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1. Compare: apart from the assignment of ‘true’ totautologies and ‘false’ to contradictions, deductive logicis silent regarding the assignment of truth values.

2. It turns out that the axiomatization that Salmon gives (p. 59) isinconsistent, and thus that by his lights no interpretation could beadmissible. His axiom A2 states:

If “\(A\) is a subclass of \(B, P(A, B) = 1\)”(read this as “the probability of \(B\), given \(A\), equals1”).

Let \(I\) be the empty class; then for all \(B, P(I, B) = 1\). But hisA3 states:

If \(B\) and \(C\) are mutually exclusive, then \(P(A, B \cup C) =\)
   \(P(A, B) + P(A, C)\).

Then for any \(X, P(I, X \cup \overline{X}) = P(I, X) + P(I,\overline{X}) = 1 + 1 = 2\), which contradicts his normalization axiomA1. Carnap (1950, 341) notes a similar inconsistency inJeffreys’ (1939) axiomatization. This problem is easily remedied— simply add the qualification in A2 that \(A\) is non-empty— but it is instructive. It suggests that we ought not take theadmissibility criterion too seriously. After all, Salmon’ssubsequent discussion of the merits and demerits of the variousinterpretations, as judged by the ascertainability and applicabilitycriteria, still stands, and that is where the real interest lies.

3. For example, we might specify that our family consists ofdistributions over the non-negative integers with a given mean,\(m\). Then it turns out that the maximum entropy distributionexists, and is geometric:

However, not just any further constraint will solve the problem. Ifinstead our family consists of distributions over the positiveintegers with finite mean, then once more there is no distributionthat achieves maximum entropy. (Intuitively, the larger the mean, themore diffuse we can make the distribution, and there is no bound onthe mean.)

4. Indeed, according to the requirement of regularity (to be discussedfurther in §3.3), one should not be certain of anything strongerthan \(T\), on pain of irrationality!

5. Some authors simplydefine ‘coherence’ asconformity to the probability calculus.

6. Still, according to some, the fair price of a bet on \(E\)measures the wrong quantity: not your probability that \(E\) willbe the case, but rather your probability that \(E\) will be thecaseand that the prize will be paid, which may be ratherless — for example, if \(E\) is unverifiable. Perhaps we shouldsay that betting behavior can be used only to measure probabilities ofpropositions of the form ‘\(E\) and it is verified that\(E\)’. For typical bets, the distinction between‘\(E\)’ and ‘\(E\) and it is verified that\(E\)’ will not matter. But if \(E\) is unverifiable,then a bet on it cannot be used to elicit the agent’s probability forit. In that case we should think of this objection as showing that thebetting interpretation is incomplete.

7. Note, however, that some authors find calibration a poor measure forevaluating degrees of belief. One probability function can be bettercalibrated than another even though the latter uniformly assignshigher probabilities to truths and lower probabilities to falsehoods— see Joyce (1998).

8. There are subtleties that I cannot go into here, including the notionof admissibility, the relativity of chances to times, and Lewis’(1994b) revised version of the Principle.

9.Interestingly, Venn has a lengthy and disparaging discussion of“Gradations of Belief” long before they became established inthe subjective interpretation. He writes:

The subjective side of Probability … seems a mere appendage ofthe objective, and affords in itself no safe ground for a science ofinference…. The conception then of the science of Probabilityas a science of the laws of belief seems to break down at everypoint. (120–121)

10. The reference class problem is analogous to the “totalevidence” problem for Carnap, discussed above. Intuitively, theright reference class is determined by all the evidence relevant to mylongevity, and it is unclear what is and is not relevant evidencewithout appeal to probabilities.

11. It should be noted that Gillies argues that Humphreys’ paradox doesnot force non-Kolmogorovian propensities on us.