From lojbab@lojban.org Thu Aug 09 13:06:50 2001 Return-Path: X-Sender: lojbab@lojban.org X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-7_3_1); 9 Aug 2001 20:06:50 -0000 Received: (qmail 94874 invoked from network); 9 Aug 2001 20:05:06 -0000 Received: from unknown (10.1.10.26) by l9.egroups.com with QMQP; 9 Aug 2001 20:05:06 -0000 Received: from unknown (HELO stmpy-3.cais.net) (205.252.14.73) by mta1 with SMTP; 9 Aug 2001 20:05:06 -0000 Received: from user.lojban.org (dynamic225.cl8.cais.net [205.177.20.225]) by stmpy-3.cais.net (8.11.1/8.11.1) with ESMTP id f79K55o75797 for ; Thu, 9 Aug 2001 16:05:05 -0400 (EDT) Message-Id: <4.3.2.7.2.20010809154754.00d2dde0@pop.cais.com> X-Sender: vir1036@pop.cais.com X-Mailer: QUALCOMM Windows Eudora Version 4.3.2 Date: Thu, 09 Aug 2001 16:03:33 -0400 To: lojban@yahoogroups.com Subject: Re: [lojban] A or B, depending on C, and related issues In-Reply-To: <99.18effd42.28a42efe@aol.com> Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed From: "Bob LeChevalier (lojbab)" At 02:22 PM 8/9/01 -0400, pycyn@aol.com wrote: >There are 256 three-placed truth functions (8 lines each capable of bing >filled in two ways, each line independently of the others). There are three >times as many ways to join three sentences using two two-place truth >functional connectives. So, it looks like there ought to be a way to express >any three-place truth functional connective using only three sentences and >two two-place connectives. But it doesn't work; too many of the reduced >forms produce the same function. As a result, in Lojban, we have often to use >three truth functions and four sentences (one repetition or denial) to >represent some relations among three sentences. Indeed, we sometimes need >even more complex forms. > From time to time we have considered either devising three-place truth >functional operators or working up non-truth-functional (officially) ways of >dealing with larger cases (threes and on up). Neither of these projects has >ever come to any official product that I can find. Actually it did. lu'a for selecting individuals from a set came *directly* from your posing this problem to me back in 1989 or so. The other members of lu'a were added later. I believe that Athelstan then demonstrated that we could match all 3 and 4 place truth functional connective truth table with no obvious limit to what we could handle in larger sizes being found. The form translated as "1 from the set {coffee, tea} AND 1 from the set {sugar, cream} is an example of this solution. sumti sets can include sets of propositions by using du'u or la'elu/li'u, which I think solves the first problem. > The first (three-place >functions) runs into serious grammatical issues, not to mention logical ones >of grouping and the like. The second typically involves a set of sentences >and a selector of some sort: "exactly one of the following three," Which was precisely how we implemented it. >Surely among the first to be dealt with would be the two >versions of "if P then Q, else R," which are also pleasantly simple: >(if P then Q) and (if not P then R)" and "(P iff Q) and (not P iff R)" While >I am sure there are easier ways to show that these are the simplest forms for >these functions, I confess to just having run all the possibilities from >disjunctive normal forms on down. I'm sure it would be interesting to note which multiple connective combinations actually have a use, and giving the set selection equivalents for them. lojbab -- lojbab lojbab@lojban.org Bob LeChevalier, President, The Logical Language Group, Inc. 2904 Beau Lane, Fairfax VA 22031-1303 USA 703-385-0273 Artificial language Loglan/Lojban: http://www.lojban.org