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 formulas 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在关于建立直觉主义乘法线性逻辑基扩展语义学的论文中所采用的方法，是Sandqvist针对直觉主义命题逻辑的工作向子结构情境的一个有趣移植。尤为引人注目的是，向子结构情境的迁移如何自然而然地为直觉主义线性逻辑的乘法片段提供了语义学。尽管Gheorghiu、Gu与Pym最终利用其基础为束蕴含逻辑建立了语义学，但这自然引发了一个问题：直觉主义线性逻辑的其余部分该如何处理？本文即提出这样一种语义学。这尤其引人关注，因为该逻辑包含一个模态连接词bang作为连接符号。捕捉以该连接词标记的公式的推理主义内容极具挑战性，本文末尾将对此进行专门讨论。