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∂AIL Dr. Patrick Hayes↓Department of Computer Science
↓University of Essex↓P.O. Box 23↓Wivenhoe Park↓Colchester↓England CO4 3SQ∞
Dear Pat:

	Thanks for your long letter.

	First, as you will see minimal inference is unrelated to optimal
fixed points.  I am rather skeptical about optimal fixed points, because
the optimal fixed point depends on the algebraic structure of the
computation space.  Thus extending the space from the integers to Gaussian
integers or %2R(%E\%12)
can change the optimal fixed point, and
therefore it is hard to see that the optimal fixed point represents
any kind of computational process.

	I can't just write down the functional equation
of %2occur%1 as an ordinary predicate, because I want to be able
to go from a recursive definition to its representation in
first order logic without having to prove anything
first.  I intend to do the proving
after getting into first order logic.  Therefore, whatever I can do
with %2occur%1, I must also be able to do with

	%Bsillier x ← ¬ sillier x%1,

but allowing

	%B∀x.(sillier x ≡ ¬sillier x)%1

is disastrous, because from it I can prove anything, i.e. the functional
equation has no interpretations at all.  However,

	%B∀x.(silliera x = not silliera x)%1

is entirely harmless.  Since we limit ourselves to continuous functionals,
solutions always exist.  By the way, you may be interested in
the enclosed note on strange continuous functionals.
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