Two Accounts of Aristotle's Logic - Nicolas Fillion

2 CRITICAL PRESENTATION OF ̷LUKASIEWICZ’S VIEW 18
The point is that arguing validly requires more than making valid inferences.
It also requires that the inferences done be known to be valid. By showing the
equivalence of those methods of proof, Aristotle addresses the latter requirement:
“[. . . ] the problem of one’s knowledge of the validity of a per impossibile
syllogism is reduced to that of our knowledge of the validity of a direct inference.”
(Lear, 1980, p. 53) Thus, it is guaranteed that the only difference between
per impossibile and direct syllogisms, within the confines of syllogistic, is one of
argumentative structure. It is impossible to account for this part of Aristotle’s
theory if syllogisms are material implications.
Proofs by Ecthesis Apart from the proofs by conversion and the proofs
per impossibile, there is still a third kind of proof used by Aristotle, namely
the proofs by exposition or ἔκθεσις. According to ̷Lukasiewicz, there are three
passages where Aristotle uses that kind of proof in Prior Analytics 14 : (1) proof
of the conversion of the E-premise, (2) the proof of the mood Darapti and (3)
the proof of the mood Bocardo. The word ‘ecthesis’, however, appears only in
(2). ̷Lukasiewicz thinks that this logical process lies outside the limits of the
syllogistic, and that new axioms are needed in their place (̷Lukasiewicz, 1957,
p. 45) Let us examine the proof of conversion:
First, let premise AB be universally privative. Now, if A belongs to
none of the Bs, then neither will B belong to any of the As. For if it
does not belong to some (for instance to C), it will not be true that
A belong to none of the Bs, since C is one of the Bs. (Pr. An., A.2,
25a14-17)
The proof by exposition requires the introduction of a new term, called the
‘exposed term’; here it is C. In his commentary, Alexander says that C is a
singular term given by perception. For ̷Lukasiewicz, this would not do since “a
proof by perception is not a logical proof” (̷Lukasiewicz, 1957, p. 60). On the
other hand, if C is a universal term, then the proof would require existential
quantifiers (since it produces an instantiation). For this, however, an extension
of syllogistic to quantification theory would be needed.
This was perhaps the reason why he dropped this kind of proof
in his final chapter (7) of Book I of the Prior Analytics, where he
sums up his systematic investigation of syllogistic. Nobody after him
understood these proofs. It was reserved for modern formal logic to
explain them by the idea of the existential quantifier. (̷Lukasiewicz,
1957, p. 67)
This issue will be discussed in more details in the next subsection about rejection.
14 See (Raymond, 2006, p. 33ff) and Smith (1982) for a detailed discussion.