Carnap’s general topic in this and the next chapter would often now be described as the “interpretation” of probability. That is, the term “probability” now tends to be attached in the first instance to certain abstract laws, the axioms of probability theory, and the question becomes how to give the term a more concrete meaning by identifying a specific concept that obeys these laws. (This is an approach to concepts that Carnap was familiar with; but, when this book was written, it was not as widely used in the case probability as it now.)

Carnap considers two such interpretations of probability, statistical probability (or “frequency”) and logical probability. Now it would be common to add two more that were around at the time the book was written but have received more attention since then, subjective (or “personal”) probability and propensities. The four can be divided roughly into two groups. Statistical probability and the idea of propensities are features of the world apart from our thought about it. As such, an empiricist like Carnap would hold that empirical research is required to establish such probabilities. Logical probability and subjective probability are features of our thought, and empirical research is not required to apply these ideas. Although Carnap introduces the idea of logical probability in ch. 2, most of his discussion of it appears in ch. 3, and we will wait to discuss subjective probability along with it.

The idea of propensities is naturally discussed along with that of statistical probability. Carnap gives a good introduction to the issues concerning the latter. The idea of propensities arise in connection with one of these issues. Carnap notes that statistical probability does not immediately apply to single instances or “single cases.” He describes an indirect way of applying the idea that was suggested by Reichenbach. Proponents of the idea of propensity went further and argued that one could speak of probabilities of single cases directly: events can have real propensities to occur or not and the measure of these propensities is a kind of probability. One of its proponents was Karl Popper (1902-1994), a philosopher of science comparable in importance to Carnap and Reichenbach; here is a description of the view from his autobiography:

Part of Popper’s motivation concerned problems in the understanding of quantum mechanics, but he also argued directly concerning the kind of example of single case probabilities Carnap mentions (see p. 27-28). He noted that in cases where the there was no actual sequence of events within which to include the single case, someone wishing to speak of statistical probability or frequency, might speak of it with respect to a “virtual sequence” defined by the conditions generating the single event (e.g., that it is a throw of a given die), and he argued that

(If you want to see more of his argument, this article is available on JSTOR.)

As part of your preparation for class, think which of these two interpretations, statistical probability and propensity, you think is the better way of understanding probability as thought of as an objective feature of the world. In the next chapter, Carnap will argue that statistical probability is the sort of probability that occurs within scientific laws (while logical probability appears the probability of those laws). Do you think statistical probability or propensity provides a better account of the concept of probability that appears in laws?