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RE: rdf as a base for other languages

From: Je'ro^me Euzenat <Jerome.Euzenat@inrialpes.fr>
Date: Sat, 9 Jun 2001 19:06:50 +0200
Message-Id: <a05010403b747d28ce613@[213.228.45.88]>
To: pat hayes <phayes@ai.uwf.edu>, "Ziv Hellman" <ziv@unicorn.com>
Cc: www-rdf-logic@w3.org
Hi all,

In his message (RE: rdf as a base for other languages) of 7/06/01,
pat hayes wrote:
>>  >I agree that would be a desirable goal. BTW, the 'A-/T-box'
>>  >terminology was originally used to distinguish assertions from
>>  >definitions (of concept vocabulary) , which isnt quite exactly the
>>  >same as the ground-fact/rule distinction.
>>
>>Could you elucidate the distinction between definitions and 
>>assertions, and explain how this differs from ground-fact/rule?
>
>Ah, now I have painted myself into a corner, since I never fully 
>understood the definition/assertion distinction myself, though it 
>seemed central to many folk (and still does). Although to be fair, 
>the idea of a definition is a pretty common one in mathematics and 
>life generally, in spite of its having no obvious logical content. 
>The intuition as I understand it is that saying that Foo is defined 
>by a certain assertion (eg a biconditional, say, but it could have 
>any logical form) is saying more than simply that the assertion is 
>true of Foo; it is saying that this condition is in some sense 'all 
>there is' to the meaning of Foo; that it completely defines the 
>meaning. This is not to say, of course, that the definition 
>completely specifies all the facts involving Foo, since the whole 
>point, usually, of defining concepts is so that they can be handily 
>used to state new facts. But it does imply a distinction between the 
>facts about Foo that are definitional in nature - that specify the 
>meaning of Foo, and moreover do so in some sense completely, ie 
>comprise a full account of that meaning - and facts about Foo that 
>are merely facts, which are stated using 'Foo' but which are not, as 
>it were, constitutive of the actual meaning. So for example, if the 
>defining condition were simply an assertion about the concept, then 
>to assert something that contradicts that definition would simply 
>generate a contradiction; but if it is taken to be definitional, 
>then one knows immediately that the contradicting assertion must be 
>false.
>
>...] Certainly it cannot be identified with anything as simple as a 
>syntactic distinction like ground-fact/quantified rule. Some 
>ontology folk argue that making the distinction logically requires 
>the use of a modal logic, so that definitions are not just true but 
>necessarily true, or that the terms so defined are 'rigid' (have the 
>same denotation in every possible world.) I have rather a jaundiced 
>view of this approach, but that is a topic which probably goes 
>beyond the purview of this mailing list. But in any case, many Krep 
>systems have tried to provide some way to make the distinction. (KIF 
>for example has an elaborate syntax for defining relations, 
>functions and so on.) The A-box/T-box distinction was one such 
>attempt. The key operational point, as I understand it, is that 
>while both the Tbox and the Abox consist of assertions, those in the 
>Tbox are cast in stone and cannot be altered, whereas those in the 
>Abox are mere data, which if they seem to contradict those in the 
>Tbox must be faulty.

Hum.
If one consider the use of these terms in description logics, this is 
a bit more precise. The TBox defines a terminology, i.e. assign terms 
(which are interpreted as sets) to names, while the ABox asserts 
formulas generally about individuals. The ABox can use the names 
assigned in the TBox, while the TBox usually do not refer to those 
interpreted as individuals in the ABox (but this is evolving).
This meets the mathematical delimitation between definitions and 
assertions (no?!).

Logically this can be cleanly defined by separating the languages. In 
model theoretic terms, this means that all terms are interpreted as 
sets (and all assertions can be reduced to inclusion between these 
sets).
There is an important consequence on that separation (that held at 
least in first DL languages): the terminology itself cannot be found 
inconsistent. This is because, it only deals with sets (e.g. the set 
of things with more than four legs and less than two legs) and the 
worst that can happen to them is to be empty (but having a whole 
theory about plenty of empty sets is not inconsistent). On the 
contrary the ABox can be inconsistent, just because it asserts things 
(e.g. that there exists something with four legs belonging to the set 
of things with at most three legs).

Nowadays, the terminologies are rather theories like you noted with 
assertions of equivalence (or subsumption) between terms (e.g. the 
set of individual with more than four legs is equivalent to the set 
of individuals with less that two legs). One could prove that no 
element belonging to the first set can belong to the second one and 
vice versa. But this does not make the theory inconsistent because it 
has at least one model: the one in which no individual exists (in 
that model both sets are empty and indeed equivalent).

If I am not wrong, the ONE-OF construct in DAML-OIL does only break 
the rule that the TBox cannot talk about individual terms. But it 
still does not assert anything about the considered individual terms 
(because all assertions are in the context of a term and if that term 
is interpreted as the empty set, nothing can be said about the 
considered individual terms).

Moreover, even the TBox cast-in-iron statement can be challenged. If 
I remember Nebel's thesis, he used the terms Terminology for TBox and 
World description for ABox. This clearly seems to denote the opposite 
standpoint: if the terminology cannot account correctly for the world 
description then, this is the Terminology which will be wrong.
But this is not a definitive position: one can take one view or 
another depending of the considered application (or depending if 
(s)he is platonist or aristotlelist, for using the big words). The 
view presented by Pat is more rigorous since a world description can 
be inconsistent in itself (without Terminology) while the Terminology 
cannot. So, it is a good bet to incriminate the World description. It 
is also more rigorous since the World description refers to the 
Terminology, so it is not a World description in itself.
-- 
  Jérôme Euzenat                  __
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Received on Saturday, 9 June 2001 13:13:07 GMT

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