Thursday, January 12, 2006
Additional reading:
Cory Casanave's paper on Data
Access
work on ontology for biological signal pathways
[127]
ß parallel discussion in
generative methodology bead thread
[127]
ß parallel discussion on Rosen
complexity
On the meaning of “axiom”
Earlier
part of this discussion
[342], [343], [344], [345], [346], [347], [348], [349], [350]
Communication from John Sowa about the meaning of “axiom”.
Paul,
I would agree that many people who talk about the Semantic Web use the terminology of math and logic in a rather loose way. But there are many of them who do have a strong background in math and logic.
In the usual translations from Euclid, the domain independent assumptions are called _axioms_, and the domain-dependent assumptions are called _postulates_.
However, most mathematicians and logicians today prefer to call all their assumptions _axioms_.
You said:
If we talk about the axioms
of First Order Logic, then how many are they? What is the list?
That depends on what methods of inference you use. With simpler rules of inference, you might need half a dozen axioms. With more complex rules of inference (such as natural deduction), there are zero axioms. In Peirce's system (which is a version of natural deduction), there is exactly one axiom, which is a blank sheet of paper. See, for example,
http://www.jfsowa.com/peirce/ms514.htm
You said:
People talk about an
ontology as a set of concepts, relationships between concepts, attributes
and/or properties of these concepts, then there are the
"axioms" and inference engines.
These axioms and inference engines seem in infinite variety.
Following is some basic terminology:
1. An axiom is an assumption, usually stated in some formal language for the purpose of automatic processing. But in more traditional math, axioms are usually stated in a highly disciplined version of a natural language supplemented with mathematical symbols.
2. A rule of inference is a kind of metalevel assumption about the methods for carrying out a proof. (That explains why you can reduce the number of axioms by increasing the number of rules of inference or vice-versa.)
3. An inference engine is a program that implements the rules of inference in order to derive a proof, either by starting from the axioms and deriving the conclusion or by negating the conclusion and deriving a contradiction.
4. A definition in _closed form_ is a definition of a symbol S by an expression X, which can be substituted for every occurrence of S.
5. An implicit definition is the adoption of some undefined symbol S and a collection of axioms that make statements about S that constrain its possible meanings.
Because of points #4 and #5, you can usually reduce the number of definitions by increasing the number of axioms or vice-versa.
You said:
I made the argument, not
accepted by the ONTAC group, that an ontology should be free of inference,
because inference is a logical entailment, and my training in
works by Rosen and others tells me that entailment is the
key issue that is not gotten right as yet. But we, our society, could really use
controlled vocabularies and concept ontologies.
You could say that an ontology is specified by a collection of axioms and definitions. Then you could consider inference to be a method of *using* the ontology. So in that sense, you could distinguish them, but on the other hand, they are related.
Bottom line: I would agree that a lot of people use many of the terms rather loosely.
John (Sowa)