On Jan 28, 2009, at 11:21 AM, Len Yabloko wrote:
[EB]>Rather than discussing the philosophical and theoretical significance of
undefined terms, I would suggest that the proponents of Category Theory
for knowledge engineering identify the aspect(s) of category theory for
which they see a specific use in knowledge engineering. Otherwise the
discussion is pointless.
Now, to that end, Len Yabloko wrote:
There is no reliable way in classical Logic to establish and confirm
the identity of any object outside of specific context.
I did not comment on this at the time as it didnt seem like this thread was likely to go anywhere, but it needs to be clarified. First, classical logic does not even refer to contexts, so let us put that particular issue to one side for now. So the claim is there is no way in classical logic to 'establish the identity' of an object. I think that is correct, but first I want to know what it is supposed to mean. What does 'establishing the identity' amount to? Can anyone give an example of this hypothetical process being carried out successfully? Suppose I tried, but failed, to establish an identity: how would I know that I had failed?
What is correct is that there is no way in pure logic to write axioms which guarantee that a given name refers to a particular thing (except possibly certain very abstract Platonic kinds of 'thing' such as the property of being an Abelian group, but that is cheating since these 'things' are themselves defined only relative to logically expressible axioms.) For example, the fact the the name "Len Yabloko" refers to you, the actual living breathing Len, cannot be captured by logical axioms. This is often referred to as the "grounding problem" in discussions of knowledge representation in philosophical AI.
However, if this is what you are referring to, Category theory is no help, as the fact that your name denotes you cannot be specified in category theory either. In fact, it cannot be specified in any purely mathematical theory or framework. So I am left wondering if indeed this is what you are referring to, or whether you are talking about a different notion altogether.
There is no universal reference scheme for 'thing' in terms of
properties. The presumption of classical logic is that terms that
denote 'things' in the UoD do just that. The presumption that distinct
terms denote different 'things' is an axiom, which a given theory may or
may not include.
I think the usefulness of any theory depends on what it preserves. Obviously axioms of any theory are preserved by that theory. Someone correctly pointed that CT theory itself can be stated in classical Logic. But you would have to create axioms for any property that you want to preserve, and such axioms may not be shared by different theories. The difference in using CT is that certain axioms, such as identity and associativity are universal.
I really do not know what you are talking about here. Of course axioms are 'universal': that is, they are (assumed to be) true everywhere. That is why they are called "axioms". This holds in all logical and mathematical frameworks. If you wish to assert associativity as a universal axiom (not a good idea, since it is often false; but if you do) then this is very easy to do in logic. Here is the axiom you need in CLIF syntax:
(forall (R x y z)(= (R x (R y z))(R (R x y) z) ))
However, as I say, this is not a good axiom, as it has many counterexamples. For example, take R to be subtraction and x y and z to be numbers.
CT, on the other hand, includes identity in the very definition of object.
Citation please. This sentence means nothing to me.
As stated it means nothing, but I think what was meant is that CT assumes that all 'objects' (in its highly technical sense of 'object', which might be glossed in English as 'mathematical object') have an identity morphism defined on them. The identity morphism is a foundational part of each category of objects. However, CT's notion of morphism should not be confused with any philosophical or even common-sense notion of "identity".
But identity is an interesting problem in logical theories, and it is
possible that this bit of the discussion is actually going somewhere.
The direction I would like it to go is actually back to the question of theory grounding,
Again, I need to know a lot more about what this is supposed to mean. I don't think it means what is often referred to as "symbol grounding", ie how to associate names with their intended denotations. (Is it??)
which I see as the law of identity preservation.
I find this remark wholly opaque.
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