How to Confirm the Conjunction of Disconfirmed Hypotheses University of Groningen How to confirm the conjunction of disconfirmed hypotheses Kuipers, T.A.F.; Atkinson, D; Peijnenburg, A.J.M. Published in: Philosophy of Science DOI: 10.1086/598164 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2009 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Kuipers, T. A. F., Atkinson, D., & Peijnenburg, A. J. M. (2009). How to confirm the conjunction of disconfirmed hypotheses. Philosophy of Science, 76(1), 1-21. https://doi.org/10.1086/598164 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 06-04-2021 https://doi.org/10.1086/598164 https://research.rug.nl/en/publications/how-to-confirm-the-conjunction-of-disconfirmed-hypotheses(c9a9d747-5977-4c89-9d5a-d898d39c7b79).html https://doi.org/10.1086/598164 Philosophy of Science, 76 (January 2009) pp. 1–21. 0031-8248/2009/7601-0002$10.00 Copyright 2009 by the Philosophy of Science Association. All rights reserved. 1 How to Confirm the Conjunction of Disconfirmed Hypotheses* David Atkinson, Jeanne Peijnenburg, and Theo Kuipers†‡ Could some evidence confirm a conjunction of two hypotheses more than it confirms either of the hypotheses separately? We show that it might, moreover under conditions that are the same for ten different measures of confirmation. Further, we demonstrate that it is even possible for the conjunction of two disconfirmed hypotheses to be confirmed by the same evidence. 1. Introduction. Alan Author has just made an important discovery. From his calculations it follows that recent evidence e supports the conjunction of two popular hypotheses, and . With great gusto he sets himselfh h1 2 to the writing of a research proposal in which he explains his idea and asks for time and money to work out all its far-reaching consequences. Alan Author’s proposal is sent to Rachel Reviewer, who—to his dismay— writes a devastating report. Ms. Reviewer first recalls what is common knowledge within the scientific community, namely that e strongly dis- confirms not only , but also as well. Then she intimates that Alanh h1 2 Author is clearly not familiar with the relevant literature; for if he were, he would have realized that any calculation that results in confirming the conjunction of two disconfirmed hypotheses must contain a mistake. At any rate, he should never have launched this preposterous idea, which will make him the laughing stock of his peers. Is Reviewer right? Did Author indeed make a blunder by assuming *Received December 2007; revised November 2008. †To contact the authors, please write to: Faculty of Philosophy, University of Groningen, Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands; e-mail: d.atkinson@ rug.nl; jeanne.peijnenburg@rug.nl; t.a.f.kuipers@rug.nl. ‡We would like to thank Igor Douven for having made most useful comments. He brought our attention to the fact that a particular class of examples of what we in Appendix B shall call the Alan Author Effect has been recently published by him (Douven 2007, 155–156). We acknowledge also the lively and helpful comments of the members of the Groningen research group PCCP (Promotion Club Cognitive Patterns). This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 2 DAVID ATKINSON ET AL. that e might confirm a conjunction of hypotheses, , given that theh ∧ h1 2 same e disconfirms and separately? In this article we will argue thath h1 2 Author may not have been mistaken. If c is some generic confirmation function, then it can happen that is greater than zero, indi-c(h ∧ h , e)1 2 cating confirmation of the conjunction of the hypotheses, while c(h , e)1 and are both less than zero, indicating disconfirmation of eachc(h , e)2 hypothesis separately. Friend Author might have written a defensible pro- posal after all. Our article can be seen as a reinforcement of recent claims made by Crupi, Fitelson, and Tentori (2008). In this very stimulating paper, the authors (henceforth CFT) introduce the two inequalities1 c(h , e) ≤ 0 and c(h , eFh ) 1 0. (1)1 2 1 Next they convincingly argue that (1) is a sufficient condition for c(h ∧ h , e) 1 c(h , e). (2)1 2 1 In other words, if e disconfirms , but confirms when is true, thenh h h1 2 1 e gives more confirmation to than to alone—although it confirmsh ∧ h h1 2 1 alone even more: .h c(h , e) 1 c(h ∧ h , e)2 2 1 2 The fact that e can give more confirmation to than to one ofh ∧ h1 2 its conjuncts is of course intriguing, since the corresponding claim for conditional probabilities is false. Under no condition whatsoever can it be true that P(h ∧ h Fe) 1 P(h Fe). (3)1 2 1 Indeed, (3) instantiates the notorious conjunction fallacy. Ever since its description by Tversky and Kahneman (1983), various explanations of the conjunction fallacy have been put forward. Fisk (2004) discusses no less than nine different explanations (ranging from an appeal to linguistic misunderstandings to the idea that people resort to some form of aver- aging process in deriving conjunctive judgments) only to conclude that none of them is convincing. As to Tversky and Kahneman’s own ac- count—based as it is on their earlier ideas about representativeness (Kah- neman and Tversky 1972, 1973)—Fisk reports that it failed several em- pirical tests (2004, 28). His overall judgment is that “an adequate account of the fallacy remains elusive” (2004, 40). As CFT see it, most of the explanations endorse the following inference: first, a formally defined attribute is identified—which, in certain condi- tions, would rank over one of its conjuncts—then the conclusionh ∧ h1 2 1. The notation “ ” is used by CFT to mean “ on the assumption thatc(h ,eFh ) c(h ,e)2 1 2 is true.” As we will further explain in n. 2, is not the same ash c(h ,eFh ) c(h ,e ∧1 2 1 2 .h )1 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 3 is immediately drawn that this attribute explains the fallacy. But, as CFT note, “pending an independent argument to the effect that in standard probabilistic (and betting) conjunction tasks participants are rationally justified in evaluating something else other than probabilities andP(h Fe)1 , we see this inference as spurious” (Crupi et al. 2008, 191).P(h ∧ h Fe)1 2 Following a suggestion by Sides et al. (2002), CFT offer a different account. They surmise that the conjunction fallacy might arise from a confusion between (2) and (3). We find this idea plausible and promising, all the more so because it appears to have solid empirical support (Tentori et al. 2007). However, we also believe that the scope of this idea can be significantly expanded. For if we replace CFT’s condition (1) by another sufficient condition, then we are able to obtain a much stronger result. In this article we will argue that if we take as condition P(h ∧ ¬h Fe) p 0 and P(¬h ∧ h Fe) p 0, (4)1 2 1 2 then it can be shown that c(h ∧ h , e) ≥ c(h , e) and c(h ∧ h , e) ≥ c(h , e). (5)1 2 1 1 2 2 There are two senses in which (5) is more general, and cuts deeper than CFT’s result (2). First, (5) implies that there is a situation in which the conjunction is more, or equally, confirmed than either of the con-h ∧ h1 2 juncts—whereas (2) implies that only one of the conjuncts receives less confirmation than the conjunction. Second, (5) is also applicable when both and are confirmed—whereas (2) requires that one of the hy-h h1 2 potheses must be disconfirmed. Hence, CFT’s explanatory net can in fact be cast much more widely than they do, since there appear to be more ways in which the occurrence of a conjunction fallacy might actually be guided by a sound assessment of a confirmation relation. Of course, this is still open to experiment—here we are simply formulating hypotheses that can be tested empirically. Condition (4) states that, if e is the case, then either both and areh h1 2 true, or neither is: . This is sufficient for (5), as we willP(h ↔ h Fe) p 11 2 prove in Section 2 and in Appendix A. However, it is by no means nec- essary, as will become clear from Section 4 and Appendix B, cf. (13). There we describe a sufficient condition under which is pos-c(h ∧ h , e)1 2 itive, but both and are negative:c(h , e) c(h , e)1 2 c(h ∧ h , e) 1 0 and c(h , e) ! 0 and c(h , e) ! 0. (6)1 2 1 2 Note that (6) describes the case in which Alan Author would be vin- dicated after all. As we will explain later, the sufficient condition for (6) is more detailed than (4). In fact, as we will see from (13), it consists of a relaxed version of (4)—in the sense that small, nonzero values of the two conditional probabilities are tolerated—plus some extra constraints. This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 4 DAVID ATKINSON ET AL. Since (6) implies (5), but not the other way around, the sufficient condition for (6) also suffices for (5). In this sense, a relaxed version of (4), together with some constraints that we will spell out later, would still be enough to deduce (5). 2. Robust Confirmation. CFT justly emphasize that their analysis is ro- bust: it holds for various specifications of the generic function , thatc(h,e) is, for various measures of confirmation (cf. Fitelson 1999). In general, a confirmation function is a measure of the increase in probabilityc(h,e) that a hypothesis acquires when some evidence for its veracity is added. It is an increasing function of the conditional probability of the hypothesis (at constant unconditional probability), and a decreasing function of the unconditional probability (at constant conditional probability). CFT list six prevailing confirmation measures that satisfy these necessary condi- tions, and they prove that their conclusions follow for any of them. We will demonstrate that our argument also goes through robustly in this sense. Indeed, we will add three more measures of confirmation to those listed by CFT, making nine measures in all. The nine measures in question are the following—where, as usual, denotes a conditional, andP(hFe) an unconditional probability:P(h) C(h, e) p P(h ∧ e) � P(h)P(e) D(h, e) p P(hFe) � P(h) S(h, e) p P(hFe) � P(hF¬e) P(hFe) � P(h) Z(h, e) p if P(hFe) ≥ P(h) P(¬h) P(hFe) � P(h) p if P(hFe) ! P(h) P(h) P(hFe) R(h, e) p log [ ]P(h) P(eFh) L(h, e) p log [ ]P(eF¬h) N(h, e) p P(eFh) � P(eF¬h) This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 5 P(eFh) � P(eF¬h) K(h, e) p P(eFh) � P(eF¬h) P(hFe) � P(hF¬e) F(h, e) p . P(hFe) � P(hF¬e) The first six of these measures have been discussed by CFT. Measure C is attributed to Carnap (1950), D to Carnap (1950) and Eells (1982), S to Christensen (1999) and Joyce (1999), Z to Crupi, Tentori and Gon- zalez (2007), R to Keynes (1921) and Milne (1996), and L to Good (1950) and Fitelson (2001). The three extra measures that we have added are N, K, and F. Measures N and K have been taken from Tentori et al. (2007), who attribute them respectively to Nozick (1981), and Kemeny and Oppenheim (1952). Note that and . F is inspired by a measureN(h, e) p S(e, h) K(h, e) p F(e, h) introduced by Fitelson (2003). The latter is in fact a measure of coherence rather than of confirmation. However, it is well known that there are close conceptual connections between coherence and confirmation, and confir- mation measures are sometimes used to indicate coherence (cf. Douven and Meijs 2006). For example, Carnap’s confirmation measure D, to which Carnap himself gives special attention in (Carnap 1962), is the favorite measure of coherence of Douven and Meijs (2007). Further, the exponent of the confirmation measure R of Keynes and Milne, , is equal to a coherence measure of Shogenji (1999).exp R(h, e) CFT succeed in showing that their inference from (1) to (2) remains valid under any of their six measures of confirmation. Similarly, we can prove that our inference from (4) to (5) remains valid as one passes from one of the nine specifications of to another. Take for example thec(h, e) case in which is specified as the Carnap-Eells measure . Wec(h, e) D(h, e) prove in Appendix A that, if our condition (4) is fulfilled, then D(h ∧ h , e) � D(h , e) p P(h ∧ ¬h ∧ ¬e) ≥ 0, (7)1 2 1 1 2 and similarly with and interchanged. Clearly, if (7) holds, thenh h1 2 , which is the first half of our conclusion (5), withD(h ∧ h , e) ≥ D(h , e)1 2 1 D substituted for c. An analogous argument applies to , and this willh 2 give us the second half of (5).2 Similarly, but with more effort, it can be shown that all the remaining specifications of c in terms of , and F will do the trick:C, S, Z, R, L, N, K 2. In terms of D, the second inequality in CFT’s condition (1) reads D(h , eFh ) {2 1 . This function is not the same asP(h Fe ∧ h ) � P(h Fh ) ≥ 0 D(h , e ∧ h ) p P(h Fe ∧2 1 2 1 2 1 2 .h ) � P(h )1 2 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 6 DAVID ATKINSON ET AL. they lead to quotients of products of probabilities that are necessarily nonnegative. In this manner we will have robustly deduced the two in- equalities of (5) from (4). Full details are given in Appendix A, where in passing we also explain under which additional conditions (5) is deducible with 1 in place of ≥.3 As said above, the condition (4) is sufficient, but by no means necessary, for (5)—for it still follows robustly if the probabilities in (4) are small but not zero, and if some extra constraints are in force. In Section 4 and in Appendix B, we will formulate upper bounds for andP(h ∧ ¬h Fe)1 2 —as well as extra constraints such that (5) still holds robustly,P(¬h ∧ h Fe)1 2 that is, under any of the nine confirmation measures in our list. But first, in Section 3, we will illustrate the validity of our inference from (4) to (5) with some examples. The purpose of this exercise is to make the inference intuitively reasonable and to explain its connection to a (particular type of) conjunction fallacy. 3. Conjunction Fallacies. Suppose a roulette wheel bearing the numbers 1 to 10 is spun in secrecy and the number that comes up is recorded by the game master. Consider two gamblers who entertain different hypoth- eses about what the number is. Hypothesis is that the number is 2h1 through 5, or perhaps 9—whereas is that it is 5 through 8, or perhapsh 2 3. We conclude that 5 1 P(h ) p P(h ) p p .1 2 10 2 Suppose next that the game master provides the clue that the number is prime. This can be treated as incoming evidence , withe p {2, 3, 5, 7} and . Since e contains four primes,h p {2, 3, 4, 5, 9} h p {3, 5, 6, 7, 8}1 2 whereas and contain only three primes apiece, we find for the con-h h1 2 ditional probabilities 3 P(h Fe) p P(h Fe) p .1 2 4 Let us now work out this example in terms of the Carnap-Eells measure 3. In Appendix C we present yet another specification of c—the tenth—under which our inference from (4) to (5) goes through. In that appendix we discuss a coherence measure of Bovens and Hartmann (2003a, 2003b), which can be treated as a com- parative measure of confirmation. Since the approach of Bovens and Hartmann is radically different from the above nine cases, we do not consider the Bovens-Hartmann measure here, but devote a separate appendix to it. See also Meijs and Douven (2005) for a critique of the Bovens-Hartmann measure, and the reply of Bovens and Hartmann (2005). This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 7 D, keeping in mind that similar results apply to any of the other nine measures. Since by definition , it is the case thatD(h, e) p P(hFe) � P(h) 3 1 1 D(h , e) p P(h Fe) � P(h ) p � p ,1 1 1 4 2 4 and similarly . However , soD(h , e) p 1/4 h ∧ h p {3, 5} P(h ∧ h Fe) p2 1 2 1 2 , which is strictly less than or ; but2/4 p 1/2 P(h Fe) P(h Fe) P(h ∧ h ) p1 2 1 2 , which is strictly less than or .2/10 p 1/5 P(h ) P(h )1 2 The Carnap degree of confirmation of ish ∧ h1 2 1 1 3 D(h ∧ h , e) p P(h ∧ h Fe) � P(h ∧ h ) p � p ,1 2 1 2 1 2 2 5 10 which is strictly larger than the degrees of confirmation of or . Soh h1 2 D(h ∧ h , e) 1 D(h , e) and D(h ∧ h , e) 1 D(h , e),1 2 1 1 2 2 and this is our result (5), with D substituted for c, and ≥ replaced by 1. It is intuitively clear why should confirme p {2, 3, 5, 7} h ∧ h p1 2 more than or , since{3, 5} h p {2, 3, 4, 5, 9} h p {3, 5, 6, 7, 8} h ∧ h1 2 1 2 contains only primes, whereas and are each ‘diluted’ by nonprimes.h h1 2 However, by the same token it also seems clear how this example can trigger the commission of a conjunction fallacy. Imagine that subjects are given the following information. A game master spins a roulette wheel. He records the number, but does not tell anybody what it is. The only thing he makes known is that the number is prime (e). Suppose that now the question is posed, What do you think is more probable? That the number recorded by the game master is 2 through 5, or perhaps 9 ( )? That it is 5 through 8, or perhaps 3h1 ( )? Or that it is 3 or 5 ( )? It is quite likely that many people wouldh h ∧ h2 1 2 choose the last option, thereby committing a conjunction fallacy (although they would have been right had the question been about confirmation rather than probability). Here is a different example. Imagine that you have a little nephew of whom you are very fond. One day you receive an e-mail from his mother, telling you that the child is suffering a severe bout of measles (e). You immediately decide to visit the boy, bringing him some of the jigsaw puzzles that you know he likes so much. What do you think is more probable to find upon your arrival? That the child has a fever ( ), thath1 he has red spots all over his body ( ), or that he has a fever and redh 2 spots ( )? Again we expect that many people will opt for the thirdh ∧ h1 2 possibility. Although this answer is fallacious, the nephew story does in- stantiate our inference from (4) to (5). Under the assumption that measles This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 8 DAVID ATKINSON ET AL. always comes with fever and red spots—so that (4) is satisfied—our story makes it true that4 c(h ∧ h , e) ≥ c(h , e) and c(h ∧ h , e) ≥ c(h , e).1 2 1 1 2 2 It should be noted that the conjunction fallacies committed in the two examples above differ from those discussed by CFT, who focus exclusively on conjunction fallacies of the Linda-type, as described by Tversky and Kahneman (1983): when hearing about a person named Linda, who is 31 years old, single, bright and outspoken, concerned about discrimination and social justice, involved in antinuclear demonstrations (e), most people find ‘Linda is a bank teller and is active in the feminist movement’ ( ) more probable than ‘Linda is a bank teller’ ( ). Linda-typeh ∧ h h1 2 1 conjunction arguments have the following characteristics in common (Crupi et al. 2008, 187–188, authors’ emphasis): (i) e is negatively (if at all) correlated with ;h1 (ii) e is positively correlated with , even conditionally on ; andh h2 1 (iii) and are mildly (if at all) negatively correlated.h h1 2 It is solely in relation to fallacies of this type that CFT submit their idea that people may actually rely on assessments of confirmation when judging probabilities. Indeed, CFT’s condition (1), which is robustly suf- ficient for their conclusion (2), is an “appropriate confirmation-theoretic rendition of (i) and (ii),” (Crupi et al. 2008, 187–188). In Section 1 we have already intimated that there are two differences between CFT’s conclusion (2) and our result (5). The first was that (5) can handle the case in which the conjunction might be more con-h ∧ h1 2 firmed than either of its conjuncts, the second that (5) also applies when both and are confirmed, that is, and are positive. Weh h c(h , e) c(h , e)1 2 1 2 are now in a position to understand that these differences stem from the dissimilarity between Linda-type fallacies on the one hand and ‘our’ con- junction fallacies on the other. Linda-type fallacies treat and asym-h h1 2 metrically in the sense that one must be disconfirmed, while the other is confirmed. In our case, by contrast, all options are possible: both may be confirmed or both disconfirmed, or indeed, the one may be confirmed and the other disconfirmed. Moreover, in Linda examples the confirma- tion degree of the conjunction lies between those of the two conjuncts. 4. The assumption that there are no measles without both fever and red spots does not imply , although it does entailc(h ∧ h , e) p c(h , e) p c(h , e) P(h ∧ h Fe) p1 2 1 2 1 2 , which in fact are all equal to 1. However, it is less probable that aP(h Fe) p P(h Fe)1 2 patient has fever and spots than that he has just one of these afflictions, if one does not conditionalize on his having measles: , andP(h ∧ h ) ! P(h ) and P(h ∧ h ) ! P(h )1 2 1 1 2 2 therefore is greater than either or .c(h ∧ h , e) c(h , e) c(h , e)1 2 1 2 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 9 In our case, on the other hand, the confirmation degree of the conjunction is greater than that of either of the conjuncts. Given this dissimilarity, it is perhaps appropriate to say that Linda-like problems are best analyzed by CFT’s inference from (1) to (2), whereas ‘our’ conjunction fallacies are best analyzed by our inference from (4) to (5). In both cases the fallacious arguments are explained by pointing to a confusion between probability judgments and confirmation assessments. And both cases satisfy the minimal constraint that the confirmation degree of the conjunction is greater than that of at least one of the conjuncts. It is to be expected that there are many different kinds of conjunction fallacy, all of which will satisfy this minimal constraint. We suspect that, for each kind of fallacy, it will be possible to reconstruct and spell out the corresponding legitimate inference that ought to take the place of the fallacious probabilistic claim. 4. How Alan Author May Be Correct. In this section we will formulate a sufficient condition for (6). That is, we will explain how Alan Author might be correct when he claims that a conjunction of two disconfirmed hypotheses can itself be confirmed. In the process, it will become clear that the rather strict condition (4) is not necessary for obtaining (5). Indeed, (5) is consistent with much looser forms of (4), namely, ones in which the conditional probabilities and areP(h ∧ ¬h Fe) P(¬h ∧ h Fe)1 2 1 2 positive and can be numerically different from one another. Granted, it is only under certain restrictions that (5) follows from weaker versions of (4). But we will show that a precise specification can be given of these restrictions, together with bounds for and .P(h ∧ ¬h Fe) P(¬h ∧ h Fe)1 2 1 2 Since we are dealing with two hypotheses and one piece of evidence, the following eight triple probabilities exhaust all the possibilities that are open to us: P(h ∧ h ∧ e) P(¬h ∧ h ∧ e)1 2 1 2 P(h ∧ ¬h ∧ e) P(¬h ∧ ¬h ∧ e)1 2 1 2 P(h ∧ h ∧ ¬e) P(¬h ∧ h ∧ ¬e)1 2 1 2 P(h ∧ ¬h ∧ ¬e) P(¬h ∧ ¬h ∧ ¬e).1 2 1 2 Note that all the (un)conditional probabilities and all the (un)condi- tional degrees of confirmation are functions of the above eight proba- bilities. Hence looking for (in)equalities between confirmation degrees and probability functions reduces itself to (in)equalities between the values of This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 10 DAVID ATKINSON ET AL. these eight triple probabilities. Because these triples are probabilities, they cannot be negative; and because there are not more triples than these eight, their sum is unity. The sum of all eight triples is one, and we shall set P(¬h ∧ ¬h ∧ ¬e)1 2 to be equal to 1 minus the sum of the remaining seven triples. These seven triples are independent of one another, subject only to the requirement that their sum be not greater than 1. Two of them, andP(h ∧ ¬h ∧ e)1 2 , are both zero under our condition (4). However, here weP(¬h ∧ h ∧ e)1 2 will relax (4) and allow them to be positive. We shall restrict our attention to the case in which these two triples are equal: z p P(h ∧ ¬h ∧ e) p P(¬h ∧ h ∧ e), (8)1 2 1 2 where need not be zero. There are now five other independent triplesz left, and to further reduce the search to manageable proportions we will give them all the same numerical value: x p P(h ∧ h ∧ e) p P(¬h ∧ ¬h ∧ e)1 2 1 2 p P(h ∧ h ∧ ¬e) p P(¬h ∧ h ∧ ¬e) (9)1 2 1 2 p P(h ∧ ¬h ∧ ¬e).1 2 We stress that the artifices (8) and (9) are purely for convenience: they limit the search from 7 to 2 dimensions. Many more possibilities remain open: no matter, our ambition is only to find a sufficient (not a necessary) condition for to be confirmed, while both and are disconfirmed.h ∧ h h h1 2 1 2 The key question that we now have to ask ourselves is, Can we find values of z and x such that (6) follows? If we can, then we will have discovered a sufficient condition for to be positive (confir-c(h ∧ h , e)1 2 mation), while both and are negative (disconfirmation).c(h , e) c(h , e)1 2 Hence, we would have shown that there is at least one way in which Alan Author could have written a defensible research proposal. Moreover, we would also have demonstrated that our general conclusion (5) does not require the rather rigorous condition (4), but is also compatible with a modified form of that condition. For if (6) were to follow from a relaxed version of (4), then so would (5), since the latter follows from (6). In Appendix B we prove that the answer to our key question above is yes. In particular, we show that if , and0 ≤ z ≤ 1/12 1 � 2z 1 � 2z ! x ! , 6 5 then (6) holds. Actually there is more, for z may be a little bigger, as large This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 11 Figure 1. Allowed region of x-z plane for Alan Author Effect. as , but then the upper limit on x decreases. If , the re-1 1/12 ≤ z ≤ 1/88 strictions on x are 1 � 2z 1 � 4z ! x ! . 6 4 A proof of what we shall call the Alan Author Theorem can be found in Appendix B. In Figure 1 we show the area in which Mr. Author’s proposal would have made sense. The straight diagonal line in Figure 1 gives the lower bound of x for values of z between 0 and , while the bent line gives1/8 the upper bound for the x-values. For every x and z between these lines, the inequalities (6) are observed. The region in which the Alan Author Effect occurs might perhaps look small, but in fact it is quite large. If , which corresponds to the strict condition (4), x can be betweenz p 0 and , as can be seen from Figure 1. Since the five triples given in1/6 1/5 (9) are all equal to one another, their sum can be between and 1,5/6 which is a large part of the whole probability space. True, at the maximum allowed value of , namely, , the allowed values for x shrink to thez 1/8 point ; but at , for instance—which corresponds to thex p 1/8 z p 1/12 kink in the bent line of Figure 1—one finds .5/36 ! x ! 1/6 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 12 DAVID ATKINSON ET AL. Figure 2. Venn diagram for h1, h2, and e. Appendix A: Proof of Confirmation Here is a compact notation for the probabilities of the elementary inter- sections of the sets , as displayed in Figure 2:h , h , e1 2 x p P(h ∧ h ∧ e)1 2 y p P(¬h ∧ ¬h ∧ e) y p P(h ∧ ¬h ∧ ¬e) y p P(¬h ∧ h ∧ ¬e)0 1 2 1 1 2 2 1 2 z p P(h ∧ h ∧ ¬e) z p P(¬h ∧ h ∧ e) z p P(h ∧ ¬h ∧ e)0 1 2 1 1 2 2 1 2 An important point is that these seven triple probabilities are nonneg- ative, that is, each one must be positive or zero. Further their sum cannot be larger than unity, for x � y � y � y � z � z � z p 1 � P(¬h ∧ ¬h ∧ ¬e) ≤ 1. (10)0 1 2 0 1 2 1 2 Under this global condition (10) the seven triple probabilities are inde- pendent variables, spanning a hypervolume in seven dimensions. The purpose of the calculations in this appendix is to show that, under the restrictions , it is the case that andz p 0 p z c(h ∧ h , e) ≥ c(h , e)1 2 1 2 1 , for all of the nine realizations of the generic measurec(h ∧ h , e) ≥ c(h , e)1 2 2 , that is, for , , , , , , , , . To do this, it will be proved, forc C D S Z R L N K F each of the measures, that can be reduced to an ex-c(h ∧ h ,e) � c(h ,e)1 2 1 pression involving the five remaining independent triple probabilities that is manifestly nonnegative. Terms that can be recognized as being non- negative are of course products of triple probabilities with a plus sign, or terms involving , where is a triple probability or the sum of two1 � S S or more of them (cf. (10)). Once that has been done, the job is finished, for then has been demonstrated;c(h ∧ h , e) ≥ c(h , e) c(h ∧ h , e) ≥1 2 1 1 2 follows immediately by interchanging the subscripts 1 and 2c(h , e)2 throughout the proof. This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 13 Let us begin with the Carnap measure . We read off from the VennC diagram that C(h ∧ h , e) � C(h , e) p x � (x � z )(x � y )1 2 1 0 0 �x � (x � y � z )(x � y )1 0 0 p (x � y )y .0 1 Since is manifestly nonnegative, we conclude that(x � y )y C(h ∧ h , e) ≥0 1 1 2 . This concludes the demonstration for this simple measure.C(h , e)1 The Carnap-Eells measure D and Christensen’s S are not much harder: C(h ∧ h , e) � C(h , e) (x � y )y1 2 1 0 1D(h ∧ h , e) � D(h , e) p p ≥ 0.1 2 1 P(e) P(e) Since , given that z1 p 0 p z2, and(x � y ) p P(e) y p P(h ∧ ¬h ∧0 1 1 2 , we have thereby proved inequality (7). Similarly,¬e) C(h ∧ h , e) � C(h , e) (x � y )y1 2 1 0 1S(h ∧ h , e) � S(h , e) p p ≥ 0.1 2 1 P(e)P(¬e) P(e)P(¬e) For the Z measure of Crupi, Tentori, and Gonzalez, we have to dis- tinguish the cases in which is larger from those in which it isP(h ∧ h Fe)1 2 smaller than . In the former case, : eitherP(h Fe) Z(h ∧ h , e) ≥ 01 1 2 , so trivially, or , and then weZ(h , e) ! 0 Z(h ∧ h , e) 1 Z(h , e) Z(h , e) ≥ 01 1 2 1 1 find NZ�Z(h ∧ h , e) � Z(h , e) p ,1 2 1 P(e)P(¬(h ∧ h ))P(¬h )1 2 1 where has the formNZ� [x � (x � z )(x � y )][1 � x � y � z ]0 0 1 0 �[x � (x � y � z )(x � y )][1 � x � z ] p y y ,1 0 0 0 0 1 which is nonnegative. The alternative is , and thenZ(h ∧ h , e) ! 01 2 D(h ∧ h , e)1 2Z(h ∧ h , e) p ,1 2 P(h ∧ h )1 2 so implies . Since , we knowZ(h ∧ h , e) ! 0 D(h ∧ h , e) ! 0 z p z p 01 2 1 2 1 2 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 14 DAVID ATKINSON ET AL. that , so . Thus and we canD(h , e) ≤ D(h ∧ h , e) D(h , e) ! 0 Z(h , e) ! 01 1 2 1 1 calculate the difference NZ�Z(h ∧ h , e) � Z(h , e) p ,1 2 1 P(e)P(h ∧ h )P(h )1 2 1 where is given byNZ� [x � (x � z )(x � y )][x � y � z ]0 0 1 0 �[x � (x � y � z )(x � y )][x � z ] p xy ,1 0 0 0 1 which is also nonnegative. This concludes the proof for the measure .Z As to the two measures involving logarithms, x � y � z y1 0 1R(h ∧ h , e) � R(h , e) p log p log 1 � ≥ 0.1 2 1 [ ] [ ]x � z x � z0 0 is a little more complicated, and we findL (1 � x � z )(x � y � z )0 1 0L(h ∧ h , e) � L(h , e) p log1 2 1 [ ](1 � x � y � z )(x � z )1 0 0 y y1 1 p log (1 � )(1 � ) ≥ 0.[ ]1 � x � y � z ) x � z1 0 0 Nozick’s measure yields NNN(h ∧ h , e) � N(h , e) p ,1 2 1 P(h )P(¬h )P(h ∧ h )P(¬(h ∧ h ))1 1 1 2 1 2 where N p [x � (x � y )(x � z ))](x � y � z )(1 � x � y � z )N 0 0 1 0 1 0 �[x � (x � y )(x � y � z )](x � z )(1 � x � z )0 1 0 0 0 p xy (1 � x � z )(1 � x � y � z ) � y y (x � z )(x � y � z ) ≥ 0.1 0 1 0 0 1 0 1 0 For the Kemeny-Oppenheim measure, K(h ∧ h , e) � K(h , e) p1 2 1 NK , [P(h ∧ h ∧ e)P(¬(h ∧ h )) � P(¬(h ∧ h ) ∧ ¬e)P(e)][P(h ∧ e)P(¬h ) � P(¬h ∧ e)P(h )]1 2 1 2 1 2 1 1 1 1 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 15 with the numerator N p [x � (x � z )(x � y )][x � (x � y � z )(x � y � 2x)]K 0 0 1 0 0 �[x � (x � y � z )(x � y )][x � (x � z )(x � y � 2x)]1 0 0 0 0 p 2xy y ≥ 0.0 1 Similarly, Fitelson’s form leads to F(h ∧ h , e) � F(h , e) p1 2 1 NF , [P(h ∧ h ∧ e)P(¬e) � P(h ∧ h ∧ ¬e)P(e)][P(h ∧ e)P(¬e) � P(h ∧ ¬e)P(e)]1 2 1 2 1 1 where this numerator is N p [x � (x � z )(x � y )][x � ( y � z � x)(x � y )]F 0 0 1 0 0 �[x � (x � y � z )(x � y )][x � (z � x)(x � y )]1 0 0 0 0 p 2xy (x � y )(1 � x � y ) ≥ 0.1 0 0 This concludes the proof that (4) is a robust sufficient condition for the validity of (5). However, by scrutinizing the forms that we have obtained for each of the expressions for , we observe that, ifc(h ∧ h , e) � c(h , e)1 2 1 we add to (4) the requirement that none of the remaining five triple prob- abilities, is zero—which is always a valid option—then thex, y , y , y , z0 1 2 0 inequalities in (5) can be replaced by strict inequalities, that is, c(h ∧ h , e) 1 c(h , e) and c(h ∧ h , e) 1 c(h , e).1 2 1 1 2 2 It is of course interesting that the equality option can be excluded so easily, and moreover robustly, that is, in a manner that works for all the measures considered. Appendix B: Disconfirmed Hypotheses In this appendix we will describe sufficient conditions for the validity of the three strict inequalities , , and . Toc(h , e) ! 0 c(h , e) ! 0 c(h ∧ h , e) 1 01 2 1 2 do this we will set and . Thus five ofx p y p y p y p z z { z p z0 1 2 0 1 2 the marked areas in the Venn diagram are equal to , while the remainingx two are equal to z. It will be required that x is nonzero, while z may be zero or nonzero, so in this appendix the condition (4) is being relaxed: it will turn out that z should be small, but need not be zero. The reason for reducing the seven-dimensional problem to a two-dimensional one is purely one of convenience. With more trouble, for instance, one could This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 16 DAVID ATKINSON ET AL. allow and to be different. However the purpose here is only to showz z1 2 that the inequalities (6) are possible, not to explore every part of the seven-dimensional hypervolume for which they hold. Consider first the Carnap confirmation function . We read offC(h, e) from the Venn diagram that C(h , e) p x � z � (x � y � z � z )(x � y � z � z )1 2 1 0 2 0 1 2 p x � z � (3x � z)(2x � 2z) p (x � z)(1 � 6x � 2z) C(h ∧ h , e) p x � (x � z )(x � y � z � z )1 2 0 0 1 2 p x � 2x(2x � 2z) p x(1 � 4x � 4z). If and , then and C(h1 ∧ h2, e)1 � 6x � 2z ! 0 1 � 4x � 4z 1 0 C(h , e) ! 01 1 0. The first inequality yields , while the second gives6x 1 1 � 2z 4x ! , which are consistent with each other if , and1 � 4z (1 � 4z)/4 1 (1 � 2z)/6 that is only possible if . When this holds,z ! 1/8 1 z 1 � ! x ! � z. (11) 6 3 4 In addition, there is the constraint ,x � y � y � y � z � z � z ≤ 10 1 2 0 1 2 which means that . Note that is possible, for then the5x � 2z ≤ 1 z p 0 inequalities simply reduce to . We see that and1/6 ! x ! 1/5 C(h , e) ! 01 are simultaneously possible, and because of the symmetryC(h ∧ h , e) 1 01 2 between and , also (indeed, with the symmetries that weh h C(h , e) ! 01 2 2 have imposed, ). As we have seen, this can occur underC(h , e) p C(h ,e)2 1 the strict condition (4), but also when this condition is relaxed. We will now show that these inequalities guarantee andc(h , e) ! 01 , also when is realized by the other measures of confir-c(h ∧ h , e) 1 0 c1 2 mation. This is obvious for , , , , and becauseD S N K F C(h, e) C(h, e) C(h, e) D(h, e) p S(h, e) p N(h, e) p P(e) P(e)P(¬e) P(h)P(¬h) C(h, e) K(h, e) p . P(h ∧ e)P(¬h) � P(¬h ∧ e)P(h) This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 17 C(h, e) F(h, e) p . P(h ∧ e)P(¬e) � P(h ∧ ¬e)P(e) Whenever is positive (or negative), , and are likewiseC D, S, N, K F positive (or negative), so the same sufficient conditions are applicable. The same applies to the measure , but we have to distinguish betweenZ two cases: C(h, e) Z(h, e) p if P(hFe) ≥ P(h), P(e)P(¬h) C(h, e) p if P(hFe) ! P(h). P(e)P(h) The first form must be used for , when and are bothh { h ∧ h C Z1 2 positive. The second form is needed for , for then and areh { h C Z1 negative. We have used the fact that , , , , and are all equivalentC D S N F up to normalization, which indeed inspired Crupi et al. (2007) to produce their -measure, which has the property that it is equal to 1 if impliesZ e , and if e implies .h �1 ¬h For the logarithmic measures and we can writeR L P(hFe) � P(h) C(h, e) R(h, e) p log 1 � p log 1 � ,[ ] [ ]P(h) P(h)P(e) and P(eFh) � P(eF¬h) C(h, e) L(h, e) p log 1 � p log 1 � .[ ] [ ]P(eF¬h) P(h)P(¬h)P(hF¬e) Evidently, when is positive, zero, or negative, is positive,C(h, e) R(h, e) zero, or negative, respectively, and the same goes for , so our resultsL(h, e) also extend to these cases. Indeed, the results apply to any (Bayesian) measure of confirmation, , for whichc(h, e) sign[c(h, e)] p sign[C(h, e)], (12) where we understand to be , 0, or , according to whethersign[b] �1 �1 is positive, zero, or negative, respectively.b In conclusion, we have proved the following robust result, which we call the Alan Author Theorem: A sufficient condition for ,c(h , e) ! 01 and isc(h , e) ! 0 c(h ∧ h , e) 1 02 1 2 1 � 2z 1 � 2z 1 � 4z 1 ! x ! min , and 0 ≤ z ≤ , (13)[ ]6 5 4 8 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 18 DAVID ATKINSON ET AL. where stands for , or any other measure thatc C, D, S, Z, R, L, N, K, F satisfies (12), and where x p P(h ∧ h ∧ e) p P(¬h ∧ ¬h ∧ e)1 2 1 2 p P(h ∧ ¬h ∧ ¬e) p P(¬h ∧ h ∧ ¬e) p P(h ∧ h ∧ ¬e)1 2 1 2 1 2 z p P(¬h ∧ h ∧ e) p P(h ∧ ¬h ∧ e).1 2 1 2 The upper bound, , is in fact already implied by the first part ofz ≤ 1/8 (13). Appendix C: Bovens and Hartmann Coherence In Section 2 it was noted that there is a close conceptual link between confirmation and coherence: measures of confirmation can be put to work as measures of coherence and vice versa. The nine measures of confir- mation that we have considered so far were all quantitative, that is, the confirmation that gives to is expressed as a number between ande h �1 .�1 Bovens and Hartmann (2003a, 2003b) have introduced a coherence measure that is comparative rather than quantitative. Given two pairs of propositions, for example and , Bovens and Hartmann de-′ ′{h, e} {h , e } scribe a condition such that, when it is fulfilled, it tells us which of the two pairs is the more coherent. An ordering of pairs is thus introduced, but it is what Bovens and Hartmann call a quasiordering. For it can happen that the condition is not fulfilled, and then the relative coherence of two different pairs is simply not defined. In more detail, the quasiordering can be explained as follows. Imagine a witness who reports on a hypothesis and some evidence about which he has heard. Let be his report concerning the hypothesis, and be hish e formulation of the evidence. Define a p P(h ∧ e)0 a p P(¬h ∧ e) � P(h ∧ ¬e)1 a p P(¬h ∧ ¬e) p 1 � a � a .2 0 1 Consider the function 2a � (1 � a )x0 0B(h, e; x) p , (14) 2a � a x � a x0 1 2 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). CONJUNCTION OF DISCONFIRMED HYPOTHESES 19 where x is a number in the interval that represents the unreliability[0, 1] of the witness, with 1 corresponding to total unreliability and 0 to total reliability. According to Bovens and Hartmann, the pair is not less′ ′{h , e } coherent than the pair if{h, e} ′ ′B(h , e ; x) ≥ B(h, e; x), Gx � [0, 1]. (15) The salient point is that this inequality must hold for all , that is, forx all possible degrees of unreliability of the witness, being the same forx both pairs and . If (15) holds with replaced by , then′ ′{h , e } {h, e} ≥ ≤ is said to be not more coherent than the pair . However, it′ ′{h , e } {h, e} can happen that (15) holds neither with ≥ nor with ≤, and then the relative coherence of the two pairs is undefined: they are not ordered in respect of their Bovens-Hartmann coherence. Bovens and Hartmann illustrate cases in which the inequality (15) holds by means of a graph that shows the left- and the right-hand sides as curves that do not intersect one another. Here we introduce, however, a simple algebraic alternative. Since , it follows from (14) thata p 1 � a � a2 0 1 2B(h, e; x) a x0 2x(1 � x) p (1 � x ) � . 1 � B(h, e; x) a a1 1 Further, since the left-hand side of this equation is a monotonic increasing function of , for fixed , it follows that (15) is equivalentB(h, e; x) � [0, 1] x to ′ 2 2a x a x0 02 2(1 � x ) � ≥ (1 � x ) � , Gx � [0, 1]. (16)′ ′a a a a1 1 1 1 With the notation , this can be rewritten2X p x ′a X a X0 0(1 � X ) � ≥ (1 � X ) � , GX � [0,1]. (17)′ ′a a a a1 1 1 1 The above inequality holds at if , and it holds at′ ′X p 0 a /a ≥ a /a0 1 0 1 if . This is a necessary and sufficient condition such that′X p 1 1/a ≥ 1/a1 1 the two straight lines in (17), to the left and right of , do not cross, and≥ therefore that (16) is true. These requirements are equivalent to ′a a1 1′a ≤ a and ≤ . (18)1 1 ′a a0 0 This content downloaded from 129.125.148.244 on July 03, 2018 00:13:02 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 20 DAVID ATKINSON ET AL. We now set , , ; and we use Bovens-Hartmann′ ′h p h h p h ∧ h e p e1 1 2 coherence as a (relative) measure of confirmation. Conditions (18) read P(¬(h ∧ h ) ∧ e) � P(h ∧ h ∧ ¬e) ≤ P(¬h ∧ e) � P(h ∧ ¬e)1 2 1 2 1 1 P(¬(h ∧ h ) ∧ e) � P(h ∧ h ∧ ¬e) P(¬h ∧ e) � P(h ∧ ¬e)1 2 1 2 1 1≤ . P(h ∧ h ∧ e) P(h ∧ e)1 2 1 On referring to the Venn diagram of Appendix A, we transcribe these conditions as follows: y � z � z � z ≤ y � y � z � z0 0 1 2 0 1 0 1 y � z � z � z y � y � z � z0 0 1 2 0 1 0 1≤ , x x � z2 which reduce respectively to z ≤ y and z (x � y � z � z � z ) ≤ xy . (19)2 1 2 0 0 1 2 1 Evidently both of these equalities are satisfied automatically if .z p 02 Under this condition we conclude that is not less highly Bovens-h ∧ h1 2 Hartmann confirmed by than is alone. 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