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

From: Pat Hayes (
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
>>  paradoxes.
>OK; I can see why this concerns you.
>This sort of concern about paradoxes has convinced me
>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 
same paradoxes as before.

>more related ramblings in...
>>  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).

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