  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.
