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Classical Logic without Bivalance

Alexander V. Gheorghiu

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 $ω$-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.

Classical Logic without Bivalance

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 -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.

Paper Structure

This paper contains 5 sections, 3 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical Logic without Bivalence
  3. Theories of Arithmetic
  4. The Consistency of Arithmetic
  5. Soundness and Completeness

Key Result

Proposition 5

If $\Gamma$ is numerically definite, then it is $\omega$-complete. That is, if $\Gamma \Vdash \varphi(\bar{n})$ for every numeral $\bar{n}$, then $\Gamma \Vdash \forall x\,\varphi(x)$.

Figures (2)

  • Figure 1: Support in a Base
  • Figure 2: Arithmetic Base $\mathfrak{A}$

Theorems & Definitions (12)

  • Definition 1: Atomic Rule
  • Definition 2: Base
  • Definition 3: Support
  • Definition 4: Numerically Definite
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Definition 7: Consistency
  • Definition 8: Peano Arithmetic
  • ...and 2 more