Two Accounts of Aristotle's Logic - Nicolas Fillion

2 CRITICAL PRESENTATION OF ̷LUKASIEWICZ’S VIEW 22
the theory of syllogism” (̷Lukasiewicz, 1957, p. 47). Concerning those proofs,
̷Lukasiewicz emphasized (1) that Aristotle’s proofs by conversion are insufficient
and need to be completed, (2) that proofs by reductio ad impossibile and by ecthesis
must replaced by axiomatic acceptation or rejection of some propositions
and (3) that laws of conversions must be accepted axiomatically. Basically, he
has “shown” that all methods of proofs can only be understood if we replace
them by a formalistic deduction in propositional logic. This is why, according
to ̷Lukasiewicz, it is only possible to understand Aristotle’s proofs if it is
understood that they necessitate propositional calculus.
Propositional logic was completely ignored by Aristotle. ̷Lukasiewicz affirms
that the first system of propositional logic was invented about half a century
after Aristotle, by the Stoics. However, the understanding of the more fundamental
character of this logical system has only been acquired in the end of the
nineteenth century, as a consequence of the 1879 event, due to the great German
logician Gottlob Frege. This theory has been later brought to perfection by the
authors of Principia Mathematica, Whitehead and Russell, who developed it so
profoundly that they made it the core of all mathematics. 18 To observe the
difference between propositional logic and syllogistic, it is sufficient to compare
Aab and p → q:
They differ in their constants, which I call functors: in the first formula
the functor reads ‘all–is’, in the second ‘if–then’. Both are
functors of two arguments which are here identical. But the main
difference lies in the arguments. In both formulae the arguments are
variables, but of a different kind: the values which may be substituted
for the variable A are terms [. . . ]. The value of the variable p
are not terms but propositions [. . . ]. This difference between termvariables
and proposition-variables is the primary difference between
the two formulae and consequently between the two systems of logic,
and, as propositions and terms belong to different semantical categories,
the difference is a fundamental one. (̷Lukasiewicz, 1957,
p. 47-8)
For ̷Lukasiewicz, in contrast with the syllogistic, which is a set of true material
implications, propositional logic is a system of rules of inference, and that makes
it the primitive logic underlying all logical reasoning.
[. . . ] Aristotle’s logic is not a primitive logical theory, and for its
foundation requires a certain earlier theory that investigates sentences
in general, without engaging in the study of the details of
their structure. (̷Lukasiewicz, 1963, p. 106)
That it is the case, however, depends on his being justified in making the propositional
additions to Aristotle’s doctrine. We must first notice that ̷Lukasiewicz
18 See ̷Lukasiewicz (1957, p. 48) for the literal text of the propaganda.