Aristotle
considers the premisses alone as constituting the Syllogism; the
conclusion is not a part thereof, but something distinct and
superadded. It appears to me that the _vulgus logicorum_ are here in
the right.]
In the Analytica Priora, we find ourselves involved, from and after
the second chapter, in the distinction of Modal propositions, the
necessary and the possible. The rules respecting the simple Assertory
propositions are thus, even from the beginning, given in conjunction
and contrast with those respecting the Modals. This is one among many
causes of the difficulty and obscurity with which the treatise is
beset. Theophrastus and Eudemus seem also to have followed their
master by giving prominence to the Modals:[11] recent expositors
avoid the difficulty, some by omitting them altogether, others by
deferring them until the simple assertory propositions have been
first made clear. I shall follow the example of these last; but it
deserves to be kept in mind, as illustrating Aristotle's point of
view, that he regards the Modals as principal varieties of the
proposition, co-ordinate in logical position with the simple
assertory.
[Footnote 11: Eudemi Fragmenta, cii.-ciii. p. 145, ed. Spengel.]
Before entering on combinations of propositions, Aristotle begins by
shewing what can be done with single propositions, in view to the
investigation or proving of truth. A single proposition may be
_converted_; that is, its subject and predicate may be made to change
places. If a proposition be true, will it be true when thus
converted, or (in other words) will its converse be true? If false,
will its converse be false? If this be not always the case, what are
the conditions and limits under which (assuming the proposition to be
true) the process of conversion leads to assured truth, in each
variety of propositions, affirmative or negative, universal or
particular? As far as we know, Aristotle was the first person that
ever put to himself this question; though the answer to it is
indispensable to any theory of the process of proving or disproving.
He answers it before he enters upon the Syllogism.
The rules which he lays down on the subject have passed into all
logical treatises. They are now familiar; and readers are apt to
fancy that there never was any novelty in them--that every one knows
them without being told. Such fancy would be illusory. These rules
are very far from being self-evident, any more than the maxims of
Contradiction and of the Excluded Middle. Not one of the rules could
have been laid down with its proper limits, until the discrimination
of propositions, both as to quality (affirmative or negative), and as
to quantity (universal or particular), had been put prominently
forward and appreciated in all its bearings. The rule for trustworthy
conversion is different for each variety of propositions. The
Universal Negative may be converted simply; that is, the predicate
may become subject, and the subject may become predicate--the
proposition being true after conversion, if it was true before. But
the Universal Affirmative cannot be thus converted simply. It admits
of conversion only in the manner called by logicians _per accidens_:
if the predicate change places with the subject, we cannot be sure
that the proposition thus changed will be true, unless the new
subject be lowered in quantity from universal to particular; _e.g._
the proposition, All men are animals, has for its legitimate converse
not, _All_ animals are men, but only, _Some_ animals are men. The
Particular Affirmative may be converted simply: if it be true that
Some animals are men, it will also be true that Some men are animals.
But, lastly, if the true proposition to be converted be a Particular
Negative, it cannot be converted at all, so as to make sure that the
converse will be true also.[12]
[Footnote 12: Aristot. Analyt. Prior. I. ii. p. 25, a. 1-26.]
Here then are four separate rules laid down, one for each variety of
propositions. The rules for the second and third variety are proved
by the rule for the first (the Universal Negative), which is thus the
basis of all. But how does Aristotle prove the rule for the Universal
Negative itself? He proceeds as follows: "If A cannot be predicated
of any one among the B's, neither can B be predicated of any one
among the A's. For if it could be predicated of any one among them
(say C), the proposition that A cannot be predicated of any B would
not be true; since C is one among the B's."[13] Here we have a proof
given which is no proof at all. If I disbelieved or doubted the
proposition to be proved, I should equally disbelieve or doubt the
proposition given to prove it. The proof only becomes valid, when you
add a farther assumption which Aristotle has not distinctly
enunciated, viz.: That if some A (_e.g._ C) is B, then some B must
also be A; which would be contrary to the fundamental supposition.
But this farther assumption cannot be granted here, because it would
imply that we already know the rule respecting the convertibility of
Particular Affirmatives, viz., that they admit of being converted
simply. Now the rule about Particular Affirmatives is afterwards
itself proved by help of the preceding demonstration respecting the
Universal Negative. As the proof stands, therefore, Aristotle
demonstrates each of these by means of the other; which is not
admissible.[14]
[Footnote 13: Ibid. p. 25, a. 15: [Greek: ei) ou)=n mêdeni\ tô=n B
to\ A u(pa/rchei, ou)de\ tô=n A ou)deni\ u(pa/rxei to\ B. ei) ga\r
tini, oi(=on tô=| G, ou)k a)lêthe\s e)/stai to\ mêdeni\ tô=n B to\ A
u(pa/rchein; to\ ga\r G tô=n B ti/ e)stin.]
Julius Pacius (p. 129) proves the Universal Negative to be
convertible _simpliciter_, by a _Reductio ad Absurdum_ cast into a
syllogism in the First figure. But it is surely unphilosophical to
employ the rules of Syllogism as a means of proving the legitimacy of
Conversion, seeing that we are forced to assume conversion in our
process for distinguishing valid from invalid syllogisms. Moreover
the _Reductio ad Absurdum_ assumes the two fundamental Maxims of
Contradiction and Excluded Middle, though these are less obvious, and
stand more in need of proof than the simple conversion of the
Universal Negative, the point that they are brought to establish.]
[Footnote 14: Waitz, in his note (p. 374), endeavours, but I think
without success, to show that Aristotle's proof is not open to the
criticism here advanced. He admits that it is obscurely indicated,
but the amplification of it given by himself still remains exposed to
the same objection.]
Even the friends and companions of Aristotle were not satisfied with
his manner of establishing this fundamental rule as to the conversion
of propositions. Eudêmus is said to have given a different proof; and
Theophrastus assumed as self-evident, without any proof, that the
Universal Negative might always be converted simply.[15] It appears
to me that no other or better evidence of it can be offered, than the
trial upon particular cases, that is to say, Induction.[16] Nothing
is gained by dividing (as Aristotle does) the whole A into parts, one
of which is C; nor can I agree with Theophrastus in thinking that
every learner would assent to it at first hearing, especially at a
time when no universal maxims respecting the logical value of
propositions had ever been proclaimed. Still less would a Megaric
dialectician, if he had never heard the maxim before, be satisfied to
stand upon an alleged _à priori_ necessity without asking for
evidence. Now there is no other evidence except by exemplifying the
formula, No A is B, in separate propositions already known to the
learner as true or false, and by challenging him to produce any one
case, in which, when it is true to say No A is B, it is not equally
true to say, No B is A; the universality of the maxim being liable to
be overthrown by any one contradictory instance.[17] If this proof
does not convince him, no better can be produced. In a short time,
doubtless, he will acquiesce in the general formula at first hearing,
and he may even come to regard it as self-evident. It will recall to
his memory an aggregate of separate cases each individually
forgotten, summing up their united effect under the same aspect, and
thus impressing upon him the general truth as if it were not only
authoritative but self-authorized.
[Footnote 15: See the Scholia of Alexander on this passage, p. 148,
a. 30-45, Brandis; Eudemi Fragm. ci.-cv. pp. 145-149, ed. Spengel.]
[Footnote 16: We find Aristotle declaring in Topica, II. viii. p.
113, b. 15, that in converting a true Universal Affirmative
proposition, the negative of the Subject of the convertend is always
true of the negative of the Predicate of the convertend; _e.g._ If
every man is an animal, every thing which is not an animal is not a
man. This is to be assumed (he says) upon the evidence of
Induction--uncontradicted iteration of particular cases, extended to
all cases universally--[Greek: lamba/nein d' e)x e)pagôgê=s, oi(=on
ei) o( a)/nthrôpos zô=|on, to\ mê\ zô=|on ou)k a)/nthrôpos; o(moi/ôs
de\ kai\ e)pi\ tô=n a)/llôn. . . . . e)pi\ pa/ntôn ou)=n to\
toiou=ton a)xiôte/on.]
The rule for the simple conversion of the Universal Negative rests
upon the same evidence of Induction, never contradicted.]
[Footnote 17: Dr. Wallis, in one of his acute controversial treatises
against Hobbes, remarks upon this as the process pursued by Euclid in
his demonstrations:--"You tell us next that an Induction, without
enumeration of all the particulars, is not sufficient to infer a
conclusion. Yes, Sir, if after the enumeration of some particulars,
there comes a general clause, _and the like in other cases_ (as here
it doth), this may pass for a proofe till there be a possibility of
giving some instance to the contrary, which here you will never be
able to doe. And if such an Induction may not pass for proofe, there
is never a proposition in Euclid demonstrated. For all along he takes
no other course, or at least grounds his Demonstrations on
Propositions no otherwise demonstrated. As, for instance, he
proposeth it in general (i. c. 1.)--_To make an equilateral triangle
on a line given_. And then he shows you how to do it upon the line A
B, which he there shows you, and leaves you to supply: _And the same,
by the like means, may be done upon any other strait line_; and then
infers his general conclusion. Yet I have not heard any man object
that the Induction was not sufficient, because he did not actually
performe it in all lines possible."--(Wallis, Due Correction to Mr.
Hobbes, Oxon. 1656, sect. v. p. 42.) This is induction by _parity of
reasoning_.
So also Aristot. Analyt. Poster. I. iv. p. 73, b. 32: [Greek: to\
katho/lou de\ u(pa/rchei to/te, o(/tan e)pi\ tou= tucho/ntos kai\
prô/tou deiknu/êtai.]]
Aristotle passes next to Affirmatives, both Universal and Particular.
First, if A can be predicated of all B, then B can be predicated of
_some_ A; for if B cannot be predicated of any A, then (by the rule
for the Universal Negative) neither can A be predicated of any B.
Again, if A can be predicated of some B, in this case also, and for
the same reason, B can be predicated of some A.[18] Here the rule for
the Universal Negative, supposed already established, is applied
legitimately to prove the rules for Affirmatives. But in the first
case, that of the Universal, it fails to prove _some_ in the sense of
_not-all_ or _some-at-most_, which is required; whereas, the rules
for both cases can be proved by Induction, like the formula about the
Universal Negative. When we come to the Particular Negative,
Aristotle lays down the position, that it does not admit of being
necessarily converted in any way. He gives no proof of this, beyond