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

William, Roche

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A Note on Confirmation and Matthew PropertiesWilliam RocheDepartment of PhilosophyTexas Christian Universitye-mail: w.roche@tcu.edu1. Introduction2. Festa’s Two Theses3. Two Replacement Theses4. ConclusionABSTRACT. There are numerous (Bayesian) confirmation measures inthe literature. Festa provides a formal characterization of a certainclass of such measures. He calls the members of this class “incre-mental measures”. Festa then introduces six rather interesting pro-perties called “Matthew properties” and puts forward two theses, he-reafter “T1” and “T2”, concerning which of the various extant incre-mental measures have which of the various Matthew properties.Festa’s discussion is potentially helpful with the problem of measu-re sensitivity. I argue, that, while Festa’s discussion is illuminatingon the whole and worthy of careful study, T1 and T2 are strictlyspeaking incorrect (though on the right track) and should be rejectedin favor of two similar but distinct theses.KEYWORDS: Confirmation, Festa, Matthew Properties, Problem ofMeasure Sensitivity.1.  IntroductionThere are numerous (Bayesian) confirmation measures in the literature. Festa(2012) provides a formal characterization of a certain class of such measures.1L&PS – Logic & Philosophy of ScienceVol. XII, No. 1, 2014, pp. 91-101© The Author 2014.Published by L&PS – Logic and Philosophy of Sciencehttp://www2.units.it/~episteme – ISSN: 1826-10431 All references to Festa are to Festa (2012).He calls the members of that class “incremental measures”.2 Each of the fol-lowing is an incremental measure:3cd (H, E) = p(H|E) – p(H)92WILLIAM ROCHE2 Below, in Section 2, I explain Festa’s formal characterization of the class of incrementalmeasures. It is worth noting now, though, that incremental measures are more than just rele-vance measures, where a measure c is a relevance measure just in case there is a neutral point nsuch that c(H,E) > / = / < n iff p(H|E) > / = / < p(H). (This characterization of relevance mea-sures 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 incre-mental measure as characterized by Festa. This is just as it should be, it seems, if an incremen-tal 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 confirmsH2. This allows that there are conceptions of confirmation distinct from the incremental con-ception (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) andJoyce (1999, Ch. 6, sec. 6.4) for relevant discussion.3 The subscripts in these measures, along with the subscripts in the measures set out belowin 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 proper-ties” and puts forward two theses, hereafter “T1” and “T2”, concerning whichof the various extant incremental measures have which of the various Matthewproperties.No two of the three measures cd, clr, and cr are ordinally equivalent to eachother (i.e., impose the same ordering on any two ordered pairs of proposi-tions).4 This is prima facie problematic in that each of the three measures hassome intuitive plausibility and yet certain results in confirmation theory invol-ving one of the measures do not carry over to (at least one of) the other twomeasures. This is the problem of measure sensitivity.5Festa’s discussion is potentially helpful with this problem. Suppose one ofthe various Matthew properties is compelling in that any adequate incremen-tal 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 crshould be rejected as inadequate (as incremental measures). This would serveto narrow down the field of potentially adequate incremental measures andthus constitute progress towards solving the problem of measure sensitivity.It turns out, however, that, while Festa’s discussion is illuminating on thewhole 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 mea-sures under consideration along with T1 and T2. In Section 3, I argue that T1and T2 should be rejected in favor of two similar but distinct theses. In Sec-tion 4, I conclude.2.  Festa’s Two ThesesFesta characterizes the class of incremental measures in terms of the followingproperties (or conditions):6Initial and Final Probability Dependence (IFPD): c(H,E) is a function ofp(H|E) and p(H).Final Probability Incrementality (FPI): Suppose p(H1) = p(H2 ). Thenc(H1,E1) > / < c(H2,E2 ) if and only if p(H1|E1) > / < p(H2|E2 ).93A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES4 Measures c and c* are ordinally equivalent to each other just in case, for any ordered pairsof 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 measuresensitivity.6 It should be understood throughout the discussion that the propositions involved in the va-rious 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 ). Supposep(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 ). Thenc(H1,E1) = c(H2,E2 ).The class of incremental measures is defined as the class of measures havingeach of IFPD, FPI, IPI, and E.It turns out that many extant confirmation measures are members of theclass of incremental measures. Festa considers, in addition to cd, clr, and cr, thefollowing:94WILLIAM ROCHEp(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 toeach other: cr is ordinally equivalent to each of cKu and cr*; clr is ordinallyequivalent to each of cor and cKO; cP is ordinally equivalent to cPl.7 This is si-gnificant in that if one measure is ordinally equivalent to another, then the onehas a given Matthew property just in case the other too has that property. I thuswant to set aside cKu, cr*, cor, cKO , and cPl and focus on cr, clr, and cP alongwith the remaining eight measures.8Take 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:95A NOTE ON CONFIRMATION AND MATTHEW PROPERTIESp(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, followingFesta, Q(H,E) is H’s predictive success with respect to E.It follows, as Festa notes, that each of the incremental measures under consi-deration 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:96WILLIAM ROCHEcd (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 con-sideration.I can now state the six Matthew properties introduced by Festa. They canbe put as follows:Matthew Independence for Positive Confirmation (MIP): For any Q > 1, ifQ 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 isheld 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 isheld 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 fi-xed, then c(H,E) is a decreasing function of p(H).Reverse Matthew Effect for Disconfirmation (RMD): For any Q < 1, if Q isheld 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 anydegree 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 confirmationjust 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 asa methodological version of the so-called Matthew effect, according towhich the rich profit more than the poor” ..., in agreement with the sen-tence—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 (greaterthan 1) with respect to some piece of evidence, and if initially the two hypothe-ses had different probabilities, then the hypothesis that initially had the higherprobability (the “richer” of the two hypotheses initially) is more strongly confir-med by (“profits” more from) the evidence than does the hypothesis that initiallyhad the lower probability (the “poorer” of the two hypotheses initially). Thus thename “Matthew Effect for Positive Confirmation” and, for consistency, the na-mes of the remaining five properties. It is clear that MIP, MEP, and RMP are pairwise mutually inconsistent inthat any measure having one of them lacks each of the other two. It is also clearthat the same is true with respect to MID, MED, and RMD. But which mea-sures 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 Q-functions 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 increasingfunction of p(H). If Q < 1 and Q is held fixed, it follows that p(H)[Q – 1] is adecreasing 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 modifica-tions are needed.97A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES3.  Two Replacement ThesesSuppose E entails ¬H so that p(H|E) = 0 = p(E|H). Then p(H|E)/p(H) = 0 re-gardless of p(H). But Q = p(E|H)/p(E) = p(H|E)/p(H). So Q = 0 regardless ofp(H). Suppose c is an incremental measure such that c(H,E) takes the mini-mum value (for c) in any case where E entails ¬H. Then it is not true that forany Q < 1, if Q is held fixed, then c(H,E) is a decreasing function of p(H), andit is not true that for any Q < 1, if Q is held fixed, then c(H,E) is an increasingfunction of p(H). So c has neither MED nor RMD.This spells trouble for T1 and T2. Suppose E entails ¬H. Then it followsthat:998WILLIAM ROCHE9 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) = 0is –∞, it follows that cSo(H,E) is undefined when p(H|E) = 0. This is less than ideal, it seems, sincethere should be a degree of confirmation even when p(H|E) = 0. It seems preferable to understandlog[ 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) = 0p(E|H) – p(E)cP (H,E) = ______________________p(E|H) + p(E) – p(H ∧ E)0 – p(E)= ___________0 + p(E) – 0= – 1Note that cπ (H,E) = –1 and cα (H,E) = 0 regardless of the values specified forπ and α respectively. It follows that cSo, clr, and cP do not have MED, that chPdoes not have RMD, that cπ does not have MED when π < 0 and does not ha-ve RMD when π > 0, and that cα does not have MED when α < 0 and does nothave RMD when α > 0. So T1b is incorrect and each of T2a, T2b, and T2c isincorrect. So T1 and T2 are incorrect.T1 and T2, though, are on the right track. They can be replaced by the fol-lowing: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.99A NOTE ON CONFIRMATION AND MATTHEW PROPERTIEScSo 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= – 1p(H|E) – p(H)cπ (H,E) = _________________________ p(H|E) + p(H) + πp(H|E)p(H)0 – p(H)= __________0 + p(H) + 0= – 1p(H|E) + αp(H)p(H|E)cα (H,E) = ___________________ p(H) + αp(H)p(H|E)0 + 0= ________p(H) + 0= 0T2* a cdb has RMP and RMD.b chP has RMP but does not have MID, MED, or RMD; chP hasRMD 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 haveMID, MED, or RMD when π > 0; cπ has MED when π < 0 in thespecial case where 1 > Q > 0; cπ has RMD when π > 0 in the spe-cial 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 haveMID, MED, or RMD when α > 0; cα has MED when α < 0 in thespecial case where 1 > Q > 0; cα has RMD when α > 0 in the spe-cial 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 andtake that value in any case where E entails H. Consider cSo for example. If Eentails H so that p(H|E) = 1, it follows that cSo(H,E) takes its maximum valueof 1 regardless of H’s prior probability. Why is it that T1 and T2 run into trou-ble in the case where E entails ¬H but do not run into trouble in the case whe-re E entails H?Return to the case where E entails ¬H. The key here is that Q equals 0 re-gardless of H’s prior probability. This means that Q can be held fixed whilep(H) increases or decreases. This in turn means that if c(H,E) takes the mini-mum value (for c) in any case where E entails ¬H, then there can be caseswhere E entails ¬H, Q is held fixed, and c(H,E) remains constant at the mini-mum value while p(H) decreases, in which case c does not have MED, and the-re can be cases where E entails ¬H, Q is held fixed, and c(H,E) remains con-stant at the minimum value while p(H) increases, in which case c does not ha-ve RMD. Things are different in the case where E entails H. Suppose c is anincremental measure such that c(H,E) takes the maximum value (for c) in anycase 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 thatp(H) too is held fixed. Hence there can be no cases where E entails H, Q is heldfixed, and c(H, E) remains constant at the maximum value while p(H) increa-ses or decreases. So no case where E entails H could show that c does not ha-ve MEP or that c does not have RMP.100WILLIAM ROCHE4.  ConclusionT1 and T2 are incorrect in some of what they imply with respect to cases whe-re E entails ¬H and thus Q = 0. They should be rejected in favor of T1* andT2*. 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 confirmationtheorists can perhaps use T1* and T2* to narrow down the field of potentiallyadequate incremental measures and so make progress towards solving the pro-blem of measure sensitivity.Acknowledgments. Thanks to an anonymous reviewer for helpful comments on anearlier version of the paper.REFERENCESATKINSON, D. (2012). Confirmation and Justification: A Commentary on Shogen-ji’s Measure. Synthese, 184, pp. 49-61.BRÖSSEL, P. (2013). The Problem of Measure Sensitivity Redux. Philosophy ofScience, 80, pp. 378-397.FESTA, R. (2012). “For unto every one that hath shall be given”. Matthew Proper-ties for Incremental Confirmation. Synthese, 184, pp. 89-100.FITELSON, B. (1999). The Plurality of Bayesian Measures of Confirmation and theProblem 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: Cam-bridge University Press.KUIPERS, T. (2000). From Instrumentalism to Constructive Realism: On Some Re-lations between Confirmation, Empirical Progress, and Truth Approximation.Dordrecht: Kluwer.SHOGENJI, T. (2012). The Degree of Epistemic Justification and the ConjunctionFallacy. Synthese, 184, pp. 29-48.101A NOTE ON CONFIRMATION AND MATTHEW PROPERTIES10 Festa (sec. 3.3.2) suggests that at least in some cases RMP is compelling.

A note on confirmation and Matthew properties

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A note on confirmation and Matthew properties

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