Probabilities as Numbers

After more than three hundred years of study, the predominant theories of probability have settled on a few core notions.
Formally, probabilities are represented as numbers. Pre-formally, they're interpreted as frequencies, degrees of belief, intermediate truth values, propensities, weights, sizes.
While no single axiomatization is universally accepted, the predominant paradigm consists of theories in which probabilities are represented numerically (usually as values between 0 and 1) and whose primary relationships to one another are determined by the way in which they combine via addition.

In this chapter we catalog a few of these theories, with the goal of understanding how each one agrees with pre-formal (i.e., non-mathematical) intuition about probability. Our objective is a pre-formal meaning explanation for the axioms of each formal system, as opposed to an interpretation of what the specific probabilities are. In addition to shedding new light on formal concepts that are likely already familiar to some readers, this discussion sets the stage for the vastly different formalism of probability introduced in Chapter 5 and developed throughout the rest of the book.

Note: This is a chapter in the author's forthcoming book {\em Probability, Intuition, and Common Sense}.