Benjamin Miller

We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation.

I argue that we cannot fully understand Aristotle’s position on political stability and state preservation in the Politics with paying close attention to his Eudemian Ethics. We learn from considering the Politics and the Eudemian Ethics in concert that even ‘correct’ regimes are unstable when citizens do not possess full virtue. Aristotle introduces his formal account of the knowledge requirements for virtue in Eudemian Ethics 8.3, and he applies these knowledge requirements as an explanation for state decline in Politics 2.9 when discussing the Spartans. If we primarily focus on the Nicomachean Ethics as Aristotle’s single essential ethical work, we will not learn the lesson he intends his readers to take away from the Spartan discussion in the Politics: that virtue requires correct understanding of the hierarchy and structure of the good life. This knowledge prevents the erosion of the virtues of character and the decline of political regimes.