From cowan@ccil.org Wed Mar 06 10:43:58 2002 Return-Path: X-Sender: cowan@mercury.ccil.org X-Apparently-To: lojban@yahoogroups.com Received: (EGP: unknown); 6 Mar 2002 18:43:58 -0000 Received: (qmail 79900 invoked from network); 6 Mar 2002 18:43:58 -0000 Received: from unknown (216.115.97.172) by m8.grp.snv.yahoo.com with QMQP; 6 Mar 2002 18:43:58 -0000 Received: from unknown (HELO mercury.ccil.org) (192.190.237.100) by mta2.grp.snv.yahoo.com with SMTP; 6 Mar 2002 18:43:57 -0000 Received: from cowan by mercury.ccil.org with local (Exim 3.12 #1 (Debian)) id 16igOQ-00040g-00 for ; Wed, 06 Mar 2002 13:44:02 -0500 Subject: Re: [lojban] Re: [jboske] Quantifiers, Existential Import To: lojban@yahoogroups.com Date: Wed, 6 Mar 2002 13:44:02 -0500 (EST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: X-eGroups-From: John Cowan From: John Cowan X-Yahoo-Group-Post: member; u=212516 X-Yahoo-Profile: johnwcowan We can understand the uselessness of I- by translating categorical logic into naive set theory. Usually, this would be seen as a retrograde move, since the problems of naive set theory are greater than those of categorial logic, but let's do it anyway. In this mapping, S and P stand for sets when on the right side of a colon. I will write SP for S intersect P, /= for inequality, and 0 for the null set. All S is P (A): SP = S Some S is P (I): SP /= 0 No S is P (E): SP = 0 Some S is not P (O): SP /= S Then existential import is simply the assertion that S /= 0, and we can understand I- as asserting that S, which may be null, has a non-null intersection with P. But plainly no set P can have a non-null intersection with 0, and so from SP /= 0 we can deduce that S /= 0. Therefore I- is false if S = 0, and to assert anything useful we need I+. -- John Cowan http://www.ccil.org/~cowan cowan@ccil.org To say that Bilbo's breath was taken away is no description at all. There are no words left to express his staggerment, since Men changed the language that they learned of elves in the days when all the world was wonderful. --_The Hobbit_