Proof-Theoretic Semantics, a Problem with Negation and Prospects for Modality

TL;DR

This paper examines proof-theoretic semantics, highlighting issues with defining negation and exploring potential rule-based accounts for modal operators, grounded in Dummett's and Prawitz's philosophical perspectives.

Contribution

It provides a detailed analysis of harmony and stability in proof-theoretic semantics and discusses the challenges of formalizing negation and modality.

Findings

Identifies a problem in defining negation within proof-theoretic semantics.

Analyzes the concepts of harmony and stability in natural deduction systems.

Explores prospects for rule-based semantics of modal operators.

Abstract

This paper discusses proof-theoretic semantics, the project of specifying the meanings of the logical constants in terms of rules of inference governing them. I concentrate on Michael Dummett's and Dag Prawitz' philosophical motivations and give precise characterisations of the crucial notions of harmony and stability, placed in the context of proving normalisation results in systems of natural deduction. I point out a problem for defining the meaning of negation in this framework and prospects for an account of the meanings of modal operators in terms of rules of inference.