tag:blogger.com,1999:blog-6769895939935803236.post798852813366792821..comments2020-11-24T18:16:36.183-08:00Comments on r-nd-m: E.T. Jaynes and Boolean algebrardmhttp://www.blogger.com/profile/13809495052049903484noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6769895939935803236.post-44691008983642856412012-08-09T06:35:54.997-07:002012-08-09T06:35:54.997-07:00I'm not feeling inclined to agree with this di...I'm not feeling inclined to agree with this distinction.<br /><br />I would classify the distinction you are making as: it's meaningful to use real valued numbers in the range [0..1] to represent something about the truthfulness of a proposition (because they can represent probabilities), but it's not meaningful to use integer valued numbers in that range to represent truthfulness.<br /><br />You might also be saying that proofs in an integer domain are invalid if they are structurally different from proofs about related concepts in a real domain?<br /><br />Anyways, if you are working with contrived propositions -- propositions which are not part of a larger systematic framework -- then the usefulness of both the propositions themselves and of numbers representing the truth value of the propositions are both dubious.<br /><br />But we can have propositions which are part of some larger system.<br /><br />For example, if we have a proposition P(x) which is true if x is prime then we can express Euler's totient function as the sum of P(x) for x <= n.rdmhttps://www.blogger.com/profile/13809495052049903484noreply@blogger.comtag:blogger.com,1999:blog-6769895939935803236.post-70490969523336035912012-08-08T15:43:11.325-07:002012-08-08T15:43:11.325-07:00Ah, thanks, I think I understand better.
AB is in...Ah, thanks, I think I understand better.<br /><br />AB is indeed odd, from a human point of view, but nevertheless perfectly sensible logically, which is all we're after.<br /><br />Re: using numbers to represent truth values, sure, that's exactly where Jaynes is going, of course. The difference is that <em>propositions</em> don't have numerical values, but the <em>probabilities</em> of the propositions certainly do. So we can't ascribe a number 0.0..1.0 to A, but we certainly can, and do, ascribe such a value to P(A). Where Jaynes further parts company from Boole is in what equations he uses then to calculate using probabilities, but I'll get to that in a future post.<br /><br />Thanks for the clarification!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6769895939935803236.post-4491995946227830412012-08-08T14:48:44.839-07:002012-08-08T14:48:44.839-07:00I meant the conjunction.
Similarly, if we allow o...I meant the conjunction.<br /><br />Similarly, if we allow ourselves to a truth value with 1 or 0, then using these numbers in any of these three cases also seems "perfectly fine, albeit contrived".<br /><br />And, of course, the advantage of using numbers to represent truth values is that we can then use numerical notation to treat statements which deal with logical propositions.rdmhttps://www.blogger.com/profile/13809495052049903484noreply@blogger.comtag:blogger.com,1999:blog-6769895939935803236.post-54904599943633627612012-08-08T14:35:46.631-07:002012-08-08T14:35:46.631-07:00Which proposition makes no sense? "Christophe...Which proposition makes no sense? "Christopher Columbus discovered America in 1492," "It will rain today," or "Christopher Columbus discovered America in 1492 and it will rain today?" All three of them seem perfectly fine, albeit contrived: they're declarative statements that can be ascribed truth values, true or false. So I'm not quite sure what you mean.Anonymousnoreply@blogger.com