William Roche

Recently there have been several attempts in formal epistemology to develop an adequate probabilistic measure of coherence. There is much to recommend probabilistic measures of coherence. They are quantitative and render formally precise a notion—coherence—notorious for its elusiveness. Further, some of them do very well, intuitively, on a variety of test cases. Siebel, however, argues that there can be no adequate probabilistic measure of coherence. Take some set of propositions A, some probabilistic measure of coherence, and a probability distribution such that all the probabilities on which A’s degree of coherence depends (according to the measure in question) are defined. Then, the argument goes, the degree to which A is coherent depends solely on the details of the distribution in question and not at all on the explanatory relations, if any, standing between the propositions in A. This is problematic, the argument continues, because, first, explanation matters for coherence, and, second, explanation cannot be adequately captured solely in terms of probability. We argue that Siebel’s argument falls short

We show that as a chain of confirmation becomes longer, confirmation dwindles under screening-off. For example, if E confirms H1, H1 confirms H2, and H1 screens off E from H2, then the degree to which E confirms H2 is less than the degree to which E confirms H1. Although there are many measures of confirmation, our result holds on any measure that satisfies the Weak Law of Likelihood. We apply our result to testimony cases, relate it to the Data-Processing Inequality in information theory, and extend it in two respects so that it covers a broader range of cases.

There are many scientific and everyday cases where each of Pr and Pr is high and it seems that Pr is high. But high probability is not transitive and so it might be in such cases that each of Pr and Pr is high and in fact Pr is not high. There is no issue in the special case where the following condition, which I call “C1”, holds: H 1 entails H 2. This condition is sufficient for transitivity in high probability. But many of the scientific and everyday cases referred to above are cases where it is not the case that H 1 entails H 2. I consider whether there are additional conditions sufficient for transitivity in high probability. I consider three candidate conditions. I call them “C2”, “C3”, and “C2&3”. I argue that C2&3, but neither C2 nor C3, is sufficient for transitivity in high probability. I then set out some further results and relate the discussion to the Bayesian requirement of coherence.

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.

We argued that explanatoriness is evidentially irrelevant in the following sense: Let H be a hypothesis, O an observation, and E the proposition that H would explain O if H and O were true. Then our claim is that Pr = Pr. We defended this screening-off thesis by discussing an example concerning smoking and cancer. Climenhaga argues that SOT is mistaken because it delivers the wrong verdict about a slightly different smoking-and-cancer case. He also considers a variant of SOT, called “SOT*”, and contends that it too gives the wrong result. We here reply to Climenhaga’s arguments and suggest that SOT provides a criticism of the widely held theory of inference called “inference to the best explanation”.

Bayesian confirmation theory is rife with confirmation measures. Many of them differ from each other in important respects. It turns out, though, that all the standard confirmation measures in the literature run counter to the so-called “Reverse Matthew Effect” (“RME” for short). Suppose, to illustrate, that H1 and H2 are equally successful in predicting E in that p(E | H1)/p(E) = p(E | H2)/p(E) > 1. Suppose, further, that initially H1 is less probable than H2 in that p(H1) < p(H2). Then by RME it follows that the degree to which E confirms H1 is greater than the degree to which it confirms H2. But by all the standard confirmation measures in the literature, in contrast, it follows that the degree to which E confirms H1 is less than or equal to the degree to which it confirms H2. It might seem, then, that RME should be rejected as implausible. Festa (2012), however, argues that there are scientific contexts in which RME holds. If Festa’s argument is sound, it follows that there are scientific contexts in which none of the standard confirmation measures in the literature is adequate. Festa’s argument is thus interesting, important, and deserving of careful examination. I consider five distinct respects in which E can be related to H, use them to construct five distinct ways of understanding confirmation measures, which I call “Increase in Probability”, “Partial Dependence”, “Partial Entailment”, “Partial Discrimination”, and “Popper Corroboration”, and argue that each such way runs counter to RME. The result is that it is not at all clear that there is a place in Bayesian confirmation theory for RME.

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

Bayesian confirmation theory is rife with confirmation measures. Zalabardo focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other three measures. Glass and McCartney, hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p and a certain other probability involving H. I then evaluate G&M’s argument in light of them.

Dretske’s theory of self-knowledge is interesting but peculiar and can seem implausible. He denies that we can know by introspection that we have thoughts, feelings, and experiences. But he allows that we can know by introspection what we think, feel, and experience. We consider two puzzles. The first puzzle, PUZZLE 1, is interpretive. Is there a way of understanding Dretske’s theory on which the knowledge affirmed by its positive side is different than the knowledge denied by its negative side? The second puzzle, PUZZLE 2, is substantive. Each of the following theses has some prima facie plausibility: there is introspective knowledge of thoughts, knowledge requires evidence, and there are no experiences of thoughts. It is unclear, though, that these claims form a consistent set. These puzzles are not unrelated. Dretske’s theory of self-knowledge is a potential solution to PUZZLE 2 in that Dretske’s theory is meant to show how,, and can all be true. We provide a solution to PUZZLE 1 by appeal to Dretske’s early work in the philosophy of language on contrastive focus. We then distinguish between “Closure” and “Transmissibility”, and raise and answer a worry to the effect that Dretske’s theory of self-knowledge runs counter to Transmissibility. These results help to secure Dretske’s theory as a viable solution to PUZZLE 2.

Some recent work in formal epistemology shows that “witness agreement” by itself implies neither an increase in the probability of truth nor a high probability of truth—the witnesses need to have some “individual credibility.” It can seem that, from this formal epistemological result, it follows that coherentist justification (i.e., doxastic coherence) is not truth-conducive. I argue that this does not follow. Central to my argument is the thesis that, though coherentists deny that there can be noninferential justification, coherentists do not deny that there can be individual credibility

Coherentists on epistemic justification claim that all justification is inferential, and that beliefs, when justified, get their justification together (not in isolation) as members of a coherent belief system. Some recent work in formal epistemology shows that “individual credibility” is needed for “witness agreement” to increase the probability of truth and generate a high probability of truth. It can seem that, from this result in formal epistemology, it follows that coherentist justification is not truth-conducive, that it is not the case that, under the requisite conditions, coherentist justification increases the probability of truth and generates a high probability of truth. I argue that this does not follow.