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Re: [ontolog-forum] FW: Lattice of theories

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Len Yabloko" <lenya@xxxxxxxxxxxxx>
Date: Sun, 18 Jan 2009 21:44:58 +0000
Message-id: <W1118117960116971232315098@webmail42>
John,     (01)

As I said there is more to it than terminology. We still need to nail the issue 
of "grounding" raised by Sean.
>LY> T-Box/A-Box distinction is highly reminiscent (to say the least)
> > of class/individual or type/instance.
>The T-box is used to define a hierarchy of types (in Aristotle's
>terminology, 'categories') and the A-box is used to make assertions
>about individual instances that belong to those categories or types.
>LY> I think these are the closest common ancestors to both
> > terminologies - not syllogism.
>Aristotle introduced the method of definition by genus and differentiae.
>For each category (or type), the genus is the supertype and the
>differentiae are the axioms that distinguish a given category from
>other categories with the same supertype.  The syllogisms are the
>patterns of reasoning about the relationships among the categories.
>For this reason, the categories are also called "categorial syllogisms".
You and Pat H. seem to both subscribe directly to Aristotle in claiming that 
there is nothing but axioms that distinguish categories/types. It follows that 
type vs. individual (aka class vs. instance) distinctions is a matter of 
convenience, T-box and A-box are just boxes. Then how do we go about 
abstraction? Are you equating generalization with abstraction?    (02)

On the other hand, your two-stage semantics which Pat H opposed so vehemently 
in recent thread, seem to suggest that abstraction precedes generalization. 
That raises a question about how to apply such generalization to concrete 
things in world. Perhaps meta-level reasoning is your answer. But how do you 
establish correspondence between statements about types (ie abstraction) and 
statements about concrete things. This is the question of grounding, which 
neither you or Pat H is interested or able to answer.        (03)

>LY> This had been discussed on this forum many times, but I still fail
> > to see how all these different branches of logic relate.
>For a survey of Aristotle's categories and syllogisms and their
>relationship to modern logic, see
>    http://www.jfsowa.com/ontology/ontoshar.htm
>    Building, Merging, and Sharing Ontologies
>    (04)

I did read it (so carefully, in fact, that I notices broken link to tutorial on 
math and logic). Interestingly the word "class" only appears in a form of 
"sub-class" and seems to be indistinguishable from type/sub-type and category. 
This is problematic because notion of class is essential for abstraction.        (05)

>LY> When describing lattice of theories you used the term "type"
> > seemingly interchangeable with "theory".  According to your
> > description the lattice of theories is simply ordering lattice
> > for types. The question then is: what operation does this order
> > corresponds to ( must be some sort of composition operator).
>I'm sorry.  I made a slip of the fingers when I wrote that note.
>The partial ordering of theories in the lattice is determined by
>generalization and specialization.  In talking about lattices
>of types or categories, I have a habit of writing 'supertype'
>and 'subtype'.  But those are not the correct terms to use for
>the relation that links the theories.  For a more detailed
>presentation of the lattice of theories, see
>    http://www.jfsowa.com/logic/theories.htm
>    Theories, Models, Reasoning, Language, and Truth
>John Sowa    (06)

I red that one too, a few times before and now. I appreciate you tackling very 
important and difficult issues. But I still have the same questions about 
abstract-concrete loop (grounding), which is absolutely essential to 
application of any theory. What am I missing?       (07)

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