Two Semantics for Zalta’s Object Theory

This paper investigates the relationship between two semantics for Edward Zalta’s Elementary Object Theory (OT): one proposed by Dana Scott and the other by Peter Aczel. We present some philosophical motivations underlying OT, characterize its second-order monadic fragment (MOT), and prove some of its theses. We define Scott and Aczel structures and establish a soundness theorem for MOT with respect to the latter. We indicate a class of Aczel structures in which a given formula is true iff it is true in all Scott structures. We also investigate two formulas: one concerning the extensionality of the identity of properties and another related to the overloading of extensions containing abstracta, meaning that if one abstract object exemplifies a property, then all abstract objects do.