Defenders of Inference to the Best Explanation (IBE) claim that explanatory factors should play an important role in empirical inference. They disagree, however, about how exactly to formulate this role. In particular, they disagree about whether to formulate IBE as an inference rule for full beliefs or for degrees of belief, and, if it is formulated as a rule for degrees of belief, how this rule relates to Bayesianism. In this essay I advance a new argument against non-Bayesian versions of IBE that arises when we are concerned with multiple levels of explanation of some phenomenon. I show that in many such cases, following IBE as an inference rule for full belief leads to deductively inconsistent beliefs, and following IBE as a non-Bayesian updating rule for degrees of belief leads to (synchronically) probabilistically incoherent degrees of belief.Proponents of Inference to the Best Explanation (IBE) claim that our inferences should give explanatory considerations a central role. Beyond this general agreement, however, they have differed on precisely how explanation should inform inference. A particular area of controversy has been the relation of IBE to Bayesianism. Should IBE be formulated in terms of full beliefs, as in traditional epistemology, or in terms of degrees of belief, as in Bayesian epistemology? If it is formulated in the latter way, is it compatible with Bayesian epistemology?In this essay I advance a new argument against non-Bayesian formulations of IBE. This includes both traditional formulations of IBE in terms of full belief and non-Bayesian formulations of IBE in terms of degrees of belief. I show that in some instances, IBE for full belief licenses deductively inconsistent inferences from the same evidence. In similar instances, following non-Bayesian IBE updating rules for degrees of belief leads to probabilistically incoherent credences.

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The epistemic probability of A given B is the degree to which B evidentially supports A, or makes A plausible. This paper is a first step in answering the question of what determines the values of epistemic probabilities. I break this question into two parts: the structural question and the substantive question. Just as an object's weight is determined by its mass and gravitational acceleration, some probabilities are determined by other, more basic ones. The structural question asks what probabilities are not determined in this way-these are the basic probabilities which determine values for all other probabilities. The substantive question asks how the values of these basic probabilities are determined. I defend an answer to the structural question on which basic probabilities are the probabilities of atomic propositions conditional on potential direct explanations. I defend this against the view, implicit in orthodox mathematical treatments of probability, that basic probabilities are the unconditional probabilities of complete worlds. I then apply my answer to the structural question to clear up common confusions in expositions of Bayesianism and shed light on the "problem of the priors."

Where E is the proposition that [If H and O were true, H would explain O], William Rocheand Elliot Sober have argued that P(H|O&E) = P(H|O). In this paper I argue that not only is this equality not generally true, it is false in the very kinds of cases that Roche and Sober focus on, involving frequency data. In fact, in such cases O raises the probability of H only given that there is an explanatory connection between them. Key WordsBayesianism; Confirmation; Evidence; Explanation; Inference to the Best Explanation; Probability; William Roche; Elliott Sober 1 I am grateful to Robert Audi, Daniel Immerman, Ted Poston, and two reviewers for Philosophy of Science for very helpful comments on earlier drafts of this paper.In two recent essays, Sober (2013, 2014) (hereafter R&S) argue that the proposition that a hypothesis H would explain an observation O is evidentially irrelevant to H. Where E says that, were H and O true, H would explain O, R&S's thesis is that (1) P(H|O&E) = P(H|O).Once we know O, R&S claim, E gives us no further evidence that H is true. In other words, O screens off E from H. Call this claim the Screening-Off Thesis (SOT) (R&S 2014: 193).In endorsing SOT, R&S are presumably not making a claim about subjective probabilities, for an agent could virtually always assign coherent subjective probabilities on which the above equality is false. I will instead read them as making a claim about epistemic probabilities, which we can understand as rationally constraining subjective probabilities.Theses similar to SOT are endorsed by other Bayesians skeptical of inference to the best explanation. For example, van Fraassen (1989: 166) famously denies that the claim that a hypothesis is explanatory can give it any probabilistic "bonus." Often, however, such skeptics do not make clear in precisely what way they think that explanation is irrelevant to confirmation. Van Fraassen does consider a precise version of inference to the best explanation, but it is an uncharitable one, on which inference to the best explanation is understood as a non-Bayesian updating rule on which good explanations get higher probabilities than Bayesian conditionalization would give them.2 R&S are thus to be 2 Some explanationists have defended this rule against van Fraassen's criticisms of it (see, e.g., Douven 2013). However, whether or not van Fraassen's original arguments against noncommended for stating a precise anti-explanationist thesis which does not mischaracterize their opponents' position. If SOT is true, there is a clear sense in which explanation is not relevant to confirmation.That said, we do need to clarify the scope of SOT before we can evaluate its significance and plausibility. It is widely acknowledged by Bayesians that all confirmation is relative to a context. In other words, we always have some background knowledge K, which may be left implicit but is always guiding our judgments of probability. While subjective Bayesians often think of this background as being part of the probability function P(.) its...

I present a cumulative case for the thesis that we only know propositions that are certain for us. I argue that this thesis can easily explain the truth of eight plausible claims about knowledge:(1) There is a qualitative difference between knowledge and non-knowledge.(2) Knowledge is valuable in a way that non-knowledge is not.(3) Subjects in Gettier cases do not have knowledge.