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C00002 00002	Notes on Approximate Theories
C00010 00003	.cb Partially Meaningful Concepts with Applications to Counterfactuals
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Notes on Approximate Theories

	Approximate theories, e.g. those whose concepts do not admit
precise definition in terms of the state of the world, have states
that only partially describe the world.  The causal laws of such
theories sometimes do not prescribe successors or prescribe successor
states less defined than the initial states.  It may be a
consequence of a theory that after some time, the theory will no
longer apply or at least that some of its objects may go away.

	An automata theoretic example might clarify the concepts.
It would be especially interesting to treat configurations in a
cellular automaton system as moving objects.

On Scott's advice

%3Scott, Dana (1970)%1: "Advice on Modal Logic" in %2Philosophical
Problems in Logic - Some Recent Developments%1, Karel Lambert (ed.),
D. Reidel Publishing Company.

	Scott's idea of indices (worlds) (points of view) seems usable.
Necessity-like operators are to be obtained by universally quantifying
over indices having a given relation to the %2referring%1 index.
Contrary to his advice, I favor introducing the indices explicitly
into a first order formalism.

	If we are to treat approximate theories, then we must
be sophisticated about what entities exist in related indices.  Thus
some indices will have entities like causes and wishes but lack
determinism.

	For cross-index correspondences,
we can use names of entities and have a relation %2corr(x1,x2,i1,i2)%1,
asserting that entity ⊗x1 in index ⊗i1 corresponds to ⊗x2 in ⊗i2, 
where the %2x%1's must be regarded as names of names or concepts or
some kind of intensional entity.
While rigid designators can easily be
declared in this formalism, a major interest will lie in
entities whose meaningfulness is restricted to a few indices.  This
correspondence relation
should may have to apply to predicates and functions as well
as to individuals.

	I don't know whether Scott proposes that theories define his
set ⊗V of virtual entities, but anyway this won't be possible in the
cases of interest to us.

EXAMPLES

	Here are some examples I would like to treat, but I am not sure
what we want to say, let alone how to express it formally.

.item←0
	#. Physical objects have locations, and smaller objects may be
located on parts of the larger objects or constitute parts of smaller
objects.  It doesn't make sense to ask the address of Washington, D.C.
We need to elaborate the argument that we cannot define the location
of an extended object as the position of its center of mass.  This
assumes more physics than a user of the concept usually knows, and if
you take general relativity seriously, it isn't precisely defined
anyway.  One could try to define a hierarchy of location sizes, but
again this isn't epistemologically plausible, i.e. it isn't plausible
that people have such a hierarchy as part of their common sense knowledge.

	#. The U.S. disapproves of the Soviet invasion of Afghanistan.
While this corresponds to the disapproval of many individuals in the
U.S., it is a distinct entity.  Namely, some theories of the international
behavior can use concepts of the opinions of nations to draw certain
conclusions without requiring a catalog giving for each nation a list
of the individuals (or their positions) whose disapproval constitutes
disapproval by the nation.  Ascribing a mental attitude to a nation
does not justify fully personifying the nation and ascribing to it
all the mental qualities of an individual person.  Common sense usage
doesn't do it.  It would be interesting to try to formalize just how
far this ascription can go.  Certainly it is used as far as ascribing
goals, beliefs and actions and supposing that certain of the actions
taken are in accordance with beliefs that they are appropriate ways
of achieving the goals.  However, these national mental attitudes
"interact" with those of individuals or groups at the very next
level of analysis, and mixed levels wherein national, group and
individual mental qualities are discussed are used in common sense
reasoning.  Of course, it may turn out that ascription of attitudes
to nations is not useful as soon as any kind of concrete analysis
is required.  It might be the most appropriate way for an Afghan
who cannot name even one American to express what he knows but
inappropriate for anyone with greater knowledge.
A sophisticated American policy analyst, who might have no reason
to say that the U.S. is angry at the S.U. might still ascribe this
belief to Afghans.

	When we use indices to represent points of view, a problem
will arise as soon as we need to express people's beliefs about
points of view.  This problem arises in any formalism.

	From the AI point of view, the chief problems with approximate
theories are what kinds of reasoning (in particular predictions) they
permit.  Existence as a predicate will be important.
.cb Partially Meaningful Concepts with Applications to Counterfactuals

	Our thesis is that many common sense concepts
are useful even though they seem to dissolve when examined closely.
The proper domain of such a %2partially meaningful
concept%1 is an %2approximate theory%1 of some aspect of the world.
The approximate theory is useful because it relates observations
that humans (or computer programs) can perform, general facts about
the world, facts about particular situations, actions that can be
performed, and the new situations that result from these actions.
However, when we try to reduce the theory to science, for example
physics, we may discover that its concepts cannot be given complete
definitions in terms of what we know about the physical world.