Definite Formulae, Negation-as-Failure, and the Base-extension Semantics of Intuitionistic Propositional Logic

TL;DR

This paper explores proof-theoretic semantics for intuitionistic propositional logic, interpreting bases as definite formulae and linking negation-as-failure in logic programming to the semantics of negation.

Contribution

It establishes the completeness of IPL for base-extension semantics using proof-search, and clarifies negation-as-failure within proof-theoretic semantics.

Findings

Supports the interpretation of negation-as-failure as denial in semantics

Proves completeness of IPL for base-extension semantics

Connects proof-theoretic semantics with logic programming principles

Abstract

Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). This semantics is given by a relation called support, explaining the meaning of the logical constants, which is parameterized by systems of rules called bases that provide the semantics of atomic propositions. In this paper, we interpret bases as collections of definite formulae and use the operational view of the latter as provided by uniform proof-search -- the proof-theoretic foundation of logic programming (LP) -- to establish the completeness of IPL for the B-eS. This perspective allows negation, a subtle issue in P-tS, to be understood in terms of the negation-as-failure protocol in LP. Specifically, while the denial of a proposition is traditionally understood as the assertion of its negation, in B-eS we may understand the denial of a proposition as the failure to find a proof of it. In this way, assertion and denial are both prime concepts in P-tS.