From pycyn@aol.com Thu Aug 09 11:24:20 2001 Return-Path: X-Sender: Pycyn@aol.com X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-7_3_1); 9 Aug 2001 18:24:20 -0000 Received: (qmail 39576 invoked from network); 9 Aug 2001 18:23:06 -0000 Received: from unknown (10.1.10.142) by m8.onelist.org with QMQP; 9 Aug 2001 18:23:06 -0000 Received: from unknown (HELO imo-r03.mx.aol.com) (152.163.225.99) by mta3 with SMTP; 9 Aug 2001 18:23:06 -0000 Received: from Pycyn@aol.com by imo-r03.mx.aol.com (mail_out_v31.9.) id r.99.18effd42 (4324) for ; Thu, 9 Aug 2001 14:22:50 -0400 (EDT) Message-ID: <99.18effd42.28a42efe@aol.com> Date: Thu, 9 Aug 2001 14:22:54 EDT Subject: A or B, depending on C, and related issues To: lojban@yahoogroups.com MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="part1_99.18effd42.28a42efe_boundary" X-Mailer: AOL 6.0 for Windows US sub 10531 From: pycyn@aol.com --part1_99.18effd42.28a42efe_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit 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. 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," for example -- the generalization of exclusive "or" -- leaving the issue of what the members of the set are exactly and what the grammar is to be. A third approach has been to look for the simplest versions of interesting cases might be. 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 not quite sure what this has to do with anything, but it arose in some other discussion -- perhaps in a more general form -- and perhaps this answer, which is easily generalizable though with rapidly lengthening results, will help. --part1_99.18effd42.28a42efe_boundary Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: 7bit 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. 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," for
example -- the generalization of exclusive "or" -- leaving the issue of what
the members of the set are exactly and what the grammar is to be.
A third approach has been to look for the simplest versions of interesting
cases might be. 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 not quite sure what this has to do with anything, but it arose in some
other discussion -- perhaps in a more general form -- and perhaps this
answer, which is easily generalizable though with rapidly lengthening
results, will help. --part1_99.18effd42.28a42efe_boundary--