On 12/15/2011 10:56 AM, joseph simpson wrote:
> Can you provide references to more specific applications and/or examples
> of the use of a Lindenbaum Lattice. (01)
I'm sorry for not responding sooner. In any case, my response to
Michael G. adds some further points. But I'd like to correct a typo
in my previous note to Michael: (02)
The supremum of two or more theories is their minimal common
generalization -- that's what Michael called the "strongest shared
subtheory". The infinimum is the maximal common specialization. (03)
For a brief summary of the definition of a lattice, you can start
with Section 7 of my summary of math & logic: (04)
http://jfsowa.com/logic/math.htm#Lattice (05)
There's a huge literature about lattices, but the Wikipedia article
on lattices is sufficient for the basic ideas. For Lindenbaum
lattices, the basic ideas follow from the observation that entailment
is the partial ordering that defines a lattice of all theories that
can be stated in a given logic L with a vocabulary V. (06)
For ontology, you can take V as the union of the vocabularies of all
the ontologies being considered. You can treat the hierarchies of
COLORE as finite subsets of the lattice. The relationship between
the lattice and the hierarchies is the same as the relationship
between the infinite set of integers and the finite subsets used
in actual computations. (07)
The critical operators on theories are the AGM operators that have
been developed for belief revision (or theory revision). The AGM
operators and postulates for belief revision are named after the
three authors who specified them: (08)
C. E. Alchourròn, P. Gärdenfors, and D. Makinson (1985).
On the logic of theory change: Partial meet contraction and
revision functions. Journal of Symbolic Logic, 50:510–530. (09)
For a brief overview of the ideas, the Wikipedia article is OK: (010)
http://en.wikipedia.org/wiki/Belief_revision (011)
For a good discussion of the theory and practice of belief revision,
see the books by the authors G and M of AGM: (012)
Gärdenfors, Peter (1988) Knowledge in Flux: Modeling the Dynamics
of Epistemic States, MIT Press, Cambridge, MA. (013)
Makinson, David (2005) Bridges from Classical to Nomonotonic Logic,
King's College Publications, London. (014)
Following is a good review article with 110 references. (Send a note
to the author for a copy, see http://pavlos.bma.upatras.gr/ ) (015)
Peppas, Pavlos (2008) Belief revision, in F. van Harmelen,
V. Lifschitz, & B. Porter, Handbook of Knowledge Representation,
Elsevier, Amsterdam, pp. 317-359. (016)
My KR book has brief overviews of both the lattice and nonmon reasoning: (017)
Sowa, John F. (2000) Knowledge Representation: Logical,
Philosophical, and Computational Foundations, Brooks/Cole
Publishing Co., Pacific Grove, CA. (018)
See pp. 94 to 97 for the lattice of theories. See pp. 373 to 394 for
an overview of nonmonotonic logics, their relationship to belief
revision, and the relationship of the lattice to the AGM operators.
To get a better idea of how nonmonotonic logic is related to belief
revision, see exercises 12 to 15 on p. 406. (019)
In that book, I extend the three AGM operators with a fourth operator,
which I called analogy in that book. I later decided to use the term
'relabeling' instead of 'analogy'. It's narrower and more descriptive. (020)
The basic idea is that renaming the vocabulary of a theory is a trivial
kind of revision -- that's why AGM didn't talk about it. But when you
consider an ontology, relabeling is essential for mapping a pattern
of axioms from one domain to another. It makes a jump from one branch
of the lattice to another. (021)
For a discussion of how the lattice of theories relates to dynamic
logic, natural language semantics, and nonmonotonic reasoning,
see slides 66 to 93 of the following presentation: (022)
http://www.jfsowa.com/talks/goal.pdf (023)
In particular, see slide 86 for the procedure for translating a proof
by Reiter's version of default logic to belief revision by the AGM
operators. That same method can be adapted to mapping any proof by
negation as failure to belief revision. (024)
I apologize to Michael for not stating that procedure formally,
but I believe that anybody with a mathematical background could
adopt Reiter's notation and systematically map a proof by his
method to belief revision by following the steps in slide 86. (025)
If anybody has any difficulty in carrying out such a formalization,
please send me a note, and I'd be happy to walk them through the
steps. (026)
John (027)
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