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class="vector-toc-link" href="#Correspondence_to_other_mathematical_frameworks"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Correspondence to other mathematical frameworks</span> </div> </a> <button aria-controls="toc-Correspondence_to_other_mathematical_frameworks-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Correspondence to other mathematical frameworks subsection</span> </button> <ul id="toc-Correspondence_to_other_mathematical_frameworks-sublist" class="vector-toc-list"> <li id="toc-Algebraic_semantics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_semantics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Algebraic semantics</span> </div> </a> <ul id="toc-Algebraic_semantics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Probability calculus</span> </div> </a> <ul id="toc-Probability_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subjective_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subjective_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Subjective logic</span> </div> </a> <ul id="toc-Subjective_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Alleged_cases_of_failure" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Alleged_cases_of_failure"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Alleged cases of failure</span> </div> </a> <ul id="toc-Alleged_cases_of_failure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Possible_fallacies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Possible_fallacies"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Possible fallacies</span> </div> </a> <ul id="toc-Possible_fallacies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul 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Available in 29 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%8A%D8%A7%D8%B3_%D8%A7%D8%B3%D8%AA%D8%AB%D9%86%D8%A7%D8%A6%D9%8A" title="قياس استثنائي – Arabic" lang="ar" hreflang="ar" data-title="قياس استثنائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D1%81_%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%81" title="Модус поненс – Bulgarian" lang="bg" hreflang="bg" data-title="Модус поненс" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Modus_ponendo_ponens" title="Modus ponendo ponens – Catalan" lang="ca" hreflang="ca" data-title="Modus ponendo ponens" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Czech" lang="cs" hreflang="cs" data-title="Modus ponens" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Danish" lang="da" hreflang="da" data-title="Modus ponens" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – German" lang="de" hreflang="de" data-title="Modus ponens" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Estonian" lang="et" hreflang="et" data-title="Modus ponens" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Modus_ponendo_ponens" title="Modus ponendo ponens – Spanish" lang="es" hreflang="es" data-title="Modus ponendo ponens" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Modus_ponendo_ponens" title="Modus ponendo ponens – Basque" lang="eu" hreflang="eu" data-title="Modus ponendo ponens" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%88%D8%B6%D8%B9_%D9%85%D9%82%D8%AF%D9%85" title="وضع مقدم – Persian" lang="fa" hreflang="fa" data-title="وضع مقدم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – French" lang="fr" hreflang="fr" data-title="Modus ponens" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EA%B1%B4_%EA%B8%8D%EC%A0%95" title="전건 긍정 – Korean" lang="ko" hreflang="ko" data-title="전건 긍정" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Croatian" lang="hr" hreflang="hr" data-title="Modus ponens" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Indonesian" lang="id" hreflang="id" data-title="Modus ponens" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/J%C3%A1kv%C3%A6%C3%B0_j%C3%A1tunarregla" title="Jákvæð játunarregla – Icelandic" lang="is" hreflang="is" data-title="Jákvæð játunarregla" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Italian" lang="it" hreflang="it" data-title="Modus ponens" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%93%D7%95%D7%A1_%D7%A4%D7%95%D7%A0%D7%A0%D7%A1" title="מודוס פוננס – Hebrew" lang="he" hreflang="he" data-title="מודוס פוננס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Hungarian" lang="hu" hreflang="hu" data-title="Modus ponens" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Dutch" lang="nl" hreflang="nl" data-title="Modus ponens" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%A2%E3%83%BC%E3%83%80%E3%82%B9%E3%83%9D%E3%83%8D%E3%83%B3%E3%82%B9" title="モーダスポネンス – Japanese" lang="ja" hreflang="ja" data-title="モーダスポネンス" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Modus_ponendo_ponens" title="Modus ponendo ponens – Polish" lang="pl" hreflang="pl" data-title="Modus ponendo ponens" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Portuguese" lang="pt" hreflang="pt" data-title="Modus ponens" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Russian" lang="ru" hreflang="ru" data-title="Modus ponens" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Simple English" lang="en-simple" hreflang="en-simple" data-title="Modus ponens" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Slovak" lang="sk" hreflang="sk" data-title="Modus ponens" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BE%D0%B4%D1%83%D1%81_%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%81" title="Модус поненс – Serbian" lang="sr" hreflang="sr" data-title="Модус поненс" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Swedish" lang="sv" hreflang="sv" data-title="Modus ponens" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/Modus_ponens" title="Modus ponens – Ukrainian" lang="uk" hreflang="uk" data-title="Modus ponens" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%82%AF%E5%AE%9A%E5%89%8D%E4%BB%B6" title="肯定前件 – Chinese" lang="zh" hreflang="zh" data-title="肯定前件" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q655742#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> 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For other uses, see <a href="/wiki/Forward_chaining" title="Forward chaining">Forward chaining</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Rule of logical inference</div> <p class="mw-empty-elt"> </p> <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><table class="infobox vcard"><caption class="infobox-title fn" style="padding-bottom:0.2em;"><em>Modus ponens</em></caption><tbody><tr><th scope="row" class="infobox-label">Type</th><td class="infobox-data"><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><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deductive</a> <a href="/wiki/Argument_form" class="mw-redirect" title="Argument form">argument form</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">Field</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">Statement</th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is true. Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> must also be true.</td></tr><tr><th scope="row" class="infobox-label">Symbolic statement</th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to Q,\;P\;\vdash \ Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>P</mi> <mspace width="thickmathspace" /> <mo>⊢</mo> <mtext> </mtext> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to Q,\;P\;\vdash \ Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab15e54594e644f0ead345aa509ddd4d234d837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.397ex; height:2.509ex;" alt="{\displaystyle P\to Q,\;P\;\vdash \ Q}"></span></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output 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li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Rule_of_inference" title="Rule of inference">Transformation rules</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%; border-bottom:1px #fefefe solid;"> <a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Conditional_proof" title="Conditional proof"><span>Implication introduction</span></a> / <a class="mw-selflink selflink"><span title="A→B,   A   ⊢   B">elimination (<i>modus ponens</i>)</span></a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction"><span title="A→B,   B→A   ⊢   A↔B">Biconditional introduction</span></a> / <a href="/wiki/Biconditional_elimination" title="Biconditional elimination"><span title="A↔B   ⊢   A→B">elimination</span></a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction"><span title="A,   B   ⊢   A∧B">Conjunction introduction</span></a> / <a href="/wiki/Conjunction_elimination" title="Conjunction elimination"><span title="A∧B   ⊢   A">elimination</span></a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction"><span title="A   ⊢   A∨B">Disjunction introduction</span></a> / <a href="/wiki/Disjunction_elimination" title="Disjunction elimination"><span title="A∨B,   A→C,   B→C   ⊢   C">elimination</span></a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism"><span title="A∨B,   ¬A   ⊢   B">Disjunctive</span></a> / <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism"><span title="A→B,   B→C   ⊢   A→C">hypothetical syllogism</span></a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma"><span title="A→P,   B→Q,   A∨B   ⊢   P∨Q">Constructive</span></a> / <a href="/wiki/Destructive_dilemma" title="Destructive dilemma"><span title="A→P,   B→Q,   ¬P∨¬Q   ⊢   ¬A∨¬B">destructive dilemma</span></a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)"><span title="A→B   ⊢   A→A∧B">Absorption</span></a> / <a href="/wiki/Modus_tollens" title="Modus tollens"><span title="A→B,   ¬B   ⊢   ¬A"><i>modus tollens</i></span></a> / <a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens"><span title="¬(A∧B),   A   ⊢   ¬B"><i>modus ponendo tollens</i></span></a></li> <li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation introduction</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_replacement" title="Rule of replacement">Rules of replacement</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <div class="hlist"> <ul><li><a href="/wiki/Associative_property#Propositional_logic" title="Associative property"><span title="A∨(B∨C)   =   (A∨B)∨C">Associativity</span></a></li> <li><a href="/wiki/Commutative_property#Propositional_logic" title="Commutative property"><span title="A∨B   =   B∨A">Commutativity</span></a></li> <li><a href="/wiki/Distributive_property#Propositional_logic" title="Distributive property"><span title="A∧(B∨C)   =   (A∧B)∨(A∧C)">Distributivity</span></a></li> <li><a href="/wiki/Double_negation" title="Double negation"><span title="¬¬A   =   A">Double negation</span></a></li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)"><span title="A→B   ⊢   ¬A∨B">Material implication</span></a></li> <li><a href="/wiki/Exportation_(logic)" title="Exportation (logic)"><span title="(A∧B)→C   ⊢   A→(B→C)">Exportation</span></a></li> <li><a href="/wiki/Tautology_(rule_of_inference)" title="Tautology (rule of inference)"><span title="A∨A   =   A">Tautology</span></a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;;background:#ddf;font-size:110%;"> <a href="/wiki/First-order_logic" title="First-order logic">Predicate logic</a></th></tr><tr><th class="sidebar-heading" style="background:#eaeaff;"> <a href="/wiki/Rule_of_inference" title="Rule of inference">Rules of inference</a></th></tr><tr><td class="sidebar-content" style="padding-top:0.15em;"> <ul><li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal generalization</a> / <a href="/wiki/Universal_instantiation" title="Universal instantiation">instantiation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential generalization</a> / <a href="/wiki/Existential_instantiation" title="Existential instantiation">instantiation</a></li></ul></td> </tr></tbody></table> <p>In <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional logic</a>, <b><span title="Latin-language text"><i lang="la">modus ponens</i></span></b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'m' in 'my'">m</span><span title="/oʊ/: 'o' in 'code'">oʊ</span><span title="'d' in 'dye'">d</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'s' in 'sigh'">s</span></span><span class="wrap"> </span><span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'p' in 'pie'">p</span><span title="/oʊ/: 'o' in 'code'">oʊ</span><span title="'n' in 'nigh'">n</span><span title="/ɛ/: 'e' in 'dress'">ɛ</span><span title="'n' in 'nigh'">n</span><span title="'z' in 'zoom'">z</span></span>/</a></span></span>; <b>MP</b>), also known as <b><span title="Latin-language text"><i lang="la">modus ponendo ponens</i></span></b> (from <a href="/wiki/Latin_language" class="mw-redirect" title="Latin language">Latin</a> 'method of putting by placing'),<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <b>implication elimination</b>, or <b>affirming the antecedent</b>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> is a <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive</a> <a href="/wiki/Argument_form" class="mw-redirect" title="Argument form">argument form</a> and <a href="/wiki/Rule_of_inference" title="Rule of inference">rule of inference</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It can be summarized as "<i>P</i> <a href="/wiki/Material_conditional" title="Material conditional">implies</a> <i>Q.</i> <i>P</i> is true. Therefore, <i>Q</i> must also be true." </p><p><i>Modus ponens</i> is a mixed <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a> and is closely related to another <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> form of argument, <i><a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a></i>. Both have apparently similar but invalid forms: <a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a> and <a href="/wiki/Denying_the_antecedent" title="Denying the antecedent">denying the antecedent</a>. <a href="/wiki/Constructive_dilemma" title="Constructive dilemma">Constructive dilemma</a> is the <a href="/wiki/Logical_disjunction" title="Logical disjunction">disjunctive</a> version of <i>modus ponens</i>. </p><p>The history of <i>modus ponens</i> goes back to <a href="/wiki/Classical_antiquity" title="Classical antiquity">antiquity</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The first to explicitly describe the argument form <i>modus ponens</i> was <a href="/wiki/Theophrastus" title="Theophrastus">Theophrastus</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> It, along with <i><a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a></i>, is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=1" title="Edit section: Explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The form of a <i>modus ponens</i> argument is a mixed <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a>, with two premises and a conclusion: </p> <ol><li>If <i>P</i>, then <i>Q</i>.</li> <li><i>P</i>.</li> <li>Therefore, <i>Q</i>.</li></ol> <p>The first premise is a <a href="/wiki/Material_conditional" title="Material conditional">conditional</a> ("if–then") claim, namely that <i>P</i> implies <i>Q</i>. The second premise is an assertion that <i>P</i>, the <a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a> of the conditional claim, is the case. From these two premises it can be logically concluded that <i>Q</i>, the <a href="/wiki/Consequent" title="Consequent">consequent</a> of the conditional claim, must be the case as well. </p><p>An example of an argument that fits the form <i>modus ponens</i>: </p> <ol><li>If today is Tuesday, then John will go to work.</li> <li>Today is Tuesday.</li> <li>Therefore, John will go to work.</li></ol> <p>This argument is <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a>, but this has no bearing on whether any of the statements in the argument are actually <a href="/wiki/Truth" title="Truth">true</a>; for <i>modus ponens</i> to be a <a href="/wiki/Soundness" title="Soundness">sound</a> argument, the premises must be true for any true instances of the conclusion. An <a href="/wiki/Argument" title="Argument">argument</a> can be valid but nonetheless unsound if one or more premises are false; if an argument is valid <i>and</i> all the premises are true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is only sound on Tuesdays (when John goes to work), but valid on every day of the week. A <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional</a> argument using <i>modus ponens</i> is said to be <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive</a>. </p><p>In single-conclusion <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculi</a>, <i>modus ponens</i> is the Cut rule. The <a href="/wiki/Cut-elimination_theorem" title="Cut-elimination theorem">cut-elimination theorem</a> for a calculus says that every proof involving Cut can be transformed (generally, by a constructive method) into a proof without Cut, and hence that Cut is <a href="/wiki/Admissible_rule" title="Admissible rule">admissible</a>. </p><p>The <a href="/wiki/Curry%E2%80%93Howard_correspondence" title="Curry–Howard correspondence">Curry–Howard correspondence</a> between proofs and programs relates <i>modus ponens</i> to <a href="/wiki/Function_application" title="Function application">function application</a>: if <i>f</i> is a function of type <i>P</i> → <i>Q</i> and <i>x</i> is of type <i>P</i>, then <i>f x</i> is of type <i>Q</i>. </p><p>In <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, <i>modus ponens</i> is often called <a href="/wiki/Forward_chaining" title="Forward chaining">forward chaining</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Formal_notation">Formal notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=2" title="Edit section: Formal notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>modus ponens</i> rule may be written in <a href="/wiki/Sequent" title="Sequent">sequent</a> notation as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to Q,\;P\;\;\vdash \;\;Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>P</mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>⊢</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to Q,\;P\;\;\vdash \;\;Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfe9d41c26581c2a6fb6d4dd49275f53d5df619" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.752ex; height:2.509ex;" alt="{\displaystyle P\to Q,\;P\;\;\vdash \;\;Q}"></span></dd></dl> <p>where <i>P</i>, <i>Q</i> and <i>P</i> → <i>Q</i> are statements (or propositions) in a formal language and <a href="/wiki/%E2%8A%A2" class="mw-redirect" title="⊢">⊢</a> is a <a href="/wiki/Metalogic" title="Metalogic">metalogical</a> symbol meaning that <i>Q</i> is a <a href="/wiki/Logical_consequence" title="Logical consequence">syntactic consequence</a> of <i>P</i> and <i>P</i> → <i>Q</i> in some <a href="/wiki/Formal_system" title="Formal system">logical system</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Justification_via_truth_table">Justification via truth table</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=3" title="Edit section: Justification via truth table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The validity of <i>modus ponens</i> in classical two-valued logic can be clearly demonstrated by use of a <a href="/wiki/Truth_table" title="Truth table">truth table</a>. </p> <table align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center;" class="wikitable"> <tbody><tr> <th><i>p</i> </th> <th><i>q</i> </th> <th><i>p</i> → <i>q</i> </th></tr> <tr> <td>T</td> <td>T</td> <td>T </td></tr> <tr style="color:red"> <td>T</td> <td>F</td> <td>F </td></tr> <tr style="color:pink"> <td>F</td> <td>T</td> <td>T </td></tr> <tr style="color:blue"> <td>F</td> <td>F</td> <td>T </td></tr></tbody></table> <p>In instances of <i>modus ponens</i> we assume as premises that <i>p</i> → <i>q</i> is true and <i>p</i> is true. Only one line of the truth table—the first—satisfies these two conditions (<i>p</i> and <i>p</i> → <i>q</i>). On this line, <i>q</i> is also true. Therefore, whenever <i>p</i> → <i>q</i> is true and <i>p</i> is true, <i>q</i> must also be true. </p> <div class="mw-heading mw-heading2"><h2 id="Status">Status</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=4" title="Edit section: Status"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While <i>modus ponens</i> is one of the most commonly used <a href="/wiki/Argument_form" class="mw-redirect" title="Argument form">argument forms</a> in logic it must not be mistaken for a logical law; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the "rule of definition" and the "rule of substitution".<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <i>Modus ponens</i> allows one to eliminate a <a href="/wiki/Material_conditional" title="Material conditional">conditional statement</a> from a <a href="/wiki/Formal_proof" title="Formal proof">logical proof or argument</a> (the antecedents) and thereby not carry these antecedents forward in an ever-lengthening string of symbols; for this reason modus ponens is sometimes called the <b>rule of detachment</b><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> or the <b>law of detachment</b>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Enderton, for example, observes that "modus ponens can produce shorter formulas from longer ones",<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and Russell observes that "the process of the inference cannot be reduced to symbols. Its sole record is the occurrence of ⊦q [the consequent] ... an inference is the dropping of a true premise; it is the dissolution of an implication".<sup id="cite_ref-auto_10-0" class="reference"><a href="#cite_note-auto-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>A justification for the "trust in inference is the belief that if the two former assertions [the antecedents] are not in error, the final assertion [the consequent] is not in error".<sup id="cite_ref-auto_10-1" class="reference"><a href="#cite_note-auto-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In other words: if one <a href="/wiki/Statement_(logic)" title="Statement (logic)">statement</a> or <a href="/wiki/Proposition" title="Proposition">proposition</a> <a href="/wiki/Material_conditional" title="Material conditional">implies</a> a second one, and the first statement or proposition is true, then the second one is also true. If <i>P</i> implies <i>Q</i> and <i>P</i> is true, then <i>Q</i> is true.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Correspondence_to_other_mathematical_frameworks">Correspondence to other mathematical frameworks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=5" title="Edit section: Correspondence to other mathematical frameworks"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Algebraic_semantics">Algebraic semantics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=6" title="Edit section: Algebraic semantics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In mathematical logic, <a href="/wiki/Algebraic_semantics_(mathematical_logic)" title="Algebraic semantics (mathematical logic)"> algebraic semantics</a> treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a <a href="/wiki/Lattice_(order)" title="Lattice (order)"> lattice</a>-like structure with a single element (the "always-true") at the top and another single element (the "always-false") at the bottom. Logical equivalence becomes identity, so that when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {(P\wedge Q)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {(P\wedge Q)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be161515235d3370fbe696fd3139d4e796b1e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.526ex; height:2.843ex;" alt="{\displaystyle \neg {(P\wedge Q)}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {P}\vee \neg {Q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {P}\vee \neg {Q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2742b9536b178ed58f26cf33e741c6e370992a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.267ex; height:2.509ex;" alt="{\displaystyle \neg {P}\vee \neg {Q}}"></span>, for instance, are equivalent (as is standard), then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c68e6647f92a05127918c8dda35f56184c5b1e1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.891ex; height:2.843ex;" alt="{\displaystyle \neg {(P\wedge Q)}=\neg {P}\vee \neg {Q}}"></span>. Logical implication becomes a matter of relative position: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> logically implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> just in case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\leq Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≤</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\leq Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/988a9d9c9076581ea7a51e0a16cdd755d5bf3bb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P\leq Q}"></span>, i.e., when either <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abc7e2c5a78e9e6cb7a2a907279953f9b4a3f52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.682ex; height:2.509ex;" alt="{\displaystyle P=Q}"></span> or else <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> lies below <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> and is connected to it by an upward path. </p><p>In this context, to say that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/038590207af1024a629c1a08c855e9ac46bf5610" 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="{\textstyle P}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></span> together imply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>—that is, to affirm <i>modus ponens</i> as valid—is to say that the highest point which lies below both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></span> lies below <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span>, i.e., that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\wedge (P\rightarrow Q)\leq Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39e185d682e36e2200579ab31b26b303a9b70ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.272ex; height:2.843ex;" alt="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}"></span>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> In the semantics for basic propositional logic, the algebra is <a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)"> Boolean</a>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> construed as the <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q=\neg {P}\vee Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo>=</mo> <mi mathvariant="normal">¬</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mo>∨</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q=\neg {P}\vee Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d18fcbec1d1f3b4670d59b70f19089639e3a915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.013ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q=\neg {P}\vee Q}"></span>. Confirming that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\wedge (P\rightarrow Q)\leq Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39e185d682e36e2200579ab31b26b303a9b70ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.272ex; height:2.843ex;" alt="{\displaystyle P\wedge (P\rightarrow Q)\leq Q}"></span> is then straightforward, because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50828a331ead18c5b93547b2ef79722650bce3c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.6ex; height:2.843ex;" alt="{\displaystyle P\wedge (P\rightarrow Q)=P\wedge Q}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\wedge Q\leq Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> <mo>≤</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\wedge Q\leq Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c680779a5b9efbdd7043b58c1aa393ce05f1eed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.103ex; height:2.509ex;" alt="{\displaystyle P\wedge Q\leq Q}"></span>. With other treatments of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span>, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted. </p> <div class="mw-heading mw-heading3"><h3 id="Probability_calculus">Probability calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=7" title="Edit section: Probability calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(P\rightarrow Q)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(P\rightarrow Q)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2f7dc3d5d4fa14d812eae3cbd60b0afc89abec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.93ex; height:2.843ex;" alt="{\displaystyle \Pr(P\rightarrow Q)=x}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(P)=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(P)=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7962bbc6ffff74f479141205a52a92472bcced8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.303ex; height:2.843ex;" alt="{\displaystyle \Pr(P)=y}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c629eb750321a80c64ad3529aded1a33e35f0bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.142ex; height:2.843ex;" alt="{\displaystyle \Pr(Q)}"></span> must lie in the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x+y-1,x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x+y-1,x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/451d2bceec60acd6fc3f90b5f28bf3a51cfb0b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.986ex; height:2.843ex;" alt="{\displaystyle [x+y-1,x]}"></span>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hailperin,_T._1996_14-0" class="reference"><a href="#cite_note-Hailperin,_T._1996-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> For the special case <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52620e42f1d8b888bc7664321a6e8bf6318d6df8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.845ex; height:2.509ex;" alt="{\displaystyle x=y=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr(Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr(Q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c629eb750321a80c64ad3529aded1a33e35f0bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.142ex; height:2.843ex;" alt="{\displaystyle \Pr(Q)}"></span> must equal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Subjective_logic">Subjective logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=8" title="Edit section: Subjective logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Modus ponens</i> represents an instance of the binomial deduction operator in <a href="/wiki/Subjective_logic" title="Subjective logic">subjective logic</a> expressed as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>⊚</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984519555dff576b7f1fe9ff2f226cc2ffee1958" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:28.198ex; height:3.676ex;" alt="{\displaystyle \omega _{Q\|P}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})\circledcirc \omega _{P}^{A}\,,}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}"></span> denotes the subjective opinion about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> as expressed by source <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, and the conditional opinion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q|P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{Q|P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbcadc50e7dbdbb35fea79b4b71a2c00305c529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.67ex; height:3.676ex;" alt="{\displaystyle \omega _{Q|P}^{A}}"></span> generalizes the logical implication <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\to Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cad5b2c2991ae1dbded560c5d875fbf49fe8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\to Q}"></span>. The deduced marginal opinion about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q\|P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{Q\|P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20dad6bcf5e3ab6771b6c23ac043f94082495db3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.034ex; height:3.676ex;" alt="{\displaystyle \omega _{Q\|P}^{A}}"></span>. The case where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}"></span> is an absolute TRUE opinion about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is equivalent to source <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> saying that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is TRUE, and the case where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}"></span> is an absolute FALSE opinion about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is equivalent to source <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> saying that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> is FALSE. The deduction operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circledcirc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊚</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circledcirc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf70a53592b87a725eabcbb2dffc880e9aa9b66c" 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 \circledcirc }"></span> of <a href="/wiki/Subjective_logic" title="Subjective logic">subjective logic</a> produces an absolute TRUE deduced opinion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q\|P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mo fence="false" stretchy="false">‖</mo> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{Q\|P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20dad6bcf5e3ab6771b6c23ac043f94082495db3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:5.034ex; height:3.676ex;" alt="{\displaystyle \omega _{Q\|P}^{A}}"></span> when the conditional opinion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{Q|P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{Q|P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fbcadc50e7dbdbb35fea79b4b71a2c00305c529" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:4.67ex; height:3.676ex;" alt="{\displaystyle \omega _{Q|P}^{A}}"></span> is absolute TRUE and the antecedent opinion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{P}^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{P}^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f71c98c7da7a716f2a6f273e4e4c07f512ed2dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.912ex; height:3.176ex;" alt="{\displaystyle \omega _{P}^{A}}"></span> is absolute TRUE. Hence, subjective logic deduction represents a generalization of both <i>modus ponens</i> and the <a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Alleged_cases_of_failure">Alleged cases of failure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=9" title="Edit section: Alleged cases of failure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Philosophers and linguists have identified a variety of cases where <i>modus ponens</i> appears to fail. <a href="/w/index.php?title=Vann_McGee&action=edit&redlink=1" class="new" title="Vann McGee (page does not exist)">Vann McGee</a>, for instance, argued that <i>modus ponens</i> can fail for conditionals whose consequents are themselves conditionals.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The following is an example: </p> <ol><li>Either <a href="/wiki/Shakespeare" class="mw-redirect" title="Shakespeare">Shakespeare</a> or <a href="/wiki/Thomas_Hobbes" title="Thomas Hobbes">Hobbes</a> wrote <i><a href="/wiki/Hamlet" title="Hamlet">Hamlet</a></i>.</li> <li>If either Shakespeare or Hobbes wrote <i>Hamlet</i>, then if Shakespeare did not do it, Hobbes did.</li> <li>Therefore, if Shakespeare did not write <i>Hamlet</i>, Hobbes did it.</li></ol> <p>Since Shakespeare did write <i>Hamlet</i>, the first premise is true. The second premise is also true, since starting with a set of possible authors limited to just Shakespeare and Hobbes and eliminating one of them leaves only the other. However, the conclusion is doubtful, since ruling out Shakespeare as the author of <i>Hamlet</i> would leave numerous possible candidates, many of them more plausible alternatives than Hobbes (if the if-thens in the inference are read as material conditionals, the conclusion comes out true simply by virtue of the false antecedent. This is one of the <a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">paradoxes of material implication</a>). </p><p>The general form of McGee-type counterexamples to <i>modus ponens</i> is simply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P,P\rightarrow (Q\rightarrow R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P,P\rightarrow (Q\rightarrow R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa231e26c737e13fdd3488af510cbd9a5afcdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.165ex; height:2.843ex;" alt="{\displaystyle P,P\rightarrow (Q\rightarrow R)}"></span>, therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\rightarrow R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\rightarrow R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76e444a574635586201a7ce3424d8777e3d6e22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.216ex; height:2.509ex;" alt="{\displaystyle Q\rightarrow R}"></span>; it is not essential that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><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}"></span> be a disjunction, as in the example given. That these kinds of cases constitute failures of <i>modus ponens</i> remains a controversial view among logicians, but opinions vary on how the cases should be disposed of.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Deontic_logic" title="Deontic logic">deontic logic</a>, some examples of conditional obligation also raise the possibility of <i>modus ponens</i> failure. These are cases where the conditional premise describes an obligation predicated on an immoral or imprudent action, e.g., "If Doe murders his mother, he ought to do so gently," for which the dubious unconditional conclusion would be "Doe ought to gently murder his mother."<sup id="cite_ref-SEP_Deontic_Logic_20-0" class="reference"><a href="#cite_note-SEP_Deontic_Logic-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> It would appear to follow that if Doe is in fact gently murdering his mother, then by <i>modus ponens</i> he is doing exactly what he should, unconditionally, be doing. Here again, <i>modus ponens'</i> failure is not a popular diagnosis but is sometimes argued for.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Possible_fallacies">Possible fallacies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=10" title="Edit section: Possible fallacies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fallacy of <a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a> is a common misinterpretation of the <i>modus ponens</i>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Condensed_detachment" title="Condensed detachment">Condensed detachment</a></li> <li><a href="/wiki/Import-export_(logic)" class="mw-redirect" title="Import-export (logic)">Import-export (logic)</a> – Principle of classical logic<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Latin_phrases" class="mw-redirect" title="Latin phrases">Latin phrases</a></li> <li><a href="/wiki/Modus_tollens" title="Modus tollens"><i>Modus tollens</i></a> – Rule of logical inference</li> <li><a href="/wiki/Modus_vivendi" title="Modus vivendi"><i>Modus vivendi</i></a> – Arrangement that allows conflicting parties to coexist in peace</li> <li><a href="/wiki/Stoic_logic" title="Stoic logic">Stoic logic</a> – System of propositional logic developed by the Stoic philosophers</li> <li><a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a> – 1895 allegorical dialogue by Lewis Carroll</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=12" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">The highest point that lies below both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is the "<a href="/wiki/Join_and_meet" title="Join and meet">meet</a>" of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\wedge Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∧</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\wedge Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6edc6915b42026ef5d46c585f7e44955f2d15ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.336ex; height:2.176ex;" alt="{\displaystyle X\wedge Y}"></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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}"></span> implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> must always be greater than or equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2914ce99dee5c9e92ccedf35d1f455d319ddf36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.158ex; height:2.509ex;" alt="{\displaystyle 1-y}"></span>, and therefore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a43835d4c846620b3e473907aaa125cccd5dffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.328ex; height:2.509ex;" alt="{\displaystyle x+y-1}"></span> will be greater than or equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>. And since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> must always be less than or equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a43835d4c846620b3e473907aaa125cccd5dffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.328ex; height:2.509ex;" alt="{\displaystyle x+y-1}"></span> must always be less than or equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFStone1996" class="citation book cs1">Stone, Jon R. (1996). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/latinforillitera0000ston"><i>Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language</i></a></span>. London: Routledge. p. <a rel="nofollow" class="external text" href="https://archive.org/details/latinforillitera0000ston/page/60">60</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-91775-1" title="Special:BookSources/0-415-91775-1"><bdi>0-415-91775-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Latin+for+the+Illiterati%3A+Exorcizing+the+Ghosts+of+a+Dead+Language&rft.place=London&rft.pages=60&rft.pub=Routledge&rft.date=1996&rft.isbn=0-415-91775-1&rft.aulast=Stone&rft.aufirst=Jon+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flatinforillitera0000ston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095354544">"Oxford reference: affirming the antecedent"</a>. <i><a href="/wiki/Oxford_Reference" class="mw-redirect" title="Oxford Reference">Oxford Reference</a></i>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Enderton 2001:110</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="/wiki/Susanne_Bobzien" title="Susanne Bobzien">Susanne Bobzien</a> (2002). "The Development of Modus Ponens in Antiquity", <i>Phronesis</i> 47, No. 4, 2002.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-ancient/#StoSyl">"Ancient Logic: Forerunners of <i>Modus Ponens</i> and <i>Modus Tollens</i>"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Alfred Tarski 1946:47. Also Enderton 2001:110ff.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Tarski 1946:47</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Modus_ponens">"Modus ponens - Encyclopedia of Mathematics"</a>. <i>encyclopediaofmath.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">5 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=encyclopediaofmath.org&rft.atitle=Modus+ponens+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FModus_ponens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Enderton 2001:111</span> </li> <li id="cite_note-auto-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-auto_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-auto_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Whitehead and Russell 1927:9</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJago2007" class="citation book cs1">Jago, Mark (2007). <i>Formal Logic</i>. Humanities-Ebooks LLP. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84760-041-7" title="Special:BookSources/978-1-84760-041-7"><bdi>978-1-84760-041-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formal+Logic&rft.pub=Humanities-Ebooks+LLP&rft.date=2007&rft.isbn=978-1-84760-041-7&rft.aulast=Jago&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span></span> </li> <li id="cite_note-Hailperin,_T._1996-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hailperin,_T._1996_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHailperin1996" class="citation book cs1">Hailperin, Theodore (1996). <i>Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications</i>. London: Associated University Presses. p. 203. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0934223459" title="Special:BookSources/0934223459"><bdi>0934223459</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sentential+Probability+Logic%3A+Origins%2C+Development%2C+Current+Status%2C+and+Technical+Applications&rft.place=London&rft.pages=203&rft.pub=Associated+University+Presses&rft.date=1996&rft.isbn=0934223459&rft.aulast=Hailperin&rft.aufirst=Theodore&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Audun Jøsang 2016:92</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Vann McGee (1985). "A Counterexample to Modus Ponens", <i>The Journal of Philosophy</i> 82, 462–471.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Sinnott-Armstrong, Moor, and Fogelin (1986). "A Defense of Modus Ponens", <i>The Journal of Philosophy</i> 83, 296–300.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">D. E. Over (1987). "Assumption and the Supposed Counterexamples to Modus Ponens", <i>Analysis</i> 47, 142–146.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Bledin (2015). "Modus Ponens Defended", <i>The Journal of Philosophy</i> 112, 462–471.</span> </li> <li id="cite_note-SEP_Deontic_Logic-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-SEP_Deontic_Logic_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-deontic/#4.5">"Deontic Logic"</a>. 21 April 2010<span class="reference-accessdate">. Retrieved <span class="nowrap">30 January</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Deontic+Logic&rft.date=2010-04-21&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-deontic%2F%234.5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span> <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i>.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">E.g., by Kolodny and MacFarlane (2010). "Ifs and Oughts", <i>The Journal of Philosophy</i> 107, 115–143.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><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.iep.utm.edu/fallacy/">"Fallacies | Internet Encyclopedia of Philosophy"</a>. <i>iep.utm.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">6 March</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=iep.utm.edu&rft.atitle=Fallacies+%7C+Internet+Encyclopedia+of+Philosophy&rft_id=https%3A%2F%2Fwww.iep.utm.edu%2Ffallacy%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=14" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Herbert B. Enderton, 2001, <i>A Mathematical Introduction to Logic Second Edition</i>, Harcourt Academic Press, Burlington MA, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-238452-3" title="Special:BookSources/978-0-12-238452-3">978-0-12-238452-3</a>.</li> <li>Audun Jøsang, 2016, <i>Subjective Logic; A formalism for Reasoning Under Uncertainty</i> Springer, Cham, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-42337-1" title="Special:BookSources/978-3-319-42337-1">978-3-319-42337-1</a></li> <li><a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> 1927 <i>Principia Mathematica to *56 (Second Edition)</i> paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN.</li> <li><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> 1946 <i>Introduction to Logic and to the Methodology of the Deductive Sciences</i> 2nd Edition, reprinted by Dover Publications, Mineola NY. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-28462-X" title="Special:BookSources/0-486-28462-X">0-486-28462-X</a> (pbk).</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Modus_ponens&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><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=Modus_ponens">"Modus ponens"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <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=Modus+ponens&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DModus_ponens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AModus+ponens" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://philpapers.org/s/modus_ponens">Modus ponens</a> at <a href="/wiki/PhilPapers" title="PhilPapers">PhilPapers</a></li> <li><i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ModusPonens.html">Modus ponens</a></i> at Wolfram MathWorld</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 solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output 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logic)">Predicate</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Connective</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Truth_function" title="Truth function">Truth function</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Well-formed formula</a></li> <li><a href="/wiki/Idempotency_of_entailment" title="Idempotency of entailment">Idempotency of entailment</a></li> <li><a href="/wiki/Logicism" title="Logicism">Logicism</a></li> <li><a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">Problem of multiple generality</a></li> <li><a href="/wiki/Associative_property" title="Associative property">Associativity</a></li> <li><a href="/wiki/Distributive_property" title="Distributive property">Distribution</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Logic.svg" class="mw-file-description"><img alt="Law of noncontradiction" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/75px-Logic.svg.png" decoding="async" width="75" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/113px-Logic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Logic.svg/150px-Logic.svg.png 2x" data-file-width="85" data-file-height="28" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classical logics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Term_logic" title="Term logic">Term</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order</a></li> <li><a href="/wiki/Second-order_logic" title="Second-order logic">Second-order</a></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Principles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Commutativity_of_conjunction" title="Commutativity of conjunction">Commutativity of conjunction</a></li> <li><a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">Excluded middle</a></li> <li><a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">Bivalence</a></li> <li><a href="/wiki/Law_of_noncontradiction" title="Law of noncontradiction">Noncontradiction</a></li> <li><a href="/wiki/Monotonicity_of_entailment" title="Monotonicity of entailment">Monotonicity of entailment</a></li> <li><a href="/wiki/Principle_of_explosion" title="Principle of explosion">Explosion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Rules</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a></li> <li><a href="/wiki/Material_implication_(rule_of_inference)" title="Material implication (rule of inference)">Material implication</a></li> <li><a href="/wiki/Transposition_(logic)" class="mw-redirect" title="Transposition (logic)">Transposition</a></li> <li><a class="mw-selflink selflink">modus ponens</a></li> <li><a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a></li> <li><a href="/wiki/Modus_ponendo_tollens" title="Modus ponendo tollens">modus ponendo tollens</a></li> <li><a href="/wiki/Constructive_dilemma" title="Constructive dilemma">Constructive dilemma</a></li> <li><a href="/wiki/Destructive_dilemma" title="Destructive dilemma">Destructive dilemma</a></li> <li><a href="/wiki/Disjunctive_syllogism" title="Disjunctive syllogism">Disjunctive syllogism</a></li> <li><a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">Hypothetical syllogism</a></li> <li><a href="/wiki/Absorption_(logic)" title="Absorption (logic)">Absorption</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Introduction</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Negation_introduction" title="Negation introduction">Negation</a></li> <li><a href="/wiki/Double_negation_introduction" class="mw-redirect" title="Double negation introduction">Double negation</a></li> <li><a href="/wiki/Existential_generalization" title="Existential generalization">Existential</a></li> <li><a href="/wiki/Universal_generalization" title="Universal generalization">Universal</a></li> <li><a href="/wiki/Biconditional_introduction" title="Biconditional introduction">Biconditional</a></li> <li><a href="/wiki/Conjunction_introduction" title="Conjunction introduction">Conjunction</a></li> <li><a href="/wiki/Disjunction_introduction" title="Disjunction introduction">Disjunction</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Elimination</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Double_negation_elimination" class="mw-redirect" title="Double negation elimination">Double negation</a></li> <li><a href="/wiki/Existential_instantiation" title="Existential instantiation">Existential</a></li> <li><a href="/wiki/Universal_instantiation" title="Universal instantiation">Universal</a></li> <li><a href="/wiki/Biconditional_elimination" title="Biconditional elimination">Biconditional</a></li> <li><a href="/wiki/Conjunction_elimination" title="Conjunction elimination">Conjunction</a></li> <li><a href="/wiki/Disjunction_elimination" title="Disjunction elimination">Disjunction</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a></li> <li><a href="/wiki/George_Boole" title="George Boole">George Boole</a></li> <li><a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li> <li><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li> <li><a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">Augustus De Morgan</a></li> <li><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a></li> <li><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li> <li><a href="/wiki/Hugh_MacColl" title="Hugh MacColl">Hugh MacColl</a></li> <li><a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a></li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peirce</a></li> <li><a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li> <li><a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Ernst Schröder</a></li> <li><a href="/wiki/Henry_M._Sheffer" title="Henry M. Sheffer">Henry M. Sheffer</a></li> <li><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a></li> <li><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a></li> <li><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a></li> <li><a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Works</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></li> <li><a href="/wiki/Function_and_Concept" title="Function and Concept">Function and Concept</a></li> <li><a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a></li> <li><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></li> <li><a href="/wiki/Tractatus_Logico-Philosophicus" title="Tractatus Logico-Philosophicus">Tractatus Logico-Philosophicus</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" 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=Modus_ponens&oldid=1233080425">https://en.wikipedia.org/w/index.php?title=Modus_ponens&oldid=1233080425</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Rules_of_inference" title="Category:Rules of inference">Rules of inference</a></li><li><a href="/wiki/Category:Latin_logical_phrases" title="Category:Latin logical phrases">Latin logical phrases</a></li><li><a href="/wiki/Category:Theorems_in_propositional_logic" title="Category:Theorems in propositional logic">Theorems in propositional logic</a></li><li><a href="/wiki/Category:Classical_logic" title="Category:Classical logic">Classical logic</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_May_2018" title="Category:Use dmy dates from May 2018">Use dmy dates from May 2018</a></li><li><a href="/wiki/Category:Articles_containing_Latin-language_text" title="Category:Articles containing Latin-language text">Articles containing Latin-language text</a></li><li><a href="/wiki/Category:Pages_displaying_short_descriptions_of_redirect_targets_via_Module:Annotated_link" title="Category:Pages displaying short descriptions of redirect targets via Module:Annotated link">Pages displaying short descriptions of redirect targets via Module:Annotated link</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 7 July 2024, at 05:04<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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