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class="vector-toc-numb">2</span> <span>Necessity</span> </div> </a> <ul id="toc-Necessity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sufficiency" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sufficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Sufficiency</span> </div> </a> <ul id="toc-Sufficiency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_between_necessity_and_sufficiency" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationship_between_necessity_and_sufficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Relationship between necessity and sufficiency</span> </div> </a> <ul id="toc-Relationship_between_necessity_and_sufficiency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simultaneous_necessity_and_sufficiency" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Simultaneous_necessity_and_sufficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Simultaneous necessity and sufficiency</span> </div> </a> <ul id="toc-Simultaneous_necessity_and_sufficiency-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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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" 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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-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%89%A0%E1%89%82%E1%8A%93_%E1%8A%A0%E1%88%B5%E1%8D%88%E1%88%8B%E1%8C%8A" title="በቂና አስፈላጊ – Amharic" lang="am" hreflang="am" data-title="በቂና አስፈላጊ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B4%D8%B1%D8%B7_%D8%B6%D8%B1%D9%88%D8%B1%D9%8A_%D9%88%D8%B4%D8%B1%D8%B7_%D9%83%D8%A7%D9%81" title="شرط ضروري وشرط كاف – Arabic" lang="ar" hreflang="ar" data-title="شرط ضروري وشرط كاف" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%AF%D8%B1%DA%A9_%D8%A7%DB%8C%D9%84%D9%87_%DA%AF%D8%A4%D8%B1%D8%B1" title="گرک ایله گؤرر – South Azerbaijani" lang="azb" hreflang="azb" data-title="گرک ایله گؤرر" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Condici%C3%B3_necess%C3%A0ria_i_suficient" title="Condició necessària i suficient – Catalan" lang="ca" hreflang="ca" data-title="Condició necessària i suficient" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%B8%D1%80%D0%BB%C4%95_%D1%82%D0%B0%D1%82%D0%B0_%C3%A7%D0%B8%D1%82%D0%B5%D0%BB%C4%95%D0%BA%D0%BB%C4%95_%D0%BC%D0%B0%D0%BB%D1%81%C4%83%D0%BB%D1%82%D0%B0%D0%B2%D1%81%D0%B5%D0%BC" title="Кирлĕ тата çителĕклĕ малсăлтавсем – Chuvash" lang="cv" hreflang="cv" data-title="Кирлĕ тата çителĕклĕ малсăлтавсем" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Nutn%C3%A1_a_posta%C4%8Duj%C3%ADc%C3%AD_podm%C3%ADnka" title="Nutná a postačující podmínka – Czech" lang="cs" hreflang="cs" data-title="Nutná a postačující podmínka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Notwendige_und_hinreichende_Bedingung" title="Notwendige und hinreichende Bedingung – German" lang="de" hreflang="de" data-title="Notwendige und hinreichende Bedingung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es badge-Q70893996 mw-list-item" title=""><a href="https://es.wikipedia.org/wiki/Condici%C3%B3n_necesaria_y_suficiente" title="Condición necesaria y suficiente – Spanish" lang="es" hreflang="es" data-title="Condición necesaria y suficiente" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Neceso_kaj_sufi%C4%89o" title="Neceso kaj sufiĉo – Esperanto" lang="eo" hreflang="eo" data-title="Neceso kaj sufiĉo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Beharrezkotasuna_eta_nahikotasuna" title="Beharrezkotasuna eta nahikotasuna – Basque" lang="eu" hreflang="eu" data-title="Beharrezkotasuna eta nahikotasuna" 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%84%D8%A7%D8%B2%D9%85_%D9%88_%DA%A9%D8%A7%D9%81%DB%8C" title="لازم و کافی – Persian" lang="fa" hreflang="fa" data-title="لازم و کافی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Condici%C3%B3n_necesaria_e_suficiente" title="Condición necesaria e suficiente – Galician" lang="gl" hreflang="gl" data-title="Condición necesaria e suficiente" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%84%EC%9A%94%EC%A1%B0%EA%B1%B4%EA%B3%BC_%EC%B6%A9%EB%B6%84%EC%A1%B0%EA%B1%B4" title="필요조건과 충분조건 – Korean" lang="ko" hreflang="ko" data-title="필요조건과 충분조건" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%B6%D5%B0%D6%80%D5%A1%D5%AA%D5%A5%D5%B7%D5%BF_%D6%87_%D5%A2%D5%A1%D5%BE%D5%A1%D6%80%D5%A1%D6%80_%D5%BA%D5%A1%D5%B5%D5%B4%D5%A1%D5%B6%D5%B6%D5%A5%D6%80" title="Անհրաժեշտ և բավարար պայմաններ – Armenian" lang="hy" hreflang="hy" data-title="Անհրաժեշտ և բավարար պայմաններ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Necessitate_e_sufficientia" title="Necessitate e sufficientia – Interlingua" lang="ia" hreflang="ia" data-title="Necessitate e sufficientia" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Condizione_necessaria_e_sufficiente" title="Condizione necessaria e sufficiente – Italian" lang="it" hreflang="it" data-title="Condizione necessaria e sufficiente" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%B0%D0%B6%D0%B5%D1%82%D1%82%D1%96_%D0%B6%D3%99%D0%BD%D0%B5_%D0%B6%D0%B5%D1%82%D0%BA%D1%96%D0%BB%D1%96%D0%BA%D1%82%D1%96_%D1%88%D0%B0%D1%80%D1%82%D1%82%D0%B0%D1%80" title="Қажетті және жеткілікті шарттар – Kazakh" lang="kk" hreflang="kk" data-title="Қажетті және жеткілікті шарттар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A7%E1%80%80%E1%80%94%E1%80%BA%E1%80%99%E1%80%AF%E1%80%81%E1%80%BB%E1%80%96%E1%80%BC%E1%80%85%E1%80%BA%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8%E1%80%94%E1%80%BE%E1%80%84%E1%80%B7%E1%80%BA_%E1%80%9C%E1%80%AF%E1%80%B6%E1%80%9C%E1%80%B1%E1%80%AC%E1%80%80%E1%80%BA%E1%80%81%E1%80%BC%E1%80%84%E1%80%BA%E1%80%B8" title="ဧကန်မုချဖြစ်ခြင်းနှင့် လုံလောက်ခြင်း – Burmese" lang="my" hreflang="my" data-title="ဧကန်မုချဖြစ်ခြင်းနှင့် လုံလောက်ခြင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Noodzakelijke_en_voldoende_voorwaarde" title="Noodzakelijke en voldoende voorwaarde – Dutch" lang="nl" hreflang="nl" data-title="Noodzakelijke en voldoende voorwaarde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Necessidade_e_sufici%C3%AAncia" title="Necessidade e suficiência – Portuguese" lang="pt" hreflang="pt" data-title="Necessidade e suficiência" 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/%D0%9D%D0%B5%D0%BE%D0%B1%D1%85%D0%BE%D0%B4%D0%B8%D0%BC%D0%BE%D0%B5_%D0%B8_%D0%B4%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%BE%D1%87%D0%BD%D0%BE%D0%B5_%D1%83%D1%81%D0%BB%D0%BE%D0%B2%D0%B8%D1%8F" title="Необходимое и достаточное условия – Russian" lang="ru" hreflang="ru" data-title="Необходимое и достаточное условия" 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/Necessary_and_sufficient_conditions" title="Necessary and sufficient conditions – Simple English" lang="en-simple" hreflang="en-simple" data-title="Necessary and sufficient conditions" 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-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%DB%8E%D9%88%DB%8C%D8%B3%D8%AA%DB%8C_%D9%88_%D8%A8%DB%95%D8%B4%DA%A9%D8%B1%D8%AF%D9%88%D9%88%DB%8C%DB%8C" title="پێویستی و بەشکردوویی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="پێویستی و بەشکردوویی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4ltt%C3%A4m%C3%A4t%C3%B6n_ja_riitt%C3%A4v%C3%A4_ehto" title="Välttämätön ja riittävä ehto – Finnish" lang="fi" hreflang="fi" data-title="Välttämätön ja riittävä ehto" 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/N%C3%B6dv%C3%A4ndiga_och_tillr%C3%A4ckliga_villkor" title="Nödvändiga och tillräckliga villkor – Swedish" lang="sv" hreflang="sv" data-title="Nödvändiga och tillräckliga villkor" 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/%D0%9D%D0%B5%D0%BE%D0%B1%D1%85%D1%96%D0%B4%D0%BD%D0%B0_%D1%96_%D0%B4%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%BD%D1%8F_%D1%83%D0%BC%D0%BE%D0%B2%D0%B0" title="Необхідна і достатня умова – Ukrainian" lang="uk" hreflang="uk" data-title="Необхідна і достатня умова" 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/C%E1%BA%A7n_v%C3%A0_%C4%91%E1%BB%A7" title="Cần và đủ – Vietnamese" lang="vi" hreflang="vi" data-title="Cần và đủ" 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-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%85%85%E5%88%86%E5%BF%85%E8%A6%81%E6%A2%9D%E4%BB%B6" title="充分必要條件 – Cantonese" lang="yue" hreflang="yue" data-title="充分必要條件" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the formal terminology in logic. For causal meanings of the terms, see <a href="/wiki/Causality" title="Causality">Causality</a>. For the concepts in statistics, see <a href="/wiki/Sufficient_statistic" title="Sufficient statistic">Sufficient statistic</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Necessary But Not Sufficient" redirects here. For the novel by Eliyahu Goldratt, see <a href="/wiki/Necessary_But_Not_Sufficient_(novel)" title="Necessary But Not Sufficient (novel)">Necessary But Not Sufficient (novel)</a>.</div> <p>In <a href="/wiki/Logic" title="Logic">logic</a> and <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>necessity</b> and <b>sufficiency</b> are terms used to describe a <a href="/wiki/Material_conditional" title="Material conditional">conditional</a> or implicational relationship between two <a href="/wiki/Statement_(logic)" title="Statement (logic)">statements</a>. For example, in the <a href="/wiki/Conditional_sentence" title="Conditional sentence">conditional statement</a>: "If <span class="texhtml mvar" style="font-style:italic;">P</span> then <span class="texhtml mvar" style="font-style:italic;">Q</span>", <span class="texhtml mvar" style="font-style:italic;">Q</span> is <b>necessary</b> for <span class="texhtml mvar" style="font-style:italic;">P</span>, because the <a href="/wiki/Truth_value" title="Truth value">truth</a> of <span class="texhtml mvar" style="font-style:italic;">Q</span> is guaranteed by the truth of <span class="texhtml mvar" style="font-style:italic;">P</span>. (Equivalently, it is impossible to have <span class="texhtml mvar" style="font-style:italic;">P</span> without <span class="texhtml mvar" style="font-style:italic;">Q</span>, or the falsity of <span class="texhtml mvar" style="font-style:italic;">Q</span> ensures the falsity of <span class="texhtml mvar" style="font-style:italic;">P</span>.)<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> Similarly, <span class="texhtml mvar" style="font-style:italic;">P</span> is <b>sufficient</b> for <span class="texhtml mvar" style="font-style:italic;">Q</span>, because <span class="texhtml mvar" style="font-style:italic;">P</span> being true always implies that <span class="texhtml mvar" style="font-style:italic;">Q</span> is true, but <span class="texhtml mvar" style="font-style:italic;">P</span> not being true does not always imply that <span class="texhtml mvar" style="font-style:italic;">Q</span> is not true.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In general, a necessary condition is one (possibly one of several conditions) that must be present in order for another condition to occur, while a sufficient condition is one that produces the said condition.<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> The assertion that a statement is a "necessary <i>and</i> sufficient" condition of another means that the former statement is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the latter is true. That is, the two statements must be either simultaneously true, or simultaneously false.<sup id="cite_ref-betz_4-0" class="reference"><a href="#cite_note-betz-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Manktelow_5-0" class="reference"><a href="#cite_note-Manktelow-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-asnina_6-0" class="reference"><a href="#cite_note-asnina-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Ordinary_English" class="mw-redirect" title="Ordinary English">ordinary English</a> (also <a href="/wiki/Natural_language" title="Natural language">natural language</a>) "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being a man is a necessary condition for being a brother, but it is not sufficient—while being a man sibling is a necessary and sufficient condition for being a brother. Any conditional statement consists of at least one sufficient condition and at least one necessary condition. </p><p>In <a href="/wiki/Analytics" title="Analytics">data analytics</a>, necessity and sufficiency can refer to different <a href="/wiki/Causality" title="Causality">causal</a> logics,<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> where <a href="/wiki/Necessary_condition_analysis" title="Necessary condition analysis">necessary condition analysis</a> and <a href="/wiki/Qualitative_comparative_analysis" title="Qualitative comparative analysis">qualitative comparative analysis</a> can be used as analytical techniques for examining necessity and sufficiency of conditions for a particular outcome of interest. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necessity_and_sufficiency&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the conditional statement, "if <i>S</i>, then <i>N</i>", the expression represented by <i>S</i> is called the <a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a>, and the expression represented by <i>N</i> is called the <a href="/wiki/Consequent" title="Consequent">consequent</a>. This conditional statement may be written in several equivalent ways, such as "<i>N</i> if <i>S</i>", "<i>S</i> only if <i>N</i>", "<i>S</i> implies <i>N</i>", "<i>N</i> is implied by <i>S</i>", <span class="texhtml"><i>S</i> → <i>N</i></span> , <span class="texhtml"><i>S</i> ⇒ <i>N</i></span> and "<i>N</i> whenever <i>S</i>".<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>In the above situation of "N whenever S," <i>N</i> is said to be a <b>necessary</b> condition for <i>S</i>. In common language, this is equivalent to saying that if the conditional statement is a true statement, then the consequent <i>N</i> <i>must</i> be true—if <i>S</i> is to be true (see third column of "<a href="/wiki/Truth_table" title="Truth table">truth table</a>" immediately below). In other words, the antecedent <i>S</i> cannot be true without <i>N</i> being true. For example, in order for someone to be called <i><b>S</b></i>ocrates, it is necessary for that someone to be <i><b>N</b></i>amed. Similarly, in order for human beings to live, it is necessary that they have air.<sup id="cite_ref-:2_9-0" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>One can also say <i>S</i> is a <b>sufficient</b> condition for <i>N</i> (refer again to the third column of the truth table immediately below). If the conditional statement is true, then if <i>S</i> is true, <i>N</i> must be true; whereas if the conditional statement is true and N is true, then S may be true or be false. In common terms, "the truth of <i>S</i> guarantees the truth of <i>N</i>".<sup id="cite_ref-:2_9-1" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> For example, carrying on from the previous example, one can say that knowing that someone is called <i><b>S</b></i>ocrates is sufficient to know that someone has a <i><b>N</b></i>ame. </p><p>A <i><b>necessary and sufficient</b></i> condition requires that both of the implications <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 S\Rightarrow N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">⇒</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\Rightarrow N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2f1463e9ddc3683b68b6b6e35999e11a415367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle S\Rightarrow N}"></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 N\Rightarrow S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">⇒</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\Rightarrow S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f23438d00f8b46cce19ac2ffa05f2ab0e2efb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle N\Rightarrow S}"></span> (the latter of which can also be written as <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 S\Leftarrow N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">⇐</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\Leftarrow N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c075d21e5f5d53a3ebe2d33014d30174c07358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle S\Leftarrow N}"></span>) hold. The first implication suggests that <i>S</i> is a sufficient condition for <i>N</i>, while the second implication suggests that <i>S</i> is a necessary condition for <i>N</i>. This is expressed as "<i>S</i> is necessary and sufficient for <i>N</i> ", "<i>S</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>N</i> ", or <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 S\Leftrightarrow N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">⇔</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\Leftrightarrow N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fae01412e38dba7c1a6612037946df6df15d14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle S\Leftrightarrow N}"></span>. </p> <table class="wikitable" style="margin:1em auto; text-align:center;"> <caption>Truth table </caption> <tbody><tr> <th scope="col" style="width:20%"><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">S</span></span> </th> <th scope="col" style="width:20%"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><span class="nobold"><span class="texhtml mvar" style="font-style:italic;">N</span></span> </th> <th scope="col" style="width:20%"><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 S\Rightarrow N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">⇒</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\Rightarrow N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af2f1463e9ddc3683b68b6b6e35999e11a415367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle S\Rightarrow N}"></span> </th> <th scope="col" style="width:20%"><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 S\Leftarrow N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">⇐</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\Leftarrow N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c075d21e5f5d53a3ebe2d33014d30174c07358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle S\Leftarrow N}"></span> </th> <th scope="col" style="width:20%"><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 S\Leftrightarrow N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">⇔</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\Leftrightarrow N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fae01412e38dba7c1a6612037946df6df15d14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.177ex; height:2.176ex;" alt="{\displaystyle S\Leftrightarrow N}"></span> </th></tr> <tr> <td>T</td> <td>T</td> <td>T</td> <td>T</td> <td>T </td></tr> <tr> <td>T</td> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F</td> <td>T</td> <td style="background:papayawhip">F </td></tr> <tr> <td style="background:papayawhip">F</td> <td>T</td> <td>T</td> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F </td></tr> <tr> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F</td> <td>T</td> <td>T</td> <td>T </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Necessity">Necessity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necessity_and_sufficiency&action=edit&section=2" title="Edit section: Necessity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Solar_eclipse_1999_4.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Solar_eclipse_1999_4.jpg/200px-Solar_eclipse_1999_4.jpg" decoding="async" width="200" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Solar_eclipse_1999_4.jpg/300px-Solar_eclipse_1999_4.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Solar_eclipse_1999_4.jpg/400px-Solar_eclipse_1999_4.jpg 2x" data-file-width="3543" data-file-height="3489" /></a><figcaption>The sun being above the horizon is a necessary condition for direct sunlight; but it is not a sufficient condition, as something else may be casting a shadow, e.g., the moon in the case of an <a href="/wiki/Solar_eclipse" title="Solar eclipse">eclipse</a>.</figcaption></figure> <p>The assertion that <i>Q</i> is necessary for <i>P</i> is colloquially equivalent to "<i>P</i> cannot be true unless <i>Q</i> is true" or "if Q is false, then P is false".<sup id="cite_ref-:2_9-2" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> By <a href="/wiki/Contraposition" title="Contraposition">contraposition</a>, this is the same thing as "whenever <i>P</i> is true, so is <i>Q</i>". </p><p>The logical relation between <i>P</i> and <i>Q</i> is expressed as "if <i>P</i>, then <i>Q</i>" and denoted "<i>P</i> ⇒ <i>Q</i>" (<i>P</i> <a href="/wiki/Logical_consequence" title="Logical consequence">implies</a> <i>Q</i>). It may also be expressed as any of "<i>P</i> only if <i>Q</i>", "<i>Q</i>, if <i>P</i>", "<i>Q</i> whenever <i>P</i>", and "<i>Q</i> when <i>P</i>". One often finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition (i.e., individually necessary and jointly sufficient<sup id="cite_ref-:2_9-3" class="reference"><a href="#cite_note-:2-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>), as shown in Example 5. </p> <dl><dt>Example 1</dt> <dd>For it to be true that "John is a bachelor", it is necessary that it be also true that he is <ol><li>unmarried,</li> <li>male,</li> <li>adult,</li></ol></dd> <dd>since to state "John is a bachelor" implies John has each of those three additional <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a>.</dd></dl> <dl><dt>Example 2</dt> <dd>For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime.</dd></dl> <dl><dt>Example 3</dt> <dd>Consider thunder, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is thunder. The thunder <i>does not cause</i> the lightning (since lightning causes thunder), but because lightning always comes with thunder, we say that thunder is necessary for lightning. (That is, in its formal sense, necessity doesn't imply causality.)</dd></dl> <dl><dt>Example 4</dt> <dd>Being at least 30 years old is necessary for serving in the U.S. Senate. If you are under 30 years old, then it is impossible for you to be a senator. That is, if you are a senator, it follows that you must be at least 30 years old.</dd></dl> <dl><dt>Example 5</dt> <dd>In <a href="/wiki/Algebra" title="Algebra">algebra</a>, for some <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>S</i> together with an <a href="/wiki/Binary_operation" title="Binary operation">operation</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 \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"></span> to form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, it is necessary 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 \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"></span> be <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>. It is also necessary that <i>S</i> include a special element <i>e</i> such that for every <i>x</i> in <i>S</i>, it is the case that <i>e</i> <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 \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"></span> <i>x</i> and <i>x</i> <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 \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"></span> <i>e</i> both equal <i>x</i>. It is also necessary that for every <i>x</i> in <i>S</i> there exist a corresponding element <i>x″</i>, such that both <i>x</i> <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 \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"></span> <i>x″</i> and <i>x″</i> <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 \star }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋆</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \star }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \star }"></span> <i>x</i> equal the special element <i>e</i>. None of these three necessary conditions by itself is sufficient, but the <a href="/wiki/Conjunction_(logic)" class="mw-redirect" title="Conjunction (logic)">conjunction</a> of the three is.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sufficiency">Sufficiency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necessity_and_sufficiency&action=edit&section=3" title="Edit section: Sufficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ICE_3_Fahlenbach.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/ICE_3_Fahlenbach.jpg/200px-ICE_3_Fahlenbach.jpg" decoding="async" width="200" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/ICE_3_Fahlenbach.jpg/300px-ICE_3_Fahlenbach.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/ICE_3_Fahlenbach.jpg/400px-ICE_3_Fahlenbach.jpg 2x" data-file-width="3389" data-file-height="2236" /></a><figcaption>That a train runs on schedule is a sufficient condition for arriving on time (if one boards the train and it departs on time, then one will arrive on time); but it is not a necessary condition, since there are other ways to travel (if the train does not run to time, one could still arrive on time through other means of transport).</figcaption></figure> <p>If <i>P</i> is sufficient for <i>Q</i>, then knowing <i>P</i> to be true is adequate grounds to conclude that <i>Q</i> is true; however, knowing <i>P</i> to be false does not meet a minimal need to conclude that <i>Q</i> is false. </p><p>The logical relation is, as before, expressed as "if <i>P</i>, then <i>Q</i>" or "<i>P</i> ⇒ <i>Q</i>". This can also be expressed as "<i>P</i> only if <i>Q</i>", "<i>P</i> implies <i>Q</i>" or several other variants. It may be the case that several sufficient conditions, when taken together, constitute a single necessary condition (i.e., individually sufficient and jointly necessary), as illustrated in example 5. </p> <dl><dt>Example 1</dt> <dd>"John is a king" implies that John is male. So knowing that John is a king is sufficient to knowing that he is a male.</dd></dl> <dl><dt>Example 2</dt> <dd>A number's being divisible by 4 is sufficient (but not necessary) for it to be even, but being divisible by 2 is both sufficient and necessary for it to be even.</dd></dl> <dl><dt>Example 3</dt> <dd>An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense that hearing thunder, and unambiguously recognizing it as such, justifies concluding that there has been a lightning bolt.</dd></dl> <dl><dt>Example 4</dt> <dd>If the U.S. Congress passes a bill, the president's signing of the bill is sufficient to make it law. Note that the case whereby the president did not sign the bill, e.g. through exercising a presidential <a href="/wiki/Veto#United_States" title="Veto">veto</a>, does not mean that the bill has not become a law (for example, it could still have become a law through a congressional <a href="/wiki/Veto_override" class="mw-redirect" title="Veto override">override</a>).</dd></dl> <dl><dt>Example 5</dt> <dd>That the center of a <a href="/wiki/Playing_card" title="Playing card">playing card</a> should be marked with a single large spade (♠) is sufficient for the card to be an ace. Three other sufficient conditions are that the center of the card be marked with a single diamond (♦), heart (♥), or club (♣). None of these conditions is necessary to the card's being an ace, but their <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a> is, since no card can be an ace without fulfilling at least (in fact, exactly) one of these conditions.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Relationship_between_necessity_and_sufficiency">Relationship between necessity and sufficiency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necessity_and_sufficiency&action=edit&section=4" title="Edit section: Relationship between necessity and sufficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Set_intersection.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Set_intersection.svg/260px-Set_intersection.svg.png" decoding="async" width="260" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/da/Set_intersection.svg/390px-Set_intersection.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/da/Set_intersection.svg/520px-Set_intersection.svg.png 2x" data-file-width="900" data-file-height="600" /></a><figcaption>Being in the purple region is sufficient for being in A, but not necessary. Being in A is necessary for being in the purple region, but not sufficient. Being in A and being in B is necessary and sufficient for being in the purple region.</figcaption></figure> <p>A condition can be either necessary or sufficient without being the other. For instance, <i>being a <a href="/wiki/Mammal" title="Mammal">mammal</a></i> (<i>N</i>) is necessary but not sufficient to <i>being human</i> (<i>S</i>), and that a number <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> <i>is rational</i> (<i>S</i>) is sufficient but not necessary 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> <i>being a <a href="/wiki/Real_number" title="Real number">real number</a></i> (<i>N</i>) (since there are real numbers that are not rational). </p><p>A condition can be both necessary and sufficient. For example, at present, "today is the <a href="/wiki/Fourth_of_July" class="mw-redirect" title="Fourth of July">Fourth of July</a>" is a necessary and sufficient condition for "today is <a href="/wiki/Independence_Day_(United_States)" title="Independence Day (United States)">Independence Day</a> in the <a href="/wiki/United_States" title="United States">United States</a>". Similarly, a necessary and sufficient condition for <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">invertibility</a> of a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> <i>M</i> is that <i>M</i> has a nonzero <a href="/wiki/Determinant" title="Determinant">determinant</a>. </p><p>Mathematically speaking, necessity and sufficiency are <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">dual</a> to one another. For any statements <i>S</i> and <i>N</i>, the assertion that "<i>N</i> is necessary for <i>S</i>" is equivalent to the assertion that "<i>S</i> is sufficient for <i>N</i>". Another facet of this duality is that, as illustrated above, conjunctions (using "and") of necessary conditions may achieve sufficiency, while disjunctions (using "or") of sufficient conditions may achieve necessity. For a third facet, identify every mathematical <a href="/wiki/Predicate_(mathematics)" class="mw-redirect" title="Predicate (mathematics)">predicate</a> <i>N</i> with the set <i>T</i>(<i>N</i>) of objects, events, or statements for which <i>N</i> holds true; then asserting the necessity of <i>N</i> for <i>S</i> is equivalent to claiming that <i>T</i>(<i>N</i>) is a <a href="/wiki/Superset" class="mw-redirect" title="Superset">superset</a> of <i>T</i>(<i>S</i>), while asserting the sufficiency of <i>S</i> for <i>N</i> is equivalent to claiming that <i>T</i>(<i>S</i>) is a <a href="/wiki/Subset" title="Subset">subset</a> of <i>T</i>(<i>N</i>). </p><p>Psychologically speaking, necessity and sufficiency are both key aspects of the classical view of concepts. Under the classical theory of concepts, how human minds represent a category X, gives rise to a set of individually necessary conditions that define X. Together, these individually necessary conditions are sufficient to be X.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This contrasts with the probabilistic theory of concepts which states that no defining feature is necessary or sufficient, rather that categories resemble a family tree structure. </p> <div class="mw-heading mw-heading2"><h2 id="Simultaneous_necessity_and_sufficiency">Simultaneous necessity and sufficiency</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necessity_and_sufficiency&action=edit&section=5" title="Edit section: Simultaneous necessity and sufficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Material_equivalence" class="mw-redirect" title="Material equivalence">Material equivalence</a></div> <p>To say that <i>P</i> is necessary and sufficient for <i>Q</i> is to say two things: </p> <ol><li>that <i>P</i> is necessary for <i>Q</i>, <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\Leftarrow 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\Leftarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda627141a82e9511bfb63ad9607fe0a32eecf36" 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\Leftarrow Q}"></span>, and that <i>P</i> is sufficient for <i>Q</i>, <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/27a57c9bc077d0b20e4f5ec006f5342cfbb18fd2" 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>.</li> <li>equivalently, it may be understood to say that <i>P</i> and <i>Q</i> is necessary for the other, <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\land Q\Rightarrow P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">⇒</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Rightarrow Q\land Q\Rightarrow P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f1b67c85c701b89957ccb696178ba4da5ba9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.978ex; height:2.509ex;" alt="{\displaystyle P\Rightarrow Q\land Q\Rightarrow P}"></span>, which can also be stated as each <i>is sufficient for</i> or <i>implies</i> the other.</li></ol> <p>One may summarize any, and thus all, of these cases by the statement "<i>P</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>Q</i>", which 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 P\Leftrightarrow 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\Leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe75af42226920bc628ac7bbd53c023928f346ba" 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\Leftrightarrow Q}"></span>, whereas cases tell us 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\Leftrightarrow 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\Leftrightarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe75af42226920bc628ac7bbd53c023928f346ba" 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\Leftrightarrow Q}"></span> is identical 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 P\Rightarrow Q\land Q\Rightarrow P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇒</mo> <mi>Q</mi> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">⇒</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Rightarrow Q\land Q\Rightarrow P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f1b67c85c701b89957ccb696178ba4da5ba9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.978ex; height:2.509ex;" alt="{\displaystyle P\Rightarrow Q\land Q\Rightarrow P}"></span>. </p><p>For example, in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a> a graph <i>G</i> is called <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite</a> if it is possible to assign to each of its vertices the color <i>black</i> or <i>white</i> in such a way that every edge of <i>G</i> has one endpoint of each color. And for any graph to be bipartite, it is a necessary and sufficient condition that it contain no odd-length <a href="/wiki/Cycle_(graph_theory)" title="Cycle (graph theory)">cycles</a>. Thus, discovering whether a graph has any odd cycles tells one whether it is bipartite and conversely. A philosopher<sup id="cite_ref-stan_11-0" class="reference"><a href="#cite_note-stan-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> might characterize this state of affairs thus: "Although the concepts of bipartiteness and absence of odd cycles differ in <a href="/wiki/Intension" title="Intension">intension</a>, they have identical <a href="/wiki/Extension_(semantics)" title="Extension (semantics)">extension</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>In mathematics, theorems are often stated in the form "<i>P</i> is true if and only if <i>Q</i> is true". </p><p>Because, as explained in previous section, necessity of one for the other is equivalent to sufficiency of the other for the first one, e.g. <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\Leftarrow 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\Leftarrow Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda627141a82e9511bfb63ad9607fe0a32eecf36" 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\Leftarrow Q}"></span> is <a href="/wiki/Logical_equivalence" title="Logical equivalence">equivalent to</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 Q\Rightarrow P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">⇒</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\Rightarrow P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048d962b1984242660104afec771ac59dab3bb81" 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 Q\Rightarrow P}"></span>, if <i>P</i> is necessary and sufficient for <i>Q</i>, then <i>Q</i> is necessary and sufficient for <i>P</i>. We can write <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\Leftrightarrow Q\equiv Q\Leftrightarrow P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">⇔</mo> <mi>Q</mi> <mo>≡</mo> <mi>Q</mi> <mo stretchy="false">⇔</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\Leftrightarrow Q\equiv Q\Leftrightarrow P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09abe443a1d9fd8661dc827e7ec7f7007b228ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.494ex; height:2.509ex;" alt="{\displaystyle P\Leftrightarrow Q\equiv Q\Leftrightarrow P}"></span> and say that the statements "<i>P</i> is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>Q</i>, is true" and "<i>Q</i> is true if and only if <i>P</i> is true" are equivalent. </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=Necessity_and_sufficiency&action=edit&section=6" 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" style="column-width: 30em;"> <ul><li><a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">Affirming the consequent</a></li> <li><a href="/wiki/Biological_tests_of_necessity_and_sufficiency" title="Biological tests of necessity and sufficiency">Biological tests of necessity and sufficiency</a></li> <li><a href="/wiki/Causality" title="Causality">Causality</a></li> <li><a href="/wiki/Closed_concept" title="Closed concept">Closed concept</a></li> <li><a href="/wiki/Denying_the_antecedent" title="Denying the antecedent">Denying the antecedent</a></li> <li><a href="/wiki/If_and_only_if" title="If and only if">If and only if</a></li> <li><a href="/wiki/Material_implication_(disambiguation)" class="mw-redirect mw-disambig" title="Material implication (disambiguation)">Material implication (disambiguation)</a></li> <li><a href="/wiki/Principle_of_sufficient_reason" title="Principle of sufficient reason">Principle of sufficient reason</a></li> <li><a href="/wiki/Wason_selection_task" title="Wason selection task">Wason selection task</a></li> <li><i><a href="/wiki/Modus_ponens" title="Modus ponens">Modus ponens</a></i></li> <li><i><a href="/wiki/Modus_tollens" title="Modus tollens">Modus tollens</a></i></li></ul></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=Necessity_and_sufficiency&action=edit&section=7" 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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://philosophy.hku.hk/think/meaning/nsc.php">"[M06] Necessity and sufficiency"</a>. <i>philosophy.hku.hk</i><span class="reference-accessdate">. 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New York: Springer. p. 247. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7487-7" title="Special:BookSources/978-1-4419-7487-7"><bdi>978-1-4419-7487-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Managing+Science%3A+Methodology+and+Organization+of+Research&rft.place=New+York&rft.pages=247&rft.pub=Springer&rft.date=2011&rft.isbn=978-1-4419-7487-7&rft.aulast=Betz&rft.aufirst=Frederick&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" class="Z3988"></span></span> </li> <li id="cite_note-Manktelow-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Manktelow_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFManktelow1999" class="citation book cs1">Manktelow, K. I. (1999). <i>Reasoning and Thinking</i>. East Sussex, UK: Psychology Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-86377-708-2" title="Special:BookSources/0-86377-708-2"><bdi>0-86377-708-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reasoning+and+Thinking&rft.place=East+Sussex%2C+UK&rft.pub=Psychology+Press&rft.date=1999&rft.isbn=0-86377-708-2&rft.aulast=Manktelow&rft.aufirst=K.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" class="Z3988"></span></span> </li> <li id="cite_note-asnina-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-asnina_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsnina,_ErikaOsis,_JanisJansone,_Asnate2013" class="citation journal cs1">Asnina, Erika; Osis, Janis & Jansone, Asnate (2013). "Formal Specification of Topological Relations". <i>Databases and Information Systems VII</i>. <b>249</b> (Databases and Information Systems VII): 175. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3233%2F978-1-61499-161-8-175">10.3233/978-1-61499-161-8-175</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Databases+and+Information+Systems+VII&rft.atitle=Formal+Specification+of+Topological+Relations&rft.volume=249&rft.issue=Databases+and+Information+Systems+VII&rft.pages=175&rft.date=2013&rft_id=info%3Adoi%2F10.3233%2F978-1-61499-161-8-175&rft.au=Asnina%2C+Erika&rft.au=Osis%2C+Janis&rft.au=Jansone%2C+Asnate&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" 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="CITEREFRichterHauff2022" class="citation journal cs1">Richter, Nicole Franziska; Hauff, Sven (2022-08-01). <a rel="nofollow" class="external text" href="https://findresearcher.sdu.dk/ws/files/199180258/1_s2.0_S1090951622000037_main.pdf">"Necessary conditions in international business research–Advancing the field with a new perspective on causality and data analysis"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of World Business</i>. <b>57</b> (5): 101310. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jwb.2022.101310">10.1016/j.jwb.2022.101310</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1090-9516">1090-9516</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+World+Business&rft.atitle=Necessary+conditions+in+international+business+research%E2%80%93Advancing+the+field+with+a+new+perspective+on+causality+and+data+analysis&rft.volume=57&rft.issue=5&rft.pages=101310&rft.date=2022-08-01&rft_id=info%3Adoi%2F10.1016%2Fj.jwb.2022.101310&rft.issn=1090-9516&rft.aulast=Richter&rft.aufirst=Nicole+Franziska&rft.au=Hauff%2C+Sven&rft_id=https%3A%2F%2Ffindresearcher.sdu.dk%2Fws%2Ffiles%2F199180258%2F1_s2.0_S1090951622000037_main.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin2004" class="citation cs2">Devlin, Keith (2004), <i>Sets, Functions and Logic / An Introduction to Abstract Mathematics</i> (3rd ed.), Chapman & Hall, pp. 22–23, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-449-1" title="Special:BookSources/978-1-58488-449-1"><bdi>978-1-58488-449-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+Functions+and+Logic+%2F+An+Introduction+to+Abstract+Mathematics&rft.pages=22-23&rft.edition=3rd&rft.pub=Chapman+%26+Hall&rft.date=2004&rft.isbn=978-1-58488-449-1&rft.aulast=Devlin&rft.aufirst=Keith&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" class="Z3988"></span></span> </li> <li id="cite_note-:2-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-:2_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:2_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:2_9-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:2_9-3"><sup><i><b>d</b></i></sup></a></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.sfu.ca/~swartz/conditions1.htm#section3">"The Concept of Necessary Conditions and Sufficient Conditions"</a>. <i>www.sfu.ca</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-02</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.sfu.ca&rft.atitle=The+Concept+of+Necessary+Conditions+and+Sufficient+Conditions&rft_id=https%3A%2F%2Fwww.sfu.ca%2F~swartz%2Fconditions1.htm%23section3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://iep.utm.edu/classical-theory-of-concepts/">"Classical Theory of Concepts, the | Internet Encyclopedia of Philosophy"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Classical+Theory+of+Concepts%2C+the+%26%23124%3B+Internet+Encyclopedia+of+Philosophy&rft_id=https%3A%2F%2Fiep.utm.edu%2Fclassical-theory-of-concepts%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANecessity+and+sufficiency" class="Z3988"></span></span> </li> <li id="cite_note-stan-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-stan_11-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-intensional/">Stanford University primer, 2006</a>.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">"Meanings, in this sense, are often called <i>intensions</i>, and things designated, <i>extensions</i>. Contexts in which extension is all that matters are, naturally, called <i>extensional</i>, while contexts in which extension is not enough are <i>intensional</i>. Mathematics is typically extensional throughout." <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/logic-intensional/">Stanford University primer, 2006</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Necessity_and_sufficiency&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 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href="https://commons.wikimedia.org/wiki/Category:Necessity_and_sufficiency" class="extiw" title="commons:Category:Necessity and sufficiency">Necessity and sufficiency</a></span>.</div></div> </div> <ul><li>Critical thinking web tutorial: <a rel="nofollow" class="external text" href="http://philosophy.hku.hk/think/meaning/nsc.php"><i>Necessary and Sufficient Conditions</i></a></li> <li>Simon Fraser University: <a rel="nofollow" class="external text" href="https://www.sfu.ca/~swartz/conditions1.htm">Concepts with examples</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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title="Inference">Inference</a></li> <li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</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/Classical_logic" title="Classical logic">Classical</a></li> <li><a href="/wiki/Informal_logic" title="Informal logic">Informal</a> <ul><li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a></li> <li><a href="/wiki/Reason" title="Reason">Reason</a></li></ul></li> <li><a href="/wiki/Mathematical_logic" 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0.25em"> <ul><li><a href="/wiki/Abductive_reasoning" title="Abductive reasoning">Abduction</a></li> <li><a href="/wiki/Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li> <li><a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li> <li><a href="/wiki/Consequent" title="Consequent">Consequent</a></li> <li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a> <ul><li><a href="/wiki/Paradox" title="Paradox">Paradox</a></li> <li><a href="/wiki/Antinomy" title="Antinomy">Antinomy</a></li></ul></li> <li><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deduction</a></li> <li><a href="/wiki/Deductive_closure" title="Deductive closure">Deductive closure</a></li> <li><a href="/wiki/Definition" title="Definition">Definition</a></li> <li><a href="/wiki/Description" title="Description">Description</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Entailment</a> <ul><li><a href="/wiki/Entailment_(linguistics)" title="Entailment (linguistics)">Linguistic</a></li></ul></li> <li><a href="/wiki/Logical_form" title="Logical form">Form</a></li> <li><a href="/wiki/Inductive_reasoning" title="Inductive reasoning">Induction</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Name" title="Name">Name</a></li> <li><a class="mw-selflink selflink">Necessity and sufficiency</a></li> <li><a href="/wiki/Premise" title="Premise">Premise</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Reference" title="Reference">Reference</a></li> <li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Truth" title="Truth">Truth</a></li> 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