Truth values and proof theory
type: Item
instance of: scholarly article
title: Truth values and proof theory
zbMATH Open document ID: 1191.03042
author: Greg Restall
published in: Studia Logica
publication date: 2009-11-16T00:00:00Z
review text: Canonical models constructed from the maximally consistent (with respect to derivability with cut) sequents provide completeness proofs for classical, intuitionistic and modal logic. The author reverses this process. In the classical case the conditions, say, for \(A\& B\in X\), \(A\&B\in Y\) in a maximally consistent sequent \(X\vdash Y\) determine standard Boolean semantics. In the intuitionistic case, two possible definitions of maximal consistency lead to two different semantics. If \(X\vdash Y\) is defined to be consistent when \(X\vdash A\) is underivable for every \(A\in Y\), then Beth semantics results. If \(X\vdash Y\) is consistent when \(X\vdash Y'\) is underivable for every \(Y'\subseteq Y\) in a multiple-succedent formulation, then Kripke semantics results. For S5, the authors considers a Kripke-style hypersequent (tableau) formulation where sequents correspond to possible worlds. It would be interesting to see whether the same result can be obtained from a cut-free sequent formulation by G. Shvarts.
Mathematics Subject Classification ID: 03F03
Mathematics Subject Classification ID: 03B20
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 5633562
zbMATH Keywords: sequent calculus
zbMATH Keywords: completeness
zbMATH Keywords: intuitionistic logic
zbMATH Keywords: modal logic
zbMATH Keywords: classical logic
zbMATH Keywords: maximal consistency
zbMATH Keywords: semantics
reviewed by: G. E. Mints
MaRDI profile type: Publication
full work available at URL: s11225-009-9197-y
OpenAlex ID: W2070877511
cites work: Q4342082
cites work: Intuitionism. An introduction
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cites work: Q5457387
DOI: 10.1007/S11225-009-9197-Y
description: scientific article; zbMATH DE number 5633562