>> IAN: The statement A=A tells us nothing.
>> It is meaningless, it is no more meaning-
>> ful than saying "A."
>
>Not meaningless, but it is a trivial truth. Have you ever tried to
>prove a mathematical theorem? A proof often amounts to fiddling with
>both sides of an equation, replacing terms with equivalent expressions,
>until you're left with something like "u + ax = u + ax" - something
>which is utterly boring in itself, but which implies the truth of the
>proposition.
>
>Deny the value of such trivial truths, and you deny the foundation of
>all logical proof.
IAN: The utility of "A=A" is to denote that "="
means "same as." This is NOT an analysis of what
identity is. The notion "=" is only meaningful
to the extent that expressions on either side
of it appear DIFFERENT, such as "2+3 = 5."
So once again, two things are meaningfully
related only to the degree that they DIFFER.
"2+3" and "5" are different expressions of
the same thing. This tells us absolutely
nothing about what identity is.
>> and a theory of identity
>> tells us what identity is.
>
>But that's circular. Since identity is a symbolic construct - you can't
>go out and dig up rocks and find an "identity" - identity can only be
>what it is defined to be; a theory follows from the definitions.
IAN: Identity is what a thing is; wether rock,
symbol, or idea, all things have an identity.
Simply saying that A is the same as A tells
us nothing about what identity is.
>> >> and logic therefore
>> >> dictates that it must be a false theory,
>> >> particularly when the identity theory,
>> >> "A=A+~A," is never shown to be false.
>> >
>> >Because it is vacuous.
>>
>> IAN: But the statement "A=A" is not??!!
>
>Nope. "A=A" distinguishes A from that which is not A. "A=A+~A" does
>not: since A+~A is the whole universe, A+~A = B+~B for any B, thus A=B
>for any A,B; any two things are equal to each other. Thus, as you say,
>this relation can never be false.
IAN: So (A=A) = (B=B), thus A=B.
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