From pycyn@aol.com Sat Aug 18 15:04:59 2001 Return-Path: X-Sender: Pycyn@aol.com X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-7_3_1); 18 Aug 2001 22:04:59 -0000 Received: (qmail 96542 invoked from network); 18 Aug 2001 22:04:59 -0000 Received: from unknown (10.1.10.26) by l9.egroups.com with QMQP; 18 Aug 2001 22:04:59 -0000 Received: from unknown (HELO imo-d10.mx.aol.com) (205.188.157.42) by mta1 with SMTP; 18 Aug 2001 22:04:58 -0000 Received: from Pycyn@aol.com by imo-d10.mx.aol.com (mail_out_v31_r1.4.) id r.9e.18fc6844 (4540) for ; Sat, 18 Aug 2001 18:04:55 -0400 (EDT) Message-ID: <9e.18fc6844.28b04087@aol.com> Date: Sat, 18 Aug 2001 18:04:55 EDT Subject: Re: [lojban] polyadic connectives To: lojban@yahoogroups.com MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="part1_9e.18fc6844.28b04087_boundary" X-Mailer: AOL 6.0 for Windows US sub 10531 From: pycyn@aol.com --part1_9e.18fc6844.28b04087_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit My sermon delivered and the world still standing (another one?), I've tried to find the notes from that ancient discussion on this issue, because 1) I never want to do those calculations again or reconstruct the program for them and 2) there were as I recall some fun cases turned up. But some notes on the present discussion. A) Lojban can express any n-adic logical connective. it just takes a while in some cases (at the worst, it involves disjunctive normal forms: representing each line of a truth table by a conjunction of the atomic sentences or their negations depedning on whether the value on the line is T or F [for another discussion: this can be said without indirect questions] and then disjoining all the respresentatives of lines on which the compound is true. Happily most cases can be simplified (but not all). B) The problem we were working on earlier was two fold i) what n-adic connectives could we express using the n atomic sentences each only once (with n-1 two-place connectives) or ii) what did we get from running together various combinations of binary connectives. As far as I can remember (others seem to ahve this better than I already) we got a strange collection, with very few of the things we expected: we could extend "or" and "and" indefinitely of course, but not "iff" or "xor" and noting like "if" (although we disagreed about what an extension of that would be like). And we got weird combinations, like "an even number true" and "all, none or 7 true," but few of the more natural "exactly 3 (of 4 or more) true" and the like, whence the device for setting up a heap of things and quantifying into (which I can't now remember nor how it works with sentences). I think we found nothing that worked like conditions, in any useful way (assuming we knew what that was, but "if then else" is presumably a paradigm). I suppose that "depending on" continues that hunt, with those heaps --or ordered versions of them -- playing some useful role (moving up CASE, then). --part1_9e.18fc6844.28b04087_boundary Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: 7bit My sermon delivered and the world still standing (another one?), I've tried
to find the notes from that ancient discussion on this issue, because 1) I
never want to do those calculations again or reconstruct the program for them
and 2) there were as I recall some fun cases turned up. But some notes on
the present discussion.
A) Lojban can express any n-adic logical connective. it just takes a while
in some cases (at the worst, it involves disjunctive normal forms:
representing each line of a truth table by a conjunction of the atomic
sentences or their negations depedning on whether the value on the line is T
or F [for another discussion: this can be said without indirect questions]
and then disjoining all the respresentatives of lines on which the compound
is true. Happily most cases can be simplified (but not all).
B) The problem we were working on earlier was two fold i) what n-adic
connectives could we express using the n atomic sentences each only once
(with n-1 two-place connectives) or ii) what did we get from running together
various combinations of binary connectives. As far as I can remember (others
seem to ahve this better than I already) we got a strange collection, with
very few of the things we expected: we could extend "or" and "and"
indefinitely of course, but not "iff" or "xor" and noting like "if" (although
we disagreed about what an extension of that would be like). And we got weird
combinations, like "an even number true" and "all, none or 7 true," but few
of the more natural "exactly 3 (of 4 or more) true" and the like, whence the
device for setting up a heap of things and quantifying into (which I can't
now remember nor how it works with sentences). I think we found nothing that
worked like conditions, in any useful way (assuming we knew what that was,
but "if then else" is presumably a paradigm). I suppose that "depending on"
continues that hunt, with those heaps --or ordered versions of them --
playing some useful role (moving up CASE, then). --part1_9e.18fc6844.28b04087_boundary--