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Difference between revisions of "Propositional function"

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A function whose arguments and values are truth values (cf. [[Truth value|Truth value]]). This term is used when the discussion is about interpretations of a formalized logical language.
 
A function whose arguments and values are truth values (cf. [[Truth value|Truth value]]). This term is used when the discussion is about interpretations of a formalized logical language.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075520/p0755201.png" /> is the set of truth values of formulas of a given language, then a propositional function is any expression of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075520/p0755202.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075520/p0755203.png" />). These functions are interpreted as propositional connectives (cf. [[Propositional connective|Propositional connective]]) that allow one to form new statements or formulas. In the classical two-valued interpretation of the set of truth values, i.e. when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075520/p0755204.png" />, such functions are also called functions of the algebra of logic.
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If $\Omega$ is the set of truth values of formulas of a given language, then a propositional function is any expression of the type $\Omega^n\to\Omega$ ($n\geq0$). These functions are interpreted as propositional connectives (cf. [[Propositional connective|Propositional connective]]) that allow one to form new statements or formulas. In the classical two-valued interpretation of the set of truth values, i.e. when $\Omega=\{0,1\}$, such functions are also called functions of the algebra of logic.
  
  
  
 
====Comments====
 
====Comments====
Propositional functions are also called truth functions. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075520/p0755205.png" />, they are also called Boolean functions.
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Propositional functions are also called truth functions. When $\Omega=\{0,1\}$, they are also called Boolean functions.
  
 
More or less equivalently, propositional functions are functions whose arguments and values are propositions.
 
More or less equivalently, propositional functions are functions whose arguments and values are propositions.

Revision as of 19:17, 7 July 2014

A function whose arguments and values are truth values (cf. Truth value). This term is used when the discussion is about interpretations of a formalized logical language.

If $\Omega$ is the set of truth values of formulas of a given language, then a propositional function is any expression of the type $\Omega^n\to\Omega$ ($n\geq0$). These functions are interpreted as propositional connectives (cf. Propositional connective) that allow one to form new statements or formulas. In the classical two-valued interpretation of the set of truth values, i.e. when $\Omega=\{0,1\}$, such functions are also called functions of the algebra of logic.


Comments

Propositional functions are also called truth functions. When $\Omega=\{0,1\}$, they are also called Boolean functions.

More or less equivalently, propositional functions are functions whose arguments and values are propositions.

References

[a1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1959) pp. 144; 226
How to Cite This Entry:
Propositional function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Propositional_function&oldid=32391
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article