A Note on Confirmation and Matthew Properties William Roche Department of Philosophy Texas Christian University e-mail: w.roche@tcu.edu 1. Introduction 2. Festa's Two Theses 3. Two Replacement Theses 4. Conclusion ABSTRACT. There are numerous (Bayesian) confirmation measures in the literature. Festa provides a formal characterization of a certain class of such measures. He calls the members of this class "incremental measures". Festa then introduces six rather interesting properties called "Matthew properties" and puts forward two theses, hereafter "T1" and "T2", concerning which of the various extant incremental measures have which of the various Matthew properties. Festa's discussion is potentially helpful with the problem of measure sensitivity. I argue, that, while Festa's discussion is illuminating on the whole and worthy of careful study, T1 and T2 are strictly speaking incorrect (though on the right track) and should be rejected in favor of two similar but distinct theses. KEYWORDS: Confirmation, Festa, Matthew Properties, Problem of Measure Sensitivity. 1. Introduction There are numerous (Bayesian) confirmation measures in the literature. Festa (2012) provides a formal characterization of a certain class of such measures.1 L&PS – Logic & Philosophy of Science Vol. XII, No. 1, 2014, pp. 91-101 © The Author 2014. Published by L&PS – Logic and Philosophy of Science http://www2.units.it/~episteme – ISSN: 1826-1043 1 All references to Festa are to Festa (2012). He calls the members of that class "incremental measures".2 Each of the following is an incremental measure:3 cd (H, E) = p(H|E) – p(H) 92 WILLIAM ROCHE 2 Below, in Section 2, I explain Festa's formal characterization of the class of incremental measures. It is worth noting now, though, that incremental measures are more than just relevance measures, where a measure c is a relevance measure just in case there is a neutral point n such that c(H,E) > / = / < n iff p(H|E) > / = / < p(H). (This characterization of relevance measures is adapted from Fitelson 1999.) Consider the following (well known) measures: cC(H,E) = p(H ∧ E) – p(H)p(E) cM(H,E) = p(E|H) – p(E) cN(H,E) = p(E|H) – p(E|¬H) cS(H,E) = p(H|E) – p(H|¬E). Each of these measures is a relevance measure as defined above. But none of them is an incremental measure as characterized by Festa. This is just as it should be, it seems, if an incremental measure is understood as a measure of the amount of increase in H's probability due to E, for on each of cC, cM, cN, and cs there can be cases where p(H1|E1) > p(H2|E2) while p(H1) < p(H2) and yet the degree to which E1 confirms H1 is less than the degree to which E2 confirms H2. This allows that there are conceptions of confirmation distinct from the incremental conception (where confirmation is a matter of the amount of increase in H's probability due to E) and in terms of which cC, cM, cN, and cs are best understood. See Hajek and Joyce (2008) and Joyce (1999, Ch. 6, sec. 6.4) for relevant discussion. 3 The subscripts in these measures, along with the subscripts in the measures set out below in Section 2, are identical to the subscripts used by Festa. p(E|H) clr (H, E) = ________ p(E|¬H) p(H|E) cr (H, E) = _________. p(H) Festa then introduces six rather interesting properties called "Matthew properties" and puts forward two theses, hereafter "T1" and "T2", concerning which of the various extant incremental measures have which of the various Matthew properties. No two of the three measures cd, clr, and cr are ordinally equivalent to each other (i.e., impose the same ordering on any two ordered pairs of propositions).4 This is prima facie problematic in that each of the three measures has some intuitive plausibility and yet certain results in confirmation theory involving one of the measures do not carry over to (at least one of) the other two measures. This is the problem of measure sensitivity.5 Festa's discussion is potentially helpful with this problem. Suppose one of the various Matthew properties is compelling in that any adequate incremental measure should have that property. Suppose it follows from T1 and T2 that, say, cd has the property in question but neither clr nor cr does. Then clr and cr should be rejected as inadequate (as incremental measures). This would serve to narrow down the field of potentially adequate incremental measures and thus constitute progress towards solving the problem of measure sensitivity. It turns out, however, that, while Festa's discussion is illuminating on the whole and worthy of careful study, T1 and T2 are strictly speaking incorrect (though on the right track). In Section 2, I set out the various incremental measures under consideration along with T1 and T2. In Section 3, I argue that T1 and T2 should be rejected in favor of two similar but distinct theses. In Section 4, I conclude. 2. Festa's Two Theses Festa characterizes the class of incremental measures in terms of the following properties (or conditions):6 Initial and Final Probability Dependence (IFPD): c(H,E) is a function of p(H|E) and p(H). Final Probability Incrementality (FPI): Suppose p(H1) = p(H2 ). Then c(H1,E1) > / < c(H2,E2 ) if and only if p(H1|E1) > / < p(H2|E2 ). 93 A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES 4 Measures c and c* are ordinally equivalent to each other just in case, for any ordered pairs of propositions <H, E> and <H', E'>, the following holds: c(H, E) > / = / < c(H', E') iff c*(H, E) > / = / < c*(H', E'). 5 See Brössel (2013) and Fitelson (1999) for helpful discussion of the problem of measure sensitivity. 6 It should be understood throughout the discussion that the propositions involved in the various probabilities are "p-normal" in that they have nonextreme unconditional probabilities (i.e., unconditional probabilities less than one and greater than zero). Initial Probability Incrementality (IPI): Suppose 0 < p(H1|E1) = p(H2|E2) < 1. Then c(H1,E1) > / < c(H2,E2 ) if and only if p(H1) < / > p(H2 ). Suppose p(H1|E1) = p(H2E2 ) = 0 or p(H1|E1) = p(H2|E2 ) = 1. Then (a) c(H1,E1) ≥ c(H2,E2 ) if p(H1) < p(H2 ) and (b) c(H1,E1) ≤ c(H2,E2) if p(H1) > p(H2 ). Equineutrality (E): Suppose p(H1|E1) = p(H1) and p(H2|E2 ) = p(H2 ). Then c(H1,E1) = c(H2,E2 ). The class of incremental measures is defined as the class of measures having each of IFPD, FPI, IPI, and E. It turns out that many extant confirmation measures are members of the class of incremental measures. Festa considers, in addition to cd, clr, and cr, the following: 94 WILLIAM ROCHE p(H|E) – p(H) cr* (H,E) = _____________ p(H|E) + p(H) o(H|E) p(H|E) p(H) cor (H,E) = _______ where o(H|E) = ________ and o(H) = ______ o(H) p(¬H|E) p(¬H) p(H|E) – p(H) cG (H,E) = _____________ 1 – p(H) p(H|E) – p(H)_____________ if p(H|E) ≥ p(H) 1 – p(H) cz = p(H|E) – p(H)_____________ if p(H|E) < p(H) p(H) log[ p(H|E) /p(H)] cSo (H,E) = ________________ – log[ p(H)] p(H|E) – p(H) cPl (H,E) = ________________________ p(H|E) + p(H) – p(H|E)p(H) { p(H|E) – p(H) chP (H,E) = ________________________ p(H|E) + p(H) + p(H|E)p(H) Some of the sixteen measures under consideration are ordinally equivalent to each other: cr is ordinally equivalent to each of cKu and cr*; clr is ordinally equivalent to each of cor and cKO; cP is ordinally equivalent to cPl. 7 This is significant in that if one measure is ordinally equivalent to another, then the one has a given Matthew property just in case the other too has that property. I thus want to set aside cKu, cr*, cor, cKO , and cPl and focus on cr, clr, and cP along with the remaining eight measures.8 Take some incremental measure c. Since c has IFPD, it follows that c(H, E) is a function of p(H|E) and p(H). But: 95 A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES p(H|E) – p(H) cπ (H,E) = _________________________ where – 2 ≤ π ≤ ∞ p(H|E) + p(H) + πp(H|E)p(H) p(H|E) + αp(H)p(H|E) cα (H,E) = ___________________ where – 1 ≤ α ≤ ∞ p(H) + αp(H)p(H|E) p(H|E) – p(H) cdb (H,E) = _____________ p(H|E)p(H) p(E|H) – p(E|¬H) cKO (H,E) = _______________ p(E|H) + p(E|¬H) p(E|H) – p(E) cP (H,E) = ______________________ p(E|H) + p(E) – p(H ∧ E) p(E|H) cKu (H,E) = _______ p(E) 7 It is straightforward to verify that cr (H,E) = cKu(H,E), cr* (H,E) = [cr (H,E) – 1] / [cr (H,E) + 1] where [n – 1] / [n + 1] is an increasing function of n for n ≥ 0, clr (H,E) = cor (H,E), cKO (H,E) = [clr (H,E) – 1] / [clr (H,E) + 1] where, again, [n – 1] / [n + 1] is an increasing function of n for n ≥ 0, and cP (H,E) = cPl (H,E). 8 There is thus no mention of cKu, cr*, cor, cKO, and cPl in T1 and T2 as formulated below. p(E|H) p(H,E) = _______ p(H). p(E) So c(H,E) is a function of Q(H,E) = p(E|H)/p(E) and p(H), where, following Festa, Q(H,E) is H's predictive success with respect to E. It follows, as Festa notes, that each of the incremental measures under consideration can be restated in terms of Q(H,E), hereafter "Q", and p(H). cd (H,E), for example, can be restated as p(H)[Q – 1]. This can be seen as follows: 96 WILLIAM ROCHE cd (H,E) = p(H|E) – p(H) p(E|H) = _______ p(H) – p(H) p(E) = p(H)[Q – 1] Festa provides a "Q-function" for each of the incremental measures under consideration. I can now state the six Matthew properties introduced by Festa. They can be put as follows: Matthew Independence for Positive Confirmation (MIP): For any Q > 1, if Q is held fixed, then c(H,E) is held fixed and thus is independent of p(H). Matthew Effect for Positive Confirmation (MEP): For any Q > 1, if Q is held fixed, then c(H,E) is an increasing function of p(H). Reverse Matthew Effect for Positive Confirmation (RMP): For any Q > 1, if Q is held fixed, then c(H,E) is a decreasing function of p(H). Matthew Independence for Disconfirmation (MID): For any Q < 1, if Q is held fixed, then c(H,E) is held fixed and thus is independent of p(H). Matthew Effect for Disconfirmation (MED): For any Q < 1, if Q is held fixed, then c(H,E) is a decreasing function of p(H). Reverse Matthew Effect for Disconfirmation (RMD): For any Q < 1, if Q is held fixed, then c(H,E) is an increasing function of p(H). Recall that (following Festa) Q is H's predictive success with respect to E. MIP can be glossed: for any degree of predictive success greater than 1, c(H,E) is independent of H's prior probability. MEP, in turn, can be glossed: for any degree of predictive success greater than 1, the greater is H's prior probability, the greater is c(H,E). And so on for RMP, MID, MED, and RMD. Why are the six Matthew properties named "Matthew" properties? Festa (referencing Kuipers 2000) writes: Kuipers ... introduces the concept of Matthew effect for confirmation just w.r.t. [MEP restricted to cases where H logically implies E]. In fact, [MEP restricted to cases where H logically implies E] "may be seen as a methodological version of the so-called Matthew effect, according to which the rich profit more than the poor" ..., in agreement with the sentence-made famous by the Gospel according to St. Matthew-that "unto every one that hath shall be given". (p. 95, emphasis original) MEP implies that if two hypotheses have the same predictive success (greater than 1) with respect to some piece of evidence, and if initially the two hypotheses had different probabilities, then the hypothesis that initially had the higher probability (the "richer" of the two hypotheses initially) is more strongly confirmed by ("profits" more from) the evidence than does the hypothesis that initially had the lower probability (the "poorer" of the two hypotheses initially). Thus the name "Matthew Effect for Positive Confirmation" and, for consistency, the names of the remaining five properties. It is clear that MIP, MEP, and RMP are pairwise mutually inconsistent in that any measure having one of them lacks each of the other two. It is also clear that the same is true with respect to MID, MED, and RMD. But which measures have which properties? T1 and T2 are meant to answer this question. They can be put like this: T1 a cr has MIP and MID. b cd, cG, cSo, clr, and cP have MEP and MED. c cz has MEP and MID. T2 a chP and cdb have RMP and RMD. b cπ has MEP and MED when π < 0, has MIP and MID when π = 0, and has RMP and RMD when π > 0. c cα has MEP and MED when α < 0, has MIP and MID when α = 0, and has RMP and RMD when α > 0. T1 and T2, I take it, are meant to follow straightforwardly from the various Qfunctions provided by Festa. Recall that the Q-function for cd (H, E) is p(H)[Q – 1]. If Q > 1 and Q is held fixed, it follows that p(H)[Q – 1] is an increasing function of p(H). If Q < 1 and Q is held fixed, it follows that p(H)[Q – 1] is a decreasing function of p(H). So, just as T1b implies, cd has MEP and MED. It turns out, however, that not all is right with T1 and T2. Some modifications are needed. 97 A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES 3. Two Replacement Theses Suppose E entails ¬H so that p(H|E) = 0 = p(E|H). Then p(H|E)/p(H) = 0 regardless of p(H). But Q = p(E|H)/p(E) = p(H|E)/p(H). So Q = 0 regardless of p(H). Suppose c is an incremental measure such that c(H,E) takes the minimum value (for c) in any case where E entails ¬H. Then it is not true that for any Q < 1, if Q is held fixed, then c(H,E) is a decreasing function of p(H), and it is not true that for any Q < 1, if Q is held fixed, then c(H,E) is an increasing function of p(H). So c has neither MED nor RMD. This spells trouble for T1 and T2. Suppose E entails ¬H. Then it follows that:9 98 WILLIAM ROCHE 9 cSo can be understood as having the range (–∞, 1]. See Shogenji (2012, p. 37) and Atkinson (2012, p. 53). But then, as the only plausible candidate value for cSo(H,E) to take when p(H|E) = 0 is –∞, it follows that cSo(H,E) is undefined when p(H|E) = 0. This is less than ideal, it seems, since there should be a degree of confirmation even when p(H|E) = 0. It seems preferable to understand log[ p(H|E) /p(H)] cSo (H,E) = ________________ – log[ p(H)] – ∞ = ___________ – log[ p(H)] = – ∞ p(E|H) clr (H,E) = ________ p(E|¬H) 0 = ________ p(E|¬H) = 0 p(E|H) – p(E) cP (H,E) = ______________________ p(E|H) + p(E) – p(H ∧ E) 0 – p(E) = ___________ 0 + p(E) – 0 = – 1 Note that cπ (H,E) = –1 and cα (H,E) = 0 regardless of the values specified for π and α respectively. It follows that cSo, clr, and cP do not have MED, that chP does not have RMD, that cπ does not have MED when π < 0 and does not have RMD when π > 0, and that cα does not have MED when α < 0 and does not have RMD when α > 0. So T1b is incorrect and each of T2a, T2b, and T2c is incorrect. So T1 and T2 are incorrect. T1 and T2, though, are on the right track. They can be replaced by the following: T1* a cr has MIP and MID. b cd and cG have MEP and MED. c cSo, clr, and cP have MEP but do not have MID, MED, or RMD; cSo, clr, and cP have MED in the special case where 1 > Q > 0. d cz has MEP and MID. 99 A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES cSo as having the range [–∞, 1] and as taking the value –∞ when p(H|E) = 0. Atkinson and Shogenji (personal communication) agree that cSo should be understood as having the range [–∞, 1]. p(H|E) – p(H) chP (H,E) = ________________________ p(H|E) + p(H) + p(H|E)p(H) 0 – p(H) = __________ 0 + p(H) + 0 = – 1 p(H|E) – p(H) cπ (H,E) = _________________________ p(H|E) + p(H) + πp(H|E)p(H) 0 – p(H) = __________ 0 + p(H) + 0 = – 1 p(H|E) + αp(H)p(H|E) cα (H,E) = ___________________ p(H) + αp(H)p(H|E) 0 + 0 = ________ p(H) + 0 = 0 T2* a cdb has RMP and RMD. b chP has RMP but does not have MID, MED, or RMD; chP has RMD in the special case where 1 > Q > 0. c cπ has MEP but does not have MID, MED, or RMD when π < 0; cπ has MIP and MID when π = 0; cπ has RMP but does not have MID, MED, or RMD when π > 0; cπ has MED when π < 0 in the special case where 1 > Q > 0; cπ has RMD when π > 0 in the special case where 1 > Q > 0. d cα has MEP but does not have MID, MED, or RMD when α < 0; cα has MIP and MID when α = 0; cα has RMP but does not have MID, MED, or RMD when α > 0; cα has MED when α < 0 in the special case where 1 > Q > 0; cα has RMD when α > 0 in the special case where 1 > Q > 0. T1* and T2* differ from T1 and T2 only with respect to cases where E entails ¬H and thus Q = 0. Some of the measures referred to in T1 and T2 have a maximum value and take that value in any case where E entails H. Consider cSo for example. If E entails H so that p(H|E) = 1, it follows that cSo(H,E) takes its maximum value of 1 regardless of H's prior probability. Why is it that T1 and T2 run into trouble in the case where E entails ¬H but do not run into trouble in the case where E entails H? Return to the case where E entails ¬H. The key here is that Q equals 0 regardless of H's prior probability. This means that Q can be held fixed while p(H) increases or decreases. This in turn means that if c(H,E) takes the minimum value (for c) in any case where E entails ¬H, then there can be cases where E entails ¬H, Q is held fixed, and c(H,E) remains constant at the minimum value while p(H) decreases, in which case c does not have MED, and there can be cases where E entails ¬H, Q is held fixed, and c(H,E) remains constant at the minimum value while p(H) increases, in which case c does not have RMD. Things are different in the case where E entails H. Suppose c is an incremental measure such that c(H,E) takes the maximum value (for c) in any case where E entails H. Suppose E entails H so that p(H|E) = 1. Then 1/p(H) = p(H|E)/p(H) = p(E|H)/p(E) = Q. But then if Q is held fixed, it follows that p(H) too is held fixed. Hence there can be no cases where E entails H, Q is held fixed, and c(H, E) remains constant at the maximum value while p(H) increases or decreases. So no case where E entails H could show that c does not have MEP or that c does not have RMP. 100 WILLIAM ROCHE 4. Conclusion T1 and T2 are incorrect in some of what they imply with respect to cases where E entails ¬H and thus Q = 0. They should be rejected in favor of T1* and T2*. The way is now clear for confirmation theorists to focus on which, if any, of the various Matthew properties are compelling.10 By doing so confirmation theorists can perhaps use T1* and T2* to narrow down the field of potentially adequate incremental measures and so make progress towards solving the problem of measure sensitivity. Acknowledgments. Thanks to an anonymous reviewer for helpful comments on an earlier version of the paper. REFERENCES ATKINSON, D. (2012). Confirmation and Justification: A Commentary on Shogenji's Measure. Synthese, 184, pp. 49-61. BRÖSSEL, P. (2013). The Problem of Measure Sensitivity Redux. Philosophy of Science, 80, pp. 378-397. FESTA, R. (2012). "For unto every one that hath shall be given". Matthew Properties for Incremental Confirmation. Synthese, 184, pp. 89-100. FITELSON, B. (1999). The Plurality of Bayesian Measures of Confirmation and the Problem of Measure Sensitivity. Philosophy of Science, 66, pp. S362-S378. HAJEK, A. and JOYCE, J. (2008). Confirmation. In S. Psillos and M. Curd (Eds.), The Routledge Companion to Philosophy of Science. London: Routledge, pp. 115-128. JOYCE, J. (1999). The Foundations of Causal Decision Theory. Cambridge: Cambridge University Press. KUIPERS, T. (2000). From Instrumentalism to Constructive Realism: On Some Relations between Confirmation, Empirical Progress, and Truth Approximation. Dordrecht: Kluwer. SHOGENJI, T. (2012). The Degree of Epistemic Justification and the Conjunction Fallacy. Synthese, 184, pp. 29-48. 101 A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES 10 Festa (sec. 3.3.2) suggests that at least in some cases RMP is compelling.