Modal logics with several operators and probability interpretations
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 \(ZF\vdash Con(PA+Con(PA)+\neg \phi)\).
sequence of extentions of Peano Arithmetic
scientific article; zbMATH DE number 4014677