Completion of the Causal Completability Problem

We give a few results concerning the notions of causal completability and causal closedness of classical probability spaces (Hofer-Szabó, Rédei & Szabó [1999], Gyenis & Rédei [2004]). Answering a question from Hofer-Szabó et al. [1999], we prove that any classical probability space has a causally closed extension. We also employ the notion of causal up-to-n-closedness (Wroński & Marczyk [2010a]) to show that any finite classical probability space with rational probabilities on the atoms of the event algebra can be extended to a finite space which is causally up-to-3-closed. Lastly, we prove that any classical probability space can be extended to a space in which all correlations between events which are logically independent modulo measure zero event have a countably infinite common cause system (for the definition of the latter notion, see Hofer-Szabó & Rédei [2004]). Collectively, these results show that it is surprisingly easy to find Reichenbach-style explanations for correlations, underlining doubts as to whether this approach can yield a philosophically relevant account of causality