TL;DR
This paper introduces a new relational semantics based on poset products that unifies and generalizes existing semantics for various substructural logics, providing a comprehensive framework with soundness and completeness guarantees.
Contribution
It presents a novel poset product-based relational semantics that unifies and extends previous semantics for multiple substructural logics, ensuring soundness and completeness.
Findings
Unifies existing semantics for H"ajek's basic logic and intuitionistic Lukasiewicz logic.
Provides sufficient conditions for soundness and completeness of the semantics.
Extends the framework to infinitely-many substructural logics.
Abstract
We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics.
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