Probability and Evidence.

Abstract

Professor Ayer writes as well as ever but, by his own high standards, this book is rather slight and slipshod. The first, main and title essay is less than ninety pages. In so short a span one would not expect justice to be done to this vast topic, and it is not. Twenty pages more are spent in discussing and rejecting Harrod's attempted refutation, in his Foundations of Inductive Logic (1956), of Hume's sceptical arguments on induction. The last thirty go on a discussion of conditionals, especially of the nontruth-functional kind usually called " subjunctive " or " counterfactual ", which are involved in characterizing the generalizations that sustain inductive inference.The topic of the second essay is slight, to my mind. Harrod's argument never was more than specious ; it does not call for such attention as Ayer gives it, especially in the light of what Ayer loaves out of his first essay. The last essay is good, clear and subtle, but neither very original, nor conclusive, nor well integrated with the rest of the book. Now Ayer does not claim these qualities for it but, like the Harrod piece, it is not just there " to bring this book up to a respectable size ", as Ayer engagingly puts it in the preface. It is also supposed to bear, and clearly does bear, on the theme of the first essay ; what is not clear in detail is how it does so. Ayer admits it leaves gaps ; another draft of the whole book might have plugged a number of them to our greater profit.

Macroscopic Observables and the Born Rule, I: Long Run Frequencies

“…If, for a general yes-no question Q and state ρ, one initially interprets ρ, Q as the probability of obtaining a positive answer to Q in the state ρ (or, more generally, interprets ρ, f as the expectation value of an observable f in a state ρ), then one still has to expand this interpretation by stipulating what notion of probability one is using [18,34]. Even if a probability equals one, as in (7), one still has to declare whether or not one adopts the so-called Necessity Thesis [34] (stating that probability one implies certainty).…”

“…Even if a probability equals one, as in (7), one still has to declare whether or not one adopts the so-called Necessity Thesis [34] (stating that probability one implies certainty). These questions cannot be answered by the mathematical formalism.…”

“…Even if a measurement of f ψ ∞ were to take place by computing the limit (14) from an infinite list x of single-case measurements, interpreting Born probabilities as limiting frequencies would face all the usual objections to the frequency interpretation of probability [16,18,20,34].…”

“…11 Except in an empty way, as in Popper's so-called propensity interpretation of probability [18,34].…”

Challenges to Bayesian Confirmation Theory

“…This topic will be addressed only in so far as it arises in broader challenges to the Bayesian approach. These debates are addressed by Sandy Zabell (This Volume) and Philip Dawid (This Volume) as well as survey volumes [Galavotti, 2005;Gillies, 2000;Mellor, 2005]. 4 Hence I apologize in advance to Bayesians who feel that their work has been mischaracterized as a challenge to Bayesianism.…”