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Re: log:forSome (Was: Model-specific identity for anon resources, and its representation: A new issue?

From: pat hayes <phayes@ai.uwf.edu>
Date: Sat, 23 Jun 2001 23:22:15 -0500
Message-Id: <v04210101b759b8a20ff3@[205.160.76.176]>
To: Graham Klyne <Graham.Klyne@Baltimore.com>
Cc: w3c-rdfcore-wg@w3.org
>At 02:53 PM 6/21/01 -0500, pat hayes wrote:
>>Existentially quantified variables do not bind to multiple values either.

Ah, I see that this may have been misleading, and apologise for being 
too quick. (It depends what 'bind to' means...) Please read on.

>
>Hmmm, there's something I'd like to better understand here.
>
>Consider something like:
>
>   child(fred) & ( exists(Y): parentOf(Y,fred) )
>
>Now, as I understand, Skolemizing this would yield something like:
>
>   child(fred) & parentOf(Y,fred)

Well, the existential variable Y would become a (new) constant name, 
say sk2356, and this would become
child(fred) & parentOf(sk2356,fred)
The fact that this name is 'new' means that there are no other 
assertions about it, ie all that you know about sk2356 is that it 
exists (since all FOL names refer to something) and that the above 
axiom is true of it, which is exactly what you knew about Y.  Now, 
there could logically have been several Y's that might have existed 
and would have satisfied the orginal existential axiom (so in this 
sense the existential variable might be said to 'bind to' more than 
one thing, though this sense of 'bind' is not that used in 
programming language semantics); the existential quantifer doesn't 
guaranteee uniqueness. The skolemised formula in effect 'picks out' 
one of them and says something about it, since every logical name 
only denotes one thing in a given interpretation. But there isnt 
really any difference in content, since you don't know which of them 
it picks out (denotes), and in fact it could denote any of them (if 
there was more than one) and still be correct.

>I guess that you're saying that in this expression Y binds to a 
>single instance of fred's parent, even though we expect there to be 
>two possible such bindings.  So the logical expression does not, of 
>itself, uniquely define the binding, even though there may be only 
>one in any instance of the expression.

Once Y has been made into a constant it has a unique referent 
(binding) in any given interpretation, yes. And the logical 
expression does not define the binding, indeed. (An existential 
assertion says a(tleast one) binding makes the sentence true, a 
universal says it is true for any binding, and a name just assumes a 
single binding, but does not specify it directly, only by whatever 
the axioms say about it.)

>Some problems, it seems to me, call for enumeration of all the 
>possible bindings (including problems of the kind RDF is likely to 
>be used with).  E.g. Who are fred's parents?  Should I conclude that 
>such problems cannot be fully specified within the framework of FOL?

No, they can be, but you need to say more than just 'parents exist' 
if you want to talk about all the parents. There are all sorts of 
ways to do this, eg you might say something like
parentOf(?x,fred) implies .....?x....
where the ?x is univerally quantified, or you might introduce sets 
and talk about the set of freds parents, or you might introduce 
explicit Mother and Father relations.

BTW, notice that 'who are fred's parents' is logically an open-ended 
query; you and I know it isn't open-ended because we know that people 
only have two parents, but theres no *logical* reason why someone 
might not have, say, infinitely many parents.

The two-parent rule could be axiomatised by saying (sorry, I will 
revert to SKIF notation)
(forall (?x)
     (exists (?y1 ?y2)
         (& (parentOf ?y1, ?x)
              (parentOf ?y2, ?x)
              (not (= y1 y2))
              (forall (?z)(implies (parentOf ?z ?x) (or (= ?z ?y1)(= ?z ?y2))))
)))
If you skolemise the inner existentials y1 and y2, by the way, the 
natural names to use for the skolem functions would be 'motherOf' and 
'fatherOf'

Pat Hayes

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Received on Monday, 25 June 2001 10:43:03 EDT

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