A Proof - Theoretic Foundation of Abortive Continuations ( Extended version )

We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce’s law without enforcing Ex Falso Quodlibet. We show that a “natural” implementation of this logic is Parigot’s classical natural deduction. We then move on to the computational side and emphasize that Parigot’s λμ corresponds to minimal classical logic. A continuation constant must be added to λμ to get full classical logic. We then map the extended λμ to a new theory of control, λC-tp, which extends Felleisen’s reduction theory. The new theory λC-tp distinguishes between aborting and throwing to a continuation and is in correspondence with a refined version of Prawitz’s natural deduction system.