tag:blogger.com,1999:blog-6642011.post109660678614701813..comments2023-10-29T10:32:36.914-04:00Comments on Philosophy, et cetera: Logic Trees and Modal IndexicalsRichard Y Chappellhttp://www.blogger.com/profile/16725218276285291235noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-6642011.post-92002924647434591762013-03-01T15:27:44.585-05:002013-03-01T15:27:44.585-05:00Actually, re my previous comment, I don't my c...Actually, re my previous comment, I don't my correction of your argument was quite right. It should be:<br /><br />1 ~(PosA -> NecPosA), i<br />2 PosA, i <br />3 ~NecPosA , i<br />4 Pos~PosA, i<br />5 Pos~PosA<br />6 ~PosA, j <br />7 PosA<br />8 ~PosA<br />X <br /><br />Per your rules, we need to deindex "Pos~PosA, i" before we can derive "~PosA, i", requiring an extra step. That gives us 8 steps, same as the original rules. So of the two arguments in my post, on your rules, one requires more steps than the original rules, and the other requires the same.Kane Bhttps://www.blogger.com/profile/04795518980472722113noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-16972456002924572792013-03-01T15:17:36.407-05:002013-03-01T15:17:36.407-05:00This is an interesting idea, but I think there are...This is an interesting idea, but I think there are some mistakes in how you've presented it. All non-modal formulas need to be indexed. Your argument should be:<br /><br />1 ~(PosA -> NecPosA), i <br />2 PosA, i <br />3 ~NecPosA, i <br />4 Pos~PosA, i <br />5 ~PosA, j <br />6 Pos A<br />7 ~PosA <br />X <br /><br />with the last two lines applying the deindexing rule to (2) and (5).<br /><br />What's wrong with not indexing all the non-modal formulas? Consider the tree for (NecA -> A):<br /><br />1 ~(NecA -> A)<br />2 NecA<br />3 ~A<br /><br />Now what? There's nothing more we can do. The problem is that the rule for Nec is "NecA => A /i (for any i)". In the above argument, we don't have an i, so we can't apply Nec.<br /><br />It should be:<br /><br />1 ~(NecA -> A), i<br />2 NecA, i<br />3 ~A, i<br />4 NecA<br />5 A, i<br />X<br /><br />So I'm not sure just how much your rules do simplify things. After all, the original rules would give us:<br /><br />1 ~(NecA -> A), i<br />2 NecA, i<br />3 ~A, i<br />4 A, i<br />X<br /><br />Which seems to be simpler... applying your rules to the same argument, we have to add the superfluous step of deindexing Nec before we can use it. My guess is that your rules probably are simpler, but I'd have to work with them a little more to be sure.Kane Bhttps://www.blogger.com/profile/04795518980472722113noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1102789735817227312004-12-11T13:28:00.000-05:002004-12-11T13:28:00.000-05:00_Possibilities and Paradox: An Introduction to Mod..._Possibilities and Paradox: An Introduction to Modal and Many-Valued Logics_ is by J.C. Beall and Bas van Fraassen. It is a good book. <br /><br /><A></A><A></A>Posted by<A><B> </B></A><A HREF="http://www.blogger.com/r?http%3A%2F%2Fpixnaps.blogspot.com%2F2004%2F10%2Flogic-trees-and-modal-indexicals.html" TITLE="mgbarber at syr dot edu">Mark</A>Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1096784678170368852004-10-03T02:24:00.000-04:002004-10-03T02:24:00.000-04:00I can't make the claim to have had Mendelsohn as a...I can't make the claim to have had Mendelsohn as a prof, but I did use one of his books for a class. I learned most of the modal stuff with the help of a book called Possibilites and Paradox (the author escapes me at the moment and I'm too lazy to Amazon it :P) which was pretty well written and covers the basics of both modal and multi-valued logics. I love this stuff. :)<br /><br />I'll have to check out the links yall posted, and I'm checking out your blogs right now. It's always a pleasure to find someone else who knows something about symbolic logics. I absolutely want to learn much more. <br /><br /><A></A><A></A>Posted by<A><B> </B></A><A HREF="http://www.blogger.com/r?http%3A%2F%2Fdanweasel.com" TITLE="danweasel at gmail dot com">Andrew</A>Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1096782099131050682004-10-03T01:41:00.000-04:002004-10-03T01:41:00.000-04:00Thanks for the links! 
Posted by RichardThanks for the links! <br /><br /><A></A><A></A>Posted by<A><B> </B></A>RichardAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1096774493374296682004-10-02T23:34:00.000-04:002004-10-02T23:34:00.000-04:00Wow, a fellow philoso-geek! BTW I've had Mendelsoh...Wow, a fellow philoso-geek! BTW I've had Mendelsohn as a professor. You can access some of his papers and material from his recent book on his website, or you could last spring anyway. Here it is:<br />http://comet.lehman.cuny.edu/mendel <br /><br /><A></A><A></A>Posted by<A><B> </B></A><A HREF="http://www.blogger.com/r?celibateinthecity.blogspot.com" TITLE="celibatecity at juno dot com">JL</A>Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1096751930293185732004-10-02T17:18:00.000-04:002004-10-02T17:18:00.000-04:00Melvin Fitting has a few books about this, e.g., "...Melvin Fitting has a few books about this, e.g., "Proof Methods for Modal and Intuitionistic Logics" and the (newer) Fitting and Mendelsohn, "First-order Modal Logic". Rajeev GorĂ© wrote a thesis on this which is available (in PostScript) at<br />http://rsise.anu.edu.au/~rpg/publications.html<br /><br />From the point of view of a logician, S5 is the most primitive modal logic in the sense that it has practically no structure. As pointed out by Andrew, the reason you index the modal formulas even in S5 is that the indexing is necessary once you pass to more interesting logics. <br /><br /><A></A><A></A>Posted by<A><B> </B></A><A HREF="http://www.blogger.com/r?http%3A%2F%2Fwww.ucalgary.ca%2F%7Erzach%2F" TITLE="rzach at ucalgary dot ca">Richard Zach</A>Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1096689564092923102004-10-01T23:59:00.000-04:002004-10-01T23:59:00.000-04:00Yeah, I had been wondering how one would go about ...Yeah, I had been wondering how one would go about converting the tree rules to apply to more primitive systems. I don't suppose you'd happen to have a handy reference (or, better still, a URL) to some place where I could learn more about this matter?Richard Y Chappellhttps://www.blogger.com/profile/16725218276285291235noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1096687768990702542004-10-01T23:29:00.000-04:002004-10-01T23:29:00.000-04:00Yeah, you're right. If you're working in S5, then...Yeah, you're right. If you're working in S5, then there is no need whatsoever for indexing statements to particular worlds. Best I can tell, indexing worlds with the tree rules is basically a hold over from working in more primitive systems (such as S4) where it is not always the case that a modal statement holds true at every world in the universe.<br /><br />In S4, where you can't do:<br /><br />Gen(A) / w1<br />w1 R w2<br />=><br />Gen(A) / w2<br /><br />It is necessary to have the indexing rules, otherwise you would get incorrect results.<br /><br />So the tree system (with indexing) is motivated, I think, so that the same system can be used for all logical systems, whether they are S4, S5, or something even more primitive. Yeah it would be cleaner to do away with indexing for S5 as much as possible, but then your system of proof falls out of step with proofs for other systems.<br /><br />Make sense?Andrewhttps://www.blogger.com/profile/06906127204087747883noreply@blogger.com