{"id":2021,"date":"2009-08-19T20:05:51","date_gmt":"2009-08-20T04:05:51","guid":{"rendered":"http:\/\/www.oeconomist.com\/blogs\/daniel\/?p=2021"},"modified":"2009-09-14T15:52:58","modified_gmt":"2009-09-14T23:52:58","slug":"certainty-and-impossibility","status":"publish","type":"post","link":"https:\/\/www.oeconomist.com\/blogs\/daniel\/?p=2021","title":{"rendered":"Certainty and Impossibility"},"content":{"rendered":"<p>Recently, when working on my next paper, I was struck by the <em>formal similarity<\/em> between two expressions.<\/p> <p>Using <img loading=\"lazy\" decoding=\"async\" src=\"wp-content\/uploads\/2009\/08\/outcact.png\" width=\"45\" height=\"22\" alt=\"(X|c)\" style=\"display: block ; margin-top: 1em ; margin-bottom: 1em ; margin-left: auto ; margin-right: auto ;\" \/> to represent <span style=\"font-style: italic ;\">outcome <var>X<\/var> of action <var>c<\/var><\/span>, and <img loading=\"lazy\" decoding=\"async\" src=\"wp-content\/uploads\/2009\/08\/equiplaus.png\" width=\"14\" height=\"13\" alt=\"~\" style=\"display: block ; margin-top: 1em ; margin-bottom: 1em ; margin-left: auto ; margin-right: auto ;\" \/> to represent <span style=\"font-style: italic ;\">a relationship of equal plausibility<\/span>, <img loading=\"lazy\" decoding=\"async\" src=\"wp-content\/uploads\/2009\/08\/certainty.png\" width=\"224\" height=\"28\" alt=\"[(X_i v X_j | c) ~ (X_i | c)] for_all X_j\" style=\"display: block ; margin-top: 1em ; margin-bottom: 1em ; margin-left: auto ; margin-right: auto ;\" \/> implies that <var>X<\/var><sub>i<\/sub> is <em>certain<\/em> given <var>c<\/var>, and <img loading=\"lazy\" decoding=\"async\" src=\"wp-content\/uploads\/2009\/08\/imposs.png\" width=\"225\" height=\"28\" alt=\"[(X_i v X_j | c) ~ (X_j | c)] for_all X_j\" style=\"display: block ; margin-top: 1em ; margin-bottom: 1em ; margin-left: auto ; margin-right: auto ;\" \/> implies that <var>X<\/var><sub>i<\/sub> is <em>impossible<\/em> given <var>c<\/var>. (The two expressions differ in the subscript of the outcome on the right-hand side of the relation.)<\/p> <p>The ideas here are simple: If no other outcome contributes plausibility, then <var>X<\/var><sub>i<\/sub> is certain; if <var>X<\/var><sub>i<\/sub> contributes no plausibility, then <var>X<\/var><sub>i<\/sub> is impossible.<\/p>","protected":false},"excerpt":{"rendered":"Recently, when working on my next paper, I was struck by the formal similarity between two expressions. Using to represent outcome X of action c, and to represent a relationship of equal plausibility, implies that Xi is certain given c, and implies that Xi is impossible given c. (The two expressions differ in the subscript [&hellip;]","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"footnotes":""},"categories":[36,4],"tags":[299,445],"class_list":["post-2021","post","type-post","status-publish","format-standard","hentry","category-economics","category-public","tag-decision-theory","tag-papers"],"_links":{"self":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts\/2021","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2021"}],"version-history":[{"count":0,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=\/wp\/v2\/posts\/2021\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2021"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2021"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oeconomist.com\/blogs\/daniel\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2021"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}