Leonardo Mazzanti

In addition to Chrysippean syllogistic analysis, the ancient sources attest to and provide examples of an unorthodox analysis conducted with a different logical principle, namely the Synthetic Principle (SP), from the canonical themata. This alternative analysis is structurally distinct from the standard one: (i) it uses indemonstrables as inference rules rather than as zero-ary rules/axioms; (ii) it is based on assumptions rather than axioms; (iii) the directionality of the proof goes from the leaves (initial sequents) to the root (sequent to be proven). A formalization of SP is here presented, and its deductive equivalence with the three themata of standard Chrysippean analysis demonstrated. Additionally, by comparing it with Gentzen’s Cut Rule and Mix Rule, it is shown that SP is, in fact, a stronger principle (at least within the Stoic system). Finally, as for the origin of this principle, it is conjectured for it to be Peripatetic, subsequent to Aristotle, based on the lexicon and the similarities between its two formulations (Sextus Empiricus and Alexander of Aphrodisias). Moreover, a debated passage from Galen suggests that the subsequent introduction of SP into Stoic logic, and thus the invention of a method of syllogistic reduction different from that of Chrysippus, is due to Antipater of Tarsus.