Running it up the flagpole to see if anyone salutes: A response to Woodward on causal and explanatory asymmetries

Does smoke cause fire or does fire cause smoke? James Woodward’s “Flagpoles anyone? Causal and explanatory asymmetries” argues that various statistical independence relations not only help us to uncover the directions of causal and explanatory relations in our world, but also are the worldly basis of causal and explanatory directions. We raise questions about Woodward’s envisioned epistemology, but our primary focus is on his metaphysics. We argue that any alleged connection between statistical (in)dependence and causal/explanatory direction is contingent, at best. The directions of causal/explanatory relations in our world seem not to depend on the statistical (in)dependence relations in our world (conceived of either as frequency patterns or as relations among chances). Thus, we doubt that statistical (in)dependence relations are the worldly basis of causal and explanatory directions.

An asymptotic property of large matrices with identically distributed Boolean independent entries

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically [Formula: see text]-diagonal and Boolean independent from each other. This paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that the random matrices considered are asymptotically Boolean independent from some of their partial transposes. The main results of the paper are based on combinatorial techniques.

Independence in randomizations

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy.