Leibniz on the Logic of Conceptual Containment and Coincidence

In a series of early essays written around 1679, Leibniz sets out to explore the logic of conceptual containment. In his more mature logical writings from the mid-1680s, however, his focus shifts away from the logic of containment to that of coincidence, or mutual containment. This shift in emphasis is indicative of the fact that Leibniz’s logic has its roots in two distinct theoretical frameworks: (i) the traditional theory of the categorical syllogism based on rules of inference such as Barbara, and (ii) equational systems of arithmetic and geometry based on the rule of substitution of equals. While syllogistic reasoning is naturally modeled in a logic of containment, substitutional reasoning of the sort performed in arithmetic and geometry is more naturally modeled in a logic of coincidence. In this paper, we argue that Leibniz’s logic of conceptual containment and his logic of coincidence can in fact be viewed as two alternative axiomatizations, in different but equally expressive languages, of one and the same logical theory. Thus, far from being incoherent, the varying syllogistic and equational themes that run throughout Leibniz’s logical writings complement one another and fit together harmoniously.