Entailment

Entailment is a relation between two statements in which the truth of the first guarantees the truth of the second. It is central to logical reasoning, linguistics, and information theory, and is used to describe a necessary connection in which one statement must hold if the other is true.

In formal logic, semantic entailment is typically written as A ⊨ B. It means that in every interpretation (model) where A is true, B is also true. Equivalently, B is true in all models that satisfy A. This notion is connected to satisfiability: if A ⊨ B, then the conjunction A ∧ not B is unsatisfiable. Semantic entailment distinguishes from simple implication, which concerns truth values within a single interpretation rather than across all models.

Syntactic entailment, written A ⊢ B, captures the idea that B can be derived from A using a fixed set of axioms and inference rules. For first-order logic, the completeness theorem states that semantic entailment and syntactic entailment coincide: A ⊨ B if and only if A ⊢ B.

In natural language processing, entailment refers to the relationship where a premise supports a conclusion. For example, Premise: "A man is playing a guitar." Hypothesis: "A man is playing an instrument." The hypothesis is entailed by the premise. In NLI tasks, entailment is contrasted with contradiction and neutrality to classify the relationship between statements.

Applications include knowledge representation, automated reasoning, formal verification, and questions answering, where determining entailment helps assess what must be true given certain information.