A Classical Linear $λ$-Calculus based on Contraposition

Abstract: We present a novel linear $λ$-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 $\typ\limp\typtwo$ and $\lneg\typtwo\limp\lneg\typ$ should have the same proofs. The introduction of a linear modus tollens rule, stating that from $\lneg\typtwo\limp\lneg\typ$ and $\typ$ we may conclude $\typtwo$, allows one to recover classical \MLL. Furthermore, a term assignment for this elimination rule,{the study of proof normalization in a $λ$-calculus with this elimination rule} prompts us to define the novel notion of contra-substitution $\tm\cos{\lvar}{\tmtwo}$. Introduced alongside linear substitution, contra-substitution denotes the term that results from grabbing'' the unique occurrence of $\lvar$ in $\tm$ andpulling'' from it, in order to turn the term $\tm$ inside out (much like a sock) and then replacing $a$ with $\tmtwo$. We call the one-sided natural deduction presentation of classical \MLL, the \CalcMLL-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 $\CalcMELL$.