Intermediate logics and the de Jongh property
type: Item
instance of: scholarly article
title: Intermediate logics and the de Jongh property
zbMATH Open document ID: 1233.03033
author: Rineke Verbrugge
author: Albert Visser
author: Dick H. J. de Jongh
published in: Archive for Mathematical Logic
publication date: 2011-03-02T00:00:00Z
review text: Dick de Jongh proved in an unpublished paper that the propositional logic of Heyting's arithmetic, \textbf{HA}, is precisely \textbf{IPC}, that means that, for any proposition \(A(p_1,\dots, p_n)\), \(\mathbf{IPC}\vdash A(p_1,\dots, p_n)\) if and and only if \(\mathbf{HA}\vdash A(B_1,\dots, B_n)\), for all arithmetical sentences \(B_1,\dots, B_n\). Propositional intermediate logics are the logics between intuitionistic propositional logic \textbf{IPC} and classical propositional logic \textbf{CPC}. An intermediate logic is said to have the finite model property, \textbf{FMP}, if there is a class of finite Kripke frames for which it is complete. In this paper, the authors generalize de Jongh's theorem for extensions of \textbf{HA} with propositional logics that have \textbf{FMP}. \textit{C. Smoryński}, in his paper [``Applications of Kripke models'', in: A. S. Troelstra (ed.), Metamathematical investigation of intuitionistic arithmetic and analysis. Berlin: Springer. Lect. Notes Math. 344, 324--391 (1973; Zbl 0275.02025)], gave two proofs for the de Jongh theorem. The authors of this paper use Smoryński's second proof, which is more complicated but more flexible for applications.
reviewed by: Mohammad Ardeshir
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03B20
Mathematics Subject Classification ID: 03F30
zbMATH DE Number: 5859290
zbMATH Keywords: intuitionistic logic
zbMATH Keywords: intermediate logic
zbMATH Keywords: Heyting arithmetic
zbMATH Keywords: de Jong prpoperty
MaRDI profile type: Publication
full work available at URL: s00153-010-0209-4
OpenAlex ID: W2295008513
cites work: Intuitionistic validity in \(T\)-normal Kripke structures
cites work: A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property
cites work: Kripke models and the intuitionistic theory of species
cites work: Q3214890
cites work: Q4179016
cites work: Q3919709
cites work: Recursive Predicates and Quantifiers
cites work: Disjunction and existence under implication in elementary intuitionistic formalisms
cites work: Eine Unableitbarkeitsbeweismethode für den Intuitionistischen Aussagenkalkül
cites work: Q5588586
cites work: Q3880811
cites work: Incompleteness in intuitionistic metamathematics
cites work: Q5622184
cites work: Q3852250
cites work: A Survey of Propositional Realizability Logic
cites work: Propositional Calculus and Realizability
cites work: Provability interpretations of modal logic
cites work: Constructivism in mathematics. An introduction. Volume I
cites work: Extension of Lifschitz' realizability to higher order arithmetic, and a solution to a problem of F. Richman
cites work: On the completenes principle: A study of provability in heyting's arithmetic and extensions
cites work: Q4250050
cites work: Rules and arithmetics
cites work: Substitutions of \(\Sigma_1^0\)-sentences: Explorations between intuitionistic propositional logic and intuitionistic arithmetic
cites work: Can We Make the Second Incompleteness Theorem Coordinate Free?
DOI: 10.1007/S00153-010-0209-4
Recommended article: Q2998377
Recommended article: Finite frames fail: how infinity works its way into the semantics of admissibility
Recommended article: Interpolation property for extensions of intuitionistic provability logic
Recommended article: Q3340837
Recommended article: Intermediate logics preserving admissible inference rules of heyting calculus
Recommended article: Arithmetical Completeness of the Intuitionistic Logic of Proofs
Recommended article: THE -PROVABILITY LOGIC OF
Recommended article: On an interpretation of second order quantification in first order intuitionistic propositional logic
Recommended article: Intermediate logics with the same disjunctionless fragment as intuitionistic logic
Recommended article: Q5874220
description: scientific article; zbMATH DE number 5859290