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id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Eduki taularen ikusgarritasuna aldatu" > <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">Eduki taularen ikusgarritasuna aldatu</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">Logika matematiko</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="Joan beste hizkuntza batean idatzitako artikulu batera. 85 hizkuntzatan eskuragarri." > <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-85" 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">85 hizkuntza</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/Wiskundige_logika" title="Wiskundige logika – afrikaansa" lang="af" hreflang="af" data-title="Wiskundige logika" data-language-autonym="Afrikaans" data-language-local-name="afrikaansa" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Mathematische_Logik" title="Mathematische Logik – Suitzako alemana" lang="gsw" hreflang="gsw" data-title="Mathematische Logik" data-language-autonym="Alemannisch" data-language-local-name="Suitzako alemana" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%92%E1%88%B3%E1%89%A3%E1%8B%8A_%E1%88%A5%E1%8A%90_%E1%8A%A0%E1%88%9D%E1%8A%AD%E1%8A%95%E1%8B%AE" title="ሒሳባዊ ሥነ አምክንዮ – amharera" lang="am" hreflang="am" data-title="ሒሳባዊ ሥነ አምክንዮ" data-language-autonym="አማርኛ" data-language-local-name="amharera" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Lochica_matematica" title="Lochica matematica – aragoiera" lang="an" hreflang="an" data-title="Lochica matematica" data-language-autonym="Aragonés" data-language-local-name="aragoiera" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="منطق رياضي – arabiera" lang="ar" hreflang="ar" data-title="منطق رياضي" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/L%C3%B3xica_matem%C3%A1tica" title="Lóxica matemática – asturiera" lang="ast" hreflang="ast" data-title="Lóxica matemática" data-language-autonym="Asturianu" data-language-local-name="asturiera" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyazi_m%C9%99ntiq" title="Riyazi məntiq – azerbaijanera" lang="az" hreflang="az" data-title="Riyazi məntiq" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BB%D0%BE%D0%B3%D1%96%D0%BA%D0%B0" title="Матэматычная логіка – bielorrusiera" lang="be" hreflang="be" data-title="Матэматычная логіка" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BB%D1%91%D0%B3%D1%96%D0%BA%D0%B0" title="Матэматычная лёгіка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матэматычная лёгіка" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математическа логика – bulgariera" lang="bg" hreflang="bg" data-title="Математическа логика" data-language-autonym="Български" data-language-local-name="bulgariera" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AF%E0%A7%81%E0%A6%95%E0%A7%8D%E0%A6%A4%E0%A6%BF%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="গাণিতিক যুক্তিবিজ্ঞান – bengalera" lang="bn" hreflang="bn" data-title="গাণিতিক যুক্তিবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matemati%C4%8Dka_logika" title="Matematička logika – bosniera" lang="bs" hreflang="bs" data-title="Matematička logika" data-language-autonym="Bosanski" data-language-local-name="bosniera" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/L%C3%B2gica_matem%C3%A0tica" title="Lògica matemàtica – katalana" lang="ca" hreflang="ca" data-title="Lògica matemàtica" data-language-autonym="Català" data-language-local-name="katalana" 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/Matematick%C3%A1_logika" title="Matematická logika – txekiera" lang="cs" hreflang="cs" data-title="Matematická logika" data-language-autonym="Čeština" data-language-local-name="txekiera" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математикăлла логика – txuvaxera" lang="cv" hreflang="cv" data-title="Математикăлла логика" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Rhesymeg_mathemateg" title="Rhesymeg mathemateg – galesa" lang="cy" hreflang="cy" data-title="Rhesymeg mathemateg" data-language-autonym="Cymraeg" data-language-local-name="galesa" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Matematisk_logik" title="Matematisk logik – daniera" lang="da" hreflang="da" data-title="Matematisk logik" data-language-autonym="Dansk" data-language-local-name="daniera" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mathematische_Logik" title="Mathematische Logik – alemana" lang="de" hreflang="de" data-title="Mathematische Logik" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%BB%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE" title="Μαθηματική λογική – greziera" lang="el" hreflang="el" data-title="Μαθηματική λογική" data-language-autonym="Ελληνικά" data-language-local-name="greziera" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Mathematical_logic" title="Mathematical logic – ingelesa" lang="en" hreflang="en" data-title="Mathematical logic" data-language-autonym="English" data-language-local-name="ingelesa" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Matematika_logiko" title="Matematika logiko – esperantoa" lang="eo" hreflang="eo" data-title="Matematika logiko" data-language-autonym="Esperanto" data-language-local-name="esperantoa" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática – gaztelania" lang="es" hreflang="es" data-title="Lógica matemática" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Matemaatiline_loogika" title="Matemaatiline loogika – estoniera" lang="et" hreflang="et" data-title="Matemaatiline loogika" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%D8%B7%D9%82_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="منطق ریاضی – persiera" lang="fa" hreflang="fa" data-title="منطق ریاضی" data-language-autonym="فارسی" data-language-local-name="persiera" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matemaattinen_logiikka" title="Matemaattinen logiikka – finlandiera" lang="fi" hreflang="fi" data-title="Matemaattinen logiikka" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Logique_math%C3%A9matique" title="Logique mathématique – frantsesa" lang="fr" hreflang="fr" data-title="Logique mathématique" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Loighic_mhatamaitici%C3%BAil" title="Loighic mhatamaiticiúil – irlandera" lang="ga" hreflang="ga" data-title="Loighic mhatamaiticiúil" data-language-autonym="Gaeilge" data-language-local-name="irlandera" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Loidig_mhatamataigeach" title="Loidig mhatamataigeach – Eskoziako gaelikoa" lang="gd" hreflang="gd" data-title="Loidig mhatamataigeach" data-language-autonym="Gàidhlig" data-language-local-name="Eskoziako gaelikoa" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/L%C3%B3xica_matem%C3%A1tica" title="Lóxica matemática – galiziera" lang="gl" hreflang="gl" data-title="Lóxica matemática" data-language-autonym="Galego" data-language-local-name="galiziera" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9C%D7%95%D7%92%D7%99%D7%A7%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="לוגיקה מתמטית – hebreera" lang="he" hreflang="he" data-title="לוגיקה מתמטית" data-language-autonym="עברית" data-language-local-name="hebreera" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%A4%E0%A4%B0%E0%A5%8D%E0%A4%95%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="गणितीय तर्कशास्त्र – hindia" lang="hi" hreflang="hi" data-title="गणितीय तर्कशास्त्र" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dka_logika" title="Matematička logika – kroaziera" lang="hr" hreflang="hr" data-title="Matematička logika" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_logika" title="Matematikai logika – hungariera" lang="hu" hreflang="hu" data-title="Matematikai logika" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D6%80%D5%A1%D5%B4%D5%A1%D5%A2%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մաթեմատիկական տրամաբանություն – armeniera" lang="hy" hreflang="hy" data-title="Մաթեմատիկական տրամաբանություն" data-language-autonym="Հայերեն" data-language-local-name="armeniera" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Logika_matematika" title="Logika matematika – indonesiera" lang="id" hreflang="id" data-title="Logika matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Matematikala_logiko" title="Matematikala logiko – idoa" lang="io" hreflang="io" data-title="Matematikala logiko" data-language-autonym="Ido" data-language-local-name="idoa" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/St%C3%A6r%C3%B0fr%C3%A6%C3%B0ileg_r%C3%B6kfr%C3%A6%C3%B0i" title="Stærðfræðileg rökfræði – islandiera" lang="is" hreflang="is" data-title="Stærðfræðileg rökfræði" data-language-autonym="Íslenska" data-language-local-name="islandiera" 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/Logica_matematica" title="Logica matematica – italiera" lang="it" hreflang="it" data-title="Logica matematica" data-language-autonym="Italiano" data-language-local-name="italiera" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E8%AB%96%E7%90%86%E5%AD%A6" title="数理論理学 – japoniera" lang="ja" hreflang="ja" data-title="数理論理学" data-language-autonym="日本語" data-language-local-name="japoniera" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Logika_mat%C3%A9matika" title="Logika matématika – javera" lang="jv" hreflang="jv" data-title="Logika matématika" data-language-autonym="Jawa" data-language-local-name="javera" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%9A%E1%83%9D%E1%83%92%E1%83%98%E1%83%99%E1%83%90" title="მათემატიკური ლოგიკა – georgiera" lang="ka" hreflang="ka" data-title="მათემატიკური ლოგიკა" data-language-autonym="ქართული" data-language-local-name="georgiera" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математикалық логика – kazakhera" lang="kk" hreflang="kk" data-title="Математикалық логика" data-language-autonym="Қазақша" data-language-local-name="kazakhera" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4_%E0%B2%A4%E0%B2%B0%E0%B3%8D%E0%B2%95%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಗಣಿತ ತರ್ಕಶಾಸ್ತ್ರ – kannada" lang="kn" hreflang="kn" data-title="ಗಣಿತ ತರ್ಕಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EB%A6%AC_%EB%85%BC%EB%A6%AC%ED%95%99" title="수리 논리학 – koreera" lang="ko" hreflang="ko" data-title="수리 논리학" data-language-autonym="한국어" data-language-local-name="koreera" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D0%BA_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математикалык логика – kirgizera" lang="ky" hreflang="ky" data-title="Математикалык логика" data-language-autonym="Кыргызча" data-language-local-name="kirgizera" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Logica_mathematica" title="Logica mathematica – latina" lang="la" hreflang="la" data-title="Logica mathematica" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Mathematesch_Logik" title="Mathematesch Logik – luxenburgera" lang="lb" hreflang="lb" data-title="Mathematesch Logik" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxenburgera" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Okusengekensonga_okw%27ekibalo(Mathematical_logic)" title="Okusengekensonga okw'ekibalo(Mathematical logic) – luganda" lang="lg" hreflang="lg" data-title="Okusengekensonga okw'ekibalo(Mathematical logic)" data-language-autonym="Luganda" data-language-local-name="luganda" class="interlanguage-link-target"><span>Luganda</span></a></li><li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://lij.wikipedia.org/wiki/Logica_matematica" title="Logica matematica – liguriera" lang="lij" hreflang="lij" data-title="Logica matematica" data-language-autonym="Ligure" data-language-local-name="liguriera" class="interlanguage-link-target"><span>Ligure</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Simbolin%C4%97_logika" title="Simbolinė logika – lituaniera" lang="lt" hreflang="lt" data-title="Simbolinė logika" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matem%C4%81tisk%C4%81_lo%C4%A3ika" title="Matemātiskā loģika – letoniera" lang="lv" hreflang="lv" data-title="Matemātiskā loģika" data-language-autonym="Latviešu" data-language-local-name="letoniera" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математичка логика – mazedoniera" lang="mk" hreflang="mk" data-title="Математичка логика" data-language-autonym="Македонски" data-language-local-name="mazedoniera" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Logik_matematik" title="Logik matematik – malaysiera" lang="ms" hreflang="ms" data-title="Logik matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática – mirandesa" lang="mwl" hreflang="mwl" data-title="Lógica matemática" data-language-autonym="Mirandés" data-language-local-name="mirandesa" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%9A%E1%80%AF%E1%80%90%E1%80%B9%E1%80%90%E1%80%AD%E1%80%97%E1%80%B1%E1%80%92" title="သင်္ချာယုတ္တိဗေဒ – birmaniera" lang="my" hreflang="my" data-title="သင်္ချာယုတ္တိဗေဒ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmaniera" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskundige_logica" title="Wiskundige logica – nederlandera" lang="nl" hreflang="nl" data-title="Wiskundige logica" data-language-autonym="Nederlands" data-language-local-name="nederlandera" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_logikk" title="Matematisk logikk – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Matematisk logikk" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematisk_logikk" title="Matematisk logikk – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Matematisk logikk" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Yaayaa_Herregaa" title="Yaayaa Herregaa – oromoera" lang="om" hreflang="om" data-title="Yaayaa Herregaa" data-language-autonym="Oromoo" data-language-local-name="oromoera" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Logika_matematyczna" title="Logika matematyczna – poloniera" lang="pl" hreflang="pl" data-title="Logika matematyczna" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D9%85%D9%86%D8%B7%D9%82" title="ریاضیاتی منطق – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ریاضیاتی منطق" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/L%C3%B3gica_matem%C3%A1tica" title="Lógica matemática – portugesa" lang="pt" hreflang="pt" data-title="Lógica matemática" data-language-autonym="Português" data-language-local-name="portugesa" 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/Logic%C4%83_matematic%C4%83" title="Logică matematică – errumaniera" lang="ro" hreflang="ro" data-title="Logică matematică" data-language-autonym="Română" data-language-local-name="errumaniera" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математическая логика – errusiera" lang="ru" hreflang="ru" data-title="Математическая логика" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Mathematical_logic" title="Mathematical logic – eskoziera" lang="sco" hreflang="sco" data-title="Mathematical logic" data-language-autonym="Scots" data-language-local-name="eskoziera" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matemati%C4%8Dka_logika" title="Matematička logika – serbokroaziera" lang="sh" hreflang="sh" data-title="Matematička logika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%B8%E0%B6%BA_%E0%B6%AD%E0%B6%BB%E0%B7%8A%E0%B6%9A%E0%B6%AB%E0%B6%BA" title="ගණිතමය තර්කණය – sinhala" lang="si" hreflang="si" data-title="ගණිතමය තර්කණය" data-language-autonym="සිංහල" data-language-local-name="sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_logic" title="Mathematical logic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical logic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematick%C3%A1_logika" title="Matematická logika – eslovakiera" lang="sk" hreflang="sk" data-title="Matematická logika" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" 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/Matemati%C4%8Dna_logika" title="Matematična logika – esloveniera" lang="sl" hreflang="sl" data-title="Matematična logika" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Logjika_matematikore" title="Logjika matematikore – albaniera" lang="sq" hreflang="sq" data-title="Logjika matematikore" data-language-autonym="Shqip" data-language-local-name="albaniera" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BB%D0%BE%D0%B3%D0%B8%D0%BA%D0%B0" title="Математичка логика – serbiera" lang="sr" hreflang="sr" data-title="Математичка логика" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematisk_logik" title="Matematisk logik – suediera" lang="sv" hreflang="sv" data-title="Matematisk logik" data-language-autonym="Svenska" data-language-local-name="suediera" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BD%D1%82%D0%B8%D2%9B%D0%B8_%D1%80%D0%B8%D1%91%D0%B7%D3%A3" title="Мантиқи риёзӣ – tajikera" lang="tg" hreflang="tg" data-title="Мантиқи риёзӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="tajikera" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="คณิตตรรกศาสตร์ – thailandiera" lang="th" hreflang="th" data-title="คณิตตรรกศาสตร์" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Lohikang_matematikal" title="Lohikang matematikal – tagaloa" lang="tl" hreflang="tl" data-title="Lohikang matematikal" data-language-autonym="Tagalog" data-language-local-name="tagaloa" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematiksel_mant%C4%B1k" title="Matematiksel mantık – turkiera" lang="tr" hreflang="tr" data-title="Matematiksel mantık" data-language-autonym="Türkçe" data-language-local-name="turkiera" 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%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%BB%D0%BE%D0%B3%D1%96%D0%BA%D0%B0" title="Математична логіка – ukrainera" lang="uk" hreflang="uk" data-title="Математична логіка" data-language-autonym="Українська" data-language-local-name="ukrainera" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D9%85%D9%86%D8%B7%D9%82" title="ریاضیاتی منطق – urdua" lang="ur" hreflang="ur" data-title="ریاضیاتی منطق" data-language-autonym="اردو" data-language-local-name="urdua" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematik_mantiq" title="Matematik mantiq – uzbekera" lang="uz" hreflang="uz" data-title="Matematik mantiq" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekera" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Logic_to%C3%A1n" title="Logic toán – vietnamera" lang="vi" hreflang="vi" data-title="Logic toán" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91" title="数理逻辑 – wu txinera" lang="wuu" hreflang="wuu" data-title="数理逻辑" data-language-autonym="吴语" data-language-local-name="wu txinera" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E9%80%BB%E8%BE%91" title="数理逻辑 – txinera" lang="zh" hreflang="zh" data-title="数理逻辑" data-language-autonym="中文" data-language-local-name="txinera" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/S%C3%B2%CD%98-l%C3%AD_su-l%C3%AD" title="Sò͘-lí su-lí – Minnan" lang="nan" hreflang="nan" data-title="Sò͘-lí su-lí" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%B8%E5%AD%B8%E9%82%8F%E8%BC%AF" title="數學邏輯 – kantonera" lang="yue" hreflang="yue" data-title="數學邏輯" data-language-autonym="粵語" data-language-local-name="kantonera" class="interlanguage-link-target"><span>粵語</span></a></li> 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="eu" dir="ltr"><p><b>Logika matematikoa</b>, <b>logika sinbolikoa</b>, <b>logika teoretikoa</b>, <b>logika formala</b> edo <b>logistika</b> ere deitua<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>, <a href="/wiki/Logika" title="Logika">logikaren</a> azterketa formal eta sinbolikoa da, eta <a href="/wiki/Matematika" title="Matematika">matematikaren</a> eta <a href="/wiki/Zientzia" title="Zientzia">zientziaren</a> arlo batzuetan duen aplikazioa. Logika formalaren teknikak matematikaren eta arrazoiketa matematikoaren eraikuntzan eta garapenean aplikatzea ulertzen du, eta, elkarrekin, logika formalaren irudikapenean eta analisian teknika matematikoak aplikatzea. Logika matematikoaren ikerketak berebiziko garrantzia izan du <a href="/w/index.php?title=Matematikaren_oinarriak&action=edit&redlink=1" class="new" title="Matematikaren oinarriak (sortu gabe)">matematikaren oinarriak</a> aztertzean. </p><p>Logika matematikoak <a href="/wiki/Inferentzia" class="mw-disambig" title="Inferentzia">inferentziaz</a> aztertzen du logika proposizionala, <a href="/wiki/Lehen_mailako_logika" title="Lehen mailako logika">lehen mailako logika</a> edo logika modala bezalako <a href="/w/index.php?title=Sistema_formal&action=edit&redlink=1" class="new" title="Sistema formal (sortu gabe)">sistema formalak</a> eraikiz. Sistema hauek, <a href="/wiki/Lengoaia_natural" class="mw-redirect" title="Lengoaia natural">lengoaia naturaletan</a> balio duten inferentzien funtsezko ezaugarriak harrapatzen dituzte, baina, <a href="/wiki/Analisi_matematiko" title="Analisi matematiko">analisi matematikoa</a> izan dezaketen egitura formalak direnez, hauei buruzko <a href="/wiki/Demostrazio_matematiko" class="mw-redirect" title="Demostrazio matematiko">demostrazio</a> zorrotzak egitea ahalbidetzen dute. </p><p>Logika matematikoa lau arlotan banatu ohi da: <a href="/w/index.php?title=Ereduen_teoria&action=edit&redlink=1" class="new" title="Ereduen teoria (sortu gabe)">ereduen teoria</a>, <a href="/w/index.php?title=Demostrazioaren_teoria&action=edit&redlink=1" class="new" title="Demostrazioaren teoria (sortu gabe)">demostrazioaren teoria</a>, <a href="/wiki/Multzoen_teoria" class="mw-redirect" title="Multzoen teoria">multzoen teoria</a> eta <a href="/wiki/Konputagarritasunaren_teoria" title="Konputagarritasunaren teoria">konputagarritasunaren teoria</a>. Frogapenaren teoria eta ereduen teoria logika matematikoaren oinarria izan ziren. Multzoen teoria <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantorek</a> <a href="/wiki/Infinitu" title="Infinitu">infinituaren</a> azterketan sortu zen, eta logika matematikoaren gai desafiatzaile eta garrantzitsuenetako askoren iturria izan da, <a href="/wiki/Cantorren_teorema" title="Cantorren teorema">Cantorren teorematik</a>, <a href="/w/index.php?title=Aukerako_axioma&action=edit&redlink=1" class="new" title="Aukerako axioma (sortu gabe)">aukerako axiomatik</a> eta <a href="/w/index.php?title=Continuumaren_hipotesia&action=edit&redlink=1" class="new" title="Continuumaren hipotesia (sortu gabe)">continuumaren hipotesiaren</a> independentziatik abiatuta, <a href="/w/index.php?title=Axioma_kardinal&action=edit&redlink=1" class="new" title="Axioma kardinal (sortu gabe)">axioma kardinal</a> handiei buruzko eztabaida modernora. Logika matematikoak lotura estuak ditu konputazio zientziekin. Konputagarritasunaren teoriak konputazioaren ideia termino logiko eta aritmetikoetan harrapatzen du. Bere lorpenik klasikoenak <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turingen</a> <i><a href="/w/index.php?title=Entscheidungsproblem&action=edit&redlink=1" class="new" title="Entscheidungsproblem (sortu gabe)">Entscheidungsproblem</a></i> delakoaren adierazezintasuna eta <a href="/w/index.php?title=Church-Turingen_tesia&action=edit&redlink=1" class="new" title="Church-Turingen tesia (sortu gabe)">Church-Turingen tesiaren</a> aurkezpena dira. Gaur egun, konputagarritasunaren teoria konplexutasun-moten arazo finenaz (noiz da eraginkortasunez konpon daitekeen arazoa?) eta <a href="/w/index.php?title=Disolbaezintasun-gradu&action=edit&redlink=1" class="new" title="Disolbaezintasun-gradu (sortu gabe)">disolbaezintasun-graduen</a> sailkapenaz arduratzen da nagusiki. </p><p>Logika matematikoak oinarrizko nozio eta objektu matematikoen definizioak ere aztertzen ditu, hala nola, multzoak, <a href="/wiki/Zenbaki" title="Zenbaki">zenbakiak</a>, demostrazioak eta <a href="/wiki/Algoritmo" title="Algoritmo">algoritmoak</a>. Logika matematikoak dedukzio-arau formalak, hizkuntza formalen adierazpen-gaitasunak eta horien propietate metalogikoak aztertzen ditu. </p><p>Oinarrizko maila batean, logikak arauak eta teknikak ematen ditu sistema formal jakin baten barruan emandako argudio bat baliozkoa den ala ez erabakitzeko. Maila aurreratu batean, logika matematikoa teoria matematikoak axiomatizatzeko aukeraz arduratzen da, haien adierazpen-gaitasuna sailkatzeaz eta sistema formaletan baliagarriak diren metodo konputazionalak garatzeaz. Frogapenaren teoria eta <a href="/w/index.php?title=Alderantzizko_matematika&action=edit&redlink=1" class="new" title="Alderantzizko matematika (sortu gabe)">alderantzizko matematika</a> logika matematiko abstraktuaren azken arrazoibideetako bi dira. Aipatu behar da logika matematikoa bere alderdi guztietan baliokideak ez diren sistema formalez arduratzen dela, eta, beraz, logika matematikoa ez da <a href="/wiki/Mundu_fisiko" class="mw-redirect" title="Mundu fisiko">mundu fisiko</a> errealeko egiak aurkitzeko metodo bat, baizik eta <a href="/wiki/Teoria_zientifiko" title="Teoria zientifiko">teoria zientifikoei</a>, bereziki matematika konbentzionalari, aplika dakizkiekeen eredu logikoen iturri posible bat. </p><p>Bestalde, logika matematikoak ez du giza <a href="/wiki/Arrazonamendu" class="mw-redirect" title="Arrazonamendu">arrazonamendu</a> orokorraren kontzeptua edo argumentu zorrotzen bidez baina zenbait zeinu edo diagramarekin lengoaia informala erabiliz demostrazio matematikoak eraikitzeko prozesu sortzailea aztertzen, baizik eta erabat formaliza daitezkeen demostrazio eta arrazonamenduena. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading3"><h3 id="XIX._Mendea">XIX. Mendea</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logika_matematiko&veaction=edit&section=1" title="Aldatu atal hau: «XIX. Mendea»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logika_matematiko&action=edit&section=1" title="Edit section's source code: XIX. Mendea"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>XIX. mendearen bigarren erditik aurrera, logika goitik behera irauli zen. 1847an, <a href="/wiki/George_Boole" title="George Boole">George Boolek</a> Logikaren analisi matematikoa izeneko tratatu labur bat argitaratu zuen, eta 1854an beste garrantzitsuago bat, <i>Pentsamenduaren legeak</i> izenekoa. Booleren ideia logika kalkulu gisa eraikitzea izan zen, non egiaren balioak F (faltsukeria) eta V (egia) bidez adierazten diren, eta horiei batuketa eta biderketa bezalako eragiketa matematikoak aplikatzen zaizkien. </p><p>XIX. mendearen azken herenean logikak bere eraldaketarik sakonena aurkitu zuen ikerketa matematiko eta logikoen eskutik, hizkuntzaren egitura sakonen, linguistikoaren ikerketaren garapenarekin batera, behin betiko zientzia formal bihurtuz. Zientzia formala da, ideiak aztertzen dituelako eta ezagutzaren beste zientzia eta arlo guztietarako tresna kontzeptuala delako. Ezagutza arrazionalen eta koherenteen multzo sistematiko baten parte da, prozesu logikoak eta matematikoak aztertzeaz arduratzen dena. </p><p>Aldi berean, <a href="/w/index.php?title=Augustus_De_Morgan&action=edit&redlink=1" class="new" title="Augustus De Morgan (sortu gabe)">Augustus De Morganek</a> 1847an bere Logika formala argitaratu zuen. Bertan, <a href="/wiki/De_Morganen_legeak" title="De Morganen legeak">De Morganen legeak</a> sartu zituen eta silogismoaren nozioa orokortzen saiatu zen. Beste zergadun ingeles garrantzitsu bat <a href="/wiki/John_Venn" title="John Venn">John Venn</a> izan zen, 1881ean <i>Logika Sinbolikoa</i> liburua argitaratu zuena, non Vennen diagrama ospetsuak sartu zituen. </p><p><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Charles Sanders Peircek</a> eta <a href="/w/index.php?title=Ernst_Schr%C3%B6der&action=edit&redlink=1" class="new" title="Ernst Schröder (sortu gabe)">Ernst Schröderrek</a> ere ekarpen garrantzitsuak egin zituzten. </p><p>Hala ere, logikaren benetako iraultza <a href="/wiki/Gottlob_Frege" title="Gottlob Frege">Gottlob Fregeren</a> eskutik etorri zen, sarritan Aristotelesekin batera historiako logikarik garrantzitsuentzat hartzen dena. 1879ko bere lanean, <i>Konzeptografian</i>, Fregek predikatuen logika eta kalkulu proposizionalaren sistema oso bat eskaintzen du lehen aldiz. Hizkuntza formalaren ideia ere garatzen du eta frogaren nozioa definitzen du. Ideia horiek oinarrizko oinarri teorikoa izan ziren konputagailuen eta konputazio-zientzien garapenerako, besteak beste. Hala eta guztiz ere, Fregeren garaikideek alde batera utzi zituzten beren ekarpenak, ziur aski egileak garatu zuen notazio korapilatsuagatik. 1893 eta 1903an, Fregek Aritmetikaren legeak bi liburukitan argitaratu zuen, non matematika osoa logikatik deduzitzen saiatu zen, proiektu logistiko bezala ezagutzen den horretan. Haren sistemak eta multzoen teorian zuen aplikazioak, ordea, kontraesana zekarren (<a href="/wiki/Russellen_paradoxa" title="Russellen paradoxa">Russellen paradoxa</a>). </p><p>Logika matematikoa <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peanok</a> diziplina honetarako emandako izena izan zen. Funtsean Aristotelesen logika da, baina aljebratik hartutako notazio berri abstraktuago baten ikuspuntutik. </p> <div class="mw-heading mw-heading3"><h3 id="XX._Mendea">XX. Mendea</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logika_matematiko&veaction=edit&section=2" title="Aldatu atal hau: «XX. Mendea»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logika_matematiko&action=edit&section=2" title="Edit section's source code: XX. Mendea"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>XX. mendean logikaren garapen itzeletako bat izan zen. XX. mendetik aurrera, logika bere berezko interesagatik aztertu zen, eta ez bakarrik <a href="/wiki/Propedeutika" title="Propedeutika">propedeutika</a> gisa zituen bertuteengatik, askoz maila abstraktuagoetan aztertu zen.⁸ </p><p>1910ean, <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russellek</a> eta <a href="/wiki/Alfred_North_Whitehead" title="Alfred North Whitehead">Alfred North Whiteheadek</a> Principia mathematica argitaratu zuten. Lan monumental horretan, matematikaren zati handi bat logikatik abiatuta lortu zuten, Frege erori zen paradoxetan erortzea saihestuz. Teoria matematikoak tautologia logikoak zirela suposatzen zen, eta programak hori erakutsi behar zuen matematika logikara murriztuz. Autoreek Fregeren meritua aitortzen dute hitzaurrean. Fregeren lanarekin kontrastean, Principia mathematicak erabateko arrakasta izan zuen, eta XX. mende osoko ez-fikziozko lan garrantzitsu eta eraginkorrenetakotzat jo zen. Principia mathematicak <a href="/wiki/Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a>renean inspiratutako notazio bat erabiltzen du, gaur egun oraindik asko erabiltzen dena. </p><p>1912an I. Lewisek <i>Conditionals</i> and <i>the Algebra of Logic</i> argitaratu zuen, Russell eta Whitehead-en <i>Printzia Mathematikoa</i>ren ondoren. 1918an <i>A Survey of Symbolic Logic</i> argitaratu zuen, eta bertan beste baldintza egokiago bat proposatu zuen hizkuntza naturalaren «bai... orduan» esamoldearen esanahia jasotzeko. Lewisek inplikazio zorrotza deitzen dio. Baldintza berriak, egiazkoa izateko, aurrekariaren eta kontsekuentearen arteko erlazio sendoagoa eskatzen du, baldintza klasikoa baino. </p><p>1920an <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbertek</a> modu esplizituan proposatu zuen Hilbert-en programa bezala ezagutua izan zen ikerketa-proiektu bat . Matematika oinarri sendo eta erabat logikoetan formulatzea nahi zuen. Proiektua Gödelen osatu gabeko teoremek ezeztatu zuten. Hilberten programaren adierazpena eta Gödelen ezeztapena logika matematikoaren bigarren esparrua, matematika logikari aplikatzea frogapenaren teoriaren forman, ezartzearen mende zeuden. Osatugabetasunaren teoremen izaera negatiboa gorabehera, Gödelen konplexutasunaren teorema, emaitza bat ereduen teorian eta matematikaren beste aplikazio bat logikan, logismo hurbilaren frogapen gisa uler daiteke: hertsiki definitutako teoria matematiko oro lehen mailako teoria batek zehatz harrapa dezake. Frege-ren proba kalkulatzea nahikoa da matematika osoa deskribatzeko, nahiz eta haren baliokidea ez izan. </p><p>Konputazio-eredu abstraktuen jatorria 1930eko hamarkadan kokatzen da (ordenagailu modernoak existitu aurretik), <a href="/wiki/Alonzo_Church" title="Alonzo Church">Alonzo Church</a>, <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>, <a href="/w/index.php?title=Stephen_Kleene&action=edit&redlink=1" class="new" title="Stephen Kleene (sortu gabe)">Stephen Kleene</a>, <a href="/w/index.php?title=Emil_Leon_Post&action=edit&redlink=1" class="new" title="Emil Leon Post (sortu gabe)">Emil Leon Post</a>, <a href="/wiki/Haskell_Curry" title="Haskell Curry">Haskell Curry</a> eta <a href="/wiki/Alan_Turing" title="Alan Turing">Alan Turing</a> logikoen lanean. Hasierako lan horiek eragin handia izan dute, bai garapen teorikoan, bai konputazioaren praktikaren alderdi askotan; izan ere, helburu orokorreko ordenagailuen existentzia, programak interpretatzeko aukera, softwarearen eta hardwarearen arteko dualtasuna eta produkzio arauetan oinarritutako egitura formalen bidezko lengoaien irudikapena aurreikusi dira. </p><p><a href="/w/index.php?title=Gerhard_Gentzen&action=edit&redlink=1" class="new" title="Gerhard Gentzen (sortu gabe)">Gerhard Gentzenek</a> <i>inferentzia logikoari buruzko ikerketak (Untersuchungen über das logische Schliessen)</i> lanean sartu zuen dedukzio naturala, 1934-1935 aldian argitaratua. </p><p>1940ko hamarkadan <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a>, bere ikasleekin batera, aljebra erlazionala garatzen hasi zen, non multzoen teoria axiomatikoa nahiz Peanoren aritmetika adieraz daitezkeen. Bere dizipuluekin batera aljebra zilindrikoak ere garatu zituen, aljebra boolearrak logika proposizionalera lehen mailako logikara doazenak. 1941ean ingelesez logikako eskuliburu ospetsuenetako bat argitaratu zuen, I<i>ntroduction to Logic and to the Methodology of Deductive Sciences</i>. </p><p><a href="/wiki/Noam_Chomsky" title="Noam Chomsky">Noam Chomskyk</a> 1956an hizkuntza formalak sortzen dituzten gramatika formalen sailkapen hierarkiko bat proposatzen du, <i>Chomskyren hierarkia</i> izenekoa. </p><p>Sistema garaikideen argitan logika aristotelikoa okerra eta osatugabea irudi dezakeen arren, <a href="/wiki/Jan_%C5%81ukasiewicz" title="Jan Łukasiewicz">Jan Lukasiewiczek</a> erakutsi zuen, zailtasun handiak izan arren, logika aristotelikoa sendoa zela, klaseen logika gisa interpretatu behar bazen ere, eta hori ez da aldaketa txikia. Horregatik, silogistikak ia ez du erabilerarik gaur egun. </p><p>Logika proposizionalaz eta predikatuen logikaz gain, XX. mendeak beste sistema formal askoren garapena ikusi zuen; horien artean, jokamolde logiko asko nabarmentzen dira. </p> <div class="mw-heading mw-heading2"><h2 id="Erreferentziak">Erreferentziak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logika_matematiko&veaction=edit&section=3" title="Aldatu atal hau: «Erreferentziak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Logika_matematiko&action=edit&section=3" title="Edit section's source code: Erreferentziak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation" id="CITEREFAgazzi,_Evandro.1986">Agazzi, Evandro..  (1986). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/56760455"><i>La lógica simbólica. </i></a> (4. ed. argitaraldia) Editorial Herder <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/84-254-0130-5" title="Berezi:BookSources/84-254-0130-5">84-254-0130-5</a>. <a href="/wiki/PubMed_Central" class="mw-redirect" title="PubMed Central">PMC</a> <a rel="nofollow" class="external text" href="http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=56760455">56760455</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2020-05-18)</small></span></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logika_matematiko&veaction=edit&section=4" title="Aldatu atal hau: «Kanpo estekak»" 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