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和Pym在其论文中为直觉主义乘法线性逻辑提供基础扩展语义的方法，是对Sandqvist为直觉主义命题逻辑所做工作向子结构情境的有趣改编。特别有趣的是，转向子结构情境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。虽然Gheorghiu、Gu和Pym最终利用其基础为束蕴涵逻辑提供了语义，但这引出了一个疑问：直觉主义线性逻辑的其余部分呢？在本文中，我提出了这样一种语义。这尤其令人感兴趣，因为该逻辑包含一个模态连接词——bang。捕捉带有该连接词的公式的推理主义内容尤其具有挑战性，本文最后将专门对此进行讨论。