Martin-Löf Clashes With Griffin, Operationally

The computational reading of proofs in intutionistic predicate logic (IPL) as λ-terms can be extended in two well-known ways. On the one hand, Martin-Löf showed how the Axiom of Choice (AC) can be interpreted as a term of dependent type theory. On the other hand, Griffin showed how Peirce’s law (which generates classical logic from intuitionistic) can be interpreted as a program using control operators, and hence we obtain a program from any classical proof. Together, these suggest the following naive computational interpretation of classical logic with AC. Take a proof in IPL + Peirce + AC and obtain a program using control effects following Griffin, using Martin-Löf’s term to translate AC. Then choose an evaluation strategy (call-by-name or call-by-value) and run the program. This paper shows that the naive approach does not work. We present a proof in IPL + Peirce + AC, which translates to a program that, when executed, regardless of the evaluation strategy chosen, gets stuck—it attempts to pattern-match a λ-abstraction. For any inhabited type σ, we can restrict AC to σ and still construct this example.