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