Re: [ontolog-forum] Validity of laws of predicate logic under CWA and NA

[Adrian Walker <adriandwalker@xxxxxxxxx>]

Hi Ankesh --

Good questions.

Unfortunately, there is more than one model theoretic semantics for negation as failure. (Apt-Blair-Walker stratified model, stable model,...)

Fortunately, I'd guess that all of your statements are indeed valid under most of the above.  

If Pat Hayes is watching, I'd expect an erudite contribution from him real soon (:-)

Hope this helps,  -- Adrian

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Adrian Walker
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On Fri, Feb 20, 2009 at 10:35 AM, Ankesh Khandelwal <ankesh@xxxxxxxxxx> wrote:

Dear Ontolog-members,

I have a knowledge base and a set of rules written under closed world
assumption that use negation only as negation as failure.

Under these circumstances are the laws of quantifier movement valid?
Laws of Quantifier movements:
1. '(all x.P(x)) --> Q' equivalent-to 'exists x.(P(x)-->Q)', provided x is
not free in Q
2. '(exisits x.P(x)) --> Q' equivalent-to 'all x.(P(x)-->Q)', provided x
is not free in Q
3. 'P --> (all x.Q(x))' equivalent-to 'all x.(P --> Q(x))', provided x is
not free in P
4. 'P --> (exists x.Q(x))' equivalent-to 'exists x.(P --> Q(x))', provided
x is not free in P

And/ Or are the following laws valid?
1. 'not(all x. P(x))' equivalent-to 'exists x.(not P(x))', where not has
the Negation as Failure semantics.
2. 'not(exists x. P(x))' equivalent-to 'all x.(not P(x))', where not has
the Negation as Failure semantics.

Thank you,
Ankesh

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