[SCL] Identity and Horrocks sentences

[pat hayes]

>On Sun, Jun 01, 2003 at 08:37:47PM -0500, Pat Hayes wrote:
>>  >I guess I don't see how this can be an HONEST rendering of (*) in
>>  >TFOL.  There is a model of this sentence on which there are many
>>  >things, as long as eq is an equivalence relation on the domain and
>>  >either everything is P and not Q, or vice versa.  That is a
>>  >DISHONEST rendering of (*)!  In particular, eq is simply an
>>  >untenable rendering of "=".  Things that are not identical can be
>>  >eq, and so any translation of (*) that renders identity as eq is
>>  >just plain wrong.  There is no wiggle room on this point, IMO.
>>
>>  Hey, guys, this is a textbook example of a classical clash of
>>  intuitions. You are both right :-) Chris is right if '=' is a logical
>>  symbol with a fixed semantics; Tamel is right if "=" is simply a
>>  binary predicate defined axiomatically (relative to a vocabulary).
>
>Well and good, but "=" isn't a mere binary predicate.  It's identity.
>Anything else is not identity.  Identity is what people want, not some
>anemic substitute.

That is one of the possible points of view. I have lived with "=" 
being a binary predicate in the past: that is in fact what it is in 
FOL (sans=). I won't rise to "mere".

>  > Seems to me that either intuition is OK, in fact, and that there is
>>  already a terminology to distinguish them: Tamel is talking about
>>  FOL, Chris is talking about FOL with equality.  We ought not to be
>>  surprised that these languages don't behave exactly similarly in
>>  matters like this, since they also don't behave exactly the same for
>>  such classical results as the upward Sk-L theorem, right?
>
>Not sure what you mean.  Upward and downward L-Sk hold for both FOL and
>FOL=.

Upward doesn't: (forall (?x ?y)(?x = ?y)) doesn't have a countable 
model in FOL=, does in plain FOL. Countable means cardinality 
aleph-0, right?

>  > I wonder, do we have strong reasons for insisting that SCL is a
>>  language with equality?
>
>Yes.

Which are?? They had better not be purely philosophical :-)

>  > Logical equality (ie equality defined semantically to be true identity
>>  in all models) does provide us with some problems, in fact, apart from
>>  this issue. For example, our somewhat 'nonstandard' use of an
>>  extension mapping gives us an intensional view of relations. If
>  > equality were only required to be a substitutive equivalence (i.e.
>  > defined axiomatically) then we could allow models of our GOFOL
>  > sublanguage in which equality was interpreted over relations
>>  extensionally, and these would be perfectly correct SCL
>>  interpretations of the sublanguage.
>
>I don't follow your argument here.  What do you mean by equality being
>interpreted over relations extensionally?

Sorry, I meant that R= S iff R and S had the same extension.

Pat
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