Entailment

Synonyms

Implication; Logical consequence

Definition

The term entailment is used in the context of logical reasoning. Formally, a logical formula T entails a formula c if and only if all models of T are also a model of c. This is usually denoted as $T\vDash c$ and means that c is a logical consequence of T or that c is implied by T.

Let us elaborate this definition for propositional clausal logic, where the formulae T could be the following expression:

flies :- bird, normal. bird :- blackbird. bird :- ostrich.

Here, the first clause or rule can be read as flies if normal and bird, that is, normal birds fly, the second and third one as stating that blackbirds, resp. ostriches, are birds. An interpretation is then an assignment of truth-values to the propositional variables. For instance, for the above domain

{ostrich, bird} {blackbird, bird, normal}

are interpretations, specified through the set of propositional variables that are true. This means that in the first interpretation, the...