Theories, Axiomatizations, and Contexts

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Message-id: <199208272157.AA05886@cs.umn.edu>
Date: Thu, 27 Aug 92 17:42:40 EDT
From: sowa@watson.ibm.com
To: srkb@isi.edu, cg@cs.umn.edu
Subject: Theories, Axiomatizations, and Contexts

I wanted to clarify some of the points in Tom Gruber's recent note:

> Here is a definition that seems to be consistent with
> standard usage and the recommendations of people I've asked who are
> authorities on such matters:
>
>    A theory is a set of sentences.   (Sentences = axioms.)
>
> We can leave open whether a set of definitions is anything else than a
> run-of-the-mill set of sentences.  Same for contexts viz theories.
> Would that suit the needs of this discussion?

I'm all in favor of using the common definitions in logic, but the
quotations from McCarthy and Hayes were not intended as complete
definitions -- it is not true that any arbitrary set of sentences is
a theory.  It is essential to add that the set of sentences that form
a theory must be closed under deduction; i.e. any sentence that is
provable from the sentences of a theory T is also a sentence of T.

Also you may have a theory with a finite number of axioms, but an
infinite number of sentences that are provable from those axioms.
So you can't write the equation "Sentences = axioms".

> Second, whether there is a distinction between a
> set of definitions and an arbitrary set of axioms (which is one of the
> formal definitions of "theory") is a longstanding, interesting, and
> currently popular RESEARCH QUESTION in the knowledge representation
> community.

It is not a research question, but a matter of convention whether
you decide to have very conservative definitions and put all the
assumptions in your axioms or whether you allow definitions that
have assumptions buried in them.  You can express exactly the same
theories either way, but the two approaches differ in how explicitly
you distinguish the underlying assumptions.

Second, there is a distinction between a theory as the deductive closure
of a set of sentences and a particular axiomatization of a theory.  It
is possible (in fact, rather common) for different people to formulate
different sets of axioms that have exactly the same consequences.  In
that case, one could say that there was exactly one theory, but two
different axiomatizations for it.

It is also important to distinguish a theory from the package that
contains the theory.  In Guha's dissertation, it is not always clear
when he is talking about the package and when he is talking about
the contents.  And when you talk about the contents, you have to
distinguish the finite number of axioms that happened to be written
in the package from the possibly infinite number of theorems that
might be provable from those axioms.

In my writings on conceptual graphs, I define a context as a
concept (i.e. a box) that contains a set of graphs (i.e. sentences
stated as conceptual graphs).  That makes it very explicit that
the context mechanism is purely and simply a packaging device.

If I want to talk about the graphs in a context, I can attach
other relations to the context box c, such as "I believe c" or
"The collection in c is closed under deduction" or "The collection
in c is inconsistent with the collection in d" or "The collection
in c expresses rules of inference for manipulating the collection
in d".  This mechanism provides the metalanguage capability for
formulating theories, packaging them, and reasoning about them.

John Sowa