tag:blogger.com,1999:blog-14117162.post3186825938380108544..comments2024-03-28T02:29:26.853-07:00Comments on Obscure and Confused Ideas: logic blegUnknownnoreply@blogger.comBlogger2125tag:blogger.com,1999:blog-14117162.post-78024143550267051452011-06-22T16:31:35.173-07:002011-06-22T16:31:35.173-07:00Fantastic -- Thanks a lot!Fantastic -- Thanks a lot!Greg Frost-Arnoldhttps://www.blogger.com/profile/08563986984421570652noreply@blogger.comtag:blogger.com,1999:blog-14117162.post-55731887916299906642011-06-22T15:28:05.522-07:002011-06-22T15:28:05.522-07:00Hi Greg!
The relevant logic R is a natural choice...Hi Greg!<br /><br />The relevant logic <b>R</b> is a natural choice. <i>p</i> → <i>p</i> is a theorem, and so is ((<i>p</i>→ <i>p</i>) & <i>q</i>) → (<i>p</i>→ <i>p</i>). However, <i>q</i> → (<i>p</i>→ <i>p</i>) is not a theorem. It isn't entailed by ((<i>p</i>→ <i>p</i>) & <i>q</i>) → (<i>p</i>→ <i>p</i>). This inference fails in logics at <b>RM3</b> and below. (The 3-valued <b>RM3</b> counterexample: let <i>p</i> take the value 1/2 and <i>q</i> take the value 1.) <br /><br />I don't know of any logics other than relevantish ones in which & is conjunction-like, → is conditional-like, and in which this inference fails, though I don't have a characterisation the class of logics in which it fails other than saying that it’s at least <b>RM3</b> and below.<br /><br />I hope that helps.Greg Restallhttps://www.blogger.com/profile/04572249717477450403noreply@blogger.com