imported>RandFreeman: Adding short description: "Expression in propositional calculus"

2025-06-24T22:33:07Z

Adding short description: "Expression in propositional calculus"

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			{{Short description\|Expression in propositional calculus}}
	In [[propositional calculus]], a '''propositional function''' or a '''predicate''' is a sentence expressed in a way that would assume the value of [[logical truth\|true]] or [[false (logic)\|false]], except that within the sentence there is a [[Variable (mathematics)\|variable]] (''x'') that is not defined or specified (thus being a [[free variable]]), which leaves the statement undetermined. The sentence may contain several such variables (e.g. ''n'' variables, in which case the function takes ''n'' arguments).		In [[propositional calculus]], a '''propositional function''' or a '''predicate''' is a sentence expressed in a way that would assume the value of [[logical truth\|true]] or [[false (logic)\|false]], except that within the sentence there is a [[Variable (mathematics)\|variable]] (''x'') that is not defined or specified (thus being a [[free variable]]), which leaves the statement undetermined. The sentence may contain several such variables (e.g. ''n'' variables, in which case the function takes ''n'' arguments).

imported>BD2412: /* Overview */clean up spacing around commas and other punctuation fixes, replaced: , → ,

2024-03-11T18:19:39Z

Overview: clean up spacing around commas and other punctuation fixes, replaced: , → ,

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In [[propositional calculus]], a '''propositional function''' or a '''predicate''' is a sentence expressed in a way that would assume the value of [[logical truth|true]] or [[false (logic)|false]], except that within the sentence there is a [[Variable (mathematics)|variable]] (''x'') that is not defined or specified (thus being a [[free variable]]), which leaves the statement undetermined. The sentence may contain several such variables (e.g. ''n'' variables, in which case the function takes ''n'' arguments).

==Overview==
As a [[Function (mathematics)|mathematical function]], ''A''(''x'') or ''A''(''x''{{sub|1}}, ''x''{{sub|2}}, ..., ''x''{{sub|''n''}}), the propositional function is abstracted from [[predicate (mathematical logic)|predicates]] or propositional forms. As an example, consider the predicate scheme, "x is hot". The substitution of any entity for ''x'' will produce a specific proposition that can be described as either true or false, even though "''x'' is hot" on its own has no value as either a true or false statement. However, when a value is assigned to ''x'', such as [[lava]], the function then has the value ''true''; while one assigns to ''x'' a value like [[ice]], the function then has the value ''false''.

Propositional functions are useful in [[set theory]] for the formation of [[set (mathematics)|sets]]. For example, in 1903 [[Bertrand Russell]] wrote in ''[[The Principles of Mathematics]]'' (page 106):
:"...it has become necessary to take ''propositional function'' as a [[primitive notion]].

Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types.<ref name="Tiles">{{cite book |last=Tiles |first=Mary |authorlink=Mary Tiles |title=The philosophy of set theory an historical introduction to Cantor's paradise |year=2004 |publisher=Dover Publications |location=Mineola, N.Y. |isbn=978-0-486-43520-6 |page=159 |url=http://store.doverpublications.com/0486435202.html |edition=Dover |accessdate=1 February 2013}}</ref>

A Propositional Function, or a predicate, in a variable ''x'' is an [[open formula]] ''p''(''x'') involving ''x'' that becomes a proposition when one gives ''x'' a definite value from the set of values it can take.

According to [[Clarence Lewis]], "A [[proposition]] is any expression which is either true or false; a propositional function is an expression, containing one or more variables, which becomes a proposition when each of the variables is replaced by some one of its values from a [[domain of discourse|discourse domain]] of individuals."<ref>[[Clarence Lewis]] (1918) ''A Survey of Symbolic Logic'', page 232, [[University of California Press]], second edition 1932, Dover edition 1960</ref> Lewis used the notion of propositional functions to introduce [[relation (mathematics)|relation]]s, for example, a propositional function of ''n'' variables is a relation of [[arity]] ''n''. The case of ''n'' = 2 corresponds to [[binary relation]]s, of which there are [[homogeneous relation]]s (both variables from the same set) and [[heterogeneous relation]]s.

== See also ==
* [[Propositional formula]]
* [[Boolean-valued function]]
* [[Formula (logic)]]
* [[Sentence (logic)]]
* [[Truth function]]
* [[Open sentence]]

== References ==
{{reflist}}

[[Category:Functions and mappings]]
[[Category:Mathematical relations]]
[[Category:Concepts in logic]]
[[Category:Predicate logic]]
[[Category:Logical expressions]]

Propositional function - Revision history

imported>RandFreeman: Adding short description: "Expression in propositional calculus"

imported>BD2412: /* Overview */clean up spacing around commas and other punctuation fixes, replaced: , → ,