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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rules_of_inference"> <div class="vector-toc-text"> 3 Rules of inference </div> </a> <button aria-controls="toc-Rules_of_inference-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Rules of inference subsection </button> <ul id="toc-Rules_of_inference-sublist" class="vector-toc-list"> <li id="toc-Prominent_rules_of_inference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prominent_rules_of_inference"> <div class="vector-toc-text"> 3.1 Prominent rules of inference </div> </a> <ul id="toc-Prominent_rules_of_inference-sublist" class="vector-toc-list"> <li id="toc-Modus_ponens" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modus_ponens"> <div class="vector-toc-text"> 3.1.1 Modus ponens </div> </a> <ul id="toc-Modus_ponens-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modus_tollens" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Modus_tollens"> <div class="vector-toc-text"> 3.1.2 Modus tollens </div> </a> <ul id="toc-Modus_tollens-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hypothetical_syllogism" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hypothetical_syllogism"> <div class="vector-toc-text"> 3.1.3 Hypothetical syllogism </div> </a> <ul id="toc-Hypothetical_syllogism-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"> 3.2 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"> 3.3 Definitory and strategic rules </div> </a> <ul id="toc-Definitory_and_strategic_rules-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Validity_and_soundness" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Validity_and_soundness"> <div class="vector-toc-text"> 4 Validity and soundness </div> </a> <ul id="toc-Validity_and_soundness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Difference_from_ampliative_reasoning" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Difference_from_ampliative_reasoning"> <div class="vector-toc-text"> 5 Difference from ampliative reasoning </div> </a> <ul id="toc-Difference_from_ampliative_reasoning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_various_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_various_fields"> <div class="vector-toc-text"> 6 In various fields </div> </a> <button aria-controls="toc-In_various_fields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle In various fields subsection </button> <ul id="toc-In_various_fields-sublist" class="vector-toc-list"> <li id="toc-Cognitive_psychology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cognitive_psychology"> <div class="vector-toc-text"> 6.1 Cognitive psychology </div> </a> <ul id="toc-Cognitive_psychology-sublist" class="vector-toc-list"> <li id="toc-Psychological_theories_of_deductive_reasoning" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Psychological_theories_of_deductive_reasoning"> <div class="vector-toc-text"> 6.1.1 Psychological theories of deductive reasoning </div> </a> <ul id="toc-Psychological_theories_of_deductive_reasoning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Intelligence" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Intelligence"> <div class="vector-toc-text"> 6.1.2 Intelligence </div> </a> <ul id="toc-Intelligence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Epistemology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Epistemology"> <div class="vector-toc-text"> 6.2 Epistemology </div> </a> <ul id="toc-Epistemology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability_logic"> <div class="vector-toc-text"> 6.3 Probability logic </div> </a> <ul id="toc-Probability_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 7 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_concepts_and_theories" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_concepts_and_theories"> <div class="vector-toc-text"> 8 Related concepts and theories </div> </a> <button aria-controls="toc-Related_concepts_and_theories-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Related concepts and theories subsection </button> <ul id="toc-Related_concepts_and_theories-sublist" class="vector-toc-list"> <li id="toc-Deductivism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deductivism"> <div class="vector-toc-text"> 8.1 Deductivism </div> </a> <ul id="toc-Deductivism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Natural_deduction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Natural_deduction"> <div class="vector-toc-text"> 8.2 Natural deduction </div> </a> <ul id="toc-Natural_deduction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometrical_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometrical_method"> <div class="vector-toc-text"> 8.3 Geometrical method </div> </a> <ul id="toc-Geometrical_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> 10 Notes and references </div> </a> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 11 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 12 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">Deductive reasoning</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" 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Available in 62 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-62" aria-hidden="true" > 62 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D9%86%D8%A8%D8%A7%D8%B7" title="استنباط – Arabic" lang="ar" hreflang="ar" data-title="استنباط" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Deduksiya" title="Deduksiya – Azerbaijani" lang="az" hreflang="az" data-title="Deduksiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%B4%D1%83%D0%BA%D1%86%D0%B8%D1%8F" title="Дедукция – Bulgarian" lang="bg" hreflang="bg" data-title="Дедукция" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Dedukcija" title="Dedukcija – Bosnian" lang="bs" hreflang="bs" data-title="Dedukcija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Raonament_deductiu" title="Raonament deductiu – Catalan" lang="ca" hreflang="ca" data-title="Raonament deductiu" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dedukce" title="Dedukce – Czech" lang="cs" hreflang="cs" data-title="Dedukce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Deduktion" title="Deduktion – Danish" lang="da" hreflang="da" data-title="Deduktion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Deduktion" title="Deduktion – German" lang="de" hreflang="de" data-title="Deduktion" 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/Deduktsioon" title="Deduktsioon – Estonian" lang="et" hreflang="et" data-title="Deduktsioon" 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%A0%CE%B1%CF%81%CE%B1%CE%B3%CF%89%CE%B3%CE%B9%CE%BA%CF%8C%CF%82_%CF%83%CF%85%CE%BB%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Παραγωγικός συλλογισμός – Greek" lang="el" hreflang="el" data-title="Παραγωγικός συλλογισμός" 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/Razonamiento_deductivo" title="Razonamiento deductivo – Spanish" lang="es" hreflang="es" data-title="Razonamiento deductivo" 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/Dedukto" title="Dedukto – Esperanto" lang="eo" hreflang="eo" data-title="Dedukto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Arrazoibide_deduktibo" title="Arrazoibide deduktibo – Basque" lang="eu" hreflang="eu" data-title="Arrazoibide deduktibo" 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/%D8%A7%D8%B3%D8%AA%D8%AF%D9%84%D8%A7%D9%84_%D8%A7%D8%B3%D8%AA%D9%86%D8%AA%D8%A7%D8%AC%DB%8C" title="استدلال استنتاجی – Persian" lang="fa" hreflang="fa" data-title="استدلال استنتاجی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Raisonnement_d%C3%A9ductif" title="Raisonnement déductif – French" lang="fr" hreflang="fr" data-title="Raisonnement déductif" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Deduci%C3%B3n" title="Dedución – Galician" lang="gl" hreflang="gl" data-title="Dedución" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ki mw-list-item"><a href="https://ki.wikipedia.org/wiki/Nguny%C4%A9r%C4%A9ko_(deduction)" title="Ngunyĩrĩko (deduction) – Kikuyu" lang="ki" hreflang="ki" data-title="Ngunyĩrĩko (deduction)" 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/%EC%97%B0%EC%97%AD" title="연역 – Korean" lang="ko" hreflang="ko" data-title="연역" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D5%A4%D5%B8%D6%82%D5%AF%D6%81%D5%AB%D5%A1" title="Դեդուկցիա – Armenian" lang="hy" hreflang="hy" data-title="Դեդուկցիա" 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%A8%E0%A4%BF%E0%A4%97%E0%A4%AE%E0%A4%A8%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95" title="निगमनात्मक तर्क – Hindi" lang="hi" hreflang="hi" data-title="निगमनात्मक तर्क" 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/Dedukcija" title="Dedukcija – Croatian" lang="hr" hreflang="hr" data-title="Dedukcija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Metode_deduksi" title="Metode deduksi – Indonesian" lang="id" hreflang="id" data-title="Metode deduksi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Aflei%C3%B0sla" title="Afleiðsla – Icelandic" lang="is" hreflang="is" data-title="Afleiðsla" 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/Deduzione" title="Deduzione – Italian" lang="it" hreflang="it" data-title="Deduzione" 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%94%D7%A1%D7%A7%D7%94_%D7%93%D7%93%D7%95%D7%A7%D7%98%D7%99%D7%91%D7%99%D7%AA" title="הסקה דדוקטיבית – Hebrew" lang="he" hreflang="he" data-title="הסקה דדוקטיבית" 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/Panalaran_d%C3%A9duksi" title="Panalaran déduksi – Javanese" lang="jv" hreflang="jv" data-title="Panalaran déduksi" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B5%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D1%82%D1%96_%D0%BF%D0%B0%D0%B9%D1%8B%D0%BC%D0%B4%D0%B0%D1%83" title="Дедуктивті пайымдау – Kazakh" lang="kk" hreflang="kk" data-title="Дедуктивті пайымдау" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B5%D0%B4%D1%83%D0%BA%D1%86%D0%B8%D1%8F" title="Дедукция – Kyrgyz" lang="ky" hreflang="ky" data-title="Дедукция" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Dedukt%C4%ABvs_sl%C4%93dziens" title="Deduktīvs slēdziens – Latvian" lang="lv" hreflang="lv" data-title="Deduktīvs slēdziens" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Dedukci%C3%B3" title="Dedukció – Hungarian" lang="hu" hreflang="hu" data-title="Dedukció" 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%94%D0%B5%D0%B4%D1%83%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Дедукција – Macedonian" lang="mk" hreflang="mk" data-title="Дедукција" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Penaakulan_deduktif" title="Penaakulan deduktif – Malay" lang="ms" hreflang="ms" data-title="Penaakulan deduktif" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Deductie" title="Deductie – Dutch" lang="nl" hreflang="nl" data-title="Deductie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%BC%94%E7%B9%B9" title="演繹 – Japanese" lang="ja" hreflang="ja" data-title="演繹" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Deduksjon_(filosofi)" title="Deduksjon (filosofi) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Deduksjon (filosofi)" 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/Deduksjon" title="Deduksjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Deduksjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Deduksiya" title="Deduksiya – Uzbek" lang="uz" hreflang="uz" data-title="Deduksiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D8%AE%D8%B1%D8%A7%D8%AC" title="استخراج – Western Punjabi" lang="pnb" hreflang="pnb" data-title="استخراج" 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/%D9%84%D9%87_%DA%A9%D9%8F%D9%84_%DA%85%D8%AE%D9%87_%D8%AC%D8%B2_%D8%AA%D9%87_(%D8%AA%D8%B9%D9%84%DB%8C%D9%84%D9%8A)_%D8%A7%D8%B3%D8%AA%D8%AF%D9%84%D8%A7%D9%84" title="له کُل څخه جز ته (تعلیلي) استدلال – Pashto" lang="ps" hreflang="ps" data-title="له کُل څخه جز ته (تعلیلي) استدلال" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozumowanie_dedukcyjne" title="Rozumowanie dedukcyjne – Polish" lang="pl" hreflang="pl" data-title="Rozumowanie dedukcyjne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Form of reasoning</div> Deductive reasoning is the process of drawing valid <a href="/wiki/Inference" title="Inference">inferences</a>. An inference is <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> if its conclusion follows <a href="/wiki/Logic" title="Logic">logically</a> from its <a href="/wiki/Premise" title="Premise">premises</a>, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and "<a href="/wiki/Socrates" title="Socrates">Socrates</a> is a man" to the conclusion "Socrates is mortal" is deductively valid. An <a href="/wiki/Argument" title="Argument">argument</a> is <a href="/wiki/Soundness" title="Soundness">sound</a> if it is valid and all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Deductive logic studies under what conditions an argument is valid. According to the <a href="/wiki/Semantic" class="mw-redirect" title="Semantic">semantic</a> approach, an argument is valid if there is no possible <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> of the argument whereby its premises are true and its conclusion is false. The <a href="/wiki/Syntactic" class="mw-redirect" title="Syntactic">syntactic</a> approach, by contrast, focuses on <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a>, that is, schemas of drawing a conclusion from a set of premises based only on their <a href="/wiki/Logical_form" title="Logical form">logical form</a>. There are various rules of inference, such as <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> and <a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a>. Invalid deductive arguments, which do not follow a rule of inference, are called <a href="/wiki/Formal_fallacies" class="mw-redirect" title="Formal fallacies">formal fallacies</a>. Rules of inference are definitory rules and contrast with strategic rules, which specify what inferences one needs to draw in order to arrive at an intended conclusion. Deductive reasoning contrasts with non-deductive or <a href="/wiki/Ampliative" title="Ampliative">ampliative</a> reasoning. For ampliative arguments, such as <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive</a> or <a href="/wiki/Abductive_reasoning" title="Abductive reasoning">abductive arguments</a>, the premises offer weaker support to their conclusion: they indicate that it is most likely, but they do not guarantee its truth. They make up for this drawback with their ability to provide genuinely new information (that is, information not already found in the premises), unlike deductive arguments. <a href="/wiki/Cognitive_psychology" title="Cognitive psychology">Cognitive psychology</a> investigates the mental processes responsible for deductive reasoning. One of its topics concerns the factors determining whether people draw valid or invalid deductive inferences. One such factor is the form of the argument: for example, people draw valid inferences more successfully for arguments of the form modus ponens than of the form modus tollens. Another factor is the content of the arguments: people are more likely to believe that an argument is valid if the claim made in its conclusion is plausible. A general finding is that people tend to perform better for realistic and concrete cases than for abstract cases. Psychological theories of deductive reasoning aim to explain these findings by providing an account of the underlying psychological processes. Mental logic theories hold that deductive reasoning is a language-like process that happens through the manipulation of representations using rules of inference. Mental model theories, on the other hand, claim that deductive reasoning involves models of possible states of the world without the medium of language or rules of inference. According to <a href="/wiki/Dual-process_theories" class="mw-redirect" title="Dual-process theories">dual-process theories</a> of reasoning, there are two qualitatively different cognitive systems responsible for reasoning. The problem of deduction is relevant to various fields and issues. <a href="/wiki/Epistemology" title="Epistemology">Epistemology</a> tries to understand how <a href="/wiki/Justification_(epistemology)" title="Justification (epistemology)">justification</a> is transferred from the <a href="/wiki/Belief" title="Belief">belief</a> in the premises to the belief in the conclusion in the process of deductive reasoning. <a href="/wiki/Probability_logic" class="mw-redirect" title="Probability logic">Probability logic</a> studies how the probability of the premises of an inference affects the probability of its conclusion. The controversial thesis of <a href="/wiki/Deductivism" class="mw-redirect" title="Deductivism">deductivism</a> denies that there are other correct forms of inference besides deduction. <a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a> is a type of proof system based on simple and self-evident rules of inference. In philosophy, the geometrical method is a way of philosophizing that starts from a small set of self-evident axioms and tries to build a comprehensive logical system using deductive reasoning. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> Deductive reasoning is the psychological process of drawing deductive <a href="/wiki/Inference" title="Inference">inferences</a>. An inference is a set of <a href="/wiki/Premise" title="Premise">premises</a> together with a conclusion. This psychological process starts from the premises and <a href="/wiki/Reason" title="Reason">reasons</a> to a conclusion based on and supported by these premises. If the reasoning was done correctly, it results in a <a href="/wiki/Validity_(logic)" title="Validity (logic)">valid</a> deduction: the truth of the premises ensures the truth of the conclusion.<a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-Houde-2">[2]</a><a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-4">[4]</a> For example, in the <a href="/wiki/Syllogistic" class="mw-redirect" title="Syllogistic">syllogistic</a> argument "all frogs are amphibians; no cats are amphibians; therefore, no cats are frogs" the conclusion is true because its two premises are true. But even arguments with wrong premises can be deductively valid if they obey this principle, as in "all frogs are mammals; no cats are mammals; therefore, no cats are frogs". If the premises of a valid <a href="/wiki/Argument" title="Argument">argument</a> are true, then it is called a <a href="/wiki/Soundness" title="Soundness">sound</a> argument.<a href="#cite_note-Evans-5">[5]</a> The relation between the premises and the conclusion of a deductive argument is usually referred to as "<a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a>". According to <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, logical consequence has 3 essential features: it is necessary, formal, and knowable <a href="/wiki/A_priori" class="mw-redirect" title="A priori">a priori</a>.<a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Tarski-7">[7]</a> It is necessary in the sense that the premises of valid deductive arguments necessitate the conclusion: it is impossible for the premises to be true and the conclusion to be false, independent of any other circumstances.<a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Tarski-7">[7]</a> Logical consequence is formal in the sense that it depends only on the form or the syntax of the premises and the conclusion. This means that the validity of a particular argument does not depend on the specific contents of this argument. If it is valid, then any argument with the same logical form is also valid, no matter how different it is on the level of its contents.<a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Tarski-7">[7]</a> Logical consequence is knowable a priori in the sense that no <a href="/wiki/Empirical" class="mw-redirect" title="Empirical">empirical</a> knowledge of the world is necessary to determine whether a deduction is valid. So it is not necessary to engage in any form of empirical investigation.<a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Tarski-7">[7]</a> Some logicians define deduction in terms of <a href="/wiki/Possible_world" title="Possible world">possible worlds</a>: A deductive inference is valid if and only if, there is no possible world in which its conclusion is false while its premises are true. This means that there are no counterexamples: the conclusion is true in all such cases, not just in most cases.<a href="#cite_note-Johnson-Laird2009-1">[1]</a> It has been argued against this and similar definitions that they fail to distinguish between valid and invalid deductive reasoning, i.e. they leave it open whether there are invalid deductive inferences and how to define them.<a href="#cite_note-Vorobej-8">[8]</a><a href="#cite_note-Wilbanks-9">[9]</a> Some authors define deductive reasoning in psychological terms in order to avoid this problem. According to Mark Vorobey, whether an argument is deductive depends on the psychological state of the person making the argument: "An argument is deductive if, and only if, the author of the argument believes that the truth of the premises necessitates (guarantees) the truth of the conclusion".<a href="#cite_note-Vorobej-8">[8]</a> A similar formulation holds that the speaker claims or intends that the premises offer deductive support for their conclusion.<a href="#cite_note-Copi1-10">[10]</a><a href="#cite_note-IEPDeductiveInductive-11">[11]</a> This is sometimes categorized as a speaker-determined definition of deduction since it depends also on the speaker whether the argument in question is deductive or not. For speakerless definitions, on the other hand, only the argument itself matters independent of the speaker.<a href="#cite_note-Wilbanks-9">[9]</a> One advantage of this type of formulation is that it makes it possible to distinguish between good or valid and bad or invalid deductive arguments: the argument is good if the author's <a href="/wiki/Belief" title="Belief">belief</a> concerning the relation between the premises and the conclusion is true, otherwise it is bad.<a href="#cite_note-Vorobej-8">[8]</a> One consequence of this approach is that deductive arguments cannot be identified by the law of inference they use. For example, an argument of the form <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> may be non-deductive if the author's beliefs are sufficiently confused. That brings with it an important drawback of this definition: it is difficult to apply to concrete cases since the intentions of the author are usually not explicitly stated.<a href="#cite_note-Vorobej-8">[8]</a> Deductive reasoning is studied in <a href="/wiki/Logic" title="Logic">logic</a>, <a href="/wiki/Psychology" title="Psychology">psychology</a>, and the <a href="/wiki/Cognitive_sciences" class="mw-redirect" title="Cognitive sciences">cognitive sciences</a>.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a> Some theorists emphasize in their definition the difference between these fields. On this view, psychology studies deductive reasoning as an empirical mental process, i.e. what happens when humans engage in reasoning.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a> But the descriptive question of how actual reasoning happens is different from the <a href="/wiki/Normative" class="mw-redirect" title="Normative">normative</a> question of how it should happen or what constitutes correct deductive reasoning, which is studied by logic.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-BritannicaPhilosophy-12">[12]</a><a href="#cite_note-IEPLogical-6">[6]</a> This is sometimes expressed by stating that, strictly speaking, logic does not study deductive reasoning but the deductive relation between premises and a conclusion known as <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a>. But this distinction is not always precisely observed in the academic literature.<a href="#cite_note-Schechter-3">[3]</a> One important aspect of this difference is that logic is not interested in whether the conclusion of an argument is sensible.<a href="#cite_note-Johnson-Laird2009-1">[1]</a> So from the premise "the printer has ink" one may draw the unhelpful conclusion "the printer has ink and the printer has ink and the printer has ink", which has little relevance from a psychological point of view. Instead, actual reasoners usually try to remove redundant or irrelevant information and make the relevant information more explicit.<a href="#cite_note-Johnson-Laird2009-1">[1]</a> The psychological study of deductive reasoning is also concerned with how good people are at drawing deductive inferences and with the factors determining their performance.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> Deductive inferences are found both in <a href="/wiki/Natural_language" title="Natural language">natural language</a> and in <a href="/wiki/Formal_logical_systems" class="mw-redirect" title="Formal logical systems">formal logical systems</a>, such as <a href="/wiki/Propositional_logic" class="mw-redirect" title="Propositional logic">propositional logic</a>.<a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-Hintikka-13">[13]</a> <div class="mw-heading mw-heading2"><h2 id="Conceptions_of_deduction">Conceptions of deduction</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=2" title="Edit section: Conceptions of deduction">edit</a>]</div> Deductive arguments differ from non-deductive arguments in that the truth of their premises ensures the truth of their conclusion.<a href="#cite_note-Stump-14">[14]</a><a href="#cite_note-RoutledgeFormalInformal-15">[15]</a><a href="#cite_note-IEPLogical-6">[6]</a> There are two important conceptions of what this exactly means. They are referred to as the <a href="/wiki/Syntactic" class="mw-redirect" title="Syntactic">syntactic</a> and the <a href="/wiki/Semantic" class="mw-redirect" title="Semantic">semantic</a> approach.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Evans-5">[5]</a> According to the syntactic approach, whether an argument is deductively valid depends only on its form, syntax, or structure. Two arguments have the same form if they use the same logical vocabulary in the same arrangement, even if their contents differ.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Evans-5">[5]</a> For example, the arguments "if it rains then the street will be wet; it rains; therefore, the street will be wet" and "if the meat is not cooled then it will spoil; the meat is not cooled; therefore, it will spoil" have the same logical form: they follow the <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>. Their form can be expressed more abstractly as "if A then B; A; therefore B" in order to make the common syntax explicit.<a href="#cite_note-Evans-5">[5]</a> There are various other valid logical forms or <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a>, like <a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a> or the <a href="/wiki/Disjunction_elimination" title="Disjunction elimination">disjunction elimination</a>. The syntactic approach then holds that an argument is deductively valid if and only if its conclusion can be deduced from its premises using a valid rule of inference.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Evans-5">[5]</a> One difficulty for the syntactic approach is that it is usually necessary to express the argument in a <a href="/wiki/Formal_language" title="Formal language">formal language</a> in order to assess whether it is valid. This often brings with it the difficulty of translating the <a href="/wiki/Natural_language" title="Natural language">natural language</a> argument into a formal language, a process that comes with various problems of its own.<a href="#cite_note-Hintikka-13">[13]</a> Another difficulty is due to the fact that the syntactic approach depends on the distinction between formal and non-formal features. While there is a wide agreement concerning the paradigmatic cases, there are also various controversial cases where it is not clear how this distinction is to be drawn.<a href="#cite_note-MacFarlane-16">[16]</a><a href="#cite_note-BritannicaPhilosophy-12">[12]</a> The semantic approach suggests an alternative definition of deductive validity. It is based on the idea that the sentences constituting the premises and conclusions have to be <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">interpreted</a> in order to determine whether the argument is valid.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Evans-5">[5]</a> This means that one ascribes semantic values to the expressions used in the sentences, such as the reference to an object for <a href="/wiki/Singular_term" title="Singular term">singular terms</a> or to a <a href="/wiki/Truth-value" class="mw-redirect" title="Truth-value">truth-value</a> for atomic sentences. The semantic approach is also referred to as the model-theoretic approach since the branch of mathematics known as <a href="/wiki/Model_theory" title="Model theory">model theory</a> is often used to interpret these sentences.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-IEPLogical-6">[6]</a> Usually, many different interpretations are possible, such as whether a singular term refers to one object or to another. According to the semantic approach, an argument is deductively valid if and only if there is no possible interpretation where its premises are true and its conclusion is false.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-IEPLogical-6">[6]</a><a href="#cite_note-Evans-5">[5]</a> Some objections to the semantic approach are based on the claim that the semantics of a language cannot be expressed in the same language, i.e. that a richer <a href="/wiki/Metalanguage" title="Metalanguage">metalanguage</a> is necessary. This would imply that the semantic approach cannot provide a universal account of deduction for language as an all-encompassing medium.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-BritannicaPhilosophy-12">[12]</a> <div class="mw-heading mw-heading2"><h2 id="Rules_of_inference">Rules of inference</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=3" title="Edit section: Rules of inference">edit</a>]</div> Deductive reasoning usually happens by applying <a href="/wiki/Rules_of_inference" class="mw-redirect" title="Rules of inference">rules of inference</a>. A rule of inference is a way or schema of drawing a conclusion from a set of premises.<a href="#cite_note-Macmillan-17">[17]</a> This happens usually based only on the <a href="/wiki/Logical_form" title="Logical form">logical form</a> of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called <a href="/wiki/Formal_fallacies" class="mw-redirect" title="Formal fallacies">formal fallacies</a>: the truth of their premises does not ensure the truth of their conclusion.<a href="#cite_note-IEPFallacies-18">[18]</a><a href="#cite_note-Stump-14">[14]</a> In some cases, whether a rule of inference is valid depends on the logical system one is using. The dominant logical system is <a href="/wiki/Classical_logic" title="Classical logic">classical logic</a> and the rules of inference listed here are all valid in classical logic. But so-called <a href="/wiki/Deviant_logic" title="Deviant logic">deviant logics</a> provide a different account of which inferences are valid. For example, the rule of inference known as <a href="/wiki/Double_negation_elimination" class="mw-redirect" title="Double negation elimination">double negation elimination</a>, i.e. that if a proposition is not not true then it is also true, is accepted in classical logic but rejected in <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>.<a href="#cite_note-Moschovakis-19">[19]</a><a href="#cite_note-MacMillanNonClassical-20">[20]</a> <div class="mw-heading mw-heading3"><h3 id="Prominent_rules_of_inference">Prominent rules of inference</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=4" title="Edit section: Prominent rules of inference">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Modus_ponens">Modus ponens</h4>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=5" title="Edit section: Modus ponens">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Modus_ponens" title="Modus ponens">Modus ponens</a></div> Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive <a href="/wiki/Rule_of_inference" title="Rule of inference">rule of inference</a>. It applies to arguments that have as first premise a <a href="/wiki/Material_conditional" title="Material conditional">conditional statement</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}">) and as second premise the antecedent (<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}">) of the conditional statement. It obtains the consequent (<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}">) of the conditional statement as its conclusion. The argument form is listed below: <ol><li><code><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></code>  (First premise is a conditional statement)</li> <li><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}">  (Second premise is the antecedent)</li> <li><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}">  (Conclusion deduced is the consequent)</li></ol> In this form of deductive reasoning, the consequent (<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}">) obtains as the conclusion from the premises of a conditional statement (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}">) and its antecedent (<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}">). However, the antecedent (<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}">) cannot be similarly obtained as the conclusion from the premises of the conditional statement (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}">) and the consequent (<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 an argument commits the <a href="/wiki/Logical_fallacy" class="mw-redirect" title="Logical fallacy">logical fallacy</a> of <a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a>. The following is an example of an argument using modus ponens: <ol><li>If it is raining, then there are clouds in the sky.</li> <li>It is raining.</li> <li>Thus, there are clouds in the sky.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Modus_tollens">Modus tollens</h4>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=6" title="Edit section: Modus tollens">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/Modus_tollens" title="Modus tollens">Modus tollens</a></div> Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b5e3033d1f0e7333fc9e708c2ea802f9b9fca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle \lnot Q}">) and as conclusion the negation of the antecedent (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a32afb77c17696c41588f6deaf9bcd7109b10c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \lnot P}">). In contrast to <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}">. (First premise is a conditional statement)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b5e3033d1f0e7333fc9e708c2ea802f9b9fca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle \lnot Q}">. (Second premise is the negation of the consequent)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lnot P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">¬</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lnot P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a32afb77c17696c41588f6deaf9bcd7109b10c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.296ex; height:2.176ex;" alt="{\displaystyle \lnot P}">. (Conclusion deduced is the negation of the antecedent)</li></ol> The following is an example of an argument using modus tollens: <ol><li>If it is raining, then there are clouds in the sky.</li> <li>There are no clouds in the sky.</li> <li>Thus, it is not raining.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Hypothetical_syllogism">Hypothetical syllogism</h4>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=7" title="Edit section: Hypothetical syllogism">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/Hypothetical_syllogism" title="Hypothetical syllogism">hypothetical syllogism</a></div> A hypothetical <a href="/wiki/Syllogism" title="Syllogism">syllogism</a> is an inference that takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow Q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86439ea857adc8eaec93c4d14270b8ba6bd2a6a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle P\rightarrow Q}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\rightarrow R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\rightarrow R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76e444a574635586201a7ce3424d8777e3d6e22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.216ex; height:2.509ex;" alt="{\displaystyle Q\rightarrow R}"></li> <li>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\rightarrow R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\rightarrow R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49c1fd9f72e6b64b996ec899e5d2a68f6ac539a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.124ex; height:2.176ex;" alt="{\displaystyle P\rightarrow R}">.</li></ol> In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in <a href="/wiki/Term_logic" title="Term logic">term logic</a>, although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition. The following is an example of an argument using a hypothetical syllogism: <ol><li>If there had been a thunderstorm, it would have rained.</li> <li>If it had rained, things would have gotten wet.</li> <li>Thus, if there had been a thunderstorm, things would have gotten wet.<a href="#cite_note-21">[21]</a></li></ol> <div class="mw-heading mw-heading3"><h3 id="Fallacies">Fallacies</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=8" title="Edit section: Fallacies">edit</a>]</div> Various formal fallacies have been described. They are invalid forms of deductive reasoning.<a href="#cite_note-IEPFallacies-18">[18]</a><a href="#cite_note-Stump-14">[14]</a> An additional aspect of them is that they appear to be valid on some occasions or on the first impression. They may thereby seduce people into accepting and committing them.<a href="#cite_note-22">[22]</a> One type of formal fallacy is <a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a>, as in "if John is a bachelor, then he is male; John is male; therefore, John is a bachelor".<a href="#cite_note-23">[23]</a> This is similar to the valid rule of inference named <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>, but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is <a href="/wiki/Denying_the_antecedent" title="Denying the antecedent">denying the antecedent</a>, 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-BritannicaThought-24">[24]</a><a href="#cite_note-25">[25]</a> This is similar to the valid rule of inference called <a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a>, the difference being that the second premise and the conclusion are switched around. Other formal fallacies include <a href="/wiki/Affirming_a_disjunct" title="Affirming a disjunct">affirming a disjunct</a>, <a href="/wiki/Denying_a_conjunct" class="mw-redirect" title="Denying a conjunct">denying a conjunct</a>, and the <a href="/wiki/Fallacy_of_the_undistributed_middle" title="Fallacy of the undistributed middle">fallacy of the undistributed middle</a>. All of them have in common that the truth of their premises does not ensure the truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true.<a href="#cite_note-IEPFallacies-18">[18]</a><a href="#cite_note-Stump-14">[14]</a> <div class="mw-heading mw-heading3"><h3 id="Definitory_and_strategic_rules">Definitory and strategic rules</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=9" title="Edit section: Definitory and strategic rules">edit</a>]</div> Rules of inferences are definitory rules: they determine whether an argument is deductively valid or not. But reasoners are usually not just interested in making any kind of valid argument. Instead, they often have a specific point or conclusion that they wish to prove or refute. So given a set of premises, they are faced with the problem of choosing the relevant rules of inference for their deduction to arrive at their intended conclusion.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-BritannicaSystems-26">[26]</a><a href="#cite_note-Pedemonte-27">[27]</a> This issue belongs to the field of strategic rules: the question of which inferences need to be drawn to support one's conclusion. The distinction between definitory and strategic rules is not exclusive to logic: it is also found in various games.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-BritannicaSystems-26">[26]</a><a href="#cite_note-Pedemonte-27">[27]</a> In <a href="/wiki/Chess" title="Chess">chess</a>, for example, the definitory rules state that <a href="/wiki/Bishop_(chess)" title="Bishop (chess)">bishops</a> may only move diagonally while the strategic rules recommend that one should control the center and protect one's <a href="/wiki/King_(chess)" title="King (chess)">king</a> if one intends to win. In this sense, definitory rules determine whether one plays chess or something else whereas strategic rules determine whether one is a good or a bad chess player.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-BritannicaSystems-26">[26]</a> The same applies to deductive reasoning: to be an effective reasoner involves mastering both definitory and strategic rules.<a href="#cite_note-Hintikka-13">[13]</a> <div class="mw-heading mw-heading2"><h2 id="Validity_and_soundness">Validity and soundness</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=10" title="Edit section: Validity and soundness">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Argument_terminology_used_in_logic_(en).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Argument_terminology_used_in_logic_%28en%29.svg/400px-Argument_terminology_used_in_logic_%28en%29.svg.png" decoding="async" width="400" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Argument_terminology_used_in_logic_%28en%29.svg/600px-Argument_terminology_used_in_logic_%28en%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Argument_terminology_used_in_logic_%28en%29.svg/800px-Argument_terminology_used_in_logic_%28en%29.svg.png 2x" data-file-width="1052" data-file-height="744" /></a><figcaption>Argument terminology</figcaption></figure> Deductive arguments are evaluated in terms of their <a href="/wiki/Validity_(logic)" title="Validity (logic)">validity</a> and <a href="/wiki/Soundness" title="Soundness">soundness</a>. An argument is valid if it is impossible for its <a href="/wiki/Premise" title="Premise">premises</a> to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false. An argument is sound if it is valid and the premises are true. It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form. The following is an example of an argument that is “valid”, but not “sound”: <ol><li>Everyone who eats carrots is a quarterback.</li> <li>John eats carrots.</li> <li>Therefore, John is a quarterback.</li></ol> The example's first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is “valid”, but not “sound”. False generalizations – such as "Everyone who eats carrots is a quarterback" – are often used to make unsound arguments. The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument. In this example, the first statement uses <a href="/wiki/Term_logic" title="Term logic">categorical reasoning</a>, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as <a href="/wiki/Term_logic" title="Term logic">term logic</a> – was developed by <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, but was superseded by <a href="/wiki/Propositional_calculus" title="Propositional calculus">propositional (sentential) logic</a> and <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a>. [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Deductive reasoning can be contrasted with <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a>, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means). <div class="mw-heading mw-heading2"><h2 id="Difference_from_ampliative_reasoning">Difference from ampliative reasoning</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=11" title="Edit section: Difference from ampliative reasoning">edit</a>]</div> Deductive reasoning is usually contrasted with non-deductive or <a href="/wiki/Ampliative" title="Ampliative">ampliative</a> reasoning.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-Backmann-28">[28]</a><a href="#cite_note-IEPArguments-29">[29]</a> The hallmark of valid deductive inferences is that it is impossible for their premises to be true and their conclusion to be false. In this way, the premises provide the strongest possible support to their conclusion.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-Backmann-28">[28]</a><a href="#cite_note-IEPArguments-29">[29]</a> The premises of ampliative inferences also support their conclusion. But this support is weaker: they are not necessarily truth-preserving. So even for correct ampliative arguments, it is possible that their premises are true and their conclusion is false.<a href="#cite_note-IEPDeductiveInductive-11">[11]</a> Two important forms of ampliative reasoning are <a href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive</a> and <a href="/wiki/Abductive_reasoning" title="Abductive reasoning">abductive reasoning</a>.<a href="#cite_note-StanfordAbduction-30">[30]</a> Sometimes the term "inductive reasoning" is used in a very wide sense to cover all forms of ampliative reasoning.<a href="#cite_note-IEPDeductiveInductive-11">[11]</a> However, in a more strict usage, inductive reasoning is just one form of ampliative reasoning.<a href="#cite_note-StanfordAbduction-30">[30]</a> In the narrow sense, inductive inferences are forms of statistical generalization. They are usually based on many individual <a href="/wiki/Observation" title="Observation">observations</a> that all show a certain pattern. These observations are then used to form a conclusion either about a yet unobserved entity or about a general law.<a href="#cite_note-MacmillanInduction-31">[31]</a><a href="#cite_note-32">[32]</a><a href="#cite_note-33">[33]</a> For abductive inferences, the premises support the conclusion because the conclusion is the best explanation of why the premises are true.<a href="#cite_note-StanfordAbduction-30">[30]</a><a href="#cite_note-Koslowski-34">[34]</a> The support ampliative arguments provide for their conclusion comes in degrees: some ampliative arguments are stronger than others.<a href="#cite_note-IEPDeductiveInductive-11">[11]</a><a href="#cite_note-StanfordInductive-35">[35]</a><a href="#cite_note-StanfordAbduction-30">[30]</a> This is often explained in terms of <a href="/wiki/Probability" title="Probability">probability</a>: the premises make it more likely that the conclusion is true.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-Backmann-28">[28]</a><a href="#cite_note-IEPArguments-29">[29]</a> Strong ampliative arguments make their conclusion very likely, but not absolutely certain. An example of ampliative reasoning is the inference from the premise "every raven in a random sample of 3200 ravens is black" to the conclusion "all ravens are black": the extensive random sample makes the conclusion very likely, but it does not exclude that there are rare exceptions.<a href="#cite_note-StanfordInductiveLogic-36">[36]</a> In this sense, ampliative reasoning is defeasible: it may become necessary to retract an earlier conclusion upon receiving new related information.<a href="#cite_note-BritannicaPhilosophy-12">[12]</a><a href="#cite_note-StanfordAbduction-30">[30]</a> Ampliative reasoning is very common in everyday discourse and the <a href="/wiki/Science" title="Science">sciences</a>.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-Bunge-37">[37]</a> An important drawback of deductive reasoning is that it does not lead to genuinely new information.<a href="#cite_note-Evans-5">[5]</a> This means that the conclusion only repeats information already found in the premises. Ampliative reasoning, on the other hand, goes beyond the premises by arriving at genuinely new information.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-Backmann-28">[28]</a><a href="#cite_note-IEPArguments-29">[29]</a> One difficulty for this characterization is that it makes deductive reasoning appear useless: if deduction is uninformative, it is not clear why people would engage in it and study it.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-D'Agostino-38">[38]</a> It has been suggested that this problem can be solved by distinguishing between surface and depth information. On this view, deductive reasoning is uninformative on the depth level, in contrast to ampliative reasoning. But it may still be valuable on the surface level by presenting the information in the premises in a new and sometimes surprising way.<a href="#cite_note-Hintikka-13">[13]</a><a href="#cite_note-Evans-5">[5]</a> A popular misconception of the relation between deduction and induction identifies their difference on the level of particular and general claims.<a href="#cite_note-Houde-2">[2]</a><a href="#cite_note-Wilbanks-9">[9]</a><a href="#cite_note-39">[39]</a> On this view, deductive inferences start from general premises and draw particular conclusions, while inductive inferences start from particular premises and draw general conclusions. This idea is often motivated by seeing deduction and induction as two inverse processes that complement each other: deduction is top-down while induction is bottom-up. But this is a misconception that does not reflect how valid deduction is defined in the field of <a href="/wiki/Logic" title="Logic">logic</a>: a deduction is valid if it is impossible for its premises to be true while its conclusion is false, independent of whether the premises or the conclusion are particular or general.<a href="#cite_note-Houde-2">[2]</a><a href="#cite_note-Wilbanks-9">[9]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-Schechter-3">[3]</a> Because of this, some deductive inferences have a general conclusion and some also have particular premises.<a href="#cite_note-Houde-2">[2]</a> <div class="mw-heading mw-heading2"><h2 id="In_various_fields">In various fields</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=12" title="Edit section: In various fields">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Cognitive_psychology">Cognitive psychology</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=13" title="Edit section: Cognitive psychology">edit</a>]</div> <a href="/wiki/Cognitive_psychology" title="Cognitive psychology">Cognitive psychology</a> studies the psychological processes responsible for deductive reasoning.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> It is concerned, among other things, with how good people are at drawing valid deductive inferences. This includes the study of the factors affecting their performance, their tendency to commit <a href="/wiki/Fallacies" class="mw-redirect" title="Fallacies">fallacies</a>, and the underlying <a href="/wiki/Cognitive_bias" title="Cognitive bias">biases</a> involved.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> A notable finding in this field is that the type of deductive inference has a significant impact on whether the correct conclusion is drawn.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-40">[40]</a><a href="#cite_note-41">[41]</a> In a meta-analysis of 65 studies, for example, 97% of the subjects evaluated <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a> inferences correctly, while the success rate for <a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a> was only 72%. On the other hand, even some fallacies like <a href="/wiki/Affirming_the_consequent" title="Affirming the consequent">affirming the consequent</a> or <a href="/wiki/Denying_the_antecedent" title="Denying the antecedent">denying the antecedent</a> were regarded as valid arguments by the majority of the subjects.<a href="#cite_note-Schechter-3">[3]</a> An important factor for these mistakes is whether the conclusion seems initially plausible: the more believable the conclusion is, the higher the chance that a subject will mistake a fallacy for a valid argument.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> An important bias is the matching bias, which is often illustrated using the <a href="/wiki/Wason_selection_task" title="Wason selection task">Wason selection task</a>.<a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-42">[42]</a><a href="#cite_note-43">[43]</a> In an often-cited experiment by <a href="/wiki/Peter_Wason" class="mw-redirect" title="Peter Wason">Peter Wason</a>, 4 cards are presented to the participant. In one case, the visible sides show the symbols D, K, 3, and 7 on the different cards. The participant is told that every card has a letter on one side and a number on the other side, and that "[e]very card which has a D on one side has a 3 on the other side". Their task is to identify which cards need to be turned around in order to confirm or refute this conditional claim. The correct answer, only given by about 10%, is the cards D and 7. Many select card 3 instead, even though the conditional claim does not involve any requirements on what symbols can be found on the opposite side of card 3.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> But this result can be drastically changed if different symbols are used: the visible sides show "drinking a beer", "drinking a coke", "16 years of age", and "22 years of age" and the participants are asked to evaluate the claim "[i]f a person is drinking beer, then the person must be over 19 years of age". In this case, 74% of the participants identified correctly that the cards "drinking a beer" and "16 years of age" have to be turned around.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> These findings suggest that the deductive reasoning ability is heavily influenced by the content of the involved claims and not just by the abstract logical form of the task: the more realistic and concrete the cases are, the better the subjects tend to perform.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Evans-5">[5]</a> Another bias is called the "negative conclusion bias", which happens when one of the premises has the form of a negative <a href="/wiki/Material_conditional" title="Material conditional">material conditional</a>,<a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-44">[44]</a><a href="#cite_note-45">[45]</a> as in "If the card does not have an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card has an A on the left". The increased tendency to misjudge the validity of this type of argument is not present for positive material conditionals, as in "If the card has an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card does not have an A on the left".<a href="#cite_note-Evans-5">[5]</a> <div class="mw-heading mw-heading4"><h4 id="Psychological_theories_of_deductive_reasoning">Psychological theories of deductive reasoning</h4>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=14" title="Edit section: Psychological theories of deductive reasoning">edit</a>]</div> Various psychological theories of deductive reasoning have been proposed. These theories aim to explain how deductive reasoning works in relation to the underlying psychological processes responsible. They are often used to explain the empirical findings, such as why human reasoners are more susceptible to some types of fallacies than to others.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-Johnson-Laird1993-46">[46]</a> An important distinction is between mental logic theories, sometimes also referred to as rule theories, and mental model theories. Mental logic theories see deductive reasoning as a <a href="/wiki/Language" title="Language">language</a>-like process that happens through the manipulation of representations.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-García-Madruga-47">[47]</a><a href="#cite_note-Johnson-Laird1993-46">[46]</a> This is done by applying syntactic rules of inference in a way very similar to how systems of <a href="#Natural_deduction">natural deduction</a> transform their premises to arrive at a conclusion.<a href="#cite_note-Johnson-Laird1993-46">[46]</a> On this view, some deductions are simpler than others since they involve fewer inferential steps.<a href="#cite_note-Schechter-3">[3]</a> This idea can be used, for example, to explain why humans have more difficulties with some deductions, like the <a href="/wiki/Modus_tollens" title="Modus tollens">modus tollens</a>, than with others, like the <a href="/wiki/Modus_ponens" title="Modus ponens">modus ponens</a>: because the more error-prone forms do not have a native rule of inference but need to be calculated by combining several inferential steps with other rules of inference. In such cases, the additional cognitive labor makes the inferences more open to error.<a href="#cite_note-Schechter-3">[3]</a> Mental model theories, on the other hand, hold that deductive reasoning involves models or <a href="/wiki/Mental_representation" title="Mental representation">mental representations</a> of possible states of the world without the medium of language or rules of inference.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-Johnson-Laird1993-46">[46]</a> In order to assess whether a deductive inference is valid, the reasoner mentally constructs models that are compatible with the premises of the inference. The conclusion is then tested by looking at these models and trying to find a counterexample in which the conclusion is false. The inference is valid if no such counterexample can be found.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a><a href="#cite_note-Johnson-Laird1993-46">[46]</a> In order to reduce cognitive labor, only such models are represented in which the premises are true. Because of this, the evaluation of some forms of inference only requires the construction of very few models while for others, many different models are necessary. In the latter case, the additional cognitive labor required makes deductive reasoning more error-prone, thereby explaining the increased rate of error observed.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-Johnson-Laird2009-1">[1]</a> This theory can also explain why some errors depend on the content rather than the form of the argument. For example, when the conclusion of an argument is very plausible, the subjects may lack the motivation to search for counterexamples among the constructed models.<a href="#cite_note-Schechter-3">[3]</a> Both mental logic theories and mental model theories assume that there is one general-purpose reasoning mechanism that applies to all forms of deductive reasoning.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-García-Madruga-47">[47]</a><a href="#cite_note-48">[48]</a> But there are also alternative accounts that posit various different special-purpose reasoning mechanisms for different contents and contexts. In this sense, it has been claimed that humans possess a special mechanism for permissions and obligations, specifically for detecting cheating in social exchanges. This can be used to explain why humans are often more successful in drawing valid inferences if the contents involve human behavior in relation to social norms.<a href="#cite_note-Schechter-3">[3]</a> Another example is the so-called <a href="/wiki/Dual_process_theory" title="Dual process theory">dual-process theory</a>.<a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-Schechter-3">[3]</a> This theory posits that there are two distinct cognitive systems responsible for reasoning. Their interrelation can be used to explain commonly observed biases in deductive reasoning. System 1 is the older system in terms of evolution. It is based on associative learning and happens fast and automatically without demanding many cognitive resources.<a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-Schechter-3">[3]</a> System 2, on the other hand, is of more recent evolutionary origin. It is slow and cognitively demanding, but also more flexible and under deliberate control.<a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-Schechter-3">[3]</a> The dual-process theory posits that system 1 is the default system guiding most of our everyday reasoning in a pragmatic way. But for particularly difficult problems on the logical level, system 2 is employed. System 2 is mostly responsible for deductive reasoning.<a href="#cite_note-Evans-5">[5]</a><a href="#cite_note-Schechter-3">[3]</a> <div class="mw-heading mw-heading4"><h4 id="Intelligence">Intelligence</h4>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=15" title="Edit section: Intelligence">edit</a>]</div> The <a href="/wiki/Ability" title="Ability">ability</a> of deductive reasoning is an important aspect of <a href="/wiki/Intelligence" title="Intelligence">intelligence</a> and many <a href="/wiki/Intelligence_test" class="mw-redirect" title="Intelligence test">tests of intelligence</a> include problems that call for deductive inferences.<a href="#cite_note-Johnson-Laird2009-1">[1]</a> Because of this relation to intelligence, deduction is highly relevant to psychology and the cognitive sciences.<a href="#cite_note-Evans-5">[5]</a> But the subject of deductive reasoning is also pertinent to the <a href="/wiki/Computer_sciences" class="mw-redirect" title="Computer sciences">computer sciences</a>, for example, in the creation of <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>.<a href="#cite_note-Johnson-Laird2009-1">[1]</a> <div class="mw-heading mw-heading3"><h3 id="Epistemology">Epistemology</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=16" title="Edit section: Epistemology">edit</a>]</div> Deductive reasoning plays an important role in <a href="/wiki/Epistemology" title="Epistemology">epistemology</a>. Epistemology is concerned with the question of <a href="/wiki/Justification_(epistemology)" title="Justification (epistemology)">justification</a>, i.e. to point out which beliefs are justified and why.<a href="#cite_note-49">[49]</a><a href="#cite_note-50">[50]</a> Deductive inferences are able to transfer the justification of the premises onto the conclusion.<a href="#cite_note-Schechter-3">[3]</a> So while logic is interested in the truth-preserving nature of deduction, epistemology is interested in the justification-preserving nature of deduction. There are different theories trying to explain why deductive reasoning is justification-preserving.<a href="#cite_note-Schechter-3">[3]</a> According to <a href="/wiki/Reliabilism" title="Reliabilism">reliabilism</a>, this is the case because deductions are truth-preserving: they are reliable processes that ensure a true conclusion given the premises are true.<a href="#cite_note-Schechter-3">[3]</a><a href="#cite_note-51">[51]</a><a href="#cite_note-52">[52]</a> Some theorists hold that the thinker has to have explicit awareness of the truth-preserving nature of the inference for the justification to be transferred from the premises to the conclusion. One consequence of such a view is that, for young children, this deductive transference does not take place since they lack this specific awareness.<a href="#cite_note-Schechter-3">[3]</a> <div class="mw-heading mw-heading3"><h3 id="Probability_logic">Probability logic</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=17" title="Edit section: Probability logic">edit</a>]</div> <a href="/wiki/Probability_logic" class="mw-redirect" title="Probability logic">Probability logic</a> is interested in how the probability of the premises of an argument affects the probability of its conclusion. It differs from classical logic, which assumes that propositions are either true or false but does not take into consideration the probability or certainty that a proposition is true or false.<a href="#cite_note-53">[53]</a><a href="#cite_note-54">[54]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=18" title="Edit section: History">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></td><td class="mbox-text"><div class="mbox-text-span">This section needs expansion. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Deductive_reasoning&action=edit&section=">adding to it</a>. (January 2015)</div></td></tr></tbody></table> <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, a <a href="/wiki/Greek_philosopher" class="mw-redirect" title="Greek philosopher">Greek philosopher</a>, started documenting deductive reasoning in the 4th century BC.<a href="#cite_note-55">[55]</a> <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, in his book <a href="/wiki/Discourse_on_Method" class="mw-redirect" title="Discourse on Method">Discourse on Method</a>, refined the idea for the <a href="/wiki/Scientific_Revolution" title="Scientific Revolution">Scientific Revolution</a>. Developing four rules to follow for proving an idea deductively, Descartes laid the foundation for the deductive portion of the <a href="/wiki/Scientific_method" title="Scientific method">scientific method</a>. Descartes' background in geometry and mathematics influenced his ideas on the truth and reasoning, causing him to develop a system of general reasoning now used for most mathematical reasoning. Similar to postulates, Descartes believed that ideas could be self-evident and that reasoning alone must prove that observations are reliable. These ideas also lay the foundations for the ideas of <a href="/wiki/Rationalism" title="Rationalism">rationalism</a>.<a href="#cite_note-56">[56]</a> <div class="mw-heading mw-heading2"><h2 id="Related_concepts_and_theories">Related concepts and theories</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=19" title="Edit section: Related concepts and theories">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Deductivism">Deductivism</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=20" title="Edit section: Deductivism">edit</a>]</div> Deductivism is a philosophical position that gives primacy to deductive reasoning or arguments over their non-deductive counterparts.<a href="#cite_note-Bermejo-Luque-57">[57]</a><a href="#cite_note-Howson-58">[58]</a> It is often understood as the evaluative claim that only deductive inferences are good or correct inferences. This theory would have wide-reaching consequences for various fields since it implies that the rules of deduction are "the only acceptable standard of <a href="/wiki/Evidence" title="Evidence">evidence</a>".<a href="#cite_note-Bermejo-Luque-57">[57]</a> This way, the rationality or correctness of the different forms of inductive reasoning is denied.<a href="#cite_note-Howson-58">[58]</a><a href="#cite_note-59">[59]</a> Some forms of deductivism express this in terms of degrees of reasonableness or probability. Inductive inferences are usually seen as providing a certain degree of support for their conclusion: they make it more likely that their conclusion is true. Deductivism states that such inferences are not rational: the premises either ensure their conclusion, as in deductive reasoning, or they do not provide any support at all.<a href="#cite_note-Stove-60">[60]</a> One motivation for deductivism is the <a href="/wiki/Problem_of_induction" title="Problem of induction">problem of induction</a> introduced by <a href="/wiki/David_Hume" title="David Hume">David Hume</a>. It consists in the challenge of explaining how or whether inductive inferences based on past experiences support conclusions about future events.<a href="#cite_note-Howson-58">[58]</a><a href="#cite_note-61">[61]</a><a href="#cite_note-Stove-60">[60]</a> For example, a chicken comes to expect, based on all its past experiences, that the person entering its coop is going to feed it, until one day the person "at last wrings its neck instead".<a href="#cite_note-62">[62]</a> According to <a href="/wiki/Karl_Popper" title="Karl Popper">Karl Popper</a>'s falsificationism, deductive reasoning alone is sufficient. This is due to its truth-preserving nature: a theory can be falsified if one of its deductive consequences is false.<a href="#cite_note-63">[63]</a><a href="#cite_note-64">[64]</a> So while inductive reasoning does not offer positive evidence for a theory, the theory still remains a viable competitor until falsified by <a href="/wiki/Empirical_evidence" title="Empirical evidence">empirical observation</a>. In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what is the case.<a href="#cite_note-Howson-58">[58]</a> <a href="/wiki/Hypothetico-deductivism" class="mw-redirect" title="Hypothetico-deductivism">Hypothetico-deductivism</a> is a closely related scientific method, according to which science progresses by formulating hypotheses and then aims to falsify them by trying to make observations that run counter to their deductive consequences.<a href="#cite_note-65">[65]</a><a href="#cite_note-66">[66]</a> <div class="mw-heading mw-heading3"><h3 id="Natural_deduction">Natural deduction</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=21" title="Edit section: Natural deduction">edit</a>]</div> The term "<a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a>" refers to a class of proof systems based on self-evident rules of inference.<a href="#cite_note-IEPNatural-67">[67]</a><a href="#cite_note-StanfordNatural-68">[68]</a> The first systems of natural deduction were developed by <a href="/wiki/Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> and <a href="/wiki/Stanislaw_Jaskowski" class="mw-redirect" title="Stanislaw Jaskowski">Stanislaw Jaskowski</a> in the 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place.<a href="#cite_note-69">[69]</a> In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as <a href="/wiki/Hilbert-style_deductive_systems" class="mw-redirect" title="Hilbert-style deductive systems">Hilbert-style deductive systems</a>, which employ axiom schemes to express <a href="/wiki/Logical_truth" title="Logical truth">logical truths</a>.<a href="#cite_note-IEPNatural-67">[67]</a> Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how <a href="/wiki/Logical_constant" title="Logical constant">logical constants</a> behave. They are often divided into <a href="/wiki/Natural_deduction#Introduction_and_elimination" title="Natural deduction">introduction rules and elimination rules</a>. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the <a href="/wiki/Formal_proof" title="Formal proof">proof</a>.<a href="#cite_note-IEPNatural-67">[67]</a><a href="#cite_note-StanfordNatural-68">[68]</a> For example, the introduction rule for the logical constant "<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 }">" (and) is "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {A,B}{(A\land B)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {A,B}{(A\land B)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/019e90567fc8bd3222ad7427ff56d7947a1a3ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.735ex; height:6.176ex;" alt="{\displaystyle {\frac {A,B}{(A\land B)}}}">". It expresses that, given the premises "<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}">" and "<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}">" individually, one may draw the conclusion "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\land B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∧</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\land B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74954195333a8593163b93a9688695b8dc74da55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.09ex; height:2.176ex;" alt="{\displaystyle A\land B}">" and thereby include it in one's proof. This way, the symbol "<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 }">" is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(A\land B)}{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>∧</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(A\land B)}{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34242edbeddf1d09bc36d6c524705ec7c511aa73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.735ex; height:5.843ex;" alt="{\displaystyle {\frac {(A\land B)}{A}}}">", which states that one may deduce the sentence "<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}">" from the premise "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\land B)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\land B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdcd2d1d13bc1f915aa141415965509a4e2b4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.899ex; height:2.843ex;" alt="{\displaystyle (A\land B)}">". Similar introduction and elimination rules are given for other logical constants, such as the propositional operator "<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 }">", the <a href="/wiki/Logical_connective" title="Logical connective">propositional connectives</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 }">" and "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53e574cc3aa5b4bf5f3f5906caf121a378eef08b" 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 \rightarrow }">", and the <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifiers</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 }">" and "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937" 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 \forall }">".<a href="#cite_note-IEPNatural-67">[67]</a><a href="#cite_note-StanfordNatural-68">[68]</a> The focus on rules of inferences instead of axiom schemes is an important feature of natural deduction.<a href="#cite_note-IEPNatural-67">[67]</a><a href="#cite_note-StanfordNatural-68">[68]</a> But there is no general agreement on how natural deduction is to be defined. Some theorists hold that all proof systems with this feature are forms of natural deduction. This would include various forms of <a href="/wiki/Sequent_calculi" class="mw-redirect" title="Sequent calculi">sequent calculi</a><a href="#cite_note-natDeduc-70">[a]</a> or <a href="/wiki/Method_of_analytic_tableaux#Tableau_calculi_and_their_properties" title="Method of analytic tableaux">tableau calculi</a>. But other theorists use the term in a more narrow sense, for example, to refer to the proof systems developed by Gentzen and Jaskowski. Because of its simplicity, natural deduction is often used for teaching logic to students.<a href="#cite_note-IEPNatural-67">[67]</a> <div class="mw-heading mw-heading3"><h3 id="Geometrical_method">Geometrical method</h3>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=22" title="Edit section: Geometrical method">edit</a>]</div> The geometrical method is a method of <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> based on deductive reasoning. It starts from a small set of <a href="/wiki/Self-evident" class="mw-redirect" title="Self-evident">self-evident</a> axioms and tries to build a comprehensive logical system based only on deductive inferences from these first <a href="/wiki/Axiom" title="Axiom">axioms</a>.<a href="#cite_note-DalyHandbook-71">[70]</a> It was initially formulated by <a href="/wiki/Baruch_Spinoza" title="Baruch Spinoza">Baruch Spinoza</a> and came to prominence in various <a href="/wiki/Rationalist" class="mw-redirect" title="Rationalist">rationalist</a> philosophical systems in the modern era.<a href="#cite_note-72">[71]</a> It gets its name from the forms of <a href="/wiki/Mathematical_proof" title="Mathematical proof">mathematical demonstration</a> found in traditional <a href="/wiki/Geometry" title="Geometry">geometry</a>, which are usually based on axioms, <a href="/wiki/Definition" title="Definition">definitions</a>, and inferred <a href="/wiki/Theorem" title="Theorem">theorems</a>.<a href="#cite_note-73">[72]</a><a href="#cite_note-74">[73]</a> An important motivation of the geometrical method is to repudiate <a href="/wiki/Philosophical_skepticism" title="Philosophical skepticism">philosophical skepticism</a> by grounding one's philosophical system on absolutely certain axioms. Deductive reasoning is central to this endeavor because of its necessarily truth-preserving nature. This way, the certainty initially invested only in the axioms is transferred to all parts of the philosophical system.<a href="#cite_note-DalyHandbook-71">[70]</a> One recurrent criticism of philosophical systems build using the geometrical method is that their initial axioms are not as self-evident or certain as their defenders proclaim.<a href="#cite_note-DalyHandbook-71">[70]</a> This problem lies beyond the deductive reasoning itself, which only ensures that the conclusion is true if the premises are true, but not that the premises themselves are true. For example, Spinoza's philosophical system has been criticized this way based on objections raised against the <a href="/wiki/Causal" class="mw-redirect" title="Causal">causal</a> axiom, i.e. that "the knowledge of an effect depends on and involves knowledge of its cause".<a href="#cite_note-75">[74]</a> A different criticism targets not the premises but the reasoning itself, which may at times implicitly assume premises that are themselves not self-evident.<a href="#cite_note-DalyHandbook-71">[70]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=23" 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 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href="/wiki/Subjective_logic" title="Subjective logic">Subjective logic</a></li> <li><a href="/wiki/Theory_of_justification" class="mw-redirect" title="Theory of justification">Theory of justification</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=24" title="Edit section: Notes and references">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-natDeduc-70"><a href="#cite_ref-natDeduc_70-0">^</a> In natural deduction, a simplified <a href="/wiki/Sequent" title="Sequent">sequent</a> consists of an environment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"> that yields (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdash }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⊢</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdash }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c0d30cf8cb7dba179e317fcde9583d842e80f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \vdash }">) a single conclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}">; a single sequent would take the form <dl><dd>"Assumptions A1, A2, A3 etc. yield Conclusion C1"; in the symbols of <a href="/wiki/Natural_deduction" title="Natural deduction">natural deduction</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma A_{1},A_{2},A_{3}...\vdash C_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>⊢</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma A_{1},A_{2},A_{3}...\vdash C_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be29100a3c78150f48c4998b2c9f9ce3aa9af3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.796ex; height:2.509ex;" alt="{\displaystyle \Gamma A_{1},A_{2},A_{3}...\vdash C_{1}}"></dd></dl> <ul><li>However if the premises were true but the conclusion were false, a hidden assumption could be intervening; alternatively, a hidden process might be coercing the form of presentation, and so forth; then the task would be to unearth the hidden factors in an ill-formed syllogism, in order to make the form valid.</li> <li>see <a href="/wiki/Deduction_theorem" title="Deduction theorem">Deduction theorem</a></li></ul> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Johnson-Laird2009-1">^ <a href="#cite_ref-Johnson-Laird2009_1-0">a</a> <a href="#cite_ref-Johnson-Laird2009_1-1">b</a> <a href="#cite_ref-Johnson-Laird2009_1-2">c</a> <a href="#cite_ref-Johnson-Laird2009_1-3">d</a> <a href="#cite_ref-Johnson-Laird2009_1-4">e</a> <a href="#cite_ref-Johnson-Laird2009_1-5">f</a> <a href="#cite_ref-Johnson-Laird2009_1-6">g</a> <a href="#cite_ref-Johnson-Laird2009_1-7">h</a> <a href="#cite_ref-Johnson-Laird2009_1-8">i</a> <a href="#cite_ref-Johnson-Laird2009_1-9">j</a> <a href="#cite_ref-Johnson-Laird2009_1-10">k</a> <a 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Retrieved 19 March 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Epistemology&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2020&rft.aulast=Steup&rft.aufirst=Matthias&rft.au=Neta%2C+Ram&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fepistemology%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecker" class="citation encyclopaedia cs1">Becker, Kelly. <a rel="nofollow" class="external text" href="https://iep.utm.edu/reliabil/">"Reliabilism"</a>. Internet Encyclopedia of Philosophy. 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Retrieved 15 March 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Natural+Deduction&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft.aulast=Indrzejczak&rft.aufirst=Andrzej&rft_id=https%3A%2F%2Fiep.utm.edu%2Fnatural-deduction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-StanfordNatural-68">^ <a href="#cite_ref-StanfordNatural_68-0">a</a> <a href="#cite_ref-StanfordNatural_68-1">b</a> <a href="#cite_ref-StanfordNatural_68-2">c</a> <a href="#cite_ref-StanfordNatural_68-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPelletierHazen2021" class="citation encyclopaedia cs1">Pelletier, Francis Jeffry; Hazen, Allen (2021). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/natural-deduction/">"Natural Deduction Systems in Logic"</a>. The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 15 March 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Natural+Deduction+Systems+in+Logic&rft.btitle=The+Stanford+Encyclopedia+of+Philosophy&rft.pub=Metaphysics+Research+Lab%2C+Stanford+University&rft.date=2021&rft.aulast=Pelletier&rft.aufirst=Francis+Jeffry&rft.au=Hazen%2C+Allen&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fnatural-deduction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-69"><a href="#cite_ref-69">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGentzen1934" class="citation journal cs1 cs1-prop-foreign-lang-source">Gentzen, Gerhard (1934). <a rel="nofollow" class="external text" href="https://gdz.sub.uni-goettingen.de/id/PPN266833020_0039?tify={%22pages%22:%5b180%5d,%22panX%22:0.559,%22panY%22:0.785,%22view%22:%22info%22,%22zoom%22:0.411}">"Untersuchungen über das logische Schließen. I"</a>. Mathematische Zeitschrift (in German). 39 (2): 176–210. <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%2FBF01201353">10.1007/BF01201353</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:121546341">121546341</a>. <q>Ich wollte nun zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens. (First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Zeitschrift&rft.atitle=Untersuchungen+%C3%BCber+das+logische+Schlie%C3%9Fen.+I&rft.volume=39&rft.issue=2&rft.pages=176-210&rft.date=1934&rft_id=info%3Adoi%2F10.1007%2FBF01201353&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121546341%23id-name%3DS2CID&rft.aulast=Gentzen&rft.aufirst=Gerhard&rft_id=https%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fid%2FPPN266833020_0039%3Ftify%3D%7B%2522pages%2522%3A%5B180%5D%2C%2522panX%2522%3A0.559%2C%2522panY%2522%3A0.785%2C%2522view%2522%3A%2522info%2522%2C%2522zoom%2522%3A0.411%7D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-DalyHandbook-71">^ <a href="#cite_ref-DalyHandbook_71-0">a</a> <a href="#cite_ref-DalyHandbook_71-1">b</a> <a href="#cite_ref-DalyHandbook_71-2">c</a> <a href="#cite_ref-DalyHandbook_71-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaly2015" class="citation book cs1">Daly, Chris (2015). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1057/9781137344557_1">"Introduction and Historical Overview"</a>. The Palgrave Handbook of Philosophical Methods. Palgrave Macmillan. pp. 1–30. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1057%2F9781137344557_1">10.1057/9781137344557_1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-137-34455-7" title="Special:BookSources/978-1-137-34455-7"><bdi>978-1-137-34455-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Introduction+and+Historical+Overview&rft.btitle=The+Palgrave+Handbook+of+Philosophical+Methods&rft.pages=1-30&rft.pub=Palgrave+Macmillan&rft.date=2015&rft_id=info%3Adoi%2F10.1057%2F9781137344557_1&rft.isbn=978-1-137-34455-7&rft.aulast=Daly&rft.aufirst=Chris&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1057%2F9781137344557_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDutton" class="citation web cs1">Dutton, Blake D. <a rel="nofollow" class="external text" href="https://iep.utm.edu/spinoza/#H2">"Spinoza, Benedict De"</a>. Internet Encyclopedia of Philosophy. Retrieved 16 March 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=Spinoza%2C+Benedict+De&rft.aulast=Dutton&rft.aufirst=Blake+D.&rft_id=https%3A%2F%2Fiep.utm.edu%2Fspinoza%2F%23H2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-73"><a href="#cite_ref-73">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldenbaum" class="citation encyclopaedia cs1">Goldenbaum, Ursula. <a rel="nofollow" class="external text" href="https://iep.utm.edu/geo-meth/">"Geometrical Method"</a>. Internet Encyclopedia of Philosophy. Retrieved 17 February 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Geometrical+Method&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft.aulast=Goldenbaum&rft.aufirst=Ursula&rft_id=https%3A%2F%2Fiep.utm.edu%2Fgeo-meth%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNadler2006" class="citation book cs1">Nadler, Steven (2006). "The geometric method". <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/abs/spinozas-ethics/geometric-method/08550AF622C78ACC388069710D37036E">Spinoza's 'Ethics': An Introduction</a>. Cambridge University Press. pp. 35–51. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-83620-3" title="Special:BookSources/978-0-521-83620-3"><bdi>978-0-521-83620-3</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+geometric+method&rft.btitle=Spinoza%27s+%27Ethics%27%3A+An+Introduction&rft.pages=35-51&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-83620-3&rft.aulast=Nadler&rft.aufirst=Steven&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fabs%2Fspinozas-ethics%2Fgeometric-method%2F08550AF622C78ACC388069710D37036E&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDoppelt2010" class="citation book cs1">Doppelt, Torin (2010). "The Truth About 1A4". <a rel="nofollow" class="external text" href="https://qspace.library.queensu.ca/bitstream/handle/1974/6052/Doppelt_Torin_201009_MA.pdf">Spinoza's Causal Axiom: A Defense</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Truth+About+1A4&rft.btitle=Spinoza%27s+Causal+Axiom%3A+A+Defense&rft.date=2010&rft.aulast=Doppelt&rft.aufirst=Torin&rft_id=https%3A%2F%2Fqspace.library.queensu.ca%2Fbitstream%2Fhandle%2F1974%2F6052%2FDoppelt_Torin_201009_MA.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=25" title="Edit section: Further reading">edit</a>]</div> <ul><li><a href="/wiki/Vincent_F._Hendricks" title="Vincent F. Hendricks">Vincent F. Hendricks</a>, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/87-991013-7-8" title="Special:BookSources/87-991013-7-8">87-991013-7-8</a></li> <li><a href="/wiki/Philip_Johnson-Laird" title="Philip Johnson-Laird">Philip Johnson-Laird</a>, <a href="/wiki/Ruth_M._J._Byrne" title="Ruth M. J. Byrne">Ruth M. J. Byrne</a>, Deduction, Psychology Press 1991, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-86377-149-1" title="Special:BookSources/978-0-86377-149-1">978-0-86377-149-1</a></li> <li>Zarefsky, David, Argumentation: The Study of Effective Reasoning Parts I and II, The Teaching Company 2002</li> <li>Bullemore, Thomas. <a rel="nofollow" class="external text" href="https://www.academia.edu/4154895/Some_Remarks_on_the_Pragmatic_Problem_of_Induction.html">The Pragmatic Problem of Induction</a>[<a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot">permanent dead link</a>‍].</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Deductive_reasoning&action=edit&section=26" title="Edit section: External links">edit</a>]</div> <style 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title="PhilPapers">PhilPapers</a></li> <li><a rel="nofollow" class="external text" href="https://www.inphoproject.org/idea/636">Deductive reasoning</a> at the <a href="/wiki/Indiana_Philosophy_Ontology_Project" class="mw-redirect" title="Indiana Philosophy Ontology Project">Indiana Philosophy Ontology Project</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="http://www.iep.utm.edu/ded-ind">"Deductive reasoning"</a>. <a href="/wiki/Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Deductive+reasoning&rft.btitle=Internet+Encyclopedia+of+Philosophy&rft_id=http%3A%2F%2Fwww.iep.utm.edu%2Fded-ind&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeductive+reasoning" 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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" 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title="Antinomy">Antinomy</a></li></ul></li> <li><a class="mw-selflink selflink">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 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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" 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0.25em"> <ul><li><a href="/wiki/Philosophical_analysis" title="Philosophical analysis">Analysis</a></li> <li><a href="/wiki/Ambiguity" title="Ambiguity">Ambiguity</a></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Belief" title="Belief">Belief</a></li> <li><a href="/wiki/Bias" title="Bias">Bias</a></li> <li><a href="/wiki/Credibility" title="Credibility">Credibility</a></li> <li><a href="/wiki/Dialectic" title="Dialectic">Dialectic</a> <ul><li><a href="/wiki/Antithesis" title="Antithesis">Antithesis</a>, <a href="/wiki/Socratic_method" title="Socratic method">Socratic method</a>, <a href="/wiki/Unity_of_opposites" title="Unity of opposites">Unity of opposites</a></li></ul></li> <li><a href="/wiki/Evidence" title="Evidence">Evidence</a></li> <li><a href="/wiki/Explanation" title="Explanation">Explanation</a></li> <li><a href="/wiki/Explanatory_power" title="Explanatory power">Explanatory power</a></li> <li><a href="/wiki/Fact" 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