Two Accounts of Aristotle's Logic - Nicolas Fillion

2 CRITICAL PRESENTATION OF ̷LUKASIEWICZ’S VIEW 16
X and M is also predicated of every N, then it is necessary for M to
belong to every X: but it was assumed not to belong to some.) And
if M belongs to every N but not to every X, then there will be a
deduction that N does not belong to every X. (The demonstration
is the same.) (Pr. An., 1.5, 27a36-b3)
This is a very concise proof that is traditionally explained along those lines: (1)
it is admitted that the premises ‘M belongs to every N’ and ‘M does not belong
to some X’ are true, and (2) we conclude from this that the conclusion ‘N does
not belong to some X’. For if it were false, its contradictory, ‘N belongs to all
X’, would be true. This proposition is the starting point of the reduction. As
it is admitted that the premise ‘M belongs to every N’ is true, we get from this
premise and the proposition ‘N belongs to every X’ the conclusion ‘M belongs
to all X’ by the mood Barbara. But this conclusion is false, for it is admitted
that its contradictory ‘M does not belong to some X’ is true. Therefore, the
starting point of our reduction, ‘N belongs to every X’, which leads to a false
conclusion, must be false, and its contradictory, ‘N does not belong to some X’,
must be true. Aristotle himself explains this method of proof in this way:
A demonstration [leading] into an impossibility differs from a probative
demonstration in that it puts as a premise what it wants
to reject by leading away into an agreed falsehood, while a probative
demonstration begins from agreed positions. More precisely,
both demonstration take two agreed premises, but one takes the
premises which the deduction is from, while the other takes one of
these premises and, as the other premise, the contradictory of the
conclusion. (Pr. An., B.14, 62b29-34)
For ̷Lukasiewicz, such an explication is anathema, for it relies on the conception
of syllogisms as (rules of) inference. This is why this “argument is only apparently
convincing; in fact, it does not prove the above syllogism” (̷Lukasiewicz,
1957, p. 54). Not, at least, if syllogisms are taken to be material implications.
As such, a syllogism is an implication which is true for all values of the variables
M, N, and X, and not only for those values that verify the premises. This is
why ̷Lukasiewicz claims that the “proof given by Aristotle is neither sufficient
nor a proof by reductio ad impossibile” (̷Lukasiewicz, 1957, p. 55). The proof of
a syllogistic mood qua implication by reductio ad impossibile can not suppose
the negation of the conclusion only:
The indirect proof of the mood Baroco should start from the negation
of this mood, and not from the negation of its conclusion, and
this negation should lead to an unconditionally false statement, and
not to a proposition that is admitted to be false under certain conditions.
(̷Lukasiewicz, 1957, p. 56)