Categorical Harmony and Paradoxes in Proof-Theoretic Semantics

Abstract

There are two camps in the theory of meaning: the referentialist one including Davidson, and the inferentialist one including Dummett and Brandom. Proof-theoretic semantics is a semantic enterprise to articulate an inferentialist account of the meaning of logical constants and inferences within the proof-theoretic tradition of Gentzen, Prawitz, and Martin-Löf, replacing Davidson’s path “from truth to meaning” by another path “from proof to meaning”. The present paper aims at contributing to developments of categorical proof-theoretic semantics, proposing the principle of categorical harmony, and thereby shedding structural light on Prior’s “tonk” and related paradoxical logical constants. Categorical harmony builds upon Lawvere’s conception of logical constants as adjoint functors, which amount to double-line rules of certain form in inferential terms. Conceptually, categorical harmony supports the iterative conception of logic. According to categorical harmony, there are intensional degrees of paradoxicality of logical constants; in the light of the intensional distinction, Russell-type paradoxical constants are maximally paradoxical, and tonk is less paradoxical. The categorical diagnosis of the tonk problem is that tonk mixes up the binary truth and falsity constants, equating truth with falsity; hence Prior’s tonk paradox is caused by equivocation, whereas Russell’s paradox is not. This tells us Prior’s tonk-type paradoxes can be resolved via disambiguation while Russell-type paradoxes cannot. Categorical harmony thus allows us to demarcate a border between tonk-type pseudo-paradoxes and Russell-type genuine paradoxes. I finally argue that categorical semantics based on the methods of categorical logic might even pave the way for reconciling and uniting the two camps.