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<title>Propositional function</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Propositional function</span></span>
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<p>In <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional calculus</a>, a <b>propositional function</b> or a <b>predicate</b> is a sentence expressed in a way that would assume the value of <a href="Logical_truth" title="Logical truth">true</a> or <a href="False_(logic)" title="False (logic)">false</a>, except that within the sentence there is a <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a> (<i>x</i>) that is not defined or specified (thus being a <a href="Free_variable" class="mw-redirect" title="Free variable">free variable</a>), which leaves the statement undetermined. The sentence may contain several such variables (e.g. <i>n</i> variables, in which case the function takes <i>n</i> arguments).
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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>
<p>As a <a href="Function_(mathematics)" title="Function (mathematics)">mathematical function</a>, <i>A</i>(<i>x</i>) or <i>A</i>(<i>x</i><sub>1</sub>, <i>x</i><sub>2</sub>, ..., <i>x</i><sub><i>n</i></sub>), the propositional function is abstracted from <a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">predicates</a> or propositional forms. As an example, consider the predicate scheme, "x is hot". The substitution of any entity for <i>x</i> will produce a specific proposition that can be described as either true or false, even though "<i>x</i> is hot" on its own has no value as either a true or false statement. However, when a value is assigned to <i>x</i>, such as <a href="Lava" title="Lava">lava</a>, the function then has the value <i>true</i>; while one assigns to <i>x</i> a value like <a href="Ice" title="Ice">ice</a>, the function then has the value <i>false</i>.
</p><p>Propositional functions are useful in <a href="Set_theory" title="Set theory">set theory</a> for the formation of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a>. For example, in 1903 <a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> wrote in <i><a href="The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></i> (page 106):
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<dl><dd>"...it has become necessary to take <i>propositional function</i> as a <a href="Primitive_notion" title="Primitive notion">primitive notion</a>.</dd></dl>
<p>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.<sup id="cite_ref-Tiles_1-0" class="reference"><a href="#cite_note-Tiles-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>A Propositional Function, or a predicate, in a variable <i>x</i> is an <a href="Open_formula" title="Open formula">open formula</a> <i>p</i>(<i>x</i>) involving <i>x</i> that becomes a proposition when one gives <i>x</i> a definite value from the set of values it can take.
</p><p>According to <a href="Clarence_Lewis" class="mw-redirect" title="Clarence Lewis">Clarence Lewis</a>, "A <a href="Proposition" title="Proposition">proposition</a> 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 <a href="Domain_of_discourse" title="Domain of discourse">discourse domain</a> of individuals."<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Lewis used the notion of propositional functions to introduce <a href="Relation_(mathematics)" title="Relation (mathematics)">relations</a>, for example, a propositional function of <i>n</i> variables is a relation of <a href="Arity" title="Arity">arity</a> <i>n</i>. The case of <i>n</i> = 2 corresponds to <a href="Binary_relation" title="Binary relation">binary relations</a>, of which there are <a href="Homogeneous_relation" title="Homogeneous relation">homogeneous relations</a> (both variables from the same set) and <a href="Heterogeneous_relation" class="mw-redirect" title="Heterogeneous relation">heterogeneous relations</a>.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>
<li><a href="Boolean-valued_function" title="Boolean-valued function">Boolean-valued function</a></li>
<li><a href="Formula_(logic)" class="mw-redirect" title="Formula (logic)">Formula (logic)</a></li>
<li><a href="Sentence_(logic)" class="mw-redirect" title="Sentence (logic)">Sentence (logic)</a></li>
<li><a href="Truth_function" title="Truth function">Truth function</a></li>
<li><a href="Open_sentence" class="mw-redirect" title="Open sentence">Open sentence</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFTiles2004" class="citation book cs1"><a href="Mary_Tiles" title="Mary Tiles">Tiles, Mary</a> (2004). <a rel="nofollow" class="external text" href="http://store.doverpublications.com/0486435202.html"><i>The philosophy of set theory an historical introduction to Cantor's paradise</i></a> (Dover ed.). Mineola, N.Y.: Dover Publications. p. 159. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43520-6</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">1 February</span> 2013</span>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="Clarence_Lewis" class="mw-redirect" title="Clarence Lewis">Clarence Lewis</a> (1918) <i>A Survey of Symbolic Logic</i>, page 232, <a href="University_of_California_Press" title="University of California Press">University of California Press</a>, second edition 1932, Dover edition 1960</span>
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