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Explore the philosophical foundations of semantic inferentialism through this colloquium lecture that examines how propositional contents are understood through their roles in material implication and incompatibility relations. Delve into the mathematical expression of reason relations using both proof-theoretic and model-theoretic approaches, while critically examining the structural assumptions underlying standard metainferential vocabularies. Discover how logical consequence properties like monotonicity and transitivity, while applicable in mathematical contexts, often fail to capture the complexity of reasoning in broader domains. Learn about innovative sequent-calculus specifications of classical logic and implication-space model-theoretic frameworks based on Girard's phase-space semantics for linear logic, designed to articulate even radically substructural reason relations. Compare top-down relational approaches that prioritize the quality of implications and their subjunctive robustness against bottom-up atomistic methods that begin with sentence truth values, understanding why the former proves more expressively powerful in substructural contexts.
