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Re: [ontolog-forum] Search engine for the ontology

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Chris Menzel <cmenzel@xxxxxxxx>
Date: Sun, 2 Mar 2008 16:00:46 -0600 (CST)
Message-id: <alpine.OSX.1.00.0803021556360.14908@xxxxxxxxxxxxxx>
On Sun, 2 Mar 2008, Chris Menzel wrote:
> ...A system S is negation complete if, for any sentence A of
> the language of S, either A or its negation ~A is a theorem of S.  Note
> that this is a purely proof theoretic notion.  A *logic* L -- that is, a
> language + semantics + proof theory -- is semantically complete if every
> logical truth of the language of L is a theorem of L.  Note that nothing
> we typically call a system of *logic* is negation complete.  For
> example, in propositional logic, no propositional variable p or its
> negation ~p is a logical theorem.  In predicate logic, neither (x)Px nor
> ~(x)Px, for example, is a theorem.    (01)

I forgot to emphasize (as most folks here probably know):  While neither
propositional nor predicate logic is negation complete (and that's a
very good thing!), both are *semantically* complete (that's also a very
good thing :-).    (02)

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