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.cb How to formalize vague concepts

	#. Consider "The red dog sees the cat" formalized as

!!a1:	%2sees(the(red(dog)),the(cat))%1

regarded as a proposition rather than a truth value.

We want to avoid committing ourselves as to the exact semantics of
any of the above entities according to the principle that formalizing
common sense concepts should not require %2knowing more science than
one actually knows%1, but nevertheless meanings should benefit from
improvements in knowledge.  The parallel principle should be adopted
that %2formalizing common sense shouldn't require solving all the
problems of philosophy in advance%1.
  Thus we wish to preserve some vagueness
in our understanding of what ⊗dog, etc. mean, but still use Tarskian
semantics.

	#. We introduce predicates ⊗true1(sentence, context),
⊗true2(sentence, context) where ({eq a1}) can serve as a
sentence argument, but the different ⊗true predicates can take
different kinds of entities as ⊗context.  We also have
⊗value1(term,context), etc. in our language, and ⊗value(the(red(dog)), context) 
will depend on the kind of context.   We include contexts of fiction
and hypothetical contexts.

	#. We are especially interested in what we might want to
say about ⊗the(red(dog)) that would hold for a wide collection
(possibly all) ⊗true predicates and ⊗value functions.

	#. In so far as we don't commit ourselves to specific ⊗true and
⊗value, we are allowing some of the necessary vagueness.  We emphasize
that the vagueness is not an accident of English, or even merely
a necessity of language; the use of vague concepts is essential
for successful thought.

	#. Circumscription and other non-monotonic reasoning goes
from general statements involving the vague concepts to ⊗models 
(in the sense of science or operations research rather than in
the narrow sense of mathematical logic).

	#. The models permit a finitization.  Thus the "cases" that
a model allows can often be explicitly listed.

	#. If we imagine the real situation to be infinitely rich,
we have mappings from finite models into reality (e.g. or perhaps i.e.
by naming) and models from reality into finite models (e.g. by
properties).  Aside to mathematicians: I have been considering an analogy with
the study of topology by singular homology (mappings from polyhedra
into topological spaces) and Cech homology (mappings from topological
spaces into polyhedra).

	#.  Possibly we can usefully treat the two
kinds of mappings separately, but it looks like we have to combine
them to get anything.  Thus when we we speak of "the color of the
dog" ⊗color(the(dog)) it is determined by ⊗value(the(dog),context) which
is an object in the world and by a COLOR mapping from an object
in the world to color names.