[Top] [All Lists]

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

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Sat, 17 Jan 2009 11:33:30 -0500
Message-id: <4972085A.5050507@xxxxxxxxxxx>
Ali, Len, Pat H, and Pat C,    (01)

I agree with the comments by Pat Hayes in this thread, but I'd
like to add a few more comments on some other points.    (02)

AH> My thoughts on the subject indicate that foundation ontologies
 > are inadequate. They each capture the biases of what people
 > consider to be foundational concept. It is difficult to identify
 > what is being said. However, I sympathize with Pat C's vision.
 > Though I think an ontology repository is more suitable than simply
 > an FO.    (03)

I agree.  I have been using Pat C's term because nobody had a better
one.  But the acronym FO could be preserved by inserting a little
preposition:  Foundation for Ontology.  Or perhaps we should make
the noun plural:  Foundation for Ontologies.    (04)

AH> The repository architecture I propose... is similar to the idea
 > of metaphor -- how is the solar system similar to an atom? We are
 > mapping substructures from one concept to another. Is there
 > something special about these patterns?    (05)

I very much agree.  See Figure 4 of the following paper:    (06)

    A Dynamic Theory of Ontology    (07)

That diagram shows an excerpt from the lattice of theories with four
operators for moving around the lattice.  The first three operators
are the AGM operators for belief revision and the fourth supports
metaphors:    (08)

  1. Contraction:  Delete axioms to move to a more general theory.    (09)

  2. Expansion:  Add axioms to move to a more specialized theory.    (010)

  3. Revision:  Perform contraction followed by expansion to move
     to a theory that is a sibling or a cousin of the previous one.    (011)

  4. Analogy:  Relabel one or more names of types, relations, or
     individuals to form an isomorphic theory in some other branch
     of the lattice.  (Mathematicians usually consider isomorphic
     theories to be identical, but for engineering applications
     it's important to distinguish theories about different, but
     similar physical systems.)    (012)

In the paper, I discussed how those operators can move from Newton's
theory of the solar system to Bohr's theory of the hydrogen atom.    (013)

AH>>> We're not combining ontologies, we're linking them in useful ways.    (014)

PH>> Well, that sounds nice, but until you clarify what you mean,
 >> its impossible to evaluate it. What does it mean to link as
 >> opposed to combine ontologies?    (015)

AH> Ok, when I say links, I mean that there is a relation (not
 > necessarily first order) that connects the two ontologies together.    (016)

These issues can be clarified by distinguishing the mathematical
formalism from the practical implementations.  The lattice of all
possible theories is an infinite mathematical structure that could
only be stored in its full glory in a Platonic heaven.  I use the
term 'moving' from theory to theory, since they all exist in that
eternal heaven.    (017)

For a finite repository, I suggest the term 'hierarchy of theories'
for any subset that may be documented in the repository.  Whether
and how the files that store those theories are linked, combined,
or moved is an implementation issue.  It's important for practical
purposes, but it doesn't affect the theoretical issues.    (018)

AH> The ontologies can be equivalent, consistent, inconsistent,
 > one contained in another, or disjoint. (Some of the preceding
 > interact with one another... i.e. disjoint ontologies are
 > trivially consistent).  These are what i mean by links. So you
 > can say that O1 is consistent with O2.    (019)

All those operations and the interactions among them are very clear
in terms of the lattice of theories:    (020)

  1. Two theories are equivalent if they correspond to the same node
     in the lattice.  The conditions for equivalence can be stated
     in proof-theoretic terms (same deductive closure) or in model-
     theoretic terms (true in exactly the same models).  For Common
     Logic the proof-theoretic and model-theoretic criteria determine
     exactly the same node in the lattice.    (021)

  2. Two theories are inconsistent if their only common specialization
     is the absurd theory at the bottom of the lattice (i.e., the
     theory in which everything is provable -- its deductive closure
     is every syntactically well-formed statement in the given logic).    (022)

  3. Two theories are consistent if they have a common specialization
     that is not the bottom of the lattice.    (023)

  4. I'm not sure what you mean by 'disjoint', but the simplest
     criterion for two theories to be "trivially consistent" is
     to have no common names of types, relations, or individuals.
     No axiom of one could contradict any axiom or conjunction of
     axioms in the other.  Therefore, their common specialization
     could be determined by taking the union of the axioms of each.    (024)

Whether you implement these operations by manipulating pointers
(URIs) or by physically moving or copying the axioms is irrelevant
to the formalism or its implications.    (025)

AH> They [3D and 4D theories] are so similar in fact that it is
 > hard for many people to even perceive the differences between
 > them. Intuitively they are indistinguishable. But therein lies
 > the problem: because our reasoning engines have no intuition,
 > and formally they (the theories) are incompatible. Intuitive
 > similarity means very little when it comes to ontologies.    (026)

Again, the lattice of all possible theories provides a clear
and precise way of analyzing the theories and determining how
to represent the similarities.  You can start with physical
observations of times and positions, which are recorded in
the same way in both.  Since all those observation statements
are the same for both, they would all be present in any common
generalization.  Whether any other generalizations would be
common to both would depend on the details of the axioms of
the 3D and 4D systems.    (027)

LY> T-Box/A-Box distinction is highly reminiscent (to say the least)
 > of class/individual or type/instance.    (028)

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.    (029)

LY> I think these are the closest common ancestors to both
 > terminologies - not syllogism.    (030)

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".    (031)

LY> This had been discussed on this forum many times, but I still fail
 > to see how all these different branches of logic relate.    (032)

For a survey of Aristotle's categories and syllogisms and their
relationship to modern logic, see    (033)

    Building, Merging, and Sharing Ontologies    (034)

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).    (035)

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    (036)

    Theories, Models, Reasoning, Language, and Truth    (037)

John Sowa    (038)

Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/  
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/  
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (039)

<Prev in Thread] Current Thread [Next in Thread>