Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such asreductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th (...) century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titledSpecimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule ofreductio ad absurdumin a purely categorical calculus in which every proposition is of the formA is B. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule ofreductio ad absurdum, but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic known as RMI$_{{}_ \to ^\neg }$. (shrink)

This paper discusses the following question: why was the term “dunatos” (“possible”) employed by Aristotle as an alternative label for imperfect syllogisms in his discussion of assertoric syllogistic? My answer ascribes to Aristotle a bottom up perspective, in which he stresses what is necessary in the premise-pairs to attain target conclusions of a given form within a given figure. I argue that “dunatos” is employed by Aristotle to stress that an imperfect syllogism is always one of the possible options to (...) attain a conclusion of a given form within a given figure. I also argue that this picture sheds some light on Aristotle’s clarifications of the final clause in his definition of syllogism. (shrink)