Abstraction Principles and the Classification of Second-Order Equivalence Relations

Abstract

This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class-those equinumerous elements of the equivalence class with equinumerous complements-can have one of only three profiles. The improvements to Fine's theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan's relative categoricity theorem. 2 We put aside concerns, like those of Quine in [18], that second-order languages are themselves not "logical" in virtue of their quantifying over higher-order objects.3 This view is associated with Tarski's identification of logical notions as those that are permutation invariant in this sense (see [22]), for a more recent defense see the work of Gila Sher (see [21]). Other relevant discussions can be found in [2] and [16]. The so-called Tarski-Sher thesis identifying logicality with permutation invariance is controversial; that permutation invariance is a necessary feature of logical notions is not (see [15]).