Re: Equality and subclass axioms

> > > >Ian Horrocks wrote:
> > > > You didn't "negate" my axiom (you can never do that), you 
>just added some
> > > > additional information (an additional constraint). Assuming it is true
> > > > that no model can allow triangles that are both three and 
>four-sided, then
> > > > this is an example of the kind of "over-constraining" that I 
>mentioned in
> > > > my email: our ontology now constrains allowable models to the 
>extent that
> > > > none can ever contain an instance of triangle (i.e., we can infer that
> > > > triangle is equivalent to the class "Nothing"). If we use a reasoner to
> > > > check the ontology generated by our crawler, then it will detect this
> > > > fact, and can alert an intelligent (possibly human) agent to 
>the fact that
> > > > there may be a problem with the axioms relating to triangle.
> > > >
> > >Jeff Heflin wrote:
> > >But how can a system know when a particular definition is
> > >"over-constrained" and when an equivalence to "Nothing" is actually
> > >intended? Is a human going have to step in every time "Nothing" is
> > >defined and say, "Yes, I really meant 'Nothing'?" I hope not, because I
> > >can see ontology integration as a frequent occurence. I think that
> > >semantic search engines will need to be able to integrate ontologies on
> > >the fly to meet the needs/context of each query issued by a user. I
> > >don't believe you can have a single integrated ontology that works for
> > >all queries.
> > Pat Hayes wrote:
> > There may be a problem of nomenclature here. "Over-constrained" in
> > this sense just means "inconsistent". In a sense Ian is right, that
> > (monotonic) logic only allows one to add information, so that it is
> > impossible to "negate" an assertion with another, if this means
> > something like 'erase' or 'nullify'. But this is slightly
> > disingenuous, since it IS possible to contradict one assertion with
> > another. If A asserts P and B asserts not-P, then we usually would
> > say that they disagree, or are contradicting each other. Translated
> > into Horrocks-talk, this means that the conjunction of their
> > assertions (P and not-P) is so over-constrained that there is no
> > possible way to interpret it as describing a state of affairs, ie
> > what A says about the world cannot be reconciled with - contradicts -
> what B says about it.

>Ian Horrocks wrote:
>I was trying to make a serious point, not to engage in disingenuous
>double-talk. In the triangle example, what A and B assert is not P and
>not-P, but "X <-> P and X <-> Q, where P -> not-Q. From this we can infer
>that there is no such thing as an X (or a P, or a Q), just because this is
>the only state of affairs in which both assertions hold. In some
>circumstances (like our triangle example) the inference may be trivial,
>and/or may conflict with our intuition; in this case we may want to
>conclude that A and B "disagree", and that the ontology is "incorrect". In
>other circumstances the inference may be non-trivial and/or consistent
>with our intuition; in this case we may want to conclude that both A and B
>were "correct", and that by combining their knowledge we have discovered
>some new and useful fact.

I didn't mean to imply that you werent being serious or were engaging 
in double-talk; please forgive any unintended offense. My point was 
only that 'overconstrained' in the sense being used here just means 
'inconsistent'. And I agree that detecting an inconsistency is a 
useful process and does not indicate a problem of some kind. There is 
no real difference between making inferences and detecting 
inconsistency. Every sentence makes a claim about what the world can 
possibly be like; drawing conclusions is the process of ruling out 
some states of affairs as impossible, and detecting a contradiction 
is discovering that there are no possible states of affairs left.

BTW, in your example, I think it is really not at all clear what one 
'should' infer. If one accepts both of A and B as reliable sources of 
truth, then one would be justified in concluding that there were no 
X's, indeed. But speaking purely pragmatically, that would seem to me 
to be an unlikely conclusion. Why would a sane agent make an explicit 
assertion about things which do not exist? It seems more likely, in 
this case, that A and B in fact disagree about the nature of X's, and 
hold rival opinions. Logic is purely neutral on this point; it only 
tells us that something has got to give, as it were. We can't believe 
in X's and also believe both A and B; but which way to resolve this 
matter is up to us to decide. (For example, it might be instructive 
to check whether A and B, seperately, would agree with the conclusion 
that there are no X's.)

Pat

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Received on Tuesday, 28 November 2000 11:00:32 UTC