[extropy-chat] Identity (was: Survival tangent)

[Ian Goddard]

John said to Heartland:

> One last thing, if you decide to respond to this 
> send me your ORIGINAL E mail.  For all I know you 
> may actually agree with me because I've never seen
> an original from you and all the E mail copies 
> floating around on the net have a different 
> identity.


 Let me offer semantic and syntactic logical proofs
that different copies of a file posses unique
identities under the classic definition of 'identity'.

 Because 'identity' is a partial ordering it is
transitive. Consequently this is a universal truth: if
x = y, then x relates to z iff y relates to z. In
second-order logic (where 'R' denotes any 'relation'
and 'Rxz' means "x has a Relation to z"): 

  AxAyAz[ (x = y) -> AR(Rxz <-> Ryz) ]

 Now, at any given moment any computer file has a
unique relation to a unique block of memory -- the
physical substrate upon which it is encoded. So given
a domain with files and memory blocks, if 'x' denotes
a file and 'z' denotes a memory block, 'Rxz' means:
"File x has a Relation to memory block z." But now we
can easily see that for any two copies (x, y) of a
file, if 'Rxz' is true, then 'Ryz' is false since each
copy has a unique relation to a unique memory block.
Therefore, the consequent above 'AR(Rxz <-> Ryz)' is
false in the given domain, and thus, since the whole
logical statement above is true in all domains (proof
below), by the truth conditions for '->' (an if-then
statement is false iff its antecedent is true and its
consequent false, but is true if both antecedent and
consequent are false), the antecedent 'x = y' must
also be false. 

 So n copies of a file represent n unique identities
(hence 'copies' is plural), even if they are all
similar in just the ways that matter. QED 


Syntactic proof:

Derive:  AxAyAz[(x = y) -> AR(Rxz <-> Ryz)]

from:  x = y

1.     x = y                       assume
2.         Rxz                     assume
3.         Ryz                     1,2 identity 
4.     Rxz -> Ryz                2-3 deduction theorem
5.         Ryz                     assume
6.         Rxz                     1,5 identity 
7.     Ryz -> Rxz                  5-6 deduct theorem
8.     Rxz <-> Ryz                 4,7 def. of '<->'
9.     AR(Rxz <-> Ryz)             8, universal gen
10. (x = y) -> AR(Rxz <-> Ryz)     1-9 deduct theorem
11. AxAyAz[(x = y) -> AR(Rxz <-> Ryz)] 10, univer gen

This proves that 11 is in fact a universal truth. So,
pointing back once again to our semantic domain of
interpretation, if any file x and any file y are the
same, then they must be encoded on exactly the same
memory block (ie, physical elements). ~Ian


http://IanGoddard.net

"Propositions show the logical form of reality. They
display it." -- Wittgenstein  


 
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