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C00002 00002	Bodies of knowledge
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Bodies of knowledge

	Suppose we want to say that R2D2 knows the laws governing the
effects of moving blocks on a table.  One way of doing it is to give a set
of axioms governing the effects of action and for expressing situations on
tables and to assert that R2D2 knows these sentences.  This won't work if
we ourselves don't know the laws in question and will say too much if we
know them but don't know that R2D2 uses the same functions and predicates
that we do and even the same axioms.  For this and other reasons, we need
to be able to refer to "bodies of knowledge" without giving them explicitly.

	What is a body of knowledge?  It is not a theory, because the
specification of a theory involves specifying a language and a class
of sentences in this language.  One approach is to regard a body of
knowledge as an equivalence class of theories under a suitable equivalence
relation.

	What is the equivalence relation?  It seems that each predicate
or function of one theory should be definable in the other, and the
axioms of each should be theorems of the other.  There will also be
a 1-1 correspondence between the sets of models of the two theories.
Namely, given a model ⊗M1 of theory ⊗T1, and the definitions of the
predicates and functions of ⊗T2 in terms of those of ⊗T1, the values
... hmm what about the domain elements themselves?
WE'LL DO CHECKER POSITIONS BY LAGRANGE AND EULER AS THE EXAMPLE.

	Certain relations between models of one theory may correspond to the
same relations  between models of the other.  For example, it seems
to me likely that if ⊗M1 is an elementary submodel of ⊗M1' in theory
⊗T1, and that ⊗M2 corresponds to ⊗M1 and ⊗M2' to ⊗M1' in the correspondence
between the models of ⊗T1 and ⊗T2, then it will turn out that
⊗M2 is an elementary submodel of ⊗M2'.  If this is so, then some
of the model theory will turn out to be properties not just of
theories but of bodies of knowledge.