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

2026-02-02

AI Summary

This work addresses the absence of a linear λ-calculus system for classical multiplicative exponential linear logic (MELL) that aligns with the propositions-as-types paradigm. It proposes an intuitionistic term-assignment-based one-sided natural deduction system, embedding the involutive nature of classical negation directly into the λ-calculus through linear negation, rejection rules, and a novel “reverse substitution” operation. The resulting classical linear λ-calculus, denoted λ_MELL, supports exponentials and is both sound and complete with respect to MELL. The system enjoys subject reduction, confluence, and strong normalization, and it uniformly encodes several existing term assignments for classical logics, thereby significantly extending the applicability of the propositions-as-types correspondence within linear logic.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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 $t\cos{a}s$. 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 $\lambda_{\rm MLL}$.

Problem

• linear lambda-calculus
• classical logic
• contraposition
• MELL
• propositions-as-types

Innovation

• linear lambda-calculus
• contraposition
• contra-substitution
• classical linear logic
• modus tollens