Entailments

Entailment is a relation between two statements or propositions in which the truth of the first guarantees the truth of the second. If A entails B, then in every situation where A is true, B is also true. Entailment is central to logic and to semantics, and it is distinguished from implication and from equivalence.

Formal definition: In model theory, A entails B, written A ⊨ B, if every model that makes A true also makes B true. In deductive systems, B is entailed by A if there is a formal proof of B from A and the axioms of the system. Entailment is a semantic notion (truth-conditional) rather than a purely syntactic one, though the two are related by completeness theorems.

In linguistics, entailment describes how the truth-conditions of sentences relate. For example, “A dog barked” entails “A dog existed.” and “The cake was eaten” entails “The cake existed.” Entailment is a component of natural language inference, where systems classify whether a hypothesis sentence is entailed by a given premise, contradicted, or neither.

Entailment interacts with background knowledge, world knowledge, and lexical semantics. It is used in knowledge representation, automated reasoning, and information retrieval to assess whether information follows from what is stated. Distinguishing entailment from related notions such as implication, presupposition, and consistency helps in formal analysis of arguments and in designing reasoning systems.