure; and
the equality, therefore, of the number of its
moods to that of the moods in the other three
figures, is not susceptible of this method of
proof.

This table will supply the grounds of a
division of categorical syllogisms into eight
systems, containing three each. Six of these
systems have one mood in each of the first
three figures. The other two are contained
entirely in the fourth. The systems are as
follows: —

Figure

1

2
A A I

3

4

5

6

First

A A A

A I I

E A E

E AO

E I O

Second

A O O

A E O

A E E

E I O

E A O

E AE

Third

0 A O

E A O

E 10

I A I

A A I

A I I

INDIRECT PROOF,

Fourth Figure.

Ill

7

8

A A I

I A I

A EO

AEE

E A 0

E I O

Each system must contain one affirmative
and two negative syllogisms. For the contra-
dictory of the conclusion of every affirmative
syllogism must constitute one of the premises
in its two complementary syllogisms, or those
contained in the same system with itself; and
as this contradictory must be a negative pro-
position, the two syllogisms that contain it
must be negative also. Hence, every affirma-
tive syllogism must have two complementary
negative syllogisms. And as the affirmative
contradictory of the conclusion of a negative
syllogism may be combined with either its
negative or its affirmative premiss, its two
complementary syllogisms must themselves
also be the one negative, and the other affir-
mative. As, then, every system contains one
affirmative, and two negative syllogisms, it is
evident that the whole number of negative
must double the number of affirmative syllo-
gisms.

SECTION VI.

SYMBOLICAL EXPRESSION FOR THE LAWS OF
CATEGORICAL SYLLOGISM.

(72.) THE conditioning laws of categorical
syllogism admit of a very simple analytical
expression from which all its properties may
be readily obtained. But the more especial
object in treating this subject mathematically,
is the exhibition of that symmetry, from which,
the equality of the number of moods that are
true in the first three figures, may be derived
a priori to all consideration of the moods
themselves. Before, however, we proceed to
make any assumptions, it is necessary to re-
mind the reader, that the usual arithmetical
interpretation of the symbols employed has no
natural connection with the subject under con-
sideration, but can merely be useful as an index
to the laws of their combination. If, for in-
stance, any sets of symbols be juxtaposed, and
equated to zero, the equation will indicate that
at least one of these factors may itself be singly
equated to zero ; and, in that case, all its con-

SYMBOLICAL. 113

stituent symbols must receive whatever inter-
pretations may arise from the transposition of
terms, or any other operations, that are for-
mally analogous to those of arithmetical alge-
bra, and have been admitted in our particular
application of the science. In the present case
the conception of homogeneity is not intro-"
duced, and it will therefore follow, that, if
several symbols are equated to several others,
each of those on one side of the equation must
be considered as equated to one on the other,
but no division of a symbol into parts will be
considered admissible.

The laws of categorical syllogism have been
already once stated, but are nevertheless re-
peated here, as a trifling alteration in the
manner of expressing those respecting quality
will be necessary for their reduction to a
mathematical form. As the latter will be re-
presented in a single equation, it will be more
convenient to throw them into one rule, which
may be stated as follows : —

(1.) At least one of the premises must have
an affirmative copula, and the quality of the
conclusion will be the same as that of the re-
maining premiss.

As this law respects the quality of the copu-
las, which will always determine the quantity
of the predicates, it may be written thus :

114 A SYLLABUS OF LOGIC.

1. The predicate of at least one of the
premises is undistributed, and the predicate of
the conclusion will be of the same quantity as
the predicate of the other premiss.

The other three laws are —

2. The middle term must be distributed
in at least one of the premises,

3. The major term must not be distributed
in the conclusion, unless it has been distributed
in the major premiss.

4. The minor term must not be distri-
buted in the conclusion, unless it has been dis-
tributed in the minor premiss.

Let u represent an undistributed term.

Let d represent a distributed term.

Also, let pl9 pz, p3, and sl9 s2, s3, represent
the predicates and subjects respectively of the
three propositions as expressed below.

SUBJECTS. PREDICATES.

Major premiss sv pi

Minor premiss sz p2

Conclusion s3 p3

where each of these symbols must = d or u.

Now, observing from rule (1), that at least
one of the predicates of the premiss (i. e. pl9 p2)
must be undistributed or = u9 we may express
the other predicate as = p} -f p^ - u.

But by the latter clause of law (1) we also
find that whatever is the quantity of that pre-

SYMBOLICAL. 115

dicate, must also be that of the predicate of
the conclusion, and we shall therefore have

^3=jPi+A-^ .... law (1);
and this equation will be a correct expression
for law (1).

Now from law (2) we find that the middle
term must be distributed (i.e.=d), in at least
one premiss, and as sl and p2 are the middle
term in the first figure, one or both of the
following equations must be true in that

figure,

(*l- d) = 0

(ft- «J) = 0,-
and we shall therefore have

(*i - <0 (ft - «0 = 0 ^w (2)

as an equation expressing the second law for
the first figure ; and as the same reasoning will
apply to all the four, we shall have

Fig. (1).... (*, _rf)(pa~rf) = 0}
2)....<ft-<*)(ft-rf)=0f
3)....(*I-£i)(*a-«0 = or

Fig. (4) .... (p, - d) (s, _ d) = 0 J

Again, from law (3) we know that either the
predicate of the conclusion, i.e. p3, must be
undistributed or = uy or else the major term
which is either sl or pL according to the figure
must be distributed or = d; and by reasoning
similar to the preceding we shall have

I 2

116 A SYLLABUS OF LOGIC.

Fig. (1) .... (Pl - d) (Ps - «) = 0

Fig. (3) ; .' .' .' (p\ ~ 2 (£ 1 3 1 S j: • • law '3'-

Fig. (4) (sl - d} (p3 — u} •= 0

In precisely the same manner we may obtain
the following expressions for law (4).

Fig.

Fig.(2)....(,2-^)(,3-w) = 0^^law(^

We shall now assume the symbols p'3 and *3
of such a nature that

^'3 +^3 = d + u (5).

and s'3 + 53 = d + u (6).

And as it has been already stated that homo-
geneity is not introduced in the conception of
our symbols, it will follow from these equations
that p'3 and *'3 are each = d or u. They will
also be respectively complementary to p2 and
s3 in such a manner that when one = u, the
other = d, and vice versa ; and they will there-
fore be of precisely the same nature as the
other six symbols, pl9p.i9 &c.

From (5) and (6) we obtain for the values of
p3, *3 respectively

P3 = d + ti - p'3

s3 = d 4- u — s'y

And substituting these values of p3 and s3 in
the equations (1), (3), and (4), we shall get

SYMBOLICAL. 117

the following sets of equations for the four
figures of categorical syllogism.

V5i - (Pz - I) v First figure.

(P\ — d} O' - d) = 0 (3) '

(4)

(Pi — d) (P2 — d) = 0 (2) .

V^J ' > Second Figure

(sl — a) (pa — d} = 0 (3) '

(s2 — d) (s'3 — d) = 0 (4)

Pi "I- P2 +*P'3 = d -f 2 w... (1)

(.,-,.,•(. f--) = 0 (2)

(Pi - d) (P3 — «0 — 0 (3) '

(p2 J. d) (/3 — d} == 0 (4)

Pi + P2 + P'a = ^ + 2 «••• (0

f FourthFigure
C^y
(pa - ^) (*'3 - <0 =0 (4)

(73). If we examine the equations to the
first three figures, we shall find each set per-

* The following is the most general symbolical expression for
the syllogistic laws.

Let x and y represent the middle term in the major and minor
premises respectively, and m and n the major and minor terms.
The equations will then be equally applicable to all the four
figures, and will assume the following forms :

(a: — d) ( y — d) = 0 (2).

(m — d) O3 - d) = 0 (3).

(n — c?) (s'3 — d) = 0 (4).

where x 4- m = p\ + s} (5).

and y -\~ n =>jp2 + 52 (6)«

118 A SYLLABUS OF LOGIC.

fectly symmetrical with respect to the other two.
For in equation (1), which is common to all
the three figures, the symbols pl9 p^ p3 are
perfectly symmetrically involved, and the only
assumptions that have been made respecting
$„ *2, $3 are that each of them must either —
d or u. Hence any interchange among the
three symbols *,, $2, *3 or the three pl9 pz, p'3
will not in the least affect the form of the
equations. But if we interchange p2 and p£ in
the equations to the first figure, we shall
obtain the equations to the second ; and if we
interchange sl and s'3 in the first figure, we
shall obtain the equation to the third ; or lastly,
if we interchange p2 an'dj0'3, also 5, and $3', in the
equations to the third figure, we shall obtain
those to the second ; and of course the same
interchanges will reproduce the first from the
second, the first from the third, &c. : hence it
follows that the sets of equations to the first
three figures are perfectly symmetrical with
regard to each other.*

If however it were required to obtain the

* Although the sets of equations to the first three figures are
perfectly symmetrical with respect to each other, yet does it by
no means follow that the corresponding equations represent the
same laws. The equation that contains the law against illicit
major in the first figure, corresponds to the similar equation in
the third ; but in the second figure it corresponds to the equation
containing the law for the distribution of the middle term ; and
similarly of the others.

SYMBOLICAL. 119

equations to the fourth figure from those to
either of the other three, it would be neces-
sary to interchange one or more of the three
symbols sl9 s2, sj with one or more of the
symbols pl9 p^ p^\ and this cannot be admitted,
as the three latter symbols are all involved in
equation (1), whereas the three former are
neither involved in that or any other cor-
responding equation ; and hence it follows
that the equations to the fourth figure are
not symmetrical with those to the other three.

(74.) As the symbols are symmetrically in-
volved in the equations to the first three
figures, we can know a priori to all other
considerations that there must be the same
number of solutions, and therefore the same
number of true moods for all of them. But as
the equations to the fourth figure do not
involve the symbols symmetrically with the
equations to the other three, we cannot say at
once that the number of their solutions must
be the same as in those to the other figures,
but can only shew that it is so by determining
the number of solutions in each case. For
this purpose it will be sufficient to investigate
separately the number of solutions for the first
and fourth figures.

We will commence by examining the first
figure. It is evident from equation (I), that

120 A SYLLABUS OF LOGIC.

one of the symbols pl9 /?2, /r must = d, and
the other two each = u ; and from equation
(3) that either pl or p^ must = d. Hence
there will be two solutions for equation (3)
(accordingly as pl or p3 — d), which, will cor-
respond to two solutions for equation (1).

But as equation (3) requires that either p{ or
p'3 should = d, it follows from equation (1)
that p2 must always = u, and therefore from
equation (2) that sl = dl9 or otherwise neither
factor in equation (2) would vanish. Hence
there is but one solution for equation (2), and
only two solutions for the equations (1 ) and (3),
as the latter are mutually dependent on each
other. Equation (4) will have three solutions,
accordingly as both together or either sepa-
rately of the symbols sz and *3' = d. And
as the solutions of equation (4) are quite in-
dependent of the symbols involved in the
equations (1), (2), (3), it follows that any one
of the three solutions of the former may be
combined with either of the two solutions of
the other three equations, and thus produce
six sets of solutions which will correspond to
the six moods that are'' sound in the first
figure.*

* Great care must be taken, in the interpretation of these equa-
tions into their corresponding moods, not to confound p'3 with py
or s'3 with 53. The values of p'3, and s'3 are first determined

SYMBOLICAL. 121

In the fourth figure we shall have three solu-
tions for equation ( 1 ), accordingly as either of
the three symbols pl9 p2, or p'3 may — d. Of
the other three equations (2), (3), and (4), that
which contains the particular symbol of
the three p}) /?2, p^ which = d, will admit of
two solutions ; but the two equations which
respectively contain those two of the three
symbols pl9 p^ p3 that are = u, will admit but
of one solution. Hence, each of the three
solutions of equation (1) may be combined
with the two solutions of one of the other
three equations, and thus produce six com-
binations which will respectively answer to
the six moods of the fourth figure.

Although we have only been able to prove
that the number of moods in the fourth figure
is equal to that in the other three, by the
numerical tentative method already given, yet
can we at least shew that no other method
is possible. For it is evident that the only
method by which such an equality can be
shewn independently of numbers, is that of
form, which has been already employed in be-

by the four equations to the figure under consideration, and
from them the values of p3 and s3 are deduced by means of the
equations

and will indicate the peculiar categorical form of the conclusion.

122 A SYLLABUS OF LOGIC.

hoof of the first three figures. But the equa-
tions to those three figures completely deter-
mine two of their six symbols (sl and p2 in the
first ; s. and p' in the second ; and p and s'

l •* 3 •* 2 3

in the third), whereas the equations to the
fourth do not determine a single one, but
admit variations in all. Hence it follows
that no artifice can bring the equations to
the fourth figure under the same form as
the equations to the other three, and conse-
quently that all formal proof of the equality
of the number of the solutions is absolutely
impossible.

(75.) Among other advantages in the sym-
bolical expression of the laws of categorical
syllogism, we may mention the facility with
which the derived secondary laws may be ob-
tained, and the peculiar fallacies exposed which
their violation entails. For instance, we know
from equation (1), that one alone of the sym-
bols pl9 pz, p'3 can = d ; and moreover from
equation (3) in the first figure, that either
pl or p^ must = d. We may therefore con-
clude that /?2 = u, that is to say, the predi-
cate of the minor premiss in the first figure
is undistributed, and the quality of its copula
affirmative. But if p2 = n, it will follow from
equation (2) in the first figure that sx = d, or
in other words that the subject of the major

SYMBOLICAL. 123

premiss is distributed, and that premiss an
universal proposition. Should either of these
secondary laws be violated, the conditions of
one of the equations (2) or (3) cannot be ful-
filled, and there must either be a fallacy of
undistributed middle, or illicit process of the
major. In the same manner it may be proved,
that in the second figure s} = d, and p'3 = u,
(or p3 = d,) in other words, that the major
premiss must be universal, and the conclu-
sion negative. But these examples are suf-
ficient to enable the reader to derive the other
laws for himself.

Truth of Premises.

(76.) To assert that ' if the premises are true,
the conclusion that is deduced from them must
be true likewise/ is a mere tautology; for
the very definition of a conclusion is ' that
proposition, the truth of which follows neces-
sarily from the truth of the premises.' But the
converse of this proposition is by no means
true, for it does not follow that if the conclu-
sion is true, the premises from which it is
deduced must be true also. A more satisfac-
tory explanation of this subject will be given
in a future section : the following, however, is

124 A SYLLABUS OF LOGIC.

an unobjectionable proof by the method of the
deductio ad absurdum.*

Let us assume that if the conclusion is true,
the premises must be true also ; it will follow
that if the premises are not true, the conclu-
sion will not be true either. Let any sound
mood in any figure be represented by the
general symbols x y %, and let a?, y, %', represent
the formal contradictories of the propositions
x, y, % respectively. Let one or both of the
premises x and y be false ; it follows from our
present hypothesis, that % must be false also.
If then, for example, x is false, and its falsity
is sufficient to ensure the falsity of *, and if
we substitute for x a proposition which merely
states that falsity, we may also substitute for *
a proposition which merely states its falsity,
and thus change the form without affecting
the soundness of the reasoning. But the con-
tradictories of x and % respectively state the
falsity of those propositions, and consequently
a syllogism of the form x y *' must represent

* This proof will only strictly apply to categorical syllogisms. It
may, however, be extended to the hypothetical and disjunctive by
converting their form into the matter of a categorical. Thus the
hypothetical, ' if A is, B is ; A is, therefore B is,' may have its form
put into the matter of a categorical in the usual way, commencing
with ' all the cases of A being,' &c,, and as this categorical syl-
logism may have false premises and true conclusion, it is evident
the hypothetical may also.

SYMBOLICAL. 125

an unobjectionable mood in the same figure as
that of x y %. And as we might have assumed
that y was false instead of x, or that both were
false together, we shall have the three moods
x1 y %', x y *', and x y «' all sound in the same
figure as that of x y *.

But contradictories differ from each other in
the quantity both of their predicates and sub-
jects. If, then, we recur to the symbolical
expression of the syllogistic laws (Art. 73), we
must find that for every solution of the equa-
tions there given, three corresponding solu-
tions may be obtained by exactly reversing the
values of *3 and p'3 together with the values
of one or both pairs of symbols ^, pl and s^ p2
in such a manner that those which = u should
= d, and vice versd. But it was shewn in Art.
74, that in the first figure the values of sl and
p2 are determined by the equations (1), (2),
and (3), and consequently neither of the pairs
of symbols s}, p{ or s2, p2 can have their values
reversed in that figure. In the second figure
we find the symbol p'3 determined by the equa-
tions (1) and (2), and in the third figure the
symbol s'3 determined by the equations (1),
(3), and (4), and consequently the changes of
values cannot take place in these two figures
any more than in the first. Again, it appears
from the equations to the fourth figure, that

126 A SYLLABUS OF LOGIC.

when *i can change its value, p{ is determined,
and when s2 can change its value, p2 is deter-
mined, and consequently in neither case can
one of the pairs ^, pl or *2, p2 both change their
values simultaneously. Rente it follows that
the conclusions at which we have arrived are
all false, and consequently that the hypothesis
which we assumed must be false also ; and that
true conclusions may be deduced from false
premises in every figure of categorical syllo-
gism.

FORMAL LOGIC.

BOOK II.
SECTION I.

LIMITATIONS UPON THE FORM OF
JUDGMENTS, &C.

(77.) In the present section, it is proposed to
trace the limitations upon the form of judgments,
the superiority of the first figure of categorical
syllogism, and the possibility of true conclu-
sions from false premises, to a common a priori
ground in the very constitution of the under-
standing itself. The ground in question is a
simple property of the understanding-concep-
tion, Substance, and may be stated as follows.
Substance, or the substratum of phenomena,
(i.e. the thing that is, — but is not phenomenon,)
can never become a predicate.

For the conception of Substance may be
defined as that which is thought as remaining
when all possible predicates have been ab-

128 A SYLLABUS OF LOGIC.

stracted from it. It is therefore impossible to
make it a predicate, for nothing is left which
can become its subject ; and were we to at-
tempt to predicate it of phenomena, we must
previously think a substratum for these phe-
nomena, and should therefore only be predicat-
ing the simple conception-substance of itself,
which is absurd.*

Law between the predicate and copula of
categorical propositions.

(78.) This property of the understanding-con-
ception, Substance, will immediately explain the
reason of the law between the predicate and
copula of categorical propositions. This law,

* It is necessary to put the reader on his guard against a cer-
tain species of categorical judgments which appear to militate
against the observations in the text, and to contain the conception
of substance in the predicate as well as subject. The predicate
in these judgments contains matter of such a nature as to deter-
mine it to a particular object, e. g. ' that man travelled with me
yesterday.' In this judgment, taken alone, I really only think of
the existing man before me, and predicate of him all that is con-
tained in my conception of his having travelled with me yesterday.
But the instant after making such a judgment as this, I might very
probably convert it in my own mind, and think first of the exist-
ing man with whom I travelled yesterday, and afterwards predi-
cate of him that he then stood before me. And the case with
which the understanding can thus at pleasure make either term a ,
subject containing the conception of substance, very naturally
produces a false semblance of that conception being in the pre-
dicate.

LIMITATIONS, &C. 129

which has already been stated in Art. 33, is
repeated here : —

Affirmative copulas have undistributed pre-
dicates ; negative copulas have distributed pre-
dicates.

For, inasmuch as the predicate cannot con-
tain the conception of substance, it is not
thought in respect of its sphere, or the things
that are contained under it, but in respect of
its matter, or the representations contained in
it. This term is accordingly a mere concep-
tion, which is never formally determined as to
its quantity, and the precise limits of its
sphere must always remain unknown. The
limits of the sphere of the subject cannot,
therefore, be exactly compared with those of
the sphere of the predicate, but the former
term must either be placed wholly in or wholly
out of the latter. If it is placed wholly in, it
is compared with only a part of the sphere of
the predicate ; if it is placed wholly out, it is
compared with the whole. From these con-
siderations it is immediately evident, that the
quantity of the predicate is always undistri-
buted in affirmative, and distributed in nega-
tive judgments.

130 A SYLLABUS OF LOGIC.

There is no formal proposition of identity.

(79.) For if the subject is single, it evidently
cannot be identical with the predicate which is a
conception. But if the subject is a conception,
it must be considered either as to its partial
representations, i. e. what is thought in it, or
else as to its sphere, i. e. what is thought under
it. In the first case the conception in the
subject must be literally the same as the con-
ception in the predicate, and the result would
be a tautology, but not a judgment : in the
second case the subject, which refers to the
sphere or aggregate of individuals, cannot of
course be identical with the predicated con-
ception, the quantum of whose sphere is
formally indeterminate. In neither case there-
fore could there be a formal proposition of
identity. It need hardly be observed, that in
the proposition ' A is identical with B,' the
identity is expressed in the matter, and not in
the form.

Figures of Categorical Syllogism-

(80.) The great advantages in elegance and
perspicuity that the first possesses over the
other three figures, have been already men-
tioned in the first book ; and the ground of
this superiority may now be derived from the

LIMITATIONS, &C. 131

considerations introduced at the commence-
ment of this section.

It has been shewn in Art. 77, that the con-
ception substance is never placed in the
predicate of a categorical proposition. Hence
a conception, which in the subject is considered
as to its sphere, (or the aggregate of indi-
viduals that are contained under it,) in the
predicate is considered as to its contents, (or
the representation that is contained in it as
a mere conception). Any change then in the
situation of a term introduces the necessity of
a change in the manner in which it is thought.
Now in the first figure, both subject and pre-
dicate of the conclusion retain the same
position which they held in the premises, and
consequently no such change in the manner of
thinking them is necessary. But in the second
figure, the major term changes its place from
subject in the major premiss to predicate in
the conclusion ; and in the third figure the
minor term changes its place from predicate in
the minor premiss to subject in the conclusion,
and consequently a change takes place in the
manner of thinking one extreme in each of
these figures. Again, in the fourth figure
neither of the extremes hold the same position
in the conclusion which they hold in the
premises, and consequently two changes in the

K2

132 A SYLLABUS OF LOGIC.

nature of the terms take place. In this man-
ner, then, it is sufficiently easy to account for
the superiority of the first, and inferiority of
the fourth, to all the other figures. For all but
the first involve the necessity of some change
in the manner of our thinking one at least of
the extremes; and the fourth involves two
such changes. If the reader will make the
experiment of different moods in different
figures, he will become immediately conscious
that their comparative merits entirely depend
on the cause alleged. Perhaps the mood
E I O is the fittest for the experiment, as it is
sound in all figures.

True Conclusions from False Premises.

(81.) True" conclusions may be logically
deduced from false premises in every correct
form of categorical syllogism.

This proposition has been formally de-
monstrated in the first book, by means of
the symbolical expression of the syllogistic
laws.

The following is another rather preferable
proof, which could not with propriety have
been introduced there, as it rests upon certain
considerations from Transcendental Logic.

As no categorical proposition is either for-
mally identical, or formally exhaustive, (which

LIMITATIONS, &C, 133

latter form would really be negatively identical,
since it would assert the identity of not B with
A,) it follows that no categorical proposition
completely determines the whole of one term
with respect to the whole of the other. For
instance, in the proposition ' No A is B,' each
term is placed in the external sphere of the
other, but whereabouts in it, is quite undeter-
mined. Again, in the proposition ' All A is B,'
B is a conception, the excess of whose sphere
above that of A is also undetermined, as it may
vary from nothing to infinity. Now the inde-
terminate parts of the premises can never be
introduced in the conclusion, which must
follow from them necessarily, and therefore
depend on the determinate alone. Hence it is
evident that the materially necessary parts of
the premises are invariably less than those
which are formally necessary. If then we
suppose an error in the indeterminate, and
therefore unavailable parts of the premises, they
will of course be false, although the conclusion,
which is materially dependent on the remaining
parts of those premises, is itself true.

It ought to be, if it is not, an axiom in an
a priori science, that the general proof should
invariably precede all reference to particular
examples. But the reasoning in the last para-
graph may now be elucidated, by taking as an

134 A SYLLABUS OF LOGIC.

example the mood A A A. The general form
of a syllogism in this mood is the following :
All B is A,
All C is B,
Therefore All C is A.

In this syllogism, the indeterminate parts of
the terms are the excess of A above B, and
of B above C, and these parts are accordingly
unavailable in the conclusion. For it is only
so far as C agrees with B, that it agrees with
what B agrees ; and in the same manner it
is only so far as B agrees with A, that C, which
agrees with B, can be concluded to agree with
A. Let us then assume an error in the com-
parison of that part of B, which exceeds C,
with A. Still A will be predicated of so much
B as agrees with C, and consequently the con-
clusion will be true, though the major premiss
is false.

The following is an example of a sound syl-
logism, of the form A A A in the first figure, in
which the major premiss is false, though the
conclusion is true : —

' All animals are quadrupeds/
' All horses are animals/
Therefore ' All horses are quadrupeds.'

We subjoin three diagrams of this form
of syllogism, in which the conclusion is true,
though one or both premises are false. In

LIMITATIONS, &C. 135

Fig. 1, both premises are false ; in Fig. 2, the
major premiss is false ; and in Fig. 3, the minor
premiss is false.

Fig. 1.

False, < All B is A.'
False, < All C is B.'
True, < All C is A.'

False, < All B is A.' True, ' All B is A.'

True, ' All C is B.'
True, < All C is A.'

False, < All C is B.'
True, « All C is A.'

These diagrams will also represent many
other forms of syllogism, in which the con-
clusion is true, though one or both premises
are false. For instance, if A be taken as the
middle term, B as the major, and C the minor,
Fig. 1 will represent the mood A E E in the
second figure with a true conclusion, though
both its premises are false. Again, if C be
taken as middle term, A as major, and B as
minor, Fig. 2 will represent the mood E I O
in the first figure having its conclusion and
minor premiss true, but its major premiss false.
And similarly a great many other cases might
be adduced, to which the above diagrams would
be equally applicable.

This manner of proof may easily be extended
to the other forms of syllogism. For instance,

136 A SYLLABUS OF LOGIC.

in the hypothetical syllogism the antecedent
contains the grounds of the truth of the con-
sequent; but as it may also contain much
more, their exact limits cannot be determined.
The truth therefore of the consequent will be
only formally, and not materially dependent on
some part of the antecedent ; and if an error
is introduced in the minor premiss of a syllo-
gism in the modus ponens, and in that part
of it which does not constitute one of the
grounds of the consequent, the result will be
a true conclusion, though one of the premises
is false : e. g. 'If the whole of the regi-
ment were going to Canada, Captain A. would

go ; but the whole of the regiment are

going to Canada ; therefore Captain A. will go/
Now the antecedent contains all the necessary
grounds for the truth of the consequent, and
a great deal more besides ; and consequently
our minor premiss, which states that the whole
regiment is going, might be false, and yet
the conclusion that ' Captain A. is going' true.

As the conclusion must always be true if
the premises are true, and will sometimes be
true when they are false, it follows that the
probability of the truth of the conclusion must
always be rather greater than the probability
of the truth of the premises.

LIMITATIONS, &C. 137

Conclusions of the Reason.

(82.) The limitation which the understanding
imposes upon the actual use of the reason, is
very well calculated to place in a clear and
strong light the distinct functions of the two
faculties in every categorical syllogism ; it is
therefore briefly re-considered here.

The reason can deduce a conclusion from
premises of the form I O in the first figure,
by precisely the same mental operation as from
the premises A A, or premises of any other
legitimate form. These propositions are of
the form

' Some B is A ;
' Some C is not B :'

from which the conclusion may be logically
deduced, that ' Some C is not some A.'

But this conclusion, though derived by
the same function of the reason as any
other legitimate conclusion (i. e. by a con-
junction in the consciousness of two acts of
the understanding), is nevertheless absolutely
worthless, as it may be predicated a priori
of any objects of which the understanding can
think. For whatever be the nature of the
hypothesis respecting the relation of C and A,
(e. g. let 'All C be identical with all A/) still
will it be true that ' Some C is not some

138 A SYLLABUS OF LOGIC.

A.' And here therefore we find the logical
reason performing its regular office in com-
plete blindness, and quite independently of
the nature of the result when considered in
reference to the understanding.

( 139

SECTION II.

MODALITY OF SYLLOGISM.

(83.) Modality has been already defined as
the determination of a judgment in respect of
its relation to truth. If the matter of the judg-
ment is merely in accordance with the a priori
laws of thinking, the judgment is, in respect
of its modality, problematical, or of the lowest
degree : if it is in accordance not only with
those laws, but also with the matter of the
senses, it is assertive, or of the second degree ;
but if one of these a priori laws becomes the
matter of the judgment, the latter is, in respect
of its modality, necessary, or of the highest
degree.*

As the formal use of the reason is quite in-
dependent of the matter of judgments, which
at the same time determines their modality, it
follows that this point of judging cannot have
any influence on the nature of the conclusion
in respect of its categorical form. But this
remark does not extend to the modality of

* Vide Arts. 25, 26, on this subject.

140 A SYLLABUS OF LOGIC.

the conclusion, which may in some measure be
determined by that of the premises, and the
extent to which one is a criterion of the other
will supply the topic of the present section.

(84.) In order to place this subject in as clear
a light as possible, it will be necessary to say
a few words on the different sides from which
the modality of a conclusion may be viewed.
And in the first place, great care must be
taken never to confound the modality of this
proposition, 'that such a conclusion follows
from such premises/ with the modality of the
conclusion considered merely as to its own
matter, and quite independently of its grounds.

The former modality is always of the highest
degree, as it constitutes the very essence of a
conclusion that it follows necessarily from its
premises ; whereas the real modality of the
conclusion depends on the peculiar nature of
its matter, and is determinable in the same
manner as that of any other judgment.

But this distinction will become more ap-
parent, when the above proposition assumes
its appropriate form of a hypothetical, in
which the conclusion is the consequent, and
the premises the antecedent. It may then be
stated thus : If such and such premises are
both true, it will follow that such a conclusion
is true. Now the relation between these

MODALITY OF SYLLOGISM. 141

judgments is unquestionably necessary, as it
consists in those laws of the reason by which
the syllogism is known to be a correct one ;
but this is no criterion whatever for the moda-
lity of the judgments themselves, which consti-
tute the matter of the hypothetical, and are
therefore only propounded problematically.
Before proceeding any further, an example
will throw some light upon the foregoing
remarks.

'All men are liars;'

' Obadiah is a man ;'
Therefore ' Obadiah is a liar/
Now the modality of this conclusion, con-
sidered in reference to the premises, is of the
highest degree, or necessary ; for the propo-
sition that ' Obadiah is a liar* follows ne-
cessarily from the two propositions that 'all
men are liars,' and that ' Obadiah is a man.'
But the modality of this conclusion considered
by itself is by no means necessary, since no
necessity is contained in the proposition that
' Obadiah is a liar,' for no contradiction is
involved in the supposition that he always tells
the truth.

As we shall have occasion to recur to these
two views of the modality of a conclusion, we
shall designate them by the terms ( proper,' or
that of any judgment considered merely as to

142 A SYLLABUS OF LOGIC.

its own nature, and 'consequential,' or that
which depends on its reference to the judg-
ments from which it is deduced.

(85.) The proper modality of a conclusion
cannot be fully determined by that of the pre-
mises, for the modality of the premises depends
on their matter, and therefore on the middle
term which occurs in each of them. But the
conclusion will be unaffected as to matter, and
therefore as to modality, by any change in the
matter of the middle term, provided its formal
quantity is preserved. Hence it follows that
a change is possible in the modality of the
premises, while that of the conclusion remains
the same, and therefore that the modality of
the one can never be fully determined by that
of the other. This will appear from the
following example. Let the two premises be -

' All animals are organic beings ;'

' All horses are animals.'

These propositions are analytical, and there-
fore necessary ; and the conclusion derived
from them, that ' All horses are organic beings/
is also analytical and necessary. But provided
the formal quantity of the middle term ' ani-
mals, be retained in each premiss, any other
matter may be substituted for it without
affecting the legitimacy of the reasoning.
Thus let the term ' cows' be substituted for

MODALITY OF SYLLOGISM. 143

the middle term e animals/ ' All cows are
organic beings ; all horses are cows ; therefore
all horses are organic beings.' Now the same
conclusion has been logically deduced from
these as from the original premises, and con-
sequently its modality is the same in each case.
But in the former syllogism the modality of
both the premises was necessary ; whereas in
the latter, one of them is so far from being
necessary that it is not even true. Hence it is
evident that the nature of the premises can
never be a complete criterion for the proper
modality of the conclusion. There is, how-
ever, one law which will enable us to arrive at
some sure knowledge on this subject, and may
be stated thus : The modality of the conclu-
sion is never of a lower degree than the lowest
in either premiss.

(86.) Previous, however, to the consideration
of this law, it will be necessary to shew why the
case of two problematical judgments must be
entirely put out of the question : for two
judgments of this modality always leave the
possibility of such a disjunctive relation exist-
ing between them, that they are never asser-
tively true simultaneously.* No conclusion,

* It is necessary to distinguish between events that are * simul-
taneously possible,' and 'possible simultaneously.' The first
signifies that ' each is at the same time possible to happen;' the

144- A SYLLABUS OF LOGIC.

therefore, can be deduced from them : but the
truth of this remark will become more evident
by an example. Let us suppose that a box
contains only black and white balls, of which
I abstract several, and put them in a bag
without observing their colour : T may then
state as problematical judgments—

' All the balls in the bag may be white ;
several black balls may be in the bag.' But I
cannot deduce from these judgments even the
problematical conclusion that ' several black
balls may be white/ although such a syllogism
in respect of its mere rational ft qm (i. e. A I I
in the first figure) would be perfectly legitimate.
It is evident in this example that a disjunctive
relation exists between the two premises, and
consequently that they can never be true simul-
taneously, though either may be true when
taken alone. As however the assertive modality
of either premiss would destroy the possibility
of such a relation, the objection will only
apply to those cases in which both are proble-
matical.

other, that « both may possibly happen at the same time.' If I put
my hand into a bag and draw out only one ball, I may draw out
either a black one or a white one, and these events are therefore
simultaneously possible. But as one ball cannot be both a black
ball and a white ball too, these events are not possible simultane-
ously. The judgments in the example in the text, and also in
every disjunctive proposition, are simultaneously possible, but not
possible simultaneously.

MODALITY OF SYLLOGISM. 145

(87.) After the exclusion of this particular
combination of premises, five others will remain
to which no such objection can be offered.
These will accordingly come under the law,
that the modality of the conclusion is never
of a lower degree than that in the weakest*
premiss.

To attempt a strict demonstration of any
law respecting the operations of the Reason,
involves the absurdity of making that faculty
both judge and defendant in its own case : for
in every proof the reason must tacitly assume
the validity of its own laws, and any pretend-
ed demonstration of them from themselves is
open to the objection of reasoning in a circle.
It is useless therefore to sue the reason in
its own court. All that can be done in a
question respecting its functions, is to place
it in as axiomatic a light as possible, and
admit as final the decision of the faculty
itself. In the present instance it is perhaps
sufficient to observe, that inasmuch as the
connection between the major arid minor
terms in the conclusion is entirely dependent
on their previous comparison with the middle
term in their respective premises, this connec-
tion must be of at least as high a degree as

* By the term ' weakest' that premiss is intended, whose mo-
dality is of at least as low a degree as that of the other premiss.

L

A SYLLABUS OF LOGIC.

the lowest between the middle term and either
extreme. And the same reasoning will also
make evident the impossibility of logically
deducing from the premises a higher degree
of modality for the conclusion, * though
such may perhaps exist. It will therefore
be necessary to introduce the distinction of
'proper' and 'derived' modality, for the pur-
pose of avoiding those long periphrases which
would otherwise be perpetually recurring.

(88.) Perhaps the only great logical error
of which Aristotle is guilty in his Analytics,
refers to this very point of the modality of
syllogism ; and as an exposure of the fallacy
will supply the best explanation of the last few
paragraphs, we shall briefly consider it here.

The following passage will be found in the
ninth chapter of the first book of the Former
Analytics : —

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* This law is precisely analogous to the mechanical axiom
that every chain is of the same strength as its weakest link.

MODALITY OF SYLLOGISM. 147

(Translation*} — It happens in some cases
that if one premiss is necessary the syllogism
becomes necessary (i. e. the conclusion of the
syllogism), not however either premiss at ran-
dom, but that which involves the major ex-
treme; just as if, for instance, B is assumed
to be A or not A, of necessity, and C is assumed
merely to be B (i. e. assertively, not necessarily);
for the premises being thus assumed, C will be
A, or will not be A, of necessity. For since
' All B is A, or is not A,' of necessity, and ' C is
a part of B/ it is evident that C will be one of
these things (i. e. ' A or not A') of necessity.

Now as merely assertive judgments can only
have empirical grounds for their truth, it
follows that their predicates are stated of the
sphere or individuals contained under the
conception of the subject, and not of the
conception itself. For a conception can only
be predicated of another conception, (prior at
least to the empirical consideration of the
sphere of the latter,) either by being a superior
conception, in which case the judgment is
analytical, or else by some a priori law ; but
in either case the judgment is necessary, and
therefore more than merely assertive.

Now, by the hypothesis, the minor premiss
in the above syllogism is assertive, and con-
sequently the middle term B is only predicated

L 2

148 A SYLLABUS OF LOGIC.

of the minor term C in respect of its sphere,
and not of its matter. But the truth of the
conclusion (considered as such) depends en-
tirely on this minor premiss, and in that
judgment, therefore, A must be predicated of C
in respect of its sphere and not its matter.
But when a conception is predicated only of
the sphere of another, the judgment cannot
be necessary, as it involves no a priori law of
thinking, and conveys no certainty that an
exception may not be found on some future
occasion. Hence the derived modality of the
conclusion that ' all C is A, or is not A,' can
never exceed the assertive or second degree,
which is that of the weakest premiss. Let us
take the following example :

' All members of Caius College are members
of the University. Several Norfolk men are
members of Caius College. Therefore several
Norfolk men are members of the University/

Now the major premiss of this syllogism is
analytical, and therefore necessary, for it con-
stitutes a part of the conception of a member of
a college that he should be a member of the
University. But the minor premiss on the
contrary is merely assertive, as it is by no
means necessary to our conception of Norfolk
men that several of them should be members
of Caius College. And the conclusion is also

MODALITY OF SYLLOGISM. 149

merely assertive, as it does not necessarily
enter into our conception of Norfolk men that
they must be members of the University.

It is evident from this example, that only
an assertive conclusion can be deduced from
a necessary major and an assertive minor
premiss. But this law regards the derived
modality alone, and leaves the proper moda-
lity quite undetermined. For a trifling alter-
ation in the example just given will render
the proper modality of the conclusion neces-
sary, and leave that of the premises unaltered.
Let the minor term be changed into ' Several
members of the University who reside in
Norfolk ;' the minor premiss will still be only
assertive, as it is not necessary that any one
of these men should be a member of Caius
College ; but the conclusion will be analytical,
and therefore its proper modality will be ne-
cessary.

But, that Aristotle never intended to say
that the conclusion from premises so assumed
might sometimes* be necessary in its own
nature, and sometimes only assertive, appears
from his giving this proof in the general
symbols A, B, C ; and also from his predi-

* The restriction contained in the word TTOTS, evidently alludes
to the limitation of the law to those cases in which the major
premiss is necessary, and the minor assertive.

150 A SYLLABUS OF LOGIC.

eating necessity of the syllogism, and not
merely of the conclusion. Neither is he merely
alluding to the consequential modality of the
conclusion, for he distinguishes between the
proper and consequential modalities more
than once, and particularly in the middle of
the following chapter. So that his evident
meaning is this : if a necessary major premiss
and an assertive minor premiss be assumed
in the first figure, that modality which can
be derived from that of the premises for the
conclusion as a judgment considered by itself,
will be necessary and not merely assertive.

(89.) A fallacy involving a breach of the
same law is to be found at the commencement
of the nineteenth chapter of the first book
of the Former Analytics.* Aristotle here lays
down as a general rule, that if one premiss
is necessary and negative, and the other
problematical and affirmative, an assertive
conclusion may be deduced from them.

" For let ' All B be necessarily not A/ and
' All C be possibly A :' if the negative premiss
is converted, ' All A will be necessarily not B ;'
but 'All C may possibly be A;' there will
therefore be a syllogism in the first figure to
the effect that ' All C is possibly not B.' But

* This passage is at page 97 of Bekker's 8vo. Edition, Oxford

reprint: 'Ei 6' »j /ULIV 1% ai/ay/ojs •• TeOgftj TO <rT£/ot}Ti/c<)j/.

MODALITY OF SYLLOGISM. 151

it is evident, moreover, that ' All C will be not
B.' For let us assume that ' Some C is B :' if
then ' All B is necessarily not A/ and ' Some
C is B,' it follows that ' Some C cannot be A,'
which is absurd, as by the original hypothesis,
'All C may be A.' Therefore, &c."

This reasoning is extremely ingenious, but
not less sophistical. It will be found to hinge
upon the very fallacy which has just been
exposed in the last paragraph. For the con-
clusion which Aristotle wishes to deduce, is
assertive, viz. ' All A is B :' he accordingly
assumes the assertive contradictory of this
conclusion as the minor premiss of the first
syllogism in an indirect proof, and from this
assertive minor, and the original necessary
major premiss, fallaciously deduces a necessary
conclusion, to the effect that, ' Some C cannot
be A.' And as this conclusion is at variance
with the other premiss that ' All C may be A,'
he infers that his assumption of the contra-
dictory of the proposition ' All A is B' was
unwarranted, and therefore concludes that
' All A is B.' Now the necessity of the con-
clusion that ' Some C cannot be A* was merely
consequential and not proper, and the legiti-
mate conclusion that ' Some C is not A' is not
at all opposed to the original minor premiss that
' All C is possibly A/ For this, as well as

152 A SYLLABUS OF LOGIC.

every other problematical conclusion, contains
the conception of the possibility of the con-
trary. The proposition ' All C is possibly A*
is therefore perfectly compatible with the
proposition that 'Some C is possibly not A/
And it must be observed that the assertive
modality of the conclusion that * Some C is
not A/ arises from the assumption of the
assertive premiss that ' Some C is B/ which
latter proposition states more than the pre-
mises would warrant in the conclusion of the
original syllogism, though it is by no means
contradictory to it. We will bring the question
to an experimentum crucis, which will render
the fallacy immediately evident. This will be
most readily effected by taking a particular
example, and carrying out the whole argu-
ment in precisely the same form as that in
which it is given by Aristotle.

Let a bag contain several balls whose colour
is unknown, and may therefore possibly be white
or black, or any other colour. The necessary
and negative premiss may then be, ' No white
balls are black balls/ and the problematical
affirmative, 'All the balls in the bag may be
white balls ;' from which Aristotle's reason-
ing would deduce as conclusion, that ( none
of the balls in the bag are black balls.' His
proof would run as follows : ' Let some of the

MODALITY OF SYLLOGISM. 153

balls in the bag be black balls; as no black
balls can be white balls, it follows that some
of the balls in the bag cannot be white balls.
But this is absurd, for by the hypothesis all
the balls in the bag may be white. There-
fore, &c.'

This species of fallacy will become yet
more apparent by stripping it of the syllo-
gism with which it is connected, and proving
by its means that every judgment which is
true problematically, must also be true asser-
tively. Let the judgment be, ' All A may pos-
sibly be B.' It will follow from this that ' All
A is B.' For if it is false that ' All A is B,' it
must necessarily be true that ( Some A is
not B.' But this is absurd, as by the hypo-
thesis ' All A may be B.' Hence it follows, that
the assumed falsity of the proposition ' All A
is B,' must itself be false, and therefore that
< All A is B.'

Both of these fallacies of Aristotle are of
considerable importance, as they violate this
fundamental principle of the modality of syl-
logism, namely, that the derived modality of
the conclusion is the same as that of the
weakest premiss.

(90.) There is however another fallacy, very
near the end of the twenty-eighth chapter of the
same book, which is introduced here rather

154 A SYLLABUS OF LOGIC.

as a curiosity than for any other reason, as
in all probability it is the only instance of a
false mood and figure in the whole work. It
is stated in this passage that the denying B of
H is exactly equivalent to identifying B with
some T, where T has been previously assumed
to represent the whole external sphere of a
certain term E, which is itself a predicate of
H. This paralogism is of the form A E E in
the first figure, which has an illicit major, or
E A E in the fourth figure, which has an illicit
minor.*

It is rather remarkable, that, notwithstand-
ing Aristotle's entire rejection of the fourth
figure, he has introduced it once in this very
chapter, in the mood A A I.f

* Page 115, To yap /urj ti^t'xo-flcu TO B, K. T. \.
f Page 112, El 8s TW H TO B TOLVTOV, K. T. \.

APPENDIX.

THE following are a few examples in which the reader
can try his skill in detecting fallacies, determining the
peculiar form of syllogisms, and supplying the sup-
pressed premises of enthymemes. The arguments alone
have been adopted from the different authors whose
names are attached, as alterations in the mode of ex-
pressing them have invariably been found necessary to
bring them a little nearer the simple syllogistic forms.
Several of the examples contain more than one syllo-
gism.

(1.) None but those who are contented with their
lot in life can justly be considered happy. But the
truly wise man will always make himself contented with
his lot in life, and therefore he may justly be consi-
dered happy.

(2.) All nations, whose commerce has been very ex-
tensive, have reached a great height in refinement and
luxury. The Romans reached a great height in refine-
ment and luxury. Therefore the Romans must have
had a very extensive commerce.

156 APPENDIX.

(3.) A really terrible enemy would never excite so
small a degree of fear that any other passion could
master the feeling. But there is no passion so weak
that it cannot master the fear of death. Therefore
death is no very terrible enemy. — Bacon.

(4.) None but the contented are happy; the good
are happy; therefore they are contented.

(5.) If there is a possibility of the existence of God,
nothing can be more evident than that men ought to
live virtuously and piously, and that vice is the most
absurd thing in nature. But that God exists is per-
fectly possible, as no demonstration can be given of
the contrary. Therefore men ought to live virtuously
and piously, and vice is the most absurd thing in
nature. — Clarke.

(6.) If all cats are animals, and all animals are organic
beings, it follows that all cats are organic beings. But
all cats are organic beings. Therefore all cats are ani-
mals, and all animals are organic beings.

(7.) A statesman should particularly avoid anything
that tends to bring him into contempt. Idle and
frivolous duels will probably have this effect, and he
should therefore particularly avoid them. — Taylor's
Statesman.

(8.) All intelligible propositions must be either true or
false. The two propositions ' Caesar is living still,' and
* Caesar is dead,' are both intelligible propositions ; there-
fore they are both true, or both false.

(9.) God acts according to laws because he knows
them, he knows them because he has made them, he has
made them because they bear a certain relation to his
wisdom and power; therefore God acts according to

APPENDIX. 157

laws because they bear a certain relation to his wisdom
and power. — Montesquieu.

(10.) Blessed are the pure in heart, for they shall see
God.

(11.) None but the good are really great, and all the
good are happy. The slaves of passion are never really
great, and therefore they are never happy.

(12.) No man who is in London can be in York. As
no person has told me where Mr. A is, for all I know to
the contrary he may be in York. Therefore I conclude
that he is not in London. For if this conclusion is false,
it must be true that he is in London/ But no man that
is in London can be in York. Therefore Mr. A is not
in York, which is contrary to the original hypothesis
that he may be in York ; and therefore the second
hypothesis, that he is in London, must be false, and
Mr. A is not in London.

(13.) All God's gifts are intended for use. Our
foresight and power over the future are God's gifts.
Therefore it is intended that we should use our fore-
sight and power over the future. But Slavery pre-
cludes all possibility of the exercise of these powers ;
it is therefore opposed to the intentions of Providence
— Channing.

(14.) Many things are more difficult than to do no-
thing. Nothing is more difficult to do than to walk
on one's head. Therefore many things are more dif-
ficult than to walk on one's head.

(15.) If God does not grant his grace upon the
same conditions to all mankind, he is a respecter of
persons. But God is no respecter of persons. There-

158 APPENDIX.

fore God grants his grace upon the same conditions
to all mankind.

(16.) He that is of God heareth God's words:
ye therefore hear them not, because ye are not of God.
— John, c. viii, v. 47. Quoted from Whately^s Logic.

(17.) It is highly probable that all persons who
have established a new religion entirely subversive of
the old, have suffered persecution. The first preachers
of Christianity were such persons. It is therefore highly
probable that they suffered persecution. — Paley's Evi-
dences.

(18.) The men who barter their eternal welfare for
temporary gratifications are very deficient in real wis-
dom. But the men who are thus deficient are by no
means few. Therefore the men who barter their eternal
welfare for temporary gratifications are by no means few.

(.19.) None but Whigs vote for Mr. B. All who vote
for Mr B. are ten-pound householders. Therefore none
but Whigs are ten-pound householders.

(20.) The waking state succeeds the sleeping, and the
sleeping succeeds the waking ; things become cold from
having been hot, and hot from having been cold ; men
can only become taller from having been shorter, and
shorter from having been taller. Thus all contraries
mutually produce, and are produced from, each other.
But the states of life and death are contrary to each
other. Therefore the state of life succeeds that of death,
as that of death does that of life.— Plato's Phcedo.

(21.) If the Mosaic account of the cosmogony is
strictly correct, the sun was not created till the fourth
day. And if the sun was not created till the fourth day,
it could not have been the cause of the alternation of day

APPENDIX. 159

and night for the first three days. But either the word
' day' is used in Scripture in a different sense to that in
which it is commonly accepted now, or else the sun must
have been the cause of the alternation of day and night
for the first three days. Hence it follows that either
the Mosaic account of the cosmogony is not strictly
correct, or else the word e day' is used in Scripture in
a different sense to that in which it is commonly ac-
cepted now.

(22.) Men who would peril their own lives and those
of their fellow-creatures on no better pretext than that of
maintaining a reputation for courage, would unquestion-
ably engage in duels upon very slight provocation.
But none except very weak men would ever peril any
human life with such an object alone, and we therefore
conclude that none but very weak men would engage in
duels upon very slight provocation,

(23.) Laws, in the widest acceptation of the word, are
the necessary relations which derive their origin from
the nature of things. The Deity, therefore, has his laws,
for he bears a certain relation to the universe as its
creator and preserver. — Montesquieu.

(24.) Suffering is a title to an excellent inheritance;
for God chastens every son whom he receives. (Quoted
literally from Jeremy Taylor's Holy Living and Dying.)
This sentence may be put under the following form : All
whom God receives, he chastens ; all who suffer, God
chastens ; therefore, all who suffer, God receives.

(25.) To do a certain evil for a problematical good is
contrary to the spirit of Christianity. But to hurry
a great criminal into the presence of his Creator, and this
for the sake of the very questionable advantage which

160 APPENDIX.

capital may have over other severe punishments in
deterring others from crime, is to do a certain evil for
a problematical good. Therefore capital punishment is
contrary to the spirit of Christianity.

ADDITION TO NOTE p. 117.

The complete elimination of the indeterminate symbols x, y, m, n
from the equations on note to p. 117, will furnish additional equa-
tions representing the general laws of Categorical Syllogism in
their simplest form : each of whose solutions will correspond to a
possible mood.

It is unnecessary to insert the process of elimination ; but it
will be seen without difficulty that the equations thus obtained
are the following : —

Quality pl + p2 + p'a = d + 2 u (1).

Laws of particular ((sl — d) (s2 — d) = CH

premises and par- < (st — d) (s'3 — d) = 0 I (2).

ticular conclusion. [(s2 — d) (s'3 — d) = OJ

I E O excluded as f

leading to illicit 1 (s} — d) (pi _ d) (p'3 — d) = 0... (3).

major. [

(1.) Implies that two of the quantities ptp2p'3 must be = u,
and a third = d, and admits therefore of 3 solutions.

(2.) Imply that any two or all three of the quantities s{ s2 s'3
must = d . . and admit therefore of 4 solutions.

The whole number of solutions therefore admissible from (1)
and (2) is 3 X 4 = 12.

Of these one is rendered inadmissible by (3). The whole num.
ber of possible moods is therefore 1 1 .

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