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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logical_consequence"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Logical consequence</span> </div> </a> <ul id="toc-Logical_consequence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Logical_notions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Logical_notions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Logical notions</span> </div> </a> <ul id="toc-Logical_notions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Selected_publications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Selected_publications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Selected publications</span> </div> </a> <ul id="toc-Selected_publications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-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"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </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"> <span class="vector-toc-numb">8</span> <span>External links</span> </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" title="Table of Contents" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Alfred Tarski</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 48 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-48" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">48 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Afrikaans" lang="af" hreflang="af" data-title="Alfred Tarski" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D9%84%D9%81%D8%B1%D9%8A%D8%AF_%D8%AA%D8%A7%D8%B1%D8%B3%D9%83%D9%8A" title="ألفريد تارسكي – Arabic" lang="ar" hreflang="ar" data-title="ألفريد تارسكي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A2%D9%84%D9%81%D8%B1%D8%AF_%D8%AA%D8%A7%D8%B1%D8%B3%DA%A9%DB%8C" title="آلفرد تارسکی – South Azerbaijani" lang="azb" hreflang="azb" data-title="آلفرد تارسکی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BB%D1%8C%D1%84%D1%80%D1%8D%D0%B4_%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D1%96" title="Альфрэд Тарскі – Belarusian" lang="be" hreflang="be" data-title="Альфрэд Тарскі" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D1%84%D1%80%D0%B5%D0%B4_%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8" title="Алфред Тарски – Bulgarian" lang="bg" hreflang="bg" data-title="Алфред Тарски" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Catalan" lang="ca" hreflang="ca" data-title="Alfred Tarski" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Czech" lang="cs" hreflang="cs" data-title="Alfred Tarski" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – German" lang="de" hreflang="de" data-title="Alfred Tarski" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Estonian" lang="et" hreflang="et" data-title="Alfred Tarski" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%86%CE%BB%CF%86%CF%81%CE%B5%CE%BD%CF%84_%CE%A4%CE%AC%CF%81%CF%83%CE%BA%CE%B9" title="Άλφρεντ Τάρσκι – Greek" lang="el" hreflang="el" data-title="Άλφρεντ Τάρσκι" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Spanish" lang="es" hreflang="es" data-title="Alfred Tarski" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Esperanto" lang="eo" hreflang="eo" data-title="Alfred Tarski" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Basque" lang="eu" hreflang="eu" data-title="Alfred Tarski" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A2%D9%84%D9%81%D8%B1%D8%AF_%D8%AA%D8%A7%D8%B1%D8%B3%DA%A9%DB%8C" title="آلفرد تارسکی – Persian" lang="fa" hreflang="fa" data-title="آلفرد تارسکی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – French" lang="fr" hreflang="fr" data-title="Alfred Tarski" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Galician" lang="gl" hreflang="gl" data-title="Alfred Tarski" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%8C%ED%94%84%EB%A0%88%ED%8A%B8_%ED%83%80%EB%A5%B4%EC%8A%A4%ED%82%A4" title="알프레트 타르스키 – Korean" lang="ko" hreflang="ko" data-title="알프레트 타르스키" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%AC%D6%86%D6%80%D5%A5%D5%A4_%D5%8F%D5%A1%D6%80%D5%BD%D5%AF%D5%AB" title="Ալֆրեդ Տարսկի – Armenian" lang="hy" hreflang="hy" data-title="Ալֆրեդ Տարսկի" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Croatian" lang="hr" hreflang="hr" data-title="Alfred Tarski" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Indonesian" lang="id" hreflang="id" data-title="Alfred Tarski" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Icelandic" lang="is" hreflang="is" data-title="Alfred Tarski" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Italian" lang="it" hreflang="it" data-title="Alfred Tarski" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%A4%D7%A8%D7%93_%D7%98%D7%A8%D7%A1%D7%A7%D7%99" title="אלפרד טרסקי – Hebrew" lang="he" hreflang="he" data-title="אלפרד טרסקי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8%D0%B9_%D0%90%D0%BB%D1%8C%D1%84%D1%80%D0%B5%D0%B4" title="Тарский Альфред – Kyrgyz" lang="ky" hreflang="ky" data-title="Тарский Альфред" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Alfreds_Tarskis" title="Alfreds Tarskis – Latvian" lang="lv" hreflang="lv" data-title="Alfreds Tarskis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Hungarian" lang="hu" hreflang="hu" data-title="Alfred Tarski" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D0%BB%D1%84%D1%80%D0%B5%D0%B4_%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8" title="Алфред Тарски – Macedonian" lang="mk" hreflang="mk" data-title="Алфред Тарски" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B5%BD%E0%B4%AB%E0%B5%8D%E0%B4%B0%E0%B4%A1%E0%B5%8D_%E0%B4%9F%E0%B4%BE%E0%B5%BC%E0%B4%B8%E0%B5%8D%E0%B4%95%E0%B4%BF" title="ആൽഫ്രഡ് ടാർസ്കി – Malayalam" lang="ml" hreflang="ml" data-title="ആൽഫ്രഡ് ടാർസ്കി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D8%A7%D9%84%D9%81%D8%B1%D9%8A%D8%AF_%D8%AA%D8%A7%D8%B1%D8%B3%D9%83%D9%89" title="الفريد تارسكى – Egyptian Arabic" lang="arz" hreflang="arz" data-title="الفريد تارسكى" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Dutch" lang="nl" hreflang="nl" data-title="Alfred Tarski" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%AB%E3%83%95%E3%83%AC%E3%83%88%E3%83%BB%E3%82%BF%E3%83%AB%E3%82%B9%E3%82%AD" title="アルフレト・タルスキ – Japanese" lang="ja" hreflang="ja" data-title="アルフレト・タルスキ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Alfred Tarski" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Uzbek" lang="uz" hreflang="uz" data-title="Alfred Tarski" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Piedmontese" lang="pms" hreflang="pms" data-title="Alfred Tarski" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Polish" lang="pl" hreflang="pl" data-title="Alfred Tarski" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Portuguese" lang="pt" hreflang="pt" data-title="Alfred Tarski" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Romanian" lang="ro" hreflang="ro" data-title="Alfred Tarski" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8%D0%B9,_%D0%90%D0%BB%D1%8C%D1%84%D1%80%D0%B5%D0%B4" title="Тарский, Альфред – Russian" lang="ru" hreflang="ru" data-title="Тарский, Альфред" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Slovak" lang="sk" hreflang="sk" data-title="Alfred Tarski" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Slovenian" lang="sl" hreflang="sl" data-title="Alfred Tarski" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BB%D1%84%D1%80%D0%B5%D0%B4_%D0%A2%D0%B0%D1%80%D1%81%D0%BA%D0%B8" title="Алфред Тарски – Serbian" lang="sr" hreflang="sr" data-title="Алфред Тарски" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Finnish" lang="fi" hreflang="fi" data-title="Alfred Tarski" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Swedish" lang="sv" hreflang="sv" data-title="Alfred Tarski" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Turkish" lang="tr" hreflang="tr" data-title="Alfred Tarski" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D1%8C%D1%84%D1%80%D0%B5%D0%B4_%D0%A2%D0%B0%D1%80%D1%81%D1%8C%D0%BA%D0%B8%D0%B9" title="Альфред Тарський – Ukrainian" lang="uk" hreflang="uk" data-title="Альфред Тарський" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Alfred_Tarski" title="Alfred Tarski – Vietnamese" lang="vi" hreflang="vi" data-title="Alfred Tarski" data-language-autonym="Tiếng Việt" 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Polish–American mathematician (1901–1983)</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Alfred Tarski</div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:AlfredTarski1968.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/AlfredTarski1968.jpeg/220px-AlfredTarski1968.jpeg" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/AlfredTarski1968.jpeg/330px-AlfredTarski1968.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/7/71/AlfredTarski1968.jpeg 2x" data-file-width="400" data-file-height="271" /></a></span><div class="infobox-caption">Tarski in 1968</div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><div style="display:inline" class="nickname">Alfred Teitelbaum</div><br /><span style="display:none">(<span class="bday">1901-01-14</span>)</span>January 14, 1901<br /><div style="display:inline" class="birthplace"><a href="/wiki/Warsaw" title="Warsaw">Warsaw</a>, <a href="/wiki/Congress_Poland" title="Congress Poland">Congress Poland</a></div></td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">October 26, 1983<span style="display:none">(1983-10-26)</span> (aged 82)<br /><div style="display:inline" class="deathplace"><a href="/wiki/Berkeley,_California" title="Berkeley, California">Berkeley, California</a>, US</div></td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category">Polish, American</td></tr><tr><th scope="row" class="infobox-label">Education</th><td class="infobox-data"><a href="/wiki/University_of_Warsaw" title="University of Warsaw">University of Warsaw</a> (Ph.D., 1924)</td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li>Work on the foundations of modern <a href="/wiki/Logic" title="Logic">logic</a></li> <li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">Semantic theory of truth</a> (<a href="/wiki/Convention_T" class="mw-redirect" title="Convention T">Convention T</a>)</li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a></li> <li>Development of <a href="/wiki/Model_theory" title="Model theory">model theory</a></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Logic of relations</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li><a href="/wiki/Tarski%27s_fixed-point_theorem" class="mw-redirect" title="Tarski's fixed-point theorem">Tarski's fixed-point theorem</a></li> <li><a href="/wiki/Tarski-style_universes" class="mw-redirect" title="Tarski-style universes">Tarski-style universes</a></li> <li><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a></li> <li><a href="/wiki/Tarski_monster_group" title="Tarski monster group">Tarski monster group</a></li> <li><a href="/wiki/Tarski%27s_circle-squaring_problem" title="Tarski's circle-squaring problem">Tarski's circle-squaring problem</a></li> <li><a href="/wiki/Tarski%E2%80%93Kuratowski_algorithm" title="Tarski–Kuratowski algorithm">Tarski–Kuratowski algorithm</a></li> <li><a href="/wiki/J%C3%B3nsson%E2%80%93Tarski_duality" class="mw-redirect" title="Jónsson–Tarski duality">Jónsson–Tarski duality</a></li> <li><a href="/wiki/Lindenbaum%E2%80%93Tarski_algebra" title="Lindenbaum–Tarski algebra">Lindenbaum–Tarski algebra</a></li></ul> </div></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a>, <a href="/wiki/Logic" title="Logic">logic</a>, <a href="/wiki/Formal_language" title="Formal language">formal language</a></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li><a href="/wiki/University_of_Warsaw" title="University of Warsaw">University of Warsaw</a> (1925–1939)</li> <li><a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a> (1942–1983)</li></ul> </div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Thesis" title="Thesis">Thesis</a></th><td class="infobox-data"><i><a rel="nofollow" class="external text" href="http://hint.org.pl/hid=F14E6">O wyrazie pierwotnym logistyki (On the Primitive Term of Logistic)</a></i> <span style="font-size:97%;">(1924)</span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Stanis%C5%82aw_Le%C5%9Bniewski" title="Stanisław Leśniewski">Stanisław Leśniewski</a></td></tr><tr><th scope="row" class="infobox-label">Doctoral students</th><td class="infobox-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li><a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a></li> <li><a href="/wiki/Haim_Gaifman" title="Haim Gaifman">Haim Gaifman</a></li> <li><a href="/wiki/Bjarni_J%C3%B3nsson" title="Bjarni Jónsson">Bjarni Jónsson</a></li> <li><a href="/wiki/Howard_Jerome_Keisler" title="Howard Jerome Keisler">Howard Jerome Keisler</a></li> <li><a href="/wiki/Roger_Maddux" title="Roger Maddux">Roger Maddux</a></li> <li><a href="/wiki/Richard_Montague" title="Richard Montague">Richard Montague</a></li> <li><a href="/wiki/Anne_C._Morel" title="Anne C. Morel">Anne C. Morel</a></li> <li><a href="/wiki/Andrzej_Mostowski" title="Andrzej Mostowski">Andrzej Mostowski</a></li> <li><a href="/wiki/Julia_Robinson" title="Julia Robinson">Julia Robinson</a></li> <li><a href="/wiki/Wanda_Szmielew" title="Wanda Szmielew">Wanda Szmielew</a></li> <li><a href="/wiki/Robert_Vaught" class="mw-redirect" title="Robert Vaught">Robert Vaught</a></li></ul> </div></td></tr><tr><th scope="row" class="infobox-label">Other notable students</th><td class="infobox-data"><a href="/wiki/Evert_Willem_Beth" title="Evert Willem Beth">Evert Willem Beth</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Alfred Tarski</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="'t' in 'tie'">t</span><span title="/ɑːr/: 'ar' in 'far'">ɑːr</span><span title="'s' in 'sigh'">s</span><span title="'k' in 'kind'">k</span><span title="/i/: 'y' in 'happy'">i</span></span>/</a></span></span>; <style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">Polish:</span> <span class="IPA nowrap" lang="pl-Latn-fonipa"><a href="/wiki/Help:IPA/Polish" title="Help:IPA/Polish">[ˈtarskʲi]</a></span>; born <b>Alfred Teitelbaum</b>;<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Alfred_Tarski_2-0" class="reference"><a href="#cite_note-Alfred_Tarski-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-oxfordreference.com_3-0" class="reference"><a href="#cite_note-oxfordreference.com-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> January 14, 1901 – October 26, 1983) was a Polish-American<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Logic" title="Logic">logician</a> and <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A prolific author best known for his work on <a href="/wiki/Model_theory" title="Model theory">model theory</a>, <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a>, and <a href="/wiki/Algebraic_logic" title="Algebraic logic">algebraic logic</a>, he also contributed to <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, <a href="/wiki/Topology" title="Topology">topology</a>, <a href="/wiki/Geometry" title="Geometry">geometry</a>, <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, <a href="/wiki/Set_theory" title="Set theory">set theory</a>, and <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">analytic philosophy</a>. </p><p>Educated in Poland at the <a href="/wiki/University_of_Warsaw" title="University of Warsaw">University of Warsaw</a>, and a member of the <a href="/wiki/Lw%C3%B3w%E2%80%93Warsaw_school" title="Lwów–Warsaw school">Lwów–Warsaw school of logic</a> and the <a href="/wiki/Warsaw_school_of_mathematics" class="mw-redirect" title="Warsaw school of mathematics">Warsaw school of mathematics</a>, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a>, from 1942 until his death in 1983.<sup id="cite_ref-FefA_6-0" class="reference"><a href="#cite_note-FefA-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>His biographers <a href="/wiki/Anita_Burdman_Feferman" title="Anita Burdman Feferman">Anita Burdman Feferman</a> and <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a> state that, "Along with his contemporary, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>, he changed the face of logic in the twentieth century, especially through his work on the concept of <a href="/wiki/Truth" title="Truth">truth</a> and the theory of models."<sup id="cite_ref-FFintro_7-0" class="reference"><a href="#cite_note-FFintro-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Life">Life</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=1" title="Edit section: Life"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Early_life_and_education">Early life and education</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=2" title="Edit section: Early life and education"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alfred Tarski was born Alfred Teitelbaum (<a href="/wiki/Polish_language" title="Polish language">Polish</a> spelling: "Tajtelbaum"), to parents who were <a href="/wiki/Polish_Jews" class="mw-redirect" title="Polish Jews">Polish Jews</a> in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw's <i><a href="/w/index.php?title=Szko%C5%82a_Mazowiecka&action=edit&redlink=1" class="new" title="Szkoła Mazowiecka (page does not exist)">Szkoła Mazowiecka</a></i>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Nevertheless, he entered the <a href="/wiki/University_of_Warsaw" title="University of Warsaw">University of Warsaw</a> in 1918 intending to study <a href="/wiki/Biology" title="Biology">biology</a>.<sup id="cite_ref-Feferman_&_Feferman,_p.26_9-0" class="reference"><a href="#cite_note-Feferman_&_Feferman,_p.26-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>After Poland regained independence in 1918, Warsaw University came under the leadership of <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a>, <a href="/wiki/Stanis%C5%82aw_Le%C5%9Bniewski" title="Stanisław Leśniewski">Stanisław Leśniewski</a> and <a href="/wiki/Wac%C5%82aw_Sierpi%C5%84ski" title="Wacław Sierpiński">Wacław Sierpiński</a> and quickly became a world-leading research institution in logic, <a href="/wiki/Foundational_mathematics" class="mw-redirect" title="Foundational mathematics">foundational mathematics</a>, and the philosophy of mathematics. Leśniewski recognized Tarski's potential as a mathematician and encouraged him to abandon biology.<sup id="cite_ref-Feferman_&_Feferman,_p.26_9-1" class="reference"><a href="#cite_note-Feferman_&_Feferman,_p.26-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński, <a href="/wiki/Stefan_Mazurkiewicz" title="Stefan Mazurkiewicz">Stefan Mazurkiewicz</a> and <a href="/wiki/Tadeusz_Kotarbi%C5%84ski" title="Tadeusz Kotarbiński">Tadeusz Kotarbiński</a>, and in 1924 became the only person ever to complete a doctorate under Leśniewski's supervision. His thesis was entitled <i>O wyrazie pierwotnym logistyki</i> (<i>On the Primitive Term of Logistic</i>; published 1923). Tarski and Leśniewski soon grew cool to each other, mainly due to the latter's increasing anti-semitism.<sup id="cite_ref-FFintro_7-1" class="reference"><a href="#cite_note-FFintro-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> However, in later life, Tarski reserved his warmest praise for Kotarbiński, which was reciprocated. </p><p>In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to "Tarski". The Tarski brothers also converted to <a href="/wiki/Roman_Catholicism" class="mw-redirect" title="Roman Catholicism">Roman Catholicism</a>, Poland's dominant religion. Alfred did so even though he was an avowed <a href="/wiki/Atheism" title="Atheism">atheist</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Career">Career</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=3" title="Edit section: Career"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>After becoming the youngest person ever to complete a doctorate at Warsaw University, Tarski taught logic at the Polish Pedagogical Institute, mathematics and logic at the university, and served as Łukasiewicz's assistant. Because these positions were poorly paid, Tarski also taught mathematics at the Third Boys’ Gimnazjum of the Trade Union of Polish Secondary-School Teachers (later the Stefan Żeromski Gimnazjum), a Warsaw secondary school, beginning in 1925.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence until his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them ground-breaking, but also did so while supporting himself primarily by teaching high-school mathematics.<sup id="cite_ref-FOOTNOTEMcFarlandMcFarlandSmith2014319_13-0" class="reference"><a href="#cite_note-FOOTNOTEMcFarlandMcFarlandSmith2014319-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background. She had worked as a courier for the army in the <a href="/wiki/Polish%E2%80%93Soviet_War" title="Polish–Soviet War">Polish–Soviet War</a>. They had two children; a son <a href="/w/index.php?title=Jan_Tarski,&action=edit&redlink=1" class="new" title="Jan Tarski, (page does not exist)">Jan Tarski,</a> who became a physicist, and a daughter Ina, who married the mathematician <a href="/wiki/Andrzej_Ehrenfeucht" title="Andrzej Ehrenfeucht">Andrzej Ehrenfeucht</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>Tarski applied for a chair of philosophy at <a href="/wiki/Lw%C3%B3w_University" class="mw-redirect" title="Lwów University">Lwów University</a>, but on <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s recommendation it was awarded to <a href="/wiki/Leon_Chwistek" title="Leon Chwistek">Leon Chwistek</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> In 1930, Tarski visited the <a href="/wiki/University_of_Vienna" title="University of Vienna">University of Vienna</a>, lectured to <a href="/wiki/Karl_Menger" title="Karl Menger">Karl Menger</a>'s colloquium, and met <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>. Thanks to a fellowship, he was able to return to Vienna during the first half of 1935 to work with Menger's research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the <a href="/wiki/Unity_of_Science" class="mw-redirect" title="Unity of Science">Unity of Science</a> movement, an outgrowth of the <a href="/wiki/Vienna_Circle" title="Vienna Circle">Vienna Circle</a>. Tarski's academic career in Poland was strongly and repeatedly impacted by his heritage. For example, in 1937, Tarski applied for a chair at <a href="/wiki/Pozna%C5%84_University" class="mw-redirect" title="Poznań University">Poznań University</a> but the chair was abolished to avoid assigning it to Tarski (who was undisputedly the strongest applicant) because he was a Jew.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Tarski's ties to the Unity of Science movement likely saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at <a href="/wiki/Harvard_University" title="Harvard University">Harvard University</a>. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet <a href="/wiki/Invasion_of_Poland" title="Invasion of Poland">invasion of Poland</a> and the outbreak of <a href="/wiki/World_War_II" title="World War II">World War II</a>. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the <a href="/wiki/Nazi" class="mw-redirect" title="Nazi">Nazi</a> threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities. </p><p>Once in the United States, Tarski held a number of temporary teaching and research positions: Harvard University (1939), <a href="/wiki/City_College_of_New_York" title="City College of New York">City College of New York</a> (1940), and thanks to a <a href="/wiki/Guggenheim_Fellowship" title="Guggenheim Fellowship">Guggenheim Fellowship</a>, the <a href="/wiki/Institute_for_Advanced_Study" title="Institute for Advanced Study">Institute for Advanced Study</a> in <a href="/wiki/Princeton,_New_Jersey" title="Princeton, New Jersey">Princeton</a> (1942), where he again met Gödel. In 1942, Tarski joined the Mathematics Department at the <a href="/wiki/University_of_California,_Berkeley" title="University of California, Berkeley">University of California, Berkeley</a>, where he spent the rest of his career. Tarski became an American citizen in 1945.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> At Berkeley, Tarski acquired a reputation as an astounding and demanding teacher, a fact noted by many observers: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>His seminars at Berkeley quickly became famous in the world of mathematical logic. His students, many of whom became distinguished mathematicians, noted the awesome energy with which he would coax and cajole their best work out of them, always demanding the highest standards of clarity and precision.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></p></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></p></blockquote> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>A charismatic leader and teacher, known for his brilliantly precise yet suspenseful expository style, Tarski had intimidatingly high standards for students, but at the same time he could be very encouraging, and particularly so to women — in contrast to the general trend. Some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the field.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></p></blockquote> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:20070206_uw_buw_hall_glowny_biblioteki.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/20070206_uw_buw_hall_glowny_biblioteki.jpg/330px-20070206_uw_buw_hall_glowny_biblioteki.jpg" decoding="async" width="330" height="440" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/20070206_uw_buw_hall_glowny_biblioteki.jpg/495px-20070206_uw_buw_hall_glowny_biblioteki.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/20070206_uw_buw_hall_glowny_biblioteki.jpg/660px-20070206_uw_buw_hall_glowny_biblioteki.jpg 2x" data-file-width="1944" data-file-height="2592" /></a><figcaption><a href="/wiki/Warsaw_University_Library" class="mw-redirect" title="Warsaw University Library">Warsaw University Library</a> – at entrance (seen from rear) are pillared statues of <a href="/wiki/Lviv-Warsaw_School_of_Logic" class="mw-redirect" title="Lviv-Warsaw School of Logic">Lwów-Warsaw School</a> philosophers (<i>right to left</i>) <a href="/wiki/Kazimierz_Twardowski" title="Kazimierz Twardowski">Kazimierz Twardowski</a>, <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a>, Alfred Tarski, <a href="/wiki/Stanis%C5%82aw_Le%C5%9Bniewski" title="Stanisław Leśniewski">Stanisław Leśniewski</a>.</figcaption></figure> <p>Tarski supervised twenty-four Ph.D. dissertations including (in chronological order) those of <a href="/wiki/Andrzej_Mostowski" title="Andrzej Mostowski">Andrzej Mostowski</a>, <a href="/wiki/Bjarni_J%C3%B3nsson" title="Bjarni Jónsson">Bjarni Jónsson</a>, <a href="/wiki/Julia_Robinson" title="Julia Robinson">Julia Robinson</a>, <a href="/wiki/Robert_Vaught" class="mw-redirect" title="Robert Vaught">Robert Vaught</a>, <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a>, <a href="/wiki/Richard_Montague" title="Richard Montague">Richard Montague</a>, <a href="/w/index.php?title=James_Donald_Monk&action=edit&redlink=1" class="new" title="James Donald Monk (page does not exist)">James Donald Monk</a>, <a href="/wiki/Haim_Gaifman" title="Haim Gaifman">Haim Gaifman</a>, <a href="/w/index.php?title=Donald_Pigozzi&action=edit&redlink=1" class="new" title="Donald Pigozzi (page does not exist)">Donald Pigozzi</a>, and <a href="/wiki/Roger_Maddux" title="Roger Maddux">Roger Maddux</a>, as well as <a href="/wiki/Chen_Chung_Chang" title="Chen Chung Chang">Chen Chung Chang</a> and <a href="/wiki/Jerome_Keisler" class="mw-redirect" title="Jerome Keisler">Jerome Keisler</a>, authors of <i>Model Theory</i> (1973),<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> a classic text in the field.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-studentlist_24-0" class="reference"><a href="#cite_note-studentlist-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> He also strongly influenced the dissertations of <a href="/wiki/Adolf_Lindenbaum" title="Adolf Lindenbaum">Adolf Lindenbaum</a>, <a href="/wiki/Dana_Scott" title="Dana Scott">Dana Scott</a>, and <a href="/w/index.php?title=Steven_Givant&action=edit&redlink=1" class="new" title="Steven Givant (page does not exist)">Steven Givant</a>. Five of Tarski's students were women, a remarkable fact given that men represented an overwhelming majority of graduate students at the time.<sup id="cite_ref-studentlist_24-1" class="reference"><a href="#cite_note-studentlist-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> However, he had extra-marital affairs with at least two of these students. After he showed another of his female student's<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (July 2023)">who?</span></a></i>]</sup> work to a male colleague<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (July 2023)">who?</span></a></i>]</sup>, the colleague published it himself, leading her to leave the graduate study and later move to a different university and a different advisor.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>Tarski lectured at <a href="/wiki/University_College,_London" class="mw-redirect" title="University College, London">University College, London</a> (1950, 1966), the <a href="/wiki/Institut_Henri_Poincar%C3%A9" title="Institut Henri Poincaré">Institut Henri Poincaré</a> in Paris (1955), the <a href="/wiki/Miller_Institute_for_Basic_Research_in_Science" class="mw-redirect" title="Miller Institute for Basic Research in Science">Miller Institute for Basic Research in Science</a> in Berkeley (1958–60), the <a href="/wiki/University_of_California_at_Los_Angeles" class="mw-redirect" title="University of California at Los Angeles">University of California at Los Angeles</a> (1967), and the <a href="/wiki/Pontifical_Catholic_University_of_Chile" title="Pontifical Catholic University of Chile">Pontifical Catholic University of Chile</a> (1974–75). Among many distinctions garnered over the course of his career, Tarski was elected to the <a href="/wiki/United_States_National_Academy_of_Sciences" class="mw-redirect" title="United States National Academy of Sciences">United States National Academy of Sciences</a>, the <a href="/wiki/British_Academy" title="British Academy">British Academy</a> and the <a href="/wiki/Royal_Netherlands_Academy_of_Arts_and_Sciences" title="Royal Netherlands Academy of Arts and Sciences">Royal Netherlands Academy of Arts and Sciences</a> in 1958,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> received <a href="/wiki/Honorary_degree" title="Honorary degree">honorary degrees</a> from the Pontifical Catholic University of Chile in 1975, from <a href="/wiki/Marseille" title="Marseille">Marseille</a>'s <a href="/wiki/Paul_C%C3%A9zanne_University" title="Paul Cézanne University">Paul Cézanne University</a> in 1977 and from the <a href="/wiki/University_of_Calgary" title="University of Calgary">University of Calgary</a>, as well as the Berkeley Citation in 1981. Tarski presided over the <a href="/wiki/Association_for_Symbolic_Logic" title="Association for Symbolic Logic">Association for Symbolic Logic</a>, 1944–46, and the International Union for the History and Philosophy of Science, 1956–57. He was also an honorary editor of <i><a href="/wiki/Algebra_Universalis" title="Algebra Universalis">Algebra Universalis</a></i>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Work_in_mathematics">Work in mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=4" title="Edit section: Work in mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tarski's mathematical interests were exceptionally broad. His collected papers run to about 2,500 pages, most of them on mathematics, not logic. For a concise survey of Tarski's mathematical and logical accomplishments by his former student Solomon Feferman, see "Interludes I–VI" in Feferman and Feferman.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p>Tarski's first paper, published when he was 19 years old, was on <a href="/wiki/Set_theory" title="Set theory">set theory</a>, a subject to which he returned throughout his life.<sup id="cite_ref-mathshistory.st-andrews.ac.uk_29-0" class="reference"><a href="#cite_note-mathshistory.st-andrews.ac.uk-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> In 1924, he and <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a> proved that, if one accepts the <a href="/wiki/Axiom_of_Choice" class="mw-redirect" title="Axiom of Choice">Axiom of Choice</a>, a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">ball</a> can be cut into a finite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the <a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a>.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>In <i>A decision method for elementary algebra and geometry</i>, Tarski showed, by the method of <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">quantifier elimination</a>, that the <a href="/wiki/First-order_theory" class="mw-redirect" title="First-order theory">first-order theory</a> of the <a href="/wiki/Real_number" title="Real number">real numbers</a> under addition and multiplication is <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>. (While this result appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, because <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a> proved in 1936 that <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> (the theory of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>) is <i>not</i> decidable. Peano arithmetic is also incomplete by <a href="/wiki/G%C3%B6del%27s_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorem</a>. In his 1953 <i>Undecidable theories</i>, Tarski et al. showed that many mathematical systems, including <a href="/wiki/Lattice_theory" class="mw-redirect" title="Lattice theory">lattice theory</a>, abstract <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>, and <a href="/wiki/Closure_algebra" class="mw-redirect" title="Closure algebra">closure algebras</a>, are all undecidable. The theory of <a href="/wiki/Abelian_group" title="Abelian group">Abelian groups</a> is decidable, but that of non-Abelian groups is not. </p><p>While teaching at the Stefan Żeromski Gimnazjum in the 1920s and 30s, Tarski often taught <a href="/wiki/Geometry" title="Geometry">geometry</a>.<sup id="cite_ref-FOOTNOTEMcFarlandMcFarlandSmith2014Section_9.2:_Teaching_geometry,_pp._179–184_31-0" class="reference"><a href="#cite_note-FOOTNOTEMcFarlandMcFarlandSmith2014Section_9.2:_Teaching_geometry,_pp._179–184-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Using some ideas of <a href="/wiki/Mario_Pieri" title="Mario Pieri">Mario Pieri</a>, in 1926 Tarski devised an original <a href="/wiki/Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> for plane <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, one considerably more concise than <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a> form a first-order theory devoid of set theory, whose individuals are <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, and having only two primitive <a href="/wiki/Finitary_relation" title="Finitary relation">relations</a>. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers. </p><p>In 1929 he showed that much of Euclidean <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a> could be recast as a second-order theory whose individuals are <i>spheres</i> (a <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notion</a>), a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containment <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partially orders</a> the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of <a href="/wiki/Mereology" title="Mereology">mereology</a> far easier to exposit than <a href="/wiki/Lesniewski" class="mw-redirect" title="Lesniewski">Lesniewski</a>'s variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p><i>Cardinal Algebras</i> studied algebras whose models include the arithmetic of <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. <i>Ordinal Algebras</i> sets out an algebra for the additive theory of <a href="/wiki/Order_type" title="Order type">order types</a>. Cardinal, but not ordinal, addition commutes. </p><p>In 1941, Tarski published an important paper on <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>, which began the work on <a href="/wiki/Relation_algebra" title="Relation algebra">relation algebra</a> and its <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a> that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of <a href="/wiki/Roger_Lyndon" title="Roger Lyndon">Roger Lyndon</a>) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> and <a href="/wiki/Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>. For an introduction to <a href="/wiki/Relation_algebra" title="Relation algebra">relation algebra</a>, see Maddux (2006). In the late 1940s, Tarski and his students devised <a href="/wiki/Cylindric_algebra" title="Cylindric algebra">cylindric algebras</a>, which are to <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> what the <a href="/wiki/Two-element_Boolean_algebra" title="Two-element Boolean algebra">two-element Boolean algebra</a> is to classical <a href="/wiki/Sentential_logic" class="mw-redirect" title="Sentential logic">sentential logic</a>. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Work_in_logic">Work in logic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=5" title="Edit section: Work in logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tarski's student, <a href="/wiki/Robert_Lawson_Vaught" title="Robert Lawson Vaught">Robert Lawson Vaught</a>, has ranked Tarski as one of the four greatest logicians of all time — along with <a href="/wiki/Aristotle" title="Aristotle">Aristotle</a>, <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a>, and <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>.<sup id="cite_ref-FFintro_7-2" class="reference"><a href="#cite_note-FFintro-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Vaught_35-0" class="reference"><a href="#cite_note-Vaught-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Restall_36-0" class="reference"><a href="#cite_note-Restall-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> However, Tarski often expressed great admiration for <a href="/wiki/Logic_of_relatives" class="mw-redirect" title="Logic of relatives">Charles Sanders Peirce</a>, particularly for his pioneering work in the <a href="/wiki/Finitary_relation" title="Finitary relation">logic of relations</a>. </p><p>Tarski produced axioms for <i>logical consequence</i> and worked on <a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive systems</a>, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics. Around 1930, Tarski developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set of <i>sentences</i>). In <a href="/wiki/Abstract_algebraic_logic" title="Abstract algebraic logic">abstract algebraic logic</a>, finitary closure operators are still studied under the name <i>consequence operator</i>, which was coined by Tarski. The set <i>S</i> represents a set of sentences, a subset <i>T</i> of <i>S</i> a theory, and cl(<i>T</i>) is the set of all sentences that follow from the theory. This abstract approach was applied to fuzzy logic (see Gerla 2000). </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only can its concepts and results be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.<sup id="cite_ref-Sinaceur_37-0" class="reference"><a href="#cite_note-Sinaceur-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies.<sup id="cite_ref-mathshistory.st-andrews.ac.uk_29-1" class="reference"><a href="#cite_note-mathshistory.st-andrews.ac.uk-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as <i>Introduction to Logic and to the Methodology of Deductive Sciences</i>.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>Tarski's 1969 "Truth and proof" considered both <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> and <a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability theorem</a>, and mulled over their consequences for the axiomatic method in mathematics. </p> <div class="mw-heading mw-heading3"><h3 id="Truth_in_formalized_languages">Truth in formalized languages</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=6" title="Edit section: Truth in formalized languages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Semantic_theory_of_truth" title="Semantic theory of truth">Semantic theory of truth</a></div> <p>In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych",<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> "Setting out a mathematical definition of truth for formal languages." The 1935 German translation was titled "Der Wahrheitsbegriff in den formalisierten Sprachen", "The concept of truth in formalized languages", sometimes shortened to "Wahrheitsbegriff". An English translation appeared in the 1956 first edition of the volume <i><a href="/w/index.php?title=Logic,_Semantics,_Metamathematics&action=edit&redlink=1" class="new" title="Logic, Semantics, Metamathematics (page does not exist)">Logic, Semantics, Metamathematics</a></i>. This collection of papers from 1923 to 1938 is an event in 20th-century <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">analytic philosophy</a>, a contribution to <a href="/wiki/Mathematical_logic" title="Mathematical logic">symbolic logic</a>, <a href="/wiki/Semantics" title="Semantics">semantics</a>, and the <a href="/wiki/Philosophy_of_language" title="Philosophy of language">philosophy of language</a>. For a brief discussion of its content, see <a href="/wiki/Convention_T" class="mw-redirect" title="Convention T">Convention T</a> (and also <a href="/wiki/T-schema" title="T-schema">T-schema</a>). </p><p>A philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a <a href="/wiki/Correspondence_theory_of_truth" title="Correspondence theory of truth">correspondence theory of truth</a>. The debate centers on how to read Tarski's condition of material adequacy for a true definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined: </p> <dl><dd>"p" is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> p.</dd></dl> <p>(where p is the proposition expressed by "p") </p><p>The debate amounts to whether to read sentences of this form, such as </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote"><p>"Snow is white" is true if and only if snow is white</p></blockquote> <p>as expressing merely a <a href="/wiki/Deflationary_theory_of_truth" title="Deflationary theory of truth">deflationary theory of truth</a> or as embodying <a href="/wiki/Truth" title="Truth">truth</a> as a more substantial property (see Kirkham 1992). </p> <div class="mw-heading mw-heading3"><h3 id="Logical_consequence">Logical consequence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=7" title="Edit section: Logical consequence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1936, Tarski published Polish and German versions of a lecture, “On the Concept of Following Logically",<sup id="cite_ref-Tarski2002_41-0" class="reference"><a href="#cite_note-Tarski2002-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> he had given the preceding year at the International Congress of Scientific Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper and corrects a number of mistranslations in Tarski (1983).<sup id="cite_ref-Tarski2002_41-1" class="reference"><a href="#cite_note-Tarski2002-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p><p>This publication set out the modern <a href="/wiki/Model_theory" title="Model theory">model-theoretic</a> definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different <a href="/wiki/Cardinal_number" title="Cardinal number">cardinalities</a>).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> This question is a matter of some debate in the philosophical literature. <a href="/wiki/John_Etchemendy" title="John Etchemendy">John Etchemendy</a> stimulated much of the discussion about Tarski's treatment of varying domains.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p><p>Tarski ends by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. "What are Logical Notions?" can thus be viewed as continuing "On the Concept of Logical Consequence".<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Logical_notions">Logical notions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=8" title="Edit section: Logical notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Alfred_Tarski.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Alfred_Tarski.jpeg/220px-Alfred_Tarski.jpeg" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Alfred_Tarski.jpeg/330px-Alfred_Tarski.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1d/Alfred_Tarski.jpeg 2x" data-file-width="400" data-file-height="277" /></a><figcaption>Alfred Tarski at Berkeley</figcaption></figure> <p>Tarski's "What are Logical Notions?" (Tarski 1986) is the published version of a talk that he gave originally in 1966 in London and later in 1973 in <a href="/wiki/Buffalo,_New_York" title="Buffalo, New York">Buffalo</a>; it was edited without his direct involvement by <a href="/wiki/John_Corcoran_(logician)" title="John Corcoran (logician)">John Corcoran</a>. It became the most cited paper in the journal <i>History and Philosophy of Logic</i>.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p><p>In the talk, Tarski proposed demarcation of logical operations (which he calls "notions") from non-logical. The suggested criteria were derived from the <a href="/wiki/Erlangen_program" title="Erlangen program">Erlangen program</a> of the 19th-century German mathematician <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>. Mautner (in 1946), and possibly<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (July 2023)">clarification needed</span></a></i>]</sup> an article by the Portuguese mathematician <a href="/wiki/Jos%C3%A9_Sebasti%C3%A3o_e_Silva" title="José Sebastião e Silva">José Sebastião e Silva</a>, anticipated Tarski in applying the Erlangen Program to logic.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> </p><p>The Erlangen program classified the various types of geometry (<a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, <a href="/wiki/Affine_geometry" title="Affine geometry">affine geometry</a>, <a href="/wiki/Topology" title="Topology">topology</a>, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> </p><p>As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a <a href="/wiki/Polygon" title="Polygon">polygon</a> from an <a href="/wiki/Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a> (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> </p><p>Tarski's proposal<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Avoid_weasel_words" class="mw-redirect" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (July 2023)">which?</span></a></i>]</sup> was to demarcate the logical notions by considering all possible one-to-one transformations (<a href="/wiki/Automorphism" title="Automorphism">automorphisms</a>) of a domain onto itself. By domain is meant the <a href="/wiki/Universe_of_discourse" class="mw-redirect" title="Universe of discourse">universe of discourse</a> of a model for the semantic theory of logic. If one identifies the <a href="/wiki/Truth_value" title="Truth value">truth value</a> True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal: </p> <ol><li><i><a href="/wiki/Truth-function" class="mw-redirect" title="Truth-function">Truth-functions</a></i>: All truth-functions are admitted by the proposal. This includes, but is not limited to, all <i>n</i>-ary truth-functions for finite <i>n</i>. (It also admits of truth-functions with any infinite number of places.)</li> <li><i>Individuals</i>: No individuals, provided the domain has at least two members.</li> <li><i>Predicates</i>: <ul><li>the one-place total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension</li> <li>two-place total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension</li> <li>the two-place identity predicate, with the set of all order-pairs <<i>a</i>,<i>a</i>> in its extension, where <i>a</i> is a member of the domain</li> <li>the two-place diversity predicate, with the set of all order pairs <<i>a</i>,<i>b</i>> where <i>a</i> and <i>b</i> are distinct members of the domain</li> <li><i>n</i>-ary predicates in general: all predicates definable from the identity predicate together with <a href="/wiki/Logical_conjunction" title="Logical conjunction">conjunction</a>, <a href="/wiki/Disjunction" class="mw-redirect" title="Disjunction">disjunction</a> and <a href="/wiki/Negation" title="Negation">negation</a> (up to any ordinality, finite or infinite)</li></ul></li> <li><i><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></i>: Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates <i>Fx</i> and <i>Gy</i>, "More(<i>x, y</i>)", which says "More things have <i>F</i> than have <i>G</i>."</li> <li><i>Set-Theoretic relations</i>: Relations such as <a href="/wiki/Inclusion_(set_theory)" class="mw-redirect" title="Inclusion (set theory)">inclusion</a>, <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> and <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> applied to <a href="/wiki/Subset" title="Subset">subsets</a> of the domain are logical in the present sense.</li> <li><i>Set membership</i>: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of <a href="/wiki/Type_theory" title="Type theory">type theory</a>, but is extralogical if set theory is set out axiomatically, as in the canonical <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>.</li> <li><i>Logical notions of higher order</i>: While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup></li></ol> <p>In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>'s and <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Whitehead</a>'s <i><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i> are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and <a href="/w/index.php?title=Steven_Givant&action=edit&redlink=1" class="new" title="Steven Givant (page does not exist)">Givant</a> (1987).<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a> and <a href="/w/index.php?title=Vann_McGee&action=edit&redlink=1" class="new" title="Vann McGee (page does not exist)">Vann McGee</a> further discussed Tarski's proposal<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Avoid_weasel_words" class="mw-redirect" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (July 2023)">which?</span></a></i>]</sup> in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with a sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard first-order logic without identity.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> </p><p><a href="/w/index.php?title=Vann_McGee&action=edit&redlink=1" class="new" title="Vann McGee (page does not exist)">Vann McGee</a> (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends first-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Selected_publications">Selected publications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=9" title="Edit section: Selected publications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Anthologies and collections</dt></dl> <ul><li>1986. <i>The Collected Papers of Alfred Tarski</i>, 4 vols. Givant, S. R., and McKenzie, R. N., eds. Birkhäuser.</li> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGivant_Steven1986" class="citation journal cs1">Givant Steven (1986). "Bibliography of Alfred Tarski". <i>Journal of Symbolic Logic</i>. <b>51</b> (4): <span class="nowrap">913–</span>41. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2273905">10.2307/2273905</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2273905">2273905</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:44369365">44369365</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Bibliography+of+Alfred+Tarski&rft.volume=51&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E913-%3C%2Fspan%3E41&rft.date=1986&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A44369365%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2273905%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2273905&rft.au=Givant+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li>1983 (1956). <i>Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski</i>, Corcoran, J., ed. Hackett. 1st edition edited and translated by <a href="/wiki/J._H._Woodger" class="mw-redirect" title="J. H. Woodger">J. H. Woodger</a>, Oxford Uni. Press.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> This collection contains translations from Polish of some of Tarski's most important papers of his early career, including <i>The Concept of Truth in Formalized Languages</i> and <i>On the Concept of Logical Consequence</i> discussed above.</li></ul> <dl><dt>Original publications of Tarski</dt> <dd></dd></dl> <ul><li>1930 Une contribution à la théorie de la mesure. Fund Math 15 (1930), 42–50.</li> <li>1930. (with <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Łukasiewicz</a>). "Untersuchungen uber den Aussagenkalkul" ["Investigations into the Sentential Calculus"], <i>Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie</i>, Vol, 23 (1930) Cl. III, pp. 31–32 in Tarski (1983): 38–59.</li> <li>1931. "Sur les ensembles définissables de nombres réels I", <i>Fundamenta Mathematicae 17</i>: 210–239 in Tarski (1983): 110–142.</li> <li>1936. <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k383668/f5.image.r=.langFR">"Grundlegung der wissenschaftlichen Semantik"</a>, <i>Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935</i>, vol. III, <i>Language et pseudo-problèmes</i>, Paris, Hermann, 1936, pp. 1–8 in Tarski (1983): 401–408.</li> <li>1936. <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k38370h/f5.image.r=.langFR">"Über den Begriff der logischen Folgerung"</a>, <i>Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935</i>, vol. VII, <i>Logique</i>, Paris: Hermann, pp. 1–11 in Tarski (1983): 409–420.</li> <li>1936 (with <a href="/wiki/Adolf_Lindenbaum" title="Adolf Lindenbaum">Adolf Lindenbaum</a>). "On the Limitations of Deductive Theories" in Tarski (1983): 384–92.</li> <li>1937. <i>Einführung in die Mathematische Logik und in die Methodologie der Mathematik</i>. Springer, Wien (Vienna).</li> <li>1994 (1941).<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> <i>Introduction to Logic and to the Methodology of Deductive Sciences</i>. Dover.</li> <li>1941. "On the calculus of relations", <i>Journal of Symbolic Logic 6</i>: 73–89.</li> <li>1944. "<a rel="nofollow" class="external text" href="http://www.ditext.com/tarski/tarski.html">The Semantical Concept of Truth and the Foundations of Semantics</a>," <i>Philosophy and Phenomenological Research 4</i>: 341–75.</li> <li>1948. <i>A decision method for elementary algebra and geometry</i>. Santa Monica CA: RAND Corp.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup></li> <li>1949. <i>Cardinal Algebras</i>. Oxford Univ. Press.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup></li> <li>1953 (with <a href="/wiki/A._Mostowski" class="mw-redirect" title="A. Mostowski">Mostowski</a> and <a href="/wiki/R._M._Robinson" class="mw-redirect" title="R. M. Robinson">Raphael Robinson</a>). <i>Undecidable theories</i>. North Holland.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup></li> <li>1956. <i>Ordinal algebras</i>. North-Holland.</li> <li>1965. "A simplified formalization of predicate logic with identity", <i>Archiv für Mathematische Logik und Grundlagenforschung 7</i>: 61-79</li> <li>1969. "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20141229081319/http://people.scs.carleton.ca/~bertossi/logic/material/tarski.pdf">Truth and Proof</a>", <i>Scientific American 220</i>: 63–77.</li> <li>1971 (with <a href="/wiki/Leon_Henkin" title="Leon Henkin">Leon Henkin</a> and <a href="/w/index.php?title=Donald_Monk&action=edit&redlink=1" class="new" title="Donald Monk (page does not exist)">Donald Monk</a>). <i>Cylindric Algebras: Part I</i>. North-Holland.</li> <li>1985 (with <a href="/wiki/Leon_Henkin" title="Leon Henkin">Leon Henkin</a> and Donald Monk). <i>Cylindric Algebras: Part II</i>. North-Holland.</li> <li>1986. "What are Logical Notions?", Corcoran, J., ed., <i>History and Philosophy of Logic 7</i>: 143–54.</li> <li>1987 (with Steven Givant). <i>A Formalization of Set Theory Without Variables</i>. Vol.41 of American Mathematical Society colloquium publications. Providence RI: American Mathematical Society. <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/978-0821810415" title="Special:BookSources/978-0821810415">978-0821810415</a>. <a rel="nofollow" class="external text" href="https://www.ams.org/bull/1989-20-02/S0273-0979-1989-15778-9/S0273-0979-1989-15778-9.pdf">Review</a></li> <li>1999 (with Steven Givant). <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/434679.html">"Tarski's system of geometry"</a>, <i>Bulletin of Symbolic Logic 5</i>: 175–214.</li> <li>2002. "On the Concept of Following Logically" (Magda Stroińska and David Hitchcock, trans.) <i>History and Philosophy of Logic 23</i>: 155–196.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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href="/wiki/Interpretability" title="Interpretability">Interpretability</a></li> <li><a href="/wiki/Weak_interpretability" title="Weak interpretability">Weak interpretability</a></li> <li><a href="/wiki/List_of_things_named_after_Alfred_Tarski" title="List of things named after Alfred Tarski">List of things named after Alfred Tarski</a></li> <li><a href="/wiki/Timeline_of_Polish_science_and_technology" title="Timeline of Polish science and technology">Timeline of Polish science and technology</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Alfred Tarski, <a rel="nofollow" class="external text" href="https://www.britannica.com/biography/Alfred-Tarski">"Alfred Tarski"</a>, <i>Encyclopædia Britannica</i>.</span> </li> <li id="cite_note-Alfred_Tarski-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Alfred_Tarski_2-0">^</a></b></span> <span class="reference-text">School of Mathematics and Statistics, University of St Andrews, <a rel="nofollow" class="external text" href="http://www-history.mcs.st-and.ac.uk/Biographies/Tarski.html">"Alfred Tarski"</a>, <i>School of Mathematics and Statistics, University of St Andrews</i>.</span> </li> <li id="cite_note-oxfordreference.com-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-oxfordreference.com_3-0">^</a></b></span> <span class="reference-text"><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.oxfordreference.com/view/10.1093/oi/authority.20110803102144812">"Alfred Tarski"</a>. <i>Oxford Reference</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Alfred+Tarski&rft.btitle=Oxford+Reference&rft_id=http%3A%2F%2Fwww.oxfordreference.com%2Fview%2F10.1093%2Foi%2Fauthority.20110803102144812&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGomez-Torrente2014" class="citation web cs1">Gomez-Torrente, Mario (March 27, 2014). <a rel="nofollow" class="external text" href="http://www.oxfordbibliographies.com/view/document/obo-9780195396577/obo-9780195396577-0186.xml">"Alfred Tarski - Philosophy - Oxford Bibliographies"</a>. <i><a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">October 24,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Oxford+University+Press&rft.atitle=Alfred+Tarski+-+Philosophy+-+Oxford+Bibliographies&rft.date=2014-03-27&rft.aulast=Gomez-Torrente&rft.aufirst=Mario&rft_id=http%3A%2F%2Fwww.oxfordbibliographies.com%2Fview%2Fdocument%2Fobo-9780195396577%2Fobo-9780195396577-0186.xml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Alfred Tarski, <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/tarski/">"Alfred Tarski"</a>, <i>Stanford Encyclopedia of Philosophy</i>.</span> </li> <li id="cite_note-FefA-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FefA_6-0">^</a></b></span> <span class="reference-text"><a href="#FefA">Feferman A.</a></span> </li> <li id="cite_note-FFintro-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-FFintro_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FFintro_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FFintro_7-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, p.1</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, pp.17-18</span> </li> <li id="cite_note-Feferman_&_Feferman,_p.26-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Feferman_&_Feferman,_p.26_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Feferman_&_Feferman,_p.26_9-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, p.26</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, p.294</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">"Most of the Socialist Party members were also in favor of assimilation, and Tarski's political allegiance was socialist at the time. So, along with its being a practical move, becoming more Polish than Jewish was an ideological statement and was approved by many, though not all, of his colleagues. As to why Tarski, a professed atheist, converted, that just came with the territory and was part of the package: if you were going to be Polish then you had to say you were Catholic." Anita Burdman Feferman, Solomon Feferman, <i>Alfred Tarski: Life and Logic</i> (2004), page 39.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcFarlandMcFarlandSmith2014" class="citation book cs1">McFarland, Andrew; McFarland, Joanna; Smith, James T. (2014). <i>Alfred Tarski: Early work in Poland — geometry and teaching</i>. Birkhäuser/Springer, New York. p. 173. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-1473-9" title="Special:BookSources/978-1-4939-1473-9"><bdi>978-1-4939-1473-9</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3307383">3307383</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alfred+Tarski%3A+Early+work+in+Poland+%E2%80%94+geometry+and+teaching&rft.pages=173&rft.pub=Birkh%C3%A4user%2FSpringer%2C+New+York&rft.date=2014&rft.isbn=978-1-4939-1473-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3307383%23id-name%3DMR&rft.aulast=McFarland&rft.aufirst=Andrew&rft.au=McFarland%2C+Joanna&rft.au=Smith%2C+James+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEMcFarlandMcFarlandSmith2014319-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMcFarlandMcFarlandSmith2014319_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcFarlandMcFarlandSmith2014">McFarland, McFarland & Smith 2014</a>, p. 319.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Feferman & Feferman (2004), pp. 239–242.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, p. 67</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, pp. 102-103</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, Chap. 5, pp. 124-149</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobert_VaughtJohn_AddisonBenson_MatesJulia_Robinson1985" class="citation web cs1">Robert Vaught; John Addison; Benson Mates; Julia Robinson (1985). <a rel="nofollow" class="external text" href="http://content.cdlib.org/xtf/view?docId=hb4d5nb20m&doc.view=frames&chunk.id=div00159&toc.depth=1&toc.id=">"Alfred Tarski, Mathematics: Berkeley"</a>. University of California (System) Academic Senate<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-12-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Alfred+Tarski%2C+Mathematics%3A+Berkeley&rft.pub=University+of+California+%28System%29+Academic+Senate&rft.date=1985&rft.au=Robert+Vaught&rft.au=John+Addison&rft.au=Benson+Mates&rft.au=Julia+Robinson&rft_id=http%3A%2F%2Fcontent.cdlib.org%2Fxtf%2Fview%3FdocId%3Dhb4d5nb20m%26doc.view%3Dframes%26chunk.id%3Ddiv00159%26toc.depth%3D1%26toc.id%3D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www-groups.dcs.st-and.ac.uk/~history/Obits/Tarski.html">Obituary in <i>Times</i>, reproduced here</a></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Gregory Moore, "Alfred Tarski" in <i><a href="/wiki/Dictionary_of_Scientific_Biography" title="Dictionary of Scientific Biography">Dictionary of Scientific Biography</a></i></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#Fef">Feferman</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Chang, C.C., and Keisler, H.J., 1973. <i>Model Theory</i>. North-Holland, Amsterdam. American Elsevier, New York.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=13347">Alfred Tarski</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></span> </li> <li id="cite_note-studentlist-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-studentlist_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-studentlist_24-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, pp. 385-386</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, pp. 177–178 and 197–201.</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00003268">"Alfred Tarski (1902 - 1983)"</a>. Royal Netherlands Academy of Arts and Sciences<span class="reference-accessdate">. Retrieved <span class="nowrap">17 July</span> 2015</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Alfred+Tarski+%281902+-+1983%29&rft.pub=Royal+Netherlands+Academy+of+Arts+and+Sciences&rft_id=http%3A%2F%2Fwww.dwc.knaw.nl%2Fbiografie%2Fpmknaw%2F%3Fpagetype%3DauthorDetail%26aId%3DPE00003268&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Tarski.html">"Alfred Tarski"</a>, <i><a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a></i>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Alfred+Tarski&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FBiographies%2FTarski.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><a href="#F-F">Feferman & Feferman</a>, pp. 43-52, 69-75, 109-123, 189-195, 277-287, 334-342</span> </li> <li id="cite_note-mathshistory.st-andrews.ac.uk-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-mathshistory.st-andrews.ac.uk_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mathshistory.st-andrews.ac.uk_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Biographies/Tarski/#:~:text=Tarski%27s%20first%20paper%20was%20published%20in%201921%20when,submitted%20his%20doctoral%20thesis%20for%20examination%20in%201923.">"Alfred Tarski"</a>. <i>mathshistory.st-andrews.ac.uk</i><span class="reference-accessdate">. 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Papers from 1923 to 1938</i> by Alfred Tarski; translated by J. H. Woodger"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>63</b> (2): <span class="nowrap">155–</span>156. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1957-10115-3">10.1090/S0002-9904-1957-10115-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+Logic%2C+semantics%2C+metamathematics.+Papers+from+1923+to+1938+by+Alfred+Tarski%3B+translated+by+J.+H.+Woodger&rft.volume=63&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E155-%3C%2Fspan%3E156&rft.date=1957&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1957-10115-3&rft.au=Halmos%2C+Paul&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1957-63-02%2FS0002-9904-1957-10115-3%2FS0002-9904-1957-10115-3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-47">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuine,_W._V.1938" class="citation journal cs1"><a href="/wiki/Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Quine, W. V.</a> (1938). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1938-44-05/S0002-9904-1938-06731-6/S0002-9904-1938-06731-6.pdf">"Review: <i>Einführung in die mathematische Logik und in die Methodologie der Mathematik</i> by Alfred Tarski. Vienna, Springer, 1937. x+166 pp"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>44</b> (5): <span class="nowrap">317–</span>318. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1938-06731-6">10.1090/s0002-9904-1938-06731-6</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+Einf%C3%BChrung+in+die+mathematische+Logik+und+in+die+Methodologie+der+Mathematik+by+Alfred+Tarski.+Vienna%2C+Springer%2C+1937.+x%2B166+pp.&rft.volume=44&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E317-%3C%2Fspan%3E318&rft.date=1938&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1938-06731-6&rft.au=Quine%2C+W.+V.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1938-44-05%2FS0002-9904-1938-06731-6%2FS0002-9904-1938-06731-6.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCurry,_Haskell_B.1942" class="citation journal cs1"><a href="/wiki/Haskell_Curry" title="Haskell Curry">Curry, Haskell B.</a> (1942). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1942-48-07/S0002-9904-1942-07698-1/S0002-9904-1942-07698-1.pdf">"Review: <i>Introduction to Logic and to the Methodology of Deductive Sciences</i> by Alfred Tarski"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>48</b> (7): <span class="nowrap">507–</span>510. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1942-07698-1">10.1090/s0002-9904-1942-07698-1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+Introduction+to+Logic+and+to+the+Methodology+of+Deductive+Sciences+by+Alfred+Tarski&rft.volume=48&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E507-%3C%2Fspan%3E510&rft.date=1942&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1942-07698-1&rft.au=Curry%2C+Haskell+B.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1942-48-07%2FS0002-9904-1942-07698-1%2FS0002-9904-1942-07698-1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcNaughton,_Robert1953" class="citation journal cs1">McNaughton, Robert (1953). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1953-59-01/S0002-9904-1953-09664-1/S0002-9904-1953-09664-1.pdf">"Review: <i>A decision method for elementary algebra and geometry</i> by A. Tarski"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>59</b> (1): <span class="nowrap">91–</span>93. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1953-09664-1">10.1090/s0002-9904-1953-09664-1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+A+decision+method+for+elementary+algebra+and+geometry+by+A.+Tarski&rft.volume=59&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E91-%3C%2Fspan%3E93&rft.date=1953&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1953-09664-1&rft.au=McNaughton%2C+Robert&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1953-59-01%2FS0002-9904-1953-09664-1%2FS0002-9904-1953-09664-1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirkhoff,_Garrett1950" class="citation journal cs1"><a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a> (1950). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1950-56-02/S0002-9904-1950-09394-X/S0002-9904-1950-09394-X.pdf">"Review: <i>Cardinal algebras</i> by A. Tarski"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>56</b> (2): <span class="nowrap">208–</span>209. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1950-09394-x">10.1090/s0002-9904-1950-09394-x</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+Cardinal+algebras+by+A.+Tarski&rft.volume=56&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E208-%3C%2Fspan%3E209&rft.date=1950&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1950-09394-x&rft.au=Birkhoff%2C+Garrett&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1950-56-02%2FS0002-9904-1950-09394-X%2FS0002-9904-1950-09394-X.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGál,_Ilse_Novak1954" class="citation journal cs1">Gál, Ilse Novak (1954). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1954-60-06/S0002-9904-1954-09858-0/S0002-9904-1954-09858-0.pdf">"Review: <i>Undecidable theories</i> by Alfred Tarski in collaboration with A. Mostowsku and R. M. Robinson"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>60</b> (6): <span class="nowrap">570–</span>572. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1954-09858-0">10.1090/S0002-9904-1954-09858-0</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+Undecidable+theories+by+Alfred+Tarski+in+collaboration+with+A.+Mostowsku+and+R.+M.+Robinson&rft.volume=60&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E570-%3C%2Fspan%3E572&rft.date=1954&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1954-09858-0&rft.au=G%C3%A1l%2C+Ilse+Novak&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1954-60-06%2FS0002-9904-1954-09858-0%2FS0002-9904-1954-09858-0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Biographical references</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="FefA" class="citation book cs1"><a href="/wiki/Anita_Burdman_Feferman" title="Anita Burdman Feferman">Feferman, Anita Burdman</a> (1999). "Alfred Tarski". <a href="/wiki/American_National_Biography" title="American National Biography"><i>American National Biography</i></a>. Vol. 21. Oxford University Press. pp. <span class="nowrap">330–</span>332. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-512800-0" title="Special:BookSources/978-0-19-512800-0"><bdi>978-0-19-512800-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Alfred+Tarski&rft.btitle=American+National+Biography&rft.pages=%3Cspan+class%3D%22nowrap%22%3E330-%3C%2Fspan%3E332&rft.pub=Oxford+University+Press&rft.date=1999&rft.isbn=978-0-19-512800-0&rft.aulast=Feferman&rft.aufirst=Anita+Burdman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="F-F" class="citation book cs1"><a href="/wiki/Anita_Burdman_Feferman" title="Anita Burdman Feferman">Feferman, Anita Burdman</a>; <a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Feferman, Solomon</a> (2004). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/alfredtarskilife0000fefe"><i>Alfred Tarski: Life and Logic</i></a></span>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-80240-6" title="Special:BookSources/978-0-521-80240-6"><bdi>978-0-521-80240-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/54691904">54691904</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alfred+Tarski%3A+Life+and+Logic&rft.pub=Cambridge+University+Press&rft.date=2004&rft_id=info%3Aoclcnum%2F54691904&rft.isbn=978-0-521-80240-6&rft.aulast=Feferman&rft.aufirst=Anita+Burdman&rft.au=Feferman%2C+Solomon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falfredtarskilife0000fefe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrost-Arnold2013" class="citation book cs1">Frost-Arnold, Greg (2013). <i>Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science</i>. Chicago: Open Court. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780812698374" title="Special:BookSources/9780812698374"><bdi>9780812698374</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Carnap%2C+Tarski%2C+and+Quine+at+Harvard%3A+Conversations+on+Logic%2C+Mathematics%2C+and+Science&rft.place=Chicago&rft.pub=Open+Court&rft.date=2013&rft.isbn=9780812698374&rft.aulast=Frost-Arnold&rft.aufirst=Greg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGivant_Steven1991" class="citation journal cs1">Givant Steven (1991). "A portrait of Alfred Tarski". <i>Mathematical Intelligencer</i>. <b>13</b> (3): <span class="nowrap">16–</span>32. <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%2Fbf03023831">10.1007/bf03023831</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:122867668">122867668</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Intelligencer&rft.atitle=A+portrait+of+Alfred+Tarski&rft.volume=13&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E16-%3C%2Fspan%3E32&rft.date=1991&rft_id=info%3Adoi%2F10.1007%2Fbf03023831&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122867668%23id-name%3DS2CID&rft.au=Givant+Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li>Patterson, Douglas. <i>Alfred Tarski: Philosophy of Language and Logic</i> (Palgrave Macmillan; 2012) 262 pages; biography focused on his work from the late-1920s to the mid-1930s, with particular attention to influences from his teachers Stanislaw Lesniewski and Tadeusz Kotarbinski.</li></ul> <dl><dt>Logic literature</dt></dl> <ul><li>The December 1986 issue of the <i>Journal of Symbolic Logic</i> surveys Tarski's work on <a href="/wiki/Model_theory" title="Model theory">model theory</a> (<a href="/wiki/Robert_Vaught" class="mw-redirect" title="Robert Vaught">Robert Vaught</a>), <a href="/wiki/Abstract_algebra" title="Abstract algebra">algebra</a> (Jonsson), <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">undecidable theories</a> (McNulty), <a href="/wiki/Algebraic_logic" title="Algebraic logic">algebraic logic</a> (Donald Monk), and <a href="/wiki/Geometry" title="Geometry">geometry</a> (Szczerba). The March 1988 issue of the same journal surveys his work on <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a> (Azriel Levy), <a href="/wiki/Real_closed_fields" class="mw-redirect" title="Real closed fields">real closed fields</a> (Lou Van Den Dries), <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable theory</a> (Doner and <a href="/wiki/Wilfrid_Hodges" title="Wilfrid Hodges">Wilfrid Hodges</a>), <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a> (Blok and Pigozzi), <a href="/wiki/Truth" title="Truth">truth</a> and <a href="/wiki/Logical_consequence" title="Logical consequence">logical consequence</a> (<a href="/wiki/John_Etchemendy" title="John Etchemendy">John Etchemendy</a>), and general <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> (Patrick Suppes). <ul><li>Blok, W. J.; Pigozzi, Don, <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2274426">"Alfred Tarski's Work on General Metamathematics"</a>, <i>The Journal of Symbolic Logic</i>, Vol. 53, No. 1 (Mar., 1988), pp. 36–50</li></ul></li> <li><a href="/wiki/Chen_Chung_Chang" title="Chen Chung Chang">Chang, C.C.</a>, and <a href="/wiki/Howard_Jerome_Keisler" title="Howard Jerome Keisler">Keisler, H.J.</a>, 1973. <i>Model Theory</i>. North-Holland, Amsterdam. American Elsevier, New York.</li> <li><a href="/wiki/John_Corcoran_(logician)" title="John Corcoran (logician)">Corcoran, John</a>, and Sagüillo, José Miguel, 2011. "The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper", <i>History and Philosophy of Logic 32</i>: 359–80. <a rel="nofollow" class="external autonumber" href="http://www.tandfonline.com/doi/abs/10.1080/01445340.2011.577145#.UksmOD_-kQs">[1]</a></li> <li>Corcoran, John, and Weber, Leonardo, 2015. "Tarski's convention T: condition beta", South American Journal of Logic. 1, 3–32.</li> <li><a href="/wiki/John_Etchemendy" title="John Etchemendy">Etchemendy, John</a>, 1999. <i>The Concept of Logical Consequence</i>. Stanford CA: CSLI Publications. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-57586-194-1" title="Special:BookSources/1-57586-194-1">1-57586-194-1</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeferman_Solomon1999" class="citation journal cs1"><a href="/wiki/Solomon_Feferman" title="Solomon Feferman">Feferman Solomon</a> (1999). <a rel="nofollow" class="external text" href="http://math.stanford.edu/~feferman/papers/logiclogicism.pdf">"Logic, Logics, and Logicism"</a> <span class="cs1-format">(PDF)</span>. <i>Notre Dame Journal of Formal Logic</i>. <b>40</b>: <span class="nowrap">31–</span>54. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1305%2Fndjfl%2F1039096304">10.1305/ndjfl/1039096304</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notre+Dame+Journal+of+Formal+Logic&rft.atitle=Logic%2C+Logics%2C+and+Logicism&rft.volume=40&rft.pages=%3Cspan+class%3D%22nowrap%22%3E31-%3C%2Fspan%3E54&rft.date=1999&rft_id=info%3Adoi%2F10.1305%2Fndjfl%2F1039096304&rft.au=Feferman+Solomon&rft_id=http%3A%2F%2Fmath.stanford.edu%2F~feferman%2Fpapers%2Flogiclogicism.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li><a href="/w/index.php?title=Gerla,_G.&action=edit&redlink=1" class="new" title="Gerla, G. (page does not exist)">Gerla, G.</a> (2000) <i>Fuzzy Logic: Mathematical Tools for Approximate Reasoning</i>. <a href="/wiki/Kluwer_Academic_Publishers" class="mw-redirect" title="Kluwer Academic Publishers">Kluwer Academic Publishers</a>.</li> <li><a href="/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a>, 2000. <i>The Search for Mathematical Roots 1870-1940</i>. Princeton Uni. Press.</li> <li><a href="/wiki/Richard_Kirkham" title="Richard Kirkham">Kirkham, Richard</a>, 1992. <i>Theories of Truth</i>. MIT Press.</li> <li><a href="/wiki/Roger_Maddux" title="Roger Maddux">Maddux, Roger D.</a>, 2006. <i>Relation Algebras</i>, vol. 150 in "Studies in Logic and the Foundations of Mathematics", Elsevier Science.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMautner_F._I.1946" class="citation journal cs1"><a href="/wiki/Friederich_Ignaz_Mautner" title="Friederich Ignaz Mautner">Mautner F. 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"An Extension of Klein's Erlanger Program: Logic as Invariant-Theory". <i>American Journal of Mathematics</i>. <b>68</b> (3): <span class="nowrap">345–</span>84. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2371821">10.2307/2371821</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2371821">2371821</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=An+Extension+of+Klein%27s+Erlanger+Program%3A+Logic+as+Invariant-Theory&rft.volume=68&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E345-%3C%2Fspan%3E84&rft.date=1946&rft_id=info%3Adoi%2F10.2307%2F2371821&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2371821%23id-name%3DJSTOR&rft.au=Mautner+F.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcGee_Van1996" class="citation journal cs1">McGee Van (1996). "Logical Operations". <i>Journal of Philosophical Logic</i>. <b>25</b> (6): <span class="nowrap">567–</span>80. <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%2Fbf00265253">10.1007/bf00265253</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:32381037">32381037</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Philosophical+Logic&rft.atitle=Logical+Operations&rft.volume=25&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E567-%3C%2Fspan%3E80&rft.date=1996&rft_id=info%3Adoi%2F10.1007%2Fbf00265253&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A32381037%23id-name%3DS2CID&rft.au=McGee+Van&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li><a href="/wiki/Karl_Popper" title="Karl Popper">Popper, Karl R.</a>, 1972, Rev. Ed. 1979, "Philosophical Comments on Tarski's Theory of Truth", with Addendum, <i>Objective Knowledge</i>, Oxford: 319–340.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinaceur_H2001" class="citation journal cs1">Sinaceur H (2001). <a rel="nofollow" class="external text" href="https://halshs.archives-ouvertes.fr/halshs-01119489">"Alfred Tarski: Semantic shift, heuristic shift in metamathematics"</a>. <i>Synthese</i>. <b>126</b>: <span class="nowrap">49–</span>65. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2Fa%3A1005268531418">10.1023/a:1005268531418</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:28783841">28783841</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Synthese&rft.atitle=Alfred+Tarski%3A+Semantic+shift%2C+heuristic+shift+in+metamathematics&rft.volume=126&rft.pages=%3Cspan+class%3D%22nowrap%22%3E49-%3C%2Fspan%3E65&rft.date=2001&rft_id=info%3Adoi%2F10.1023%2Fa%3A1005268531418&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A28783841%23id-name%3DS2CID&rft.au=Sinaceur+H&rft_id=https%3A%2F%2Fhalshs.archives-ouvertes.fr%2Fhalshs-01119489&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlfred+Tarski" class="Z3988"></span></li> <li>Smith, James T., 2010. "Definitions and Nondefinability in Geometry", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a> 117:475–89.</li> <li><a href="/wiki/Jan_Wole%C5%84ski" title="Jan Woleński">Wolenski, Jan</a>, 1989. <i>Logic and Philosophy in the Lvov–Warsaw School</i>. Reidel/Kluwer.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alfred_Tarski&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Alfred_Tarski" class="extiw" title="commons:Category:Alfred Tarski">Alfred Tarski</a></span>.</div></div> </div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Alfred_Tarski" class="extiw" title="q:Special:Search/Alfred Tarski">Alfred Tarski</a></b></i>.</div></div> </div> <ul><li><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a>: <ul><li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/tarski-truth/">Tarski's Truth Definitions</a> by <a href="/wiki/Wilfred_Hodges" class="mw-redirect" title="Wilfred Hodges">Wilfred Hodges</a>.</li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/tarski/">Alfred Tarski</a> by Mario Gómez-Torrente.</li> <li><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/logic-algebraic-propositional/">Algebraic Propositional Logic</a> by Ramon Jansana. 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class="uid"><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=8034&url_prefix=https://opac.vatlib.it/auth/detail/&id=495/172234">Vatican</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007313877905171">Israel</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://opac.kbr.be/LIBRARY/doc/AUTHORITY/14741228">Belgium</a></span></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Academics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://ci.nii.ac.jp/author/DA0077541X?l=en">CiNii</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.mathgenealogy.org/id.php?id=13347">Mathematics Genealogy Project</a></span></li><li><span 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