The modal logic of pure provability
type: Item
instance of: scholarly article
title: The modal logic of pure provability
zbMATH Open document ID: 0713.03009
author: Samuel R. Buss
published in: Notre Dame Journal of Formal Logic
publication date: 1990-01-01T00:00:00Z
review text: The author considers a provability-type interpretation of modal formulas treating \(\square\) as ``provable in all extensions''. More precisely, let T be a consistent propositional theory extending classical logic (in the same language). For a modal propositional formula \(\phi\), the predicate \(T\vDash_{PP}\phi\) is defined inductively: 1) if \(\phi\) is \(\square\)-free, then \(T\vDash_{PP}\phi\) iff \(T\vdash \phi;\) 2) if \(\phi =\square \psi\) then \(T\vDash_{PP}\phi\) iff \(S\vDash_{PP}\psi\) for any consistent \(S\supseteq T;\) 3) if \(\phi =\eta (\square \psi_ 1,...,\square \psi_ k)\), \(\eta\) being \(\square\)-free, \(\theta_ i=p\vee \neg p\) if \(T\vDash \square \psi_ i\), and \(\theta_ i=p\wedge \neg p\) otherwise, then \(T\vDash_{PP}\phi\) iff \(T\vdash \eta (\theta_ 1,...,\theta_ k).\) The pure provability theory is defined thus: \(PP=\{\phi |\forall T T\vDash_{PP}\phi \}\); this is not a logic in the usual sense because it is not substitution-closed. The author proves completeness of PP w.r.t. a Kripke model, constructs an (infinite) axiomatic system for PP, and gives a deciding procedure. PP differs from arithmetically complete provability logic as well as from non-monotonic systems treating possibility as ``consistency'' [cf. \textit{D. McDermott} and \textit{J. Doyle}, Artif. Intell. 13, 41-72 (1980; Zbl 0435.68074)]. However the author hopes to investigate applications to non-monotonic logics in the future.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03F40
Mathematics Subject Classification ID: 03B60
zbMATH DE Number: 4172949
zbMATH Keywords: consistency
zbMATH Keywords: propositional theory
zbMATH Keywords: modal logic
zbMATH Keywords: S4
zbMATH Keywords: Grzegorczyk axiom
zbMATH Keywords: Kripke model
zbMATH Keywords: provability logic
zbMATH Keywords: non-monotonic logics
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093635417
OpenAlex ID: W1996093895
DOI: 10.1305/NDJFL/1093635417
description: scientific article; zbMATH DE number 4172949