entails

Entails is a relation between statements in logic and related fields, describing when one statement or a set of statements follows from another. In its semantic sense, a sentence B is entailed by a set A if every interpretation that makes all sentences in A true also makes B true. This is often written as A ⊨ B. For a single formula A, A ⊨ B means B follows from A in every model where A is true.

Syntactic entailment is a closely related notion used within a formal deductive system. It is written Γ ⊢ φ, meaning φ can be derived from the set of premises Γ using the rules of a specified proof system. For many standard logics, completeness theorems state that semantic entailment and syntactic entailment coincide: Γ ⊨ φ if and only if Γ ⊢ φ.

Entailment and truth are related but distinct concepts. An entailment does not require A and B to be true in every possible world; rather, B must be true in every world where A is true. Entailment is monotonic: adding new premises to A cannot invalidate an existing entailment, though it may generate new ones.

Examples help illustrate the idea. From the premises "All humans are mortal" and "Socrates is a human," one can entail "Socrates is mortal." In natural language, entailment can reflect semantic connections, such as "John owns a car" entailing "John owns a vehicle" if the context treats cars as vehicles.

Applications of entailment span formal logic, knowledge representation, automated reasoning, and natural language processing. Related concepts include validity (a sentence that is true in every model) and satisfiability (the existence of a model in which a set of sentences is true).