The semantic conception of proof

TL;DR

This paper formalizes the semantic understanding of proof, axiomatizes proof relations, introduces a self-referential calculus with liar sentences, and explores interpreting formal systems with provability operators.

Contribution

It provides a formal axiomatization of the semantic conception of proof and introduces a consistent self-referential propositional calculus.

Findings

Axiomatization of proof relation and provability operator

Introduction of a consistent self-referential calculus

Analysis of interpreting formal systems with provability operators

Abstract

We analyze the informal semantic conception of proof and axiomatize the proof relation and the provability operator. A self referential propositional calculus which admits provable liar type sentences is introduced and proven consistent. We also investigate the problem of interpreting arbitrary formal systems in systems which include a provability operator.