Relation in which one set of sentences semantically guarantees the truth of another sentence in every interpretation: a consequence relation between premises and conclusion across all models of the premises.

Principle

A sentence φ is a semantic entailment of a set Γ (Γ |= φ) when every model that makes all sentences in Γ true also makes φ true; entailment formalizes semantic preservation from premises to conclusion.

Demonstration

From {All humans are mortal, Socrates is human} we semantically entail Socrates is mortal, because in every model where the two premises hold the conclusion must hold; in propositional clause logic, {p → q, p} entails q.

Misapplication

Treating entailment as mere syntactic derivability without regard to soundness: a proof system might not preserve semantic entailment if it is unsound, or might fail to derive entailments if it is incomplete.

Consequence

Entailment yields reliable inferences: once semantic entailment is established, conclusions follow in all models of the premises and can be used to transfer truth from assumptions to derived facts.

Reversal

Failure of entailment means there exists at least one model of the premises where the conclusion is false; contraposition or refutation by countermodel are typical reversals used diagnostically.

Boundary

Entailment depends on the chosen logic, signature, and semantics; nonmonotonic logics change entailment behavior (adding premises can defeat previous entailments), unlike classical monotonic entailment.

Semantic Tension

Entailment vs implication: implication is a formula-level connective inside the language, while entailment is a meta-level relation between sets of formulas and formulas; they interact but are distinct notions.

Synthesis

Entailment captures semantic guarantee: whenever premises are true in a model, the entailed conclusion is also true; it formalizes dependable logical consequence across admissible models.