The approach taken by Gheorghiu, Gu and Pym in their paper on giving a base-extension semantics for Intuitionistic Multiplicative Linear Logic is an interesting adaptation of the work of Sandqvist for IPL to the substructural setting. What is particularly interesting is how naturally the move to the substructural setting provided a semantics for the multiplicative fragment of intuitionistic linear logic. Whilst ultimately the Gheorghiu, Gu and Pym used their foundations to provide a semantics for bunched implication logic, it begs the question, what of the rest of intuitionistic linear logic? In this paper, I present just such a semantics. This is particularly of interest as this logic has as a connective the bang, a modal connective. Capturing the inferentialist content of formulae marked with this connective is particularly challenging and a discussion is dedicated to this at the end of the paper.

翻译：Gheorghiu、Gu与Pym在其论文中为直觉乘法线性逻辑建立基扩展语义学的方法，是对Sandqvist为直觉命题逻辑所做工作在子结构框架下的一次有趣改编。尤为引人注目的是，这种向子结构框架的迁移如何自然地为直觉线性逻辑的乘法片段提供了语义依据。虽然Gheorghiu、Gu与Pym最终基于其基础理论为汇聚蕴涵逻辑构建了语义学，但这自然引出一个问题：直觉线性逻辑的其余部分又当如何？本文正是提出这样一种语义学。值得特别关注的是，该逻辑包含一个模态连接词——bang算子（!）。捕捉带有该连接词的公式的推理论内容尤为具有挑战性，本文末尾将专门就此展开讨论。