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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