Logical Pluralism via Mathematical Convergence

Abstract We propose a novel approach to logical pluralism based on algebra-valued models of set theory, systematically demonstrating how both classical and non-classical set theories can be constructed within a unified mathematical framework. Our approach extends Shapiro’s eclectic pluralism to the specific setting of algebra-valued models. This leads us to formulate two notions of logical pluralism: liberal pluralism, which recognizes as legitimate any logic that underpins a set theory capable of capturing a significant portion of mathematical practice, and strict pluralism, which further requires that the resulting set theory satisfies a convergence constraint, ensuring its mathematical expressiveness is comparable to that of classical set theory. We argue that this latter notion of pluralism represents a departure from previous accounts, as it is based on convergence rather than divergence among classical and non-classical mathematical theories.