• Salmon’s discussion of the hypothetico-deductive method beginning on p. 75 raises some of the same issues as Giere but from the point of view of someone far less sympathetic to that approach to confirmation. (Salmon’s comment that the H-D account is “woefully inadequate,” p. 76, nicely matches Giere’s comment, regarding views like ones Salmon advocates later, that “no [such] model of science … will prove adequate,” p. 277.) We won’t be able to spend much time on this discussion, but do note the diagram on p. 76; this picture of deduction from a hypothesis will be useful to have in mind at a number of points later in the course.

• The remainder of §2 (pp. 77-80) is where Salmon presents the Bayesian approach to confirmation. This is likely to occupy the bulk of our time on Salmon, and I’ve added a few explanatory notes on his discussion to the end of this guide.

• At the end of §2 and in §3, Salmon addresses the role of something like the “prior probabilities” of the Bayesian approach in a variety of views about science. He begins and ends with the idea of an ascription of plausibility to hypotheses that both helps to explain and is grounded in the historical development of science. In between, he considers the place of prior probabilities in a variety of positions on science. In the course of this he sketches interpretations of probability—the “logical probability” of people like Carnap, the “frequency theory” that Salmon supports, and the “subjective” or “personal” probability favored by many Bayesians. Along with the idea of “propensity” noted by Giere (on his p. 279), these form the key recent ways of understanding what probability is.

Some notes on Salmon’s discussion of Bayes’s theorem

The two equations displayed below restate Salmon’s equations on p. 78 with several changes. First, I’ve relabeled his letters A, B, and C to K, H, and O, respectively (for “background knowledge,” “hypothesis,” and “observation”) to suggest the role of these terms in the case of theoretical confirmation. Second, I’ve moved K (i.e., his A) from the first argument to a subscript; the quantity P_K obeys the same laws as P (though, of course, their values are likely to be different—e.g., P_K(H,O) = P(K & H,O), and these are different from P(H,O) unless K is not statistically relevant to O given H). Finally, I’ve used the first of these equations to state the denominator of the second in a simpler way:

Salmon uses the more complicated denominator in his statement of Bayes’s theorem (his 2nd equation) to show that the posterior probability P_K(O,H) depends on only the prior probability P_K(H) and the two likelihoods P_K(H,O) and P_K(~H,O). The dependence on these three factors can be seen even more clearly by considering Bayes’s theorem for odds,* something that is possible whenever neither the prior probability of ~H nor the probability of O given ~H is 0. Under these conditions, the theorem for odds is this:

(which results from dividing the probability theorem for H by the same theorem for ~H). This form of the theorem permits the simple statement that the posterior odds are the result of multiplying the prior odds by the ratio of the likelihoods.†