Fitelson (1999) demonstrates that the validity of various arguments within Bayesian confirmation theory depends on which confirmation measure is adopted. The present paper adds to the results set out in Fitelson (1999), expanding on them in two principal respects. First, it considers more confirmation measures. Second, it shows that there are important arguments within Bayesian confirmation theory and that there is no confirmation measure that renders them all valid. Finally, the paper reviews the ramifications that this "strengthened problem of measure (...) sensitivity" has for Bayesian confirmation theory and discusses whether it points at pluralism about notions of confirmation. (shrink)

It is well known that the probabilistic relation of confirmation is not transitive in that even if E confirms H1 and H1 confirms H2, E may not confirm H2. In this paper we distinguish four senses of confirmation and examine additional conditions under which confirmation in different senses becomes transitive. We conduct this examination both in the general case where H1 confirms H2 and in the special case where H1 also logically entails H2. Based on these analyses, we argue that (...) the Screening-Off Condition is the most important condition for transitivity in confirmation because of its generality and ease of application. We illustrate our point with the example of Moore’s “proof” of the existence of a material world, where H1 logically entails H2, the Screening-Off Condition holds, and confirmation in all four senses turns out to be transitive. (shrink)

Probability ratio and likelihood ratio measures of inductive support and related notions have appeared as theoretical tools for probabilistic approaches in the philosophy of science, the psychology of reasoning, and artificial intelligence. In an effort of conceptual clarification, several authors have pursued axiomatic foundations for these two families of measures. Such results have been criticized, however, as relying on unduly demanding or poorly motivated mathematical assumptions. We provide two novel theorems showing that probability ratio and likelihood ratio measures can be (...) axiomatized in a way that overcomes these difficulties. (shrink)

Igor Douven establishes several new intransitivity results concerning evidential support. I add to Douven’s very instructive discussion by establishing two further intransitivity results and a transitivity result.

Probabilistic support is not transitive. There are cases in which x probabilistically supports y , i.e., Pr( y | x ) > Pr( y ), y , in turn, probabilistically supports z , and yet it is not the case that x probabilistically supports z . Tomoji Shogenji, though, establishes a condition for transitivity in probabilistic support, that is, a condition such that, for any x , y , and z , if Pr( y | x ) > Pr( y (...) ), Pr( z | y ) > Pr( z ), and the condition in question is satisfied, then Pr( z | x ) > Pr( z ). I argue for a second and weaker condition for transitivity in probabilistic support. This condition, or the principle involving it, makes it easier (than does the condition Shogenji provides) to establish claims of probabilistic support, and has the potential to play an important role in at least some areas of philosophy. (shrink)

Bayesian epistemology suggests various ways of measuring the support that a piece of evidence provides a hypothesis. Such measures are defined in terms of a subjective probability assignment, pr, over propositions entertained by an agent. The most standard measure (where “H” stands for “hypothesis” and “E” stands for “evidence”) is: the difference measure: d(H,E) = pr(H/E) - pr(H).0 This may be called a “positive (probabilistic) relevance measure” of confirmation, since, according to it, a piece of evidence E qualitatively confirms a (...) hypothesis H if and only if pr(H/E) > pr(H), where qualitative disconfirmation is characterized by replacing “>” with “ “ with “=”. Other more or less standard positive relevance measures that have been proposed are: the log-ratio measure: r(H,E) = log[pr(H/E)/pr(H)] and the log-likelihood-ratio measure: l(H,E) = log[pr(E/H)/pr(E/~H)]. (shrink)

This article shows that a slight variation of the argument in Milne 1996 yields the log‐likelihood ratio l rather than the log‐ratio measure r as “the one true measure of confirmation. ” *Received December 2006; revised December 2007. †To contact the author, please write to: Formal Epistemology Research Group, Zukunftskolleg and Department of Philosophy, University of Konstanz, P.O. Box X906, 78457 Konstanz, Germany; e‐mail: franz.huber@uni‐konstanz.de.

Three confirmation principles discussed by Hempel are the Converse Consequence Condition, the Special Consequence Condition and the Entailment Condition. Le Morvan (1999) has argued that, when the choice among confirmation principles is just about them, it is the Converse Consequence Condition that must be rejected. In this paper, I make this argument definitive. In doing that, I will provide an indisputable proof that the simple conjunction of the Converse Consequence Condition and the Entailment Condition yields a disastrous consequence.

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. (shrink)

The old evidence problem affects any probabilistic confirmation measure based on comparing pr(H/E) and pr(H). The article argues for the following points: (1) measures based on likelihood ratios also suffer old evidence difficulties; (2) the less-discussed synchronic old evidence problem is, in an important sense, the most acute; (3) prominent attempts to solve or dissolve the synchronic problem fail; (4) a little-discussed variant of the standard measure avoids the problem, in an appealing way; and (5) this measure nevertheless reveals a (...) different problem for probabilistic confirmation measures, a problem that is unlikely to lend itself to formal solution. (shrink)

Carnap's inductive logic (or confirmation) project is revisited from an "increase in firmness" (or probabilistic relevance) point of view. It is argued that Carnap's main desiderata can be satisfied in this setting, without the need for a theory of "logical probability." The emphasis here will be on explaining how Carnap's epistemological desiderata for inductive logic will need to be modified in this new setting. The key move is to abandon Carnap's goal of bridging confirmation and credence, in favor of bridging (...) confirmation and evidential support. (shrink)

This paper presents a new analysis of C.G. Hempel’s conditions of adequacy for any relation of confirmation [Hempel C. G. (1945). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press, pp. 3–51.], differing from the one Carnap gave in §87 of his [1962. Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press.]. Hempel, it is argued, felt the need for two concepts of confirmation: one aiming at true hypotheses and another (...) aiming at informative hypotheses. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative hypotheses. I then show that one can have Hempel’s cake and eat it too. There is a logic that takes into account both of these two conflicting aspects. According to this logic, a sentence H is an acceptable hypothesis for evidence E if and only if H is both sufficiently plausible given E and sufficiently informative about E. Finally, the logic sheds new light on Carnap’s analysis. (shrink)

There is a plethora of confirmation measures in the literature. Zalabardo considers four such measures: PD, PR, LD, and LR. He argues for LR and against each of PD, PR, and LD. First, he argues that PR is the better of the two probability measures. Next, he argues that LR is the better of the two likelihood measures. Finally, he argues that LR is superior to PR. I set aside LD and focus on the trio of PD, PR, and LR. (...) The question I address is whether Zalabardo succeeds in showing that LR is superior to each of PD and PR. I argue that the answer is negative. I also argue, though, that measures such as PD and PR, on one hand, and measures such as LR, on the other hand, are naturally understood as explications of distinct senses of confirmation. (shrink)

The debate on probabilistic measures of coherence flourishes for about 15 years now. Initiated by papers that have been published around the turn of the millennium, many different proposals have since then been put forward. This contribution is partly devoted to a reassessment of extant coherence measures. Focusing on a small number of reasonable adequacy constraints I show that (i) there can be no coherence measure that satisfies all constraints, and that (ii) subsets of these adequacy constraints motivate two different (...) classes of coherence measures. These classes do not coincide with the common distinction between coherence as mutual support and coherence as relative set-theoretic overlap. Finally, I put forward arguments to the effect that for each such class of coherence measures there is an outstanding measure that outperforms all other extant proposals. One of these measures has recently been put forward in the literature, while the other one is based on a novel probabilistic measure of confirmation. (shrink)

This paper synthesizes confirmation by instances and confirmation by successful predictions, and thereby the Hempelian and the hypothetico-deductive traditions in confirmation theory. The merger of these two approaches is subsequently extended to the piecemeal confirmation of entire theories. It is then argued that this synthetic account makes a useful contribution from both a historical and a systematic perspective.

Several forms of symmetry in degrees of evidential support areconsidered. Some of these symmetries are shown not to hold in general. This has implications for the adequacy of many measures of degree ofevidential support that have been proposed and defended in the philosophical literature.

It is well known that the probabilistic relation of confirmation is not transitive in that even if E confirms H1 and H1 confirms H2, E may not confirm H2. In this paper we distinguish four senses of confirmation and examine additional conditions under which confirmation in different senses becomes transitive. We conduct this examination both in the general case where H1 confirms H2 and in the special case where H1 also logically entails H2. Based on these analyses, we argue that (...) the Screening-Off Condition is the most important condition for transitivity in confirmation because of its generality and ease of application. We illustrate our point with the example of Moore’s ‘‘proof’’ of the existence of a material world, where H1 logically entails H2, the Screening-Off Condition holds, and confirmation in all four senses turns out to be transitive. (shrink)

Epistemologists and philosophers of science have often attempted to express formally the impact of a piece of evidence on the credibility of a hypothesis. In this paper we will focus on the Bayesian approach to evidential support. We will propose a new formal treatment of the notion of degree of confirmation and we will argue that it overcomes some limitations of the currently available approaches on two grounds: (i) a theoretical analysis of the confirmation relation seen as an extension of (...) logical deduction and (ii) an empirical comparison of competing measures in an experimental inquiry concerning inductive reasoning in a probabilistic setting. (shrink)

In this paper, I offer a proof that a disastrous conclusion (namely, that any observation report confirms any hypothesis) may be derived directly from two principles of qualitative confirmation which Carl Hempel called the "Converse Consequence Condition" and the "Entailment Condition." I then discuss three strategies which a defender of the Converse Consequence Condition may deploy to save this principle.

In the recent literature on confirmation there are two leading approaches to the provision of a probabilistic measure of the degree to which a hypothesis is confirmed by evidence. The first is to construe the degree to which evidence E confirms hypothesis H as a function that is directly proportional to p and inversely proportional to p . I shall refer to this as the probability approach. The second approach construes the notion as a function that is directly proportional to (...) the true-positive rate – the probability of the evidence if the hypothesis is true, p – and inversely proportional to the false-positive rate – the probability of the evidence if the hypothesis is false, p . These reverse conditional probabilities – of the evidence on the truth or falsehood of the hypothesis – are sometimes known as likelihoods. I shall refer to the approach to confirmation that uses them as the likelihood approach.For each of these approaches, there are two obvious options as to how to define the confirmation function. The first is to define it as the difference between the magnitude that is treated as increasing confirmation or p ) and the magnitude that is treated as decreasing confirmation or p ). The second is to define confirmation as the quotient of these two magnitudes.This yields four different measures of confirmation, represented in the following table: Differences RatiosProbabilities PD = p – p1 PR = Graphic2Likelihoods LD = p – p3 LR = Graphic4All these measures agree on whether or not E provides some support for H. Intuitively, we want to say that E provides some support for H just in case learning that E is true would make us assign a higher probability to H, i.e. when p(H …. (shrink)

This paper presents an outline of a new theory of relevant deduction which arose from the purpose of solving paradoxes in various fields of analytic philosophy. In distinction to relevance logics, this approach does not replace classical logic by a new one, but distinguishes between relevance and validity. It is argued that irrelevant arguments are, although formally valid, nonsensical and even harmful in practical applications. The basic idea is this: a valid deduction is relevant iff no subformula of the conclusion (...) is replaceable on some of its occurrences by any other formula salva validitate of the deduction. The paper first motivates the approach by showing that four paradoxes seemingly very distant from each other have a common source. Then the exact definition of relevant deduction is given and its logical properties are investigated. An extension to relevance of premises is discussed. Finally the paper presents an overview of its applications in philosophy of science, ethics, cognitive psychology and artificial intelligence. (shrink)

Plausibly, when we adopt a probabilistic standpoint any measure Cb of the degree to which evidence e confirms hypothesis h relative to background knowledge b should meet these five desiderata: Cb > 0 when P > P < 0 when P < P; Cb = 0 when P = P. Cb is some function of the values P and P assume on the at most sixteen truth-functional combinations of e and h. If P < P and P = P then (...) Cb ≤ Cb; if P = P and P < P then Cb ≥ Cb. Cb – Cb is fully determined by Cb and Cbe – Cbe; if Cb = 0 then Cb + Cbe = 0. If P = P then Cb = Cb. (shrink)

Over the years several non-equivalent probabilistic measures of coherence have been discussed in the philosophical literature. In this paper we examine these measures with respect to their empirical adequacy. Using test cases from the coherence literature as vignettes for psychological experiments we investigate whether the measures can predict the subjective coherence assessments of the participants. It turns out that the participants’ coherence assessments are best described by Roche’s coherence measure based on Douven and Meijs’ average mutual support approach and the (...) conditional probability. (shrink)

In Gemes (1990) I examined certain formal versions of hypothetico-deductivism (H-D) showing that they have the unacceptable consequence that "Abe is a white raven" confirms "All ravens are black"! In Gemes (1992) I developed a new notion of content that could save H-D from this bizarre consequence. In this paper, I argue that more traditional formulations of H-D also need recourse to this new notion of content. I present a new account of the vexing notion of the natural axiomatization of (...) a theory. The notion is used to construct a form of H-D that allows for the type of selective confirmation without which Glymour (1980a,b) claims H-D is hopeless. (shrink)

Confirmation of a hypothesis by evidence can be measured by one of the so far known incremental measures of confirmation. As we show, incremental measures can be formally defined as the measures of confirmation satisfying a certain small set of basic conditions. Moreover, several kinds of incremental measure may be characterized on the basis of appropriate structural properties. In particular, we focus on the so-called Matthew properties: we introduce a family of six Matthew properties including the reverse Matthew effect; we (...) further prove that incremental measures endowed with reverse Matthew effect are possible; finally, we shortly consider the problem of the plausibility of Matthew properties. (shrink)

How can we determine the adequacy of a probabilistic coherence measure? A widely accepted approach to this question besides formulating adequacy constraints is to employ paradigmatic test cases consisting of a scenario providing a joint probability distribution over some specified set of propositions coupled with a normative coherence assessment for this set. However, despite the popularity of the test case approach, a systematic evaluation of the proposed test cases is still missing. This paper’s aim is to change this. Using a (...) custom written computer program for the necessary probabilistic calculations a large number of coherence measures in an extensive collection of test cases is examined. The result is a detailed overview of the test case performance of any probabilistic coherence measures proposed so far. It turns out that none of the popular coherence measures such as Shogenji’s, Glass’ and Olsson’s, Fitelson’s or Douven and Meijs’ but two rather unnoticed measures perform best. This, however, does not mean that the other measures can be rejected straightforwardly. Instead, the results presented here are to be understood as a contribution among others to the project of finding adequate probabilistic coherence measures. (shrink)