The Mathematical
Analysis of Logic By
GEORGE BOOLE
I.I
NIVER51TY OF TORONTO
UNiV
OF TORONTO
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The Mathematical
Analysis of Logic By
GEORGE BOOLE
I.I
NIVER51TY OF TORONTO
UNiV
OF TORONTO
^
THE MATHEMATICAL ANALYSIS OF LOGIC
THE MATHEMATICAL ANALYSIS OF LOGIC BEING AN ESSAY TOWARDS A CALCULUS OF DEDUCTIVE REASONING
By
GEORGE BOOLE
PHILOSOPHICAL LIBRARY
NEW YORK
Published in the United States of America 1948, by the Philosophical Library, Inc., 15 East 40th Street, New York, N.Y.
THE MATHEMATICAL ANALYSIS
OF LOGIC,
BEING AN ESSAY TOWARDS A CALCULUS OF DEDUCTIVE REASONING.
BY GEORGE BOOLE.
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ARISTOTLE, Anal.
CAMBRIDGE
Posf., lib.
:
MACMILLAN, BARCLAY, & MACMILLAN LONDON: GEORGE BELL. 1847
i.
;
cap. xi.
o
/
PRINTED IN ENGLAND BY
HENDERSON & SPALDING LONDON. W.I
PREFACE. IN presenting
this
Work
deem
public notice, I
to
irrelevant to observe, that speculations similar to those it
at different periods,
records have,
In the spring of the present year then
to the question
my
occupied
my
moved between
attention Sir
W.
it
not
which
thoughts.
was directed
Hamilton and
Professor De Morgan; and I was induced by the interest which it inspired, to resume the almostforgotten thread of
former inquiries.
It
appeared
to
me
that,
although Logic
of quantity,* it might be viewed with reference to the idea had also another and a deeper system of relations. If it was lawful to regard it from without, as connecting itself through the
medium
of
Number
was lawful
it
facts of
also
to
regard
it
from within,
based upon
as
another order which have their abode in the consti
of the Mind.
tution
with the intuitions of Space and Time,
which
inquiries
it
The
results
and of the
of this view,
suggested, are embodied, in the following
Treatise. It
the
is
not generally
mode
in
which
permitted to
an Author
his production shall be
to
judged
;
prescribe
but there
two conditions which I may venture to require of those who shall undertake to estimate the merits of this performance. are
The
no preconceived notion of the impossibility of its objects shall be permitted to interfere with that candour and impartiality which the investigation of Truth demands first
is,
that
;
the second shall
not
is,
that their
judgment of the system as a whole
be founded either upon the examination of only * See p. 42.
PREFACE. it, or upon the measure of its conformity with any received system, considered as a standard of reference from
a part of
which appeal
theorems which
It is in the general
denied.
is
occupy the latter chapters of this work, results to which there no existing counterpart, that the claims of the method, as
is
a Calculus of Deductive Reasoning, are most fully set forth.
What may
be the
final estimate
of the value of the system,
have neither the wish nor the right to anticipate. The is not simply determined by its truth
I
estimation of a theory It also
depends upon the importance of
extent of
be
left
its
applications;
human
to the arbitrariness of
of the application
Logic were
its
subject,
to
the science
of the subject permits
danger carried
upon "
:
a principle
which has
Whenever
the nature
the reasoning process to
contrary case
while in the
;
should be so constructed, that there shall be
it
the greatest possible obstacle to a
mere mechanical use of
In one respect, the science of Logic the perfection of
method
its
of the speculative truth of
employment of
be without
on mechanically, the language should be con
structed on as mechanical principles as possible
common
is its
differs
from
principles.
mind an
* Mill s
LINCOLN,
athletic
and
it
the
to the rigour
who knows
and warfare which imparts vigour, and teaches it to contend
to rely
toil
upon
itself in
System of Logic, Ratiocinative
Oct. 29, 1847.
it."*
others;
To supersede
reason, or to subject
the value of that intellectual
difficulties
all
chiefly valuable as an evidence
of technical forms, would be the last desire of one
the
of
solely a question of Notation, I should be content
been stated by an able living writer
with
still
If the utility
Opinion.
of Mathematical forms
to rest the defence of this attempt
to
and the
beyond which something must
emergencies.
and Inductive, Vol.
II.
p. 292.
MATHEMATICAL ANALYSIS OF
LOGIC.
INTRODUCTION. THEY who
are acquainted with the present state of the theory
of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a ques tion on the properties of numbers, under another, that of
a geometrical problem, and under a third, that of a problem This principle is indeed of fundamental of dynamics or optics. it and may with safety be affirmed, that the recent importance ;
advances of pure analysis have been much assisted by the it has exerted in directing the current of
influence which investigation.
But the doctrine
full recognition of the consequences of this important has been, in some measure, retarded by accidental
circumstances.
It
has
happened
in
every
known form
analysis, that the elements to be determined have
of
been con
ceived as measurable by comparison with some fixed standard. The predominant idea has been that of magnitude, or more strictly,
of numerical ratio.
The expression of magnitude, B
or
4
INTRODUCTION.
of operations for for
upon magnitude, has been
the
express
object
which the symbols of Analysis have been invented, and which their laws have been investigated. Thus the ab
stractions of the
modern Analysis, not
less
than the ostensive
diagrams of the ancient Geometry, have encouraged the notion, that Mathematics are essentially, as well as actually, the Science of Magnitude. The consideration of that view which has already been stated, as embodying the true principle of the Algebra of Symbols,
would, however, lead us
to infer that this conclusion is
by no
means necessary.
If every existing interpretation is shewn to involve the idea of magnitude, it is only by induction that we
can assert that no other interpretation is possible. And it may be doubted whether our experience is sufficient to render such an induction legitimate. The history of pure Analysis is, it may be said, too recent to permit us to set limits to the extent of its
Should we grant
to the inference a high degree and with reason, maintain the still, sufficiency of the definition to which the principle already stated would lead us. We might justly assign it as the definitive
applications.
of probability,
we might
character of a true Calculus, that
it
is
a
method
resting
upon
the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent
That
forms of Analysis a quan titative interpretation is assigned, is the result of the circum stances by which those forms were determined, and is not to
interpretation.
to the existing
be construed into a universal condition of Analysis.
It is
upon
the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place the acknowledged forms of Mathematical Analysis, re gardless that in its object and in its instruments it must at
among
present stand alone. That which renders Logic possible, is the existence in our minds of general notions, our ability to conceive of a class,
and
to designate its individual
members by
a
common name.
INTRODUCTION.
The theory
of Logic
is
O
thus intimately connected with that of
A
successful attempt to express logical propositions laws of whose combinations should be founded the by symbols, laws of the mental processes which they represent, the upon
Language.
be a step toward a philosophical language. But a view which we need not here follow into detail.*
would, so this
is
far,
Assuming the notion of a
we
class,
are able, from
ceivable collection of objects, to separate
which belong from the rest.
contemplate them apart Such, or a similar act of election, we may con
to the
given
ceive to be repeated. sideration
those
by
any con
a mental act, those
may
be
and
class,
The group
still
to
of individuals left under con
further limited,
among them which belong
to
by mentally selecting some other recognised class,
as well as to the one before contemplated.
And
this process
may be repeated with other elements of distinction, until we arrive at an individual possessing all the distinctive characters which we have taken
into account,
and a member,
at the
same
we have enumerated. It is in fact which we employ whenever, in common
time, of every class which a
method
language,
more
similar to this
we accumulate
descriptive epithets for the sake of
precise definition.
Now
the several mental operations which in the above case
we have supposed
to
be performed, are subject
It is possible to assign relations
to peculiar laws.
among them, whether
as re
spects the repetition of a given operation or the succession of different ones, or It
is,
for
some other
particular,
which are never
violated.
example, true that the result of two successive acts
* This view
is
is well expressed in one of Blanco White s Letters Logic is most part a collection of technical rules founded on classification. The Syllogism is nothing but a result of the classification of things, which the mind naturally and necessarily forms, in forming a language. All abstract terms are or rather the labels of the classes which the mind has settled." classifications Memoirs of the Rev. Joseph Blanco White, vol. n. p. 163. See also, for a "
:
for the
;
very
lucid introduction, Dr.
Latham
s First
Outlines of Logic applied to Language^
Becker s German Grammar, 8$c. Extreme Nominalists make Logic entirely dependent upon language. For the opposite view, see Cudworth s Eternal and Immutable Morality, Book iv. Chap. in. JB2
6
INTRODUCTION.
unaffected by the order in which they are performed ; and there are at least two other laws which will be pointed out in the
proper place. These will perhaps to some appear so obvious as to be ranked among necessary truths, and so little important as to be undeserving of special notice. And probably they are noticed for the first time in this Essay. Yet it may with con fidence be asserted, that
they were other than they are, the entire mechanism of reasoning, nay the very laws and constitu tion of the
human
might indeed
if
intellect,
exist, but
it
would be
vitally
changed.
A
Logic
would no longer be the Logic we
possess.
Such are the elementary laws upon the existence of which, and upon their capability of exact symbolical expression, the method of the following Essay is founded ; and it is presumed that the object which it seeks to attain will be thought to have been very fully accomplished. Every logical proposition, whether categorical or hypothetical, will be found to be capable of exact and rigorous expression, and not only will the laws of
conversion and of syllogism be thence deducible, but the resolu tion of the most complex systems of propositions, the separation of any proposed element, and the expression of its value in terms of the remaining elements, with every subsidiary rela tion involved. Every process will represent deduction, every
mathematical consequence will express a logical inference. The method will even permit us to express arbi generality of the trary operations of the intellect, and thus lead to the demon stration of general theorems in logic analogous, in no slight
No degree, to the general theorems of ordinary mathematics. inconsiderable part of the pleasure which we derive from the the interpretation of external nature, application of analysis to from the conceptions which it enables us to form of the The general formula to universality of the dominion of law. arises
which we
are conducted
seem
to give to that
element a visible
of particular cases presence, and the multitude extent of apply, demonstrate the
its
sway.
Even
to
which they
the
symmetry
INTRODUCTION. of their
expression
analytical
may
no fanciful
in
sense
be
harmony and its consistency. Now I the same sources of do not presume say to what extent The measure of pleasure are opened in the following Essay.
deemed
indicative of
its
to
may be
that extent
left to
the estimate of those
who
shall
think
But I may venture to subject worthy of their study. intellectual of assert that such occasions gratification are not the
The
here wanting.
laws
we have
to
examine are the laws of
one of the most important of our mental faculties. The mathe matics we have to construct are the mathematics of the human
Nor
intellect.
from
of notice. regard to its interpretation, undeserving even a remarkable exemplification, in its general
all
There
are the form and character of the method, apart
is
theorems, of that species of excellence which consists in free dom from exception. And. this is observed where, in the cor
such a character responding cases of the received mathematics, there is that who think few that The is no means apparent.
by
in analysis
which renders
it
deserving of attention for
its
own
worth while to study it under a form in which be solved and every solution interpreted. can every equation Nor will it lessen the interest of this study to reflect that every will notice in the form of the Calculus peculiarity which they sake,
may
find
it
feature in the constitution of their represents a corresponding
own minds. would be premature to speak of the value which this method may possess as an instrument of scientific investigation. I speak here with reference to the theory of reasoning, and to It
the principle of a true classification of the forms and cases of aim of these investigations Logic considered as a Science.* The
was
in
the
first
instance confined to the expression of the
received logic, and to the forms of the Aristotelian arrangement, * WJiately s Elements of Logic. Indeed Strictly a Science" also "an Art." unless we are willing, with ought we not to reg.ord all Art as applied Science ? Plato, "the multitude/ to consider Art as "guessing and aiming well" "
;
;
Phikbus.
INTRODUCTION. soon became apparent that restrictions were thus intro duced, which were purely arbitrary and had no foundation in
but
it
These were noted as they occurred, and the nature of things. When it became neces will be discussed in the proper place. which sary to consider the subject of hypothetical propositions (in still more, when an for the was demanded general theorems of the interpretation Calculus, it was found to be imperative to dismiss all regard for
comparatively less has been done), and
precedent and authority, and to interrogate the method itself for an expression of the just limits of its application. Still, how
was no special effort to arrive at novel results. among those which at the time of their discovery appeared such, it may be proper to notice the following. ever, there
A logical
But to
be
according to the method of this Essay, the expressible by an equation the form of which determines rules of conversion and of transformation, to which the given proposition
proposition
is
subject.
simple conversion,
is
is,
Thus
the law of what logicians term
determined by the
fact,
that the corre
sponding equations are symmetrical, that they are unaffected by a mutual change of place, in those symbols which correspond to
the convertible classes.
The received laws
of conversion
were thus determined, and afterwards another system, which is thought to be more elementary, and more general. See Chapter,
On the Conversion of Propositions. The premises of a syllogism being expressed by equations, the elimination of a common symbol between them leads to a third equation which expresses the conclusion, this conclusion being
always the most general possible, whether Aristotelian or not. Among the cases in which no inference was possible, it was found, that there were two distinct forms of the final equation. It was a considerable time before the explanation of this fact was discovered, but it was at length seen to depend upon the
presence or absence of a true the premises. is
The
medium
distinction
illustrated in the Chapter,
On
which
of comparison between is
Syllogisms.
thought
to
be new
INTRODUCTION. character of the disjunctive conclusion of a hypothetical syllogism, is very clearly pointed out in the examples of this species of argument.
The nonexclusive
problems illustrated in the chapter, On and the Solution of Elective Equations is conceived to be new
The
class of logical
:
,
is believed that the method of that chapter affords the means of a perfect analysis of any conceivable system of propositions, an end toward which the rules for the conversion of a single it
the first step. categorical proposition are but However, upon the originality of these or
any of these views,
conscious that I possess too slight an acquaintance with the literature of logical science, and especially with its older lite I
am
me to speak with confidence. be not inappropriate, before concluding these obser may a few remarks upon the general question of the offer vations, to rature, to permit It
use of symbolical language in the mathematics. Objections have lately been very strongly urged against this practice, on the ground, that by obviating the necessity of thought, and substituting a reference to general formulae in the room of faculties. personal effort, it tends to weaken the reasoning
Now
the question of the use of symbols may be considered in two distinct points of view. First, it may be considered with reference to the progress of scientific discovery, and secondly, with reference to its bearing upon the discipline of the intellect.
And
with respect to the first view, it may be observed that as it is one fruit of an accomplished labour, that it sets us at arduous toils, so it is a necessary liberty to engage in more result of an advanced state of science, that we are permitted,
and even called upon, to proceed which we before contemplated.
If through the advancing
obvious.
we
find that the
afford
the
no longer a
remedy
tracks,
to
to
is,
to
higher problems, than those
The
proceed
ample
to
seek for difficulties
inference
is
power of scientific methods,
pursuits on which sufficiently
practical
we were once engaged,
field for intellectual effort,
higher inquiries, and, in new And such is, yet unsubdued.
10
INTRODUCTION.
indeed,
the
law of scientific progress. We must be abandon the hope of further conquest, or to
actual
content, either to
employ such
aids of symbolical language, as are proper to the
stage of progress, at to
commit ourselves
which we have arrived. to
so near to the boundaries of possible
the apprehension, that scope will inventive faculties.
In
Nor need we
We have
such a course.
knowledge,
fail
fear
not yet arrived as to suggest
for the exercise of the
second, and scarcely less momentous ques discussing tion of the influence of the use of symbols upon the discipline the
of the intellect, an important distinction ought to be made. It is of most material whether those are consequence, symbols
used with a
understanding of their meaning, with a perfect comprehension of that which renders their use lawful, and an full
expand the abbreviated forms of reasoning which they devolopment or whether they are mere unsuggestive characters, the use of which is suffered
ability to
induce, into their full syllogistic
to rest
upon
;
authority.
The answer which must be given
to the question proposed,
one or the other of these suppositions In the former case an intellectual discipline of a
will differ according as the is
admitted.
high order
is
provided, an exercise not only of reason, but of In the latter case there is no
the faculty of generalization.
mental discipline whatever. It were perhaps the best security against the danger of an unreasoning reliance upon symbols,
on the one hand, and a neglect of their just claims on the other, that each subject of applied mathematics should be treated in the spirit of the
application
methods which were known
was made, but
have assumed.
in the best
The order
would thus bear some
at the
time
when
the
form which those methods
of attainment in the individual
mind
relation to the actual order of scientific
discovery, and the
more
would be offered
to
abstract methods of the higher analysis such minds only, as were prepared to
receive them.
The
relation in
which
this
Essay stands
at
once
to
Logic and
11
INTRODUCTION. to
may
Mathematics,
which has
lately
as to
some notice of the question the relative value of the two
One
of the chief objections which
further justify
been revived,
studies in a liberal education.
have been urged against the study of Mathematics in general, is but another form of that which has been already considered with
And
respect to the use of symbols in particular.
it
need not here
be further dwelt upon, than to notice, that if it avails anything, The it applies with an equal force against the study of Logic. canonical forms of the Aristotelian syllogism are really symbol ; only the symbols are less perfect of their kind than those
ical
of mathematics.
argument, they
If they are as truly
employed
an
supersede the exercise of reason, as does
Whether men
a reference to a formula of analysis.
make
to test the validity of
do, in the
use of the Aristotelian canons, except as present day, a special illustration of the rules of Logic, may be doubted ; yet it
this
cannot be questioned that
when
the authority of Aristotle was
dominant in the schools of Europe, such applications were habit And our argument only requires the admission, ually made. that the case
is
possible.
But the question before us has been argued upon higher grounds. Regarding Logic as a branch of Philosophy, and de fining Philosophy as the
research of tigation
causes,"
of the
"science
and assigning
"
why,
(TO
existence," and "the main business the inves
of a real as its
while Mathematics display Sir W. Hamilton has contended,
Slori,),"
that, (TO oYl)," only the not simply, that the superiority rests with the study of Logic, but that the study of Mathematics is at once dangerous and use "
The trained him
less.*
pursuits of the mathematician
"
have not only not
to that acute scent, to that delicate,
almost instinc
which, in the twilight of probability, the search and discrimination of its finer facts demand; they have gone to cloud
tive, tact
his vision, to indurate his touch, to all
iron chain of demonstration,
and
left
but the blazing light, the
him out of the narrow con
fines of his science, to a passive credulity in *
Edinburgh Review, vol. LXII. p. 409, and Letter
to
any premises, or A.
De Morgan, Esq.
to
INTRODUCTION. an absolute incredulity in In support of these and of other both and argument charges, copious authority are adduced.* I shall not attempt a complete discussion of the topics which all."
are suggested by these remarks. My object is not controversy, and the observations which follow are offered not in the spirit
of antagonism, but in the hope of contributing to the formation of just views upon an important subject. Of Sir W. Hamilton
impossible to speak otherwise than with that respect which
it is is
due
genius and learning. Philosophy is then described as the science of a real existence
and
to
the research
"
word
ton
s
why."
the ancient writers.
that
no doubt may
rest
upon
further said, that philosophy These definitions are common
it is
cause,
mainly investigates the
among
And
of causes.
the meaning of the
Thus Seneca, one of Sir W. Hamil The philosopher seeks LXXXVIIT., "
authorities, Epistle
and knows the causes of natural things, of which the mathe matician searches out and computes the numbers and the mea It may be remarked, in passing, that in whatever the belief has prevailed, that the business of philosophy degree sures."
is
immediately with
science
whose object
causes; is
in
the same degree has every
the investigation of laws, been lightly
esteemed. Thus the Epistle to which we have referred, bestows, by contrast with Philosophy, a separate condemnation on Music
and Grammar, on Mathematics and Astronomy, although Mathematics only that Sir W. Hamilton has quoted.
it
is
that of
Now we
might take our stand upon the conviction of many thoughtful and reflective minds, that in the extent of the mean The business of ing above stated, Philosophy is impossible. true Science, they conclude,
nature of Being, the
mode
is
with laws and phenomena.
The
of the operation of Cause, the why,
* The arguments are in general better than the authorities. Many writers quoted in condemnation of mathematics (Aristo, Seneca, Jerome, Augustine, Cornelius Agrippa, &c.) have borne a no less explicit testimony against other The treatise of the last named sciences, nor least of all, against that of logic. writer
De
Vanitate Scientiantm,
Vide cap. en.
must surely have been
referred to
by mistake.
INTRODUCTION.
13
they hold to be beyond the reach of our intelligence.
But we
do not require the vantageground of this position; nor is it doubted that whether the aim of Philosophy is attainable or not, the desire which impels us to the attempt is an instinct of our
Let
higher nature. the
baffled "
"
science of a real that
which
not a hopeless one;
is
existence,"
for
kernel"
be granted that the problem which has
it
of ages,
efforts
and
"
that the
the research of
causes,"
"
is
Philosophy
human
not transcend the limits of the
still
militant,"
I
intellect.
am
do
then
according to this view of the nature of Philosophy, Logic forms no part of it. On the principle of
compelled
to assert, that
we ought no longer to associate Logic and but Metaphysics, Logic and Mathematics. Should any one after what has been said, entertain a doubt a true classification,
upon
must
this point, I
refer
him
to the
afforded in the following Essay.
He
like
truths,
Geometry upon axiomatic
structed
upon
evidence which will be
will there see
and
Logic resting theorems con
its
that general doctrine of symbols,
which
tutes the foundation of the recognised Analysis.
consti
In the Logic
of Aristotle he will be led to view a collection of the formulae of the science, expressed
by another, but, (it is thought) less scheme of symbols. I feel bound to contend for the absolute exactness of this parallel. It is no escape from the con clusion to which it points to assert, that Logic not only constructs perfect
a science, but also inquires into the origin
own "
"
principles,
It is
a distinction which
wholly beyond the domain of
to inquire into the origin
Review, page 415. tion be maintained
clear
and the nature of
denied
to
their
principles."
But upon what ground can such a ?
What
definition of the to allow
distinc
term Science will
such differences
?
application of this conclusion to the question before us
and decisive.
The mental
discipline which
the study of Logic, as an exact science, as that afforded
its
Mathematics.
mathematicians," it is said,
and nature of
be found sufficiently arbitrary
The
is
by the study of Analysis.
is,
is
afforded by in species, the same is
14 Is
INTRODUCTION. then contended that either Logic or Mathematics can
it
supply a perfect discipline to the Intellect
The most
?
careful
and unprejudiced examination of this question leads me to doubt whether such a position can be maintained. The exclusive claims of either must, I believe, be abandoned, nor can any others, par taking of a like exclusive character, be admitted in their room.
an important observation, which has more than once been made, that it is one thing to arrive at correct premises, and another It is
thing to deduce logical conclusions, and that the business of life depends more upon the former than upon the latter. The study of the exact sciences
may
teach us the one, and
some general preparation of knowledge and of attainment of the other, but
it
action, in the field of Practical Logic, the arena of
we
may
give us
practice for the
union of thought with
to the
is
it
Human
Life,
and more perfect accomplishment. I desire here to express my conviction, that with the ad vance of our knowledge of all true science, an everincreasing
that
are to look for
harmony
will
its
be found
The view winch
fuller
to prevail
among
separate branches.
its
leads to the rejection of one, ought, if con
sistent, to lead to
the rejection of others.
And
indeed
many
of the authorities which have been quoted against the study of Mathematics, are even more explicit in their condemnation of "
Logic.
Natural
science,"
says the Chian Aristo,
Logical science does not concern are founded (as they often are)
us."
When
"
is
above us,
such conclusions
upon a deep conviction of the
preeminent value and importance of the study of Morals, we admit the premises, but must demur to the inference. For it has been well said by an ancient writer, that it is the charac teristic of the liberal sciences, not that they conduct us to Virtue, "
and Melancthon s senti but that they prepare us for Virtue has in abeunt studia mores," ment, passed into a proverb. ;"
"
Moreover, there
is
votaries of truth
a
may
common ground upon which
all
sincere
meet, exchanging with each other the
The works of the language of Flamsteed s appeal to Newton, will be better Providence understood Eternal through your "
labors and
mine."
15
(
)
FIRST PRINCIPLES.
LET
us employ the symbol 1, or unity, to represent the Universe, and let us understand it as comprehending every
conceivable class of objects whether actually existing or not, being premised that the same individual may be found in
it
more than one quality in
class,
inasmuch
common with
as
it
may
possess
other individuals.
more than one
Let us employ the
Z, to represent the individual members of classes, applying to every member of one class, as members of that to every member of another class as particular class, and
letters
X, Y,
X
Y
members
of such class, and so on, according to the received lan
guage of
treatises
Further
let
on Logic.
us conceive a class of symbols x, y,
z,
possessed
of the following character.
The symbol x
operating upon any subject comprehending
individuals or classes, shall be supposed to select from that
Xs which
In like manner the symbol shall be supposed to select from y, operating upon any subject, subject
it
all
all
the
it
contains.
individuals of the class
Y which
are comprised in
it,
and
so on.
When verse) to
no subject is expressed, we shall suppose 1 (the Uni be the subject understood, so that we shall have x = x
(1),
the meaning of either term being the selection from the Universe of all the Xs which it contains, and the result of the operation
16
FIRST PRINCIPLES.
being in common language, the class X, each member is an X.
From
i.
e.
the class of which
these premises
it will follow, that the product xy will represent, in succession, the selection of the class Y, and the selection from the class of such individuals of the class as
Y
are contained in
both
Xs and
represent a
it,
Ys.
X
the result being the class whose And in like manner the
members
product xyz
compound operation
ments are the selection of the such individuals of the class
of which the successive ele
class
Y as
are will
Z, the selection from are contained in
it,
it
of
and the
from the result thus obtained of all the individuals of which it contains, the final result being the class common to X, Y, and Z.
selection
X
the class
From
the nature of the operation which the symbols x, y,
are conceived to represent,
symbols.
An
we
shall designate
them
z,
as elective
expression in which they are involved will be and an equation of which the mem
called an elective function,
bers are elective functions, will be termed an elective equation. It will not be necessary that we should here enter into the analysis of that mental operation
which we have represented by the elective symbol. It is not an aqt of Abstraction according to the common acceptation of that term, because we never lose sight of the concrete, but it may probably be referred .to an ex ercise of the faculties of Comparison and Attention. Our present concern is rather with the laws of combination and of succession,
by which
its
results are governed,
and of these
it
will suffice to
notice the following.
The
result of an act of election is independent of the or of the subject. classification grouping Thus it is indifferent whether from a group of objects con sidered as a whole, we select the class X, or whether we divide 1st.
the group into two parts, select the Xs from them separately, and then connect the results in one aggregate conception. may express this law mathematically by the equation
We
x (u +
v)
= xu + xv,
17
FIRST PRINCIPLES. representing the undivided subject, component parts of it.
u
i
v
2nd.
It is indifferent in
and u and
what order two successive
the
v
acts of
election are performed.
Whether from
the class of animals
we
select sheep,
and from
the sheep those which are horned, or whether from the class of animals we select the horned, and from these such as are sheep,
the result
is
In either case we arrive
unaffected.
at the class
horned sheep.
The
symbolical expression of this law
is
xy = yx.
The result of a given act of election performed twice, number of times in succession, is the result of the same
3rd.
or any
performed once. If from a group of objects we select the Xs, we obtain a class If we repeat the operation of which all the members are Xs. act
no further change will ensue Thus we have
on
this class
we
take the whole.
xx = s*
or
:
in selecting the
Xs
x,
= x
;
and supposing the same operation to be n times performed, we have x n = X)
which
is
the mathematical expression of the law above stated.*
The laws we have
established under the symbolical forms
x (u +
*
The
office of
in the class
X.
= xu + xv xy = yx *n = * v)
the elective symbol
Let the class
whatever the class
Y may be,
X
we
office
least of its
is
(3), individuals comprehended
the
known
"
;
then,
y.
which x performs is now equivalent to the symbol interpretations, and the index law (3) gives
+ which
=
),
(2),
be supposed to embrace the universe have xy
The
x, is to select
(1
= +,
property of that symbol.
f
,
in
one at
18
F1KST PRINCIPLES.
From
are sufficient for the basis of a Calculus. it
the
first
of these,
appears that elective symbols are distributive, from the second
that they are commutative/ properties
which they possess
in
common with symbols of quantity, and in virtue of which, all the processes of common algebra are applicable to the present The one and sufficient axiom involved in this appli system. cation
is
that equivalent operations
performed upon equivalent
subjects produce equivalent results.*
The
third law (3)
we
shall
denominate the index law.
It is
symbols, and will be found of great impor tance in enabling us to reduce our results to forms meet for peculiar to elective
interpretation.
From
the circumstance that the processes of algebra
applied to the present system, it is
may be
not to be inferred that the
interpretation of an elective equation will be unaffected by such The expression of a truth cannot be negatived by processes. * It is generally asserted by writers on Logic, that all reasoning ultimately What depends on an application of the dictum of Aristotle, de omni et nullo. ever is predicated universally of any class of things, may be predicated in like "
comprehended in that class." But it is agreed that this not immediately applicable in all cases, and that in a majority of What are the instances, a certain previous process of reduction is necessary. elements involved in that process of reduction? Clearly they are as much
manner
of any thing
dictum
is
a part of general reasoning as the dictum itself. Another mode of considering the subject resolves
all reasoning into an appli cation of one or other of the following canons, viz. 1 If two terms agree with one and the same third, they agree with each .
other. 2.
If one term agrees,
and another disagrees, with one and the same
third,
these two disagree with each other. But the application of these canons depends on mental acts equivalent to have to those which are involved in the beforenamed process of reduction.
We
select individuals
from
classes, to convert propositions, &c., before
we can
avail
of the process of reasoning is insuffi cient, which does not represent, as well the laws of the operation which the mind performs in that process, as the primary truths which it recognises and
ourselves of their guidance.
Any account
applies. It is presumed that the laws in question are adequately represented by the fundamental equations of the present Calculus. The proof of this will be found
in its
and of exhibiting in the results of capability of expressing propositions, that may be arrived at by ordinary reasoning. processes, every result
its
FIRST PRINCIPLES.
The equation are equivalent, member x, and we have
may be Y and Z
a legitimate operation, but it y = z implies that the classes for
member.
Multiply
it
by a
factor
xy =
limited.
xz,
which expresses that the individuals which are common and Z, and vice and are also common to classes
X
Y
19
X
to the versd.
This is a perfectly legitimate inference, but the fact which it declares is a less general one than was asserted in the original proposition.
OF EXPRESSION AND INTERPRETATION.
A
Proposition is a sentence which either affirms or denies, No creature is independent.
as,
All
men
are
mortal,
A Proposition has necessarily two terms,
as men, mortal; the former of which, the one spoken of, is called the subject the latter, or that which is affirmed or denied of the These are connected together subject, the predicate. by the copula is, or is not, or by some other modiEcation of the substantive verb. The substantive verb is the only verb recognised in Logic ; all others are resolvable by means of the verb to be and a participle or adjective, e.g. The Komans conquered"; the word conquered is both copula and predicate, being equivalent to "were (copula) victorious" (predicate). ;
"
A
Proposition must either be affirmative or negative, and must be also either universal or particular. Thus we reckon in all, four kinds of pure categorical Propositions. Universal affirmative,
1st.
usually represented by A,
Ex. 2nd.
All
Ex. 3rd.
4th.
Xs
are Ys.
Universalnegative, usually represented
by E,
NoXsareYs.
Particularaffirmative, usually represented
by Ex. Some Xs are Ys.
Particularnegative, usually represented
I,
by O,*
Ex. Some Xs are not Ys. 1.
To
express the
class,
notX, that
is,
the class
including
individuals that are not Xs.
all
The But
class
X and the class notX together make the Universe. and the class X determined by the
the Universe
is 1,
symbol x 3 therefore the the symbol 1  x. *
The above
and Whately.
is
taken, with
is
class
notX
little variation,
will
be determined by
from the Treatises of Aldrich
OF EXPRESSION AND INTERPRETATION.
Hence
the office of the symbol
will
subject
select
to
be,
from
 x attached to a given
1
all
it
notXs which
the
it
contains.
And
product xy expresses the entire  x) class whose members are both Xs and Ys, the symbol y (1 will represent the class whose members are Ys but not Xs, in like
and the symbol are neither
Xs
(1
As
the
all
obvious
that
Universe
all
x) (1

all
whose members
Xs,
are Ys.
found in the
exist are
is
class
Y,
it
is
Ys, and from at once from the
all
the same as to select
Xs.
x
or
To
Xs
out of the Universe
select
Hence
3.
y) the entire class
the Proposition, All
Xs which to
these to select

nor Ys.
To express
2.
as the
manner,
(1

xy =
x,
=
0,
y)
(4).
No Xs
express the Proposition,
are Ys.
Xs are Ys, is the same as to assert that Now there are no terms common to the classes X and Y. all individuals common to those classes are represented by xy. Hence the Proposition that No Xs are Ys, is represented by To
assert that
no
the equation
xy =
(5).
Some Xs
4.
To
If
some Xs are Ys, there are some terms common to the and Y. Let those terms constitute a separate class
V,
to
express the Proposition,
are Ys.
X
classes
v,
0,
which there
shall
correspond a separate elective symbol
then v
And
as v includes all
we can
= xy,
terms
indifferently interpret
(6).
common it,
as
to the classes
Some Xs,
or
X
and Y,
Some Ys.
02
OF EXPRESSION AND INTERPRETATION.
Z 5. .
To express
In the
last
the Proposition, Some Xs are not Ys. equation write 1 y for y, and we have
the interpretation of v being indifferently
Some Xs
or
Some
notYs.
The above
equations involve the complete theory of cate gorical Propositions, and so far as respects the employment of analysis for the deduction of logical inferences, nothing
But
more
may be satisfactory to notice some par ticular forms deducible from the third and fourth equations, and can be desired.
it
susceptible of similar application.
If
we
multiply the equation (6) by x,
vx = x*y = xy by
Comparing with
(6),
we
we have
(3).
find v = tx,
or

v (1
And
x) = 0,
multiplying (6) by y,
(8).
and reducing
in a similar
manner,
we have v
= vy,
v (1  y} = 0,
or
Comparing
(8)
and
(9),
vx = ty =
And
(9).
v,
(10).
further comparing (8) and (9) with (4),
we have
as the
equivalent of this system of equations the Propositions
All All
The system
Vs Vs
are
Xs\
are
YsJ
(10) might be used to replace (6), or the single
equation
vx = vy,
(11),
vx the interpretation, Some Xs, and to vy the interpretation, Some Ys. But it will be observed that
might be used, assigning
to
23
ON EXPRESSION AND INTERPRETATION.
system does not express quite so much as the single equa from which it is derived. Both, indeed, express the Proposition, Some Xs are Ys, but the system (10) does not this
tion (6),
imply that the and Y. to
class
V
includes all the terms that are
common
X
In like manner, from the equation (7) which expresses the Proposition Some Xs are not Ys, we may deduce the system ra?
= c(l  y) =
(12),
t>,

Since in which the interpretation of v (1 y) is Some notYs. in this case vy = 0, we must of course be careful not to in terpret vy as If
we
Some Ys.
multiply the
first
equation of the system (12),
ex
by
y,
(I
viz.
y),
we have
..
which
v

is
a
form that
vxy =
ty (1
vxy =
0,

y);
(13),
will occasionally present itself.
necessary to revert to the primitive
pret this, for the condition that vx represents
us by virtue of
(5), that its
It is not
equation in order to inter
Some Xs, shews
import will be
Some Xs the subject comprising all the
are not Ys,
Xs
that are found in the class
Universally in these cases, difference of
V.
fosm implies a dif
ference of interpretation with respect to the auxiliary symbol
and each form
is
interpretable by
r,
itself.
Further, these differences do not introduce into the Calculus a needless perplexity.
a precision
be seen that they give conclusions, which could not
It will hereafter
and a definiteness
to
its
otherwise be secured. Finally,
we may remark
that
all
the equations by which
are deducible from
any one general equation, expressing any one general Proposition, from which those particular Propositions arc necessary deductions. particular truths are expressed,
OF EXPRESSION AND INTERPRETATION.
24
This has been partially shewn already, but exemplified in the following scheme.
x =
The general equation implies that the classes
member
;
X
fully
y,
Y
and
much more
it is
are equivalent,
member
for
that every individual belonging to the one, belongs
Multiply the equation by x, and
to the other also.
we have
= xy x = xy,
z* /.
;
which implies, by (4), that all Xs are Ys. equation by y, and we have in like manner
Multiply the same
y = xy* the import of which
is,
that all
Ys
are Xs.
Take
equations, the latter for instance, and writing (i
we may quantity, will be
regard is
it
sought
as
to
 *) = y
it
either of these
under the form
o,
an equation in which y, an be expressed in terms of x.
shewn when we come
unknown
Now
it
to treat of the Solution of Elective
Equations (and the result may here be verified by substitution) that the most general solution of this equation is
y =
*>*,
which implies that All Ys are Xs, and Multiply by x and we have = vx, vy
that
Some Xs
are Ys.
y
which
indifferently implies that
some Ys are Xs and some
Xs
are Ys, being the particular form at which we before arrived. For convenience of reference the above and some other results
have been
classified
in
the annexed Table,
the
first
column of which contains propositions, the second equations, and the third the conditions of final interpretation. It is to be observed, that the auxiliary equations which are given in this
column are not independent
:
they are implied
either
in the equations of the second column, or in the condition for
OF EXPRESSION AND INTERPRETATION. the interpretation of
v.
But
it
has been thought better to write
And it is them separately, for greater ease and convenience. forms different three further to be borne in mind, that although of the particular proposi are given for the expression of each in the first form. included tions, everything is really
TABLE. The
class
X
The
class
notX
All
Xs
are
Ys i
All
Ys
are
Xs J
All
Xs
are
Ys
No Xs
are
Ys
/vx i
NoYsareXs
~
vx =
SomenotXsareYsr C
Some Xs
are
Ys 
v = some
v = xy
or vx = vy
= y) [or vx (1
C
Some Xs
are not Ys ] I
=
v (1  *) =
,
^
 some Xs
v (1  x)
=

x (1 v y) or vx = v (1  y) or vxy =
0.
some notXs 0.
Xs
or
some Ys
vx = some Xs, vy = some Ys = 0, v (1  y) = 0. v x) (1
v = some Xs, or some notYs vx = some Xs, v (1  y) = some notYs v (1  x) = 0, vy = 0.
OF THE CONVERSION OF PROPOSITIONS.
A
Proposition is said to be converted when its terms are transposed nothing more is done, this is called simple conversion e.g.
;
when
;
No virtuous man is a tyrant, No tyrant is a virtuous man.
is
converted into
Logicians also recognise conversion per accidens, or by limitation, e.g. All birds are animals, is converted into
Some animals
And
Every poet
He who is
E
are birds.
conversion by contraposition or negation, as is
a
not a
man of genius, converted into man of genius is not a poet.
In one of these three ways every Proposition may be illatively converted, and I simply, A and O by negation, A and E by limitation.
The primary
canonical
viz.
forms already determined for the
expression of Propositions, are
Xs
are Ys,
No Xs
are Ys,
All
On
x
(1

=
0,
xy =
0,
y)
Some Xs
are Ys,
v = xy,
Some Xs
are not Ys,
v
we
=x
(1
A. .
.
.
.
E. I.

y) ... .0.
E
and I are sym examining metrical with respect to x and y, so that x being changed into y, and y into x, the equations remain Hence E and I these,
perceive that
unchanged.
may be
interpreted into
No Ys
are Xs,
Some Ys
are Xs,
Thus we have the known
respectively. that particular affirmative
admit of simple conversion.
rule of the Logicians,
and universal negative Propositions
OF THE CONVERSION OF PROPOSITIONS.
The equations
Now obtained
be written in the forms
are precisely the forms
these if
A and O may
we had 
in those equations
which we should have changed x into
1

y,
in
which would have represented the changing the original Propositions of the Xs into notYs, and the
Ys
into notXs, the resulting Propositions being
and y into
1
x,
All notYs are notXs,
Some notYs
are not notXs (a).
Or we may, by simply inverting the order of the second member of 0, and writing it in the form v = (1  y)
interpret
it
by
is really
another form of
are Xs,
Hence
(a).
follows the rule,
and particular negative Propositions
that universal affirmative
admit of negative conversion,
by
t
I into
Some notYs which
x
factors in the
or, as
it is
also
termed, conversion
contraposition.
The
equations
A
and E, written in the forms (1

x=
0,
yz=
0,
y)
give on solution the respective forms
x = vy, x = v the correctness of which
may
(1

y),
be shewn by substituting these
values of x in the equations to which they belong, and observing that those equations are satisfied quite independently of the nature
of the symbol
v.
The
first
solution
Some Ys and the second
may be
are Xs,
into
Some notYs
are Xs.
interpreted into
OF THE CONVERSION OF PROPOSITIONS.
23
From which
it
appears that universalaffirmative, and universal
negative Propositions are convertible by limitation, or, as
it
has
been termed, per accidens. The above are the laws of Conversion recognized by Abp. Whately. Writers differ however as to the admissibility of negative conversion. The question depends on whether we will consent to use such terms as notX, notY. Agreeing with those
who
faulty
and defective.
think that such terms ought to be admitted, even although they change the kind of the Proposition, I am con strained to observe that the present classification of them is into All
Ys
No Xs
Thus the conversion of
are notXs, though perfectly legitimate,
cognised in the above scheme.
It
may
are Ys,
is
not re
therefore be proper to
examine the subject somewhat more fully. Should we endeavour, from the system of equations we have obtained, to deduce the laws not only of the conversion, but also of the general transformation of propositions, we should be led to recognise the following distinct elements, each connected
with a distinct mathematical process. 1st.
or
The negation into X.
of a term,
i. e.
the changing of
X into notX,
notX 2nd.
The
of a
translation
Proposition
we should change Xs are Ys into Some Xs
from one kind
to
another, as if
All
which would be lawful All
Xs
are
Ys
;
are
A
Ys
into I,
or
into
No Xs
are
A
Y.
into
E,
which would be unlawful. 3rd.
The simple conversion
The
conditions in obedience to which these processes
lawfully be performed,
of a Proposition.
may be deduced from
which Propositions are expressed. We have All Xs are Ys x
No Xs
are
Ys
(\

may
the equations by
y) =
0.
A,
xy =
0.
E.
OF THE CONVERSION OF PROPOSITIONS. Write
E
in the
form
by
0 y)} A into
All
Xs
*{i and
it is
so that
interpretable
o,
are notYs,
we may change
No Xs In like manner
Ys
are
A
into All
interpreted by
No Xs so that
=
Xs
E
are notYs.
gives
are notYs,
we may change All
From
Xs
are
Ys
into
we have
these cases
affirmative Proposition
No Xs
are notYs.
the following Rule
:
A universal
convertible into a universalnegative,
is
and, vice versd, by negation of the predicate.
Again, we have
Some Xs
are
Some Xs
are not
These equations only presence of the term
Ys
v
Ys
= x
....
from those
differ
= xy,
last
(1
y).
considered by the
The same reasoning
v.

therefore applies,
and we have the Rule
A particularaffirmative ticularnegative, and
convertible into a par by negation of the predicate. is
proposition
vice versd,
Assuming the universal Propositions All
Xs
No Xs Multiplying by
v,
we
x
are Ysare

(\
Ys
y}
=
xy =
find
vx(\  y) = oxy 
0,
0,
which are interpretable into
Some Xs
are
Some Xs
are not Ys.
Ys
1, .
.
O.
0, 0.
OF THE CONVERSION OF PROPOSITIONS.
30
Hence
a universalaffirmative
convertible into a particular
is
and a universalnegative into
affirmative,
a particularnegative
without negation of subject or predicate.
Combining the above with the already proved rule of simple conversion,
we
arrive at the following system of independent
laws of transformation. 1st.
An
may be changed
affirmative Proposition
into
its
cor
responding negative (A into E, or I into O), and vice versa,
by negation of the
A
2nd.
predicate. /
universal Proposition
may be changed
sponding particular Proposition, (A 3rd.
into I, or
E
into
its
corre
into O).
In a particularaffirmative, or universalnegative Propo may be mutually converted.
sition, the terms
Wherein negation and
vice versd,
and
of a term
is
the changing of
X into notX,
not to be understood as affecting the kind
is
of the Proposition. Every lawful transformation
is
reducible to the above rules.
Thus we have All
Xs
No Xs No
are Ys, are notYs
notYs are
Xs
by
1st rule,
by 3rd
rule,
All notYs are notXs by 1st rule,
which
is
an example of negative conversion.
No Xs
are Ys,
No Ys
are
All
which
is
Ys
Xs
3rd rule,
are notXs
1st rule,
the case already deduced.
Again,
OF SYLLOGISMS.
A
Syllogism consists of three Propositions, the last of which, called the is a logical consequence of the two former, called the premises
conclusion,
;
Prennscs,
Ys
are Xs. z& All Zs are Xs.
(All
\^
Conclusion,
^^
Every syllogism has three and only three terms, whereof that which is the subject of the conclusion is called the minor term, the predicate of the conclusion, the major term, and the remaining term common to both premises, Thus, in ths above formula,
the middle term.
major term,
Y
Z
is
the minor term,
X
the
the middle term.
The figure of a syllogism consists in the situation of the middle term with The varieties of figure are exhibited respect to the terms of the conclusion. the annexed scheme.
m
2nd
3rd Fig.
4th Fig.
YX
XY
YX
XY
ZY ZX
ZY ZX
YZ
YZ ZX
1st Fig.
When we
Fig.
ZX
designate the three propositions of a syllogism by their usual I, O), and in their actual order, we are said to determine
symbols (A, E, the
mood
of the syllogism.
illustration, belongs to the
Thus the syllogism given above, by way
mood
AAA
The moods of all syllogisms commonly received by the vowels in the following mnemonic verses. Fig.
1.
Fig.
2.
Fig.
3.
as valid,
are represented
bArbArA, cElArEnt, dArll, fErlO que pr roris. cEsArE, cAmEstrEs, fEstlnQ bArOkO, secunda?. Tertia dArAptl, dlsAmls, dAtlsI, fElAptOn,
bOkArdO, fErlsO, habet Fig. 4.
of
in the first figure.
:
quarta insuper addit.
brAmAntlp, cAmEnEs, dlmArls, fEsapO, frEsIsOn.
TriE equation by which cerning the
classes
symbols x and
y,
X
we
and Y,
express any Proposition con is
an equation between the
and the equation by which we express any
OF SYLLOGISMS.
Y
and Z, is an equation Proposition concerning the classes and z. from two such equations the If between symbols y
we
eliminate y, the result,
equation between x and
do not vanish, will be an
if it
and
*,
will
constitute
the third
be interpretable
X and Z.
Proposition concerning the classes
And
it
into
a
will then
or Conclusion, of a Syllogism,
member,
of which the two given Propositions are the premises.
The
result of the elimination of
y from the equations
y+*o, =
0,
ab  a b =
0,
ay is
the equation
+
Now the equations of Propositions being of the first order with reference to each of the variables involved, all the cases of elimination which we shall have to consider, will be re d
ducible to the above case,
the
replaced by functions of x,
and the auxiliary symbol
As
z,
constants
a, b,
,
b
,
being
v.
the choice of equations for the expression of our premises, the only restriction is, that the equations must not both be of the form ay = 0, for in such cases elimination would to
be impossible. necessary to
we
When
both equations are of
solve one of them,
choose for this purpose.
the form xy
=
is
it
this form, it is
indifferent
which we
If that
select
which is
of
0, its solution is
y if
and
(!*),
(16),
of the form (1  x) y = 0, the solution will be
y =
vx,
(17),
and these are the only cases which can
The reason
arise.
of this exception will appear in the sequel.
For the sake of uniformity we
shall,
in the
expression of
confine ourselves to the forms particular propositions,
DX = 0y,
vx = v
(1

y\
Some Xs
are Ys,
Some Xs
are not Ys,
S3
OF SYLLOGISMS.
a closer analogy with (16) and (17), than the other forms which might be used.
These have
Between the forms about
to
be developed, and the Aristotelian
canons, some points of difference will occasionally be observed, of which it may be proper to forewarn the reader.
To
the right understanding of these it is proper to remark, that the essential structure of a Syllogism is, in some measure, arbitrary. Supposing the order of the premises to be fixed,
and the
major and the minor term to be purely a matter of choice which of
distinction of the
thereby determined,
it
is
the two shall have precedence in the Conclusion.
have settled is
it
this
question in
that this
clear,
is
Logicians favour of the minor term, but
a convention.
Had
that the major term should have the first clusion, a logical
been agreed place in the con it
scheme might have been constructed,
less
convenient in some cases than the existing one, but superior
What
in others.
Convenience but
it is
Now
to
it
lost in barbara, it
would gain
in Iramantip.
perhaps in favour of the adopted arrangement,* be remembered that it is merely an arrangement. is
method we shall exhibit, not having reference one scheme of arrangement more than to another, will
to
the
always give the more general conclusion, regard being paid only to its abstract lawfulness, considered as a result of pure reasoning.
us the
to
And
therefore
we
shall
sometimes have presented
of conclusions, which
spectacle
a logician
would
pronounce informal, but never of such as a reasoning being would account false.
The
Aristotelian canons, however, beside restricting the order
of the terms of a conclusion, limit their this limitation is of
nature also;
and
more consequence than the former.
We
may, by a change of *
figure, replace the particular conclusion
The contrary view was maintained by Hobbes. The question Hallam s Introduction to the Literature of Europe,
fairly discussed in
is
very
vol. in.
In the rhetorical use of Syllogism, the advantage appears to rest with the rejected form.
p. 309.
04
ON SYLLOGISMS.
of Iramantipy
by the general conclusion of barbara; but cannot thus reduce to rule such inferences, as
Some notXs Yet
we
are not Ys.
there are cases in which such inferences
may
lawfully
be drawn, and in unrestricted argument they are of frequent Now if an inference of this, or of any other occurrence. lawful in
is
kind,
it
itself,
be exhibited in the results
will
of our method.
We
may by
restricting the
canon of interpretation confine
our expressed results within the limits of the scholastic logic; but this would only be to restrict ourselves to the use of a part of the conclusions to which our analysis entitles us.
The
classification
and we which
name
shall
it
we
shall
adopt will be purely mathematical,
afterwards consider the logical arrangement to It will
corresponds.
be
sufficient, for reference, to
the premises and the Figure in which they are found.
CLASS
1st.
Forms
in
which
v does not enter.
Those which admit of an inference are
AE; EA, Ex. order),
Fig. 2;
A A, A A,
All
Ys
A A, AE,
Fig.
1,
Fig.
4.
Fig.
are Xs,
y
(1

z (1 
Eliminating y by (lo)
x) = 0,
or (1  x)
y)=
or
1 ;
..
mode
0,
zy
y=
0,
=
Q.
 z
we have z (1  x)
convenient
Fig.
and, by mutation of premises (change of
All Zs are Ys,
A
AA, EA,
4.
=
0,
All Zs are Xs.
of effecting the elimination,
is
to write
the equation of the premises, so that y shall appear only as a factor of one member in the first equation, and only as a factor of the opposite member in the second equation, and
then to multiply the equations, omitting the
we
shall adopt.
y.
This method
OF SYLLOGISMS.
Ex.
AE,
Fig. 2, and,
All
Xs
No
Zs are Ys,
35
E A,
by mutation of premises, x (1  y) = zy =
are Ys,
x
or
0,
=
Fig,
2.
xy
= zy
0,
;*r=0 .*.
The only
case in
which there
is
No
no inference
AllXsareYs,
*(l/)=0,
All Zs are Ys,
z (1  y) = o,
Zs are Xs.
A A,
is
Fig. 2,
x = xy zy = z 2 = #Z .
When
CLASS 2nd.
v
is
.
0=0.
introduced by the solution of an
equation.
The lawful cases directly or indirectly* determinable Aristotelian Rules are AE, A, AE, EA, Fig. 1;
A
EA,
by the Fig. 3;
Fig. 4.
The
lawful cases not so determinable, are
EE,
Fig.
and, by mutation of premises,
EA,
Fig 2; EE, Fig. 3; EE, Fig.
Ex.
AE,
Fig.
1,
All
Ys
No
Zs are Ys,
are
Xs,
y
1
;
EE,
4.
(1

zy
= x)
0,
y = vx
=0,
= zy
Fig.
4.
(a)
= vzz :.
Some Xs
are not Zs.
The reason why we cannot are notXs,
is
that
by
into Some Zs interpret vzz = the very terms of the first equation (a)
the interpretation of vx is fixed, as Some Xs v is regarded ; as the representative of Some, only with reference to the class *
X.
We
say
directly or indirectly,
mutation or conversion of premises being
some instances required. Thus, AE (fig. 1) is resolvable by Fesapo (fig. 4), or by Ferio (fig. 1). Aristotle and his followers rejected the fourth figure as only a modification of the first, but this being a mere question of form, either scheme may be termed Aristotelian. in
36
OF SYLLOGISMS.
For the reason of our employing a solution of one of the Had primitive equations, see the remarks on (16) and (17).
we
solved the second equation instead of the
we should
first,
have had
(la?)y=0, tj(l*0 =
y,
(a),
v(\z) (l*) =
0,
(),
Some notZs
.*.
Here
to
it is
be observed, that the second equation
the meaning of v(\ 2), as of the result (b)
Y
are Xs.
Some
notZs.
that all the notZs
is,
The full meaning which are found in
X, and
are found in the class
is
evident that
the
class
this
could not have been expressed in any other way.
Ex.
AA,
Fig.
All
Ys
are Xs,
y (1
All
Ys
are Zs,
y
2.
(a) fixes
it
3.
(1

a?)
2)
=
0,
=
0,
y = vx Q = y(lz) = vx(\ :.
Some Xs
z)
are Zs.
solved the second equation, we should have had The form of the final equation as our result, Some Zs are Xs.
Had we
particularizes
remark
The is
is
what Xs or what Zs are referred
to,
and
this
general.
EE, Fig. 1, and, by mutation, EE, Fig. 4, a of lawful case not determinable by the Aris an example following,
totelian Rules.
No Ys No Zs
xy = = zy
are Xs, are Ys,
0, 0,
:.
CLASS
3rd.
When
but not introduced by
v
is
met with
solution.
= xy y = v (1 = v (1 
Some notZs
2)
2)
x
are not Xs.
in one of the equations,
OF SYLLOGISMS.
37
The
lawful cases determinable directly or indirectly by the Aristotelian Rules, are AI, El, Fig. 1 ; AO, El, OA, IE,
AI, AO, El, EO, IA, IE, OA, OE, Fig. 3; IA, IE,
Fig. 2; 4.
Fig.
Those not
The I A,
so determinable are
OA, AI, El, AO, Ex.
1.
EO,
;
Fig. 4.
possible, are
is
AO, EO,
AI, EO, IA, OE, Fig. 2; OA, OE,
Fig. 1; 4.
Fig.
AI, Fig. All
1
Fig.
which no inference
cases in
IE,
OE,
Ys
1,
and, by mutation,
are Xs,
Some Zs
y
I A, Fig. 4.

(1
are Ys,
x) =
= vy
vz
vz(l 
/.
Ex.
Xs
All
AO,
2.
Fig. 2, and,
are Ys,
Some Zs
*)= Some Zs are Xs,
by mutation, OA, Fig.
x = xy
#(ly)=o,
are not Ys,
vz =
2.
vy = v(\z)
v(ly\
tx = vx(\z) vzz =
Some Zs
:.
The
interpretation of vz as
observed,
senting the proposition
The
OE,
Some Zs
cases not determinable
Fig.
1,
Some
the equation vz = v
in
and, by mutation,
Some Ys
No
are not
(1
Zs,

is
(l
by the
EO,
#)>
but
it
will
be
Aristotelian Rules are
Fig.
4.
vy = v
Xs,
Zs are Ys,
equation of the

it
are not Ys.

(1
x)
o = Zy
/.
The
implied,
y) considered as repre
= v
c
are not Xs.
(1

Some notXs
x) z
are not Zs.
premiss here permits us to interpret does not enable us to interpret vz. first
D2
38
OF SYLLOGISMS.
Of
which no inference
in
cases
is
possible,
we
take
as
examples
AO,
1,
Fig.
and, by mutation,
AllYsareXs, Some Zs are not Ys,
OA,
Fig. 4,
y(\x)=Q
2/(lz)=0, vz  v (1  y)
v(l z) = vy
(a)
i>(l*)(laO0
)
= since the auxiliary equation in this case
it is satisfactory to
vz as
Some
do
Zs, but
it
so.
The equation
classes
it is
=
0.
(If),
(a), it is seen, defines

z), so that we might and content ourselves with
does not define v
stop at the result of elimination
saying, that
z)
not necessary to perform this reduction, but
it is
Practically

v (1
is
(1
not interpretable into a relation between the
X and Z.
Take
as
IA, Fig.
a second
example AT, Fig.
and, by mutation,
2,
2,
AllXsareYs, Some Zs are Ys,
x = xy vy = vz
s(ly)=0, vz = vy,
vz = vxz
z)x=Q
0(1
= the auxiliary equation in this case being 0(1
Indeed in every case in
this class,

*)=
0,
0.
which no inference
in
the form is possible, the result of elimination is reducible to = 0. Examples therefore need not be multiplied.
CLASS
No
1st.
When
inference
tinction class.
4th.
among The two
When
is
v enters into both equations,
possible in
any
case,
but there exists a
the unlawful cases which
is
dis
peculiar to this
divisions are,
the
result of elimination
to the form auxiliary equations
=
0.
is
The
reducible
by the
cases are II, OI,
39
OF SYLLOGISMS. Fig. 1;
II,
00,
Fig. 2
II,
;
00,
10, 01,
Fig. 3;
Fig.
When
2nd.
the result of elimination
=
auxiliary equations to the form
The
are 1O,
cases
OO,
is
not reducible by the
0.
10, OI, Fig. 2; OI,
Fig. 1;
Let us take
an example of the former case, II, Fig.
as
Some Xs
are Ys,
vx = vy,
Some Zs
are Ys,
vz
=
Now
v
v y,
y = vz
x
Substituting
vx =
=
vz
v,
v
.
we have vv = vv
=
..
As an example
,
0.
of the latter case, let us take 10, Fig.
Some Ys are Xs, Some Zs are not Ys,
vy
v (1

v (1
y),
reducible.
The above which
it
classification is
We
now
shall
=vy
z}
vv
x
is,
that the classes v
indeed evident.
purely founded on mathematical
inquire what
cases of the first class
is the*
logical division
comprehend
two universal premises,
be drawn.
We
the second,
a
all
those in
universal conclusion
see that they include
barbara and celarent in the in
as is
z)
corresponds.
lawful
which, from
may
interpretation
common member,
have nc
distinctions.
Its

 x) = 0, v (1  z) = 0, (1 this form that all similar
Now the auxiliary equations being v It is to the above reduces to vv = 0. cases are
1 ,
vy = vx
= vx,
vz =
vv (I
The
vv z
the auxiliary equations v (1  x) = 0, v (1  z) = 0,
give
v
3.
vx = vy
vv
to
00,
4.
Fig.
and
10,
II,
4.
first figure,
the premises of of cesare and camcstrcs
and of bramantip and camcnes in the fourth.
40
OF SYLLOGISMS,
The premises of bramantip
are included, because they admit of an universal conclusion, not in the same although figure.
The
lawful cases of the second class
a particular
conclusion only
are those in
which
deducible from two universal
is
premises.
The
lawful cases of the third class
conclusion
are those in
deducible from two premises, universal and the other particular.
The
which a
one of which
is
is
fourth class has no lawful cases.
Among
the cases in which no inference of any kind
is
pos
we
find six in the fourth class distinguishable from the others by the circumstance, that the result of elimination does sible,
not assume the form
Some Ys
=
0.
The
cases are
are Xs, f Some Ys are not Xs,] f Some Xs are Ys, Zs are not Zs are not \Some Ys, j (Some Zs are not Ys,/ Ys,J \Some f
and
the
"\
"I
three
others
which are
obtained
by mutation of
premises. It might be presumed that some logical peculiarity would be found to answer to the mathematical peculiarity which we
have noticed, and in fact there exists a very remarkable one. If we examine each pair of premises in the above scheme, we shall find that there is virtually
no middle term,
of comparison, in any of them. the individuals spoken of in the
belong
to
the class
Y, but
i.
Thus, in the first
premiss
those spoken
e.
no medium
first
example,
are asserted to
of in
the second
premiss are virtually asserted to belong to the class notY: nor can we by any lawful transformation or conversion alter
The comparison will one premiss, and with the
this state of things.
the class
Y
in
still
class
be made with
notY in the
other.
Now a
in every case beside the above six, there will be found
middle term, either expressed or implied.
of the most difficult cases.
I
select
two
41
OF SYLLOGISMS. In
AO,
Fig.
viz.
1,
All
Ys
are Xs,
Some Zs
we
are not Ys,
have, by negative conversion of the
first
premiss,
All notXs are notYs,
Some Zs and the middle term
is
Again, in EO, Fig.
now
seen to be notY.
1,
No Ys
.
are not Ys,
are
Some Zs a proved conversion
of the
Xs,
are not Ys, first
premiss (see Conversion of
Propositions), gives
All
Xs
are notYs,
Some Zs
are notYs,
and the middle term, the true medium of comparison,
is
plainly
although as the notYs in the one premiss may be different from those in the other, no conclusion can be drawn. notY,
The mathematical
condition in question, therefore, the irre= final the of to the form equation 0, ducibility adequately represents the logical condition of there being no middle term, or
common medium I
am
not aware
of comparison, in the given premises. that
the
distinction
occasioned
by the
presence or absence of a middle term, in the strict sense here understood, has been noticed by logicians before.
The
dis
though real and deserving attention, is indeed by no means an obvious one, and it would have been unnoticed
tinction,
in the present instance but for the peculiarity of
its
mathe
matical expression.
What
appears to be novel in the above case
is
the proof
of the existence of combinations of premises in which there
OF SYLLOGISMS.
When such a medium absolutely no medium of comparison. of comparison, or true middle term, does exist, the condition
is
that
its
ceed
in
quantification
both premises
together
shewn by Professor De Morgan
clearly
ex
shall
and
quantification as a single whole, has been ably
its
to
be necessary to
And lawful inference (Cambridge Memoirs, Vol. vm. Part 3). this is undoubtedly the true principle of the Syllogism, viewed from the standingpoint of Arithmetic. I have said that
it
would be
possible to impose conditions
of interpretation which should restrict the results of this cal culus to the Aristotelian forms. Those conditions would be,
That we should agree not
1st.
to interpret the
forms v(l 
x),
v(lz). That we should agree
2ndly.
to reject
which the order of the terms should
every interpretation in
violate the Aristotelian rule.
Or, instead of the second condition,
it
might be agreed
that,
the conclusion being determined, the order of the premises should,
if
necessary, be changed, so as to
make
the syllogism
formal.
From
the general character of the system
it is
indeed plain,
may be made to represent any conceivable scheme of logic, by imposing the conditions proper to the case con that
it
templated.
We have
found it, in a certain class of cases, to be necessary the two equations expressive of universal Propo to replace sitions,
that
it
done * It
by their solutions; and it may be proper to remark, would have been allowable in all instances to have
this,*
may we
Barbara,
so
that every case of the Syllogism, without ex
be satisfactory to illustrate this statement by an example.
should have All
Ys
are Xs,
y
All Zs are Ys,
z
z .
.
= vx =vy = vv x
All Zs are Xs.
In
43
OF SYLLOGISMS. ception,
might have been treated by equations comprised in
the general forms
y = vx, y = v (1 vy = vx, _ = vy v (i Or,
we may
y vx = or y + vx  v = = vy vx
or x),
 = vy + vx v
x),
multiply the resulting equation by
1
.
.
.
.
A,
.
.
. .
E,
.
.
.
.
.
.
.
.
 x, which
I,
O.
gives
.(l.)O, conclusion, All Zs are Xs. additional examples of the application of the system of equations in the text to the demonstration of general theorems, may not be inappropriate.
whence the same
Some
Let y be the term to be eliminated, and let x stand indifferently for either of the other symbols, then each of the equations of the premises of any given syllogism may be put in the form
+
ay if
the premiss
is
affirmative,
(a)
0,
and in the form
ay if it is
=
bx
+ 6(1*) =
0,
(/3)
and b being either constant, or of the form v. To prove us examine each kind of proposition, making y successively
negative, a
this in detail, let
subject and predicate.
A, All Ys are Xs, All Xs are Ys, E,
I,
No Ys No Xs
y x
are Xs, are Ys,
Some Xs Some Ys
y
The
affirmative equations (y), equations (), (tj) and (0), to (/3).
vy
(
ab z
Hence,
.
_
a bx
if
there
drawn from an
conclusion
= is
,
a conclusion,
affirmative
it is
affirmative.
and a negative proposition
is
negative.
By
and
(a)
(/3),
we have
for the given propositions
ay
ay + .
which
is
of the form
A
3rd.
(/3)
.

a bx
b
conclusion draicn
+ 
ab (1
Hence the
.
(1
bx
0,
*)
=
0,
z)
=
0,
conclusion, if there
is
one,
is
negative.
from two
negative premises will involve a negation, predicate, and will therefore be inadmissible in
(noX, notZ) in both subject and the Aristotelian system, though just in itself.
For the premises being
+b ay + b ay
the conclusion will be ab (1
which

2)
(1 (1


a?)
z)
a b (1
= = 
0, 0,
ar)
=
0,
only interpretable into a proposition that has a negation in each term. 4th. Taking into account those syllogisms only, in ichich the conclusion is the most general, that can be deduced from the premises, if, in an Aristotelian is
minor premises be changed in quality (from affirmative to negative negative to affirmative), whether it be changed in quantity or not, no con clusion will be deducible in the same figure. syllogism, the
or
from
An
Aristotelian proposition does not admit a term of the form notZ in the Now on changing the quantity of the minor proposition of a syllogism, it from the general form
subject, we transfer
ay to the general
form
+
bz
=
0,
+ & (1  *) = 0, And therefore, in the
ay
see (a) and (/3), or vice versd. there will be a change from z to
equation of the conclusion,
But this is equivalent to the change of Z into notZ, or notZ into Z. Now the subject of the original conclusion must have involved a Z and not a notZ, therefore the subject of the new conclusion will involve a notZ, and the conclusion will not be admissible in the Aristotelian forms, except a change of Figure.
1
*,,
or vice versd.
by conversion, which would render necessary
Now the conclusions of this calculus are always the most general that can be drawn, and therefore the above demonstration must not be supposed to extend to a syllogism, in which a particular conclusion is deduced, when a universal one is possible. This is the case with bramantip only, among the Aristotelian forms, and therefore the transformation of bramantip into camenes, and vice versd, is
the case of restriction contemplated in the preliminary statement of the
theorem.
45
OF SYLLOGISMS. from those in which
must.
it
But
demonstration of
for the
above system
the certain general properties of the Syllogism,
from
is,
its
and from the mutual analogy of
simplicity,
forms, very convenient.
We
shall
apply
its
to the following
it
theorem.*
is
Given the three propositions of a Syllogism, prove that there but one order in which they can be legitimately arranged,
and determine that order. All the forms above given for the expression of propositions, are particular cases of the general form,
a + bx + cy =
0.
con
an Aristotelian syllogism, we substitute If for the minor premiss of in the same figure. deducible is tradictory, no conclusion case of bramantip, all the others It is here only necessary to examine the last the proposition. being determined by have AO, On changing the minor of bramantip to its contradictory, we its
5th.
Fig.
4,
and
this admits of
no legitimate inference.
Hence the theorem is true without exception. may in like manner be proved.
Many other
general theorems
* This communicated by the Rev. Charles Graves, elegant theorem was to whom the Fellow and Professor of Mathematics in Trinity College, Dublin, for a very his acknowledgments record grateful Author desires further to for some new of the former portion of this work, and examination judicious method. The example of Reduction ad impossible applications of the among the number
is
following
:
Reducend Mood, All Xs Baroko
1
are Ys,
Some Zs
axe not
Ys
Some Zs
are not
Xs
Reduct Mood, All Xs are Ys All Zs are Xs Barbara All Zs are
Ys

y
=
t>
w=v vz 1
* (\
* (1


y
x)
y)
 .r)  y)  x) (1
(1 (1
= vv = V (1 = * = 0.
*)
be the contradictory of the The conclusion of the reduct mood is &c. It may just be remarked that the Whence, minor premiss. suppressed one mathematical test of contradictory propositions is, that on eliminating seen to
the other elective symbol vanishes. symbol between their equations, involves no difficulty. Bokardo and Baroko of reduction The ostensive elective
= vy for the Professor Graves suggests the employment of the equation a: that on and are Xs remarks, All Ys, primary expression of the Proposition  y) = 0, the equation from 1  y, we obtain .r (1 members both by multiplying which we set out in the text, and of which the previous one is a solution.
46
OF SYLLOGISMS.
Assume then
for the premises of the given syllogism, the
equations
then, eliminating
a + bx + cy =
0,
(18),
a + bz+ cy =
0,
(19),
we
shall
have for the conclusion
y>
ad  a c +
bc x  b cz = 0,
(20).
Now
taking this as one of our premises, and either of the original equations, suppose (18), as the other, if by elimination of a common term x, between them, we can obtain a result
equivalent to the remaining premiss (19), it will appear that there are more than one order in which the Propositions may
be lawfully written is
;
but
if
otherwise, one arrangement only
lawful.
Effecting then the elimination, 1
be (a + b z
+
c
we have
y}=
0,
(21),
Now on equivalent to (19) multiplied by a factor be. the of this value in factor the examining equations A, E, I, O, we find it in each case to be v or  v. But it is evident, which
is
given Proposition be mul derived from another equa factor,
that if an equation expressing a tiplied by an extraneous tion,
its
will
interpretation
Thus there
impossible.
either
will either
be
limited
be no result
or
rendered
at all, or the
result will be a limitation of the remaining Proposition. If,
however, one of the original equations were
x =
the factor be would be ation
of the
or
y, 1,
other premiss.
x  y =
0,
and would not
Hence
if
limit the interpret
the
first
member
of
syllogism should be understood to represent the double proposition All Xs are Ys, and All Ys are Xs, it would be a
indifferent written.
in
what order the remaining Propositions were
47
OF SYLLOGISMS.
more general form of the above investigation would be, the equations express the premises by
A to
a + bx + cy + dxy = a +b z + cy + d zy =
0,
(22),
0,
(23).
we should
After the double elimination of y and x (be
and
it

ad} (a +
bz +
would be seen that the
cy + d
zy)
factor be
=
find
;
 ad must in every
of meaning. case either vanish or express a limitation
The determination ficiently obvious.
of the order of the Propositions
is
suf
OF HYPOTHETICALS.
A
hypothetical Proposition is defined to be two or more categorical* united by a copula (or conjunction), and the different kinds of hypothetical Propositions are named from their respective conjunctions, viz. conditional (if)
disjunctive
(either, or), &c.
is
In conditionals, that categorical Proposition from which the other called the antecedent, that which results from it the consequent.
Of the 1st.
conditional syllogism there are two, and only
The
constructive,
two
results
formula?.
A is B, then C is D, A is B, therefore C is D.
If
But 2nd.
The
Destructive, If
But C
is
A is B, then
C
not D, therefore
is
D,
A is not B.
A
dilemma is a complex conditional syllogism, with several antecedents in the major, and a disjunctive minor.
IF we examine either of the forms of conditional syllogism above given, we shall see that the validity of the argument does not depend upon any considerations which have reference the terms A, B, C, D, considered as the representatives of individuals or 6f classes. may, in fact, represent the is B, C is D, Propositions and by the arbitrary symbols to
We
A
respectively,
following
X
Y
and express our syllogisms in such forms
as the
:
X But X If
Y is true, is true, therefore Y is true. is
Thus, what we have
true, then
to consider is not objects
and
classes
of objects, but the truths of Propositions, namely, of those
49
OF HYPOTHKTICALS,
which are embodied
in the terms of
elementary Propositions our hypothetical premises.
we the symbols X, Y, Z, representative of Propositions, elective symbols x, y, z, in the following appropriate the
To may
sense. shall comprehend hypothetical Universe, 1, circumstances. of able cases and conjunctures
The
The
elective
symbol x attached
such cases shall select is
true,
If
and similarly
we
conceiv
to
any subject expressive those cases in which the Proposition
for
Y
of
X
and Z.
contemplation of a given pro and hold in abeyance every other consideration,
confine ourselves to the
X,
position
all
then two cases only are conceivable, viz. and secondly that it Proposition is true,
first is
that the given
false*
As
these
of the Proposition, and cases together make up the Universe elective symbol x, the latter as the former is determined by the is
determined by the symbol
1

x.
But if other considerations are admitted, each of these cases less extensive, the will be resolvable into others, individually that a Proposition is either true or false, * It was upon the obvious principle future events, endeavoured that the Stoics, applying it to assertions respecting to their argument, that been has It Fate. of replied doctrine the to establish which is id quod the precise meaning of an abuse of the word true, involves true nor false." Copleston res est. An assertion respecting the future is neither however, pre on Necessity and Predestination, p. 36. Were the Stoic axiom, a given event will take place, sented under the form, It is either certain that fail to meet the difficulty. would above the not will it reply that ; or certain settle the The proper answer would be, that no merely verbal definition can of Nature. When we is the actual course and constitution what question, take place, or certain that affirm that it is either certain that an event will that the order of events is necessary, it will not take place, we tacitly assume that the state of things that the Future is but an evolution of the Present so But this (at least as re be. shall which that which is, completely determines Exhibited is the very question at issue. moral of conduct the agents) spects abuse of terms, an involve not does Stoic the reasoning under its proper form, "
;
but a petitio principii. of the doctrine of Necessity It should be added, that enlightened advocates and through the as end the appointed only in in the present day, viewing which are the reproa means, justly repudiate those practical 01 consequences of Fatalism.
50
ON HYPOTHETICALS.
number
of which will depend
upon the number of foreign con Thus if we associate the Propositions X number of conceivable cases will be found as
siderations admitted.
and Y, the
total
exhibited in the following scheme. Cases.
1st
2nd 3rd 4th If
Elective expressions.
X true, Y true X true, Y false X false, Y true X false, Y false
we add
x (I
xy (1
y)
 z) y
*)(!
(1
the elective expressions for the two
above cases the sum
is
x,
which
is
y)
(24)
first
of the
the elective symbol appro being true independently
X
more general case of of any consideration of and if we add the elective expres ; sions in the two last cases together, the result is 1  x which y priate to the
Y
is
of
the elective expression appropriate to the
X being
more general
case
false.
Thus the extent of the hypothetical Universe does not at all depend upon the number of circumstances which are taken
And
it is
to
those circumstances
may
be, the
into account.
be noted that however few or
sum
many
of the elective expressions
representing every conceivable case will be unity. Thus let us consider the three Propositions, X, It rains, Y, It hails,
Z, It freezes.
The
possible cases are the following
Cases.
:
Elective expressions.
and freezes, xyz and hails, but does not freeze xy (1  z) rains and freezes, but does not hail xz (1  y) freezes and hails, but does not rain yz (1  x)
1st
It rains, hails,
2nd
It rains
3rd
It
4th
It


5th
It rains, but neither hails nor freezes
6th
but neither rains nor freezes y (1  x) (1  z) It freezes, but neither hails nor rains z (I  x)(l y)  x)(l It neither rains, hails, nor freezes (1 y) (1
7th 8th
x
(1
y) (1
z)
It hails,
1
= sum
z)
51
OF HYPOTHET1CALS. Expression of Hypothetical Propositions.
X
To
is true. express that a given Proposition 1  x selects those cases in which the Proposi But if the Proposition is true, there are no is false.
The symbol tion
X
such cases in
its
hypothetical Universe, therefore 1
 x =
0,
x =
1,
or
(25).
X
is false. To express that a given Proposition The elective symbol x selects all those cases in
Proposition
is
and therefore
true,
x =
And
in every case,
0,
if
the Proposition
which the
is false,
(26).
having determined the elective expression
assert the truth of that appropriate to a given Proposition, we elective the expression to unity, and Proposition by equating its
by equating the same expression to 0. and Y, are simulta To express that two Propositions, falsehood
X
neously true.
The
elective
symbol appropriate
the equation sought
To
express that
to this case is xy, therefore
is
xy =
(27).
1,
two Propositions,
X and Y, are simultaneously
false.
The
condition will obviously be
(!*)(! y)= x + y
or
To
is
Y
is
*>
0,
(28).
is to
X
true, or the
two Propositions not true, that they are both false.
either one or the other of
assert that
it is
the elective expression appropriate to their both being  x} (1 the equation required is y), therefore
false is (1
(1
or
is
true.
assert that
true,
Now
xy =
express that either the Proposition
Proposition
To

*)(! sO=0, x + y  xy
=
1,
(29).
OF HYPOTHETICALS.
And, by
indirect considerations of this kind,
may every
dis
junctive Proposition, however numerous its members, be ex But the following general Rule will usually be pressed. preferable.
RULE. Consider what are
those distinct and mutually exclusive it is which in the statement of the given Propo of implied sition, that some one of them is true, and equate the sum of their
cases
elective expressions to unity.
This will give the equation of the
given Proposition.
For the sum of the
elective expressions for all distinct con
ceivable cases will be unity.
and
some all
Now all these cases being mutually
being asserted in the given Proposition that one case out of a given set of them is true, it follows that
exclusive,
it
which are not included in that
set are false,
elective expressions are severally equal to 0.
and
that their
Hence
the
sum
of the elective expressions for the remaining cases, viz. those included in the given set, will be unity. Some one of those
and as they are mutually more than one should be true.
cases will therefore be true, it
is
impossible that
exclusive,
Whence
the Rule in question.
And
in the application of this
Rule
it is
to
be observed, that
if the cases contemplated in the given disjunctive Proposition
are not mutually exclusive, they
must be resolved
into,
an equi
valent series of cases which are mutually exclusive. if
we
Either
X
Thus, viz.
take the Proposition of the preceding example, is true, and assume that the two true, or
Y
is
members of this Proposition are not exclusive, insomuch that in the enumeration of possible cases, we must reckon that of the Propositions
X
and
exclusive cases which
Y
fill
being both true, then the mutually up the Universe of the Proposition,
with their elective expressions, are 1st,
2nd, 3rd,
X true and Y false, Y true and X false, X true and Y true,
x (I  y),  x\ y(\ xy,
OF HYPOTHETIC A LS.
and the sum of these
elective expressions equated to unity gives
x + y xy =
we suppose
1.
(30),
members of
the disjunctive to be con cases the only Proposition to be exclusive, then sidered are
But
as before*
1st,
2nd,
if
the
X true, Y false, Y true, X false,
x y
(1 (1

and the sum of these elective expressions equated
xThe
To
Ixy + y =
1,
y),
 x\ to 0, gives
(31).
subjoined examples will further illustrate this method.
express the Proposition, Either
X
is
Y
not true, or
is
not
true, the members being exclusive.
The mutually 1st,
2nd,
exclusive cases are
X not true, Y true, Y not true, X true,
and the sum of these equated
to
is
(1
x
(1
#), y),
unity gives
x  2xy + y = which

y
(32),
1,
the same as (31), and in fact the Propositions which
they represent are equivalent.
To
X
express the Proposition, Either members not being exclusive.
is
not true, or
Y
is
not
true, the
To
we must add
the cases contemplated in the last Example,
the following, viz.
X not true, Y not true, The sum
(i

y)
+ y or
To is

x) (1

y).
of the elective expressions gives
#
Y
(1
 *) + 0, xy
 *)
(!

y) =
^
(33).
X
express the disjunctive Proposition, Either Z is true, the members being exclusive.
is
true, or
true, or
E 2
OF HYPOTHETIC A LS.
Here
the mutually exclusive cases are
1st,
X true, Y false,
Z
2nd,
Y
3rd,
Z
X false, Y false,
Z
true,
false,
X false,
true,
and the sum of the
x(\  y)
false,
y
(1

(1
z) (1

2),
x),
* (1  a) (1  y),
elective expressions equated to 1, gives,
upon reduction, x + y +
 2 (xy + yz + zz) 4 Say* =
z
The
of
the same Proposition, expression are in no sense exclusive, will be

(1
And
it

x) (1

y) (1
=
z)
0,
1,
(34).
when
the
members
(35).
easy to see that our method will apply to the
is
expression of any similar Proposition, subject to any specified
To
express the
whose members amount and character of exclusion.
conditional Proposition,
X
If
is
are
true,
Y
is true.
Here
it
cases of
by the
is
Y
implied that
the cases of
The former
being true.
elective
all
symbol
and the
x,
X
being true, are
cases being determined
latter
by
y,
we
have, in
virtue of (4),
*(ly)=0,
(36).
To
express the conditional Proposition, If not true.
The
equation
is
be true,
Y
is
obviously
*y0, this is equivalent to (33),
X
X
and in
(37); fact the disjunctive Proposition,
Y not true, and the conditional not true, are equivalent. true, Y Proposition, If X To express that If X not true, Y not true. Either
is
not true, or is
is
is
is
In (36) write
1
is
 x for x, and (i
I

 *) y
y
for y,
o.
we have
55
OF HYPOTHETICALS.
The resuhs which we have obtained admit in
many
ticular
Let
different ways.
of verification
it suffice to take for
examination the equation
x2xy
+
y=l,
(38),
which expresses the conditional Proposition, Either or
Y
is
more par
true, the
members being
X
First, let the Proposition stituting,
X
is
true,
in this case exclusive.
z=\, and
be true, then
sub
we have 
1
= 2y + y
which implies that let
Secondly,
Y
X
y =
0,
be not true, then x =
y =
Y
y =
or
0,
not true.
is
gives
which implies that

/.
1,
is
i
(39),
9
In like manner
true.
with the assumptions that
and the equation
0,
Y is true,
or that
x = Again, in virtue of the property the equation in the form x 1  Ixy +
y*
=
Y
x, y*
=
we may proceed is false.
y,
we may
write
1,
and extracting the square root, we have x  y = 1, (40), and or
this
as when represents the actual case; for, is respectively false or true, we have
X
is
true
false, Y
x =
/.
There
will
be no
y = x  y
1
or 0, or
=
1
1,
or 
1.
of other cases. difficulty in the analysis
Examples of Hypothetical Syllogism. of every form of hypothetical Syllogism will consist in forming the equations of the premises, and eliminating the symbol or symbols which are found in more than one of them. The result will express the conclusion.
The treatment
56
OF HYPOTHETICALS. 1st.
Disjunctive Syllogism.
Either
But
X
X
is
is
true,
Y is not
Therefore
X
Either
But
X
is
Y is
true, or
true,
Y
true, or
x + y  2 xy = x 1
true (exclusive),
x
true (not exclusive),
is
y=
/.
.
not true, Therefore is true,
+
is
Y
1
.*.
y xy = = x y= 1
1
2nd. Constructive Conditional Syllogism.
X X
If
is
But
true, is
Y is true,
true,
Y
Therefore
is
..
true,
1
x (1  y) = x = I or y = y =
1.
3rd. Destructive Conditional Syllogism.
If
X
But
is
true,
Y
is
x
true,

(I
y)
Y is not true,
Therefore
X
is
not true,
=
y = x=
..
Simple Constructive Dilemma, the minor premiss ex
4th. clusive.
X
Y If Z is true, Y If
is
But Either
From x and
is
true,
X
is
is
x
true,
Z
is
true,
In whatever way we
y) = 0,
we have
the equations (41), (42), (43),
z.

z (1  y) = x\z 2xz =
true,
true, or
(1
(41),
0,
(42),
1,
(43).
to eliminate
effect this, the result is
yii whence 5th.
it
appears that the Proposition
Y
Complex Constructive Dilemma,
is
true.
the minor premiss not
exclusive.
X
If If
"W
is is
Either
From
true,
Y
true,
Z
X
is
is is
x
true,
W
is
x
true,
these equations, eliminating x,
y +

z

yz =
1,
y)
=
w (1 z) = + w  xw = 
true,
true, or
(l
we have
0,
0, 1.
57
OF HYPOTHETICALS. which expresses the Conclusion, Either the
Y
is
true, or
Z
is
true,
members being nonexclusive. Complex Destructive Dilemma, the minor premiss ex
6th. clusive.
X
If
is
W
If
is
From
true,
Y
Either
result
true,
is
these
Y
is
true,
Z
is
true,
x(\  y) = w (1  2) =
Z
not true, or
is
we must
equations
is
xw =
eliminate
arid
y
1
.
The
z.
Qj
X
which expresses the Conclusion, Either not true, the members not being exclusive.
Complex Destructive Dilemma,
7th.
y + z 2yz =
not true,
is
not true, or
Y
is
the minor premiss not
exclusive.
X
If
is
W
If
is
Either
On
true, true,
Y
is
true,
Z
is
true,
x(\  y} = 10 ( 1  z) =
Y is not true, or Z
elimination of y and
is
0.
we have
z,
xw 
0,
which indicates the same Conclusion It
yz =
not true,
as the previous
example.
.
appears from these and similar cases, that whether the of the minor premiss of a Dilemma are exclusive
members
or not, the
members of
The above which
the (disjunctive) Conclusion are never
This fact has perhaps escaped the notice of logicians.
exclusive.
are the principal forms of hypothetical Syllogism
logicians have recognised.
It would be easy, however, the list, especially by blending of the disjunctive and the conditional character in the same Proposition, of which
to
extend the
the following If
X
But
is
Y
is
an example.
true, then either
Y
is
true, or
Z
is
true,
x(\yz + yz)=Q is
y=
not true,
Therefore If
X
is
true,
Z
is
true,
/.
x(\ 
o z)
=
0.
58
OF HYPOTHETICALS.
That which logicians term a Causal Proposition is properly a conditional Syllogism, the major premiss of which is sup pressed.
The
X
assertion that the Proposition
Proposition Y
is
is
true,
is
true, because the
equivalent to the assertion,
Y
The
is true, Proposition Therefore the Proposition
X
is
true;
and these are the minor premiss and conclusion of the con ditional Syllogism,
If
Y is true, X is true, Y is true,
But
Therefore
X
is
true.
And
thus causal Propositions are seen to be included in the applications of our general method.
Note, that there
a family of disjunctive and conditional
is
Propositions, which do
not, of right, belong to the class
con
sidered in this Chapter. Such are those in which the force of the disjunctive or conditional particle is expended upon the predicate of the Proposition, as if, speaking of the inhabitants of a particular island, we should say, that they are all either Europeans or Asiatics; meaning, that it is true of each indi vidual, that he
the
appropriate
is
Europeans, and z
to Asiatics,
then the equation of the above
is
Proposition
x = xy + to
European or an Asiatic. If we symbol x to the inhabitants, y to
either a
elective
or
xz,
z(lyz)=0,
which we might add the condition yz =
(a);
0, since
no Europeans
The nature
of the symbols x, y, z, indicates that the Proposition belongs to those which we have before de are Asiatics.
Very different from the above is the signated as Categorical. all the inhabitants are Europeans, or they Either Proposition, are
all Asiatics.
positions.
The
sent Chapter;
Here the case
is
disjunctive particle separates
Pro
that contemplated in (31) of the pre
and the symbols by
which
it
is
expressed,
59
OF HYPOTHETICALS. although subject to the same laws as those of
(a),
have a
totally
different interpretation.*
The distinction is real and important. Every Proposition which language can express may be represented by elective symbols, and the laws of combination of those symbols are in cases the
same
but in one
class of instances the
symbols have reference to collections of objects, in the other, to the all
;
truths of constituent Propositions. * Some writers, among whom is Dr. Latham (First Outlines), regard it as the exclusive office of a conjunction to connect Propositions, not words. In this view I am not able to agree. The Proposition, Every animal is either rational or irrational, cannot be resolved into, Either every animal is rational, or every animal is irrational. The former belongs to pure categoricals, the latter to
such conversions would seem to be hypotheticals. In singular Propositions, allowable. This animal is either rational irrational, is equivalent to, Either t>r
this
animal
is
rational, or it is irrational.
would almost
This peculiarity of singular Pro
justify our ranking them, a separate class, as Ramus and his followers did.
positions
though truly universals, in
PROPERTIES OF ELECTIVE FUNCTIONS.
symbols combine according to the laws of quantity, we may, by Maclaurin s theorem, expand a given
SINCE
elective
(x), in
function
ascending powers of x,
excepted.
0(*)=
Now
if in
(44)
we make x =
0(l) = 0(0) +
1,
we have
(0)+^
)
+
&c.,
whence
Substitute this value for the coefficient of x in the second
member
of (45), and (x)
=
we have*
(0)
+ (0 (1) 
(0)} x,
(46),
* Although this and the following theorems have only been proved for those forms of functions which are expansible by Maclaurin s theorem, they may be regarded as true for all forms whatever this will appear from the applications. The reason seems to be that, as it is only through the one form of expansion ;
that elective functions
become
interpretable,
no conflicting interpretation
is
possible.
The development of (x) may mula for expansion in factorials,
also
be determined thus.
By
the
known
for
PROPERTIES OF ELECTIVE FUNCTIONS. which we
shall also (*)
employ under the form
=(!)*+
Every function of x, are alone involved,
The
order.
(U
0(0) (1*),
(47).
which integer powers of that symbol by this theorem reducible to the first
in
is
$
quantities
of the function
(0),
we
(1),
moduli
shall call the
are of great importance in the theory of elective functions, as will appear from the succeeding
They
(x).
Propositions.
PROP.
Any two
1.
functions
(x),
$
are
(x),
equivalent,
whose corresponding moduli are equal. This
is
a plain consequence of the last Proposition. For since
it
is
(x)
evident that
=
if
(0) +
(0)
(48).
under required to determine the conditions
it
which the following equation
)}"*(!),
X
is satisfied, viz.
once gives
at (0)
^(
(!)
1
)
=
X( 1 )
>
(#), be proved by substituting for coefficients the and equating expanded forms,
also
may
tne ^ r
<j>
of the resulting equation properly reduced.
All the above theorems
may be extended
to functions of
For, as different elective
more
symbols combine
than one symbol. with each other according to the same laws as symbols of quan a given function with reference to any we can first tity,
particular
expand symbol which
with reference
to
and then expand the result so on in succession, the and any other symbol, it
contains,
order of the expansions being quite indifferent. Thus the given function being (xy) we have
= (xy)
and expanding the
(ay)
=
$
(00) 4
coefficients
{
r
+ ^r
= It
^r
t (J
f
.
( Ol ) ary.
^ +
^r(
^ (,)
would not be
a: 8)
(1
+
*,
difficult
to
{
{
,
=
\Jr.

(00.
(...)