{"id":2589,"date":"2019-02-23T13:33:08","date_gmt":"2019-02-23T10:33:08","guid":{"rendered":"http:\/\/spektrum.fi\/spektraklet\/?p=2589"},"modified":"2022-01-04T16:18:00","modified_gmt":"2022-01-04T13:18:00","slug":"dricker-du-sa-dricker-jag","status":"publish","type":"post","link":"http:\/\/spektrum.fi\/spektraklet\/dricker-du-sa-dricker-jag\/","title":{"rendered":"Dricker du s\u00e5 dricker jag"},"content":{"rendered":"\n<p>Det \u00e4r dags f\u00f6r en till ramble om matematisk logik.<br>Observera f\u00f6ljande p\u00e5st\u00e5ende:<br><em>&#8221;I&nbsp;varje&nbsp;bar&nbsp;finns&nbsp;en&nbsp;person&nbsp;s\u00e5&nbsp;att&nbsp;om&nbsp;han&nbsp;dricker,&nbsp;dricker&nbsp;alla.&#8221;<\/em><\/p>\n\n\n\n<p>\u00c4r p\u00e5st\u00e5endet sant eller falskt? Intuitivt l\u00e5ter det ju som total nonsense, men meningen kan granskas exakt med hj\u00e4lp av <em>predikatlogik<\/em>. <\/p>\n\n\n\n<p>D\u00e5 vi betecknar personer i baren med <em>x<\/em> och <em>y<\/em>, f\u00e5s f\u00f6ljande symboliska form:<br><img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cexists+x+%28D%28x%29+%5Cto+%5Cforall+y+D%28y%29%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\exists x (D(x) \\to \\forall y D(y))' title='\\exists x (D(x) \\to \\forall y D(y))' class='latex' \/><br><\/p>\n\n\n\n<p>Kort f\u00f6rklaring: <\/p>\n\n\n\n<ul><li> <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cexists+x&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\exists x' title='\\exists x' class='latex' \/> betyder &#8221;det existerar x s\u00e5 att&#8230;&#8221; <\/li><li> <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cforall+y&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\forall y' title='\\forall y' class='latex' \/> betyder &#8221;f\u00f6r alla y g\u00e4ller&#8230;&#8221; <\/li><li> <img src='http:\/\/s0.wp.com\/latex.php?latex=D%28x%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='D(x)' title='D(x)' class='latex' \/> inneb\u00e4r &#8221;x dricker&#8221; <\/li><li> <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cto&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\to' title='\\to' class='latex' \/> \u00e4r en <em>implikation<\/em>, som har f\u00f6ljande sanningstabell: <\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" loading=\"lazy\" width=\"323\" height=\"137\" src=\"https:\/\/spektrum.fi\/spektraklet\/wp-content\/uploads\/2019\/02\/sanningstabell.png\" alt=\"\" class=\"wp-image-2596\" srcset=\"http:\/\/spektrum.fi\/spektraklet\/wp-content\/uploads\/2019\/02\/sanningstabell.png 323w, http:\/\/spektrum.fi\/spektraklet\/wp-content\/uploads\/2019\/02\/sanningstabell-300x127.png 300w\" sizes=\"(max-width: 323px) 100vw, 323px\" \/><\/figure>\n\n\n\n<p>Tv\u00e5 distinkta situationer g\u00e4ller nu f\u00f6r baren:<br>Om alla i baren dricker, kan vi v\u00e4lja vilken som helst person <img src='http:\/\/s0.wp.com\/latex.php?latex=y&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='y' title='y' class='latex' \/>. D\u00e5 \u00e4r <img src='http:\/\/s0.wp.com\/latex.php?latex=y&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='y' title='y' class='latex' \/> en s\u00e5dan person, att om han dricker, dricker alla. <br>Om det f\u00f6reg\u00e5ende <em>inte<\/em> g\u00e4ller, finns det en person <img src='http:\/\/s0.wp.com\/latex.php?latex=x&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x' title='x' class='latex' \/> i baren som inte dricker. Nu \u00e4r b\u00e5da p\u00e5st\u00e5endena <img src='http:\/\/s0.wp.com\/latex.php?latex=D%28x%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='D(x)' title='D(x)' class='latex' \/> och <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cforall+y+D%28y%29&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\forall y D(y)' title='\\forall y D(y)' class='latex' \/> falska, s\u00e5 enligt sanningstabellen ovan \u00e4r implikationen sann.<\/p>\n\n\n\n<p>Tolkat i predikatlogik \u00e4r allts\u00e5 p\u00e5st\u00e5endet alltid sant, dvs. vi har en tautologi. Detta kan \u00e4ven verifieras exakt t.ex. med Tarskis sanningsdefinition.<\/p>\n\n\n\n<p> Men beakta nu f\u00f6ljande situation med tre personer p\u00e5 en bar:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" loading=\"lazy\" width=\"446\" height=\"249\" src=\"https:\/\/spektrum.fi\/spektraklet\/wp-content\/uploads\/2019\/02\/tidsschema.png\" alt=\"\" class=\"wp-image-2599\" srcset=\"http:\/\/spektrum.fi\/spektraklet\/wp-content\/uploads\/2019\/02\/tidsschema.png 446w, http:\/\/spektrum.fi\/spektraklet\/wp-content\/uploads\/2019\/02\/tidsschema-300x167.png 300w\" sizes=\"(max-width: 446px) 100vw, 446px\" \/><\/figure>\n\n\n\n<p>Alla personerna dricker vid n\u00e5got skede, men ingen av dem f\u00e5r alla andra att dricka samtidigt. Nu verkar p\u00e5st\u00e5endet igen inte st\u00e4mma, what gives?<\/p>\n\n\n\n<p>Vad vi nyss har diskuterat \u00e4r <a href=\"https:\/\/en.wikipedia.org\/wiki\/Drinker_paradox\">Drinker Paradox<\/a>, som i sj\u00e4lva verket inte \u00e4r en paradox, men illustrerar hur matematisk logik inte alltid st\u00e4mmer \u00f6verens med naturligt spr\u00e5k. Skillnaden ligger i hur implikationer tolkas: i naturligt spr\u00e5k \u00e4r en implikation inte meningsfull ifall premissen \u00e4r falsk. D\u00e4remot har logikens s.k. <em><a href=\"https:\/\/sv.wikipedia.org\/wiki\/Materiell_implikation\">materiella&nbsp;implikation<\/a><\/em> ingen s\u00e5dan begr\u00e4nsning: en levande person som p\u00e5st\u00e5r &#8221;om jag \u00e4r d\u00f6d lever jag f\u00f6revigt&#8221; skulle tala sanning enligt denna modell.<\/p>\n\n\n\n<p>D\u00e5 vi \u00e4nnu \u00e5terg\u00e5r till baren och figuren ovan, m\u00e4rker vi att logik inte tar tidsdimensionen i beaktande. P\u00e5st\u00e5endet g\u00e4ller bara som en materiell implikation d\u00e5 en specifik tidpunkt fixeras. Detta \u00e4r meningsfullt, eftersom kunder kan anl\u00e4nda till och l\u00e4mna baren, och i synnerhet kan vi inte tala om drickande personer ifall baren \u00e4r tom.<\/p>\n\n\n\n<p>Den materiella implikationen \u00e4r inte on\u00f6dig eller meningsl\u00f6s inom matematik, men tolkat inom en vanlig mening kan vi formulera v\u00e4ldigt underh\u00e5llande &#8221;sanna&#8221; p\u00e5st\u00e5enden. Om du vill hitta p\u00e5 egna: ers\u00e4tt <img src='http:\/\/s0.wp.com\/latex.php?latex=x&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x' title='x' class='latex' \/> och <img src='http:\/\/s0.wp.com\/latex.php?latex=y&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='y' title='y' class='latex' \/> med andra personer eller f\u00f6rem\u00e5l, och <img src='http:\/\/s0.wp.com\/latex.php?latex=D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='D' title='D' class='latex' \/> med n\u00e5gon annan egenskap \u00e4n &#8221;dricker&#8221;,  s\u00e5 f\u00e5r du t.ex.<\/p>\n\n\n\n<p><em>&#8221;I&nbsp;varje&nbsp;godisp\u00e5se&nbsp;finns&nbsp;en&nbsp;karamell&nbsp;s\u00e5&nbsp;att&nbsp;om&nbsp;den&nbsp;\u00e4r&nbsp;choklad,&nbsp;<\/em><br><em>\u00e4r&nbsp;alla&nbsp;karameller&nbsp;choklad.&#8221;<\/em><\/p>\n\n\n\n<p><em>&#8221;I&nbsp;varje&nbsp;\u00e4mnesf\u00f6rening&nbsp;finns&nbsp;en&nbsp;person&nbsp;s\u00e5&nbsp;att&nbsp;om&nbsp;han&nbsp;\u00e4r&nbsp;vegan,&nbsp;<\/em><br><em>\u00e4r&nbsp;alla&nbsp;veganer.&#8221;<\/em><\/p>\n\n\n\n<p><em>&#8221;I&nbsp;varje&nbsp;m\u00e4nniskokropp&nbsp;finns&nbsp;ett&nbsp;ben&nbsp;s\u00e5&nbsp;att&nbsp;om&nbsp;det&nbsp;benet&nbsp;bryts,&nbsp;<\/em><br><em>bryts&nbsp;alla&nbsp;ben.&#8221;<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Det \u00e4r dags f\u00f6r en till ramble om matematisk logik.Observera f\u00f6ljande p\u00e5st\u00e5ende:&#8221;I&nbsp;varje&nbsp;bar&nbsp;finns&nbsp;en&nbsp;person&nbsp;s\u00e5&nbsp;att&nbsp;om&nbsp;han&nbsp;dricker,&nbsp;dricker&nbsp;alla.&#8221; \u00c4r p\u00e5st\u00e5endet sant eller falskt? Intuitivt l\u00e5ter det ju som total nonsense, men meningen kan granskas exakt med hj\u00e4lp av predikatlogik. D\u00e5 vi betecknar personer i baren med x och y, f\u00e5s f\u00f6ljande symboliska form: Kort f\u00f6rklaring: betyder &#8221;det existerar x s\u00e5 att&#8230;&#8221; &hellip; <a href=\"http:\/\/spektrum.fi\/spektraklet\/dricker-du-sa-dricker-jag\/\" class=\"more-link\">Forts\u00e4tt l\u00e4sa <span class=\"screen-reader-text\">Dricker du s\u00e5 dricker jag<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":21,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[142,42],"tags":[145,95,57],"_links":{"self":[{"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts\/2589"}],"collection":[{"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/users\/21"}],"replies":[{"embeddable":true,"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/comments?post=2589"}],"version-history":[{"count":16,"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts\/2589\/revisions"}],"predecessor-version":[{"id":2614,"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/posts\/2589\/revisions\/2614"}],"wp:attachment":[{"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/media?parent=2589"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/categories?post=2589"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/spektrum.fi\/spektraklet\/wp-json\/wp\/v2\/tags?post=2589"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}