Re: Blank nodes, "leaning", and the LEM

On Wed, Mar 23, 2011 at 3:21 PM, David Booth <david@dbooth.org> wrote:

> On Wed, 2011-03-23 at 08:38 -0500, Pat Hayes wrote:
> > [ . . . ]
> > OK, consider the two sentences
> >
> > A:  %E% x @ P(x)
> > B: %E% z, y @ P(z) & P(y)
> >
> > Suppose A is true. Then there is something X such that P is true of X.
> > Is the second sentence true under these circumstances? Yes, because
> > that X can be the value for both z and y, and it makes both conjuncts
> > true, and so the conjunction is true. Now suppose B is true: is A
> > true? Obviously yes. Ergo, A and B each entail the other. Ergo, they
> > are logically equivalent.
>
> Right.  And just to elaborate, the reason that the above equivalence may
> not be obvious at first glance is because two different variable names
> ("z" and "y") were used in B, so the reader may erroneously make a
> "unique name assumption" that z != y.  But in fact, B has no requirement
> that z != y.
>

Actually, as somebody pointed out to me off list, quantified variables like
x, y, and z above are not assigned values in a model, so such equalities and
inequalities are not even meaningful.  I knew that but managed to forget it.

But this did lead me to realize my real question is whether existentially
quantified variables should be construed as having intensional sense.  This
seems like a variant of Frege's problem: does "a = b" have the same meaning
as "a = a"?  The answer is no, and it looks to me like the same
considerations should apply to RDF terms and expressions: <a b _:x> and <a b
_:y> have different senses, since _:x and _:y have different intensions -
you could say the reader is justified in making the "unique intension
assumption".  After all, that's the way natural language works - "Pedro owns
x and y" means he owns two things.  Then again I've never really thought
about what how quantification works if the variables are taken to have
intensional sense; I guess the variables would have to range over concepts
rather than individuals.  Or is it the sentence that has intensional sense?
 In any case if intensions matter - and it looks to me like they must matter
for RDF terms - then it seems like a purely extensional model theory
wouldn't work.  Or at least "leaning" based on existential quantification
would not work.

-Gregg