From pycyn@aol.com Mon Jun 26 07:13:25 2000 Return-Path: Received: (qmail 14460 invoked from network); 26 Jun 2000 14:13:23 -0000 Received: from unknown (10.1.10.26) by m1.onelist.org with QMQP; 26 Jun 2000 14:13:23 -0000 Received: from unknown (HELO imo-r20.mx.aol.com) (152.163.225.162) by mta1 with SMTP; 26 Jun 2000 14:13:22 -0000 Received: from Pycyn@aol.com by imo-r20.mx.aol.com (mail_out_v27.10.) id a.e6.73cf772 (4534) for ; Mon, 26 Jun 2000 10:13:11 -0400 (EDT) Message-ID: Date: Mon, 26 Jun 2000 10:13:11 EDT Subject: Re: [lojban] RE:Trivalent Logics To: lojban@egroups.com MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: AOL 3.0 16-bit for Windows sub 41 From: pycyn@aol.com In a message dated 00-06-25 22:26:50 EDT, xorxes writes: << I tried assigning (-1,1,0) to {nai} but it becomes too different from the binary meaning of {nai}. In any case, {cu'i} = (0,1,-1) is -1. What would that be? A counter-negation? >> What you tried for {nai} was a Post or circular negation, that is +1 to each value. It places an important role in the theory of many-value logics, as the preferred negation when it comes to looking for minimal connective sets. The Sheffer function is always this negation of the min function and it is this negation that gives the analog of double negation (triple negation in the trivalent case) -- you go round the circle one complete turn by doing n negations. Your {nai} always gives double negation, even for zillion-valued logics. Although I can't find anyone who messed with it, your {cu'i} is also a circular negation, just going the other direction. I wonder if it might not play a role in defining the other adequate single connective, the Peirce (amphec) function, NOR. Might that be max(x,y)-1? It is in the 2 case, which doesn't prove much, and looks at a glance to be in 3 and also to be adequate. The rest of the system looks very promising (Hell, very done, except for usage thoughts -- especially for the binary connectives).