Difference between revisions of "Logical consequence"
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''of a given set of premises'' | ''of a given set of premises'' | ||
− | A proposition that is true for any interpretation of the non-logical symbols (that is, the names (cf. [[ | + | 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. [[ | + | 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==== | ====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> | + | <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> | ||
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====References==== | ====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> | + | <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> |
Latest revision as of 21:24, 8 December 2016
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) |
Logical consequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_consequence&oldid=39932