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.

References

[1]	H. Rasiowa, "The mathematics of metamathematics" , Polska Akad. Nauk (1963)
[2]	K. Gödel, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls" Monatsh. Math. Phys. , 37 (1930) pp. 349–360

Comments

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 $A$" .

References

[a1]	P.T. Johnstone, "Notes on logic and set theory" , Cambridge Univ. Press (1987)
[a2]	A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974)