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class="vector-toc-numb">3.1</span> <span>Example from Nāgārjuna's Mūlamadhyamakakārikā</span> </div> </a> <ul id="toc-Example_from_Nāgārjuna's_Mūlamadhyamakakārikā-sublist" class="vector-toc-list"> <li id="toc-13:5" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#13:5"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>13:5</span> </div> </a> <ul id="toc-13:5-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Principle_of_non-contradiction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Principle_of_non-contradiction"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Principle of non-contradiction</span> </div> </a> <ul id="toc-Principle_of_non-contradiction-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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-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">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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Available in 37 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-37" 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">37 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/%D8%A8%D8%B1%D9%87%D8%A7%D9%86_%D8%A7%D9%84%D8%AE%D9%84%D9%81" title="برهان الخلف – Arabic" lang="ar" hreflang="ar" data-title="برهان الخلف" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Azerbaijani" lang="az" hreflang="az" data-title="Reductio ad absurdum" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D0%B0%D0%B2%D1%8F%D0%B4%D0%B7%D0%B5%D0%BD%D0%BD%D0%B5_%D0%B4%D0%B0_%D0%B0%D0%B1%D1%81%D1%83%D1%80%D0%B4%D1%83" title="Давядзенне да абсурду – Belarusian" lang="be" hreflang="be" data-title="Давядзенне да абсурду" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B5%D0%B6%D0%B4%D0%B0%D0%BD%D0%B5_%D0%B4%D0%BE_%D0%B0%D0%B1%D1%81%D1%83%D1%80%D0%B4" title="Довеждане до абсурд – Bulgarian" lang="bg" hreflang="bg" data-title="Довеждане до абсурд" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Bosnian" lang="bs" hreflang="bs" data-title="Reductio ad absurdum" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Reducci%C3%B3_a_l%27absurd" title="Reducció a l'absurd – Catalan" lang="ca" hreflang="ca" data-title="Reducció a l'absurd" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Absurd_(logik)" title="Absurd (logik) – Danish" lang="da" hreflang="da" data-title="Absurd (logik)" 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/Reductio_ad_absurdum" title="Reductio ad absurdum – German" lang="de" hreflang="de" data-title="Reductio ad absurdum" 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/Reductio_ad_absurdum" title="Reductio ad absurdum – Estonian" lang="et" hreflang="et" data-title="Reductio ad absurdum" 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/Reductio_ad_absurdum" title="Reductio ad absurdum – Spanish" lang="es" hreflang="es" data-title="Reductio ad absurdum" 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/Reductio_ad_absurdum" title="Reductio ad absurdum – Basque" lang="eu" hreflang="eu" data-title="Reductio ad absurdum" 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/%D8%AA%D8%B9%D9%84%DB%8C%D9%82_%D8%A8%D9%87_%D9%85%D8%AD%D8%A7%D9%84" title="تعلیق به محال – Persian" lang="fa" hreflang="fa" data-title="تعلیق به محال" 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/Raisonnement_par_l%27absurde" title="Raisonnement par l'absurde – French" lang="fr" hreflang="fr" data-title="Raisonnement par l'absurde" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Croatian" lang="hr" hreflang="hr" data-title="Reductio ad absurdum" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dimostrazione_per_assurdo" title="Dimostrazione per assurdo – Italian" lang="it" hreflang="it" data-title="Dimostrazione per assurdo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Latin" lang="la" hreflang="la" data-title="Reductio ad absurdum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Privedimas_iki_absurdo" title="Privedimas iki absurdo – Lithuanian" lang="lt" hreflang="lt" data-title="Privedimas iki absurdo" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Hungarian" lang="hu" hreflang="hu" data-title="Reductio ad absurdum" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B5%D0%B4%D1%83%D0%B2%D0%B0%D1%9A%D0%B5_%D0%B4%D0%BE_%D0%B0%D0%BF%D1%81%D1%83%D1%80%D0%B4" title="Доведување до апсурд – Macedonian" lang="mk" hreflang="mk" data-title="Доведување до апсурд" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B8%B0%E8%AC%AC%E6%B3%95_(%E4%BF%AE%E8%BE%9E%E5%AD%A6)" title="帰謬法 (修辞学) – Japanese" lang="ja" hreflang="ja" data-title="帰謬法 (修辞学)" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Reductio ad absurdum" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Reductio ad absurdum" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Dimostrassion_p%C3%ABr_assurd" title="Dimostrassion për assurd – Piedmontese" lang="pms" hreflang="pms" data-title="Dimostrassion për assurd" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Portuguese" lang="pt" hreflang="pt" data-title="Reductio ad absurdum" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Argumentum_ad_absurdum" title="Argumentum ad absurdum – Romanian" lang="ro" hreflang="ro" data-title="Argumentum ad absurdum" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5_%D0%B4%D0%BE_%D0%B0%D0%B1%D1%81%D1%83%D1%80%D0%B4%D0%B0" title="Доведение до абсурда – Russian" lang="ru" hreflang="ru" data-title="Доведение до абсурда" 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/Reductio_ad_absurdum" title="Reductio ad absurdum – Simple English" lang="en-simple" hreflang="en-simple" data-title="Reductio ad absurdum" 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/Reductio_ad_absurdum_(z%C3%A1kon)" title="Reductio ad absurdum (zákon) – Slovak" lang="sk" hreflang="sk" data-title="Reductio ad absurdum (zákon)" 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%A1%D0%B2%D0%BE%D1%92%D0%B5%D1%9A%D0%B5_%D0%BD%D0%B0_%D0%B0%D0%BF%D1%81%D1%83%D1%80%D0%B4" title="Свођење на апсурд – Serbian" lang="sr" hreflang="sr" data-title="Свођење на апсурд" 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-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Reductio ad absurdum" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Finnish" lang="fi" hreflang="fi" data-title="Reductio ad absurdum" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Swedish" lang="sv" hreflang="sv" data-title="Reductio ad absurdum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Turkish" lang="tr" hreflang="tr" data-title="Reductio ad absurdum" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%BE%D0%B2%D0%B5%D0%B4%D0%B5%D0%BD%D0%BD%D1%8F_%D0%B4%D0%BE_%D0%B0%D0%B1%D1%81%D1%83%D1%80%D0%B4%D1%83" title="Доведення до абсурду – Ukrainian" lang="uk" hreflang="uk" data-title="Доведення до абсурду" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A9p_ph%E1%BA%A3n_ch%E1%BB%A9ng" title="Phép phản chứng – Vietnamese" lang="vi" hreflang="vi" data-title="Phép phản chứng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum – Zazaki" lang="diq" hreflang="diq" data-title="Reductio ad absurdum" data-language-autonym="Zazaki" data-language-local-name="Zazaki" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%B8%E8%AC%AC%E6%B3%95" title="歸謬法 – Chinese" lang="zh" hreflang="zh" data-title="歸謬法" 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/Q14402006#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div 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argues towards an older pensive white Christian cleric in black." src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/John_Pettie_-_Reductio_Ad_Absurdum.jpg/260px-John_Pettie_-_Reductio_Ad_Absurdum.jpg" decoding="async" width="260" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/John_Pettie_-_Reductio_Ad_Absurdum.jpg/390px-John_Pettie_-_Reductio_Ad_Absurdum.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/John_Pettie_-_Reductio_Ad_Absurdum.jpg/520px-John_Pettie_-_Reductio_Ad_Absurdum.jpg 2x" data-file-width="798" data-file-height="552" /></a><figcaption><span title="Latin-language text"><i lang="la">Reductio ad absurdum</i></span>, painting by <a href="/wiki/John_Pettie" title="John Pettie">John Pettie</a> exhibited at the <a href="/wiki/Royal_Academy" class="mw-redirect" title="Royal Academy">Royal Academy</a> in 1884</figcaption></figure> <p>In <a href="/wiki/Logic" title="Logic">logic</a>, <b><span title="Latin-language text"><i lang="la">reductio ad absurdum</i></span></b> (<a href="/wiki/Latin" title="Latin">Latin</a> for "reduction to absurdity"), also known as <b><span title="Latin-language text"><i lang="la">argumentum ad absurdum</i></span></b> (<a href="/wiki/Latin" title="Latin">Latin</a> for "argument to absurdity") or <i><b>apagogical arguments</b></i>, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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><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><sup id="cite_ref-IEP_4-0" class="reference"><a href="#cite_note-IEP-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>This argument form traces back to <a href="/wiki/Ancient_Greek_philosophy" title="Ancient Greek philosophy">Ancient Greek philosophy</a> and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. Formally, the proof technique is captured by an axiom for "Reductio ad Absurdum", normally given the abbreviation RAA, which is expressible in <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>. This axiom is the introduction rule for negation (see <i><a href="/wiki/Negation_introduction" title="Negation introduction">negation introduction</a></i>) and it is sometimes named to make this connection clear. It is a consequence of the related mathematical proof technique called <i><a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The "absurd" conclusion of a <i>reductio ad absurdum</i> argument can take a range of forms, as these examples show: </p> <ul><li>The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find people falling off the edge.</li> <li>There is no smallest positive <a href="/wiki/Rational_number" title="Rational number">rational number</a> <span class="nowrap">⁠<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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>⁠</span>. If there were, then <span class="nowrap">⁠<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/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba306d94f7204705700f3131e99d0f1d9c80ad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle q/2}"></span>⁠</span> would be also a rational number, it would be positive, and we would have <span class="nowrap">⁠<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/2<q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo><</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q/2<q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0701336921f0044dfb029466fc16328326d1fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.562ex; height:2.843ex;" alt="{\displaystyle q/2<q}"></span>⁠</span>. This contradicts the hypothetical minimality of <span class="nowrap">⁠<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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>⁠</span> among positive rational numbers, so we conclude that there is no such smallest positive rational number.</li></ul> <p>The first example argues that denial of the premise would result in a ridiculous conclusion, against the evidence of our senses (<a href="/wiki/Empirical_evidence" title="Empirical evidence">empirical evidence</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> The second example is a mathematical <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> (also known as an indirect proof<sup id="cite_ref-:1_6-0" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>), which argues that the denial of the premise would result in a <a href="/wiki/Contradiction" title="Contradiction">logical contradiction</a> (there is a "smallest" number and yet there is a number smaller than it).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Greek_philosophy">Greek philosophy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=2" title="Edit section: Greek philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Reductio ad absurdum</i> was used throughout <a href="/wiki/Greek_philosophy" class="mw-redirect" title="Greek philosophy">Greek philosophy</a>. The earliest example of a <span title="Latin-language text"><i lang="la">reductio</i></span> argument can be found in a satirical poem attributed to <a href="/wiki/Xenophanes_of_Colophon" class="mw-redirect" title="Xenophanes of Colophon">Xenophanes of Colophon</a> (c. 570 – c. 475 <a href="/wiki/Common_Era" title="Common Era">BCE</a>).<sup id="cite_ref-Daigle_8-0" class="reference"><a href="#cite_note-Daigle-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Criticizing <a href="/wiki/Homer" title="Homer">Homer</a>'s attribution of human faults to the gods, Xenophanes states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and ox bodies.<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> The gods cannot have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false. </p><p>Greek mathematicians proved fundamental propositions using <i>reductio ad absurdum</i>. <a href="/wiki/Euclid_of_Alexandria" class="mw-redirect" title="Euclid of Alexandria">Euclid of Alexandria</a> (mid-4th – mid-3rd centuries BCE) and <a href="/wiki/Archimedes_of_Syracuse" class="mw-redirect" title="Archimedes of Syracuse">Archimedes of Syracuse</a> (c. 287 – c. 212 BCE) are two very early examples.<sup id="cite_ref-Euclid_10-0" class="reference"><a href="#cite_note-Euclid-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>The earlier dialogues of <a href="/wiki/Plato" title="Plato">Plato</a> (424–348 BCE), relating the discourses of <a href="/wiki/Socrates" title="Socrates">Socrates</a>, raised the use of <span title="Latin-language text"><i lang="la">reductio</i></span> arguments to a formal dialectical method (<span title="Latin-language text"><i lang="la"><a href="/wiki/Method_of_elenchus" class="mw-redirect" title="Method of elenchus">elenchus</a></i></span>), also called the <a href="/wiki/Socratic_method" title="Socratic method">Socratic method</a>.<sup id="cite_ref-Bobzien_11-0" class="reference"><a href="#cite_note-Bobzien-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Typically, Socrates' opponent would make what would seem to be an innocuous assertion. In response, Socrates, via a step-by-step train of reasoning, bringing in other background assumptions, would make the person admit that the assertion resulted in an absurd or contradictory conclusion, forcing him to abandon his assertion and adopt a position of <a href="/wiki/Aporia" title="Aporia">aporia</a>.<sup id="cite_ref-:1_6-1" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The technique was also a focus of the work of <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> (384–322 BCE), particularly in his <i><a href="/wiki/Prior_Analytics" title="Prior Analytics">Prior Analytics</a></i> where he referred to it as demonstration to the impossible (<style data-mw-deduplicate="TemplateStyles:r1248666159">.mw-parser-output .tfd-dated{font-size:85%}.mw-parser-output .tfd-default{border-bottom:1px solid var(--border-color-base,#a2a9b1);clear:both;text-align:center}.mw-parser-output .tfd-tiny{font-weight:bold}.mw-parser-output .tfd-inline{border:1px solid var(--border-color-base,#a2a9b1)}.mw-parser-output .tfd-sidebar{border-bottom:1px solid var(--border-color-base,#a2a9b1);text-align:center;position:relative}@media(min-width:640px){.mw-parser-output .tfd-sidebar{clear:right;float:right;width:22em}}</style><span class="tfd tfd-dated tfd-tiny"><a href="/wiki/Wikipedia:Templates_for_discussion/Log/2024_November_12#Template:Lang-grc-gre" title="Wikipedia:Templates for discussion/Log/2024 November 12">‹See Tfd›</a></span><a href="/wiki/Greek_language" title="Greek language">Greek</a>: <span title="Ancient Greek (to 1453)-language text"><span lang="grc">ἡ εἰς τὸ ἀδύνατον ἀπόδειξις</span></span>, <small><a href="/wiki/Literal_translation" title="Literal translation">lit.</a></small> "demonstration to the impossible", 62b).<sup id="cite_ref-IEP_4-1" class="reference"><a href="#cite_note-IEP-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Another example of this technique is found in the <a href="/wiki/Sorites_paradox" title="Sorites paradox">sorites paradox</a>, where it was argued that if 1,000,000 grains of sand formed a heap, and removing one grain from a heap left it a heap, then a single grain of sand (or even no grains) forms a heap.<sup id="cite_ref-FOOTNOTEHydeRaffman2018_12-0" class="reference"><a href="#cite_note-FOOTNOTEHydeRaffman2018-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Buddhist_philosophy">Buddhist philosophy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=3" title="Edit section: Buddhist philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Much of <a href="/wiki/Madhyamaka" title="Madhyamaka">Madhyamaka</a> <a href="/wiki/Buddhist_philosophy" title="Buddhist philosophy">Buddhist philosophy</a> centers on showing how various <a href="/wiki/Essentialism" title="Essentialism">essentialist</a> ideas have absurd conclusions through <i>reductio ad absurdum</i> arguments (known as <i>prasaṅga</i>, "consequence" in Sanskrit). In the <a href="/wiki/M%C5%ABlamadhyamakak%C4%81rik%C4%81" title="Mūlamadhyamakakārikā">Mūlamadhyamakakārikā</a>, <a href="/wiki/N%C4%81g%C4%81rjuna" class="mw-redirect" title="Nāgārjuna">Nāgārjuna</a>'s <i>reductio ad absurdum</i> arguments are used to show that any theory of substance or essence was unsustainable and therefore, phenomena (<i>dharmas</i>) such as change, causality, and sense perception were empty (<i>sunya</i>) of any essential existence. Nāgārjuna's main goal is often seen by scholars as refuting the essentialism of certain Buddhist <a href="/wiki/Abhidharma" title="Abhidharma">Abhidharma</a> schools (mainly <a href="/wiki/Vaibh%C4%81%E1%B9%A3ika" title="Vaibhāṣika"><i>Vaibhasika</i></a>) which posited theories of <i><a href="/wiki/Svabhava" title="Svabhava">svabhava</a></i> (essential nature) and also the Hindu <a href="/wiki/Nyaya" title="Nyaya">Nyāya</a> and <a href="/wiki/Vaisheshika" title="Vaisheshika">Vaiśeṣika</a> schools which posited a theory of ontological substances (<i>dravyatas</i>).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example_from_Nāgārjuna's_Mūlamadhyamakakārikā"><span id="Example_from_N.C4.81g.C4.81rjuna.27s_M.C5.ABlamadhyamakak.C4.81rik.C4.81"></span>Example from Nāgārjuna's Mūlamadhyamakakārikā</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=4" title="Edit section: Example from Nāgārjuna's Mūlamadhyamakakārikā"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 13.5, Nagarjuna wishes to demonstrate consequences of the presumption that things essentially, or inherently, exist, pointing out that if a "young man" exists in himself then it follows he cannot grow old (because he would no longer be a "young man"). As we attempt to separate the man from his properties (youth), we find that everything is subject to momentary change, and are left with nothing beyond the merely arbitrary convention that such entities as "young man" depend upon. </p> <div class="mw-heading mw-heading4"><h4 id="13:5">13:5</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=5" title="Edit section: 13:5"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd>A thing itself does not change.</dd> <dd>Something different does not change.</dd> <dd>Because a young man does not grow old.</dd> <dd>And because an old man does not grow old either.<sup id="cite_ref-FOOTNOTEGarfield1995210_14-0" class="reference"><a href="#cite_note-FOOTNOTEGarfield1995210-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Principle_of_non-contradiction">Principle of non-contradiction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=6" title="Edit section: Principle of non-contradiction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aristotle clarified the connection between contradiction and falsity in his <a href="/wiki/Principle_of_non-contradiction" class="mw-redirect" title="Principle of non-contradiction">principle of non-contradiction</a>, which states that a proposition cannot be both true and false.<sup id="cite_ref-Ziembiński_15-0" class="reference"><a href="#cite_note-Ziembiński-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ferguson1_16-0" class="reference"><a href="#cite_note-Ferguson1-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> That is, a proposition <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 its negation <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 \lnot Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b5e3033d1f0e7333fc9e708c2ea802f9b9fca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle \lnot Q}"></span> (not-<i>Q</i>) cannot both be true. Therefore, if a proposition and its negation can both be derived logically from a premise, it can be concluded that the premise is false. This technique, known as indirect proof or <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a>,<sup id="cite_ref-:1_6-2" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> has formed the basis of <span title="Latin-language text"><i lang="la">reductio ad absurdum</i></span> arguments in formal fields such as <a href="/wiki/Logic" title="Logic">logic</a> and mathematics. </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=Reductio_ad_absurdum&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Appeal_to_ridicule" title="Appeal to ridicule">Appeal to ridicule</a></li> <li><a href="/wiki/Argument_from_fallacy" title="Argument from fallacy">Argument from fallacy</a></li> <li><a href="/wiki/Contraposition" title="Contraposition">Contraposition</a></li> <li><a href="/wiki/List_of_Latin_phrases" title="List of Latin phrases">List of Latin phrases</a></li> <li><a href="/wiki/Mathematical_proof" title="Mathematical proof">Mathematical proof</a></li> <li><a href="/wiki/Prasangika" class="mw-redirect" title="Prasangika">Prasangika</a></li> <li><a href="/wiki/Slippery_slope" title="Slippery slope">Slippery slope</a></li> <li><a href="/wiki/Straw_man" title="Straw man">Strawman</a></li></ul> </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=Reductio_ad_absurdum&action=edit&section=8" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFHydeRaffman2018" class="citation encyclopaedia cs1">Hyde, Dominic; Raffman, Diana (2018). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/sorites-paradox/">"Sorites Paradox"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i> (Summer 2018 ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Sorites+Paradox&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.edition=Summer+2018&rft.date=2018&rft.aulast=Hyde&rft.aufirst=Dominic&rft.au=Raffman%2C+Diana&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fsorites-paradox%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarfield1995" class="citation cs2">Garfield, Jay L. (1995), <i>The Fundamental Wisdom of the Middle Way</i>, Oxford: Oxford University Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fundamental+Wisdom+of+the+Middle+Way&rft.place=Oxford&rft.pub=Oxford+University+Press&rft.date=1995&rft.aulast=Garfield&rft.aufirst=Jay+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></li> <li>Pasti, Mary. Reductio Ad Absurdum: An Exercise in the Study of Population Change. United States, Cornell University, Jan., 1977.</li> <li>Daigle, Robert W.. The Reductio Ad Absurdum Argument Prior to Aristotle. N.p., San Jose State University, 1991.</li></ul> <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=Reductio_ad_absurdum&action=edit&section=9" title="Edit section: References"><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-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_1-0">^</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://www.britannica.com/topic/reductio-ad-absurdum">"Reductio ad absurdum | logic"</a>. <i>Encyclopedia Britannica</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">21 July</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reductio+ad+absurdum&rft.btitle=The+Internet+Encyclopedia+of+Philosophy&rft.au=Nicholas+Rescher&rft_id=http%3A%2F%2Fwww.utm.edu%2Fresearch%2Fiep%2Fr%2Freductio.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeLancey2017" class="citation cs2">DeLancey, Craig (2017-03-27), <a rel="nofollow" class="external text" href="https://milnepublishing.geneseo.edu/concise-introduction-to-logic/chapter/8-reductio-ad-absurdum/">"8. 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Retrieved <span class="nowrap">2019-11-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ThoughtCo&rft.atitle=Reductio+Ad+Absurdum+in+Argument&rft.aulast=Nordquist&rft.aufirst=Richard&rft_id=https%3A%2F%2Fwww.thoughtco.com%2Freductio-ad-absurdum-argument-1691903&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoward-SnyderHoward-SnyderWasserman2012" class="citation book cs1">Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan (30 March 2012). <i>The Power of Logic</i> (5th ed.). 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(1991). <a rel="nofollow" class="external text" href="http://scholarworks.sjsu.edu/cgi/viewcontent.cgi?article=1228&context=etd_theses">"The reductio ad absurdum argument prior to Aristotle"</a>. <i>Master's Thesis</i>. San Jose State Univ<span class="reference-accessdate">. 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Retrieved <span class="nowrap">2021-08-31</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Literary+Devices&rft.atitle=Reductio+ad+Absurdum+-+Definition+%26+Examples&rft.date=2014-05-18&rft_id=https%3A%2F%2Fliterarydevices.net%2Freductio-ad-absurdum%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></span> </li> <li id="cite_note-Euclid-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Euclid_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJoyce1996" class="citation web cs1">Joyce, David (1996). <a rel="nofollow" class="external text" href="https://mathcs.clarku.edu/~djoyce/elements/bookI/propI6.html">"Euclid's Elements: Book I"</a>. <i>Euclid's Elements</i>. Department of Mathematics and Computer Science, Clark University<span class="reference-accessdate">. Retrieved <span class="nowrap">December 23,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Euclid%27s+Elements&rft.atitle=Euclid%27s+Elements%3A+Book+I&rft.date=1996&rft.aulast=Joyce&rft.aufirst=David&rft_id=https%3A%2F%2Fmathcs.clarku.edu%2F~djoyce%2Felements%2FbookI%2FpropI6.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></span> </li> <li id="cite_note-Bobzien-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bobzien_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBobzien2006" class="citation encyclopaedia cs1">Bobzien, Susanne (2006). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-ancient/#NonModSyl">"Ancient Logic"</a>. <i>Stanford Encyclopedia of Philosophy</i>. 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Retrieved <span class="nowrap">August 22,</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ancient+Logic&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.pub=The+Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2006&rft.aulast=Bobzien&rft.aufirst=Susanne&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-ancient%2F%23NonModSyl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEHydeRaffman2018-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHydeRaffman2018_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHydeRaffman2018">Hyde & Raffman 2018</a>.</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">Wasler, Joseph. <i>Nagarjuna in Context.</i> New York: Columibia University Press. 2005, pgs. 225-263.</span> </li> <li id="cite_note-FOOTNOTEGarfield1995210-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGarfield1995210_14-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGarfield1995">Garfield 1995</a>, p. 210.</span> </li> <li id="cite_note-Ziembiński-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ziembiński_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZiembiński2013" class="citation book cs1">Ziembiński, Zygmunt (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LOfsCAAAQBAJ&q=%22principle+of+non-contradiction%22&pg=PA95"><i>Practical Logic</i></a>. Springer. p. 95. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9401756044" title="Special:BookSources/978-9401756044"><bdi>978-9401756044</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Logic&rft.pages=95&rft.pub=Springer&rft.date=2013&rft.isbn=978-9401756044&rft.aulast=Ziembi%C5%84ski&rft.aufirst=Zygmunt&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLOfsCAAAQBAJ%26q%3D%2522principle%2Bof%2Bnon-contradiction%2522%26pg%3DPA95&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></span> </li> <li id="cite_note-Ferguson1-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ferguson1_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFergusonPriest2016" class="citation book cs1">Ferguson, Thomas Macaulay; Priest, Graham (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2Q5nDAAAQBAJ&q=%22principle+of+non-contradiction%22&pg=PT146"><i>A Dictionary of Logic</i></a>. Oxford University Press. p. 146. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0192511553" title="Special:BookSources/978-0192511553"><bdi>978-0192511553</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Dictionary+of+Logic&rft.pages=146&rft.pub=Oxford+University+Press&rft.date=2016&rft.isbn=978-0192511553&rft.aulast=Ferguson&rft.aufirst=Thomas+Macaulay&rft.au=Priest%2C+Graham&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D2Q5nDAAAQBAJ%26q%3D%2522principle%2Bof%2Bnon-contradiction%2522%26pg%3DPT146&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reductio_ad_absurdum&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/24px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/32px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> The dictionary definition of <a href="https://en.wiktionary.org/wiki/reductio_ad_absurdum" class="extiw" title="wiktionary:reductio ad absurdum"><i>per impossibile</i></a> at Wiktionary</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/reductio/">"Reductio ad absurdum"</a>. <i><a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reductio+ad+absurdum&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Freductio%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReductio+ad+absurdum" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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