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In `F33f`_`[propositional calculus`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Propositional_calculus]`_`f, a `!propositional function`! or a `!predicate`! is a sentence expressed in a way that would assume the value of `F33f`_`[true`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logical_truth]`_`f or `F33f`_`[false`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=False_(logic)]`_`f, except that within the sentence there is a `F33f`_`[variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Variable_(mathematics)]`_`f (`*x`*) that is not defined or specified (thus being a `F33f`_`[free variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Free_variable]`_`f), which leaves the statement undetermined. The sentence may contain several such variables (e.g. `*n`* variables, in which case the function takes `*n`* arguments).

>>Contents

• `F0af`_`[Overview`#overview]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Overview

As a `F33f`_`[mathematical function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f, `*A`*(`*x`*) or `*A`*(`*x`*1, `*x`*2, ..., `*x`*`*n`*), the propositional function is abstracted from `F33f`_`[predicates`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Predicate_(mathematical_logic)]`_`f 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 `F33f`_`[lava`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lava]`_`f, the function then has the value `*true`*; while one assigns to `*x`* a value like `F33f`_`[ice`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ice]`_`f, the function then has the value `*false`*.

Propositional functions are useful in `F33f`_`[set theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Set_theory]`_`f for the formation of `F33f`_`[sets`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Set_(mathematics)]`_`f. For example, in 1903 `F33f`_`[Bertrand Russell`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bertrand_Russell]`_`f wrote in `*`F33f`_`[The Principles of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=The_Principles_of_Mathematics]`_`f`* (page 106):

"...it has become necessary to take `*propositional function`* as a `F33f`_`[primitive notion`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Primitive_notion]`_`f.

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.`:cite-ref-tiles-1-0[`F5bf`_`[1`#cite-note-tiles-1]`_`f]

A Propositional Function, or a predicate, in a variable `*x`* is an `F33f`_`[open formula`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Open_formula]`_`f `*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 `F33f`_`[Clarence Lewis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Clarence_Lewis]`_`f, "A `F33f`_`[proposition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Proposition]`_`f 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 `F33f`_`[discourse domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Domain_of_discourse]`_`f of individuals."`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f] Lewis used the notion of propositional functions to introduce `F33f`_`[relations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Relation_(mathematics)]`_`f, for example, a propositional function of `*n`* variables is a relation of `F33f`_`[arity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arity]`_`f `*n`*. The case of `*n`* = 2 corresponds to `F33f`_`[binary relations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Binary_relation]`_`f, of which there are `F33f`_`[homogeneous relations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Homogeneous_relation]`_`f (both variables from the same set) and `F33f`_`[heterogeneous relations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heterogeneous_relation]`_`f.

>>See also

• `F33f`_`[Propositional formula`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Propositional_formula]`_`f

• `F33f`_`[Boolean-valued function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boolean-valued_function]`_`f

• `F33f`_`[Formula (logic)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Formula_(logic)]`_`f

• `F33f`_`[Sentence (logic)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sentence_(logic)]`_`f

• `F33f`_`[Truth function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Truth_function]`_`f

• `F33f`_`[Open sentence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Open_sentence]`_`f

>>References

`:cite-note-tiles-1`!1.`! `F0af`_`[↑`#cite-ref-tiles-1-0]`_`f `:citereftiles2004`a`F33f`_`[Tiles, Mary`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mary_Tiles]`_`f (2004). `*The philosophy of set theory an historical introduction to Cantor's paradise`* (Dover ed.). Mineola, N.Y.: Dover Publications. p. 159. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-486-43520-6. Retrieved 1 February 2013.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `F33f`_`[Clarence Lewis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Clarence_Lewis]`_`f (1918) `*A Survey of Symbolic Logic`*, page 232, `F33f`_`[University of California Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=University_of_California_Press]`_`f, second edition 1932, Dover edition 1960

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