Branden Fitelson

Corroborating Testimony, Probability and Surprise’, Erik J. Olsson ascribes to L. Jonathan Cohen the claims that if two witnesses provide us with the same information, then the less probable the information is, the more confident we may be that the information is true (C), and the stronger the information is corroborated (C*). We question whether Cohen intends anything like claims (C) and (C*). Furthermore, he discusses the concurrence of witness reports within a context of independent witnesses, whereas the witnesses in Olsson's model are not independent in the standard sense. We argue that there is much more than, in Olsson's words, ‘a grain of truth’ to claim (C), both on his own characterization as well as on Cohen's characterization of the witnesses. We present an analysis for independent witnesses in the contexts of decision-making under risk and decision-making under uncertainty and generalize the model for n witnesses. As to claim (C*), Olsson's argument is contingent on the choice of a particular measure of corroboration and is not robust in the face of alternative measures. Finally, we delimit the set of cases to which Olsson's model is applicable. 1 Claim (C) examined for Olsson's characterization of the relationship between the witnesses 2 Claim (C) examined for two or more independent witnesses 3 Robustness and multiple measures of corroboration 4 Discussion.

Naive deductivist accounts of confirmation have the undesirable consequence that if E confirms H, then E also confirms the conjunction H·X, for any X—even if X is completely irrelevant to E and H. Bayesian accounts of confirmation may appear to have the same problem. In a recent article in this journal Fitelson (2002) argued that existing Bayesian attempts to resolve of this problem are inadequate in several important respects. Fitelson then proposes a new‐and‐improved Bayesian account that overcomes the problem of irrelevant conjunction, and does so in a more general setting than past attempts. We will show how to simplify and improve upon Fitelson's solution.

With the inclusion of an e ective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study by various researchers. Is the formula XCB = e(x e(e(e(x y) e(z y)) z)) a single axiom for the classical equivalential calculus when the rules of inference consist of detachment (modus ponens) and substitution? Where the function e represents equivalence, this calculus can be axiomatized quite naturally with the formulas (x x), e(e(x y) e(y x)), and e(e(x y) e(e(y z) e(x z))), which correspond to reexivity, symmetry, and transitivity, respectively. (We note that e(x x) is dependent on the other two axioms.) Heretofore, thirteen shortest single axioms for classical equivalence of length eleven had been discovered, and XCB was the only remaining formula of that length whose status was undetermined. To show that XCB is indeed such a single axiom, we focus on the rule of condensed detachment, a rule that captures detachment together with an appropriately general, but restricted, form of substitution. The proof we present in this paper consists of twenty- ve applications of condensed detachment, completing with the deduction of transitivity followed by a deduction of symmetry. We also discuss some factors that may explain in part why XCB resisted relinquishing its treasure for so long. Our approach relied on diverse strategies applied by the automated reasoning program OTTER. Thus ends the search for shortest single axioms for the equivalential calculus.

This is a high quality, concise collection of articles on the foundations of probability and statistics. Its editor, Richard Swinburne, has collected five papers by contemporary leaders in the field, written a pretty thorough and even-handed introductory essay, and placed a very clean and accessible version of Reverend Thomas Bayes’s famous essay (“An Essay Towards the Solving a Problem in the Doctrine of Chances”) at the end, as an Appendix (with a brief historical introduction by the noted statistician G.A. Barnard). I will briefly discuss each of the five papers in the volume, with an emphasis on certain issues arising from the use of probability as a tool for thinking about evidence.

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a "middle way" between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the "Monty Hall" problem

Contemporary Bayesian confirmation theorists measure degree of (incremental) confirmation using a variety of non-equivalent relevance measures. As a result, a great many of the arguments surrounding quantitative Bayesian confirmation theory are implicitly sensitive to choice of measure of confirmation. Such arguments are enthymematic, since they tacitly presuppose that certain relevance measures should be used (for various purposes) rather than other relevance measures that have been proposed and defended in the philosophical literature. I present a survey of this pervasive class of Bayesian confirmation-theoretic enthymemes, and a brief analysis of some recent attempts to resolve the problem of measure sensitivity