Countable Additivity in the Philosophical Foundations of Probability

Abstract

I study an open problem in the current philosophy of science: should countable additivity be an axiom of probability? Having examined the available philosophical arguments, I reach the conclusion that they all, on both sides of the debate, sometimes openly but sometimes not, crucially rely on two deep intuitions which are simply incompatible: one regards additivity, the other regards (the possibility of) uniformity between probability values. Given that any argument for or against the principle of countable additivity must contrast one of these two intuitions, this explains why the debate is still open, and will most likely stay that way. Finally, I examine a recent attempt at solving the deadlock, which makes use of non-standard analysis, at the price of losing real-valued probabilities and our usual idea of sum.