Non-Classical Approaches to the Brandenburger-Keisler Paradox

In this paper, we consider a well-known epistemic game theoretical paradox, called the Brandenburger-Keisler Paradox, and provide various alternative models in which the paradoxical statement becomes satisfiable. For this task, first, we resort to various non-classical logical frameworks, and reformulate the paradoxical statement in them. We discuss the paradox in non-well-founded set theory and in paraconsistent (inconsistency-friendly) logic. By constructing models which satisfy the paradoxical sentence, we provide a richer toolkit which can be used in epistemic game theoretical formalisms, and suggest that the choice of classical and traditional models in epistemic game theory seems rather arbitrary. Second, we suggest a different formulation of the paradox which requires models with higher cardinality. We achieve this by constructing a Yablo-like version of the paradox which turns out > ω categorical.