{"id":68346,"date":"2026-06-04T00:00:17","date_gmt":"2026-06-03T21:00:17","guid":{"rendered":"https:\/\/bilimvegelecek.com.tr\/?p=68346"},"modified":"2026-06-04T00:51:01","modified_gmt":"2026-06-03T21:51:01","slug":"p-q-onermesinin-degerindeki-tuhafliginin-giderilmesi","status":"publish","type":"post","link":"https:\/\/bilimvegelecek.com.tr\/index.php\/2026\/06\/04\/p-q-onermesinin-degerindeki-tuhafliginin-giderilmesi","title":{"rendered":"p \u27f6 q \u00f6nermesinin de\u011ferindeki tuhafl\u0131\u011f\u0131n\u0131n giderilmesi"},"content":{"rendered":"

Zafer Ercan<\/strong><\/p>\n

19. y\u00fczy\u0131lda Leopold Kronecker (1823-1891) \u201csay\u0131lar\u0131 (do\u011fal say\u0131lar) Tanr\u0131 in\u015fa etti\u201d derken, Kurt G\u00f6del (1906-1978) \u201cDo\u011fal say\u0131lar\u0131 in\u015fa eden bir sistem tutarl\u0131ysa kendi sistemi i\u00e7erisinde tutarl\u0131 oldu\u011funu kan\u0131tlayamaz\u201d\u0131 kan\u0131tlad\u0131. Ve bunun yan\u0131nda, yiyecek bollu\u011fu ve yeme becerisinde olmas\u0131n\u0131n a\u00e7l\u0131ktan \u00f6lmesine engel olamayaca\u011f\u0131n\u0131 da; \u00e7\u00fcnk\u00fc tak\u0131nt\u0131l\u0131yd\u0131. Bu veriler alt\u0131nda, Tanr\u0131\u2019n\u0131n her \u015feye muktedir oldu\u011fu do\u011fruysa Kronecker yan\u0131l\u0131yor olmal\u0131yd\u0131. Kronecker do\u011fruysa \u201cTanr\u0131 her \u015feye muktedir de\u011fildir\u201d \u00e7\u0131kar\u0131m\u0131 ortaya \u00e7\u0131k\u0131yordu.<\/p>\n

B\u00f6ylesi bir ortamda, Hilbert\u2019in \u201c1000 y\u0131l sonra dirilip gelsem, soraca\u011f\u0131m ilk soru \u2018Riemann Hipotezi ne durumda?\u2019 olurdu\u201dya el y\u00fckseltip, 0=1 oldu\u011fu g\u00f6sterildi mi?\u201d sorusu olabilirdi. Ayr\u0131ca, konu edilecek \u00e7\u0131kar\u0131m\u0131n d\u00fczeyi, \u201cAli \u00e7al\u0131\u015fkan ve Ali \u00e7al\u0131\u015fkan oldu\u011funda Veli de \u00e7al\u0131\u015fkan oluyorsa Ali \u00e7al\u0131\u015fkand\u0131r\u201d seviyesinde olacak.<\/p>\n

\u00d6nerme, teknik olarak, sonlu say\u0131da sembol\u00fcn belirli bir dizilimidir ve fonksiyon terimiyle ifade edilir. Konu\u015fma dilinde ise \u00f6nerme, do\u011fru ya da yanl\u0131\u015f de\u011ferlerinden sadece biriyle donat\u0131lm\u0131\u015f bir ifade olarak tan\u0131mlanabilir. Yani her \u00f6nermenin Do\u011fru (1 ile g\u00f6sterilir) ya da Yanl\u0131\u015f (0 ile g\u00f6sterilir) bir de\u011feri vard\u0131r. p ile g\u00f6sterilen bir \u00f6nermenin de\u011feri d(p) ile ifade edilir. \u00d6nermelerin do\u011fru ve yanl\u0131\u015f ile temsil edilmesi George Boole taraf\u0131ndan 1854\u2019te yay\u0131nlanan An Investigation of the Laws of Thought<\/em> adl\u0131 eserinde yer alm\u0131\u015ft\u0131r.<\/p>\n

Gottlob Frege, 1879\u2019da yay\u0131nlanan Begriffsschrift<\/em> adl\u0131 eserinde, mant\u0131\u011f\u0131 sembolik bir dil ile ifade ederek, ko\u015fullu \u00f6nermeleri ve t\u00fcretme kurallar\u0131n\u0131 modern matematiksel mant\u0131k ba\u011flam\u0131nda sistemle\u015ftirmi\u015ftir. Bu \u00fcretim genel olarak iki temele indirgenebilir:<\/p>\n

A: <\/strong>p<\/em> bir \u00f6nerme ise \u00acp ile g\u00f6sterilen ve p<\/em>\u2019nin de\u011fili olarak adland\u0131r\u0131lan bir \u00f6nerme vard\u0131r. Bu \u00f6nermelerin do\u011fruluk de\u011ferleri aras\u0131ndaki ili\u015fki:<\/p>\n

d(\u00acp)=1-d(p)<\/p>\n

Bu tan\u0131m tepeden inme de\u011fildir; ya\u015fam\u0131n do\u011fal ak\u0131\u015f\u0131na uyumludur.<\/p>\n

Bir di\u011fer \u00f6nerme bi\u00e7imi ise \u015fu \u015fekildedir:<\/p>\n

B: <\/strong>p<\/em> ve q<\/em> iki \u00f6nermeyi g\u00f6steren semboller olmak \u00fczere \u201cise ba\u011fla\u00e7l\u0131 ( p<\/em> ise q<\/em> ) olarak adland\u0131r\u0131lan ve p\u2192 q\u00a0ile g\u00f6sterilen bir ba\u015fka \u00f6nermedir. Bu \u00f6nermelerin de\u011ferleri aras\u0131ndaki ili\u015fki,<\/p>\n

d(p\u27f6 q)=1-d(p)+d(p)d(q)<\/p>\n

olarak verilir. Bu e\u015fitlik \u00f6ncelikli olarak, Bertand Russel ve Alferd North Whitehead\u2019\u0131n 1910-1913 tarihli Principia Mathematica <\/em>eserinde tablolarla yer alm\u0131\u015f ve yukar\u0131da verilen e\u015fitlik 1930\u2019l\u0131 y\u0131llarda Emil Post, Clarence Lewis, Quine taraf\u0131ndan verilmi\u015ftir. Bu e\u015fitlik iki \u00f6z \u00fczerine kurulmu\u015ftur.<\/p>\n

Birincisi<\/strong>: Bu do\u011fruluk ili\u015fkisine g\u00f6re, $p$ \u00f6nermesinin de\u011feri yanl\u0131\u015f ise, q<\/em>\u2019n\u0131n de\u011feri ne olursa olsun, p\u27f6q\u00a0\u00f6nermesinin de\u011feri do\u011fru oluyor ki; bu biraz tuhafl\u0131k bar\u0131nd\u0131r\u0131yor. Ger\u00e7ekten de bu, \u201cDo\u011fru \u00e7al\u0131\u015fan bir d\u00fczenekte yanl\u0131\u015f bir girdi sonucunda ortaya \u00e7\u0131kan her \u015fey do\u011frudur\u201d sonucunu \u00e7a\u011fr\u0131\u015ft\u0131r\u0131yor. Bunun sonucunda, \u00f6rne\u011fin, \u201cdo\u011fru \u00e7al\u0131\u015fan bir hukuk d\u00fczeninde adalet bakan\u0131n\u0131n Ak\u0131n G\u00fcrlek olmas\u0131 nedeniyle ortaya \u00e7\u0131kan her karar do\u011fru olacakt\u0131r\u201d gibi beklenmedik bir sonu\u00e7 ve gerginlik yarat\u0131yor.<\/p>\n

\u0130kincisi<\/strong>: D\u00fc\u015f\u00fcncenin atomik yap\u0131s\u0131n\u0131n \u00f6nermeler \u00fczerine kurulu bir temel \u00e7\u0131kar\u0131m kural\u0131 oldu\u011fu s\u00f6ylenebilir: Bu \u00e7\u0131kar\u0131m kural\u0131 iki \u00f6nc\u00fcl ve bir sonu\u00e7tan olu\u015fur. Bu kural, \u00f6nermeler mant\u0131\u011f\u0131nda (s\u0131f\u0131r dereceli mant\u0131kta, modus ponens<\/strong> olarak bilinir ve sembolik dilde \u015f\u00f6yle ifade edilir:<\/p>\n

p\u00a0<\/em>ve q p\u27f6q ise q<\/em><\/p>\n

Yani,<\/p>\n

p ve p\u27f6q<\/em><\/p>\n

\u00f6nermeleri do\u011fru ise q<\/em> \u00f6nermesi do\u011frudur,<\/p>\n

olup, B<\/strong>\u2019de verilen e\u015fitli\u011fin \u00f6zel bir hali olup, bu hayat\u0131n ak\u0131\u015f\u0131na uyumlu olmas\u0131n\u0131n \u00e7ok \u00e7ok \u00f6tesindedir. Bu, sembolik olarak,<\/p>\n

p,p\u27f6 q\u22a2 q<\/p>\n

ile g\u00f6sterilir. Bu ifadenin k\u00f6keni, insanl\u0131\u011f\u0131 binlerce y\u0131l k\u00f6lesi yapan,<\/p>\n

– Ya\u011fmur ya\u011fd\u0131.<\/p>\n

– Ya\u011fmur ya\u011farsa yerler \u0131slan\u0131r.<\/p>\n

O halde yerler \u0131slakt\u0131r, bi\u00e7iminde ifade edilen bir \u00e7\u0131kar\u0131m\u0131d\u0131r. \u0130\u015fte bu, d\u00fc\u015f\u00fcncenin atomik yap\u0131s\u0131d\u0131r. Bu \u00e7\u0131kar\u0131m kullan\u0131larak Kurt G\u00f6del \u015funu kan\u0131tl\u0131yordu: Do\u011fal say\u0131lar\u0131 tan\u0131mlayabilen bir adam tutarl\u0131ysa kendi d\u00fcnyas\u0131nda (sisteminde) tutarl\u0131 oldu\u011funu kan\u0131tlayamaz. Ama elde edilen bu sonu\u00e7 bu sonu\u00e7tan \u201c0=1 \u00a0olamaz\u201d sonucu elde edilemez.<\/p>\n

Verilen p ve q \u00f6nermeleri \u00fczerinden, sadece ko\u015fullu \u00f6nerme de\u011fil, ba\u015fka \u00f6ner-
\nmeler de t\u00fcretilebilir. \u00d6rne\u011fin, \u201cve (\u02c4)\u201d, \u201cveya (\u02c5)\u201d ve ba\u011fla\u00e7lar\u0131yla da a\u015fa\u011f\u0131daki
\n\u00f6nermeler olu\u015fturulabilir.<\/p>\n

p<\/span>\u02c4<\/span>q<\/span><\/p>\n

p<\/span>\u02c5<\/span>q<\/span><\/p>\n

Bu \u00f6nermelerin de\u011ferlerinin,<\/span><\/p>\n

d<\/span>(<\/span>p<\/span>\u02c4<\/span>q<\/span>) = <\/span>d(p)d(q)<\/span><\/p>\n

d<\/span>(<\/span>p<\/span>\u02c5<\/span>q<\/span>) = <\/span>d(p)+d(q)<\/span>\u2013<\/span>d(p)d(q)<\/span><\/p>\n

olarak tan\u0131mlanmas\u0131na hi\u00e7bir ak\u0131l sahibi <\/span>itiraz edemez.<\/span><\/p>\n

p <\/span>yanl\u0131\u015f ise, yani, <\/span>d<\/span>(<\/span>p<\/span>) = 0 ise <\/span>q<\/span>\u2019n\u0131n de<\/span>\u011feri ne olursa olsun, ister do\u011fru ister yanl\u0131\u015f, p\u2192q \u00f6nermesinin de\u011feri do\u011frudur, <\/span>yani d(p\u2192q)=1 olur. Bu tahaf bir durumdur. Ama bu tuhafl\u0131k, \u201ciki \u00f6nermenin denk <\/span>(\u2261 ile g\u00f6sterilir ve e\u015fitmi\u015f gibi d\u00fc\u015f\u00fcn\u00fclebilir) olmas\u0131\u201dn\u0131n ne anlama geldi\u011finin bilindi\u011fi ya da tahmin edilebilece\u011fi varsay\u0131<\/span>m\u0131yla, her biri hayat\u0131n ak\u0131\u015f\u0131na uygun olan <\/span>\u015fu ad\u0131mlar at\u0131larak giderilebilir.<\/span><\/p>\n

p\u2192q \u00f6nermesinin de\u011fili, yani \u00ac(p\u2192q) <\/span>p<\/span>\u2019nin de\u011fili ile <\/span>q<\/span>\u2019nin bir arada, yani \u201cve <\/span>(ve)\u201d baglay\u0131c\u0131yla yan yana gelmesi\u201ddir, <\/span>yani bunlar denk olup, \u015fu bi\u00e7imde yaz\u0131labilir.<\/span><\/p>\n

\u00ac(p\u2192q) \u2261 (\u00acp)\u02c4 q<\/span><\/p>\n

d<\/span>(\u00ac(p\u2192q)) = d((\u00acp)\u02c4 q)<\/span><\/p>\n

d((\u00acp)\u02c4 q) = d(\u00acp)d(q) = (1-d(p))d(q)<\/span><\/p>\n

d(\u00ac(p\u2192q)) = (1-d(p))d(q) <\/span>p\u2192q \u2261 \u00ac\u00ac(p\u2192q)<\/span><\/p>\n

d(p\u2192q) = d(\u00ac\u00ac(p\u2192q))=1-d(\u00ac(p\u2192q)) = <\/span>1-(1-d(p))d(q) = 1-d(p)+d(p)d(q).<\/span><\/p>\n

Tuhafl\u0131k giderildi. Ve b\u00f6ylece, p\u2192q <\/span>\u00f6nermesinin de\u011ferini g\u00f6n\u00fcl rahatl\u0131\u011f\u0131yla <\/span>kullan\u0131labilir.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Zafer Ercan 19. y\u00fczy\u0131lda Leopold Kronecker (1823-1891) \u201csay\u0131lar\u0131 (do\u011fal say\u0131lar) Tanr\u0131 in\u015fa etti\u201d derken, Kurt G\u00f6del (1906-1978) \u201cDo\u011fal say\u0131lar\u0131 in\u015fa eden bir sistem tutarl\u0131ysa kendi sistemi i\u00e7erisinde tutarl\u0131 oldu\u011funu kan\u0131tlayamaz\u201d\u0131 kan\u0131tlad\u0131. Ve bunun yan\u0131nda, yiyecek bollu\u011fu ve yeme becerisinde olmas\u0131n\u0131n a\u00e7l\u0131ktan \u00f6lmesine engel olamayaca\u011f\u0131n\u0131 da; \u00e7\u00fcnk\u00fc tak\u0131nt\u0131l\u0131yd\u0131. Bu veriler alt\u0131nda, Tanr\u0131\u2019n\u0131n her \u015feye muktedir oldu\u011fu […]<\/p>\n","protected":false},"author":726,"featured_media":68510,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[10536,38,535,510],"tags":[208,8755],"class_list":["post-68346","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-266-sayi","category-dergi-sayilari","category-forum","category-surekli-bolumler","tag-matematik","tag-zafer-ercan"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/68346","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/users\/726"}],"replies":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/comments?post=68346"}],"version-history":[{"count":6,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/68346\/revisions"}],"predecessor-version":[{"id":68518,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/posts\/68346\/revisions\/68518"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/media\/68510"}],"wp:attachment":[{"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/media?parent=68346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/categories?post=68346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bilimvegelecek.com.tr\/index.php\/wp-json\/wp\/v2\/tags?post=68346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}