Substructural Routes to Variable Inclusion

This paper examines a range of logical systems within the family of variable inclusion logics—also known as containment logics. We focus on those logics that restrict classically valid inferences to ones meeting specific variable inclusion constraints, hence called variable inclusion companions of classical logic. These constraints can be seen as enforcing varying degrees of relevance between premises and conclusions, placing these systems within the broader tradition of relevance logics. We review established companions of Classical Logic, including Weak Kleene logics (Bochvar, 1938, Halldén, 1949), pure variable inclusion logics (Paoli et al., 2021), uniform Weak Kleene logics (Borzi and Zirattu, forth.), and their intersection—pure uniform logics. We then introduce two new systems named analytic and synthetic companions of Classical Logic. We characterize their consequence relations and interpret them, as their names suggest, as validating only analytic and synthetic inferences. Lastly, we argue that these systems more effectively address the irrelevance cases that motivated earlier proposals, by correctly tracing them to the Monotonicity of the consequence relation.