A Classical Linear $\lambda$-Calculus based on Contraposition

Abstract

We present a novel linear $\lambda$-calculus for Classical Multiplicative Exponential Linear Logic (\MELL) along the lines of the propositions-as-types paradigm. Starting from the standard term assignment for Intuitionistic Multiplicative Linear Logic (IMLL), we observe that if we incorporate linear negation, its involutive nature implies that both $A\multimap B$ and $B^\perp\multimap A^\perp$ should have the same proofs. The introduction of a linear modus tollens rule, stating that from $B^\perp\multimap A^\perp$ and $A$ we may conclude $B$, allows one to recover classical MLL. Furthermore, a term assignment for this elimination rule, {the study of proof normalization in a $\lambda$-calculus with this elimination rule} prompts us to define the novel notion of contra-substitution $t \{ a \backslash\!\backslash s \}$. Introduced alongside linear substitution, contra-substitution denotes the term that results from "grabbing" the unique occurrence of $a$ in $t$ and "pulling" from it, in order to turn the term $t$ inside out (much like a sock) and then replacing $a$ with $s$. We call the one-sided natural deduction presentation of classical MLL, the $\lambda_{\rm MLL}$-calculus. Guided by the behavior of contra-substitution in the presence of the exponentials, we extend it to a similar presentation for MELL. We prove that this calculus is sound and complete with respect to MELL and that it satisfies the standard properties of a typed programming language: subject reduction, confluence and strong normalization. Moreover, we show that several well-known term assignments for classical logic can be encoded in $\lambda_{\rm MLL}$.