Richard Whately.

Elements of logic : comprising the substance of the article in the Encyclopaedia metropolitana ; with additions, &c. online

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> expressed, to which the definition given by Aldrich, for
;' instance, would not apply ; so that he appears to employ
*y '^ " syllogism " as synonymous with ''argument." But be-
''( sides that it is clearer and more convenient, when we
/have these two words at hand, to employ them in the two
/senses respectively which we want to express, the truth


Since, then, an argument is an expression Definition of
/in which "from something laid down and
granted as true (i, e, the Premises) something
I else (i, e, the Conclusion) heyond this must he
admitted to be true, as following necessarily (or
resulting) from the other ; and since Logic is
wholly concerned in the use of language, it
follows that a Syllogism (which is an argument
stated in a regular logical form) must be
" an argument so expressed, that the con- Definition of
clusiveness of it is manifest from the mere
• fo7xe of the expression,' i, e. without consider-
ing the meaning of the terms : e, g, in this
syllogism, '' Y is X, Z is Y, therefore Z is X :"
the conclusion is inevitable, whatever terms
X, Y, and Z, respectively are understood to
c stand for. And to this form all legitimate
arguments may ultimately be brought.

is, that in so doing I have actually conformed to Aldrich's

practice : for he generally, if not always, employs the

term syllogism in the very sense to which I have confined

(' it : viz. to denote an argument stated in regular logical

S form ; as, e. g. in a part of his work (omitted in the late

{ editions) In which he is objecting to a certain pretended

\ syllogism in the work of another writer, he says, " valet

certe argumentum ; syllogismus tamen est falsissimus," &c.

Now (waiving the exception that might be taken at this

use of *' falsissimus" nothing being, strictly, true or false,

but a proposition) it is plain that he limits the word

( "syllogism" to the sense in which it is here defined, and

'' is consequently inconsistent with his own definition of it.



§ 2.

Aristotle's The rule or axiom (commonly called " dic-
tum de omni et nullo") by which Aristotle
explains the validity of this argument, is this :
^' whatever is predicated of a term distributed,
whether affirmatively or negatively, may he pre-
dicated in like manner of every thing contained
under it" Thus, in the examples above, X is
predicated of Y distributed, and Z is contained
under Y (i. e, is its subject ;) therefore X is
predicated of Z : so " all tyrants," Sf'c. (p. 74.)
This rule may be ultimately applied to all
arguments ; (and their validity ultimately rests
on their conformity thereto) but it cannot be
directly and immediately applied to all even of
pure categorical syllogisms ; for the sake of
■A brevity, therefore, some other axioms are
commonly applied in practice, tb avoid the
- occasional tediousness of reducing all syllo-
gisms to that form in which Aristotle's dictum
is applicable.*

• Instead of following Aldrich's arrangement, in laying
down first the canons which apply to all the figures of
categorical syllogisms, and then going back to the "dic-
tum of Aristotle " which applies to only one of them, I
have pursued what appears a simpler and more philo-
sophical arrangement, and more likely to impress on the
learner's mind a just view of the science; viz. 1st. to
give the rule (Aristotle's dictum) which applies to the


We will speak first of pure categorical
syllogisms ; and the axioms or canons by
which their vahdity is to be explained : viz.
first, if two terms agree with one and the same
third, they agree with each other: secondly,
if one term agrees and another disagrees with
one and the same third, these two disagree with
each other. On the former of these canons
rests the validity of affirmative conclusions ;
on the latter, of negative: for no categorical
syllogism can be faulty which does not violate
these canons; none correct which does : hence
on these two canons are built the rules or
cautions which are to be observed with respect
to syllogisms, for the purpose of ascertaining
whether those canons have been strictly ob-
served or not.

1st. Every syllogism has three, and only
three terms: viz, the middle term, and the
two terms (or extremes, as they are commonly
called) of the Conclusion or Question. Of

most clearly and regularly-constructed argument, the

Syllogism in the first figure, to which all reasoning may

be reduced ; then the canons applicable tp all categoricals ;

then, those belonging to the hypotheticals ; and lastly, to

treat of the Sorites; which is improperly placed by

(Aldrich before the hypotheticals. By this plan the pro-

ivince of strict Logic is extended as far it can be ; every

1 kind of argument which is of a syllogistic character, and

/accordingly directly cognizable by the rules of logic,

being enumerated in natural order.


these, 1st, the subject of the conclusion is
called the minor term; 2d, its predicate, the
major term ; and 3d, the middle term, is that
with which each of them is separately com-
pared, in order to judge of their agreement
or disagreement with each other. If there-
fore there were two middle terms, the ex-
tremes, (or terms of the conclusion) not being
both compared to the same, could not be
conclusively compared to each other.

2d. Every syllogism has three, and only
three propositions ; viz. 1st, the major premiss
(in which the major term is compared with the
middle:) 2d, the minor premiss (in which the
minor term is compared with the middle ;) and
3d, the Conclusion, in which the Minor term
is compared with the Major.

3d. Note, that if the middle term is amhi-
guous, there are in reality two middle terms, in
sense, though but one in sound. An am-
biguous middle term is either an equivocal
term used in different senses in the two pre-
mises ; (e,g.

" Li^ht is contrary to darkness ;
Feathers are light ; therefore
Feathers are contrary to darkness :")

or a term not distributed: for as it is then
used to stand for a part only of its signijicates,
it may happen that one of the extremes may



have been compared with one part of it, and
the other with another part of it ; e. g.

" White is a colour,

Black is a colour ; therefore

Black is white." Again,

** Some animals are beasts,

Some animals are birds ; therefore

Some birds are beasts."

The middle term therefore must be distri-
buted once, at least, in the premises ; (i, e, by
being the subject of an universal, or predicate
of a negative. Chap. ii. § 2. p. 63,) and once is
sufficient ; since if one extreme has been
compared to a ^art of the middle term, and
another to the whole of it, they must have
been both compared to the same,

4th. iVo term must be distributed in the con-
clusion which was not distributed in one of the
premises; for that (which is called an illicit
process, either of the Major or the Minor
term) would be to employ the whole of a
term in the Conclusion, when you had em-
ployed only a part of it in the Premiss ; and
thus, in reality, to introduce a fourth term :

" All quadrupeds are animals,

A bird is not a quadruped ; therefore

It is not an animal." — Illicit process of the major.
■{ . ■

5th. From negative premises you can infer

nothing. For in them the Middle is pro-
nounced to disagree with both extremes; not.


to agree with both ; or, to agree with one, and
disagree with the other; therefore they can-
not be compared together ; e, g,

" A fish is not a quadruped;"

" A bird is not a quadruped," proves nothing.

6th. If one premiss he negative, the conch-
sion must he negative ; for in that premiss the
middle term is pronounced to disagree with
one of the extremes, and in the other premiss
(which of course is affirmative by the pre-
ceding rule) to agree with the other extreme ;
therefore the extremes disagreeing with each
other, the conclusion is negative. In the
same manner it may be shown, that to prove
a negative conclusion one of the Premises must
he a negative,

* By these six rules all Syllogisms are to be
^ tried ; and from them it will be evident ; 1st,
that nothing can he proved from two particular
Premises; (for you will then have either the
middle Term undistributed, or an illicit pro-
cess: e. g,

* Aldrich has given twelve rules, which I found might
more conveniently be reduced to six. No syllogism can
be faulty which violates none of these six rules. It is
much less perplexing to a learner not to lay down as a
distinct rule, that, e. g. against particular premises ; which
is properly a result of the foregoing ; since a syllogism
with two particular premises would offend against either
R. 3. or R. 4.


*' Some animals are sagacious :
Some beasts are not sagacious :
Some beasts are not animals.")

And, for the same reason, 2dly, that if one
of the Premises be particular, the Conclusion
must be particular ; e, g,

" All who fight bravely deserve reward ;

Some soldiers fight bravely ;" you can only infer that

** Some soldiers deserve reward :"

for to infer a universal Conclusion would be
an illicit process of the minor. But from two
universal Premises you cannot always infer a
universal Conclusion; e,g,

" All gold is precious,

All gold is a mineral : therefore

Some mineral is precious."*

And even when we can infer a universal,
we are always at liberty to infer a particular ;
since what is predicated of all may of course be
predicated of some.

Of Moods.


: When we designate the three propositions

of a syllogism in their order, according to

* Aldrich, by a strange oversight, has so expressed
himself as to imply (though he could hardly mean it) that
we always may, if we will, infer a universal conclusion
from two universal premises.



their respective quantity and quality (L e, their
symbols) we are said to determine the mood
of the syllogism ; e. g, the example just above,
" all gold, Sfc." is in the mood A, A, I. As
there are four kinds of propositions, and three
propositions in each syllogism, all the possible
ways of combining these four, (A, E, I, O,) by
threes, are sixty-four. For any one of these
four may be the major premiss, each of these
four majors may have four different minors,
and of these sixteen pairs of premises, each
may have four different conclusions. 4x4
(=16) X 4 = 64. This is a mere arithmetical
calculation of the moods, without any regard
to the logical rules : for many of these moods
are inadmissible in practice, from violating
some of those rules ; e, g, the mood E, E, E,
must be rejected as having negaitve premises ;
I, O, O, for particular premises; and many
others for the same faults ; to which must be
added I, E, O, for an illicit process of the
major, in every figure. By examination then
of all, it will be found that, of the sixty-four,
there remain but eleven moods which can be
used in a legitimate syllogism, viz. A, A, A,
A, A, I, A, E, E, A, E, O, A, I, I, A, O, O,
E, A, E, E, A, O, E, I, O, I, A, I, O, A, O.


Of Figure,


j^ The Figure of a syllogism consists in the
N., situation of the Middle term with respect to
the Extremes of the Conclusion, (i, e, the major
and minor term,) When the Middle term is
made the subject of the major premiss, and the
predicate of the minor, that is called the first
U Figure; (which is far the most natural and
clear of all, as to this alone Aristotle's Dictum
may be at once applied.) In the second Figure
yj the Middle term is the predicate of both pre-
mises : in the third, the subject of both : in the
"" fourth the predicate of the Major premiss, and
the subject of the Minor, (This is the most
awkward and unnatural of all, being the very
reverse of the first.) Note, that the proper
order is to place the Major premiss ^r^^, and
the Minor second; but this does not constitute
the Major and Minor premises ; for that pre-
miss (wherever placed) is the Major, which
contains the major term, and the Minor, the
minor (v. R. 2. p. 78.) Each of the allowable
moods mentioned above will not be allowable
in every Figure ; since it may violate some of
the foregoing rules, in one Figure, though not
in another : e. g, I, A, I, is an allowable mood
in the third Figure ; but in the first it would



have an undistributed middle.^ So A, E, E,
would in the first Figure have an illicit process
of the major, but is allowable in the second ;
and A, A, A, which in the first Figure is allow-
able, would in the third have an illicit process
of the minor : all which may be ascertained by
trying the different Moods in each figure, as
per scheme.

Let X represent the major term, Z the
minor, Y the middle.

1st Fig.

2d Fig.

3d Fig.

4th Fig.

Y, X,

X, Y,

Y, X,


Z, Y,

Z, Y,

Y, Z,

Y, Z,


Z, X,

Z, X,

Z, X.

The Terms alone being here stated, the
quantity and quality of each Proposition (and
consequently the Mood of the whole Syllo-
gism) is left to be filled up : (i, e, between
Y and X, we may place either a negative or
affirmative Copula : and we may prefix either
a universal or particular sign to Y.) By apply-
ing the Moods then to each Figure, it will be
found that each Figure will admit six Moods

r A'

♦ €. g. Some restraint is salutary : all restraint is un-

( — 1

pleasant : something unpleasant is salutary. Again: Some


herbs are fit for food : nightshade is an herb : some

nightshade is fit for food.


only, as not violating the rules against undis-
tributed middle y and against illicit process : and
of the Moods so admitted, several (though
vahd) are useless, as having a particular Con-
clusion, when a universal might have been
drawn ; e. g. A, A, I, in the first Figure,

"All human creatures are entitled to liberty;
All slaves are human creatures ; therefore
Some slaves are entitled to liberty."

Of the twenty-four Moods, then, (six in
each Figure) five are for this reason neg-
lected: for the remaining nineteen, logicians
have devised names to distinguish both the
Mood itself, and the Figure in which it is.
found ; since when one Mood (i, e, one m
itself, without regard to Figure) occurs in
two different Figures, (as E, A, E, in the
first and second) the mere letters denoting
the mood would not inform us concerning
the figure. In these names, then, the three
.vowels denote the propositions of which the
) Syllogism is composed : the consonants (be-
,> sides their other uses, of which hereafter)
/serve to keep in mind the Figure of the
( Syllogism.

Fig. 1. bArbArA, cElArEnt, dArll, fErlOque prio-

Fig. 2. cEsArE, cAmEstrEs, fEstInO, bArOkO,*

* Or, Fakoro, see § 7.


( tenia, dArAptI, dlsAmls, dAtlsI, f ElAptOn,
Fig. 3. < bOkArdO,* f ErIsO, habet : quarta insuper

( addit.
Fig. 4. brAmAntIp, cAmEnEs, dImArls, fEsApo,

By a careful study of these mnemonic lines
(which must be committed to memory) you
will perceive that A can only be proved in
the first Figure, in which also every other
Proposition may be proved; that the second
proves only negatives; the third only parti-
\ culars ; that the first Figure requires the
/ major premiss to be universal, and the minor,
affirmative, ^c; with many other such ob-
servations, which will readily be made, (on
trial of several Syllogisms, in different Moods)
and the reasons for which will be found in
the foregoing rules : e, g, to show why the
second figure has only negative Conclusions,
we have only to consider, that in it the mid-
dle term being the predicate in both premises,
would not be distributed unless one premiss
were negative ; (Chap. ii. § 2.) therefore the
Conclusion must be negative also, by Chap,
iii. § 2, Rule 6. One Mood in each figure
may suffice in this place by way of example :

First, Barbara, viz, (bAr.) " Every Y is X ;
(bA) every Z is Y; therefore (rA) every Z
is X : " e, g. let the major term (which is

* Or, Dokamo, see § 7.


represented by X) be " one who possesses all
virtue ;" the minor term (Z) ^' every man who
possesses one virtue ;" and the middle term
(Y) *' every one who possesses prudence ;"
and you will have the celebrated argument of
Aristotle, Eth, sixth book, to prove that the
virtues are inseparable ; viz,

*' He who possesses prudence, possesses all virtue ;
He who possesses one virtue, must possess prudence ;

He who possesses one, possesses all." .

Second, Camestres, (cAm) '^ every X is Y ;
(Es) no Z is Y ; (trES) no Z is X." Let the
major term (X) be " true philosophers," the
minor (Z) " the Epicureans ;" the middle (Y)
^^ reckoning virtue a good in itself;" and this
will be part of the reasoning of Cicero, Off.
book first and third, against the Epicureans.

Third, Darapti, viz. (dA) '' every Y is X ;
{rA'p) every Y is Z ; therefore {tT) Some Z is
X :" e.g.

*' Prudence has for its object the benefit of individuals ;
but prudence is a virtue : therefore some virtue has for
its object the benefit of the individual,"

is part of Adam Smith's reasoning {Moral
Sentiments) against Hutcheson and others,
who placed all virtue in benevolence.

Fourth, Camenes, viz. (cAm) " every X is Y ;
CE?iJ no Y is Z ; therefore (Es) no Z is X :"
e. g.


" Whatever is expedient, is conformable to nature ;
Whatever is conformable to nature, is not hurtful to

society ; therefore
What is hurtful to society is never expedient."

is part of Cicero's argument in Off. Lib. iii. ;
but it is an inverted and clumsy way of
stating what would much more naturally fall
into the first Figure ; for if you examine the
Propositions of a Syllogism in the fourth
Figure, beginning at the Conclusion, you will
see that as the major term is predicated of the
minor, so is the minor of the middle, and that
again of the major ; so that the major appears
to be merely 'predicated of itself. Hence the
five Moods in this Figure are seldom or never
used ; some one of the fourteen (moods with
names) in the first three Figures; being the
forms into which all arguments may most
readily be thrown ; but of these, the four in
the first Figure are the clearest and most
natural ; as to them Aristotle's dictum will
immediately apply.* And as it is on this dictum

* With respect to the use of the first three Figures
(for the fourth is never employed but by an accidental
awkwardness of expression) it may be remarked, that the
First is that into which an argument will be found to fall
the most naturally, except in the following cases :— First,
When we have to disprove something that has been main-
tained, or is likely to be believed, our arguments will
usually be found to take most conveniently the form of
the Second Figure : viz. we prove that the thing we are



that all Reasoning ultimately depends, so all

arguments may be in one way or other

brought into some one of these four Moods ;

/ and a Syllogism is, in that case, said to be

^- reduced: (L e, to the first Jigure,) These four

\ are called the perfect moods, and all the rest

< imperfect,

speaking of cannot belong to such a Class, either because
it wants what belongs to the whole of that Class, (Cesare)
or because it has something of which that Class is desti-
tute; (Camestres) e.g. "No impostor would have warned
his followers, as Jesus did, of the persecutions they would
have to submit to :" and again, " An enthusiast would
have expatiated, which Jesus and his followers did not,
on the particulars of a future state."

The same observations will apply, mutatis mutandis^
when a Particular conclusion is sought, as in Festino and

The arguments used in the process called the "Ab-
scissio Infiniti," will in general be the most easily referred
to this Figure. See Chap. v. § 1. subsection 6.

The Third Figure is, of course, the one employed when
the Middle term is Singular, since a Singular term can
only be a Subject. This is also the form into which most
arguments will naturally fall that are used to establish
an objection (Enstasis of Aristotle) to an opponent's Pre-
miss, when his argument is such as to require that premiss
to be Universal. It might be called, therefore, the
Enstatic Figure. E. G. If any one contends that " this
or that doctrine ought not to be admitted, because it
cannot be explained or comprehended," his suppressed
major premiss may be refuted by the argument that " the
connexion of the Body and Soul cannot be explained or
comprehended," &c.

A great part of the reasoning of Butler's Analogy may
be exhibited in this form.


Ostenswe Reduction.
§ 5.

In reducing a Syllogism, we are not, of
(^ course, allowed to introduce any new Term
/ or Proposition, having nothing granted but
the truth of the Premises ; but these Pre-
mises are allowed to be illatively converted
(because the truth of any Proposition implies
that of its illative converse) or transposed: by
taking advantage of this liberty, where there
is need, we deduce (in Figure 1st,) from the
i Premises originally given, either the very same
) Conclusion as the original one, or another
> from which the original Conclusion follows by
illative conversion ; e, g, Darapti,

" All wits are dreaded ;

All wits are admired ;

Some who are admired are dreaded,"

into Darii, by converting by limitation (per
accidens) the minor Premiss.

*' All wits are dreaded;

Some who are admired are wits ; therefore

Some who are admired are dreaded."


" All true philosophers account virtue a good in itself;
The advocates of pleasure do not account, ^c.
Therefore they are not true philosophers,"


reduced to Celarent, by simply converting the
minor, and then transposing the Premises.

" Those who account virtue a good in itself, are not

advocates of pleasure ;
All true philosophers account virtue, ^c, : therefore
No true philosophers are advocates of pleasure."

This Conclusion may be illatively converted
into the original one.

Baroko;'' e, g. Reduction by

means of

" Every true patriot is a friend to religion; by^negaUon.

Some great statesmen are not friends to religion ;
Some great statesmen are not true patriots,"

to Ferio, by converting the major hy negatio?i,
(contraposition), vide Chap. ii. § 4.

" He who is not a friend to religion, is not a true patriot :
Some great statesmen, Sj-c."

and the rest of the Syllogism remains the
same : only that the minor Premiss must be
considered as affirmative, because you take
'' not-a-friend-to-religion," as the middle term.
In the same manner Bokardof to Darii ; e.g,

" Some slaves are not discontented ;

All slaves are wronged ; therefore

Some who are wronged are not discontented."

Convert the major by negation (contrapo-
sition) and then transpose them; the Con-
clusion will be the converse hy negation of the

* Or Fakoro, considered i, e. as Festino.
t Or Dokamo, considered i. e, as Disamis.


original one, which therefore may be inferred
from it ; e, g,

** All slaves are wronged ;

Some who are not discontented are slaves ;

Some who are not discontented are wronged."

In these ways (by what is called Ostensive
Reduction, because you prove, in the first
figure, either the veri/ same Conclusion as be-
fore, or one which implies it) all the imperfect
Moods may be reduced to the four perfect
ones. But there is also another way, called

Reductio ad impossibile,


By which we prove (in the first figure) not
directly that the original Conclusion is true,
but that it cannot be false ; i. e, that an ab-
surdity would follow from the supposition of
its being false ; e, g,

" All true patriots are friends to religion ;
Some great statesmen are not friends to religion ;
Some great statesmen are not true patriots."

If this Conclusion be not true, its contra-
dictory must be true ; viz,

'* All great statesmen are true patriots."
Let this then be assumed, in the place of the


Online LibraryRichard WhatelyElements of logic : comprising the substance of the article in the Encyclopaedia metropolitana ; with additions, &c. → online text (page 7 of 26)