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[ontolog-forum] Fwd: Ontology and Category Theory

To: "ontolog-forum" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Len Yabloko" <lenya@xxxxxxxxxxxxx>
Date: Thu, 29 Jan 2009 15:39:27 +0000
Message-id: <W435457809280981233243567@webmail26>
Pat,     (01)

Thank you for detailed response to my message which had unfortunately been 
jammed, chewed and replicated by ontolog-forum SW bug as Peter explained.     (02)

>On Jan 28, 2009, at 11:21 AM, Len Yabloko wrote:
>> Ed,
>> [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.
>> I agree.
>>> 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.
[PH]>I did not comment on this at the time as it didnt seem like this  
>thread was likely to go anywhere,    (03)

Now I am not so sure it was a bug :-)    (04)

> 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?    (05)

In ordinary life their are plenty of examples of 'establishing identity' from 
simple "hello" to solving and prosecuting crime. But formal meaning of identity 
in CT is (in my non-mathematical mind) a any procedure that performs identity 
morphism according to definition of Category (cited below).    (06)

>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.
>    (07)

Again (in my engineering mind) if "I" is defined as object that belongs to some 
Category (along with "You" and every other participant in this forum), then 
performing identity morphism procedure on any 'composition' of objects (such as 
this forum is) will have the same effect as 'composition' of separately morphed 
and then composed objects. This (I believe) is what axiom of associativity 
suggests (please correct me if I am wrong)     (08)

>>> 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.
>    (09)

I understand that not every operator needs to associative. But according to 
definition of Category there must be some (at least one - identity) morphisms 
that are associative. Those and only those define Category by providing 'law' 
of composition that always preserves identity and other properties entailed.    (010)

>>>> 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".
>    (011)

I think the intention (not intension) of the term "identity" in CT is the same 
as in any philosophical or even common-sense notion of "identity". However, 
unlike the later it can be formally proven, and therefore "grounded" in quite 
ordinary sense of having solid ground under(attached to) it.    (012)

>>> But identity is an interesting problem in logical theories, and it is
>>> possible that this bit of the discussion is actually going somewhere.
>>> -Ed
>> 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??)    (013)

No. My understanding of "symbol grounding" is of attempt to turn symbols into 
Category with associativity guarantied by whatever grammar is used to compose 
expressions. This attempt had failed (as fa as I know) for reasons that no 
grammar serves that purpose (speaking informally)     (014)

>> which I see as the law of identity preservation.
>I find this remark wholly opaque.
>    (015)

I hope I made it more transparent in my comments above.    (016)

>Pat Hayes
>> I believe that in this sense CT provides a natural framework for  
>> grounding.
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>    (017)

Len    (018)

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