From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/177 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Horizontal line notation. Date: Tue, 17 Mar 2009 12:36:12 -0700 Message-ID: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1237382659 4929 80.91.229.12 (18 Mar 2009 13:24:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 18 Mar 2009 13:24:19 +0000 (UTC) To: categories@mta.ca Original-X-From: categories@mta.ca Wed Mar 18 14:25:36 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1Ljvlr-00087K-Cy for gsmc-categories@m.gmane.org; Wed, 18 Mar 2009 14:25:23 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ljv4K-0004Ho-Ng for categories-list@mta.ca; Wed, 18 Mar 2009 09:40:24 -0300 Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:177 Archived-At: David Ellerman wrote in part: >The horizontal line notation was introduced by Gerhard Gentzen in his >logical sequent calculus. If a set of formulas S implies a and b, then S >implies a/\b and vice-versa which was written as: S |- a,b _________ S |- a/\b >Going from top to bottom was conjunction-introduction, and going in the >other direction was conjunction-elimination. But much of sequent calculus is irreversible. If nothing else, structural rules such as weakening: S |- b -------- S:a |- b where S:a is the list or set S followed by a. (Thank all the Haskell recently for making me write it like this.) This is valid downwards, but not upwards. I see sequent-calculus people using a double line if it's good both ways: S |- a=>b =========== S:a |- b (This is a propositional version of the example in the original question.) Incidentally, your first example I would write as S |- a; S |- b ================ S |- a/\b because if you have a set, such as {a,b}, of propositions on the right, then this is interpreted disjunctively (not conjunctively as on the left). But I suppose that you can be lax about this if you agree beforehand that you'll only be doing "intuitionistic" sequents (that is those with only one statement on the right). --Toby