{"id":3219,"date":"2012-08-29T07:09:58","date_gmt":"2012-08-29T06:09:58","guid":{"rendered":"http:\/\/emilkirkegaard.dk\/en\/?p=3219"},"modified":"2012-08-29T07:09:58","modified_gmt":"2012-08-29T06:09:58","slug":"a-small-thing-about-enumerative-induction","status":"publish","type":"post","link":"https:\/\/emilkirkegaard.dk\/en\/2012\/08\/a-small-thing-about-enumerative-induction\/","title":{"rendered":"A small thing about enumerative induction"},"content":{"rendered":"<p>A thing occurred to me while i was reviewing the idea of <a href=\"http:\/\/en.wikipedia.org\/wiki\/Enumerative_induction\">enumerative induction<\/a> becus i mentioned it to a friend of mine. The thing is that such inductions are often presented in the a way like this:<\/p>\n<p style=\"padding-left: 30px;\">A particular thing of the type F is also of the type G<br \/>\nHere is another one<br \/>\nAnd another<br \/>\n&#8230;<br \/>\nThus, all things of type F are also of type G<\/p>\n<p>Suppose we formalize it in a simple lazy way:<\/p>\n<p style=\"padding-left: 30px;\">(\u2203!x<sub>1<\/sub>)(F x<sub>1<\/sub>\u2227G x<sub>1<\/sub>)<br \/>\n(\u2203!x<sub>2<\/sub>)(F x<sub>2<\/sub>\u2227G x<sub>2<\/sub>)<br \/>\n&#8230;<br \/>\n(\u2203!x<sub>n<\/sub>)(F x<sub>n<\/sub>\u2227G x<sub>n<\/sub>)<br \/>\n\u22a2 (\u2200x)(Fx\u2192Gx)<\/p>\n<p>But wait, one might as well draw the conclusion:<\/p>\n<p style=\"padding-left: 30px;\">\u00a0\u22a2 (\u2200x)(Gx\u2192Fx)<\/p>\n<p>Surely that follows just as well. However, if we keep this in mind when reviewing the typical example, then we get a result that differs from normal:<\/p>\n<p style=\"padding-left: 30px;\">This swan is white.<br \/>\nSo is this one.<br \/>\nAnd this one.<br \/>\n&#8230;<br \/>\nSo, all swans are white.<\/p>\n<p>But following the above, we might as well just draw this conclusion:<\/p>\n<p style=\"padding-left: 30px;\">So, all white things are swans.<\/p>\n<p>I see no way to block that inference while letting the other thru, for the simple reason that <em>F<\/em> and <em>G<\/em> above are arbitrary. In second-order predicate logic, it looks something like this (with greek capital letters for predicate variables):<\/p>\n<p style=\"padding-left: 30px;\">(\u2203\u03a8)(\u2203\u03a9)(\u2203!x<sub>1<\/sub>)(\u03a8x<sub>1<\/sub>\u2227\u03a9x<sub>1<\/sub>)<br \/>\n(\u2203\u03a8)(\u2203\u03a9)(\u2203!x<sub>2<\/sub>)(\u03a8x<sub>2<\/sub>\u2227\u03a9x<sub>2<\/sub>)<br \/>\n&#8230;<br \/>\n(\u2203\u03a8)(\u2203\u03a9)(\u2203!x<sub>n<\/sub>)(\u03a8x<sub>n<\/sub>\u2227\u03a9x<sub>n<\/sub>)<br \/>\n\u22a2 (\u2203\u03a8)(\u2203\u03a9) (\u2200x)(\u03a8x\u2192\u03a9x)<br \/>\n\u22a2 (\u2203\u03a8)(\u2203\u03a9) (\u2200x)(\u03a9x\u2192\u03a8x)<\/p>\n<p>Altho, the formalizations above are broken in a slight way. They don&#8217;t capture the fact that the predicates in each premise have to be the same. So, one wud have to do it something like this:<\/p>\n<p style=\"padding-left: 30px;\">(\u2203\u03a8)(\u2203\u03a9)(\u2203!x<sub>1<\/sub>)(\u2203!x<sub>2<\/sub>)&#8230;(\u2203!x<sub>n<\/sub>)(\u03a8x<sub>1<\/sub>\u2227\u03a9x<sub>1<\/sub>)\u2227(\u03a8x<sub>2<\/sub>\u2227\u03a9x<sub>2<\/sub>)\u2227&#8230;\u2227(\u2203!x<sub>n<\/sub>)(\u03a8x<sub>n<\/sub>\u2227\u03a9x<sub>n<\/sub>)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A thing occurred to me while i was reviewing the idea of enumerative induction becus i mentioned it to a friend of mine. The thing is that such inductions are often presented in the a way like this: A particular thing of the type F is also of the type G Here is another one [&hellip;]<\/p>\n","protected":false},"author":17,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-3219","post","type-post","status-publish","format-standard","hentry","category-logic-philosophy","entry"],"_links":{"self":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/3219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/comments?post=3219"}],"version-history":[{"count":1,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/3219\/revisions"}],"predecessor-version":[{"id":3220,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/3219\/revisions\/3220"}],"wp:attachment":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media?parent=3219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/categories?post=3219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/tags?post=3219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}