🤖 AI Summary
Prior proof-theoretic semantics for intuitionistic linear logic (ILL) were restricted to the multiplicative fragment and lacked a inferentialist account of the exponential modality “!”.
Method: This paper introduces a novel proof-theoretic semantics for full ILL—including “!”—based on base extensions, providing the first systematic inferentialist interpretation of “!” that respects resource-sensitive reasoning.
Contribution/Results: Within this framework, we establish soundness and completeness for ILL, precisely characterizing the structural role of “!” in resource-aware deduction. Our semantics fully covers all logical connectives and the “!” modality, thereby closing a longstanding theoretical gap in the proof-theoretic semantics of ILL. Moreover, the base-extension methodology offers a scalable paradigm for developing inferentialist semantics in substructural logics, advancing foundational work on meaning-as-use in resource-conscious systems.
📝 Abstract
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.