The Best of All Possible Leibnizian Completeness Theorems

Leibniz developed several arithmetical interpretations of the assertoric syllogistic in a series of drafts from April 1679. In this article, I present what I take to be one of his most mature articulations of the arithmetical semantics from that series. I show that the assertoric syllogistic can be characterized exactly not only in the full divisibility lattice, as Leibniz implicitly suggests, but in a certain four-element sublattice thereof. This refinement is also shown to be optimal in the sense that the assertoric syllogistic is not complete with respect to any smaller sublattice using Leibniz's truth conditions.