From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2325 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: topos logic arising nicely Date: Wed, 4 Jun 2003 08:20:39 -0700 Message-ID: <20030604152039.GC11425@math-rs-n01.ucr.edu> References: <200306032014.WAA05082@fb04305.mathematik.tu-darmstadt.de> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241018576 3560 80.91.229.2 (29 Apr 2009 15:22:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:56 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jun 5 16:09:32 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 05 Jun 2003 16:09:32 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19O001-0006re-00 for categories-list@mta.ca; Thu, 05 Jun 2003 16:02:09 -0300 Content-Disposition: inline In-Reply-To: <200306032014.WAA05082@fb04305.mathematik.tu-darmstadt.de> User-Agent: Mutt/1.4i Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 16 Original-Lines: 29 Xref: news.gmane.org gmane.science.mathematics.categories:2325 Archived-At: Thomas Streicher wrote: >In Higher Order Logic one may >well assume inhabitedness of all types when these are built up from >N (and 1) via x,-> and P(-). In higher order logic one cannot form >subtypes in the logical sense and that's the only way how one can >build types that aren't inhabited. That may be the only way that one can *construct* such a type, hence the only way that one can *prove* that such a type exists, but how do you know that some unspecified type variable \sigma doesn't refer to an uninhabited type to begin with? The answer will depend on the interpretation, I suppose; some logics simply aren't applicable to some semantics. >Of course, even if type sigma is inhabited from > (1) \forall x:\sigma. A(x)->B >one must not conclude > (2) \exists x:\sigma. A(x) /\ B >BUT only > (3) \exists x:\sigma. A(x) -> B Of course. Only with \exists x:\sigma. A(x) will (2) follow. -- Toby