# Re: semantic paradoxes (was Re: how to handle DAML+OIL syntax in theRDF model theory)

From: Pat Hayes ([email protected])
Date: 12/04/01

```>[this conversation has gone somewhat far afield;
>let me know if I should stop using joint-committee for it.]
>
>"Peter F. Patel-Schneider" wrote:
>>
>>  The problem with including meaningful syntax in interpretations is coming
>>  up with a set of semantic rules that actually work, i.e., don't produce
>
>OK; I can see why this concerns you.
>
>that we need to drop "P or not P" from the axioms (er.. axiom schemas)
>in whatever logic(s) we use for the Semantic Web.

That's a classical way out that doesn't really work. That is, it
works for the 'negation' paradox, but its easy to construct an
equally awkward case that breaks the logic even when it is weakened
in this way. But in any case, think about the consequences of
dropping 'P or not P'. That is a terrible thing to do to an
assertional logic, if you take it seriously. It amounts to saying
that P might be neither true nor false, so it can have some other
value. So you have in effect made all of Web logic into a
three-valued logic. That gets you effectively nothing in
functionality, but weakens the logic in all kinds of ways. It
basically just makes life more complicated to no purpose. If that
third value has a name (and it is usually the truthvalue of (not(P or
(not P)) , if negation works at all) then you can develop all the

>more related ramblings in...
>   http://www.w3.org/DesignIssues/Logic.html
>
>
>>  But when we try to finish this interpretation by determining CEXT(a) we are
>>  in trouble.  If something is in CEXT(a) then, by the semantic rules for
>>  complementOf, it has to not be in CEXT(a).  If something is not in CEXT(a)
>>  then, by the semantic rules for complementOf, it has to be in CEXT(a).
>
>That's the sort of thing that motivates getting rid of "P or not P"...
>at least from axioms. I wonder if it allows us to un-answer
>questions like this in the semantics.

Not all questions like this, unless the logic has no notion of
falsehood at all (like LCF).

Pat
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