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<title>Metalanguage</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Metalanguage</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Metalanguage_(disambiguation)" class="mw-disambig" title="Metalanguage (disambiguation)">Metalanguage (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Metalinguistics" title="Metalinguistics">metalinguistics</a>.</div>

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<p>In <a href="Logic" title="Logic">logic</a> and <a href="Linguistics" title="Linguistics">linguistics</a>, a <b>metalanguage</b> is a language used to describe another language, often called the <i>object language</i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, <a href="Quotation_mark" title="Quotation mark">quotation marks</a>, or writing on a separate line. The structure of sentences and phrases in a metalanguage can be described by a <a href="Metasyntax" title="Metasyntax">metasyntax</a>.<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> For example, to say that the word "noun" can be used as a noun in a sentence, one could write <i>"noun" is a <noun></i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_metalanguage">Types of metalanguage</h2></div>

<p>There are a variety of recognized types of metalanguage, including <i>embedded</i>, <i>ordered</i>, and <i>nested</i> (or <i>hierarchical</i>) metalanguages.

</p>

<div class="mw-heading mw-heading3"><h3 id="Embedded"> Embedded</h3></div>

<p>An <i>embedded metalanguage</i> is a language formally, naturally and firmly fixed in an object language. This idea is found in <a href="Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a>'s book, <i><a href="G%C3%B6del%2C_Escher%2C_Bach" title="Gödel, Escher, Bach">Gödel, Escher, Bach</a></i>, in a discussion of the relationship between formal languages and <a href="Number_theory" title="Number theory">number theory</a>: "... it is in the nature of any formalization of number theory that its metalanguage is embedded within it."<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>It occurs in natural, or informal, languages, as well—such as in English, where words such as <i>noun</i>, <i>verb</i>, or even <i>word</i> describe features and concepts pertaining to the English language itself.

</p>

<div class="mw-heading mw-heading3"><h3 id="Ordered">Ordered</h3></div>

<p>An <i>ordered metalanguage</i> is analogous to an <a href="Ordered_logic" class="mw-redirect" title="Ordered logic">ordered logic</a>. An example of an ordered metalanguage is the construction of one metalanguage to discuss an object language, followed by the creation of another metalanguage to discuss the first, etc.

</p>

<div class="mw-heading mw-heading3"><h3 id="Nested">Nested</h3></div>

<p>A <i>nested</i> (or <i>hierarchical</i>) <i>metalanguage</i> is similar to an ordered metalanguage in that each level represents a greater degree of abstraction. However, a nested metalanguage differs from an ordered one in that each level includes the one below.

</p><p>The <a href="Paradigmatic" class="mw-redirect" title="Paradigmatic">paradigmatic</a> example of a nested metalanguage comes from the <a href="Scientific_classification" class="mw-redirect" title="Scientific classification">Linnean taxonomic system</a> in biology. Each level in the system incorporates the one below it. The language used to discuss genus is also used to discuss species; the one used to discuss orders is also used to discuss genera, etc., up to kingdoms.

</p>

<div class="mw-heading mw-heading2"><h2 id="In_natural_language">In natural language</h2></div>

<p>Natural language combines nested and ordered metalanguages. In a natural language there is an infinite regress of metalanguages, each with more specialized vocabulary and simpler syntax.

</p><p>Designating the language now as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

</math></span><img src="./db742b8c210fc611329a4c2dcc3af4b4e1a110cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{0}}" loading="lazy"></span>, the grammar of the language is a discourse in the metalanguage <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation>

</semantics>

</math></span><img src="./0e79dc1b001f8b923df475ed14de023cbc456013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" loading="lazy"></span>, which is a sublanguage<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> nested within <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

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<ul><li>The grammar of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation>

</semantics>

</math></span><img src="./0e79dc1b001f8b923df475ed14de023cbc456013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" loading="lazy"></span>, which has the form of a factual description, is a discourse in the meta–metalanguage <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation>

</semantics>

</math></span><img src="./c6a952cfe42c86b7741f55a817da0e251793a358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}" loading="lazy"></span>, which is also a sublanguage of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

<li>The grammar of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation>

</semantics>

</math></span><img src="./c6a952cfe42c86b7741f55a817da0e251793a358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}" loading="lazy"></span>, which has the form of a theory describing the syntactic structure of such factual descriptions, is stated in the meta–meta–metalanguage <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{3}}</annotation>

</semantics>

</math></span><img src="./23c881465e19ffb2fc3c562245e73dc1144cd830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{3}}" loading="lazy"></span>, which likewise is a sublanguage of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{3}}</annotation>

</semantics>

</math></span><img src="./23c881465e19ffb2fc3c562245e73dc1144cd830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{3}}" loading="lazy"></span> has the form of a metatheory describing the syntactic structure of theories stated in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation>

</semantics>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{4}}</annotation>

</semantics>

</math></span><img src="./633d409da9ca3fdc946caf2c0b8ddc932c0870df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{4}}" loading="lazy"></span> and succeeding metalanguages have the same grammar as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{3}}</annotation>

</semantics>

<p>Since all of these metalanguages are sublanguages of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

</math></span><img src="./db742b8c210fc611329a4c2dcc3af4b4e1a110cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{0}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation>

</semantics>

</math></span><img src="./0e79dc1b001f8b923df475ed14de023cbc456013.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}" loading="lazy"></span> is a nested metalanguage, but <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation>

</semantics>

</math></span><img src="./c6a952cfe42c86b7741f55a817da0e251793a358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}" loading="lazy"></span> and sequel are ordered metalanguages.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Since all these metalanguages are sublanguages of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L_{0}}</annotation>

</semantics>

</p><p>Metalanguages of formal systems all resolve ultimately to natural language, the 'common parlance' in which mathematicians and logicians converse to define their terms and operations and 'read out' their formulae.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_expressions">Types of expressions</h2></div>

<p>There are several entities commonly expressed in a metalanguage. In logic usually the object language that the metalanguage is discussing is a <a href="Formal_language" title="Formal language">formal language</a>, and very often the metalanguage as well.

</p>

<div class="mw-heading mw-heading3"><h3 id="Deductive_systems">Deductive systems</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Deductive_system" class="mw-redirect" title="Deductive system">Deductive system</a></div>

<p>A <i>deductive system</i> (or, <i>deductive apparatus</i> of a <a href="Formal_system" title="Formal system">formal system</a>) consists of the <a href="Axiom" title="Axiom">axioms</a> (or <a href="Axiom_schema" title="Axiom schema">axiom schemata</a>) and <a href="Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a> that can be used to <a href="Formal_proof" title="Formal proof">derive</a> the <a href="Theorem" title="Theorem">theorems</a> of the system.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Metavariables">Metavariables</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Metavariable_(logic)" class="mw-redirect" title="Metavariable (logic)">Metavariable (logic)</a></div>

<p>A <i>metavariable</i> (or <i>metalinguistic</i> or <i>metasyntactic</i> variable) is a <a href="Symbol_(formal)" title="Symbol (formal)">symbol</a> or set of symbols in a metalanguage which stands for a symbol or set of symbols in some object language. For instance, in the sentence:

</p>

<dl><dd>Let <i>A</i> and <i>B</i> be arbitrary <a href="Well-formed_formula" title="Well-formed formula">formulas</a> of a <a href="Formal_language" title="Formal language">formal language</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

<p>The symbols <i>A</i> and <i>B</i> are not symbols of the object language <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span>, they are metavariables in the metalanguage (in this case, English) that is discussing the object language <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</p>

<div class="mw-heading mw-heading3"><h3 id="Metatheories_and_metatheorems">Metatheories and metatheorems</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Metatheory" title="Metatheory">Metatheory</a> and <a href="Metatheorem" title="Metatheorem">Metatheorem</a></div>

<p>A <i>metatheory</i> is a <a href="Theory" title="Theory">theory</a> whose subject matter is some other theory (a theory about a theory). <a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">Statements</a> made in the metatheory about the theory are called <a href="Metatheorem" title="Metatheorem">metatheorems</a>. A <i>metatheorem</i> is a <a href="Truth" title="Truth">true</a> statement about a <a href="Formal_system" title="Formal system">formal system</a> expressed in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a <a href="Metatheory" title="Metatheory">metatheory</a>, and may reference concepts that are present in the <a href="Metatheory" title="Metatheory">metatheory</a> but not the object theory.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Interpretations">Interpretations</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation (logic)</a></div>

<p>An <i>interpretation</i> is an <a href="Valuation_(logic)" title="Valuation (logic)">assignment</a> of meanings to the <a href="Symbol_(formal)" title="Symbol (formal)">symbols</a> and <a href="Word" title="Word">words</a> of a language.

</p>

<div class="mw-heading mw-heading2"><h2 id="Role_in_metaphor">Role in metaphor</h2></div>

<p>Michael J. Reddy (1979) argues that much of the language we use to talk about language is conceptualized and structured by what he refers to as the <a href="Conduit_metaphor" title="Conduit metaphor">conduit metaphor</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This paradigm operates through two distinct, related frameworks.

</p><p>The <i>major framework</i> views language as a sealed pipeline between people:

</p>

<caption>Major framework

</caption>

<th>Stage</th>

<th>Description</th>

<th>Example

</th></tr>

<tr>

<td>Language transfers people's thoughts and feelings (<a href="Mental_content" class="mw-redirect" title="Mental content">mental content</a>) to others</td>

<td>Try to get your thoughts across better

</td></tr>

<tr>

<td>Speakers and writers insert their mental content into words</td>

<td>You have to put each concept into words more carefully

</td></tr>

<tr>

<td>Words are containers</td>

<td>That sentence was filled with emotion

</td></tr>

<tr>

<td>Listeners and readers extract mental content from words</td>

<td>Let me know if you find any new sensations in the poem

</td></tr></tbody></table>

<p>The <i>minor framework</i> views language as an open pipe spilling mental content into the void:

</p>

<caption>Minor framework

</caption>

<th>Stage</th>

<th>Description</th>

<th>Example

</th></tr>

<tr>

<td>Speakers and writers eject mental content into an external space</td>

<td>Get those ideas out where they can do some good

</td></tr>

<tr>

<td>Mental content is reified (viewed as concrete) in this space</td>

<td>That concept has been floating around for decades

</td></tr>

<tr>

<td>Listeners and readers extract mental content from this space</td>

<td>Let me know if you find any good concepts in the essay

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Metaprogramming">Metaprogramming</h2></div>

<p>Computers follow programs, sets of instructions in a formal language. The development of a <a href="Programming_language" title="Programming language">programming language</a> involves the use of a metalanguage. The act of working with metalanguages in programming is known as <i><a href="Metaprogramming" title="Metaprogramming">metaprogramming</a></i>.

</p><p><a href="Backus%E2%80%93Naur_form" title="Backus–Naur form">Backus–Naur form</a>, developed in the 1960s by John Backus and Peter Naur, is one of the earliest metalanguages used in computing. Examples of modern-day programming languages which commonly find use in metaprogramming include <a href="ML_(programming_language)" title="ML (programming language)">ML</a>, <a href="Lisp_(programming_language)" title="Lisp (programming language)">Lisp</a>, <a href="M4_(computer_language)" title="M4 (computer language)">m4</a>, and <a href="Yacc" title="Yacc">Yacc</a>.

</p>

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<ul><li><a href="Category_theory" title="Category theory">Category theory</a> – General theory of mathematical structures</li>

<li><a href="Formal_language" title="Formal language">Formal language</a> – Sequence of words formed by specific rules</li>

<li><a href="Jakobson's_functions_of_language" title="Jakobson's functions of language">Jakobson's functions of language</a> – Theory of language</li>

<li><a href="Language-oriented_programming" title="Language-oriented programming">Language-oriented programming</a></li>

<li><a href="Meta-communication" title="Meta-communication">Meta-communication</a> – Communication about how information is meant to be interpreted</li>

<li><a href="Metaethics" title="Metaethics">Metaethics</a> – Branch of ethics seeking to understand ethical properties</li>

<li><a href="Metafiction" title="Metafiction">Metafiction</a> – Genre of fiction about fiction</li>

<li><a href="Metagraphy" class="mw-redirect" title="Metagraphy">Metagraphy</a> – Experimental form of visual communication<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Metamathematics" title="Metamathematics">Metamathematics</a> – Study of mathematics itself</li>

<li><a href="Metalinguistic_abstraction" title="Metalinguistic abstraction">Metalinguistic abstraction</a> – Principle in computer science of domain-specific languages for problem solving</li>

<li><a href="Metalinguistic_awareness" title="Metalinguistic awareness">Metalinguistic awareness</a> – Ability to consciously reflect on the nature of language</li>

<li><a href="Metalocutionary_act" title="Metalocutionary act">Metalocutionary act</a> – Concept in linguistic pragmatics</li>

<li><a href="Metaphilosophy" title="Metaphilosophy">Metaphilosophy</a> – Investigation of the nature of philosophy</li>

<li><a href="Natural_semantic_metalanguage" title="Natural semantic metalanguage">Natural semantic metalanguage</a> – Linguistic theory of semantic description</li>

<li><a href="Nested_quotation" title="Nested quotation">Nested quotation</a></li>

<li><a href="Paralanguage" title="Paralanguage">Paralanguage</a> – Communication of additional meaning, nuance, or emotion in speech</li>

<li><a href="Self-reference" title="Self-reference">Self-reference</a> – Sentence, idea or formula that refers to itself</li>

<li><a href="Use%E2%80%93mention_distinction" title="Use–mention distinction">Use–mention distinction</a> – Distinction between using a word and mentioning it</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Dictionaries">Dictionaries</h2></div>

<ul><li>Audi, R. 1996. <i>The Cambridge Dictionary of Philosophy</i>. Cambridge: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</li>

<li>Baldick, C. 1996. <i>Oxford Concise Dictionary of Literary Terms</i>. Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</li>

<li><a href="J._A._Cuddon" title="J. A. Cuddon">Cuddon, J. A.</a> 1999. <i>The Penguin Dictionary of Literary Terms and Literary Theory</i>. London: <a href="Penguin_Books" title="Penguin Books">Penguin Books</a>.</li>

<li>Honderich, T. 1995. <i><a href="The_Oxford_Companion_to_Philosophy" title="The Oxford Companion to Philosophy">The Oxford Companion to Philosophy</a></i>. Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</li>

<li>Matthews, P. H. 1997. <i>The Concise Oxford Dictionary of Linguistics</i>. Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-280008-4</bdi>.</li>

<li>McArthur, T. 1996. <i>The Concise Oxford Companion to the English Language</i>. Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">2010. <i>Cambridge Advanced Learner's Dictionary</i>. Cambridge: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. Dictionary online. Available from <a rel="nofollow" class="external free" href="http://dictionary.cambridge.org/dictionary/british/metalanguage">http://dictionary.cambridge.org/dictionary/british/metalanguage</a> Internet. Retrieved 20 November 2010</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">van Wijngaarden, A., et al. "<a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-642-95279-1_2">Language and metalanguage</a>." Revised Report on the Algorithmic Language Algol 68. Springer, Berlin, Heidelberg, 1976. 17-35.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="Douglas_Hofstadter" title="Douglas Hofstadter">Hofstadter, Douglas</a>. 1980. <a href="G%C3%B6del%2C_Escher%2C_Bach" title="Gödel, Escher, Bach"><i>Gödel, Escher, Bach: An Eternal Golden Braid</i></a>. New York: Vintage Books <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-14-017997-6</bdi></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFHarris1991" class="citation book cs1"><a href="Zellig_Harris" title="Zellig Harris">Harris, Zellig S.</a> (1991). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryoflanguage00harr"><i>A theory of language and information: A mathematical approach</i></a></span>. Oxford: Clarendon Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryoflanguage00harr/page/272">272</a>–318. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-824224-6</bdi>.</cite></span>

</li>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFBorel1928" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="%C3%89mile_Borel" title="Émile Borel">Borel, Félix Édouard Justin Émile</a> (1928). <i>Leçons sur la theorie des fonctions</i> (in French) (3 ed.). Paris: Gauthier-Villars & Cie. p. 160.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFHunter1996" class="citation book cs1"><a href="Geoffrey_Hunter_(logician)" title="Geoffrey Hunter (logician)">Hunter, Geoffrey</a> (1996) [1971]. <i>Metalogic: An Introduction to the Metatheory of Standard First-Order Logic</i>. University of California Press (published 1973). p. 7. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780520023567</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36312727">36312727</a>.</cite> (<a rel="nofollow" class="external text" href="https://archive.org/details/metalogicintrodu0000hunt">accessible to patrons with print disabilities</a>)</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="George_Ritzer" title="George Ritzer">Ritzer, George</a>. 1991. <i>Metatheorizing in Sociology</i>. New York: Simon Schuster <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-669-25008-2</bdi></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Reddy, Michael J. 1979. The conduit metaphor: A case of frame conflict in our language about language. In Andrew Ortony (ed.), <i>Metaphor and Thought</i>. Cambridge: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://pespmc1.vub.ac.be/METALARE.html">Metalanguage</a>, <i><a href="Principia_Cybernetica" title="Principia Cybernetica">Principia Cybernetica</a></i>.</li>

<li><a rel="nofollow" class="external text" href="http://lists.village.virginia.edu/lists_archive/Humanist/v20/0091.html">Willard McCarty (submitted 2006) Problematic Metaphors</a>, <i>Humanist Discussion Group</i>, Vol. 20, No. 92.</li></ul>

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</style><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom" title="Axiom">Axiom</a>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="First-order_logic" title="First-order logic">First-order logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li>

<li><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)<br> and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>

<li><a href="Tarski's_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li>

<li><a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li>

<li>Cantor's <a href="Cantor's_theorem" title="Cantor's theorem">theorem,</a> <a href="Cantor's_paradox" title="Cantor's paradox">paradox</a> and <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li>

<li><a href="Compactness_theorem" title="Compactness theorem">Compactness</a></li>

<li><a href="Halting_problem" title="Halting problem">Halting problem</a></li>

<li><a href="Lindstr%C3%B6m's_theorem" title="Lindström's theorem">Lindström's</a></li>

<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li>

<li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional95" scope="row" class="navbox-group" style="width:1%"><a href="Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical logic</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li>

<li><a href="Consistency" title="Consistency">Consistency</a>

<ul><li><a href="Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li>

<li><a href="Argument" title="Argument">Argument</a></li>

<li><a href="Soundness" title="Soundness">Soundness</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li>

<li><a href="Syllogism" title="Syllogism">Syllogism</a></li>

<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</a></li>

<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>

<li><a href="Boolean_function" title="Boolean function">Boolean functions</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connectives</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>

<li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>

<li><a href="Truth_table" title="Truth table">Truth tables</a></li>

<li><a href="Many-valued_logic" title="Many-valued logic">Many-valued logic</a>

<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="First-order_logic" title="First-order logic">First-order</a>

<li><a href="Second-order_logic" title="Second-order logic">Second-order</a>

<ul><li><a href="Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li>

<li><a href="Higher-order_logic" title="Higher-order logic">Higher-order</a></li>

<li><a href="Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>

<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li>(<a href="Urelement" title="Urelement">Ur-</a>)<a href="Element_(mathematics)" title="Element (mathematics)">Element</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Extensionality" title="Extensionality">Extensionality</a></li>

<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>

<li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a>

<ul><li><a href="Equivalence_relation" title="Equivalence relation">equivalence</a></li>

<li><a href="Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li>

<li>Set operations:

<ul><li><a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">union</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></li>

<li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Power_set" title="Power set">power set</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Inhabited_set" title="Inhabited set">Inhabited</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li>

<li><a href="Universe_(mathematics)" title="Universe (mathematics)">Universe</a>

<ul><li><a href="Constructible_universe" title="Constructible universe">constructible</a></li>

<li><a href="Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li>

<li><a href="Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="Map_(mathematics)" title="Map (mathematics)">Map</a>

<ul><li><a href="Domain_of_a_function" title="Domain of a function">domain</a></li>

<li><a href="Codomain" title="Codomain">codomain</a></li>

<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li>

<li><a href="Injective_function" title="Injective function">In</a>/<a href="Surjective_function" title="Surjective function">Sur</a>/<a href="Bijection" title="Bijection">Bi</a>-jection</li>

<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li>

<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>

<li><a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li>

<li><a href="Enumeration" title="Enumeration">Enumeration</a></li>

<li><a href="Large_cardinal" title="Large cardinal">Large cardinal</a>

<ul><li><a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li>

<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li>

<li><a href="Operation_(mathematics)" title="Operation (mathematics)">Operation</a>

<ul><li><a href="Binary_operation" title="Binary operation">binary</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a>

<ul><li><a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li>

<li><a href="General_set_theory" title="General set theory">General</a></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="New_Foundations" title="New Foundations">New Foundations</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li>

<li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li>

<li><a href="Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li>

<li><a href="Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Formal_system" title="Formal system">Formal systems</a> (<a href="List_of_formal_systems" title="List of formal systems"><span style="font-size: 85%;">list</span></a>),<br><a href="Formal_language" title="Formal language">language</a> and <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li>

<li><a href="Arity" title="Arity">Arity</a></li>

<li><a href="Automata_theory" title="Automata theory">Automata</a></li>

<li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a></li>

<li><a href="Expression_(mathematics)" title="Expression (mathematics)">Expression</a>

<ul><li><a href="Ground_expression" title="Ground expression">ground</a></li></ul></li>

<li><a href="Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a>

<ul><li><a href="Extension_by_definitions" class="mw-redirect" title="Extension by definitions">by definition</a></li>

<li><a href="Conservative_extension" title="Conservative extension">conservative</a></li></ul></li>

<li><a href="Finitary_relation" title="Finitary relation">Relation</a></li>

<li><a href="Formation_rule" title="Formation rule">Formation rule</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Grammar</a></li>

<li><a href="Well-formed_formula" title="Well-formed formula">Formula</a>

<ul><li><a href="Atomic_formula" title="Atomic formula">atomic</a></li>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li>

<li><a href="Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li>

<li><a href="Formal_language" title="Formal language">Language</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connective</a>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a>

<ul><li><a href="Functional_predicate" title="Functional predicate">functional</a></li>

<li><a href="Predicate_variable" title="Predicate variable">variable</a></li>

<li><a href="Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a>

<ul><li><a href="Atomic_sentence" title="Atomic sentence">atomic</a></li>

<li><a href="Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li>

<li><a href="Signature_(logic)" title="Signature (logic)">Signature</a></li>

<li><a href="String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Symbol_(formal)" title="Symbol (formal)">Symbol</a>

<ul><li><a href="Uninterpreted_function" title="Uninterpreted function">function</a></li>

<li><a href="Logical_constant" title="Logical constant">logical/constant</a></li>

<li><a href="Non-logical_symbol" title="Non-logical symbol">non-logical</a></li>

<li><a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="Axiomatic_system" title="Axiomatic system">axiomatic<br>systems</a> <span style="font-size: 85%;">(<a href="List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>of <a href="True_arithmetic" title="True arithmetic">arithmetic</a>:

<ul><li><a href="Peano_axioms" title="Peano axioms">Peano</a></li>

<li><a href="Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li>

<li><a href="Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li>

<li><a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li>

<li><a href="Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li>

<li><a href="Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li>

<li>of the <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a>

<ul><li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li>

<li>of <a href="Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a>

<ul><li><a href="Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li>

<li><a href="Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li>

<li>of <a href="Foundations_of_geometry" title="Foundations of geometry">geometry</a>:

<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>:

<ul><li><a href="Euclid's_Elements" title="Euclid's Elements"><i>Elements</i></a></li>

<li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's</a></li>

<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li>

<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul>

<ul><li><i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proof_theory" title="Proof theory">Proof theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Natural_deduction" title="Natural deduction">Natural deduction</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Rule_of_inference" title="Rule of inference">Rule of inference</a></li>

<li><a href="Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Formal_system" title="Formal system">Systems</a>

<ul><li><a href="Axiomatic_system" title="Axiomatic system">axiomatic</a></li>

<li><a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li>

<li><a href="Hilbert_system" title="Hilbert system">Hilbert</a>

<li><a href="Complete_theory" title="Complete theory">Complete theory</a></li>

<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li>

<li><a href="Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li>

<li><a href="Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li>

<li><a href="Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li>

<li><a href="Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Model_theory" title="Model theory">Model theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a>

<ul><li><a href="Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li>

<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a>

<ul><li><a href="Elementary_equivalence" title="Elementary equivalence">equivalence</a></li>

<li><a href="Finite_model_theory" title="Finite model theory">finite</a></li>

<li><a href="Saturated_model" title="Saturated model">saturated</a></li>

<li><a href="Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li>

<li><a href="Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li>

<li><a href="Non-standard_model" title="Non-standard model">Non-standard model</a>

<ul><li><a href="Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li>

<li><a href="Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a>

<ul><li><a href="Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li>

<li><a href="Categorical_theory" title="Categorical theory">Categorical theory</a></li>

<li><a href="Model_complete_theory" title="Model complete theory">Model complete theory</a></li>

<li><a href="Satisfiability" title="Satisfiability">Satisfiability</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li>

<li><a href="Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a>

<ul><li><a href="Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li>

<li><a href="Tarski's_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li>

<li><a href="Kripke's_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li>

<li><a href="T-schema" title="T-schema">T-schema</a></li>

<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>

<li><a href="Truth_predicate" title="Truth predicate">Truth predicate</a></li>

<li><a href="Truth_value" title="Truth value">Truth value</a></li>

<li><a href="Ultraproduct" title="Ultraproduct">Ultraproduct</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Church_encoding" title="Church encoding">Church encoding</a></li>

<li><a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li>

<li><a href="Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li>

<li><a href="Computable_function" title="Computable function">Computable function</a></li>

<li><a href="Computable_set" title="Computable set">Computable set</a></li>

<li><a href="Decision_problem" title="Decision problem">Decision problem</a>

<ul><li><a href="Decidability_(logic)" title="Decidability (logic)">decidable</a></li>

<li><a href="Undecidable_problem" title="Undecidable problem">undecidable</a></li>

<li><a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li>

<li><a href="Recursion" title="Recursion">Recursion</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li>

<li><a href="Turing_machine" title="Turing machine">Turing machine</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_logic" title="Abstract logic">Abstract logic</a></li>

<li><a href="Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

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<li><a href="Category_of_sets" title="Category of sets">Category of sets</a></li>

<li><a href="History_of_logic" title="History of logic">History of logic</a></li>

<li><a href="History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a>

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<li><a href="Logicism" title="Logicism">Logicism</a></li>

<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>

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