From pycyn@aol.com Mon Feb 18 06:50:08 2002 Return-Path: X-Sender: Pycyn@aol.com X-Apparently-To: lojban@yahoogroups.com Received: (EGP: mail-8_0_2); 18 Feb 2002 14:50:08 -0000 Received: (qmail 23353 invoked from network); 18 Feb 2002 14:50:08 -0000 Received: from unknown (216.115.97.167) by m4.grp.snv.yahoo.com with QMQP; 18 Feb 2002 14:50:08 -0000 Received: from unknown (HELO imo-m10.mx.aol.com) (64.12.136.165) by mta1.grp.snv.yahoo.com with SMTP; 18 Feb 2002 14:50:08 -0000 Received: from Pycyn@aol.com by imo-m10.mx.aol.com (mail_out_v32.5.) id r.92.217cbcdd (4013) for ; Mon, 18 Feb 2002 09:49:50 -0500 (EST) Message-ID: <92.217cbcdd.29a26e8d@aol.com> Date: Mon, 18 Feb 2002 09:49:49 EST Subject: Re: [lojban] Re: [jboske] RE: Anything but tautologies To: lojban@yahoogroups.com MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="part1_92.217cbcdd.29a26e8d_boundary" X-Mailer: AOL 7.0 for Windows US sub 118 From: pycyn@aol.com X-Yahoo-Group-Post: member; u=2455001 X-Yahoo-Profile: kaliputra X-Yahoo-Message-Num: 13344 --part1_92.217cbcdd.29a26e8d_boundary Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit In a message dated 2/18/2002 1:58:17 AM Central Standard Time, lojbab@lojban.org writes: > The usage I had in mind in writing the place structure (in English, since I > don't pretend to know how to effectively convey the mathematical concepts > well enough to make the example useful). > > f(x) [x1] is a function on domain x2 integers with range x3 even integers > by expression/rule x4 li f(x)= x*2 > > change the domain x2 to real numbers and the range likewise, and call it > g(x). the rule x4 has not changed but the function x1 is a different > function. Therefore x1 and x4 cannot be the same argument. > Will {li f(x)= x*2} be well-formed? Or, perhaps, how is it to be parsed? It is possible to argue that the rule has changed, since it now multiplies different things and thus involves a different multiplication function. Otherwise, we need some notion of what sort of a thing a rule is. We have taken it as a function spelled out as an equation or some such thing in terms of more fundamental functions, but computing the same value for each argument as does the original. But then, by extensionality, they are in fact the same function, arrived at in different ways (the two descriptions have different senses, but the same referent). What is the alternative that avoids this "duplication" (not, I stress, an inefficient one -- quite the contrary) and still does its job? --part1_92.217cbcdd.29a26e8d_boundary Content-Type: text/html; charset="US-ASCII" Content-Transfer-Encoding: 7bit In a message dated 2/18/2002 1:58:17 AM Central Standard Time, lojbab@lojban.org writes:


The usage I had in mind in writing the place structure (in English, since I
don't pretend to know how to effectively convey the mathematical concepts
well enough to make the example useful).

f(x) [x1] is a function on domain x2 integers with range x3 even integers
by expression/rule x4 li f(x)= x*2

change the domain x2 to real numbers and the range likewise, and call it
g(x).  the rule x4 has not changed but the function x1 is a different
function.  Therefore x1 and x4 cannot be the same argument.


Will {li f(x)= x*2} be well-formed? Or, perhaps, how is it to be parsed?
It is possible to argue that the rule has changed, since it now multiplies different things and thus involves a different multiplication function.  Otherwise, we need some notion of what sort of a thing a rule is.  We have taken it as a function spelled out as an equation or some such thing in terms of more fundamental functions, but computing the same value for each argument as does the original.  But then, by extensionality, they are in fact the same function, arrived at in different ways (the two descriptions have different senses, but the same referent).  What is the alternative that avoids this "duplication" (not, I stress, an inefficient one -- quite the contrary) and still does its job? --part1_92.217cbcdd.29a26e8d_boundary--