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class="vector-toc-list"> <li id="toc-Premises_and_conclusions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Premises_and_conclusions"> <div class="vector-toc-text"> 2.1.1 Premises and conclusions </div> </a> <ul id="toc-Premises_and_conclusions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Internal_structure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Internal_structure"> <div class="vector-toc-text"> 2.1.2 Internal structure </div> </a> <ul id="toc-Internal_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_truth" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Logical_truth"> <div class="vector-toc-text"> 2.1.3 Logical truth </div> </a> <ul id="toc-Logical_truth-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Truth_tables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Truth_tables"> <div class="vector-toc-text"> 2.1.4 Truth tables </div> </a> <ul id="toc-Truth_tables-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arguments_and_inferences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arguments_and_inferences"> <div class="vector-toc-text"> 2.2 Arguments and inferences </div> </a> <ul id="toc-Arguments_and_inferences-sublist" class="vector-toc-list"> <li id="toc-Deductive" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Deductive"> <div class="vector-toc-text"> 2.2.1 Deductive </div> </a> <ul id="toc-Deductive-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ampliative" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ampliative"> <div class="vector-toc-text"> 2.2.2 Ampliative </div> </a> <ul id="toc-Ampliative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fallacies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fallacies"> <div class="vector-toc-text"> 2.3 Fallacies </div> </a> <ul id="toc-Fallacies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definitory_and_strategic_rules" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definitory_and_strategic_rules"> <div class="vector-toc-text"> 2.4 Definitory and strategic rules </div> </a> <ul id="toc-Definitory_and_strategic_rules-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_systems"> <div class="vector-toc-text"> 2.5 Formal systems </div> </a> <ul id="toc-Formal_systems-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Systems_of_logic" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Systems_of_logic"> <div class="vector-toc-text"> 3 Systems of logic </div> </a> <button aria-controls="toc-Systems_of_logic-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Systems of logic subsection </button> <ul id="toc-Systems_of_logic-sublist" class="vector-toc-list"> <li id="toc-Aristotelian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aristotelian"> <div class="vector-toc-text"> 3.1 Aristotelian </div> </a> <ul id="toc-Aristotelian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classical" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical"> <div class="vector-toc-text"> 3.2 Classical </div> </a> <ul id="toc-Classical-sublist" class="vector-toc-list"> <li id="toc-Propositional_logic" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Propositional_logic"> <div class="vector-toc-text"> 3.2.1 Propositional logic </div> </a> <ul id="toc-Propositional_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-First-order_logic" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#First-order_logic"> <div class="vector-toc-text"> 3.2.2 First-order logic </div> </a> <ul id="toc-First-order_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extended" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extended"> <div class="vector-toc-text"> 3.3 Extended </div> </a> <ul id="toc-Extended-sublist" class="vector-toc-list"> <li id="toc-Modal_logic" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modal_logic"> <div class="vector-toc-text"> 3.3.1 Modal logic </div> </a> <ul id="toc-Modal_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher_order_logic" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Higher_order_logic"> <div class="vector-toc-text"> 3.3.2 Higher order logic </div> </a> <ul id="toc-Higher_order_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Deviant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deviant"> <div class="vector-toc-text"> 3.4 Deviant </div> </a> <ul id="toc-Deviant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Informal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Informal"> <div class="vector-toc-text"> 3.5 Informal </div> </a> <ul id="toc-Informal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Areas_of_research" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Areas_of_research"> <div class="vector-toc-text"> 4 Areas of research </div> </a> <button aria-controls="toc-Areas_of_research-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Areas of research subsection </button> <ul id="toc-Areas_of_research-sublist" class="vector-toc-list"> <li id="toc-Philosophy_of_logic_and_philosophical_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Philosophy_of_logic_and_philosophical_logic"> <div class="vector-toc-text"> 4.1 Philosophy of logic and philosophical logic </div> </a> <ul id="toc-Philosophy_of_logic_and_philosophical_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metalogic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metalogic"> <div class="vector-toc-text"> 4.2 Metalogic </div> </a> <ul id="toc-Metalogic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_logic"> <div class="vector-toc-text"> 4.3 Mathematical logic </div> </a> <ul id="toc-Mathematical_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_logic"> <div class="vector-toc-text"> 4.4 Computational logic </div> </a> <ul id="toc-Computational_logic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formal_semantics_of_natural_language" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_semantics_of_natural_language"> <div class="vector-toc-text"> 4.5 Formal semantics of natural language </div> </a> <ul id="toc-Formal_semantics_of_natural_language-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Epistemology_of_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Epistemology_of_logic"> <div class="vector-toc-text"> 4.6 Epistemology of logic </div> </a> <ul id="toc-Epistemology_of_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 5 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </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"> 6 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 7 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 7.1 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 7.2 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> 7.3 Bibliography </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </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"> 8 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 id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Logic</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 164 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-164" aria-hidden="true" > 164 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Logika" title="Logika – Afrikaans" lang="af" hreflang="af" data-title="Logika" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Logik" title="Logik – Alemannic" lang="gsw" hreflang="gsw" data-title="Logik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%B5%E1%8A%90_%E1%8A%A0%E1%88%9D%E1%8A%AD%E1%8A%95%E1%8B%AE" title="ስነ አምክንዮ – Amharic" lang="am" hreflang="am" data-title="ስነ አምክንዮ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.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" title="तर्कशास्त्र – Angika" lang="anp" hreflang="anp" data-title="तर्कशास्त्र" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Arabic" lang="ar" hreflang="ar" data-title="منطق" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Lochica" title="Lochica – Aragonese" lang="an" hreflang="an" data-title="Lochica" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.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%AB%D6%82%D5%B6" title="Տրամաբանութիւն – Western Armenian" lang="hyw" hreflang="hyw" data-title="Տրամաբանութիւն" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target">Արեւմտահայերէն</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%A4%E0%A7%B0%E0%A7%8D%E0%A6%95%E0%A6%B6%E0%A6%BE%E0%A6%B8%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="তৰ্কশাস্ত্ৰ – Assamese" lang="as" hreflang="as" data-title="তৰ্কশাস্ত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/L%C3%B3xica" title="Lóxica – Asturian" lang="ast" hreflang="ast" data-title="Lóxica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Kuaarape" title="Kuaarape – Guarani" lang="gn" hreflang="gn" data-title="Kuaarape" data-language-autonym="Avañe'ẽ" data-language-local-name="Guarani" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C9%99ntiq" title="Məntiq – Azerbaijani" lang="az" hreflang="az" data-title="Məntiq" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%DB%8C%D9%82" title="منطیق – South Azerbaijani" lang="azb" hreflang="azb" data-title="منطیق" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</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%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="যুক্তিবিজ্ঞান – Bangla" lang="bn" hreflang="bn" data-title="যুক্তিবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Su-l%C3%AD" title="Su-lí – Minnan" lang="nan" hreflang="nan" data-title="Su-lí" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Bashkir" lang="ba" hreflang="ba" data-title="Логика" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D1%96%D0%BA%D0%B0" title="Логіка – Belarusian" lang="be" hreflang="be" data-title="Логіка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%91%D0%B3%D1%96%D0%BA%D0%B0" title="Лёгіка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лёгіка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Lohika" title="Lohika – Central Bikol" lang="bcl" hreflang="bcl" data-title="Lohika" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Bulgarian" lang="bg" hreflang="bg" data-title="Логика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Logik" title="Logik – Bavarian" lang="bar" hreflang="bar" data-title="Logik" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Logika" title="Logika – Bosnian" lang="bs" hreflang="bs" data-title="Logika" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Poelloniezh" title="Poelloniezh – Breton" lang="br" hreflang="br" data-title="Poelloniezh" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A3%D1%88%D0%B0%D1%80_%D1%83%D1%85%D0%B0%D0%B0%D0%BD" title="Ушар ухаан – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Ушар ухаан" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/L%C3%B2gica" title="Lògica – Catalan" lang="ca" hreflang="ca" data-title="Lògica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Chuvash" lang="cv" hreflang="cv" data-title="Логика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Logika" title="Logika – Czech" lang="cs" hreflang="cs" data-title="Logika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cbk-zam mw-list-item"><a href="https://cbk-zam.wikipedia.org/wiki/Logica" title="Logica – Chavacano" lang="cbk" hreflang="cbk" data-title="Logica" data-language-autonym="Chavacano de Zamboanga" data-language-local-name="Chavacano" class="interlanguage-link-target">Chavacano de Zamboanga</a></li><li class="interlanguage-link interwiki-co mw-list-item"><a href="https://co.wikipedia.org/wiki/Logica" title="Logica – Corsican" lang="co" hreflang="co" data-title="Logica" data-language-autonym="Corsu" data-language-local-name="Corsican" class="interlanguage-link-target">Corsu</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhesymeg" title="Rhesymeg – Welsh" lang="cy" hreflang="cy" data-title="Rhesymeg" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Logik" title="Logik – Danish" lang="da" hreflang="da" data-title="Logik" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Logik" title="Logik – German" lang="de" hreflang="de" data-title="Logik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Loogika" title="Loogika – Estonian" lang="et" hreflang="et" data-title="Loogika" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9B%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE" title="Λογική – Greek" lang="el" hreflang="el" data-title="Λογική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Spanish" lang="es" hreflang="es" data-title="Lógica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Logiko" title="Logiko – Esperanto" lang="eo" hreflang="eo" data-title="Logiko" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Extremaduran" lang="ext" hreflang="ext" data-title="Lógica" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Logika" title="Logika – Basque" lang="eu" hreflang="eu" data-title="Logika" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Persian" lang="fa" hreflang="fa" data-title="منطق" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Logic" title="Logic – Fiji Hindi" lang="hif" hreflang="hif" data-title="Logic" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logique" title="Logique – French" lang="fr" hreflang="fr" data-title="Logique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Logika" title="Logika – Western Frisian" lang="fy" hreflang="fy" data-title="Logika" data-language-autonym="Frysk" data-language-local-name="Western Frisian" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Loighic" title="Loighic – Irish" lang="ga" hreflang="ga" data-title="Loighic" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%B3xica" title="Lóxica – Galician" lang="gl" hreflang="gl" data-title="Lóxica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E5%AD%B8" title="邏輯學 – Gan" lang="gan" hreflang="gan" data-title="邏輯學" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/%C5%A8go_(logic)" title="Ũgo (logic) – Kikuyu" lang="ki" hreflang="ki" data-title="Ũgo (logic)" data-language-autonym="Gĩkũyũ" data-language-local-name="Kikuyu" class="interlanguage-link-target">Gĩkũyũ</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%85%BC%EB%A6%AC%ED%95%99" title="논리학 – Korean" lang="ko" hreflang="ko" data-title="논리학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-haw mw-list-item"><a href="https://haw.wikipedia.org/wiki/K%C5%ABpili" title="Kūpili – Hawaiian" lang="haw" hreflang="haw" data-title="Kūpili" data-language-autonym="Hawaiʻi" data-language-local-name="Hawaiian" class="interlanguage-link-target">Hawaiʻi</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%B8%D6%82%D5%B6" title="Տրամաբանություն – Armenian" lang="hy" hreflang="hy" data-title="Տրամաբանություն" 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" title="तर्कशास्त्र – Hindi" lang="hi" hreflang="hi" data-title="तर्कशास्त्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Logika" title="Logika – Croatian" lang="hr" hreflang="hr" data-title="Logika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Logiko" title="Logiko – Ido" lang="io" hreflang="io" data-title="Logiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/%C7%B8z%C3%B9" title="Ǹzù – Igbo" lang="ig" hreflang="ig" data-title="Ǹzù" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target">Igbo</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Lohika" title="Lohika – Iloko" lang="ilo" hreflang="ilo" data-title="Lohika" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logika" title="Logika – Indonesian" lang="id" hreflang="id" data-title="Logika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Logica" title="Logica – Interlingua" lang="ia" hreflang="ia" data-title="Logica" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Logica" title="Logica – Interlingue" lang="ie" hreflang="ie" data-title="Logica" data-language-autonym="Interlingue" data-language-local-name="Interlingue" class="interlanguage-link-target">Interlingue</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-logic" title="I-logic – Xhosa" lang="xh" hreflang="xh" data-title="I-logic" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/R%C3%B6kfr%C3%A6%C3%B0i" title="Rökfræði – Icelandic" lang="is" hreflang="is" data-title="Rökfræði" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Logica" title="Logica – Italian" lang="it" hreflang="it" data-title="Logica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9C%D7%95%D7%92%D7%99%D7%A7%D7%94" title="לוגיקה – Hebrew" lang="he" hreflang="he" data-title="לוגיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Logika" title="Logika – Javanese" lang="jv" hreflang="jv" data-title="Logika" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kbp mw-list-item"><a href="https://kbp.wikipedia.org/wiki/C%C9%94l%C9%A9y%C9%9B" title="Cɔlɩyɛ – Kabiye" lang="kbp" hreflang="kbp" data-title="Cɔlɩyɛ" data-language-autonym="Kabɩyɛ" data-language-local-name="Kabiye" class="interlanguage-link-target">Kabɩyɛ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9A%E1%83%9D%E1%83%92%E1%83%98%E1%83%99%E1%83%90" title="ლოგიკა – Georgian" lang="ka" hreflang="ka" data-title="ლოგიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Kazakh" lang="kk" hreflang="kk" data-title="Логика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-kw mw-list-item"><a href="https://kw.wikipedia.org/wiki/Lojyk" title="Lojyk – Cornish" lang="kw" hreflang="kw" data-title="Lojyk" data-language-autonym="Kernowek" data-language-local-name="Cornish" class="interlanguage-link-target">Kernowek</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mantiki" title="Mantiki – Swahili" lang="sw" hreflang="sw" data-title="Mantiki" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Lojik" title="Lojik – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Lojik" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Mentiq" title="Mentiq – Kurdish" lang="ku" hreflang="ku" data-title="Mentiq" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Kyrgyz" lang="ky" hreflang="ky" data-title="Логика" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lld mw-list-item"><a href="https://lld.wikipedia.org/wiki/Logica" title="Logica – Ladin" lang="lld" hreflang="lld" data-title="Logica" data-language-autonym="Ladin" data-language-local-name="Ladin" class="interlanguage-link-target">Ladin</a></li><li class="interlanguage-link interwiki-lbe mw-list-item"><a href="https://lbe.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D0%B8%D0%BA%D1%8C" title="Мантикь – Lak" lang="lbe" hreflang="lbe" data-title="Мантикь" data-language-autonym="Лакку" data-language-local-name="Lak" class="interlanguage-link-target">Лакку</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logica" title="Logica – Latin" lang="la" hreflang="la" data-title="Logica" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Lo%C4%A3ika" title="Loģika – Latvian" lang="lv" hreflang="lv" data-title="Loģika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Logik" title="Logik – Luxembourgish" lang="lb" hreflang="lb" data-title="Logik" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Logika" title="Logika – Lithuanian" lang="lt" hreflang="lt" data-title="Logika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lfn mw-list-item"><a href="https://lfn.wikipedia.org/wiki/Lojica" title="Lojica – Lingua Franca Nova" lang="lfn" hreflang="lfn" data-title="Lojica" data-language-autonym="Lingua Franca Nova" data-language-local-name="Lingua Franca Nova" class="interlanguage-link-target">Lingua Franca Nova</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/logji" title="logji – Lojban" lang="jbo" hreflang="jbo" data-title="logji" data-language-autonym="La .lojban." data-language-local-name="Lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Logega" title="Logega – Lombard" lang="lmo" hreflang="lmo" data-title="Logega" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Logika" title="Logika – Hungarian" lang="hu" hreflang="hu" data-title="Logika" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Macedonian" lang="mk" hreflang="mk" data-title="Логика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/L%C3%B4jika" title="Lôjika – Malagasy" lang="mg" hreflang="mg" data-title="Lôjika" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B5%BC%E0%B4%95%E0%B5%8D%E0%B4%95%E0%B4%B6%E0%B4%BE%E0%B4%B8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="തർക്കശാസ്ത്രം – Malayalam" lang="ml" hreflang="ml" data-title="തർക്കശാസ്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.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" title="तर्कशास्त्र – Marathi" lang="mr" hreflang="mr" data-title="तर्कशास्त्र" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9A%E1%83%9D%E1%83%92%E1%83%98%E1%83%99%E1%83%90" title="ლოგიკა – Mingrelian" lang="xmf" hreflang="xmf" data-title="ლოგიკა" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Egyptian Arabic" lang="arz" hreflang="arz" data-title="منطق" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target">مصرى</a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Mazanderani" lang="mzn" hreflang="mzn" data-title="منطق" data-language-autonym="مازِرونی" data-language-local-name="Mazanderani" class="interlanguage-link-target">مازِرونی</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logik" title="Logik – Malay" lang="ms" hreflang="ms" data-title="Logik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-min mw-list-item"><a href="https://min.wikipedia.org/wiki/Logika" title="Logika – Minangkabau" lang="min" hreflang="min" data-title="Logika" data-language-autonym="Minangkabau" data-language-local-name="Minangkabau" class="interlanguage-link-target">Minangkabau</a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/L%C3%B3gica" title="Lógica – Mirandese" lang="mwl" hreflang="mwl" data-title="Lógica" data-language-autonym="Mirandés" data-language-local-name="Mirandese" class="interlanguage-link-target">Mirandés</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA" title="Логик – Mongolian" lang="mn" hreflang="mn" data-title="Логик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9A%E1%80%AF%E1%80%90%E1%80%B9%E1%80%90%E1%80%AD%E1%80%97%E1%80%B1%E1%80%92" title="ယုတ္တိဗေဒ – Burmese" lang="my" hreflang="my" data-title="ယုတ္တိဗေဒ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Vakasama" title="Vakasama – Fijian" lang="fj" hreflang="fj" data-title="Vakasama" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="Fijian" class="interlanguage-link-target">Na Vosa Vakaviti</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Logica" title="Logica – Dutch" lang="nl" hreflang="nl" data-title="Logica" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95" title="तर्क – Newari" lang="new" hreflang="new" data-title="तर्क" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target">नेपाल भाषा</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" title="論理学 – Japanese" lang="ja" hreflang="ja" data-title="論理学" 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-ce mw-list-item"><a href="https://ce.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Chechen" lang="ce" hreflang="ce" data-title="Логика" data-language-autonym="Нохчийн" data-language-local-name="Chechen" class="interlanguage-link-target">Нохчийн</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Loogik" title="Loogik – Northern Frisian" lang="frr" hreflang="frr" data-title="Loogik" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-pih mw-list-item"><a href="https://pih.wikipedia.org/wiki/Lojik" title="Lojik – Norfuk / Pitkern" lang="pih" hreflang="pih" data-title="Lojik" data-language-autonym="Norfuk / Pitkern" data-language-local-name="Norfuk / Pitkern" class="interlanguage-link-target">Norfuk / Pitkern</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Logikk" title="Logikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Logikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Logikk" title="Logikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Logikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Logike" title="Logike – Novial" lang="nov" hreflang="nov" data-title="Logike" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Logica" title="Logica – Occitan" lang="oc" hreflang="oc" data-title="Logica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Yaaya(logic)" title="Yaaya(logic) – Oromo" lang="om" hreflang="om" data-title="Yaaya(logic)" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Mantiq" title="Mantiq – Uzbek" lang="uz" hreflang="uz" data-title="Mantiq" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%A4%E0%A8%B0%E0%A8%95_%E0%A8%B8%E0%A8%BC%E0%A8%BE%E0%A8%B8%E0%A8%A4%E0%A8%B0" title="ਤਰਕ ਸ਼ਾਸਤਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਤਰਕ ਸ਼ਾਸਤਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AA%D9%8F%DA%A9" title="تُک – Western Punjabi" lang="pnb" hreflang="pnb" data-title="تُک" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%B3%D9%88%D9%84%D9%BE%D9%88%D9%87%D9%86%D9%87" title="سولپوهنه – Pashto" lang="ps" hreflang="ps" data-title="سولپوهنه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Lajik" title="Lajik – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Lajik" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%8F%E1%9E%80%E1%9F%92%E1%9E%80%E1%9E%9C%E1%9E%B7%E1%9E%87%E1%9F%92%E1%9E%87%E1%9E%B6" title="តក្កវិជ្ជា – Khmer" lang="km" hreflang="km" data-title="តក្កវិជ្ជា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="Khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/L%C3%B2gica" title="Lògica – Piedmontese" lang="pms" hreflang="pms" data-title="Lògica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-tpi mw-list-item"><a href="https://tpi.wikipedia.org/wiki/Lajik" title="Lajik – Tok Pisin" lang="tpi" hreflang="tpi" data-title="Lajik" data-language-autonym="Tok Pisin" data-language-local-name="Tok Pisin" class="interlanguage-link-target">Tok Pisin</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Logik" title="Logik – Low German" lang="nds" hreflang="nds" data-title="Logik" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logika" title="Logika – Polish" lang="pl" hreflang="pl" data-title="Logika" 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/L%C3%B3gica" title="Lógica – Portuguese" lang="pt" hreflang="pt" data-title="Lógica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-crh mw-list-item"><a href="https://crh.wikipedia.org/wiki/Mant%C4%B1q" title="Mantıq – Crimean Tatar" lang="crh" hreflang="crh" data-title="Mantıq" data-language-autonym="Qırımtatarca" data-language-local-name="Crimean Tatar" class="interlanguage-link-target">Qırımtatarca</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Logic%C4%83" title="Logică – Romanian" lang="ro" hreflang="ro" data-title="Logică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Sullwa" title="Sullwa – Quechua" lang="qu" hreflang="qu" data-title="Sullwa" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9B%D0%BE%D2%91%D1%96%D0%BA%D0%B0" title="Лоґіка – Rusyn" lang="rue" hreflang="rue" data-title="Лоґіка" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Russian" lang="ru" hreflang="ru" data-title="Логика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Yakut" lang="sah" hreflang="sah" data-title="Логика" data-language-autonym="Саха тыла" data-language-local-name="Yakut" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sat mw-list-item"><a href="https://sat.wikipedia.org/wiki/%E1%B1%A1%E1%B1%A9%E1%B1%A0%E1%B1%9B%E1%B1%A4" title="ᱡᱩᱠᱛᱤ – Santali" lang="sat" hreflang="sat" data-title="ᱡᱩᱠᱛᱤ" data-language-autonym="ᱥᱟᱱᱛᱟᱲᱤ" data-language-local-name="Santali" class="interlanguage-link-target">ᱥᱟᱱᱛᱟᱲᱤ</a></li><li class="interlanguage-link interwiki-sc mw-list-item"><a href="https://sc.wikipedia.org/wiki/L%C3%B2gica" title="Lògica – Sardinian" lang="sc" hreflang="sc" data-title="Lògica" data-language-autonym="Sardu" data-language-local-name="Sardinian" class="interlanguage-link-target">Sardu</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Logic" title="Logic – Scots" lang="sco" hreflang="sco" data-title="Logic" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logjika" title="Logjika – Albanian" lang="sq" hreflang="sq" data-title="Logjika" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/L%C3%B2ggica" title="Lòggica – Sicilian" lang="scn" hreflang="scn" data-title="Lòggica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Logic" title="Logic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Logic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82" title="منطق – Sindhi" lang="sd" hreflang="sd" data-title="منطق" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Logika" title="Logika – Slovak" lang="sk" hreflang="sk" data-title="Logika" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Logika" title="Logika – Slovenian" lang="sl" hreflang="sl" data-title="Logika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%84%DB%86%DA%98%DB%8C%DA%A9" title="لۆژیک – Central Kurdish" lang="ckb" hreflang="ckb" data-title="لۆژیک" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Логика – Serbian" lang="sr" hreflang="sr" data-title="Логика" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Logika" title="Logika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Logika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Logika" title="Logika – Sundanese" lang="su" hreflang="su" data-title="Logika" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Logiikka" title="Logiikka – Finnish" lang="fi" hreflang="fi" data-title="Logiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Logik" title="Logik – Swedish" lang="sv" hreflang="sv" data-title="Logik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Lohika" title="Lohika – Tagalog" lang="tl" hreflang="tl" data-title="Lohika" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8F%E0%AE%B0%E0%AE%A3%E0%AE%AE%E0%AF%8D" title="ஏரணம் – Tamil" lang="ta" hreflang="ta" data-title="ஏரணம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-shi mw-list-item"><a href="https://shi.wikipedia.org/wiki/Tam%E1%BA%93la" title="Tamẓla – Tachelhit" lang="shi" hreflang="shi" data-title="Tamẓla" data-language-autonym="Taclḥit" data-language-local-name="Tachelhit" class="interlanguage-link-target">Taclḥit</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tame%E1%BA%93la" title="Tameẓla – Kabyle" lang="kab" hreflang="kab" data-title="Tameẓla" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D1%8B%D0%B9%D0%BA" title="Мантыйк – Tatar" lang="tt" hreflang="tt" data-title="Мантыйк" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="ตรรกศาสตร์ – Thai" lang="th" hreflang="th" data-title="ตรรกศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D0%B8%D2%9B" title="Мантиқ – Tajik" lang="tg" hreflang="tg" data-title="Мантиқ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mant%C4%B1k" title="Mantık – Turkish" lang="tr" hreflang="tr" data-title="Mantık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Mantyk" title="Mantyk – Turkmen" lang="tk" hreflang="tk" data-title="Mantyk" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target">Türkmençe</a></li><li class="interlanguage-link interwiki-tyv mw-list-item"><a href="https://tyv.wikipedia.org/wiki/%D0%91%D0%B8%D0%BB%D0%B8%D0%B3%D0%BB%D0%B5%D0%BB" title="Билиглел – Tuvinian" lang="tyv" hreflang="tyv" data-title="Билиглел" data-language-autonym="Тыва дыл" data-language-local-name="Tuvinian" class="interlanguage-link-target">Тыва 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Click here for more information."><img alt="Featured article" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/20px-Cscr-featured.svg.png" decoding="async" width="20" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/30px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/40px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></div></div> <div id="mw-indicator-spoken-icon" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/File:En-Logic-article.ogg" title="Listen to this article"><img alt="Listen to this article" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/20px-Sound-icon.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/30px-Sound-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/40px-Sound-icon.svg.png 2x" data-file-width="128" data-file-height="96" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Formal_logic&redirect=no" class="mw-redirect" title="Formal logic">Formal logic</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Study of correct reasoning</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">This article is about the study of correct reasoning. For other uses, see <a href="/wiki/Logic_(disambiguation)" class="mw-disambig" title="Logic (disambiguation)">Logic (disambiguation)</a> and <a href="/wiki/Logician_(disambiguation)" class="mw-disambig" title="Logician (disambiguation)">Logician (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Logical" and "Logically" redirect here. For the rapper, see <a href="/wiki/Logic_(rapper)" title="Logic (rapper)">Logic (rapper)</a>. For the British company, see <a href="/wiki/Logically_(company)" title="Logically (company)">Logically (company)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Modus_ponendo_ponens.png" class="mw-file-description"><img alt="Depiction of inference using modus ponens" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Modus_ponendo_ponens.png/220px-Modus_ponendo_ponens.png" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Modus_ponendo_ponens.png/330px-Modus_ponendo_ponens.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Modus_ponendo_ponens.png/440px-Modus_ponendo_ponens.png 2x" data-file-width="682" data-file-height="562" /></a><figcaption>Logic studies valid forms of inference like <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>.</figcaption></figure> Logic is the study of correct <a href="/wiki/Logical_reasoning" title="Logical reasoning">reasoning</a>. It includes both <a class="mw-selflink-fragment" href="#Formal_logic">formal</a> and <a href="/wiki/Informal_logic" title="Informal logic">informal logic</a>. Formal logic is the study of <a href="/wiki/Validity_(logic)" title="Validity (logic)">deductively valid</a> inferences or <a href="/wiki/Logical_truth" title="Logical truth">logical truths</a>. It examines how conclusions follow from <a href="/wiki/Premise" title="Premise">premises</a> based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with <a href="/wiki/Informal_fallacies" class="mw-redirect" title="Informal fallacies">informal fallacies</a>, <a href="/wiki/Critical_thinking" title="Critical thinking">critical thinking</a>, and <a href="/wiki/Argumentation_theory" title="Argumentation theory">argumentation theory</a>. Informal logic examines arguments expressed in <a href="/wiki/Natural_language" title="Natural language">natural language</a> whereas formal logic uses <a href="/wiki/Formal_language" title="Formal language">formal language</a>. When used as a <a href="/wiki/Countable_noun" class="mw-redirect" title="Countable noun">countable noun</a>, the term "a logic" refers to a specific logical <a href="/wiki/Formal_system" title="Formal system">formal system</a> that articulates a <a href="/wiki/Proof_system" class="mw-redirect" title="Proof system">proof system</a>. Logic plays a central role in many fields, such as <a href="/wiki/Philosophy" title="Philosophy">philosophy</a>, <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, and <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work".<a href="#cite_note-FOOTNOTEVelleman20068,_103-1">[1]</a> Premises and conclusions express <a href="/wiki/Proposition" title="Proposition">propositions</a> or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by <a href="/wiki/Logical_connective" title="Logical connective">logical vocabulary</a> like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"> (<a href="/wiki/Logical_conjunction" title="Logical conjunction">and</a>) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> (<a href="/wiki/Material_conditional" title="Material conditional">if...then</a>). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">Deductive arguments</a> have the strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for <a href="/wiki/Ampliative" title="Ampliative">ampliative</a> arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive</a> and <a href="/wiki/Abductive_reasoning" title="Abductive reasoning">abductive</a> arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens.<a href="#cite_note-FOOTNOTEVickers2022-2">[2]</a> Abductive arguments are <a href="/wiki/Inference" title="Inference">inferences</a> to the best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer.<a href="#cite_note-FOOTNOTENunes20112066–2069-3">[3]</a> Arguments that fall short of the standards of correct reasoning often embody <a href="/wiki/Fallacies" class="mw-redirect" title="Fallacies">fallacies</a>. Systems of logic are theoretical frameworks for assessing the correctness of arguments. Logic has been studied since <a href="/wiki/Ancient_history" title="Ancient history">antiquity</a>. Early approaches include <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a>, <a href="/wiki/Stoic_logic" title="Stoic logic">Stoic logic</a>, <a href="/wiki/Nyaya" title="Nyaya">Nyaya</a>, and <a href="/wiki/Mohism" title="Mohism">Mohism</a>. Aristotelian logic focuses on reasoning in the form of <a href="/wiki/Syllogism" title="Syllogism">syllogisms</a>. It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>. Today, the most commonly used system is <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>. It consists of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> and <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>. Propositional logic only considers logical relations between full propositions. First-order logic also takes the internal parts of propositions into account, like <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a> and <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifiers</a>. Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a>, <a href="/wiki/Ethics" title="Ethics">ethics</a>, and <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>. Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Logic&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> The word "logic" originates from the Greek word "logos", which has a variety of translations, such as <a href="/wiki/Reason" title="Reason">reason</a>, <a href="/wiki/Discourse" title="Discourse">discourse</a>, or <a href="/wiki/Language" title="Language">language</a>.<a href="#cite_note-FOOTNOTEPépin2004LogosOnline_Etymology_Staff-4">[4]</a> Logic is traditionally defined as the study of the <a href="/wiki/Laws_of_thought" class="mw-redirect" title="Laws of thought">laws of thought</a> or <a href="/wiki/Logical_reasoning" title="Logical reasoning">correct reasoning</a>,<a href="#cite_note-FOOTNOTEHintikka2019lead_section,_§Nature_and_varieties_of_logic-5">[5]</a> and is usually understood in terms of <a href="/wiki/Inference" title="Inference">inferences</a> or <a href="/wiki/Argument" title="Argument">arguments</a>. Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences.<a href="#cite_note-FOOTNOTEHintikka2019§Nature_and_varieties_of_logicHaack19781–10Philosophy_of_logicsSchlesingerKeren-PortnoyParush2001220-6">[6]</a> An argument is a set of premises together with a conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion.<a href="#cite_note-FOOTNOTEHintikkaSandu200613Audi1999bPhilosophy_of_logicMcKeon-7">[7]</a> These general characterizations apply to logic in the widest sense, i.e., to both <a class="mw-selflink-fragment" href="#Formal_logic">formal</a> and <a href="/wiki/Informal_logic" title="Informal logic">informal logic</a> since they are both concerned with assessing the correctness of arguments.<a href="#cite_note-FOOTNOTEBlairJohnson200093–95Craig1996Formal_and_informal_logic-8">[8]</a> Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic.<sup id="cite_ref-FOOTNOTECraig1996Formal_and_informal_logicBarnes2007274Planty-Bonjour2012[httpsbooksgooglecombooksid0EpFBgAAQBAJpgPA62_62]Rini2010[httpsbooksgooglecombooksidvard024vjFgCpgPA26_26]_9-0" class="reference"><a href="#cite_note-FOOTNOTECraig1996Formal_and_informal_logicBarnes2007274Planty-Bonjour2012[httpsbooksgooglecombooksid0EpFBgAAQBAJpgPA62_62]Rini2010[httpsbooksgooglecombooksidvard024vjFgCpgPA26_26]-9">[9]</a> <div class="mw-heading mw-heading3"><h3 id="Formal_logic">Formal logic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=2" title="Edit section: Formal logic">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Formal_system" title="Formal system">Formal system</a></div> Formal logic is also known as symbolic logic and is widely used in <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>. It uses a <a href="/wiki/Formal_system" title="Formal system">formal</a> approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the <a href="/wiki/Logical_form" title="Logical form">logical form</a> of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content.<a href="#cite_note-FOOTNOTEMacFarlane2017Corkum2015753–767BlairJohnson200093–95Magnus200512–41.6_Formal_languages-10">[10]</a> Formal logic is interested in deductively <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false.<a href="#cite_note-FOOTNOTEMcKeonCraig1996Formal_and_informal_logic-11">[11]</a> For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a <a href="/wiki/Rule_of_inference" title="Rule of inference">rule of inference</a>.<a href="#cite_note-FOOTNOTEHintikkaSandu200613-12">[12]</a> For example, <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> is a rule of inference according to which all arguments of the form "(1) p, (2) if p then q, (3) therefore q" are valid, independent of what the terms p and q stand for.<a href="#cite_note-FOOTNOTEMagnus2005Proofs,_p._102-13">[13]</a> In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of <a href="/wiki/Logical_truth" title="Logical truth">logical truths</a>.<a href="#cite_note-FOOTNOTEHintikkaSandu200613–16Makridis20221–2RuncoPritzker1999155-14">[14]</a> A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all <a href="/wiki/Possible_world" title="Possible world">possible worlds</a> and under all <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretations</a> of its non-logical terms, like the claim "either it is raining, or it is not".<a href="#cite_note-FOOTNOTEGómez-Torrente2019Magnus20051.5_Other_logical_notions,_p._10-15">[15]</a> These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q" is a logical truth.<a href="#cite_note-FOOTNOTEHintikkaSandu200616-16">[16]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:First-order_logic.png" class="mw-file-description"><img alt="Visualization of how to translate an English sentence into first-order logic" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/First-order_logic.png/350px-First-order_logic.png" decoding="async" width="350" height="85" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/First-order_logic.png/525px-First-order_logic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/First-order_logic.png/700px-First-order_logic.png 2x" data-file-width="2094" data-file-height="510" /></a><figcaption>Formal logic needs to translate natural language arguments into a formal language, like first-order logic, to assess whether they are valid. In this example, the letter "c" represents Carmen while the letters "M" and "T" stand for "Mexican" and "teacher". The symbol "∧" has the meaning of "and".</figcaption></figure> Formal logic uses <a href="/wiki/Formal_language" title="Formal language">formal languages</a> to express and analyze arguments.<a href="#cite_note-FOOTNOTEHonderich2005logic,_informalCraig1996Formal_and_informal_logicJohnson1999265–268-17">[17]</a> They normally have a very limited vocabulary and exact <a href="/wiki/Syntax" title="Syntax">syntactic rules</a>. These rules specify how their symbols can be combined to construct sentences, so-called <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formulas</a>.<a href="#cite_note-FOOTNOTECraig1996Formal_languages_and_systemsSimpson200814-18">[18]</a> This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether a given argument is valid.<a href="#cite_note-FOOTNOTECraig1996Formal_languages_and_systems-19">[19]</a> Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be <a href="/wiki/Logic_translation#Natural_language_formalization" title="Logic translation">translated into formal language</a> before their validity can be assessed.<a href="#cite_note-FOOTNOTEHintikkaSandu200622–3Magnus20058–91.4_Deductive_validityJohnson1999267-20">[20]</a> The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning the rules of inference they accept as valid and the formal languages used to express them.<a href="#cite_note-FOOTNOTEHaack19781–2,_4Philosophy_of_logicsHintikkaSandu200616–17Jacquette2006Introduction:_Philosophy_of_logic_today,_pp._1–12-21">[21]</a> Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic.<a href="#cite_note-FOOTNOTEHaack19781–2,_4Philosophy_of_logicsJacquette20061–12Introduction:_Philosophy_of_logic_today-22">[22]</a> For example, it has been suggested that only <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">logically complete</a> systems, like <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>, qualify as logics. For such reasons, some theorists deny that <a href="/wiki/Higher-order_logic" title="Higher-order logic">higher-order logics</a> are logics in the strict sense.<a href="#cite_note-FOOTNOTEHaack19785–7,_9Philosophy_of_logicsHintikkaSandu200631–2Haack1996229–30-23">[23]</a> <div class="mw-heading mw-heading3"><h3 id="Informal_logic">Informal logic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=3" title="Edit section: Informal logic">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/Informal_logic" title="Informal logic">Informal logic</a></div> When understood in a wide sense, logic encompasses both formal and informal logic.<a href="#cite_note-FOOTNOTEHaack19781–10Philosophy_of_logicsGroarke2021lead_section;_1.1_Formal_and_Informal_Logic-24">[24]</a> Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse.<a href="#cite_note-FOOTNOTEJohnson2014228–9-25">[25]</a> Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments.<a href="#cite_note-FOOTNOTEGroarke2021lead_section;_1._HistoryAudi1999aInformal_logicJohnson1999265–274-26">[26]</a> In this regard, it considers problems that formal logic on its own is unable to address.<a href="#cite_note-FOOTNOTECraig1996Formal_and_informal_logicJohnson1999267-27">[27]</a> Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies.<a href="#cite_note-FOOTNOTEBlairJohnson200093–97Craig1996Formal_and_informal_logic-28">[28]</a> Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition.<a href="#cite_note-FOOTNOTEJohnson1999265–270van_EemerenGarssenKrabbeSnoeck_Henkemans1–45Informal_Logic-29">[29]</a> The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, informal logic studies arguments that are in informal or natural language.<a href="#cite_note-FOOTNOTEGroarke20211.1_Formal_and_Informal_LogicAudi1999aInformal_logicHonderich2005logic,_informal-30">[30]</a> Formal logic can only examine them indirectly by translating them first into a formal language while informal logic investigates them in their original form.<a href="#cite_note-FOOTNOTEBlairJohnson200093–107Groarke2021lead_section;_1.1_Formal_and_Informal_Logicvan_EemerenGrootendorstJohnsonPlantin169-31">[31]</a> On this view, the argument "Birds fly. Tweety is a bird. Therefore, Tweety flies." belongs to natural language and is examined by informal logic. But the formal translation "(1) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x(Bird(x)\to Flies(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi>i</mi> <mi>r</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>F</mi> <mi>l</mi> <mi>i</mi> <mi>e</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x(Bird(x)\to Flies(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47470c39cb95609466e0ced564f7554e6aed5c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.565ex; height:2.843ex;" alt="{\displaystyle \forall x(Bird(x)\to Flies(x))}">; (2) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Bird(Tweety)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>i</mi> <mi>r</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>w</mi> <mi>e</mi> <mi>e</mi> <mi>t</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Bird(Tweety)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7410bba34fafd2dd9cf92e72eb6bf6864d31f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.103ex; height:2.843ex;" alt="{\displaystyle Bird(Tweety)}">; (3) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Flies(Tweety)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>l</mi> <mi>i</mi> <mi>e</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mi>w</mi> <mi>e</mi> <mi>e</mi> <mi>t</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Flies(Tweety)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b341702eeb3e76d6d1ddedf05c4ed81c6b0993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.682ex; height:2.843ex;" alt="{\displaystyle Flies(Tweety)}">" is studied by formal logic.<a href="#cite_note-FOOTNOTEOaksfordChater200747-32">[32]</a> The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent.<a href="#cite_note-FOOTNOTECraig1996Formal_and_informal_logicWalton19872–3,_6–81._A_new_model_of_argumentEngel198259–922._The_medium_of_language-33">[33]</a> Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation. In this sense, it includes questions about the role of <a href="/wiki/Rationality" title="Rationality">rationality</a>, <a href="/wiki/Critical_thinking" title="Critical thinking">critical thinking</a>, and the psychology of argumentation.<a href="#cite_note-FOOTNOTEBlairJohnson1987147–51-34">[34]</a> Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic.<a href="#cite_note-FOOTNOTEFalikowskiMills202298Weddle2011383–836._Informal_logic_and_the_eductive-inductive_distinctionBlair201147-35">[35]</a> Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive argument</a> from the empirical observation that "all ravens I have seen so far are black" to the conclusion "all ravens are black".<a href="#cite_note-FOOTNOTEVickers2022Nunes20112066–9Logical_Reasoning_and_Learning-36">[36]</a> A further approach is to define informal logic as the study of <a href="/wiki/Informal_fallacies" class="mw-redirect" title="Informal fallacies">informal fallacies</a>.<a href="#cite_note-FOOTNOTEJohnson2014181Johnson1999267BlairJohnson1987147–51-37">[37]</a> Informal fallacies are incorrect arguments in which errors are present in the content and the <a href="/wiki/Context_(language_use)" class="mw-redirect" title="Context (language use)">context</a> of the argument.<a href="#cite_note-FOOTNOTEVleet2010ix–xIntroductionDowdenStump-38">[38]</a> A <a href="/wiki/False_dilemma" title="False dilemma">false dilemma</a>, for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us".<a href="#cite_note-FOOTNOTEMaltbyDayMacaskill2007564Dowden-39">[39]</a> Some theorists state that formal logic studies the general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive <a href="/wiki/Concept" title="Concept">concepts</a> into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of <a href="/wiki/Epistemology" title="Epistemology">epistemology</a> for the assessment of arguments.<a href="#cite_note-FOOTNOTECraig1996Formal_and_informal_logicJohnson1999265–270-40">[40]</a> <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2>[<a href="/w/index.php?title=Logic&action=edit&section=4" title="Edit section: Basic concepts">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Premises,_conclusions,_and_truth">Premises, conclusions, and truth</h3>[<a href="/w/index.php?title=Logic&action=edit&section=5" title="Edit section: Premises, conclusions, and truth">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Premises_and_conclusions">Premises and conclusions</h4>[<a href="/w/index.php?title=Logic&action=edit&section=6" title="Edit section: Premises and conclusions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Premise" title="Premise">Premise</a> and <a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></div> Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicHonderich2005philosophical_logic-41">[41]</a> For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be <a href="/wiki/Truth-bearer" title="Truth-bearer">truth-bearers</a>.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicHonderich2005philosophical_logic-41">[41]</a><a href="#cite_note-43">[a]</a> This means that they have a <a href="/wiki/Truth_value" title="Truth value">truth value</a>: they are either true or false. Contemporary philosophy generally sees them either as <a href="/wiki/Proposition" title="Proposition">propositions</a> or as <a href="/wiki/Sentence_(linguistics)" title="Sentence (linguistics)">sentences</a>.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logic-44">[43]</a> Propositions are the <a href="/wiki/Denotation" title="Denotation">denotations</a> of sentences and are usually seen as <a href="/wiki/Abstract_object" class="mw-redirect" title="Abstract object">abstract objects</a>.<a href="#cite_note-FOOTNOTEFalgueraMartínez-VidalRosen2021Tondl2012111-45">[44]</a> For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grün" but both express the same proposition.<a href="#cite_note-FOOTNOTEOlkowskiPirovolakis2019[httpsbooksgooglecombooksidFhaGDwAAQBAJpgPT65_65–66]-46">[45]</a> Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, <a href="/wiki/Naturalism_(philosophy)" title="Naturalism (philosophy)">philosophical naturalists</a> usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logic-44">[43]</a> These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like the symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicPietroski2021-47">[46]</a> Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as <a href="/wiki/Psychologism" title="Psychologism">psychologism</a>. It was discussed at length around the turn of the 20th century but it is not widely accepted today.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicKusch2020Rush20141–10,_189–190-48">[47]</a> <div class="mw-heading mw-heading4"><h4 id="Internal_structure">Internal structure</h4>[<a href="/w/index.php?title=Logic&action=edit&section=7" title="Edit section: Internal structure">edit</a>]</div> Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex.<a href="#cite_note-FOOTNOTEKing2019Pickel20202991–3006-49">[48]</a> A complex proposition has other propositions as its constituents, which are linked to each other through <a href="/wiki/Logical_connective" title="Logical connective">propositional connectives</a> like "and" or "if...then". Simple propositions, on the other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like <a href="/wiki/Singular_term" title="Singular term">singular terms</a> and <a href="/wiki/Predicate_(grammar)" title="Predicate (grammar)">predicates</a>.<a href="#cite_note-FOOTNOTEHonderich2005philosophical_logic-50">[49]</a><a href="#cite_note-FOOTNOTEKing2019Pickel20202991–3006-49">[48]</a> For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by the propositional connective "and".<a href="#cite_note-FOOTNOTEHonderich2005philosophical_logic-50">[49]</a> Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using <a href="/wiki/Truth_function" title="Truth function">truth-functional</a> propositional connectives, their truth only depends on the truth values of their parts.<a href="#cite_note-FOOTNOTEHonderich2005philosophical_logic-50">[49]</a><a href="#cite_note-FOOTNOTEPickel20202991–3006-51">[50]</a> But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects.<a href="#cite_note-FOOTNOTEHonderich2005philosophical_logicCraig1996Philosophy_of_logicMichaelsonReimer2019-52">[51]</a> Whether the simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by <a href="/wiki/Theories_of_reference" class="mw-redirect" title="Theories of reference">theories of reference</a>.<a href="#cite_note-FOOTNOTEMichaelsonReimer2019-53">[52]</a> <div class="mw-heading mw-heading4"><h4 id="Logical_truth">Logical truth</h4>[<a href="/w/index.php?title=Logic&action=edit&section=8" title="Edit section: Logical truth">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/Logical_truth" title="Logical truth">Logical truth</a></div> Some complex propositions are true independently of the substantive meanings of their parts.<a href="#cite_note-FOOTNOTEHintikka2019§Nature_and_varieties_of_logicMacFarlane2017-54">[53]</a> In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like the simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it.<a href="#cite_note-FOOTNOTEGómez-Torrente2019MacFarlane2017Honderich2005philosophical_logic-55">[54]</a> This means that it is true under all interpretations of its non-logical terms. In some <a href="/wiki/Modal_logic" title="Modal logic">modal logics</a>, this means that the proposition is true in all possible worlds.<a href="#cite_note-FOOTNOTEGómez-Torrente2019Jago201441-56">[55]</a> Some theorists define logic as the study of logical truths.<a href="#cite_note-FOOTNOTEHintikkaSandu200616-16">[16]</a> <div class="mw-heading mw-heading4"><h4 id="Truth_tables">Truth tables</h4>[<a href="/w/index.php?title=Logic&action=edit&section=9" title="Edit section: Truth tables">edit</a>]</div> <a href="/wiki/Truth_table" title="Truth table">Truth tables</a> can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for the truth values "true" and "false".<a href="#cite_note-FOOTNOTEMagnus200535–383._Truth_tablesAngell1964164HallO'Donnell2000[httpsbooksgooglecombooksidyP4MJ36C4ZgCpgPA48_48]-57">[56]</a> The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\land q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∧</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\land q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4904e68d8180ec4c12e20fabc15017b60b098b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\land q}">" uses the logical connective <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"> (<a href="/wiki/Logical_conjunction" title="Logical conjunction">and</a>). It could be used to express a sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> ("yesterday was Sunday") and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099107443792f5fec9bebe39b919a690db7198c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.204ex; margin-bottom: -0.376ex; width:1.55ex; height:1.176ex;" alt="{\displaystyle \lnot }"> (<a href="/wiki/Negation" title="Negation">not</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∨</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lor }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab47f6b1f589aedcf14638df1d63049d233d851a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \lor }"> (<a href="/wiki/Logical_disjunction" title="Logical disjunction">or</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1daab843254cfcb23a643070cf93f3badc4fbbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \to }"> (<a href="/wiki/Material_conditional" title="Material conditional">if...then</a>), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"> (<a href="/wiki/Sheffer_stroke" title="Sheffer stroke">Sheffer stroke</a>).<a href="#cite_note-FOOTNOTEMagnus200535–453._Truth_tablesAngell1964164-58">[57]</a> Given the conditional proposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\to q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fccb3827df1efe7930f9d1febd15d2359971b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\to q}">, one can form truth tables of its <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\to p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">→</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\to p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0cb04539b19dc2826d0215f22383710a565599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.853ex; height:2.176ex;" alt="{\displaystyle q\to p}">, its <a href="/wiki/Inverse_(logic)" title="Inverse (logic)">inverse</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot p\to \lnot q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot p\to \lnot q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ddaf4545f34c586fe44965b121a2f050f8f17b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.954ex; height:2.176ex;" alt="{\displaystyle \lnot p\to \lnot q}">), and its <a href="/wiki/Contrapositive_(logic)" class="mw-redirect" title="Contrapositive (logic)">contrapositive</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot q\to \lnot p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>q</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">¬</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot q\to \lnot p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234339d9722ddb73bdfe059073d84108bdffccc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.954ex; height:2.176ex;" alt="{\displaystyle \lnot q\to \lnot p}">). Truth tables can also be defined for more complex expressions that use several propositional connectives.<a href="#cite_note-FOOTNOTETarski199440-59">[58]</a> <table class="wikitable" style="margin:1em; text-align:center;"> <caption>Truth table of various expressions </caption> <tbody><tr> <th style="width:15%">p </th> <th style="width:15%">q </th> <th style="width:15%">p ∧ q </th> <th style="width:15%">p ∨ q </th> <th style="width:15%">p → q </th> <th style="width:15%">¬p → ¬q </th> <th style="width:15%">p <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \uparrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">↑</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \uparrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb20b28c74cdaa09e1f101d426441da1996072f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \uparrow }"> q </th></tr> <tr> <td>T</td> <td>T</td> <td>T</td> <td>T</td> <td>T</td> <td>T</td> <td style="background:papayawhip">F </td></tr> <tr> <td>T</td> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F</td> <td>T</td> <td style="background:papayawhip">F</td> <td>T</td> <td>T </td></tr> <tr> <td style="background:papayawhip">F</td> <td>T</td> <td style="background:papayawhip">F</td> <td>T</td> <td>T</td> <td style="background:papayawhip">F</td> <td>T </td></tr> <tr> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F</td> <td style="background:papayawhip">F</td> <td>T</td> <td>T</td> <td>T </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Arguments_and_inferences">Arguments and inferences</h3>[<a href="/w/index.php?title=Logic&action=edit&section=10" title="Edit section: Arguments and inferences">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Argument" title="Argument">Argument</a> and <a href="/wiki/Inference" title="Inference">inference</a></div> Logic is commonly defined in terms of arguments or inferences as the study of their correctness.<a href="#cite_note-FOOTNOTEHintikka2019lead_section,_§Nature_and_varieties_of_logicAudi1999bPhilosophy_of_logic-60">[59]</a> An argument is a set of premises together with a conclusion.<a href="#cite_note-FOOTNOTEBlackburn2008argumentStairs2017343-61">[60]</a> An inference is the process of reasoning from these premises to the conclusion.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logic-44">[43]</a> But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on the other hand, are true or false depending on whether they are in accord with reality. In formal logic, a <a href="/wiki/Soundness_(logic)" class="mw-redirect" title="Soundness (logic)">sound</a> argument is an argument that is both correct and has only true premises.<a href="#cite_note-FOOTNOTECopiCohenRodych2019[httpsbooksgooglecombooksid38bADwAAQBAJpgPA30_30]-62">[61]</a> Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For a complex argument to be successful, each link of the chain has to be successful.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logic-44">[43]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Argument_terminology.svg" class="mw-file-description"><img alt="Diagram of argument terminology used in logic" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Argument_terminology.svg/400px-Argument_terminology.svg.png" decoding="async" width="400" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Argument_terminology.svg/600px-Argument_terminology.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Argument_terminology.svg/800px-Argument_terminology.svg.png 2x" data-file-width="1052" data-file-height="744" /></a><figcaption><a href="/wiki/Argument" title="Argument">Argument</a> terminology used in logic</figcaption></figure> Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different <a href="/wiki/Method_of_reasoning" class="mw-redirect" title="Method of reasoning">types of reasoning</a>.<a href="#cite_note-FOOTNOTEHintikkaSandu200620Backmann2019235–255IEP_Staff-63">[62]</a> The strongest form of support corresponds to <a href="/wiki/Deductive_reasoning" title="Deductive reasoning">deductive reasoning</a>. But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases, the term ampliative or inductive reasoning is used.<a href="#cite_note-FOOTNOTEHintikkaSandu200616Backmann2019235–255IEP_Staff-64">[63]</a> Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic.<a href="#cite_note-FOOTNOTEGroarke20211.1_Formal_and_Informal_LogicWeddle2011383–836._Informal_logic_and_the_eductive-inductive_distinctionvan_EemerenGarssen2009191-65">[64]</a> <div class="mw-heading mw-heading4"><h4 id="Deductive">Deductive</h4>[<a href="/w/index.php?title=Logic&action=edit&section=11" title="Edit section: Deductive">edit</a>]</div> A deductively valid argument is one whose premises guarantee the truth of its conclusion.<a href="#cite_note-FOOTNOTEMcKeonCraig1996Formal_and_informal_logic-11">[11]</a> For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs" is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises.<a href="#cite_note-FOOTNOTEEvans20058._Deductive_Reasoning,_[httpsbooksgooglecombooksidznbkHaC8QeMCpgPA169_p._169]-66">[65]</a> According to an influential view by <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, deductive arguments have three essential features: (1) they are formal, i.e. they depend only on the form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by <a href="/wiki/Logical_necessity" class="mw-redirect" title="Logical necessity">logical necessity</a> for the given propositions, independent of any other circumstances.<a href="#cite_note-FOOTNOTEMcKeon-67">[66]</a> Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference.<a href="#cite_note-FOOTNOTEHintikkaSandu200613–4-68">[67]</a> Rules of inference specify the form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid.<a href="#cite_note-FOOTNOTEHintikkaSandu200613–4Blackburn2016rule_of_inference-69">[68]</a> The modus ponens is a prominent rule of inference. It has the form "p; if p, then q; therefore q".<a href="#cite_note-FOOTNOTEBlackburn2016rule_of_inference-70">[69]</a> Knowing that it has just rained (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">) and that after rain the streets are wet (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\to q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">→</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\to q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fccb3827df1efe7930f9d1febd15d2359971b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.942ex; height:2.176ex;" alt="{\displaystyle p\to q}">), one can use modus ponens to deduce that the streets are wet (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}">).<a href="#cite_note-FOOTNOTEDickMüller2017157-71">[70]</a> The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false.<a href="#cite_note-FOOTNOTEHintikkaSandu200613Backmann2019235–255Douven2021-72">[71]</a> Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in the premises.<a href="#cite_note-FOOTNOTEHintikkaSandu200614D'AgostinoFloridi2009271–315-73">[72]</a> But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on the depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs.<a href="#cite_note-FOOTNOTEHintikkaSandu200614Sagüillo201475–88Hintikka1970135–152-74">[73]</a> <div class="mw-heading mw-heading4"><h4 id="Ampliative">Ampliative</h4>[<a href="/w/index.php?title=Logic&action=edit&section=12" title="Edit section: Ampliative">edit</a>]</div> Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with a certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth.<a href="#cite_note-FOOTNOTEHintikkaSandu200613–6Backmann2019235–255IEP_Staff-75">[74]</a> This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to <a href="/wiki/Non-monotonic_logic" title="Non-monotonic logic">non-monotonicity</a> and <a href="/wiki/Defeasible_reasoning" title="Defeasible reasoning">defeasibility</a>: it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn.<a href="#cite_note-FOOTNOTERocci201726HintikkaSandu200613,_16Douven2021-76">[75]</a> Ampliative reasoning plays a central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between.<a href="#cite_note-FOOTNOTEIEP_StaffDouven2021Hawthorne2021-77">[76]</a> The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term "<a href="/wiki/Inductive_reasoning" title="Inductive reasoning">induction</a>" to cover all forms of non-deductive arguments.<a href="#cite_note-FOOTNOTEIEP_StaffHawthorne2021Wilbanks2010107–124-78">[77]</a> But in a more narrow sense, induction is only one type of ampliative argument alongside <a href="/wiki/Abductive_reasoning" title="Abductive reasoning">abductive arguments</a>.<a href="#cite_note-FOOTNOTEDouven2021-79">[78]</a> Some philosophers, like Leo Groarke, also allow conductive arguments<a href="#cite_note-80">[b]</a> as another type.<a href="#cite_note-FOOTNOTEGroarke20214.1_AV_CriteriaPossin2016563–593-81">[79]</a> In this narrow sense, induction is often defined as a form of statistical generalization.<a href="#cite_note-FOOTNOTEScottMarshall2009analytic_inductionHoudeCamacho2003Induction-82">[80]</a> In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains.<a href="#cite_note-FOOTNOTEBorchert2006bInduction-83">[81]</a> In this sense, one may infer that "all elephants are gray" based on one's past observations of the color of elephants.<a href="#cite_note-FOOTNOTEDouven2021-79">[78]</a> A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray.<a href="#cite_note-FOOTNOTEBorchert2006bInduction-83">[81]</a> Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference.<a href="#cite_note-FOOTNOTEDouven2021-79">[78]</a> Abductive inference may or may not take statistical observations into consideration. In either case, the premises offer support for the conclusion because the conclusion is the best <a href="/wiki/Explanation" title="Explanation">explanation</a> of why the premises are true.<sup id="cite_ref-FOOTNOTEDouven2021Koslowski2017[httpswwwtaylorfranciscomlocsedit1043249781315725697-20abductive-reasoning-explanation-barbara-koslowski_Abductive_reasoning_and_explanation]_84-0" class="reference"><a href="#cite_note-FOOTNOTEDouven2021Koslowski2017[httpswwwtaylorfranciscomlocsedit1043249781315725697-20abductive-reasoning-explanation-barbara-koslowski_Abductive_reasoning_and_explanation]-84">[82]</a> In this sense, abduction is also called the inference to the best explanation.<a href="#cite_note-FOOTNOTECummings2010Abduction,_p._1-85">[83]</a> For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean the table. This conclusion is justified because it is the best explanation of the current state of the kitchen.<a href="#cite_note-FOOTNOTEDouven2021-79">[78]</a> For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it is not the best or most likely explanation.<sup id="cite_ref-FOOTNOTEDouven2021Koslowski2017[httpswwwtaylorfranciscomlocsedit1043249781315725697-20abductive-reasoning-explanation-barbara-koslowski_Abductive_reasoning_and_explanation]_84-1" class="reference"><a href="#cite_note-FOOTNOTEDouven2021Koslowski2017[httpswwwtaylorfranciscomlocsedit1043249781315725697-20abductive-reasoning-explanation-barbara-koslowski_Abductive_reasoning_and_explanation]-84">[82]</a><a href="#cite_note-FOOTNOTECummings2010Abduction,_p._1-85">[83]</a> <div class="mw-heading mw-heading3"><h3 id="Fallacies">Fallacies</h3>[<a href="/w/index.php?title=Logic&action=edit&section=13" title="Edit section: Fallacies">edit</a>]</div> Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as <a href="/wiki/Fallacy" title="Fallacy">fallacies</a>. Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion.<a href="#cite_note-FOOTNOTEHansen2020Chatfield2017194-86">[84]</a> So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though the conclusion is true. Some theorists, like <a href="/wiki/John_Stuart_Mill" title="John Stuart Mill">John Stuart Mill</a>, give a more restrictive definition of fallacies by additionally requiring that they appear to be correct.<a href="#cite_note-FOOTNOTEWalton198771._A_new_model_of_argumentHansen2020-87">[85]</a> This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them.<a href="#cite_note-FOOTNOTEHansen2020-88">[86]</a> However, this reference to appearances is controversial because it belongs to the field of <a href="/wiki/Psychology" title="Psychology">psychology</a>, not logic, and because appearances may be different for different people.<a href="#cite_note-FOOTNOTEHansen2020Walton1987633._Logic_of_propositions-89">[87]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg" class="mw-file-description"><img alt="Poster from 1901" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg/330px-Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg" decoding="async" width="330" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg/495px-Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg/660px-Young_America%27s_dilemma_-_Dalrymple._LCCN2010651418.jpg 2x" data-file-width="8615" data-file-height="5396" /></a><figcaption>Young America's dilemma: Shall I be wise and great, or rich and powerful? (poster from 1901) This is an example of a <a href="/wiki/False_dilemma" title="False dilemma">false dilemma</a>: an <a href="/wiki/Informal_fallacy" title="Informal fallacy">informal fallacy</a> using a disjunctive premise that excludes viable alternatives.</figcaption></figure> Fallacies are usually divided into <a href="/wiki/Formal_fallacy" title="Formal fallacy">formal</a> and informal fallacies.<a href="#cite_note-FOOTNOTEVleet2010ix–xIntroductionDowdenStump-38">[38]</a> For formal fallacies, the source of the error is found in the form of the argument. For example, <a href="/wiki/Denying_the_antecedent" title="Denying the antecedent">denying the antecedent</a> is one type of formal fallacy, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore Othello is not male".<a href="#cite_note-FOOTNOTESternbergStone2012327–356-90">[88]</a> But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the content or the context of the argument.<a href="#cite_note-FOOTNOTEWalton19872–41._A_new_model_of_argumentDowdenHansen2020-91">[89]</a> Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity, the ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark".<a href="#cite_note-FOOTNOTEEngel198259–922._The_medium_of_languageMackie1967Stump-92">[90]</a> Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise.<a href="#cite_note-FOOTNOTEStumpEngel1982143–2124._Fallacies_of_presumption-93">[91]</a> In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it.<a href="#cite_note-FOOTNOTEStumpMackie1967-94">[92]</a> <div class="mw-heading mw-heading3"><h3 id="Definitory_and_strategic_rules">Definitory and strategic rules</h3>[<a href="/w/index.php?title=Logic&action=edit&section=14" title="Edit section: Definitory and strategic rules">edit</a>]</div> The main focus of most logicians is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference.<a href="#cite_note-FOOTNOTEHintikkaSandu200620-95">[93]</a> They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach a given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In <a href="/wiki/Chess" title="Chess">chess</a>, for example, the definitory rules dictate that <a href="/wiki/Bishop_(chess)" title="Bishop (chess)">bishops</a> may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's <a href="/wiki/King_(chess)" title="King (chess)">king</a>.<a href="#cite_note-FOOTNOTEHintikkaSandu200620Pedemonte20181–17Hintikka2023-96">[94]</a> It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning.<a href="#cite_note-FOOTNOTEHintikkaSandu200620-95">[93]</a> <div class="mw-heading mw-heading3"><h3 id="Formal_systems">Formal systems</h3>[<a href="/w/index.php?title=Logic&action=edit&section=15" title="Edit section: Formal systems">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/Formal_system" title="Formal system">Formal system</a></div> A formal system of logic consists of a formal language together with a set of <a href="/wiki/Axiom" title="Axiom">axioms</a> and a <a href="/wiki/Proof_system" class="mw-redirect" title="Proof system">proof system</a> used to draw inferences from these axioms.<a href="#cite_note-FOOTNOTEBorisAlexander201774Cook2009124-97">[95]</a> In logic, axioms are statements that are accepted without proof. They are used to justify other statements.<a href="#cite_note-FOOTNOTEFlotyński2020[httpsbooksgooglecombooksidEC4NEAAAQBAJpgPA39_39]BerlemannMangold2009[httpsbooksgooglecombooksidXUGN9tKTIiYCpgPA194_194]-98">[96]</a> Some theorists also include a <a href="/wiki/Semantics_of_logic" title="Semantics of logic">semantics</a> that specifies how the expressions of the formal language relate to real objects.<a href="#cite_note-FOOTNOTEGensler2006xliiiFontJansana20178-99">[97]</a> Starting in the late 19th century, many new formal systems have been proposed.<a href="#cite_note-FOOTNOTEHaack19781–10Philosophy_of_logicsHintikkaSandu200631–32Jacquette20061–12Introduction:_Philosophy_of_logic_today-100">[98]</a> A formal language consists of an alphabet and syntactic rules. The alphabet is the set of basic symbols used in <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a>. The syntactic rules determine how these symbols may be arranged to result in well-formed formulas.<a href="#cite_note-FOOTNOTEMooreCarling198253Enderton2001[httpsbooksgooglecombooksiddVncCl_EtUkCpgPA12_12–13]Sentential_Logic-101">[99]</a> For instance, the syntactic rules of <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a> determine that "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\land Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}">" is a well-formed formula but "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e1393b3f36b277a9aa8490660cb150921741a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle \land Q}">" is not since the logical conjunction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \land }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∧</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \land }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6823e5a222eb3ca49672818ac3d13ec607052c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \land }"> requires terms on both sides.<a href="#cite_note-FOOTNOTELeporeCumming20125-102">[100]</a> A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of <a href="/wiki/Conjunction_introduction" title="Conjunction introduction">conjunction introduction</a> states that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\land Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}"> follows from the premises <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}">. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including <a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a> and <a href="/wiki/Sequent_calculus" title="Sequent calculus">sequent calculi</a>.<a href="#cite_note-FOOTNOTEWasilewska2018145–6RathjenSieg2022-103">[101]</a> A semantics is a system for <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a> expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance, the semantics for <a href="/wiki/Classical_logic" title="Classical logic">classical</a> propositional logic assigns the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\land Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}"> the denotation "true" whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true.<a href="#cite_note-FOOTNOTESider201034–42ShapiroKouri_Kissel2022Bimbo20168–9-104">[102]</a> A system of logic is <a href="/wiki/Soundness" title="Soundness">sound</a> when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly.<a href="#cite_note-FOOTNOTERestallStandefer202391Enderton2001[httpsbooksgooglecombooksiddVncCl_EtUkCpgPA131_131–146]Chapter_2.5van_Dalen1994Chapter_1.5-105">[103]</a> <div class="mw-heading mw-heading2"><h2 id="Systems_of_logic">Systems of logic</h2>[<a href="/w/index.php?title=Logic&action=edit&section=16" title="Edit section: Systems of logic">edit</a>]</div> Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a> was treated as the canon of logic in the Western world,<a href="#cite_note-FOOTNOTEJacquette20061–12Introduction:_Philosophy_of_logic_todaySmith2022Groarke-106">[104]</a> but modern developments in this field have led to a vast proliferation of logical systems.<a href="#cite_note-FOOTNOTEHaack19961._'Alternative'_in_'Alternative_Logic'-107">[105]</a> One prominent categorization divides modern formal logical systems into <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a>, extended logics, and <a href="/wiki/Deviant_logic" title="Deviant logic">deviant logics</a>.<a href="#cite_note-FOOTNOTEHaack19781–10Philosophy_of_logicsHaack19961._'Alternative'_in_'Alternative_Logic'Wolf1978327–340-108">[106]</a> <div class="mw-heading mw-heading3"><h3 id="Aristotelian">Aristotelian</h3>[<a href="/w/index.php?title=Logic&action=edit&section=17" title="Edit section: Aristotelian">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/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a></div> <a href="/wiki/Aristotelian_logic" class="mw-redirect" title="Aristotelian logic">Aristotelian logic</a> encompasses a great variety of topics. They include <a href="/wiki/Metaphysics" title="Metaphysics">metaphysical</a> theses about <a href="/wiki/Ontology" title="Ontology">ontological</a> categories and problems of scientific explanation. But in a more narrow sense, it is identical to <a href="/wiki/Term_logic" title="Term logic">term logic</a> or syllogistics. A <a href="/wiki/Syllogism" title="Syllogism">syllogism</a> is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a <a href="/wiki/Subject_(grammar)" title="Subject (grammar)">subject</a>, a predicate, and a <a href="/wiki/Copula_(linguistics)" title="Copula (linguistics)">copula</a> connecting the subject to the predicate.<a href="#cite_note-FOOTNOTESmith2022GroarkeBobzien2020-109">[107]</a> For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is".<a href="#cite_note-FOOTNOTEGroarke-110">[108]</a> The subject and the predicate are the terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using a logical connective like "and" to form a new complex proposition.<a href="#cite_note-FOOTNOTESmith2022Magnus20052.2_Connectives-111">[109]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Square_of_opposition,_set_diagrams.svg" class="mw-file-description"><img alt="Diagram of the square of opposition" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Square_of_opposition%2C_set_diagrams.svg/310px-Square_of_opposition%2C_set_diagrams.svg.png" decoding="async" width="310" height="401" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Square_of_opposition%2C_set_diagrams.svg/465px-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/620px-Square_of_opposition%2C_set_diagrams.svg.png 2x" data-file-width="479" data-file-height="620" /></a><figcaption>The <a href="/wiki/Square_of_opposition" title="Square of opposition">square of opposition</a> is often used to visualize the relations between the four basic <a href="/wiki/Categorical_propositions" class="mw-redirect" title="Categorical propositions">categorical propositions</a> in Aristotelian logic. It shows, for example, that the propositions "All S are P" and "Some S are not P" are contradictory, meaning that one of them has to be true while the other is false.</figcaption></figure> In Aristotelian logic, the subject can be universal, particular, indefinite, or singular. For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates".<a href="#cite_note-FOOTNOTESmith2022Bobzien2020HintikkaSpade[httpswwwbritannicacomtopichistory-of-logicAristotle_Aristotle]-112">[110]</a> Aristotelian logic only includes predicates for simple <a href="/wiki/Property_(philosophy)" title="Property (philosophy)">properties</a> of entities. But it lacks predicates corresponding to <a href="/wiki/Relations_(philosophy)" class="mw-redirect" title="Relations (philosophy)">relations</a> between entities.<a href="#cite_note-FOOTNOTEWesterståhl1989577–585-113">[111]</a> The predicate can be linked to the subject in two ways: either by affirming it or by denying it.<a href="#cite_note-FOOTNOTESmith2022Groarke-114">[112]</a> For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case.<a href="#cite_note-FOOTNOTESmith2022Hurley20154._Categorical_SyllogismsCopiCohenRodych2019[httpsbooksgooglecombooksid38bADwAAQBAJpgPA187_6._Categorical_Syllogisms]-115">[113]</a> Thus, these three propositions contain three predicates, referred to as major term, minor term, and middle term.<a href="#cite_note-FOOTNOTEGroarkeHurley20154._Categorical_SyllogismsCopiCohenRodych2019[httpsbooksgooglecombooksid38bADwAAQBAJpgPA187_6._Categorical_Syllogisms]-116">[114]</a> The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how the propositions are formed.<a href="#cite_note-FOOTNOTESmith2022Groarke-114">[112]</a><a href="#cite_note-FOOTNOTEHurley20154._Categorical_Syllogisms-117">[115]</a> For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid.<a href="#cite_note-FOOTNOTESpriggs201220–2-118">[116]</a> <div class="mw-heading mw-heading3"><h3 id="Classical">Classical</h3>[<a href="/w/index.php?title=Logic&action=edit&section=18" title="Edit section: Classical">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/Classical_logic" title="Classical logic">Classical logic</a></div> <a href="/wiki/Classical_logic" title="Classical logic">Classical logic</a> is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in the sense that it is based on basic logical intuitions shared by most logicians.<a href="#cite_note-FOOTNOTEHintikka2019§Nature_and_varieties_of_logic,_§Alternative_logicsHintikkaSandu200627–8Bäck2016317-119">[117]</a> These intuitions include the <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a>, the <a href="/wiki/Double_negation_elimination" class="mw-redirect" title="Double negation elimination">double negation elimination</a>, the <a href="/wiki/Principle_of_explosion" title="Principle of explosion">principle of explosion</a>, and the bivalence of truth.<a href="#cite_note-FOOTNOTEShapiroKouri_Kissel2022-120">[118]</a> It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are the contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future.<a href="#cite_note-FOOTNOTEBurgess20091._Classical_logic-121">[119]</a> Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like <a href="/wiki/Ethics" title="Ethics">ethics</a> or epistemology that lie beyond the scope of mathematics.<a href="#cite_note-FOOTNOTEJacquette20061–12Introduction:_Philosophy_of_logic_todayBorchert2006cLogic,_Non-ClassicalGoble2001Introduction-122">[120]</a> <div class="mw-heading mw-heading4"><h4 id="Propositional_logic">Propositional logic</h4>[<a href="/w/index.php?title=Logic&action=edit&section=19" title="Edit section: Propositional logic">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/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></div> Propositional logic comprises formal systems in which formulae are built from <a href="/wiki/Atomic_propositions" class="mw-redirect" title="Atomic propositions">atomic propositions</a> using <a href="/wiki/Logical_connectives" class="mw-redirect" title="Logical connectives">logical connectives</a>. For instance, propositional logic represents the <a href="/wiki/Conjunction_(logic)" class="mw-redirect" title="Conjunction (logic)">conjunction</a> of two atomic propositions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"> as the complex formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\land Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>∧</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\land Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5690bb4822d8c821a00cfe3c6644b046a884af4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.166ex; height:2.509ex;" alt="{\displaystyle P\land Q}">. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component.<a href="#cite_note-FOOTNOTEBrody2006535–536-123">[121]</a> Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition.<a href="#cite_note-FOOTNOTEKlement1995b-124">[122]</a> <div class="mw-heading mw-heading4"><h4 id="First-order_logic">First-order logic</h4>[<a href="/w/index.php?title=Logic&action=edit&section=20" title="Edit section: First-order logic">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:BS-12-Begriffsschrift_Quantifier1-svg.svg" class="mw-file-description"><img alt="Symbol introduced by Gottlob Frege for the universal quantifier" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/BS-12-Begriffsschrift_Quantifier1-svg.svg/220px-BS-12-Begriffsschrift_Quantifier1-svg.svg.png" decoding="async" width="220" height="66" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/BS-12-Begriffsschrift_Quantifier1-svg.svg/330px-BS-12-Begriffsschrift_Quantifier1-svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/BS-12-Begriffsschrift_Quantifier1-svg.svg/440px-BS-12-Begriffsschrift_Quantifier1-svg.svg.png 2x" data-file-width="512" data-file-height="154" /></a><figcaption><a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>'s <a href="/wiki/Begriffschrift" class="mw-redirect" title="Begriffschrift">Begriffschrift</a> introduced the notion of quantifier in a graphical notation, which here represents the judgment that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x.F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>.</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x.F(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba23799cf40a6e9b0a427c429dc89fbb8a74bfee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.536ex; height:2.843ex;" alt="{\displaystyle \forall x.F(x)}"> is true.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></div> First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates the internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, <a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a>, which refer to properties and relations, and quantifiers, which treat notions like "some" and "all".<a href="#cite_note-FOOTNOTEShapiroKouri_Kissel2022Honderich2005philosophical_logicMichaelsonReimer2019-125">[123]</a> For example, to express the proposition "this raven is black", one may use the predicate <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}"> for the property "black" and the singular term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> referring to the raven to form the expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf985b9cc5e851d080440261106d69b6d61dd88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.622ex; height:2.843ex;" alt="{\displaystyle B(r)}">. To express that some objects are black, the existential quantifier <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }"> is combined with the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> to form the proposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists xB(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists xB(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4426e051732eddd927ca9b665c90a8b7e0b8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.525ex; height:2.843ex;" alt="{\displaystyle \exists xB(x)}">. First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists xB(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists xB(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4426e051732eddd927ca9b665c90a8b7e0b8e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.525ex; height:2.843ex;" alt="{\displaystyle \exists xB(x)}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf985b9cc5e851d080440261106d69b6d61dd88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.622ex; height:2.843ex;" alt="{\displaystyle B(r)}">.<a href="#cite_note-FOOTNOTENolt2021Magnus20054_Quantified_logic-126">[124]</a> <div class="mw-heading mw-heading3"><h3 id="Extended">Extended</h3>[<a href="/w/index.php?title=Logic&action=edit&section=21" title="Edit section: Extended">edit</a>]</div> Extended logics are logical systems that accept the basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like <a href="/wiki/Metaphysics" title="Metaphysics">metaphysics</a>, <a href="/wiki/Ethics" title="Ethics">ethics</a>, and <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>.<a href="#cite_note-FOOTNOTEBunninYu2009[httpsbooksgooglecombooksidM7ZFEAAAQBAJpgPA179_179]Garson2023[httpsplatostanfordeduentrieslogic-modal_Introduction]-127">[125]</a> <div class="mw-heading mw-heading4"><h4 id="Modal_logic">Modal logic</h4>[<a href="/w/index.php?title=Logic&action=edit&section=22" title="Edit section: Modal logic">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/Modal_logic" title="Modal logic">Modal logic</a></div> <a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a> is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Diamond }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◊</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Diamond }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5e6f31a91de53f443e2cd4c7478693a1a6a57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.55ex; height:2.343ex;" alt="{\displaystyle \Diamond }"> expresses that something is possible while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"> expresses that something is necessary.<a href="#cite_note-FOOTNOTEGarson2023Sadegh-Zadeh2015983-128">[126]</a> For example, if the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eafeec3a494b40022d4f4402ec11aff5a45ca2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.664ex; height:2.843ex;" alt="{\displaystyle B(s)}"> stands for the sentence "Socrates is a banker" then the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Diamond B(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◊</mi> <mi>B</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Diamond B(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77c5d10926155e9fedc321c80588d3272d495e19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.214ex; height:2.843ex;" alt="{\displaystyle \Diamond B(s)}"> articulates the sentence "It is possible that Socrates is a banker".<a href="#cite_note-FOOTNOTEFitch201417-129">[127]</a> To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something is necessary, then it is also possible. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Diamond A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◊</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Diamond A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98e1c8a6d2e420ba59226c86cf7a81787fbbf450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.293ex; height:2.343ex;" alt="{\displaystyle \Diamond A}"> follows from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a5cd9c3308f520a7b3c53b9f828a97b7eb697d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.551ex; height:2.176ex;" alt="{\displaystyle \Box A}">. Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a5cd9c3308f520a7b3c53b9f828a97b7eb697d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.551ex; height:2.176ex;" alt="{\displaystyle \Box A}"> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot \Diamond \lnot A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>◊</mi> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot \Diamond \lnot A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac9a1e59ec369a33da38f90189a5a2255ffd53d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.394ex; height:2.343ex;" alt="{\displaystyle \lnot \Diamond \lnot A}">.<a href="#cite_note-FOOTNOTEGarson2023CarnielliPizzi20083Benthem-130">[128]</a> Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, <a href="/wiki/Deontic_logic" title="Deontic logic">deontic logic</a> concerns the field of ethics and introduces symbols to express the ideas of <a href="/wiki/Obligation" title="Obligation">obligation</a> and <a href="/wiki/Permission_(philosophy)" title="Permission (philosophy)">permission</a>, i.e. to describe whether an agent has to perform a certain action or is allowed to perform it.<a href="#cite_note-FOOTNOTEGarson2023-131">[129]</a> The modal operators in <a href="/wiki/Temporal_logic" title="Temporal logic">temporal modal logic</a> articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time.<a href="#cite_note-FOOTNOTEGarson2023-131">[129]</a> In epistemology, <a href="/wiki/Epistemic_modal_logic" title="Epistemic modal logic">epistemic modal logic</a> is used to represent the ideas of <a href="/wiki/Knowledge" title="Knowledge">knowing</a> something in contrast to merely <a href="/wiki/Belief" title="Belief">believing</a> it to be the case.<a href="#cite_note-FOOTNOTERendsvigSymons2021-132">[130]</a> <div class="mw-heading mw-heading4"><h4 id="Higher_order_logic">Higher order logic</h4>[<a href="/w/index.php?title=Logic&action=edit&section=23" title="Edit section: Higher order logic">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/Higher-order_logic" title="Higher-order logic">Higher-order logic</a></div> <a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order logics</a> extend classical logic not by using modal operators but by introducing new forms of quantification.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicVäänänen2021Ketland2005Second_Order_Logic-133">[131]</a> Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists x(Apple(x)\land Sweet(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mi>p</mi> <mi>p</mi> <mi>l</mi> <mi>e</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>S</mi> <mi>w</mi> <mi>e</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x(Apple(x)\land Sweet(x))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91553ca9cfa2cb1d9476194df2acbd86689b0d74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.321ex; height:2.843ex;" alt="{\displaystyle \exists x(Apple(x)\land Sweet(x))}">" (some apples are sweet) is an example of the <a href="/wiki/Existential_quantification" title="Existential quantification">existential quantifier</a> "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ed842b6b90b2fdd825320cf8e5265fa937b583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }">" applied to the individual variable "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">". In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists Q(Q(Mary)\land Q(John))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mi>a</mi> <mi>r</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>J</mi> <mi>o</mi> <mi>h</mi> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists Q(Q(Mary)\land Q(John))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0558c050f0f79e86630ebac840b936b5b368f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.027ex; height:2.843ex;" alt="{\displaystyle \exists Q(Q(Mary)\land Q(John))}">". In this case, the existential quantifier is applied to the predicate variable "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}">".<sup id="cite_ref-FOOTNOTEAudi1999bPhilosophy_of_logicVäänänen2021DaintithWright2008[httpswwwencyclopediacomcomputingdictionaries-thesauruses-pictures-and-press-releasespredicate-calculus_Predicate_calculus]_134-0" class="reference"><a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicVäänänen2021DaintithWright2008[httpswwwencyclopediacomcomputingdictionaries-thesauruses-pictures-and-press-releasespredicate-calculus_Predicate_calculus]-134">[132]</a> The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logic-44">[43]</a> But it has drawbacks in regard to its meta-logical properties and ontological implications, which is why first-order logic is still more commonly used.<a href="#cite_note-FOOTNOTEAudi1999bPhilosophy_of_logicKetland2005Second_Order_Logic-135">[133]</a> <div class="mw-heading mw-heading3"><h3 id="Deviant">Deviant</h3>[<a href="/w/index.php?title=Logic&action=edit&section=24" title="Edit section: Deviant">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/Deviant_logic" title="Deviant logic">Deviant logic</a></div> <a href="/wiki/Deviant_logic" title="Deviant logic">Deviant logics</a> are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue.<a href="#cite_note-FOOTNOTEHaack19961._'Alternative'_in_'Alternative_Logic'Wolf1978327–340-136">[134]</a> <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">Intuitionistic logic</a> is a restricted version of classical logic.<a href="#cite_note-FOOTNOTEMoschovakis2022Borchert2006cLogic,_Non-Classical-137">[135]</a> It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that <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}"> follows from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot \lnot A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot \lnot A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631eaaaa1c4f2d430a0ecd582b0b98b20d514129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.844ex; height:2.176ex;" alt="{\displaystyle \lnot \lnot A}">. This is a valid rule of inference in classical logic but it is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a>. It states that for every sentence, either it or its negation is true. This means that every proposition of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\lor \lnot A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∨</mo> <mi mathvariant="normal">¬</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\lor \lnot A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42bc39c90e40298464f5f82377cb9e7ed52e938f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.619ex; height:2.176ex;" alt="{\displaystyle A\lor \lnot A}"> is true.<a href="#cite_note-FOOTNOTEMoschovakis2022Borchert2006cLogic,_Non-Classical-137">[135]</a> These deviations from classical logic are based on the idea that truth is established by verification using a proof. Intuitionistic logic is especially prominent in the field of <a href="/wiki/Constructivism_(philosophy_of_mathematics)" title="Constructivism (philosophy of mathematics)">constructive mathematics</a>, which emphasizes the need to find or construct a specific example to prove its existence.<a href="#cite_note-FOOTNOTEBorchert2006cLogic,_Non-ClassicalBridgesIshiharaRathjenSchwichtenberg202373–74Friend2014101-138">[136]</a> <a href="/wiki/Multi-valued_logics" class="mw-redirect" title="Multi-valued logics">Multi-valued logics</a> depart from classicality by rejecting the <a href="/wiki/Principle_of_bivalence" title="Principle of bivalence">principle of bivalence</a>, which requires all propositions to be either true or false. For instance, <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a> and <a href="/wiki/Stephen_Cole_Kleene" title="Stephen Cole Kleene">Stephen Cole Kleene</a> both proposed <a href="/wiki/Ternary_logic" class="mw-redirect" title="Ternary logic">ternary logics</a> which have a third truth value representing that a statement's truth value is indeterminate.<a href="#cite_note-FOOTNOTESider2010Chapter_3.4Gamut19915.5Zegarelli201030-139">[137]</a> These logics have been applied in the field of linguistics. <a href="/wiki/Fuzzy_logics" class="mw-redirect" title="Fuzzy logics">Fuzzy logics</a> are multivalued logics that have an infinite number of "degrees of truth", represented by a <a href="/wiki/Real_number" title="Real number">real number</a> between 0 and 1.<a href="#cite_note-FOOTNOTEHájek2006-140">[138]</a> <a href="/wiki/Paraconsistent_logic" title="Paraconsistent logic">Paraconsistent logics</a> are logical systems that can deal with contradictions. They are formulated to avoid the principle of explosion: for them, it is not the case that anything follows from a contradiction.<a href="#cite_note-FOOTNOTEBorchert2006cLogic,_Non-ClassicalPriestTanakaWeber2018Weber-141">[139]</a> They are often motivated by <a href="/wiki/Dialetheism" title="Dialetheism">dialetheism</a>, the view that contradictions are real or that reality itself is contradictory. <a href="/wiki/Graham_Priest" title="Graham Priest">Graham Priest</a> is an influential contemporary proponent of this position and similar views have been ascribed to <a href="/wiki/Georg_Wilhelm_Friedrich_Hegel" title="Georg Wilhelm Friedrich Hegel">Georg Wilhelm Friedrich Hegel</a>.<a href="#cite_note-FOOTNOTEPriestTanakaWeber2018WeberHaack1996Introduction-142">[140]</a> <div class="mw-heading mw-heading3"><h3 id="Informal">Informal</h3>[<a href="/w/index.php?title=Logic&action=edit&section=25" title="Edit section: Informal">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/Informal_logic" title="Informal logic">Informal logic</a></div> <a href="/wiki/Informal_logic" title="Informal logic">Informal logic</a> is usually carried out in a less systematic way. It often focuses on more specific issues, like investigating a particular type of fallacy or studying a certain aspect of argumentation. Nonetheless, some frameworks of informal logic have also been presented that try to provide a systematic characterization of the correctness of arguments.<a href="#cite_note-FOOTNOTEHansen2020Korb200441–42,_48Ritola2008335-143">[141]</a> The pragmatic or dialogical approach to informal logic sees arguments as <a href="/wiki/Speech_act" title="Speech act">speech acts</a> and not merely as a set of premises together with a conclusion.<a href="#cite_note-FOOTNOTEHansen2020Korb200443–44Ritola2008335-144">[142]</a> As speech acts, they occur in a certain context, like a <a href="/wiki/Dialogue" title="Dialogue">dialogue</a>, which affects the standards of right and wrong arguments.<a href="#cite_note-FOOTNOTEWalton19872–31._A_new_model_of_argumentRitola2008335-145">[143]</a> A prominent version by <a href="/wiki/Douglas_N._Walton" title="Douglas N. Walton">Douglas N. Walton</a> understands a dialogue as a game between two players. The initial position of each player is characterized by the propositions to which they are committed and the conclusion they intend to prove. Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own conclusion.<a href="#cite_note-FOOTNOTEWalton19873–4,_18–221._A_new_model_of_argument-146">[144]</a> This is achieved by making arguments: arguments are the moves of the game.<a href="#cite_note-FOOTNOTEWalton19873–4,_11,_181._A_new_model_of_argumentRitola2008335-147">[145]</a> They affect to which propositions the players are committed. A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows from them. This is usually not possible straight away. For this reason, it is normally necessary to formulate a sequence of arguments as intermediary steps, each of which brings the opponent a little closer to one's intended conclusion. Besides these positive arguments leading one closer to victory, there are also negative arguments preventing the opponent's victory by denying their conclusion.<a href="#cite_note-FOOTNOTEWalton19873–4,_18–221._A_new_model_of_argument-146">[144]</a> Whether an argument is correct depends on whether it promotes the progress of the dialogue. Fallacies, on the other hand, are violations of the standards of proper argumentative rules.<a href="#cite_note-FOOTNOTEHansen2020Walton19873–4,_18–223._Logic_of_propositions-148">[146]</a> These standards also depend on the type of dialogue. For example, the standards governing the scientific discourse differ from the standards in business negotiations.<a href="#cite_note-FOOTNOTERitola2008335-149">[147]</a> The epistemic approach to informal logic, on the other hand, focuses on the epistemic role of arguments.<a href="#cite_note-FOOTNOTEHansen2020Korb200443,_54–55-150">[148]</a> It is based on the idea that arguments aim to increase our knowledge. They achieve this by linking justified beliefs to beliefs that are not yet justified.<a href="#cite_note-FOOTNOTESiegelBiro1997277–292-151">[149]</a> Correct arguments succeed at expanding knowledge while fallacies are epistemic failures: they do not justify the belief in their conclusion.<a href="#cite_note-FOOTNOTEHansen2020Korb200441–70-152">[150]</a> For example, the <a href="/wiki/Fallacy_of_begging_the_question" class="mw-redirect" title="Fallacy of begging the question">fallacy of begging the question</a> is a fallacy because it fails to provide independent justification for its conclusion, even though it is deductively valid.<a href="#cite_note-FOOTNOTEMackie1967SiegelBiro1997277–292-153">[151]</a> In this sense, logical normativity consists in epistemic success or rationality.<a href="#cite_note-FOOTNOTESiegelBiro1997277–292-151">[149]</a> The <a href="/wiki/Bayesian_epistemology" title="Bayesian epistemology">Bayesian approach</a> is one example of an epistemic approach.<a href="#cite_note-FOOTNOTEHansen2020MooreCromby201660-154">[152]</a> Central to Bayesianism is not just whether the agent believes something but the degree to which they believe it, the so-called credence. Degrees of belief are seen as <a href="/wiki/Subjective_probability" class="mw-redirect" title="Subjective probability">subjective probabilities</a> in the believed proposition, i.e. how certain the agent is that the proposition is true.<a href="#cite_note-FOOTNOTEOlsson2018431–442Bayesian_EpistemologyHájekLin2017207–232HartmannSprenger2010609–620Bayesian_Epistemology-155">[153]</a> On this view, reasoning can be interpreted as a process of changing one's credences, often in reaction to new incoming information.<a href="#cite_note-FOOTNOTEShermer2022136-156">[154]</a> Correct reasoning and the arguments it is based on follow the laws of probability, for example, the <a href="/wiki/Bayesian_epistemology#Principle_of_conditionalization" title="Bayesian epistemology">principle of conditionalization</a>. Bad or irrational reasoning, on the other hand, violates these laws.<a href="#cite_note-FOOTNOTEKorb200441–42,_44–46HájekLin2017207–232Talbott2016-157">[155]</a> <div class="mw-heading mw-heading2"><h2 id="Areas_of_research">Areas of research</h2>[<a href="/w/index.php?title=Logic&action=edit&section=26" title="Edit section: Areas of research">edit</a>]</div> Logic is studied in various fields. In many cases, this is done by applying its formal method to specific topics outside its scope, like to ethics or computer science.<a href="#cite_note-FOOTNOTEHintikka2019§Logic_and_other_disciplinesHaack19781–10Philosophy_of_logics-158">[156]</a> In other cases, logic itself is made the subject of research in another discipline. This can happen in diverse ways. For instance, it can involve investigating the philosophical assumptions linked to the basic concepts used by logicians. Other ways include interpreting and analyzing logic through mathematical structures as well as studying and comparing abstract properties of formal logical systems.<a href="#cite_note-FOOTNOTEHintikka2019lead_section,_§Features_and_problems_of_logicGödel1984447–469Russell's_mathematical_logicMonk19761–9Introduction-159">[157]</a> <div class="mw-heading mw-heading3"><h3 id="Philosophy_of_logic_and_philosophical_logic">Philosophy of logic and philosophical logic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=27" title="Edit section: Philosophy of logic and philosophical logic">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a> and <a href="/wiki/Philosophical_logic" title="Philosophical logic">Philosophical logic</a></div> Philosophy of logic is the philosophical discipline studying the scope and nature of logic.<a href="#cite_note-FOOTNOTEHintikka2019lead_section,_§Nature_and_varieties_of_logicAudi1999bPhilosophy_of_logic-60">[59]</a> It examines many presuppositions implicit in logic, like how to define its basic concepts or the metaphysical assumptions associated with them.<a href="#cite_note-FOOTNOTEJacquette20061–12Introduction:_Philosophy_of_logic_today-160">[158]</a> It is also concerned with how to classify logical systems and considers the <a href="/wiki/Ontological" class="mw-redirect" title="Ontological">ontological</a> commitments they incur.<a href="#cite_note-FOOTNOTEHintikka2019§Problems_of_ontology-161">[159]</a> Philosophical logic is one of the areas within the philosophy of logic. It studies the application of logical methods to philosophical problems in fields like metaphysics, ethics, and epistemology.<a href="#cite_note-FOOTNOTEJacquette20061–12Introduction:_Philosophy_of_logic_todayBurgess20091._Classical_logic-162">[160]</a> This application usually happens in the form of <a href="#Extended">extended</a> or <a href="#Deviant">deviant logical systems</a>.<a href="#cite_note-FOOTNOTEGoble2001IntroductionHintikkaSandu200631–32-163">[161]</a> <div class="mw-heading mw-heading3"><h3 id="Metalogic">Metalogic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=28" title="Edit section: Metalogic">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/Metalogic" title="Metalogic">Metalogic</a></div> Metalogic is the field of inquiry studying the properties of formal logical systems. For example, when a new formal system is developed, metalogicians may study it to determine which formulas can be proven in it. They may also study whether an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> could be developed to find a proof for each formula and whether every provable formula in it is a tautology. Finally, they may compare it to other logical systems to understand its distinctive features. A key issue in metalogic concerns the relation between syntax and semantics. The syntactic rules of a formal system determine how to deduce conclusions from premises, i.e. how to formulate proofs. The semantics of a formal system governs which sentences are true and which ones are false. This determines the validity of arguments since, for valid arguments, it is impossible for the premises to be true and the conclusion to be false. The relation between syntax and semantics concerns issues like whether every valid argument is provable and whether every provable argument is valid. Metalogicians also study whether logical systems are complete, sound, and <a href="/wiki/Consistency" title="Consistency">consistent</a>. They are interested in whether the systems are <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a> and what <a href="/wiki/Expressive_power_(computer_science)" title="Expressive power (computer science)">expressive power</a> they have. Metalogicians usually rely heavily on abstract mathematical reasoning when examining and formulating metalogical proofs. This way, they aim to arrive at precise and general conclusions on these topics.<a href="#cite_note-FOOTNOTEGensler2006xliii–xlivSider20104–6Schagrin-164">[162]</a> <div class="mw-heading mw-heading3"><h3 id="Mathematical_logic">Mathematical logic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=29" title="Edit section: Mathematical logic">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/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Bertrand_Russell_1949.jpg" class="mw-file-description"><img alt="Photograph of Bertrand Russell" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Bertrand_Russell_1949.jpg/220px-Bertrand_Russell_1949.jpg" decoding="async" width="220" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Bertrand_Russell_1949.jpg/330px-Bertrand_Russell_1949.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/Bertrand_Russell_1949.jpg/440px-Bertrand_Russell_1949.jpg 2x" data-file-width="1583" data-file-height="1997" /></a><figcaption>Bertrand Russell made various contributions to mathematical logic.<a href="#cite_note-FOOTNOTEIrvine2022-165">[163]</a></figcaption></figure> The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include <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/Set_theory" title="Set theory">set theory</a>, and <a href="/wiki/Computability_theory" title="Computability theory">computability theory</a>.<a href="#cite_note-FOOTNOTELi2010ixRautenberg201015Quine19811Stolyar19842-166">[164]</a> Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations of mathematics</a>.<a href="#cite_note-FOOTNOTEStolyar19843–6-167">[165]</a> The latter was a major concern in early 20th-century mathematical logic, which pursued the program of <a href="/wiki/Logicism" title="Logicism">logicism</a> pioneered by philosopher-logicians such as Gottlob Frege, <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whitehead</a>, and <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>. Mathematical theories were supposed to be logical <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautologies</a>, and their program was to show this by means of a reduction of mathematics to logic. Many attempts to realize this program failed, from the crippling of Frege's project in his Grundgesetze by <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>, to the defeat of <a href="/wiki/Hilbert%27s_program" title="Hilbert's program">Hilbert's program</a> by <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorems</a>.<a href="#cite_note-FOOTNOTEHintikkaSpade[httpswwwbritannicacomtopichistory-of-logicGodels-incompleteness-theorems_Gödel's_incompleteness_theorems]Linsky20114Richardson199815-168">[166]</a> Set theory originated in the study of the infinite by <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, and it has been the source of many of the most challenging and important issues in mathematical logic. They include <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">Cantor's theorem</a>, the status of the <a href="/wiki/Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">Axiom of Choice</a>, the question of the independence of the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>, and the modern debate on <a href="/wiki/Large_cardinal" title="Large cardinal">large cardinal</a> axioms.<a href="#cite_note-FOOTNOTEBagaria2021Cunningham-169">[167]</a> Computability theory is the branch of mathematical logic that studies effective procedures to solve calculation problems. One of its main goals is to understand whether it is possible to solve a given problem using an algorithm. For instance, given a certain claim about the positive integers, it examines whether an algorithm can be found to determine if this claim is true. Computability theory uses various theoretical tools and models, such as <a href="/wiki/Turing_machines" class="mw-redirect" title="Turing machines">Turing machines</a>, to explore this type of issue.<a href="#cite_note-FOOTNOTEBorchert2006aComputability_TheoryLearyKristiansen2015195-170">[168]</a> <div class="mw-heading mw-heading3"><h3 id="Computational_logic">Computational logic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=30" title="Edit section: Computational logic">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Computational_logic" title="Computational logic">Computational logic</a> and <a href="/wiki/Logic_in_computer_science" title="Logic in computer science">Logic in computer science</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TransistorANDgate.png" class="mw-file-description"><img alt="Diagram of an AND gate using transistors" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/TransistorANDgate.png/220px-TransistorANDgate.png" decoding="async" width="220" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/TransistorANDgate.png/330px-TransistorANDgate.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7e/TransistorANDgate.png 2x" data-file-width="365" data-file-height="531" /></a><figcaption>Conjunction (AND) is one of the basic operations of Boolean logic. It can be electronically implemented in several ways, for example, by using two <a href="/wiki/Transistor" title="Transistor">transistors</a>.</figcaption></figure> Computational logic is the branch of logic and <a href="/wiki/Computer_science" title="Computer science">computer science</a> that studies how to implement mathematical reasoning and logical formalisms using computers. This includes, for example, <a href="/wiki/Automatic_theorem_prover" class="mw-redirect" title="Automatic theorem prover">automatic theorem provers</a>, which employ rules of inference to construct a proof step by step from a set of premises to the intended conclusion without human intervention.<a href="#cite_note-FOOTNOTEPaulson20181–14Castaño20182WileGossRoesner2005447-171">[169]</a> <a href="/wiki/Logic_programming" title="Logic programming">Logic programming</a> languages are designed specifically to express facts using logical formulas and to draw inferences from these facts. For example, <a href="/wiki/Prolog" title="Prolog">Prolog</a> is a logic programming language based on predicate logic.<sup id="cite_ref-FOOTNOTEClocksinMellish2003237–238,_252–255,_257The_Relation_of_Prolog_to_LogicDaintithWright2008[httpswwwencyclopediacomcomputingdictionaries-thesauruses-pictures-and-press-releaseslogic-programming-languages_Logic_Programming_Languages]_172-0" class="reference"><a href="#cite_note-FOOTNOTEClocksinMellish2003237–238,_252–255,_257The_Relation_of_Prolog_to_LogicDaintithWright2008[httpswwwencyclopediacomcomputingdictionaries-thesauruses-pictures-and-press-releaseslogic-programming-languages_Logic_Programming_Languages]-172">[170]</a> Computer scientists also apply concepts from logic to problems in computing. The works of <a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a> were influential in this regard. He showed how <a href="/wiki/Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> can be used to understand and implement computer circuits.<a href="#cite_note-FOOTNOTEO'Regan201649CalderbankSloane2001768-173">[171]</a> This can be achieved using electronic <a href="/wiki/Logic_gates" class="mw-redirect" title="Logic gates">logic gates</a>, i.e. electronic circuits with one or more inputs and usually one output. The truth values of propositions are represented by voltage levels. In this way, logic functions can be simulated by applying the corresponding voltages to the inputs of the circuit and determining the value of the function by measuring the voltage of the output.<a href="#cite_note-FOOTNOTEDaintithWright2008[httpswwwencyclopediacomcomputingdictionaries-thesauruses-pictures-and-press-releaseslogic-gate_Logic_Gate]-174">[172]</a> <div class="mw-heading mw-heading3"><h3 id="Formal_semantics_of_natural_language">Formal semantics of natural language</h3>[<a href="/w/index.php?title=Logic&action=edit&section=31" title="Edit section: Formal semantics of natural language">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/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></div> Formal semantics is a subfield of logic, <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>, and the <a href="/wiki/Philosophy_of_language" title="Philosophy of language">philosophy of language</a>. The discipline of <a href="/wiki/Semantics" title="Semantics">semantics</a> studies the meaning of language. Formal semantics uses formal tools from the fields of symbolic logic and mathematics to give precise theories of the meaning of <a href="/wiki/Natural_language" title="Natural language">natural language</a> expressions. It understands meaning usually in relation to <a href="/wiki/Truth_condition" title="Truth condition">truth conditions</a>, i.e. it examines in which situations a sentence would be true or false. One of its central methodological assumptions is the <a href="/wiki/Principle_of_compositionality" title="Principle of compositionality">principle of compositionality</a>. It states that the meaning of a complex expression is determined by the meanings of its parts and how they are combined. For example, the meaning of the verb phrase "walk and sing" depends on the meanings of the individual expressions "walk" and "sing". Many theories in formal semantics rely on model theory. This means that they employ set theory to construct a model and then interpret the meanings of expression in relation to the elements in this model. For example, the term "walk" may be interpreted as the set of all individuals in the model that share the property of walking. Early influential theorists in this field were <a href="/wiki/Richard_Montague" title="Richard Montague">Richard Montague</a> and <a href="/wiki/Barbara_Partee" class="mw-redirect" title="Barbara Partee">Barbara Partee</a>, who focused their analysis on the English language.<a href="#cite_note-FOOTNOTEJanssenZimmermann20213–4Partee2016King2009557–8AloniDekker2016[httpsbooksgooglecombooksidltSgDAAAQBAJpgPT22_22–23]-175">[173]</a> <div class="mw-heading mw-heading3"><h3 id="Epistemology_of_logic">Epistemology of logic</h3>[<a href="/w/index.php?title=Logic&action=edit&section=32" title="Edit section: Epistemology of logic">edit</a>]</div> The epistemology of logic studies how one knows that an argument is valid or that a proposition is logically true.<a href="#cite_note-FOOTNOTEWarren20206._The_Epistemology_of_LogicSchechter-176">[174]</a> This includes questions like how to justify that modus ponens is a valid rule of inference or that contradictions are false.<a href="#cite_note-FOOTNOTEWarren20206._The_Epistemology_of_Logic-177">[175]</a> The traditionally dominant view is that this form of logical understanding belongs to knowledge <a href="/wiki/A_priori_and_a_posteriori" title="A priori and a posteriori">a priori</a>.<a href="#cite_note-FOOTNOTESchechter-178">[176]</a> In this regard, it is often argued that the <a href="/wiki/Mind" title="Mind">mind</a> has a special faculty to examine relations between pure ideas and that this faculty is also responsible for apprehending logical truths.<a href="#cite_note-FOOTNOTEGómez-Torrente2019-179">[177]</a> A similar approach understands the rules of logic in terms of <a href="/wiki/Conventionalism" title="Conventionalism">linguistic conventions</a>. On this view, the laws of logic are trivial since they are true by definition: they just express the meanings of the logical vocabulary.<a href="#cite_note-FOOTNOTEWarren20206._The_Epistemology_of_LogicGómez-Torrente2019Warren20201._What_is_Conventionalism-180">[178]</a> Some theorists, like <a href="/wiki/Hilary_Putnam" title="Hilary Putnam">Hilary Putnam</a> and <a href="/wiki/Penelope_Maddy" title="Penelope Maddy">Penelope Maddy</a>, object to the view that logic is knowable a priori. They hold instead that logical truths depend on the <a href="/wiki/Empirical" class="mw-redirect" title="Empirical">empirical</a> world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world. According to this view, they may be explored by studying general patterns of the <a href="/wiki/Fundamental_sciences" class="mw-redirect" title="Fundamental sciences">fundamental sciences</a>. For example, it has been argued that certain insights of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> refute the <a href="/wiki/Principle_of_distributivity" title="Principle of distributivity">principle of distributivity</a> in classical logic, which states that the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land (B\lor C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>∨</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land (B\lor C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df394b2737c9b42f114b26dbce84dfa4917b1a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.248ex; height:2.843ex;" alt="{\displaystyle A\land (B\lor C)}"> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)\lor (A\land C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∨</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)\lor (A\land C)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f564623cc81c802fdbd0f0d4cdf60d4d7a474922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.383ex; height:2.843ex;" alt="{\displaystyle (A\land B)\lor (A\land C)}">. This claim can be used as an empirical argument for the thesis that <a href="/wiki/Quantum_logic" title="Quantum logic">quantum logic</a> is the correct logical system and should replace classical logic.<a href="#cite_note-FOOTNOTEChua2017631–636Wilce2021Putnam1969216–241-181">[179]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Logic&action=edit&section=33" title="Edit section: History">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/History_of_logic" title="History of logic">History of logic</a></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:322px;max-width:322px"><div class="trow"><div class="tsingle" style="width:163px;max-width:163px"><div class="thumbimage" style="height:215px;overflow:hidden"><a href="/wiki/File:Aristotle_Altemps_Inv8575.jpg" class="mw-file-description"><img alt="Bust of Aristotle" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/161px-Aristotle_Altemps_Inv8575.jpg" decoding="async" width="161" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/242px-Aristotle_Altemps_Inv8575.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Aristotle_Altemps_Inv8575.jpg/322px-Aristotle_Altemps_Inv8575.jpg 2x" data-file-width="1700" data-file-height="2275" /></a></div></div><div class="tsingle" style="width:155px;max-width:155px"><div class="thumbimage" style="height:215px;overflow:hidden"><a href="/wiki/File:Avicenne_-_Avicenna_-_Ibn_Sina_(980-1037)_CIPB2067.jpg" class="mw-file-description"><img alt="Portrait of Avicenna" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Avicenne_-_Avicenna_-_Ibn_Sina_%28980-1037%29_CIPB2067.jpg/153px-Avicenne_-_Avicenna_-_Ibn_Sina_%28980-1037%29_CIPB2067.jpg" decoding="async" width="153" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Avicenne_-_Avicenna_-_Ibn_Sina_%28980-1037%29_CIPB2067.jpg/230px-Avicenne_-_Avicenna_-_Ibn_Sina_%28980-1037%29_CIPB2067.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Avicenne_-_Avicenna_-_Ibn_Sina_%28980-1037%29_CIPB2067.jpg/306px-Avicenne_-_Avicenna_-_Ibn_Sina_%28980-1037%29_CIPB2067.jpg 2x" data-file-width="1908" data-file-height="2692" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:189px;max-width:189px"><div class="thumbimage" style="height:227px;overflow:hidden"><a href="/wiki/File:Guillaume_Occam.jpg" class="mw-file-description"><img alt="Portrait of William of Ockham" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Guillaume_Occam.jpg/187px-Guillaume_Occam.jpg" decoding="async" width="187" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/Guillaume_Occam.jpg/281px-Guillaume_Occam.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/Guillaume_Occam.jpg/374px-Guillaume_Occam.jpg 2x" data-file-width="738" data-file-height="900" /></a></div></div><div class="tsingle" style="width:129px;max-width:129px"><div class="thumbimage" style="height:227px;overflow:hidden"><a href="/wiki/File:Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_(01-1).JPG" class="mw-file-description"><img alt="Bust showing Gottlob Frege" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_%2801-1%29.JPG/127px-Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_%2801-1%29.JPG" decoding="async" width="127" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_%2801-1%29.JPG/191px-Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_%2801-1%29.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_%2801-1%29.JPG/254px-Wismar_Marienkirche_Bronzeb%C3%BCste_Gottlob_Frege_%2801-1%29.JPG 2x" data-file-width="428" data-file-height="768" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Top row: <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, who established the canon of western philosophy;<a href="#cite_note-FOOTNOTEGroarke-110">[108]</a> and <a href="/wiki/Avicenna" title="Avicenna">Avicenna</a>, who replaced Aristotelian logic in <a href="/wiki/Logic_in_Islamic_philosophy" title="Logic in Islamic philosophy">Islamic discourse</a>.<a href="#cite_note-FOOTNOTELagerlund2018-182">[180]</a> Bottom row: <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a>, a major figure of medieval scholarly thought;<a href="#cite_note-FOOTNOTESpadePanaccio2019-183">[181]</a> and <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, one of the founders of modern symbolic logic.<a href="#cite_note-FOOTNOTEHaaparanta2009[httpsbooksgooglecombooksid0jXavKsArnICpgPA4_4–6]1._IntroductionHintikkaSpadeModern_logic,_Logic_since_1900-184">[182]</a></div></div></div></div> Logic was developed independently in several cultures during antiquity. One major early contributor was <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, who developed term logic in his <a href="/wiki/Organon" title="Organon">Organon</a> and <a href="/wiki/Prior_Analytics" title="Prior Analytics">Prior Analytics</a>.<a href="#cite_note-FOOTNOTEKline1972"A_major_achievement_of_Aristotle_was_the_founding_of_the_science_of_logic",_p._53Łukasiewicz19577LiuGuo202315-185">[183]</a> He was responsible for the introduction of the <a href="/wiki/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a><a href="#cite_note-FOOTNOTELear198034-186">[184]</a> and temporal modal logic.<a href="#cite_note-FOOTNOTEKnuuttila198071FisherGabbayVila2005119-187">[185]</a> Further innovations include inductive logic<a href="#cite_note-FOOTNOTEBerman2009133-188">[186]</a> as well as the discussion of new logical concepts such as <a href="/wiki/Terminology" title="Terminology">terms</a>, <a href="/wiki/Predicable" title="Predicable">predicables</a>, syllogisms, and propositions. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century.<a href="#cite_note-FOOTNOTEFredeGroarke-189">[187]</a> It has now been superseded by later work, though many of its key insights are still present in modern systems of logic.<a href="#cite_note-FOOTNOTEEwald2019Smith2022-190">[188]</a> <a href="/wiki/Avicenna" title="Avicenna">Ibn Sina</a> (Avicenna) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of <a href="/wiki/Logic_in_Islamic_philosophy" title="Logic in Islamic philosophy">logic in the Islamic world</a>.<a href="#cite_note-FOOTNOTEHasse2008Lagerlund2018-191">[189]</a> It influenced Western medieval writers such as <a href="/wiki/Albertus_Magnus" title="Albertus Magnus">Albertus Magnus</a> and <a href="/wiki/William_of_Ockham" title="William of Ockham">William of Ockham</a>.<a href="#cite_note-FOOTNOTEWashell1973445–50KnealeKneale1962229,_266-192">[190]</a> Ibn Sina wrote on the hypothetical syllogism<a href="#cite_note-FOOTNOTEGoodman2003155-193">[191]</a> and on the <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional calculus</a>.<a href="#cite_note-FOOTNOTEGoodman1992188-194">[192]</a> He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic.<a href="#cite_note-FOOTNOTEHintikkaSpade[httpswwwbritannicacomtopichistory-of-logicMedieval-logicref65928_Arabic_Logic]-195">[193]</a> He also made use of inductive logic, such as <a href="/wiki/Mill%27s_Methods" title="Mill's Methods">his methods</a> of agreement, difference, and concomitant variation, which are critical to the <a href="/wiki/Scientific_method" title="Scientific method">scientific method</a>.<a href="#cite_note-FOOTNOTEGoodman2003155-193">[191]</a> <a href="/wiki/Fakhr_al-Din_al-Razi" title="Fakhr al-Din al-Razi">Fakhr al-Din al-Razi</a> was another influential Muslim logician. He criticized Aristotelian syllogistics and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill.<a href="#cite_note-FOOTNOTEIqbal201399–115The_Spirit_of_Muslim_Culture-196">[194]</a> During the <a href="/wiki/Middle_Ages" title="Middle Ages">Middle Ages</a>, many translations and interpretations of Aristotelian logic were made. The works of <a href="/wiki/Boethius" title="Boethius">Boethius</a> were particularly influential. Besides translating Aristotle's work into Latin, he also produced textbooks on logic.<a href="#cite_note-FOOTNOTEMarenbon2021Introduction;_3._The_Logical_Text-BooksHintikkaSpade-197">[195]</a> Later, the works of Islamic philosophers such as Ibn Sina and <a href="/wiki/Averroes" title="Averroes">Ibn Rushd</a> (Averroes) were drawn on. This expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries. In 1323, William of Ockham's influential <a href="/wiki/Sum_of_Logic" title="Sum of Logic">Summa Logicae</a> was released. It is a comprehensive treatise on logic that discusses many basic concepts of logic and provides a systematic exposition of types of propositions and their truth conditions.<a href="#cite_note-FOOTNOTEHintikkaSpadeHasse2008SpadePanaccio2019-198">[196]</a> In Chinese philosophy, the <a href="/wiki/School_of_Names" title="School of Names">School of Names</a> and <a href="/wiki/Mohism" title="Mohism">Mohism</a> were particularly influential. The School of Names focused on the use of language and on paradoxes. For example, <a href="/wiki/Gongsun_Long" title="Gongsun Long">Gongsun Long</a> proposed the <a href="/wiki/White_horse_paradox" class="mw-redirect" title="White horse paradox">white horse paradox</a>, which defends the thesis that a white horse is not a horse. The school of Mohism also acknowledged the importance of language for logic and tried to relate the ideas in these fields to the realm of ethics.<a href="#cite_note-FOOTNOTEWillman2022Rošker2015301–309-199">[197]</a> In India, the study of logic was primarily pursued by the schools of <a href="/wiki/Nyaya" title="Nyaya">Nyaya</a>, <a href="/wiki/Buddhism" title="Buddhism">Buddhism</a>, and <a href="/wiki/Jainism" title="Jainism">Jainism</a>. It was not treated as a separate academic discipline and discussions of its topics usually happened in the context of epistemology and theories of dialogue or argumentation.<a href="#cite_note-FOOTNOTESarukkaiChakraborty2022117–8-200">[198]</a> In Nyaya, inference is understood as a source of knowledge (<a href="/wiki/Pram%C4%81%E1%B9%87a" class="mw-redirect" title="Pramāṇa">pramāṇa</a>). It follows the perception of an object and tries to arrive at conclusions, for example, about the cause of this object.<a href="#cite_note-FOOTNOTEDastiLead_section;_1b._InferenceMills2018[httpsbooksgooglecombooksidsGhqDwAAQBAJpgPA121_121]-201">[199]</a> A similar emphasis on the relation to epistemology is also found in Buddhist and Jainist schools of logic, where inference is used to expand the knowledge gained through other sources.<a href="#cite_note-FOOTNOTEEmmanuel2015320–2Vidyabhusana1988221-202">[200]</a> Some of the later theories of Nyaya, belonging to the <a href="/wiki/Navya-Ny%C4%81ya" title="Navya-Nyāya">Navya-Nyāya</a> school, resemble modern forms of logic, such as Gottlob Frege's distinction between <a href="/wiki/Sense_and_reference" title="Sense and reference">sense and reference</a> and his definition of number.<a href="#cite_note-FOOTNOTEChakrabarti1976554–563-203">[201]</a> The syllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in the foundations of mathematics stimulated the development of modern symbolic logic.<a href="#cite_note-FOOTNOTEGroarkeHaaparanta2009[httpsbooksgooglecombooksid0jXavKsArnICpgPA3_3–5]1._Introduction-204">[202]</a> Many see Gottlob Frege's <a href="/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a> as the birthplace of modern logic. <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>'s idea of a <a href="/wiki/Characteristica_universalis" title="Characteristica universalis">universal formal language</a> is often considered a forerunner. Other pioneers were <a href="/wiki/George_Boole" title="George Boole">George Boole</a>, who invented <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a> as a mathematical system of logic, and <a href="/wiki/Charles_Peirce" class="mw-redirect" title="Charles Peirce">Charles Peirce</a>, who developed the <a href="/wiki/Logic_of_relatives" class="mw-redirect" title="Logic of relatives">logic of relatives</a>. Alfred North Whitehead and Bertrand Russell, in turn, condensed many of these insights in their work <a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a>. Modern logic introduced novel concepts, such as <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, quantifiers, and relational predicates. A hallmark of modern symbolic logic is its use of formal language to precisely codify its insights. In this regard, it departs from earlier logicians, who relied mainly on natural language.<a href="#cite_note-FOOTNOTEHaaparanta2009[httpsbooksgooglecombooksid0jXavKsArnICpgPA4_4–6]HintikkaSpadeModern_logic,_Logic_since_1900-205">[203]</a> Of particular influence was the development of first-order logic, which is usually treated as the standard system of modern logic.<a href="#cite_note-FOOTNOTEEwald2019-206">[204]</a> Its analytical generality allowed the formalization of mathematics and drove the investigation of <a href="/wiki/Set_theory" title="Set theory">set theory</a>. It also made Alfred Tarski's approach to <a href="/wiki/Model_theory" title="Model theory">model theory</a> possible and provided the foundation of modern mathematical logic.<a href="#cite_note-FOOTNOTEEwald2019Schreiner202122-207">[205]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Logic&action=edit&section=34" 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> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Glossary_of_logic" title="Glossary of logic">Glossary of logic</a></li> <li><a href="/wiki/Outline_of_logic" title="Outline of logic">Outline of logic</a> – Overview of and topical guide to logic</li> <li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a> – Analysis of facts to form a judgment</li> <li><a href="/wiki/List_of_logic_journals" title="List of logic journals">List of logic journals</a></li> <li><a href="/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a> – List of symbols used to express logical relations</li> <li><a href="/wiki/List_of_logicians" title="List of logicians">List of logicians</a></li> <li><a href="/wiki/Logic_puzzle" title="Logic puzzle">Logic puzzle</a> – Puzzle deriving from the mathematics field of deduction</li> <li><a href="/wiki/Logical_reasoning" title="Logical reasoning">Logical reasoning</a> – Process of drawing correct inferences</li> <li><a href="/wiki/Logos" title="Logos">Logos</a> – Concept in philosophy, religion, rhetoric, and psychology</li> <li><a href="/wiki/Vector_logic" title="Vector logic">Vector logic</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Logic&action=edit&section=35" title="Edit section: References">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3>[<a href="/w/index.php?title=Logic&action=edit&section=36" 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-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-43"><a href="#cite_ref-43">^</a> However, there are some forms of logic, like <a href="/wiki/Imperative_logic" title="Imperative logic">imperative logic</a>, where this may not be the case.<a href="#cite_note-FOOTNOTEHaack197451-42">[42]</a> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> Conductive arguments present reasons in favor of a conclusion without claiming that the reasons are strong enough to decisively support the conclusion. </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=Logic&action=edit&section=37" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEVelleman20068,_103-1"><a href="#cite_ref-FOOTNOTEVelleman20068,_103_1-0">^</a> <a href="#CITEREFVelleman2006">Velleman 2006</a>, pp. 8, 103. </li> <li id="cite_note-FOOTNOTEVickers2022-2"><a href="#cite_ref-FOOTNOTEVickers2022_2-0">^</a> <a href="#CITEREFVickers2022">Vickers 2022</a>. </li> <li id="cite_note-FOOTNOTENunes20112066–2069-3"><a href="#cite_ref-FOOTNOTENunes20112066–2069_3-0">^</a> <a href="#CITEREFNunes2011">Nunes 2011</a>, pp. 2066–2069. </li> <li id="cite_note-FOOTNOTEPépin2004LogosOnline_Etymology_Staff-4"><a href="#cite_ref-FOOTNOTEPépin2004LogosOnline_Etymology_Staff_4-0">^</a> <a href="#CITEREFPépin2004">Pépin 2004</a>, Logos; <a href="#CITEREFOnline_Etymology_Staff">Online Etymology Staff</a>. </li> <li id="cite_note-FOOTNOTEHintikka2019lead_section,_§Nature_and_varieties_of_logic-5"><a href="#cite_ref-FOOTNOTEHintikka2019lead_section,_§Nature_and_varieties_of_logic_5-0">^</a> <a href="#CITEREFHintikka2019">Hintikka 2019</a>, lead section, §Nature and varieties of logic. </li> <li id="cite_note-FOOTNOTEHintikka2019§Nature_and_varieties_of_logicHaack19781–10Philosophy_of_logicsSchlesingerKeren-PortnoyParush2001220-6"><a href="#cite_ref-FOOTNOTEHintikka2019§Nature_and_varieties_of_logicHaack19781–10Philosophy_of_logicsSchlesingerKeren-PortnoyParush2001220_6-0">^</a> <a href="#CITEREFHintikka2019">Hintikka 2019</a>, §Nature and varieties of logic; <a href="#CITEREFHaack1978">Haack 1978</a>, pp. 1–10, Philosophy of logics; <a href="#CITEREFSchlesingerKeren-PortnoyParush2001">Schlesinger, Keren-Portnoy & Parush 2001</a>, p. 220. </li> <li id="cite_note-FOOTNOTEHintikkaSandu200613Audi1999bPhilosophy_of_logicMcKeon-7"><a href="#cite_ref-FOOTNOTEHintikkaSandu200613Audi1999bPhilosophy_of_logicMcKeon_7-0">^</a> <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, p. 13; <a href="#CITEREFAudi1999b">Audi 1999b</a>, Philosophy of logic; <a href="#CITEREFMcKeon">McKeon</a>. </li> <li id="cite_note-FOOTNOTEBlairJohnson200093–95Craig1996Formal_and_informal_logic-8"><a href="#cite_ref-FOOTNOTEBlairJohnson200093–95Craig1996Formal_and_informal_logic_8-0">^</a> <a href="#CITEREFBlairJohnson2000">Blair & Johnson 2000</a>, pp. 93–95; <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic. </li> <li id="cite_note-FOOTNOTECraig1996Formal_and_informal_logicBarnes2007274Planty-Bonjour2012[httpsbooksgooglecombooksid0EpFBgAAQBAJpgPA62_62]Rini2010[httpsbooksgooglecombooksidvard024vjFgCpgPA26_26]-9"><a href="#cite_ref-FOOTNOTECraig1996Formal_and_informal_logicBarnes2007274Planty-Bonjour2012[httpsbooksgooglecombooksid0EpFBgAAQBAJpgPA62_62]Rini2010[httpsbooksgooglecombooksidvard024vjFgCpgPA26_26]_9-0">^</a> <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic; <a href="#CITEREFBarnes2007">Barnes 2007</a>, p. 274; <a href="#CITEREFPlanty-Bonjour2012">Planty-Bonjour 2012</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0EpFBgAAQBAJ&pg=PA62">62</a>; <a href="#CITEREFRini2010">Rini 2010</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vard024vjFgC&pg=PA26">26</a>. </li> <li id="cite_note-FOOTNOTEMacFarlane2017Corkum2015753–767BlairJohnson200093–95Magnus200512–41.6_Formal_languages-10"><a href="#cite_ref-FOOTNOTEMacFarlane2017Corkum2015753–767BlairJohnson200093–95Magnus200512–41.6_Formal_languages_10-0">^</a> <a href="#CITEREFMacFarlane2017">MacFarlane 2017</a>; <a href="#CITEREFCorkum2015">Corkum 2015</a>, pp. 753–767; <a href="#CITEREFBlairJohnson2000">Blair & Johnson 2000</a>, pp. 93–95; <a href="#CITEREFMagnus2005">Magnus 2005</a>, pp. 12–4, 1.6 Formal languages. </li> <li id="cite_note-FOOTNOTEMcKeonCraig1996Formal_and_informal_logic-11">^ <a href="#cite_ref-FOOTNOTEMcKeonCraig1996Formal_and_informal_logic_11-0">a</a> <a href="#cite_ref-FOOTNOTEMcKeonCraig1996Formal_and_informal_logic_11-1">b</a> <a href="#CITEREFMcKeon">McKeon</a>; <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic. </li> <li id="cite_note-FOOTNOTEHintikkaSandu200613-12"><a href="#cite_ref-FOOTNOTEHintikkaSandu200613_12-0">^</a> <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, p. 13. </li> <li id="cite_note-FOOTNOTEMagnus2005Proofs,_p._102-13"><a href="#cite_ref-FOOTNOTEMagnus2005Proofs,_p._102_13-0">^</a> <a href="#CITEREFMagnus2005">Magnus 2005</a>, Proofs, p. 102. </li> <li id="cite_note-FOOTNOTEHintikkaSandu200613–16Makridis20221–2RuncoPritzker1999155-14"><a href="#cite_ref-FOOTNOTEHintikkaSandu200613–16Makridis20221–2RuncoPritzker1999155_14-0">^</a> <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, pp. 13–16; <a href="#CITEREFMakridis2022">Makridis 2022</a>, pp. 1–2; <a href="#CITEREFRuncoPritzker1999">Runco & Pritzker 1999</a>, p. 155. </li> <li id="cite_note-FOOTNOTEGómez-Torrente2019Magnus20051.5_Other_logical_notions,_p._10-15"><a href="#cite_ref-FOOTNOTEGómez-Torrente2019Magnus20051.5_Other_logical_notions,_p._10_15-0">^</a> <a href="#CITEREFGómez-Torrente2019">Gómez-Torrente 2019</a>; <a href="#CITEREFMagnus2005">Magnus 2005</a>, 1.5 Other logical notions, p. 10. </li> <li id="cite_note-FOOTNOTEHintikkaSandu200616-16">^ <a href="#cite_ref-FOOTNOTEHintikkaSandu200616_16-0">a</a> <a href="#cite_ref-FOOTNOTEHintikkaSandu200616_16-1">b</a> <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, p. 16. </li> <li id="cite_note-FOOTNOTEHonderich2005logic,_informalCraig1996Formal_and_informal_logicJohnson1999265–268-17"><a href="#cite_ref-FOOTNOTEHonderich2005logic,_informalCraig1996Formal_and_informal_logicJohnson1999265–268_17-0">^</a> <a href="#CITEREFHonderich2005">Honderich 2005</a>, logic, informal; <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic; <a href="#CITEREFJohnson1999">Johnson 1999</a>, pp. 265–268. </li> <li id="cite_note-FOOTNOTECraig1996Formal_languages_and_systemsSimpson200814-18"><a href="#cite_ref-FOOTNOTECraig1996Formal_languages_and_systemsSimpson200814_18-0">^</a> <a href="#CITEREFCraig1996">Craig 1996</a>, Formal languages and systems; <a href="#CITEREFSimpson2008">Simpson 2008</a>, p. 14. </li> <li id="cite_note-FOOTNOTECraig1996Formal_languages_and_systems-19"><a href="#cite_ref-FOOTNOTECraig1996Formal_languages_and_systems_19-0">^</a> <a href="#CITEREFCraig1996">Craig 1996</a>, Formal languages and systems. </li> <li id="cite_note-FOOTNOTEHintikkaSandu200622–3Magnus20058–91.4_Deductive_validityJohnson1999267-20"><a href="#cite_ref-FOOTNOTEHintikkaSandu200622–3Magnus20058–91.4_Deductive_validityJohnson1999267_20-0">^</a> <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, pp. 22–3; <a href="#CITEREFMagnus2005">Magnus 2005</a>, pp. 8–9, 1.4 Deductive validity; <a href="#CITEREFJohnson1999">Johnson 1999</a>, p. 267. </li> <li id="cite_note-FOOTNOTEHaack19781–2,_4Philosophy_of_logicsHintikkaSandu200616–17Jacquette2006Introduction:_Philosophy_of_logic_today,_pp._1–12-21"><a href="#cite_ref-FOOTNOTEHaack19781–2,_4Philosophy_of_logicsHintikkaSandu200616–17Jacquette2006Introduction:_Philosophy_of_logic_today,_pp._1–12_21-0">^</a> <a href="#CITEREFHaack1978">Haack 1978</a>, pp. 1–2, 4, Philosophy of logics; <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, pp. 16–17; <a href="#CITEREFJacquette2006">Jacquette 2006</a>, Introduction: Philosophy of logic today, pp. 1–12. </li> <li id="cite_note-FOOTNOTEHaack19781–2,_4Philosophy_of_logicsJacquette20061–12Introduction:_Philosophy_of_logic_today-22"><a href="#cite_ref-FOOTNOTEHaack19781–2,_4Philosophy_of_logicsJacquette20061–12Introduction:_Philosophy_of_logic_today_22-0">^</a> <a href="#CITEREFHaack1978">Haack 1978</a>, pp. 1–2, 4, Philosophy of logics; <a href="#CITEREFJacquette2006">Jacquette 2006</a>, pp. 1–12, Introduction: Philosophy of logic today. </li> <li id="cite_note-FOOTNOTEHaack19785–7,_9Philosophy_of_logicsHintikkaSandu200631–2Haack1996229–30-23"><a href="#cite_ref-FOOTNOTEHaack19785–7,_9Philosophy_of_logicsHintikkaSandu200631–2Haack1996229–30_23-0">^</a> <a href="#CITEREFHaack1978">Haack 1978</a>, pp. 5–7, 9, Philosophy of logics; <a href="#CITEREFHintikkaSandu2006">Hintikka & Sandu 2006</a>, pp. 31–2; <a href="#CITEREFHaack1996">Haack 1996</a>, pp. 229–30. </li> <li id="cite_note-FOOTNOTEHaack19781–10Philosophy_of_logicsGroarke2021lead_section;_1.1_Formal_and_Informal_Logic-24"><a href="#cite_ref-FOOTNOTEHaack19781–10Philosophy_of_logicsGroarke2021lead_section;_1.1_Formal_and_Informal_Logic_24-0">^</a> <a href="#CITEREFHaack1978">Haack 1978</a>, pp. 1–10, Philosophy of logics; <a href="#CITEREFGroarke2021">Groarke 2021</a>, lead section; 1.1 Formal and Informal Logic. </li> <li id="cite_note-FOOTNOTEJohnson2014228–9-25"><a href="#cite_ref-FOOTNOTEJohnson2014228–9_25-0">^</a> <a href="#CITEREFJohnson2014">Johnson 2014</a>, pp. 228–9. </li> <li id="cite_note-FOOTNOTEGroarke2021lead_section;_1._HistoryAudi1999aInformal_logicJohnson1999265–274-26"><a href="#cite_ref-FOOTNOTEGroarke2021lead_section;_1._HistoryAudi1999aInformal_logicJohnson1999265–274_26-0">^</a> <a href="#CITEREFGroarke2021">Groarke 2021</a>, lead section; 1. History; <a href="#CITEREFAudi1999a">Audi 1999a</a>, Informal logic; <a href="#CITEREFJohnson1999">Johnson 1999</a>, pp. 265–274. </li> <li id="cite_note-FOOTNOTECraig1996Formal_and_informal_logicJohnson1999267-27"><a href="#cite_ref-FOOTNOTECraig1996Formal_and_informal_logicJohnson1999267_27-0">^</a> <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic; <a href="#CITEREFJohnson1999">Johnson 1999</a>, p. 267. </li> <li id="cite_note-FOOTNOTEBlairJohnson200093–97Craig1996Formal_and_informal_logic-28"><a href="#cite_ref-FOOTNOTEBlairJohnson200093–97Craig1996Formal_and_informal_logic_28-0">^</a> <a href="#CITEREFBlairJohnson2000">Blair & Johnson 2000</a>, pp. 93–97; <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic. </li> <li id="cite_note-FOOTNOTEJohnson1999265–270van_EemerenGarssenKrabbeSnoeck_Henkemans1–45Informal_Logic-29"><a href="#cite_ref-FOOTNOTEJohnson1999265–270van_EemerenGarssenKrabbeSnoeck_Henkemans1–45Informal_Logic_29-0">^</a> <a href="#CITEREFJohnson1999">Johnson 1999</a>, pp. 265–270; <a href="#CITEREFvan_EemerenGarssenKrabbeSnoeck_Henkemans">van Eemeren et al.</a>, pp. 1–45, Informal Logic. </li> <li id="cite_note-FOOTNOTEGroarke20211.1_Formal_and_Informal_LogicAudi1999aInformal_logicHonderich2005logic,_informal-30"><a href="#cite_ref-FOOTNOTEGroarke20211.1_Formal_and_Informal_LogicAudi1999aInformal_logicHonderich2005logic,_informal_30-0">^</a> <a href="#CITEREFGroarke2021">Groarke 2021</a>, 1.1 Formal and Informal Logic; <a href="#CITEREFAudi1999a">Audi 1999a</a>, Informal logic; <a href="#CITEREFHonderich2005">Honderich 2005</a>, logic, informal. </li> <li id="cite_note-FOOTNOTEBlairJohnson200093–107Groarke2021lead_section;_1.1_Formal_and_Informal_Logicvan_EemerenGrootendorstJohnsonPlantin169-31"><a href="#cite_ref-FOOTNOTEBlairJohnson200093–107Groarke2021lead_section;_1.1_Formal_and_Informal_Logicvan_EemerenGrootendorstJohnsonPlantin169_31-0">^</a> <a href="#CITEREFBlairJohnson2000">Blair & Johnson 2000</a>, pp. 93–107; <a href="#CITEREFGroarke2021">Groarke 2021</a>, lead section; 1.1 Formal and Informal Logic; <a href="#CITEREFvan_EemerenGrootendorstJohnsonPlantin">van Eemeren et al.</a>, p. 169. </li> <li id="cite_note-FOOTNOTEOaksfordChater200747-32"><a href="#cite_ref-FOOTNOTEOaksfordChater200747_32-0">^</a> <a href="#CITEREFOaksfordChater2007">Oaksford & Chater 2007</a>, p. 47. </li> <li id="cite_note-FOOTNOTECraig1996Formal_and_informal_logicWalton19872–3,_6–81._A_new_model_of_argumentEngel198259–922._The_medium_of_language-33"><a href="#cite_ref-FOOTNOTECraig1996Formal_and_informal_logicWalton19872–3,_6–81._A_new_model_of_argumentEngel198259–922._The_medium_of_language_33-0">^</a> <a href="#CITEREFCraig1996">Craig 1996</a>, Formal and informal logic; <a href="#CITEREFWalton1987">Walton 1987</a>, pp. 2–3, 6–8, 1. A new model of argument; <a href="#CITEREFEngel1982">Engel 1982</a>, pp. 59–92, 2. The medium of language. </li> <li id="cite_note-FOOTNOTEBlairJohnson1987147–51-34"><a href="#cite_ref-FOOTNOTEBlairJohnson1987147–51_34-0">^</a> <a href="#CITEREFBlairJohnson1987">Blair & Johnson 1987</a>, pp. 147–51. </li> <li id="cite_note-FOOTNOTEFalikowskiMills202298Weddle2011383–836._Informal_logic_and_the_eductive-inductive_distinctionBlair201147-35"><a href="#cite_ref-FOOTNOTEFalikowskiMills202298Weddle2011383–836._Informal_logic_and_the_eductive-inductive_distinctionBlair201147_35-0">^</a> <a href="#CITEREFFalikowskiMills2022">Falikowski & Mills 2022</a>, p. 98; <a href="#CITEREFWeddle2011">Weddle 2011</a>, pp. 383–8, 36. Informal logic and the eductive-inductive distinction; <a href="#CITEREFBlair2011">Blair 2011</a>, p. 47. </li> <li id="cite_note-FOOTNOTEVickers2022Nunes20112066–9Logical_Reasoning_and_Learning-36"><a href="#cite_ref-FOOTNOTEVickers2022Nunes20112066–9Logical_Reasoning_and_Learning_36-0">^</a> <a href="#CITEREFVickers2022">Vickers 2022</a>; <a href="#CITEREFNunes2011">Nunes 2011</a>, pp. 2066–9, Logical Reasoning and Learning. </li> <li id="cite_note-FOOTNOTEJohnson2014181Johnson1999267BlairJohnson1987147–51-37"><a href="#cite_ref-FOOTNOTEJohnson2014181Johnson1999267BlairJohnson1987147–51_37-0">^</a> <a href="#CITEREFJohnson2014">Johnson 2014</a>, p. 181; <a href="#CITEREFJohnson1999">Johnson 1999</a>, p. 267; <a href="#CITEREFBlairJohnson1987">Blair & Johnson 1987</a>, pp. 147–51. </li> <li id="cite_note-FOOTNOTEVleet2010ix–xIntroductionDowdenStump-38">^ <a href="#cite_ref-FOOTNOTEVleet2010ix–xIntroductionDowdenStump_38-0">a</a> <a href="#cite_ref-FOOTNOTEVleet2010ix–xIntroductionDowdenStump_38-1">b</a> <a href="#CITEREFVleet2010">Vleet 2010</a>, pp. ix–x, Introduction; 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Introduction. </li> <li id="cite_note-FOOTNOTEHaaparanta2009[httpsbooksgooglecombooksid0jXavKsArnICpgPA4_4–6]HintikkaSpadeModern_logic,_Logic_since_1900-205"><a href="#cite_ref-FOOTNOTEHaaparanta2009[httpsbooksgooglecombooksid0jXavKsArnICpgPA4_4–6]HintikkaSpadeModern_logic,_Logic_since_1900_205-0">^</a> <a href="#CITEREFHaaparanta2009">Haaparanta 2009</a>, pp. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0jXavKsArnIC&pg=PA4">4–6</a>; <a href="#CITEREFHintikkaSpade">Hintikka & Spade</a>, Modern logic, Logic since 1900. </li> <li id="cite_note-FOOTNOTEEwald2019-206"><a href="#cite_ref-FOOTNOTEEwald2019_206-0">^</a> <a href="#CITEREFEwald2019">Ewald 2019</a>. </li> <li id="cite_note-FOOTNOTEEwald2019Schreiner202122-207"><a href="#cite_ref-FOOTNOTEEwald2019Schreiner202122_207-0">^</a> <a href="#CITEREFEwald2019">Ewald 2019</a>; <a href="#CITEREFSchreiner2021">Schreiner 2021</a>, p. 22. </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3>[<a href="/w/index.php?title=Logic&action=edit&section=38" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAloniDekker2016" class="citation book cs1">Aloni, Maria; Dekker, Paul (7 July 2016). The Cambridge Handbook of Formal Semantics. Cambridge University Press. pp. 22–23. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-316-55273-5" title="Special:BookSources/978-1-316-55273-5"><bdi>978-1-316-55273-5</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+Cambridge+Handbook+of+Formal+Semantics&rft.pages=22-23&rft.pub=Cambridge+University+Press&rft.date=2016-07-07&rft.isbn=978-1-316-55273-5&rft.aulast=Aloni&rft.aufirst=Maria&rft.au=Dekker%2C+Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAngell1964" class="citation book cs1">Angell, Richard B. (1964). Reasoning and Logic. 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Retrieved 29 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Philosophy+of+logic&rft.btitle=The+Cambridge+Dictionary+of+Philosophy&rft.pages=679-681&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-1-107-64379-6&rft.aulast=Audi&rft.aufirst=Robert&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FAUDTCD-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBackmann2019" class="citation journal cs1">Backmann, Marius (1 June 2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12136-018-0371-6">"Varieties of Justification—How (Not) to Solve the Problem of Induction"</a>. Acta Analytica. 34 (2): 235–255. <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%2Fs12136-018-0371-6">10.1007/s12136-018-0371-6</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/1874-6349">1874-6349</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:125767384">125767384</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Analytica&rft.atitle=Varieties+of+Justification%E2%80%94How+%28Not%29+to+Solve+the+Problem+of+Induction&rft.volume=34&rft.issue=2&rft.pages=235-255&rft.date=2019-06-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125767384%23id-name%3DS2CID&rft.issn=1874-6349&rft_id=info%3Adoi%2F10.1007%2Fs12136-018-0371-6&rft.aulast=Backmann&rft.aufirst=Marius&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs12136-018-0371-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBagaria2021" class="citation web cs1">Bagaria, Joan (2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/set-theory/">"Set Theory"</a>. 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Clarendon Press. p. 274. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-151574-3" title="Special:BookSources/978-0-19-151574-3"><bdi>978-0-19-151574-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Truth%2C+etc.%3A+Six+Lectures+on+Ancient+Logic&rft.pages=274&rft.pub=Clarendon+Press&rft.date=2007-01-25&rft.isbn=978-0-19-151574-3&rft.aulast=Barnes&rft.aufirst=Jonathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenthem" class="citation web cs1">Benthem, Johan van. <a rel="nofollow" class="external text" href="https://iep.utm.edu/modal-lo/#H1">"Modal Logic: Contemporary View: 1. Modal Notions and Reasoning Patterns: a First Pass"</a>. Internet Encyclopedia of Philosophy. 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John Wiley & Sons. p. 194. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-75443-6" title="Special:BookSources/978-0-470-75443-6"><bdi>978-0-470-75443-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cognitive+Radio+and+Dynamic+Spectrum+Access&rft.pages=194&rft.pub=John+Wiley+%26+Sons&rft.date=2009-07-10&rft.isbn=978-0-470-75443-6&rft.aulast=Berlemann&rft.aufirst=Lars&rft.au=Mangold%2C+Stefan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerman2009" class="citation book cs1">Berman, Harold J. (1 July 2009). Law and Revolution, the Formation of the Western Legal Tradition. 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Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-954143-0" title="Special:BookSources/978-0-19-954143-0"><bdi>978-0-19-954143-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220108194809/https://www.oxfordreference.com/view/10.1093/oi/authority.20110803100432990">Archived</a> from the original on 8 January 2022. Retrieved 8 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=rule+of+inference&rft.btitle=The+Oxford+Dictionary+of+Philosophy&rft.pub=Oxford+University+Press&rft.date=2016-03-24&rft.isbn=978-0-19-954143-0&rft.aulast=Blackburn&rft.aufirst=Simon&rft_id=https%3A%2F%2Fwww.oxfordreference.com%2Fview%2F10.1093%2Foi%2Fauthority.20110803100432990&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlairJohnson1987" class="citation journal cs1"><a href="/wiki/J._Anthony_Blair" title="J. Anthony Blair">Blair, J. Anthony</a>; Johnson, Ralph H. (1987). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BLATCS">"The Current State of Informal Logic"</a>. Informal Logic. 9 (2): 147–51. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.22329%2Fil.v9i2.2671">10.22329/il.v9i2.2671</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211230194638/https://philpapers.org/rec/BLATCS">Archived</a> from the original on 30 December 2021. Retrieved 2 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Informal+Logic&rft.atitle=The+Current+State+of+Informal+Logic&rft.volume=9&rft.issue=2&rft.pages=147-51&rft.date=1987&rft_id=info%3Adoi%2F10.22329%2Fil.v9i2.2671&rft.aulast=Blair&rft.aufirst=J.+Anthony&rft.au=Johnson%2C+Ralph+H.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBLATCS&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlairJohnson2000" class="citation journal cs1"><a href="/wiki/J._Anthony_Blair" title="J. Anthony Blair">Blair, J. Anthony</a>; Johnson, Ralph H. (2000). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BLAILA-3">"Informal Logic: An Overview"</a>. Informal Logic. 20 (2): 93–107. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.22329%2Fil.v20i2.2262">10.22329/il.v20i2.2262</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211209195317/https://philpapers.org/rec/BLAILA-3">Archived</a> from the original on 9 December 2021. Retrieved 29 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Informal+Logic&rft.atitle=Informal+Logic%3A+An+Overview&rft.volume=20&rft.issue=2&rft.pages=93-107&rft.date=2000&rft_id=info%3Adoi%2F10.22329%2Fil.v20i2.2262&rft.aulast=Blair&rft.aufirst=J.+Anthony&rft.au=Johnson%2C+Ralph+H.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBLAILA-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlair2011" class="citation book cs1"><a href="/wiki/J._Anthony_Blair" title="J. Anthony Blair">Blair, J. Anthony</a> (20 October 2011). Groundwork in the Theory of Argumentation: Selected Papers of J. Anthony Blair. Springer Science & Business Media. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-007-2363-4" title="Special:BookSources/978-94-007-2363-4"><bdi>978-94-007-2363-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Groundwork+in+the+Theory+of+Argumentation%3A+Selected+Papers+of+J.+Anthony+Blair&rft.pages=47&rft.pub=Springer+Science+%26+Business+Media&rft.date=2011-10-20&rft.isbn=978-94-007-2363-4&rft.aulast=Blair&rft.aufirst=J.+Anthony&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBobzien2020" class="citation web cs1">Bobzien, Susanne (2020). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-ancient/#Ari">"Ancient Logic: 2. Aristotle"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180828102117/https://plato.stanford.edu/entries/logic-ancient/#Ari">Archived</a> from the original on 28 August 2018. Retrieved 3 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Ancient+Logic%3A+2.+Aristotle&rft.date=2020&rft.aulast=Bobzien&rft.aufirst=Susanne&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-ancient%2F%23Ari&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorchert2006a" class="citation book cs1">Borchert, Donald, ed. (2006a). "Computability Theory". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BORMEO">Macmillan Encyclopedia of Philosophy Volume 2</a> (2nd ed.). Macmillan. pp. 372–390. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-865782-0" title="Special:BookSources/978-0-02-865782-0"><bdi>978-0-02-865782-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Computability+Theory&rft.btitle=Macmillan+Encyclopedia+of+Philosophy+Volume+2&rft.pages=372-390&rft.edition=2nd&rft.pub=Macmillan&rft.date=2006&rft.isbn=978-0-02-865782-0&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBORMEO&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorchert2006b" class="citation book cs1">Borchert, Donald (2006b). "Induction". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BORMEO">Macmillan Encyclopedia of Philosophy Volume 4</a> (2nd ed.). Macmillan. pp. 635–648. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-865784-4" title="Special:BookSources/978-0-02-865784-4"><bdi>978-0-02-865784-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210112065913/https://philpapers.org/rec/BORMEO">Archived</a> from the original on 12 January 2021. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Induction&rft.btitle=Macmillan+Encyclopedia+of+Philosophy+Volume+4&rft.pages=635-648&rft.edition=2nd&rft.pub=Macmillan&rft.date=2006&rft.isbn=978-0-02-865784-4&rft.aulast=Borchert&rft.aufirst=Donald&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBORMEO&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorchert2006c" class="citation book cs1">Borchert, Donald (2006c). "Logic, Non-Classical". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BORMEO">Macmillan Encyclopedia of Philosophy Volume 5</a> (2nd ed.). Macmillan. pp. 485–492. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-865785-1" title="Special:BookSources/978-0-02-865785-1"><bdi>978-0-02-865785-1</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210112065913/https://philpapers.org/rec/BORMEO">Archived</a> from the original on 12 January 2021. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Logic%2C+Non-Classical&rft.btitle=Macmillan+Encyclopedia+of+Philosophy+Volume+5&rft.pages=485-492&rft.edition=2nd&rft.pub=Macmillan&rft.date=2006&rft.isbn=978-0-02-865785-1&rft.aulast=Borchert&rft.aufirst=Donald&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBORMEO&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorisAlexander2017" class="citation book cs1">Boris, Kulik; Alexander, Fridman (30 November 2017). N-ary Relations for Logical Analysis of Data and Knowledge. IGI Global. p. 74. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-5225-2783-1" title="Special:BookSources/978-1-5225-2783-1"><bdi>978-1-5225-2783-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=N-ary+Relations+for+Logical+Analysis+of+Data+and+Knowledge&rft.pages=74&rft.pub=IGI+Global&rft.date=2017-11-30&rft.isbn=978-1-5225-2783-1&rft.aulast=Boris&rft.aufirst=Kulik&rft.au=Alexander%2C+Fridman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBridgesIshiharaRathjenSchwichtenberg2023" class="citation book cs1">Bridges, Douglas; Ishihara, Hajime; Rathjen, Michael; Schwichtenberg, Helmut (30 April 2023). Handbook of Constructive Mathematics. Cambridge University Press. pp. 73–4. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-316-51086-5" title="Special:BookSources/978-1-316-51086-5"><bdi>978-1-316-51086-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Constructive+Mathematics&rft.pages=73-4&rft.pub=Cambridge+University+Press&rft.date=2023-04-30&rft.isbn=978-1-316-51086-5&rft.aulast=Bridges&rft.aufirst=Douglas&rft.au=Ishihara%2C+Hajime&rft.au=Rathjen%2C+Michael&rft.au=Schwichtenberg%2C+Helmut&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrody2006" class="citation book cs1">Brody, Boruch A. (2006). Encyclopedia of Philosophy. Vol. 5. Donald M. Borchert (2nd ed.). Thomson Gale/Macmillan Reference US. pp. 535–536. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-865780-6" title="Special:BookSources/978-0-02-865780-6"><bdi>978-0-02-865780-6</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/61151356">61151356</a>. <q>The two most important types of logical calculi are propositional (or sentential) calculi and functional (or predicate) calculi. A propositional calculus is a system containing propositional variables and connectives (some also contain propositional constants) but not individual or functional variables or constants. In the extended propositional calculus, quantifiers whose operator variables are propositional variables are added.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Encyclopedia+of+Philosophy&rft.pages=535-536&rft.edition=2nd&rft.pub=Thomson+Gale%2FMacmillan+Reference+US&rft.date=2006&rft_id=info%3Aoclcnum%2F61151356&rft.isbn=978-0-02-865780-6&rft.aulast=Brody&rft.aufirst=Boruch+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBunninYu2009" class="citation book cs1">Bunnin, Nicholas; Yu, Jiyuan (27 January 2009). The Blackwell Dictionary of Western Philosophy. John Wiley & Sons. p. 179. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4051-9112-8" title="Special:BookSources/978-1-4051-9112-8"><bdi>978-1-4051-9112-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Blackwell+Dictionary+of+Western+Philosophy&rft.pages=179&rft.pub=John+Wiley+%26+Sons&rft.date=2009-01-27&rft.isbn=978-1-4051-9112-8&rft.aulast=Bunnin&rft.aufirst=Nicholas&rft.au=Yu%2C+Jiyuan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgess2009" class="citation book cs1"><a href="/wiki/John_P._Burgess" title="John P. Burgess">Burgess, John P.</a> (2009). "1. Classical logic". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BURPL-3">Philosophical Logic</a>. Princeton, NJ: Princeton University Press. pp. 1–12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-15633-0" title="Special:BookSources/978-0-691-15633-0"><bdi>978-0-691-15633-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211216143954/https://philpapers.org/rec/BURPL-3">Archived</a> from the original on 16 December 2021. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.+Classical+logic&rft.btitle=Philosophical+Logic&rft.pages=1-12&rft.pub=Princeton%2C+NJ%3A+Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-15633-0&rft.aulast=Burgess&rft.aufirst=John+P.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBURPL-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBäck2016" class="citation book cs1">Bäck, Allan T. (2016). Aristotle's Theory of Predication. Brill. p. 317. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-90-04-32109-0" title="Special:BookSources/978-90-04-32109-0"><bdi>978-90-04-32109-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Aristotle%27s+Theory+of+Predication&rft.pages=317&rft.pub=Brill&rft.date=2016&rft.isbn=978-90-04-32109-0&rft.aulast=B%C3%A4ck&rft.aufirst=Allan+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCalderbankSloane2001" class="citation journal cs1">Calderbank, Robert; Sloane, Neil J. A. (April 2001). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F35071223">"Claude Shannon (1916–2001)"</a>. Nature. 410 (6830): 768. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F35071223">10.1038/35071223</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/1476-4687">1476-4687</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/11298432">11298432</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:4402158">4402158</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Claude+Shannon+%281916%E2%80%932001%29&rft.volume=410&rft.issue=6830&rft.pages=768&rft.date=2001-04&rft.issn=1476-4687&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4402158%23id-name%3DS2CID&rft_id=info%3Apmid%2F11298432&rft_id=info%3Adoi%2F10.1038%2F35071223&rft.aulast=Calderbank&rft.aufirst=Robert&rft.au=Sloane%2C+Neil+J.+A.&rft_id=https%3A%2F%2Fdoi.org%2F10.1038%252F35071223&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarnielliPizzi2008" class="citation book cs1">Carnielli, Walter; Pizzi, Claudio (2008). 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Springer Science & Business Media. p. 191. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-9165-0" title="Special:BookSources/978-1-4020-9165-0"><bdi>978-1-4020-9165-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pondering+on+Problems+of+Argumentation%3A+Twenty+Essays+on+Theoretical+Issues&rft.pages=191&rft.pub=Springer+Science+%26+Business+Media&rft.date=2009&rft.isbn=978-1-4020-9165-0&rft.aulast=van+Eemeren&rft.aufirst=Frans+H.&rft.au=Garssen%2C+Bart&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_EemerenGarssenKrabbeSnoeck_Henkemans" class="citation book cs1">van Eemeren, Frans H.; Garssen, Bart; Krabbe, Erik C. W.; Snoeck Henkemans, A. Francisca; Verheij, Bart; Wagemans, Jean H. M. (2021). <a rel="nofollow" class="external text" href="https://link.springer.com/referenceworkentry/10.1007%2F978-94-007-6883-3_7-1">"Informal Logic"</a>. Handbook of Argumentation Theory. Springer Netherlands. pp. 1–45. <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%2F978-94-007-6883-3_7-1">10.1007/978-94-007-6883-3_7-1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-94-007-6883-3" title="Special:BookSources/978-94-007-6883-3"><bdi>978-94-007-6883-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211231172324/https://link.springer.com/referenceworkentry/10.1007/978-94-007-6883-3_7-1">Archived</a> from the original on 31 December 2021. 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Fundamentals of Argumentation Theory: A Handbook of Historical Backgrounds and Contemporary Developments. Routledge. p. 169. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-136-68804-1" title="Special:BookSources/978-1-136-68804-1"><bdi>978-1-136-68804-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Argumentation+Theory%3A+A+Handbook+of+Historical+Backgrounds+and+Contemporary+Developments&rft.pages=169&rft.pub=Routledge&rft.date=2013&rft.isbn=978-1-136-68804-1&rft.aulast=van+Eemeren&rft.aufirst=Frans+H.&rft.au=Grootendorst%2C+Rob&rft.au=Johnson%2C+Ralph+H.&rft.au=Plantin%2C+Christian&rft.au=Willard%2C+Charles+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEmmanuel2015" class="citation book cs1">Emmanuel, Steven M. (2015). A Companion to Buddhist Philosophy. John Wiley & Sons. pp. 320–2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-14466-3" title="Special:BookSources/978-1-119-14466-3"><bdi>978-1-119-14466-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Companion+to+Buddhist+Philosophy&rft.pages=320-2&rft.pub=John+Wiley+%26+Sons&rft.date=2015&rft.isbn=978-1-119-14466-3&rft.aulast=Emmanuel&rft.aufirst=Steven+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnderton2001" class="citation book cs1"><a href="/wiki/Herbert_Enderton" title="Herbert Enderton">Enderton, Herbert</a> (2001). A Mathematical Introduction to Logic. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-238452-3" title="Special:BookSources/978-0-12-238452-3"><bdi>978-0-12-238452-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Mathematical+Introduction+to+Logic&rft.pub=Elsevier&rft.date=2001&rft.isbn=978-0-12-238452-3&rft.aulast=Enderton&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngel1982" class="citation book cs1">Engel, S. Morris (1982). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/ENGWGR">With Good Reason an Introduction to Informal Fallacies</a>. 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Retrieved 11 March 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Modal+Logic&rft.date=2023&rft.aulast=Garson&rft.aufirst=James&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-modal%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGensler2006" class="citation book cs1">Gensler, Harry J. (2006). The A to Z of Logic. Scarecrow Press. pp. xliii–xliv. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4617-3182-5" title="Special:BookSources/978-1-4617-3182-5"><bdi>978-1-4617-3182-5</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+A+to+Z+of+Logic&rft.pages=xliii-xliv&rft.pub=Scarecrow+Press&rft.date=2006&rft.isbn=978-1-4617-3182-5&rft.aulast=Gensler&rft.aufirst=Harry+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoble2001" class="citation book cs1">Goble, Lou (2001). "Introduction". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/GOBTBG-2">The Blackwell Guide to Philosophical Logic</a>. Wiley-Blackwell. pp. 1–8. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-631-20692-7" title="Special:BookSources/978-0-631-20692-7"><bdi>978-0-631-20692-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207184959/https://philpapers.org/rec/GOBTBG-2">Archived</a> from the original on 7 December 2021. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction&rft.btitle=The+Blackwell+Guide+to+Philosophical+Logic&rft.pages=1-8&rft.pub=Wiley-Blackwell&rft.date=2001&rft.isbn=978-0-631-20692-7&rft.aulast=Goble&rft.aufirst=Lou&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FGOBTBG-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodman1992" class="citation book cs1">Goodman, Lenn Evan (1992). Avicenna. Routledge. p. 188. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-01929-3" title="Special:BookSources/978-0-415-01929-3"><bdi>978-0-415-01929-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Avicenna&rft.pages=188&rft.pub=Routledge&rft.date=1992&rft.isbn=978-0-415-01929-3&rft.aulast=Goodman&rft.aufirst=Lenn+Evan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoodman2003" class="citation book cs1">Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press. p. 155. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-513580-0" title="Special:BookSources/978-0-19-513580-0"><bdi>978-0-19-513580-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Islamic+Humanism&rft.pages=155&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=978-0-19-513580-0&rft.aulast=Goodman&rft.aufirst=Lenn+Evan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGroarke" class="citation web cs1">Groarke, Louis F. <a rel="nofollow" class="external text" href="https://iep.utm.edu/aris-log/">"Aristotle: Logic"</a>. Internet Encyclopedia of Philosophy. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211229235433/https://iep.utm.edu/aris-log/">Archived</a> from the original on 29 December 2021. Retrieved 1 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Internet+Encyclopedia+of+Philosophy&rft.atitle=Aristotle%3A+Logic&rft.aulast=Groarke&rft.aufirst=Louis+F.&rft_id=https%3A%2F%2Fiep.utm.edu%2Faris-log%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGroarke2021" class="citation web cs1">Groarke, Leo (2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-informal/">"Informal Logic"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220112030519/https://plato.stanford.edu/entries/logic-informal/">Archived</a> from the original on 12 January 2022. Retrieved 31 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Informal+Logic&rft.date=2021&rft.aulast=Groarke&rft.aufirst=Leo&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-informal%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGómez-Torrente2019" class="citation web cs1">Gómez-Torrente, Mario (2019). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logical-truth/">"Logical Truth"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211002190110/https://plato.stanford.edu/entries/logical-truth/">Archived</a> from the original on 2 October 2021. Retrieved 22 November 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Logical+Truth&rft.date=2019&rft.aulast=G%C3%B3mez-Torrente&rft.aufirst=Mario&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogical-truth%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGödel1984" class="citation book cs1">Gödel, Kurt (1984). "Russell's mathematical logic". In Benacerraf, Paul; Putnam, Hilary (eds.). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/abs/philosophy-of-mathematics/russells-mathematical-logic/4D82F215FABFE06149D03EF1EF5BE7E4">Philosophy of Mathematics: Selected Readings</a> (2nd ed.). Cambridge University Press. pp. 447–469. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-29648-9" title="Special:BookSources/978-0-521-29648-9"><bdi>978-0-521-29648-9</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220111091740/https://www.cambridge.org/core/books/abs/philosophy-of-mathematics/russells-mathematical-logic/4D82F215FABFE06149D03EF1EF5BE7E4">Archived</a> from the original on 11 January 2022. Retrieved 9 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Russell%27s+mathematical+logic&rft.btitle=Philosophy+of+Mathematics%3A+Selected+Readings&rft.pages=447-469&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1984&rft.isbn=978-0-521-29648-9&rft.aulast=G%C3%B6del&rft.aufirst=Kurt&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fabs%2Fphilosophy-of-mathematics%2Frussells-mathematical-logic%2F4D82F215FABFE06149D03EF1EF5BE7E4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHájek2006" class="citation web cs1">Hájek, Petr (3 September 2006). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/Archives/Win2012/entries/logic-fuzzy/">"Fuzzy Logic"</a>. Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 19 July 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=Fuzzy+Logic&rft.date=2006-09-03&rft.aulast=H%C3%A1jek&rft.aufirst=Petr&rft_id=https%3A%2F%2Fplato.stanford.edu%2FArchives%2FWin2012%2Fentries%2Flogic-fuzzy%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHájekLin2017" class="citation journal cs1">Hájek, Alan; Lin, Hanti (2017). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/HJEATO">"A Tale of Two Epistemologies?"</a>. Res Philosophica. 94 (2): 207–232. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.11612%2Fresphil.1540">10.11612/resphil.1540</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:160029122">160029122</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220104182746/https://philpapers.org/rec/HJEATO">Archived</a> from the original on 4 January 2022. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Res+Philosophica&rft.atitle=A+Tale+of+Two+Epistemologies%3F&rft.volume=94&rft.issue=2&rft.pages=207-232&rft.date=2017&rft_id=info%3Adoi%2F10.11612%2Fresphil.1540&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A160029122%23id-name%3DS2CID&rft.aulast=H%C3%A1jek&rft.aufirst=Alan&rft.au=Lin%2C+Hanti&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FHJEATO&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallO'Donnell2000" class="citation book cs1">Hall, Cordelia; O'Donnell, John (2000). Discrete Mathematics Using a Computer. Springer Science & Business Media. p. 48. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-089-7" title="Special:BookSources/978-1-85233-089-7"><bdi>978-1-85233-089-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+Mathematics+Using+a+Computer&rft.pages=48&rft.pub=Springer+Science+%26+Business+Media&rft.date=2000&rft.isbn=978-1-85233-089-7&rft.aulast=Hall&rft.aufirst=Cordelia&rft.au=O%27Donnell%2C+John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHoudeCamacho2003" class="citation book cs1">Houde, R.; Camacho, L. (2003). "Induction". <a rel="nofollow" class="external text" href="https://www.encyclopedia.com/science-and-technology/computers-and-electrical-engineering/electrical-engineering/induction">New Catholic Encyclopedia</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7876-4004-0" title="Special:BookSources/978-0-7876-4004-0"><bdi>978-0-7876-4004-0</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220108171720/https://www.encyclopedia.com/science-and-technology/computers-and-electrical-engineering/electrical-engineering/induction">Archived</a> from the original on 8 January 2022. Retrieved 8 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Induction&rft.btitle=New+Catholic+Encyclopedia&rft.date=2003&rft.isbn=978-0-7876-4004-0&rft.aulast=Houde&rft.aufirst=R.&rft.au=Camacho%2C+L.&rft_id=https%3A%2F%2Fwww.encyclopedia.com%2Fscience-and-technology%2Fcomputers-and-electrical-engineering%2Felectrical-engineering%2Finduction&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaack1974" class="citation book cs1"><a href="/wiki/Susan_Haack" title="Susan Haack">Haack, Susan</a> (1974). Deviant Logic: Some Philosophical Issues. CUP Archive. p. 51. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-20500-9" title="Special:BookSources/978-0-521-20500-9"><bdi>978-0-521-20500-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deviant+Logic%3A+Some+Philosophical+Issues&rft.pages=51&rft.pub=CUP+Archive&rft.date=1974&rft.isbn=978-0-521-20500-9&rft.aulast=Haack&rft.aufirst=Susan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaack1978" class="citation book cs1"><a href="/wiki/Susan_Haack" title="Susan Haack">Haack, Susan</a> (1978). "1. 'Philosophy of logics'". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/HAAPOL-2">Philosophy of Logics</a>. London and New York: Cambridge University Press. pp. 1–10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-29329-7" title="Special:BookSources/978-0-521-29329-7"><bdi>978-0-521-29329-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207200551/https://philpapers.org/rec/HAAPOL-2">Archived</a> from the original on 7 December 2021. Retrieved 29 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.+%27Philosophy+of+logics%27&rft.btitle=Philosophy+of+Logics&rft.pages=1-10&rft.pub=London+and+New+York%3A+Cambridge+University+Press&rft.date=1978&rft.isbn=978-0-521-29329-7&rft.aulast=Haack&rft.aufirst=Susan&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FHAAPOL-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaack1996" class="citation book cs1"><a href="/wiki/Susan_Haack" title="Susan Haack">Haack, Susan</a> (1996). Deviant Logic, Fuzzy Logic: Beyond the Formalism. University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-31133-3" title="Special:BookSources/978-0-226-31133-3"><bdi>978-0-226-31133-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Deviant+Logic%2C+Fuzzy+Logic%3A+Beyond+the+Formalism&rft.pub=University+of+Chicago+Press&rft.date=1996&rft.isbn=978-0-226-31133-3&rft.aulast=Haack&rft.aufirst=Susan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaaparanta2009" class="citation book cs1">Haaparanta, Leila (2009). "1. Introduction". The Development of Modern Logic. Oxford University Press. pp. 4–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-513731-6" title="Special:BookSources/978-0-19-513731-6"><bdi>978-0-19-513731-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.+Introduction&rft.btitle=The+Development+of+Modern+Logic&rft.pages=4-6&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-513731-6&rft.aulast=Haaparanta&rft.aufirst=Leila&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHansen2020" class="citation web cs1">Hansen, Hans (2020). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/fallacies/">"Fallacies"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210329182946/https://plato.stanford.edu/entries/fallacies/">Archived</a> from the original on 29 March 2021. Retrieved 18 March 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Fallacies&rft.date=2020&rft.aulast=Hansen&rft.aufirst=Hans&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Ffallacies%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartmannSprenger2010" class="citation book cs1">Hartmann, Stephan; Sprenger, Jan (2010). "Bayesian Epistemology". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/BOVSIO">The Routledge Companion to Epistemology</a>. London: Routledge. pp. 609–620. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-415-96219-3" title="Special:BookSources/978-0-415-96219-3"><bdi>978-0-415-96219-3</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210516095047/https://philpapers.org/rec/BOVSIO">Archived</a> from the original on 16 May 2021. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bayesian+Epistemology&rft.btitle=The+Routledge+Companion+to+Epistemology&rft.pages=609-620&rft.pub=London%3A+Routledge&rft.date=2010&rft.isbn=978-0-415-96219-3&rft.aulast=Hartmann&rft.aufirst=Stephan&rft.au=Sprenger%2C+Jan&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FBOVSIO&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHasse2008" class="citation web cs1">Hasse, Dag Nikolaus (2008). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/arabic-islamic-influence/">"Influence of Arabic and Islamic Philosophy on the Latin West"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 19 July 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Influence+of+Arabic+and+Islamic+Philosophy+on+the+Latin+West&rft.date=2008&rft.aulast=Hasse&rft.aufirst=Dag+Nikolaus&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Farabic-islamic-influence%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHawthorne2021" class="citation web cs1">Hawthorne, James (2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-inductive/">"Inductive Logic"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220121081805/https://plato.stanford.edu/entries/logic-inductive/">Archived</a> from the original on 21 January 2022. Retrieved 6 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Inductive+Logic&rft.date=2021&rft.aulast=Hawthorne&rft.aufirst=James&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Flogic-inductive%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintikka2019" class="citation web cs1"><a href="/wiki/Jaakko_Hintikka" title="Jaakko Hintikka">Hintikka, Jaakko J.</a> (2019). <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/philosophy-of-logic">"Philosophy of logic"</a>. Encyclopædia Britannica. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150428101732/http://www.britannica.com/EBchecked/topic/346240/philosophy-of-logic">Archived</a> from the original on 28 April 2015. Retrieved 21 November 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclop%C3%A6dia+Britannica&rft.atitle=Philosophy+of+logic&rft.date=2019&rft.aulast=Hintikka&rft.aufirst=Jaakko+J.&rft_id=https%3A%2F%2Fwww.britannica.com%2Ftopic%2Fphilosophy-of-logic&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintikka2023" class="citation web cs1"><a href="/wiki/Jaakko_Hintikka" title="Jaakko Hintikka">Hintikka, Jaakko J.</a> (2023). <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/logic/Logical-systems">"Logical systems"</a>. 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Noûs. 4 (2): 135–152. <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%2F2214318">10.2307/2214318</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/0029-4624">0029-4624</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/2214318">2214318</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=No%C3%BBs&rft.atitle=Information%2C+Deduction%2C+and+the+A+Priori&rft.volume=4&rft.issue=2&rft.pages=135-152&rft.date=1970&rft.issn=0029-4624&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2214318%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2214318&rft.aulast=Hintikka&rft.aufirst=Jaakko&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintikkaSandu2006" class="citation book cs1"><a href="/wiki/Jaakko_Hintikka" title="Jaakko Hintikka">Hintikka, Jaakko</a>; Sandu, Gabriel (2006). "What is Logic?". In Jacquette, D. (ed.). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/JAAWIL">Philosophy of Logic</a>. North Holland. pp. 13–39. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51541-4" title="Special:BookSources/978-0-444-51541-4"><bdi>978-0-444-51541-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207235525/https://philpapers.org/rec/JAAWIL">Archived</a> from the original on 7 December 2021. Retrieved 29 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=What+is+Logic%3F&rft.btitle=Philosophy+of+Logic&rft.pages=13-39&rft.pub=North+Holland&rft.date=2006&rft.isbn=978-0-444-51541-4&rft.aulast=Hintikka&rft.aufirst=Jaakko&rft.au=Sandu%2C+Gabriel&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FJAAWIL&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHintikkaSpade" class="citation web cs1"><a href="/wiki/Jaakko_Hintikka" title="Jaakko Hintikka">Hintikka, Jaakko J.</a>; Spade, Paul Vincent. <a rel="nofollow" class="external text" href="https://www.britannica.com/topic/history-of-logic">"History of logic"</a>. Encyclopædia Britannica. 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Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-926479-7" title="Special:BookSources/978-0-19-926479-7"><bdi>978-0-19-926479-7</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210129082636/https://philpapers.org/rec/HONTOC-2">Archived</a> from the original on 29 January 2021. Retrieved 2 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Oxford+Companion+to+Philosophy&rft.pub=Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-926479-7&rft.aulast=Honderich&rft.aufirst=Ted&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FHONTOC-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHurley2015" class="citation book cs1">Hurley, Patrick J. (2015). "4. Categorical Syllogisms". Logic: The Essentials. Wadsworth. pp. 189–237. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-305-59041-0" title="Special:BookSources/978-1-305-59041-0"><bdi>978-1-305-59041-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4.+Categorical+Syllogisms&rft.btitle=Logic%3A+The+Essentials&rft.pages=189-237&rft.pub=Wadsworth&rft.date=2015&rft.isbn=978-1-305-59041-0&rft.aulast=Hurley&rft.aufirst=Patrick+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIEP_Staff" class="citation web cs1">IEP Staff. <a rel="nofollow" class="external text" href="https://iep.utm.edu/ded-ind/">"Deductive and Inductive Arguments"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100528032124/https://iep.utm.edu/ded-ind/">Archived</a> from the original on 28 May 2010. 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Stanford University Press. pp. 99–115. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8047-8686-7" title="Special:BookSources/978-0-8047-8686-7"><bdi>978-0-8047-8686-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Spirit+of+Muslim+Culture&rft.btitle=The+Reconstruction+of+Religious+Thought+in+Islam&rft.pages=99-115&rft.pub=Stanford+University+Press&rft.date=2013&rft.isbn=978-0-8047-8686-7&rft.aulast=Iqbal&rft.aufirst=Mohammad&rft_id=http%3A%2F%2Fwww.allamaiqbal.com%2Fworks%2Fprose%2Fenglish%2Freconstruction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIrvine2022" class="citation web cs1">Irvine, Andrew David (2022). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/russell/">"Bertrand Russell"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 29 September 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Bertrand+Russell&rft.date=2022&rft.aulast=Irvine&rft.aufirst=Andrew+David&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Frussell%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacquette2006" class="citation book cs1">Jacquette, Dale (2006). "Introduction: Philosophy of logic today". <a rel="nofollow" class="external text" href="https://philpapers.org/rec/JACPOL">Philosophy of Logic</a>. North Holland. pp. 1–12. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-51541-4" title="Special:BookSources/978-0-444-51541-4"><bdi>978-0-444-51541-4</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207184932/https://philpapers.org/rec/JACPOL">Archived</a> from the original on 7 December 2021. Retrieved 29 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction%3A+Philosophy+of+logic+today&rft.btitle=Philosophy+of+Logic&rft.pages=1-12&rft.pub=North+Holland&rft.date=2006&rft.isbn=978-0-444-51541-4&rft.aulast=Jacquette&rft.aufirst=Dale&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FJACPOL&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJago2014" class="citation book cs1">Jago, Mark (2014). The Impossible: An Essay on Hyperintensionality. OUP Oxford. p. 41. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-101915-9" title="Special:BookSources/978-0-19-101915-9"><bdi>978-0-19-101915-9</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+Impossible%3A+An+Essay+on+Hyperintensionality&rft.pages=41&rft.pub=OUP+Oxford&rft.date=2014&rft.isbn=978-0-19-101915-9&rft.aulast=Jago&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJanssenZimmermann2021" class="citation web cs1">Janssen, Theo M. V.; Zimmermann, Thomas Ede (2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/montague-semantics/">"Montague Semantics"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. pp. 3–4. Retrieved 10 March 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Montague+Semantics&rft.pages=3-4&rft.date=2021&rft.aulast=Janssen&rft.aufirst=Theo+M.+V.&rft.au=Zimmermann%2C+Thomas+Ede&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fmontague-semantics%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohnson1999" class="citation journal cs1">Johnson, Ralph H. (1999). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/JOHTRB-2">"The Relation Between Formal and Informal Logic"</a>. Argumentation. 13 (3): 265–274. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1007789101256">10.1023/A:1007789101256</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:141283158">141283158</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207184706/https://philpapers.org/rec/JOHTRB-2">Archived</a> from the original on 7 December 2021. 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University of Windsor. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-920233-71-9" title="Special:BookSources/978-0-920233-71-9"><bdi>978-0-920233-71-9</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+Rise+of+Informal+Logic%3A+Essays+on+Argumentation%2C+Critical+Thinking%2C+Reasoning+and+Politics&rft.pub=University+of+Windsor&rft.date=2014-07-15&rft.isbn=978-0-920233-71-9&rft.aulast=Johnson&rft.aufirst=Ralph+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKetland2005" class="citation book cs1">Ketland, Jeffrey (2005). 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Retrieved 20 November 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Internet+Encyclopedia+of+Philosophy&rft.atitle=Logical+Consequence&rft.aulast=McKeon&rft.aufirst=Matthew&rft_id=https%3A%2F%2Fiep.utm.edu%2Flogcon%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichaelsonReimer2019" class="citation web cs1">Michaelson, Eliot; Reimer, Marga (2019). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/reference/">"Reference"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207184705/https://plato.stanford.edu/entries/reference/">Archived</a> from the original on 7 December 2021. 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Eurasia Journal of Mathematics, Science and Technology Education. 14 (9): 1–17. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.29333%2Fejmste%2F92562">10.29333/ejmste/92562</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/1305-8215">1305-8215</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:126245285">126245285</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211207195246/https://www.ejmste.com/article/strategic-vs-definitory-rules-their-role-in-abductive-argumentation-and-their-relationship-with-5539">Archived</a> from the original on 7 December 2021. 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Broadview Press. p. 14. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-77048-495-5" title="Special:BookSources/978-1-77048-495-5"><bdi>978-1-77048-495-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essentials+of+Symbolic+Logic&rft.pages=14&rft.edition=3rd&rft.pub=Broadview+Press&rft.date=2008&rft.isbn=978-1-77048-495-5&rft.aulast=Simpson&rft.aufirst=R.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2022" class="citation web cs1">Smith, Robin (2022). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/aristotle-logic/">"Aristotle's Logic"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 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Metaphysics Research Lab, Stanford University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=William+of+Ockham&rft.date=2019&rft.aulast=Spade&rft.aufirst=Paul+Vincent&rft.au=Panaccio%2C+Claude&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fockham%2F%23SummLogi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpriggs2012" class="citation book cs1">Spriggs, John (2012). GSN – The Goal Structuring Notation: A Structured Approach to Presenting Arguments. 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Informal Logic. 32 (3): 327–356. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.22329%2Fil.v32i3.3681">10.22329/il.v32i3.3681</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220228123240/https://philpapers.org/rec/STODTA">Archived</a> from the original on 28 February 2022. 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Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-008615-2" title="Special:BookSources/978-0-19-008615-2"><bdi>978-0-19-008615-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Shadows+of+Syntax%3A+Revitalizing+Logical+and+Mathematical+Conventionalism&rft.pub=Oxford+University+Press&rft.date=2020&rft.isbn=978-0-19-008615-2&rft.aulast=Warren&rft.aufirst=Jared&rft_id=https%3A%2F%2Fglobal.oup.com%2Facademic%2Fproduct%2Fshadows-of-syntax-9780190086152&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWashell1973" class="citation journal cs1">Washell, Richard F. (1973). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/WASLLA-3">"Logic, Language, and Albert the Great"</a>. Journal of the History of Ideas. 34 (3): 445–50. <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%2F2708963">10.2307/2708963</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/2708963">2708963</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+History+of+Ideas&rft.atitle=Logic%2C+Language%2C+and+Albert+the+Great&rft.volume=34&rft.issue=3&rft.pages=445-50&rft.date=1973&rft_id=info%3Adoi%2F10.2307%2F2708963&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2708963%23id-name%3DJSTOR&rft.aulast=Washell&rft.aufirst=Richard+F.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FWASLLA-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWasilewska2018" class="citation book cs1">Wasilewska, Anita (2018). Logics for Computer Science: Classical and Non-Classical. Springer. pp. 145–6. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-92591-2" title="Special:BookSources/978-3-319-92591-2"><bdi>978-3-319-92591-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logics+for+Computer+Science%3A+Classical+and+Non-Classical&rft.pages=145-6&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-319-92591-2&rft.aulast=Wasilewska&rft.aufirst=Anita&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeber" class="citation web cs1">Weber, Zach. <a rel="nofollow" class="external text" href="https://iep.utm.edu/para-log/">"Paraconsistent Logic"</a>. Internet Encyclopedia of Philosophy. Retrieved 12 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Internet+Encyclopedia+of+Philosophy&rft.atitle=Paraconsistent+Logic&rft.aulast=Weber&rft.aufirst=Zach&rft_id=https%3A%2F%2Fiep.utm.edu%2Fpara-log%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeddle2011" class="citation book cs1">Weddle, Perry (2011). "Chapter 36. Informal logic and the eductive-inductive distinction". <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9783110867718.383/html">Across the Lines of Disciplines</a>. De Gruyter Mouton. pp. 383–388. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110867718.383">10.1515/9783110867718.383</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-086771-8" title="Special:BookSources/978-3-11-086771-8"><bdi>978-3-11-086771-8</bdi></a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211231172343/https://www.degruyter.com/document/doi/10.1515/9783110867718.383/html">Archived</a> from the original on 31 December 2021. Retrieved 2 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+36.+Informal+logic+and+the+eductive-inductive+distinction&rft.btitle=Across+the+Lines+of+Disciplines&rft.pages=383-388&rft.pub=De+Gruyter+Mouton&rft.date=2011&rft_id=info%3Adoi%2F10.1515%2F9783110867718.383&rft.isbn=978-3-11-086771-8&rft.aulast=Weddle&rft.aufirst=Perry&rft_id=https%3A%2F%2Fwww.degruyter.com%2Fdocument%2Fdoi%2F10.1515%2F9783110867718.383%2Fhtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWesterståhl1989" class="citation journal cs1">Westerståhl, Dag (1989). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/WESASA">"Aristotelian Syllogisms and Generalized Quantifiers"</a>. Studia Logica. 48 (4): 577–585. <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%2FBF00370209">10.1007/BF00370209</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:32089424">32089424</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220104182746/https://philpapers.org/rec/WESASA">Archived</a> from the original on 4 January 2022. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studia+Logica&rft.atitle=Aristotelian+Syllogisms+and+Generalized+Quantifiers&rft.volume=48&rft.issue=4&rft.pages=577-585&rft.date=1989&rft_id=info%3Adoi%2F10.1007%2FBF00370209&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32089424%23id-name%3DS2CID&rft.aulast=Westerst%C3%A5hl&rft.aufirst=Dag&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FWESASA&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilbanks2010" class="citation journal cs1">Wilbanks, Jan J. (1 March 2010). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s10503-009-9131-5">"Defining Deduction, Induction, and Validity"</a>. Argumentation. 24 (1): 107–124. <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%2Fs10503-009-9131-5">10.1007/s10503-009-9131-5</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/1572-8374">1572-8374</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:144481717">144481717</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220108171721/https://link.springer.com/article/10.1007/s10503-009-9131-5">Archived</a> from the original on 8 January 2022. Retrieved 8 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Argumentation&rft.atitle=Defining+Deduction%2C+Induction%2C+and+Validity&rft.volume=24&rft.issue=1&rft.pages=107-124&rft.date=2010-03-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144481717%23id-name%3DS2CID&rft.issn=1572-8374&rft_id=info%3Adoi%2F10.1007%2Fs10503-009-9131-5&rft.aulast=Wilbanks&rft.aufirst=Jan+J.&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs10503-009-9131-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilce2021" class="citation web cs1">Wilce, Alexander (2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/qt-quantlog/#RealQuanLogi">"Quantum Logic and Probability Theory: 2.1 Realist Quantum Logic"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 11 March 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Quantum+Logic+and+Probability+Theory%3A+2.1+Realist+Quantum+Logic&rft.date=2021&rft.aulast=Wilce&rft.aufirst=Alexander&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fqt-quantlog%2F%23RealQuanLogi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWileGossRoesner2005" class="citation book cs1">Wile, Bruce; Goss, John; Roesner, Wolfgang (2005). Comprehensive Functional Verification: The Complete Industry Cycle. Elsevier. p. 447. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-047664-3" title="Special:BookSources/978-0-08-047664-3"><bdi>978-0-08-047664-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Comprehensive+Functional+Verification%3A+The+Complete+Industry+Cycle&rft.pages=447&rft.pub=Elsevier&rft.date=2005&rft.isbn=978-0-08-047664-3&rft.aulast=Wile&rft.aufirst=Bruce&rft.au=Goss%2C+John&rft.au=Roesner%2C+Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillman2022" class="citation web cs1">Willman, Marshall D. (2022). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/chinese-logic-language/">"Logic and Language in Early Chinese Philosophy"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Introduction. Retrieved 11 March 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Stanford+Encyclopedia+of+Philosophy&rft.atitle=Logic+and+Language+in+Early+Chinese+Philosophy&rft.pages=Introduction&rft.date=2022&rft.aulast=Willman&rft.aufirst=Marshall+D.&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fchinese-logic-language%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWolf1978" class="citation journal cs1">Wolf, Robert G. (1978). <a rel="nofollow" class="external text" href="https://philpapers.org/rec/WOLARL">"Are Relevant Logics Deviant?"</a>. Philosophia. 7 (2): 327–340. <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%2FBF02378819">10.1007/BF02378819</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:143697796">143697796</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211216143955/https://philpapers.org/rec/WOLARL">Archived</a> from the original on 16 December 2021. Retrieved 4 January 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophia&rft.atitle=Are+Relevant+Logics+Deviant%3F&rft.volume=7&rft.issue=2&rft.pages=327-340&rft.date=1978&rft_id=info%3Adoi%2F10.1007%2FBF02378819&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A143697796%23id-name%3DS2CID&rft.aulast=Wolf&rft.aufirst=Robert+G.&rft_id=https%3A%2F%2Fphilpapers.org%2Frec%2FWOLARL&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZegarelli2010" class="citation book cs1">Zegarelli, Mark (2010). Logic For Dummies. John Wiley & Sons. p. 30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-05307-2" title="Special:BookSources/978-1-118-05307-2"><bdi>978-1-118-05307-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Logic+For+Dummies&rft.pages=30&rft.pub=John+Wiley+%26+Sons&rft.date=2010&rft.isbn=978-1-118-05307-2&rft.aulast=Zegarelli&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogic" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Logic&action=edit&section=39" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235611614">.mw-parser-output .spoken-wikipedia{border:1px solid #a2a9b1;background-color:var(--background-color-interactive-subtle,#f8f9fa);margin:0.5em 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logic">Outline</a></li> <li><a href="/wiki/History_of_logic" title="History of logic">History</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major fields</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/Logic_in_computer_science" title="Logic in computer science">Computer science</a></li> <li><a href="/wiki/Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Classical_logic" title="Classical logic">Classical</a></li> <li><a href="/wiki/Informal_logic" title="Informal logic">Informal</a> <ul><li><a href="/wiki/Critical_thinking" title="Critical thinking">Critical thinking</a></li> <li><a href="/wiki/Reason" title="Reason">Reason</a></li></ul></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical</a></li> <li><a href="/wiki/Non-classical_logic" title="Non-classical logic">Non-classical</a></li> <li><a href="/wiki/Philosophical_logic" title="Philosophical logic">Philosophical</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</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/Argumentation_theory" title="Argumentation theory">Argumentation</a></li> <li><a href="/wiki/Metalogic" title="Metalogic">Metalogic</a></li> <li><a href="/wiki/Metamathematics" title="Metamathematics">Metamathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set</a></li></ul> </div></td></tr></tbody></table><div> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</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/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="Philosophy" 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:Philosophy_topics" title="Template:Philosophy topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Philosophy_topics" title="Template talk:Philosophy topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Philosophy_topics" title="Special:EditPage/Template:Philosophy topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Philosophy" style="font-size:114%;margin:0 4em"><a href="/wiki/Philosophy" title="Philosophy">Philosophy</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Branches" style="font-size:114%;margin:0 4em">Branches</div></th></tr><tr><td colspan="2" class="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:6.8em"><a href="/wiki/Outline_of_philosophy#Branches" title="Outline of philosophy">Branches</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Applied_philosophy" title="Applied philosophy">Applied philosophy</a></li> <li><a class="mw-selflink selflink">Logic</a></li> <li><a href="/wiki/Metaphilosophy" title="Metaphilosophy">Metaphilosophy</a></li> <li><a href="/wiki/Philosophy_of_information" title="Philosophy of information">Philosophy of information</a></li> <li><a href="/wiki/Philosophy_of_language" title="Philosophy of language">Philosophy of language</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Philosophy_of_religion" title="Philosophy of religion">Philosophy of religion</a></li> <li><a href="/wiki/Philosophy_of_science" title="Philosophy of science">Philosophy of science</a></li> <li><a href="/wiki/Political_philosophy" title="Political philosophy">Political philosophy</a></li> <li><a href="/wiki/Practical_philosophy" title="Practical philosophy">Practical philosophy</a></li> <li><a href="/wiki/Social_philosophy" title="Social philosophy">Social philosophy</a></li> <li><a href="/wiki/Theoretical_philosophy" title="Theoretical philosophy">Theoretical philosophy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Aesthetics" title="Aesthetics">Aesthetics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aesthetic_emotions" title="Aesthetic emotions">Aesthetic response</a></li> <li><a href="/wiki/Formalism_(art)" title="Formalism (art)">Formalism</a></li> <li><a href="/wiki/Institutional_theory_of_art" class="mw-redirect" title="Institutional theory of art">Institutionalism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Epistemology" title="Epistemology">Epistemology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Empiricism" title="Empiricism">Empiricism</a></li> <li><a href="/wiki/Fideism" title="Fideism">Fideism</a></li> <li><a href="/wiki/Naturalized_epistemology" title="Naturalized epistemology">Naturalism</a></li> <li><a href="/wiki/Epistemological_particularism" title="Epistemological particularism">Particularism</a></li> <li><a href="/wiki/Rationalism" title="Rationalism">Rationalism</a></li> <li><a href="/wiki/Philosophical_skepticism" title="Philosophical skepticism">Skepticism</a></li> <li><a href="/wiki/Solipsism" title="Solipsism">Solipsism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Ethics" title="Ethics">Ethics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Consequentialism" title="Consequentialism">Consequentialism</a></li> <li><a href="/wiki/Deontology" title="Deontology">Deontology</a></li> <li><a href="/wiki/Virtue_ethics" title="Virtue ethics">Virtue</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Free_will" title="Free will">Free will</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compatibilism" title="Compatibilism">Compatibilism</a></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a> <ul><li><a href="/wiki/Hard_determinism" title="Hard determinism">Hard</a></li></ul></li> <li><a href="/wiki/Incompatibilism" title="Incompatibilism">Incompatibilism</a> <ul><li><a href="/wiki/Hard_incompatibilism" class="mw-redirect" title="Hard incompatibilism">Hard</a></li></ul></li> <li><a href="/wiki/Libertarianism_(metaphysics)" title="Libertarianism (metaphysics)">Libertarianism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Metaphysics" title="Metaphysics">Metaphysics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atomism" title="Atomism">Atomism</a></li> <li><a href="/wiki/Mind%E2%80%93body_dualism" title="Mind–body dualism">Dualism</a></li> <li><a href="/wiki/Idealism" title="Idealism">Idealism</a></li> <li><a href="/wiki/Monism" title="Monism">Monism</a></li> <li><a href="/wiki/Metaphysical_naturalism" title="Metaphysical naturalism">Naturalism</a></li> <li><a href="/wiki/Philosophical_realism" title="Philosophical realism">Realism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Philosophy_of_mind" title="Philosophy of mind">Mind</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Behaviorism" title="Behaviorism">Behaviorism</a></li> <li><a href="/wiki/Eliminative_materialism" title="Eliminative materialism">Eliminativism</a></li> <li><a href="/wiki/Emergentism" title="Emergentism">Emergentism</a></li> <li><a href="/wiki/Epiphenomenalism" title="Epiphenomenalism">Epiphenomenalism</a></li> <li><a href="/wiki/Functionalism_(philosophy_of_mind)" title="Functionalism (philosophy of mind)">Functionalism</a></li> <li><a href="/wiki/Objectivity_(philosophy)" class="mw-redirect" title="Objectivity (philosophy)">Objectivism</a></li> <li><a href="/wiki/Subjectivism" title="Subjectivism">Subjectivism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Norm_(philosophy)" title="Norm (philosophy)">Normativity</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Moral_absolutism" title="Moral absolutism">Absolutism</a></li> <li><a href="/wiki/Moral_particularism" title="Moral particularism">Particularism</a></li> <li><a href="/wiki/Relativism" title="Relativism">Relativism</a></li> <li><a href="/wiki/Moral_nihilism" title="Moral nihilism">Nihilism</a></li> <li><a href="/wiki/Moral_skepticism" title="Moral skepticism">Skepticism</a></li> <li><a href="/wiki/Moral_universalism" title="Moral universalism">Universalism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Ontology" title="Ontology">Ontology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Action_theory_(philosophy)" title="Action theory (philosophy)">Action</a></li> <li><a href="/wiki/Event_(philosophy)" title="Event (philosophy)">Event</a></li> <li><a href="/wiki/Process_philosophy" title="Process philosophy">Process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Reality" title="Reality">Reality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anti-realism" title="Anti-realism">Anti-realism</a></li> <li><a href="/wiki/Conceptualism" title="Conceptualism">Conceptualism</a></li> <li><a href="/wiki/Idealism" title="Idealism">Idealism</a></li> <li><a href="/wiki/Materialism" title="Materialism">Materialism</a></li> <li><a href="/wiki/Naturalism_(philosophy)" title="Naturalism (philosophy)">Naturalism</a></li> <li><a href="/wiki/Nominalism" title="Nominalism">Nominalism</a></li> <li><a href="/wiki/Physicalism" title="Physicalism">Physicalism</a></li> <li><a href="/wiki/Philosophical_realism" title="Philosophical realism">Realism</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="By_era" style="font-size:114%;margin:0 4em">By era</div></th></tr><tr><td colspan="2" class="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:6.8em"><a href="/wiki/History_of_philosophy" title="History of philosophy">By era</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ancient_philosophy" title="Ancient philosophy">Ancient</a></li> <li><a href="/wiki/Western_philosophy" title="Western philosophy">Western</a> <ul><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/Early_modern_philosophy" title="Early modern philosophy">Early modern</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></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Ancient_philosophy" title="Ancient philosophy">Ancient</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:6.6em;font-weight: normal;"><a href="/wiki/Chinese_philosophy" title="Chinese philosophy">Chinese</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Agriculturalism" title="Agriculturalism">Agriculturalism</a></li> <li><a href="/wiki/Confucianism" title="Confucianism">Confucianism</a></li> <li><a href="/wiki/Legalism_(Chinese_philosophy)" title="Legalism (Chinese philosophy)">Legalism</a></li> <li><a href="/wiki/School_of_Names" title="School of Names">Logicians</a></li> <li><a href="/wiki/Mohism" title="Mohism">Mohism</a></li> <li><a href="/wiki/School_of_Naturalists" title="School of Naturalists">Chinese naturalism</a></li> <li><a href="/wiki/Taoism" title="Taoism">Taoism</a></li> <li><a href="/wiki/Yangism" title="Yangism">Yangism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Ancient_Greek_philosophy" title="Ancient Greek philosophy">Greco-</a><a href="/wiki/Ancient_Roman_philosophy" title="Ancient Roman philosophy">Roman</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pre-Socratic_philosophy" title="Pre-Socratic philosophy">Presocratic</a> <ul><li><a href="/wiki/Ionian_School_(philosophy)" class="mw-redirect" title="Ionian School (philosophy)">Ionians</a></li> <li><a href="/wiki/Pythagoreanism" title="Pythagoreanism">Pythagoreans</a></li> <li><a href="/wiki/Eleatics" title="Eleatics">Eleatics</a></li> <li><a href="/wiki/Atomism" title="Atomism">Atomists</a></li> <li><a href="/wiki/Sophist" title="Sophist">Sophists</a></li></ul></li> <li><a href="/wiki/Cyrenaics" title="Cyrenaics">Cyrenaics</a></li> <li><a href="/wiki/Cynicism_(philosophy)" title="Cynicism (philosophy)">Cynicism</a></li> <li><a href="/wiki/Eretrian_school" title="Eretrian school">Eretrian school</a></li> <li><a href="/wiki/Megarian_school" title="Megarian school">Megarian school</a></li> <li><a href="/wiki/Platonic_Academy" title="Platonic Academy">Academy</a></li> <li><a href="/wiki/Peripatetic_school" title="Peripatetic school">Peripatetic school</a></li> <li><a href="/wiki/Hellenistic_philosophy" title="Hellenistic philosophy">Hellenistic philosophy</a> <ul><li><a href="/wiki/Pyrrhonism" title="Pyrrhonism">Pyrrhonism</a></li> <li><a href="/wiki/Stoicism" title="Stoicism">Stoicism</a></li> <li><a href="/wiki/Epicureanism" title="Epicureanism">Epicureanism</a></li> <li><a href="/wiki/Academic_Skepticism" class="mw-redirect" title="Academic Skepticism">Academic Skepticism</a></li></ul></li> <li><a href="/wiki/Middle_Platonism" title="Middle Platonism">Middle Platonism</a></li> <li><a href="/wiki/School_of_the_Sextii" title="School of the Sextii">School of the Sextii</a></li> <li><a href="/wiki/Neopythagoreanism" title="Neopythagoreanism">Neopythagoreanism</a></li> <li><a href="/wiki/Second_Sophistic" title="Second Sophistic">Second Sophistic</a></li> <li><a href="/wiki/Neoplatonism" title="Neoplatonism">Neoplatonism</a></li> <li><a href="/wiki/Church_Fathers" title="Church Fathers">Church Fathers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Indian_philosophy" title="Indian philosophy">Indian</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hindu_philosophy" title="Hindu philosophy">Hindu</a> <ul><li><a href="/wiki/Samkhya" title="Samkhya">Samkhya</a></li> <li><a href="/wiki/Nyaya" title="Nyaya">Nyaya</a></li> <li><a href="/wiki/Vaisheshika" title="Vaisheshika">Vaisheshika</a></li> <li><a href="/wiki/Yoga_Sutras_of_Patanjali" title="Yoga Sutras of Patanjali">Yoga</a></li> <li><a href="/wiki/M%C4%ABm%C4%81%E1%B9%83s%C4%81" title="Mīmāṃsā">Mīmāṃsā</a></li> <li><a href="/wiki/%C4%80j%C4%ABvika" title="Ājīvika">Ājīvika</a></li> <li><a href="/wiki/Aj%C3%B1ana" title="Ajñana">Ajñana</a></li> <li><a href="/wiki/Charvaka" title="Charvaka">Cārvāka</a></li></ul></li> <li><a href="/wiki/Jain_philosophy" title="Jain philosophy">Jain</a> <ul><li><a href="/wiki/Anekantavada" title="Anekantavada">Anekantavada</a></li> <li><a href="/wiki/Sy%C4%81dv%C4%81da" class="mw-redirect" title="Syādvāda">Syādvāda</a></li></ul></li> <li><a href="/wiki/Buddhist_philosophy" title="Buddhist philosophy">Buddhist</a> <ul><li><a href="/wiki/Abhidharma" title="Abhidharma">Abhidharma</a></li> <li><a href="/wiki/Sarvastivada" title="Sarvastivada">Sarvāstivadā</a></li> <li><a href="/wiki/Pudgalavada" title="Pudgalavada">Pudgalavada</a></li> <li><a href="/wiki/Sautr%C4%81ntika" title="Sautrāntika">Sautrāntika</a></li> <li><a href="/wiki/Madhyamaka" title="Madhyamaka">Madhyamaka</a></li> <li><a href="/wiki/Svatantrika%E2%80%93Prasa%E1%B9%85gika_distinction" title="Svatantrika–Prasaṅgika distinction">Svatantrika and Prasangika</a></li> <li><a href="/wiki/%C5%9A%C5%ABnyat%C4%81" title="Śūnyatā">Śūnyatā</a></li> <li><a href="/wiki/Yogachara" title="Yogachara">Yogacara</a></li> <li><a href="/wiki/Tibetan_Buddhism" title="Tibetan Buddhism">Tibetan</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Iranian_philosophy" title="Iranian philosophy">Persian</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mazdakism" title="Mazdakism">Mazdakism</a></li> <li><a href="/wiki/Mithraism" title="Mithraism">Mithraism</a></li> <li><a href="/wiki/Zoroastrianism" title="Zoroastrianism">Zoroastrianism</a></li> <li><a href="/wiki/Zurvanism" title="Zurvanism">Zurvanism</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Medieval_philosophy" title="Medieval philosophy">Medieval</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:6.6em;font-weight: normal;">East Asian</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Xuanxue" title="Xuanxue">Neotaoism</a></li> <li><a href="/wiki/Tiantai" title="Tiantai">Tiantai</a></li> <li><a href="/wiki/Huayan" title="Huayan">Huayan</a></li> <li><a href="/wiki/Chan_Buddhism" title="Chan Buddhism">Chan</a></li> <li><a href="/wiki/Zen" title="Zen">Zen</a></li> <li><a href="/wiki/Neo-Confucianism" title="Neo-Confucianism">Neo-Confucianism</a></li> <li><a href="/wiki/Korean_Confucianism" title="Korean Confucianism">Korean Confucianism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Western_philosophy" title="Western philosophy">European</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Christian_philosophy" title="Christian philosophy">Christian</a></li> <li><a href="/wiki/Augustinianism" title="Augustinianism">Augustinianism</a></li> <li><a href="/wiki/Scholasticism" title="Scholasticism">Scholasticism</a></li> <li><a href="/wiki/Thomism" title="Thomism">Thomism</a></li> <li><a href="/wiki/Scotism" title="Scotism">Scotism</a></li> <li><a href="/wiki/Occamism" title="Occamism">Occamism</a></li> <li><a href="/wiki/Renaissance_humanism" title="Renaissance humanism">Renaissance humanism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">Indian</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Vedanta" title="Vedanta">Vedanta</a> <ul><li><a href="/wiki/Achintya_Bheda_Abheda" title="Achintya Bheda Abheda">Acintya bheda abheda</a></li> <li><a href="/wiki/Advaita_Vedanta" title="Advaita Vedanta">Advaita</a></li> <li><a href="/wiki/Bhedabheda" title="Bhedabheda">Bhedabheda</a></li> <li><a href="/wiki/Dvaita_Vedanta" title="Dvaita Vedanta">Dvaita</a></li> <li><a href="/wiki/Nimbarka_Sampradaya" title="Nimbarka Sampradaya">Nimbarka Sampradaya</a></li> <li><a href="/wiki/Shuddhadvaita" title="Shuddhadvaita">Shuddhadvaita</a></li> <li><a href="/wiki/Vishishtadvaita" title="Vishishtadvaita">Vishishtadvaita</a></li></ul></li> <li><a href="/wiki/Navya-Ny%C4%81ya" title="Navya-Nyāya">Navya-Nyāya</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Islamic_philosophy" title="Islamic philosophy">Islamic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aristotelianism" title="Aristotelianism">Aristotelianism</a></li> <li><a href="/wiki/Averroism" title="Averroism">Averroism</a></li> <li><a href="/wiki/Avicennism" title="Avicennism">Avicennism</a></li> <li><a href="/wiki/Illuminationism" title="Illuminationism">Illuminationism</a></li> <li><a href="/wiki/Kalam" title="Kalam">ʿIlm al-Kalām</a></li> <li><a href="/wiki/Sufi_philosophy" title="Sufi philosophy">Sufi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Jewish_philosophy" title="Jewish philosophy">Jewish</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Judeo-Islamic_philosophies_(800%E2%80%931400)" title="Judeo-Islamic philosophies (800–1400)">Judeo-Islamic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Modern_philosophy" title="Modern philosophy">Modern</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anarchism" title="Anarchism">Anarchism</a></li> <li><a href="/wiki/Classical_Realism" title="Classical Realism">Classical Realism</a></li> <li><a href="/wiki/Collectivism_and_individualism" class="mw-redirect" title="Collectivism and individualism">Collectivism</a></li> <li><a href="/wiki/Conservatism" title="Conservatism">Conservatism</a></li> <li><a href="/wiki/Determinism" title="Determinism">Determinism</a></li> <li><a href="/wiki/Mind%E2%80%93body_dualism" title="Mind–body dualism">Dualism</a></li> <li><a href="/wiki/Edo_neo-Confucianism" title="Edo neo-Confucianism">Edo neo-Confucianism</a></li> <li><a href="/wiki/Empiricism" title="Empiricism">Empiricism</a></li> <li><a href="/wiki/Existentialism" title="Existentialism">Existentialism</a></li> <li><a href="/wiki/Foundationalism" title="Foundationalism">Foundationalism</a></li> <li><a href="/wiki/Historicism" title="Historicism">Historicism</a></li> <li><a href="/wiki/Holism" title="Holism">Holism</a></li> <li><a href="/wiki/Humanism" title="Humanism">Humanism</a> <ul><li><a href="/wiki/Antihumanism" title="Antihumanism">Anti-</a></li></ul></li> <li><a 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href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</a></li> <li><a href="/wiki/Philosophy_of_Friedrich_Nietzsche" title="Philosophy of Friedrich Nietzsche">Nietzscheanism</a></li> <li><a href="/wiki/Spinozism" class="mw-redirect" title="Spinozism">Spinozism</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.8em"><a href="/wiki/Contemporary_philosophy" title="Contemporary philosophy">Contemporary</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="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:6.6em;font-weight: normal;"><a href="/wiki/Analytic_philosophy" title="Analytic philosophy">Analytic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Applied_ethics" title="Applied ethics">Applied ethics</a></li> <li><a href="/wiki/Analytical_feminism" title="Analytical feminism">Analytic feminism</a></li> <li><a href="/wiki/Analytical_Marxism" title="Analytical Marxism">Analytical Marxism</a></li> <li><a href="/wiki/Communitarianism" title="Communitarianism">Communitarianism</a></li> <li><a href="/wiki/Consequentialism" title="Consequentialism">Consequentialism</a></li> <li><a href="/wiki/Critical_rationalism" title="Critical rationalism">Critical rationalism</a></li> <li><a href="/wiki/Experimental_philosophy" title="Experimental philosophy">Experimental philosophy</a></li> <li><a href="/wiki/Falsifiability" title="Falsifiability">Falsificationism</a></li> <li><a href="/wiki/Foundationalism" title="Foundationalism">Foundationalism</a> / <a href="/wiki/Coherentism" title="Coherentism">Coherentism</a></li> <li><a href="/wiki/Internalism_and_externalism" title="Internalism and externalism">Internalism and externalism</a></li> <li><a href="/wiki/Logical_positivism" title="Logical positivism">Logical positivism</a></li> <li><a href="/wiki/Legal_positivism" title="Legal positivism">Legal positivism</a></li> <li><a href="/wiki/Meta-ethics" class="mw-redirect" title="Meta-ethics">Meta-ethics</a></li> <li><a href="/wiki/Moral_realism" title="Moral realism">Moral realism</a></li> <li><a href="/wiki/Naturalized_epistemology" title="Naturalized epistemology">Quinean naturalism</a></li> <li><a href="/wiki/Normative_ethics" title="Normative ethics">Normative ethics</a></li> <li><a href="/wiki/Ordinary_language_philosophy" title="Ordinary language philosophy">Ordinary language philosophy</a></li> <li><a href="/wiki/Postanalytic_philosophy" title="Postanalytic philosophy">Postanalytic philosophy</a></li> <li><a href="/wiki/Quietism_(philosophy)" title="Quietism (philosophy)">Quietism</a></li> <li><a href="/wiki/John_Rawls" title="John Rawls">Rawlsian</a></li> <li><a href="/wiki/Reformed_epistemology" title="Reformed epistemology">Reformed epistemology</a></li> <li><a href="/wiki/Systemics" title="Systemics">Systemics</a></li> <li><a href="/wiki/Scientism" title="Scientism">Scientism</a></li> <li><a href="/wiki/Scientific_realism" title="Scientific realism">Scientific realism</a></li> <li><a href="/wiki/Scientific_skepticism" title="Scientific skepticism">Scientific skepticism</a></li> <li><a href="/wiki/Transactionalism" title="Transactionalism">Transactionalism</a></li> <li><a href="/wiki/Utilitarianism#Developments_in_the_20th_century" title="Utilitarianism">Contemporary utilitarianism</a></li> <li><a href="/wiki/Vienna_Circle" title="Vienna Circle">Vienna Circle</a></li> <li><a href="/wiki/Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Wittgensteinian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Continental_philosophy" title="Continental philosophy">Continental</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Critical_theory" title="Critical theory">Critical theory</a></li> <li><a href="/wiki/Deconstruction" title="Deconstruction">Deconstruction</a></li> <li><a href="/wiki/Existentialism" title="Existentialism">Existentialism</a></li> <li><a href="/wiki/Feminist_philosophy" title="Feminist philosophy">Feminist</a></li> <li><a href="/wiki/Frankfurt_School" title="Frankfurt School">Frankfurt School</a></li> <li><a href="/wiki/Hermeneutics" title="Hermeneutics">Hermeneutics</a></li> <li><a href="/wiki/Neo-Marxism" title="Neo-Marxism">Neo-Marxism</a></li> <li><a href="/wiki/New_historicism" title="New historicism">New Historicism</a></li> <li><a href="/wiki/Phenomenology_(philosophy)" title="Phenomenology (philosophy)">Phenomenology</a></li> <li><a href="/wiki/Posthumanism" title="Posthumanism">Posthumanism</a></li> <li><a href="/wiki/Postmodern_philosophy" title="Postmodern philosophy">Postmodernism</a></li> <li><a href="/wiki/Post-structuralism" title="Post-structuralism">Post-structuralism</a></li> <li><a href="/wiki/Social_constructionism" title="Social constructionism">Social constructionism</a></li> <li><a href="/wiki/Structuralism" title="Structuralism">Structuralism</a></li> <li><a href="/wiki/Western_Marxism" title="Western Marxism">Western Marxism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kyoto_School" title="Kyoto School">Kyoto School</a></li> <li><a href="/wiki/Objectivism" title="Objectivism">Objectivism</a></li> <li><a href="/wiki/Postcritique" title="Postcritique">Postcritique</a></li> <li><a href="/wiki/Russian_cosmism" title="Russian cosmism">Russian cosmism</a></li> <li><a href="/wiki/List_of_philosophies" title="List of philosophies">more...</a></li></ul> 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0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/African_philosophy" title="African philosophy">African</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ubuntu_philosophy" title="Ubuntu philosophy">Bantu</a></li> <li><a href="/wiki/Ancient_Egyptian_philosophy" title="Ancient Egyptian philosophy">Egyptian</a></li> <li><a href="/wiki/Ethiopian_philosophy" title="Ethiopian philosophy">Ethiopian</a></li> <li><a href="/wiki/Africana_philosophy" title="Africana philosophy">Africana</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Eastern_philosophy" title="Eastern philosophy">Eastern</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Buddhist_philosophy" title="Buddhist philosophy">Buddhist</a></li> <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">Indonesian</a></li> <li><a href="/wiki/Japanese_philosophy" title="Japanese philosophy">Japanese</a></li> <li><a href="/wiki/Korean_philosophy" title="Korean philosophy">Korean</a></li> <li><a href="/wiki/Philosophy_in_Taiwan" title="Philosophy in Taiwan">Taiwanese</a></li> <li><a href="/wiki/Vietnamese_philosophy" title="Vietnamese philosophy">Vietnamese</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;"><a href="/wiki/Middle_Eastern_philosophy" title="Middle Eastern philosophy">Middle Eastern</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 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href="/wiki/History_of_philosophy_in_Poland" title="History of philosophy in Poland">Polish</a></li> <li><a href="/wiki/List_of_Slovene_philosophers" title="List of Slovene philosophers">Slovene</a></li> <li><a href="/wiki/Spanish_philosophy" title="Spanish philosophy">Spanish</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.6em;font-weight: normal;">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Indigenous_American_philosophy" title="Indigenous American philosophy">Amerindian</a></li> <li><a href="/wiki/Aztec_philosophy" title="Aztec philosophy">Aztec</a></li> <li><a href="/wiki/Romanian_philosophy" title="Romanian philosophy">Romanian</a></li> <li><a href="/wiki/Russian_philosophy" title="Russian philosophy">Russian</a></li> <li><a href="/wiki/Yugoslav_philosophy" title="Yugoslav philosophy">Yugoslav</a></li></ul> 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[\"CITEREFBorchert2006c\"] = 1,\n [\"CITEREFBorisAlexander2017\"] = 1,\n [\"CITEREFBridgesIshiharaRathjenSchwichtenberg2023\"] = 1,\n [\"CITEREFBrody2006\"] = 1,\n [\"CITEREFBunninYu2009\"] = 1,\n [\"CITEREFBurgess2009\"] = 1,\n [\"CITEREFBäck2016\"] = 1,\n [\"CITEREFCalderbankSloane2001\"] = 1,\n [\"CITEREFCarnielliPizzi2008\"] = 1,\n [\"CITEREFCastaño2018\"] = 1,\n [\"CITEREFChakrabarti1976\"] = 1,\n [\"CITEREFChatfield2017\"] = 1,\n [\"CITEREFChua2017\"] = 1,\n [\"CITEREFClocksinMellish2003\"] = 1,\n [\"CITEREFCook2009\"] = 1,\n [\"CITEREFCopiCohenRodych2019\"] = 1,\n [\"CITEREFCorkum2015\"] = 1,\n [\"CITEREFCraig1996\"] = 1,\n [\"CITEREFCummings2010\"] = 1,\n [\"CITEREFCunningham\"] = 1,\n [\"CITEREFD\u0026#039;AgostinoFloridi2009\"] = 1,\n [\"CITEREFDaintithWright2008\"] = 1,\n [\"CITEREFDasti\"] = 1,\n [\"CITEREFDickMüller2017\"] = 1,\n [\"CITEREFDouven2021\"] = 1,\n [\"CITEREFDowden\"] = 1,\n [\"CITEREFEmmanuel2015\"] = 1,\n [\"CITEREFEnderton2001\"] = 1,\n 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1,\n [\"CITEREFHintikka1970\"] = 1,\n [\"CITEREFHintikka2019\"] = 1,\n [\"CITEREFHintikka2023\"] = 1,\n [\"CITEREFHintikkaSandu2006\"] = 1,\n [\"CITEREFHintikkaSpade\"] = 1,\n [\"CITEREFHonderich2005\"] = 1,\n [\"CITEREFHoudeCamacho2003\"] = 1,\n [\"CITEREFHurley2015\"] = 1,\n [\"CITEREFHájek2006\"] = 1,\n [\"CITEREFHájekLin2017\"] = 1,\n [\"CITEREFIEP_Staff\"] = 1,\n [\"CITEREFIqbal2013\"] = 1,\n [\"CITEREFIrvine2022\"] = 1,\n [\"CITEREFJacquette2006\"] = 1,\n [\"CITEREFJago2014\"] = 1,\n [\"CITEREFJanssenZimmermann2021\"] = 1,\n [\"CITEREFJohnson1999\"] = 1,\n [\"CITEREFJohnson2014\"] = 1,\n [\"CITEREFKetland2005\"] = 1,\n [\"CITEREFKing2009\"] = 1,\n [\"CITEREFKing2019\"] = 1,\n [\"CITEREFKlement1995b\"] = 1,\n [\"CITEREFKline1972\"] = 1,\n [\"CITEREFKnealeKneale1962\"] = 1,\n [\"CITEREFKnuuttila1980\"] = 1,\n [\"CITEREFKorb2004\"] = 1,\n [\"CITEREFKoslowski2017\"] = 1,\n [\"CITEREFKusch2020\"] = 1,\n [\"CITEREFLagerlund2018\"] = 1,\n [\"CITEREFLear1980\"] = 1,\n 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