Wayne, Horwich, and Evidential Diversity Branden Fitelson† August 29, 1997 Wayne (1995) critiques the Bayesian explication of the confirma- tional significance of evidential diversity (CSED) offered by Horwich (1982). Presently, I argue that Wayne’s reconstruction of Horwich’s account of CSED is uncharitable. As a result, Wayne’s criticisms ultimately present no real problem for Horwich. I try to provide a more faithful and charitable rendition of Horwich’s account of CSED. Unfortunately, even when Horwich’s approach is charitably reconstructed, it is still not completely satisfying. This paper is an updated and expanded version of Fitelson (1996). †Thanks to Ellery Eells, Malcolm Forster, Leslie Graves, Geoffrey Hellman, Mike Kruse, Patrick Maher, and an anonymous Philosophy of Science referee for their helpful comments and suggestions on earlier versions of this paper. 1 Introduction Wayne (1995) gives one reconstruction of Horwich’s (1982) Bayesian account of the value of evidential diversity. He then shows that there are counterexamples to this reconstruction of Horwich’s explication of CSED. Such counterexamples would undermine Horwich’s account of CSED, if Wayne’s reconstruction were a charitable one. Presently, I argue that Wayne’s reconstruction of Horwich’s account of CSED is uncharitable. As a result, his criticisms are not genuine problems for Horwich. This does not mean that Horwich’s explication of CSED — charitably re- constructed — is unproblematic. On the contrary, after my analysis of Wayne’s critique, I discuss several remaining problems for Horwich’s account. In the end, I conclude that Horwich’s Bayesian explication of CSED is inadequate. In the final section of the paper, I briefly discuss two recent alternative Bayesian explications of CSED, including a refreshing new account developed in Fitelson (1997a) which is based on a Bayesian account of independent induc- tive support and its relationship to the diversity and confirmational power of collections of evidence. 2 Wayne’s reconstruction of Horwich’s account In a typical confirmation theoretic context C, we have a hypothesis under test H1 and n−1 competing hypotheses H2, . . . ,Hn, where the n hypotheses are as- sumed to be mutually exclusive and exhaustive. Wayne’s (1995) reconstruction of Horwich’s (1982) explication of CSED involves the following three proposi- tions concerning such contexts: (H1) One collection of evidence E1 is more confirmationally diverse (c-diverse) than another collection of evidence E2 in context C iff (∀i 6= 1)[Pr(E1|Hi&KC) < Pr(E2|Hi&KC)].1 The intuition behind H1 is that the more c-diverse collection of evidence is supposed to “rule-out most plausible alternatives” to the hypothesis under test. It is for this reason that Horwich’s account has been called ‘eliminativist’.2 (H2) E1 confirms H more strongly than E2 confirms H if and only if r(H,E1) > r(H,E2), where the ratio measure of degree of con- firmation r(H,E) is defined as follows: r(H,E) =df Pr(H|E) Pr(H) . 1Where, the proposition KC encodes the background knowledge in confirmational contextC. Hereafter, I will, for simplicity’s sake, drop explicit reference to KC in probability statements. It is to be understood, of course, that we are uniformly conditioning Pr on KC, whenever we make a Bayesian confirmational comparison. 2Notice that my H1 is sufficient but not necessary for E1’s “ruling-out” most alternatives to H1 in C. As we’ll see below, this added strength is needed to shore-up Horwich’s formal account of CSED. See the Appendix for all technical details. 1 (H3) For every confirmational context C, if E1 is more c-diverse than E2 in C, then E1 confirms H1 (i.e., the hypothesis under test in C) more strongly than E2 confirms H1 in C. According to Wayne (1995), H3 captures the kernel of Horwich’s account of CSED. In the next section, we will look at a counterexample to H3 due to Wayne (1995). 3 Wayne’s counterexample to H3 Wayne (1995, page 119) asks us to: . . . consider a simple context Cw in which only three hypotheses have substantial prior probabilities, Pr(H1) = 0.2, Pr(H2) = 0.2, Pr(H3) = 0.6, and two data sets E1 and E2 such that: Pr(E1|H1) = 0.2 Pr(E2|H1) = 0.6 Pr(E1|H2) = 0.4 Pr(E2|H2) = 0.5 Pr(E1|H3) = 0.4 Pr(E2|H3) = 0.6 This is plainly a paradigm case of H1: for all Hi, Pr(E1|Hi) is significantly less than Pr(E2|Hi). Yet, a straightforward substitution shows that H3 is violated! Thus, we obtain the counterintuitive result that the similar evidence lends a greater boost to the hypothesis under test than does the diverse evidence . . . Horwich’s account fails to reproduce our most basic intuition about diverse evidence.3 Wayne is right about Cw in the following two respects.4 (1) In Cw, E1 is more c-diverse than E2. (2) In Cw, E2 confirms H1 more strongly than E1 confirms H1, according to the ratio measure r. Hence, Cw is a legitimate counterexample to H3. In the next section, I will discuss some aspects of Wayne’s example that he neglects to mention. Then, I will reflect on what the existence of this counterexample implies — and doesn’t imply — about Horwich’s account of CSED. 4 Why Wayne’s counterexample is not salient 4.1 What Wayne doesn’t say about his counterexample Here is a fact about Wayne’s counterexample to H3 that he neglects to mention. (3) In Cw, E2 confirms H1; whereas, E1 disconfirms H1. 3I have taken the liberty of translating this passage from Wayne (1995) into my notation. 4See the Appendix for all proofs and counterexamples. 2 Wayne has certainly described a confirmational context Cw in which a less c-diverse data set confirms the hypothesis under test more strongly than a more c-diverse data set does. But, as it turns out, Cw is also a context in which the more c-diverse evidence disconfirms the hypothesis under test; whereas, the less c-diverse evidence confirms the hypothesis under test. What does this mean? 4.2 Charitably reconstructing Horwich’s account As far as I can tell, (3) shows that H3 must not be what Horwich has in mind in his explication of CSED. Surely, Horwich would not want to say that more c-diverse disconfirmatory evidence should confirm more strongly than less c- diverse confirmatory evidence. To say the least, this would not be in the spirit of the Bayesian definition of confirmation. A more charitable reconstruction of Horwich’s account of CSED should add a suitable probabilistic ceteris paribus clause to H3. In such a reconstruction, Wayne’s H3 might be replaced by: (H′3) If CP then H3. Where CP is an appropriate probabilistic ceteris paribus clause. Wayne’s counterexample teaches us that, at the very least, CP should entail: (CP1) Both E1 and E2 confirm H1 in C. Indeed, CP1 would avoid the counterexample raised by Wayne. Moreover, it would insure that H′3 does not contradict the Bayesian definition of confirmation (as Wayne’s H3 does). Interestingly, CP1 is not a sufficient ceteris paribus clause. For, CP1 does not entail H3.5 We will need to make CP substantially stronger than CP1 in order to make H′3 a theorem of the mathematical theory of probability. There are many ways to define sufficient ceteris paribus clauses in this sense.6 Here is one such proposal that I think remains faithful to what Horwich has in mind: (CP∗) CP1, and Pr(E1|H1) = Pr(E2|H1) in C. CP∗ says that E1 and E2 both confirm H1 in C, and that E1 and E2 are ‘C-commensurate’, in the sense that the hypothesis under test has the same like- lihood (i.e., goodness of fit) with respect to both E1 and E2 in C. This ceteris 5For a relevant counterexample, see the Appendix. 6Hellman (1997) proposes the following alternative sufficient ceteris paribus clause: (CP†) CP1 , and Pr(E1|H1) − Pr(E1) = Pr(E2|H1) − Pr(E2) in C. It is true that CP† is sufficient for H3. However, CP† is clearly not the kind of Bayesian proposal that Horwich (1982) has in mind. In Horwich’s canonical examples, it is typically assumed that Pr(E1|H1) = Pr(E2|H1) (see below for more on this point). Moreover, Horwich wants sets of evidence with greater c-diversity to have lesser prior probability (e.g., Horwich wants Pr(E1) < Pr(E2) in his canonical example). These two constraints jointly entail that CP† does not hold. So, while Hellman’s alternative makes sense from a generic Bayesian point of view, it is not a faithful reconstruction of Horwich’s Bayesian explication of CSED. 3 paribus clause seems to be implicit in Horwich’s depiction of the kinds of con- firmational contexts he has in mind. Figure 1 shows the kind of confirmational contexts and comparisons that Horwich (1982, pages 119–120) uses as canonical illustrations of his account of CSED. E2 E1 = = H1 C Figure 1: A Canonical Horwichian Example of CSED Figure 1 depicts a canonical confirmation theoretic context C in which the hypothesis under test H1 fits two data sets E1 and E2 equally well, in accordance with CP∗. Moreover, E1 is more intuitively diverse (i-diverse) than E2, since the abscissa values of E1 are more spread-out than the abscissa values of E2. 7 Horwich (1982) seems only to be claiming that — other things being equal (those other things being the likelihoods Pr(E1|H1) and Pr(E2|H1)) — more diverse8 sets of evidence (e.g., E1) will confirm the hypothesis under test (e.g., H1) more strongly than less diverse sets of evidence (e.g., E2) will. This is an appropriate time to state the following theorem: (H3∗) If CP∗, then H3. Since H3∗ is a theorem of the mathematical theory of probability, this re- construction of Horwich’s account is guaranteed to be immune to any formal counterexamples. In this sense, our present reconstruction of Horwich’s account is a charitable one. However, even this charitable reconstruction of Horwich’s account of CSED has its problems. In the next section, I will briefly discuss some of my remaining worries about Horwich’s account of CSED. 7This notion of the ‘intuitive diversity’ (i-diversity) of a data set is never precisely defined by Horwich (1982). But, in canonical curve-fitting contexts, the ‘intuitive diversity’ of a data set should boil down to some measure of the spread (or variance) of its abscissa values. 8I am being intentionally vague here about which kind of diversity Horwich has in mind. I think Horwich has i-diversity in mind; but, he clearly wants this relationship to obtain also with respect to c-diversity. I’ll try to resolve this important tension below. 4 5 Remaining worries about Horwich’s account 5.1 The relationship between i-diversity and c-diversity Horwich’s H1 says that a more c-diverse set of evidence E1 will tend to “rule- out more of the plausible alternative hypotheses Hj 6=1” than a less c-diverse set of evidence E2 will. But, when Horwich gives his canonical curve-fitting examples, he appeals to an intuitive sense of diversity (i-diversity) which does not obviously correspond to the formal, confirmational diversity specified in H1. At this point, as natural question to ask is: “What is the relationship between i-diversity and c-diversity, anyway?” Ideally, we would like the following general correspondence to obtain between the i-diversity and c-diversity of data sets: (H4) If E1 is more i-diverse than E2 in C, then E1 is more c-diverse than E2 in C. If H4 were generally true (i.e., true for all C), then all of the intuitive examples of CSED would automatically translate into formal examples of CSED with just the right mathematical properties. And, Horwich’s formal account of CSED (i.e., H1–H′3) would be vindicated by its ability to match our intuitions about CSED in all cases. Unfortunately, things don’t work out quite this nicely. It turns out that H4 is not generally true. To see this, let’s reconsider Horwich’s canonical example of CSED, depicted in Figure 1. In this example, E1 is more i-diverse and more c-diverse than E2 in C. It is obvious why E1 is more i-diverse than E2 in C (just inspect the spread of the abcissa values of E1 vs E2). However, it is not so obvious why E1 is more c-diverse than E2 in C. Horwich claims that E1 tends to rule-out more of the plausible alternatives to H1 than E2 does. I think it is more perspicuous to say instead that E1 tends to rule-out more of the simple alternatives to H1 than E2 does. 9 Horwich doesn’t say exactly how we should measure the ‘relative simplicity’ of competing hypotheses. We can make some sense out of Horwich’s canonical example, if we make the following plausible and common assumption about how to measure the simplicity of a polynomial hypothesis in a curve-fitting context: (H5) The simplicity of a polynomial hypothesis H is equal to the di- mensionality of the smallest (non-trivial) family of polynomial 9Horwich (1982, pages 121-122) and Horwich (1993, pages 66–67) explains that his account of CSED depends on a substantive Bayesian understanding of the simplicity of statistical hypotheses. Given our reconstruction of Horwich’s account of CSED, we can see vividly why this is so. Horwich seems to be assuming that simple hypotheses have some kind of a priori probabilistic advantage over complex hypotheses. This kind of assumption is known as a simplicity postulate. Simplicity postulates are a well-known source of controversy in Bayesian philosophy of science. I won’t dwell here on the problematical nature of simplicity postulates, since I think they are a problem for a rather large class of Bayesian accounts of CSED. For an interesting discussion of simplicity postulates in Bayesian confirmation theory, see Popper (1992, Appendix *viii ). See, especially, Forster (1995) for a detailed critique of the simplicity postulate in the context of curve-fitting. 5 functions of which H is a member.10 If we characterize simplicity in this way, we can explain why E1 tends to rule-out more of the simple alternatives to H1 than E2 does in the canonical example depicted in Figure 1. Figure 2 gives us way to picture what’s going on in Horwich’s canonical example in a rather illuminating and explanatory way.11 C H1 Figure 2: Why Horwich’s canonical example has the right formal properties The lightly shaded area in Figure 2 corresponds to the set of linear hypothe- ses that are consistent with the data; and, the darkly shaded region corresponds to the set of parabolic hypotheses that are consistent with the data. Now, if we were to ‘spread-out’ the abscissa values of the data set in Figure 2 — while keeping the likelihood with respect to H1 constant, in accordance with CP∗ — the resulting, more intuitively diverse, data set would end up ruling-out more of the plausible (i.e., simpler) alternative hypotheses than the original data set does. This is because the shaded region (whose area is roughly proportional to the number of simple alternatives to that are consistent with the data) will shrink as we spread out the data set along the linear H1. So, in such an example, it is plausible to expect that the more intuitively diverse data set will also be 10 This is a standard way of measuring the simplicity of hypotheses in curve-fitting contexts. Take, for instance, the curve H: y = x2 + 2x. The smallest (non-trivial) family of polynomials containing H is family PAR: y = ax2 + bx + c (where a, b, and c are adjustable parameters ). The dimensionality of PAR is 3 (which is also the number of freely adjustable parameters in PAR). Hence, the ‘simplicity value’ of H is 3. As a rule, then, lower dimensionality families contain simpler curves. 11 Thanks to Malcolm Forster for generating this informative graphic and allowing me to use it for this purpose. 6 more confirmationally diverse in the formal sense of H1. However, this will not generally be the case. In general, whether or not H4 holds will depend on how complex the hypothesis under test is. We can imagine situations in which the hypothesis under test is sufficiently complex relative to its competitors in C. In such situations, increasing the spread (or i-diversity) of a data set (in accordance with CP∗) may not automatically increase its confirmational diversity.12 To see this, consider a confirmational context C′ in which the hypothesis under test H1 is a highly complex curve, and has only one competitor in C′: a linear hypothesis H2. Now, assume that some data E2 set falls exactly on H1 in such a way that is inconsistent with H2. If we spread out E2 in just the right way — in accordance with CP∗ — to form a more intuitively diverse data set E1, we may end up with a data set that is not more confirmationally diverse than E2. In fact, depending on how complex H1 is (and how cleverly we choose to spread out E2 along H1), E1 may turn-out to be less c-diverse than E2. For instance, E1 might just happen to fall exactly on the linear alternative H2. This kind of ‘non-canonical’ confirmational context — in which a more i-diverse data set turns out to be less c-diverse — is pictured below in Figure 3. E2 E1 = = H1 H2 C ′ Figure 3: Why the truth of H4 depends on the complexity of H1 To sum up: Horwich’s formal sense of confirmational diversity only corre- sponds to his intuitive sense of diversity in contexts where the hypothesis under test is a relatively simple hypothesis. If the hypothesis under test is sufficiently complex relative to its competitors, then the connection between Horwich’s for- mal definition of diversity (in H1) and the intuitive notion of diversity seen in Horwich’s canonical curve-fitting contexts breaks down. Because this connec- tion is essential to the general success of Horwich’s approach to explicate our pre-theoretic intuitions about CSED, Horwich’s account would seem — at best — to provide an incomplete explication of CSED. 12 Thanks to Patrick Maher for getting me to see this point clearly. 7 5.2 Horwich’s choice of measure of confirmation Horwich presupposes that the quotient measure r is an adequate Bayesian mea- sure of degree of confirmation. Several recent authors have argued that r is an inadequate Bayesian measure of confirmation.13 While I find some of these rea- sons for rejecting r rather convincing, I have ‘independent’ reasons for thinking that r is inadequate. One problem with r is that it is not appropriately sensi- tive to the distinction between independent and dependent inductive support for a hypothesis. In Fitelson (1997b), I investigate how various measures of confir- mation handle (or — in the case of r — fail to handle) the distinction between independent and dependent inductive support. There, I argue that r is not an adequate measure of degree of confirmation, since it fails to properly distinguish between dependent and independent pieces of evidence for a hypothesis.14 Luckily for Horwich, however, our reconstruction of his account of CSED is not sensitive to this unfortunate choice of measure of confirmation. It turns out that H3∗ remains true for a wide variety of non-equivalent relevance measures of confirmation (see the Appendix).15 6 Alternative accounts and future directions Carnap (1962) insists that any good Bayesian confirmation theory must be able to adequately explicate CSED. Since then, several Bayesian philosophers of sci- ence have answered the call. The two most well-known and popular Bayesian explications of CSED that have appeared in recent years seem to be Horwich’s ‘eliminativist’ explication, and the so-called ‘correlation’ or ‘similarity’ explica- tion outlined by Howson and Urbach (1993) and Earman (1992). The ‘correla- tion’ approach tries to cache-out the intuition that ‘diverse’ evidence tends not to be highly self -correlated. I find the ‘correlation’ account more intuitively appealing than Horwich’s, because it attempts to unpack ‘diversity’ in terms of some kind of independence of the evidence. I carefully examine the ‘correlation’ explication of CSED in Fitelson (1997a). There, I argue that evidential diversity cannot be properly understood in terms of probabilistic independence of the ev- idence simpliciter, as ‘correlation’ theorists would have us believe. Rather, the increased boost in confirmation provided by more ‘diverse’ collections of evi- dence is really a consequence of the elements of such collections of evidence providing independent inductive support for the hypothesis under test. In Fitelson (1997a), I develop a Bayesian account of independent induc- tive support, and I use it to construct a refreshing new Bayesian explication of 13 See, for instance, Gillies (1986), Good (1984), Rosenkrantz (1981), and Schum (1994). 14 I also argue in Fitelson (1997b) that the difference measure is inadequate. In fact, I argue in Fitelson (1997b) that only a small class of likelihood-based measures can adequately cope with independent inductive support. This is bad news for many recent philosophers of science whose resolutions of various problems in confirmation theory have presupposed inadequate measures of confirmation. 15 In this sense, Horwich’s account is more robust than the so-called ‘correlation’ account of Howson and Urbach (1993). In Fitelson (1997a), I discuss the measure sensitivity of Howson and Urbach’s ‘correlation’ explication of CSED. 8 CSED. Unlike Horwich’s ‘eliminativist’ account and/or Howson and Urbach’s ‘correlation’ account, my explication of CSED requires no additional probabilis- tic ‘ceteris paribus clauses’. Furthermore, my account — unlike Howson and Urbach’s — holds true for all Bayesian relevance measures of confirmation. 9 Appendix: Proofs and Counterexamples A Proofs of (1), (2), and (3) A.1 Proof of (1) The task at hand is to prove: (1) In Cw, E1 is more c-diverse than E2. Proof. Recall that, in Wayne’s counterexample context Cw, the hypothesis un- der test H1 has only two competitors with non-negligible priors: H2 and H3. Moreover, Wayne stipulates that, in Cw, both: Pr(E1|H2) = 0.4 < Pr(E2|H2) = 0.5, and Pr(E1|H3) = 0.4 < Pr(E2|H3) = 0.6. In conjunction with the characterization of c-diversity given in H1, these two facts about Cw yield the desired result. A.2 Proof of (2) We need to demonstrate that: (2) In Cw, E2 confirms H1 more strongly than E1 confirms H1, according to the ratio measure r (i.e., r(H1,E2) > r(H1,E1)). Proof. Wayne’s description of Cw, together with Bayes’s Theorem, and the def- inition of r reported in H2 yields: r(H1,E1) = Pr(E1|H1)∑ i Pr(E1|Hi) · Pr(Hi) = 0.2 (0.2 · 0.2) + (0.4 · 0.2) + (0.4 · 0.6) ≈ 0.555, and r(H1,E2) = Pr(E2|H1)∑ i Pr(E2|Hi) · Pr(Hi) = 0.6 (0.6 · 0.2) + (0.5 · 0.2) + (0.6 · 0.6) ≈ 1.034. Hence, we have r(H1,E2) > r(H1,E1) in Cw, as desired. 10 A.3 Proof of (3) Next, we will prove: (3) In Cw, E2 confirms H1; whereas, E1 disconfirms H1. Proof. According to Bayesian confirmation theory, E confirmsdisconfirms H if and only if r(H,E) ≷ 1. This fact about Bayesian confirmation theory, in conjunction with the following two facts about Cw (both of which were proved in the preceding section of this Appendix): r(H1,E1) ≈ 0.555 < 1, and r(H1,E2) ≈ 1.034 > 1, yields the desired result. B Proof of Theorem H3∗ In this section, we will not only prove Horwich’s H3∗, which presupposes the ratio measure r of degree of confirmation; we will also show that H3∗ remains true even if we use the difference measure d or the likelihood ratio measure l — instead of r — to measure degree of confirmation. Where, the alternative measures of confirmation d and l are defined as follows.16 d(H,E) =df Pr(H|E) − Pr(H) l(H,E) =df Pr(E|H) Pr(E|H̄) This will establish the measure invariance (or measure insensitivity ) of our present reconstruction of Horwich’s explication of CSED. B.1 Proof of Horwich ’s H3∗ (the measure r version) Below, we prove the following Horwichian version of H3∗, which presupposes the measure r of degree of confirmation: (H3r∗) If the following probabilistic ‘ceteris paribus clause’ is satisfied (CP∗) Pr(E1|H1) = Pr(E2|H1), then: E1 is more c-diverse than E2 ⇓ r(H1,E1) > r(H1,E2) 16 I use overbars to express negations of propositions (i.e., ‘X̄’ stands for ‘not -X’. 11 Proof. The definition of r, together with Bayes’s Theorem, entails: r(H1,E1) = Pr(E1|H1) Pr(E1|H1) · Pr(H1) + ∑ i 6=1 Pr(E1|Hi) · Pr(Hi) Applying Horwich’s probabilistic ‘ceteris paribus clause’ CP∗, yields the follow- ing equation.17 r(H1,E1) = Pr(E2|H1) Pr(E2|H1) · Pr(H1) + ∑ i 6=1 Pr(E1|Hi) · Pr(Hi) Assuming that E1 is more c-diverse than E2 (in the H1 sense), we then have: r(H1,E1) > Pr(E2|H1) Pr(E2|H1) · Pr(H1) + ∑ i 6=1 Pr(E2|Hi) · Pr(Hi) Applying the definition of r and Bayes’s Theorem gives: r(H1,E1) > r(H1,E2), which completes the proof of H3r∗. B.2 Proof of the measure d version of H3∗ Below, we prove the following alternative version of H3∗, which presupposes the difference measure d of degree of confirmation: (H3d∗) If the following probabilistic ‘ceteris paribus clause’ is satisfied (CP∗) Pr(E1|H1) = Pr(E2|H1), then: E1 is more c-diverse than E2 ⇓ d(H1,E1) > d(H1,E2) Proof. It turns out that H3d∗ is a straightforward corollary of H3r∗. Simple algebra shows that d(H1,E1) > d(H1,E2) if and only if r(H1,E1) > r(H1,E2). Hence, any sufficient condition for r(H1,E1) > r(H1,E2) is automatically a sufficient condition for d(H1,E1) > d(H1,E2), and vice versa. 17 Notice that the assumption Pr(E1|H1) = Pr(E2|H1) in C is, by itself, sufficient for H3∗. However, while we make no use of the first conjunct of CP∗ in our proof of H3∗, we still must include the assumption that both E1 and E2 confirm H1 in C, in order to avoid spurious counterexamples of the kind constructed by Wayne. 12 B.3 Proof of the measure l version of H3∗ Below, we prove the following alternative version of H3∗, which presupposes the likelihood ratio measure l of degree of confirmation: (H3l∗) If the following probabilistic ‘ceteris paribus clause’ is satisfied (CP∗) Pr(E1|H1) = Pr(E2|H1), then: E1 is more c-diverse than E2 ⇓ l(H1,E1) > l(H1,E2) Proof. By the definition of l, we have the following biconditional: l(H1,E1) > l(H1,E2) ⇐⇒ Pr(E1|H1) Pr(E1|H̄1) > Pr(E2|H1) Pr(E2|H̄1) If we assume that the ‘ceteris paribus clause’ (CP∗) is satisfied, this becomes: l(H1,E1) > l(H1,E2) ⇐⇒ 1 Pr(E1|H̄1) > 1 Pr(E2|H̄1) Applying a little algebra, we then have the following result: (CP∗) =⇒ [ l(H1,E1) > l(H1,E2) ⇐⇒ Pr(E1|H̄1) < Pr(E2|H̄1) ] (∗) Now, from the nature of confirmational contexts, we know that H̄1 is logically equivalent to ∨ Hi 6=1, where the Hi 6=1 are mutually exclusive. Hence, from the probability calculus, we may infer both: Pr(E1|H̄1) = ∑ i 6=1 Pr(E1|Hi) · Pr(Hi)∑ i 6=1 Pr(Hi) , and Pr(E2|H̄1) = ∑ i 6=1 Pr(E2|Hi) · Pr(Hi)∑ i 6=1 Pr(Hi) . From which (with some algebraic manipulation), we may obtain: (∀i 6= 1)[Pr(E1|Hi) < Pr(E2|Hi)] =⇒ Pr(E1|H̄1) < Pr(E2|H̄1) (∗∗) But, the antecedent of (∗∗) just says that E1 is more c-diverse than E2, in the sense of H1. Therefore, (∗) and (∗∗) jointly entail H3l∗. 13 C Counterexample to CP1 =⇒H3 In this section, we show (by generating a concrete counterexample) that: CP1 ; H3. Proof. Consider a simple context18 Cw1 in which only three hypotheses have substantial prior probabilities, Pr(H1) = 0.2, Pr(H2) = 0.2, Pr(H3) = 0.6, and two data sets E1 and E2 such that: Pr(E1|H1) = 0.41 Pr(E2|H1) = 0.6 Pr(E1|H2) = 0.4 Pr(E2|H2) = 0.5 Pr(E1|H3) = 0.4 Pr(E2|H3) = 0.6 This is plainly a case in which E1 is more c-diverse than E2, in the sense of H1: for all Hi, Pr(E1|Hi) is significantly less than Pr(E2|Hi). Moreover, this is also a case in which the probabilistic ‘ceteris paribus clause’ CP1 holds. As the following calculations show, both E1 and E2 confirm H1 in Cw1 . Pr(H1|E1) = Pr(E1|H1) · Pr(H1)∑ i Pr(E1|Hi) · Pr(Hi) = 0.41 · 0.2 (0.41 · 0.2) + (0.4 · 0.2) + (0.4 · 0.6) ≈ 0.204 > Pr(H1) = 0.2, and Pr(H1|E2) = Pr(E2|H1) · Pr(H1)∑ i Pr(E2|Hi) · Pr(Hi) = 0.6 · 0.2 (0.6 · 0.2) + (0.5 · 0.2) + (0.6 · 0.6) ≈ 0.207 > Pr(H1) = 0.2. Finally, Cw1 is such that E2 (the less c-diverse collection of evidence) confirms H1 more strongly than E1 (the more c-diverse collection of evidence), according to all three Bayesian relevance measures r, d, and l. This follows from the fact that Pr(H1|E2) > Pr(H1|E1) in Cw1 (see above), and the proofs given in the previous section concerning the sufficiency of Pr(H1|E2) > Pr(H1|E1) for c(H1,E2) > c(H1,E1), where c is any of the three Bayesian relevance measures of confirmation r, d, or l. Therefore, Cw1 is a counterexample to CP1 =⇒H3. 18 Note: Cw1 is just a slight modification of Wayne’s Cw. 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