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data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Language </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" href="/w/index.php?title=Special:UserLogin&returnto=Negation" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Negation&action=edit" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Edit </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <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 negation in linguistics, see <a href="/wiki/Affirmation_and_negation" title="Affirmation and negation">Affirmation and negation</a>. For other uses, see <a href="/wiki/Negation_(disambiguation)" class="mw-disambig" title="Negation (disambiguation)">Negation (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style>In <a href="/wiki/Logic" title="Logic">logic</a>, negation, also called the logical not or logical complement, is an <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> that takes a <a href="/wiki/Proposition_(mathematics)" class="mw-redirect" title="Proposition (mathematics)">proposition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> to another proposition "not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">", written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathord {\sim }}P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathord {\sim }}P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daca81525792364a725ec311d9d762a8a6956a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.554ex; height:2.176ex;" alt="{\displaystyle {\mathord {\sim }}P}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5e1bed5bc42d4e46dd9e5c7d2fc327927b87169" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.025ex; height:3.009ex;" alt="{\displaystyle {\overline {P}}}">. It is interpreted intuitively as being true when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is false, and false when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is true.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is "Spot runs", then "not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">" is "Spot does not run". <table class="infobox"><caption class="infobox-title" style="background:navy; color:white;">Negation</caption><tbody><tr><th colspan="2" class="infobox-above">NOT</th></tr><tr><td colspan="2" class="infobox-image"><a href="/wiki/File:Venn10.svg" class="mw-file-description" title="Venn diagram of Negation"><img alt="Venn diagram of Negation" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Venn10.svg/150px-Venn10.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Venn10.svg/225px-Venn10.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Venn10.svg/300px-Venn10.svg.png 2x" data-file-width="280" data-file-height="280"></a></td></tr><tr><th scope="row" class="infobox-label">Definition</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot {x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/561558609a1ef8421895e7953c96ad487e960300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.88ex; height:1.676ex;" alt="{\displaystyle \lnot {x}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Truth_table" title="Truth table">Truth table</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (01)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>01</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (01)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0849a25948c03d17713210f17d69290cb574cb01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle (01)}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Logic_gate" title="Logic gate">Logic gate</a></th><td class="infobox-data"><a href="/wiki/File:NOT_ANSI.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/NOT_ANSI.svg/70px-NOT_ANSI.svg.png" decoding="async" width="70" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/NOT_ANSI.svg/105px-NOT_ANSI.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/NOT_ANSI.svg/140px-NOT_ANSI.svg.png 2x" data-file-width="100" data-file-height="50"></a></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;">Normal forms</th></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Disjunctive_normal_form" title="Disjunctive normal form">Disjunctive</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot {x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/561558609a1ef8421895e7953c96ad487e960300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.88ex; height:1.676ex;" alt="{\displaystyle \lnot {x}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Conjunctive_normal_form" title="Conjunctive normal form">Conjunctive</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot {x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot {x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/561558609a1ef8421895e7953c96ad487e960300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.88ex; height:1.676ex;" alt="{\displaystyle \lnot {x}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Zhegalkin_polynomial" title="Zhegalkin polynomial">Zhegalkin polynomial</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\oplus x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>⊕</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\oplus x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6360c75948cd75ab65f4b9691f620640bb6abd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle 1\oplus x}"></td></tr><tr><th colspan="2" class="infobox-header" style="background:navy; color:white;"><a href="/wiki/Post%27s_lattice" title="Post's lattice">Post's lattices</a></th></tr><tr><th scope="row" class="infobox-label">0-preserving</th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label">1-preserving</th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Monotonic_function" title="Monotonic function">Monotone</a></th><td class="infobox-data">no</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Affine_transformation" title="Affine transformation">Affine</a></th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label">Self-dual</th><td class="infobox-data">yes</td></tr><tr><td colspan="2" class="infobox-navbar"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"> Negation is a <a href="/wiki/Unary_operation" title="Unary operation">unary</a> <a href="/wiki/Logical_connective" title="Logical connective">logical connective</a>. It may furthermore be applied not only to propositions, but also to <a href="/wiki/Notion_(philosophy)" class="mw-redirect" title="Notion (philosophy)">notions</a>, <a href="/wiki/Truth_value" title="Truth value">truth values</a>, or <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">semantic values</a> more generally. In <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, negation is normally identified with the <a href="/wiki/Truth_function" title="Truth function">truth function</a> that takes truth to falsity (and vice versa). In <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>, according to the <a href="/wiki/Brouwer%E2%80%93Heyting%E2%80%93Kolmogorov_interpretation" title="Brouwer–Heyting–Kolmogorov interpretation">Brouwer–Heyting–Kolmogorov interpretation</a>, the negation of a proposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is the proposition whose proofs are the refutations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">. An operand of a negation is a negand,<a href="#cite_note-:21-3">[3]</a> or negatum.<a href="#cite_note-:21-3">[3]</a> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Definition">1 Definition</a></li> <li class="toclevel-1 tocsection-2"><a href="#Notation">2 Notation</a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Precedence">2.1 Precedence</a></li> </ul> </li> <li class="toclevel-1 tocsection-4"><a href="#Properties">3 Properties</a> <ul> <li class="toclevel-2 tocsection-5"><a href="#Double_negation">3.1 Double negation</a></li> <li class="toclevel-2 tocsection-6"><a href="#Distributivity">3.2 Distributivity</a></li> <li class="toclevel-2 tocsection-7"><a href="#Linearity">3.3 Linearity</a></li> <li class="toclevel-2 tocsection-8"><a href="#Self_dual">3.4 Self dual</a></li> <li class="toclevel-2 tocsection-9"><a href="#Negations_of_quantifiers">3.5 Negations of quantifiers</a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Rules_of_inference">4 Rules of inference</a></li> <li class="toclevel-1 tocsection-11"><a href="#Programming_language_and_ordinary_language">5 Programming language and ordinary language</a></li> <li class="toclevel-1 tocsection-12"><a href="#Kripke_semantics">6 Kripke semantics</a></li> <li class="toclevel-1 tocsection-13"><a href="#See_also">7 See also</a></li> <li class="toclevel-1 tocsection-14"><a href="#References">8 References</a></li> <li class="toclevel-1 tocsection-15"><a href="#Further_reading">9 Further reading</a></li> <li class="toclevel-1 tocsection-16"><a href="#External_links">10 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Definition">Definition</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=1" title="Edit section: Definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> Classical negation is an <a href="/wiki/Logical_operation" class="mw-redirect" title="Logical operation">operation</a> on one <a href="/wiki/Logical_value" class="mw-redirect" title="Logical value">logical value</a>, typically the value of a <a href="/wiki/Proposition" title="Proposition">proposition</a>, that produces a value of true when its operand is false, and a value of false when its operand is true. Thus if statement <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is true, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (pronounced "not P") would then be false; and conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is true, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  would be false. The <a href="/wiki/Truth_table" title="Truth table">truth table</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is as follows: <dl><dd><table class="wikitable" style="text-align:center; background-color: #ddffdd;"> <tbody><tr bgcolor="#ddeeff"> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td></tr> <tr> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">True</td> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">False </td></tr> <tr> <td style="background: #FFE3E3; color: black; vertical-align: middle; text-align: center;" class="table-no2">False</td> <td style="background:#bfd; color:black; vertical-align:middle; text-align:center;" class="table-yes2">True </td></tr></tbody></table></dd></dl> Negation can be defined in terms of other logical operations. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.168ex; height:2.176ex;" alt="{\displaystyle P\rightarrow \bot }"></noscript><span class="lazy-image-placeholder" style="width: 7.168ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" data-alt="{\displaystyle P\rightarrow \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" 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 \rightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" data-alt="{\displaystyle \rightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" 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 \bot }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" data-alt="{\displaystyle \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="/wiki/False_(logic)" title="False (logic)">absolute falsehood</a>). Conversely, one can define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" 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 \bot }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" data-alt="{\displaystyle \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\land \neg Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\land \neg Q}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdaf497bd9328824019118ed8188eccc249803c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.81ex; height:2.509ex;" alt="{\displaystyle Q\land \neg Q}"></noscript><span class="lazy-image-placeholder" style="width: 7.81ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdaf497bd9328824019118ed8188eccc249803c6" data-alt="{\displaystyle Q\land \neg Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for any proposition Q (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="/wiki/Logical_conjunction" title="Logical conjunction">logical conjunction</a>). The idea here is that any <a href="/wiki/Contradiction" title="Contradiction">contradiction</a> is false, and while these ideas work in both classical and intuitionistic logic, they do not work in <a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">paraconsistent logic</a>, where contradictions are not necessarily false. As a further example, negation can be defined in terms of NAND and can also be defined in terms of NOR. Algebraically, classical negation corresponds to <a href="/wiki/Complement_(order_theory)" class="mw-redirect" title="Complement (order theory)">complementation</a> in a <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a>, and intuitionistic negation to pseudocomplementation in a <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a>. These algebras provide a <a href="/wiki/Algebraic_semantics_(mathematical_logic)" title="Algebraic semantics (mathematical logic)">semantics</a> for classical and intuitionistic logic. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Notation">Notation</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=2" title="Edit section: Notation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> The negation of a proposition p is notated in different ways, in various contexts of discussion and fields of application. The following table documents some of these variants: <table class="wikitable"> <tbody><tr style="background:paleturquoise"> <th>Notation </th> <th>Plain text </th> <th>Vocalization </th></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg p}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b198c79234d926cbee42c0f271d903ea55dc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.72ex; height:2.009ex;" alt="{\displaystyle \neg p}"></noscript><span class="lazy-image-placeholder" style="width: 2.72ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b198c79234d926cbee42c0f271d903ea55dc21" data-alt="{\displaystyle \neg p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="text-align:center"><style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style>¬p , <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">7p<a href="#cite_note-4">[4]</a> </td> <td>Not p </td></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathord {\sim }}p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mrow> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathord {\sim }}p}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26fe3458fbf75f8454c1b27de390ad876ef54ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.978ex; height:2.009ex;" alt="{\displaystyle {\mathord {\sim }}p}"></noscript><span class="lazy-image-placeholder" style="width: 2.978ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26fe3458fbf75f8454c1b27de390ad876ef54ccc" data-alt="{\displaystyle {\mathord {\sim }}p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">~p </td> <td>Not p </td></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -p}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233ea0d764a4823bcf8b9a31b2f25f3966e77845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.978ex; height:2.343ex;" alt="{\displaystyle -p}"></noscript><span class="lazy-image-placeholder" style="width: 2.978ex;height: 2.343ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/233ea0d764a4823bcf8b9a31b2f25f3966e77845" data-alt="{\displaystyle -p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">-p </td> <td>Not p </td></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Np}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Np}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d86ac82f9845ca629f9b3496c0d82908fb60f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.233ex; height:2.509ex;" alt="{\displaystyle Np}"></noscript><span class="lazy-image-placeholder" style="width: 3.233ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d86ac82f9845ca629f9b3496c0d82908fb60f5" data-alt="{\displaystyle Np}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td> </td> <td>En p </td></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p'}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e623e3163571a220ed60ecb31aa78c24104b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.944ex; height:2.843ex;" alt="{\displaystyle p'}"></noscript><span class="lazy-image-placeholder" style="width: 1.944ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e623e3163571a220ed60ecb31aa78c24104b85" data-alt="{\displaystyle p'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">p' </td> <td><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li>p prime,</li><li>p complement</li></ul></div> </td></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {p}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3884eccbf972d435831da42dd4154e192d550e16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.032ex; width:1.316ex; height:2.676ex;" alt="{\displaystyle {\overline {p}}}"></noscript><span class="lazy-image-placeholder" style="width: 1.316ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3884eccbf972d435831da42dd4154e192d550e16" data-alt="{\displaystyle {\overline {p}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734"> ̅p </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li>p bar,</li><li>Bar p</li></ul></div> </td></tr> <tr> <td style="text-align:center"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle !p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>!</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle !p}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/937175f4715102052885063560a251ad55aa2e65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.816ex; height:2.509ex;" alt="{\displaystyle !p}"></noscript><span class="lazy-image-placeholder" style="width: 1.816ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/937175f4715102052885063560a251ad55aa2e65" data-alt="{\displaystyle !p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </td> <td style="text-align:center"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734">!p </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li>Bang p</li><li>Not p</li></ul></div> </td></tr> </tbody></table> The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Np}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Np}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d86ac82f9845ca629f9b3496c0d82908fb60f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.233ex; height:2.509ex;" alt="{\displaystyle Np}"></noscript><span class="lazy-image-placeholder" style="width: 3.233ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48d86ac82f9845ca629f9b3496c0d82908fb60f5" data-alt="{\displaystyle Np}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="/wiki/Polish_notation#Polish_notation_for_logic" title="Polish notation">Polish notation</a>. In <a href="/wiki/Set_theory#Basic_concepts_and_notation" title="Set theory">set theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \setminus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo class="MJX-variant">∖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \setminus }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e20e45087a97f0448fc3d4bc27b060084830f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle \setminus }"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e20e45087a97f0448fc3d4bc27b060084830f4" data-alt="{\displaystyle \setminus }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is also used to indicate 'not in the set of': <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\setminus A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\setminus A}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584717f0e3e8f7f356c27423329325fcb9192e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.72ex; height:2.843ex;" alt="{\displaystyle U\setminus A}"></noscript><span class="lazy-image-placeholder" style="width: 5.72ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584717f0e3e8f7f356c27423329325fcb9192e36" data-alt="{\displaystyle U\setminus A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the set of all members of U that are not members of A. Regardless how it is notated or <a href="/wiki/List_of_logic_symbols" title="List of logic symbols">symbolized</a>, the negation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  can be read as "it is not the case that P", "not that P", or usually more simply as "not P". <div class="mw-heading mw-heading3"><h3 id="Precedence">Precedence</h3> <a role="button" href="/w/index.php?title=Negation&action=edit&section=3" title="Edit section: Precedence" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </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/Logical_connective#Order_of_precedence" title="Logical connective">Logical connective § Order of precedence</a></div> As a way of reducing the number of necessary parentheses, one may introduce <a href="/wiki/Precedence_rule" class="mw-redirect" title="Precedence rule">precedence rules</a>: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> <mi>R</mi> </mrow> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fc0c4ae9cfddb8e178c7f59f2e77bd549afec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.177ex; height:2.509ex;" alt="{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}"></noscript><span class="lazy-image-placeholder" style="width: 17.177ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fc0c4ae9cfddb8e178c7f59f2e77bd549afec5" data-alt="{\displaystyle P\vee Q\wedge {\neg R}\rightarrow S}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is short for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09942e077a17c2e657e603c4a66cbe03b12ba42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.251ex; height:2.843ex;" alt="{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S.}"></noscript><span class="lazy-image-placeholder" style="width: 23.251ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09942e077a17c2e657e603c4a66cbe03b12ba42" data-alt="{\displaystyle (P\vee (Q\wedge (\neg R)))\rightarrow S.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Here is a table that shows a commonly used precedence of logical operators.<a href="#cite_note-5">[5]</a> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>Operator</th> <th>Precedence </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \neg }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 1.176ex;vertical-align: 0.204ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa78fd02085d39aa58c9e47a6d4033ce41e02fad" data-alt="{\displaystyle \neg }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td>1 </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" data-alt="{\displaystyle \land }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td>2 </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.009ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" data-alt="{\displaystyle \lor }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td>3 </td></tr> <tr> <td><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><noscript><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 }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" data-alt="{\displaystyle \to }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td>4 </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \leftrightarrow }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" 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 \leftrightarrow }"></noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.843ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046b918c43e05caf6624fe9b676c69ec9cd6b892" data-alt="{\displaystyle \leftrightarrow }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </td> <td>5 </td></tr></tbody></table> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="Properties">Properties</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=4" title="Edit section: Properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <div class="mw-heading mw-heading3"><h3 id="Double_negation">Double negation</h3> <a role="button" href="/w/index.php?title=Negation&action=edit&section=5" title="Edit section: Double negation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Within a system of <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, double negation, that is, the negation of the negation of a proposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , is <a href="/wiki/Logically_equivalent" class="mw-redirect" title="Logically equivalent">logically equivalent</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Expressed in symbolic terms, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg P\equiv P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>≡</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg P\equiv P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbf8c22800dfe80e6343d84718c229b41aff1d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.69ex; height:2.176ex;" alt="{\displaystyle \neg \neg P\equiv P}"></noscript><span class="lazy-image-placeholder" style="width: 9.69ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbf8c22800dfe80e6343d84718c229b41aff1d9" data-alt="{\displaystyle \neg \neg P\equiv P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>, a proposition implies its double negation, but not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> of period two. However, in <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>, the weaker equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg \neg P\equiv \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>≡</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg \neg P\equiv \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17089637b8b9487eac521d8650f18e10e1abd327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.791ex; height:2.176ex;" alt="{\displaystyle \neg \neg \neg P\equiv \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 12.791ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17089637b8b9487eac521d8650f18e10e1abd327" data-alt="{\displaystyle \neg \neg \neg P\equiv \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  does hold. This is because in intuitionistic logic, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is just a shorthand for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.168ex; height:2.176ex;" alt="{\displaystyle P\rightarrow \bot }"></noscript><span class="lazy-image-placeholder" style="width: 7.168ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" data-alt="{\displaystyle P\rightarrow \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and we also have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow \neg \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow \neg \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc01898cc16ec321ac4b565e2fabbd988cf70f18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.206ex; height:2.176ex;" alt="{\displaystyle P\rightarrow \neg \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 10.206ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc01898cc16ec321ac4b565e2fabbd988cf70f18" data-alt="{\displaystyle P\rightarrow \neg \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Composing that last implication with triple negation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg P\rightarrow \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg P\rightarrow \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4439d01da18810c6313f85f867c21ff42c64d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.268ex; height:2.176ex;" alt="{\displaystyle \neg \neg P\rightarrow \bot }"></noscript><span class="lazy-image-placeholder" style="width: 10.268ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4439d01da18810c6313f85f867c21ff42c64d2" data-alt="{\displaystyle \neg \neg P\rightarrow \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.168ex; height:2.176ex;" alt="{\displaystyle P\rightarrow \bot }"></noscript><span class="lazy-image-placeholder" style="width: 7.168ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" data-alt="{\displaystyle P\rightarrow \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  . As a result, in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable. This result is known as <a href="/wiki/Double-negation_translation" title="Double-negation translation">Glivenko's theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Distributivity">Distributivity</h3> <a role="button" href="/w/index.php?title=Negation&action=edit&section=6" title="Edit section: Distributivity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a> provide a way of <a href="/wiki/Distributive_property" title="Distributive property">distributing</a> negation over <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a> and <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (P\lor Q)\equiv (\neg P\land \neg Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∨</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (P\lor Q)\equiv (\neg P\land \neg Q)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7733a90cbe00f1903dfb0e39c260e014c9a32e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.701ex; height:2.843ex;" alt="{\displaystyle \neg (P\lor Q)\equiv (\neg P\land \neg Q)}"></noscript><span class="lazy-image-placeholder" style="width: 23.701ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7733a90cbe00f1903dfb0e39c260e014c9a32e1a" data-alt="{\displaystyle \neg (P\lor Q)\equiv (\neg P\land \neg Q)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , and</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (P\land Q)\equiv (\neg P\lor \neg Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg (P\land Q)\equiv (\neg P\lor \neg Q)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b34ce92fe822e82a92535a6d2afb8ac1d840a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.701ex; height:2.843ex;" alt="{\displaystyle \neg (P\land Q)\equiv (\neg P\lor \neg Q)}"></noscript><span class="lazy-image-placeholder" style="width: 23.701ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b34ce92fe822e82a92535a6d2afb8ac1d840a6" data-alt="{\displaystyle \neg (P\land Q)\equiv (\neg P\lor \neg Q)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Linearity">Linearity</h3> <a role="button" href="/w/index.php?title=Negation&action=edit&section=7" title="Edit section: Linearity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊕</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oplus }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b16e2bdaefee9eed86d866e6eba3ac47c710f60" data-alt="{\displaystyle \oplus }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  denote the logical <a href="/wiki/Xor" class="mw-redirect" title="Xor">xor</a> operation. In <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, a linear function is one such that: If there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},a_{1},\dots ,a_{n}\in \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},a_{1},\dots ,a_{n}\in \{0,1\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224cb631d2e9cbcf4d7f496df0123fa74be206c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.753ex; height:2.843ex;" alt="{\displaystyle a_{0},a_{1},\dots ,a_{n}\in \{0,1\}}"></noscript><span class="lazy-image-placeholder" style="width: 21.753ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224cb631d2e9cbcf4d7f496df0123fa74be206c9" data-alt="{\displaystyle a_{0},a_{1},\dots ,a_{n}\in \{0,1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(b_{1},b_{2},\dots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \dots \oplus (a_{n}\land b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊕</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∧</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(b_{1},b_{2},\dots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \dots \oplus (a_{n}\land b_{n})}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da68343e4aaa71e605d556df9e0612fef2372f2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.031ex; height:2.843ex;" alt="{\displaystyle f(b_{1},b_{2},\dots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \dots \oplus (a_{n}\land b_{n})}"></noscript><span class="lazy-image-placeholder" style="width: 50.031ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da68343e4aaa71e605d556df9e0612fef2372f2d" data-alt="{\displaystyle f(b_{1},b_{2},\dots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \dots \oplus (a_{n}\land b_{n})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},b_{2},\dots ,b_{n}\in \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1},b_{2},\dots ,b_{n}\in \{0,1\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87cb0f3f85beecfefaa6b41810a8b7c2d80e35f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.056ex; height:2.843ex;" alt="{\displaystyle b_{1},b_{2},\dots ,b_{n}\in \{0,1\}}"></noscript><span class="lazy-image-placeholder" style="width: 21.056ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d87cb0f3f85beecfefaa6b41810a8b7c2d80e35f" data-alt="{\displaystyle b_{1},b_{2},\dots ,b_{n}\in \{0,1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Another way to express this is that each variable always makes a difference in the <a href="/wiki/Truth-value" class="mw-redirect" title="Truth-value">truth-value</a> of the operation, or it never makes a difference. Negation is a linear logical operator. <div class="mw-heading mw-heading3"><h3 id="Self_dual">Self dual</h3> <a role="button" href="/w/index.php?title=Negation&action=edit&section=8" title="Edit section: Self dual" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, a self dual function is a function such that: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a_{1},\dots ,a_{n})=\neg f(\neg a_{1},\dots ,\neg a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi mathvariant="normal">¬</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(a_{1},\dots ,a_{n})=\neg f(\neg a_{1},\dots ,\neg a_{n})}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dab6fe5a6356c7666bb42c3201cd803cb2d3a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.746ex; height:2.843ex;" alt="{\displaystyle f(a_{1},\dots ,a_{n})=\neg f(\neg a_{1},\dots ,\neg a_{n})}"></noscript><span class="lazy-image-placeholder" style="width: 33.746ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dab6fe5a6356c7666bb42c3201cd803cb2d3a8" data-alt="{\displaystyle f(a_{1},\dots ,a_{n})=\neg f(\neg a_{1},\dots ,\neg a_{n})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},\dots ,a_{n}\in \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},\dots ,a_{n}\in \{0,1\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0eed02a7d3fc2ee57a5a62960bd40d181ec3385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.435ex; height:2.843ex;" alt="{\displaystyle a_{1},\dots ,a_{n}\in \{0,1\}}"></noscript><span class="lazy-image-placeholder" style="width: 18.435ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0eed02a7d3fc2ee57a5a62960bd40d181ec3385" data-alt="{\displaystyle a_{1},\dots ,a_{n}\in \{0,1\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Negation is a self dual logical operator. <div class="mw-heading mw-heading3"><h3 id="Negations_of_quantifiers">Negations of quantifiers</h3> <a role="button" href="/w/index.php?title=Negation&action=edit&section=9" title="Edit section: Negations of quantifiers" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> In <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, there are two quantifiers, one is the universal quantifier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }"></noscript><span class="lazy-image-placeholder" style="width: 1.293ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" data-alt="{\displaystyle \forall }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (means "for all") and the other is the existential quantifier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }"></noscript><span class="lazy-image-placeholder" style="width: 1.293ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" data-alt="{\displaystyle \exists }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (means "there exists"). The negation of one quantifier is the other quantifier (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93041045df4feb6cce740a7773e31dd9576c1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.212ex; height:2.843ex;" alt="{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}"></noscript><span class="lazy-image-placeholder" style="width: 21.212ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93041045df4feb6cce740a7773e31dd9576c1e0" data-alt="{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd301c4f840a19ec7bff3606ed9160b011bd19d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.212ex; height:2.843ex;" alt="{\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)}"></noscript><span class="lazy-image-placeholder" style="width: 21.212ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd301c4f840a19ec7bff3606ed9160b011bd19d8" data-alt="{\displaystyle \neg \exists xP(x)\equiv \forall x\neg P(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ). For example, with the predicate P as "x is mortal" and the domain of x as the collection of all humans, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall xP(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall xP(x)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25873948fc98344950ea1b91f88dd52239cf9c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.507ex; height:2.843ex;" alt="{\displaystyle \forall xP(x)}"></noscript><span class="lazy-image-placeholder" style="width: 7.507ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25873948fc98344950ea1b91f88dd52239cf9c87" data-alt="{\displaystyle \forall xP(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  means "a person x in all humans is mortal" or "all humans are mortal". The negation of it is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93041045df4feb6cce740a7773e31dd9576c1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.212ex; height:2.843ex;" alt="{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}"></noscript><span class="lazy-image-placeholder" style="width: 21.212ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93041045df4feb6cce740a7773e31dd9576c1e0" data-alt="{\displaystyle \neg \forall xP(x)\equiv \exists x\neg P(x)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , meaning "there exists a person x in all humans who is not mortal", or "there exists someone who lives forever". </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Rules_of_inference">Rules of inference</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=10" title="Edit section: Rules of inference" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Double_negation" title="Double negation">Double negation</a></div> There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negation in a <a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a> setting is to take as primitive rules of inference negation introduction (from a derivation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  to both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></noscript><span class="lazy-image-placeholder" style="width: 1.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" data-alt="{\displaystyle Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg Q}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad34798abb0bbbc063c906e459f103a09b1660e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle \neg Q}"></noscript><span class="lazy-image-placeholder" style="width: 3.389ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad34798abb0bbbc063c906e459f103a09b1660e" data-alt="{\displaystyle \neg Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ; this rule also being called <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a>), negation elimination (from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></noscript><span class="lazy-image-placeholder" style="width: 1.838ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" data-alt="{\displaystyle Q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ; this rule also being called ex falso quodlibet), and double negation elimination (from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d7b8bafb9762c3a07924b6c06be6e08cb5680f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.846ex; height:2.176ex;" alt="{\displaystyle \neg \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 4.846ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d7b8bafb9762c3a07924b6c06be6e08cb5680f" data-alt="{\displaystyle \neg \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ). One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination. Negation introduction states that if an absurdity can be drawn as conclusion from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  must not be the case (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from an absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" 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 \bot }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f282c7bc331cc3bfcf1c57f1452cc23c022f58de" data-alt="{\displaystyle \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In this case the rule says that from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity. Typically the intuitionistic negation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \neg P}"></noscript><span class="lazy-image-placeholder" style="width: 3.296ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eb0d6c8752f8c7256d69c62e77dfe4c466dbe58" data-alt="{\displaystyle \neg P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></noscript><span class="lazy-image-placeholder" style="width: 1.745ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" data-alt="{\displaystyle P}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow \bot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">⊥</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow \bot }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.168ex; height:2.176ex;" alt="{\displaystyle P\rightarrow \bot }"></noscript><span class="lazy-image-placeholder" style="width: 7.168ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21af5d568c2fd7384626959ef1f5985e2f59c298" data-alt="{\displaystyle P\rightarrow \bot }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Then negation introduction and elimination are just special cases of implication introduction (<a href="/wiki/Conditional_proof" title="Conditional proof">conditional proof</a>) and elimination (<a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>). In this case one must also add as a primitive rule ex falso quodlibet. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Programming_language_and_ordinary_language">Programming language and ordinary language</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=11" title="Edit section: Programming language and ordinary language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable selfref">"!vote" redirects here. For use of !votes in Wikipedia discussions, see <a href="/wiki/Wikipedia:Polling_is_not_a_substitute_for_discussion#Not-votes" title="Wikipedia:Polling is not a substitute for discussion">Wikipedia:Polling is not a substitute for discussion § Not-votes</a>.</div> As in mathematics, negation is used in <a href="/wiki/Computer_science" title="Computer science">computer science</a> to construct logical statements. <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>if (!(r == t)) { /*...statements executed when r does NOT equal t...*/ } </pre></div> The <a href="/wiki/Exclamation_mark" title="Exclamation mark">exclamation mark</a> "<code>!</code>" signifies logical NOT in <a href="/wiki/B_(programming_language)" title="B (programming language)">B</a>, <a href="/wiki/C_Programming_Language" class="mw-redirect" title="C Programming Language">C</a>, and languages with a C-inspired syntax such as <a href="/wiki/C%2B%2B" title="C++">C++</a>, <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, <a href="/wiki/JavaScript" title="JavaScript">JavaScript</a>, <a href="/wiki/Perl" title="Perl">Perl</a>, and <a href="/wiki/PHP" title="PHP">PHP</a>. "<code>NOT</code>" is the operator used in <a href="/wiki/ALGOL_60" title="ALGOL 60">ALGOL 60</a>, <a href="/wiki/BASIC" title="BASIC">BASIC</a>, and languages with an ALGOL- or BASIC-inspired syntax such as <a href="/wiki/Pascal_programming_language" class="mw-redirect" title="Pascal programming language">Pascal</a>, <a href="/wiki/Ada_programming_language" class="mw-redirect" title="Ada programming language">Ada</a>, <a href="/wiki/Eiffel_(programming_language)" title="Eiffel (programming language)">Eiffel</a> and <a href="/wiki/Seed7" title="Seed7">Seed7</a>. Some languages (C++, Perl, etc.) provide more than one operator for negation. A few languages like <a href="/wiki/PL/I" title="PL/I">PL/I</a> and <a href="/wiki/Ratfor" title="Ratfor">Ratfor</a> use <code>¬</code> for negation. Most modern languages allow the above statement to be shortened from <code>if (!(r == t))</code> to <code>if (r != t)</code>, which allows sometimes, when the compiler/interpreter is not able to optimize it, faster programs. In computer science there is also bitwise negation. This takes the value given and switches all the <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary</a> 1s to 0s and 0s to 1s. See <a href="/wiki/Bitwise_operation" title="Bitwise operation">bitwise operation</a>. This is often used to create <a href="/wiki/Signed_number_representations" title="Signed number representations">ones' complement</a> or "<code>~</code>" in C or C++ and <a href="/wiki/Two%27s_complement" title="Two's complement">two's complement</a> (just simplified to "<code>-</code>" or the negative sign since this is equivalent to taking the arithmetic negative value of the number) as it basically creates the opposite (negative value equivalent) or mathematical complement of the value (where both values are added together they create a whole). To get the absolute (positive equivalent) value of a given integer the following would work as the "<code>-</code>" changes it from negative to positive (it is negative because "<code>x < 0</code>" yields true) <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>unsigned int abs(int x) { if (x < 0) return -x; else return x; } </pre></div> To demonstrate logical negation: <div class="mw-highlight mw-highlight-lang-cpp mw-content-ltr" dir="ltr"><pre>unsigned int abs(int x) { if (!(x < 0)) return x; else return -x; } </pre></div> Inverting the condition and reversing the outcomes produces code that is logically equivalent to the original code, i.e. will have identical results for any input (depending on the compiler used, the actual instructions performed by the computer may differ). In C (and some other languages descended from C), double negation (<code>!!x</code>) is used as an idiom to convert <code>x</code> to a canonical Boolean, ie. an integer with a value of either 0 or 1 and no other. Although any integer other than 0 is logically true in C and 1 is not special in this regard, it is sometimes important to ensure that a canonical value is used, for example for printing or if the number is subsequently used for arithmetic operations.<a href="#cite_note-6">[6]</a> The convention of using <code>!</code> to signify negation occasionally surfaces in ordinary written speech, as computer-related <a href="/wiki/Slang" title="Slang">slang</a> for not. For example, the phrase <code>!voting</code> means "not voting". Another example is the phrase <code>!clue</code> which is used as a synonym for "no-clue" or "clueless".<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Kripke_semantics">Kripke semantics</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=12" title="Edit section: Kripke semantics" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> In <a href="/wiki/Kripke_semantics" title="Kripke semantics">Kripke semantics</a> where the semantic values of formulae are sets of <a href="/wiki/Possible_world" title="Possible world">possible worlds</a>, negation can be taken to mean <a href="/wiki/Set-theoretic_complement" class="mw-redirect" title="Set-theoretic complement">set-theoretic complementation</a>[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] (see also <a href="/wiki/Possible_world_semantics" class="mw-redirect" title="Possible world semantics">possible world semantics</a> for more). </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=13" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-7 collapsible-block" id="mf-section-7"> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Affirmation_and_negation" title="Affirmation and negation">Affirmation and negation</a> (grammatical polarity)</li> <li><a href="/wiki/Ampheck" class="mw-redirect" title="Ampheck">Ampheck</a></li> <li><a href="/wiki/Apophasis" title="Apophasis">Apophasis</a></li> <li><a href="/wiki/Binary_opposition" title="Binary opposition">Binary opposition</a></li> <li><a href="/wiki/Bitwise_NOT" class="mw-redirect" title="Bitwise NOT">Bitwise NOT</a></li> <li><a href="/wiki/Contraposition" title="Contraposition">Contraposition</a></li> <li><a href="/wiki/Cyclic_negation" title="Cyclic negation">Cyclic negation</a></li> <li><a href="/wiki/Negation_as_failure" title="Negation as failure">Negation as failure</a></li> <li><a href="/wiki/NOT_gate" class="mw-redirect" title="NOT gate">NOT gate</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=14" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-8 collapsible-block" id="mf-section-8"> <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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <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="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Negation.html">"Negation"</a>. mathworld.wolfram.com. Retrieved 2 September 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Negation&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FNegation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.math.toronto.edu/preparing-for-calculus/3_logic/we_3_negation.html">"Logic and Mathematical Statements - Worked Examples"</a>. www.math.toronto.edu. Retrieved 2 September 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.toronto.edu&rft.atitle=Logic+and+Mathematical+Statements+-+Worked+Examples&rft_id=https%3A%2F%2Fwww.math.toronto.edu%2Fpreparing-for-calculus%2F3_logic%2Fwe_3_negation.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"> </li> <li id="cite_note-:21-3">^ <a href="#cite_ref-:21_3-0">a</a> <a href="#cite_ref-:21_3-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeall2010" class="citation book cs1">Beall, Jeffrey C. (2010). Logic: the basics. The basics (1. publ ed.). London: Routledge. p. 57. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-203-85155-5" title="Special:BookSources/978-0-203-85155-5"><bdi>978-0-203-85155-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic%3A+the+basics&rft.place=London&rft.series=The+basics&rft.pages=57&rft.edition=1.+publ&rft.pub=Routledge&rft.date=2010&rft.isbn=978-0-203-85155-5&rft.aulast=Beall&rft.aufirst=Jeffrey+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> Used as makeshift in early typewriter publications, e.g. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_E._Ladner1975" class="citation journal cs1">Richard E. Ladner (January 1975). "The circuit value problem is log space complete for P". ACM SIGACT News. 7 (101): 18–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F990518.990519">10.1145/990518.990519</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGACT+News&rft.atitle=The+circuit+value+problem+is+log+space+complete+for+P&rft.volume=7&rft.issue=101&rft.pages=18-20&rft.date=1975-01&rft_id=info%3Adoi%2F10.1145%2F990518.990519&rft.au=Richard+E.+Ladner&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'DonnellHallPage2007" class="citation cs2">O'Donnell, John; Hall, Cordelia; Page, Rex (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KKxyQQWQam4C&pg=PA120">Discrete Mathematics Using a Computer</a>, Springer, p. 120, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781846285981" title="Special:BookSources/9781846285981"><bdi>9781846285981</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+Using+a+Computer&rft.pages=120&rft.pub=Springer&rft.date=2007&rft.isbn=9781846285981&rft.aulast=O%27Donnell&rft.aufirst=John&rft.au=Hall%2C+Cordelia&rft.au=Page%2C+Rex&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKKxyQQWQam4C%26pg%3DPA120&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988">. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEgan" class="citation web cs1">Egan, David. <a rel="nofollow" class="external text" href="https://dev-notes.eu/2019/10/Double-Negation-Operator-Convert-to-Boolean-in-C/">"Double Negation Operator Convert to Boolean in C"</a>. Dev Notes.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Dev+Notes&rft.atitle=Double+Negation+Operator+Convert+to+Boolean+in+C&rft.aulast=Egan&rft.aufirst=David&rft_id=https%3A%2F%2Fdev-notes.eu%2F2019%2F10%2FDouble-Negation-Operator-Convert-to-Boolean-in-C%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="/wiki/Eric_S._Raymond" title="Eric S. Raymond">Raymond, Eric</a> and Steele, Guy. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g80P_4v4QbIC&pg=PA18&lpg=PA18">The New Hacker's Dictionary</a>, p. 18 (MIT Press 1996). </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Munat, Judith. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UOPXXYslemYC&pg=PA148&lpg=PA148">Lexical Creativity, Texts and Context</a>, p. 148 (John Benjamins Publishing, 2007). </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="Further_reading">Further reading</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=15" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-9 collapsible-block" id="mf-section-9"> <ul><li><a href="/wiki/Dov_Gabbay" title="Dov Gabbay">Gabbay, Dov</a>, and Wansing, Heinrich, eds., 1999. What is Negation?, <a href="/wiki/Kluwer" class="mw-redirect" title="Kluwer">Kluwer</a>.</li> <li><a href="/wiki/Laurence_R._Horn" title="Laurence R. Horn">Horn, L.</a>, 2001. A Natural History of Negation, <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>.</li> <li><a href="/wiki/G._H._von_Wright" class="mw-redirect" title="G. H. von Wright">G. H. von Wright</a>, 1953–59, "On the Logic of Negation", Commentationes Physico-Mathematicae 22.</li> <li>Wansing, Heinrich, 2001, "Negation", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic, <a href="/wiki/Wiley-Blackwell" title="Wiley-Blackwell">Blackwell</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTettamantiManentiDella_RosaFalini2008" class="citation journal cs1">Tettamanti, Marco; Manenti, Rosa; Della Rosa, Pasquale A.; Falini, Andrea; Perani, Daniela; Cappa, Stefano F.; Moro, Andrea (2008). "Negation in the brain: Modulating action representation". NeuroImage. 43 (2): 358–367. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.neuroimage.2008.08.004">10.1016/j.neuroimage.2008.08.004</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18771737">18771737</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:17658822">17658822</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NeuroImage&rft.atitle=Negation+in+the+brain%3A+Modulating+action+representation&rft.volume=43&rft.issue=2&rft.pages=358-367&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17658822%23id-name%3DS2CID&rft_id=info%3Apmid%2F18771737&rft_id=info%3Adoi%2F10.1016%2Fj.neuroimage.2008.08.004&rft.aulast=Tettamanti&rft.aufirst=Marco&rft.au=Manenti%2C+Rosa&rft.au=Della+Rosa%2C+Pasquale+A.&rft.au=Falini%2C+Andrea&rft.au=Perani%2C+Daniela&rft.au=Cappa%2C+Stefano+F.&rft.au=Moro%2C+Andrea&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Negation&action=edit&section=16" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div><section class="mf-section-10 collapsible-block" id="mf-section-10"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornWansing" class="citation encyclopaedia cs1">Horn, Laurence R.; Wansing, Heinrich. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/negation/">"Negation"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Negation&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.aulast=Horn&rft.aufirst=Laurence+R.&rft.au=Wansing%2C+Heinrich&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fnegation%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Negation">"Negation"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Negation&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNegation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/NOT.html">NOT</a>, on <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li></ul> <dl><dt><a href="/wiki/Truth_table" title="Truth table">Tables of Truth</a> of composite clauses</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ramsey/Logic/NotAnd.html">"Table of truth for a NOT clause applied to an END sentence"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000301195359/http://www.math.hawaii.edu/~ramsey/Logic/NotAnd.html">Archived</a> from the original on 1 March 2000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Table+of+truth+for+a+NOT+clause+applied+to+an+END+sentence&rft_id=http%3A%2F%2Fwww.math.hawaii.edu%2F~ramsey%2FLogic%2FNotAnd.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ramsey/Logic/NotAnd.html">"NOT clause of an END sentence"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000301195359/http://www.math.hawaii.edu/~ramsey/Logic/NotAnd.html">Archived</a> from the original on 1 March 2000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NOT+clause+of+an+END+sentence&rft_id=http%3A%2F%2Fwww.math.hawaii.edu%2F~ramsey%2FLogic%2FNotAnd.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ramsey/Logic/NotOr.html">"NOT clause of an OR sentence"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000117134708/http://www.math.hawaii.edu/~ramsey/Logic/NotOr.html">Archived</a> from the original on 17 January 2000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NOT+clause+of+an+OR+sentence&rft_id=http%3A%2F%2Fwww.math.hawaii.edu%2F~ramsey%2FLogic%2FNotOr.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.hawaii.edu/~ramsey/Logic/NotIfThen.html">"NOT clause of an IF...THEN period"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000301223435/http://www.math.hawaii.edu/~ramsey/Logic/NotIfThen.html/">Archived</a> from the original on 1 March 2000.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=NOT+clause+of+an+IF...THEN+period&rft_id=http%3A%2F%2Fwww.math.hawaii.edu%2F~ramsey%2FLogic%2FNotIfThen.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANegation" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px 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height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Negation&oldid=1258013704">https://en.wikipedia.org/w/index.php?title=Negation&oldid=1258013704</a>"</div></div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"> <a class="last-modified-bar" href="/w/index.php?title=Negation&action=history"> <div class="post-content last-modified-bar__content"> Last edited on 17 November 2024, at 18:15 </div> </a> <div class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%81%D9%8A_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="نفي (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="نفي (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0nkar" title="İnkar – Azerbaijani" lang="az" hreflang="az" data-title="İnkar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D1%82%D1%80%D0%B8%D1%86%D0%B0%D0%BD%D0%B8%D0%B5" title="Отрицание – Bulgarian" lang="bg" hreflang="bg" data-title="Отрицание" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Negaci%C3%B3_l%C3%B2gica" title="Negació lògica – Catalan" lang="ca" hreflang="ca" data-title="Negació lògica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Negace" title="Negace – Czech" lang="cs" hreflang="cs" data-title="Negace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Negation" title="Negation – Danish" lang="da" hreflang="da" data-title="Negation" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Negation" title="Negation – German" lang="de" hreflang="de" data-title="Negation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Eitus" title="Eitus – Estonian" lang="et" hreflang="et" data-title="Eitus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Negasi%C3%B2un_(matem%C3%A0tica)" title="Negasiòun (matemàtica) – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Negasiòun (matemàtica)" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Negaci%C3%B3n_l%C3%B3gica" title="Negación lógica – Spanish" lang="es" hreflang="es" data-title="Negación lógica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Logika_neo" title="Logika neo – Esperanto" lang="eo" hreflang="eo" data-title="Logika neo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Ukapen" title="Ukapen – Basque" lang="eu" hreflang="eu" data-title="Ukapen" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%82%DB%8C%D8%B6" title="نقیض – Persian" lang="fa" hreflang="fa" data-title="نقیض" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/N%C3%A9gation_logique" title="Négation logique – French" lang="fr" hreflang="fr" data-title="Négation logique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Negaci%C3%B3n_(l%C3%B3xica)" title="Negación (lóxica) – Galician" lang="gl" hreflang="gl" data-title="Negación (lóxica)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%EC%A0%95_(%EB%85%BC%EB%A6%AC%ED%95%99)" title="부정 (논리학) – Korean" lang="ko" hreflang="ko" data-title="부정 (논리학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BA%D5%AD%D5%BF%D5%B8%D6%82%D5%B4_(%D5%BF%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Ժխտում (տրամաբանություն) – Armenian" lang="hy" hreflang="hy" data-title="Ժխտում (տրամաբանություն)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</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%B7%E0%A5%87%E0%A4%A7_(%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95)" title="निषेध (तर्क) – Hindi" lang="hi" hreflang="hi" data-title="निषेध (तर्क)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Negacija" title="Negacija – Croatian" lang="hr" hreflang="hr" data-title="Negacija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Negasi" title="Negasi – Indonesian" lang="id" hreflang="id" data-title="Negasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Negazione_(matematica)" title="Negazione (matematica) – Italian" lang="it" hreflang="it" data-title="Negazione (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9C%D7%90_(%D7%9C%D7%95%D7%92%D7%99%D7%A7%D7%94)" title="לא (לוגיקה) – Hebrew" lang="he" hreflang="he" data-title="לא (לוגיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D1%96%D1%81%D1%82%D0%B5%D1%83" title="Терістеу – Kazakh" lang="kk" hreflang="kk" data-title="Терістеу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Negatio" title="Negatio – Latin" lang="la" hreflang="la" data-title="Negatio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Neg%C3%A1ci%C3%B3" title="Negáció – Hungarian" lang="hu" hreflang="hu" data-title="Negáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B5%D0%B3%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Негација – Macedonian" lang="mk" hreflang="mk" data-title="Негација" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Negasi" title="Negasi – Malay" lang="ms" hreflang="ms" data-title="Negasi" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logische_negatie" title="Logische negatie – Dutch" lang="nl" hreflang="nl" data-title="Logische negatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%90%A6%E5%AE%9A" title="否定 – Japanese" lang="ja" hreflang="ja" data-title="否定" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Negasjon" title="Negasjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Negasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Negassion" title="Negassion – Piedmontese" lang="pms" hreflang="pms" data-title="Negassion" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Negacja" title="Negacja – Polish" lang="pl" hreflang="pl" data-title="Negacja" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Nega%C3%A7%C3%A3o" title="Negação – Portuguese" lang="pt" hreflang="pt" data-title="Negação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D1%82%D1%80%D0%B8%D1%86%D0%B0%D0%BD%D0%B8%D0%B5" title="Отрицание – Russian" lang="ru" hreflang="ru" data-title="Отрицание" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Negacioni" title="Negacioni – Albanian" lang="sq" hreflang="sq" data-title="Negacioni" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logical_negation" title="Logical negation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logical negation" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Neg%C3%A1cia_(logika)" title="Negácia (logika) – Slovak" lang="sk" hreflang="sk" data-title="Negácia (logika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Negacija" title="Negacija – Slovenian" lang="sl" hreflang="sl" data-title="Negacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%DB%95%D8%B1%DB%8E%DA%A9%D8%B1%D8%AF%D9%86" title="نەرێکردن – Central Kurdish" lang="ckb" hreflang="ckb" data-title="نەرێکردن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D1%87%D0%BA%D0%B0_%D0%BD%D0%B5%D0%B3%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Логичка негација – Serbian" lang="sr" hreflang="sr" data-title="Логичка негација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Logi%C4%8Dka_negacija" title="Logička negacija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Logička negacija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Negation" title="Negation – Swedish" lang="sv" hreflang="sv" data-title="Negation" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Negasyon" title="Negasyon – Tagalog" lang="tl" hreflang="tl" data-title="Negasyon" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%99%E0%B8%B4%E0%B9%80%E0%B8%AA%E0%B8%98" title="นิเสธ – Thai" lang="th" hreflang="th" data-title="นิเสธ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%98%D0%BD%D0%BA%D0%BE%D1%80" title="Инкор – Tajik" lang="tg" hreflang="tg" data-title="Инкор" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/T%C3%BCmleme_(mant%C4%B1k)" title="Tümleme (mantık) – Turkish" lang="tr" hreflang="tr" data-title="Tümleme (mantık)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BF%D0%B5%D1%80%D0%B5%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Заперечення – Ukrainian" lang="uk" hreflang="uk" data-title="Заперечення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E9%9D%9E" title="邏輯非 – Cantonese" lang="yue" hreflang="yue" data-title="邏輯非" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%80%BB%E8%BE%91%E9%9D%9E" title="逻辑非 – Chinese" lang="zh" hreflang="zh" data-title="逻辑非" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> </div> <div class="minerva-footer-logo"><img src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" alt="Wikipedia" width="120" height="18" style="width: 7.5em; height: 1.125em;"/> </div> <ul id="footer-info" class="footer-info hlist hlist-separated"> <li id="footer-info-lastmod"> This page was last edited on 17 November 2024, at 18:15 (UTC).</li> <li id="footer-info-copyright">Content is available under <a class="external" rel="nofollow" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">CC BY-SA 4.0</a> unless otherwise noted.</li> </ul> <ul id="footer-places" class="footer-places hlist hlist-separated"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a 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Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-12-23T14:50:39Z","dateModified":"2024-11-17T18:15:00Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/73\/Venn10.svg","headline":"operation that takes a proposition p to another proposition \"not p\", written \u00acp, which is interpreted intuitively as being true when p is false, and false when p is true; unary (single-argument) logical connective"}</script><script>(window.NORLQ=window.NORLQ||[]).push(function(){var 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