Categorical Semantics

Abstract

In “Tonk, Plonk and Plink” Belnap [1962] defended the claim that natural deduction rules define the meaning of a connective, provided at least that those rules form a conservative extension of the structural rules of deduction. In this paper we will investigate a stronger criterion of success for defining the meaning of a connective. Each connective comes with an intended interpretation, (for example, the intended interpretation of & is recorded by the truth table for &). For a set of rules to define the meaning of a connective, we would expect it to be categorical, i.e., we expect (roughly) that it force the intended interpretation of the connective on all its “models”. To put it another way, rules define a connective when they are strong enough to eliminate any non-standard interpretations.¹

The demand that rules be categorical is a severe test for defining a connective. It is at least in the spirit, however, of Belnap’s syntactic consistency and completeness conditions. It is not difficult to show that when a semantics ∥S∥ obeys the (very weak) property that every atomic valuation has an extension in some set of valuations V obeying ∥S∥, then any system categorical for ∥S∥ meets Belnap’s conservative extension requirement. It is also possible to show that as long as there is a set of valuations V obeying ∥S∥ which is unique (in the sense that each atomic valuation has a unique extension in V) then any categorical system S defines unique roles of inference for the connectives. So our semantical requirements vindicate Belnap’s purely syntactic requirements.