Clarifications on Countable Additivity

Seidenfeld, Kadane, Schervish, and Stern (henceforth SKSS), in “Finite Additivity, Countable Additivity, and the Comparative Principle” (2024) raise some important mathematical issues around the principles I developed in “Why Countable Additivity” (2013) (and one developed by Stewart and Nielsen (2021), another response to my paper). Some of SKSS’s results are phrased in dense mathematical language. In this note, I try to explain the important aspects of their results in ways that are more understandable, to help clarify their dialectical force. I show why I am dialectically unmoved by many of the points that they make. I still accept merely countably-additive probability as the best theory of probability for epistemology. However, I take their objection to my attempted extension of this argument to a comparative (rather than numerical) framework as a very significant one.