Aristotle
one single exemplification: If some animal is not a man, you are not
thereby warranted in asserting the converse, that some man is not an
animal.[19] It is plain that such an exemplification is only an
appeal to Induction: you produce one particular example, which is
entering on the track of Induction; and one example alone is
sufficient to establish the negative of an universal proposition.[20]
The converse of a Particular Negative is not in all cases true,
though it may be true in many cases.
[Footnote 18: Aristot. Analyt. Prior. I. ii. p. 25, a. 17-22.]
[Footnote 19: Ibid. p. 25, a. 22-26.]
[Footnote 20: Though some may fancy that the rule for converting the
Universal Negative is intuitively known, yet every one must see that
the rule for converting the Universal Affirmative is not thus
self-evident, or derived from natural intuition. In fact, I believe
that every learner at first hears it with great surprise. Some are
apt to fancy that the Universal Affirmative (like the Particular
Affirmative) may be converted _simply_. Indeed this error is not
unfrequently committed in actual reasoning; all the more easily,
because there is a class of cases (with subject and predicate
co-extensive) where the converse of the Universal Affirmative _is_
really true. Also, in the case of the Particular Negative, there are
many true propositions in which the simple converse is true. A novice
might incautiously generalize upon those instances, and conclude that
both were convertible simply. Nor could you convince him of his error
except by producing examples in which, when a true proposition of
this kind is converted simply, the resulting converse is notoriously
false. The appeal to various separate cases is the only basis on
which we can rest for testing the correctness or incorrectness of all
these maxims proclaimed as universal.]
From one proposition taken singly, no new proposition can be
inferred; for purposes of inference, two propositions at least are
required.[21] This brings us to the rules of the Syllogism, where two
propositions as premisses conduct us to a third which necessarily
follows from them; and we are introduced to the well-known three
Figures with their various Modes.[22] To form a valid Syllogism,
there must be three terms and no more; the two, which appear as
Subject and Predicate of the conclusion, are called the _minor_ term
(or minor extreme) and the _major_ term (or major extreme)
respectively; while the third or _middle_ term must appear in each of
the premisses, but not in the conclusion. These terms are called
_extremes_ and _middle_, from the position which they occupy in every
perfect Syllogism--that is in what Aristotle ranks as the First among
the three figures. In _his_ way of enunciating the Syllogism, this
middle position formed a conspicuous feature; whereas the modern
arrangement disguises it, though the denomination _middle_ term is
still retained. Aristotle usually employs letters of the alphabet,
which he was the first to select as abbreviations for exposition;[23]
and he has two ways (conforming to what he had said in the first
chapter of the present treatise) of enunciating the modes of the
First figure. In one way, he begins with the major extreme (Predicate
of the conclusion): A may be predicated of all B, B may be predicated
of all C; therefore, A may be predicated of all C (Universal
Affirmative). Again, A cannot be predicated of any B, B can be
predicated of all C; therefore, A cannot be predicated of any C
(Universal Negative). In the other way, he begins with the minor term
(Subject of the conclusion): C is in the whole B, B is in the whole
A; therefore, C is in the whole A (Universal Affirmative). And, C is
in the whole B, B is not in the whole A; therefore, C is not in the
whole A (Universal Negative). We see thus that in Aristotle's way of
enunciating the First figure, the middle term is really placed
between the two extremes,[24] though this is not so in the Second and
Third figures. In the modern way of enunciating these figures, the
middle term is never placed between the two extremes; yet the
denomination _middle_ still remains.
[Footnote 21: Analyt. Prior. I. xv. p. 34, a. 17; xxiii. p. 40, b.
35; Analyt. Poster. I. iii. p. 73, a. 7.]
[Footnote 22: Aristot. Analyt. Prior. I. iv. p. 25, b. 26, seq.]
[Footnote 23: M. Barthélemy St. Hilaire (Logique d'Aristote, vol. ii.
p. 7, n.), referring to the examples of Conversion in chap. ii.,
observes:--"Voici le prémier usage des lettres représentant des
idées; c'est un procédé tout à fait algébrique, c'est à dire, de
généralisation. Déjà, dans l'Herméneia, ch. 13, § 1 et suiv.,
Aristote a fait usage de tableaux pour représenter sa pensée
relativement à la consécution des modales. Il parle encore
spécialement de figures explicatives, liv. 2. des Derniers
Analytiques, ch. 17, § 7. Vingt passages de l'Histoire des Animaux
attestent qu'il joignait des dessins à ses observations et à ses
théories zoologiques. Les illustrations pittoresques datent donc de
fort loin. L'emploi symbolique des lettres a été appliqué aussi par
Aristote à la Physique. Il l'avait emprunté, sans doute, aux procédés
des mathématiciens."
We may remark, however, that when Aristotle proceeds to specify those
combinations of propositions which _do not_ give a valid conclusion,
he is not satisfied with giving letters of the alphabet; he superadds
special illustrative examples (Analyt. Prior. I. v. p. 27, a. 7, 12,
34, 38).]
[Footnote 24: Aristot. Analyt. Prior. I. iv. p. 25, b. 35: [Greek:
kalô= de\ _me/son_, o(\ kai\ au)to\ e)n a)/llô| kai\ a)/llo e)n
tou/tô| e)sti/n, o(\ kai\ tê=| the/sei gi/netai me/son.]]
The Modes of each figure are distinguished by the different character
and relation of the two premisses, according as these are either
affirmative or negative, either universal or particular. Accordingly,
there are four possible varieties of each, and sixteen possible modes
or varieties of combinations between the two. Aristotle goes through
most of the sixteen modes, and shows that in the first Figure there
are only four among them that are legitimate, carrying with them a
necessary conclusion. He shows, farther, that in all the four there
are two conditions observed, and that both these conditions are
indispensable in the First figure:--(1) The major proposition must be
universal, either affirmative or negative; (2) The minor proposition
must be affirmative, either universal or particular or indefinite.
Such must be the character of the premisses, in the first Figure,
wherever the conclusion is valid and necessary; and _vice versâ_, the
conclusion will be valid and necessary, when such is the character of
the premisses.[25]
[Footnote 25: Aristot. Analyt. Prior. I. iv. p. 26, b. 26, et sup.]
In regard to the four valid modes (_Barbara_, _Celarent_, _Darii_,
_Ferio_, as we read in the scholastic Logic) Aristotle declares at
once in general language that the conclusion follows necessarily;
which he illustrates by setting down in alphabetical letters the
skeleton of a syllogism in _Barbara_. If A is predicated of all B,
and B of all C, A must necessarily be predicated of all C. But he
does not justify it by any real example; he produces no special
syllogism with real terms, and with a conclusion known beforehand to
be true. He seems to think that the general doctrine will be accepted
as evident without any such corroboration. He counts upon the
learner's memory and phantasy for supplying, out of the past
discourse of common life, propositions conforming to the conditions
in which the symbolical letters have been placed, and for not
supplying any contradictory examples. This might suffice for a
treatise; but we may reasonably believe that Aristotle, when teaching
in his school, would superadd illustrative examples; for the doctrine
was then novel, and he is not unmindful of the errors into which
learners often fall spontaneously.[26]
[Footnote 26: Analyt. Poster. I. xxiv. p. 85, b. 21.]
When he deals with the remaining or invalid modes of the First
figure, his manner of showing their invalidity is different, and in
itself somewhat curious. "If (he says) the major term is affirmed of
all the middle, while the middle is denied of all the minor, no
necessary consequence follows from such being the fact, nor will
there be any syllogism of the two extremes; for it is equally
possible, either that the major term may be affirmed of all the
minor, or that it may be denied of all the minor; so that no
conclusion, either universal or particular, is necessary in all
cases."[27] Examples of such double possibility are then exhibited:
first, of three terms arranged in two propositions (A and E), in
which, from the terms specially chosen, the major happens to be truly
affirmable of all the minor; so that the third proposition is an
universal Affirmative:--
Major and }
} Animal is predicable of every Man;
Middle. }
Middle and }
} Man is not predicable of any Horse;
Minor }
Major and }
} Animal is predicable of every Horse.
Minor }
Next, a second example is set out with new terms, in which the major
happens not to be truly predicable of any of the minor; thus
exhibiting as third proposition an universal Negative:--
Major and }
} Animal is predicable of every Man;
Middle. }
Middle and }
} Man is not predicable of any Stone;
Minor }
Major and }
} Animal is not predicable of any Stone.
Minor }
Here we see that the full exposition of a syllogism is indicated with
real terms common and familiar to every one; alphabetical symbols
would not have sufficed, for the learner must himself recognize the
one conclusion as true, the other as false. Hence we are taught that,
after two premisses thus conditioned, if we venture to join together
the major and minor so as to form a pretended conclusion, we may in
some cases obtain a true proposition universally Affirmative, in
other cases a true proposition universally Negative. Therefore
(Aristotle argues) there is no one necessary conclusion, the same in
all cases, derivable from such premisses; in other words, this mode
of syllogism is invalid and proves nothing. He applies the like
reasoning to all the other invalid modes of the first Figure; setting
them aside in the same way, and producing examples wherein double and
opposite conclusions (improperly so called), both true, are obtained
in different cases from the like arrangement of premisses.
[Footnote 27: Analyt. Prior. I. iv. p. 26, a. 2, seq.]
This mode of reasoning plainly depends upon an appeal to prior
experience. The validity or invalidity of each mode of the First
figure is tested by applying it to different particular cases, each
of which is familiar and known to the learner _aliunde_; in one case,
the conjunction of the major and minor terms in the third proposition
makes an universal Affirmative which he knows to be true; in another
case, the like conjunction makes an universal Negative, which he also
knows to be true; so that there is no one _necessary_ (_i.e._ no one
uniform and trustworthy) conclusion derivable from such
premisses.[28] In other words, these modes of the First figure are
not valid or available in form; the negation being sufficiently
proved by one single undisputed example.
[Footnote 28: Though M. Barthélemy St. Hilaire (note, p. 19) declares
Aristotle's exposition to be a model of analysis, it appears to me
that the grounds for disallowing this invalid mode of the First
figure (A--E--A, or A--E--E) are not clearly set forth by Aristotle
himself, while they are rendered still darker by some of his best
commentators. Thus Waitz says (p. 381): "Per exempla allata probat