From owner-cse663-fa06-list@LISTSERV.BUFFALO.EDU Tue Sep 12 09:24:20 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 k8CDOJmw021026 for ; Tue, 12 Sep 2006 09:24:19 -0400 (EDT) Received: from front2.acsu.buffalo.edu (coldfront.acsu.buffalo.edu [128.205.6.89]) by ares.cse.buffalo.edu (8.13.6/8.13.6) with SMTP id k8CDOGiF055271 for ; Tue, 12 Sep 2006 09:24:16 -0400 (EDT) Received: (qmail 12111 invoked from network); 12 Sep 2006 13:24:16 -0000 Received: from mailscan7.acsu.buffalo.edu (128.205.6.158) by front2.acsu.buffalo.edu with SMTP; 12 Sep 2006 13:24:16 -0000 Received: (qmail 12041 invoked from network); 12 Sep 2006 13:24:16 -0000 Received: from deliverance.acsu.buffalo.edu (128.205.7.57) by front2.acsu.buffalo.edu with SMTP; 12 Sep 2006 13:24:16 -0000 Received: (qmail 12974 invoked from network); 12 Sep 2006 13:24:09 -0000 Received: from listserv.buffalo.edu (128.205.7.35) by deliverance.acsu.buffalo.edu with SMTP; 12 Sep 2006 13:24:09 -0000 Received: by LISTSERV.BUFFALO.EDU (LISTSERV-TCP/IP release 14.5) with spool id 464475 for CSE663-FA06-LIST@LISTSERV.BUFFALO.EDU; Tue, 12 Sep 2006 09:24:09 -0400 Delivered-To: cse663-fa06-list@listserv.buffalo.edu Received: (qmail 13270 invoked from network); 12 Sep 2006 13:24:09 -0000 Received: from mailscan3.acsu.buffalo.edu (128.205.6.135) by listserv.buffalo.edu with SMTP; 12 Sep 2006 13:24:09 -0000 Received: (qmail 4943 invoked from network); 12 Sep 2006 13:24:08 -0000 Received: from castor.cse.buffalo.edu (128.205.32.14) by smtp5.acsu.buffalo.edu with SMTP; 12 Sep 2006 13:24:08 -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 k8CDO8N3021010 for ; Tue, 12 Sep 2006 09:24:08 -0400 (EDT) Received: (from rapaport@localhost) by castor.cse.Buffalo.EDU (8.13.6/8.12.9/Submit) id k8CDO8tc021009 for cse663-fa06-list@listserv.buffalo.edu; Tue, 12 Sep 2006 09:24:08 -0400 (EDT) X-UB-Relay: (castor.cse.buffalo.edu) X-PM-EL-Spam-Prob: : 7% Message-ID: <200609121324.k8CDO8tc021009@castor.cse.Buffalo.EDU> Date: Tue, 12 Sep 2006 09:24:08 -0400 Reply-To: CSE 663 - Advanced Knowledge Representation - Fall 2006 Sender: CSE 663 - Advanced Knowledge Representation - Fall 2006 From: "William J. Rapaport" Subject: CSE 663: FREEDOM & BONDAGE 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 1335; Body=0 Fuz1=0 Fuz2=0 X-Virus-Scanned: ClamAV 0.88.3/1870/Tue Sep 12 08:28:49 2006 on ares.cse.buffalo.edu X-Virus-Status: Clean Status: R Content-Length: 1763 ------------------------------------------------------------------------ Subject: CSE 663: FREEDOM AND BONDAGE ------------------------------------------------------------------------ Here is a more precise definition of "free" and "bound" occurrences of variables in quantified wffs of FOL. Digression: In what follows, (1) "Q" will be any quantifier, either the universal quantifier (here written as "A") or the existential quantifier (here written as "E"), (2) "v" will be a metavariable ranging over variables, and (3) "Z" and "Y" will be a metavariables ranging over predicates (I should use "alpha" and "beta", but I can't in this font). (4) "is(df)" means: is by definition (5) "=df" means: is defined as First, there are two slightly different definitions of "scope" floating around in the literature. S1: Let Qv[Z] be a wff. Then Z is(df) the scope of (that occurrence of) Q. S2: Let Qv[Z] be a wff. Then Qv[Z] is the scope of (that occurrence of) Q. B&L do not define "scope"! But their definition is consistent with S2 (whereas I had given you S1 in lecture). Following S2, we can then define freedom and bondage of variable occurrences as follows (these definitions are based on: Kalish, Donald; Montague, Richard; & Mar, Gary (1980), _Logic: Techniques of Formal Reasoning, 2nd Edition_ (New York: Harcourt Brace Jovanovich). Def: Let Y,Z be wffs. Then an occurrence of a variable v is bound in Z =df that occurrence stands within an occurrence in Z of a wff of the form Qv[Y]. Def: Let Z be a wff. Then an occurrence of a variable v is free in Z =df that occurrence stands within Z but is not bound in Z.