{"id":3264,"date":"2012-09-28T12:11:36","date_gmt":"2012-09-28T11:11:36","guid":{"rendered":"http:\/\/emilkirkegaard.dk\/en\/?p=3264"},"modified":"2012-10-02T04:23:04","modified_gmt":"2012-10-02T03:23:04","slug":"3264","status":"publish","type":"post","link":"https:\/\/emilkirkegaard.dk\/en\/2012\/09\/3264\/","title":{"rendered":"Incomplete formal proof that the KK-principle is wrong"},"content":{"rendered":"

(KK)<\/span><\/span><\/span><\/span><\/a>If one knows that p, then one knows that one knows that p.<\/span><\/span><\/span><\/p>\n


\n<\/strong><\/span><\/span><\/span>Definitions<\/strong><\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>A<\/span><\/span><\/span>0<\/span><\/span><\/span>is the proposition that 1+1=2.<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>A<\/span><\/span><\/span>1<\/span><\/span><\/span>is the proposition that Emil knows that 1+1=2.<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>A<\/span><\/span><\/span>2<\/span><\/span><\/span>is the proposition that Emil knows that Emil knows that 1+1=2.<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>\u2026<\/strong><\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>A<\/span><\/span><\/span>n<\/span><\/span><\/span>is the proposition that Emil knows that Emil knows that \u2026 that 1+1=2.<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>Where \u201c…\u201d is filled by \u201cthat Emil knows\u201d repeated the number of times in the subscript of A.<\/span><\/span><\/span><\/p>\n

Argument<\/strong><\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>1. Assumption for RAA<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>(\u2200P\u2200x)Kx(P)\u2192Kx(Kx(P)))<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>For any proposition, P, and any person, x, if x knows that P, then x knows that x knows that P.<\/span><\/span><\/span><\/p>\n

2. Premise<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>Ke(A<\/span><\/span><\/span>0<\/span><\/span><\/span>)<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>Emil knows that A<\/span><\/span><\/span>0<\/span><\/span><\/span>.<\/span><\/span><\/span><\/p>\n

3. Premise<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>(\u2203S<\/span><\/span><\/span>1<\/span><\/span><\/span>)(A<\/span><\/span><\/span>0<\/span><\/span><\/span>\u2208<\/strong><\/span><\/span><\/span>S<\/span><\/span><\/span>1<\/span><\/span><\/span>\u2227<\/strong><\/span><\/span><\/span>A<\/span><\/span><\/span>1<\/span><\/span><\/span>\u2208<\/strong><\/span><\/span><\/span>S<\/span><\/span><\/span>1<\/span><\/span><\/span>\u2227<\/strong><\/span><\/span><\/span>…\u2227A<\/span><\/span><\/span>n<\/span><\/span><\/span>\u2208<\/strong><\/span><\/span><\/span>S<\/span><\/span><\/span>1<\/span><\/span><\/span>)\u2227|S<\/span><\/span><\/span>1<\/span><\/span><\/span>|=\u221e\u2227S<\/span><\/span><\/span>1<\/span><\/span><\/span>=S<\/span><\/span><\/span>A<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>There is a set, S<\/span><\/span><\/span>1<\/span><\/span><\/span>, such that A<\/span><\/span><\/span>0<\/span><\/span><\/span>belongs to S<\/span><\/span><\/span>1<\/span><\/span><\/span>, and A<\/span><\/span><\/span>1<\/span><\/span><\/span>belongs to S<\/span><\/span><\/span>1<\/span><\/span><\/span>, and \u2026 and A<\/span><\/span><\/span>n<\/span><\/span><\/span>belongs to S<\/span><\/span><\/span>1<\/span><\/span><\/span>, and the cardinality of S<\/span><\/span><\/span>1<\/span><\/span><\/span>is infinite, and S<\/span><\/span><\/span>1<\/span><\/span><\/span>is identicla to S<\/span><\/span><\/span>A<\/span><\/span><\/span>.<\/span><\/span><\/span><\/p>\n

4. Inference from (1), (2), and (3)<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>(\u2200P)P\u2208S<\/span><\/span><\/span>A<\/span><\/span><\/span>\u2192<\/strong><\/span><\/span><\/span>Ke(P)<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>For any proposition, P, if P belongs to S<\/span><\/span><\/span>A<\/span><\/span><\/span>, then Emil knows that P.<\/span><\/span><\/span><\/p>\n

5. Premise<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>\u00ac(\u2200P)P\u2208S<\/span><\/span><\/span>A<\/span><\/span><\/span>\u2192<\/strong><\/span><\/span><\/span>Ke(P)<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>It is not the case that, for any proposition, P, if P belongs to S<\/span><\/span><\/span>A<\/span><\/span><\/span>, then Emil knows that P.<\/span><\/span><\/span><\/p>\n

6. Inference from (1-5), RAA<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>\u00ac(\u2200P\u2200x)Kx(P)\u2192Kx(Kx(P)))<\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>It is not the case that, for any proposition, P, and any person, x, if x knows that P, then x knows that x knows that P.<\/span><\/span><\/span><\/p>\n

Proving it<\/strong><\/span><\/span><\/span>
\n<\/strong><\/span><\/span><\/span>Proving that it is valid formally is sort of difficult as it requires a system with set theory, predicate logic with quantification over propositions. The above sketch should be enough for whoever doubts the formal validity.<\/span><\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

(KK)If one knows that p, then one knows that one knows that p. Definitions A0is the proposition that 1+1=2. A1is the proposition that Emil knows that 1+1=2. A2is the proposition that Emil knows that Emil knows that 1+1=2. \u2026 Anis the proposition that Emil knows that Emil knows that \u2026 that 1+1=2. Where \u201c…\u201d is […]<\/p>\n","protected":false},"author":17,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[27,11],"tags":[],"class_list":["post-3264","post","type-post","status-publish","format-standard","hentry","category-epistemology","category-logic-philosophy","entry"],"_links":{"self":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/3264","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=3264"}],"version-history":[{"count":6,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/3264\/revisions"}],"predecessor-version":[{"id":3275,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/3264\/revisions\/3275"}],"wp:attachment":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media?parent=3264"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/categories?post=3264"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/tags?post=3264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}