FOM: reverse logic

[Harvey Friedman]

Simpson 1:33PM 10/18/98 writes:

>"Reverse logic" strikes me as a novel idea, one that I had never
>considered before.  But at first glance it seems much less compelling
>than reverse mathematics, simply because the axioms of classical logic
>is much less questionable than the set-existence axioms which are the
>stock-in-trade of reverse mathematics.

Reverse logic is an old subject. Of course, it hasn't been pursued nearly
as systematically as reverse mathematics. For instance, Errett Bishop was
concerned with whether or not certain principles implied the "limited
principle of omniscience" or "the law of excluded middle of a certain
kind," etcetera. Even in arithmetic, certain benchmarks were explicitly
pointed out. E.g., law of excluded middle for sigma-0-1 sentences, Markov's
principle not(forall integers x)(R(x)) implies (therexists x)(notR(x)),
where R is primitive recursive, Church's Thesis as a scheme (forall
x)(therexists y)(A(x,y)) implies (therexists a recursive f)(forall
x)(A(x,f(x)), etcetera. And a number of scattered results were obtained.

For a number of reasons, which are not so easy to pinpoint, this does not
seem to be nearly as fertile an idea as reverse mathematics, but I think
can be made to connect up with it in a serious way.