entailed

Entailed describes a relationship in logic and semantics where one claim follows from another. Specifically, a sentence B is entailed by a set of premises Γ if B must be true in every interpretation in which all sentences in Γ are true. This semantic notion is about truth in all possible worlds; it is closely related to the idea of logical consequence. The syntactic counterpart is provability: B is entailed by Γ if B can be derived from Γ using a formal proof system.

Formally, Γ entails B is written as Γ ⊨ B. If every model of Γ also makes B true, then Γ ⊨ B holds. In deductive systems, B is syntactically entailed by Γ if Γ ⊢ B. In sound and complete logical systems, semantic entailment and syntactic entailment coincide, though distinctions between them arise in incomplete or non-classical logics. An everyday way to think about it is: if the premises are true, the conclusion must also be true.

Examples help clarify the idea. From the premises Socrates is a man and all men are mortal, one can conclude Socrates is mortal; hence “Socrates is mortal” is entailed by the premises. In lexical semantics, entailment occurs at the word or phrase level too: bachelor entails man, since being a bachelor implies being a man.

Entailment underpins argument evaluation, knowledge representation, and natural language understanding. It is distinguished from non-entailment (where the conclusion does not necessarily follow) and from contradiction (where the conclusion is necessarily false given the premises).