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composition of logical relations
Here is a simple counterexample to composition of logical
relations. Let A be the full set-theoretic
hierarchy over some infinite ground type, and B the term model
over inf many variables at each type. For simplicity, why don't
we assume A and B are identical at base type, and let R be
the identity relation at base type. The composition of
R and R-inverse at base type is the identity relation on
A, so if R composed with R-inverse (at all types) is logical,
it will have to be the identity relation on every type of A.
To see that this fails, look at R at the first functional type.
For f in A, we can have R(f,g) only if f and g are the same
function. Since the term model is countable, there will be many
f in A which are not in B, and so the composition of R and R-inverse
at a functional type will be much less that all of A.
John