Entailment

Definition

Entailment

For a set of formulas and a formula , entailment is the relation

Here, ranges over interpretations of the chosen logic. means that every formula in is true in .

Unlike derivability (), entailment concerns truth in every model, not permitted inference steps.

Propositional logic

Definition

Entailment (Propositional Logic)

Let and be propositional formulas. The premises entail , written

if every interpretation that makes all premises true also makes true:

Here is the truth value assigns to , and denotes the truth value true. For a single premise this reads : entails .

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First-order logic

Definition

Entailment (First-Order Logic)

Let be formulas. Then entail , written

if for every structure and every variable assignment :

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