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−  An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607201.png" />place function defined on the set of truth values (cf. [[Truth valueTruth value]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607202.png" /> and taking values in this set. With every [[Logical operationlogical operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607203.png" /> is associated a logical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607204.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607205.png" /> are truth values, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607206.png" /> is the truth value of the proposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607208.png" /> are propositions such that the truth value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l0607209.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l06072010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l06072011.png" />.  +  {{TEXdone}} 
 +  An $n$place function defined on the set of truth values (cf. [[Truth valueTruth value]]) $\{\text T,\text F\}$ and taking values in this set. With every [[Logical operationlogical operation]] $\mathfrak A$ is associated a logical function $f_\mathfrak A$: If $V_1,\ldots,V_n$ are truth values, then $f_\mathfrak A(V_1,\ldots,V_n)$ is the truth value of the proposition $\mathfrak A(P_1,\ldots,P_n)$, where $P_1,\ldots,P_n$ are propositions such that the truth value of $P_i$ is equal to $V_i$, $i=1,\ldots,n$. 
   
−  A logical function is sometimes defined as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l06072012.png" />place function defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l06072013.png" /> and taking values in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060720/l06072014.png" />. Such functions are used in mathematical logic as an analogue of the concept of a [[Predicatepredicate]].  +  A logical function is sometimes defined as an $n$place function defined on a set $M$ and taking values in the set $\{\text T,\text F\}$. Such functions are used in mathematical logic as an analogue of the concept of a [[Predicatepredicate]]. 
 +  
 +  [[Category:Logic and foundations]] 
Latest revision as of 16:55, 2 November 2014
An $n$place function defined on the set of truth values (cf. Truth value) $\{\text T,\text F\}$ and taking values in this set. With every logical operation $\mathfrak A$ is associated a logical function $f_\mathfrak A$: If $V_1,\ldots,V_n$ are truth values, then $f_\mathfrak A(V_1,\ldots,V_n)$ is the truth value of the proposition $\mathfrak A(P_1,\ldots,P_n)$, where $P_1,\ldots,P_n$ are propositions such that the truth value of $P_i$ is equal to $V_i$, $i=1,\ldots,n$.
A logical function is sometimes defined as an $n$place function defined on a set $M$ and taking values in the set $\{\text T,\text F\}$. Such functions are used in mathematical logic as an analogue of the concept of a predicate.
How to Cite This Entry:
Logical function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_function&oldid=19211
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics  ISBN 1402006098.
See original article