Matematiksel Mant\u0131k<\/p>\n
\u00c7a\u011fda\u015f mant\u0131\u011f\u0131n ve \u00e7a\u011fda\u015f felsefenin kurucusu Alman mant\u0131k\u00e7\u0131s\u0131 Gottlob Frege, “Matematik mant\u0131\u011f\u0131n uygulama alan\u0131d\u0131r” g\u00f6r\u00fc\u015f\u00fcnden hareketle matemati\u011fin, mant\u0131\u011f\u0131n aksiyomatik sistemi \u00fczerine kurulabilece\u011fini d\u00fc\u015f\u00fcnm\u00fc\u015ft\u00fcr. Bu d\u00fc\u015f\u00fcnceden hareket ederek aritmeti\u011fin temelleri konusundaki felsefi \u00e7al\u0131\u015fmalar\u0131 i\u00e7in bir mant\u0131k sistemi geli\u015ftirmi\u015fti.\u00c7a\u011fda\u015f mant\u0131\u011f\u0131n ve \u00e7a\u011fda\u015f felsefenin kurucusu Alman mant\u0131k\u00e7\u0131s\u0131 Gottlob Frege, “Matematik mant\u0131\u011f\u0131n uygulama alan\u0131d\u0131r” g\u00f6r\u00fc\u015f\u00fcnden hareketle matemati\u011fin, mant\u0131\u011f\u0131n aksiyomatik sistemi \u00fczerine kurulabilece\u011fini d\u00fc\u015f\u00fcnm\u00fc\u015ft\u00fcr. Bu d\u00fc\u015f\u00fcnceden hareket ederek aritmeti\u011fin temelleri konusundaki felsefi \u00e7al\u0131\u015fmalar\u0131 i\u00e7in bir mant\u0131k sistemi geli\u015ftirmi\u015fti.<\/p>\n
Daha sonra, Frege’nin \u00e7al\u0131\u015fmalar\u0131na dayanarak, Russell ve Whitehead 1910-1913 y\u0131llar\u0131 aras\u0131nda Principia Mathematica ad\u0131n\u0131 verdikleri eserde matemati\u011fi mant\u0131\u011fa indirgeyerek formel bir sistem haline getirmeye \u00e7al\u0131\u015ft\u0131lar. Fakat matemati\u011fin formel hale getirilemeyece\u011fini G\u00f6del 1933’te yay\u0131nlad\u0131\u011f\u0131 bir kitab\u0131ndaki (\u00dcber die unentsheidbare Saetze der Principia Mathematica und verwander Systeme) me\u015fhur teoremiyle g\u00f6sterdi.<\/p>\n
Alan Robinson, 1967’de \u00e7\u00f6z\u00fcl\u00fcm teorem ispatlama y\u00f6ntemini geli\u015ftirdi. Bu y\u00f6ntem 1972’de A. Colmaurer taraf\u0131ndan ilk mant\u0131k programlama dilinin (Prolog) geli\u015ftirilmesine yol a\u00e7t\u0131. Bu dil 1975’te D. Warren taraf\u0131ndan \u201cWarren Abstract Machine\u201d (WAM) olarak uguland\u0131. Ki\u015fisel bilgisayarlar \u00fczerinde ilk uygulamalar 1980’lerde ortaya \u00e7\u0131kt\u0131.<\/p>\n
Daha sonra, Frege’nin \u00e7al\u0131\u015fmalar\u0131na dayanarak, Russell ve Whitehead 1910-1913 y\u0131llar\u0131 aras\u0131nda Principia Mathematica ad\u0131n\u0131 verdikleri eserde matemati\u011fi mant\u0131\u011fa indirgeyerek formel bir sistem haline getirmeye \u00e7al\u0131\u015ft\u0131lar. Fakat matemati\u011fin formel hale getirilemeyece\u011fini G\u00f6del 1933’te yay\u0131nlad\u0131\u011f\u0131 bir kitab\u0131ndaki (\u00dcber die unentsheidbare Saetze der Principia Mathematica und verwander Systeme) me\u015fhur teoremiyle g\u00f6sterdi.<\/p>\n
Alan Robinson, 1967’de \u00e7\u00f6z\u00fcl\u00fcm teorem ispatlama y\u00f6ntemini geli\u015ftirdi. Bu y\u00f6ntem 1972’de A. Colmaurer taraf\u0131ndan ilk mant\u0131k programlama dilinin (Prolog) geli\u015ftirilmesine yol a\u00e7t\u0131. Bu dil 1975’te D. Warren taraf\u0131ndan \u201cWarren Abstract Machine\u201d (WAM) olarak uguland\u0131. Ki\u015fisel bilgisayarlar \u00fczerinde ilk uygulamalar 1980’lerde ortaya \u00e7\u0131kt\u0131.<\/p>\n
\u00d6NERMELER MANTI\u011eI<\/p>\n
Formel sistemler \u015fu elemanlardan meydana gelir:<\/p>\n
Tan\u0131mlanmam\u0131\u015f terimler
\n Tan\u0131mlar
\n T\u00fcretme kurallar\u0131
\n Aksiyomlard\u0131r
\n Teoremler
\n Formel mant\u0131\u011f\u0131n tan\u0131mlanmam\u0131\u015f terimleri olarak, basit \u00f6nerme ( P ) ve mant\u0131ksal ba\u011flar (de\u011fil, ve, veya, e\u011fer-ise, e\u011fer ve ancak-ise) g\u00f6sterilebilir.<\/p>\n
Tan\u0131mlanan terimlere \u00f6rnek olarak bile\u015fik \u00f6nerme kavram\u0131n\u0131 g\u00f6sterilebilir. Asl\u0131nda yukar\u0131da verilen mant\u0131ksal ba\u011flar bir tek mant\u0131ksal ba\u011f yard\u0131m\u0131yla tan\u0131mlanabilir.<\/p>\n
g\u00f6n\u00fcl isterdiki daha ayr\u0131nt\u0131l\u0131 ve g\u00fczel bi \u00f6nermeler olsun ama yok : (<\/p>\n
Olumsuzu [de\u011fi\u015ftir]Bir \u00f6nerme \u201cde\u011fil\u201d eki ile kar\u015f\u0131t ifadeye \u00e7evrilebilir; buna de\u011filleme denir.<\/p>\n
\u00d6rnek: “bu g\u00fcn g\u00fcnlerden sal\u0131: Bu g\u00fcn g\u00fcnlerden sal\u0131 degil.<\/p>\n
Ayr\u0131l\u0131m [de\u011fi\u015ftir]\u0130ki veya daha fazla basit \u00f6nermeden \u201cveya\u201d (ya da) mant\u0131ksal ba\u011f\u0131n\u0131 kullanarak bilesik \u00f6nermeler kurulabilir.<\/p>\n
\u00d6rnek: \u201cBug\u00fcn Ar\u00e7elik veya Teleta\u015f’tan ziyaret\u00e7iler gelecek.\u201d<\/p>\n
\u015eartl\u0131 c\u00fcmle [de\u011fi\u015ftir]Ayn\u0131 \u015fekilde, iki veya daha fazla say\u0131da \u00f6nermeden (e\u011fer-ise) ba\u011f\u0131n\u0131 kullanarak \u015fartl\u0131 \u00f6nermeler kurulabilir.<\/p>\n
\u00d6rnek: \u201cE\u011fer ya\u011fmur ya\u011f\u0131yor ise, hava bulutludur.\u201d<\/p>\n
Bazen \u201ce\u011fer-ise\u201d ba\u011f\u0131 yerine do\u011fal dilde \u201cgerektirir\u201d ba\u011f\u0131n\u0131 da kullanabiliyoruz.<\/p>\n
\u00d6rnek: \u201cYa\u011fmurun ya\u011f\u0131yor olmas\u0131 havan\u0131n bulutlu olmas\u0131n\u0131 gerektirir.\u201d<\/p>\n
\u00c7ift \u015fartl\u0131 \u00f6nermeler [de\u011fi\u015ftir]Yine, \u201ce\u011fer ve ancak-ise\u201d ba\u011f\u0131n\u0131 kullanarak birden fazla \u00f6nermeden \u00e7ift \u015fartl\u0131 \u00f6nermeler kurulabilir. Bu t\u00fcr \u00f6nermeler do\u011fal dilde daha az kullan\u0131lmas\u0131na ra\u011fmen, fizik ve matematikte s\u0131k s\u0131k kullan\u0131lmaktad\u0131r.<\/p>\n
\u00d6rnek: \u201cE\u011fer ve ancak \u00e7al\u0131\u015fanlar \u00fccretlerde a\u015f\u0131r\u0131 art\u0131\u015f talep ederlerse enflasyon d\u00fc\u015fmez.\u201d<\/p>\n
\u00c7ikita muz u yazan ajdar\u0131n dedi\u011fi gibi sen ate\u015fsin ben barut olarakda bilinir ve e\u011fer enflasyon d\u00fc\u015fmezse \u00e7al\u0131\u015fanlar \u00fccretlerde a\u015f\u0131r\u0131 art\u0131\u015f talep ederler.\u201d<\/p>\n
Cebirde oldu\u011fu gibi, sembolik veya matematiksel mant\u0131kta da, \u00f6nermeler yerine \u00f6nermesel de\u011fi\u015fkenler kullan\u0131l\u0131r (P, Q, R, S, T harfleri gibi).<\/p>\n
Mant\u0131ksal ba\u011flar [de\u011fi\u015ftir]Mant\u0131ksal ba\u011flar :<\/p>\n
: de\u011fil
\n : ve
\n : veya
\n : e\u011fer-ise
\n : e\u011fer ancak-ise<\/p>\n
Totoloji
\n Bir \u00f6nermesel form\u00fcl\u00fcn (veya bile\u015fik \u00f6nermenin) do\u011fruluk cetvelindeki son de\u011ferlendirme s\u00fctunundaki b\u00fct\u00fcn de\u011ferler \u201cdo\u011fru\u201d \u00e7\u0131k\u0131yorsa, bu \u00f6nermesel form\u00fcle \u201ctotoloji\u201d denir.
\n \u00c7eli\u015fki
\n Bir \u00f6nermesel form\u00fcl\u00fcn (veya bile\u015fik \u00f6nermenin) do\u011fruluk cetvelindeki son de\u011ferlendirme s\u00fctunundaki b\u00fct\u00fcn de\u011ferler \u201cyanl\u0131\u015f\u201d \u00e7\u0131k\u0131yorsa bu \u00f6nermesel form\u00fcle \u201c\u00e7eli\u015fki\u201d denir.
\n Bazen do\u011fruluk
\n Bir \u00f6nermesel form\u00fcl\u00fcn (veya bile\u015fik \u00f6nermenin) do\u011fruluk cetvelindeki son de\u011ferlendirme s\u00fctunundaki de\u011ferlerden baz\u0131lar\u0131 \u201cdo\u011fru\u201d baz\u0131lar\u0131 \u201cyanl\u0131\u015f\u201d \u00e7\u0131k\u0131yorsa bu \u00f6nermesel form\u00fcle \u201cbazen do\u011fru\u201d denir.
\n Tutarl\u0131l\u0131k
\n Bir bile\u015fik \u00f6nermeye \u201cve\u201d ekiyle ba\u015fka bir \u00f6nerme eklendi\u011fi zaman bir \u00e7eli\u015fki ortaya \u00e7\u0131km\u0131yorsa, eklenen \u00f6nerme \u00f6ncekiyle tutarl\u0131d\u0131r denir.
\n Ge\u00e7erlilik
\n Bir A1, A2, …, An \u00f6nerme dizisindeki b\u00fct\u00fcn A\u2019lar do\u011fru oldu\u011fu zaman bir B h\u00fckm\u00fc de do\u011fru oluyorsa B\u2019ye A1, A2, …, An \u00f6nermelerinin ge\u00e7erli sonucudur denir. Ge\u00e7erlilik \u015fu \u015fekilde g\u00f6sterilir:
\n A1, A2, …, An |= B.
\n Mant\u0131ksal \u0130\u00e7erik
\n Bir bile\u015fik \u00f6nermeyi yanl\u0131\u015f yapan \u015fartlar\u0131n say\u0131s\u0131n\u0131n b\u00fct\u00fcn \u015fartlar\u0131n say\u0131s\u0131na oran\u0131 ne kadar b\u00fcy\u00fckse, o \u00f6nermenin mant\u0131ksal i\u00e7eri\u011fi o kadar fazlad\u0131r. \u00c7eli\u015fkinin mant\u0131ksal i\u00e7eri\u011finden bahsedilemez (\u00e7\u00fcnk\u00fc yoktur.).(–>bu durumda \u00e7eli\u015fki i\u00e7in mant\u0131ksal i\u00e7erik 1\/1 olmas\u0131 beklenir. buna g\u00f6re ilk c\u00fcmle ile bahsedilen tan\u0131m tersi olarak d\u00fc\u015f\u00fcn\u00fclmesi gerekmektedir =>d\u00fczeltmedir, \u015fayet hata yok ise siliniz?) <\/p>\n
Y\u00dcKLEMELER MANTI\u011eI<\/p>\n
\u00d6nermeler mant\u0131\u011f\u0131n\u0131n t\u00fcretim kurallar\u0131 matematik i\u00e7in yeterli olmad\u0131\u011f\u0131 gibi g\u00fcndelik dil i\u00e7in de yeterli de\u011fildir. Mesela, klasik mant\u0131kta “Her asal say\u0131 bir do\u011fal say\u0131d\u0131r” ve “3 asal say\u0131d\u0131r” \u00f6nc\u00fcllerinden, “3 do\u011fal say\u0131d\u0131r” sonucunu \u00e7\u0131karabiliyoruz. Fakat bu ak\u0131l y\u00fcr\u00fctmenin do\u011frulu\u011fu, \u00f6nermeler mant\u0131\u011f\u0131n\u0131n kurallar\u0131 \u00e7er\u00e7evesi i\u00e7inde kan\u0131tlanamaz. Bunun nedeni de \u015fudur: \u00d6nermeler mant\u0131\u011f\u0131 bile\u015fik \u00f6nermeler i\u00e7indeki basit \u00f6nermeler aras\u0131ndaki mant\u0131ksal ba\u011flara ve basit \u00f6nermelerin do\u011fruluk de\u011ferlerine g\u00f6re bile\u015fik \u00f6nermelerin do\u011fruluklar\u0131n\u0131 inceler. Di\u011fer bir deyi\u015fle, \u00f6nermeler mant\u0131\u011f\u0131 bir \u00f6nermeyi bir\u00e7ok maksat i\u00e7in yeterli ayr\u0131nt\u0131da analiz etmez.<\/p>\n
\u0130\u015fte, terimler, y\u00fcklemler ve niceleyiciler diye isimlendirece\u011fimiz mant\u0131ksal kavramlar yard\u0131m\u0131yla g\u00fcndelik dili ve matemati\u011fin dilini b\u00fcy\u00fck \u00f6l\u00e7\u00fcde sembolize edebiliriz.<\/p>\n
Y\u00fcklemler mant\u0131\u011f\u0131nda da ayn\u0131 matematikte oldu\u011fu gibi, sabitler ve de\u011fi\u015fkenler kullan\u0131l\u0131r. Biraz \u00f6nce bahsedilen “terimleri” iki s\u0131n\u0131fa ay\u0131rabiliriz: Bireysel de\u011fi\u015fkenler, bireysel sabitler. Bireysel sabitlere \u00f6rnek olarak birey oldu\u011funu bildi\u011fimiz varl\u0131klar\u0131 sayabiliriz: \u201cG\u00f6khan\u201d, \u201cTekir\u201d, \u201cg\u00fcl\u201d gibi. Bunlar yerine de \u201cinsan\u201d, \u201chayvan\u201d, \u201cbitki\u201d kavramlar\u0131n\u0131n \u00e7er\u00e7eveleri i\u00e7inde olmak \u00fczere x, y, z, de\u011fi\u015fken sembollerini kullanabiliyoruz.<\/p>\n
Matematikte de\u011fi\u015fkenler genellikle say\u0131lar veya fonksiyonlar olabilir. Y\u00fcklemler mant\u0131\u011f\u0131nda ise bireysel terimler de\u011fi\u015fken olabildi\u011fi gibi, y\u00fcklemler de sabit veya de\u011fi\u015fken olabilir. Y\u00fcklemsel sabitlere \u00f6rnek olarak \u00f6nermeler i\u00e7inde yer alan y\u00fcklemleri g\u00f6sterebiliriz: \u201csay\u0131\u201d, \u201cmeyve\u201d, \u201cuydu\u201d, \u201csert\u201d gibi. Buna g\u00f6re,<\/p>\n
7 bir asal say\u0131d\u0131r.
\n Elma bir t\u00fcr meyvedir.
\n Miranda, Nept\u00fcn’\u00fcn uydusudur.
\n Demir sert bir metaldir.<\/p>\n
…c\u00fcmleleri i\u00e7inde “7”, “elma”, “Miranda”, “Nept\u00fcn” ve “demir” bireysel sabitler, \u201casal say\u0131, \u201cmeyve\u201d, \u201cuydu\u201d ve \u201csert metal\u201d de y\u00fcklemsel sabitlerdir.<\/p>\n
Y\u00fcklemsel ifadelerde y\u00fcklemler yukar\u0131daki \u00f6rneklerde g\u00f6r\u00fcld\u00fc\u011f\u00fc gibi bir veya iki terimli (veya arg\u00fcmanl\u0131) olabildi\u011fi gibi, daha fazla say\u0131da arg\u00fcman da i\u00e7erebilirler. Mesela: \u201cBeril, Ak\u0131n ve \u015eebnem’nin \u00f6n\u00fcnde oturuyor\u201d dedi\u011fimiz zaman, burada \u201c\u00f6n\u00fcnde oturuyor\u201d ifadesini y\u00fcklem olarak; Beril, Ak\u0131n ve \u015eebnem isimlerini de bireysel sabitler olarak alm\u0131\u015f oluyoruz.<\/p>\n
Y\u00fcklemsel ifadeler y\u00fcklemin ald\u0131\u011f\u0131 terim say\u0131s\u0131na g\u00f6re \u015fu genel bi\u00e7imlerde g\u00f6sterilebilirler:<\/p>\n
P(a), Q(b,c), R(d,e,f), …<\/p>\n
Bu ifadelerde, hemen g\u00f6r\u00fclebilece\u011fi gibi, bireysel sabitler yerine x, y, z gibi de\u011fi\u015fkenler koyarsak,<\/p>\n
P(x), Q(b,y), R(z,e,f)<\/p>\n
…gibi de\u011fi\u015fken terimli y\u00fcklemsel ifadeler elde ederiz…<\/p>\n","protected":false},"excerpt":{"rendered":"
Matematiksel Mant\u0131k \u00c7a\u011fda\u015f mant\u0131\u011f\u0131n ve \u00e7a\u011fda\u015f felsefenin kurucusu Alman mant\u0131k\u00e7\u0131s\u0131 Gottlob Frege, “Matematik mant\u0131\u011f\u0131n uygulama alan\u0131d\u0131r” g\u00f6r\u00fc\u015f\u00fcnden hareketle matemati\u011fin, mant\u0131\u011f\u0131n aksiyomatik sistemi \u00fczerine kurulabilece\u011fini d\u00fc\u015f\u00fcnm\u00fc\u015ft\u00fcr. Bu d\u00fc\u015f\u00fcnceden hareket ederek aritmeti\u011fin temelleri konusundaki felsefi \u00e7al\u0131\u015fmalar\u0131 i\u00e7in bir mant\u0131k sistemi geli\u015ftirmi\u015fti.\u00c7a\u011fda\u015f mant\u0131\u011f\u0131n ve \u00e7a\u011fda\u015f felsefenin kurucusu Alman mant\u0131k\u00e7\u0131s\u0131 Gottlob Frege, “Matematik mant\u0131\u011f\u0131n uygulama alan\u0131d\u0131r” g\u00f6r\u00fc\u015f\u00fcnden hareketle matemati\u011fin, …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[7417,7415,7416],"class_list":["post-3183","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-mantiksal-icerik","tag-matematiksel-mantik","tag-totoloji"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=3183"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3183\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=3183"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=3183"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=3183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}