On finite linear intermediate predicate logics
type: Item
instance of: scholarly article
title: On finite linear intermediate predicate logics
zbMATH Open document ID: 0671.03017
DOI: 10.1007/BF00671568
author: Hiroakira Ono
published in: Studia Logica
publication date: 1988-01-01T00:00:00Z
review text: The paper deals with first-order predicate intermediate logics. For each \(n\geq 1\) let \(LP_ n^+\) be the logic of all Kripke frames (with varying domains) of height n. S. Yokota found a finite axiomatization for \(LP^+_ n\) (unpublished): \(LP^+_ n=HPC+\tilde P_ n\); here HPC is Heyting predicate calculus, and \(\tilde P_ 1=\forall x(p_ 1(x)\vee \neg p_ 1(x))\), \(\tilde P_ n=\forall x(p_ n(x)\vee (p_ n(x)\supset \tilde P_{n-1}))\). The author considers logics \(S^+_ n=LP^+_ n+(p\supset q)\vee (q\supset p)\) and constructs corresponding cut-free sequent calculi \(GS_ n\). \(GS_ n\) is proved to be strongly Kripke-complete: a sequent is provable in \(GS_ n\) iff it is valid in any linear Kripke frame of height n. It is worth noticing that completeness theorems for \(S^+_ n\) and \(LP^+_ n\) are non-trivial because these logics are not just quantified versions of corresponding propositional logics (quantified versions of propositional intermediate logics of finite slices are known to be Kripke-incomplete [the author, Rep. Math. Logic 15, 41-58 (1983; Zbl 0519.03016)].
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03F05
zbMATH DE Number: 4097348
zbMATH Keywords: first-order predicate intermediate logics
zbMATH Keywords: Kripke frames
zbMATH Keywords: cut-free sequent calculi
zbMATH Keywords: strongly Kripke-complete
MaRDI profile type: Publication
cites work: Q5582318
cites work: Q3684108
cites work: Q3669380
cites work: Q3830979
cites work: Axiomatization of the First‐Order Intermediate Logics of Bounded Kripkean Heights II
cites work: Q5202651
full work available at URL: https://doi.org/10.1007/bf00671568
OpenAlex ID: W1978651524
description: scientific article; zbMATH DE number 4097348