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Re: semantics of daml:equivalentTo [was: Comments on Annotated DAML 1.6]

From: pat hayes <phayes@ai.uwf.edu>
Date: Fri, 13 Oct 2000 16:19:07 -0500
Message-Id: <v0421010bb60d14c1e502@[]>
To: Jeff Heflin <heflin@cs.umd.edu>
Cc: www-rdf-logic@w3.org
>Maybe I'm a little thick-headed (I don't have the experience that Peter
>and Pat have in semantics), but I don't really see the difference in
>their opinions. First, let me try to restate what they think I said:
>Pat: equivalentTo(X,Y) means that X and Y refer to same conceptual
>thing, i.e., they have the same denotation
>Peter: equivalentTo(X,Y) means that X and Y have the same definition
>Could someone please explain how X and Y could denote the same thing,
>but not have the same definition?

Oh, thats easy. Jim Hendler=DAML Project manager, but not by 
definition. The morning star and the evening star is the classical 
example. Life is just FULL of cases where we *discover* that two 
things are the same, but they werent declared to be by definition. So 
for a more mundane example, suppose it turns out that (the person who 
withdrew $1000 dollars from this bank account last Tuesday) = (the 
person with SS number 568-76-9831). This obviously doesnt follow from 
any definitions, but its still something which might be true and 
which you might want to be able to express.

Logicians often say that an identity-by-definition is a necessary 
equality, ie X and Y are identical if they couldnt possibly not be 
equal. Thats more like saying that the expressions substituted for X 
and Y have the same meaning. One of them being the definition of the 
other is one way to say they have the same meaning.

>Does the confusion lie in whether we
>consider X and Y to be symbols vs. definitions?

>In any case, if there
>really is as a difference between what Pat and Peter said, then I'm
>inclined to agree with Pat. I tend to think of ontology as providing a
>set of symbols and logical definitions for those symbols.
>In an attempt to be clear, let me take a shot at formalizing my notion
>of equivalentTo:
>Let D=a domain of concepts
>Let V=the set of DAML symbols
>Assume an interpretation function I:V->D
>Then if X,Y are elements of V, equivalentTo(X,Y) means I(X) = I(Y).

I'd phrase it thus: I(equivalentTo(X,Y))=True just when I(X)=I(Y). In 
other words, it means that *in that interpretation*. Necessary 
equality (identity) is more like:  equivalentTo(X,Y) means that for 
all I, I(X) = I(Y).

>course, since we're playing on the Web, we have to modify this a little
>bit: just b/c someone says equivalentTo(X,Y) doesn't make it true.

Yes, but just because someone says ANYTHING it doesnt make it true, 
right? Nothing to do with equivalence in particular.

>Rather, maybe the definition should be "if an agent accepts
>equivalentTo(X,Y) then the agent must accept I(X)=I(Y)."

Peter's stronger version (which is indeed what is often meant by 
'equivalent' rather than just 'equal') specifies what to do when 
someone also says something else about X  and whether one can always 
substitute Y for X in some other expression. For example, with the 
simple equality interpretation it would be fine to say that X 
equivalentTo Y and also that X  equivalentTo Z. The implication would 
be that Y was equal to Z, but there would be no inconsistency. But 
with Peter's stricter 'definitional' interpretation this would be 
illegal (unless Z and Y were identical expressions, in which case it 
would be pointless.) Or, another test: with the simple 
interpretation, if X equivalentTo Y then it would follow that Y 
equivalentTo X.

I have no particular brief for either interpretation, let me hasten 
to add. I only want the meaning to be clear. My financial future 
depends on being able to express quite a lot of facts in this DAM 
Language, and I'd like to know enough about what it means that I know 
what I'm saying in it.


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Received on Friday, 13 October 2000 17:16:11 UTC

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