{"id":364,"date":"2012-03-01T14:19:04","date_gmt":"2012-03-01T20:19:04","guid":{"rendered":"http:\/\/linesoftangency.wordpress.com\/?p=364"},"modified":"2012-03-01T14:19:04","modified_gmt":"2012-03-01T20:19:04","slug":"conditional-response","status":"publish","type":"post","link":"http:\/\/blog.chrislusto.com\/?p=364","title":{"rendered":"Conditional Response"},"content":{"rendered":"<p style=\"text-align:justify;\">Aside from being entertaining, these <a href=\"http:\/\/www.directv.com\" target=\"_blank\">DIRECTV<\/a> commercials offer at least two important lessons about logic.<\/p>\n<p style=\"text-align:justify;\">[youtube=http:\/\/www.youtube.com\/watch?v=c-zG5U0v3gU&amp;feature=relmfu]<\/p>\n<p style=\"text-align:justify;\">For starters, let's name the propositions listed in the video:<\/p>\n<blockquote>\n<ul style=\"text-align:justify;\">\n<li><strong>q: your cable is on the fritz<\/strong><\/li>\n<li><strong>r: you get frustrated<\/strong><\/li>\n<li><strong><em>s<\/em>: your daughter imitates<\/strong><\/li>\n<li><strong>t: your daughter gets thrown out of school<\/strong><\/li>\n<li><strong>u: your daughter meets undesirables<\/strong><\/li>\n<li><strong>v: your daughter ties the knot with undesirables<\/strong><\/li>\n<li><strong>w:\u00a0 you get a grandson with a dog collar<\/strong><\/li>\n<\/ul>\n<\/blockquote>\n<p style=\"text-align:justify;\">So the ad takes us through the following sequence of conditional statements:<\/p>\n<p style=\"text-align:center;\">$latex begin{array}{lcl} q &amp; longrightarrow &amp; r \\ r &amp; longrightarrow &amp; s \\ s &amp; longrightarrow &amp; t \\ t &amp; longrightarrow &amp; u \\ u &amp; longrightarrow &amp; v \\ v &amp; longrightarrow &amp; w end{array}&amp;s=2$<\/p>\n<p style=\"text-align:justify;\">Let's be generous and accept that each statement, individually, is true.\u00a0 Then we're led sequentially along a nice string of propositions, beginning at <em>q<\/em> and ending with <em>w<\/em>.\u00a0 Actually, there's one more tacit proposition, <em><strong>p<\/strong><\/em><strong>: you have cable<\/strong>. \u00a0So the commercial's (implicit + explicit) logic looks something like this:<\/p>\n<p style=\"text-align:center;\">$latex p longrightarrow q longrightarrow r longrightarrow s longrightarrow t longrightarrow u longrightarrow v longrightarrow w&amp;s=2$<\/p>\n<p style=\"text-align:justify;\">And therein our first logic lesson: <strong>conditional statements respect transitivity<\/strong>.\u00a0 We can follow an unbroken path of propositions all the way from <em>p<\/em> to <em>w<\/em>, which means we can replace that whole string of implications with the statement, \"If you have cable, then you'll get a grandson with a dog collar.\"\u00a0 Symbolically:<\/p>\n<p style=\"text-align:center;\">$latex p longrightarrow w&amp;s=2$<\/p>\n<p style=\"text-align:justify;\">We've accepted all the statements along the way, so we accept this one as well, which is both funny and logically sound.\u00a0 DIRECTV has successfully made fun of the cable companies, and we've had a chuckle.\u00a0 And if the commercial were to end there, everything would be hunky dory.\u00a0 But it <em>doesn't<\/em> end there.\u00a0 It ends on the line, \"Don't have a grandson with a dog collar; get rid of cable...\"\u00a0 Which is to say, \"If you don't have cable, you won't have a grandson with a dog collar.\"\u00a0 Or...<\/p>\n<p style=\"text-align:center;\">$latex neg p longrightarrow neg w&amp;s=2$<\/p>\n<p style=\"text-align:justify;\">But that's incorrect!\u00a0 And that's our second lesson: the technical name for this fallacy is <a href=\"http:\/\/en.wikipedia.org\/wiki\/Denying_the_antecedent\" target=\"_blank\">denying the antecedent<\/a>, or the <strong>inverse error<\/strong>.\u00a0 To give you a more intuitive example, consider the propositions:<\/p>\n<blockquote>\n<ul style=\"text-align:justify;\">\n<li><strong>p: you are a dog<\/strong><\/li>\n<li><strong>q: you are a mammal<\/strong><\/li>\n<\/ul>\n<\/blockquote>\n<p style=\"text-align:justify;\">$latex p longrightarrow q&amp;s=2$: \"If you are a dog, then you are a mammal.\"\u00a0 <strong>True<\/strong>.<\/p>\n<p style=\"text-align:justify;\">$latex neg p longrightarrow neg q&amp;s=2$: \"If you are not a dog, then you are not a mammal.\"\u00a0 <strong>Obviously false<\/strong>.<\/p>\n<p style=\"text-align:justify;\">It might very well be true that having cable leads to a grandson with a dog collar, but that certainly doesn't mean getting rid of cable is enough to avoid one.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aside from being entertaining, these DIRECTV commercials offer at least two important lessons about logic. [youtube=http:\/\/www.youtube.com\/watch?v=c-zG5U0v3gU&amp;feature=relmfu] For starters, let's name the propositions listed in the video: q: your cable is on the fritz r: you get frustrated s: your daughter imitates t: your daughter gets thrown out of school u: your daughter meets undesirables v: [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[4,26,32],"class_list":["post-364","post","type-post","status-publish","format-standard","hentry","category-math-musing","tag-advertising","tag-logic","tag-math"],"_links":{"self":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/364","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=364"}],"version-history":[{"count":0,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=\/wp\/v2\/posts\/364\/revisions"}],"wp:attachment":[{"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=364"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=364"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blog.chrislusto.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}