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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. [[Name|Name]]) of objects, functions, predicates) for which the premises are true. If a proposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607001.png" /> is a logical consequence of a set of propositions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607002.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607003.png" /> logically implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607004.png" />, or that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607005.png" /> follows logically from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607006.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607007.png" /> is a set of statements of a formalized first-order logico-mathematical language (cf. [[Logico-mathematical calculus|Logico-mathematical calculus]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607008.png" /> is a proposition of this language, then the relation  "A is a logical consequence of G"  means that any model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l0607009.png" /> is a model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070010.png" />. This relation is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070011.png" />. The [[Gödel completeness theorem|Gödel completeness theorem]] of classical predicate calculus implies that the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070012.png" /> coincides with the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070013.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070014.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070015.png" /> is deducible from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070016.png" /> by the methods of classical predicate calculus.
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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====
 
====References====
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====Comments====
 
====Comments====
The phrase  "semantic entailmentsemantic entailment"  is sometimes used instead of  "logical consequence" ; thus, the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070017.png" /> is read as  "G semantically entails A" . The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060700/l06070018.png" /> is similarly read as  "G syntactically entails A" .
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The phrase  "semantic entailment"  is sometimes used instead of  "logical consequence" ; thus, the expression $\Gamma\vDash A$ is read as  "G semantically entails A" . The expression $\Gamma\vdash A$ is similarly read as  "G syntactically entails A" .
  
 
====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>

Revision as of 21:48, 14 April 2014

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 G" 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 "G semantically entails A" . The expression $\Gamma\vdash A$ is similarly read as "G 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)
How to Cite This Entry:
Logical consequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_consequence&oldid=19217
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article