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 λμ 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.

1. Introduction

The goal of this paper is to formulate a unified logic that combines classical, intuitionistic and affine-linear logics (restricting contraction but allowing weakening). The unification is achieved semantically, proof-theoretically, and in terms of the computational interpretation of proofs. The connectives of this logic are similar to those of linear logic but contractions are controlled in a very different way. The system we introduce here, Affine Control Logic (ACL), is an extension of our PCL system presented in [17], which only combined classical and intuitionistic logics. This system also descends from previous attempts at formulating unified logics, including LU [8] and our own LKU [15]. These systems were based on linear logic. Linear logic embeds both classical and intuitionistic logics, but it is limited in its ability to mix them. For example, the interpretation of intuitionistic implication as !A −◦ B is a crucial component of linear logic. However, this interpretation is not compatible with the fragment that interprets classical logic. Consider ?((!A −◦ B) ⊕ C) (equivalently ?(!A −◦ B)&?C): here we are attempting to write an intuitionistic implication as a subformula of a classical disjunction. However, the strength of intuitionistic implication disintegrates: it is possible for A to escape its scope and be used in the derivation of C: the intuitionistic meaning and proof structure of !A −◦ B would not survive such a mixture.