Confluence for classical logic through the distinction between values and computations

We apply an idea originated in the theory of programming languages—monadic meta-language with a distinction between values and computations—in the design of a calculus of cut-elimination for classical logic. The cut-elimination calculus we obtain comprehends the call-by-name and call-byvalue fragments of Curien-Herbelin’s λ μμ̃-calculus without losing confluence, and is based on a distinction of “modes” in the proof expressions and “mode” annotations in types. Modes resemble colors and polarities, but are quite different: we give meaning to them in terms of a monadic meta-language where the distinction between values and computations is fully explored. This metalanguage is a refinement of the classical monadic language previously introduced by the authors, and is also developed in the paper.