Classical Logic without Bivalance

TL;DR

This paper presents a non-bivalent classical logic framework that addresses anti-realism, handles omega-incompleteness, and provides a simple consistency proof for Peano Arithmetic without transfinite methods.

Contribution

It introduces a semantics for classical logic without bivalence, enabling a new approach to metamathematics and foundational proofs.

Findings

Handles omega-incompleteness effectively

Makes induction a meaning-constitutive principle

Provides an elementary consistency proof for Peano Arithmetic

Abstract

Sandqvis's semantics for classical logic without bivalence resolves the question of an anti-realist account of classical reasoning after Dummett. This paper applies the framework to the essential questions of metamathematics. The system intuitively handles $\omega$-incompleteness, makes induction meaning-constitutive, and yields an elementary consistency proof for Peano Arithmetic using only ordinary induction on the natural numbers, with no appeal to transfinite ordinals or recognition-transcendent truth.