SUO: *Date 21 Apr 2002 -- Theory Query
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Theory Query
JA = Jon Awbrey
PC = Pat Cassidy
Pat,
I drafted three or four attempts this week
to respond to your comments on this question,
but I've been of three or four minds about it
and couldn't sort it out. But here's a try.
Just to remember what it was about, I'll copy some fragments of the question:
JA: With special reference to those speculating
about sundry "lattices of theories" (LOT's):
JA: 1. What is a theory?
JA: 2. What does a lattice of theories look like?
JA: I'm asking these questions in the context of proposals that we
take the model theory of first order predicate logic seriously.
JA: Consider this standard definition of a theory, the only one I know,
such as makes sense within the favored frame of first-order logics:
| A '(first-order) theory' T of $L$ is a collection of sentences of $L$.
|
| T is said to be 'closed' iff it is closed under the |= relation. Etc.
|
| Chang & Keisler, 'Model Theory', page 36.
JA: Notice that the very definition of a theory is stated
relative to a given first-order predicate language $L$.
Until such a common language has been established, all
talk of lattices or other orders is just so much Babel.
JA: In particular, I am skeptical that you can even find,
without begging the question of intercommunicability,
any definition of a lattice of theories that remains
invariant over the choice of a language in which the
theories are expressed, indeed, where the finding of
the envisioned common language of comparison is just
another way of stating the initial problem to be met.
PC: The way I am using "theory" is in the
sense of an ontological theory, which is
a "theory of the real world" -- one which
addresses the question of which formally
expressed concepts and relations best
represent the objects and processes
of the real world for the purpose
of automated reasoning.
Sounds good.
PC: In order to be such a "theory", an ontology must have specific
widely recognized real-world objects and processes and events
which are **instances** of some concept in the "theory".
Different ontological theories will have some of the
specific objects (e.g. my Honda, Albert Einstein,
the Second World War) defined as instances of
different formal concepts in the ontologies.
The lattice will need to specify at which
point the ontologies diverge.
PC: This definition specifically excludes mathematical theories
which do not purport to represent objects or events on the
real world. For the definition you propose:
CK: A '(first-order) theory' T of $L$ is a collection of sentences of $L$.
PC: I would view this as a mathematical theory, unless some of
those sentences describe real-world objects as instances of
some concepts in the theory.
People have been proposing that we take first order predicate logic
seriously as a universal representational framework and that we use
the formal semantics or model theory that normally comes along with
it to solve many of our problems with meaning.
More and more people seem to be warming up to the appealing intuitive
notion that we might use a "lattice of theories" as an "architectonic"
principle for organizing our ontological theories.
So I took that combination of ideas seriously, and started doing
a little bit of spade work, since I had some relevant courses in
all of this stuff some years ago, and was wondering when I would
get to make any use of it.
Now, my attitude toward first order logic is a mixed bag,
because I see in it a "spirit and letter" problem, where
the general idea is all good but the specific implements
can be things not up to the spirit, and it has been not
an easy thing to get folks to recognize the difference.
But, I found that I could view "first order logical languages" (FOLL's) --
notated that way to remind me of their concrete and plural aspect, plus
all of the additional symbols and variables that are added in order to
talk about constants, functions, and relations -- in the same way that
I would view "finite state languages" or "context free languages",
namely, not everything is covered by them, but when something is,
it is definitely to your advantage to know it, as you'd be
wasting power to use anything more powerful in that case.
So that gives me a modus vivendi.
In short, that definition is not something that I made up,
but comes straight out of a standard text on the variety
of logic and model theory that others have been proposing.
PC: Two different ontologies will be inconsistent if,
for example, they each have some specific object (e.g.
the IBM corporation) as an instance of different concepts
that are specified as disjoint.
PC: A subsumption lattice does not necessarily have inconsistent theories,
but it may. If the theories are not inconsistent, they may be used as
a convenient way to organize a large ontology so that users may create
by selection of theories (modules, in this case) the smallest ontology
that will suit their purposes.
I doubt if this covers all the ways that two theories (sets of sentences)
could be inconsistent with each other, but it sounds to me like like you
just said that one of them contains the sentence "IBM is A" and the other
contains the sentence "IBM is Z", and at that stage I am left wondering
just where it is supposed to be written that "no A is Z"? That is to
say, what is one trying to represent by arranging it so that sentences
fall into this or that theory? The way the world is? Or the way that
different observers think it is?
The question that I was really trying to get at with my question was:
"Given the arrays of experiences that we have of the world and that
precede our conceptualization of it, what kind of process leads us
to form concepts like A and Z and to judge them mutually exclusive
in the first place?" Because it is then, and not before, that we
can put the letters "A" and "Z" into the alphabet of our logical
language, and not until then that we can even begin to reason by
means of deduction on the basis of these concepts. The question
is about the form of reasoning that ends in settling a definition.
That is the hard part of the problem -- the rest is just epilogue.
JA: Notice that the very definition of a theory is stated
relative to a given first-order predicate language $L$.
Until such a common language has been established, all
talk of lattices or other orders is just so much Babel.
That is, finding the language in which you 'can' compare
theories 'is' the whole problem of intercommunicability.
JA: In particular, I am skeptical that you can even find,
without begging the question of intercommunicability,
any definition of a lattice of theories that remains
invariant over the choice of a language in which the
theories are expressed, indeed, where the finding of
the envisioned common language of comparison is just
another way of stating the initial problem to be met.
Here, $L$ is just a list of symbols that are subject to purely
logical constraints, the axioms. There is not as much meaning,
or even as much disambiguation in that as a lot of people seem
to think. If you want more than that from a theory of meaning
and objective reference then you have move along the shelf and
pick another brand. Here's mine:
| Consider what effects that might conceivably
| have practical bearings you conceive the
| objects of your conception to have. Then,
| your conception of those effects is the
| whole of your conception of the object.
|
| Charles Sanders Peirce, The Maxim of Pragmatism, CP 5.438
This principle of interpretation makes a connection between thought and action,
not a simplistic connection, but still a very real one, and it is in acting on
the concepts that we believe are good that we come into contact with an object
reality that informs us whether we got it even approximately right yet, or not.
Jon Awbrey
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