We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e., closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic, and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic and applies the guidelines of the multi-type methodology in the design of display calculi.

我们为具有指数的直觉、双直觉和经典线性逻辑引入了适当的显示演算，这些逻辑是可靠的、完整的、保守的，并且具有割消元和子公式特性。基于相同的设计，我们引入了具有指数的 Lambek 微积分的变体，旨在捕获交换和结合性的受控应用。适当性（即，规则中所有参数部分的统一替换下的闭包）是本提案的主要技术新颖性，既可以最顺利地证明切割消除，也可以为大量公理扩展开发总体和模块化处理以及直觉、双直觉和经典线性逻辑与指数的扩展。我们的建议建立在线性逻辑的代数和阶理论分析的基础上，并在显示演算的设计中应用多类型方法的指导方针。