Algebraic axiomatization of tense intuitionistic logic
type: Item
instance of: scholarly article
title: Algebraic axiomatization of tense intuitionistic logic
zbMATH Open document ID: 1260.03113
author: Ivan Chajda
published in: Central European Journal of Mathematics
publication date: 2012-01-16T00:00:00Z
review text: The paper studies the tense intuitionistic logics. The algebraic models of such logics are Heyting algebras endowed with tense operators. The operators \(G\) (it is always going to be the case that) and \(H\) (it has always been the case that) can be defined by the following conditions: 1. \(G(1) = 1\) and \(H(1) = 1\); 2. \(G(x \to y) \leq G(x) \to G(y)\) and \(H(x \to y) \leq H(x) \to H(y)\); 3. \(G(x \land y) \leq G(x) \to G(y)\) and \(H(x \land y) \leq H(x) \to H(y)\); 4. \(G(x \lor y) \leq G(x) \to G(y)\) and \(H(x \lor y) \leq H(x) \to H(y)\); 5. \(x \leq G(P(x))\) and \(x \leq H(F(x))\), where \(P(x) = H(x^\ast)^\ast\) and \(F = G(x^\ast)^\ast\) and \(x^\ast = x \to 0\). The justification for such an algebraic interpretation of the tense operators is given by the following theorem. Theorem. Let \(\mathcal{L} = (L; \lor, \land, \to)\) be a complete relatively pseudocomplemented lattice and \((T, \leq)\) be a frame. For \(p \in \mathcal{L}^T\) we define coordinatewise \(G(p)(x) = \bigwedge\{ p(y) ; x \leq y \} \text{ and } H(p)(x) = \bigwedge\{ p(y) ; y \leq x \}\). Then \(G,H\) are tense operators on \(\mathcal{L}^T\) satisfying the above conditions and such that \(G(0)=0=H(0)\).
reviewed by: Alex Citkin
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 03B44
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 06D15
zbMATH DE Number: 5998006
zbMATH Keywords: tense intuitionistic logic
zbMATH Keywords: algebraic semantics
zbMATH Keywords: Heyting algebra
zbMATH Keywords: tense operator
zbMATH Keywords: complete lattice
zbMATH Keywords: Brouwerian lattice
zbMATH Keywords: relative pseudocomplementation
MaRDI profile type: Publication
full work available at URL: s11533-011-0063-6
OpenAlex ID: W2051677278
cites work: Q5532570
cites work: Tense operators on basic algebras
cites work: Q4342094
cites work: Q5293965
cites work: Dynamic effect algebras
cites work: Q5389310
cites work: Q5445179
cites work: Intuitionistic tense and modal logic
cites work: Intuitionism. An introduction
cites work: Q5734410
cites work: Mathematics behind fuzzy logic
cites work: Constructive modal logics. I
DOI: 10.2478/S11533-011-0063-6
Recommended article: An algebraic axiomatization of the Ewald's intuitionistic tense logic
Recommended article: The sequent systems and algebraic semantics of intuitionistic tense logics
Recommended article: An algebraic study of tense logics with linear time
Recommended article: Q3344178
Recommended article: A strongly complete axiomatization of intuitionistic temporal logic
Recommended article: Tense logics over lattices
Recommended article: Intuitionistic tense and modal logic
Recommended article: Axiomatic systems and topological semantics for intuitionistic temporal logic
Recommended article: Q4853317
Recommended article: Dynamic order algebras as an axiomatization of modal and tense logics
description: scientific article; zbMATH DE number 5998006