Constrained consequence (Q1941731)

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.