From pycyn@aol.com Thu Jul 11 15:34:25 2002 Return-Path: X-Sender: Pycyn@aol.com X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-8_0_7_4); 11 Jul 2002 22:34:25 -0000 Received: (qmail 54522 invoked from network); 11 Jul 2002 22:34:25 -0000 Received: from unknown (66.218.66.216) by m8.grp.scd.yahoo.com with QMQP; 11 Jul 2002 22:34:25 -0000 Received: from unknown (HELO imo-d08.mx.aol.com) (205.188.157.40) by mta1.grp.scd.yahoo.com with SMTP; 11 Jul 2002 22:34:24 -0000 Received: from Pycyn@aol.com by imo-d08.mx.aol.com (mail_out_v32.21.) id r.147.114677f4 (18708) for ; Thu, 11 Jul 2002 18:33:57 -0400 (EDT) Message-ID: <147.114677f4.2a5f61d5@aol.com> Date: Thu, 11 Jul 2002 18:33:57 EDT Subject: DeMorgan and fractional quantifiers To: lojban@yahoogroups.com MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="part1_147.114677f4.2a5f61d5_boundary" X-Mailer: AOL 7.0 for Windows US sub 10509 From: pycyn@aol.com X-Yahoo-Group-Post: member; u=2455001 X-Yahoo-Profile: kaliputra --part1_147.114677f4.2a5f61d5_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit Suppose we want to to shift the quantifier through {loi broda na brode}. Since {loi broda} is covertly {pisu'o loi broda} we cannot just leave the {loi broda} unaltered, anymore than we can move from {lo broda na brode} to {lo broda naku brode} without change of meaning. In the latter case, we have to move to {ro broda naku brode}. The change is a normal DeMorgan quantification shift -- it reverses ({ro broda naku} goes to {lo broda na}) and applies to the universal as well ({ro broda na} is equivalent to {lo broda naku}). Ignoring a number of technical problems that rarely affect actual cases. Then, does {loi broda na} go over to {piro loi broda naku}? Yes, but the reverse does not work, nor does {piro loi broda na} go over to {loi broda naku}. The first works because {piro} is one way of realizing {pisu'o} and we've said that no way of realizing {pisu'o loi broda} is brode, so piro loi broda is not either. To get the right results, we have to view {loi broda} not merely as {piso'u loi broda}, a part of the mass, but recognize that it is one or several such parts, {su'o lo piso'u loi broda}. Shifting negation then works on the outside quantifier only: {loi broda na} is equivalent to {ro lo pisu'o loi broda naku} (and nothing here can be dropped) and {loi broda naku} is equivalent to {ro lo pisuo loi broda na}. On the other hand, as xorxes says, {piro loi broda} refers (unusually for Lojban) to an individual and thus is transparent to negation: {piro loi broda na} is equivalent to {piro loi broda naku}. These same manuevers apply regardless of what {pisu'o} and {piro} are applied to (individuals, masses or sets). The move with {pisu'o} applies as well to all cases with "regular" numbers in place of {su'o}, the transformation is on {su'o lo pi-n loi}. (The move from {pi-n l broda na} to {piro l broda naku} does not work of course. It seems that {loi broda na brode} also implies {pino loi broda cu brode} , which seems to work as {no lo pisu'o loi broda}. But this is open to several possible (but, I think -- and hope -- rejectable) criticisms, so it can be discussed for a while. And you thought a logical language would have a simple logic! --part1_147.114677f4.2a5f61d5_boundary Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: 7bit Suppose we want to to shift the quantifier through {loi broda na brode}. Since {loi broda} is covertly {pisu'o loi broda} we cannot just leave the {loi broda} unaltered, anymore than we can move from {lo broda na brode} to {lo broda naku brode} without change of meaning. In the latter case, we have to move to {ro broda naku brode}. The change is a normal DeMorgan quantification shift -- it reverses ({ro broda naku} goes to {lo broda na}) and applies to the universal as well ({ro broda na} is equivalent to {lo broda naku}). Ignoring a number of technical problems that rarely affect actual cases.

Then, does {loi broda na} go over to {piro loi broda naku}? Yes, but the reverse does not work, nor does {piro loi broda na} go over to {loi broda naku}. The first works because {piro} is one way of realizing {pisu'o} and we've said that no way of realizing {pisu'o loi broda} is brode, so piro loi broda is not either. To get the right results, we have to view {loi broda} not merely as {piso'u loi broda}, a part of the mass, but recognize that it is one or several such parts, {su'o lo piso'u loi broda}. Shifting negation then works on the outside quantifier only: {loi broda na} is equivalent to {ro lo pisu'o loi broda naku} (and nothing here can be dropped) and {loi broda naku} is equivalent to {ro lo pisuo loi broda na}.

On the other hand, as xorxes says, {piro loi broda} refers (unusually for Lojban) to an individual and thus is transparent to negation: {piro loi broda na} is equivalent to {piro loi broda naku}.

These same manuevers apply regardless of what {pisu'o} and {piro} are applied to (individuals, masses or sets). The move with {pisu'o} applies as well to all cases with "regular" numbers in place of {su'o}, the transformation is on {su'o lo pi-n loi}. (The move from {pi-n l broda na} to {piro l broda naku} does not work of course.
It seems that {loi broda na brode} also implies {pino loi broda cu brode} , which seems to work as {no lo pisu'o loi broda}. But this is open to several possible (but, I think -- and hope -- rejectable) criticisms, so it can be discussed for a while.

And you thought a logical language would have a simple logic! --part1_147.114677f4.2a5f61d5_boundary--