From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2330 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: topos logic arising nicely Date: Thu, 5 Jun 2003 13:08:14 +0200 (CEST) Message-ID: <200306051108.NAA00738@fb04209.mathematik.tu-darmstadt.de> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018584 3624 80.91.229.2 (29 Apr 2009 15:23:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:04 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Jun 5 16:28:08 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 05 Jun 2003 16:28:08 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19O0L0-000125-00 for categories-list@mta.ca; Thu, 05 Jun 2003 16:23:50 -0300 X-Mailer: ELM [version 2.4ME+ PL66 (25)] X-MailScanner: Found to be clean Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 21 Original-Lines: 38 Xref: news.gmane.org gmane.science.mathematics.categories:2330 Archived-At: Toby Bartels wrote > 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. Certainly, if you allow type variables then the problem crops up. I don't really see the point why one would like to have type variables unless one can perform constructions with types in a uniform way, e.g. when considering logic on top of system F, system F\omega or even on top of a dependent type theory (possibly with an impredicative universe). I guess that in case of HOL type variables were rather motivated by the analogy to functional languages with their "shallow" polymorphism. The point of my mail rather was that so-called topos logic admits subtype formation, i.e. that {x:A|phi(x)} is a type whenever \phi is a predicate on A. This, of course, allows one to introduce types with undecided inhabitedness. See W.Phoa's Edinburgh lecture notes for a calculus extending HOL with subtypes (or Bart Jacob's book). But I would be surprised if HOL has subtype formation as from a logical point of view subtypes are neither necessary nor convenient. Adding subtypes is only necessary for getting a topos out of a model of HOL. Thomas