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properties of truth-functional interpretations</span> </div> </a> <button aria-controls="toc-General_properties_of_truth-functional_interpretations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle General properties of truth-functional interpretations subsection</span> </button> <ul id="toc-General_properties_of_truth-functional_interpretations-sublist" class="vector-toc-list"> <li id="toc-Logical_connectives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logical_connectives"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Logical connectives</span> </div> </a> <ul id="toc-Logical_connectives-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interpretation_of_a_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interpretation_of_a_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Interpretation of a theory</span> </div> </a> <ul id="toc-Interpretation_of_a_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretations_for_propositional_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interpretations_for_propositional_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Interpretations for propositional logic</span> </div> </a> <ul id="toc-Interpretations_for_propositional_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First-order_logic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#First-order_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>First-order logic</span> </div> </a> <button aria-controls="toc-First-order_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle First-order logic subsection</span> </button> <ul id="toc-First-order_logic-sublist" class="vector-toc-list"> <li id="toc-Formal_languages_for_first-order_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_languages_for_first-order_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Formal languages for first-order logic</span> </div> </a> <ul id="toc-Formal_languages_for_first-order_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretations_of_a_first-order_language" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretations_of_a_first-order_language"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Interpretations of a first-order language</span> </div> </a> <ul id="toc-Interpretations_of_a_first-order_language-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_of_a_first-order_interpretation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_of_a_first-order_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Example of a first-order interpretation</span> </div> </a> <ul id="toc-Example_of_a_first-order_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-empty_domain_requirement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-empty_domain_requirement"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Non-empty domain requirement</span> </div> </a> <ul id="toc-Non-empty_domain_requirement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpreting_equality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpreting_equality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Interpreting equality</span> </div> </a> <ul id="toc-Interpreting_equality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Many-sorted_first-order_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Many-sorted_first-order_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Many-sorted first-order logic</span> </div> </a> <ul id="toc-Many-sorted_first-order_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Higher-order_predicate_logics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-order_predicate_logics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Higher-order predicate logics</span> </div> </a> <ul id="toc-Higher-order_predicate_logics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-classical_interpretations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Non-classical_interpretations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Non-classical interpretations</span> </div> </a> <ul id="toc-Non-classical_interpretations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Intended_interpretations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Intended_interpretations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Intended interpretations</span> </div> </a> <button aria-controls="toc-Intended_interpretations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Intended interpretations subsection</span> </button> <ul id="toc-Intended_interpretations-sublist" class="vector-toc-list"> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Example</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_concepts_of_interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_concepts_of_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Other concepts of interpretation</span> </div> </a> <ul id="toc-Other_concepts_of_interpretation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Interpretaci%C3%B3_(l%C3%B2gica)" title="Interpretació (lògica) – Catalan" lang="ca" hreflang="ca" data-title="Interpretació (lògica)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Interpretation_(Logik)" title="Interpretation (Logik) – German" lang="de" hreflang="de" data-title="Interpretation (Logik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Interpretaci%C3%B3n_(l%C3%B3gica)" title="Interpretación (lógica) – Spanish" lang="es" hreflang="es" data-title="Interpretación (lógica)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Interpr%C3%A9tation_(logique)" title="Interprétation (logique) – French" lang="fr" hreflang="fr" data-title="Interprétation (logique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%9A%E0%A4%A8_(%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95)" title="निर्वचन (तर्क) – Hindi" lang="hi" hreflang="hi" data-title="निर्वचन (तर्क)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Inhlakahlo" title="Inhlakahlo – Zulu" lang="zu" hreflang="zu" data-title="Inhlakahlo" data-language-autonym="IsiZulu" data-language-local-name="Zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Interpretazione_(logica)" title="Interpretazione (logica) – Italian" lang="it" hreflang="it" data-title="Interpretazione (logica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/%C3%89rtelmez%C3%A9s_(logika)" title="Értelmezés (logika) – Hungarian" lang="hu" hreflang="hu" data-title="Értelmezés (logika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Interpretatie_(logica)" title="Interpretatie (logica) – Dutch" lang="nl" hreflang="nl" data-title="Interpretatie (logica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Interpreta%C3%A7%C3%A3o_(l%C3%B3gica)" title="Interpretação (lógica) – Portuguese" lang="pt" hreflang="pt" data-title="Interpretação (lógica)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%DA%A4%DB%95_(%D9%84%DB%86%DA%98%DB%8C%DA%A9)" title="ڕاڤە (لۆژیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ڕاڤە (لۆژیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tulkinta_(logiikka)" title="Tulkinta (logiikka) – Finnish" lang="fi" hreflang="fi" data-title="Tulkinta (logiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D0%B5%D1%80%D0%BF%D1%80%D0%B5%D1%82%D0%B0%D1%86%D1%96%D1%8F_(%D0%BB%D0%BE%D0%B3%D1%96%D0%BA%D0%B0)" title="Інтерпретація (логіка) – Ukrainian" lang="uk" hreflang="uk" data-title="Інтерпретація (логіка)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A9%AE%E9%87%8B" title="詮釋 – Cantonese" lang="yue" hreflang="yue" data-title="詮釋" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%A7%A3%E9%87%8B_(%E9%82%8F%E8%BC%AF)" title="解釋 (邏輯) – Chinese" lang="zh" hreflang="zh" data-title="解釋 (邏輯)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q523607#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Assignment of meaning to the symbols of a formal language</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Interpretation_(disambiguation)" class="mw-redirect mw-disambig" title="Interpretation (disambiguation)">Interpretation (disambiguation)</a>.</div> <p>An <b>interpretation</b> is an assignment of <a href="/wiki/Meaning_(philosophy)" title="Meaning (philosophy)">meaning</a> to the <a href="/wiki/Symbol_(formal)" title="Symbol (formal)">symbols</a> of a <a href="/wiki/Formal_language" title="Formal language">formal language</a>. Many formal languages used in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Logic" title="Logic">logic</a>, and <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a> are defined in solely <a href="/wiki/Syntax" title="Syntax">syntactic</a> terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called <a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">formal semantics</a>. </p><p>The most commonly studied formal logics are <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>, <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a> and their <a href="/wiki/Modal_logic" title="Modal logic">modal</a> analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that provides the <a href="/wiki/Extension_(predicate_logic)" title="Extension (predicate logic)">extension</a> of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate <i>T</i> (for "tall") and assign it the extension {<i>a</i>} (for "Abraham Lincoln"). All our interpretation does is assign the extension {a} to the non-logical constant <i>T</i>, and does not make a claim about whether <i>T</i> is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though <i>we</i> may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function. </p><p>An interpretation often (but not always) provides a way to determine the <a href="/wiki/Truth_value" title="Truth value">truth values</a> of <a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentences</a> in a language. If a given interpretation assigns the value True to a sentence or <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">theory</a>, the interpretation is called a <a href="/wiki/Model_(model_theory)" class="mw-redirect" title="Model (model theory)">model</a> of that sentence or theory. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_languages">Formal languages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=1" title="Edit section: Formal languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Formal_language" title="Formal language">Formal language</a></div> <p>A formal language consists of a possibly infinite set of <i>sentences</i> (variously called <i>words</i> or <i><a href="/wiki/Well_formed_formula" class="mw-redirect" title="Well formed formula">formulas</a></i>) built from a fixed set of <i>letters</i> or <i>symbols</i>. The inventory from which these letters are taken is called the <i><a href="/wiki/Alphabet_(computer_science)" class="mw-redirect" title="Alphabet (computer science)">alphabet</a></i> over which the language is defined. To distinguish the strings of symbols that are in a formal language from arbitrary strings of symbols, the former are sometimes called <i><a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulæ</a></i> (wff). The essential feature of a formal language is that its syntax can be defined without reference to interpretation. For example, we can determine that (<i>P</i> or <i>Q</i>) is a well-formed formula even without knowing whether it is true or false. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A formal language <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1cc103563219127f59aec7ed9327a3595566dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}"></span> can be defined with the alphabet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =\{\triangle ,\square \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">△</mi> <mo>,</mo> <mi>◻</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\{\triangle ,\square \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675953072fb785b439b306969896942d85eec76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.819ex; height:2.843ex;" alt="{\displaystyle \alpha =\{\triangle ,\square \}}"></span>, and with a word being in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1cc103563219127f59aec7ed9327a3595566dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}"></span> if it begins with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> and is composed solely of the symbols <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455831d58fa08f311b934d324adcff89a868b4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \square }"></span>. </p><p>A possible interpretation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1cc103563219127f59aec7ed9327a3595566dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}"></span> could assign the decimal digit '1' to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d909fe94e8277a4c44a50853cb7dbbf0fa3148ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle \triangle }"></span> and '0' to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455831d58fa08f311b934d324adcff89a868b4e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \square }"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \triangle \square \triangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">△</mi> <mi>◻</mi> <mi mathvariant="normal">△</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \triangle \square \triangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e909b127abaf0e7e5bcd369f34ec89ccfc041d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.94ex; height:2.176ex;" alt="{\displaystyle \triangle \square \triangle }"></span> would denote 101 under this interpretation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {W}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">W</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {W}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1cc103563219127f59aec7ed9327a3595566dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.176ex;" alt="{\displaystyle {\mathcal {W}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Logical_constants">Logical constants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=3" title="Edit section: Logical constants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the specific cases of propositional logic and predicate logic, the formal languages considered have alphabets that are divided into two sets: the logical symbols (<a href="/wiki/Logical_constant" title="Logical constant">logical constants</a>) and the non-logical symbols. The idea behind this terminology is that <i>logical</i> symbols have the same meaning regardless of the subject matter being studied, while <i>non-logical</i> symbols change in meaning depending on the area of investigation. </p><p>Logical constants are always given the same meaning by every interpretation of the standard kind, so that only the meanings of the non-logical symbols are changed. Logical constants include quantifier symbols ∀ ("all") and ∃ ("some"), symbols for <a href="/wiki/Logical_connective" title="Logical connective">logical connectives</a> ∧ ("and"), ∨ ("or"), ¬ ("not"), parentheses and other grouping symbols, and (in many treatments) the equality symbol =. </p> <div class="mw-heading mw-heading2"><h2 id="General_properties_of_truth-functional_interpretations">General properties of truth-functional interpretations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=4" title="Edit section: General properties of truth-functional interpretations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many of the commonly studied interpretations associate each sentence in a formal language with a single truth value, either True or False. These interpretations are called <i>truth functional</i>;<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute"><span title="The article 'Truth-functional' gives a more restricted definition: the truth-value of a compound sentence should be a function of the truth-value of its sub-sentences. (September 2015)">dubious</span></a> – <a href="/wiki/Talk:Interpretation_(logic)#Dubious" title="Talk:Interpretation (logic)">discuss</a></i>]</sup> they include the usual interpretations of propositional and first-order logic. The sentences that are made true by a particular assignment are said to be <i><a href="/wiki/Satisfiable" class="mw-redirect" title="Satisfiable">satisfied</a></i> by that assignment. </p><p>In <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, no sentence can be made both true and false by the same interpretation, although this is not true of glut logics such as LP.<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> Even in classical logic, however, it is possible that the truth value of the same sentence can be different under different interpretations. A sentence is <i><a href="/wiki/Consistency" title="Consistency">consistent</a></i> if it is true under at least one interpretation; otherwise it is <i>inconsistent</i>. A sentence φ is said to be <i>logically valid</i> if it is satisfied by every interpretation (if φ is satisfied by every interpretation that satisfies ψ then φ is said to be a <i><a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a></i> of ψ). </p> <div class="mw-heading mw-heading3"><h3 id="Logical_connectives">Logical connectives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=5" title="Edit section: Logical connectives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of the logical symbols of a language (other than quantifiers) are <a href="/wiki/Logical_connective" title="Logical connective">truth-functional connectives</a> that represent truth functions — functions that take truth values as arguments and return truth values as outputs (in other words, these are operations on truth values of sentences). </p><p>The truth-functional connectives enable compound sentences to be built up from simpler sentences. In this way, the truth value of the compound sentence is defined as a certain truth function of the truth values of the simpler sentences. The connectives are usually taken to be <a href="/wiki/Logical_constant" title="Logical constant">logical constants</a>, meaning that the meaning of the connectives is always the same, independent of what interpretations are given to the other symbols in a formula. </p><p>This is how we define logical connectives in propositional logic: </p> <ul><li>¬Φ is True <a href="/wiki/Iff" class="mw-redirect" title="Iff">iff</a> Φ is False.</li> <li>(Φ ∧ Ψ) is True iff Φ is True and Ψ is True.</li> <li>(Φ ∨ Ψ) is True iff Φ is True or Ψ is True (or both are True).</li> <li>(Φ → Ψ) is True iff ¬Φ is True or Ψ is True (or both are True).</li> <li>(Φ ↔ Ψ) is True iff (Φ → Ψ) is True and (Ψ → Φ) is True.</li></ul> <p>So under a given interpretation of all the sentence letters Φ and Ψ (i.e., after assigning a truth-value to each sentence letter), we can determine the truth-values of all formulas that have them as constituents, as a function of the logical connectives. The following table shows how this kind of thing looks. The first two columns show the truth-values of the sentence letters as determined by the four possible interpretations. The other columns show the truth-values of formulas built from these sentence letters, with truth-values determined recursively. </p> <table class="wikitable" style="text-align:center; margin: 1em auto;"> <caption>Logical connectives </caption> <tbody><tr> <th>Interpretation</th> <th>Φ</th> <th>Ψ</th> <th>¬Φ</th> <th>(Φ ∧ Ψ)</th> <th>(Φ ∨ Ψ)</th> <th>(Φ → Ψ)</th> <th>(Φ ↔ Ψ) </th></tr> <tr> <td>#1</td> <td>T</td> <td>T</td> <td>F</td> <td>T</td> <td>T</td> <td>T</td> <td>T </td></tr> <tr> <td>#2</td> <td>T</td> <td>F</td> <td>F</td> <td>F</td> <td>T</td> <td>F</td> <td>F </td></tr> <tr> <td>#3</td> <td>F</td> <td>T</td> <td>T</td> <td>F</td> <td>T</td> <td>T</td> <td>F </td></tr> <tr> <td>#4</td> <td>F</td> <td>F</td> <td>T</td> <td>F</td> <td>F</td> <td>T</td> <td>T </td></tr></tbody></table> <p>Now it is easier to see what makes a formula logically valid. Take the formula <i>F</i>: (Φ ∨ ¬Φ). If our interpretation function makes Φ True, then ¬Φ is made False by the negation connective. Since the disjunct Φ of <i>F</i> is True under that interpretation, <i>F</i> is True. Now the only other possible interpretation of Φ makes it False, and if so, ¬Φ is made True by the negation function. That would make <i>F</i> True again, since one of <i>F</i>s disjuncts, ¬Φ, would be true under this interpretation. Since these two interpretations for <i>F</i> are the only possible logical interpretations, and since <i>F</i> comes out True for both, we say that it is logically valid or tautologous. </p> <div class="mw-heading mw-heading2"><h2 id="Interpretation_of_a_theory">Interpretation of a theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=6" title="Edit section: Interpretation of a theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory (mathematical logic)</a></div> <p>An <i>interpretation of a theory</i> is the relationship between a theory and some subject matter when there is a <a href="/wiki/Many-to-one" class="mw-redirect" title="Many-to-one">many-to-one</a> correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a <i>full interpretation</i>, otherwise it is called a <i>partial interpretation</i>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Interpretations_for_propositional_logic">Interpretations for propositional logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=7" title="Edit section: Interpretations for propositional logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formal language for <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, <a href="/wiki/Propositional_variable" title="Propositional variable">propositional variables</a>) and logical connectives. The only <a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical symbols</a> in a formal language for propositional logic are the propositional symbols, which are often denoted by capital letters. To make the formal language precise, a specific set of propositional symbols must be fixed. </p><p>The standard kind of interpretation in this setting is a function that maps each propositional symbol to one of the <a href="/wiki/Truth_value" title="Truth value">truth values</a> true and false. This function is known as a <i>truth assignment</i> or <i>valuation</i> function. In many presentations, it is literally a truth value that is assigned, but some presentations assign <a href="/wiki/Truthbearer" class="mw-redirect" title="Truthbearer">truthbearers</a> instead. </p><p>For a language with <i>n</i> distinct propositional variables there are 2<sup><i>n</i></sup> distinct possible interpretations. For any particular variable <i>a</i>, for example, there are 2<sup>1</sup>=2 possible interpretations: 1) <i>a</i> is assigned <b>T</b>, or 2) <i>a</i> is assigned <b>F</b>. For the pair <i>a</i>, <i>b</i> there are 2<sup>2</sup>=4 possible interpretations: 1) both are assigned <b>T</b>, 2) both are assigned <b>F</b>, 3) <i>a</i> is assigned <b>T</b> and <i>b</i> is assigned <b>F</b>, or 4) <i>a</i> is assigned <b>F</b> and <i>b</i> is assigned <b>T</b>. </p><p>Given any truth assignment for a set of propositional symbols, there is a unique extension to an interpretation for all the propositional formulas built up from those variables. This extended interpretation is defined inductively, using the truth-table definitions of the logical connectives discussed above. </p> <div class="mw-heading mw-heading2"><h2 id="First-order_logic">First-order logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=8" title="Edit section: First-order logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unlike propositional logic, where every language is the same apart from a choice of a different set of propositional variables, there are many different first-order languages. Each first-order language is defined by a <a href="/wiki/Signature_(mathematical_logic)" class="mw-redirect" title="Signature (mathematical logic)">signature</a>. The signature consists of a set of non-logical symbols and an identification of each of these symbols as either a constant symbol, a function symbol, or a <a href="/wiki/Predicate_symbol" class="mw-redirect" title="Predicate symbol">predicate symbol</a>. In the case of function and predicate symbols, a <a href="/wiki/Natural_number" title="Natural number">natural number</a> <a href="/wiki/Arity" title="Arity">arity</a> is also assigned. The alphabet for the formal language consists of logical constants, the equality relation symbol =, all the symbols from the signature, and an additional infinite set of symbols known as variables. </p><p>For example, in the language of <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, there are constant symbols 0 and 1, two binary function symbols + and ·, and no binary relation symbols. (Here the equality relation is taken as a logical constant.) </p><p>Again, we might define a first-order language <b>L</b>, as consisting of individual symbols a, b, and c; predicate symbols F, G, H, I and J; variables x, y, z; no function letters; no sentential symbols. </p> <div class="mw-heading mw-heading3"><h3 id="Formal_languages_for_first-order_logic">Formal languages for first-order logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=9" title="Edit section: Formal languages for first-order logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a signature σ, the corresponding formal language is known as the set of σ-formulas. Each σ-formula is built up out of atomic formulas by means of logical connectives; atomic formulas are built from terms using predicate symbols. The formal definition of the set of σ-formulas proceeds in the other direction: first, terms are assembled from the constant and function symbols together with the variables. Then, terms can be combined into an atomic formula using a predicate symbol (relation symbol) from the signature or the special predicate symbol "=" for equality (see the section "<a href="#Interpreting_equality">Interpreting equality"</a> below). Finally, the formulas of the language are assembled from atomic formulas using the logical connectives and quantifiers. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretations_of_a_first-order_language">Interpretations of a first-order language</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=10" title="Edit section: Interpretations of a first-order language"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">Interpretation function</a></div> <p>To ascribe meaning to all sentences of a first-order language, the following information is needed. </p> <ul><li>A <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> <i>D</i>, usually required to be non-empty (see below).</li> <li>For every constant symbol, an element of <i>D</i> as its interpretation.</li> <li>For every <i>n</i>-ary function symbol, an <i>n</i>-ary function from <i>D</i> to <i>D</i> as its interpretation (that is, a function <i>D<sup>n</sup></i> → <i>D</i>).</li> <li>For every <i>n</i>-ary predicate symbol, an <i>n</i>-ary relation on <i>D</i> as its interpretation (that is, a subset of <i>D<sup>n</sup></i>).</li></ul> <p>An object carrying this information is known as a <a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">structure</a> (of signature σ), or σ-structure, or <i>L</i>-structure (of language L), or as a "model". </p><p>The information specified in the interpretation provides enough information to give a truth value to any atomic formula, after each of its <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free variables</a>, if any, has been replaced by an element of the domain. The truth value of an arbitrary sentence is then defined inductively using the <a href="/wiki/T-schema" title="T-schema">T-schema</a>, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, <span class="nowrap">φ ∧ ψ</span> is satisfied if and only if both φ and ψ are satisfied. </p><p>This leaves the issue of how to interpret formulas of the form <span class="nowrap">∀ <i>x</i> φ(<i>x</i>)</span> and <span class="nowrap">∃ <i>x</i> φ(<i>x</i>)</span>. The domain of discourse forms the <a href="/wiki/Quantifier_(logic)#Range_of_quantification" title="Quantifier (logic)">range</a> for these quantifiers. The idea is that the sentence <span class="nowrap">∀ <i>x</i> φ(<i>x</i>)</span> is true under an interpretation exactly when every substitution instance of φ(<i>x</i>), where <i>x</i> is replaced by some element of the domain, is satisfied. The formula <span class="nowrap">∃ <i>x</i> φ(<i>x</i>)</span> is satisfied if there is at least one element <i>d</i> of the domain such that φ(<i>d</i>) is satisfied. </p><p>Strictly speaking, a substitution instance such as the formula φ(<i>d</i>) mentioned above is not a formula in the original formal language of φ, because <i>d</i> is an element of the domain. There are two ways of handling this technical issue. The first is to pass to a larger language in which each element of the domain is named by a constant symbol. The second is to add to the interpretation a function that assigns each variable to an element of the domain. Then the T-schema can quantify over variations of the original interpretation in which this variable assignment function is changed, instead of quantifying over substitution instances. </p><p>Some authors also admit <a href="/wiki/Propositional_variable" title="Propositional variable">propositional variables</a> in first-order logic, which must then also be interpreted. A propositional variable can stand on its own as an atomic formula. The interpretation of a propositional variable is one of the two truth values <i>true</i> and <i>false.</i><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Because the first-order interpretations described here are defined in <a href="/wiki/Set_theory" title="Set theory">set theory</a>, they do not associate each predicate symbol with a property<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> (or relation), but rather with the extension of that property (or relation). In other words, these first-order interpretations are <a href="/wiki/Extensional_definition" class="mw-redirect" title="Extensional definition">extensional</a><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> not <a href="/wiki/Intensional_definition" class="mw-redirect" title="Intensional definition">intensional</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Example_of_a_first-order_interpretation">Example of a first-order interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=11" title="Edit section: Example of a first-order interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An example of interpretation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9730a0ada0426927ff64141eb9f505eca132d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}"></span> of the language <b>L</b> described above is as follows. </p> <ul><li>Domain: A chess set</li> <li>Individual constants: a: The white King, b: The black Queen, c: The white King's pawn</li> <li>F(x): x is a piece</li> <li>G(x): x is a pawn</li> <li>H(x): x is black</li> <li>I(x): x is white</li> <li>J(x, y): x can capture y</li></ul> <p>In the interpretation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9730a0ada0426927ff64141eb9f505eca132d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}"></span> of L: </p> <ul><li>the following are true sentences: F(a), G(c), H(b), I(a), J(b, c),</li> <li>the following are false sentences: J(a, c), G(a).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Non-empty_domain_requirement">Non-empty domain requirement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=12" title="Edit section: Non-empty domain requirement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As stated above, a first-order interpretation is usually required to specify a nonempty set as the domain of discourse. The reason for this requirement is to guarantee that equivalences such as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\phi \lor \exists x\psi )\leftrightarrow \exists x(\phi \lor \psi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>∨</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo>∨</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\phi \lor \exists x\psi )\leftrightarrow \exists x(\phi \lor \psi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720ee16b96a0c334a783307743922617101191b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.086ex; height:2.843ex;" alt="{\displaystyle (\phi \lor \exists x\psi )\leftrightarrow \exists x(\phi \lor \psi ),}"></span> where <i>x</i> is not a free variable of φ, are logically valid. This equivalence holds in every interpretation with a nonempty domain, but does not always hold when empty domains are permitted. For example, the equivalence <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\forall y(y=y)\lor \exists x(x=x)]\equiv \exists x[\forall y(y=y)\lor x=x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>≡</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">[</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\forall y(y=y)\lor \exists x(x=x)]\equiv \exists x[\forall y(y=y)\lor x=x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1103d5e6440cb56fa1c1cfeb394b2f4306fc9ae3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.754ex; height:2.843ex;" alt="{\displaystyle [\forall y(y=y)\lor \exists x(x=x)]\equiv \exists x[\forall y(y=y)\lor x=x]}"></span> fails in any structure with an empty domain. Thus the proof theory of first-order logic becomes more complicated when empty structures are permitted. However, the gain in allowing them is negligible, as both the intended interpretations and the interesting interpretations of the theories people study have non-empty domains.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Empty relations do not cause any problem for first-order interpretations, because there is no similar notion of passing a relation symbol across a logical connective, enlarging its scope in the process. Thus it is acceptable for relation symbols to be interpreted as being identically false. However, the interpretation of a function symbol must always assign a well-defined and total function to the symbol. </p> <div class="mw-heading mw-heading3"><h3 id="Interpreting_equality">Interpreting equality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=13" title="Edit section: Interpreting equality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equality relation is often treated specially in first order logic and other predicate logics. There are two general approaches. </p><p>The first approach is to treat equality as no different than any other binary relation. In this case, if an equality symbol is included in the signature, it is usually necessary to add various axioms about equality to axiom systems (for example, the substitution axiom saying that if <i>a</i> = <i>b</i> and <i>R</i>(<i>a</i>) holds then <i>R</i>(<i>b</i>) holds as well). This approach to equality is most useful when studying signatures that do not include the equality relation, such as the signature for <a href="/wiki/Set_theory" title="Set theory">set theory</a> or the signature for <a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a> in which there is only an equality relation for numbers, but not an equality relation for set of numbers. </p><p>The second approach is to treat the equality relation symbol as a logical constant that must be interpreted by the real equality relation in any interpretation. An interpretation that interprets equality this way is known as a <i>normal model</i>, so this second approach is the same as only studying interpretations that happen to be normal models. The advantage of this approach is that the axioms related to equality are automatically satisfied by every normal model, and so they do not need to be explicitly included in first-order theories when equality is treated this way. This second approach is sometimes called <i>first order logic with equality</i>, but many authors adopt it for the general study of first-order logic without comment. </p><p>There are a few other reasons to restrict study of first-order logic to normal models. First, it is known that any first-order interpretation in which equality is interpreted by an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> and satisfies the substitution axioms for equality can be cut down to an <a href="/wiki/Elementary_substructure" class="mw-redirect" title="Elementary substructure">elementarily equivalent</a> interpretation on a subset of the original domain. Thus there is little additional generality in studying non-normal models. Second, if non-normal models are considered, then every consistent theory has an infinite model; this affects the statements of results such as the <a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a>, which are usually stated under the assumption that only normal models are considered. </p> <div class="mw-heading mw-heading3"><h3 id="Many-sorted_first-order_logic">Many-sorted first-order logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=14" title="Edit section: Many-sorted first-order logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A generalization of first order logic considers languages with more than one <i>sort</i> of variables. The idea is different sorts of variables represent different types of objects. Every sort of variable can be quantified; thus an interpretation for a many-sorted language has a separate domain for each of the sorts of variables to range over (there is an infinite collection of variables of each of the different sorts). Function and relation symbols, in addition to having arities, are specified so that each of their arguments must come from a certain sort. </p><p>One example of many-sorted logic is for planar <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This should probably refer to a particular axiomatization that the author has in mind. Tarski's axiomatization uses only a single sort, namely points. (June 2022)">clarification needed</span></a></i>]</sup>. There are two sorts; points and lines. There is an equality relation symbol for points, an equality relation symbol for lines, and a binary incidence relation <i>E</i> which takes one point variable and one line variable. The intended interpretation of this language has the point variables range over all points on the <a href="/wiki/Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, the line variable range over all lines on the plane, and the incidence relation <i>E</i>(<i>p</i>,<i>l</i>) holds if and only if point <i>p</i> is on line <i>l</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Higher-order_predicate_logics">Higher-order predicate logics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=15" title="Edit section: Higher-order predicate logics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A formal language for <a href="/wiki/Higher-order_logic" title="Higher-order logic">higher-order predicate logic</a> looks much the same as a formal language for first-order logic. The difference is that there are now many different types of variables. Some variables correspond to elements of the domain, as in first-order logic. Other variables correspond to objects of higher type: subsets of the domain, functions from the domain, functions that take a subset of the domain and return a function from the domain to subsets of the domain, etc. All of these types of variables can be quantified. </p><p>There are two kinds of interpretations commonly employed for higher-order logic. <i>Full semantics</i> require that, once the domain of discourse is satisfied, the higher-order variables range over all possible elements of the correct type (all subsets of the domain, all functions from the domain to itself, etc.). Thus the specification of a full interpretation is the same as the specification of a first-order interpretation. <i>Henkin semantics</i>, which are essentially multi-sorted first-order semantics, require the interpretation to specify a separate domain for each type of higher-order variable to range over. Thus an interpretation in Henkin semantics includes a domain <i>D</i>, a collection of subsets of <i>D</i>, a collection of functions from <i>D</i> to <i>D</i>, etc. The relationship between these two semantics is an important topic in higher order logic. </p> <div class="mw-heading mw-heading2"><h2 id="Non-classical_interpretations">Non-classical interpretations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=16" title="Edit section: Non-classical interpretations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The interpretations of propositional logic and predicate logic described above are not the only possible interpretations. In particular, there are other types of interpretations that are used in the study of <a href="/wiki/Non-classical_logic" title="Non-classical logic">non-classical logic</a> (such as <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>), and in the study of modal logic. </p><p>Interpretations used to study non-classical logic include <a href="/w/index.php?title=Topological_model&action=edit&redlink=1" class="new" title="Topological model (page does not exist)">topological models</a>, <a href="/wiki/Boolean-valued_model" title="Boolean-valued model">Boolean-valued models</a>, and <a href="/wiki/Kripke_model" class="mw-redirect" title="Kripke model">Kripke models</a>. <a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a> is also studied using Kripke models. </p> <div class="mw-heading mw-heading2"><h2 id="Intended_interpretations">Intended interpretations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=17" title="Edit section: Intended interpretations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many formal languages are associated with a particular interpretation that is used to motivate them. For example, the first-order signature for set theory includes only one binary relation, ∈, which is intended to represent set membership, and the domain of discourse in a first-order theory of the natural numbers is intended to be the set of natural numbers. </p><p>The intended interpretation is called the <i>standard model</i> (a term introduced by <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a> in 1960).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In the context of <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>, it consists of the natural numbers with their ordinary arithmetical operations. All models that are <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the one just given are also called standard; these models all satisfy the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a>. There are also <a href="/wiki/Peano_axioms#Nonstandard_models" title="Peano axioms">non-standard models of the (first-order version of the) Peano axioms</a>, which contain elements not correlated with any natural number. </p><p>While the intended interpretation can have no explicit indication in the strictly formal <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">syntactical rules</a>, it naturally affects the choice of the <a href="/wiki/Formal_grammar" title="Formal grammar">formation</a> and <a href="/wiki/Transformation_rules" class="mw-redirect" title="Transformation rules">transformation rules</a> of the syntactical system. For example, <a href="/wiki/Primitive_notion" title="Primitive notion">primitive signs</a> must permit expression of the concepts to be modeled; <a href="/wiki/Sentential_formula" class="mw-redirect" title="Sentential formula">sentential formulas</a> are chosen so that their counterparts in the intended interpretation are <a href="/wiki/Meaning_(linguistics)" class="mw-redirect" title="Meaning (linguistics)">meaningful</a> <a href="/wiki/Declarative_sentence" class="mw-redirect" title="Declarative sentence">declarative sentences</a>; <a href="/wiki/Axiom" title="Axiom">primitive sentences</a> need to come out as <a href="/wiki/Truth" title="Truth">true</a> <a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentences</a> in the interpretation; <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a> must be such that, if the sentence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a124c17f5bab114652d405f59d06e52172d11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.069ex; width:2.245ex; height:2.843ex;" alt="{\displaystyle {\mathcal {I}}_{j}}"></span> is directly <a href="/wiki/Formal_proof" title="Formal proof">derivable</a> from a sentence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee6e4227826836a1a6dad3186ce63a13866c257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.069ex; width:2.135ex; height:2.509ex;" alt="{\displaystyle {\mathcal {I}}_{i}}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}_{i}\to {\mathcal {I}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}_{i}\to {\mathcal {I}}_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f52342d2be7d8d8d557a7ad01d4155821282e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.069ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle {\mathcal {I}}_{i}\to {\mathcal {I}}_{j}}"></span> turns out to be a true sentence, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"></span> meaning <a href="/wiki/Material_conditional" title="Material conditional">implication</a>, as usual. These requirements ensure that all <a href="/wiki/Formal_proof" title="Formal proof">provable</a> sentences also come out to be true.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Most formal systems have many more models than they were intended to have (the existence of <a href="/wiki/Non-standard_model" title="Non-standard model">non-standard models</a> is an example). When we speak about 'models' in <a href="/wiki/Empirical_science" class="mw-redirect" title="Empirical science">empirical sciences</a>, we mean, if we want <a href="/wiki/Reality" title="Reality">reality</a> to be a model of our science, to speak about an <i>intended model</i>. A model in the empirical sciences is an <i>intended factually-true descriptive interpretation</i> (or in other contexts: a non-intended arbitrary interpretation used to clarify such an intended factually-true descriptive interpretation.) All models are interpretations that have the same <a href="/wiki/Domain_of_discourse" title="Domain of discourse">domain of discourse</a> as the intended one, but other <a href="/wiki/Value_assignment" class="mw-redirect" title="Value assignment">assignments</a> for <a href="/wiki/Non-logical_constant" class="mw-redirect" title="Non-logical constant">non-logical constants</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Every chapter is written by a different author. (September 2015)">page needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=18" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a simple formal system (we shall call this one <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> <msup> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a8dd711a545583eab7b71b7cf80d57ee84344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.129ex; height:2.509ex;" alt="{\displaystyle {\mathcal {FS'}}}"></span>) whose alphabet α consists only of three symbols <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\blacksquare ,\bigstar ,\blacklozenge \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>◼</mi> <mo>,</mo> <mi>★</mi> <mo>,</mo> <mi>⧫</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\blacksquare ,\bigstar ,\blacklozenge \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c3a45dcbdf371128a134b28e54cfabd436e4e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.945ex; height:2.843ex;" alt="{\displaystyle \{\blacksquare ,\bigstar ,\blacklozenge \}}"></span> and whose formation rule for formulas is: </p> <dl><dd>'Any string of symbols of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> <msup> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a8dd711a545583eab7b71b7cf80d57ee84344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.129ex; height:2.509ex;" alt="{\displaystyle {\mathcal {FS'}}}"></span> which is at least 6 symbols long, and which is not infinitely long, is a formula of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> <msup> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a8dd711a545583eab7b71b7cf80d57ee84344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.129ex; height:2.509ex;" alt="{\displaystyle {\mathcal {FS'}}}"></span>. Nothing else is a formula of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> <msup> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a8dd711a545583eab7b71b7cf80d57ee84344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.129ex; height:2.509ex;" alt="{\displaystyle {\mathcal {FS'}}}"></span>.'</dd></dl> <p>The single <a href="/wiki/Axiom_schema" title="Axiom schema">axiom schema</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {FS'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> <msup> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> <mo>′</mo> </msup> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {FS'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a8dd711a545583eab7b71b7cf80d57ee84344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.129ex; height:2.509ex;" alt="{\displaystyle {\mathcal {FS'}}}"></span> is: </p> <dl><dd>" <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare \ \bigstar \ast \blacklozenge \ \blacksquare \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> <mtext> </mtext> <mi>★</mi> <mo>∗</mo> <mi>⧫</mi> <mtext> </mtext> <mi>◼</mi> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare \ \bigstar \ast \blacklozenge \ \blacksquare \ast }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f71274d22cecf7459ceeea1e1aa158dbc38634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.879ex; height:2.343ex;" alt="{\displaystyle \blacksquare \ \bigstar \ast \blacklozenge \ \blacksquare \ast }"></span> " (where " <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ast }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ast }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1858484bef51b1435c2b986c728a81788051803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \ast }"></span> " is a <a href="/wiki/Metasyntactic_variable" title="Metasyntactic variable">metasyntactic variable</a> standing for a finite string of " <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8733090f2d787d03101c3e16dc3f6404f0e7dd4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \blacksquare }"></span> "s )</dd></dl> <p>A formal proof can be constructed as follows: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> <mtext> </mtext> <mi>★</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>⧫</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77b8eac53307c667d2f2c3073c60d84a935da4f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.88ex; height:2.343ex;" alt="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> <mtext> </mtext> <mi>★</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>⧫</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a642ee77a54f644550090bec66bd7aa85c04391b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.657ex; height:2.343ex;" alt="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> <mtext> </mtext> <mi>★</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>⧫</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6115bb86b6f98d86bca9b463e16b9d31751bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.435ex; height:2.343ex;" alt="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }"></span></li></ol> <p>In this example the theorem produced " <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◼</mi> <mtext> </mtext> <mi>★</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>⧫</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> <mtext> </mtext> <mi>◼</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d6115bb86b6f98d86bca9b463e16b9d31751bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.435ex; height:2.343ex;" alt="{\displaystyle \blacksquare \ \bigstar \ \blacksquare \ \blacksquare \ \blacksquare \ \blacklozenge \ \blacksquare \ \blacksquare \ \blacksquare \ \blacksquare }"></span> " can be interpreted as meaning "One plus three equals four." A different interpretation would be to read it backwards as "Four minus three equals one."<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="The material near this tag failed verification of its source citation(s). (December 2024)">failed verification</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Other_concepts_of_interpretation">Other concepts of interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=19" title="Edit section: Other concepts of interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are other uses of the term "interpretation" that are commonly used, which do not refer to the assignment of meanings to formal languages. </p><p>In <a href="/wiki/Model_theory" title="Model theory">model theory</a>, a structure <i>A</i> is said to interpret a structure <i>B</i> if there is a definable subset <i>D</i> of <i>A</i>, and definable relations and functions on <i>D</i>, such that <i>B</i> is isomorphic to the structure with domain <i>D</i> and these functions and relations. In some settings, it is not the domain <i>D</i> that is used, but rather <i>D</i> modulo an equivalence relation definable in <i>A</i>. For additional information, see <a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">Interpretation (model theory)</a>. </p><p>A theory <i>T</i> is said to interpret another theory <i>S</i> if there is a finite <a href="/wiki/Extension_by_definition" class="mw-redirect" title="Extension by definition">extension by definitions</a> <i>T</i>′ of <i>T</i> such that <i>S</i> is contained in <i>T</i>′. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Conceptual_model" title="Conceptual model">Conceptual model</a></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free variables</a> and <a href="/wiki/Name_binding" title="Name binding">Name binding</a></li> <li><a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li> <li><a href="/wiki/Herbrand_interpretation" title="Herbrand interpretation">Herbrand interpretation</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">Interpretation (model theory)</a></li> <li><a href="/wiki/Logical_system" class="mw-redirect" title="Logical system">Logical system</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a></li> <li><a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a></li> <li><a href="/wiki/Model_theory" title="Model theory">Model theory</a></li> <li><a href="/wiki/Satisfiable" class="mw-redirect" title="Satisfiable">Satisfiable</a></li> <li><a href="/wiki/Truth" title="Truth">Truth</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=21" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Sometimes called the "universe of discourse"</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">The extension of a property (also called an attribute) is a set of individuals, so a property is a unary relation. E.g. The properties "yellow" and "prime" are unary relations.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">See also <a href="/wiki/Extension_(predicate_logic)" title="Extension (predicate logic)">Extension (predicate logic)</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=22" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="/wiki/Graham_Priest" title="Graham Priest">Priest, Graham</a>, 2008. <i><a href="/wiki/An_Introduction_to_Non-Classical_Logic" title="An Introduction to Non-Classical Logic">An Introduction to Non-Classical Logic: from If to Is</a>,</i> 2nd ed. Cambridge University Press.</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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHaskell_Curry1963" class="citation book cs1"><a href="/wiki/Haskell_Curry" title="Haskell Curry">Haskell Curry</a> (1963). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/foundationsofmat0000unse_o5q2"><i>Foundations of Mathematical Logic</i></a></span>. Mcgraw Hill. p. 48.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Mathematical+Logic&rft.pages=48&rft.pub=Mcgraw+Hill&rft.date=1963&rft.au=Haskell+Curry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffoundationsofmat0000unse_o5q2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMates1972" class="citation cs2"><a href="/wiki/Benson_Mates" title="Benson Mates">Mates, Benson</a> (1972), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/elementarylogic00mate/page/56"><i>Elementary Logic, Second Edition</i></a></span>, New York: <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/elementarylogic00mate/page/56">56</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-501491-X" title="Special:BookSources/0-19-501491-X"><bdi>0-19-501491-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Logic%2C+Second+Edition&rft.place=New+York&rft.pages=56&rft.pub=Oxford+University+Press&rft.date=1972&rft.isbn=0-19-501491-X&rft.aulast=Mates&rft.aufirst=Benson&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementarylogic00mate%2Fpage%2F56&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHailperin1953" class="citation cs2">Hailperin, Theodore (1953), "Quantification theory and empty individual-domains", <i><a href="/wiki/The_Journal_of_Symbolic_Logic" class="mw-redirect" title="The Journal of Symbolic Logic">The Journal of Symbolic Logic</a></i>, <b>18</b> (3), <a href="/wiki/Association_for_Symbolic_Logic" title="Association for Symbolic Logic">Association for Symbolic Logic</a>: <span class="nowrap">197–</span>200, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2267402">10.2307/2267402</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2267402">2267402</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0057820">0057820</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40988137">40988137</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=Quantification+theory+and+empty+individual-domains&rft.volume=18&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E197-%3C%2Fspan%3E200&rft.date=1953&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40988137%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0057820%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2267402%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2267402&rft.aulast=Hailperin&rft.aufirst=Theodore&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuine1954" class="citation cs2"><a href="/wiki/Willard_Quine" class="mw-redirect" title="Willard Quine">Quine, W. V.</a> (1954), "Quantification and the empty domain", <i>The Journal of Symbolic Logic</i>, <b>19</b> (3), Association for Symbolic Logic: <span class="nowrap">177–</span>179, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2268615">10.2307/2268615</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2268615">2268615</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0064715">0064715</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:27053902">27053902</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Symbolic+Logic&rft.atitle=Quantification+and+the+empty+domain&rft.volume=19&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E177-%3C%2Fspan%3E179&rft.date=1954&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A27053902%23id-name%3DS2CID&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0064715%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2268615%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2268615&rft.aulast=Quine&rft.aufirst=W.+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoland_Müller2009" class="citation book cs1">Roland Müller (2009). "The Notion of a Model". In Anthonie Meijers (ed.). <i>Philosophy of technology and engineering sciences</i>. Handbook of the Philosophy of Science. Vol. 9. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51667-1" title="Special:BookSources/978-0-444-51667-1"><bdi>978-0-444-51667-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Notion+of+a+Model&rft.btitle=Philosophy+of+technology+and+engineering+sciences&rft.series=Handbook+of+the+Philosophy+of+Science&rft.pub=Elsevier&rft.date=2009&rft.isbn=978-0-444-51667-1&rft.au=Roland+M%C3%BCller&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRudolf_Carnap1958" class="citation book cs1"><a href="/wiki/Rudolf_Carnap" title="Rudolf Carnap">Rudolf Carnap</a> (1958). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontosy00carn"><i>Introduction to Symbolic Logic and its Applications</i></a></span>. New York: Dover publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486604534" title="Special:BookSources/9780486604534"><bdi>9780486604534</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Symbolic+Logic+and+its+Applications&rft.place=New+York&rft.pub=Dover+publications&rft.date=1958&rft.isbn=9780486604534&rft.au=Rudolf+Carnap&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontosy00carn&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHans_Freudenthal1960" class="citation book cs1"><a href="/wiki/Hans_Freudenthal" title="Hans Freudenthal">Hans Freudenthal</a>, ed. (Jan 1960). <i>The Concept and the Role of the Model in Mathematics and Natural and Social Sciences (Colloquium proceedings)</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-010-3669-6" title="Special:BookSources/978-94-010-3669-6"><bdi>978-94-010-3669-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Concept+and+the+Role+of+the+Model+in+Mathematics+and+Natural+and+Social+Sciences+%28Colloquium+proceedings%29&rft.pub=Springer&rft.date=1960-01&rft.isbn=978-94-010-3669-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHunter1996" class="citation book cs1"><a href="/wiki/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). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780520023567" title="Special:BookSources/9780520023567"><bdi>9780520023567</bdi></a>. <a href="/wiki/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><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metalogic%3A+An+Introduction+to+the+Metatheory+of+Standard+First-Order+Logic&rft.pub=University+of+California+Press&rft.date=1996&rft_id=info%3Aoclcnum%2F36312727&rft.isbn=9780520023567&rft.aulast=Hunter&rft.aufirst=Geoffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInterpretation+%28logic%29" class="Z3988"></span> (<a rel="nofollow" class="external text" href="https://archive.org/details/metalogicintrodu0000hunt">accessible to patrons with print disabilities</a>)</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Interpretation_(logic)&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-classical/#4">Stanford Enc. 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aria-labelledby="Mathematical_logic326" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic326" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/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="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_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="/wiki/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="/wiki/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="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/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="/wiki/Propositional_calculus" 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="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/List_of_first-order_theories" title="List of first-order theories"><span style="font-size:85%;">list</span></a></li></ul></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/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="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/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="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/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="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/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="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/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="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/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="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size:85%;">list</span></a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/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="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/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="/wiki/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="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/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="/wiki/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 class="mw-selflink selflink">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda 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