PART I: A Proof-Theoretic Approach to the Semantics of Classical Linear Logic

Linear logic (LL) is a resource-aware, abstract logic programming language that refines both classical and intuitionistic logic. The semantics of linear logic is typically presented in one of two ways: by associating each formula with the set of all contexts that can be used to prove it (e.g. phase semantics) or by assigning meaning directly to proofs (e.g. coherence spaces).

This work proposes a different approach by adopting a proof-theoretic perspective. More specifically, we employ base-extension semantics (BeS) to characterise proofs through the notion of base support.

PART II: The Modal Cube Revisited: Semantics without Worlds

We present a non-deterministic semantic framework for all modal logics in the modal cube, extending prior works by Kearns and others. Our approach introduces modular and uniform multi-valued non-deterministic matrices (Nmatrices) for each logic. The semantics is grounded in an eight-valued system and provides a sound and complete decision procedure for each modal logic, extending and refining earlier semantics as particular cases. Additionally, we propose a novel model-theoretic perspective that links our framework to relational (Kripke-style) semantics, addressing longstanding conjectures regarding the correspondence between modal axioms and semantic conditions within non-deterministic settings. The result is a philosophically robust and technically modular alternative to standard possible-world semantics.