View source for Logical consequence

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''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|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|Logico-mathematical calculus]]) and $A$ is a proposition of this language, then the relation  "A is a logical consequence of G"  means that any model for $\Gamma$ is a model for $A$. This relation is denoted by $\Gamma\vDash A$. The [[Gödel completeness theorem|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Rasiowa,   "The mathematics of metamathematics" , Polska Akad. Nauk  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Gödel,   "Die Vollständigkeit der Axiome des logischen Funktionenkalküls"  ''Monatsh. Math. Phys.'' , '''37'''  (1930)  pp. 349–360</TD></TR></table>



====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.T. Johnstone,   "Notes on logic and set theory" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grzegorczyk,   "An outline of mathematical logic" , Reidel  (1974)</TD></TR></table>