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

On Mar 24, 2011, at 12:10 PM, Gregg Reynolds wrote:

> 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.  

In a word, no. Modern logic is extensional. There have been many approaches to an exact analysis of intensions (in Frege's sense and many other senses.) This is a huge area with no clear consensus even on the right approach, let alone a single widely accepted mechanism or logic. It is completely out of scope for mechanization or the semantic web. 

> 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.

That is one idea, but not a very useful one for most quantifications. If I say there are three ingredients in the stew, I usually mean to refer to ingredients rather than concepts of ingredients.

>  Or is it the sentence that has intensional sense?

Good question. Some would say it has to be a complete utterance.

>  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.

Oh, it certainly *works*. Whether it is adequate in some sense may be open to discussion, but not that it *works*.

Pat

> 
> -Gregg
> 
> 

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