Leibniz as Aristotle and Boole Conflated

Leibniz signifies one of the most difficult and confusing phases within the history of logic. He is the greatest logician of the early modern period as well as one of the greatest thinkers ever, and yet his profoundest achievements as a logician were magnificent outlines of a grand program that was doomed to failure. He was a prophet whose insistence on the immediate actualizing of ideals had forced his followers to notice the impossibility of doing so. He tried to spell out and explicate Aristotelian essentialism by constructing a proof apparatus (a semi-formal language of sorts)75 that would enable the computation of everything known, and everything that could in principle be known, by trivial substitutions of synonyms that lead to identities of the form A = A. He maintained that such proof apparatus is within close reach that a few years of collaborative work of the “republic of letters” would suffice to construct it. It would bring an end to all controversies (including theological disputes), he promised, by reducing them to elementary arithmetic calculations. Let me explain very briefly why his attempts to achieve these ideals were so magnificent and why they were doomed to failure.

Leibniz was a Modern who revered his scholastic predecessors. Like many of them he was well aware of many imperfections of the purportedly prefect Aristotelian logic: he studied them with a peculiar mix of critical independence and dogmatic awe. He rejected, for example, the validity of some of the inferences whose validity rests upon existential import, but he did not reject all of them and his reasons for his rejections seem to us today somewhat fanciful, perhaps even capricious. He held that an account of relations is indispensable for any complete logic, and considered the old Aristotelian system too stringent to deal with them properly. Yet he did not try to give relations a place of their own within a new logic, but rather simply to force their reduction to the traditional subject-copula-predicate form. He tried to determine the validity of inferences whose validity depends on relations by their reduction to inferences of the subject-copula-predicate form. This could not succeed, of course. But it did produce his well-known notion of Monad, whose properties are relations with infinitely many arguments, no less.