Constrained consequence
type: Item
instance of: scholarly article
title: Constrained consequence
zbMATH Open document ID: 1278.03030
author: Katarina Britz
author: Johannes Heidema
author: Ivan Varzinczak
published in: Logica Universalis
publication date: 2013-03-21T00:00:00Z
review text: Let \(\models\) be a classical consequence relation (semantic entailment) and let \(\alpha \models \beta\) mean \(\mathrm{Mod}(\alpha) \subseteq \mathrm{Mod}(\beta)\), where \(\mathrm{Mod}(\alpha)\) is the set of all (classical) models of \(\alpha\). A constraint entailment relation is a consequence relation \(|\!\!\!<\) satisfying the following five properties: {Reflexivity:} \(\alpha\) \(|\!\!\!<\) \(\alpha\). {Infra-classicality:} If \(\alpha\) \(|\!\!\!<\) \(\beta\) then \(\alpha \models \beta\). {Equivalence:} If \(\alpha \equiv \beta\) and \(\alpha\) \(|\!\!\!<\) \(\gamma\) then \(\beta\) \(|\!\!\!<\) \(\gamma\). {Semantical interpolation:} If \(\alpha \models \beta\), \(\beta \models \gamma\), and \(\alpha\) \(|\!\!\!<\) \(\gamma\), then \(\alpha\) \(|\!\!\!<\) \(\beta\) and \(\beta\) \(|\!\!\!<\) \(\gamma\). {Generalized disjunction:} If \(\alpha\) \(|\!\!\!<\) \(\beta\) and \(\gamma\) \(|\!\!\!<\) \(\delta\), then \(\alpha \vee \gamma\) \(|\!\!\!<\) \(\beta \vee \delta\). The paper studies the constraint consequence relation defined by \(\alpha {|\!\!\!<} \beta\) if and only if \(\alpha \models \beta\) and \(\beta \models \bullet\alpha\), where \(\bullet\) is a propositional operator satisfying the two conditions \(\alpha \models \bullet\alpha\) and \(\alpha \models \beta\) yields \(\bullet\alpha \models \bullet\beta\). The resulting consequence relation is infra-classical and nonmonotonic and can be used in the presence of inconsistencies.
reviewed by: Alex Citkin
Mathematics Subject Classification ID: 03B20
Mathematics Subject Classification ID: 03B22
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B53
Mathematics Subject Classification ID: 03B60
zbMATH DE Number: 6147860
zbMATH Keywords: constrained consequences
zbMATH Keywords: weakening operator
zbMATH Keywords: infra-classical entailment
zbMATH Keywords: nonmonotonic reasoning
zbMATH Keywords: modal logic
zbMATH Keywords: action
zbMATH Keywords: causality
zbMATH Keywords: abduction
zbMATH Keywords: diagnosis
zbMATH Keywords: paradoxes
MaRDI profile type: Publication
full work available at URL: s11787-011-0037-2
OpenAlex ID: W3192684197
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DOI: 10.1007/S11787-011-0037-2
description: scientific article; zbMATH DE number 6147860