{"id":2506,"date":"2011-11-01T00:58:34","date_gmt":"2011-10-31T23:58:34","guid":{"rendered":"http:\/\/emilkirkegaard.dk\/en\/?p=2506"},"modified":"2013-09-01T17:24:23","modified_gmt":"2013-09-01T16:24:23","slug":"three-logicians-walk-into-a-bar-a-formal-explanation","status":"publish","type":"post","link":"https:\/\/emilkirkegaard.dk\/en\/2011\/11\/three-logicians-walk-into-a-bar-a-formal-explanation\/","title":{"rendered":"Three logicians walk into a bar: a formal explanation"},"content":{"rendered":"

\"\"<\/a><\/p>\n

http:\/\/spikedmath.com\/445.html<\/a><\/p>\n

—–<\/p>\n

Let E refer to the propositions expressed by the sentence:<\/p>\n

\u201cEveryone wants beer\u201d<\/p>\n

\u201ceveryone\u201d refers to the three people on the right. Let’s call them \u201ca\u201d, \u201cb\u201d, \u201cc\u201d from left to right. Let \u201cWx\u201d =df \u201cx wants beer\u201d.<\/p>\n

We could try to show this but let’s just take it intuitively that the following holds:<\/p>\n

Everyone wants beer \u2194 a wants beer and b wants beer and c wants beer<\/p>\n

Formalizing we get:<\/p>\n

E\u2194(Wa\u2227Wb\u2227Wc)<\/p>\n

Now, assume that to begin with, a, b, c does not know whether the two others want beer or not. This is technically ‘left open’ in the comic, but it is not irrelevant.<\/p>\n

Now, assume every person knows if he wants beer or not, or rather, he either knows that he wants beer, or he knows that he does not want beer. Without this, it doesn’t work either. Introducing \u201cKx(P)\u201d to mean \u201cK knows that P\u201d and formalizing:<\/p>\n

\u2200xKx(Wx)\u2228Kx(\u00acWx)<\/p>\n

Now, interpreting \u201clogicians\u201d to be a group of people being perfect at making inferences in at least this case. Such people are sometimes called \u201cideally rational\u201d or similar. They never make a wrong inference and never miss an inference.<\/p>\n

Let’s think about a’s position as he is first to answer the question. If he knows that he wants beer, then he does not know the truth of E. But if he knows that he does not want beer, he can know that E is false. Because: he is part of everyone, so if he does not want beer, it is not the case that everyone wants beer. If a is being truthful etc., he has to answer either \u201cI don’t know\u201d or \u201cNo\u201d.<\/p>\n

Now b is in almost the same position as a. Obviously, if a has already answered \u201cNo\u201d, there is no reason to respond or at least he should respond the same. If a’s answer is \u201cI don’t know\u201d, b still lacks information about whether or not c wants beer [Wc or \u00acWc1<\/sup><\/a>]. Likewise, b knows his own state, so he knows either that he wants beer or that he doesn’t want beer. If he knows that he doesn’t, he can infer that E is false, similarly to a. If he knows that he does, then he can’t infer anything about E, and has to answer \u201cI don’t know\u201d.<\/p>\n

Now, c has all the information he needs to answer either \u201cYes\u201d or \u201cNo\u201d. Obviously, if any of the previous answers are \u201cNo\u201d, then he should also answer \u201cNo\u201d or simply not answer. If both earlier answers are \u201cI don’t know\u201d, he can infer that both a and b want beer. He also knows his own state. If he does not want beer, he will answer \u201cNo\u201d. If he does, he can infer that E is true, and thus answer \u201cYes\u201d.<\/p>\n

—-<\/p>\n

\"\"<\/a><\/p>\n

Pretty similar comic which does need set theory to properly formalize: http:\/\/mrburkemath.blogspot.com\/2011\/05\/coffee-logic.html<\/a><\/p>\n

Notes<\/h3>\n

1<\/a>Which could also be interpret as: To go to the bathroom or not go to the bathroom? :D<\/p>\n","protected":false},"excerpt":{"rendered":"

http:\/\/spikedmath.com\/445.html —– Let E refer to the propositions expressed by the sentence: \u201cEveryone wants beer\u201d \u201ceveryone\u201d refers to the three people on the right. Let’s call them \u201ca\u201d, \u201cb\u201d, \u201cc\u201d from left to right. Let \u201cWx\u201d =df \u201cx wants beer\u201d. We could try to show this but let’s just take it intuitively that the following […]<\/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":[209,425,701,1752],"class_list":["post-2506","post","type-post","status-publish","format-standard","hentry","category-logic-philosophy","tag-comic","tag-formalization","tag-logic","tag-puzzle","entry"],"_links":{"self":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/2506","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=2506"}],"version-history":[{"count":3,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/2506\/revisions"}],"predecessor-version":[{"id":3944,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/2506\/revisions\/3944"}],"wp:attachment":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media?parent=2506"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/categories?post=2506"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/tags?post=2506"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}