CATEGORICAL SYLLOGISMS
Sec. 1. The type of logical, deductive, mediate, categorical Inference is a
Syllogism directly conformable with the _Dictum_: as--
All carnivores (M) are excitable (P);
Cats (S) are carnivores (M):
.'. Cats (S) are excitable (P).
In this example P is predicated of M, a term distributed; in which term,
M, S is given as included; so that P may be predicated of S.
Many arguments, however, are of a type superficially different from the
above: as--
No wise man (P) fears death (M);
Balbus (S) fears death (M):
.'. Balbus (S) is not a wise man (P).
In this example, instead of P being predicated of M, M is predicated of
P, and yet S is given as included not in P, but in M. The divergence of
such a syllogism from the _Dictum_ may, however, be easily shown to be
superficial by writing, instead of _No wise man fears death_, the
simple, converse, _No man who fears death is wise_.
Again:
Some dogs (M) are friendly to man (P);
All dogs (M) are carnivores (S):
.'. Some carnivores (S) are friendly to man (P).
Here P is predicated of M undistributed; and instead of S being included
in M, M is included in S: so that the divergence from the type of
syllogism to which the _Dictum_ directly applies is still greater than
in the former case. But if we transpose the premises, taking first
All dogs (M) are carnivores (P),
then P is predicated of M distributed; and, simply converting the other
premise, we get--
Some things friendly to man (S) are dogs (M):
whence it follows that--
Some things friendly to man (S) are carnivores (P);
and this is the simple converse of the original conclusion.
Once more:
No pigs (P) are philosophers (M);
Some philosophers (M) are hedonists (S):
.'. Some hedonists (S) are not pigs (P).
In this case, instead of P being predicated of M distributed, M is
predicated of P distributed; and instead of S (or part of it) being
included in M, we are told that some M is included in S. Still there is
no real difficulty. Simply convert both the premises, and we have:
No philosophers (M) are pigs (P);
Some hedonists (S) are philosophers (M).
Whence the same conclusion follows; and the whole syllogism plainly
conforms directly to the _Dictum_.
Such departures as these from the normal syllogistic form are said to
constitute differences of Figure (see Sec. 2); and the processes by which
they are shown to be unessential differences are called Reduction (see Sec.
6).
Sec. 2. Figure is determined by the position of the Middle Term in the
premises; of which position there are four possible variations. The
middle term may be subject of the major premise, and predicate of the
minor, as in the first example above; and this position, being directly
conformable to the requirements of the _Dictum_, is called the First
Figure. Or the middle term may be predicate of both premises, as in the
second of the above examples; and this is called the Second Figure. Or
the middle term may be subject of both premises, as in the third of the
above examples; and this is called the Third Figure. Or, finally, the
middle term may be predicate of the major premise, and subject of the
minor, as in the fourth example given above; and this is the Fourth
Figure.
It may facilitate the recollection of this most important point if we
schematise the figures thus:
I. II. III. IV.
M---P P---M M---P P---M
\ | | /
\ | | /
\ | | /
S---M S---M M---S M---S
The horizontal lines represent the premises, and at the angles formed
with them by the slanting or by the perpendicular lines the middle term
occurs. The schema of Figure IV. resembles Z, the last letter of the
alphabet: this helps one to remember it in contrast with Figure I.,
which is thereby also remembered. Figures II. and III. seem to stand
back to back.
Sec. 3. The Moods of each Figure are the modifications of it which arise
from different combinations of propositions according to quantity and
quality. In Figure I., for example, four Moods are recognised: A.A.A.,
E.A.E., A.I.I., E.I.O.
A. All M is P;
A. All S is M:
A. .'. All S is P.
E. No M is P;
A. All S is M:
E. .'. No S is P.
A. All M is P;
I. Some S is M:
I. .'. Some S is P.
E. No M is P;
I. Some S is M:
O. .'. Some S is not P.
Now, remembering that there are four Figures, and four kinds of
propositions (A. I. E. O.), each of which propositions may be major
premise, minor premise, or conclusion of a syllogism, it appears that in
each Figure there may be 64 Moods, and therefore 256 in all. On
examining these 256 Moods, however, we find that only 24 of them are
valid (i.e., of such a character that the conclusion strictly follows
from the premises), whilst 5 of these 24 are needless, because their
conclusions are 'weaker' or less extensive than the premises warrant;
that is to say, they are particular when they might be universal. Thus,
in Figure I., besides the above 4 Moods, A.A.I. and E.A.O. are valid in
the sense of being conclusive; but they are superfluous, because
included in A.A.A. and E.A.E. Omitting, then, these 5 needless Moods,
which are called 'Subalterns' because their conclusions are subaltern
(chap. vii. Sec. 2) to those of other Moods, there remain 19 Moods that are
valid and generally recognised.
Sec. 4. How these 19 Moods are determined must be our next inquiry. There
are several ways more or less ingenious and interesting; but all depend
on the application, directly or indirectly, of the Six Canons, which
were shown in the last chapter to be the conditions of Mediate
Inference.
(1) One way is to begin by finding what Moods of Figure I. conform to
the _Dictum_. Now, the _Dictum_ requires that, in the major premise, P
be predicated of a term distributed, from which it follows that no Mood
can be valid whose major premise is particular, as in I.A.I. or O.A.O.
Again, the _Dictum_ requires that the minor premise be affirmative
("with which term another is identified"); so that no Mood can be valid
whose minor premise is negative, as in A.E.E. or A.O.O. By such
considerations we find that in Figure I., out of 64 Moods possible, only
six are valid, namely, those above-mentioned in Sec. 3, including the two
subalterns. The second step of this method is to test the Moods of the
Second, Third, and Fourth Figures, by trying whether they can be reduced
to one or other of the four Moods of the First (as briefly illustrated
in Sec. 1, and to be further explained in Sec. 6).
(2) Another way is to take the above six General or Common Canons, and
to deduce from them Special Canons for testing each Figure: an
interesting method, which, on account of its length, will be treated of
separately in the next section.
(3) Direct application of the Common Canons is, perhaps, the simplest
plan. First write out the 64 Moods that are possible without regard to
Figure, and then cross out those which violate any of the Canons or
Corollaries, thus:
[Transcriber's Note: Moods surrounded with square brackets were crossed
out in the original text.]
AAA, [AAE] (6th Can. b). AAI. [AAO] (6th Can. b).
[AEA] (6th Can. a) AEE, [AEI] (6th Can. a) AEO,
[AIA] (Cor. ii.) [AIE] (6th Can. b) AII, [AIO] (6th Can. b)
[AOA] (6th Can. a) [AOE] (Cor. ii.) [AOI] (6th Can. a) AOO.
Whoever has the patience to go through the remaining 48 Moods will
discover that of the whole 64 only 11 are valid, namely:
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.
These 11 Moods have next to be examined in each Figure, and if valid in
every Figure there will still be 44 moods in all. We find, however, that
in the First Figure, A.E.E., A.E.O., A.O.O. involve illicit process of
the major term (3rd Can.); I.A.I., O.A.O. involve undistributed Middle
(4th Can.); and A.A.I., E.A.O. are subalterns. In the Second Figure all
the affirmative Moods, A.A.A., A.A.I., A.I.I., I.A.I., involve
undistributed Middle; O.A.O. gives illicit process of the major term;
and A.E.O., E.A.O. are subalterns. In the Third Figure, A.A.A., E.A.E.,
involve illicit process of the minor term (3rd Can.); A.E.E., A.E.O.,
A.O.O., illicit process of the major term. In the Fourth Figure, A.A.A.
and E.A.E. involve illicit process of the minor term; A.I.I., A.O.O.,
undistributed Middle; O.A.O. involves illicit process of the major term;
and A.E.O. is subaltern.
Those moods of each Figure which, when tried by these tests, are not
rejected, are valid, namely:
Fig. I.--A.A.A., E.A.E., A.I.I., E.I.O. (A.A.I., E.A.O., Subaltern);
Fig. II.--E.A.E., A.E.E., E.I.O., A.O.O. (E.A.O., A.E.O., Subaltern);
Fig. III.--A.A.I., I.A.I., A.I.I., E.A.O., O.A.O., E.I.O.;
Fig. IV.--A.A.I., A.E.E., I.A.I., E.A.O., E.I.O. (A.E.O., Subaltern).
Thus, including subaltern Moods, there are six valid in each Figure. In
Fig. III. alone there is no subaltern Mood, because in that Figure there
can be no universal conclusion.
Sec. 5. Special Canons of the several Figures, deduced from the Common
Canons, enable us to arrive at the same result by a somewhat different
course. They are not, perhaps, necessary to the Science, but afford a
very useful means of enabling one to thoroughly appreciate the character
of formal syllogistic reasoning. Accordingly, the proof of each rule
will be indicated, and its elaboration left to the reader. There is no
difficulty, if one bears in mind that Figure is determined by the
position of the middle term.
Fig. I., Rule (a): _The minor premise must be affirmative_.
For, if not, in negative Moods there will be illicit process of the
major term. Applying this rule to the eleven possible Moods given in Sec.
4, as remaining after application of the Common Canons, it eliminates
A.E.E., A.E.O., A.O.O.
(b) _The major premise must be universal_.
For, if not, the minor premise being affirmative, the middle term will
be undistributed. This rule eliminates I.A.I., O.A.O.; leaving six
Moods, including two subalterns.
Fig. II. (a) _One premise must be negative._
For else neither premise will distribute the middle term. This rule
eliminates A.A.A., A.A.I., A.I.I., I.A.I.
(b) _The major premise must be universal._
For else, the conclusion being negative, there will be illicit process
of the major term. This eliminates I.A.I., O.A.O.; leaving six Moods,
including two subalterns.
Fig. III. (a) _The minor premise must be affirmative._
For else, in negative moods there will be illicit process of the major
term. This rule eliminates A.E.E., A.E.O., A.O.O.
(b) _The conclusion must be particular._
For, if not, the minor premise being affirmative, there will be illicit
process of the minor term. This eliminates A.A.A., A.E.E., E.A.E.;
leaving six Moods.
Fig. IV. (a) _When the major premise is affirmative, the minor must be
universal._
For else the middle term is undistributed. This eliminates A.I.I.,
A.O.O.
(b) _When the minor premise is affirmative the conclusion must be
particular._
Otherwise there will be illicit process of the minor term. This
eliminates A.A.A., E.A.E.
(c) _When either premise is negative, the major must be universal._
For else, the conclusion being negative, there will be illicit process
of the major term. This eliminates O.A.O.; leaving six Moods, including
one subaltern.
Sec. 6. Reduction is either--(1) Ostensive or (2) Indirect. Ostensive
Reduction consists in showing that an argument given in one Mood can
also be stated in another; the process is especially used to show that
the Moods of the second, third, and fourth Figures are equivalent to one
or another Mood of the first Figure. It thus proves the validity of the
former Moods by showing that they also essentially conform to the
_Dictum_, and that all Categorical Syllogisms are only superficial
varieties of one type of proof.
To facilitate Reduction, the recognised Moods have all had names given
them; which names, again, have been strung together into mnemonic verses
of great force and pregnancy:
Barbara, Celarent, Darii, Ferioque prioris:
Cesare, Camestres, Festino, Baroco, secundae:
Tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet: Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
In the above verses the names of the Moods of Fig. I. begin with the
first four consonants B, C, D, F, in alphabetical order; and the names
of all other Moods likewise begin with these letters, thus signifying
(except in Baroco and Bocardo) the mood of Fig. I., to which each is
equivalent, and to which it is to be reduced: as Bramantip to Barbara,
Camestres to Celarent, and so forth.
The vowels A, E, I, O, occurring in the several names, give the quantity
and quality of major premise, minor premise, and conclusion in the usual
order.
The consonants s and p, occurring after a vowel, show that the
proposition which the vowel stands for is to be converted either (s)
simply or (p) _per accidens_; except where s or p occurs after the third
vowel of a name, the conclusion: then it refers not to the conclusion of
the given Mood (say Disamis), but to the conclusion of that Mood of the
first Figure to which the given Mood is reduced (Darii).
M (_mutare_, metathesis) means 'transpose the premises' (as of
Ca_m_estres).
C means 'substitute the contradictory of the conclusion for the
foregoing premise,' a process of the Indirect Reduction to be presently
explained (see Baroco, Sec. 8).
The other consonants, r, n, t (with b and d, when not initial),
occurring here and there, have no mnemonic significance.
What now is the problem of Reduction? The difference of Figures depends
upon the position of the Middle Term. To reduce a Mood of any other
Figure to the form of the First, then, we must so manipulate its
premises that the Middle Term shall be subject of the major premise and
predicate of the minor premise.
Now in Fig. II. the Middle Term is predicate of both premises; so that
the minor premise may need no alteration, and to convert the major
premise may suffice. This is the case with Cesare, which reduces to
Celarent by simply converting the major premise; and with Festino, which
by the same process becomes Ferio. In Camestres, however, the minor
premise is negative; and, as this is impossible in Fig. I., the premises
must be transposed, and the new major premise must be simply converted:
then, since the transposition of the premises will have transposed the
terms of the conclusion (according to the usual reading of syllogisms),
the new conclusion must be simply converted in order to prove the
validity of the original conclusion. The process may be thus represented
(_s.c._ meaning 'simply convert')
Camestres. Celarent.
All P is M; ----\ /---> No M is S;
\ c/
\/
s/\
/ \
No S is M: ----/ \---> All P is M:
s.c.
.'. No S is P. <----------- No P is S.
The Ostensive Reduction of Baroco also needs special explanation; for as
it used to be reduced indirectly, its name gives no indication of the
ostensive process. To reduce it ostensively let us call it Faksnoko,
where k means 'obvert the foregoing premise.' By thus obverting (k) and
simply converting (s) (in sum, contrapositing) the major premise, and
obverting the minor premise, we get a syllogism in Ferio, thus:
Baroco or Faksnoko. Ferio.
_contrap_
All P is M; -----------------------> No m (not-M) is P;
_obv_
Some S is not M: -----------------------> Some S is m (not-M):
.'. Some S is not P. .'. Some S is not P.
In Fig. III. the middle term is subject of both premises; so that, to
reduce its Moods to the First Figure, it may be enough to convert the
minor premise. This is the case with Darapti, Datisi, Felapton, and
Ferison. But, with Disamis, since the major premise must in the First
Figure be universal, we must transpose the premises, and then simply
convert the new minor premise; and, lastly, since the major and minor
terms have now changed places, we must simply convert the new conclusion
in order to verify the old one. Thus:
Disamis. Darii.
Some M is P; ----\ /---> All M is S;
\s./
\/
/\c.
/ \
All M is S: ----/ \---> Some P is M:
s.c.
.'. Some S is P. <------------- .'. Some P is S.
Bocardo, like Baroco, indicates by its name the indirect process. To
reduce it ostensively let its name be Doksamrosk, and proceed thus:
Bocardo or Doksamrosk. Darii.
Some M is not P; ----------\ /---------> All M is S;
\ /
\/
/\ _contrap_
/ \
All M is S: ----------/ \---------> Some p (not-P) is M:
_convert & obvert_
.'. Some S is not P. <------------------------- .'. Some p (not-P) is S.
In Fig. IV. the position of the middle term is, in both premises, the
reverse of what it is in the First Figure; we may therefore reduce its
Moods either by transposing the premises, as with Bramantip, Camenes,
and Dimaris; or by converting both premises, the course pursued with
Fesapo and Fresison. It may suffice to illustrate by the case of
Bramantip:
Bramantip. Barbara.
All P is M; ----------\ /------> All M is S;
\/
/\
All M is S: ----------/ \------> All P is M:
convert per acc.
.'. Some S is P. <----------------------- .'. All P is S.
This case shows that a final significant consonant (s, p, or sk) in the
name of any Mood refers to the conclusion of the new syllogism in the
First Figure; since p in Bramantip cannot refer to that Mood's own
conclusion in I.; which, being already particular, cannot be converted
_per accidens_.
Finally, in Fig. I., Darii and Ferio differ respectively from Barbara
and Celarent only in this, that their minor premises, and consequently
their conclusions, are subaltern to the corresponding propositions of
the universal Moods; a difference which seems insufficient to give them
rank as distinct forms of demonstration. And as for Barbara and
Celarent, they are easily reducible to one another by obverting their
major premises and the new conclusions, thus:
Barbara. Celarent.
obv.
All M is P; -----------------------> No M is p (not-P);
All S is M: -----------------------> All S is M:
obv.
.'. All S is P. <------------------- .'. No S is p (not-P).
There is, then, only one fundamental syllogism.
Sec. 7. A new version of the mnemonic lines was suggested in _Mind_ No. 27,
with the object of (1) freeing them from all meaningless letters, (2)
showing by the name of each Mood the Figure to which it belongs, (3)
giving names to indicate the ostensive reduction of Baroco and Bocardo.
To obtain the first two objects, _l_ is used as the mark of Fig. I., _n_
of Fig II., _r_ of Fig. III., _t_ of Fig. IV. The verses (to be scanned
discreetly) are as follows:
Balala, Celalel, Dalii, Felioque prioris:
{Faksnoko}
Cesane, Camenes, Fesinon, { } secundae:
{ Banoco,}
Tertia, Darapri, Drisamis, Darisi, Ferapro,
Doksamrosk}
}, Ferisor habet: Quarta insuper addit.
Bocaro }
Bamatip, Cametes, Dimatis, Fesapto, Fesistot.
De Morgan praised the old verses as "more full of meaning than any
others that ever were made"; and in defence of the above alteration it
may be said that they now deserve that praise still more.
Sec. 8. Indirect reduction is the process of proving a Mood to be valid by
showing that the supposition of its invalidity involves a contradiction.
Take Baroco, and (since the doubt as to its validity is concerned not
with the truth of the premises, but with their relation to the
conclusion) assume the premises to be true. Then, if the conclusion be
false, its contradictory is true. The conclusion being in O., its
contradictory will be in A. Substituting this A. for the minor premise
of Baroco, we have the premises of a syllogism in Barbara, which will be
found to give a conclusion in A., contradictory of the original minor
premise; thus:
Baroco. Barbara.
All P is M; -----------------> All P is M;
Some S is not M: <-----\ /-----> All S is P:
\ /
contradictory \/
/\ contradictory
/ \
.'. Some S is not P ------/ \------ .'. All S is M.
But the original minor premise, _Some S is not M_, is true by
hypothesis; and therefore the conclusion of Barbara, _All S is M_, is
false. This falsity cannot, however, be due to the form of Barbara,
which we know to be valid; nor to the major premise, which, being taken
from Baroco, is true by hypothesis: it must, therefore, lie in the minor
premise of Barbara, _All S is P_; and since this is contradictory of the
conclusion of Baroco _Some S is not P_, that conclusion was true.
Similarly, with Bocardo, the Indirect Reduction proceeds by substituting
for the major premise the contradictory of the conclusion; thus again
obtaining the premises of a syllogism in Barbara, whose conclusion is
contradictory of the original major premise. Hence the initial B in
Baroco and Bocardo: it points to a syllogism in Barbara as the means of
Indirect Reduction (_Reductio ad impossibile_).
Any other Mood may be reduced indirectly: as, for example, Dimaris. If
this is supposed to be invalid and the conclusion false, substitute the
contradictory of the conclusion for the major premise, thus obtaining
the premises of Celarent:
Dimaris. Celarent.
contradictory
Some P is M; <--------- --------> No S is P;
\ /
\/
All M is S: -----------/\---------> All M is S:
/ \
contradictory/ \
.'. Some S is P. ----------- -------- .'. No M is P}
} simply converted
.'. No P is M}
The conclusion of Celarent, simply converted, contradicts the original
major premise of Dimaris, and is therefore false. Therefore the major
premise of Celarent is false, and the conclusion of Dimaris is true. We
might, of course, construct mnemonic names for the Indirect Reduction of
all the Moods: the name of Dimaris would then be Cicari.
Sec. 9. The need or use of any Figure but the First has been much discussed
by Logicians. Since, in actual debate, arguments are rarely stated in
syllogistic form, and, therefore, if reduced to that form for closer
scrutiny, generally have to be treated with some freedom; why not always
throw them at once into the First Figure? That Figure has manifest
advantages: it agrees directly with the _Dictum_; it gives conclusions
in all four propositional forms, and therefore serves every purpose of
full affirmation or denial, of showing agreement or difference (total or
partial), of establishing the contradictories of universal statements;
and it is the only Figure in which the subject and predicate of the
conclusion occupy the same positions in the premises, so that the course
of argument has in its mere expression an easy and natural flow.
Still, the Second Figure also has a very natural air in some kinds of
negative arguments. The parallelism of the two premises, with the middle
term as predicate in both, brings out very forcibly the necessary
difference between the major and minor terms that is involved in their
opposite relations to the middle term. _P is not, whilst S is, M_, says
Cesare: that drives home the conviction that _S is not P_. Similarly in
Camestres: _Deer do, oxen do not, shed their horns_. What is the
conclusion?
The Third Figure, again, furnishes in Darapti and Felapton, the most
natural forms of stating arguments in which the middle term is singular:
Socrates was truthful;
Socrates was a Greek:
.'. Some Greek was truthful.
Reducing this to Fig I., we should get for the minor premise, _Some
Greek was Socrates_: which is certainly inelegant. Still, it might be
urged that, in relation to proof, elegance is an extraneous
consideration. And as for the other advantage claimed for Fig.
III.--that, as it yields only particular conclusions, it is useful in
establishing contradictories against universals--for that purpose none
of its Moods can be better than Darii or Ferio.
As for Fig. IV., no particular advantage has been claimed for it. It is
of comparatively late recognition (sometimes called the 'Galenian,'
after Galen, its supposed discoverer); and its scientific claim to exist
at all is disputed. It is said to be a mere inversion of Fig. I.; which
is not true in any sense in which Figs. II. and III. may not be
condemned as partial inversions of Fig. I., and as having therefore
still less claim to recognition. It is also said to invert the order of
thought; as if thought had only one order, or as if the order of thought
had anything to do with Formal Logic. Surely, if distinction of Figure
be recognised at all, the Fourth Figure is scientifically necessary,
because it is inevitably generated by an analysis of the possible
positions of the middle term.
Sec. 10. Is Reduction necessary, however; or have not all the Figures equal
and independent validity? In one sense not only every Figure but each
Mood has independent validity: for any one capable of abstract thinking
sees its validity by direct inspection; and this is true not only of the
abstract Moods, but very frequently of particular concrete arguments.
But science aims at unifying knowledge; and after reducing all possible
arguments that form categorical syllogisms to the nineteen Moods, it is
another step in the same direction to reduce these Moods to one form.
This is the very nature of science: and, accordingly, the efforts of
some Logicians to expound separate principles of each Figure seem to be
supererogatory. Grant that they succeed; and what can the next step be,
but either to reduce these principles to the _Dictum_, or the _Dictum_
and the rest to one of these principles? Unless this can be done there
is no science of Formal Logic. If it is done, what is gained by reducing
the principles of the other Figures to the _Dictum_, instead of the
Moods of the other Figures to those of the first Figure? It may,
perhaps, be said that to show (1) that the Moods of the second, third,
and fourth Figures flow from their own principles (though, in fact,
these principles are laboriously adapted to the Moods); and (2) that
these principles may be derived from the _Dictum_, is the more
uncompromisingly gradual and regular method: but is not Formal Logic
already sufficiently encumbered with formalities?
Sec. 11. Euler's diagrams are used to illustrate the syllogism, though not
very satisfactorily, thus:
Barbara--
[Illustration: FIG. 5.]
Celarent--
[Illustration: FIG. 6.]
Darii--
[Illustration: FIG. 7.]
Remembering that 'Some' means 'It may be all,' it is plain that any one
of these diagrams in Fig. 7, or the one given above for Barbara, may
represent the denotative relations of P, M and S in Darii; though no
doubt the diagram we generally think of as representing Darii is No. 1
in Fig. 7.
Remembering that A may be U, and that, therefore, wherever A occurs
there may be only one circle for S and P, these syllogisms may be
represented by only two circles, and Barbara by only one.
Ferio--
[Illustration: FIG. 8.]
Here, again, probably, we generally think of No. 1 as the diagram
representing Ferio; but 2, or 3, or that given above for Celarent, is
compatible with the premises.
If instead of dealing with M, P, and S, a concrete example be taken of
Darii or Ferio, a knowledge of the facts of the case will show what
diagram is suitable to it. But, then, surely it must be possible to do
without the diagram. These diagrams, of course, can be used to
illustrate Moods of the other Figures.