Finite Additivity, Complete Additivity, and the Comparative Principle

Abstract

AbstractIn the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) Existence: Settings where no countably additive probability exists though finitely additive probabilities do. (2) Complete Additivity: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the Comparative principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the Comparative principle [$${\mathcal{C}}$$ C ]. For numerical probabilities, principle $${\mathcal{C}}$$ C contrasts pairs, P1 and P2, defined over a common partition $$\prod$$ ∏  = {ai: i ∈ I} of measurable events. $${\mathcal{C}}$$ C requires that no P1 may be pointwise dominated, i.e., no (finitely additive) probability P2 exists such that for each i ∈ I, P2(ai) > P1(ai). By design, the cardinality of $$\prod$$ ∏ is not limited in $${\mathcal{C}}$$ C , which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle $${\mathcal{C}}$$ C in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle $${\mathcal{C}}$$ C is subject to each of the above concerns,  (1) and (2). Also, Easwaran considers principle $${\mathcal{C}}$$ C with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle $${\mathcal{C}}$$ C with qualitative probabilities and show that $${\mathcal{C}}$$ C restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle $${\mathcal{C}}$$ C .