Modal logics with several operators and probability interpretations
type: Item
instance of: scholarly article
title: Modal logics with several operators and probability interpretations
zbMATH Open document ID: 0625.03007
DOI: 10.1007/BF02764872
author: Tim Carlson
published in: Israel Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The purpose of the paper is to study modal logics of provability in a sequence of extentions of Peano Arithmetic PA. The main result describes n-modal logics of provability in theories \(T_ 1,...,T_ n\) which contain PA and satisfy the following condition: \(T_ i\) contains \(Refl(T_ j)\) if \(i<j\) (Refl(T) is the schema of sentences of the form \(\Pr_ T(\ulcorner \phi \urcorner)\to \phi)\). These logics are proved to be decidable. Thus the paper proves the generalization of Smorynski's conjecture about decidability of the set of valid bimodal formulas with \(T_ 1=PA\) and \(T_ 2=ZF\). The appropriate Kripke-style semantics is developed. The paper contains an application: there is a Boolean combination of \(\Sigma_ 1\) formulas \(\phi\) such that \(PA+Con(PA)\vdash Con(PA+\phi)\) and \(PA+Con(PA)\vdash Con(PA+\neg \phi)\) while \(ZF\vdash Con(PA+Con(PA)+\phi)\) and \(ZF\vdash Con(PA+Con(PA)+\neg \phi)\).
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03F07
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 4014677
zbMATH Keywords: sequence of extentions of Peano Arithmetic
zbMATH Keywords: n-modal logics of provability
zbMATH Keywords: decidability
zbMATH Keywords: Kripke-style semantics
reviewed by: Sergei Artemov
MaRDI profile type: Publication
cites work: One hundred and two problems in mathematical logic
cites work: Solution of a problem of Leon Henkin
cites work: Q5537599
cites work: Provability interpretations of modal logic
description: scientific article; zbMATH DE number 4014677