Paradox of Confirmation

After long arguments between positivism and falsificationism, the verification of universal hypotheses was replaced with the confirmation of uncertain major premises. Unfortunately, Hemple proposed the Raven Paradox. Then, Carnap used the increment of logical probability as the confirmation measure. So far, many confirmation measures have been proposed. Measure F proposed by Kemeny and Oppenheim among them possesses symmetries and asymmetries proposed by Elles and Fitelson, monotonicity proposed by Greco et al., and normalizing property suggested by many researchers. Based on the (...) semantic information theory, a measure b* similar to F is derived from the medical test. Like the likelihood ratio, measures b* and F can only indicate the quality of channels or the testing means instead of the quality of probability predictions. Furthermore, it is still not easy to use b*, F, or another measure to clarify the Raven Paradox. For this reason, measure c* similar to the correct rate is derived. Measure c* supports the Nicod Criterion and undermines the Equivalence Condition, and hence, can be used to eliminate the Raven Paradox. An example indicates that measures F and b* are helpful for diagnosing the infection of Novel Coronavirus, whereas most popular confirmation measures are not. Another example reveals that all popular confirmation measures cannot be used to explain that a black raven can confirm “Ravens are black” more strongly than a piece of chalk. Measures F, b*, and c* indicate that the existence of fewer counterexamples is more important than more positive examples’ existence, and hence, are compatible with Popper’s falsification thought. (shrink)

We object to standard, simple random sampling resolutions of the raven paradox on the grounds that they relevantly diverge from scientific practice. In response, we develop a stratified random sampling model. It provides a better fit and apparently rehabilitates simple random sampling resolutions as legitimate idealizations of that practice. However, neither simple nor stratified models fare well with a second concern, the objection from potential bias. In response, we develop a third model on which we systematically check kinds of ways (...) in which disconfirming cases—non-black ravens—might be caused. This provides a novel resolution of the paradox that handles both objections. Suggestively, this third approach resembles Inference to the Best Explanation (IBE) and relates confirmation of the generalization to confirmation of an associated law. We give it an objective Bayesian formalization and discuss the compatibility of Bayesianism and IBE. (shrink)

Hypothetico-deductive (H-D) confirmation builds on the idea that confirming evidence consists of successful predictions that deductively follow from the hypothesis under test. This article reviews scope, history and recent development of the venerable H-D account: First, we motivate the approach and clarify its relationship to Bayesian confirmation theory. Second, we explain and discuss the tacking paradoxes which exploit the fact that H-D confirmation gives no account of evidential relevance. Third, we review several recent proposals that aim at a sounder and (...) more comprehensive formulation of H-D confirmation. Finally, we conclude that the reputation of hypothetico-deductive confirmation as outdated and hopeless is undeserved: not only can the technical problems be addressed satisfactorily, the hypothetico-deductive method is also highly relevant for scientific practice. (shrink)

Freitag (2015) and Schramm (2014) have proposed different, although converging, solutions of Goodman’s New Riddle of Induction. Answering their proposals, Dorst (2016 and 2018) has used the fictitious character of a ‘grue-speaker’ as his principal device for criticizing counterfactual-based treatments of the Riddle. In this paper, I argue that Dorst’s arguments fail: On the observation of no other than green emeralds, the ‘grue-speaker’ cannot use the symmetry between the ‘green’- and ‘grue’-languages for claiming ‘grue’- instead of ‘green’-evidence, and the counterfactuals (...) involved (explicitly by Schramm and implicitly by Freitag) remain unaffected by Dorst’s proposal for how to evaluate them. (shrink)

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. (shrink)

Historically, Nelson Goodman’s paradox involving the predicates ‘grue’ and ‘bleen’ has been taken to furnish a serious blow to Carl Hempel’s theory of confirmation in particular and to purely formal theories of confirmation in general. In this paper, I argue that Goodman’s paradox is no more serious of a threat to Hempel’s theory of confirmation than is Hempel’s own paradox of the ravens. I proceed by developing a suggestion from R. D. Rosenkrantz into an argument for the conclusion that these (...) paradoxes are, in fact, equivalent. My argument, if successful, is of both historical and philosophical interest. Goodman himself maintained that Hempel’s theory of confirmation was capable of handling the paradox of the ravens. And Hempel eventually conceded that Goodman’s paradox showed that there could be no adequate, purely syntactical theory of confirmation. The conclusion of my argument entails, by contrast, that Hempel’s theory of confirmation is incapable of handling Goodman’s paradox if and only if it is incapable of handling the paradox of the ravens. It also entails that for any adequate solution to one of these paradoxes, there is a corresponding and equally adequate solution to the other. (shrink)

Object of this reply. A chap like myself, who struggles along with an amateur's box of logical tools, is bound to feel uneasy when his arguments are probed by the kind of logical precision-instruments which Professor Hempel manipulates so effortlessly. Yet after painstakingly working over his technical arguments, and after appealing for expert assistance on matters outside my competence,1 I have reached the surprising and agreeable conclusion that my argument stands intact and that Professor Hempel's criticisms reveal once more the (...) hopelessness of the position he defends—or rather, of the position he now defends. (shrink)

This paper starts by indicating the analysis of Hempel's conditions of adequacy for any relation of confirmation (Hempel, 1945) as presented in Huber (submitted). There I argue contra Carnap (1962, Section 87) that Hempel felt the need for two concepts of confirmation: one aiming at plausible theories and another aiming at informative theories. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative theories. The main part of the paper (...) consists in working out the claim that one can have Hempel's cake and eat it too - in the sense that there is a logic of theory assessment that takes into account both of the two conflicting aspects of plausibility and informativeness. According to the semantics of this logic, a is an acceptable theory for evidence β if and only if a is both sufficiently plausible given β and sufficiently informative about β. This is spelt out in terms of ranking functions (Spohn, 1988) and shown to represent the syntactically specified notion of an assessment relation. The paper then compares these acceptability relations to explanatory and confirmatory consequence relations (Flach, 2000) as well as to nonmonotonic consequence relations (Kraus et al., 1990). It concludes by relating the plausibility-informativeness approach to Carnap's positive relevance account, thereby shedding new light on Carnap's analysis as well as solving another problem of confirmation theory. (shrink)

The concepts of coherence and confirmation are closely intertwined: according to a prominent proposal coherence is nothing but mutual confirmation. Accordingly, it should come as no surprise that both are confronted with similar problems. As regards Bayesian confirmation measures these are illustrated by the problem of tacking by conjunction. On the other hand, Bayesian coherence measures face the problem of belief individuation. In this paper we want to outline the benefit of an approach to coherence and confirmation based on content (...) elements. It will be shown that the resulting concepts, called genuine coherence and genuine confirmation, can be used in order to solve the two mentioned problems. In a final section we present some results on degrees of genuine coherence and genuine confirmation. (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)

"All ravens are black" is logically but not confirmationally equivalent with "all non-black things are non-ravens." But this is impossible, given that logical equivalence guarantees confirmational equivalence. In this paper, this paradox is solved.

To solve the highly counterintuitive paradox of confirmation represented by the statement, “A pair of red shoes confirms that all ravens are black,” Hempel employed a strategy that retained the equivalence condition but abandoned Nicod’s irrelevance condition. However, his use of the equivalence condition is fairly ad hoc, raising doubts about its applicability to this problem. Furthermore, applying the irrelevance condition from Nicod’s criterion does not necessarily lead to paradoxes, nor does discarding it prevent the emergence of paradoxes. Hempel’s approach (...) fails to adequately resolve the paradox. (shrink)

Confirmation and falsification are different strategies for testing theories and characterizing the outcomes of those tests. Roughly speaking, confirmation is the act of using evidence or reason to verify or certify that a statement is true, definite, or approximately true, whereas falsification is the act of classifying a statement as false in the light of observation reports. After expounding the intellectual history behind confirmation and falsificationism, reaching back to Plato and Aristotle, I survey some of the main controversial issues and (...) arguments that pertain to the choice between these strategies: the Raven Paradox, the Duhem/Quine problem and the Grue Paradox. Finally, I outline an evolutionary criticism of inductive Bayesian approaches based on my assumption of doxastic involuntarism. (shrink)