Unified Semantics and Proof System for Classical, Intuitionistic and Affine Logics

This paper modifies our previous work in combining classical logic with intuitionistic logic [16, 17] to also include affine linear logic, resulting in a system we call Affine Control Logic. A propositional system with six binary connectives is defined and given a phase space interpretation. Choosing classical, intuitionistic or affine reasoning is entirely dependent on the subformula property. Moreover, the connectives of these logics can mix without restriction. We give a sound and complete sequent calculus that requires novel proof transformations for cut elimination. Compared to linear logic, classical fragments of proofs are better isolated from non-classical fragments. One of our goals is to allow non-classical restrictions to coexist with computational interpretations of classical logic such as found in the lambda mu calculus. In fact, we show that the transition between different modes of proof, classical, intuitionistic and affine, can be interpreted by delimited control operators. We also discuss how to extend the definition of focused proofs to this logic.