First, a brief historical trace of the developments in confirmation theory leading up to Goodman's infamous "grue" paradox is presented. Then, Goodman's argument is analyzed from both Hempelian and Bayesian perspectives. A guiding analogy is drawn between certain arguments against classical deductive logic, and Goodman's "grue" argument against classical inductive logic. The upshot of this analogy is that the "New Riddle" is not as vexing as many commentators have claimed. Specifically, the analogy reveals an intimate connection between Goodman's problem, and (...) the "problem of old evidence". Several other novel aspects of Goodman's argument are also discussed. (shrink)

According to Hempel's paradox, evidence (E) that an object is a nonblack nonraven confirms the hypothesis (H) that every raven is black. According to the standard Bayesian solution, E does confirm H but only to a minute degree. This solution relies on the almost never explicitly defended assumption that the probability of H should not be affected by evidence that an object is nonblack. I argue that this assumption is implausible, and I propose a way out for Bayesians. Introduction Hempel's (...) paradox, the standard Bayesian solution, and the disputed assumption Attempts to defend the disputed assumption Attempts to refute the disputed assumption A way out for Bayesians Conclusion. (shrink)

The Paradox of the Ravens (a.k.a,, The Paradox of Confirmation) is indeed an old chestnut. A great many things have been written and said about this paradox and its implications for the logic of evidential support. The first part of this paper will provide a brief survey of the early history of the paradox. This will include the original formulation of the paradox and the early responses of Hempel, Goodman, and Quine. The second part of the paper will describe attempts (...) to resolve the paradox within a Bayesian framework, and show how to improve upon them. This part begins with a discussion of how probabilistic methods can help to clarify the statement of the paradox itself. And it describes some of the early responses to probabilistic explications. We then inspect the assumptions employed by traditional (canonical) Bayesian approaches to the paradox. These assumptions may appear to be overly strong. So, drawing on weaker assumptions, we formulate a new-and-improved Bayesian confirmation-theoretic resolution of the Paradox of the Ravens. (shrink)

Hempel first introduced the paradox of confirmation in (Hempel 1937). Since then, a very extensive literature on the paradox has evolved (Vranas 2004). Much of this literature can be seen as responding to Hempel’s subsequent discussions and analyses of the paradox in (Hempel 1945). Recently, it was noted that Hempel’s intuitive (and plausible) resolution of the paradox was inconsistent with his official theory of confirmation (Fitelson & Hawthorne 2006). In this article, we will try to explain how this inconsistency affects (...) the historical dialectic about the paradox and how it illuminates the nature of confirmation. In the end, we will argue that Hempel’s intuitions about the paradox of confirmation were (basically) correct, and that it is his theory that should be rejected, in favor of a (broadly) Bayesian account of confirmation. (shrink)

Many philosophers agree that Hume was not simply objecting to inductive inferences on the grounds of their logical invalidity and that his description of our inductive behaviour was inadequate, but none the less regard his argument against induction as irrefutable. I argue that this constellation of opinions contains a serious tension. In the light of the tension, I re-examine Hume’s actual sceptical argument and show that the argument as it stands is valid but unsound. I argue that it can only (...) be converted into a sound one if our inductive behaviour can be characterized as a process of rule-governed ampliation. Drawing on some Bayesian ideas, I argue that our inductive behaviour probably cannot be characterized in that way, so our immunity from Hume is secure. Finally, I compare my response to Hume’s argument with some other well known responses. (shrink)

To clarify and illuminate the place of probability in science Ellery Eells and James H. Fetzer have brought together some of the most distinguished philosophers ...

Hempel's paradox of the ravens arises from the inconsistency of three prima facie plausible principles of confirmation. This paper uses Carnapian inductive logic to (a) identify which of the principles is false, (b) give insight into why this principle is false, and (c) identify a true principle that is sufficiently similar to the false one that failure to distinguish the two might explain why the false principle is prima facie plausible. This solution to the paradox is compared with a variety (...) of other responses and is shown to differ from all of them. (shrink)

I argue that the standard Bayesian solution to the ravens paradox— generally accepted as the most successful solution to the paradox—is insufficiently general. I give an instance of the paradox which is not solved by the standard Bayesian solution. I defend a new, more general solution, which is compatible with the Bayesian account of confirmation. As a solution to the paradox, I argue that the ravens hypothesis ought not to be held equivalent to its contrapositive; more interestingly, I argue that (...) how we formally represent hypotheses ought to vary with the context of inquiry. This explains why the paradox is compelling, while dealing with standard objections to holding hypotheses inequivalent to their contrapositives. (shrink)

Several authors, e.g. Patrick Suppes and I. J. Good, have recently argued that the paradox of confirmation can be resolved within the developing subjective Bayesian account of inductive reasoning. The aim of this paper is to show that the paradox can also be resolved by the rival orthodox account of hypothesis testing currently employed by most statisticians and scientists. The key to the orthodox statistical resolution is the rejection of a generalized version of Hempel's instantiation condition, namely, the condition that (...) a PQ is inductively relevant to the hypothesis $(x)(Px\supset Qx)$ even in the absence of all further information. Though their reasons differ, it turns out that Bayesian and orthodox statisticians agree that this condition lies at the heart of the paradox. (shrink)