Explanation, confirmation, and Hempel's paradox William Roche Department of Philosophy, Texas Christian University, Fort Worth, TX, USA, e-mail: w.roche@tcu.edu ABSTRACT: Hempel's Converse Consequence Condition (CCC), Entailment Condition (EC), and Special Consequence Condition (SCC) have some prima facie plausibility when taken individually. Hempel, though, shows that they have no plausibility when taken together, for together they entail that E confirms H for any propositions E and H. This is "Hempel's paradox". It turns out that Hempel's argument would fail if one or more of CCC, EC, and SCC were modified in terms of explanation. This opens up the possibility that Hempel's paradox can be solved by modifying one or more of CCC, EC, and SCC in terms of explanation. I explore this possibility by modifying CCC and SCC in terms of explanation and considering whether CCC and SCC so modified are correct. I also relate that possibility to Inference to the Best Explanation. KEYWORDS: Bayesian causal networks; confirmation; Converse Consequence Condition; explanation; Hempel; Hempel's paradox; Inference to the Best Explanation; screening-off; Special Consequence Condition 1 Introduction Each of the following conditions has some prima facie plausibility: Converse Consequence Condition (CCC): For any propositions E, H*, and H, if (i) E confirms H* and (ii) H* is entailed by H, then E confirms H. Entailment Condition (EC): For any propositions E and H, if E entails H, then E confirms H. Special Consequence Condition (SCC): For any propositions E, H*, and H, if (i) E confirms H* and (ii) H* entails H, then E confirms H. The same is true of the condition: 2 Non-Triviality Condition (NTC): For some propositions E and H, E does not confirm H. Hempel (1965a), though, shows (in effect) that CCC, EC, and SCC together entail that NTC is false: Hempel's Argument (1) E entails E for any proposition E. Thus (2) E confirms E for any proposition E. [by (1) and EC] (3) E is entailed by E & H for any propositions E and H. Thus (4) E confirms E & H for any propositions E and H. [by (2), (3), and CCC] (5) E & H entails H for any propositions E and H. Thus (6) E confirms H for any propositions E and H. [by (4), (5), and SCC] So, given that there is no questioning NTC (on any legitimate sense of confirmation), one or more of CCC, EC, and SCC should be rejected.1 This is "Hempel's paradox".2 An intriguing possibility is that one or more of CCC, EC, and SCC should be modified in terms of explanation. Suppose, for example, CCC is modified so that "entailed" is replaced by "entailed and explained". Then (3) in Hempel's Argument would need to be modified accordingly. (3), though, would be false if it were so modified. For, as is uncontroversial, it is false that E is explained by E & H for any propositions E and H. Perhaps, then, Hempel's paradox can be solved by modifying one or more of CCC, EC, and SCC in terms of explanation. It would not be enough-to solve Hempel's paradox-to simply modify one or more of CCC, EC, and SCC in terms of explanation so that Hempel's Argument fails. The new condition or conditions would need to be correct. Does Hempel's paradox admit of a satisfactory solution? Are conditions such as CCC, EC, and SCC correct when modified in terms of explanation? These are the main questions of the paper. 1 The names "Converse Consequence Condition", "Entailment Condition", and "Special Consequence Condition" are Hempel's (1965a, pp. 31-32). Hempel formulates CCC, EC, and SCC in terms of sentences (as opposed to propositions) where E is an observation report. This difference in formulation is inconsequential for my purposes. The name "Non-Triviality Condition" is mine. Hempel (1965a, p. 32) discusses NTC but does not give it a name. 2 This paradox is of course distinct from the ravens paradox (which sometimes goes by the name "Hempel's paradox"). See Hempel (1965a). 3 My interest in the second question is due in part to my interest in the first question. If the answer to the second question is affirmative, then perhaps so too is the answer to the first question. There is more. Inference to the Best Explanation (IBE) is standardly construed in terms of categorical beliefs. The core idea can be put as follows: Idea 1 (I1): Inferences to categorical beliefs (at least some of them) should be guided by explanatory considerations. This idea has a counterpart concerning degrees of belief: Idea 2 (I2): Changes in degrees of belief (at least some of them) should be guided by explanatory considerations. I2 has received much attention in recent years.3 One issue discussed is whether I2 runs counter to Bayesian conditionalization. Some researchers, for example, flesh out I2 by developing an explanation-based alternative to Bayesian conditionalization. But there are alternative, and less radical, ways in which I2 might be true. Suppose the answer to the second of the two main questions of the paper is affirmative and so conditions such as CCC, EC, and SCC are correct when modified in terms of explanation. Suppose, further, conditions such as CCC, EC, and SCC should be understood in terms of changes in degrees of belief. Then it follows that I2 is true. The remainder of the paper is organized as follows. In Section 2, I take a step towards solving Hempel's paradox by, in part, disambiguating CCC, EC, and SCC and showing that EC is true whereas CCC and SCC are false. In Section 3, I modify CCC and SCC in terms of explanation. The new conditions are CCCE and SCCE. I also note that CCCE and SCCE are correct only if explanatory relations place constraints on probability distributions. In Section 4, I consider three obvious candidate constraints (placed by explanatory relations on probability distributions). I argue that CCCE and SCCE are incorrect when understood in terms of those candidate constraints. In Section 5, I turn to Schupbach's (this volume) articulation and defense of IBE in terms of power (explanatory power). I use it to construct a fourth candidate constraint. I argue that it too is no help for CCCE and SCCE. In Section 6, I set out the basics of Bayesian causal networks and construct a fifth candidate constraint. This constraint improves on the four 3 I have in mind here the growing literature on Bayesianism and IBE. See, e.g., Douven (1999, 2011a), Douven and Wenmackers (forthcoming), Huemer (2009), Iranzo (2008), Lipton (2001, 2004), McCain and Poston (2014), Okasha (2000), Poston (2014, Ch. 7), Psillos (2004), Roche and Sober (2013, 2014), Salmon (2001a, 2001b), Tregear (2004), van Fraassen (1989), and Weisberg (2009). 4 constraints considered in Section 4 and Section 5 in a certain key respect. I argue, though, that in the end the fifth candidate constraint too does not help with CCCE and SCCE. In Section 7, I return to Hempel's paradox and set out my preferred solution. In Section 8, I conclude. 2 A step towards solving Hempel's paradox It is standard in Bayesian confirmation theory to distinguish between absolute confirmation and incremental confirmation.4 The distinction is this: Absolute Confirmation: For any propositions E and H, E absolutely confirms H if and only if Pr(H | E) > t where t is the threshold for high probability and 1 > t ≥ 0.5. Incremental Confirmation: For any propositions E and H, E incrementally confirms H if and only if Pr(H | E) > Pr(H). Note that I have in mind probabilities understood as "degrees of belief" (or "rational degrees of belief"). Suppose the sense of confirmation at issue is absolute confirmation. Then CCC should be rejected while EC and SCC should be accepted. First, suppose a card is randomly drawn from a standard (and well-shuffled) deck of cards. Let E be the proposition that the card drawn is a Heart, H* be the proposition that the card drawn is a Red, and H be the proposition that the card drawn is a Diamond. E confirms H*, since Pr(H* | E) = 1 > t. H* is entailed by H. But E does not confirm H, since Pr(H | E) = 0 < 0.5 ≤ t. So CCC is false. Second, for any propositions E and H, if E entails H, then Pr(H | E) = 1 > t. So EC is true. Third, for any propositions E, H*, and H, if H* entails H, then Pr(H | E) ≥ Pr(H* | E). It follows that for any propositions E, H*, and H, if Pr(H* | E) > t and H* entails H, then Pr(H | E) > t. So SCC is true.5 Suppose, instead, the sense of confirmation at issue is incremental confirmation. Then each of CCC, EC, and SCC should be rejected. Return to the card case above. First, note 4 See Carnap (1962, Preface to 2nd ed.) on "concepts of firmness" and "concepts of increase in firmness". 5 Hempel (1965a) holds that CCC should be rejected while EC and SCC should be accepted. It does not follow, however, that he has in mind absolute confirmation. There is reason to believe, in fact, that he does not have in mind absolute confirmation. His "satisfaction criterion of confirmation" (1965a, sec. 9), which is motivated in part by appeal to EC and SCC, is obviously inadequate when absolute confirmation is at issue. Let E be the proposition that Tweety is a raven and Tweety is black. Let H be the proposition that all ravens are black. Suppose Pr(H) is low. Hempel's satisfaction criterion of confirmation implies that E confirms H. Clearly, though, Pr(H | E) is low (not high) and thus is less than t. See Carnap (1962, sec. 87) and Huber (2008) for discussion of how to understand Hempel on confirmation. 5 that E confirms H*, since Pr(H* | E) = 1 > 1/2 = Pr(H*), and H* is entailed by H, but E does not confirm H, since Pr(H | E) = 0 < 1/4 = Pr(H). So CCC is false. Second, let E be some contingent proposition and H be a logically true proposition. Then E entails H but Pr(H | E) = 1 = Pr(H). So EC is false. Third, let E be the proposition that the card drawn is not a Heart, H* be the proposition that the card drawn is a Diamond, and H be the proposition that the card drawn is a Red. Then E confirms H*, since Pr(H* | E) = 1/3 > 1/4 = Pr(H*), and H* entails H, but E does not confirm H, since Pr(H | E) = 1/3 < 1/2 = Pr(H). So SCC is false. The situation is a bit different if the sense of confirmation at issue is incremental confirmation and CCC, EC, and SCC are understood as restricted to propositions with non-extreme unconditional probabilities (i.e., unconditional probabilities greater than 0 and less than 1). CCC and SCC are still false, but EC is true. This is because for any propositions E and H with non-extreme unconditional probabilities, if E entails H, then Pr(H | E) = 1 > Pr(H). Hence CCC and SCC should be rejected while EC should be accepted. There are senses of confirmation (or evidential support) in addition to absolute confirmation and incremental confirmation.6 But none of them is such that EC should be rejected while CCC and SCC should be accepted.7 And none of them is such that SCC should be rejected while CCC and EC should be accepted.8,9 6 See Douven (2011b), Roche (2012b, 2015), and Roche and Shogenji (2014a) for discussion. 7 Suppose CCC and SCC are true. Take some propositions E and H* such that E confirms H*. Then: (1) E confirms H*. (2) H* is entailed by H* & H for any propositions H* and H. Thus (3) E confirms H* & H for any propositions E, H*, and H such that E confirms H*. [by (1), (2), and CCC] (4) H* & H entails H for any propositions H* and H. Thus (5) E confirms H for any propositions E, H*, and H such that E confirms H*. [by (3), (4), and SCC] Thus one or both of CCC and SCC should be rejected. This argument is noted in Skyrms (1966, p. 238). 8 Suppose CCC and EC are true. Then: (1) E entails E ∨ H for any propositions E and H. Thus (2) E confirms E ∨ H for any propositions E and H. [by (1) and EC] (3) E ∨ H is entailed by H for any propositions E and H. Thus (4) E confirms H for any propositions E and H. [by (2), (3), and CCC] Thus one or both of CCC and EC should be rejected. This argument would fail if EC were understood as restricted to propositions with non-extreme unconditional probabilities. Suppose H = ~E. Then E entails E 6 I leave it for future investigation whether there are (interesting) senses of confirmation on which CCC and EC should be rejected while SCC should be accepted, or on which EC and SCC should be rejected while CCC should be accepted. I want to focus on confirmation in the sense of incremental confirmation. This is in part because of my interest in whether I2 is true. So hereafter, unless otherwise noted, all talk of confirmation should be understood in terms of incremental confirmation. No condition concerning incremental confirmation has any plausibility unless it is understood as restricted to propositions with non-extreme unconditional probabilities. So hereafter, unless otherwise noted, all talk of confirmation should be understood as restricted to propositions with non-extreme unconditional probabilities. Given the focus on confirmation in the sense of incremental confirmation, and given the restriction to propositions with non-extreme unconditional probabilities, it follows that EC is true whereas CCC and SCC are false. Thus CCC and SCC should be rejected while EC should be accepted. This is a step towards solving Hempel's paradox. But it is not enough. Each of CCC and SCC, it seems, contains a kernel of truth in that many cases where (i) E confirms H* and (ii) H* is entailed by or entails H are cases where, it seems, E confirms H. Some such cases are cases where E is a piece of observational evidence and H* and H are scientific hypotheses one of which is more general than the other. Other such cases are more ordinary. Suppose, for example, a card is randomly drawn from a standard and wellshuffled deck of cards. Let E be the proposition that Smith testified that the card drawn is ∨ H, but, since E ∨ H has an extreme unconditional probability (of 1), it does not follow by EC that E confirms E ∨ H. The argument, though, can be readily modified to show that CCC and EC together entail an absurdity along the lines of the negation of NTC. See Moretti (2003) and Skyrms (1966, p. 238). Le Morvan (1999) gives an alternative argument for the thesis that CCC and EC together entail that NTC is false. Le Morvan's argument, like the argument above, would fail if EC were understood as restricted to propositions with non-extreme unconditional probabilities. But, it seems, Le Morvan's argument, unlike the argument above, cannot be readily modified to show that CCC and EC together entail an absurdity along the lines of the negation of NTC. See Moretti (2003) for discussion. 9 The same is true with respect to SCC and the condition: Converse Entailment Condition (CEC): For any propositions E and H, if H entails E, then E confirms H. Suppose CEC and SCC are true. Take some propositions E and H. Then: (1) E & H entails E for any propositions E and H. Thus (2) E confirms E & H for any propositions E and H. [by (1) and CEC] (3) E & H entails H for any propositions E and H. Thus (4) E confirms H for any propositions E and H. [by (2), (3), and SCC] Thus one or both of CEC and SCC should be rejected. This argument is noted in Tuomela (1976). 7 a Heart, H* be the proposition that the card drawn is a Heart, and H be the proposition that the card drawn is the Jack of Hearts. Then, given certain rather natural ways of filling in the details, E confirms H*, H* is entailed by H, and, as per CCC, E confirms H. Now let E be the proposition that Smith testified that the card drawn is the Jack of Hearts, H* be the proposition that the card drawn is the Jack of Hearts, and H be the proposition that the card drawn is a Jack, and the case is such that E confirms H*, H* entails H, and, as per SCC, E confirms H. So I want to find some adequate replacement conditions for CCC and SCC (conditions similar to them in content but not open to counterexample). I turn now to the possibility that CCC and SCC should be modified in terms of explanation. 3 CCCE and SCCE Consider: CCCE: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* is entailed by H, and (iii) H* is explained by H, then E confirms H. SCCE: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* entails H, and (iii) H* explains H, then E confirms H. CCCE is CCC but with the added condition that H* is explained by H. SCCE, in turn, is SCC but with the added condition that H* explains H.10 The expressions "explained by" and "explains" in CCCE and SCCE should be understood as short for "potentially explained by" and "potentially explains". The latter expressions, in turn, should be understood so that H* can be potentially explained by or potentially explain H even if neither H* nor H is true.11 If all explanation is deductive, then CCCE and SCCE are equivalent to the following: CCCE*: For any propositions E, H*, and H, if (i) E confirms H* and (ii) H* is explained by H, then E confirms H. 10 Brody (1968, 1974) considers a condition similar to CCCE but (a) without the constraint that H* is entailed by H and (b) restricted to confirmation by positive instances. See Gower (1973), Koslow (1970), Martin (1972, 1975), and Tuomela (1976) for relevant discussion. 11 See Lipton (2004, Ch. 4) for helpful discussion of the distinction between actual and potential explanations. 8 SCCE*: For any propositions E, H*, and H, if (i) E confirms H* and (ii) H* explains H, then E confirms H. It is a matter of controversy, though, whether all explanation is deductive.12 So I shall assume just that some explanation is deductive and focus on CCCE and SCCE.13 There is a clear respect in which CCCE and SCCE improve on CCC and SCC. If Hempel's Argument were modified in terms of CCCE and SCCE, then it would have a false premise. It would read: Hempel's Argument* (1) E entails E for any proposition E. Thus (2) E confirms E for any proposition E. [by (1) and EC] (3) E is entailed and explained by E & H for any propositions E and H. Thus (4) E confirms E & H for any propositions E and H. [by (2), (3), and CCCE] (5) E & H entails and explains H for any propositions E and H. Thus (6) E confirms H for any propositions E and H. [by (4), (5), and SCCE] Each of (3) and (5) is false. Is it the case, though, that CCCE and SCCE are correct? The answer is negative if explanatory relations place no constraints on probability distributions. This can be seen as follows. Take some H* and H such that H* is explained by H or vice versa. Suppose explanatory relations place no constraints on probability distributions. Then the fact that H* is explained by H or vice versa does not rule out any probability distributions and thus does not rule out any probability distributions on which CCC or SCC fails. It follows that neither CCCE and SCCE is correct. The key question, then, is this: What constraints, if any, do explanatory relations place on probability distributions? 12 For helpful discussion, and for additional references, see Salmon (1998, Ch. 9). 13 The assumption is not that there are cases of explanation where the explanans explains the explanandum by virtue of entailing the explanandum. The assumption is simply that there are cases of explanation where in fact the explanans entails the explanandum. 9 4 Entailment, high probability, and increase in probability There are some obvious candidate constraints (placed by explanatory relations on probability distributions). Consider (where, as above, t is the threshold for high probability and 1 > t ≥ 0.5): (c1) For any propositions E and H, H explains E only if H entails E and thus Pr(E | H) = 1. (c2) For any propositions E and H, H explains E only if Pr(E | H) > t. (c3) For any propositions E and H, H explains E only if Pr(E | H) > Pr(E). Each of (c1), (c2), and (c3) has some intuitive appeal. And each of them can be found in the literature.14 Are CCCE or SCCE correct when understood in terms of (c1), (c2) and (c3)? Suppose the only (non-trivial) constraints placed by explanatory relations on probability distributions are (c1), (c2), and (c3) (and their implications). Then whether CCCE and SCCE are correct hinges on whether the following are correct: CCCEc1-c3: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* is entailed by H, and (iii) Pr(H* | H) = 1, Pr(H* | H) > t, and Pr(H* | H) > Pr(H*), then E confirms H. SCCEc1-c3: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* entails H, and (iii) Pr(H | H*) = 1, Pr(H | H*) > t, and Pr(H | H*) > Pr(H), then E confirms H. But CCCEc1-c3 is equivalent to CCC, and SCCEc1-c3 is equivalent to SCC. This follows from the fact that in each case condition (iii) holds if condition (ii) holds. So, since CCC and SCC are incorrect, it follows that so too are CCCEc1-c3 and SCCEc1-c3. The same is true, of course, if some but not all of (c1), (c2), and (c3) are set aside. And for the same reason: the entailment condition holds only if the posterior probability in question equals 1, is greater than t, and is greater than the prior probability in question. I turn now to a fourth candidate constraint. 14 (c1) and (c2) can be found in Hempel (1965b). (c3) can be found in Salmon (1965). For discussion of the rather extensive extant literature on explanation, and for further references, see Salmon (2006) and Woodward (2014). 10 5 Explanatory power Schupbach (this volume) sets out and defends a version of IBE on which explanatoriness is fleshed out in terms of power (explanatory power), where the degree to which H has power over E is measured by: e(E,H ) = Pr(H | E)− Pr(H |~ E) Pr(H | E)+ Pr(H |~ E) e(E, H)'s range is from -1 to 1 (inclusive). H's power over E is positive if e(E, H) > 0. H has no power over E if e(E, H) = 0. H's power over E is negative if e(E, H) < 0. It is not important for my purposes whether Schupbach's defense of his version of IBE, which includes his defense of e, succeeds. But a certain part of that defense is important for my purposes. Schupbach notes that positive power places constraints (some of which are rather significant) on probability distributions. If e(E, H) > 0, then: (a) Pr(H | E)− Pr(H |~ E) Pr(H | E)+ Pr(H |~ E) > 0 (b) Pr(H | E)− Pr(H |~ E) > 0 (c) Pr(E |H ) Pr(E) > Pr(~ E |H ) Pr(~ E) (d) Pr(E |H )− Pr(E |H )Pr(E) > Pr(E)− Pr(E |H )Pr(E) (e) Pr(E |H ) > Pr(E) (f) Pr(E |H ) > Pr(E |~ H ) (g) Pr(H | E) > Pr(H ) It is worth noting that (a)-(g) follow from positive power as measured by all of the main measures of power in the literature.15 Now consider: (c4) For any propositions E and H, H explains E only if e(E, H) > 0. 15 Here I have in mind the measures noted in Crupi and Tentori (2012) and Schupbach (2011). 11 Suppose the only (non-trivial) constraint placed by explanatory relations on probability distributions is (c4) (and its implications). Then whether CCCE and SCCE are correct hinges on whether the following are correct: CCCEc4: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* is entailed by H, and (iii) e(H*, H) > 0, then E confirms H. SCCEc4: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* entails H, and (iii) e(H, H*) > 0, then E confirms H. Are CCCEc4 and SCCEc4 correct? It is true that positive power places constraints on probability distributions. It is true in particular that if e(E, H) > 0, then (a)-(g) above all hold. It is also true, though, that any constraints placed on probability distributions by e(H*, H)'s being positive are also placed on probability distributions by H*'s being entailed by H, and that, similarly, any constraints placed on probability distributions by e(H, H*)'s being positive are also placed on probability distributions by H*'s entailing H. This is because any case where H* is entailed by H is a case where e(H*, H) is positive (in fact equal to 1), and because any case where H* entails H is a case where e(H, H*) is positive (in fact equal to 1). It follows that CCCEc4 is equivalent to CCC and that SCCEc4, in turn, is equivalent to SCC. So CCCEc4 and SCCEc4, as with CCC and SCC, are incorrect.16 The situation is this. It is not enough, for CCCE and SCCE to be correct, that explanatory relations place some constraints on probability distributions. It needs to be the case that they place constraints on probability distributions over and above the constraints already placed on probability distributions by entailment relations. This is why CCCEc1-c3, SCCEc1-c3, CCCEc4, and SCCEc4 all fail. The candidate constraint constructed in the next section improves on (c1)-(c4) in that it provides a way of understanding CCCE and SCCE on which they are not equivalent to CCC and SCC. I turn now to that candidate constraint. 16 It does not help, of course, to modify CCC and SCC in terms of all four of the candidate constraints on the table at this point. The resulting conditions-CCCEc1-c4 and SCCEc1-c4-are equivalent to CCC and SCC and thus are incorrect. 12 6 Causation and screening-off It is not implausible prima facie that some explanations are non-causal in that the explanans phenomenon is not a cause of the explanandum phenomenon.17 Clearly, though, many explanations are causal. I want to focus on CCCE and SCCE understood so that the kind of explanation at issue is causal. I want to do this because, arguably, causes screen-off in a sense to be explained below and because screening-off places constraints on probability distributions over and above the constraints already placed on probability distributions by entailment. In Section 6.1, I explain the basics of Bayesian causal networks. In Section 6.2, I construct a fifth candidate constraint placed by explanatory relations on probability distributions. This constraint, (c5), involves screening-off. In Section 6.3, I return to CCC and SCC and examine why they fail. In Section 6.4, I evaluate CCCE and SCCE when understood in terms of (c5). 6.1 Bayesian causal networks A Bayesian causal network consists of a set of variables V = {V1, V2, ..., Vn}, a directed acyclic graph G over V, and a probability distribution Pr over V. G consists of nodes and directed edges (or arrows). Each node is a variable in V (and each variable in V is a node). Each directed edge connects exactly two nodes. A directed edge from one node to another indicates that the former is a direct cause of the latter. One node is a parent of another (and the latter is a child of the former) if and only if there is a directed edge from the former to the latter. One node is an ancestor of another (and the latter is a descendant of the former) if and only if there is a directed path (or series of directed edges), aligned tip-to-tail linking intermediate nodes, from the former to the latter. G is acyclic in that no node is an ancestor of itself. An example of V (taken from Neapolitan 2004) is the set {B, C, F, H, L}, where: Variable Values B b1 = bronchitis is present b2 = bronchitis is not present C c1 = chest X-ray is positive c2 = chest X-ray is not positive F f1 = fatigue is present f2 = fatigue is not present H h1 = there is a history of smoking 17 See Lipton (2009) for discussion. 13 h2 = there is not a history of smoking L l1 = lung cancer is present l2 = lung cancer is not present Here each variable is binary. But this is not required in general. An example of G (also taken from Neapolitan 2004) is the directed acyclic graph: Here H is a parent of each of B and L, B is a parent of F, L is a parent of each of F and C, and though H is not a parent of F or of C, H is an ancestor of F and of C. It is crucial to note that not just any probability distribution is admissible in a Bayesian causal network. Pr should be such that for any variable Vi in V with parents (direct causes) and non-descendants (non-effects) in G, Vi's parents screen-off Vi's nondescendants from Vi, that is, render Vi's non-descendants probabilistically irrelevant to Vi. This assumption is (a version of a condition) called "the causal Markov condition". Take the example above. Each of L and C is a non-descendant of B. So, by the causal Markov condition, Pr should be such that each of L and C is screened-off from B by H in that Pr(bi | hj & lk) = Pr(bi | hj) for any i, j, k and Pr(bi | hj & ck) = Pr(bi | hj) for any i, j, k. If, say, it is given that h1 (there is a history of smoking), then Pr should be such that none of l1 (lung cancer is present), l2 (lung cancer is not present), c1 (chest X-ray is positive), and c2 (chest X-ray is not positive) has any impact on the probability of b1 (bronchitis is present) or the probability of b2 (bronchitis is not present). 14 It might seem strange to talk of variables as causes. Bear in mind, though, that there are ways of understanding Bayesian causal networks on which variables are not causes. Consider, say, the directed edge in the example above from H to B. This edge need not be understood as indicating that H is a direct cause of B. It could instead be understood as indicating that a given subject's having (or not having) a history of smoking is a direct cause of her having (or not having) bronchitis. This in no way runs counter to the idea that causes (in singular causation) are events.18 Much more can be said about all this.19 The important point for my purposes, though, is the idea that causal relations and screening-off walk hand in hand. This idea is important in that screening-off places rather significant constraints on probability distributions. 6.2 (c5) Consider: (c5) For any propositions E and H, H explains E only if there are partitions of propositions Γ, Γ*, and Γ** such that (a) H is a member of Γ, (b) E is a member of Γ*, (c) Γ** is a set of non-descendants of E, and (d) Γ screens-off Γ** from Γ*. This condition is like the causal Markov condition except that it is framed in terms of explanation (as opposed to causation) and partitions (as opposed to variables).20 Return to the example above where V = {B, C, F, H, L}. It is straightforward to reinterpret that example in terms of (c5). Let P be the patient at issue. Take H, B, L, and the fact that H screens-off L from B. H is the partition {P has a history of smoking, P does not have a history of smoking}, B is the partition {P has bronchitis, P does not have bronchitis}, and L is the partition {P has lung cancer, P does not have lung cancer}. H screens-off L from B in that given either member of the partition {P has a history of smoking, P does not have a history of smoking}, neither member of the partition {P has lung cancer, P does not have lung cancer} has any impact on the probability of either member of the partition {P has bronchitis, P does not have bronchitis}. 18 See Ehring (2009) for discussion of the relata of causation. 19 For helpful introductory discussions of Bayesian causal networks, and for additional references, see Hitchcock (2009, 2012) and Williamson (2009). 20 A similar condition can be motivated by appeal to Salmon's Statistical-Relevance (S-R) model of explanation (1971). See Woodward (2014) for discussion of that model. 15 Now suppose the only (non-trivial) constraints placed by explanatory relations on probability distributions are (c5) and its implications. Then whether CCCE and SCCE are correct hinges on whether the following are correct: CCCEc5: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* is entailed by H, and (iii) there are partitions Γ, Γ*, and Γ** such that (a) H is a member of Γ, (b) H* is a member of Γ*, (c) Γ** is a set of non-descendants of H*, and (d) Γ screens-off Γ** from Γ*, then E confirms H. SCCEc5: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* entails H, and (iii) there are partitions Γ, Γ*, and Γ** such that (a) H* is a member of Γ, (b) H is a member of Γ*, (c) Γ** is a set of non-descendants of H, and (d) Γ screens-off Γ** from Γ*, then E confirms H. CCCEc5 improves on CCCEc1-c3 and CCCEc4 in that CCCEc5 is not equivalent to CCC. SCCEc5, in turn, improves on SCCEc1-c3 and SCCEc4 in that SCCEc5 is not equivalent to SCC. But are CCCEc5 and SCCEc5 correct? I turn now to the issue of why CCC and SCC fail. Getting clear on that issue will help in evaluating CCCEc5 and SCCEc5. 6.3 Why CCC and SCC fail Take any three propositions E, H*, and H. Then (by a proof given in Shogenji forthcoming) it follows that: (1) Pr(H | E)− Pr(H ) = Pr(H |H*)− Pr(H )[ ] Pr(H* | E)− Pr(H*)[ ]+ Pr(H |~ H*)− Pr(H )[ ] Pr(~ H* | E)− Pr(~ H*)[ ]+ Pr(H* | E) Pr(H |H *&E)− Pr(H |H*)[ ]+ Pr(~ H* | E) Pr(H |~ H *&E)− Pr(H |~ H*)[ ] ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ Let the first addend on the right side of (1) be "A", the second addend on the right side of (1) be "B", the third addend on the right side of (1) be "C", and the fourth addend on the right side of (1) be "D". Any case where the antecedent of CCC or SCC holds is a case where A and B are positive. It does not follow, though, that any case where the antecedent of CCC or SCC holds is a case where the left side of (1) is positive, for it might be that in some such cases the sum of C and D is negative and greater than or equal to in absolute value the sum of A and B. 16 Suppose, for example, a card is randomly drawn from a standard and well-shuffled deck of cards. Let E be the proposition that the card drawn is a Club, the Ace of Spades, or the Ace of Hearts, H* be the proposition that the card drawn is not a Spade, and H be the proposition that the card drawn is a Red. Then the antecedent of CCC holds and thus the sum of A and B in (1) is positive: (2) 2 3 − 1 2 ⎡ ⎣⎢ ⎤ ⎦⎥ 14 15 − 3 4 ⎡ ⎣⎢ ⎤ ⎦⎥ + 0 − 1 2 ⎡ ⎣⎢ ⎤ ⎦⎥ 1 15 − 1 4 ⎡ ⎣⎢ ⎤ ⎦⎥ = 11 90 The sum of C and D in (1), though, is negative: (3) 14 15 ⎡ ⎣⎢ ⎤ ⎦⎥ 1 14 − 2 3 ⎡ ⎣⎢ ⎤ ⎦⎥ + 1 15 ⎡ ⎣⎢ ⎤ ⎦⎥ 0 − 0[ ] = − 5 9 The right side of (3) is greater than in absolute value the right side of (2) and thus the left side of (1) is negative. So this case is a counterexample to CCC. Now let E be the proposition that the card drawn is not a Heart, H* be the proposition that the card drawn is a Diamond, and H be the proposition that the card drawn is a Red. Then the antecedent of SCC holds and thus the sum of A and B in (1) is positive: (4) 1− 1 2 ⎡ ⎣⎢ ⎤ ⎦⎥ 1 3 − 1 4 ⎡ ⎣⎢ ⎤ ⎦⎥ + 1 3 − 1 2 ⎡ ⎣⎢ ⎤ ⎦⎥ 2 3 − 3 4 ⎡ ⎣⎢ ⎤ ⎦⎥ = 1 18 The sum of C and D in (1), however, is negative: (5) 1 3 ⎡ ⎣⎢ ⎤ ⎦⎥ 1−1[ ]+ 2 3 ⎡ ⎣⎢ ⎤ ⎦⎥ 0 − 1 3 ⎡ ⎣⎢ ⎤ ⎦⎥ = − 2 9 Since the right side of (5) is greater than in absolute value the right side of (4), it follows that the left side of (1) is negative. So this case is a counterexample to SCC. It is clear, then, why CCC and SCC fail. Their antecedents leave it open that the sum of C and D in (1) is negative and greater than or equal to in absolute value the sum of A and B. The crucial question vis-à-vis CCCEc5 and SCCEc5 is whether the same is true of their antecedents. If yes, then CCCEc5 and SCCEc5, as with CCC and SCC, are open to counterexample. If no, then CCCEc5 and SCCEc5, unlike CCC and SCC, hold without exception. 17 6.4 Why CCCEc5 and SCCEc5 fail First, consider the following variant of (1): (1*) Pr(H* | E)− Pr(H*) = Pr(H* |H )− Pr(H*)[ ] Pr(H | E)− Pr(H )[ ]+ Pr(H* |~ H )− Pr(H*)[ ] Pr(~ H | E)− Pr(~ H )[ ]+ Pr(H | E) Pr(H* |H & E)− Pr(H* |H )[ ]+ Pr(~ H | E) Pr(H* |~ H & E)− Pr(H* |~ H )[ ] ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ It follows from (1*) that: (6) Pr(H* | E)− Pr(H*) = Pr(H* |H )− Pr(H* |~ H )[ ] Pr(H | E)− Pr(H )[ ]+ Pr(H | E) Pr(H* |H & E)− Pr(H* |H )[ ]+ Pr(~ H | E) Pr(H* |~ H & E)− Pr(H* |~ H )[ ] ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ Let the first addend on the right side of (6) be "A*", the second addend on the right side of (6) be "B*", and the third addend on the right side of (6) be "C*". Suppose the antecedent of CCCEc5 holds and thus the left side of (6) is positive and the first multiplicand in A* is positive. Suppose Γ = {H, ~H} and E is included in Γ** (a set of non-descendants of H*). It follows that: (7) Pr(H* |H & E) = Pr(H* |H ) (8) Pr(H* |~ H & E) = Pr(H* |~ H ) By (7) and (8) it follows that the sum of B* and C* in (6) equals zero. Hence, as the left side of (6) is positive and the first multiplicand in A* in (6) is positive, it follows that the second multiplicand in A* in (6) is positive and so E confirms H. Hence it is not the case that the sum of C and D in (1) is negative and greater than or equal to in absolute value the sum of A and B in (1). Next, suppose the antecedent of SCCEc5 holds. Suppose Γ = {H*, ~H*} and E is included in Γ** (a set of non-descendants of H). Then: (9) Pr(H |H *&E) = Pr(H |H*) 18 (10) Pr(H |~ H *&E) = Pr(H |~ H*) By (9) and (10) it follows that each of C and D in (1) equals zero and thus the left side of (1) equals the sum of A and B in (1). Given that the first two conditions in SCCEc5 hold and thus the sum of A and B in (1) is positive, it follows that the left side of (1) is positive and thus E confirms H. There are conditions, then, under which CCCEc5 and SCCEc5 hold without exception. CCCEc5 holds without exception under the condition that Γ = {H, ~H} and E is included in Γ** (a set of non-descendants of H*). SCCEc5, in turn, holds without exception under the condition that Γ = {H*, ~H*} and E is included in Γ** (a set of non-descendants of H). There is a potential problem however. The antecedents of CCCEc5 and SCCEc5 leave it open that E is not included in Γ**. Do CCCEc5 and SCCEc5 hold without exception even when E is not included in Γ**?21 It turns out that the answer is negative and that because of this the antecedents of CCCEc5 and SCCEc5 leave it open that the sum of C and D in (1) is negative and greater than or equal to in absolute value the sum of A and B in (1) (see Appendix for proof). Hence CCCEc5 and SCCEc5, as with CCC and SCC, are incorrect. CCCEc5 and SCCEc5 could be modified by adding a condition to their antecedents to the effect that E is included in Γ**. This would shield them from counterexample. But then they would be open to a different objection. Their ranges of application would be too restrictive. Neither CCCEc5 nor SCCEc5 would have application in cases where E is not included in Γ**. It does not follow that there is no plausible way of understanding explanation on which an explanatory relation between H* and H suffices to close off the possibility that the sum of C and D in (1) is negative and greater than in absolute value the sum of A and B in (1). I suspect, though, that this is the case. I want to set aside CCCE and SCCE and turn to a different approach. 21 The antecedents of CCCEc4 and SCCEc4 leave it open not just that E is not included in Γ** but also that the screening-off partition is not binary. So another question is whether CCCEc4 and SCCEc4 hold without exception even when the screening-off partition is not binary. I want to set aside this question and focus on the question of whether CCCEc4 and SCCEc4 hold without exception even when E is not included in Γ**. If, as I argue below, the answer to this question is negative, then it does not matter for my purposes whether CCCEc4 and SCCEc4 hold without exception even when the screening-off partition is not binary. 19 7 A solution to Hempel's paradox It follows from (1) in Section 6.3 that: CCCSO: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* is entailed by H, and (iii) {H*, ~H*} screens-off E from H in that Pr(H | H* & E) ≥ Pr(H | H*) and Pr(H | ~H* & E) ≥ Pr(H | ~H*), then E confirms H. SCCSO: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* entails H, and (iii) {H*, ~H*} screens-off E from H in that Pr(H | H* & E) ≥ Pr(H | H*) and Pr(H | ~H* & E) ≥ Pr(H | ~H*), then E confirms H. Any case where (i) and (ii) in CCCSO or SCCSO hold is a case where A and B in (1) are positive. So, given that any case where (iii) in CCCSO or SCCSO holds is a case where C and D in (1) are non-negative, any case where the antecedent of CCCSO or SCCSO holds is a case where the left side of (1) is positive. Hence CCCSO and SCCSO hold without exception. Note that with both CCCSO and SCCSO the screening-off partition involves the middle proposition H* in the chain: E, H*, H. Note also that with both CCCSO and SCCSO the screening-off at issue is negative-impact screening-off opposed to no-impact screeningoff. Thus the use of "≥" instead of "=".22 I gave a case in Section 2 where a card is randomly drawn from a standard and wellshuffled deck of cards, E is the proposition that Smith testified that the card drawn is a Heart, H* is the proposition that the card drawn is a Heart, and H is the proposition that 22 CCCSO and SCCSO are special cases of a more general condition: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* confirms H, and (iii) {H*, ~H*} screens-off E from H in that Pr(H | H* & E) ≥ Pr(H | H*) and Pr(H | ~H* & E) ≥ Pr(H | ~H*), then E confirms H. This condition is stronger than the condition: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* confirms H, and (iii) {H*, ~H*} screens-off E from H in that Pr(H | H* & E) = Pr(H | H*) and Pr(H | ~H* & E) = Pr(H | ~H*), then E confirms H. The first of these conditions is established in Roche (2012a). The second is established in Shogenji (2003). See Roche (2014) for discussion of the first condition in the context of peer disagreement. See Roche and Shogenji (2014a) for discussion of the first condition in the context of Moore's proof of the existence of a material world. See Sober (2015, Ch. 5) for discussion of the first condition in the context of the problem of evil. See Roche and Shogenji (2014b) for discussion of the second condition in the context of degree of confirmation. 20 the card drawn is the Jack of Hearts. I also gave a case where instead E is the proposition that Smith testified that the card drawn is the Jack of Hearts, H* is the proposition that the card drawn is the Jack of Hearts, and H is the proposition that the card drawn is a Jack. In each case, given certain rather natural ways of filling in the details, E confirms H*, H* is entailed by or entails H, and {H*, ~H*} screens-off E from H in that Pr(H | H* & E) ≥ Pr(H | H*) and Pr(H | ~H* & E) ≥ Pr(H | ~H*). So, given CCCSO and SCCSO, in each case E confirms H. Is it the case that {H*, ~H*} screens-off E from H if and only if {H, ~H} screens-off E from H*? And, regardless, is it the case that CCCSO and SCCSO would hold without exception if they were modified so that the screening-off partition involved the last, as opposed to the middle, proposition H? The answer to each question is no. But the following conditions hold without exception: CCCSO*: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* is entailed by H, and (iii) {H, ~H} screens-off E from H* in that Pr(H* | H & E) ≤ Pr(H* | H) and Pr(H* | ~H & E) ≤ Pr(H* | ~H), then E confirms H. SCCSO*: For any propositions E, H*, and H, if (i) E confirms H*, (ii) H* entails H, and (iii) {H, ~H} screens-off E from H* in that Pr(H* | H & E) ≤ Pr(H* | H) and Pr(H* | ~H & E) ≤ Pr(H* | ~H), then E confirms H. Recall (6) from Section 6.4. Any case where the antecedent of CCCSO* or the antecedent of SCCSO* holds is a case where the left side of (6) is positive, the first multiplicand in A* in is positive, and the sum of B* and C* is non-positive. It follows that any case where the antecedent of CCCSO* or the antecedent of SCCSO* holds is a case where the second multiplicand in A* is positive and thus E confirms H.23 Note that the screening-off at issue in CCCSO* and SCCSO* is positive-impact screening-off as opposed to negative-impact screening-off. If, instead, the screening-off at issue in CCCSO* and SCCSO* were negative-impact screening-off, then CCCSO* and SCCSO* would be open to counterexample.24 CCCSO, SCCSO, CCCSO*, and SCCSO* together have a rather large range of application. There are cases where H* is entailed by H, but there are also cases where H* entails H. There are cases where the screening-off partition involves the middle proposition H* and the screening-off is negative-impact screening-off, but there are also 23 CCCSO* and SCCSO* are due essentially to Tomoji Shogenji (personal correspondence). 24 Similarly, if the screening-off at issue in CCCSO and SCCSO were positive-impact screening-off, then CCCSO and SCCSO would be open to counterexample. 21 cases where the screening-off partition involves the last proposition H and the screeningoff is positive-impact screening-off. The situation is this. CCC and SCC are false. But each, it seems, contains a kernel of truth. This is borne out by CCCSO, SCCSO, CCCSO*, and SCCSO*. My solution to Hempel's paradox is now complete. EC should be accepted while CCC and SCC should be rejected in favor of CCCSO, SCCSO, CCCSO*, and SCCSO*. My solution is similar to the proposed explanation-based solution considered in Section 6 in that it involves the notion of screening-off. But with my solution on hand there is simply no need to appeal to explanation. 8 Conclusion I noted in Section 1 that the core idea behind IBE can be put as follows: I1: Inferences to categorical beliefs (at least some of them) should be guided by explanatory considerations. I also noted that this idea, which concerns categorical beliefs, has a counterpart concerning degrees of belief: I2: Changes in degrees of belief (at least some of them) should be guided by explanatory considerations. One way to flesh out I2 is to modify conditions such as CCC, EC, and SCC in terms of explanation. There is no need, though, to modify EC in terms of explanation, for EC holds without exception (given the assumption that E and H have non-extreme unconditional probabilities). And, it seems, neither CCC nor SCC is correct when understood in terms of explanation. It does not follow, of course, that I2 is incorrect. There are other ways to flesh out I2. McCain and Poston (2014, this volume), for instance, flesh out I2 in terms of the notion of resiliency.25 The overall picture, however, is clearer. If changes in degrees of belief should be guided by explanatory considerations, then, it seems, this is not because conditions such as CCC, EC, and SCC should be modified in terms of explanation. 25 See Roche and Sober (2014) for discussion. 22 Acknowledgments Thanks to Kevin McCain, Ted Poston, and Michael Roche for helpful comments on prior versions of the paper. Special thanks to Tomoji Shogenji for extensive and very helpful input on the project. Appendix A Counterexample to CCCEc4 Consider the following probability distribution: E H* H P Pr E H* H P Pr T T T T 1 29 F T T T 2 39 T T T F 407473 2841072 F T T F 8247278569811 16455562695936 T T F T 1 156 F T F T 1 531 T T F F 1 314 F T F F 1 128 T F T T 0 F F T T 0 T F T F 0 F F T F 0 T F F T 2 37 F F F T 3 56 T F F F 5 592 F F F F 320066869 2383827712 It can be readily verified that: (a) Pr(H* | E) = 0.75 > 0.749662 ≈ Pr(H*) (b) Pr(H* | H) = 1 (c) Pr(H* | H & P) = Pr(H* | H) = 1 (d) Pr(H* | H & ~P) = Pr(H* | H) = 1 (e) Pr(H* | ~H & P) = Pr(H* | ~H) ≈ 0.072 (f) Pr(H* | ~H & ~P) = Pr(H* | ~H) ≈ 0.072 (g) [Pr(H | H*) – Pr(H)][Pr(H* | E) – Pr(H*)] ≈ 0.0000823 (h) [Pr(H | ~H*) – Pr(H)][Pr(~H* | E) – Pr(~H*)] ≈ 0.000246 (i) Pr(H* | E) [Pr(H | H* & E) – Pr(H | H*)] ≈ –0.0191 (j) Pr(~H* | E) [Pr(H | ~H* & E) – Pr(H | ~H*)] = 0 23 Given (a), E confirms H*. Suppose, consistent with (b), H entails H*. Let Γ = {H, ~H}, Γ* = {H*, ~H*}, and Γ** = {P, ~P}, where P and ~P are non-descendants of H*, and where E is a descendant of H* and thus is not a member of Γ**. Then, given (c)-(f), there are partitions Γ, Γ*, and Γ** such that H is a member of Γ, H* is a member of Γ*, Γ** is a set of non-descendants of H*, and Γ screens-off Γ** from Γ*. Thus the antecedent of CCCEc4 holds. But, given (g)-(j), it is not the case that E confirms H. Thus the consequent of CCCEc4 does not hold. Thus CCCEc4 is false. QED B Counterexample to SCCEc4 Consider the following probability distribution: E H* H P Pr E H* H P Pr T T T T 1 16 F T T T 5 43 T T T F 1 8 F T T F 1888104014513 6010675728640 T T F T 0 F T F T 0 T T F F 0 F T F F 0 T F T T 1 310 F F T T 1 11 T F T F 2 41 F F T F 1 5 T F F T 1 146 F F F T 1 256 T F F F 27051 7422640 F F F F 157449879 6353779840 It can be readily verified that: (k) Pr(H* | E) = 0.75 > 0.618 ≈ Pr(H*) (l) Pr(H | H*) = 1 (m) Pr(H | H* & P) = Pr(P | H*) = 1 (n) Pr(H | H* & ~P) = Pr(P | H*) = 1 (o) Pr(H | ~H* & P) = Pr(H | ~H*) ≈ 0.897 (p) Pr(H | ~H* & ~P) = Pr(H | ~H*) ≈ 0.897 (q) [Pr(H | H*) – Pr(H)][Pr(H* | E) – Pr(H*)] ≈ 0.00518 (r) [Pr(H | ~H*) – Pr(H)][Pr(~H* | E) – Pr(~H*)] ≈ 0.00837 (s) Pr(H* | E) [Pr(H | H* & E) – Pr(H | H*)] ≈ 0 (t) Pr(~H* | E) [Pr(H | ~H* & E) – Pr(H | ~H*)] = –0.0163 24 Given (k), E confirms H*. Suppose, consistent with (l), H* entails H. Let Γ = {H*, ~H*}, Γ* = {H, ~H}, and Γ** = {P, ~P}, where P and ~P are non-descendants of H, and where E is a descendant of H and thus is not a member of Γ**. Then, given (m)-(p), there are partitions Γ, Γ*, and Γ** such that H* is a member of Γ, H is a member of Γ*, Γ** is a set of non-descendants of H, and Γ screens-off Γ** from Γ*. Thus the antecedent of SCCEc4 holds. But, given (q)-(t), it is not the case that E confirms H. Thus the consequent of SCCEc4 does not hold. Thus SCCEc4 is false. QED References Brody, B. (1968). Confirmation and explanation. Journal of Philosophy, 65, 282-299. Brody, B. (1974). More confirmation and explanation. Philosophical Studies, 26, 73-75. Carnap, R. (1962). Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press. Crupi, V., and Tentori, K. (2012). A second look at the logic of explanatory power (with two novel representation theorems). Philosophy of Science, 79, 365-385. Douven, I. (1999). Inference to the best explanation made coherent. Philosophy of Science, 66 (Proceedings), S424-S435. Douven, I. (2011a). Abduction. In E. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (spring ed.). URL = <http://plato.stanford.edu/archives/spr2011/entries/abduction/>. Douven, I. (2011b). Further results on the intransitivity of evidential support. Review of Symbolic Logic, 4, 487-497. Douven, I., and Wenmackers, S. (forthcoming). Inference to the best explanation versus Bayes's rule in a social setting. British Journal for the Philosophy of Science. Ehring, D. (2009). Causal relata. In H. Beebee, C. Hitchcock, and P. Menzies (Eds.), The Oxford handbook of causation (pp. 387-413). Oxford: Oxford University Press. Gower, B. (1973). Martin on explanation and confirmation. Analysis, 33, 107-109. Hempel, C. (1965a). Studies in the logic of confirmation. In C. Hempel, Aspects of scientific explanation and other essays in the philosophy of science (pp. 3-46). New York: Free Press. Hempel, C. (1965b). Aspects of scientific explanation. In C. Hempel, Aspects of scientific explanation and other essays in the philosophy of science (pp. 331-496). New York: Free Press. Hitchcock, C. (2009). Causal modelling. In H. Beebee, C. Hitchcock, and P. Menzies (Eds.), The Oxford handbook of causation (pp. 299-314). Oxford: Oxford University Press. 25 Hitchcock, C. (2012). Probabilistic causation. In E. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (winter ed.). URL = <http://plato.stanford.edu/entries/causation-probabilistic/>. Huber, F. (2008). Hempel's logic of confirmation. Philosophical Studies, 139, 181-189. Huemer, M. (2009). Explanationist aid for the theory of inductive logic. British Journal for the Philosophy of Science, 60, 345-375. Iranzo, V. (2008). Bayesianism and inference to the best explanation. Theoria, 23, 89106. Koslow, A. (1970). Comment. Minnesota Studies in the Philosophy of Science, 5, 104107. Le Morvan, P. (1999). The Converse Consequence Condition and Hempelian qualitative confirmation. Philosophy of Science, 66, 448-454. Lipton, P. (2001). Is explanation a guide to inference? A reply to Wesley C. Salmon. In G. Hon and S. Rakover (Eds.), Explanation: Theoretical approaches and applications (pp. 93-120). Dordrecht: Kluwer. Lipton, P. (2004). Inference to the best explanation (2nd ed.). London: Routledge. Lipton, P. (2009). Causation and explanation. In H. Beebee, C. Hitchcock, and P. Menzies (Eds.), The Oxford handbook of causation (pp. 619-631). Oxford: Oxford University Press. Martin, M. (1972). Confirmation and explanation. Analysis, 32, 167-169. Martin, M. (1975). Explanation and confirmation again. Analysis, 36, 41-42. McCain, K., and Poston, T. (2014). Why explanatoriness is evidentially irrelevant. Thought, 3, 145-153. McCain, K., and Poston, T. (this volume). ??? Moretti, L. (2003). Why the Converse Consequence Condition cannot be accepted. Analysis, 63, 297-300. Neapolitan, R. (2004). Learning Bayesian networks. Upper Saddle River: Pearson Prentice Hall. Okasha, S. (2000). Van Fraassen's critique of inference to the best explanation. Studies in the History and. Philosophy of Science, 31, 691-710. Poston, T. (2014). Reason and explanation: A defense of explanatory coherentism. New York: Palgrave Macmillan. Psillos, S. (2004). Inference to the best explanation and Bayesianism: Comments on Ilkka Niiniluoto's "Truth-seeking by abduction". In F. Stadler (Ed.), Induction and deduction in the sciences (pp. 83-91). Dordrecht: Kluwer. Roche, W. (2012a). A weaker condition for transitivity in probabilistic support. European Journal for Philosophy of Science, 2, 111-118. 26 Roche, W. (2012b). Transitivity and intransitivity in evidential support: Some further results. Review of Symbolic Logic, 5, 259-268. Roche, W. (2014). Evidence of evidence is evidence under screening-off. Episteme, 11, 119-124. Roche, W. (2015). Evidential support, transitivity, and screening-off. Review of Symbolic Logic, 8, 785-806. Roche, W., and Shogenji, T. (2014a). Confirmation, transitivity, and Moore: The Screening-Off Approach. Philosophical Studies, 168, 797-817. Roche, W., and Shogenji, T. (2014b). Dwindling confirmation. Philosophy of Science, 81, 114-137. Roche, W., and Sober, E. (2013). Explanatoriness is evidentially irrelevant, or inference to the best explanation meets Bayesian confirmation theory. Analysis, 73, 659-668. Roche, W., and Sober, E. (2014). Explanatoriness and evidence: A reply to McCain and Poston. Thought, 3, 193-199. Salmon, W. (1965). The status of prior probabilities in statistical explanation. Philosophy of Science, 32, 137-146. Salmon, W. (1971). Statistical explanation. In W. Salmon, R. Jeffrey, and J. Greeno, Statistical explanation and statistical relevance (pp. 29-87). Pittsburgh: University of Pittsburgh Press. Salmon, W. (1998). Causality and explanation. Oxford: Oxford University Press. Salmon, W. (2001a). Explanation and confirmation: A Bayesian critique of inference to the best explanation. In G. Hon and S. Rakover (Eds.), Explanation: Theoretical approaches and applications (pp. 61-91). Dordrecht: Kluwer. Salmon, W. (2001b). Reflections of a bashful Bayesian: A reply to Lipton. In G. Hon and S. Rakover (Eds.), Explanation: Theoretical approaches and applications (pp. 121136). Dordrecht: Kluwer. Salmon, W. (2006). Four decades of scientific explanation. Pittsburgh: University of Pittsburgh Press. Schupbach, J. (2011). Comparing probabilistic measures of explanatory power. Philosophy of Science, 78, 813-829. Schupbach, J. (this volume). Inferenence to the best explanation, cleaned up and made respectable. Shogenji, T. (2003). A condition for transitivity in probabilistic support. British Journal for the Philosophy of Science, 54, 613-616. Shogenji, T. (forthcoming). Mediated confirmation. British Journal for the Philosophy of Science. Skyrms, B. (1966). Nomological necessity and the paradoxes of confirmation. Philosophy of Science, 33, 230-249. 27 Sober, E. (2015). Ockham's razors: A user's manual. Cambridge: Cambridge University Press. Tregear, M. (2004). Utilising explanatory factors in induction? British Journal for the Philosophy of Science, 55, 505-519. Tuomela, R. (1976). Confirmation, explanation, and the paradoxes of transitivity. In R. Bogdan (Ed.), Local induction (pp. 319-328). Dordrecht: D. Reidel. van Fraassen, B. (1989). Laws and symmetry. Oxford: Oxford University Press. Weisberg, J. (2009). Locating IBE in the Bayesian framework. Synthese, 167, 125-143. Williamson, J. (2009). Probabilistic theories. In H. Beebee, C. Hitchcock, and P. Menzies (Eds.), The Oxford handbook of causation (pp. 185-212). Oxford: Oxford University Press. Woodward, J. (2014). Scientific explanation. In E. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (winter ed.). URL = <http://plato.stanford.edu/entries/scientific-explanation/>.