[extropy-chat] Identity

[Ian Goddard]

Michael M. Butler wrote:

> For some reason, I feel obliged to reminisce:
> 
> "We may note that in these experiments the symbol
>                                  =
>                            stands for
>                       'is confused with'."
>
> G. Spencer Brown, "Laws of Form" (quote is only 
> approximate: I sold my copy years ago).



 There are generally two semantic interpretations of
'=', one being 'is similar to'. In that case the left-
and right- hand symbols in 'a = b' point to two
different referents and '=' denotes some important set
of similarities between the two referents.

 The strong interpretation of '=' asserts that the
left- and right-hand symbols point to the *same*
referent. So given the statement 'a = b', the
interpretation of 'a' is the thing it points to, which
is often denoted by 'I(a)' and the referent of 'b' by
'I(b)'. So in the strong interpretation of '=', the
statement 'a = b' is true just in case I(a) is the
same thing as I(b). 'I' is an interpretation function
mapping symbols in a language to objects in a domain
of discourse such as:

              I
     'cat' -------> (an actual cat) = I(cat)
      'c'  ------->  I(c)
      'e'  ------->  I(e)
              .
              .
              .

 A semantic interpretation of strong '=' runs as
follows, [*] where 'I' maps symbols to a domain of
discourse 'D' (note that uses of '=' in the following
indented metalanguage expressions switch from object-
to meta-language, and are as presented in the cited
text [*]):

  I(=) = {<d,e> in D^2 | d = e}

That means: the interpretation of '=' is a set of
ordered pairs <d,e> in D^2 (the cross product of D)
such that d = e. So I(=) is a subset of D^2. Now,
taking the simpler of two explanatory routes in the
text:

  V[M](a = b) = 1 iff <I(a),I(b)> is in I(=)
  iff I(a) = I(b)

That means: a valuation function (V) on a given model
(M) maps the statement 'a = b' to 'true' (ie, to '1')
iff the ordered pair containing the interpretation of
'a' and the interpretation of 'b' (denoted as an
ordered pair by '<I(a),I(b)>' ) is in the set of
identical pairs (ie, is in I(=)), which is the case
iff I(a) is the same object as I(b), ie: I(a) = I(b). 

 Short n simple: the statement 'The King = Elvis
Presley' is true just in case I(The King) is the same
entity as I(Elvis Presley) in some domain of
discourse. ~Ian

_____________________________________________________
[*] Gamut, LTF. "Logic, Language, and Meaning, Volume
1: Introduction to Logic." University of Chicago
Press, 1991.
http://www.press.uchicago.edu/cgi-bin/hfs.cgi/00/7087.ctl


 
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