Title: Meaning, Structure, Pluralism: A Co-Determination Theory of Connective Meaning

Abstract:

In this talk, I propose a proof-theoretic theory of the meaning of logical
connectives in a sequent calculus setting. I defend the account by applying it to the debate on logical pluralism.

When do two logical connectives mean the same? Use-conditional theorists of meaning usually reply: when we use the two connectives in the same way.

When we do proofs, we use the connectives according to their proof rules. Hence, two logical connectives carry the same meaning iff they are governed by the same proof rules. This use-conditional approach to logical semantics is called proof-theoretic semantics.

According to Restall (2014), connective meaning is determined by those parts of the connective-defining rules which are immediate to the introduction/elimination of the connectives, i.e., the active/principal formulae. However, recent work by Bogdan Dicher (2016) shows that there are proof rules which share active/principal formulae but induce different connective use. To solve this problem, he proposes the co-determination thesis, according to which some non-immediate (i.e., structural) components of the proof rules also (co-) determine meaning.

At this point, however, co-determination is a mere thesis. In this talk, I develop it into a systematic theory of connective meaning. Via a series of Belnapian harmony proofs in minimal deductive contexts, I prove the exact minimal structural properties co-determining connective meaning for all connectives in the LK-family. This includes classical, intuitionistic, dual-intuitionistic, and lattice logic.

Ultimately, I argue that we should adopt this account by applying it to the debate on logical pluralism. This way, I contend that we can overcome some of the shortcomings of Beall and Restall’s famous logical pluralism, as well as Restall’s 2014 proof-theoretic rendition thereof.