##  [Entailment](/entailment-0) 

 Definition

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

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

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

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

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

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

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

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

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.