From owner-cse663-fa06-list@LISTSERV.BUFFALO.EDU Mon Dec 4 13:46:17 2006 Received: from ares.cse.buffalo.edu (ares.cse.Buffalo.EDU [128.205.32.79]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id kB4IkHpH008934 for ; Mon, 4 Dec 2006 13:46:17 -0500 (EST) Received: from front1.acsu.buffalo.edu (warmfront.acsu.buffalo.edu [128.205.6.88]) by ares.cse.buffalo.edu (8.13.6/8.13.6) with SMTP id kB4IkCNH066306 for ; Mon, 4 Dec 2006 13:46:12 -0500 (EST) Received: (qmail 4203 invoked from network); 4 Dec 2006 18:46:12 -0000 Received: from mailscan1.acsu.buffalo.edu (128.205.6.133) by front1.acsu.buffalo.edu with SMTP; 4 Dec 2006 18:46:11 -0000 Received: (qmail 26918 invoked from network); 4 Dec 2006 18:46:08 -0000 Received: from deliverance.acsu.buffalo.edu (128.205.7.57) by front2.acsu.buffalo.edu with SMTP; 4 Dec 2006 18:46:08 -0000 Received: (qmail 2170 invoked from network); 4 Dec 2006 18:46:04 -0000 Received: from listserv.buffalo.edu (128.205.7.35) by deliverance.acsu.buffalo.edu with SMTP; 4 Dec 2006 18:46:04 -0000 Received: by LISTSERV.BUFFALO.EDU (LISTSERV-TCP/IP release 14.5) with spool id 2158812 for CSE663-FA06-LIST@LISTSERV.BUFFALO.EDU; Mon, 4 Dec 2006 13:46:04 -0500 Delivered-To: cse663-fa06-list@listserv.buffalo.edu Received: (qmail 2567 invoked from network); 4 Dec 2006 18:46:04 -0000 Received: from mailscan7.acsu.buffalo.edu (128.205.6.158) by listserv.buffalo.edu with SMTP; 4 Dec 2006 18:46:04 -0000 Received: (qmail 234 invoked from network); 4 Dec 2006 18:46:02 -0000 Received: from castor.cse.buffalo.edu (128.205.32.14) by smtp3.acsu.buffalo.edu with SMTP; 4 Dec 2006 18:46:02 -0000 Received: from castor.cse.Buffalo.EDU (rapaport@localhost [127.0.0.1]) by castor.cse.Buffalo.EDU (8.13.6/8.12.10) with ESMTP id kB4Ik2vQ008921 for ; Mon, 4 Dec 2006 13:46:02 -0500 (EST) Received: (from rapaport@localhost) by castor.cse.Buffalo.EDU (8.13.6/8.12.9/Submit) id kB4Ik2HN008920 for cse663-fa06-list@listserv.buffalo.edu; Mon, 4 Dec 2006 13:46:02 -0500 (EST) X-UB-Relay: (castor.cse.buffalo.edu) X-PM-EL-Spam-Prob: : 7% Message-ID: <200612041846.kB4Ik2HN008920@castor.cse.Buffalo.EDU> Date: Mon, 4 Dec 2006 13:46:02 -0500 Reply-To: CSE 663 - Advanced Knowledge Representation - Fall 2006 Sender: CSE 663 - Advanced Knowledge Representation - Fall 2006 From: "William J. Rapaport" Subject: quantified modal logic To: CSE663-FA06-LIST@LISTSERV.BUFFALO.EDU Precedence: list List-Help: , List-Unsubscribe: List-Subscribe: List-Owner: X-UB-Relay: (castor.cse.buffalo.edu) X-DCC-Buffalo.EDU-Metrics: castor.cse.Buffalo.EDU 1029; Body=0 Fuz1=0 Fuz2=0 X-Spam-Status: No, score=-2.6 required=5.0 tests=AWL,BAYES_00 autolearn=unavailable version=3.1.7 X-Spam-Checker-Version: SpamAssassin 3.1.7 (2006-10-05) on ares.cse.buffalo.edu X-Virus-Scanned: ClamAV 0.88.6/2277/Mon Dec 4 12:10:23 2006 on ares.cse.buffalo.edu X-Virus-Status: Clean Status: R Content-Length: 2672 Here's another problem with quantified modal logic that I didn't have time to discuss in class. Because of the limitations of an ASCII keyboard, I'll use the following notation: A for the universal quantifier > for the material conditional iff for the material biconditional a for alpha (a generic proposition) a(x) for alpha containing the free variable x L for the necessity "box"; some systems of modal logic actually use "L" for necessity and "M" for possibility ======================================================================== "Leibniz's Law" is an attempt (based on some writings of the 17th-century rationalist philosopher and co-inventor of calculus, G.W.Leibniz) to define equality: (LL) AxAy[x=y > a(x) iff a(y)], for any a I.e., if x and y are identical...(question to ponder: How can TWO things be identical? If "they" are identical, there's only one of them :-) ... anyway: if x and y are identical, then something is true about one iff it's true about the other. I.e., identical things share all and only the same properties.(*) Now: Take "a" to be: being necessarily equal to x, i.e., L(x=__) Then an instance of (LL) is: (LL1) AxAy[x=y > L(x=x) iff L(x=y)] So far, so good. But: |- x=x (i.e., "x=x" is a theorem of FOL) So, |-L(x=x) (by the Rule of Necessitation) So far, so good, again. But now it follows that: AxAy[x=y > L(x=y)] (from (LL1) by Universal Instantiation and MP twice) What does that say? It says that if "two" things just happen to be identical, then "they" are *necessarily* identical. Suppose x = the morning star and y = the evening star. As it happens, the morning star = the evening star. But, logically, we now see that this was *necessary*, which seems counterintuitive to say the least. ------------------------------------------------------------------------ (*) By the way, the converse of (LL) is: AxAy[A(x) iff A(y) > x=y] i.e., if "two" things share all and only the same properties, then "they" are identical, i.e., there is really only one thing, not two. Here's a puzzle: The Two Spheres: Consider a possible universe that contains only two planets, both of which have all and only the same properties. Yet there are two, not one. How can this be? For discussion, see: Black, Max (1952), "The Identity of Indiscernibles", _Mind_, Vol. 51; reprinted in Max Black (1954), _Problems of Analysis: Philosophical Essays_ (Westport, CT: Greenwood Press), and in Michael J. Loux (ed.) (1970), _Universals and Particulars: Readings in Ontology_ (Garden City, NY: Doubleday/Anchor): 204-216. online at jstor.org (accessible from buffalo.edu machines): http://tinyurl.com/y8vwgx