直言三段论 - 维基百科，自由的百科全书

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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-语气和格式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#语气和格式"> <div class="vector-toc-text"> 1 语气和格式 </div> </a> <ul id="toc-语气和格式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-有效性" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#有效性"> <div class="vector-toc-text"> 2 有效性 </div> </a> <button aria-controls="toc-有效性-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 开关有效性子章节 </button> <ul id="toc-有效性-sublist" class="vector-toc-list"> <li id="toc-有效三段論式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#有效三段論式"> <div class="vector-toc-text"> 2.1 有效三段論式 </div> </a> <ul id="toc-有效三段論式-sublist" 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href="#第1格"> <div class="vector-toc-text"> 3.1.1 第1格 </div> </a> <ul id="toc-第1格-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-第2格" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#第2格"> <div class="vector-toc-text"> 3.1.2 第2格 </div> </a> <ul id="toc-第2格-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-第3格" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#第3格"> <div class="vector-toc-text"> 3.1.3 第3格 </div> </a> <ul id="toc-第3格-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-增补的论式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#增补的论式"> <div class="vector-toc-text"> 3.2 增补的论式 </div> </a> <ul id="toc-增补的论式-sublist" class="vector-toc-list"> <li id="toc-第4格" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#第4格"> <div class="vector-toc-text"> 3.2.1 第4格 </div> </a> <ul id="toc-第4格-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-结论弱化的论式" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#结论弱化的论式"> <div class="vector-toc-text"> 3.2.2 结论弱化的论式 </div> </a> <ul id="toc-结论弱化的论式-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-谓词演算公式的注解" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#谓词演算公式的注解"> <div class="vector-toc-text"> 3.3 谓词演算公式的注解 </div> </a> <ul id="toc-谓词演算公式的注解-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-24論式圖示" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#24論式圖示"> <div class="vector-toc-text"> 4 24論式圖示 </div> </a> <ul id="toc-24論式圖示-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> 5 参见 </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-註解" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註解"> <div class="vector-toc-text"> 6 註解 </div> </a> <ul id="toc-註解-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-引用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#引用"> <div class="vector-toc-text"> 7 引用 </div> </a> <ul id="toc-引用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部連結" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部連結"> <div class="vector-toc-text"> 8 外部連結 </div> </a> <ul id="toc-外部連結-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > 开关目录 </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">直言三段论</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。63种语言可用" > <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-63" aria-hidden="true" > 63种语言 </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Syllogismus" title="Syllogismus – 瑞士德语" lang="gsw" hreflang="gsw" data-title="Syllogismus" data-language-autonym="Alemannisch" data-language-local-name="瑞士德语" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%8A%D8%A7%D8%B3_(%D9%85%D9%86%D8%B7%D9%82)" title="قياس (منطق) – 阿拉伯语" lang="ar" hreflang="ar" data-title="قياس (منطق)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Sillogizm" title="Sillogizm – 阿塞拜疆语" lang="az" hreflang="az" data-title="Sillogizm" data-language-autonym="Azərbaycanca" data-language-local-name="阿塞拜疆语" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%BE%D0%B3%D0%B8%D0%B7%D1%8A%D0%BC" title="Силогизъм – 保加利亚语" lang="bg" hreflang="bg" data-title="Силогизъм" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Sullogismos" title="Sullogismos – 布列塔尼语" lang="br" hreflang="br" data-title="Sullogismos" data-language-autonym="Brezhoneg" data-language-local-name="布列塔尼语" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sil%C2%B7logisme" title="Sil·logisme – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Sil·logisme" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%82%DB%8C%D8%A7%D8%B3" title="قیاس – 中库尔德语" lang="ckb" hreflang="ckb" data-title="قیاس" data-language-autonym="کوردی" data-language-local-name="中库尔德语" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sylogismus" title="Sylogismus – 捷克语" lang="cs" hreflang="cs" data-title="Sylogismus" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Syllogisme" title="Syllogisme – 丹麦语" lang="da" hreflang="da" data-title="Syllogisme" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="优良条目"><a href="https://de.wikipedia.org/wiki/Syllogismus" title="Syllogismus – 德语" lang="de" hreflang="de" data-title="Syllogismus" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BB%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Συλλογισμός – 希腊语" lang="el" hreflang="el" data-title="Συλλογισμός" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Syllogism" title="Syllogism – 英语" lang="en" hreflang="en" data-title="Syllogism" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Silogismo" title="Silogismo – 世界语" lang="eo" hreflang="eo" data-title="Silogismo" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Silogismo" title="Silogismo – 西班牙语" lang="es" hreflang="es" data-title="Silogismo" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/S%C3%BCllogism" title="Süllogism – 爱沙尼亚语" lang="et" hreflang="et" data-title="Süllogism" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Silogismo" title="Silogismo – 巴斯克语" lang="eu" hreflang="eu" data-title="Silogismo" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%DB%8C%D8%A7%D8%B3" title="قیاس – 波斯语" lang="fa" hreflang="fa" data-title="قیاس" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Syllogismi" title="Syllogismi – 芬兰语" lang="fi" hreflang="fi" data-title="Syllogismi" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syllogisme" title="Syllogisme – 法语" lang="fr" hreflang="fr" data-title="Syllogisme" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Siloxismo" title="Siloxismo – 加利西亚语" lang="gl" hreflang="gl" data-title="Siloxismo" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9C%D7%95%D7%92%D7%99%D7%96%D7%9D" title="סילוגיזם – 希伯来语" lang="he" hreflang="he" data-title="סילוגיזם" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AF%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A5%8D%E0%A4%AF" title="न्यायवाक्य – 印地语" lang="hi" hreflang="hi" data-title="न्यायवाक्य" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Silogizam" title="Silogizam – 克罗地亚语" lang="hr" hreflang="hr" data-title="Silogizam" data-language-autonym="Hrvatski" data-language-local-name="克罗地亚语" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szillogizmus" title="Szillogizmus – 匈牙利语" lang="hu" hreflang="hu" data-title="Szillogizmus" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8D%D5%AB%D5%AC%D5%AC%D5%B8%D5%A3%D5%AB%D5%BD%D5%BF%D5%AB%D5%AF%D5%A1" title="Սիլլոգիստիկա – 亚美尼亚语" lang="hy" hreflang="hy" data-title="Սիլլոգիստիկա" data-language-autonym="Հայերեն" data-language-local-name="亚美尼亚语" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Syllogismo" title="Syllogismo – 国际语" lang="ia" hreflang="ia" data-title="Syllogismo" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Silogisme" title="Silogisme – 印度尼西亚语" lang="id" hreflang="id" data-title="Silogisme" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Silogismo" title="Silogismo – 伊多语" lang="io" hreflang="io" data-title="Silogismo" data-language-autonym="Ido" data-language-local-name="伊多语" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%B6khenda" title="Rökhenda – 冰岛语" lang="is" hreflang="is" data-title="Rökhenda" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sillogismo" title="Sillogismo – 意大利语" lang="it" hreflang="it" data-title="Sillogismo" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%89%E6%AE%B5%E8%AB%96%E6%B3%95" title="三段論法 – 日语" lang="ja" hreflang="ja" data-title="三段論法" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9E%D0%B9_%D2%9B%D0%BE%D1%80%D1%8B%D1%82%D1%8B%D0%BD%D0%B4%D1%8B" title="Ой қорытынды – 哈萨克语" lang="kk" hreflang="kk" data-title="Ой қорытынды" data-language-autonym="Қазақша" data-language-local-name="哈萨克语" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%BC%EB%8B%A8%EB%85%BC%EB%B2%95" title="삼단논법 – 韩语" lang="ko" hreflang="ko" data-title="삼단논법" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%BB%D0%BE%D0%B3%D0%B8%D0%B7%D0%BC" title="Силлогизм – 柯尔克孜语" lang="ky" hreflang="ky" data-title="Силлогизм" data-language-autonym="Кыргызча" data-language-local-name="柯尔克孜语" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Syllogismus" title="Syllogismus – 拉丁语" lang="la" hreflang="la" data-title="Syllogismus" data-language-autonym="Latina" data-language-local-name="拉丁语" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Syllogismus" title="Syllogismus – 卢森堡语" lang="lb" hreflang="lb" data-title="Syllogismus" data-language-autonym="Lëtzebuergesch" data-language-local-name="卢森堡语" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Silogizmas" title="Silogizmas – 立陶宛语" lang="lt" hreflang="lt" data-title="Silogizmas" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%BE%D0%B3%D0%B8%D0%B7%D0%B0%D0%BC" title="Силогизам – 马其顿语" lang="mk" hreflang="mk" data-title="Силогизам" data-language-autonym="Македонски" data-language-local-name="马其顿语" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A5%8D%E0%A4%AF" title="संवाक्य – 马拉地语" lang="mr" hreflang="mr" data-title="संवाक्य" data-language-autonym="मराठी" data-language-local-name="马拉地语" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Syllogismus" title="Syllogismus – 低地德语" lang="nds" hreflang="nds" data-title="Syllogismus" data-language-autonym="Plattdüütsch" data-language-local-name="低地德语" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Syllogisme" title="Syllogisme – 荷兰语" lang="nl" hreflang="nl" data-title="Syllogisme" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Syllogisme" title="Syllogisme – 书面挪威语" lang="nb" hreflang="nb" data-title="Syllogisme" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Silogisme" title="Silogisme – 諾維亞文" lang="nov" hreflang="nov" data-title="Silogisme" data-language-autonym="Novial" data-language-local-name="諾維亞文" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sillogisme" title="Sillogisme – 奥克语" lang="oc" hreflang="oc" data-title="Sillogisme" data-language-autonym="Occitan" data-language-local-name="奥克语" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sylogizm" title="Sylogizm – 波兰语" lang="pl" hreflang="pl" data-title="Sylogizm" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Silogism" title="Silogism – 皮埃蒙特文" lang="pms" hreflang="pms" data-title="Silogism" data-language-autonym="Piemontèis" data-language-local-name="皮埃蒙特文" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Silogismo" title="Silogismo – 葡萄牙语" lang="pt" hreflang="pt" data-title="Silogismo" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Silogism" title="Silogism – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Silogism" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%BB%D0%BE%D0%B3%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0" title="Силлогистика – 俄语" lang="ru" hreflang="ru" data-title="Силлогистика" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Syllogism" title="Syllogism – Simple English" lang="en-simple" hreflang="en-simple" data-title="Syllogism" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Sylogizmus" title="Sylogizmus – 斯洛伐克语" lang="sk" hreflang="sk" data-title="Sylogizmus" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克语" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Silogizem" title="Silogizem – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Silogizem" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Silogjizmi" title="Silogjizmi – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Silogjizmi" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%BE%D0%B3%D0%B8%D0%B7%D0%B0%D0%BC" title="Силогизам – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Силогизам" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Syllogism" title="Syllogism – 瑞典语" lang="sv" hreflang="sv" data-title="Syllogism" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Silohismo" title="Silohismo – 他加禄语" lang="tl" hreflang="tl" data-title="Silohismo" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tas%C4%B1m" title="Tasım – 土耳其语" lang="tr" hreflang="tr" data-title="Tasım" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BB%D0%BE%D0%B3%D1%96%D0%B7%D0%BC" title="Силогізм – 乌克兰语" lang="uk" hreflang="uk" data-title="Силогізм" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%DB%8C%D8%A7%D8%B3_(%D9%85%D9%86%D8%B7%D9%82)" title="قیاس (منطق) – 乌尔都语" lang="ur" hreflang="ur" data-title="قیاس (منطق)" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sillogizm" title="Sillogizm – 乌兹别克语" lang="uz" hreflang="uz" data-title="Sillogizm" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="乌兹别克语" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Tam_%C4%91o%E1%BA%A1n_lu%E1%BA%ADn" title="Tam đoạn luận – 越南语" lang="vi" hreflang="vi" data-title="Tam đoạn luận" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA" title="直言三段论 – 吴语" lang="wuu" hreflang="wuu" 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data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr">直言三段论是所有<a href="/wiki/%E5%89%8D%E6%8F%90" title="前提">前提</a>都是<a href="/wiki/%E7%9B%B4%E8%A8%80%E5%91%BD%E9%A1%8C" title="直言命題">直言命题</a>的<a href="/wiki/%E6%BC%94%E7%BB%8E%E6%8E%A8%E7%90%86" title="演绎推理">演绎推理</a>。前兩個<a href="/wiki/%E5%91%BD%E9%A2%98" title="命题">命題</a>被分别称为大前提和小前提<a href="#cite_note-1">[1]</a>。如果這個三段論是<a href="/wiki/%E6%9C%89%E6%95%88%E7%9A%84" class="mw-redirect" title="有效的">有效的</a>，這兩個前提邏輯上蕴涵了最後的命題，它叫做結論。結論的真實性建立在前提的真實性和它們之間的聯繫之上：<a href="/wiki/%E4%B8%AD%E9%A1%B9" title="中项">中項</a>在前提中必須<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>（distribute）至少一次，形成在結論中的主詞和謂词之間的連接。例如： <blockquote style="float: none; margin: 0.7em 0.3em; border: solid 1px #E0CB52; border-left: solid 5px #E0CB52; padding: 0.3em 0.7em; background-color: #FCFAEE;"> <div> <dl><dd>所有生物都會死。</dd> <dd>所有人都是生物。</dd> <dd>所以，所有人都會死。</dd></dl> </div></blockquote> 這裡的中項“生物”在大前提中周延，大項“會死者”在大前提和結論中都不周延，小項“人”在小前提和結論中周延；這個三段論符合<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>規則：中項至少在一個前提中<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>。一些直言<a href="/wiki/%E4%B8%89%E6%AE%B5%E8%AB%96" title="三段論">三段論</a>不是有效的，例如： <blockquote style="float: none; margin: 0.7em 0.3em; border: solid 1px #E0CB52; border-left: solid 5px #E0CB52; padding: 0.3em 0.7em; background-color: #FCFAEE;"> <div> <dl><dd>所有鳥都有翅膀。</dd> <dd>所有人都不是鳥。</dd> <dd>所以，没有人有翅膀。</dd></dl> </div></blockquote> 即使此例子的兩個前提和結論都是正確的，中項“鳥”在大前提和小前提中周延，大項“有翅膀”在結論中周延，小項“人”在小前提和結論中周延；此三段論卻是一種<a href="/wiki/%E5%A4%A7%E8%A9%9E%E4%B8%8D%E7%95%B6" title="大詞不當">大項不當</a><a href="/wiki/%E5%BD%A2%E5%BC%8F%E8%AC%AC%E8%AA%A4" title="形式謬誤">謬誤</a>，將結論“沒有人有翅膀”理解為同樣表達的“所有人沒有翅膀”如此一來方便了解其中的谬误；此三段論不有效的原因是它不符合另一個<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>規則：在結論中周延的詞項，在前提中也必須周延。在該三段論中大項“有翅膀”在結論被否定了，也就是說表達了人沒有“有翅膀”，大項在此周延，但在大前提中未周延，因為在大前提中“有翅膀”並沒有涉及該項的所有個體。 <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="语气和格式">语气和格式</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=1" title="编辑章节：语气和格式">编辑</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Square_of_opposition,_set_diagrams.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Square_of_opposition%2C_set_diagrams.svg/300px-Square_of_opposition%2C_set_diagrams.svg.png" decoding="async" width="300" height="388" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Square_of_opposition%2C_set_diagrams.svg/450px-Square_of_opposition%2C_set_diagrams.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Square_of_opposition%2C_set_diagrams.svg/600px-Square_of_opposition%2C_set_diagrams.svg.png 2x" data-file-width="479" data-file-height="620" /></a><figcaption><a href="/wiki/%E5%AF%B9%E7%AB%8B%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="对立四边形">對立四邊形</a>圖，揭示傳統邏輯四種命題語氣的關係，紅色表示非空，黑色表示空。</figcaption></figure> 三段論有如下典型形式： <dl><dd>大前提：所有M是P。</dd> <dd>小前提：所有S是M。</dd> <dd>結論：所有S是P。</dd></dl> 其中S代表結論的<a href="/wiki/%E4%B8%BB%E8%A9%9E" class="mw-redirect" title="主詞">主詞</a>（Subject），P代表結論的<a href="/wiki/%E8%AC%82%E8%A9%9E" class="mw-redirect" title="謂詞">謂詞</a>（Predicate），M代表中詞（Middle）。 三段論的命題可分為<a href="/wiki/%E5%85%A8%E7%A7%B0%E9%87%8F%E5%8C%96" title="全称量化">全称</a>（universal）、<a href="/wiki/%E5%AD%98%E5%9C%A8%E9%87%8F%E5%8C%96" title="存在量化">特称</a>（particular），及肯定、否定，組合起來有以下四類語氣（Mood）： <dl><dd><table class="wikitable"> <tbody><tr> <th>類型</th> <th>代號</th> <th>形式</th> <th>範例 </th></tr> <tr> <td>全稱肯定型</td> <td><tt>A</tt>（SaP）</td> <td>所有S是P</td> <td>所有人是會死的 </td></tr> <tr> <td>全稱否定型</td> <td><tt>E</tt>（SeP）</td> <td>沒有S是P</td> <td>沒有人是完美的 </td></tr> <tr> <td>特稱肯定型</td> <td><tt>I</tt>（SiP）</td> <td>有些S是P</td> <td>有些人是健康的 </td></tr> <tr> <td>特稱否定型</td> <td><tt>O</tt>（SoP）</td> <td>有些S不是P</td> <td>有些人不是健康的 </td></tr></tbody></table></dd></dl> 三段論中，結論中的謂詞稱作大詞（P，或稱大項），包含大詞在內的前提稱作大前提；結論中的主詞稱作小詞（S，或稱小項），包含小詞在內的前提稱作小前提；沒有出現在結論，卻在兩個前提重複出現的稱作中詞（M，或稱中項）。大詞、中詞、小詞依不同排列方式，可分成四種格（Figure）： <dl><dd><table class="wikitable" style="font-family:Arial;"> <tbody><tr> <th></th> <th>第1格</th> <th>第2格</th> <th>第3格</th> <th>第4格 </th></tr> <tr> <th>大前提 </th> <td>M-P</td> <td>P-M</td> <td>M-P</td> <td>P-M </td></tr> <tr> <th>小前提 </th> <td>S-M</td> <td>S-M</td> <td>M-S</td> <td>M-S </td></tr> <tr> <th>結論 </th> <td>S-P</td> <td>S-P</td> <td>S-P</td> <td>S-P </td></tr> </tbody></table></dd></dl> 將以上整合在一起，三段論的大前提、小前提、結論分別可為<tt>A</tt>、<tt>E</tt>、<tt>I</tt>、<tt>O</tt>型命題之一，又可分為4格，故總共有256種三段論（若考慮大前提與小前提對調，便有512種，但邏輯上是相同的）。 三段論依語氣與格的分類縮寫，例如AAA-1（也可以寫成1-AAA）代表「大前提為A型，小前提為A型，結論為A型，第1格」的三段論。 此外，三段論的四種格之间可相互转换： <ul><li>第1格：对换大前提的主词和谓词的位置就变成第2格，对换小前提的主词和谓词的位置就变成第3格。</li> <li>第2格：对换大前提的主词和谓词的位置就变成第1格，对换小前提的主词和谓词的位置就变成第4格。</li> <li>第3格：对换大前提的主词和谓词的位置就变成第4格，对换小前提的主词和谓词的位置就变成第1格。</li> <li>第4格：对换大前提的主词和谓词的位置就变成第3格，对换小前提的主词和谓词的位置就变成第2格。</li></ul> <tt>E</tt>和<tt>I</tt>命题对换主词和谓词的位置而保持同原命题等价。<tt>A</tt>命题和<tt>O</tt>命题不能对换主词和谓词的位置，但是可以采用<a href="/wiki/%E7%9B%B4%E6%8E%A5%E6%8E%A8%E7%90%86" title="直接推理">直接推理</a>中的“对置法”。A命题还可以在确实主词有元素存在的前提下，转换成弱于原命题的<tt>I</tt>命题后再对换主词和谓词的位置。 <div class="mw-heading mw-heading2"><h2 id="有效性">有效性</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=2" title="编辑章节：有效性">编辑</a>]</div> 考虑各种直言三段论的有效性將是非常冗长耗時的。前人想出了三个可供选择的方法来找出有效性。方法之一是记住下一章节中列出的所有論式。 還可以通过构造<a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏图</a>的方法得到有效形式。因为有三种项，文氏图需要三个交叠的圓圈来表示每一个类。首先，为小项构造一个圓圈。临近小项的圓圈的是同小項有着交叠的大项的圓圈。在这两个圓圈之上是中项的圓圈。它应当在三个位置有着交叠：大项，小项和大项与小项交叠的地方。一個三段论是有效的，其必然条件是通过图解两个前提得出结论的真实性。永不图解结论，因为结论必须从前提推导出来。总是首先图解全称命题。这是通过对一个类在另一个类中没有成员的区域加黑影来实现的。所以在前面例子的AAA-1形式中大前提“所有M是P”中，对M不与P交叠的所有区域加黑影，包括M与S交叠的部分。接着对小前提重复同样的过程。从这两个前提中可推导出在类S中所有成员也是类P的成员。但是，不能推出类P的所有成员都是类S的成员。 <dl><dd><figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:Modus_Barbara.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Modus_Barbara.svg/400px-Modus_Barbara.svg.png" decoding="async" width="400" height="520" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Modus_Barbara.svg/600px-Modus_Barbara.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Modus_Barbara.svg/800px-Modus_Barbara.svg.png 2x" data-file-width="718" data-file-height="934" /></a><figcaption></figcaption></figure></dd></dl> 作为文氏圖方法的另一个例子，考虑形式EIO-1的三段论。它的大前提是“没有M是P”，它的小前提是“有些S是M”，它的结论是“有些S不是P”。这个三段论的大项是P，它的小项是S，它的中项是M。大前提在图中通过对交集M ∩ P加阴影表示。小前提不能通过对任何区域加黑影表示。转而，我们可以在交集S ∩ M的非黑影部分使用x符号来表示“有些S是M”。（注意：黑影区域和<a href="/wiki/%E5%AD%98%E5%9C%A8%E9%87%8F%E5%8C%96" title="存在量化">存在量化</a>区域是互斥的）。接着因为存在符号位于S内但在P外，所以结论“存在一些S不是P”是正确的。 <dl><dd><figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:Modus_Ferio.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Ferio.svg/400px-Modus_Ferio.svg.png" decoding="async" width="400" height="520" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Ferio.svg/600px-Modus_Ferio.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Ferio.svg/800px-Modus_Ferio.svg.png 2x" data-file-width="718" data-file-height="934" /></a><figcaption></figcaption></figure></dd></dl> 本文最後一節列出了所有24個有效論式的文氏圖。 最后一种方法是记住下面非形式表述的幾條规则以避免<a href="/wiki/%E8%AC%AC%E8%AB%96" class="mw-redirect" title="謬論">謬論</a>。尽管文氏图对于诠释目的是好工具，有人更喜欢用這些规则来检验有效性。 基本規則： <ol><li>結論中<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>的詞必須在前提中<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>（謬誤：<a href="/wiki/%E5%A4%A7%E8%A9%9E%E4%B8%8D%E7%95%B6" title="大詞不當">大詞不當</a>、<a href="/wiki/%E5%B0%8F%E8%A9%9E%E4%B8%8D%E7%95%B6" title="小詞不當">小詞不當</a>）。</li> <li>中詞必須<a href="/wiki/%E5%91%A8%E5%BB%B6_(%E5%93%B2%E5%AD%B8%E6%A6%82%E5%BF%B5)" title="周延 (哲學概念)">周延</a>至少一次（謬誤：<a href="/wiki/%E4%B8%AD%E8%A9%9E%E4%B8%8D%E5%91%A8%E5%BB%B6" title="中詞不周延">中詞不周延</a>）。</li> <li>結論中否定命題的數目必須和前提中否定命題的數目相等： <ol><li>二前提皆肯定，則結論必須為肯定（謬誤：<a href="/wiki/%E8%82%AF%E5%AE%9A%E5%89%8D%E6%8F%90%E6%8E%A8%E5%BE%97%E5%90%A6%E5%AE%9A%E7%B5%90%E8%AB%96" title="肯定前提推得否定結論">肯定前提推得否定結論</a>）。</li> <li>一前提是否定，則結論必須為否定（謬誤：<a href="/wiki/%E5%90%A6%E5%AE%9A%E5%89%8D%E6%8F%90%E6%8E%A8%E5%BE%97%E8%82%AF%E5%AE%9A%E7%B5%90%E8%AB%96" title="否定前提推得肯定結論">否定前提推得肯定結論</a>）。</li> <li>二前提皆否定，則三段論必無效（謬誤：<a href="/wiki/%E6%8E%92%E5%AE%83%E5%89%8D%E6%8F%90%E8%AC%AC%E8%AA%A4" class="mw-redirect" title="排它前提謬誤">排它前提謬誤</a>）。</li></ol></li> <li>結論中特稱命題的數目必須和前提中特稱命題的數目相等： <ol><li>二前提皆全稱，則結論必須為全稱。</li> <li>一前提是特稱，則結論必須為特稱。</li> <li>二前提皆特稱，則三段論必無效。</li></ol></li></ol> 若一個三段論式滿足以上的所有規則，就必定有效。 其他檢查： <ul><li>如果語境上不能假設所有提及的集合非<a href="/wiki/%E7%A9%BA%E9%9B%86%E5%90%88" class="mw-redirect" title="空集合">空</a>，部分推論將會無效（謬誤：<a href="/wiki/%E5%AD%98%E5%9C%A8%E8%AC%AC%E8%AA%A4" class="mw-redirect" title="存在謬誤">存在謬誤</a>）。</li> <li>必須包含嚴格的三個詞，不多不少。且須注意所有關鍵詞和結構的語義是否一致（謬誤：<a href="/wiki/%E5%9B%9B%E8%A9%9E%E8%AC%AC%E8%AA%A4" title="四詞謬誤">四詞謬誤</a>、<a href="/wiki/%E6%AD%A7%E7%BE%A9%E8%AC%AC%E8%AA%A4" title="歧義謬誤">歧義謬誤</a>）。</li></ul> <div class="mw-heading mw-heading3"><h3 id="有效三段論式">有效三段論式</h3>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=3" title="编辑章节：有效三段論式">编辑</a>]</div> 唯有第一格的所有有效三段論式的結論涵蓋了<tt>A</tt>、<tt>E</tt>、<tt>I</tt>、<tt>O</tt>全部四種命題，第二格的所有有效三段論式皆為否定結論（<tt>E</tt>或<tt>O</tt>），第三格的所有有效三段論式皆為特稱結論（<tt>I</tt>或<tt>O</tt>），第四格的所有有效三段論式皆為否定結論或特稱結論（<tt>E</tt>、<tt>I</tt>或<tt>O</tt>）。下面表格中加下劃線者必須假設所有提及的集合非空才有效。 <dl><dd><table class="wikitable"> <tbody><tr> <th>第1格 </th> <th>第2格 </th> <th>第3格 </th> <th>第4格 </th></tr> <tr> <td>AAA </td> <td>AEE </td> <td>AAI </td> <td>AAI </td></tr> <tr> <td>EAE </td> <td>EAE </td> <td>EAO </td> <td>EAO </td></tr> <tr> <td>AII </td> <td>AOO </td> <td>AII </td> <td>AEE </td></tr> <tr> <td>EIO </td> <td>EIO </td> <td>EIO </td> <td>EIO </td></tr> <tr> <td>AAI </td> <td>AEO </td> <td>IAI </td> <td>IAI </td></tr> <tr> <td>EAO </td> <td>EAO </td> <td>OAO </td> <td>AEO </td></tr></tbody></table></dd></dl> 在全部256種三段論式中，有24種有效，但是如果不能確定所有提及的集合為非空，則只有15種有效。 <div class="mw-heading mw-heading3"><h3 id="常犯的無效三段論式">常犯的無效三段論式</h3>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=4" title="编辑章节：常犯的無效三段論式">编辑</a>]</div> <pre>1-AEE, 1-AEO, 1-EEA, 1-EEE, 1-EEI, 1-AIA, 1-IAA, 1-IAI, 1-III, 1-AOO, 1-OAO, 1-IEO 2-AAA, 2-AAI, 2-AII, 2-IAI, 2-OAO, 2-IEO, 2-EOI, 2-OEI, 2-IOO, 2-OIO 3-AAA, 3-AEE, 3-EAE, 3-AEO, 3-AOO, 3-AIA, 3-IAA, 3-III, 3-EOI, 3-OEI, 3-IEO 4-AAA, 4-EAE, 4-AII, 4-IEO </pre> <div class="mw-heading mw-heading2"><h2 id="三段论式列表">三段论式列表</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=5" title="编辑章节：三段论式列表">编辑</a>]</div> 总共有19个有效的论式，算结论弱化（全称弱化为特称）的5个论式則為24個有效论式，其中每一格刚好各有6個有效论式。為便於記憶，中世纪的学者將這些有效論式分別取了對應的<a href="/wiki/%E6%8B%89%E4%B8%81%E8%AA%9E" class="mw-redirect" title="拉丁語">拉丁語</a>名字，每個名字的加了下劃線的<a href="/wiki/%E5%85%83%E9%9F%B3" title="元音">元音</a>即是對應的語氣： <dl><dd><table class="wikitable" style="font-family:Arial;"> <tbody><tr> <th>第1格 </th> <th>第2格 </th> <th>第3格 </th> <th>第4格 </th></tr> <tr> <td>Barbara </td> <td>Camestres </td> <td>Darapti </td> <td>Bamalip </td></tr> <tr> <td>Celarent </td> <td>Cesare </td> <td>Felapton </td> <td>Fesapo </td></tr> <tr> <td>Darii </td> <td>Baroco </td> <td>Datisi </td> <td>Calemes </td></tr> <tr> <td>Ferio </td> <td>Festino </td> <td>Ferison </td> <td>Fresison </td></tr> <tr> <td>Barbari </td> <td>Camestros </td> <td>Disamis   </td> <td>Dimaris </td></tr> <tr> <td>Celaront </td> <td>Cesaro </td> <td>Bocardo </td> <td>Calemos </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="经典三段论式">经典三段论式</h3>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=6" title="编辑章节：经典三段论式">编辑</a>]</div> 下面列出的是<a href="/wiki/%E4%BA%9A%E9%87%8C%E5%A3%AB%E5%A4%9A%E5%BE%B7" title="亚里士多德">亚里士多德</a>的《<a href="/w/index.php?title=%E5%89%8D%E5%88%86%E6%9E%90%E7%AF%87&action=edit&redlink=1" class="new" title="前分析篇（页面不存在）">前分析篇</a>》中关于前3个格的14个三段论式。 <div class="mw-heading mw-heading4"><h4 id="第1格">第1格</h4>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=7" title="编辑章节：第1格">编辑</a>]</div> <ul><li>AAA（Barbara）</li></ul> <tt> </tt>所有M是P。 <tt> </tt>所有S是M。 <tt>∴</tt>所有S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \forall x(S(x)\rightarrow M(x))}{\forall x(S(x)\rightarrow P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \forall x(S(x)\rightarrow M(x))}{\forall x(S(x)\rightarrow P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed32a09520a5a25c600c0c66312656150a0fec66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.257ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \forall x(S(x)\rightarrow M(x))}{\forall x(S(x)\rightarrow P(x))}}}"> <ul><li>EAE（Celarent）</li></ul> <tt> </tt>没有M是P。 <tt> </tt>所有S是M。 <tt>∴</tt>没有S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\quad \forall x(S(x)\rightarrow M(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\quad \forall x(S(x)\rightarrow M(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5b079099e3d71011a8eb20cfe0b88eeb33f9e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.485ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\quad \forall x(S(x)\rightarrow M(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}"> <ul><li>AII（Darii）</li></ul> <tt> </tt>所有M是P。 <tt> </tt>有些S是M。 <tt>∴</tt>有些S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \exists x(S(x)\land M(x))}{\exists x(S(x)\land P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \exists x(S(x)\land M(x))}{\exists x(S(x)\land P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c18ede5218988db0032e494295a0d4e371c9063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.226ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \exists x(S(x)\land M(x))}{\exists x(S(x)\land P(x))}}}"> <ul><li>EIO（Ferio）</li></ul> <tt> </tt>没有M是P。 <tt> </tt>有些S是M。 <tt>∴</tt>有些S不是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\quad \exists x(S(x)\land M(x))}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\quad \exists x(S(x)\land M(x))}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e09217b313f0f5a3a7d025d2da790fa64f22b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.454ex; height:7.176ex;" alt="{\displaystyle {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\quad \exists x(S(x)\land M(x))}{\exists x(S(x)\land \lnot P(x))}}}"> <div class="mw-heading mw-heading4"><h4 id="第2格">第2格</h4>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=8" title="编辑章节：第2格">编辑</a>]</div> <ul><li>AEE（Camestres）</li></ul> <tt> </tt>所有P是M。 <tt> </tt>没有S是M。 <tt>∴</tt>没有S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\cfrac {{\begin{matrix}\quad \\\forall x(P(x)\rightarrow M(x))\end{matrix}}\qquad {\cfrac {\forall x(S(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot S(x))}}}{\forall x(M(x)\rightarrow \lnot S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\cfrac {\forall x(P(x)\rightarrow \lnot S(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\cfrac {{\begin{matrix}\quad \\\forall x(P(x)\rightarrow M(x))\end{matrix}}\qquad {\cfrac {\forall x(S(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot S(x))}}}{\forall x(M(x)\rightarrow \lnot S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\cfrac {\forall x(P(x)\rightarrow \lnot S(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1590a4e2e46aac626f8c6286a0b409f13b536ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:46.232ex; height:18.009ex;" alt="{\displaystyle {\cfrac {\cfrac {{\begin{matrix}\quad \\\forall x(P(x)\rightarrow M(x))\end{matrix}}\qquad {\cfrac {\forall x(S(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot S(x))}}}{\forall x(M(x)\rightarrow \lnot S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\cfrac {\forall x(P(x)\rightarrow \lnot S(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}}"> （AEE-2是AEE-4的等价形式。这种形式还有其他推导方法。）<a href="#cite_note-2">[2]</a> <ul><li>EAE（Cesare）</li></ul> <tt> </tt>没有P是M。 <tt> </tt>所有S是M。 <tt>∴</tt>没有S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\forall x(S(x)\rightarrow M(x))\end{matrix}}}{\forall x(S(x)\rightarrow \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\forall x(S(x)\rightarrow M(x))\end{matrix}}}{\forall x(S(x)\rightarrow \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69bee0c91ef6877ee987fb9a1924c8741c258e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.395ex; height:10.676ex;" alt="{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\forall x(S(x)\rightarrow M(x))\end{matrix}}}{\forall x(S(x)\rightarrow \lnot P(x))}}}"> （EAE-2是EAE-1的等价形式。） <ul><li>AOO（Baroco）</li></ul> <tt> </tt>所有P是M。 <tt> </tt>有些S不是M。 <tt>∴</tt>有些S不是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))}{\forall x(P(x)\rightarrow \lnot (\lnot M(x)))}}{\forall x((\lnot M(x))\rightarrow \lnot P(x))}}\quad {\cfrac {\exists x(S(x)\land \lnot M(x))}{\exists x(S(x)\land (\lnot M(x)))}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))}{\forall x(P(x)\rightarrow \lnot (\lnot M(x)))}}{\forall x((\lnot M(x))\rightarrow \lnot P(x))}}\quad {\cfrac {\exists x(S(x)\land \lnot M(x))}{\exists x(S(x)\land (\lnot M(x)))}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0088e1cc5db8aeb9ddc515b449177c2ed4d8126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.681ex; height:14.343ex;" alt="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))}{\forall x(P(x)\rightarrow \lnot (\lnot M(x)))}}{\forall x((\lnot M(x))\rightarrow \lnot P(x))}}\quad {\cfrac {\exists x(S(x)\land \lnot M(x))}{\exists x(S(x)\land (\lnot M(x)))}}}{\exists x(S(x)\land \lnot P(x))}}}"> （这种形式还有其他推导方法。）<a href="#cite_note-3">[3]</a> <ul><li>EIO（Festino）</li></ul> <tt> </tt>没有P是M。 <tt> </tt>有些S是M。 <tt>∴</tt>有些S不是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\exists x(S(x)\land M(x))\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\exists x(S(x)\land M(x))\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e99df6c5fca6c97ec2ffbce4bfb9b53379e3f9f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.364ex; height:10.676ex;" alt="{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\exists x(S(x)\land M(x))\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}"> （EIO-2是EIO-1的等价形式。） <div class="mw-heading mw-heading4"><h4 id="第3格">第3格</h4>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=9" title="编辑章节：第3格">编辑</a>]</div> <ul><li>AAI（Darapti）</li></ul> <tt> </tt>所有M是P。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S是P。 （这种形式需要假定有些M确实存在。）<a href="#cite_note-4">[4]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\\quad \\\forall x(M(x)\rightarrow P(x))\end{matrix}}\ {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\\quad \\\forall x(M(x)\rightarrow P(x))\end{matrix}}\ {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de435f9e280ba22d8bd5774177f0237ccfe02cda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.867ex; height:18.009ex;" alt="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\\quad \\\forall x(M(x)\rightarrow P(x))\end{matrix}}\ {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land P(x))}}}"> <ul><li>EAO（Felapton）</li></ul> <tt> </tt>没有M是P。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S不是P。 （这种形式需要假定有些M确实存在。）<a href="#cite_note-#1-5">[5]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\\quad \\\forall x(M(x)\rightarrow \lnot P(x))\end{matrix}}\,{\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\\quad \\\forall x(M(x)\rightarrow \lnot P(x))\end{matrix}}\,{\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32443fee7cd1fd0aeee8cc2501441d51073b822a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.224ex; height:18.009ex;" alt="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\\quad \\\forall x(M(x)\rightarrow \lnot P(x))\end{matrix}}\,{\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land \lnot P(x))}}}"> <ul><li>AII（Datisi）</li></ul> <tt> </tt>所有M是P。 <tt> </tt>有些M是S。 <tt>∴</tt>有些S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow P(x))\end{matrix}}\qquad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow P(x))\end{matrix}}\qquad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7470fdcc968566dc7c67ed0f212329506935c17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.814ex; height:10.676ex;" alt="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow P(x))\end{matrix}}\qquad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land P(x))}}}"> （AII-3是AII-1的等价形式。） <ul><li>EIO（Ferison）</li></ul> <tt> </tt>没有M是P。 <tt> </tt>有些M是S。 <tt>∴</tt>有些S不是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow \lnot P(x))\end{matrix}}\quad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow \lnot P(x))\end{matrix}}\quad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3acb4491f307a5fd8b993f3d10f4f6526a44a26c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.041ex; height:10.676ex;" alt="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow \lnot P(x))\end{matrix}}\quad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land \lnot P(x))}}}"> （EIO-3是EIO-1的等价形式。） <ul><li>IAI（Disamis）</li></ul> <tt> </tt>有些M是P。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\cfrac {{\cfrac {\exists x(M(x)\land P(x))}{\exists x(P(x)\land M(x))}}\qquad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow S(x))\qquad \exists x(P(x)\land M(x))}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\cfrac {{\cfrac {\exists x(M(x)\land P(x))}{\exists x(P(x)\land M(x))}}\qquad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow S(x))\qquad \exists x(P(x)\land M(x))}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4ae5493ea47c6537394a30cccf246f9dba09c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:43.65ex; height:18.009ex;" alt="{\displaystyle {\cfrac {\cfrac {{\cfrac {\exists x(M(x)\land P(x))}{\exists x(P(x)\land M(x))}}\qquad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow S(x))\qquad \exists x(P(x)\land M(x))}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}"> （IAI-3是IAI-4的等价形式。） <ul><li>OAO（Bocardo）</li></ul> <tt> </tt>有些M不是P。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S不是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\cfrac {{\cfrac {\exists x(M(x)\land \lnot P(x))}{\exists x((\lnot P(x))\land M(x))}}\quad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow S(x))\quad \exists x((\lnot P(x))\land M(x))}}{\cfrac {\exists x((\lnot P(x))\land S(x))}{\exists x(S(x)\land \lnot P(x))}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\cfrac {{\cfrac {\exists x(M(x)\land \lnot P(x))}{\exists x((\lnot P(x))\land M(x))}}\quad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow S(x))\quad \exists x((\lnot P(x))\land M(x))}}{\cfrac {\exists x((\lnot P(x))\land S(x))}{\exists x(S(x)\land \lnot P(x))}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34682974e323f22142db91ae8f6159987228c8e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:44.687ex; height:18.009ex;" alt="{\displaystyle {\cfrac {\cfrac {{\cfrac {\exists x(M(x)\land \lnot P(x))}{\exists x((\lnot P(x))\land M(x))}}\quad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow S(x))\quad \exists x((\lnot P(x))\land M(x))}}{\cfrac {\exists x((\lnot P(x))\land S(x))}{\exists x(S(x)\land \lnot P(x))}}}}"> （这种形式还有其他推导方法。）<a href="#cite_note-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="增补的论式">增补的论式</h3>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=10" title="编辑章节：增补的论式">编辑</a>]</div> 第4格由亞里士多德的學生<a href="/wiki/%E6%B3%B0%E5%A5%A7%E5%BC%97%E6%8B%89%E6%96%AF%E6%89%98%E6%96%AF" title="泰奧弗拉斯托斯">泰奧弗拉斯托斯</a>補充<a href="#cite_note-7">[7]</a>。 <div class="mw-heading mw-heading4"><h4 id="第4格">第4格</h4>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=11" title="编辑章节：第4格">编辑</a>]</div> <ul><li>AAI（Bamalip）</li></ul> <tt> </tt>所有P是M。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S是P。 （这种形式需要假定有些P确实存在。） <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\forall x(P(x)\rightarrow S(x))}}\quad {\cfrac {\exists xP(x)}{\exists x(P(x)\land P(x))}}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\forall x(P(x)\rightarrow S(x))}}\quad {\cfrac {\exists xP(x)}{\exists x(P(x)\land P(x))}}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4029cf8bfa1630de55831c2ffce555845723e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:63.871ex; height:18.009ex;" alt="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\forall x(P(x)\rightarrow S(x))}}\quad {\cfrac {\exists xP(x)}{\exists x(P(x)\land P(x))}}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}"> <ul><li>EAO（Fesapo）</li></ul> <tt> </tt>没有P是M。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S不是P。 （这种形式需要假定有些M确实存在。）<a href="#cite_note-#2-8">[8]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\end{matrix}}\,{\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\end{matrix}}\,{\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d3e1e1dfa59fc7431c88a9d13c95496ec7065d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.06ex; height:18.009ex;" alt="{\displaystyle {\cfrac {{\begin{matrix}\quad \\\quad \\{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\end{matrix}}\,{\cfrac {{\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}\quad {\cfrac {\exists xM(x)}{\exists x(M(x)\land M(x))}}}{\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}}{\exists x(S(x)\land \lnot P(x))}}}"> （EAO-4是EAO-3的等价形式。） <ul><li>AEE（Calemes）</li></ul> <tt> </tt>所有P是M。 <tt> </tt>没有M是S。 <tt>∴</tt>没有S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \forall x(M(x)\rightarrow \lnot S(x))}{\forall x(M(x)\rightarrow \lnot S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\cfrac {\forall x(P(x)\rightarrow \lnot S(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \forall x(M(x)\rightarrow \lnot S(x))}{\forall x(M(x)\rightarrow \lnot S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\cfrac {\forall x(P(x)\rightarrow \lnot S(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e7fb5b45bfe5c059056c083a212594218256fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:44.644ex; height:14.343ex;" alt="{\displaystyle {\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \forall x(M(x)\rightarrow \lnot S(x))}{\forall x(M(x)\rightarrow \lnot S(x))\qquad \forall x(P(x)\rightarrow M(x))}}{\cfrac {\forall x(P(x)\rightarrow \lnot S(x))}{\forall x(S(x)\rightarrow \lnot P(x))}}}}"> <ul><li>EIO（Fresison）</li></ul> <tt> </tt>没有P是M。 <tt> </tt>有些M是S。 <tt>∴</tt>有些S不是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b34db746d6b82df8bbee93404e4d3e73d6a14fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.449ex; height:10.676ex;" alt="{\displaystyle {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\cfrac {\exists x(M(x)\land S(x))}{\exists x(S(x)\land M(x))}}}{\exists x(S(x)\land \lnot P(x))}}}"> （EIO-4是EIO-1的等价形式。） <ul><li>IAI（Dimaris）</li></ul> <tt> </tt>有些P是M。 <tt> </tt>所有M是S。 <tt>∴</tt>有些S是P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {\cfrac {\exists x(P(x)\land M(x))\qquad \forall x(M(x)\rightarrow \ S(x))}{\forall x(M(x)\rightarrow \ S(x))\qquad \exists x(P(x)\land M(x))}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mtext> </mtext> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {\cfrac {\exists x(P(x)\land M(x))\qquad \forall x(M(x)\rightarrow \ S(x))}{\forall x(M(x)\rightarrow \ S(x))\qquad \exists x(P(x)\land M(x))}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/683157ef880a851251be01def7e6f4549ea3e910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:42.643ex; height:14.343ex;" alt="{\displaystyle {\cfrac {\cfrac {\exists x(P(x)\land M(x))\qquad \forall x(M(x)\rightarrow \ S(x))}{\forall x(M(x)\rightarrow \ S(x))\qquad \exists x(P(x)\land M(x))}}{\cfrac {\exists x(P(x)\land S(x))}{\exists x(S(x)\land P(x))}}}}"> <div class="mw-heading mw-heading4"><h4 id="结论弱化的论式">结论弱化的论式</h4>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=12" title="编辑章节：结论弱化的论式">编辑</a>]</div> 歷史上，AAI-3、EAO-3、AAI-4、EAO-4的拉丁語名字中有字母“p”，用来指示出这些论式通過引入了某个词项确实有元素存在的前提，将一个<tt>A</tt>命题弱化成了<tt>I</tt>命题。后人认为它們不是直言的即不是无条件的，这个问题被称为<a href="/wiki/%E5%AD%98%E5%9C%A8%E6%80%A7%E5%BC%95%E5%85%A5%E9%97%AE%E9%A2%98" class="mw-redirect" title="存在性引入问题">存在性引入问题</a>。 在假定结论的主词确定有成员存在的前提下，可将论式中的结论<tt>A</tt>弱化为结论<tt>I</tt>，结论<tt>E</tt>弱化为结论<tt>O</tt>，它们也可以被增补为有效论式，从而得到所有可能的24有效论式。结论弱化论式有5个：AAI-1（Barbari），即弱化的AAA-1；EAO-1（Celaront），即弱化的EAE-1；AEO-2（Camestros），即弱化的AEE-2；EAO-2（Cesaro），即弱化的EAE-2；AEO-4（Calemos），即弱化的AEE-4。AAI-1的结论同于AII-1的结论，EAO-1、EAO-2的结论同于EIO-1的结论，AEO-2、AEO-4的结论同于AOO-2的结论，需要注意结论弱化论式原来的结论依然成立。 <div class="mw-heading mw-heading3"><h3 id="谓词演算公式的注解">谓词演算公式的注解</h3>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=13" title="编辑章节：谓词演算公式的注解">编辑</a>]</div> 按照<a href="/wiki/%E5%B8%83%E5%B0%94%E9%80%BB%E8%BE%91" title="布尔逻辑">布尔逻辑</a>和<a href="/wiki/%E9%9B%86%E5%90%88%E4%BB%A3%E6%95%B0" title="集合代数">集合代数</a>的观点，三段论可以解释为：<a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>（<a href="/wiki/%E7%B1%BB_(%E6%95%B0%E5%AD%A6)" title="类 (数学)">类</a>）<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,S\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>S</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,S\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80de9e29ab9c15889689ff799b09548547d6c8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.273ex; height:2.176ex;" alt="{\displaystyle \,S\,}">和集合<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">有某种<a href="/wiki/%E4%BA%8C%E5%85%83%E5%85%B3%E7%B3%BB" title="二元关系">二元关系</a>，并且集合<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">和集合<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,P\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>P</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,P\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91614855859e9d2e1df5e74c8d4fe6fb1a9b20d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.52ex; height:2.176ex;" alt="{\displaystyle \,P\,}">有某种二元关系，从而推论出集合<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,S\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>S</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,S\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80de9e29ab9c15889689ff799b09548547d6c8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.273ex; height:2.176ex;" alt="{\displaystyle \,S\,}">和集合<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,P\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>P</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,P\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91614855859e9d2e1df5e74c8d4fe6fb1a9b20d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.52ex; height:2.176ex;" alt="{\displaystyle \,P\,}">是否存在进而为何种可确定的二元关系。两个集合之间的二元关系用直言命题可确定的有四种： <ul><li><tt>A</tt>（全称肯定）命题：所有<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">的元素是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的元素，确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">“<a href="/wiki/%E5%8C%85%E5%90%AB" class="mw-redirect" title="包含">包含</a>于”<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的关系，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的<a href="/wiki/%E5%AD%90%E9%9B%86" title="子集">子集</a>，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">的<a href="/wiki/%E8%B6%85%E9%9B%86" class="mw-redirect" title="超集">超集</a>，这是一种<a href="/wiki/%E5%81%8F%E5%BA%8F%E5%85%B3%E7%B3%BB" title="偏序关系">偏序关系</a>，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,L\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9ee80f5d8f948f5995ad5cd6d9c4c20a956cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.357ex; height:2.176ex;" alt="{\displaystyle \,L\,}">包含于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">，並且<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">包含于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">，則<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,L\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9ee80f5d8f948f5995ad5cd6d9c4c20a956cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.357ex; height:2.176ex;" alt="{\displaystyle \,L\,}">包含于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">。<tt>A</tt>命题允许两个推理方向，从元素属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">推出它属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">，从元素不属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">推出它不属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">。<tt>A</tt>命题确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">减<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的<a href="/wiki/%E5%B7%AE%E9%9B%86" class="mw-redirect" title="差集">差集</a>是<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>。</li> <li><tt>E</tt>（全称否定）命题：所有<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">的元素不是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的元素，确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">是“<a href="/wiki/%E4%B8%8D%E4%BA%A4%E9%9B%86" title="不交集">无交集</a>”的关系，这是一种<a href="/wiki/%E5%AF%B9%E7%A7%B0%E5%85%B3%E7%B3%BB" title="对称关系">对称关系</a>，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">无交集于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">，同于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">无交集于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">。<tt>E</tt>命题允许两个推理方向，从元素属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">推出它不属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">，从元素属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">推出它不属于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">。<tt>E</tt>命题确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>是<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>。</li> <li><tt>I</tt>（特称肯定）命题：有些<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">的元素是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的元素，确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">是“有<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>”的关系，这是一种<a href="/wiki/%E5%AF%B9%E7%A7%B0%E5%85%B3%E7%B3%BB" title="对称关系">对称关系</a>，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">有交集于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">，同于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">有交集于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">。<tt>I</tt>命题确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的<a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a>不是<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>。</li> <li><tt>O</tt>（特称否定）命题：有些<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">的元素不是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的元素，确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">“不<a href="/wiki/%E5%8C%85%E5%90%AB" class="mw-redirect" title="包含">包含</a>于”<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的关系。<tt>O</tt>命题确定了<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,M\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>M</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,M\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ab62b678b50c63bdaa4367ee1faeaeebc2d00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.216ex; height:2.176ex;" alt="{\displaystyle \,M\,}">减<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,N\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,N\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e546b9564ac9b89b7699116fa86e6fa16a40cd24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.838ex; height:2.176ex;" alt="{\displaystyle \,N\,}">的<a href="/wiki/%E5%B7%AE%E9%9B%86" class="mw-redirect" title="差集">差集</a>不是<a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a>。</li></ul> 两个全称命题可以推出一个新的全称命题，一个全称命题和一个特称命题可以推出一个新的特称命题，两个特称命题无法推理。<tt>A</tt>命题可以和所有四种命题組合。<tt>E</tt>命题还可以和<tt>I</tt>命题組合，两个否定命题和<tt>IE</tt>组合，不能得出屬於四種命題之一的結論。故而有效的論式，要在<tt>AA</tt>、<tt>AE</tt>、<tt>EA</tt>、<tt>AI</tt>、<tt>IA</tt>、<tt>AO</tt>、<tt>OA</tt>、<tt>EI</tt>這8種組合乘以4種格，共32種情況中找出。 <tt>AA</tt>组合中AAA-1是直接推出的；第4格<tt>AA</tt>組合推论出谓词包含於主词的关系，这不是四种命题之一，只能在谓词确实有元素存在的前提下弱化为AAI-4。<tt>AE</tt>组合中AEE-4是直接推出的，<tt>EA</tt>组合中EAE-1是直接推出的。第3格<tt>AA</tt>組合和<tt>EA</tt>組合，在中項確定有元素存在的前提下，形成AAI-3和EAO-3。AAA-1、AAI-4、AAI-3没有等价者。通過對換其前提<tt>E</tt>命題中主詞和謂詞的位置，從AEE-4得出其等價者AEE-2，從EAE-1的得出其等價者EAE-2，從EAO-3得出其等價者EAO-4。 AII-1、IAI-4是直接推出的，通過對換其前提<tt>I</tt>命題中主詞和謂詞的位置，從AII-1得出其等價者AII-3，從IAI-4得出其等價者IAI-3。AOO-2和OAO-3在歷史上採用了<a href="/wiki/%E5%8F%8D%E8%AD%89%E6%B3%95" title="反證法">反證法</a>，这里采用了<a href="/wiki/%E7%9B%B4%E6%8E%A5%E6%8E%A8%E7%90%86" title="直接推理">直接推理</a>中的“对置法”，AOO-2、OAO-3沒有等價者。EIO-1是直接推出的，通過對換其前提<tt>E</tt>命题<a href="/w/index.php?title=%E5%8F%8A/%E6%88%96&action=edit&redlink=1" class="new" title="及/或（页面不存在）">及/或</a>（英语：<a href="https://en.wikipedia.org/wiki/And/or" class="extiw" title="en:And/or">And/or</a>）<tt>I</tt>命題中主詞和謂詞的位置，從EIO-1得出其等價者EIO-2、EIO-3、EIO-4。 <div class="mw-heading mw-heading2"><h2 id="24論式圖示">24論式圖示</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=14" title="编辑章节：24論式圖示">编辑</a>]</div> 下表以<a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏圖</a>展示24個有效直言三段論，不同欄表示不同的前提，不同外框顏色表示不同的結論，需要<a href="/wiki/%E5%AD%98%E5%9C%A8%E6%80%A7%E9%A0%90%E8%A8%AD" class="mw-redirect" title="存在性預設">存在性預設</a>的推理以虛線與斜體字標示。 <table class="wikitable" style="background: #FFF;width:98%; text-align:center;"> <tbody><tr> <td rowspan="2" style="background: #AAA;">格 </td> <td style="border:0;border-left:2px solid #999;">A ∧ A </td> <td style="border:0;"> </td> <td style="border:0;border-left:2px solid #999;">A ∧ E </td> <td style="border:0;"> </td> <td style="border:0;"> </td> <td style="border:0;"> </td> <td style="border:0;border-left:2px solid #999;">A ∧ I </td> <td style="border:0;"> </td> <td style="border:0;border-left:2px solid #999;">A ∧ O </td> <td style="border:0;"> </td> <td style="border:0;border-left:2px solid #999;">E ∧ I </td></tr> <tr> <td>AAA</td> <td>AAI</td> <td>AEE</td> <td>AEO</td> <td>EAE</td> <td>EAO</td> <td>AII</td> <td>IAI</td> <td>AOO</td> <td>OAO</td> <td>EIO </td></tr> <tr> <td style="background: #AAA;">1 </td> <td style="outline-offset:-5px;outline:2px solid #8F8;border-left:2px solid #999;"><a href="/wiki/File:Modus_Barbara.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Modus_Barbara.svg/80px-Modus_Barbara.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Modus_Barbara.svg/120px-Modus_Barbara.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Modus_Barbara.svg/160px-Modus_Barbara.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Barbara.svg" title="File:Modus Barbara.svg">Barbara</a> </td> <td style="outline-offset:-5px;outline:2px dashed #88F;"><a href="/wiki/File:Modus_Barbari.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Modus_Barbari.svg/76px-Modus_Barbari.svg.png" decoding="async" width="76" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Modus_Barbari.svg/114px-Modus_Barbari.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Modus_Barbari.svg/152px-Modus_Barbari.svg.png 2x" data-file-width="718" data-file-height="986" /></a> <a href="/wiki/File:Modus_Barbari.svg" title="File:Modus Barbari.svg">Barbari</a> </td> <td style="border-left:2px solid #999;"> </td> <td> </td> <td style="outline-offset:-5px;outline:2px solid #F88;"><a href="/wiki/File:Modus_Celarent.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Modus_Celarent.svg/80px-Modus_Celarent.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Modus_Celarent.svg/120px-Modus_Celarent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Modus_Celarent.svg/160px-Modus_Celarent.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Celarent.svg" title="File:Modus Celarent.svg">Celarent</a> </td> <td style="outline-offset:-5px;outline:2px dashed #FC4;"><a href="/wiki/File:Modus_Celaront.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Modus_Celaront.svg/76px-Modus_Celaront.svg.png" decoding="async" width="76" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Modus_Celaront.svg/114px-Modus_Celaront.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Modus_Celaront.svg/152px-Modus_Celaront.svg.png 2x" data-file-width="718" data-file-height="986" /></a> <a href="/wiki/File:Modus_Celaront.svg" title="File:Modus Celaront.svg">Celaront</a> </td> <td style="outline-offset:-5px;outline:2px solid #88F; border-left:2px solid #999;"><a href="/wiki/File:Modus_Darii.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Modus_Darii.svg/80px-Modus_Darii.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Modus_Darii.svg/120px-Modus_Darii.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Modus_Darii.svg/160px-Modus_Darii.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Darii.svg" title="File:Modus Darii.svg">Darii</a> </td> <td> </td> <td style="border-left:2px solid #999;"> </td> <td> </td> <td style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"><a href="/wiki/File:Modus_Ferio.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Ferio.svg/80px-Modus_Ferio.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Ferio.svg/120px-Modus_Ferio.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Ferio.svg/160px-Modus_Ferio.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Ferio.svg" title="File:Modus Ferio.svg">Ferio</a> </td></tr> <tr> <td style="background: #AAA;">2 </td> <td style="border-left:2px solid #999;"> </td> <td> </td> <td style="outline-offset:-5px;outline:2px solid #F88;border-left:2px solid #999;"><a href="/wiki/File:Modus_Camestres.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Modus_Camestres.svg/80px-Modus_Camestres.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Modus_Camestres.svg/120px-Modus_Camestres.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Modus_Camestres.svg/160px-Modus_Camestres.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Camestres.svg" title="File:Modus Camestres.svg">Camestres</a> </td> <td style="outline-offset:-5px;outline:2px dashed #FC4;"><a href="/wiki/File:Modus_Camestros.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Modus_Camestros.svg/76px-Modus_Camestros.svg.png" decoding="async" width="76" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Modus_Camestros.svg/114px-Modus_Camestros.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/Modus_Camestros.svg/152px-Modus_Camestros.svg.png 2x" data-file-width="718" data-file-height="986" /></a> <a href="/wiki/File:Modus_Camestros.svg" title="File:Modus Camestros.svg">Camestros</a> </td> <td style="outline-offset:-5px;outline:2px solid #F88;"><a href="/wiki/File:Modus_Cesare.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Modus_Cesare.svg/80px-Modus_Cesare.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Modus_Cesare.svg/120px-Modus_Cesare.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Modus_Cesare.svg/160px-Modus_Cesare.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Cesare.svg" title="File:Modus Cesare.svg">Cesare</a> </td> <td style="outline-offset:-5px;outline:2px dashed #FC4;"><a href="/wiki/File:Modus_Cesaro.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Modus_Cesaro.svg/80px-Modus_Cesaro.svg.png" decoding="async" width="80" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Modus_Cesaro.svg/120px-Modus_Cesaro.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Modus_Cesaro.svg/160px-Modus_Cesaro.svg.png 2x" data-file-width="718" data-file-height="986" /></a> <a href="/wiki/File:Modus_Cesaro.svg" title="File:Modus Cesaro.svg">Cesaro</a> </td> <td style="border-left:2px solid #999;"> </td> <td> </td> <td style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"><a href="/wiki/File:Modus_Baroco.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Modus_Baroco.svg/80px-Modus_Baroco.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Modus_Baroco.svg/120px-Modus_Baroco.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Modus_Baroco.svg/160px-Modus_Baroco.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Baroco.svg" title="File:Modus Baroco.svg">Baroco</a> </td> <td> </td> <td style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"><a href="/wiki/File:Modus_Festino.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Modus_Festino.svg/80px-Modus_Festino.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Modus_Festino.svg/120px-Modus_Festino.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cf/Modus_Festino.svg/160px-Modus_Festino.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Festino.svg" title="File:Modus Festino.svg">Festino</a> </td></tr> <tr> <td style="background: #AAA;">3 </td> <td style="border-left:2px solid #999;"> </td> <td style="outline-offset:-5px;outline:2px dashed #88F;"><a href="/wiki/File:Modus_Darapti.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Modus_Darapti.svg/66px-Modus_Darapti.svg.png" decoding="async" width="66" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Modus_Darapti.svg/99px-Modus_Darapti.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Modus_Darapti.svg/132px-Modus_Darapti.svg.png 2x" data-file-width="718" data-file-height="1094" /></a> <a href="/wiki/File:Modus_Darapti.svg" title="File:Modus Darapti.svg">Darapti</a> </td> <td style="border-left:2px solid #999;"> </td> <td> </td> <td> </td> <td style="outline-offset:-5px;outline:2px dashed #FC4;"><a href="/wiki/File:Modus_Felapton.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Modus_Felapton.svg/66px-Modus_Felapton.svg.png" decoding="async" width="66" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Modus_Felapton.svg/99px-Modus_Felapton.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Modus_Felapton.svg/132px-Modus_Felapton.svg.png 2x" data-file-width="718" data-file-height="1094" /></a> <a href="/wiki/File:Modus_Felapton.svg" title="File:Modus Felapton.svg">Felapton</a> </td> <td style="outline-offset:-5px;outline:2px solid #88F;border-left:2px solid #999;"><a href="/wiki/File:Modus_Datisi.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Modus_Datisi.svg/80px-Modus_Datisi.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Modus_Datisi.svg/120px-Modus_Datisi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Modus_Datisi.svg/160px-Modus_Datisi.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Datisi.svg" title="File:Modus Datisi.svg">Datisi</a> </td> <td style="outline-offset:-5px;outline:2px solid #88F;"><a href="/wiki/File:Modus_Disamis.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Modus_Disamis.svg/80px-Modus_Disamis.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Modus_Disamis.svg/120px-Modus_Disamis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Modus_Disamis.svg/160px-Modus_Disamis.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Disamis.svg" title="File:Modus Disamis.svg">Disamis</a> </td> <td style="border-left:2px solid #999;"> </td> <td style="outline-offset:-5px;outline:2px solid #FC4;"><a href="/wiki/File:Modus_Bocardo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Modus_Bocardo.svg/80px-Modus_Bocardo.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Modus_Bocardo.svg/120px-Modus_Bocardo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Modus_Bocardo.svg/160px-Modus_Bocardo.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Bocardo.svg" title="File:Modus Bocardo.svg">Bocardo</a> </td> <td style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"><a href="/wiki/File:Modus_Ferison.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Modus_Ferison.svg/80px-Modus_Ferison.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Modus_Ferison.svg/120px-Modus_Ferison.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Modus_Ferison.svg/160px-Modus_Ferison.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Ferison.svg" title="File:Modus Ferison.svg">Ferison</a> </td></tr> <tr> <td style="background: #AAA;">4 </td> <td style="border-left:2px solid #999;"> </td> <td style="outline-offset:-5px;outline:2px dashed #88F;"><a href="/wiki/File:Modus_Bamalip.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Modus_Bamalip.svg/76px-Modus_Bamalip.svg.png" decoding="async" width="76" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Modus_Bamalip.svg/114px-Modus_Bamalip.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Modus_Bamalip.svg/152px-Modus_Bamalip.svg.png 2x" data-file-width="718" data-file-height="986" /></a> <a href="/wiki/File:Modus_Bamalip.svg" title="File:Modus Bamalip.svg">Bamalip</a> </td> <td style="outline-offset:-5px;outline:2px solid #F88;border-left:2px solid #999;"><a href="/wiki/File:Modus_Calemes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Modus_Calemes.svg/80px-Modus_Calemes.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Modus_Calemes.svg/120px-Modus_Calemes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Modus_Calemes.svg/160px-Modus_Calemes.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Calemes.svg" title="File:Modus Calemes.svg">Calemes</a> </td> <td style="outline-offset:-5px;outline:2px dashed #FC4;"><a href="/wiki/File:Modus_Calemos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Modus_Calemos.svg/76px-Modus_Calemos.svg.png" decoding="async" width="76" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Modus_Calemos.svg/114px-Modus_Calemos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Modus_Calemos.svg/152px-Modus_Calemos.svg.png 2x" data-file-width="718" data-file-height="986" /></a> <a href="/wiki/File:Modus_Calemos.svg" title="File:Modus Calemos.svg">Calemos</a> </td> <td> </td> <td style="outline-offset:-5px;outline:2px dashed #FC4;"><a href="/wiki/File:Modus_Fesapo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Modus_Fesapo.svg/66px-Modus_Fesapo.svg.png" decoding="async" width="66" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Modus_Fesapo.svg/99px-Modus_Fesapo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Modus_Fesapo.svg/132px-Modus_Fesapo.svg.png 2x" data-file-width="718" data-file-height="1094" /></a> <a href="/wiki/File:Modus_Fesapo.svg" title="File:Modus Fesapo.svg">Fesapo</a> </td> <td style="border-left:2px solid #999;"> </td> <td style="outline-offset:-5px;outline:2px solid #88F;"><a href="/wiki/File:Modus_Dimatis.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Modus_Dimatis.svg/80px-Modus_Dimatis.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Modus_Dimatis.svg/120px-Modus_Dimatis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Modus_Dimatis.svg/160px-Modus_Dimatis.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Dimatis.svg" title="File:Modus Dimatis.svg">Dimatis</a> </td> <td style="border-left:2px solid #999;"> </td> <td> </td> <td style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"><a href="/wiki/File:Modus_Fresison.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Fresison.svg/80px-Modus_Fresison.svg.png" decoding="async" width="80" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Fresison.svg/120px-Modus_Fresison.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Modus_Fresison.svg/160px-Modus_Fresison.svg.png 2x" data-file-width="718" data-file-height="934" /></a> <a href="/wiki/File:Modus_Fresison.svg" title="File:Modus Fresison.svg">Fresison</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="参见">参见</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=15" title="编辑章节：参见">编辑</a>]</div> <ul><li><a href="/wiki/%E7%9B%B4%E6%8E%A5%E6%8E%A8%E7%90%86" title="直接推理">直接推理</a></li> <li><a href="/wiki/%E4%BC%A0%E7%BB%9F%E9%80%BB%E8%BE%91" title="传统逻辑">传统逻辑</a></li> <li><a href="/wiki/%E8%B0%93%E8%AF%8D%E6%BC%94%E7%AE%97" class="mw-redirect" title="谓词演算">谓词演算</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="註解">註解</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=16" title="编辑章节：註解">编辑</a>]</div> <div class="reflist columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <cite class="citation book">中国社会科学院语言研究所词典编辑室. <a rel="nofollow" class="external text" href="https://archive.org/details/modern-chinese-dictionary_7th-edition">现代汉语词典</a> 2016年9月第七版. 商务印书馆. 2016: <a rel="nofollow" class="external text" href="https://archive.org/details/modern-chinese-dictionary_7th-edition/page/n43">1121</a>-1122 [2020-07-05]. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-7-100-12450-8" title="Special:网络书源/978-7-100-12450-8">ISBN 978-7-100-12450-8</a> （中文（大陆简体））. <q>.......【三段论】.......由大前提和小前提推出结论。如“凡金属都能导电”（大前提），“铜是金属”（小前提），“所以铜能导电”（结论）。.......</q></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&rft.au=%E4%B8%AD%E5%9B%BD%E7%A4%BE%E4%BC%9A%E7%A7%91%E5%AD%A6%E9%99%A2%E8%AF%AD%E8%A8%80%E7%A0%94%E7%A9%B6%E6%89%80%E8%AF%8D%E5%85%B8%E7%BC%96%E8%BE%91%E5%AE%A4&rft.btitle=%E7%8E%B0%E4%BB%A3%E6%B1%89%E8%AF%AD%E8%AF%8D%E5%85%B8&rft.date=2016&rft.edition=2016%E5%B9%B49%E6%9C%88%E7%AC%AC%E4%B8%83%E7%89%88&rft.genre=book&rft.isbn=978-7-100-12450-8&rft.pages=1121-1122&rft.pub=%E5%95%86%E5%8A%A1%E5%8D%B0%E4%B9%A6%E9%A6%86&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmodern-chinese-dictionary_7th-edition&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  引文格式1维护：未识别语文类型 (<a href="/wiki/Category:%E5%BC%95%E6%96%87%E6%A0%BC%E5%BC%8F1%E7%BB%B4%E6%8A%A4%EF%BC%9A%E6%9C%AA%E8%AF%86%E5%88%AB%E8%AF%AD%E6%96%87%E7%B1%BB%E5%9E%8B" title="Category:引文格式1维护：未识别语文类型">link</a>) </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> 这个论式还可以推导为： <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))}{\forall x(P(x)\rightarrow \lnot (\lnot M(x)))}}{\forall x((\lnot M(x))\rightarrow \lnot P(x))}}\quad {\cfrac {\forall x(S(x)\rightarrow \lnot M(x))}{\forall x(S(x)\rightarrow (\lnot M(x)))}}}{\forall x(S(x)\rightarrow \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))}{\forall x(P(x)\rightarrow \lnot (\lnot M(x)))}}{\forall x((\lnot M(x))\rightarrow \lnot P(x))}}\quad {\cfrac {\forall x(S(x)\rightarrow \lnot M(x))}{\forall x(S(x)\rightarrow (\lnot M(x)))}}}{\forall x(S(x)\rightarrow \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/687bccd57443afbf5ae6c1277e089bf8aa9a4377" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.713ex; height:14.343ex;" alt="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(P(x)\rightarrow M(x))}{\forall x(P(x)\rightarrow \lnot (\lnot M(x)))}}{\forall x((\lnot M(x))\rightarrow \lnot P(x))}}\quad {\cfrac {\forall x(S(x)\rightarrow \lnot M(x))}{\forall x(S(x)\rightarrow (\lnot M(x)))}}}{\forall x(S(x)\rightarrow \lnot P(x))}}}"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> 这个论式还可以采用<a href="/wiki/%E5%8F%8D%E8%AF%81%E6%B3%95" class="mw-redirect" title="反证法">反证法</a>来推导： <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\cfrac {{\cfrac {{\begin{matrix}\quad \\\forall x(P(x)\rightarrow M(x))\end{matrix}}\quad {\cfrac {\lnot (\exists x(S(x)\land \lnot P(x)))}{\forall x(S(x)\rightarrow P(x))}}}{\forall x(S(x)\rightarrow M(x))}}\quad {\cfrac {\exists x(S(x)\land \lnot M(x))}{\exists x((\lnot M(x))\land S(x))}}}{\cfrac {\exists x((\lnot M(x))\land M(x))}{\bot }}}\\\implies &{\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \exists x(S(x)\land \lnot M(x))}{\exists x(S(x)\land \lnot P(x))}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\cfrac {{\cfrac {{\begin{matrix}\quad \\\forall x(P(x)\rightarrow M(x))\end{matrix}}\quad {\cfrac {\lnot (\exists x(S(x)\land \lnot P(x)))}{\forall x(S(x)\rightarrow P(x))}}}{\forall x(S(x)\rightarrow M(x))}}\quad {\cfrac {\exists x(S(x)\land \lnot M(x))}{\exists x((\lnot M(x))\land S(x))}}}{\cfrac {\exists x((\lnot M(x))\land M(x))}{\bot }}}\\\implies &{\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \exists x(S(x)\land \lnot M(x))}{\exists x(S(x)\land \lnot P(x))}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9185a7fc43cf92e77f843afe6f0a9d3c796bd205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.804ex; margin-bottom: -0.2ex; width:75.139ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}&{\cfrac {{\cfrac {{\begin{matrix}\quad \\\forall x(P(x)\rightarrow M(x))\end{matrix}}\quad {\cfrac {\lnot (\exists x(S(x)\land \lnot P(x)))}{\forall x(S(x)\rightarrow P(x))}}}{\forall x(S(x)\rightarrow M(x))}}\quad {\cfrac {\exists x(S(x)\land \lnot M(x))}{\exists x((\lnot M(x))\land S(x))}}}{\cfrac {\exists x((\lnot M(x))\land M(x))}{\bot }}}\\\implies &{\cfrac {\forall x(P(x)\rightarrow M(x))\qquad \exists x(S(x)\land \lnot M(x))}{\exists x(S(x)\land \lnot P(x))}}\end{aligned}}}"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> 直接結論是：所有M是P且S。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow (P(x)\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land P(x)))}}\qquad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow (P(x)\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land P(x)))}}\qquad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99645e05c202d5473f0aab9cb9e3cbd034834ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.53ex; height:14.343ex;" alt="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(M(x)\rightarrow P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow (P(x)\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land P(x)))}}\qquad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land P(x))}}}"> </li> <li id="cite_note-#1-5"><a href="#cite_ref-#1_5-0">^</a> 直接結論是：所有M是S且非P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow ((\lnot P(x))\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land \lnot P(x)))}}\quad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow ((\lnot P(x))\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land \lnot P(x)))}}\quad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8282619ce38e35a35b90d6d4b14f914ae410e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.757ex; height:14.343ex;" alt="{\displaystyle {\cfrac {{\cfrac {\cfrac {\forall x(M(x)\rightarrow \lnot P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\forall x(M(x)\rightarrow ((\lnot P(x))\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land \lnot P(x)))}}\quad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> 这个论式还可以采用<a href="/wiki/%E5%8F%8D%E8%AF%81%E6%B3%95" class="mw-redirect" title="反证法">反证法</a>来推导： <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\cfrac {{\cfrac {{\cfrac {\lnot (\exists x(S(x)\land \lnot P(x)))}{\forall x(S(x)\rightarrow P(x))}}\quad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow P(x))}}\quad {\cfrac {\exists x(M(x)\land \lnot P(x))}{\exists x((\lnot P(x))\land M(x))}}}{\cfrac {\exists x((\lnot P(x))\land P(x))}{\bot }}}\\\implies &{\cfrac {\exists x(M(x)\land \lnot P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\exists x(S(x)\land \lnot P(x))}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">¬</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊥</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="2em" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\cfrac {{\cfrac {{\cfrac {\lnot (\exists x(S(x)\land \lnot P(x)))}{\forall x(S(x)\rightarrow P(x))}}\quad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow P(x))}}\quad {\cfrac {\exists x(M(x)\land \lnot P(x))}{\exists x((\lnot P(x))\land M(x))}}}{\cfrac {\exists x((\lnot P(x))\land P(x))}{\bot }}}\\\implies &{\cfrac {\exists x(M(x)\land \lnot P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\exists x(S(x)\land \lnot P(x))}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965ff5d71ecb75995d93da0bba92e3b2c0522a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.804ex; margin-bottom: -0.2ex; width:75.139ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}&{\cfrac {{\cfrac {{\cfrac {\lnot (\exists x(S(x)\land \lnot P(x)))}{\forall x(S(x)\rightarrow P(x))}}\quad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow P(x))}}\quad {\cfrac {\exists x(M(x)\land \lnot P(x))}{\exists x((\lnot P(x))\land M(x))}}}{\cfrac {\exists x((\lnot P(x))\land P(x))}{\bot }}}\\\implies &{\cfrac {\exists x(M(x)\land \lnot P(x))\qquad \forall x(M(x)\rightarrow S(x))}{\exists x(S(x)\land \lnot P(x))}}\end{aligned}}}"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> 在<a href="/wiki/%E4%BA%9E%E9%87%8C%E5%A3%AB%E5%A4%9A%E5%BE%B7" class="mw-redirect" title="亞里士多德">亞里士多德</a>《<a href="/w/index.php?title=%E5%89%8D%E5%88%86%E6%9E%90%E7%AF%87&action=edit&redlink=1" class="new" title="前分析篇（页面不存在）">前分析篇</a>》裡關於AEE-2的論證中，對小前提進行對換主詞與謂詞位置之後，得出第4格的AEE-4，亞里士多德稱之為再次得到了第1格，沒有因為大項和小項位置顛倒而專門稱之為第4格。在亞里士多德的定義中第1格為中項既是一個前提的主詞又是另一個前提的謂詞。第4格中有4個論式是其他格的等價形式、1個論式是結論弱化形式，因此亞里士多德三段論體系並無缺失。 </li> <li id="cite_note-#2-8"><a href="#cite_ref-#2_8-0">^</a> 直接結論是：所有M是S且非P。 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\cfrac {{\cfrac {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow ((\lnot P(x))\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land \lnot P(x)))}}\quad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mspace width="1em" /> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\cfrac {{\cfrac {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow ((\lnot P(x))\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land \lnot P(x)))}}\quad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05e7fb0cd75830953e9b6004aafda5511526acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.345ex; height:18.009ex;" alt="{\displaystyle {\cfrac {{\cfrac {\cfrac {{\cfrac {\forall x(P(x)\rightarrow \lnot M(x))}{\forall x(M(x)\rightarrow \lnot P(x))}}\qquad {\begin{matrix}\quad \\\forall x(M(x)\rightarrow S(x))\end{matrix}}}{\forall x(M(x)\rightarrow ((\lnot P(x))\land S(x)))}}{\forall x(M(x)\rightarrow (S(x)\land \lnot P(x)))}}\quad {\begin{matrix}\quad \\\exists xM(x)\end{matrix}}}{\exists x(S(x)\land \lnot P(x))}}}"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="引用">引用</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=17" title="编辑章节：引用">编辑</a>]</div> <ul><li><a href="/wiki/Aristotle" class="mw-redirect" title="Aristotle">Aristotle</a>, <a href="/w/index.php?title=Prior_Analytics&action=edit&redlink=1" class="new" title="Prior Analytics（页面不存在）">Prior Analytics</a>. transl. <a href="/w/index.php?title=Robin_Smith&action=edit&redlink=1" class="new" title="Robin Smith（页面不存在）">Robin Smith</a>（Hackett, 1989）<a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0872200647" class="internal mw-magiclink-isbn">ISBN 0-87220-064-7</a>.</li> <li><a href="/w/index.php?title=Simon_Blackburn&action=edit&redlink=1" class="new" title="Simon Blackburn（页面不存在）">Blackburn, Simon</a>, 1996. "Syllogism" in the Oxford Dictionary of Philosophy. Oxford University Press. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0192831348" class="internal mw-magiclink-isbn">ISBN 0-19-283134-8</a>.</li> <li>Broadie, Alexander, 1993. Introduction to Medieval Logic. Oxford University Press. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0198240260" class="internal mw-magiclink-isbn">ISBN 0-19-824026-0</a>.</li> <li><a href="/w/index.php?title=Irving_Copi&action=edit&redlink=1" class="new" title="Irving Copi（页面不存在）">Irving Copi</a>, 1969. Introduction to Logic, 3rd ed. Macmillan Company.</li> <li><a href="/w/index.php?title=Charles_Leonard_Hamblin&action=edit&redlink=1" class="new" title="Charles Leonard Hamblin（页面不存在）">Hamblin, Charles L.</a>, 1970. Fallacies, Methuen : London, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0416700705" class="internal mw-magiclink-isbn">ISBN 0-416-70070-5</a>. Cf. on validity of syllogisms: "A simple set of rules of validity was finally produced in the later Middle Ages, based on the concept of Distribution.“</li> <li><a href="/wiki/Jan_%C5%81ukasiewicz" class="mw-redirect" title="Jan Łukasiewicz">Jan Łukasiewicz</a>, 1987 (1957). Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. New York: Garland Publishers. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0824069242" class="internal mw-magiclink-isbn">ISBN 0824069242</a>. OCLC 15015545.</li></ul> <div class="mw-heading mw-heading2"><h2 id="外部連結">外部連結</h2>[<a href="/w/index.php?title=%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&section=18" title="编辑章节：外部連結">编辑</a>]</div> <ul><li><cite class="citation encyclopaedia">Robin Smith. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/aristotle-logic/">Aristotle's Logic</a>. <a href="/wiki/%E7%88%B1%E5%BE%B7%E5%8D%8E%C2%B7%E6%89%8E%E5%B0%94%E5%A1%94" title="爱德华·扎尔塔">扎尔塔, 爱德华·N</a> (编). 《<a href="/wiki/%E5%8F%B2%E4%B8%B9%E4%BD%9B%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹佛哲學百科全書">斯坦福哲学百科全书</a>》.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&rft.atitle=Aristotle%27s+Logic&rft.au=Robin+Smith&rft.btitle=%E3%80%8A-%7Bzh-hans%3A%E6%96%AF%E5%9D%A6%E7%A6%8F%3B+zh-tw%3A%E5%8F%B2%E4%B8%B9%E4%BD%9B%3Bzh-mo%3A%E5%8F%B2%E4%B8%B9%E7%A6%8F%3Bzh-hk%3A%E5%8F%B2%E4%B8%B9%E7%A6%8F%7D-%E5%93%B2%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6%E3%80%8B&rft.genre=bookitem&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Faristotle-logic%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><cite class="citation encyclopaedia">Terence Parsons. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/square/">The Traditional Square of Opposition</a>. <a href="/wiki/%E7%88%B1%E5%BE%B7%E5%8D%8E%C2%B7%E6%89%8E%E5%B0%94%E5%A1%94" title="爱德华·扎尔塔">扎尔塔, 爱德华·N</a> (编). 《<a href="/wiki/%E5%8F%B2%E4%B8%B9%E4%BD%9B%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹佛哲學百科全書">斯坦福哲学百科全书</a>》.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&rft.atitle=The+Traditional+Square+of+Opposition&rft.au=Terence+Parsons&rft.btitle=%E3%80%8A-%7Bzh-hans%3A%E6%96%AF%E5%9D%A6%E7%A6%8F%3B+zh-tw%3A%E5%8F%B2%E4%B8%B9%E4%BD%9B%3Bzh-mo%3A%E5%8F%B2%E4%B8%B9%E7%A6%8F%3Bzh-hk%3A%E5%8F%B2%E4%B8%B9%E7%A6%8F%7D-%E5%93%B2%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6%E3%80%8B&rft.genre=bookitem&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fsquare%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><cite class="citation encyclopaedia">Henrik Lagerlund. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/medieval-syllogism/">Medieval Theories of the Syllogism</a>. <a href="/wiki/%E7%88%B1%E5%BE%B7%E5%8D%8E%C2%B7%E6%89%8E%E5%B0%94%E5%A1%94" title="爱德华·扎尔塔">扎尔塔, 爱德华·N</a> (编). 《<a href="/wiki/%E5%8F%B2%E4%B8%B9%E4%BD%9B%E5%93%B2%E5%AD%B8%E7%99%BE%E7%A7%91%E5%85%A8%E6%9B%B8" title="史丹佛哲學百科全書">斯坦福哲学百科全书</a>》.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9B%B4%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA&rft.atitle=Medieval+Theories+of+the+Syllogism&rft.au=Henrik+Lagerlund&rft.btitle=%E3%80%8A-%7Bzh-hans%3A%E6%96%AF%E5%9D%A6%E7%A6%8F%3B+zh-tw%3A%E5%8F%B2%E4%B8%B9%E4%BD%9B%3Bzh-mo%3A%E5%8F%B2%E4%B8%B9%E7%A6%8F%3Bzh-hk%3A%E5%8F%B2%E4%B8%B9%E7%A6%8F%7D-%E5%93%B2%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6%E3%80%8B&rft.genre=bookitem&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fmedieval-syllogism%2F&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><a rel="nofollow" class="external text" href="http://www.formalontology.it/aristotle-syllogism-categorical.htm">Aristotle's Prior Analytics: the Theory of Categorical Syllogism</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20100416120736/http://www.formalontology.it/aristotle-syllogism-categorical.htm">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） an annotated bibliography on Aristotle's syllogistic</li> <li><a rel="nofollow" class="external text" href="http://www.humanities.mq.edu.au/Ockham/x52t06.html">Abbreviatio Montana</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20100507181444/http://www.humanities.mq.edu.au/Ockham/x52t06.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） article by Prof. R. J. Kilcullen of Macquarie University on the medieval classification of syllogisms.</li> <li><a rel="nofollow" class="external text" href="http://www.multicians.org/thvv/petrus-hispanius.html">The Figures of the Syllogism</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20100328202209/http://www.multicians.org/thvv/petrus-hispanius.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） is a brief table listing the forms of the syllogism.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20090817004242/http://www.fibonicci.co.uk/syllogisms">www.fibonicci.co.uk/syllogisms</a> some fun syllogism tests/quizzes</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110304101320/http://www.understandingthemind.org/syllogism.pdf">Syllogistic Reasoning in Buddhism - Example & Worksheet</a></li></ul> <table style="margin:0 auto;" align="center" width="80%" class="toccolours"> <tbody><tr> <td align="center" style="background:#f3f9ff"><a href="/wiki/%E4%BC%A0%E7%BB%9F%E9%80%BB%E8%BE%91" title="传统逻辑">传统逻辑</a>：<a href="/wiki/%E4%B8%89%E6%AE%B5%E8%AB%96" title="三段論">三段論</a> </td></tr> <tr> <td align="center">形式：<a class="mw-selflink selflink">直言三段论</a> | <a href="/wiki/%E9%80%89%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA" title="选言三段论">选言三段论</a> | <a href="/wiki/%E5%81%87%E8%A8%80%E4%B8%89%E6%AE%B5%E8%AE%BA" title="假言三段论">假言三段论</a> | <a href="/w/index.php?title=%E5%A4%8D%E5%90%88%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&redlink=1" class="new" title="复合三段论（页面不存在）">复合三段论</a> | <a href="/wiki/%E6%BA%96%E4%B8%89%E6%AE%B5%E8%AB%96" title="準三段論">準三段論</a> | <a href="/w/index.php?title=%E7%BB%9F%E8%AE%A1%E4%B8%89%E6%AE%B5%E8%AE%BA&action=edit&redlink=1" class="new" title="统计三段论（页面不存在）">统计三段论</a> </td></tr> <tr> <td align="center">其他：<a href="/wiki/%E5%AF%B9%E7%AB%8B%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="对立四边形">对立四边形</a> | <a href="/wiki/%E5%B8%83%E5%B0%94%E4%B8%89%E6%AE%B5%E8%AE%BA" title="布尔三段论">布尔三段论</a> | <a href="/w/index.php?title=%E4%B8%89%E6%AE%B5%E8%AE%BA%E8%B0%AC%E8%AE%BA&action=edit&redlink=1" class="new" title="三段论谬论（页面不存在）">三段论谬论</a> </td></tr></tbody></table> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.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 dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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title="数理逻辑">数理逻辑</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">基本概念</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%85%AC%E7%90%86" title="公理">公理</a> <ul><li><a href="/wiki/%E5%85%AC%E7%90%86%E5%88%97%E8%A1%A8" title="公理列表">列表</a></li></ul></li> <li><a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">势</a></li> <li><a href="/wiki/%E4%B8%80%E9%98%B6%E9%80%BB%E8%BE%91" title="一阶逻辑">一阶逻辑</a></li> <li><a href="/w/index.php?title=%E5%BD%A2%E5%BC%8F%E8%AF%81%E6%B3%95&action=edit&redlink=1" class="new" title="形式证法（页面不存在）">形式证法</a>（英语：<a href="https://en.wikipedia.org/wiki/Formal_proof" class="extiw" title="en:Formal proof">Formal proof</a>）</li> <li><a href="/wiki/%E9%82%8F%E8%BC%AF%E8%AA%9E%E7%BE%A9%E5%AD%B8" title="邏輯語義學">邏輯語義學</a></li> <li><a href="/wiki/%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80" title="数学基础">数学基础</a></li> <li><a href="/wiki/%E4%BF%A1%E6%81%AF%E8%AE%BA" title="信息论">信息论</a></li> <li><a href="/wiki/%E8%95%B4%E6%B6%B5" title="蕴涵">蕴涵</a></li> <li><a href="/wiki/%E7%BB%93%E6%9E%84_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="结构 (数理逻辑)">结构</a></li> <li><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a></li> <li><a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">形式理论</a></li> <li><a href="/wiki/%E7%B1%BB%E5%9E%8B%E8%AE%BA" title="类型论">类型论</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">定理 （<a href="/w/index.php?title=Category:%E6%95%B0%E5%AD%A6%E5%9F%BA%E7%A1%80%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="Category:数学基础定理（页面不存在）">列表</a>（英语：<a href="https://en.wikipedia.org/wiki/Category:Theorems_in_the_foundations_of_mathematics" class="extiw" title="en:Category:Theorems in the foundations of mathematics">Category:Theorems in the foundations of mathematics</a>）及<a href="/w/index.php?title=%E9%9B%86%E5%90%88%E8%AE%BA%E6%82%96%E8%AE%BA&action=edit&redlink=1" class="new" title="集合论悖论（页面不存在）">悖论</a>（英语：<a href="https://en.wikipedia.org/wiki/Paradoxes_of_set_theory" class="extiw" title="en:Paradoxes of set theory">Paradoxes of set theory</a>））</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%93%A5%E5%BE%B7%E5%B0%94%E5%AE%8C%E5%A4%87%E6%80%A7%E5%AE%9A%E7%90%86" title="哥德尔完备性定理">哥德尔完备性定理</a>及<a href="/wiki/%E5%93%A5%E5%BE%B7%E5%B0%94%E4%B8%8D%E5%AE%8C%E5%A4%87%E5%AE%9A%E7%90%86" title="哥德尔不完备定理">哥德尔不完备定理</a></li> <li><a href="/wiki/%E5%A1%94%E6%96%AF%E5%9F%BA%E4%B8%8D%E5%8F%AF%E5%AE%9A%E7%BE%A9%E5%AE%9A%E7%90%86" title="塔斯基不可定義定理">塔斯基不可定義定理</a></li> <li><a href="/wiki/%E5%B7%B4%E6%8B%BF%E8%B5%AB-%E5%A1%94%E6%96%AF%E5%9F%BA%E5%AE%9A%E7%90%86" title="巴拿赫-塔斯基定理">巴拿赫-塔斯基定理</a></li> <li>康托尔 <a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E5%AE%9A%E7%90%86" title="康托尔定理">定理</a>、<a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94%E6%82%96%E8%AE%BA" title="康托尔悖论">悖论</a>和<a href="/wiki/%E5%B0%8D%E8%A7%92%E8%AB%96%E8%AD%89%E6%B3%95" title="對角論證法">對角論證法</a></li> <li><a href="/wiki/%E7%B4%A7%E8%87%B4%E6%80%A7%E5%AE%9A%E7%90%86" title="紧致性定理">紧致性定理</a></li> <li><a href="/wiki/%E5%81%9C%E6%9C%BA%E9%97%AE%E9%A2%98" title="停机问题">停机问题</a></li> <li><a href="/w/index.php?title=%E6%9E%97%E5%BE%B7%E6%96%AF%E7%89%B9%E4%BC%A6%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="林德斯特伦定理（页面不存在）">林德斯特伦定理</a>（英语：<a href="https://en.wikipedia.org/wiki/Lindstr%C3%B6m%27s_theorem" class="extiw" title="en:Lindström's theorem">Lindström's theorem</a>）</li> <li><a href="/wiki/%E5%8B%92%E6%96%87%E6%B5%B7%E5%A7%86%E2%80%93%E6%96%AF%E7%A7%91%E4%BC%A6%E5%AE%9A%E7%90%86" title="勒文海姆–斯科伦定理">勒文海姆–斯科伦定理</a></li> <li><a href="/wiki/%E7%BD%97%E7%B4%A0%E6%82%96%E8%AE%BA" title="罗素悖论">罗素悖论</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%80%BB%E8%BE%91" title="逻辑">逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="传统逻辑" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E4%BC%A0%E7%BB%9F%E9%80%BB%E8%BE%91" title="传统逻辑">传统逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%82%8F%E8%BC%AF%E7%9C%9F%E7%90%86" title="邏輯真理">邏輯真理</a></li> <li><a href="/wiki/%E6%81%86%E7%9C%9F%E5%BC%8F" title="恆真式">恆真式</a></li> <li><a href="/wiki/%E5%91%BD%E9%A2%98" title="命题">命题</a></li> <li><a href="/wiki/%E6%8E%A8%E7%90%86" title="推理">推理</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E7%AD%89%E4%BB%B7" title="逻辑等价">逻辑等价</a></li> <li><a href="/wiki/%E4%B8%80%E8%87%B4%E6%80%A7_(%E9%82%8F%E8%BC%AF)" title="一致性 (邏輯)">一致性</a> <ul><li><a href="/w/index.php?title=%E7%9B%B8%E5%90%8C%E4%B8%80%E8%87%B4%E6%80%A7&action=edit&redlink=1" class="new" title="相同一致性（页面不存在）">相同一致性</a>（英语：<a href="https://en.wikipedia.org/wiki/Equiconsistency" class="extiw" title="en:Equiconsistency">Equiconsistency</a>）</li></ul></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E8%AE%BA%E8%AF%81" title="逻辑论证">逻辑论证</a></li> <li><a href="/wiki/%E5%8F%AF%E9%9D%A0%E6%80%A7%E5%AE%9A%E7%90%86" title="可靠性定理">可靠性定理</a></li> <li><a href="/wiki/%E6%9C%89%E6%95%88%E6%80%A7" title="有效性">有效性</a></li> <li><a class="mw-selflink selflink">直言三段论</a></li> <li><a href="/wiki/%E5%AF%B9%E7%AB%8B%E5%9B%9B%E8%BE%B9%E5%BD%A2" title="对立四边形">对立四边形</a></li> <li><a href="/wiki/%E6%96%87%E6%B0%8F%E5%9B%BE" title="文氏图">文氏图</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91" title="命题逻辑">命题逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%80%BB%E8%BE%91%E4%BB%A3%E6%95%B0" title="逻辑代数">逻辑代数</a></li> <li><a href="/wiki/%E5%B8%83%E5%B0%94%E5%87%BD%E6%95%B0" title="布尔函数">布尔函数</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E8%BF%90%E7%AE%97%E7%AC%A6" title="逻辑运算符">逻辑运算符</a></li> <li><a href="/wiki/%E5%91%BD%E9%A2%98%E9%80%BB%E8%BE%91" title="命题逻辑">命题逻辑</a></li> <li><a href="/wiki/%E5%85%AC%E5%BC%8F_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="公式 (数理逻辑)">命题公式</a></li> <li><a href="/wiki/%E7%9C%9F%E5%80%BC%E8%A1%A8" title="真值表">真值表</a></li> <li><a href="/wiki/%E5%A4%9A%E5%80%BC%E9%80%BB%E8%BE%91" title="多值逻辑">多值逻辑</a> <ul><li><a href="/wiki/%E4%B8%89%E5%80%BC%E9%80%BB%E8%BE%91" title="三值逻辑">三值</a></li> <li><a href="/w/index.php?title=%E6%9C%89%E9%99%90%E5%80%BC%E9%80%BB%E8%BE%91&action=edit&redlink=1" class="new" title="有限值逻辑（页面不存在）">有限值</a>（英语：<a href="https://en.wikipedia.org/wiki/Finite-valued_logic" class="extiw" title="en:Finite-valued logic">Finite-valued logic</a>）</li> <li><a href="/wiki/%E6%97%A0%E9%99%90%E5%80%BC%E9%80%BB%E8%BE%91" title="无限值逻辑">无限值</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E7%BB%8F%E5%85%B8%E9%80%BB%E8%BE%91" title="经典逻辑">经典逻辑</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BB%8F%E5%85%B8%E9%80%BB%E8%BE%91" title="经典逻辑">经典逻辑</a></li> <li><a href="/wiki/%E4%B8%80%E9%98%B6%E9%80%BB%E8%BE%91" title="一阶逻辑">一阶逻辑</a></li> <li><a href="/wiki/%E4%BA%8C%E9%9A%8E%E9%82%8F%E8%BC%AF" title="二階邏輯">二階邏輯</a> <ul><li><a href="/w/index.php?title=%E4%B8%80%E5%85%83%E4%BA%8C%E9%98%B6%E9%80%BB%E8%BE%91&action=edit&redlink=1" class="new" title="一元二阶逻辑（页面不存在）">一元</a>（英语：<a href="https://en.wikipedia.org/wiki/Monadic_second-order_logic" class="extiw" title="en:Monadic second-order logic">Monadic second-order logic</a>）</li></ul></li> <li><a href="/wiki/%E9%AB%98%E9%98%B6%E9%80%BB%E8%BE%91" title="高阶逻辑">高阶逻辑</a></li> <li><a href="/wiki/%E8%87%AA%E7%94%B1%E9%80%BB%E8%BE%91" title="自由逻辑">自由逻辑</a></li> <li><a href="/wiki/%E9%87%8F%E5%8C%96_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="量化 (数理逻辑)">量化</a></li> <li><a href="/w/index.php?title=%E8%B0%93%E8%AF%8D_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="谓词 (数理逻辑)（页面不存在）">谓词</a>（英语：<a href="https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)" class="extiw" title="en:Predicate (mathematical logic)">Predicate (mathematical logic)</a>）</li> <li><a href="/wiki/%E4%B8%80%E5%85%83%E8%B0%93%E8%AF%8D%E6%BC%94%E7%AE%97" title="一元谓词演算">一元谓词演算</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E5%90%88%E8%AE%BA" title="集合论">集合论</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a> <ul><li><a href="/w/index.php?title=%E9%81%97%E4%BC%A0%E9%9B%86&action=edit&redlink=1" class="new" title="遗传集（页面不存在）">遗传集</a>（英语：<a href="https://en.wikipedia.org/wiki/Hereditary_set" class="extiw" title="en:Hereditary set">Hereditary set</a>）</li></ul></li> <li><a href="/wiki/%E7%B1%BB_(%E6%95%B0%E5%AD%A6)" title="类 (数学)">类</a></li> <li>（<a href="/wiki/%E5%9F%BA%E6%9C%AC%E5%85%83%E7%B4%A0" title="基本元素">基本</a>）<a href="/wiki/%E5%85%83%E7%B4%A0_(%E6%95%B8%E5%AD%B8)" title="元素 (數學)">元素</a></li> <li><a href="/wiki/%E6%9C%89%E5%BA%8F%E5%AF%B9" title="有序对">有序对</a></li> <li><a href="/wiki/%E5%BA%8F%E6%95%B0" title="序数">序数</a></li> <li><a href="/wiki/%E5%AD%90%E9%9B%86" title="子集">子集</a></li> <li><a href="/wiki/%E7%9B%B8%E7%AD%89" title="相等">相等</a></li> <li><a href="/wiki/%E5%A4%96%E5%BB%B6%E6%80%A7" title="外延性">外延性</a></li> <li><a href="/wiki/%E5%8A%9B%E8%BF%AB" title="力迫">力迫</a></li> <li><a href="/wiki/%E5%85%B3%E7%B3%BB_(%E6%95%B0%E5%AD%A6)" title="关系 (数学)">关系</a> <ul><li><a href="/wiki/%E7%AD%89%E4%BB%B7%E5%85%B3%E7%B3%BB" title="等价关系">等价关系</a></li> <li><a href="/wiki/%E9%9B%86%E5%90%88%E5%88%92%E5%88%86" title="集合划分">集合划分</a></li></ul></li> <li>集合运算 <ul><li><a href="/wiki/%E4%BA%A4%E9%9B%86" title="交集">交集</a></li> <li><a href="/wiki/%E5%B9%B6%E9%9B%86" title="并集">并集</a></li> <li><a href="/wiki/%E8%A1%A5%E9%9B%86" title="补集">补集</a></li> <li><a href="/wiki/%E7%AC%9B%E5%8D%A1%E5%84%BF%E7%A7%AF" title="笛卡儿积">笛卡儿积</a></li> <li><a href="/wiki/%E5%86%AA%E9%9B%86" title="冪集">冪集</a></li> <li><a href="/w/index.php?title=%E9%9B%86%E5%90%88%E7%9A%84%E5%90%8C%E4%B8%80%E6%80%A7%E5%92%8C%E5%85%B3%E7%B3%BB%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="集合的同一性和关系列表（页面不存在）">同一性</a>（英语：<a href="https://en.wikipedia.org/wiki/List_of_set_identities_and_relations" class="extiw" title="en:List of set identities and relations">List of set identities and relations</a>）</li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学)">集合</a>种类</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%AF%E6%95%B8%E9%9B%86" title="可數集">可數集</a></li> <li><a href="/wiki/%E4%B8%8D%E5%8F%AF%E6%95%B8%E9%9B%86" title="不可數集">不可數集</a></li> <li><a href="/wiki/%E7%A9%BA%E9%9B%86" title="空集">空集</a></li> <li><a href="/w/index.php?title=%E5%B1%85%E9%9B%86&action=edit&redlink=1" class="new" title="居集（页面不存在）">居集</a>（英语：<a href="https://en.wikipedia.org/wiki/Inhabited_set" class="extiw" title="en:Inhabited set">Inhabited set</a>）</li> <li><a href="/wiki/%E5%8D%95%E5%85%83%E7%B4%A0%E9%9B%86%E5%90%88" title="单元素集合">单元素集合</a></li> <li><a href="/wiki/%E6%9C%89%E9%99%90%E9%9B%86%E5%90%88" title="有限集合">有限集合</a></li> <li><a href="/wiki/%E6%97%A0%E9%99%90%E9%9B%86%E5%90%88" title="无限集合">无限集合</a></li> <li><a href="/wiki/%E4%BC%A0%E9%80%92%E9%9B%86%E5%90%88" title="传递集合">传递集合</a></li> <li><a href="/w/index.php?title=%E8%B6%85%E6%BB%A4%E5%AD%90_(%E9%9B%86%E5%90%88%E8%AE%BA)&action=edit&redlink=1" class="new" title="超滤子 (集合论)（页面不存在）">超滤子</a>（英语：<a href="https://en.wikipedia.org/wiki/Ultrafilter_(set_theory)" class="extiw" title="en:Ultrafilter (set theory)">Ultrafilter (set theory)</a>）</li> <li><a href="/wiki/%E9%80%92%E5%BD%92%E9%9B%86%E5%90%88" title="递归集合">递归集合</a></li> <li><a href="/wiki/%E6%A8%A1%E7%B3%8A%E9%9B%86" title="模糊集">模糊集</a></li> <li><a href="/wiki/%E5%85%A8%E9%9B%86" title="全集">全集</a> <ul><li><a href="/w/index.php?title=%E5%93%A5%E5%BE%B7%E5%B0%94%E5%8F%AF%E6%9E%84%E9%80%A0%E5%85%A8%E9%9B%86&action=edit&redlink=1" class="new" title="哥德尔可构造全集（页面不存在）">可构造全集</a>（英语：<a href="https://en.wikipedia.org/wiki/Constructible_universe" class="extiw" title="en:Constructible universe">Constructible universe</a>）</li> <li><a href="/w/index.php?title=%E6%A0%BC%E4%BC%A6%E8%BF%AA%E5%85%8B%E5%85%A8%E9%9B%86&action=edit&redlink=1" class="new" title="格伦迪克全集（页面不存在）">格伦迪克全集</a>（英语：<a href="https://en.wikipedia.org/wiki/Grothendieck_universe" class="extiw" title="en:Grothendieck universe">Grothendieck universe</a>）</li> <li><a href="/wiki/%E5%86%AF%C2%B7%E8%AF%BA%E4%BC%8A%E6%9B%BC%E5%85%A8%E9%9B%86" title="冯·诺伊曼全集">冯·诺伊曼全集</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">映射</a>与<a href="/wiki/%E5%8A%BF_(%E6%95%B0%E5%AD%A6)" title="势 (数学)">势</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%87%BD%E6%95%B0" title="函数">函数</a>、<a href="/wiki/%E6%98%A0%E5%B0%84" title="映射">映射</a> <ul><li><a href="/wiki/%E5%AE%9A%E4%B9%89%E5%9F%9F" title="定义域">定义域</a></li> <li><a href="/wiki/%E5%88%B0%E8%BE%BE%E5%9F%9F" title="到达域">到达域</a></li> <li><a href="/wiki/%E5%83%8F_(%E6%95%B8%E5%AD%B8)" title="像 (數學)">像</a></li></ul></li> <li><a href="/wiki/%E5%8D%95%E5%B0%84" title="单射">单射</a>、<a href="/wiki/%E6%BB%A1%E5%B0%84" title="满射">满射</a>、<a href="/wiki/%E5%8F%8C%E5%B0%84" title="双射">双射</a></li> <li><a href="/wiki/%E5%BA%B7%E6%89%98%E5%B0%94-%E4%BC%AF%E6%81%A9%E6%96%AF%E5%9D%A6-%E6%96%BD%E7%BD%97%E5%BE%B7%E5%AE%9A%E7%90%86" title="康托尔-伯恩斯坦-施罗德定理">康托尔-伯恩斯坦-施罗德定理</a></li> <li><a href="/wiki/%E5%90%8C%E6%9E%84" title="同构">同构</a></li> <li><a href="/wiki/%E5%93%A5%E5%BE%B7%E5%B0%94%E6%95%B0" title="哥德尔数">哥德尔数</a></li> <li><a href="/wiki/%E5%88%97%E4%B8%BE%E6%B3%95_(%E9%9B%86%E5%90%88%E8%AE%BA)" title="列举法 (集合论)">列举法</a></li> <li><a href="/wiki/%E5%A4%A7%E5%9F%BA%E6%95%B0" title="大基数">大基数</a> <ul><li><a href="/wiki/%E4%B8%8D%E5%8F%AF%E9%81%94%E5%9F%BA%E6%95%B8" title="不可達基數">不可達基數</a></li></ul></li> <li><a href="/wiki/%E9%98%BF%E5%88%97%E5%A4%AB%E6%95%B8" title="阿列夫數">阿列夫數</a></li> <li><a href="/wiki/%E8%BF%90%E7%AE%97" title="运算">运算</a> <ul><li><a href="/wiki/%E4%BA%8C%E5%85%83%E8%BF%90%E7%AE%97" title="二元运算">二元运算</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">集合理论</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">策梅洛-弗兰克尔 (ZFC)</a> <ul><li><a href="/wiki/%E9%80%89%E6%8B%A9%E5%85%AC%E7%90%86" title="选择公理">选择公理</a></li> <li><a href="/wiki/%E8%BF%9E%E7%BB%AD%E7%BB%9F%E5%81%87%E8%AE%BE" title="连续统假设">连续统假设</a></li></ul></li> <li><a href="/w/index.php?title=%E5%B9%BF%E4%B9%89%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="广义集合论（页面不存在）">广义集合论 (GST)</a>（英语：<a href="https://en.wikipedia.org/wiki/General_set_theory" class="extiw" title="en:General set theory">General set theory</a>）</li> <li><a href="/w/index.php?title=%E5%85%8B%E9%87%8C%E6%99%AE%E5%85%8B-%E6%99%AE%E6%8B%89%E5%85%8B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="克里普克-普拉克集合论（页面不存在）">克里普克-普拉克 (KP)</a>（英语：<a href="https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory" class="extiw" title="en:Kripke–Platek set theory">Kripke–Platek set theory</a>）</li> <li><a href="/w/index.php?title=%E8%8E%AB%E5%B0%94%E6%96%AF-%E5%87%AF%E5%88%A9%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="莫尔斯-凯利集合论（页面不存在）">莫尔斯-凯利集合论 (MK)</a>（英语：<a href="https://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theory" class="extiw" title="en:Morse–Kelley set theory">Morse–Kelley set theory</a>）</li> <li><a href="/wiki/%E6%9C%B4%E7%B4%A0%E9%9B%86%E5%90%88%E8%AE%BA" title="朴素集合论">朴素集合论</a></li> <li><a href="/wiki/%E6%96%B0%E5%9F%BA%E7%A1%80%E9%9B%86%E5%90%88%E8%AE%BA" title="新基础集合论">新基础集合论</a></li> <li><a href="/w/index.php?title=%E5%A1%94%E6%96%AF%E5%9F%BA-%E6%A0%BC%E7%BD%97%E6%BB%95%E8%BF%AA%E5%85%8B%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="塔斯基-格罗滕迪克集合论（页面不存在）">塔斯基-格罗滕迪克 (TG)</a>（英语：<a href="https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory" class="extiw" title="en:Tarski–Grothendieck set theory">Tarski–Grothendieck set theory</a>）</li> <li><a href="/wiki/%E5%86%AF%E8%AF%BA%E4%BC%8A%E6%9B%BC-%E5%8D%9A%E5%86%85%E6%96%AF-%E5%93%A5%E5%BE%B7%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="冯诺伊曼-博内斯-哥德尔集合论">冯·诺伊曼-博内斯-哥德尔 (NBG)</a></li> <li><a href="/w/index.php?title=%E5%BB%BA%E6%9E%84%E5%BC%8F%E9%9B%86%E5%90%88%E8%AE%BA&action=edit&redlink=1" class="new" title="建构式集合论（页面不存在）">建构式集合论</a>（英语：<a href="https://en.wikipedia.org/wiki/Constructive_set_theory" class="extiw" title="en:Constructive set theory">Constructive set theory</a>）</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/w/index.php?title=%E8%AF%AD%E6%B3%95_(%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="语法 (逻辑)（页面不存在）">句法</a>（英语：<a href="https://en.wikipedia.org/wiki/Syntax_(logic)" class="extiw" title="en:Syntax (logic)">Syntax (logic)</a>）及<a href="/wiki/%E5%BD%A2%E5%BC%8F%E8%AF%AD%E8%A8%80" title="形式语言">语言</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%AD%97%E6%AF%8D%E8%A1%A8_(%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%A7%91%E5%AD%A6)" title="字母表 (计算机科学)">字母表</a></li> <li><a href="/wiki/%E5%85%83%E6%95%B0" title="元数">元数</a></li> <li><a href="/wiki/%E8%87%AA%E5%8B%95%E6%A9%9F%E7%90%86%E8%AB%96" title="自動機理論">自動機理論</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86%E6%A8%A1%E5%BC%8F" title="公理模式">公理模式</a></li> <li><a href="/wiki/%E8%A1%A8%E9%81%94%E5%BC%8F" title="表達式">表達式</a> <ul><li><a href="/w/index.php?title=%E5%9F%BA%E7%A1%80%E8%A1%A8%E8%BE%BE%E5%BC%8F&action=edit&redlink=1" class="new" title="基础表达式（页面不存在）">基础表达式</a>（英语：<a href="https://en.wikipedia.org/wiki/Ground_expression" class="extiw" title="en:Ground expression">Ground expression</a>）</li></ul></li> <li><a href="/w/index.php?title=%E9%80%9A%E8%BF%87%E6%96%B0%E7%9A%84%E5%B8%B8%E6%95%B0%E5%92%8C%E5%87%BD%E6%95%B0%E5%90%8D%E7%A7%B0%E8%BF%9B%E8%A1%8C%E6%89%A9%E5%B1%95&action=edit&redlink=1" class="new" title="通过新的常数和函数名称进行扩展（页面不存在）">扩展</a>（英语：<a href="https://en.wikipedia.org/wiki/Extension_by_new_constant_and_function_names" class="extiw" title="en:Extension by new constant and function names">Extension by new constant and function names</a>）</li> <li><a href="/wiki/%E5%85%B3%E7%B3%BB_(%E6%95%B0%E5%AD%A6)" title="关系 (数学)">关系</a></li> <li>形式 <ul><li><a href="/wiki/%E5%BD%A2%E5%BC%8F%E6%96%87%E6%B3%95" title="形式文法">文法</a></li> <li><a href="/wiki/%E5%BD%A2%E5%BC%8F%E8%AF%AD%E8%A8%80" title="形式语言">语言</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">证明</a></li> <li><a href="/wiki/%E5%BD%A2%E5%BC%8F%E7%B3%BB%E7%B5%B1" title="形式系統">系统</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">理论</a></li></ul></li> <li><a href="/w/index.php?title=%E5%BD%A2%E6%88%90%E8%A7%84%E5%88%99&action=edit&redlink=1" class="new" title="形成规则（页面不存在）">形成规则</a>（英语：<a href="https://en.wikipedia.org/wiki/Formation_rule" class="extiw" title="en:Formation rule">Formation rule</a>）</li> <li><a href="/wiki/%E5%90%88%E5%BC%8F%E5%85%AC%E5%BC%8F" title="合式公式">合式公式</a> <ul><li><a href="/wiki/%E5%8E%9F%E5%AD%90%E5%85%AC%E5%BC%8F" title="原子公式">原子公式</a></li> <li><a href="/wiki/%E5%8F%A5%E5%AD%90_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="句子 (数理逻辑)">封闭式</a></li> <li><a href="/w/index.php?title=%E5%9F%BA%E6%9C%AC%E5%BC%8F&action=edit&redlink=1" class="new" title="基本式（页面不存在）">基本式</a>（英语：<a href="https://en.wikipedia.org/wiki/Ground_formula" class="extiw" title="en:Ground formula">Ground formula</a>）</li> <li><a href="/wiki/%E5%BC%80%E6%94%BE%E5%8F%A5%E5%AD%90" title="开放句子">开放式</a></li></ul></li> <li><a href="/wiki/%E8%87%AA%E7%94%B1%E5%8F%98%E9%87%8F%E5%92%8C%E7%BA%A6%E6%9D%9F%E5%8F%98%E9%87%8F" title="自由变量和约束变量">自由变量和约束变量</a></li> <li><a href="/wiki/%E5%85%83%E8%AA%9E%E8%A8%80" title="元語言">元語言</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E8%BF%90%E7%AE%97%E7%AC%A6" title="逻辑运算符">逻辑运算符</a> <ul><li><a href="/wiki/%E9%80%BB%E8%BE%91%E9%9D%9E" title="逻辑非">¬</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E6%88%96" title="逻辑或">∨</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E4%B8%8E" title="逻辑与">∧</a></li> <li><a href="/wiki/%E5%AE%9E%E8%B4%A8%E6%9D%A1%E4%BB%B6" title="实质条件">→</a></li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E5%8F%8C%E6%9D%A1%E4%BB%B6" class="mw-redirect" title="逻辑双条件">↔</a></li> <li><a href="/w/index.php?title=%E9%80%BB%E8%BE%91%E7%9B%B8%E7%AD%89&action=edit&redlink=1" class="new" title="逻辑相等（页面不存在）">逻辑相等</a>（英语：<a href="https://en.wikipedia.org/wiki/Logical_equality" class="extiw" title="en:Logical equality">Logical equality</a>）</li></ul></li> <li><a href="/w/index.php?title=%E8%B0%93%E8%AF%8D_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="谓词 (数理逻辑)（页面不存在）">谓词</a>（英语：<a href="https://en.wikipedia.org/wiki/Predicate_(mathematical_logic)" class="extiw" title="en:Predicate (mathematical logic)">Predicate (mathematical logic)</a>） <ul><li><a href="/wiki/%E6%B3%9B%E5%87%BD%E8%B0%93%E8%AF%8D" title="泛函谓词">泛函谓词</a></li> <li><a href="/wiki/%E8%B0%93%E8%AF%8D%E5%8F%98%E9%87%8F" title="谓词变量">谓词变量</a></li> <li><a href="/wiki/%E5%91%BD%E9%A2%98%E5%8F%98%E9%87%8F" title="命题变量">命题变量</a></li></ul></li> <li><a href="/wiki/%E9%87%8F%E5%8C%96_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="量化 (数理逻辑)">量化</a> <ul><li><a href="/wiki/%E5%AD%98%E5%9C%A8%E9%87%8F%E5%8C%96" title="存在量化">∃</a></li> <li><a href="/wiki/%E5%94%AF%E4%B8%80%E9%87%8F%E5%8C%96" title="唯一量化">!</a></li> <li><a href="/wiki/%E5%85%A8%E7%A7%B0%E9%87%8F%E5%8C%96" title="全称量化">∀</a></li> <li><a href="/w/index.php?title=%E9%87%8F%E8%AF%8D%E7%BA%A7%E5%88%AB&action=edit&redlink=1" class="new" title="量词级别（页面不存在）">级别</a>（英语：<a href="https://en.wikipedia.org/wiki/Quantifier_rank" class="extiw" title="en:Quantifier rank">Quantifier rank</a>）</li></ul></li> <li><a href="/wiki/%E5%8F%A5%E5%AD%90_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="句子 (数理逻辑)">句子</a> <ul><li><a href="/wiki/%E5%8E%9F%E5%AD%90%E5%8F%A5%E5%AD%90" title="原子句子">原子句子</a></li></ul></li> <li><a href="/w/index.php?title=%E7%AD%BE%E5%90%8D_(%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="签名 (逻辑)（页面不存在）">逻辑签名</a>（英语：<a href="https://en.wikipedia.org/wiki/Signature_(logic)" class="extiw" title="en:Signature (logic)">Signature (logic)</a>）</li> <li><a href="/wiki/%E5%AD%97%E7%AC%A6%E4%B8%B2" title="字符串">字符串</a></li> <li><a href="/w/index.php?title=%E6%9B%BF%E6%8D%A2%E6%B3%95_(%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="替换法 (逻辑)（页面不存在）">替换法</a>（英语：<a href="https://en.wikipedia.org/wiki/Substitution_(logic)" class="extiw" title="en:Substitution (logic)">Substitution (logic)</a>）</li> <li><a href="/wiki/%E9%80%BB%E8%BE%91%E7%AC%A6%E5%8F%B7%E8%A1%A8" title="逻辑符号表">逻辑符号</a> <ul><li><a href="/wiki/%E5%87%BD%E6%95%B0%E7%AC%A6%E5%8F%B7" class="mw-redirect" title="函数符号">函数符号</a></li> <li><a href="/w/index.php?title=%E9%80%BB%E8%BE%91%E5%B8%B8%E9%87%8F&action=edit&redlink=1" class="new" title="逻辑常量（页面不存在）">逻辑常量</a>（英语：<a href="https://en.wikipedia.org/wiki/Logical_constant" class="extiw" title="en:Logical constant">Logical constant</a>）</li> <li><a href="/w/index.php?title=%E9%9D%9E%E9%80%BB%E8%BE%91%E7%AC%A6%E5%8F%B7&action=edit&redlink=1" class="new" title="非逻辑符号（页面不存在）">非逻辑符号</a>（英语：<a href="https://en.wikipedia.org/wiki/Non-logical_symbol" class="extiw" title="en:Non-logical symbol">Non-logical symbol</a>）</li> <li><a href="/wiki/%E8%AE%8A%E6%95%B8" title="變數">變數</a></li></ul></li> <li><a href="/w/index.php?title=%E9%80%BB%E8%BE%91%E6%9C%AF%E8%AF%AD&action=edit&redlink=1" class="new" title="逻辑术语（页面不存在）">逻辑术语</a>（英语：<a href="https://en.wikipedia.org/wiki/Term_(logic)" class="extiw" title="en:Term (logic)">Term (logic)</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">公理系统</a>示例 （<a href="/w/index.php?title=%E4%B8%80%E9%98%B6%E7%90%86%E8%AE%BA%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="一阶理论列表（页面不存在）">列表</a>（英语：<a href="https://en.wikipedia.org/wiki/List_of_first-order_theories" class="extiw" title="en:List of first-order theories">List of first-order theories</a>））</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/w/index.php?title=%E5%AE%9E%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="实算术（页面不存在）">实算术</a>（英语：<a href="https://en.wikipedia.org/wiki/True_arithmetic" class="extiw" title="en:True arithmetic">True arithmetic</a>） <ul><li><a href="/wiki/%E7%9A%AE%E4%BA%9A%E8%AF%BA%E5%85%AC%E7%90%86" title="皮亚诺公理">皮亚诺公理</a></li> <li><a href="/w/index.php?title=%E4%BA%8C%E9%98%B6%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="二阶算术（页面不存在）">二阶</a>（英语：<a href="https://en.wikipedia.org/wiki/Second-order_arithmetic" class="extiw" title="en:Second-order arithmetic">Second-order arithmetic</a>）</li> <li><a href="/w/index.php?title=%E5%88%9D%E7%AD%89%E5%87%BD%E6%95%B0%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="初等函数算术（页面不存在）">初等函数</a>（英语：<a href="https://en.wikipedia.org/wiki/Elementary_function_arithmetic" class="extiw" title="en:Elementary function arithmetic">Elementary function arithmetic</a>）</li> <li><a href="/w/index.php?title=%E5%8E%9F%E5%A7%8B%E9%80%92%E5%BD%92%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="原始递归算术（页面不存在）">原始递归</a>（英语：<a href="https://en.wikipedia.org/wiki/Primitive_recursive_arithmetic" class="extiw" title="en:Primitive recursive arithmetic">Primitive recursive arithmetic</a>）</li> <li><a href="/w/index.php?title=%E7%BD%97%E5%AE%BE%E9%80%8A%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="罗宾逊算术（页面不存在）">罗宾逊算术</a>（英语：<a href="https://en.wikipedia.org/wiki/Robinson_arithmetic" class="extiw" title="en:Robinson arithmetic">Robinson arithmetic</a>）</li> <li><a href="/w/index.php?title=%E6%96%AF%E7%A7%91%E5%8B%92%E5%A7%86%E7%AE%97%E6%9C%AF&action=edit&redlink=1" class="new" title="斯科勒姆算术（页面不存在）">斯科勒姆算术</a>（英语：<a href="https://en.wikipedia.org/wiki/Skolem_arithmetic" class="extiw" title="en:Skolem arithmetic">Skolem arithmetic</a>）</li></ul></li> <li><a href="/wiki/%E5%AF%A6%E6%95%B8%E7%9A%84%E6%A7%8B%E9%80%A0" title="實數的構造">實數的構造</a> <ul><li><a href="/w/index.php?title=%E5%A1%94%E5%B0%94%E6%96%AF%E5%9F%BA%E7%9A%84%E5%AE%9E%E6%95%B0%E5%85%AC%E7%90%86%E5%8C%96&action=edit&redlink=1" class="new" title="塔尔斯基的实数公理化（页面不存在）">塔尔斯基公理化</a>（英语：<a href="https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals" class="extiw" title="en:Tarski's axiomatization of the reals">Tarski's axiomatization of the reals</a>）</li></ul></li> <li><a href="/wiki/%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0" title="布尔代数">布尔代数</a> <ul><li><a href="/w/index.php?title=%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0%E7%9A%84%E6%AD%A3%E5%88%99%E5%AE%9A%E4%B9%89&action=edit&redlink=1" class="new" title="布尔代数的正则定义（页面不存在）">正则定义</a>（英语：<a href="https://en.wikipedia.org/wiki/Boolean_algebras_canonically_defined" class="extiw" title="en:Boolean algebras canonically defined">Boolean algebras canonically defined</a>）</li> <li><a href="/w/index.php?title=%E5%B8%83%E5%B0%94%E4%BB%A3%E6%95%B0%E7%9A%84%E6%9C%80%E5%B0%8F%E5%85%AC%E7%90%86&action=edit&redlink=1" class="new" title="布尔代数的最小公理（页面不存在）">最小公理</a>（英语：<a href="https://en.wikipedia.org/wiki/Minimal_axioms_for_Boolean_algebra" class="extiw" title="en:Minimal axioms for Boolean algebra">Minimal axioms for Boolean algebra</a>）</li></ul></li> <li><a href="/w/index.php?title=%E5%87%A0%E4%BD%95%E5%9F%BA%E7%A1%80&action=edit&redlink=1" class="new" title="几何基础（页面不存在）">几何</a>（英语：<a href="https://en.wikipedia.org/wiki/Foundations_of_geometry" class="extiw" title="en:Foundations of geometry">Foundations of geometry</a>） <ul><li><a href="/wiki/%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="欧几里得几何">欧几里得几何</a></li> <li><a href="/wiki/%E5%87%A0%E4%BD%95%E5%8E%9F%E6%9C%AC" title="几何原本">《原本》</a></li> <li><a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E5%85%AC%E7%90%86" title="希尔伯特公理">希尔伯特公理</a></li> <li><a href="/wiki/%E9%9D%9E%E6%AC%A7%E5%87%A0%E9%87%8C%E5%BE%97%E5%87%A0%E4%BD%95" title="非欧几里得几何">非欧几里得几何</a></li> <li><a href="/w/index.php?title=%E5%A1%94%E5%B0%94%E6%96%AF%E5%9F%BA%E5%85%AC%E7%90%86&action=edit&redlink=1" class="new" title="塔尔斯基公理（页面不存在）">塔尔斯基公理</a>（英语：<a href="https://en.wikipedia.org/wiki/Tarski%27s_axioms" class="extiw" title="en:Tarski's axioms">Tarski's axioms</a>）</li></ul></li> <li>《<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%8E%9F%E7%90%86" title="数学原理">数学原理</a>》</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E8%AF%81%E6%98%8E%E8%AE%BA" title="证明论">证明论</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">形式证明</a></li> <li><a href="/wiki/%E8%87%AA%E7%84%B6%E6%BC%94%E7%BB%8E" title="自然演绎">自然演绎</a></li> <li><a href="/wiki/%E8%95%B4%E6%B6%B5" title="蕴涵">蕴涵</a></li> <li><a href="/wiki/%E6%8E%A8%E7%90%86%E8%A7%84%E5%88%99" title="推理规则">推理规则</a></li> <li><a href="/wiki/%E7%9B%B8%E7%BB%A7%E5%BC%8F%E6%BC%94%E7%AE%97" title="相继式演算">相继式演算</a></li> <li><a href="/wiki/%E5%AE%9A%E7%90%86" title="定理">定理</a></li> <li>系统 <ul><li><a href="/wiki/%E5%BD%A2%E5%BC%8F%E7%B3%BB%E7%B5%B1" title="形式系統">形式</a></li> <li><a href="/wiki/%E5%85%AC%E7%90%86%E7%B3%BB%E7%BB%9F" title="公理系统">公理</a></li> <li><a href="/wiki/%E5%BD%A2%E5%BC%8F%E7%B3%BB%E7%B5%B1" title="形式系統">演绎</a></li> <li><a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E6%BC%94%E7%BB%8E%E7%B3%BB%E7%BB%9F" title="希尔伯特演绎系统">希尔伯特演绎系统</a> <ul><li><a href="/w/index.php?title=%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%B3%BB%E7%BB%9F%E5%88%97%E8%A1%A8&action=edit&redlink=1" class="new" title="希尔伯特系统列表（页面不存在）">列表</a>（英语：<a href="https://en.wikipedia.org/wiki/List_of_Hilbert_systems" class="extiw" title="en:List of Hilbert systems">List of Hilbert systems</a>）</li></ul></li></ul></li> <li><a href="/w/index.php?title=%E5%AE%8C%E5%A4%87%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="完备理论（页面不存在）">完备理论</a>（英语：<a href="https://en.wikipedia.org/wiki/Complete_theory" class="extiw" title="en:Complete theory">Complete theory</a>）</li> <li><a href="/wiki/%E7%AD%96%E6%A2%85%E6%B4%9B-%E5%BC%97%E5%85%B0%E5%85%8B%E5%B0%94%E9%9B%86%E5%90%88%E8%AE%BA" title="策梅洛-弗兰克尔集合论">ZFC</a>系统的<a href="/wiki/%E7%8D%A8%E7%AB%8B%E6%80%A7_(%E6%95%B8%E7%90%86%E9%82%8F%E8%BC%AF)" title="獨立性 (數理邏輯)">独立性</a>（英语：<a href="https://en.wikipedia.org/wiki/Independence_(mathematical_logic)" class="extiw" title="en:Independence (mathematical logic)">Independence (mathematical logic)</a>） <ul><li><a href="/wiki/ZFC%E7%B3%BB%E7%B5%B1%E7%84%A1%E6%B3%95%E7%A2%BA%E5%AE%9A%E7%9A%84%E5%91%BD%E9%A1%8C%E5%88%97%E8%A1%A8" title="ZFC系統無法確定的命題列表">列表</a></li></ul></li> <li><a href="/w/index.php?title=%E4%B8%8D%E5%8F%AF%E8%83%BD%E8%AF%81%E6%98%8E&action=edit&redlink=1" class="new" title="不可能证明（页面不存在）">不可能证明</a>（英语：<a href="https://en.wikipedia.org/wiki/Proof_of_impossibility" class="extiw" title="en:Proof of impossibility">Proof of impossibility</a>）</li> <li><a href="/w/index.php?title=%E5%BA%8F%E6%95%B0%E5%88%86%E6%9E%90&action=edit&redlink=1" class="new" title="序数分析（页面不存在）">序数分析</a>（英语：<a href="https://en.wikipedia.org/wiki/Ordinal_analysis" class="extiw" title="en:Ordinal analysis">Ordinal analysis</a>）</li> <li><a href="/wiki/%E9%80%86%E6%95%B0%E5%AD%A6" title="逆数学">逆数学</a></li> <li><a href="/w/index.php?title=%E8%87%AA%E6%81%B0%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="自恰理论（页面不存在）">自恰理论</a>（英语：<a href="https://en.wikipedia.org/wiki/Self-verifying_theories" class="extiw" title="en:Self-verifying theories">Self-verifying theories</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%A8%A1%E5%9E%8B%E8%AE%BA" title="模型论">模型论</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E8%A7%A3%E9%87%8B_(%E9%82%8F%E8%BC%AF)" title="解釋 (邏輯)">解释</a></li> <li><a href="/wiki/%E7%BB%93%E6%9E%84_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)" title="结构 (数理逻辑)">结构</a> <ul><li><a href="/wiki/%E5%88%9D%E7%AD%89%E7%AD%89%E4%BB%B7" title="初等等价">初等等价</a></li> <li><a href="/w/index.php?title=%E6%9C%89%E9%99%90%E6%A8%A1%E5%9E%8B%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="有限模型理论（页面不存在）">有限模型</a>（英语：<a href="https://en.wikipedia.org/wiki/Finite_model_theory" class="extiw" title="en:Finite model theory">Finite model theory</a>）</li> <li><a href="/wiki/%E9%A3%BD%E5%92%8C%E6%A8%A1%E5%9E%8B" title="飽和模型">飽和模型</a></li> <li><a href="/wiki/%E5%AD%90%E7%BB%93%E6%9E%84" title="子结构">子结构</a></li></ul></li> <li><a href="/wiki/%E9%9D%9E%E6%A0%87%E5%87%86%E6%A8%A1%E5%9E%8B" title="非标准模型">非标准模型</a> <ul><li><a href="/w/index.php?title=%E9%9D%9E%E6%A0%87%E5%87%86%E7%9A%84%E7%AE%97%E6%9C%AF%E6%A8%A1%E5%9E%8B&action=edit&redlink=1" class="new" title="非标准的算术模型（页面不存在）">算术</a>（英语：<a href="https://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic" class="extiw" title="en:Non-standard model of arithmetic">Non-standard model of arithmetic</a>）</li></ul></li> <li><a href="/w/index.php?title=%E7%BB%93%E6%9E%84%E5%9B%BE_(%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="结构图 (数理逻辑)（页面不存在）">结构图</a>（英语：<a href="https://en.wikipedia.org/wiki/Diagram_(mathematical_logic)" class="extiw" title="en:Diagram (mathematical logic)">Diagram (mathematical logic)</a>） <ul><li><a href="/w/index.php?title=%E5%9F%BA%E6%9C%AC%E7%BB%93%E6%9E%84%E5%9B%BE&action=edit&redlink=1" class="new" title="基本结构图（页面不存在）">基本图</a>（英语：<a href="https://en.wikipedia.org/wiki/Elementary_diagram" class="extiw" title="en:Elementary diagram">Elementary diagram</a>）</li></ul></li> <li><a href="/w/index.php?title=%E5%88%86%E7%B1%BB%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="分类理论（页面不存在）">分类理论</a>（英语：<a href="https://en.wikipedia.org/wiki/Categorical_theory" class="extiw" title="en:Categorical theory">Categorical theory</a>）</li> <li><a href="/w/index.php?title=%E5%AE%8C%E5%A4%87%E6%A8%A1%E5%9E%8B%E8%AE%BA&action=edit&redlink=1" class="new" title="完备模型论（页面不存在）">完备模型论</a>（英语：<a href="https://en.wikipedia.org/wiki/Model_complete_theory" class="extiw" title="en:Model complete theory">Model complete theory</a>）</li> <li><a href="/w/index.php?title=%E5%8F%AF%E6%BB%A1%E8%B6%B3%E6%80%A7&action=edit&redlink=1" class="new" title="可满足性（页面不存在）">可满足性</a>（英语：<a href="https://en.wikipedia.org/wiki/Satisfiability" class="extiw" title="en:Satisfiability">Satisfiability</a>）</li> <li><a href="/wiki/%E9%82%8F%E8%BC%AF%E8%AA%9E%E7%BE%A9%E5%AD%B8" title="邏輯語義學">邏輯語義學</a></li> <li><a href="/w/index.php?title=%E5%BC%BA%E5%BA%A6_(%E6%A8%A1%E5%9E%8B%E8%AE%BA)&action=edit&redlink=1" class="new" title="强度 (模型论)（页面不存在）">强度</a>（英语：<a href="https://en.wikipedia.org/wiki/Strength_(mathematical_logic)" class="extiw" title="en:Strength (mathematical logic)">Strength (mathematical logic)</a>）</li> <li><a href="/wiki/%E7%9C%9F%E7%90%86" title="真理">真理</a> <ul><li><a href="/wiki/%E7%9C%9F%E7%90%86%E7%9A%84%E8%AF%AD%E4%B9%89%E7%90%86%E8%AE%BA" title="真理的语义理论">语义理论</a></li> <li><a href="/wiki/%E7%9C%9F%E7%90%86%E7%9A%84%E8%AF%AD%E4%B9%89%E7%90%86%E8%AE%BA" title="真理的语义理论">塔尔斯基</a></li> <li><a href="/wiki/%E7%9C%9F%E7%90%86%E7%9A%84%E8%AF%AD%E4%B9%89%E7%90%86%E8%AE%BA" title="真理的语义理论">克里普克</a></li></ul></li> <li><a href="/wiki/T-%E6%A8%A1%E5%BC%8F" title="T-模式">T-模式</a></li> <li><a href="/w/index.php?title=%E8%BD%AC%E7%A7%BB%E5%8E%9F%E5%88%99&action=edit&redlink=1" class="new" title="转移原则（页面不存在）">转移原则</a>（英语：<a href="https://en.wikipedia.org/wiki/Transfer_principle" class="extiw" title="en:Transfer principle">Transfer principle</a>）</li> <li><a href="/w/index.php?title=%E7%9C%9F%E7%90%86%E8%B0%93%E8%AF%8D&action=edit&redlink=1" class="new" title="真理谓词（页面不存在）">真理谓词</a>（英语：<a href="https://en.wikipedia.org/wiki/Truth_predicate" class="extiw" title="en:Truth predicate">Truth predicate</a>）</li> <li><a href="/wiki/%E7%9C%9F%E5%80%BC" title="真值">真值</a></li> <li><a href="/wiki/%E5%9E%8B_(%E6%A8%A1%E5%9E%8B%E8%AE%BA)" title="型 (模型论)">型</a></li> <li><a href="/wiki/%E8%B6%85%E7%A9%8D" title="超積">超積</a></li> <li><a href="/wiki/%E6%9C%89%E6%95%88%E6%80%A7" title="有效性">有效性</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%8F%AF%E8%AE%A1%E7%AE%97%E6%80%A7%E7%90%86%E8%AE%BA" title="可计算性理论">可计算性理论</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E9%82%B1%E5%A5%87%E6%95%B0" title="邱奇数">邱奇数</a></li> <li><a href="/wiki/%E9%82%B1%E5%A5%87-%E5%9B%BE%E7%81%B5%E8%AE%BA%E9%A2%98" title="邱奇-图灵论题">邱奇-图灵论题</a></li> <li><a href="/wiki/%E9%80%92%E5%BD%92%E5%8F%AF%E6%9E%9A%E4%B8%BE%E9%9B%86%E5%90%88" title="递归可枚举集合">递归可枚举集合</a></li> <li><a href="/wiki/%E5%8F%AF%E8%AE%A1%E7%AE%97%E5%87%BD%E6%95%B0" title="可计算函数">可计算函数</a></li> <li><a href="/wiki/%E9%80%92%E5%BD%92%E9%9B%86%E5%90%88" title="递归集合">递归集合</a></li> <li><a href="/wiki/%E6%B1%BA%E5%AE%9A%E6%80%A7%E5%95%8F%E9%A1%8C" title="決定性問題">決定性問題</a> <ul><li><a href="/w/index.php?title=%E5%8F%AF%E5%86%B3%E5%AE%9A%E6%80%A7_(%E9%80%BB%E8%BE%91)&action=edit&redlink=1" class="new" title="可决定性 (逻辑)（页面不存在）">可决定性</a>（英语：<a href="https://en.wikipedia.org/wiki/Decidability_(logic)" class="extiw" title="en:Decidability (logic)">Decidability (logic)</a>）</li> <li><a href="/wiki/%E4%B8%8D%E5%8F%AF%E5%88%A4%E5%AE%9A%E9%97%AE%E9%A2%98" title="不可判定问题">不可决定性</a></li> <li><a href="/wiki/P_(%E8%A4%87%E9%9B%9C%E5%BA%A6)" title="P (複雜度)">P</a></li> <li><a href="/wiki/NP_(%E8%A4%87%E9%9B%9C%E5%BA%A6)" title="NP (複雜度)">NP</a></li> <li><a href="/wiki/P/NP%E9%97%AE%E9%A2%98" title="P/NP问题">P/NP问题</a></li></ul></li> <li><a href="/wiki/%E6%9F%AF%E6%B0%8F%E5%A4%8D%E6%9D%82%E6%80%A7" title="柯氏复杂性">柯氏复杂性</a></li> <li><a href="/wiki/%CE%9B%E6%BC%94%E7%AE%97" title="Λ演算">Λ演算</a></li> <li><a href="/wiki/%E5%8E%9F%E5%A7%8B%E9%80%92%E5%BD%92%E5%87%BD%E6%95%B0" title="原始递归函数">原始递归函数</a></li> <li><a href="/wiki/%E9%80%92%E5%BD%92" title="递归">递归</a></li> <li><a href="/wiki/%E9%80%92%E5%BD%92%E9%9B%86%E5%90%88" title="递归集合">递归集合</a></li> <li><a href="/wiki/%E5%9B%BE%E7%81%B5%E6%9C%BA" title="图灵机">图灵机</a></li> <li><a href="/wiki/%E7%B1%BB%E5%9E%8B%E8%AE%BA" title="类型论">类型论</a></li></ul> </div></td></tr><tr><th scope="row" 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直言三段论 - 维基百科，自由的百科全书