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Date: Mon, 18 Feb 2002 09:49:49 EST
Subject: Re: [lojban] Re: [jboske] RE: Anything but tautologies
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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?
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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?
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