Entailing

Entailing is a relation between statements or propositions in logic, semantics, and linguistics. If a statement A entails another statement B, then in every interpretation or possible world where A is true, B is also true. In formal notation, A entails B is written as A ⊨ B (or T ⊨ φ relative to a theory T). Entailment is directional: A can entail B without B entailing A.

In natural language, entailment captures when the truth of one sentence guarantees the truth of another. For example, the sentence “The box contains a gift” entails “The box contains something.” The converse does not necessarily hold, and two sentences can be compatible without one entailing the other.

Key properties include monotonicity and transitivity. Entailment is monotone: adding further premises cannot make an entailed conclusion false. It is transitive: if A entails B and B entails C, then A entails C. These properties support deductive reasoning and structured inference.

Relationships to related notions vary by context. Entailment differs from presupposition, which concerns background assumptions that may survive or fail under certain contexts. In formal logic, entailment aligns with the notion of logical consequence, whereas in everyday language it is studied within semantic theories and pragmatics.

Applications are prominent in semantic theory and artificial intelligence. In NLP, recognizing textual entailment involves determining whether a hypothesis is entailed by a given text. Entailment concepts underpin reasoning systems, knowledge representation, and proof systems, providing a foundational relation for deriving conclusions from information.