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id="toc-Logic_in_India-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hindu_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hindu_logic"> <div class="vector-toc-text"> 1.2 Hindu logic </div> </a> <ul id="toc-Hindu_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Origin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Origin"> <div class="vector-toc-text"> 1.3 Origin </div> </a> <ul id="toc-Origin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Before_Gautama" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Before_Gautama"> <div class="vector-toc-text"> 1.4 Before Gautama </div> </a> <ul id="toc-Before_Gautama-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dattatreya" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dattatreya"> <div class="vector-toc-text"> 1.5 Dattatreya </div> </a> <ul id="toc-Dattatreya-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Medhatithi_Gautama" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Medhatithi_Gautama"> <div class="vector-toc-text"> 1.6 Medhatithi Gautama </div> </a> <ul id="toc-Medhatithi_Gautama-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Panini" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Panini"> <div class="vector-toc-text"> 1.7 Panini </div> </a> <ul id="toc-Panini-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nyaya-Vaisheshika" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nyaya-Vaisheshika"> <div class="vector-toc-text"> 1.8 Nyaya-Vaisheshika </div> </a> <ul id="toc-Nyaya-Vaisheshika-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jain_Logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jain_Logic"> <div class="vector-toc-text"> 1.9 Jain Logic </div> </a> <ul id="toc-Jain_Logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Buddhist_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Buddhist_logic"> <div class="vector-toc-text"> 1.10 Buddhist logic </div> </a> <ul id="toc-Buddhist_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nagarjuna" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nagarjuna"> <div class="vector-toc-text"> 1.11 Nagarjuna </div> </a> <ul id="toc-Nagarjuna-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dignaga" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dignaga"> <div class="vector-toc-text"> 1.12 Dignaga </div> </a> <ul id="toc-Dignaga-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logic_in_China" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic_in_China"> <div class="vector-toc-text"> 1.13 Logic in China </div> </a> <ul id="toc-Logic_in_China-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Logic_in_the_West" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Logic_in_the_West"> <div class="vector-toc-text"> 2 Logic in the West </div> </a> <button aria-controls="toc-Logic_in_the_West-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Logic in the West subsection </button> <ul id="toc-Logic_in_the_West-sublist" class="vector-toc-list"> <li id="toc-Prehistory_of_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prehistory_of_logic"> <div class="vector-toc-text"> 2.1 Prehistory of logic </div> </a> <ul id="toc-Prehistory_of_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ancient_Greece_before_Aristotle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ancient_Greece_before_Aristotle"> <div class="vector-toc-text"> 2.2 Ancient Greece before Aristotle </div> </a> <ul id="toc-Ancient_Greece_before_Aristotle-sublist" class="vector-toc-list"> <li id="toc-Thales" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Thales"> <div class="vector-toc-text"> 2.2.1 Thales </div> </a> <ul id="toc-Thales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pythagoras" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Pythagoras"> <div class="vector-toc-text"> 2.2.2 Pythagoras </div> </a> <ul id="toc-Pythagoras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heraclitus_and_Parmenides" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Heraclitus_and_Parmenides"> <div class="vector-toc-text"> 2.2.3 Heraclitus and Parmenides </div> </a> <ul id="toc-Heraclitus_and_Parmenides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plato" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Plato"> <div class="vector-toc-text"> 2.2.4 Plato </div> </a> <ul id="toc-Plato-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Aristotle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aristotle"> <div class="vector-toc-text"> 2.3 Aristotle </div> </a> <ul id="toc-Aristotle-sublist" class="vector-toc-list"> <li id="toc-The_Organon" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_Organon"> <div class="vector-toc-text"> 2.3.1 The Organon </div> </a> <ul id="toc-The_Organon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stoics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stoics"> <div class="vector-toc-text"> 2.4 Stoics </div> </a> <ul id="toc-Stoics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Medieval_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Medieval_logic"> <div class="vector-toc-text"> 3 Medieval logic </div> </a> <button aria-controls="toc-Medieval_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Medieval logic subsection </button> <ul id="toc-Medieval_logic-sublist" class="vector-toc-list"> <li id="toc-Logic_in_the_Middle_East" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic_in_the_Middle_East"> <div class="vector-toc-text"> 3.1 Logic in the Middle East </div> </a> <ul id="toc-Logic_in_the_Middle_East-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logic_in_medieval_Europe" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic_in_medieval_Europe"> <div class="vector-toc-text"> 3.2 Logic in medieval Europe </div> </a> <ul id="toc-Logic_in_medieval_Europe-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Traditional_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Traditional_logic"> <div class="vector-toc-text"> 4 Traditional logic </div> </a> <button aria-controls="toc-Traditional_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Traditional logic subsection </button> <ul id="toc-Traditional_logic-sublist" class="vector-toc-list"> <li id="toc-The_textbook_tradition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_textbook_tradition"> <div class="vector-toc-text"> 4.1 The textbook tradition </div> </a> <ul id="toc-The_textbook_tradition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logic_in_Hegel's_philosophy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic_in_Hegel's_philosophy"> <div class="vector-toc-text"> 4.2 Logic in Hegel's philosophy </div> </a> <ul id="toc-Logic_in_Hegel's_philosophy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logic_and_psychology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic_and_psychology"> <div class="vector-toc-text"> 4.3 Logic and psychology </div> </a> <ul id="toc-Logic_and_psychology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rise_of_modern_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Rise_of_modern_logic"> <div class="vector-toc-text"> 5 Rise of modern logic </div> </a> <ul id="toc-Rise_of_modern_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Modern_logic"> <div class="vector-toc-text"> 6 Modern logic </div> </a> <button aria-controls="toc-Modern_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Modern logic subsection </button> <ul id="toc-Modern_logic-sublist" class="vector-toc-list"> <li id="toc-Embryonic_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Embryonic_period"> <div class="vector-toc-text"> 6.1 Embryonic period </div> </a> <ul id="toc-Embryonic_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_period"> <div class="vector-toc-text"> 6.2 Algebraic period </div> </a> <ul id="toc-Algebraic_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logicist_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logicist_period"> <div class="vector-toc-text"> 6.3 Logicist period </div> </a> <ul id="toc-Logicist_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metamathematical_period" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metamathematical_period"> <div class="vector-toc-text"> 6.4 Metamathematical period </div> </a> <ul id="toc-Metamathematical_period-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logic_after_WWII" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logic_after_WWII"> <div class="vector-toc-text"> 6.5 Logic after WWII </div> </a> <ul id="toc-Logic_after_WWII-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 8 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 9 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 10 External links </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <label 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Available in 19 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" > 19 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%A7%D8%B1%D9%8A%D8%AE_%D8%A7%D9%84%D9%85%D9%86%D8%B7%D9%82" title="تاريخ المنطق – Arabic" lang="ar" hreflang="ar" data-title="تاريخ المنطق" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AF%E0%A7%81%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF%E0%A6%AC%E0%A6%BF%E0%A6%A6%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%B0_%E0%A6%87%E0%A6%A4%E0%A6%BF%E0%A6%B9%E0%A6%BE%E0%A6%B8" title="যুক্তিবিদ্যার ইতিহাস – Bangla" lang="bn" hreflang="bn" data-title="যুক্তিবিদ্যার ইতিহাস" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geschichte_der_Logik" title="Geschichte der Logik – German" lang="de" hreflang="de" data-title="Geschichte der Logik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Historia_de_la_l%C3%B3gica" title="Historia de la lógica – Spanish" lang="es" hreflang="es" data-title="Historia de la lógica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Histoire_de_la_logique" title="Histoire de la logique – French" lang="fr" hreflang="fr" data-title="Histoire de la logique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%85%BC%EB%A6%AC%EC%82%AC" title="논리사 – Korean" lang="ko" hreflang="ko" data-title="논리사" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%A1%D5%B6_%D5%BA%D5%A1%D5%BF%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Տրամաբանության պատմություն – Armenian" lang="hy" hreflang="hy" data-title="Տրամաբանության պատմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0_%E0%A4%95%E0%A4%BE_%E0%A4%87%E0%A4%A4%E0%A4%BF%E0%A4%B9%E0%A4%BE%E0%A4%B8" title="तर्कशास्त्र का इतिहास – Hindi" lang="hi" hreflang="hi" data-title="तर्कशास्त्र का इतिहास" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/A_logika_t%C3%B6rt%C3%A9nete" title="A logika története – Hungarian" lang="hu" hreflang="hu" data-title="A logika története" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geschiedenis_van_de_logica" title="Geschiedenis van de logica – Dutch" lang="nl" hreflang="nl" data-title="Geschiedenis van de logica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%AB%96%E7%90%86%E5%AD%A6%E3%81%AE%E6%AD%B4%E5%8F%B2" title="論理学の歴史 – Japanese" lang="ja" hreflang="ja" data-title="論理学の歴史" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%AF_%D9%85%D9%86%D8%B7%D9%82_%D8%AA%D8%A7%D8%B1%DB%8C%D8%AE%DA%86%D9%87" title="د منطق تاریخچه – Pashto" lang="ps" hreflang="ps" data-title="د منطق تاریخچه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Historia_logiki" title="Historia logiki – Polish" lang="pl" hreflang="pl" data-title="Historia logiki" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Hist%C3%B3ria_da_l%C3%B3gica" title="História da lógica – Portuguese" lang="pt" hreflang="pt" data-title="História da lógica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Istoria_logicii" title="Istoria logicii – Romanian" lang="ro" hreflang="ro" data-title="Istoria logicii" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D1%81%D1%82%D0%BE%D1%80%D0%B8%D1%8F_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B8" title="История логики – Russian" lang="ru" hreflang="ru" data-title="История логики" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Logiikan_historia" title="Logiikan historia – Finnish" lang="fi" hreflang="fi" data-title="Logiikan historia" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mant%C4%B1k_tarihi" title="Mantık tarihi – Turkish" lang="tr" hreflang="tr" data-title="Mantık tarihi" 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sidebar-phi-above"> <div class="hlist"> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/10px-Socrates.png" decoding="async" width="10" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/15px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/21px-Socrates.png 2x" data-file-width="326" data-file-height="500" /> <a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></li> <li><a href="/wiki/Wikipedia:Contents/Philosophy_and_thinking" title="Wikipedia:Contents/Philosophy and thinking">Contents</a></li> <li><a href="/wiki/Outline_of_philosophy" title="Outline of philosophy">Outline</a></li> <li><a href="/wiki/Index_of_philosophy" title="Index of philosophy">Lists</a></li> <li><a href="/wiki/Glossary_of_philosophy" title="Glossary of philosophy">Glossary</a></li> <li><a href="/wiki/History_of_philosophy" title="History of philosophy">History</a></li> <li><a href="/wiki/Category:Philosophy" title="Category:Philosophy">Categories</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Category:Philosophical_schools_and_traditions" title="Category:Philosophical schools and traditions">Philosophies</a></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Category:Philosophy_by_period" title="Category:Philosophy by period">By period</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Ancient_philosophy" title="Ancient philosophy">Ancient</a> <ul><li><a href="/wiki/Ancient_Egyptian_philosophy" title="Ancient Egyptian philosophy">Ancient Egyptian</a></li> <li><a href="/wiki/Ancient_Greek_philosophy" title="Ancient Greek philosophy">Ancient Greek</a></li></ul></li> <li><a href="/wiki/Medieval_philosophy" title="Medieval philosophy">Medieval</a></li> <li><a href="/wiki/Renaissance_philosophy" title="Renaissance philosophy">Renaissance</a></li> <li><a href="/wiki/Modern_philosophy" title="Modern philosophy">Modern</a></li> <li><a href="/wiki/Contemporary_philosophy" title="Contemporary philosophy">Contemporary</a> <ul><li><a href="/wiki/Analytic_philosophy" title="Analytic philosophy">Analytic</a></li> <li><a href="/wiki/Continental_philosophy" title="Continental philosophy">Continental</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Outline_of_philosophy#Philosophic_traditions_by_region" title="Outline of philosophy">By region</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/African_philosophy" title="African philosophy">African</a> <ul><li><a href="/wiki/Ancient_Egyptian_philosophy" title="Ancient Egyptian philosophy">Egypt</a></li> <li><a href="/wiki/Ethiopian_philosophy" title="Ethiopian philosophy">Ethiopia</a></li> <li><a href="/wiki/Ubuntu_philosophy" title="Ubuntu philosophy">South Africa</a></li></ul></li> <li><a href="/wiki/Eastern_philosophy" title="Eastern philosophy">Eastern philosophy</a> <ul><li><a href="/wiki/Chinese_philosophy" title="Chinese philosophy">Chinese</a></li> <li><a href="/wiki/Indian_philosophy" title="Indian philosophy">Indian</a></li> <li><a href="/wiki/Indonesian_philosophy" title="Indonesian philosophy">Indonesia</a></li> <li><a href="/wiki/Japanese_philosophy" title="Japanese philosophy">Japan</a></li> <li><a href="/wiki/Korean_philosophy" title="Korean philosophy">Korea</a></li> <li><a href="/wiki/Vietnamese_philosophy" title="Vietnamese philosophy">Vietnam</a></li></ul></li> <li><a href="/wiki/Indigenous_American_philosophy" title="Indigenous American philosophy">Indigenous American</a> <ul><li><a href="/wiki/Aztec_philosophy" title="Aztec 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title="Ontology">Ontology</a></li> <li><a href="/wiki/Phenomenology_(philosophy)" title="Phenomenology (philosophy)">Phenomenology</a></li> <li><a href="/wiki/Political_philosophy" title="Political philosophy">Political</a></li> <li><a href="/wiki/Philosophy_of_religion" title="Philosophy of religion">Religion</a></li> <li><a href="/wiki/Philosophy_of_science" title="Philosophy of science">Science</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Lists_of_philosophers" title="Lists of philosophers">Philosophers</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/List_of_aestheticians" title="List of aestheticians">Aesthetic philosophers</a></li> <li><a href="/wiki/List_of_epistemologists" title="List of epistemologists">Epistemologists</a></li> <li><a href="/wiki/List_of_ethicists" title="List of ethicists">Ethicists</a></li> <li><a href="/wiki/List_of_logicians" title="List of logicians">Logicians</a></li> <li><a href="/wiki/List_of_metaphysicians" title="List of metaphysicians">Metaphysicians</a></li> <li><a href="/wiki/List_of_philosophers_of_mind" title="List of philosophers of mind">Philosophers of mind</a></li> <li><a href="/wiki/Index_of_sociopolitical_thinkers" title="Index of sociopolitical thinkers">Social and political philosophers</a></li> <li><a href="/wiki/Women_in_philosophy" title="Women in philosophy">Women in philosophy</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Philosophy_sidebar" title="Template:Philosophy sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Philosophy_sidebar" title="Template talk:Philosophy sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Philosophy_sidebar" title="Special:EditPage/Template:Philosophy sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> The history of logic deals with the study of the development of the science of valid <a href="/wiki/Inference" title="Inference">inference</a> (<a href="/wiki/Logic" title="Logic">logic</a>). Formal logics developed in ancient times in <a href="/wiki/Indian_logic" title="Indian logic">India</a>, <a href="/wiki/Logic_in_China" title="Logic in China">China</a>, and <a href="/wiki/Greek_philosophy" class="mw-redirect" title="Greek philosophy">Greece</a>. Greek methods, particularly <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a> (or term logic) as found in the <a href="/wiki/Organon" title="Organon">Organon</a>, found wide application and acceptance in Western science and mathematics for millennia.<a href="#cite_note-Boehner_p._xiv-1">[1]</a> The <a href="/wiki/Stoicism" title="Stoicism">Stoics</a>, especially <a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a>, began the development of <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a>. <a href="/wiki/Christian_philosophy" title="Christian philosophy">Christian</a> and <a href="/wiki/Logic_in_Islamic_philosophy" title="Logic in Islamic philosophy">Islamic</a> philosophers such as <a href="/wiki/Boethius" title="Boethius">Boethius</a> (died 524), <a href="/wiki/Avicenna" title="Avicenna">Avicenna</a> (died 1037), <a href="/wiki/Thomas_Aquinas" title="Thomas Aquinas">Thomas Aquinas</a> (died 1274) and <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a> (died 1347) further developed Aristotle's logic in the <a href="/wiki/Medieval_philosophy#High_Middle_Ages" title="Medieval philosophy">Middle Ages</a>, reaching a high point in the mid-fourteenth century, with <a href="/wiki/Jean_Buridan" title="Jean Buridan">Jean Buridan</a>. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at least one historian of logic regards this time as barren.<a href="#cite_note-ReferenceA-2">[2]</a> <a href="/wiki/Empirical_methods" class="mw-redirect" title="Empirical methods">Empirical methods</a> ruled the day, as evidenced by Sir <a href="/wiki/Francis_Bacon" title="Francis Bacon">Francis Bacon</a>'s <a href="/wiki/Novum_Organon" class="mw-redirect" title="Novum Organon">Novum Organon</a> of 1620. Logic revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formal discipline which took as its exemplar the exact method of <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> used in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a hearkening back to the Greek tradition.<a href="#cite_note-3">[3]</a> The development of the modern "symbolic" or "mathematical" logic during this period by the likes of <a href="/wiki/George_Boole" title="George Boole">Boole</a>, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a>, and <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a> is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human <a href="/wiki/Intellectual_history" title="Intellectual history">intellectual history</a>.<a href="#cite_note-Oxford_Companion_p._500-4">[4]</a> Progress in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> in the first few decades of the twentieth century, particularly arising from the work of <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a> and <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski</a>, had a significant impact on <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">analytic philosophy</a> and <a href="/wiki/Philosophical_logic" title="Philosophical logic">philosophical logic</a>, particularly from the 1950s onwards, in subjects such as <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>, <a href="/wiki/Temporal_logic" title="Temporal logic">temporal logic</a>, <a href="/wiki/Deontic_logic" title="Deontic logic">deontic logic</a>, and <a href="/wiki/Relevance_logic" title="Relevance logic">relevance logic</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Logic_in_the_East">Logic in the East</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=1" title="Edit section: Logic in the East">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Logic_in_India">Logic in India</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=2" title="Edit section: Logic in India">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Indian_logic" title="Indian logic">Indian logic</a></div> <div class="mw-heading mw-heading3"><h3 id="Hindu_logic">Hindu logic</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=3" title="Edit section: Hindu logic">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Origin">Origin</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=4" title="Edit section: Origin">edit</a>]</div> The <a href="/wiki/Nasadiya_Sukta" title="Nasadiya Sukta">Nasadiya Sukta</a> of the <a href="/wiki/Rigveda" title="Rigveda">Rigveda</a> (<a href="/wiki/Mandala_10" title="Mandala 10">RV 10</a>.129) contains <a href="/wiki/Ontological" class="mw-redirect" title="Ontological">ontological</a> speculation in terms of various logical divisions that were later recast formally as the four circles of <a href="/wiki/Tetralemma" title="Tetralemma">catuskoti</a>: "A", "not A", "A and 'not A'", and "not A and not not A". <style data-mw-deduplicate="TemplateStyles:r1023981488">@media all and (max-width:720px){.mw-parser-output .rquote{width:auto!important;float:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote rquote" style="float: right; width: 33%;">Who really knows? Who will here proclaim it? Whence was it produced? Whence is this creation? The gods came afterwards, with the creation of this universe. Who then knows whence it has arisen?<div class="templatequotecite">— <cite><a href="/wiki/Nasadiya_Sukta" title="Nasadiya Sukta">Nasadiya Sukta</a>, concerns the <a href="/wiki/Origin_of_the_universe" class="mw-redirect" title="Origin of the universe">origin of the universe</a>, <a href="/wiki/Rig_Veda" class="mw-redirect" title="Rig Veda">Rig Veda</a>, 10:129-6 <a href="#cite_note-Kramer1986-5">[5]</a><a href="#cite_note-Christian2011-6">[6]</a><a href="#cite_note-Singh2008-7">[7]</a></cite></div></blockquote> Logic began independently in <a href="/wiki/Ancient_India" class="mw-redirect" title="Ancient India">ancient India</a> and continued to develop to early modern times without any known influence from Greek logic.<a href="#cite_note-8">[8]</a> <div class="mw-heading mw-heading3"><h3 id="Before_Gautama">Before Gautama</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=5" title="Edit section: Before Gautama">edit</a>]</div> Though the origins in India of public debate (pariṣad), one form of rational inquiry, are not clear, we know that public debates were common in preclassical India, for they are frequently alluded to in various <a href="/wiki/Upanishads" title="Upanishads">Upaniṣads</a> and in the early Buddhist literature. Public debate is not the only form of public deliberations in preclassical India. Assemblies (pariṣad or <a href="/wiki/Sabh%C4%81" title="Sabhā">sabhā</a>) of various sorts, comprising relevant experts, were regularly convened to deliberate on a variety of matters, including administrative, legal and religious matters. <div class="mw-heading mw-heading3"><h3 id="Dattatreya">Dattatreya</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=6" title="Edit section: Dattatreya">edit</a>]</div> A philosopher named Dattatreya is stated in the <a href="/wiki/Bhagavata_Purana" title="Bhagavata Purana">Bhagavata purana</a> to have taught Anviksiki to Aiarka, Prahlada and others. It appears from the <a href="/wiki/Markandeya_Purana" title="Markandeya Purana">Markandeya purana</a> that the Anviksiki-vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy. Dattatreya expounded the philosophical side of Anviksiki and not its logical aspect.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> <div class="mw-heading mw-heading3"><h3 id="Medhatithi_Gautama">Medhatithi Gautama</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=7" title="Edit section: Medhatithi Gautama">edit</a>]</div> While the teachers mentioned before dealt with some particular topics of Anviksiki, the credit of founding the Anviksiki in its special sense of a science is to be attributed to <a href="/wiki/Indian_logic" title="Indian logic">Medhatithi Gautama</a> (c. 6th century BC). Guatama founded the <a href="/wiki/Anviksiki" title="Anviksiki">anviksiki</a> school of logic.<a href="#cite_note-11">[11]</a> The <a href="/wiki/Mahabharata" title="Mahabharata">Mahabharata</a> (12.173.45), around the 5th century BC, refers to the anviksiki and tarka schools of logic. <div class="mw-heading mw-heading3"><h3 id="Panini">Panini</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=8" title="Edit section: Panini">edit</a>]</div> <a href="/wiki/P%C4%81%E1%B9%87ini" title="Pāṇini">Pāṇini</a> (c. 5th century BC) developed a form of logic (to which <a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> has some similarities) for his formulation of <a href="/wiki/Vyakarana" class="mw-redirect" title="Vyakarana">Sanskrit grammar</a>. Logic is described by <a href="/wiki/Chanakya" title="Chanakya">Chanakya</a> (c. 350–283 BC) in his <a href="/wiki/Arthashastra" title="Arthashastra">Arthashastra</a> as an independent field of inquiry.<a href="#cite_note-12">[12]</a> <div class="mw-heading mw-heading3"><h3 id="Nyaya-Vaisheshika">Nyaya-Vaisheshika</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=9" title="Edit section: Nyaya-Vaisheshika">edit</a>]</div> Two of the six Indian schools of thought deal with logic: <a href="/wiki/Nyaya" title="Nyaya">Nyaya</a> and <a href="/wiki/Vaisheshika" title="Vaisheshika">Vaisheshika</a>. The <a href="/wiki/Ny%C4%81ya_S%C5%ABtras" title="Nyāya Sūtras">Nyāya Sūtras</a> of <a href="/wiki/Aksapada_Gautama" class="mw-redirect" title="Aksapada Gautama">Aksapada Gautama</a> (c. 2nd century AD) constitute the core texts of the Nyaya school, one of the six orthodox schools of <a href="/wiki/Hindu" class="mw-redirect" title="Hindu">Hindu</a> philosophy. This <a href="/wiki/Philosophical_realism" title="Philosophical realism">realist</a> school developed a rigid five-member schema of <a href="/wiki/Inference" title="Inference">inference</a> involving an initial premise, a reason, an example, an application, and a conclusion.<a href="#cite_note-13">[13]</a> The <a href="/wiki/Idealism" title="Idealism">idealist</a> <a href="/wiki/Buddhist_philosophy" title="Buddhist philosophy">Buddhist philosophy</a> became the chief opponent to the Naiyayikas. <div class="mw-heading mw-heading3"><h3 id="Jain_Logic">Jain Logic</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=10" title="Edit section: Jain Logic">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg/220px-%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg" decoding="async" width="220" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg/330px-%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg/440px-%E0%A4%89%E0%A4%AE%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%AE%E0%A5%80_%E0%A4%86%E0%A4%9A%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%AF%E0%A4%9C%E0%A5%80.jpg 2x" data-file-width="2182" data-file-height="2441" /></a><figcaption>Umaswati (2nd century AD), author of first Jain work in Sanskrit, <a href="/wiki/Tattvartha_Sutra" title="Tattvartha Sutra">Tattvārthasūtra</a>, expounding the <a href="/wiki/Jain_philosophy" title="Jain philosophy">Jain philosophy</a> in a most systematized form acceptable to all sects of Jainism.</figcaption></figure> <a href="/wiki/Jainism" title="Jainism">Jains</a> made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning the nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable. The Jains have doctrines of <a href="/wiki/Relativism" title="Relativism">relativity</a> used for logic and reasoning: <ul><li><a href="/wiki/Anekantavada" title="Anekantavada">Anekāntavāda</a> – the theory of relative pluralism or manifoldness;</li> <li><a href="/wiki/Syadvada" class="mw-redirect" title="Syadvada">Syādvāda</a> – the theory of conditioned predication and;</li> <li><a href="/w/index.php?title=Nayavada&action=edit&redlink=1" class="new" title="Nayavada (page does not exist)">Nayavāda</a> – The theory of partial standpoints.</li></ul> These <a href="/wiki/Jain_philosophy" title="Jain philosophy">Jain philosophical</a> concepts made most important contributions to the ancient <a href="/wiki/Indian_philosophy" title="Indian philosophy">Indian philosophy</a>, especially in the areas of skepticism and relativity. <a rel="nofollow" class="external autonumber" href="http://www.jainworld.com/jainbooks/firstep-2/indianjaina-1-2.htm">[4]</a><a href="#cite_note-14">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Buddhist_logic">Buddhist logic</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=11" title="Edit section: Buddhist logic">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Nagarjuna">Nagarjuna</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=12" title="Edit section: Nagarjuna">edit</a>]</div> <a href="/wiki/Nagarjuna" title="Nagarjuna">Nagarjuna</a> (c. 150–250 AD), the founder of the <a href="/wiki/Madhyamaka" title="Madhyamaka">Madhyamaka</a> ("Middle Way") developed an analysis known as the <a href="/wiki/Catu%E1%B9%A3ko%E1%B9%ADi" title="Catuṣkoṭi">catuṣkoṭi</a> (Sanskrit), a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the four possibilities of a proposition, P: <ol><li>P; that is, being.</li> <li>not P; that is, not being.</li> <li><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg/220px-Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg" decoding="async" width="220" height="256" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg/330px-Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg/440px-Eight_Patriarchs_of_the_Shingon_Sect_of_Buddhism_Nagarjuna_Cropped.jpg 2x" data-file-width="2059" data-file-height="2400" /></a><figcaption>Painting of Nāgārjuna from the Shingon Hassozō, a series of scrolls authored by the <a href="/wiki/Shingon" class="mw-redirect" title="Shingon">Shingon</a> school of Buddhism. Japan, <a href="/wiki/Kamakura_period" title="Kamakura period">Kamakura period</a> (13th–14th century)</figcaption></figure>P and not P; that is, being and not being.</li> <li>not (P or not P); that is, neither being nor not being.<div class="paragraphbreak" style="margin-top:0.5em"></div>Under <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>, <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a> would imply that this case is equivalent to the third case (P and not P), and would be therefore superfluous, with only 3 actual cases to consider.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Dignaga">Dignaga</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=13" title="Edit section: Dignaga">edit</a>]</div> However, <a href="/wiki/Dign%C4%81ga" title="Dignāga">Dignāga</a> (c 480–540 AD) is sometimes said to have developed a formal syllogism,<a href="#cite_note-15">[15]</a> and it was through him and his successor, <a href="/wiki/Dharmakirti" title="Dharmakirti">Dharmakirti</a>, that <a href="/wiki/Buddhist_logic" class="mw-redirect" title="Buddhist logic">Buddhist logic</a> reached its height; it is contested whether their analysis actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference-warranting relation, "<a href="/wiki/Vyapti" title="Vyapti">vyapti</a>", also known as invariable concomitance or pervasion.<a href="#cite_note-Matilal-16">[16]</a> To this end, a doctrine known as "apoha" or differentiation was developed.<a href="#cite_note-17">[17]</a> This involved what might be called inclusion and exclusion of defining properties. Dignāga's famous "wheel of reason" (<a href="/wiki/Hetucakra" title="Hetucakra">Hetucakra</a>) is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive.<a href="#cite_note-18">[18]</a> <div class="mw-heading mw-heading3"><h3 id="Logic_in_China">Logic in China</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=14" title="Edit section: Logic in China">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Logic_in_China" title="Logic in China">Logic in China</a></div> In China, a contemporary of <a href="/wiki/Confucius" title="Confucius">Confucius</a>, <a href="/wiki/Mozi" title="Mozi">Mozi</a>, "Master Mo", is credited with founding the <a href="/wiki/Mohism" title="Mohism">Mohist school</a>, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the <a href="/wiki/School_of_Names" title="School of Names">Logicians</a>, are credited by some scholars for their early investigation of <a href="/wiki/Formal_logic" class="mw-redirect" title="Formal logic">formal logic</a>. Due to the harsh rule of <a href="/wiki/Legalism_(Chinese_philosophy)" title="Legalism (Chinese philosophy)">Legalism</a> in the subsequent <a href="/wiki/Qin_dynasty" title="Qin dynasty">Qin dynasty</a>, this line of investigation disappeared in China until the introduction of Indian philosophy by <a href="/wiki/Buddhism" title="Buddhism">Buddhists</a>. <div class="mw-heading mw-heading2"><h2 id="Logic_in_the_West">Logic in the West</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=15" title="Edit section: Logic in the West">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Prehistory_of_logic">Prehistory of logic</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=16" title="Edit section: Prehistory of logic">edit</a>]</div> Valid reasoning has been employed in all periods of human history. However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with <a href="/wiki/Geometry" title="Geometry">geometry</a>, which originally meant the same as "land measurement".<a href="#cite_note-19">[19]</a> The <a href="/wiki/Ancient_Egypt" title="Ancient Egypt">ancient Egyptians</a> discovered <a href="/wiki/Egyptian_mathematics" class="mw-redirect" title="Egyptian mathematics">geometry</a>, including the formula for the volume of a <a href="/wiki/Frustum" title="Frustum">truncated pyramid</a>.<a href="#cite_note-Kneale3-20">[20]</a> <a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Ancient Babylon</a> was also skilled in mathematics. <a href="/wiki/Esagil-kin-apli" title="Esagil-kin-apli">Esagil-kin-apli</a>'s medical Diagnostic Handbook in the 11th century BC was based on a logical set of <a href="/wiki/Axiom" title="Axiom">axioms</a> and assumptions,<a href="#cite_note-Stol-99-21">[21]</a> while <a href="/wiki/Babylonian_astronomy" title="Babylonian astronomy">Babylonian astronomers</a> in the 8th and 7th centuries BC employed an <a href="/wiki/Internal_logic" class="mw-redirect" title="Internal logic">internal logic</a> within their predictive planetary systems, an important contribution to the <a href="/wiki/Philosophy_of_science" title="Philosophy of science">philosophy of science</a>.<a href="#cite_note-Brown-22">[22]</a> <div class="mw-heading mw-heading3"><h3 id="Ancient_Greece_before_Aristotle">Ancient Greece before Aristotle</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=17" title="Edit section: Ancient Greece before Aristotle">edit</a>]</div> While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a>. Both <a href="/wiki/Thales" class="mw-redirect" title="Thales">Thales</a> and <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a> of the <a href="/wiki/Pre-Socratic_philosophers" class="mw-redirect" title="Pre-Socratic philosophers">Pre-Socratic philosophers</a> seemed aware of geometric methods. Fragments of early proofs are preserved in the works of Plato and Aristotle,<a href="#cite_note-23">[23]</a> and the idea of a deductive system was probably known in the Pythagorean school and the <a href="/wiki/Platonic_Academy" title="Platonic Academy">Platonic Academy</a>.<a href="#cite_note-Kneale3-20">[20]</a> The proofs of <a href="/wiki/Euclid_of_Alexandria" class="mw-redirect" title="Euclid of Alexandria">Euclid of Alexandria</a> are a paradigm of Greek geometry. The three basic principles of geometry are as follows: <ul><li>Certain propositions must be accepted as true without demonstration; such a proposition is known as an <a href="/wiki/Axiom" title="Axiom">axiom</a> of geometry.</li> <li>Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> or a "derivation" of the proposition.</li> <li>The proof must be formal; that is, the derivation of the proposition must be independent of the particular subject matter in question.<a href="#cite_note-Kneale3-20">[20]</a></li></ul> Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called <a href="/wiki/Dissoi_logoi" title="Dissoi logoi">dissoi logoi</a>, probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity.<a href="#cite_note-24">[24]</a> In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the <a href="/wiki/Rhetoric" title="Rhetoric">Rhetoricians</a> or Orators and the <a href="/wiki/Sophists" class="mw-redirect" title="Sophists">Sophists</a>, who used arguments to defend or attack a thesis, both in legal and political contexts.<a href="#cite_note-25">[25]</a> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Thales%27_Theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Thales%27_Theorem.svg/130px-Thales%27_Theorem.svg.png" decoding="async" width="130" height="114" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Thales%27_Theorem.svg/195px-Thales%27_Theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Thales%27_Theorem.svg/260px-Thales%27_Theorem.svg.png 2x" data-file-width="200" data-file-height="175" /></a><figcaption>Thales Theorem</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Thales">Thales</h4>[<a href="/w/index.php?title=History_of_logic&action=edit&section=18" title="Edit section: Thales">edit</a>]</div> It is said Thales, most widely regarded as the first philosopher in the <a href="/wiki/Greek_philosophy" class="mw-redirect" title="Greek philosophy">Greek tradition</a>,<a href="#cite_note-26">[26]</a><a href="#cite_note-CPM-27">[27]</a> measured the height of the <a href="/wiki/Pyramids" class="mw-redirect" title="Pyramids">pyramids</a> by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering <a href="/wiki/Thales%27_theorem" class="mw-redirect" title="Thales' theorem">Thales' theorem</a> just as Pythagoras had the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>.<a href="#cite_note-28">[28]</a> Thales is the first known individual to use <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a> applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed.<a href="#cite_note-29">[29]</a> <a href="/wiki/Indian_mathematics" title="Indian mathematics">Indian</a> and Babylonian mathematicians knew his theorem for special cases before he proved it.<a href="#cite_note-30">[30]</a> It is believed that Thales learned that an angle inscribed in a <a href="/wiki/Semicircle" title="Semicircle">semicircle</a> is a right angle during his travels to <a href="/wiki/Babylon" title="Babylon">Babylon</a>.<a href="#cite_note-31">[31]</a> <div class="mw-heading mw-heading4"><h4 id="Pythagoras">Pythagoras</h4>[<a href="/w/index.php?title=History_of_logic&action=edit&section=19" title="Edit section: Pythagoras">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg/180px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg.png" decoding="async" width="180" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg/270px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg/360px-Illustration_to_Euclid%27s_proof_of_the_Pythagorean_theorem.svg.png 2x" data-file-width="500" data-file-height="540" /></a><figcaption>Proof of the Pythagorean Theorem in Euclid's Elements</figcaption></figure> Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales.<a href="#cite_note-32">[32]</a> The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC.<a href="#cite_note-Kneale3-20">[20]</a> Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize form rather than matter.<a href="#cite_note-33">[33]</a> <div class="mw-heading mw-heading4"><h4 id="Heraclitus_and_Parmenides">Heraclitus and Parmenides</h4>[<a href="/w/index.php?title=History_of_logic&action=edit&section=20" title="Edit section: Heraclitus and Parmenides">edit</a>]</div> The writing of <a href="/wiki/Heraclitus" title="Heraclitus">Heraclitus</a> (c. 535 – c. 475 BC) was the first place where the word <a href="/wiki/Logos" title="Logos">logos</a> was given special attention in ancient Greek philosophy,<a href="#cite_note-34">[34]</a> Heraclitus held that everything changes and all was fire and conflicting opposites, seemingly unified only by this Logos. He is known for his obscure sayings. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">This logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this logos, humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice what they do when awake, just as they forget what they do while asleep.<div class="templatequotecite">— <cite><a href="/wiki/Diels-Kranz" class="mw-redirect" title="Diels-Kranz">Diels-Kranz</a>, 22B1</cite></div></blockquote> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Busto_di_Parmenide_(cropped).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Busto_di_Parmenide_%28cropped%29.jpg/160px-Busto_di_Parmenide_%28cropped%29.jpg" decoding="async" width="160" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Busto_di_Parmenide_%28cropped%29.jpg/240px-Busto_di_Parmenide_%28cropped%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Busto_di_Parmenide_%28cropped%29.jpg/320px-Busto_di_Parmenide_%28cropped%29.jpg 2x" data-file-width="1410" data-file-height="1712" /></a><figcaption>Parmenides has been called the discoverer of logic.</figcaption></figure> In contrast to Heraclitus, <a href="/wiki/Parmenides" title="Parmenides">Parmenides</a> held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many.<a href="#cite_note-35">[35]</a> "X is not" must always be false or meaningless. What exists can in no way not exist. Our sense perceptions with its noticing of generation and destruction are in grievous error. Instead of sense perception, Parmenides advocated logos as the means to Truth. He has been called the discoverer of logic,<a href="#cite_note-36">[36]</a><a href="#cite_note-37">[37]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">For this view, that That Which Is Not exists, can never predominate. You must debar your thought from this way of search, nor let ordinary experience in its variety force you along this way, (namely, that of allowing) the eye, sightless as it is, and the ear, full of sound, and the tongue, to rule; but (you must) judge by means of the Reason (<a href="/wiki/Logos" title="Logos">Logos</a>) the much-contested proof which is expounded by me.<div class="templatequotecite">— <cite>B 7.1–8.2</cite></div></blockquote> <a href="/wiki/Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a>, a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">reductio ad absurdum</a>. This is the technique of drawing an obviously false (that is, "absurd") conclusion from an assumption, thus demonstrating that the assumption is false.<a href="#cite_note-38">[38]</a> Therefore, Zeno and his teacher are seen as the first to apply the art of logic.<a href="#cite_note-39">[39]</a> Plato's dialogue <a href="/wiki/Parmenides_(dialogue)" title="Parmenides (dialogue)">Parmenides</a> portrays Zeno as claiming to have written a book defending the <a href="/wiki/Monism" title="Monism">monism</a> of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used this method to develop his <a href="/wiki/Zeno%27s_Paradoxes" class="mw-redirect" title="Zeno's Paradoxes">paradoxes</a> in his arguments against motion. Such dialectic reasoning later became popular. The members of this school were called "dialecticians" (from a Greek word meaning "to discuss"). <div class="mw-heading mw-heading4"><h4 id="Plato">Plato</h4>[<a href="/w/index.php?title=History_of_logic&action=edit&section=21" title="Edit section: Plato">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Let no one ignorant of geometry enter here.<div class="templatequotecite">— <cite>Inscribed over the entrance to Plato's Academy.</cite></div></blockquote> <figure typeof="mw:File/Thumb"><a href="/wiki/File:MANNapoli_124545_plato%27s_academy_mosaic.jpg" class="mw-file-description"><img alt="Mosaic: seven men standing under a tree" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/MANNapoli_124545_plato%27s_academy_mosaic.jpg/200px-MANNapoli_124545_plato%27s_academy_mosaic.jpg" decoding="async" width="200" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/MANNapoli_124545_plato%27s_academy_mosaic.jpg/300px-MANNapoli_124545_plato%27s_academy_mosaic.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/MANNapoli_124545_plato%27s_academy_mosaic.jpg/400px-MANNapoli_124545_plato%27s_academy_mosaic.jpg 2x" data-file-width="2392" data-file-height="2441" /></a><figcaption><a href="/wiki/Plato%27s_Academy_mosaic" title="Plato's Academy mosaic">Plato's Academy mosaic</a></figcaption></figure> None of the surviving works of the great fourth-century philosopher <a href="/wiki/Plato" title="Plato">Plato</a> (428–347 BC) include any formal logic,<a href="#cite_note-40">[40]</a> but they include important contributions to the field of <a href="/wiki/Philosophical_logic" title="Philosophical logic">philosophical logic</a>. Plato raises three questions: <ul><li>What is it that can properly be called true or false?</li> <li>What is the nature of the connection between the assumptions of a valid argument and its conclusion?</li> <li>What is the nature of definition?</li></ul> The first question arises in the dialogue <a href="/wiki/Theaetetus_(dialogue)" title="Theaetetus (dialogue)">Theaetetus</a>, where Plato identifies thought or opinion with talk or discourse (logos).<a href="#cite_note-41">[41]</a> The second question is a result of Plato's <a href="/wiki/Theory_of_Forms" class="mw-redirect" title="Theory of Forms">theory of Forms</a>. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called <a href="/wiki/Universals" class="mw-redirect" title="Universals">universals</a>, namely an abstract entity common to each set of things that have the same name. In both the <a href="/wiki/The_Republic_(Plato)" class="mw-redirect" title="The Republic (Plato)">Republic</a> and the <a href="/wiki/Sophist_(dialogue)" title="Sophist (dialogue)">Sophist</a>, Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between "forms".<a href="#cite_note-42">[42]</a> The third question is about <a href="/wiki/Definition" title="Definition">definition</a>. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics.<a href="#cite_note-43">[43]</a> What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, in particular Aristotle's notion of the <a href="/wiki/Essence" title="Essence">essence</a> of a thing.<a href="#cite_note-44">[44]</a> <div class="mw-heading mw-heading3"><h3 id="Aristotle">Aristotle</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=22" title="Edit section: Aristotle">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Term_logic" title="Term logic">Term logic</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Aristotle_Altemps_Inv8575.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/160px-Aristotle_Altemps_Inv8575.jpg" decoding="async" width="160" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/240px-Aristotle_Altemps_Inv8575.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/320px-Aristotle_Altemps_Inv8575.jpg 2x" data-file-width="1700" data-file-height="2275" /></a><figcaption>Aristotle</figcaption></figure> The logic of <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, and particularly his theory of the <a href="/wiki/Syllogism" title="Syllogism">syllogism</a>, has had an enormous influence in <a href="/wiki/Western_thought" class="mw-redirect" title="Western thought">Western thought</a>.<a href="#cite_note-45">[45]</a> Aristotle was the first logician to attempt a systematic analysis of <a href="/wiki/Logical_syntax" class="mw-redirect" title="Logical syntax">logical syntax</a>, of noun (or <a href="/wiki/Terminology" title="Terminology">term</a>), and of verb. He was the first formal logician, in that he demonstrated the principles of reasoning by employing variables to show the underlying <a href="/wiki/Logical_form" title="Logical form">logical form</a> of an argument.<a href="#cite_note-46">[46]</a> He sought relations of dependence which characterize necessary inference, and distinguished the <a href="/wiki/Validity_(logic)" title="Validity (logic)">validity</a> of these relations, from the truth of the premises. He was the first to deal with the principles of <a href="/wiki/Principle_of_contradiction" class="mw-redirect" title="Principle of contradiction">contradiction</a> and <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">excluded middle</a> in a systematic way.<a href="#cite_note-Bochenski_p._63-47">[47]</a> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Aristoteles_Logica_1570_Biblioteca_Huelva.jpg" class="mw-file-description"><img alt="Front cover of book, titled "Aristotelis Logica", with an illustration of eagle on a snake" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Aristoteles_Logica_1570_Biblioteca_Huelva.jpg/240px-Aristoteles_Logica_1570_Biblioteca_Huelva.jpg" decoding="async" width="240" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Aristoteles_Logica_1570_Biblioteca_Huelva.jpg/360px-Aristoteles_Logica_1570_Biblioteca_Huelva.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/Aristoteles_Logica_1570_Biblioteca_Huelva.jpg/480px-Aristoteles_Logica_1570_Biblioteca_Huelva.jpg 2x" data-file-width="3191" data-file-height="2311" /></a><figcaption>Aristotle's logic was still influential in the <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a>.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="The_Organon">The Organon</h4>[<a href="/w/index.php?title=History_of_logic&action=edit&section=23" title="Edit section: The Organon">edit</a>]</div> His logical works, called the <a href="/wiki/Organon" title="Organon">Organon</a>, are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is: <ul><li><a href="/wiki/Categories_(Aristotle)" title="Categories (Aristotle)">The Categories</a>, a study of the ten kinds of primitive term.</li> <li><a href="/wiki/Topics_(Aristotle)" title="Topics (Aristotle)">The Topics</a> (with an appendix called <a href="/wiki/On_Sophistical_Refutations" class="mw-redirect" title="On Sophistical Refutations">On Sophistical Refutations</a>), a discussion of dialectics.</li> <li><a href="/wiki/De_Interpretatione" class="mw-redirect" title="De Interpretatione">On Interpretation</a>, an analysis of simple <a href="/wiki/Categorical_proposition" title="Categorical proposition">categorical propositions</a> into simple terms, negation, and signs of quantity.</li> <li><a href="/wiki/Prior_Analytics" title="Prior Analytics">The Prior Analytics</a>, a formal analysis of what makes a <a href="/wiki/Syllogism" title="Syllogism">syllogism</a> (a valid argument, according to Aristotle).</li> <li><a href="/wiki/Posterior_Analytics" title="Posterior Analytics">The Posterior Analytics</a>, a study of scientific demonstration, containing Aristotle's mature views on logic.</li></ul> <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/180px-Square_of_opposition%2C_set_diagrams.svg.png" decoding="async" width="180" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Square_of_opposition%2C_set_diagrams.svg/270px-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/360px-Square_of_opposition%2C_set_diagrams.svg.png 2x" data-file-width="479" data-file-height="620" /></a><figcaption>This diagram shows the contradictory relationships between <a href="/wiki/Categorical_proposition" title="Categorical proposition">categorical propositions</a> in the <a href="/wiki/Square_of_opposition" title="Square of opposition">square of opposition</a> of <a href="/wiki/Term_logic" title="Term logic">Aristotelian logic</a>.</figcaption></figure> These works are of outstanding importance in the history of logic. In the Categories, he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical work <a href="/wiki/Metaphysics_(Aristotle)" title="Metaphysics (Aristotle)">Metaphysics</a>, which itself had a profound influence on Western thought. He also developed a theory of non-formal logic (i.e., the theory of <a href="/wiki/Logical_fallacy" class="mw-redirect" title="Logical fallacy">fallacies</a>), which is presented in Topics and Sophistical Refutations.<a href="#cite_note-Bochenski_p._63-47">[47]</a> On Interpretation contains a comprehensive treatment of the notions of <a href="/wiki/Square_of_opposition" title="Square of opposition">opposition</a> and conversion; chapter 7 is at the origin of the <a href="/wiki/Square_of_opposition" title="Square of opposition">square of opposition</a> (or logical square); chapter 9 contains the beginning of <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>. The Prior Analytics contains his exposition of the "syllogism", where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system. <div class="mw-heading mw-heading3"><h3 id="Stoics">Stoics</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=24" title="Edit section: Stoics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stoic_logic" title="Stoic logic">Stoic logic</a></div> The other great school of Greek logic is that of the <a href="/wiki/Stoicism" title="Stoicism">Stoics</a>.<a href="#cite_note-48">[48]</a> Stoic logic traces its roots back to the late 5th century BC philosopher <a href="/wiki/Euclid_of_Megara" title="Euclid of Megara">Euclid of Megara</a>, a pupil of <a href="/wiki/Socrates" title="Socrates">Socrates</a> and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "<a href="/wiki/Megarian_school" title="Megarian school">Megarians</a>", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were <a href="/wiki/Diodorus_Cronus" title="Diodorus Cronus">Diodorus Cronus</a> and <a href="/wiki/Philo_the_Dialectician" title="Philo the Dialectician">Philo</a>, who were active in the late 4th century BC. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Chrysippos_BM_1846.jpg" class="mw-file-description"><img alt="Stone bust of a bearded, grave-looking man" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Chrysippos_BM_1846.jpg/160px-Chrysippos_BM_1846.jpg" decoding="async" width="160" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Chrysippos_BM_1846.jpg/240px-Chrysippos_BM_1846.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Chrysippos_BM_1846.jpg/320px-Chrysippos_BM_1846.jpg 2x" data-file-width="2640" data-file-height="3300" /></a><figcaption><a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a> of Soli</figcaption></figure> The Stoics adopted the Megarian logic and systemized it. The most important member of the school was <a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a> (c. 278 – c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive.<a href="#cite_note-49">[49]</a><a href="#cite_note-50">[50]</a> Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently <a href="/wiki/Diogenes_La%C3%ABrtius" class="mw-redirect" title="Diogenes Laërtius">Diogenes Laërtius</a>, <a href="/wiki/Sextus_Empiricus" title="Sextus Empiricus">Sextus Empiricus</a>, <a href="/wiki/Galen" title="Galen">Galen</a>, <a href="/wiki/Aulus_Gellius" title="Aulus Gellius">Aulus Gellius</a>, <a href="/wiki/Alexander_of_Aphrodisias" title="Alexander of Aphrodisias">Alexander of Aphrodisias</a>, and <a href="/wiki/Cicero" title="Cicero">Cicero</a>.<a href="#cite_note-51">[51]</a> Three significant contributions of the Stoic school were (i) their account of <a href="/wiki/Modal_logic" title="Modal logic">modality</a>, (ii) their theory of the <a href="/wiki/Material_conditional" title="Material conditional">Material conditional</a>, and (iii) their account of <a href="/wiki/Meaning_(philosophy_of_language)" class="mw-redirect" title="Meaning (philosophy of language)">meaning</a> and <a href="/wiki/Truth" title="Truth">truth</a>.<a href="#cite_note-52">[52]</a> <ul><li>Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between <a href="/wiki/Potentiality_and_actuality_(Aristotle)" class="mw-redirect" title="Potentiality and actuality (Aristotle)">potentiality and actuality</a>.<a href="#cite_note-53">[53]</a> Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false.<a href="#cite_note-54">[54]</a> Diodorus is also famous for what is known as his <a href="/wiki/Master_argument_(Diodorus_Cronus)" class="mw-redirect" title="Master argument (Diodorus Cronus)">Master argument</a>, which states that each pair of the following 3 propositions contradicts the third proposition:</li></ul> <dl><dd><ul><li>Everything that is past is true and necessary.</li> <li>The impossible does not follow from the possible.</li> <li>What neither is nor will be is possible.</li></ul></dd> <dd>Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true.<a href="#cite_note-55">[55]</a> Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible.<a href="#cite_note-56">[56]</a></dd></dl> <ul><li>Conditional statements. The first logicians to debate <a href="/wiki/Material_conditional" title="Material conditional">conditional statements</a> were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true <a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">antecedent</a> and a false <a href="/wiki/Consequent" title="Consequent">consequent</a>. Precisely, let T0 and T1 be true statements, and let F0 and F1 be false statements; then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the case that a false statement is asserted to follow from a true statement):</li></ul> <dl><dd><ul><li>If T0, then T1</li> <li>If F0, then T0</li> <li>If F0, then F1</li></ul></dd> <dd>The following conditional does not meet this requirement, and is therefore a false statement according to Philo: <ul><li>If T0, then F0</li></ul></dd> <dd>Indeed, Sextus says "According to [Philo], there are three ways in which a conditional may be true, and one in which it may be false."<a href="#cite_note-sextus-adv-math-57">[57]</a> Philo's criterion of truth is what would now be called a <a href="/wiki/Truth-functional" class="mw-redirect" title="Truth-functional">truth-functional</a> definition of "if ... then"; it is the definition used in <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">modern logic</a>.</dd></dl> <dl><dd>In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion.<a href="#cite_note-sextus-adv-math-57">[57]</a><a href="#cite_note-58">[58]</a><a href="#cite_note-59">[59]</a> A century later, the <a href="/wiki/Stoicism" title="Stoicism">Stoic</a> philosopher <a href="/wiki/Chrysippus" title="Chrysippus">Chrysippus</a> attacked the assumptions of both Philo and Diodorus.</dd></dl> <ul><li>Meaning and truth. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer to modern <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>.<a href="#cite_note-60">[60]</a> The Stoics distinguished between utterance (phone), which may be noise, speech (lexis), which is articulate but which may be meaningless, and discourse (logos), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a lekton, is something real; this corresponds to what is now called a proposition. Sextus says that according to the Stoics, three things are linked together: that which signifies, that which is signified, and the object; for example, that which signifies is the word Dion, and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself.<a href="#cite_note-61">[61]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Medieval_logic">Medieval logic</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=25" title="Edit section: Medieval logic">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Logic_in_the_Middle_East">Logic in the Middle East</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=26" title="Edit section: Logic in the Middle East">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Logic_in_Islamic_philosophy" title="Logic in Islamic philosophy">Logic in Islamic philosophy</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Avicennism#Avicennian_logic" title="Avicennism">Avicennian logic</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Canon-Avicenna-small.jpg" class="mw-file-description"><img alt="Arabic text in pink and blue" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Canon-Avicenna-small.jpg/220px-Canon-Avicenna-small.jpg" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/81/Canon-Avicenna-small.jpg 1.5x" data-file-width="229" data-file-height="228" /></a><figcaption>A text by <a href="/wiki/Avicenna" title="Avicenna">Avicenna</a>, founder of <a href="/wiki/Avicennism#Avicennian_logic" title="Avicennism">Avicennian logic</a> </figcaption></figure> The works of <a href="/wiki/Al-Kindi" title="Al-Kindi">Al-Kindi</a>, <a href="/wiki/Al-Farabi" title="Al-Farabi">Al-Farabi</a>, <a href="/wiki/Avicenna" title="Avicenna">Avicenna</a>, <a href="/wiki/Al-Ghazali" title="Al-Ghazali">Al-Ghazali</a>, <a href="/wiki/Averroes" title="Averroes">Averroes</a> and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West.<a href="#cite_note-62">[62]</a> <a href="/wiki/Al-Farabi" title="Al-Farabi">Al-Farabi</a> (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of <a href="/wiki/Future_contingent" class="mw-redirect" title="Future contingent">future contingents</a>, the number and relation of the categories, the relation between <a href="/wiki/Logic" title="Logic">logic</a> and <a href="/wiki/Grammar" title="Grammar">grammar</a>, and non-Aristotelian forms of <a href="/wiki/Inference" title="Inference">inference</a>.<a href="#cite_note-Britannica-63">[63]</a> Al-Farabi also considered the theories of <a href="/wiki/Conditional_syllogism" class="mw-redirect" title="Conditional syllogism">conditional syllogisms</a> and <a href="/wiki/Analogy" title="Analogy">analogical inference</a>, which were part of the <a href="/wiki/Stoicism" title="Stoicism">Stoic</a> tradition of logic rather than the Aristotelian.<a href="#cite_note-64">[64]</a> <a href="/wiki/Maimonides" title="Maimonides">Maimonides</a> (1138-1204) wrote a Treatise on Logic (Arabic: Maqala Fi-Sinat Al-Mantiq), referring to Al-Farabi as the "second master", the first being Aristotle. <a href="/wiki/Avicenna" title="Avicenna">Ibn Sina</a> (Avicenna) (980–1037) was the founder of <a href="/wiki/Avicennian_logic" class="mw-redirect" title="Avicennian logic">Avicennian logic</a>, which replaced Aristotelian logic as the dominant system of logic in the Islamic world,<a href="#cite_note-Hasse-65">[65]</a> and also had an important influence on Western medieval writers such as <a href="/wiki/Albertus_Magnus" title="Albertus Magnus">Albertus Magnus</a>.<a href="#cite_note-66">[66]</a> Avicenna wrote on the <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a><a href="#cite_note-Goodman-67">[67]</a> and on the <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a>, which were both part of the Stoic logical tradition.<a href="#cite_note-68">[68]</a> He developed an original "temporally modalized" syllogistic theory, involving <a href="/wiki/Temporal_logic" title="Temporal logic">temporal logic</a> and <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>.<a href="#cite_note-Britannica-63">[63]</a> He also made use of <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive logic</a>, such as the <a href="/wiki/Mill%27s_Methods" title="Mill's Methods">methods of agreement, difference, and concomitant variation</a> which are critical to the <a href="/wiki/Scientific_method" title="Scientific method">scientific method</a>.<a href="#cite_note-Goodman-67">[67]</a> One of Avicenna's ideas had a particularly important influence on Western logicians such as <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a>: Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio; in medieval logic and <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>, this is a sign in the mind that naturally represents a thing.<a href="#cite_note-69">[69]</a> This was crucial to the development of Ockham's <a href="/wiki/Conceptualism" title="Conceptualism">conceptualism</a>: A universal term (e.g., "man") does not signify a thing existing in reality, but rather a sign in the mind (intentio in intellectu) which represents many things in reality; Ockham cites Avicenna's commentary on Metaphysics V in support of this view.<a href="#cite_note-70">[70]</a> <a href="/wiki/Fakhr_al-Din_al-Razi" title="Fakhr al-Din al-Razi">Fakhr al-Din al-Razi</a> (b. 1149) criticised Aristotle's "<a href="/wiki/Syllogism" title="Syllogism">first figure</a>" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by <a href="/wiki/John_Stuart_Mill" title="John Stuart Mill">John Stuart Mill</a> (1806–1873).<a href="#cite_note-Iqbal-71">[71]</a> Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a <a href="/wiki/Logic_in_Islamic_philosophy#Post-Avicennian_logic" title="Logic in Islamic philosophy">Post-Avicennian logic</a>. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of <a href="/wiki/Concept" title="Concept">conceptions</a> and <a href="/wiki/Grammar_of_Assent" title="Grammar of Assent">assents</a>. In response to this tradition, <a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">Nasir al-Din al-Tusi</a> (1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.<a href="#cite_note-Stanford-72">[72]</a> The <a href="/wiki/Illuminationist_philosophy" class="mw-redirect" title="Illuminationist philosophy">Illuminationist school</a> was founded by <a href="/wiki/Shahab_al-Din_Suhrawardi" class="mw-redirect" title="Shahab al-Din Suhrawardi">Shahab al-Din Suhrawardi</a> (1155–1191), who developed the idea of "decisive necessity", which refers to the reduction of all modalities (necessity, <a href="/wiki/Logical_possibility" title="Logical possibility">possibility</a>, <a href="/wiki/Contingency_(philosophy)" title="Contingency (philosophy)">contingency</a> and <a href="/wiki/Epistemic_possibility" title="Epistemic possibility">impossibility</a>) to the single mode of necessity.<a href="#cite_note-73">[73]</a> <a href="/wiki/Ibn_al-Nafis" title="Ibn al-Nafis">Ibn al-Nafis</a> (1213–1288) wrote a book on Avicennian logic, which was a commentary of Avicenna's Al-Isharat (The Signs) and Al-Hidayah (The Guidance).<a href="#cite_note-Roubi-74">[74]</a> <a href="/wiki/Ibn_Taymiyyah" class="mw-redirect" title="Ibn Taymiyyah">Ibn Taymiyyah</a> (1263–1328), wrote the Ar-Radd 'ala al-Mantiqiyyin, where he argued against the usefulness, though not the validity, of the <a href="/wiki/Syllogism" title="Syllogism">syllogism</a><a href="#cite_note-75">[75]</a> and in favour of <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a>.<a href="#cite_note-Iqbal-71">[71]</a> Ibn Taymiyyah also argued against the certainty of <a href="/wiki/Syllogism" title="Syllogism">syllogistic arguments</a> and in favour of <a href="/wiki/Analogy" title="Analogy">analogy</a>; his argument is that concepts founded on <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">induction</a> are themselves not certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments.<a href="#cite_note-76">[76]</a><a href="#cite_note-Sowa-77">[77]</a> This model of analogy has been used in the recent work of <a href="/wiki/John_F._Sowa" title="John F. Sowa">John F. Sowa</a>.<a href="#cite_note-Sowa-77">[77]</a> The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied.<a href="#cite_note-78">[78]</a> However, "thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period.<a href="#cite_note-Stanford-72">[72]</a> <div class="mw-heading mw-heading3"><h3 id="Logic_in_medieval_Europe">Logic in medieval Europe</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=27" title="Edit section: Logic in medieval Europe">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Britoquestionsonoldlogic.jpg" class="mw-file-description"><img alt="Top left corner of early printed text, with an illuminated S, beginning "Sicut dicit philosophus"" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Britoquestionsonoldlogic.jpg/220px-Britoquestionsonoldlogic.jpg" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Britoquestionsonoldlogic.jpg/330px-Britoquestionsonoldlogic.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/3/37/Britoquestionsonoldlogic.jpg 2x" data-file-width="390" data-file-height="305" /></a><figcaption><a href="/wiki/Radulphus_Brito" title="Radulphus Brito">Brito's</a> questions on the Old Logic</figcaption></figure> "Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in <a href="/wiki/Middle_Ages" title="Middle Ages">medieval Europe</a> throughout roughly the period 1200–1600.<a href="#cite_note-Boehner_p._xiv-1">[1]</a> For centuries after Stoic logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the <a href="/wiki/Dark_Ages_(historiography)" title="Dark Ages (historiography)">Dark Ages</a>, the main source was the work of the Christian philosopher <a href="/wiki/Boethius" title="Boethius">Boethius</a>, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics.<a href="#cite_note-Kneale198-79">[79]</a> Until the twelfth century, the only works of Aristotle available in the West were the Categories, On Interpretation, and Boethius's translation of the <a href="/wiki/Isagoge" title="Isagoge">Isagoge</a> of <a href="/wiki/Porphyry_(philosopher)" title="Porphyry (philosopher)">Porphyry</a> (a commentary on the Categories). These works were known as the "Old Logic" (Logica Vetus or Ars Vetus). An important work in this tradition was the Logica Ingredientibus of <a href="/wiki/Peter_Abelard" title="Peter Abelard">Peter Abelard</a> (1079–1142). His direct influence was small,<a href="#cite_note-80">[80]</a> but his influence through pupils such as <a href="/wiki/John_of_Salisbury" title="John of Salisbury">John of Salisbury</a> was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.<a href="#cite_note-81">[81]</a> The proof for the <a href="/wiki/Principle_of_explosion" title="Principle of explosion">principle of explosion</a>, also known as the principle of Pseudo-Scotus, the law according to which any proposition can be proven from a contradiction (including its negation), was first given by the 12th century French logician <a href="/wiki/William_of_Soissons" title="William of Soissons">William of Soissons</a>. By the early thirteenth century, the remaining works of Aristotle's Organon, including the <a href="/wiki/Prior_Analytics" title="Prior Analytics">Prior Analytics</a>, <a href="/wiki/Posterior_Analytics" title="Posterior Analytics">Posterior Analytics</a>, and the <a href="/wiki/Sophistical_Refutations" title="Sophistical Refutations">Sophistical Refutations</a> (collectively known as the <a href="/wiki/Logica_Nova" class="mw-redirect" title="Logica Nova">Logica Nova</a> or "New Logic"), had been recovered in the West.<a href="#cite_note-82">[82]</a> Logical work until then was mostly paraphrasis or commentary on the work of Aristotle.<a href="#cite_note-83">[83]</a> The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were:<a href="#cite_note-84">[84]</a> <ul><li>The theory of <a href="/wiki/Supposition_theory" title="Supposition theory">supposition</a>. Supposition theory deals with the way that predicates (e.g., 'man') range over a domain of individuals (e.g., all men).<a href="#cite_note-85">[85]</a> In the proposition 'every man is an animal', does the term 'man' range over or 'supposit for' men existing just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual? Some medievalists have argued that this idea is a precursor of modern <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>.<a href="#cite_note-86">[86]</a> "The theory of supposition with the associated theories of copulatio (sign-capacity of adjectival terms), ampliatio (widening of referential domain), and distributio constitute one of the most original achievements of Western medieval logic".<a href="#cite_note-87">[87]</a></li> <li>The theory of <a href="/wiki/Syncategorematic_term" title="Syncategorematic term">syncategoremata</a>. Syncategoremata are terms which are necessary for logic, but which, unlike categorematic terms, do not signify on their own behalf, but 'co-signify' with other words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on.</li> <li>The theory of <a href="/wiki/Logical_consequence" title="Logical consequence">consequences</a>. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if ... then'. For example, 'if a man runs, then God exists' (Si homo currit, Deus est).<a href="#cite_note-88">[88]</a> A fully developed theory of consequences is given in Book III of <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a>'s work <a href="/wiki/Summa_Logicae" class="mw-redirect" title="Summa Logicae">Summa Logicae</a>. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern <a href="/wiki/Material_conditional" title="Material conditional">material implication</a> and <a href="/wiki/Logical_implication" class="mw-redirect" title="Logical implication">logical implication</a> respectively. Similar accounts are given by <a href="/wiki/Jean_Buridan" title="Jean Buridan">Jean Buridan</a> and <a href="/wiki/Albert_of_Saxony_(philosopher)" title="Albert of Saxony (philosopher)">Albert of Saxony</a>.</li></ul> The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as <a href="/wiki/John_of_St_Thomas" class="mw-redirect" title="John of St Thomas">John of St Thomas</a>), the Metaphysical Disputations of <a href="/wiki/Francisco_Suarez" class="mw-redirect" title="Francisco Suarez">Francisco Suarez</a> (1548–1617), and the Logica Demonstrativa of <a href="/wiki/Giovanni_Girolamo_Saccheri" title="Giovanni Girolamo Saccheri">Giovanni Girolamo Saccheri</a> (1667–1733). <div class="mw-heading mw-heading2"><h2 id="Traditional_logic">Traditional logic</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=28" title="Edit section: Traditional logic">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="The_textbook_tradition">The textbook tradition</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=29" title="Edit section: The textbook tradition">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fennerartoflogic-small.jpg" class="mw-file-description"><img alt="Frontispiece, with title beginning "The Artes of Logike and Rethorike, plainlie set foorth in the English tounge, easie to be learned and practised"." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Fennerartoflogic-small.jpg/220px-Fennerartoflogic-small.jpg" decoding="async" width="220" height="314" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Fennerartoflogic-small.jpg/330px-Fennerartoflogic-small.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Fennerartoflogic-small.jpg/440px-Fennerartoflogic-small.jpg 2x" data-file-width="549" data-file-height="783" /></a><figcaption><a href="/wiki/Dudley_Fenner" title="Dudley Fenner">Dudley Fenner</a>'s Art of Logic (1584)</figcaption></figure> Traditional logic generally means the textbook tradition that begins with <a href="/wiki/Antoine_Arnauld" title="Antoine Arnauld">Antoine Arnauld</a>'s and <a href="/wiki/Pierre_Nicole" title="Pierre Nicole">Pierre Nicole</a>'s Logic, or the Art of Thinking, better known as the <a href="/wiki/Port-Royal_Logic" title="Port-Royal Logic">Port-Royal Logic</a>.<a href="#cite_note-89">[89]</a> Published in 1662, it was the most influential work on logic after Aristotle until the nineteenth century.<a href="#cite_note-Buroker_xxiii-90">[90]</a> The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval <a href="/wiki/Term_logic" title="Term logic">term logic</a>. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that.<a href="#cite_note-Buroker_xxiii-90">[90]</a> The Port-Royal introduces the concepts of <a href="/wiki/Extension_(semantics)" title="Extension (semantics)">extension</a> and <a href="/wiki/Intension" title="Intension">intension</a>. The account of <a href="/wiki/Proposition" title="Proposition">propositions</a> that <a href="/wiki/John_Locke" title="John Locke">Locke</a> gives in the Essay is essentially that of the Port-Royal: "Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree."<a href="#cite_note-91">[91]</a> <a href="/wiki/Dudley_Fenner" title="Dudley Fenner">Dudley Fenner</a> helped popularize <a href="/wiki/Ramist" class="mw-redirect" title="Ramist">Ramist</a> logic, a reaction against Aristotle. Another influential work was the <a href="/wiki/Novum_Organum" title="Novum Organum">Novum Organum</a> by <a href="/wiki/Francis_Bacon" title="Francis Bacon">Francis Bacon</a>, published in 1620. The title translates as "new instrument". This is a reference to <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>'s work known as the <a href="/wiki/Organon" title="Organon">Organon</a>. In this work, Bacon rejects the syllogistic method of Aristotle in favor of an alternative procedure "which by slow and faithful toil gathers information from things and brings it into understanding".<a href="#cite_note-92">[92]</a> This method is known as <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a>, a method which starts from empirical observation and proceeds to lower axioms or propositions; from these lower axioms, more general ones can be induced. For example, in finding the cause of a phenomenal nature such as heat, three lists should be constructed: <ul><li>The presence list: a list of every situation where heat is found.</li> <li>The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat.</li> <li>The variability list: a list of every situation where heat can vary.</li></ul> Then, the form nature (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which varies by degree in every situation of the variability list. Other works in the textbook tradition include <a href="/wiki/Isaac_Watts" title="Isaac Watts">Isaac Watts</a>'s Logick: Or, the Right Use of Reason (1725), <a href="/wiki/Richard_Whately" title="Richard Whately">Richard Whately</a>'s Logic (1826), and <a href="/wiki/John_Stuart_Mill" title="John Stuart Mill">John Stuart Mill</a>'s A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection<a href="#cite_note-93">[93]</a> influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany.<a href="#cite_note-94">[94]</a> <div class="mw-heading mw-heading3"><h3 id="Logic_in_Hegel's_philosophy">Logic in Hegel's philosophy</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=30" title="Edit section: Logic in Hegel's philosophy">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:G.W.F._Hegel_(by_Sichling,_after_Sebbers).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/G.W.F._Hegel_%28by_Sichling%2C_after_Sebbers%29.jpg/220px-G.W.F._Hegel_%28by_Sichling%2C_after_Sebbers%29.jpg" decoding="async" width="220" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/G.W.F._Hegel_%28by_Sichling%2C_after_Sebbers%29.jpg/330px-G.W.F._Hegel_%28by_Sichling%2C_after_Sebbers%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/G.W.F._Hegel_%28by_Sichling%2C_after_Sebbers%29.jpg/440px-G.W.F._Hegel_%28by_Sichling%2C_after_Sebbers%29.jpg 2x" data-file-width="500" data-file-height="573" /></a><figcaption>Georg Wilhelm Friedrich Hegel</figcaption></figure> <a href="/wiki/G.W.F._Hegel" class="mw-redirect" title="G.W.F. Hegel">G.W.F. Hegel</a> indicated the importance of logic to his philosophical system when he condensed his extensive <a href="/wiki/Science_of_Logic" title="Science of Logic">Science of Logic</a> into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences. The "Shorter" or "Encyclopaedia" Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories—Hegel begins with "Pure Being" and "Pure Nothing"—to the "<a href="/wiki/Absolute_(philosophy)" title="Absolute (philosophy)">Absolute</a>", the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity"); this compulsion is, supposedly, not a matter of individual psychology, because it arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the "Absolute"—indeed of rationality itself. The method by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian <a href="/wiki/Dialectic" title="Dialectic">dialectic</a>. Although Hegel's Logic has had little impact on mainstream logical studies, its influence can be seen elsewhere: <ul><li><a href="/wiki/Carl_von_Prantl" class="mw-redirect" title="Carl von Prantl">Carl von Prantl</a>'s Geschichte der Logik im Abendland (1855–1867).<a href="#cite_note-95">[95]</a></li> <li>The work of the <a href="/wiki/British_Idealism" class="mw-redirect" title="British Idealism">British Idealists</a>, such as F. H. Bradley's Principles of Logic (1883).</li> <li>The economic, political, and philosophical studies of <a href="/wiki/Karl_Marx" title="Karl Marx">Karl Marx</a>, and in the various schools of <a href="/wiki/Marxism" title="Marxism">Marxism</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Logic_and_psychology">Logic and psychology</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=31" title="Edit section: Logic and psychology">edit</a>]</div> Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of <a href="/wiki/Psychology" title="Psychology">psychology</a>.<a href="#cite_note-96">[96]</a> The German psychologist <a href="/wiki/Wilhelm_Wundt" title="Wilhelm Wundt">Wilhelm Wundt</a>, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking."<a href="#cite_note-97">[97]</a> This view was widespread among German philosophers of the period: <ul><li><a href="/wiki/Theodor_Lipps" title="Theodor Lipps">Theodor Lipps</a> described logic as "a specific discipline of psychology".<a href="#cite_note-98">[98]</a></li> <li><a href="/wiki/Christoph_von_Sigwart" title="Christoph von Sigwart">Christoph von Sigwart</a> understood logical necessity as grounded in the individual's compulsion to think in a certain way.<a href="#cite_note-99">[99]</a></li> <li><a href="/wiki/Benno_Erdmann" title="Benno Erdmann">Benno Erdmann</a> argued that "logical laws only hold within the limits of our thinking".<a href="#cite_note-100">[100]</a></li></ul> Such was the dominant view of logic in the years following Mill's work.<a href="#cite_note-101">[101]</a> This psychological approach to logic was rejected by <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>. It was also subjected to an extended and destructive critique by <a href="/wiki/Edmund_Husserl" title="Edmund Husserl">Edmund Husserl</a> in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming".<a href="#cite_note-102">[102]</a> Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that <a href="/wiki/Skepticism" title="Skepticism">skepticism</a> and <a href="/wiki/Relativism" title="Relativism">relativism</a> were unavoidable consequences. Such criticisms did not immediately extirpate what is called "<a href="/wiki/Psychologism" title="Psychologism">psychologism</a>". For example, the American philosopher <a href="/wiki/Josiah_Royce" title="Josiah Royce">Josiah Royce</a>, while acknowledging the force of Husserl's critique, remained "unable to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa.<a href="#cite_note-103">[103]</a> <div class="mw-heading mw-heading2"><h2 id="Rise_of_modern_logic">Rise of modern logic</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=32" title="Edit section: Rise of modern logic">edit</a>]</div> The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic.<a href="#cite_note-ReferenceA-2">[2]</a> The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. The development of the modern "symbolic" or "mathematical" logic during this period is the most significant in the 2000-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.<a href="#cite_note-Oxford_Companion_p._500-4">[4]</a> A number of features distinguish modern logic from the old Aristotelian or traditional logic, the most important of which are as follows:<a href="#cite_note-104">[104]</a> Modern logic is fundamentally a calculus whose rules of operation are determined only by the shape and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the "success" of mathematics, in that there had been no prolonged dispute about any truly mathematical result. <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">C. S. Peirce</a> noted<a href="#cite_note-105">[105]</a> that even though a mistake in the evaluation of a definite integral by <a href="/wiki/Laplace" class="mw-redirect" title="Laplace">Laplace</a> led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a>. He argued that a truly "exact" logic would depend upon mathematical, i.e., "diagrammatic" or "iconic" thought. "Those who follow such methods will ... escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also "constructive" rather than "abstractive"; i.e., rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity), it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medieval logicians called "<a href="/wiki/Syncategorematic_term" title="Syncategorematic term">syncategoremata</a>") and the categoric terms are expressed in symbols. <div class="mw-heading mw-heading2"><h2 id="Modern_logic">Modern logic</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=33" title="Edit section: Modern logic">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a></div> The development of modern logic falls into roughly five periods:<a href="#cite_note-106">[106]</a> <ul><li>The embryonic period from <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed.</li> <li>The algebraic period from <a href="/wiki/Boole" class="mw-redirect" title="Boole">Boole</a>'s Analysis to <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Schröder</a>'s Vorlesungen. In this period, there were more practitioners, and a greater continuity of development.</li> <li>The <a href="/wiki/Logicist" class="mw-redirect" title="Logicist">logicist</a> period from the <a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a> of <a href="/wiki/Frege" class="mw-redirect" title="Frege">Frege</a> to the <a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a> of <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a> and <a href="/wiki/A._N._Whitehead" class="mw-redirect" title="A. N. Whitehead">Whitehead</a>. The aim of the "logicist school" was to incorporate the logic of all mathematical and scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists were <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Frege</a>, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Russell</a>, and the early <a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgenstein</a>.<a href="#cite_note-107">[107]</a> It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the <a href="/wiki/Antinomy" title="Antinomy">antinomies</a> which had been an obstacle to earlier progress.</li> <li>The metamathematical period from 1910 to the 1930s, which saw the development of <a href="/wiki/Metalogic" title="Metalogic">metalogic</a>, in the <a href="/wiki/Finitist" class="mw-redirect" title="Finitist">finitist</a> system of <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, and the non-finitist system of <a href="/wiki/Leopold_L%C3%B6wenheim" title="Leopold Löwenheim">Löwenheim</a> and <a href="/wiki/Skolem" class="mw-redirect" title="Skolem">Skolem</a>, the combination of logic and metalogic in the work of <a href="/wiki/G%C3%B6del" class="mw-redirect" title="Gödel">Gödel</a> and <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski</a>. Gödel's <a href="/wiki/Incompleteness_theorem" class="mw-redirect" title="Incompleteness theorem">incompleteness theorem</a> of 1931 was one of the greatest achievements in the history of logic. Later in the 1930s, Gödel developed the notion of <a href="/wiki/Set-theoretic_constructibility" class="mw-redirect" title="Set-theoretic constructibility">set-theoretic constructibility</a>.</li> <li>The period after World War II, when <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> branched into four inter-related but separate areas of research: <a href="/wiki/Model_theory" title="Model theory">model theory</a>, <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, and <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and its ideas and methods began to influence <a href="/wiki/Philosophy" title="Philosophy">philosophy</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Embryonic_period">Embryonic period</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=34" title="Edit section: Embryonic period">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gottfried_Wilhelm_Leibniz,_Bernhard_Christoph_Francke.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/170px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg" decoding="async" width="170" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/255px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg/340px-Gottfried_Wilhelm_Leibniz%2C_Bernhard_Christoph_Francke.jpg 2x" data-file-width="4486" data-file-height="5538" /></a><figcaption>Leibniz</figcaption></figure> The idea that inference could be represented by a purely mechanical process is found as early as <a href="/wiki/Ramon_Llull" title="Ramon Llull">Raymond Llull</a>, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the <a href="/wiki/Oxford_Calculators" title="Oxford Calculators">Oxford Calculators</a><a href="#cite_note-108">[108]</a> led to a method of using letters instead of writing out logical calculations (calculationes) in words, a method used, for instance, in the Logica magna by <a href="/wiki/Paul_of_Venice" title="Paul of Venice">Paul of Venice</a>. Three hundred years after Llull, the English philosopher and logician <a href="/wiki/Thomas_Hobbes" title="Thomas Hobbes">Thomas Hobbes</a> suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction.<a href="#cite_note-109">[109]</a> The same idea is found in the work of <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>, who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages are subject to "countless ambiguities" and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words;<a href="#cite_note-110">[110]</a> hence, he proposed to identify an <a href="/wiki/Alphabet_of_human_thought" title="Alphabet of human thought">alphabet of human thought</a> comprising fundamental concepts which could be composed to express complex ideas,<a href="#cite_note-111">[111]</a> and create a <a href="/wiki/Calculus_ratiocinator" title="Calculus ratiocinator">calculus ratiocinator</a> that would make all arguments "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate."<a href="#cite_note-112">[112]</a> <a href="/wiki/Joseph_Diaz_Gergonne" class="mw-redirect" title="Joseph Diaz Gergonne">Gergonne</a> (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning of the symbols involved.<a href="#cite_note-113">[113]</a> <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano</a> anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables:<a href="#cite_note-114">[114]</a><blockquote>Hence I say that propositions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}">,... are deducible from propositions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">,... with respect to variable parts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">,..., if every class of ideas whose substitution for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">,... makes all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">,... true, also makes all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}">,... true. Occasionally, since it is customary, I shall say that propositions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}">,... follow, or can be inferred or derived, from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">,.... Propositions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">,... I shall call the premises, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}">,... the conclusions.</blockquote>This is now known as <a href="/wiki/Semantic_validity" class="mw-redirect" title="Semantic validity">semantic validity</a>. <div class="mw-heading mw-heading3"><h3 id="Algebraic_period">Algebraic period</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=35" title="Edit section: Algebraic period">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:George_Boole_color.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/George_Boole_color.jpg/140px-George_Boole_color.jpg" decoding="async" width="140" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/George_Boole_color.jpg/210px-George_Boole_color.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/George_Boole_color.jpg/280px-George_Boole_color.jpg 2x" data-file-width="600" data-file-height="800" /></a><figcaption>George Boole</figcaption></figure> Modern logic begins with what is known as the "algebraic school", originating with Boole and including <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce</a>, <a href="/wiki/William_Stanley_Jevons" title="William Stanley Jevons">Jevons</a>, <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Schröder</a>, and <a href="/wiki/John_Venn" title="John Venn">Venn</a>.<a href="#cite_note-115">[115]</a> Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">De Morgan</a> (1847) is its immediate precursor.<a href="#cite_note-116">[116]</a> The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in <a href="/wiki/Lincoln,_Lincolnshire" class="mw-redirect" title="Lincoln, Lincolnshire">Lincoln, Lincolnshire</a>.<a href="#cite_note-117">[117]</a> For example, let x and y stand for classes, let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration.<a href="#cite_note-Kneale_p._407-118">[118]</a> An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation.<a href="#cite_note-119">[119]</a> The theory of elective functions and their "development" is essentially the modern idea of <a href="/wiki/Truth-function" class="mw-redirect" title="Truth-function">truth-functions</a> and their expression in <a href="/wiki/Disjunctive_normal_form" title="Disjunctive normal form">disjunctive normal form</a>.<a href="#cite_note-Kneale_p._407-118">[118]</a> Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between "primary propositions" which are the subject of syllogistic theory, and "secondary propositions", which are the subject of propositional logic, and showed how under different "interpretations" the same algebraic system could represent both. An example of a primary proposition is "All inhabitants are either Europeans or Asiatics." An example of a secondary proposition is "Either all inhabitants are Europeans or they are all Asiatics."<a href="#cite_note-120">[120]</a> These are easily distinguished in modern predicate logic, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system.<a href="#cite_note-121">[121]</a> In his Symbolic Logic (1881), <a href="/wiki/John_Venn" title="John Venn">John Venn</a> used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a "logical machine" which he showed to the <a href="/wiki/Royal_Society" title="Royal Society">Royal Society</a> the following year.<a href="#cite_note-Kneale_p._407-118">[118]</a> In 1885 <a href="/wiki/Allan_Marquand" title="Allan Marquand">Allan Marquand</a> proposed an electrical version of the machine that is still extant (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20080908073359/http://finelib.princeton.edu/instruction/wri172_demonstration.php">picture at the Firestone Library</a>). <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Charles_Sanders_Peirce.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Charles_Sanders_Peirce.jpg/160px-Charles_Sanders_Peirce.jpg" decoding="async" width="160" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Charles_Sanders_Peirce.jpg/240px-Charles_Sanders_Peirce.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Charles_Sanders_Peirce.jpg/320px-Charles_Sanders_Peirce.jpg 2x" data-file-width="434" data-file-height="582" /></a><figcaption>Charles Sanders Peirce</figcaption></figure> The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify <a href="/wiki/Exclusive_or" title="Exclusive or">exclusive or</a>, which allowed Boole's system to be greatly simplified.<a href="#cite_note-122">[122]</a> This was usefully exploited by Schröder when he set out theorems in parallel columns in his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "<a href="/wiki/Logical_NOR" title="Logical NOR">neither ... nor ...</a>" and equally well "<a href="/wiki/Sheffer_stroke" title="Sheffer stroke">not both ... and ...</a>",<a href="#cite_note-123">[123]</a> however, like many of Peirce's innovations, this remained unknown or unnoticed until <a href="/wiki/Henry_M._Sheffer" title="Henry M. Sheffer">Sheffer</a> rediscovered it in 1913.<a href="#cite_note-124">[124]</a> Boole's early work also lacks the idea of the <a href="/wiki/Logical_sum" class="mw-redirect" title="Logical sum">logical sum</a> which originates in Peirce (1867), <a href="/wiki/Ernst_Schr%C3%B6der_(mathematician)" title="Ernst Schröder (mathematician)">Schröder</a> (1877) and Jevons (1890),<a href="#cite_note-125">[125]</a> and the concept of <a href="/wiki/Inclusion_(logic)" title="Inclusion (logic)">inclusion</a>, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870). <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Boolean_multiples_of_2_3_5.svg" class="mw-file-description"><img alt="Coloured diagram of 4 interlocking sets" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Boolean_multiples_of_2_3_5.svg/250px-Boolean_multiples_of_2_3_5.svg.png" decoding="async" width="250" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Boolean_multiples_of_2_3_5.svg/375px-Boolean_multiples_of_2_3_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Boolean_multiples_of_2_3_5.svg/500px-Boolean_multiples_of_2_3_5.svg.png 2x" data-file-width="537" data-file-height="343" /></a><figcaption>Boolean multiples</figcaption></figure> The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schröder's monumental Vorlesungen über die Algebra der Logik ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce.<a href="#cite_note-126">[126]</a> Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic <a href="/wiki/John_Corcoran_(logician)" title="John Corcoran (logician)">John Corcoran</a> in an accessible introduction to Laws of Thought.<a href="#cite_note-127">[127]</a> Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.<a href="#cite_note-128">[128]</a> According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat—from assessing validity to solving equations—and 3) expanding the range of applications it could handle—e.g. from propositions having only two terms to those having arbitrarily many. More specifically, Boole agreed with what <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations—by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle". <div class="mw-heading mw-heading3"><h3 id="Logicist_period">Logicist period</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=36" title="Edit section: Logicist period">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Young_frege.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Young_frege.jpg/160px-Young_frege.jpg" decoding="async" width="160" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Young_frege.jpg/240px-Young_frege.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Young_frege.jpg/320px-Young_frege.jpg 2x" data-file-width="458" data-file-height="619" /></a><figcaption>Gottlob Frege.</figcaption></figure> After Boole, the next great advances were made by the German mathematician <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>. Frege's objective was the program of <a href="/wiki/Logicism" title="Logicism">Logicism</a>, i.e. demonstrating that arithmetic is identical with logic.<a href="#cite_note-k435-129">[129]</a> Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or <a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a> is important.<a href="#cite_note-k435-129">[129]</a> Frege also tried to show that the concept of <a href="/wiki/Number" title="Number">number</a> can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, <a href="/wiki/J._S._Mill" class="mw-redirect" title="J. S. Mill">J. S. Mill</a> as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination."<a href="#cite_note-130">[130]</a> Frege's first work, the Begriffsschrift ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (modus ponens and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this.<a href="#cite_note-131">[131]</a> The most significant innovation, however, was his explanation of the <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifier</a> in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man".<a href="#cite_note-132">[132]</a> At the outset Frege abandons the traditional "concepts subject and predicate", replacing them with argument and function respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention".<a href="#cite_note-133">[133]</a> Frege argued that the quantifier expression "all men" does not have the same logical or semantic form as "all men", and that the universal proposition "every A is B" is a complex proposition involving two functions, namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;x{\big (}A(x)\rightarrow B(x){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;x{\big (}A(x)\rightarrow B(x){\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128aeef992fcaa726f6d2418d9a25bff6fa11cbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.796ex; height:3.176ex;" alt="{\displaystyle \forall \;x{\big (}A(x)\rightarrow B(x){\big )}}"></dd></dl> In English, "for all x, if Ax then Bx". Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. If "all mammals" were the logical subject of the sentence "all mammals are land-dwellers", then to negate the whole sentence we would have to negate the predicate to give "all mammals are not land-dwellers". But this is not the case.<a href="#cite_note-134">[134]</a> This functional analysis of ordinary-language sentences later had a great impact on philosophy and <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>. This means that in Frege's calculus, Boole's "primary" propositions can be represented in a different way from "secondary" propositions. "All inhabitants are either men or women" is <figure typeof="mw:File/Thumb"><a href="/wiki/File:BS-13-Begriffsschrift_Quantifier2-svg.svg" class="mw-file-description"><img alt="Straight line with bend; text "x" over bend; text "F(x)" to the right of the line." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/BS-13-Begriffsschrift_Quantifier2-svg.svg/130px-BS-13-Begriffsschrift_Quantifier2-svg.svg.png" decoding="async" width="130" height="36" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/BS-13-Begriffsschrift_Quantifier2-svg.svg/195px-BS-13-Begriffsschrift_Quantifier2-svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/BS-13-Begriffsschrift_Quantifier2-svg.svg/260px-BS-13-Begriffsschrift_Quantifier2-svg.svg.png 2x" data-file-width="512" data-file-height="142" /></a><figcaption><a href="/wiki/Frege" class="mw-redirect" title="Frege">Frege</a>'s "Concept Script"</figcaption></figure> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;x{\Big (}I(x)\rightarrow {\big (}M(x)\lor W(x){\big )}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;x{\Big (}I(x)\rightarrow {\big (}M(x)\lor W(x){\big )}{\Big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d072c127af9486afdcaf068dedca643ce27466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.835ex; height:4.843ex;" alt="{\displaystyle \forall \;x{\Big (}I(x)\rightarrow {\big (}M(x)\lor W(x){\big )}{\Big )}}"></dd></dl> whereas "All the inhabitants are men or all the inhabitants are women" is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;x{\big (}I(x)\rightarrow M(x){\big )}\lor \forall \;x{\big (}I(x)\rightarrow W(x){\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>I</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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>∨</mo> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;x{\big (}I(x)\rightarrow M(x){\big )}\lor \forall \;x{\big (}I(x)\rightarrow W(x){\big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/095ef3baf6185ab9d2e72eea165f6c04600ce48f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.382ex; height:3.176ex;" alt="{\displaystyle \forall \;x{\big (}I(x)\rightarrow M(x){\big )}\lor \forall \;x{\big (}I(x)\rightarrow W(x){\big )}}"></dd></dl> As Frege remarked in a critique of Boole's calculus: <dl><dd>"The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it."<a href="#cite_note-135">[135]</a></dd></dl> As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient <a href="/wiki/Problem_of_multiple_generality" title="Problem of multiple generality">problem of multiple generality</a>. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \;x{\Big (}G(x)\rightarrow \exists \;y{\big (}B(y)\land K(x,y){\big )}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \;x{\Big (}G(x)\rightarrow \exists \;y{\big (}B(y)\land K(x,y){\big )}{\Big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79dfeef6ef90c367707761c29ee7b7bf8a38577" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.551ex; height:4.843ex;" alt="{\displaystyle \forall \;x{\Big (}G(x)\rightarrow \exists \;y{\big (}B(y)\land K(x,y){\big )}{\Big )}}"></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Giuseppe_Peano.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Giuseppe_Peano.jpg/120px-Giuseppe_Peano.jpg" decoding="async" width="120" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Giuseppe_Peano.jpg/180px-Giuseppe_Peano.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Giuseppe_Peano.jpg/240px-Giuseppe_Peano.jpg 2x" data-file-width="580" data-file-height="734" /></a><figcaption>Peano</figcaption></figure> means that to every girl there corresponds some boy (any one will do) who the girl kissed. But <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists \;x{\Big (}B(x)\land \forall \;y{\big (}G(y)\rightarrow K(y,x){\big )}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mspace width="thickmathspace" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mspace width="thickmathspace" /> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists \;x{\Big (}B(x)\land \forall \;y{\big (}G(y)\rightarrow K(y,x){\big )}{\Big )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a003e33003c1b4457421a1e4f6aff95b4da238c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.551ex; height:4.843ex;" alt="{\displaystyle \exists \;x{\Big (}B(x)\land \forall \;y{\big (}G(y)\rightarrow K(y,x){\big )}{\Big )}}"></dd></dl> means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the <a href="/wiki/Ancestral_relation" title="Ancestral relation">ancestral relation</a>, of the <a href="/wiki/Injective_function" title="Injective function">many-to-one relation</a>, and of <a href="/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>.<a href="#cite_note-136">[136]</a> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Ernst_Zermelo_1900s.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ernst_Zermelo_1900s.jpg/130px-Ernst_Zermelo_1900s.jpg" decoding="async" width="130" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ernst_Zermelo_1900s.jpg/195px-Ernst_Zermelo_1900s.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Ernst_Zermelo_1900s.jpg/260px-Ernst_Zermelo_1900s.jpg 2x" data-file-width="623" data-file-height="831" /></a><figcaption>Ernst Zermelo</figcaption></figure> This period overlaps with the work of what is known as the "mathematical school", which included <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a>, <a href="/wiki/Moritz_Pasch" title="Moritz Pasch">Pasch</a>, <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Peano</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo</a>, <a href="/wiki/Edward_Vermilye_Huntington" title="Edward Vermilye Huntington">Huntington</a>, <a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a> and <a href="/wiki/Arend_Heyting" title="Arend Heyting">Heyting</a>. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was <a href="/wiki/Hilbert%27s_Program" class="mw-redirect" title="Hilbert's Program">Hilbert's Program</a>, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard <a href="/wiki/Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> is named the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a> eponymously. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder.<a href="#cite_note-137">[137]</a> The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>. This proved Frege's <a href="/wiki/Naive_set_theory" title="Naive set theory">naive set theory</a> led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not).<a href="#cite_note-138">[138]</a> This contradiction is now known as <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>. One important method of resolving this paradox was proposed by <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a>.<a href="#cite_note-139">[139]</a> <a href="/wiki/Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a> was the first <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>. It was developed into the now-canonical <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZF). Russell's paradox symbolically is as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Let </mtext> </mrow> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mo>∉</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>, then </mtext> </mrow> <mi>R</mi> <mo>∈</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>R</mi> <mo>∉</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1083e5691d2b959d103e2a6c3a9585a1b25b0438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.889ex; height:2.843ex;" alt="{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}"></dd></dl> The monumental <a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a>, a three-volume work on the <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>, written by Russell and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a> and published 1910–1913 also included an attempt to resolve the paradox, by means of an elaborate <a href="/wiki/System_of_types" class="mw-redirect" title="System of types">system of types</a>: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "<a href="/wiki/Set_of_all_sets" class="mw-redirect" title="Set of all sets">set of all sets</a>". The Principia was an attempt to derive all mathematical truths from a well-defined set of <a href="/wiki/Axiom" title="Axiom">axioms</a> and <a href="/wiki/Inference_rule" class="mw-redirect" title="Inference rule">inference rules</a> in <a href="/wiki/Mathematical_logic" title="Mathematical logic">symbolic logic</a>. <div class="mw-heading mw-heading3"><h3 id="Metamathematical_period">Metamathematical period</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=37" title="Edit section: Metamathematical period">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Kurt_g%C3%B6del.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Kurt_g%C3%B6del.jpg/130px-Kurt_g%C3%B6del.jpg" decoding="async" width="130" height="166" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Kurt_g%C3%B6del.jpg/195px-Kurt_g%C3%B6del.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/4/42/Kurt_g%C3%B6del.jpg 2x" data-file-width="212" data-file-height="270" /></a><figcaption>Kurt Gödel</figcaption></figure> The names of <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a> and <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski</a> dominate the 1930s,<a href="#cite_note-140">[140]</a> a crucial period in the development of <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a>—the study of mathematics using mathematical methods to produce <a href="/wiki/Metatheory" title="Metatheory">metatheories</a>, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given <a href="/wiki/First-order_logic" title="First-order logic">first-order sentence</a> is <a href="/wiki/Provability_logic" title="Provability logic">deducible</a> if and only if it is logically valid—i.e. it is true in every <a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">structure</a> for its language. This is known as <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a>. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an <a href="/wiki/Effective_method" title="Effective method">effective procedure</a> such as an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> or computer program is capable of proving all facts about the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of <a href="/wiki/Set-theoretic_constructibility" class="mw-redirect" title="Set-theoretic constructibility">set-theoretic constructibility</a>, as part of his proof that the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> and the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> are consistent with <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>. In <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> developed <a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a> and the <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculus</a>. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>, while the latter was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form.<a href="#cite_note-141">[141]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AlfredTarski1968.jpeg" class="mw-file-description"><img alt="Balding man, with bookshelf in background" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/AlfredTarski1968.jpeg/200px-AlfredTarski1968.jpeg" decoding="async" width="200" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/AlfredTarski1968.jpeg/300px-AlfredTarski1968.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/71/AlfredTarski1968.jpeg 2x" data-file-width="400" data-file-height="271" /></a><figcaption>Alfred Tarski</figcaption></figure> <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, a pupil of <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Łukasiewicz</a>, is best known for his definition of truth and <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a>, and the semantic concept of <a href="/wiki/Open_sentence" class="mw-redirect" title="Open sentence">logical satisfaction</a>. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his <a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic theory of truth</a>: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the <a href="/wiki/Metalanguage" title="Metalanguage">metalanguage</a>, which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the <a href="/wiki/T-schema" title="T-schema">T-schema</a>) between phrases in the object language and elements of an <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a>. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of <a href="/wiki/Model_theory" title="Model theory">model theory</a>.<a href="#cite_note-142">[142]</a> Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">completeness</a>, <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidability</a>, <a href="/wiki/Consistency" title="Consistency">consistency</a> and <a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">definability</a>. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".<a href="#cite_note-143">[143]</a> <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> and <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> proposed formal models of computability, giving independent negative solutions to Hilbert's <a href="/wiki/Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a> in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the <a href="/wiki/Halting_problem" title="Halting problem">halting problem</a> as a key example of a mathematical problem without an algorithmic solution. Church's system for computation developed into the modern <a href="/wiki/%CE%9B-calculus" class="mw-redirect" title="Λ-calculus">λ-calculus</a>, while the <a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a> became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the <a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a> that any deterministic <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> and <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> are <a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a>. Later work by <a href="/wiki/Emil_Post" class="mw-redirect" title="Emil Post">Emil Post</a> and <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a> in the 1940s extended the scope of computability theory and introduced the concept of <a href="/wiki/Degrees_of_unsolvability" class="mw-redirect" title="Degrees of unsolvability">degrees of unsolvability</a>. The results of the first few decades of the twentieth century also had an impact upon <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">analytic philosophy</a> and <a href="/wiki/Philosophical_logic" title="Philosophical logic">philosophical logic</a>, particularly from the 1950s onwards, in subjects such as <a href="/wiki/Modal_logic" title="Modal logic">modal logic</a>, <a href="/wiki/Temporal_logic" title="Temporal logic">temporal logic</a>, <a href="/wiki/Deontic_logic" title="Deontic logic">deontic logic</a>, and <a href="/wiki/Relevance_logic" title="Relevance logic">relevance logic</a>. <div class="mw-heading mw-heading3"><h3 id="Logic_after_WWII">Logic after WWII</h3>[<a href="/w/index.php?title=History_of_logic&action=edit&section=38" title="Edit section: Logic after WWII">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Kripke.JPG" class="mw-file-description"><img alt="Man with a beard and straw hat on a beach" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Kripke.JPG/220px-Kripke.JPG" decoding="async" width="220" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Kripke.JPG/330px-Kripke.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d4/Kripke.JPG 2x" data-file-width="366" data-file-height="349" /></a><figcaption><a href="/wiki/Saul_Kripke" title="Saul Kripke">Saul Kripke</a></figcaption></figure> After World War II, <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a> branched into four inter-related but separate areas of research: <a href="/wiki/Model_theory" title="Model theory">model theory</a>, <a href="/wiki/Proof_theory" title="Proof theory">proof theory</a>, <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>, and <a href="/wiki/Set_theory" title="Set theory">set theory</a>.<a href="#cite_note-144">[144]</a> In set theory, the method of <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a> revolutionized the field by providing a robust method for constructing models and obtaining independence results. <a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a> introduced this method in 1963 to prove the independence of the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> and the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a> from <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>.<a href="#cite_note-145">[145]</a> His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic. Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as <a href="/wiki/Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>.<a href="#cite_note-146">[146]</a> The <a href="/wiki/Turing_degree" title="Turing degree">priority method</a>, discovered independently by <a href="/wiki/Albert_Muchnik" title="Albert Muchnik">Albert Muchnik</a> and <a href="/wiki/Richard_Friedberg" class="mw-redirect" title="Richard Friedberg">Richard Friedberg</a> in the 1950s, led to major advances in the understanding of the <a href="/wiki/Degrees_of_unsolvability" class="mw-redirect" title="Degrees of unsolvability">degrees of unsolvability</a> and related structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of <a href="/wiki/Constructive_analysis" title="Constructive analysis">constructive analysis</a> and <a href="/wiki/Computable_analysis" title="Computable analysis">computable analysis</a> were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of <a href="/wiki/Reverse_mathematics" title="Reverse mathematics">reverse mathematics</a>. A separate branch of computability theory, <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, was also characterized in logical terms as a result of investigations into <a href="/wiki/Descriptive_complexity" class="mw-redirect" title="Descriptive complexity">descriptive complexity</a>. Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title Contributions to the theory of models. In the 1960s, <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a> used model-theoretic techniques to develop calculus and analysis based on <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">infinitesimals</a>, a problem that first had been proposed by Leibniz. In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the <a href="/wiki/Realizability" title="Realizability">realizability</a> method invented by <a href="/wiki/Georg_Kreisel" title="Georg Kreisel">Georg Kreisel</a> and Gödel's <a href="/wiki/Dialectica_interpretation" title="Dialectica interpretation">Dialectica interpretation</a>. This work inspired the contemporary area of <a href="/wiki/Proof_mining" title="Proof mining">proof mining</a>. The <a href="/wiki/Curry%E2%80%93Howard_correspondence" title="Curry–Howard correspondence">Curry–Howard correspondence</a> emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and <a href="/wiki/Typed_lambda_calculus" title="Typed lambda calculus">typed lambda calculi</a> used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in <a href="/wiki/Ordinal_analysis" title="Ordinal analysis">ordinal analysis</a> and the study of independence results in arithmetic such as the <a href="/wiki/Paris%E2%80%93Harrington_theorem" title="Paris–Harrington theorem">Paris–Harrington theorem</a>. This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, <a href="/wiki/Tense_logic" class="mw-redirect" title="Tense logic">tense logic</a> is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher <a href="/wiki/Arthur_Prior" title="Arthur Prior">Arthur Prior</a> played a significant role in its development in the 1960s. <a href="/wiki/Modal_logic" title="Modal logic">Modal logics</a> extend the scope of formal logic to include the elements of <a href="/wiki/Linguistic_modality" class="mw-redirect" title="Linguistic modality">modality</a> (for example, <a href="/wiki/Logical_possibility" title="Logical possibility">possibility</a> and <a href="/wiki/Necessary_and_sufficient_conditions#Necessary_conditions" class="mw-redirect" title="Necessary and sufficient conditions">necessity</a>). The ideas of <a href="/wiki/Saul_Kripke" title="Saul Kripke">Saul Kripke</a>, particularly about <a href="/wiki/Possible_world" title="Possible world">possible worlds</a>, and the formal system now called <a href="/wiki/Kripke_semantics" title="Kripke semantics">Kripke semantics</a> have had a profound impact on <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">analytic philosophy</a>.<a href="#cite_note-147">[147]</a> His best known and most influential work is <a href="/wiki/Naming_and_Necessity" title="Naming and Necessity">Naming and Necessity</a> (1980).<a href="#cite_note-148">[148]</a> <a href="/wiki/Deontic_logic" title="Deontic logic">Deontic logics</a> are closely related to modal logics: they attempt to capture the logical features of <a href="/wiki/Obligation" title="Obligation">obligation</a>, <a href="/wiki/Permission_(philosophy)" title="Permission (philosophy)">permission</a> and related concepts. Although some basic novelties <a href="/wiki/Syncretism" title="Syncretism">syncretizing</a> mathematical and philosophical logic were shown by <a href="/wiki/Bernard_Bolzano#Metaphysics" title="Bernard Bolzano">Bolzano</a> in the early 1800s, it was <a href="/wiki/Ernst_Mally" title="Ernst Mally">Ernst Mally</a>, a pupil of <a href="/wiki/Alexius_Meinong" title="Alexius Meinong">Alexius Meinong</a>, who was to propose the first formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a>. Another logical system founded after World War II was <a href="/wiki/Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a> by Azerbaijani mathematician <a href="/wiki/Lotfi_Asker_Zadeh" class="mw-redirect" title="Lotfi Asker Zadeh">Lotfi Asker Zadeh</a> in 1965. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=39" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /><a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></li></ul> <ul><li><a href="/wiki/History_of_deductive_reasoning" class="mw-redirect" title="History of deductive reasoning">History of deductive reasoning</a></li> <li><a href="/wiki/History_of_inductive_reasoning" class="mw-redirect" title="History of inductive reasoning">History of inductive reasoning</a></li> <li><a href="/wiki/History_of_abductive_reasoning" class="mw-redirect" title="History of abductive reasoning">History of abductive reasoning</a></li> <li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">History of the function concept</a></li> <li><a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a></li> <li><a href="/wiki/Philosophy#History" title="Philosophy">History of Philosophy</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">Timeline of mathematical logic</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=40" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-Boehner_p._xiv-1">^ <a href="#cite_ref-Boehner_p._xiv_1-0">a</a> <a href="#cite_ref-Boehner_p._xiv_1-1">b</a> Boehner p. xiv </li> <li id="cite_note-ReferenceA-2">^ <a href="#cite_ref-ReferenceA_2-0">a</a> <a href="#cite_ref-ReferenceA_2-1">b</a> Oxford Companion p. 498; Bochenski, Part I Introduction, passim </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFrege" class="citation book cs1">Frege, Gottlob. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180920172024/http://www.naturalthinker.net/trl/texts/Frege,Gottlob/Frege,%20Gottlob%20-%20The%20Foundations%20of%20Arithmetic%20(1953)%202Ed_%207.0-2.5%20LotB.pdf">The Foundations of Arithmetic</a> (PDF). p. 1. Archived from <a rel="nofollow" class="external text" href="http://www.naturalthinker.net/trl/texts/Frege,Gottlob/Frege,%20Gottlob%20-%20The%20Foundations%20of%20Arithmetic%20%281953%29%202Ed_%207.0-2.5%20LotB.pdf">the original</a> (PDF) on 2018-09-20. Retrieved 2016-02-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Foundations+of+Arithmetic&rft.pages=1&rft.aulast=Frege&rft.aufirst=Gottlob&rft_id=http%3A%2F%2Fwww.naturalthinker.net%2Ftrl%2Ftexts%2FFrege%2CGottlob%2FFrege%2C%2520Gottlob%2520-%2520The%2520Foundations%2520of%2520Arithmetic%2520%25281953%2529%25202Ed_%25207.0-2.5%2520LotB.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-Oxford_Companion_p._500-4">^ <a href="#cite_ref-Oxford_Companion_p._500_4-0">a</a> <a href="#cite_ref-Oxford_Companion_p._500_4-1">b</a> Oxford Companion p. 500 </li> <li id="cite_note-Kramer1986-5"><a href="#cite_ref-Kramer1986_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKramer1986" class="citation book cs1">Kramer, Kenneth (January 1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RzUAu-43W5oC&pg=PA34">World Scriptures: An Introduction to Comparative Religions</a>. Paulist Press. pp. 34–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8091-2781-8" title="Special:BookSources/978-0-8091-2781-8"><bdi>978-0-8091-2781-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=World+Scriptures%3A+An+Introduction+to+Comparative+Religions&rft.pages=34-&rft.pub=Paulist+Press&rft.date=1986-01&rft.isbn=978-0-8091-2781-8&rft.aulast=Kramer&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRzUAu-43W5oC%26pg%3DPA34&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-Christian2011-6"><a href="#cite_ref-Christian2011_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChristian2011" class="citation book cs1">Christian, David (2011-09-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7RdVmDjwTtQC&pg=PA18">Maps of Time: An Introduction to Big History</a>. University of California Press. pp. 18–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-520-95067-2" title="Special:BookSources/978-0-520-95067-2"><bdi>978-0-520-95067-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maps+of+Time%3A+An+Introduction+to+Big+History&rft.pages=18-&rft.pub=University+of+California+Press&rft.date=2011-09-01&rft.isbn=978-0-520-95067-2&rft.aulast=Christian&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D7RdVmDjwTtQC%26pg%3DPA18&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-Singh2008-7"><a href="#cite_ref-Singh2008_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingh2008" class="citation book cs1">Singh, Upinder (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=H3lUIIYxWkEC&pg=PA206">A History of Ancient and Early Medieval India: From the Stone Age to the 12th Century</a>. Pearson Education India. pp. 206–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-317-1120-0" title="Special:BookSources/978-81-317-1120-0"><bdi>978-81-317-1120-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Ancient+and+Early+Medieval+India%3A+From+the+Stone+Age+to+the+12th+Century&rft.pages=206-&rft.pub=Pearson+Education+India&rft.date=2008&rft.isbn=978-81-317-1120-0&rft.aulast=Singh&rft.aufirst=Upinder&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DH3lUIIYxWkEC%26pg%3DPA206&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Bochenski p. 446 </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVidyabhusana1921" class="citation book cs1">Vidyabhusana, S. 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(1921). <a rel="nofollow" class="external text" href="http://archive.org/details/in.ernet.dli.2015.213362">History Of Indian Logic</a>. p. 11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+Of+Indian+Logic.&rft.pages=11&rft.date=1921&rft.aulast=Vidyabhusana&rft.aufirst=S.+C.&rft_id=http%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.213362&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhusana1921" class="citation book cs1">Bhusana, Satis Chandra Vidya (1921). <a rel="nofollow" class="external text" href="http://archive.org/details/in.ernet.dli.2015.489008">A History Of Indian Logic</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+Of+Indian+Logic&rft.date=1921&rft.aulast=Bhusana&rft.aufirst=Satis+Chandra+Vidya&rft_id=http%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.489008&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> S. C. Vidyabhusana (1971). A History of Indian Logic: Ancient, Mediaeval, and Modern Schools, pp. 17–21. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> R. P. Kangle (1986). The Kautiliya Arthashastra (1.2.11). Motilal Banarsidass. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> Bochenski p. 417 and passim </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaneri2002" class="citation journal cs1">Ganeri, Jonardon (2002). <a rel="nofollow" class="external text" href="https://www.academia.edu/2146233">"Jaina Logic and the Philosophical Basis of Pluralism"</a>. History and Philosophy of Logic. 23 (4): 267–281. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0144534021000051505">10.1080/0144534021000051505</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0144-5340">0144-5340</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:170089234">170089234</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=History+and+Philosophy+of+Logic&rft.atitle=Jaina+Logic+and+the+Philosophical+Basis+of+Pluralism&rft.volume=23&rft.issue=4&rft.pages=267-281&rft.date=2002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A170089234%23id-name%3DS2CID&rft.issn=0144-5340&rft_id=info%3Adoi%2F10.1080%2F0144534021000051505&rft.aulast=Ganeri&rft.aufirst=Jonardon&rft_id=https%3A%2F%2Fwww.academia.edu%2F2146233&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> Bochenski pp. 431–437 </li> <li id="cite_note-Matilal-16"><a href="#cite_ref-Matilal_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatilal1998" class="citation book cs1">Matilal, Bimal Krishna (1998). The Character of Logic in India. Albany, New York, USA: State University of New York Press. pp. 12, 18. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780791437407" title="Special:BookSources/9780791437407"><bdi>9780791437407</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Character+of+Logic+in+India&rft.place=Albany%2C+New+York%2C+USA&rft.pages=12%2C+18&rft.pub=State+University+of+New+York+Press&rft.date=1998&rft.isbn=9780791437407&rft.aulast=Matilal&rft.aufirst=Bimal+Krishna&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Bochenksi p. 441 </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Matilal, 17 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Kneale, p. 2 </li> <li id="cite_note-Kneale3-20">^ <a href="#cite_ref-Kneale3_20-0">a</a> <a href="#cite_ref-Kneale3_20-1">b</a> <a href="#cite_ref-Kneale3_20-2">c</a> <a href="#cite_ref-Kneale3_20-3">d</a> Kneale p. 3 </li> <li id="cite_note-Stol-99-21"><a href="#cite_ref-Stol-99_21-0">^</a> H. F. J. Horstmanshoff, Marten Stol, Cornelis Tilburg (2004), Magic and Rationality in Ancient Near Eastern and Graeco-Roman Medicine, p. 99, <a href="/wiki/Brill_Publishers" title="Brill Publishers">Brill Publishers</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-04-13666-5" title="Special:BookSources/90-04-13666-5">90-04-13666-5</a>. </li> <li id="cite_note-Brown-22"><a href="#cite_ref-Brown_22-0">^</a> D. Brown (2000), Mesopotamian Planetary Astronomy-Astrology , Styx Publications, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-5693-036-2" title="Special:BookSources/90-5693-036-2">90-5693-036-2</a>. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> Heath, Mathematics in Aristotle, cited in Kneale, p. 5 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> Kneale, p. 16 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/346217/history-of-logic#toc65918">"History of logic"</a>. britannica.com. Retrieved 2018-04-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=britannica.com&rft.atitle=History+of+logic&rft_id=http%3A%2F%2Fwww.britannica.com%2FEBchecked%2Ftopic%2F346217%2Fhistory-of-logic%23toc65918&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, Metaphysics Alpha, 983b18. </li> <li id="cite_note-CPM-27"><a href="#cite_ref-CPM_27-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith1870" class="citation book cs1">Smith, William (1870). <a rel="nofollow" class="external text" href="https://archive.org/stream/dictionaryofgree03smituoft#page/1016">Dictionary of Greek and Roman biography and mythology</a>. Boston, Little. p. 1016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Greek+and+Roman+biography+and+mythology&rft.pages=1016&rft.pub=Boston%2C+Little&rft.date=1870&rft.aulast=Smith&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fdictionaryofgree03smituoft%23page%2F1016&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> T. Patronis & D. Patsopoulos <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160303171258/http://journals.tc-library.org/index.php/hist_math_ed/article/viewFile/189/184">The Theorem of Thales: A Study of the naming of theorems in school Geometry textbooks</a>. <a href="/wiki/Patras_University" class="mw-redirect" title="Patras University">Patras University</a>. Archived from <a rel="nofollow" class="external text" href="http://journals.tc-library.org/index.php/hist_math_ed/article/viewFile/189/184">the original</a> on 2016-03-03. Retrieved 2012-02-12.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theorem+of+Thales%3A+A+Study+of+the+naming+of+theorems+in+school+Geometry+textbooks&rft.pub=Patras+University&rft_id=http%3A%2F%2Fjournals.tc-library.org%2Findex.php%2Fhist_math_ed%2Farticle%2FviewFile%2F189%2F184&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> (<a href="#CITEREFBoyer1991">Boyer 1991</a>, "Ionia and the Pythagoreans" p. 43) </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> de Laet, Siegfried J. (1996). History of Humanity: Scientific and Cultural Development. <a href="/wiki/UNESCO" title="UNESCO">UNESCO</a>, Volume 3, p. 14. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/92-3-102812-X" title="Special:BookSources/92-3-102812-X">92-3-102812-X</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> Boyer, Carl B. and <a href="/wiki/Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a> (2010). A History of Mathematics. John Wiley and Sons, Chapter IV. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-470-63056-6" title="Special:BookSources/0-470-63056-6">0-470-63056-6</a> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> C. B. Boyer (1968) </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamuel_Enoch_Stumpf" class="citation book cs1">Samuel Enoch Stumpf. Socrates to Sartre. p. 11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Socrates+to+Sartre&rft.pages=11&rft.au=Samuel+Enoch+Stumpf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> F.E. Peters, Greek Philosophical Terms, New York University Press, 1967. </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornford1957" class="citation book cs1">Cornford, Francis MacDonald (1957) [1939]. <a rel="nofollow" class="external text" href="https://www.bard.edu/library/arendt/pdfs/Cornford-Parmenides.pdf">Plato and Parmenides: Parmenides' Way of Truth and Plato's Parmenides translated with an introduction and running commentary</a> (PDF). Liberal Arts Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Plato+and+Parmenides%3A+Parmenides%27+Way+of+Truth+and+Plato%27s+Parmenides+translated+with+an+introduction+and+running+commentary&rft.pub=Liberal+Arts+Press&rft.date=1957&rft.aulast=Cornford&rft.aufirst=Francis+MacDonald&rft_id=https%3A%2F%2Fwww.bard.edu%2Flibrary%2Farendt%2Fpdfs%2FCornford-Parmenides.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._J._Hollingdale1974" class="citation book cs1">R. J. Hollingdale (1974). Western Philosophy: an introduction. p. 73.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Western+Philosophy%3A+an+introduction&rft.pages=73&rft.date=1974&rft.au=R.+J.+Hollingdale&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCornford1912" class="citation book cs1">Cornford, Francis MacDonald (1912). <a rel="nofollow" class="external text" href="https://www.wilbourhall.org/pdfs/From_religion_to_philosophy.pdf">From religion to philosophy: A study in the origins of western speculation</a> (PDF). Longmans, Green and Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+religion+to+philosophy%3A+A+study+in+the+origins+of+western+speculation&rft.pub=Longmans%2C+Green+and+Co.&rft.date=1912&rft.aulast=Cornford&rft.aufirst=Francis+MacDonald&rft_id=https%3A%2F%2Fwww.wilbourhall.org%2Fpdfs%2FFrom_religion_to_philosophy.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> Kneale p. 15 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=DPoqAAAAMAAJ&pg=PA170">"The Numismatic Circular"</a>. 2018-04-02. Retrieved 2018-04-02 – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Numismatic+Circular&rft.date=2018-04-02&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDPoqAAAAMAAJ%26pg%3DPA170&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> Kneale p. 17 </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> "forming an opinion is talking, and opinion is speech that is held not with someone else or aloud but in silence with oneself" Theaetetus 189E–190A </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> Kneale p. 20. For example, the proof given in the Meno that the square on the diagonal is double the area of the original square presumably involves the forms of the square and the triangle, and the necessary relation between them </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> Kneale p. 21 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> Zalta, Edward N. "<a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/aristotle-logic/#Def">Aristotle's Logic</a>". <a href="/wiki/Stanford_University" title="Stanford University">Stanford University</a>, 18 March 2000. Retrieved 13 March 2010. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> See e.g. <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/aristotle-logic/">Aristotle's logic</a>, Stanford Encyclopedia of Philosophy </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSowa2000" class="citation book cs1">Sowa, John F. (2000). Knowledge representation: logical, philosophical, and computational foundations. Pacific Grove: Brooks/Cole. p. 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-94965-7" title="Special:BookSources/0-534-94965-7"><bdi>0-534-94965-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/38239202">38239202</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Knowledge+representation%3A+logical%2C+philosophical%2C+and+computational+foundations&rft.place=Pacific+Grove&rft.pages=2&rft.pub=Brooks%2FCole&rft.date=2000&rft_id=info%3Aoclcnum%2F38239202&rft.isbn=0-534-94965-7&rft.aulast=Sowa&rft.aufirst=John+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-Bochenski_p._63-47">^ <a href="#cite_ref-Bochenski_p._63_47-0">a</a> <a href="#cite_ref-Bochenski_p._63_47-1">b</a> Bochenski p. 63 </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> "Throughout later antiquity two great schools of logic were distinguished, the Peripatetic which was derived from Aristotle, and the Stoic which was developed by Chrysippus from the teachings of the Megarians" – Kneale p. 113 </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> Oxford Companion, article "Chrysippus", p. 134 </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <a rel="nofollow" class="external autonumber" href="http://plato.stanford.edu/entries/logic-ancient/">[1]</a> Stanford Encyclopedia of Philosophy: <a href="/wiki/Susanne_Bobzien" title="Susanne Bobzien">Susanne Bobzien</a>, Ancient Logic </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> K. Hülser, Die Fragmente zur Dialektik der Stoiker, 4 vols, Stuttgart 1986–1987 </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> Kneale 117–158 </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> Metaphysics Eta 3, 1046b 29 </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <a href="/wiki/Boethius" title="Boethius">Boethius</a>, Commentary on the Perihermenias, Meiser p. 234 </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <a href="/wiki/Epictetus" title="Epictetus">Epictetus</a>, Dissertationes ed. Schenkel ii. 19. I. </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> Alexander p. 177 </li> <li id="cite_note-sextus-adv-math-57">^ <a href="#cite_ref-sextus-adv-math_57-0">a</a> <a href="#cite_ref-sextus-adv-math_57-1">b</a> Sextus Empiricus, Adv. Math. viii, Section 113 </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> Sextus Empiricus, Hypotyp. ii. 110, comp. </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> Cicero, Academica, ii. 47, de Fato, 6. </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> See e.g. Lukasiewicz p. 21 </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> Sextus Bk viii., Sections 11, 12 </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> See e.g. <a rel="nofollow" class="external text" href="http://www.rep.routledge.com/article/H057">Routledge Encyclopedia of Philosophy Online Version 2.0</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220606082214/https://www.rep.routledge.com/articles/islamic-philosophy;jsessionid=B31B033F077DD5E68E09CC9D35C02105">Archived</a> 2022-06-06 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, article 'Islamic philosophy' </li> <li id="cite_note-Britannica-63">^ <a href="#cite_ref-Britannica_63-0">a</a> <a href="#cite_ref-Britannica_63-1">b</a> <a rel="nofollow" class="external text" href="http://www.britannica.com/ebc/article-65928">History of logic: Arabic logic</a>, <a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Encyclopædia Britannica</a>. </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeldman1964" class="citation journal cs1">Feldman, Seymour (1964-11-26). "Rescher on Arabic Logic". The Journal of Philosophy. 61 (22). Journal of Philosophy, Inc.: 724–734. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2023632">10.2307/2023632</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-362X">0022-362X</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2023632">2023632</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Philosophy&rft.atitle=Rescher+on+Arabic+Logic&rft.volume=61&rft.issue=22&rft.pages=724-734&rft.date=1964-11-26&rft.issn=0022-362X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2023632%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2023632&rft.aulast=Feldman&rft.aufirst=Seymour&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> [726]. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLongSedley1987" class="citation book cs1">Long, A. A.; Sedley, D. N. (1987). The Hellenistic Philosophers. Vol 1: Translations of the principal sources with philosophical commentary. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-27556-3" title="Special:BookSources/0-521-27556-3"><bdi>0-521-27556-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Hellenistic+Philosophers.+Vol+1%3A+Translations+of+the+principal+sources+with+philosophical+commentary&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=1987&rft.isbn=0-521-27556-3&rft.aulast=Long&rft.aufirst=A.+A.&rft.au=Sedley%2C+D.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-Hasse-65"><a href="#cite_ref-Hasse_65-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHasse2008" class="citation encyclopaedia cs1">Hasse, Dag Nikolaus (2008-09-19). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/arabic-islamic-influence/">"Influence of Arabic and Islamic Philosophy on the Latin West"</a>. <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>. Retrieved 2009-10-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Influence+of+Arabic+and+Islamic+Philosophy+on+the+Latin+West&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2008-09-19&rft.aulast=Hasse&rft.aufirst=Dag+Nikolaus&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Farabic-islamic-influence%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> Richard F. Washell (1973), "Logic, Language, and Albert the Great", Journal of the History of Ideas 34 (3), pp. 445–450 [445]. </li> <li id="cite_note-Goodman-67">^ <a href="#cite_ref-Goodman_67-0">a</a> <a href="#cite_ref-Goodman_67-1">b</a> Goodman, Lenn Evan (2003), Islamic Humanism, p. 155, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-513580-6" title="Special:BookSources/0-19-513580-6">0-19-513580-6</a>. </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> Goodman, Lenn Evan (1992); Avicenna, p. 188, <a href="/wiki/Routledge" title="Routledge">Routledge</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-01929-X" title="Special:BookSources/0-415-01929-X">0-415-01929-X</a>. </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> Kneale p. 229 </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> Kneale: p. 266; Ockham: <a href="/wiki/Summa_Logicae" class="mw-redirect" title="Summa Logicae">Summa Logicae</a> i. 14; Avicenna: Avicennae Opera Venice 1508 f87rb </li> <li id="cite_note-Iqbal-71">^ <a href="#cite_ref-Iqbal_71-0">a</a> <a href="#cite_ref-Iqbal_71-1">b</a> <a href="/wiki/Muhammad_Iqbal" title="Muhammad Iqbal">Muhammad Iqbal</a>, <a href="/wiki/The_Reconstruction_of_Religious_Thought_in_Islam" title="The Reconstruction of Religious Thought in Islam">The Reconstruction of Religious Thought in Islam</a>, "The Spirit of Muslim Culture" (<a href="/wiki/Cf." title="Cf.">cf.</a> <a rel="nofollow" class="external autonumber" href="http://www.allamaiqbal.com/works/prose/english/reconstruction">[2]</a> and <a rel="nofollow" class="external autonumber" href="http://www.witness-pioneer.org/vil/Books/MI_RRTI/chapter_05.htm">[3]</a>) </li> <li id="cite_note-Stanford-72">^ <a href="#cite_ref-Stanford_72-0">a</a> <a href="#cite_ref-Stanford_72-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTony_Street2008" class="citation encyclopaedia cs1">Tony Street (2008-07-23). <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/arabic-islamic-language">"Arabic and Islamic Philosophy of Language and Logic"</a>. <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>. Retrieved 2008-12-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Arabic+and+Islamic+Philosophy+of+Language+and+Logic&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.date=2008-07-23&rft.au=Tony+Street&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Farabic-islamic-language&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> Lotfollah Nabavi, <a rel="nofollow" class="external text" href="http://public.ut.ac.ir/html/fac/lit/articles.html">Sohrevardi's Theory of Decisive Necessity and kripke's QSS System</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080126100838/http://public.ut.ac.ir/html/fac/lit/articles.html">Archived</a> 2008-01-26 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Journal of Faculty of Literature and Human Sciences. </li> <li id="cite_note-Roubi-74"><a href="#cite_ref-Roubi_74-0">^</a> Abu Shadi Al-Roubi (1982), "Ibn Al-Nafis as a philosopher", Symposium on Ibn al-Nafis, Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait (<a href="/wiki/Cf." title="Cf.">cf.</a> <a rel="nofollow" class="external text" href="http://www.islamset.com/isc/nafis/drroubi.html">Ibn al-Nafis As a Philosopher</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080206072116/http://www.islamset.com/isc/nafis/drroubi.html">Archived</a> 2008-02-06 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Encyclopedia of Islamic World). </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> See pp. 253–254 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreet2005" class="citation book cs1">Street, Tony (2005). "Logic". In Peter Adamson; Richard C. Taylor (eds.). The Cambridge Companion to Arabic Philosophy. Cambridge University Press. pp. 247–265. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-52069-0" title="Special:BookSources/978-0-521-52069-0"><bdi>978-0-521-52069-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logic&rft.btitle=The+Cambridge+Companion+to+Arabic+Philosophy&rft.pages=247-265&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=978-0-521-52069-0&rft.aulast=Street&rft.aufirst=Tony&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRuth_Mas1998" class="citation journal cs1">Ruth Mas (1998). <a rel="nofollow" class="external text" href="http://www.colorado.edu/ReligiousStudies/faculty/mas/LOGIC.pdf">"Qiyas: A Study in Islamic Logic"</a> (PDF). Folia Orientalia. 34: 113–128. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0015-5675">0015-5675</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Folia+Orientalia&rft.atitle=Qiyas%3A+A+Study+in+Islamic+Logic&rft.volume=34&rft.pages=113-128&rft.date=1998&rft.issn=0015-5675&rft.au=Ruth+Mas&rft_id=http%3A%2F%2Fwww.colorado.edu%2FReligiousStudies%2Ffaculty%2Fmas%2FLOGIC.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-Sowa-77">^ <a href="#cite_ref-Sowa_77-0">a</a> <a href="#cite_ref-Sowa_77-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_F._SowaArun_K._Majumdar2003" class="citation conference cs1"><a href="/wiki/John_F._Sowa" title="John F. Sowa">John F. Sowa</a>; Arun K. Majumdar (2003). <a rel="nofollow" class="external text" href="http://www.jfsowa.com/pubs/analog.htm">"Analogical reasoning"</a>. Conceptual Structures for Knowledge Creation and Communication, Proceedings of ICCS 2003. Berlin: Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Analogical+reasoning&rft.btitle=Conceptual+Structures+for+Knowledge+Creation+and+Communication%2C+Proceedings+of+ICCS+2003&rft.place=Berlin&rft.pub=Springer-Verlag&rft.date=2003&rft.au=John+F.+Sowa&rft.au=Arun+K.+Majumdar&rft_id=http%3A%2F%2Fwww.jfsowa.com%2Fpubs%2Fanalog.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988">, pp. 16–36 </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> <a href="/wiki/Nicholas_Rescher" title="Nicholas Rescher">Nicholas Rescher</a> and Arnold vander Nat, "The Arabic Theory of Temporal Modal Syllogistic", in George Fadlo Hourani (1975), Essays on Islamic Philosophy and Science, pp. 189–221, <a href="/wiki/State_University_of_New_York_Press" class="mw-redirect" title="State University of New York Press">State University of New York Press</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87395-224-3" title="Special:BookSources/0-87395-224-3">0-87395-224-3</a>. </li> <li id="cite_note-Kneale198-79"><a href="#cite_ref-Kneale198_79-0">^</a> Kneale p. 198 </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> Stephen Dumont, article "Peter Abelard" in Gracia and Noone p. 492 </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> Kneale, pp. 202–203 </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> See e.g. Kneale p. 225 </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> Boehner p. 1 </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> Boehner pp. 19–76 </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> Boehner p. 29 </li> <li id="cite_note-86"><a href="#cite_ref-86">^</a> Boehner p. 30 </li> <li id="cite_note-87"><a href="#cite_ref-87">^</a> Ebbesen 1981 </li> <li id="cite_note-88"><a href="#cite_ref-88">^</a> Boehner pp. 54–55 </li> <li id="cite_note-89"><a href="#cite_ref-89">^</a> Oxford Companion p. 504, article "Traditional logic" </li> <li id="cite_note-Buroker_xxiii-90">^ <a href="#cite_ref-Buroker_xxiii_90-0">a</a> <a href="#cite_ref-Buroker_xxiii_90-1">b</a> Buroker xxiii </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> (Locke, An Essay Concerning Human Understanding, IV. 5. 6) </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> Farrington, 1964, 89 </li> <li id="cite_note-93"><a href="#cite_ref-93">^</a> N. Abbagnano, "Psychologism" in P. Edwards (ed) The Encyclopaedia of Philosophy, MacMillan, 1967 </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> Of the German literature in this period, Robert Adamson wrote "Logics swarm as bees in springtime..."; Robert Adamson, A Short History of Logic, Wm. Blackwood & Sons, 1911, page 242 </li> <li id="cite_note-95"><a href="#cite_ref-95">^</a> Carl von Prantl (1855–1867), Geschichte von Logik in Abendland, Leipzig: S. Hirzl, anastatically reprinted in 1997, Hildesheim: Georg Olds. </li> <li id="cite_note-96"><a href="#cite_ref-96">^</a> See e.g. <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/psychologism">Psychologism</a>, Stanford Encyclopedia of Philosophy </li> <li id="cite_note-97"><a href="#cite_ref-97">^</a> Wilhelm Wundt, Logik (1880–1883); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, pp. 115–116. </li> <li id="cite_note-98"><a href="#cite_ref-98">^</a> Theodor Lipps, Grundzüge der Logik (1893); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 40 </li> <li id="cite_note-99"><a href="#cite_ref-99">^</a> Christoph von Sigwart, Logik (1873–1878); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 51 </li> <li id="cite_note-100"><a href="#cite_ref-100">^</a> Benno Erdmann, Logik (1892); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 96 </li> <li id="cite_note-101"><a href="#cite_ref-101">^</a> Dermot Moran, "Introduction"; Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. xxi </li> <li id="cite_note-102"><a href="#cite_ref-102">^</a> Michael Dummett, "Preface"; Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. xvii </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> Josiah Royce, "Recent Logical Enquiries and their Psychological Bearings" (1902) in John J. McDermott (ed) The Basic Writings of Josiah Royce Volume 2, Fordham University Press, 2005, p. 661 </li> <li id="cite_note-104"><a href="#cite_ref-104">^</a> Bochenski, p. 266 </li> <li id="cite_note-105"><a href="#cite_ref-105">^</a> Peirce 1896 </li> <li id="cite_note-106"><a href="#cite_ref-106">^</a> See Bochenski p. 269 </li> <li id="cite_note-107"><a href="#cite_ref-107">^</a> Oxford Companion p. 499 </li> <li id="cite_note-108"><a href="#cite_ref-108">^</a> Edith Sylla (1999), "Oxford Calculators", in The Cambridge Dictionary of Philosophy, Cambridge, Cambridgeshire: Cambridge. </li> <li id="cite_note-109"><a href="#cite_ref-109">^</a> El. philos. sect. I de corp 1.1.2. </li> <li id="cite_note-110"><a href="#cite_ref-110">^</a> Bochenski p. 274 </li> <li id="cite_note-111"><a href="#cite_ref-111">^</a> Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., The Cambridge Companion to Leibniz. Cambridge Univ. Press. </li> <li id="cite_note-112"><a href="#cite_ref-112">^</a> Wiener, Philip, 1951. Leibniz: Selections. Scribner. </li> <li id="cite_note-113"><a href="#cite_ref-113">^</a> Essai de dialectique rationelle, 211n, quoted in Bochenski p. 277. </li> <li id="cite_note-114"><a href="#cite_ref-114">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBolzano1972" class="citation book cs1">Bolzano, Bernard (1972). George, Rolf (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oA1NDDirneQC&q=%22deducible%20from%20propositions%22&pg=PA209">The Theory of Science: Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik</a>. Translated by Rolf, George. <a href="/wiki/University_of_California_Press" title="University of California Press">University of California Press</a>. p. 209. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-52001787-0" title="Special:BookSources/978-0-52001787-0"><bdi>978-0-52001787-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Science%3A+Die+Wissenschaftslehre+oder+Versuch+einer+Neuen+Darstellung+der+Logik&rft.pages=209&rft.pub=University+of+California+Press&rft.date=1972&rft.isbn=978-0-52001787-0&rft.aulast=Bolzano&rft.aufirst=Bernard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DoA1NDDirneQC%26q%3D%2522deducible%2520from%2520propositions%2522%26pg%3DPA209&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-115"><a href="#cite_ref-115">^</a> See e.g. Bochenski p. 296 and passim </li> <li id="cite_note-116"><a href="#cite_ref-116">^</a> Before publishing, he wrote to <a href="/wiki/Augustus_De_Morgan" title="Augustus De Morgan">De Morgan</a>, who was just finishing his work Formal Logic. De Morgan suggested they should publish first, and thus the two books appeared at the same time, possibly even reaching the bookshops on the same day. cf. Kneale p. 404 </li> <li id="cite_note-117"><a href="#cite_ref-117">^</a> Kneale p. 404 </li> <li id="cite_note-Kneale_p._407-118">^ <a href="#cite_ref-Kneale_p._407_118-0">a</a> <a href="#cite_ref-Kneale_p._407_118-1">b</a> <a href="#cite_ref-Kneale_p._407_118-2">c</a> Kneale p. 407 </li> <li id="cite_note-119"><a href="#cite_ref-119">^</a> Boole (1847) p. 16 </li> <li id="cite_note-120"><a href="#cite_ref-120">^</a> Boole 1847 pp. 58–59 </li> <li id="cite_note-121"><a href="#cite_ref-121">^</a> Beaney p. 11 </li> <li id="cite_note-122"><a href="#cite_ref-122">^</a> Kneale p. 422 </li> <li id="cite_note-123"><a href="#cite_ref-123">^</a> Peirce, "A Boolian Algebra with One Constant", 1880 MS, Collected Papers v. 4, paragraphs 12–20, reprinted Writings v. 4, pp. 218–221. Google <a rel="nofollow" class="external text" href="https://archive.org/details/writingsofcharle0002peir">Preview</a>. </li> <li id="cite_note-124"><a href="#cite_ref-124">^</a> Trans. Amer. Math. Soc., xiv (1913), pp. 481–488. This is now known as the <a href="/wiki/Sheffer_stroke" title="Sheffer stroke">Sheffer stroke</a> </li> <li id="cite_note-125"><a href="#cite_ref-125">^</a> Bochenski 296 </li> <li id="cite_note-126"><a href="#cite_ref-126">^</a> See CP III </li> <li id="cite_note-127"><a href="#cite_ref-127">^</a> <a href="/wiki/George_Boole" title="George Boole">George Boole</a>. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by J. Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review. 24 (2004) 167–169. </li> <li id="cite_note-128"><a href="#cite_ref-128">^</a> JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288. </li> <li id="cite_note-k435-129">^ <a href="#cite_ref-k435_129-0">a</a> <a href="#cite_ref-k435_129-1">b</a> Kneale p. 435 </li> <li id="cite_note-130"><a href="#cite_ref-130">^</a> Jevons, The Principles of Science, London 1879, p. 156, quoted in Grundlagen 15 </li> <li id="cite_note-131"><a href="#cite_ref-131">^</a> Beaney p. 10 – the completeness of Frege's system was eventually proved by <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a> in 1934 </li> <li id="cite_note-132"><a href="#cite_ref-132">^</a> See for example the argument by the medieval logician <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a> that singular propositions are universal, in <a href="/wiki/Summa_Logicae" class="mw-redirect" title="Summa Logicae">Summa Logicae</a> III. 8 (??) </li> <li id="cite_note-133"><a href="#cite_ref-133">^</a> <a href="#CITEREFFrege1879">Frege 1879</a> in <a href="#CITEREFvan_Heijenoort1967">van Heijenoort 1967</a>, p. 7 </li> <li id="cite_note-134"><a href="#cite_ref-134">^</a> "On concept and object" p. 198; Geach p. 48 </li> <li id="cite_note-135"><a href="#cite_ref-135">^</a> BLC p. 14, quoted in Beaney p. 12 </li> <li id="cite_note-136"><a href="#cite_ref-136">^</a> See e.g. <a rel="nofollow" class="external text" href="http://www.utm.edu/research/iep/f/frege.htm">The Internet Encyclopedia of Philosophy</a>, article "Frege" </li> <li id="cite_note-137"><a href="#cite_ref-137">^</a> Van Heijenoort 1967, p. 83 </li> <li id="cite_note-138"><a href="#cite_ref-138">^</a> See e.g. Potter 2004 </li> <li id="cite_note-139"><a href="#cite_ref-139">^</a> Zermelo 1908 </li> <li id="cite_note-140"><a href="#cite_ref-140">^</a> Feferman 1999 p. 1 </li> <li id="cite_note-141"><a href="#cite_ref-141">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGirardTaylorLafont1990" class="citation book cs1"><a href="/wiki/Jean-Yves_Girard" title="Jean-Yves Girard">Girard, Jean-Yves</a>; Taylor, Paul; Lafont, Yves (1990) [1989]. <a rel="nofollow" class="external text" href="https://archive.org/details/proofstypes0000gira">Proofs and Types</a>. Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-37181-3" title="Special:BookSources/0-521-37181-3"><bdi>0-521-37181-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Types&rft.pub=Cambridge+University+Press+%28Cambridge+Tracts+in+Theoretical+Computer+Science%2C+7%29&rft.date=1990&rft.isbn=0-521-37181-3&rft.aulast=Girard&rft.aufirst=Jean-Yves&rft.au=Taylor%2C+Paul&rft.au=Lafont%2C+Yves&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fproofstypes0000gira&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-142"><a href="#cite_ref-142">^</a> Feferman and Feferman 2004, p. 122, discussing "The Impact of Tarski's Theory of Truth". </li> <li id="cite_note-143"><a href="#cite_ref-143">^</a> Feferman 1999, p. 1 </li> <li id="cite_note-144"><a href="#cite_ref-144">^</a> See e.g. Barwise, Handbook of Mathematical Logic </li> <li id="cite_note-145"><a href="#cite_ref-145">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1964" class="citation journal cs1">Cohen, Paul J. (1964). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611">"The Independence of the Continuum Hypothesis, II"</a>. Proceedings of the National Academy of Sciences of the United States of America. 51 (1): 105–110. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PNAS...51..105C">1964PNAS...51..105C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.51.1.105">10.1073/pnas.51.1.105</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/72252">72252</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611">300611</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16591132">16591132</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=The+Independence+of+the+Continuum+Hypothesis%2C+II&rft.volume=51&rft.issue=1&rft.pages=105-110&rft.date=1964&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC300611%23id-name%3DPMC&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F72252%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1964PNAS...51..105C&rft_id=info%3Apmid%2F16591132&rft_id=info%3Adoi%2F10.1073%2Fpnas.51.1.105&rft.aulast=Cohen&rft.aufirst=Paul+J.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC300611&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> </li> <li id="cite_note-146"><a href="#cite_ref-146">^</a> Many of the foundational papers are collected in The Undecidable (1965) edited by Martin Davis </li> <li id="cite_note-147"><a href="#cite_ref-147">^</a> Jerry Fodor, "<a rel="nofollow" class="external text" href="http://www.lrb.co.uk/v26/n20/jerry-fodor/waters-water-everywhere">Water's water everywhere</a>", London Review of Books, 21 October 2004 </li> <li id="cite_note-148"><a href="#cite_ref-148">^</a> See Philosophical Analysis in the Twentieth Century: Volume 2: The Age of Meaning, Scott Soames: "Naming and Necessity is among the most important works ever, ranking with the classical work of Frege in the late nineteenth century, and of Russell, Tarski and Wittgenstein in the first half of the twentieth century". Cited in Byrne, Alex and Hall, Ned. 2004. 'Necessary Truths'. Boston Review October/November 2004 </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=41" title="Edit section: References">edit</a>]</div> <dl><dt>Primary Sources</dt></dl> <ul><li><a href="/wiki/Alexander_of_Aphrodisias" title="Alexander of Aphrodisias">Alexander of Aphrodisias</a>, In Aristotelis An. Pr. Lib. I Commentarium, ed. Wallies, Berlin, C.I.A.G. vol. II/1, 1882.</li> <li>Avicenna, Avicennae Opera Venice 1508.</li> <li><a href="/wiki/Boethius" title="Boethius">Boethius</a> Commentary on the Perihermenias, Secunda Editio, ed. Meiser, Leipzig, Teubner, 1880.</li> <li><a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bolzano, Bernard</a> Wissenschaftslehre, (1837) 4 Bde, Neudr., hrsg. W. Schultz, Leipzig I–II 1929, III 1930, IV 1931 (Theory of Science, four volumes, translated by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014).</li> <li>Bolzano, Bernard Theory of Science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell – <a href="/wiki/D._Reidel_Publishing_Company" class="mw-redirect" title="D. Reidel Publishing Company">D. Reidel Publishing Company</a>, Dordrecht and Boston 1973).</li> <li><a href="/wiki/George_Boole" title="George Boole">Boole, George</a> (1847) The Mathematical Analysis of Logic (Cambridge and London); repr. in Studies in Logic and Probability, ed. <a href="/wiki/Rush_Rhees" title="Rush Rhees">R. Rhees</a> (London 1952).</li> <li>Boole, George (1854) The Laws of Thought (London and Cambridge); repr. as Collected Logical Works. Vol. 2, (Chicago and London: <a href="/wiki/Open_Court_Publishing_Company" title="Open Court Publishing Company">Open Court</a>, 1940).</li> <li><a href="/wiki/Epictetus" title="Epictetus">Epictetus</a>, Epicteti Dissertationes ab Arriano digestae, edited by Heinrich Schenkl, Leipzig, Teubner. 1894.</li> <li>Frege, G., Boole's Logical Calculus and the Concept Script, 1882, in Posthumous Writings transl. P. Long and R. White 1969, pp. 9–46.</li> <li><a href="/wiki/Joseph_Diaz_Gergonne" class="mw-redirect" title="Joseph Diaz Gergonne">Gergonne, Joseph Diaz</a>, (1816) Essai de dialectique rationelle, in <a href="/wiki/Annales_de_math%C3%A9matiques_pures_et_appliqu%C3%A9es" class="mw-redirect" title="Annales de mathématiques pures et appliquées">Annales de mathématiques pures et appliquées</a> 7, 1816/1817, 189–228.</li> <li>Jevons, W. S. The Principles of Science, London 1879.</li> <li>Ockham's Theory of Terms: Part I of the <a href="/wiki/Summa_Logicae" class="mw-redirect" title="Summa Logicae">Summa Logicae</a>, translated and introduced by Michael J. Loux (Notre Dame, IN: <a href="/wiki/University_of_Notre_Dame_Press" title="University of Notre Dame Press">University of Notre Dame Press</a> 1974). Reprinted: South Bend, IN: St. Augustine's Press, 1998.</li> <li>Ockham's Theory of Propositions: Part II of the Summa Logicae, translated by Alfred J. Freddoso and Henry Schuurman and introduced by Alfred J. Freddoso (Notre Dame, IN: University of Notre Dame Press, 1980). Reprinted: South Bend, IN: St. Augustine's Press, 1998.</li> <li><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, C. S.</a>, (1896), "The Regenerated Logic", The Monist, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pa0LAAAAIAAJ">vol. VII</a>, No. 1, p <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pa0LAAAAIAAJ&pg=PA19">pp. 19</a>–40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425–455). Internet Archive <a rel="nofollow" class="external text" href="https://archive.org/details/monistquart07hegeuoft">The Monist 7</a>.</li> <li><a href="/wiki/Sextus_Empiricus" title="Sextus Empiricus">Sextus Empiricus</a>, Against the Logicians. (Adversus Mathematicos VII and VIII). Richard Bett (trans.) Cambridge: Cambridge University Press, 2005. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-53195-0" title="Special:BookSources/0-521-53195-0">0-521-53195-0</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZermelo1908" class="citation journal cs1"><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Zermelo, Ernst</a> (1908). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170908192040/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1">"Untersuchungen über die Grundlagen der Mengenlehre I"</a>. Mathematische Annalen. 65 (2): 261–281. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01449999">10.1007/BF01449999</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120085563">120085563</a>. Archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1">the original</a> on 2017-09-08. Retrieved 2013-09-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Untersuchungen+%C3%BCber+die+Grundlagen+der+Mengenlehre+I&rft.volume=65&rft.issue=2&rft.pages=261-281&rft.date=1908&rft_id=info%3Adoi%2F10.1007%2FBF01449999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120085563%23id-name%3DS2CID&rft.aulast=Zermelo&rft.aufirst=Ernst&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DPPN235181684_0065%26DMDID%3DDMDLOG_0018%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> English translation in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Heijenoort1967" class="citation book cs1"><a href="/wiki/Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a> (1967). "Investigations in the foundations of set theory". From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Harvard Univ. Press. pp. 199–215. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-674-32449-7" title="Special:BookSources/978-0-674-32449-7"><bdi>978-0-674-32449-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Investigations+in+the+foundations+of+set+theory&rft.btitle=From+Frege+to+G%C3%B6del%3A+A+Source+Book+in+Mathematical+Logic%2C+1879%E2%80%931931&rft.series=Source+Books+in+the+History+of+the+Sciences&rft.pages=199-215&rft.pub=Harvard+Univ.+Press&rft.date=1967&rft.isbn=978-0-674-32449-7&rft.aulast=van+Heijenoort&rft.aufirst=Jean&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrege1879" class="citation book cs1">Frege, Gottlob (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Begriffsschrift%2C+a+formula+language%2C+modeled+upon+that+of+arithmetic%2C+for+pure+thought&rft.date=1879&rft.aulast=Frege&rft.aufirst=Gottlob&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"> translated in van Heijenoort 1967.</li></ul> <dl><dt>Secondary Sources</dt></dl> <ul><li><a href="/wiki/Jon_Barwise" title="Jon Barwise">Barwise, Jon</a>, (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam, North Holland, 1982 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-86388-1" title="Special:BookSources/978-0-444-86388-1">978-0-444-86388-1</a> .</li> <li>Beaney, Michael, The Frege Reader, London: Blackwell 1997.</li> <li><a href="/wiki/J%C3%B3zef_Maria_Boche%C5%84ski" title="Józef Maria Bocheński">Bochenski</a>, I. M., A History of Formal Logic, Indiana, Notre Dame University Press, 1961.</li> <li><a href="/wiki/Philotheus_Boehner" title="Philotheus Boehner">Boehner, Philotheus</a>, Medieval Logic, Manchester 1950.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyer1991" class="citation cs2"><a href="/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, C.B.</a> (1991) [1989], <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye">A History of Mathematics</a> (2nd ed.), New York: Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.place=New+York&rft.edition=2nd&rft.pub=Wiley&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=C.B.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"></li> <li>Buroker, Jill Vance (transl. and introduction), A. Arnauld, P. Nicole Logic or the Art of Thinking, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, 1996, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-48249-6" title="Special:BookSources/0-521-48249-6">0-521-48249-6</a>.</li> <li><a href="/wiki/Alonzo_Church" title="Alonzo Church">Church, Alonzo</a>, 1936–1938. "A bibliography of symbolic logic". Journal of Symbolic Logic 1: 121–218; 3:178–212.</li> <li><a href="/wiki/Everard_de_Jong" title="Everard de Jong">de Jong, Everard</a> (1989), <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a>'s "Logical Treatises" and <a href="/wiki/Giacomo_Zabarella" class="mw-redirect" title="Giacomo Zabarella">Giacomo Zabarella</a>'s "Opera Logica": A Comparison, PhD dissertation, Washington, DC: Catholic University of America.</li> <li>Ebbesen, Sten "Early supposition theory (12th–13th Century)" Histoire, Épistémologie, Langage 3/1: 35–48 (1981).</li> <li>Farrington, B., The Philosophy of <a href="/wiki/Francis_Bacon" title="Francis Bacon">Francis Bacon</a>, Liverpool 1964.</li> <li>Feferman, Anita B. (1999). "Alfred Tarski". <a href="/wiki/American_National_Biography" title="American National Biography">American National Biography</a>. 21. <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. pp. 330–332. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-512800-0" title="Special:BookSources/978-0-19-512800-0">978-0-19-512800-0</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="F-F" class="citation book cs1">Feferman, Anita B.; <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Feferman, Solomon</a> (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/alfredtarskilife0000fefe">Alfred Tarski: Life and Logic</a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-80240-6" title="Special:BookSources/978-0-521-80240-6"><bdi>978-0-521-80240-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/54691904">54691904</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alfred+Tarski%3A+Life+and+Logic&rft.pub=Cambridge+University+Press&rft.date=2004&rft_id=info%3Aoclcnum%2F54691904&rft.isbn=978-0-521-80240-6&rft.aulast=Feferman&rft.aufirst=Anita+B.&rft.au=Feferman%2C+Solomon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falfredtarskilife0000fefe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"></li> <li><a href="/wiki/Dov_Gabbay" title="Dov Gabbay">Gabbay, Dov</a> and <a href="/wiki/John_Woods_(logician)" title="John Woods (logician)">John Woods</a>, eds, Handbook of the History of Logic 2004. 1. Greek, Indian and Arabic logic; 2. Mediaeval and Renaissance logic; 3. The rise of modern logic: from Leibniz to Frege; 4. British logic in the Nineteenth century; 5. Logic from Russell to Church; 6. Sets and extensions in the Twentieth century; 7. Logic and the modalities in the Twentieth century; 8. The many-valued and nonmonotonic turn in logic; 9. Computational Logic; 10. Inductive logic; 11. Logic: A history of its central concepts; <a href="/wiki/Elsevier" title="Elsevier">Elsevier</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-51611-5" title="Special:BookSources/0-444-51611-5">0-444-51611-5</a>.</li> <li>Geach, P. T. Logic Matters, Blackwell 1972.</li> <li>Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-513580-6" title="Special:BookSources/0-19-513580-6">0-19-513580-6</a>.</li> <li>Goodman, Lenn Evan (1992). Avicenna. Routledge, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-415-01929-X" title="Special:BookSources/0-415-01929-X">0-415-01929-X</a>.</li> <li><a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a>, 2000. The Search for Mathematical Roots 1870–1940. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>.</li> <li>Gracia, J. G. and Noone, T. B., A Companion to Philosophy in the Middle Ages, London 2003.</li> <li><a href="/wiki/Leila_Haaparanta" title="Leila Haaparanta">Haaparanta, Leila</a> (ed.) 2009. The Development of Modern Logic Oxford University Press.</li> <li><a href="/wiki/T._L._Heath" class="mw-redirect" title="T. L. Heath">Heath, T. L.</a>, 1949. Mathematics in Aristotle, Oxford University Press.</li> <li>Heath, T. L., 1931, A Manual of Greek Mathematics, Oxford (<a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>).</li> <li>Honderich, Ted (ed.). <a href="/wiki/The_Oxford_Companion_to_Philosophy" title="The Oxford Companion to Philosophy">The Oxford Companion to Philosophy</a> (New York: Oxford University Press, 1995) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-866132-0" title="Special:BookSources/0-19-866132-0">0-19-866132-0</a>.</li> <li><a href="/wiki/William_Kneale_(logician)" class="mw-redirect" title="William Kneale (logician)">Kneale, William</a> and Martha, 1962. The development of logic. Oxford University Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-824773-7" title="Special:BookSources/0-19-824773-7">0-19-824773-7</a>.</li> <li><a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Lukasiewicz</a>, Aristotle's Syllogistic, Oxford University Press 1951.</li> <li>Potter, Michael (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FxRoPuPbGgUC&q=logic">Set Theory and its Philosophy</a>, Oxford University Press.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=History_of_logic&action=edit&section=42" title="Edit section: External links">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://www.historyoflogic.com">The History of Logic from Aristotle to Gödel</a> with annotated bibliographies on the history of logic</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBobzien" class="citation encyclopaedia cs1">Bobzien, Susanne. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-ancient/">"Ancient Logic"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Ancient+Logic&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.aulast=Bobzien&rft.aufirst=Susanne&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-ancient%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChatti" class="citation encyclopaedia cs1">Chatti, Saloua. <a rel="nofollow" class="external text" href="http://www.iep.utm.edu/av-logic">"Avicenna (Ibn Sina): Logic"</a>. <a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Avicenna+%28Ibn+Sina%29%3A+Logic&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft.aulast=Chatti&rft.aufirst=Saloua&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fav-logic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpruyt" class="citation encyclopaedia cs1">Spruyt, Joke. <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/peter-spain/">"Peter of Spain"</a>. In <a href="/wiki/Edward_N._Zalta" title="Edward N. Zalta">Zalta, Edward N.</a> (ed.). <a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Peter+of+Spain&rft.btitle=Stanford+Encyclopedia+of+Philosophy&rft.aulast=Spruyt&rft.aufirst=Joke&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fpeter-spain%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHistory+of+logic" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://pvspade.com/Logic/docs/thoughts1_1a.pdf">Paul Spade's "Thoughts Words and Things"</a> – An Introduction to Late Mediaeval Logic and Semantic Theory (PDF)</li> <li><a rel="nofollow" class="external text" href="http://humbox.ac.uk/5497/">Open Access pdf download; Insights, Images, Bios, and links for 178 logicians</a> by David Marans</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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0.25em"> <ul><li><a href="/wiki/Abductive_reasoning" title="Abductive reasoning">Abduction</a></li> <li><a href="/wiki/Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li> <li><a href="/wiki/Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li> <li><a href="/wiki/Consequent" title="Consequent">Consequent</a></li> <li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a> <ul><li><a href="/wiki/Paradox" title="Paradox">Paradox</a></li> <li><a href="/wiki/Antinomy" title="Antinomy">Antinomy</a></li></ul></li> <li><a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deduction</a></li> <li><a href="/wiki/Deductive_closure" title="Deductive closure">Deductive closure</a></li> <li><a href="/wiki/Definition" title="Definition">Definition</a></li> <li><a href="/wiki/Description" title="Description">Description</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Entailment</a> <ul><li><a href="/wiki/Entailment_(linguistics)" title="Entailment (linguistics)">Linguistic</a></li></ul></li> <li><a href="/wiki/Logical_form" title="Logical form">Form</a></li> <li><a href="/wiki/Inductive_reasoning" title="Inductive reasoning">Induction</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Name" title="Name">Name</a></li> <li><a href="/wiki/Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li> <li><a href="/wiki/Premise" title="Premise">Premise</a></li> <li><a href="/wiki/Probability" title="Probability">Probability</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Reference" title="Reference">Reference</a></li> <li><a href="/wiki/Statement_(logic)" title="Statement (logic)">Statement</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Truth" title="Truth">Truth</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="/wiki/Index_of_logic_articles" title="Index of logic articles">topics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li> <li><a href="/wiki/List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li> <li><a href="/wiki/List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_logicians" title="List of logicians">Logicians</a></li> <li><a href="/wiki/List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li> <li><a href="/wiki/List_of_paradoxes" title="List of paradoxes">Paradoxes</a></li> <li><a href="/wiki/List_of_fallacies" title="List of fallacies">Fallacies</a></li> <li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">Logic symbols</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/18px-Socrates.png" decoding="async" width="18" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/27px-Socrates.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Socrates.png/36px-Socrates.png 2x" data-file-width="326" data-file-height="500" /> <a href="/wiki/Portal:Philosophy" title="Portal:Philosophy">Philosophy portal</a></li> <li><a href="/wiki/Category:Logic" title="Category:Logic">Category</a></li> <li><a href="/wiki/Wikipedia:WikiProject_Logic" title="Wikipedia:WikiProject Logic">WikiProject</a> (<a href="/wiki/Wikipedia_talk:WikiProject_Logic" title="Wikipedia talk:WikiProject Logic">talk</a>)</li> <li><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Special:Recentchangeslinked&target=Template:Logic&hidebots=0">changes</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="History_of_science" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:History_of_science" title="Template:History of science"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_science" class="mw-redirect" title="Template talk:History of science"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_science" title="Special:EditPage/Template:History of science"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="History_of_science" style="font-size:114%;margin:0 4em"><a href="/wiki/History_of_science" title="History of science">History of science</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sociology_of_the_history_of_science" title="Sociology of the history of science">Theories and sociology</a></li> <li><a href="/wiki/Historiography_of_science" title="Historiography of science">Historiography</a></li> <li><a href="/wiki/History_of_pseudoscience" title="History of pseudoscience">Pseudoscience</a></li> <li><a href="/wiki/History_and_philosophy_of_science" title="History and philosophy of science">History and philosophy of science</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/80px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg" decoding="async" width="80" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/120px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg/160px-Johannes-kepler-tabulae-rudolphinae-google-arts-culture.jpg 2x" data-file-width="3992" data-file-height="5880" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By era</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Science_in_the_ancient_world" title="Science in the ancient world">Ancient world</a></li> <li><a href="/wiki/Science_in_classical_antiquity" title="Science in classical antiquity">Classical Antiquity</a></li> <li><a href="/wiki/European_science_in_the_Middle_Ages" title="European science in the Middle Ages">Medieval European</a></li> <li><a href="/wiki/History_of_science_in_the_Renaissance" class="mw-redirect" title="History of science in the Renaissance">Renaissance</a></li> <li><a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a></li> <li><a href="/wiki/Science_in_the_Age_of_Enlightenment" title="Science in the Age of Enlightenment">Age of Enlightenment</a></li> <li><a href="/wiki/Romanticism_in_science" title="Romanticism in science">Romanticism</a></li> <li><a href="/wiki/19th_century_in_science" title="19th century in science">19th century in science</a></li> <li><a href="/wiki/20th_century_in_science" title="20th century in science">20th century in science</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By culture</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_science_and_technology_in_Africa" title="History of science and technology in Africa">African</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Argentina" title="History of science and technology in Argentina">Argentine</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Brazil" class="mw-redirect" title="History of science and technology in Brazil">Brazilian</a></li> <li><a href="/wiki/Byzantine_science" title="Byzantine science">Byzantine</a></li> <li><a href="/wiki/History_of_science_and_technology_in_France" class="mw-redirect" title="History of science and technology in France">French</a></li> <li><a href="/wiki/History_of_science_and_technology_in_China" title="History of science and technology in China">Chinese</a></li> <li><a href="/wiki/History_of_science_and_technology_in_the_Indian_subcontinent" class="mw-redirect" title="History of science and technology in the Indian subcontinent">Indian</a></li> <li><a href="/wiki/Science_in_the_medieval_Islamic_world" title="Science in the medieval Islamic world">Medieval Islamic</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Japan" title="History of science and technology in Japan">Japanese</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Korea" title="History of science and technology in Korea">Korean</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Mexico" title="History of science and technology in Mexico">Mexican</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Russia" class="mw-redirect" title="History of science and technology in Russia">Russian</a></li> <li><a href="/wiki/History_of_science_and_technology_in_Spain" title="History of science and technology in Spain">Spanish</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_natural_science" class="mw-redirect" title="History of natural science">Natural sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_astronomy" title="History of astronomy">Astronomy</a></li> <li><a href="/wiki/History_of_biology" title="History of biology">Biology</a></li> <li><a href="/wiki/History_of_chemistry" title="History of chemistry">Chemistry</a></li> <li><a href="/wiki/Outline_of_Earth_sciences#History_of_Earth_science" title="Outline of Earth sciences">Earth science</a></li> <li><a href="/wiki/History_of_physics" title="History of physics">Physics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_mathematics" title="History of mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">Algebra</a></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a></li> <li><a class="mw-selflink selflink">Logic</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a></li> <li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_the_social_sciences" title="History of the social sciences">Social sciences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_anthropology" title="History of anthropology">Anthropology</a></li> <li><a href="/wiki/History_of_archaeology" title="History of archaeology">Archaeology</a></li> <li><a href="/wiki/History_of_economic_thought" title="History of economic thought">Economics</a></li> <li><a href="/wiki/History" title="History">History</a></li> <li><a href="/wiki/History_of_political_science" title="History of political science">Political science</a></li> <li><a href="/wiki/History_of_psychology" title="History of psychology">Psychology</a></li> <li><a href="/wiki/History_of_sociology" title="History of sociology">Sociology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_technology" title="History of technology">Technology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_agricultural_science" title="History of agricultural science">Agricultural science</a></li> <li><a href="/wiki/History_of_computer_science" title="History of computer science">Computer science</a></li> <li><a href="/wiki/History_of_materials_science" title="History of materials science">Materials science</a></li> <li><a href="/wiki/History_of_engineering" title="History of engineering">Engineering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_medicine" title="History of medicine">Medicine</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_medicine" title="History of medicine">Human medicine</a></li> <li><a href="/wiki/History_of_veterinary_medicine" class="mw-redirect" title="History of veterinary medicine">Veterinary medicine</a></li> <li><a href="/wiki/History_of_anatomy" title="History of anatomy">Anatomy</a></li> <li><a href="/wiki/History_of_neuroscience" title="History of neuroscience">Neuroscience</a></li> <li><a href="/wiki/History_of_neurology_and_neurosurgery" title="History of neurology and neurosurgery">Neurology and neurosurgery </a></li> <li><a href="/wiki/History_of_nutrition" class="mw-redirect" title="History of nutrition">Nutrition</a></li> <li><a href="/wiki/History_of_pathology" title="History of pathology">Pathology</a></li> <li><a href="/wiki/History_of_pharmacy" title="History of pharmacy">Pharmacy</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3" style="margin-right:0.5em; padding:0.1em 0 0.4em;line-height:1.7em;"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/List_of_timelines#Science" title="List of timelines">Timelines</a></li> <li><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /></a> <a href="/wiki/Portal:History_of_science" title="Portal:History of science">Portal</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:History_of_science" title="Category:History of science">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="History_of_mathematics_(timeline)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:History_of_mathematics" title="Template:History of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:History_of_mathematics" title="Template talk:History of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:History_of_mathematics" title="Special:EditPage/Template:History of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="History_of_mathematics_(timeline)" style="font-size:114%;margin:0 4em"><a href="/wiki/History_of_mathematics" title="History of mathematics">History of mathematics</a> (<a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">timeline</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By topic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_algebra" title="History of algebra">Algebra</a> <ul><li><a href="/wiki/Timeline_of_algebra" title="Timeline of algebra">timeline</a></li></ul></li> <li>Algorithms <ul><li><a href="/wiki/Timeline_of_algorithms" title="Timeline of algorithms">timeline</a></li></ul></li> <li><a href="/wiki/History_of_arithmetic" class="mw-redirect" title="History of arithmetic">Arithmetic</a> <ul><li><a href="/wiki/Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a> <ul><li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li> <li><a href="/wiki/History_of_Grandi%27s_series" title="History of Grandi's series">Grandi's series</a></li></ul></li> <li>Category theory <ul><li><a href="/wiki/Timeline_of_category_theory_and_related_mathematics" title="Timeline of category theory and related mathematics">timeline</a></li> <li><a href="/wiki/History_of_topos_theory" title="History of topos theory">Topos theory</a></li></ul></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">Functions</a> <ul><li><a href="/wiki/History_of_logarithms" title="History of logarithms">Logarithms</a></li></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a> <ul><li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li> <li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li> <li><a href="/wiki/History_of_group_theory" title="History of group theory">Group theory</a></li> <li><a href="/wiki/History_of_information_theory" title="History of information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">timeline</a></li></ul></li> <li><a class="mw-selflink selflink">Logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">Math notation</a></li> <li>Number theory <ul><li><a href="/wiki/Timeline_of_number_theory" title="Timeline of number theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a> <ul><li><a href="/wiki/Timeline_of_probability_and_statistics" title="Timeline of probability and statistics">timeline</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li></ul></li> <li>Topology <ul><li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">Manifolds</a> <ul><li><a href="/wiki/Timeline_of_manifolds" title="Timeline of manifolds">timeline</a></li></ul></li> <li><a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">Separation axioms</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeral systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric</a></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">Ancient</a></li> <li><a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">Hindu-Arabic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By ancient cultures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Mesopotamia</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Ancient Egypt</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greece</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">China</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">India</a></li> <li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Medieval Islamic world</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Controversies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brouwer%E2%80%93Hilbert_controversy" title="Brouwer–Hilbert controversy">Brouwer–Hilbert</a></li> <li><a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">Over Cantor's theory</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton</a></li> <li><a href="/wiki/Hobbes%E2%80%93Wallis_controversy" title="Hobbes–Wallis controversy">Hobbes–Wallis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Women in mathematics <ul><li><a href="/wiki/Timeline_of_women_in_mathematics" title="Timeline of women in mathematics">timeline</a></li></ul></li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</a> <ul><li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">timeline</a></li></ul></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future of mathematics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" 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