[SCL] Identity and Horrocks sentences

[Chris Menzel]

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.

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

Yes.

> 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?

-chris