Logical consequence

of a given set of premises

A proposition that is true for any interpretation of the non-logical symbols (that is, the names (cf. Name) of objects, functions, predicates) for which the premises are true. If a proposition $A$ is a logical consequence of a set of propositions $\Gamma$, one says that $\Gamma$ logically implies $A$, or that $A$ follows logically from $\Gamma$.

If $\Gamma$ is a set of statements of a formalized first-order logico-mathematical language (cf. Logico-mathematical calculus) and $A$ is a proposition of this language, then the relation "$A$ is a logical consequence of $\Gamma$" means that any model for $\Gamma$ is a model for $A$. This relation is denoted by $\Gamma\vDash A$. The Gödel completeness theorem of classical predicate calculus implies that the relation $\Gamma\vDash A$ coincides with the relation $\Gamma\vdash A$, that is, $\Gamma\vDash A$ if and only if $A$ is deducible from $\Gamma$ by the methods of classical predicate calculus.

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The phrase "semantic entailment" is sometimes used instead of "logical consequence"; thus, the expression $\Gamma\vDash A$ is read as "$\Gamma$ semantically entails $A$" . The expression $\Gamma\vdash A$ is similarly read as "$\Gamma$ syntactically entails $A$" .

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