Editing Propositional function (section)

==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.