From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6986 Path: news.gmane.org!not-for-mail From: Vaughan Pratt Newsgroups: gmane.science.mathematics.categories Subject: Re: Empty algebras Date: Sun, 23 Oct 2011 11:04:10 -0700 Message-ID: References: <4EA1807A.1060802@cs.bham.ac.uk> Reply-To: Vaughan Pratt NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1319459258 6289 80.91.229.12 (24 Oct 2011 12:27:38 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 24 Oct 2011 12:27:38 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Mon Oct 24 14:27:30 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RIJck-0005W2-3s for gsmc-categories@m.gmane.org; Mon, 24 Oct 2011 14:27:26 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50096) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RIJbQ-0001If-Oi; Mon, 24 Oct 2011 09:26:04 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RIJbO-00025Q-Od for categories-list@mlist.mta.ca; Mon, 24 Oct 2011 09:26:02 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6986 Archived-At: On 10/22/2011 6:03 AM, Steve Vickers wrote: >> Even simpler: P(a) --> (exists) x. P(x) is a theorem of the pure >> predicate calculus, which is "clearly" false in the empty universe >> when P is taken to be the identically true predicate. > > No, that's wrong. You may have overlooked the scare quotes. > Your formula P(a) --> (exists) x. P(x) is valid in the empty carrier, I agree, but I said "false," not "invalid." > because it is (vacuously) true under every interpretation of the free > variable a. This would only make sense if you identify truth and validity. The formula is vacuously *valid* under every interpretation. > To avoid the vacuity and get a falsehood you have to > quantify out the free variable, as > > ((all) a. P(a)) --> ((exists) x. P(x)) Your "have to" here brings to mind Dan Dennett's chapter "Qualia Disqualified" in his 1991 book "Consciousness Explained." Concerning "if a tree falls in the forest and there is no one to hear it, does it make a sound?", Dennett says "The answer is left as an exercise for the reader." Dennett appears to reject the possibility of two answers depending on whether "sound" is considered a physioacoustic or psychoacoustic phenomenon on the ground that it can't be the latter since (if I understand his argument on pages 370-411) there is no coherent notion of the latter. (How could anyone doubt this after 40 pages?) Dennett's argument in turn brings to mind the argument that, since we can't pin down climate sensitivity to better than a range of 1.5 to 5 degrees per doubling of CO2, global warming can't be real. What we have here is the logical counterpart of "no one to hear the sound" as "no interpretation to witness the truth." The distinction between physioacoustic and psychoacoustic becomes that between physio-logical and psycho-logical, truth vs. validity. But did the tree actually fall? Is "P(a) --> (exists) x. P(x)" (context should make clear that those are not intended as scare quotes) actually false as I claim? Enter the scare quotes on "clearly," and enter Tarski pointing out that truth can't be defined. For this sentence it's not impossible to define truth, it's just that there's no canonical definition, since in the empty universe the truth of P(a) becomes an incoherent notion, leaving one at liberty to define it however you wish. Correct me if I've misunderstood but you (following Mostowski?) appear to have chosen to assign "true" to every non-closed formula, which stops the bleeding, a victory for first aid. CCA, concrete cylindric algebras, namely subalgebras of the Boolean algebra 2^D^V, handle all this automatically when D = 0. When the formula contains no variables at all, free or bound (i.e. propositional calculus, V=0), D^V = 0^0 = 1 and B = 2 as appropriate for propositional calculus. That is, absent both individuals *and* variables, CCA automatically becomes ordinary propositional calculus. But if V is nonempty then D^V = 0 and B = 2^0 = 1, the inconsistent Boolean algebra. *CCA automatically recognizes the inconsistency of trying to define truth in the empty universe when the formula contains variables.* With CCA there is no need to staunch the bleeding because the patient was not injured in the first place. With other semantics, which invariably disallow B = 1, YMMV as they say. On 10/22/2011 3:36 PM, Dusko Pavlovic wrote: > maybe it's time that we all recall lawvere's "Adjunctions in > foundations", that appeared some 35 years ago. in its best times, > categorical logic offered powerful tools to expand our logical > intuitions. The version of my proof (that CCA handles this problem in stride) that I told to Andreka and Nemeti assumed a fixed V, per the usual convention among cylindric algebraists (whatever their radius). Knowing how strongly Bill feels about cylindric algebra, and anticipating that he would surely object to my argument in that form, I tweaked it slightly by introducing a master set X of variables so as to be completely consistent not only with Bill's nice way of handling quantifiers, which requires allowing V to vary, but even the most egregious ways including those in the Cylindric Algebra volumes. You will hopefully have noticed that the only property of interpretations I_B that I required was that they interpret propositional formulas (V = 0) so as to make T(L) agree with standard propositional logic. The proof does not depend on how quantification is defined, and is just as sound for Bill's approach as for anyone else's. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]