On Class Hierarchies

In her seminal article ‘Proper Classes’, Penelope Maddy introduced a novel theory of classes validating the naïve comprehension rules. The theory is based on a step-by-step construction of the extension and anti-extension of the membership predicate, which mirrors Kripke’s construction of the extension and anti-extension of the truth predicate. Maddy’s theory has been criticized by Øystein Linnebo for its ‘rampant indeterminacy’ and for making identity among classes too fine-grained. In this paper, I present a theory of classes which, while building on Maddy’s theory, avoids its rampant indeterminacy and allows for identity among classes to be suitably coarsegrained. I begin by presenting a bilateral natural deduction system for Maddy’s theory, which improves on her axiomatization in several respects. I then go on to show how to avoid the rampant indeterminacy by using supervaluational schemes in the construction of the extension and anti-extension of the membership predicate and how to augment the proof theory with corresponding, motivated rules. It turns out that whilst a van Fraassen-style supervaluational scheme suffices to avoid the basic problem of rampant indeterminacy, a supervaluational scheme based on maximally consistent extensions is needed for a proper treatment of identity.