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Re: Equality and subclass axioms

From: pat hayes <phayes@ai.uwf.edu>
Date: Sun, 26 Nov 2000 13:34:38 -0600
Message-Id: <v04210100b64710c00756@[205.160.76.86]>
To: Jim Hendler <jhendler@darpa.mil>
Cc: www-rdf-logic@w3.org
>At 16:24 +0000 11/25/00, Ian Horrocks wrote:
>>I have heard some worries expressed about the effect of having
>>equality (if and only if) axioms in an ontology. For example, we could
>>state in our ontology that an object is a Triangle if and only if it
>>is in the intersection of the classes Polygon and ThreeSidedThing (in
>>this case it is sometimes said that being in the intersection of the
>>classes Polygon and ThreeSidedThing is a necessary and sufficient
>>condition for being a Triangle). The semantics of this axiom are that
>>in every satisfying "interpretation" (i.e., in every model of the
>>world that conforms to the structural constraint imposed by the
>>axiom), the set of objects that are Triangles must be equal to the
>>intersection of the set of objects that are Polygons and the set of
>>objects that are ThreeSidedThings.
>>
>>As far as I can understand it, the worry is as to what will happen if
>>another equality axiom w.r.t. Triangle is added to the ontology, e.g.,
>>that an object is a Triangle if and only if it is in the intersection
>>of the classes Polygon and ThreeAngledThing. This doesn't cause any
>>problem: the set of objects that are Triangles must be equal to the
>>intersection of the set of objects that are Polygons and the set of
>>objects that are ThreeAngledThings, and from the transitivity of
>>equality we can of course also infer that the intersection of the set
>>of objects that are Polygons and the set of objects that are
>>ThreeSidedThings is equal to the intersection of the set of objects
>>that are Polygons and the set of objects that are ThreeAngledThings.
>>
>>As with other ontological axioms, these kinds of axiom give structure
>>to the domain of discourse by restricting the set of valid models. Of
>>course it is possible to restrict the set so tightly that some (or
>>even all) classes are empty in all valid models, but this can happen
>>with or without equality axioms. If I am allowed to add a plug for
>>reasoning at this point, I would say that this is an example of how it
>>can be useful as it makes it possible for a tool to draw the users
>>attention to such an occurrence, which may indicate an error in the
>>design of the ontology.
>>
>>Regards, Ian
>
>Ian-
> Here's my worry, which is controllable, but definitely worth 
>thinking about.  You, being a rational person, come about with a 
>rule that says triangles are three-sided things.  I, being an 
>irrational person, have an axiom stating triangles are 4-sided.  We 
>both use if and only if rules.  Some web crawler comes, scrapes both 
>of our rules into the same knowledge base -- and then what happens? 
>Do we get two kinds of triangles or no kinds of triangles.

If the web crawler tries to believe both of you, it should conclude 
that there are no triangles (or maybe that three=four, but lets 
suppose it knows enough to reject that possible conclusion.) And that 
seems exactly right: if, indeed, you believe that the things with 
three sides are also things with four sides then you can figure out 
that there arent any of them (because if there were then three would 
equal four).

What exactly is the problem with this? I would be more worried if the 
web-crawler DIDNT notice that there was a contradiction here.

>Obviously, no model can allow triangles that are  both  three and 
>four-sided - so I would assume by throwing in my irrational axiom 
>I've somehow "negated" yours.

Yes, you have. But people will negate - ie contradict - each other, 
and this is a fact which will arise as long as anyone can say 
anything substantive; it has nothing particular to do with 
if-and-only-if rules.  In fact it can be that there can be N people 
who all say something such that any N-1 of them can be taken to 
agree, but they all contradict the Nth, and there is (literally) no 
way to tell, logically, which particular subset of them is 'right'. 
Logic, by its very nature, is powerless to adjudicate between 
opposing views: it simply detects inconsistency.

>Notice if we're not using If and only if, this problem doesn't come 
>up (i.e. IF three sides and IF 4 sided can concurrently occur).

Oh yes it does. It arises if we allow disjunctions (which are 
logically equivalent to implications, as Im sure you know) and 
negations; it arises, in fact, whenever it is possible to express any 
kind of contradiction. The only way to avoid it is to make it 
impossible for anyone to ever disagree with anyone else, by for 
example only allowing positive logic (no negations.) The problem with 
logics this weak is that it is very difficult to draw useful 
conclusions in them.

The process of drawing a conclusion, on the one hand, and detecting a 
contradiction, on the other, are very close: in full first-order 
logic they are basically the same process (you prove that B follows 
from A by showing that (A and notB) is inconsistent). So any logic 
which can support a useful amount of conclusion-drawing is also 
probably going to make it possible to express a contradiction. But in 
any case, I think that people should be able to contradict each 
other. God knows, people do in fact disagree about things.

> I think in general we who are worrying about web semantics will 
>either need something that tags axioms to their ontologies or 
>something that allows very permissive semantics.  Notice that I'm 
>not bothered by having something that uses the term triangle for two 
>different things - because both intended extensions are in there 
>(so, for a "search engine" we'd just be getting extra answers and 
>we'd use some rules to eliminate the ones we don't like).  I'm more 
>worried about something that let's someone whose beliefs differ from 
>mine do something that causes my inferences to fail.

But that isnt going to happen unless you *accept* his beliefs. Your 
hypothetical web-crawler got into its pickle because it *believed* 
everything it found from two different sources. More fool it, but 
there's nothing wrong with the logic involved. Gullibility isn't a 
logical issue.

> Again, these situations are sure to come up on something as 
>permissive as the net (imagine the pro-choice and anti-abortion 
>folks trying to defeat each others' theorem provers)

To be fair, your scenario is more like some hapless theorem-prover 
reading both pro-choice and anti-abortion inputs and then deriving a 
contradiction. But surely you would expect it to, wouldnt you? Would 
you prefer that it didnt even notice that there was a disagreement 
here?

> - but we need to decide at some point how to tag or mark or 
>distinguish or something where axioms come from, or something like 
>that.

I'm confused. Isnt this exactly what DAML does already, by 
'importing' from ontologies identified by URIs? In fact in DAML, 
every single symbol has a URI indicating its source. It would be 
difficult to imagine a more 'tagged' language than this.

Pat

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Received on Sunday, 26 November 2000 14:33:33 UTC

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