Aristotle
der Logik, sect. 121, p. 326), consider this to be a defect in
Aristotle.]
In the last chapter of the first book of the Analytica Priora,
Aristotle returns to the point which we have already considered in
the treatise De Interpretatione, viz. what is really a _negative_
proposition; and how the adverb of negation must be placed in order
to constitute one. We must place this adverb immediately before the
copula and in conjunction with the copula: we must not place it after
the copula and in conjunction with the predicate; for, if we do so,
the proposition resulting will not be negative but affirmative
([Greek: e)k metathe/seôs], by transposition, according to the
technical term introduced afterwards by Theophrastus). Thus of the
four propositions:
1. Est bonum. 2. Non est bonum.
4. Non est non bonum. 3. Est non bonum.
No. 1 is affirmative; No. 3 is affirmative ([Greek: e)k
metathe/seôs]); Nos. 2 and 4 are negative. Wherever No. 1 is
predicable, No. 4 will be predicable also; wherever No. 3 is
predicable, No. 2 will be predicable also--but in neither case _vice
versâ_.[89] Mistakes often flow from incorrectly setting out the two
contradictories.
[Footnote 89: Analyt. Prior. I. xlvi. p. 51, b. 5, ad finem. See
above, Chap. IV. p. 118, seq.]
CHAPTER VI.
ANALYTICA PRIORA II.
The Second Book of the Analytica Priora seems conceived with a view
mainly to Dialectic and Sophistic, as the First Book bore more upon
Demonstration.[1] Aristotle begins the Second Book by shortly
recapitulating what he had stated in the First; and then proceeds to
touch upon some other properties of the Syllogism. Universal
syllogisms (those in which the conclusion is universal) he says, have
always more conclusions than one; particular syllogisms sometimes,
but not always, have more conclusions than one. If the conclusion be
universal, it may always be converted--_simply_, when it is negative,
or _per accidens_, when it is affirmative; and its converse thus
obtained will be proved by the same premisses. If the conclusion be
particular, it will be convertible simply when affirmative, and its
converse thus obtained will be proved by the same premisses; but it
will not be convertible at all when negative, so that the conclusion
proved will be only itself singly.[2] Moreover, in the universal
syllogisms of the First figure (_Barbara_, _Celarent_), any of the
particulars comprehended under the minor term may be substituted in
place of the minor term as subject of the conclusion, and the proof
will hold good in regard to them. So, again, all or any of the
particulars comprehended in the middle term may be introduced as
subject of the conclusion in place of the minor term; and the
conclusion will still remain true. In the Second figure, the change
is admissible only in regard to those particulars comprehended under
the subject of the conclusion or minor term, and not (at least upon
the strength of the syllogism) in regard to those comprehended under
the middle term. Finally, wherever the conclusion is particular, the
change is admissible, though not by reason of the syllogism in regard
to particulars comprehended under the middle term; it is not
admissible as regards the minor term, which is itself particular.[3]
[Footnote 1: This is the remark of the ancient Scholiasts. See Schol.
p. 188, a. 44, b. 11.]
[Footnote 2: Analyt. Prior. II. i. p. 53, a. 3-14.]
[Footnote 3: Analyt. Prior. II. i. p. 53, a. 14-35. M. Barthélemy St.
Hilaire, following Pacius, justly remarks (note, p. 203 of his
translation) that the rule as to particulars breaks down in the cases
of _Baroco_, _Disamis_, and _Bocardo_.
On the chapter in general he remarks (note, p. 204):--"Cette théorie
des conclusions diverses, soit patentes soit cachées, d'un même
syllogisme, est surtout utile en dialectique, dans la discussion; où
il faut faire la plus grande attention à ce qu'on accorde à
l'adversaire, soit explicitement, soit implicitement." This
illustrates the observation cited in the preceding note from the
Scholiasts.]
Aristotle has hitherto regarded the Syllogism with a view to its
_formal_ characteristics: he now makes an important observation which
bears upon its _matter_. Formally speaking, **the two premisses are
always assumed to be true; but in any real case of syllogism (form
and matter combined) it is possible that either one or both may be
false. Now, Aristotle remarks that if both the premisses are true
(the syllogism being correct in form), the conclusion must of
necessity be true; but that if either or both the premisses are
false, the conclusion need not necessarily be false likewise. The
premisses being false, the conclusion may nevertheless be true; but
it will not be true because of or by reason of the premisses.[4]
[Footnote 4: Analyt. Prior. II. ii. p. 53, b. 5-10: [Greek: e)x
a)lêthô=n me\n ou)=n ou)k e)/sti pseu=dos sullogi/sasthai, e)k
pseudô=n d' e)/stin a)lêthe/s, plê\n ou) dio/ti a)ll' o(/ti; tou=
ga\r dio/ti ou)k e)/stin e)k pseudô=n sullogismo/s; di' ê(\n d'
ai)ti/an, e)n toi=s e(pome/nois lechthê/setai.]
The true conclusion is not true by reason of these false premisses,
but by reason of certain other premisses which are true, and which
may be produced to demonstrate it. Compare Analyt. Poster. I. ii. p.
71, b. 19.]
First, he would prove that if the premisses be true, the conclusion
must be true also; but the proof that he gives does not seem more
evident than the _probandum_ itself. Assume that if A exists, B must
exist also: it follows from hence (he argues) that if B does not
exist, neither can A exist; which he announces as a _reductio ad
absurdum_, seeing that it contradicts the fundamental supposition of
the existence of A.[5] Here the _probans_ is indeed equally evident
with the _probandum_, but not at all more evident; one who disputes
the latter, will dispute the former also. Nothing is gained in the
way of proof by making either of them dependent on the other. Both of
them are alike self-evident; that is, if a man hesitates to admit
either of them, you have no means of removing his scruples except by
inviting him to try the general maxim upon as many particular cases
as he chooses, and to see whether it does not hold good without a
single exception.
[Footnote 5: Ibid. II. ii. p. 53, b. 11-16.]
In regard to the case here put forward as illustration, Aristotle has
an observation which shows his anxiety to maintain the characteristic
principles of the Syllogism; one of which principles he had declared
to be--That nothing less than three terms and two propositions, could
warrant the inferential step from premisses to conclusion. In the
present case he assumed, If A exists, then B must exist; giving only
one premiss as ground for the inference. This (he adds) does not
contravene what has been laid down before; for A in the case before
us represents two propositions conceived in conjunction.[6] Here he
has given the type of hypothetical reasoning; not recognizing it as a
variety _per se_, nor following it out into its different forms (as
his successors did after him), but resolving it into the categorical
syllogism.[7] He however conveys very clearly the cardinal principle
of all hypothetical inference--That if the antecedent be true, the
consequent must be true also, but not _vice versâ_; if the consequent
be false, the antecedent must be false also, but not _vice versâ_.
[Footnote 6: Analyt. Prior. II. ii. p. 53, b. 16-25. [Greek: to\
ou)=n A ô(/sper e(\n kei=tai, du/o prota/seis sullêphthei=sai.]]
[Footnote 7: Aristotle, it should be remarked, uses the word [Greek:
katêgoriko/s], not in the sense which it subsequently acquired, as
the antithesis of [Greek: u(pothetiko/s] in application to the
proposition and syllogism, but in the sense of affirmative as opposed
to [Greek: sterêtiko/s].]
Having laid down the principle, that the conclusion may be true,
though one or both the premisses are false, Aristotle proceeds, at
great length, to illustrate it in its application to each of the
three syllogistic figures.[8] No portion of the Analytica is traced
out more perspicuously than the exposition of this most important
logical doctrine.
[Footnote 8: Analyt. Prior. II. ii.-iv. p. 53, b. 26-p. 57, b. 17. At
the close (p. 57, a. 36-b. 17), the general doctrine is summed up.]
It is possible (he then continues, again at considerable length) to
invert the syllogism and to demonstrate _in a circle_. That is, you
may take the conclusion as premiss for a new syllogism, together with
one of the old premisses, transposing its terms; and thus you may
demonstrate the other premiss. You may do this successively, first
with the major, to demonstrate the minor; next, with the minor, to
demonstrate the major. Each of the premisses will thus in turn be
made a demonstrated conclusion; and the circle will be complete. But
this can be done perfectly only in _Barbara_, and when, besides, all
the three terms of the syllogism reciprocate with each other, or are
co-extensive in import; so that each of the two premisses admits of
being simply converted. In all other cases, the process of circular
demonstration, where possible at all, is more or less imperfect.[9]
[Footnote 9: Ibid. II. v.-viii. p. 57, b. 18-p. 59, a. 35.]
Having thus shown under what conditions the conclusion can be
employed for the demonstration of the premisses, Aristotle proceeds
to state by what transformation it can be employed for the refutation
of them. This he calls _converting_ the syllogism; a most
inconvenient use of the term _convert_ ([Greek: a)ntistre/phein]),
since he had already assigned to that same term more than one other
meaning, distinct and different, in logical procedure.[10] What it
here means is _reversing_ the conclusion, so as to exchange it either
for its contrary, or for its contradictory; then employing this
reversed proposition as a new premiss, along with one of the previous
premisses, so as to disprove the other of the previous
premisses--_i.e._ to prove its contrary or contradictory. The result
will here be different, according to the manner in which the
conclusion is reversed; according as you exchange it for its contrary
or its contradictory. Suppose that the syllogism demonstrated is: A
belongs to all B, B belongs to all C; _Ergo_, A belongs to all C
(_Barbara_). Now, if we reverse this conclusion by taking its
_contrary_, A belongs to no C, and if we combine this as a new premiss
with the major of the former syllogism, A belongs to all B, we shall
obtain as a conclusion B belongs to no C; which is the _contrary_ of
the minor, in the form _Camestres_. If, on the other hand, we reverse
the conclusion by taking its _contradictory_, A does not belong to all
C, and combine this with the same major, we shall have as conclusion,
B does not belong to all C; which is the _contradictory_ of the minor,
and in the form _Baroco_: though in the one case as in the other the
minor is disproved. The major is _contradictorily_ disproved, whether
it be the contrary or the contradictory of the conclusion that is
taken along with the minor to form the new syllogism; but still the
form varies from _Felapton_ to _Bocardo_. Aristotle shows farther how
the same process applies to the other modes of the First, and to the
modes of the Second and Third figures.[11] The new syllogism,
obtained by this process of reversal, is always in a different figure
from the syllogism reversed. Thus syllogisms in the First figure are
reversed by the Second and Third; those in the second, by the First
and Third; those in the Third, by the First and Second.[12]
[Footnote 10: Schol. (ad Analyt. Prior. p. 59, b. 1), p. 190, b. 20,
Brandis. Compare the notes of M. Barthélemy St. Hilaire, pp. 55,
242.]