# Re: Can we say the DL knowledge base is only composed of axioms?

From: Ioachim Drugus <sw@semanticsoft.net>
Date: Sat, 07 Jul 2007 19:16:59 -0700
Message-ID: <4690491B.3060101@semanticsoft.net>
To: Jie Bao <baojie@cs.iastate.edu>
CC: Leon Essence <leon.essence@gmail.com>, semantic-web@w3.org
```
I believe, Leon wants
(1) To find out whether *objects* and *statements* are reducible to each
other, and only then
(2) Whether in DL a knowledge base can be composed only of axioms
The direct answer to (2) is simple - "yes", and the complete answer to
(2) is "yes, but it is better to keep expressivity". Probably, Leon
knows both answers to (2). So, I assume Leon meant (1), which is not an
easy question to answer. Below, I try to give my explanations.

Generally, you can work with a pure logic of statements where you talk,
or "predicate",  about objects - this kind of logic is called
*assertoric logic* or due to "predication" -" logic of predicates". You
can also work with a pure logic of objects, some of which represent
statements  - in mathematical logic, this kind of logic is said to be
*combinatory logic* ("lambda conversion" is part of it). Because you can
work with only one of these logics, they regard the two logics as
reducible to each other.

RDF is pure assertoric logic and you can express anything but it takes
long expressions to do this. To improve expressivity, RDFS and further
OWL introduced (object) classes and operations over objects.  Semantic
web logic is based on DL, which provides both types of logic via two
types of things it uses with and its TBox and ABox. So, in Semantic Web
we should not discard any of the two kinds of logic.

But I did not answer the main question on reducibility of *objects* and
*statements* to each other. I believe, they are reducible to each other
only *within the context of logic*. But when we also include the agent -
they call this "pragmatics"-, then they are *irreducible* to each other.
Logically, objects and statements are reducible to each other, but
pragmatically they are not. For Semantic Web, pragmatics is important,
because the web is a *web of agents*.

Here is a very short explaination why the two notions are irreducible to
each other with respect to an agent. If A can be *re-presented* as B,
this does not mean that A *is* B. If an agent has a presentation and he
is not told whether this is re-presentation of an object or
re-presentation of a statement, he cannot find this out on its own.
Therefore, an agent will get into a *confusion* - say, software will
encounter an *exceptional situation* which needs to be dealt with by a
programmer. To work without exceptional situations, we need to regard
*objects* and *statements* as different types of phenomena. More details
on this I gave in a paper which, I hope, will appear in the publications
of WI07, where I am also planning to give a talk on this.

Jie Bao wrote:
>
> http://en.wikipedia.org/wiki/Axiom#Logical_axioms
>
> I believe the answer is yes.
>
> Concepts, roles, individual and constructs are basic elements of
> axioms, which may form TBox, ABox or RBox axioms.
>
> On 7/7/07, Leon Essence <leon.essence@gmail.com> wrote:
>> Hi, dear colleagues,
>>
>> Recently I read some papers of Description Logic, I am confused by
>> something
>>
>> As we know, a DL knowledge base consists of concepts, roles, individual,
>> constructs, TBox and ABox,
>> The TBox contains axioms that make statements about how concepts and
>> roles
>> related to each other,
>> the ABox contian statements that asserts properties of individuals by
>> relating concepts and roles to them,
>>
>> Is the statement in ABox a kind of axiom?
>> What's the definition of axiom?
>> Can we say the DL knowledge base is only composed of axioms?
>>
>> can someone help me figure it out,
>> Thank you very much!
>>
>>
>> --
>> Best Regards!
>>
>> Jun Fang
>
>
```
Received on Sunday, 8 July 2007 02:17:02 UTC

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