Generalized Entailments

Abstract

Abstract.A semantic relation between a family of sets of formulas and a set of formulas, dubbed generalized entailment, and its subrelation, called constructive generalized entailment, are defined and examined. Entailment construed in the usual way and multiple-conclusion entailment can be viewed as special cases of generalized entailment. The concept of constructive generalized entailment, in turn, enables an explication of some often used notion of interrogative entailment, and coincides with inquisitive entailment at the propositional level. Some interconnections between constructive generalized entailment and Inferential Erotetic Logic are also analysed.Keywords: entailment; families of sets; logic of questions Basic intuitionsAs for logic, entailment is most often conceived of as a relation between a set of well-formed formulas (wffs for short) on the one hand, and a single wff on the other. Entailment ensures transmission of truth: a wff A entailed by a set of wffs X must be true if only all the wffs in X are true. What 'must' means here depends on a logic under consideration, and similarly for 'truth.' The transmission of truth principle falls under the general schema: where A stands, depending on a case, for: 'valuation' (of an appropriate kind), 'model', 'intended model', 'world of a model', and so forth.Entailment understood in the standard way exhibits a kind of asymmetry: what is entailed is a single wff, while what is entailing it is a set of wffs. If, for some reasons, you prefer symmetry over the lack of it,