Maximality in modal logic
type: Item
instance of: scholarly article
title: Maximality in modal logic
zbMATH Open document ID: 0624.03013
DOI: 10.1016/0168-0072(87)90068-6
author: Harvey M. Friedman
author: Robert C. Flagg
published in: Annals of Pure and Applied Logic
publication date: 1987-01-01T00:00:00Z
review text: A system FP of propositional logic is maximal for the corresponding system FA of arithmetic if the underivability of \(A(p_ 1,...,p_ n)\) in FP implies underivability in FA of \(A(B_ 1,...,B_ n)\) for some sentences \(B_ 1,...,B_ n\) of FA. The authors prove maximality for Grzegorczyk's modal logic Grz and for S4. For Grz the proof is modelled after \textit{C. Smorynski}'s argument (Applications of Kripke models, in \textit{A. Troelstra}'s volume [Lecture Notes Math. 344 (1973; Zbl 0275.02025)]). Falsifying sentences \(B_ i\) are constructed in rather natural way from \(\Sigma^ 0_ 1\)-sentences completely independent over PA. The proof for S4 is much more complicated and may be even more important than the final result. The main novel feature is verification of the induction axiom in the constructed Kripke model. A version of iterated Ramsey theorem and a translation of modal arithmetic formulas in non-modal ones are developed to achieve this.
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 4012573
zbMATH Keywords: maximality
zbMATH Keywords: Grzegorczyk's modal logic Grz
zbMATH Keywords: S4
zbMATH Keywords: Kripke model
zbMATH Keywords: iterated Ramsey theorem
zbMATH Keywords: modal arithmetic formulas
reviewed by: G. E. Mints
MaRDI profile type: Publication
full work available at URL: 0168-0072(87)90068-6
OpenAlex ID: W2059928120
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description: scientific article; zbMATH DE number 4012573