Abstract
Bayesian confirmation theory is rife with confirmation measures. Many of them differ from each other in important respects. It turns out, though, that all the standard confirmation measures in the literature run counter to the so-called “Reverse Matthew Effect” (“RME” for short). Suppose, to illustrate, that \(H_{1}\) and \(H_{2}\) are equally successful in predicting E in that \(p\left( {E\,|\,H_1 } \right) /p\left( E \right) =p\left( {E\,|\,H_2 } \right) /p\left( E \right) >1\). Suppose, further, that initially \(H_{1}\) is less probable than \(H_{2}\) in that \(p(H_{1}) < p(H_{2})\). Then by RME it follows that the degree to which E confirms \(H_{1}\) is greater than the degree to which it confirms \(H_{2}\). But by all the standard confirmation measures in the literature, in contrast, it follows that the degree to which E confirms \(H_{1}\) is less than or equal to the degree to which it confirms \(H_{2}\). It might seem, then, that RME should be rejected as implausible. Festa (Synthese 184:89–100, 2012), however, argues that there are scientific contexts in which RME holds. If Festa’s argument is sound, it follows that there are scientific contexts in which none of the standard confirmation measures in the literature is adequate. Festa’s argument is thus interesting, important, and deserving of careful examination. I consider five distinct respects in which E can be related to H, use them to construct five distinct ways of understanding confirmation measures, which I call “Increase in Probability”, “Partial Dependence”, “Partial Entailment”, “Partial Discrimination”, and “Popper Corroboration”, and argue that each such way runs counter to RME. The result is that it is not at all clear that there is a place in Bayesian confirmation theory for RME.
Similar content being viewed by others
Notes
Here and throughout the sense of confirmation at issue is such that E confirms H if and only if \(p(H\,{\vert }\, E) > p(H)\), or equivalently \(p(H \,{\vert }\, E) > p(H \,{\vert }\, \lnot E)\), or equivalently \(p(E \,{\vert }\, H) > p(E)\), or equivalently \(p(E \,{\vert }\, H) > p(E \,{\vert }\, \lnot H)\). This sense of confirmation stands in contrast to so-called “absolute confirmation”. The latter, unlike the former, is a matter of high probability. See Douven (2011), Roche (2012, 2015), and Roche and Shogenji (2014) for further discussion of different senses of confirmation (or evidential support).
ME, MI, and RME can be reformulated in terms of a single hypothesis and a single piece of evidence (see Festa 2012 and Roche 2014). ME, for example, can be reformulated like this: If (i) \(p(E \,{\vert }\, H)/p(E) > 1\) and (ii) \(p(E \,{\vert }\, H)/p(E)\) is held fixed, then \(c(H,\, E)\) is an increasing function of p(H).
Let c and \(c^*\) be confirmation measures. Then c and \(c^*\) are ordinally equivalent to each other if and only if the following holds for any ordered pairs of propositions \(\langle H_{1}\), \(E_{1}\rangle \) and \(\langle H_{2}\), \(E_{2}\rangle : c(H_{1},\, E_{1}) > / = / < c(H_{2},\, E_{2})\) if and only if \(c^*(H_{1},\, E_{1}) > / = / < c^*(H_{2},\,E_{2})\).
This is prima facie problematic. Many, if not all of, \(c_{1},\, c_{2}/c_{6},\,c_{3},\,c_{4},\, c_{5},\,c_{7}\), and \(c_{8}\) have some intuitive plausibility. But certain results in Bayesian confirmation theory involving some such measures fail to carry over to at least some of the others. This is “the problem of measure sensitivity”. See Brössel (2013) and Fitelson (1999) for helpful discussion.
Each of \(c_{1}\) and \(c_{8}\) meets ME whereas each of \(c_{2}\) and \(c_{6}\) meets MI. This is noted in Festa (2012) and Roche (2014). None of \(c_{3}\), \(c_{4}\), \(c_{5}\), and \(c_{7}\) meets ME, MI, or RME. This can be verified on Mathematica using PrSAT (developed by Branden Fitelson in collaboration with Jason Alexander and Ben Blum). See Fitelson (2008) for discussion of PrSAT.
It is worth noting in this regard that:
$$\begin{aligned}&\displaystyle c_9 \left( {H_1 ,E} \right) \approx 0.99996<0.99998\approx c_9 \left( {H_2 ,E} \right)&\\&\displaystyle c_{10} \left( {H_1 ,E} \right) =33334<99998\approx c_{10} \left( {H_2 ,E} \right)&\\&\displaystyle c_{11} \left( {H_1 ,E} \right) =99999<9999900000=c_{11} \left( {H_2 ,E} \right)&\end{aligned}$$So by \(c_{9}\) the degree to which E confirms \(H_{1}\) is slightly less than the degree to which E confirms \(H_{2}\) whereas both by \(c_{10}\) and by \(c_{11}\) the degree to which E confirms \(H_{1}\) is much less than the degree to which E confirms \(H_{2}\).
It is not essential that \(p(H_{1} \,{\vert }\, E)\) equals unity on Distribution D. Any value very close to unity would suffice.
Festa (2012, Sect. 3.3) considers the so-called “Problem of Irrelevant Conjunction” in the context of evaluating ME, MI, and RME. His main point can be put as follows: If \(H_{1}\) is a “genuine” hypothesis and \(H_{2}\) is an “irrelevant” or “nonsensical” hypothesis such as the hypothesis that the moon is made of green cheese, then \( H_{1}\, \& \,H_{2}\) is not a genuine hypothesis and thus should be set aside when evaluating ME, MI, and RME. I take it that this point has no application in the case above (where \(H_{1}\) is the proposition that all ravens are black and \(H_{2}\) is the proposition that Smith testified that Tweety is dark brown). For, in that case, \(H_{2}\) is neither irrelevant nor nonsensical.
The same is true with respect to MI.
MI entails that \(c(H_{1},\, E)=c(H_{2},\, E)\). So PDe also runs counter to MI.
The title of Crupi and Tentori (2013) is “Confirmation as partial entailment: A representation theorem in inductive logic”.
MI entails that \(c(H_{1},\, E)=c(H_{2},\,E)\). So PE also runs counter to MI.
MI entails that \(c(H_{1},\, E)=c(H_{2},\, E)\). So PDi also runs counter to MI.
If PC6 were not so understood, then Popper would be wrong that his preferred corroboration measure (\(c_{13}\) below) meets PC6. Further, it is clear from the surrounding discussion that Popper has in mind cases where E confirms each of \(H_{1}\) and \(H_{2}\).
See Popper (1954) for a different but similar set of adequacy conditions on corroboration measures (though there Popper speaks in terms of “confirmation” as opposed to “corroboration”). One notable difference is that PC5 is not included in the earlier set of adequacy conditions. See Díez (2011) and Sprenger (forthcoming) for discussion of the earlier set of adequacy conditions.
Popper (1954) initially suggests a different measure. It can be put as follows:
$$\begin{aligned} c_{14} (H,E)=\left[ {\frac{p(E\,|\,H)-p(E)}{p(E\,|\,H)+\Pr (E)}} \right] \left[ {1+p(H)p(H\,|\,E)} \right] \end{aligned}$$This measure is not ordinally equivalent to \(c_{13}\) in that there are cases where \(c_{13}(H_{1},\, E_{1})> c_{13}(H_{2},\, E_{2})\) but \(c_{14}(H_{1},\, E_{1}) \le c_{14}(H_{2},\, E_{2})\). It can be shown, though, and is noted by Popper (1983, p. 251), that \(c_{14}\), as with \(c_{13}\), meets each of PC1–PC6.
MI entails that \(c(H_{1},\, E)=c(H_{2},\, E)\). So PC also runs counter to MI.
Sober (2015, p. 96) notes in effect that this is true in the special case where (a) \(H_{1}\) entails \(H_{2}\) but not vice versa and (b) each of \(H_{1}\) and \(H_{2}\) entails E. Festa (2012, Sect. 3, p. 97) goes farther and notes that it is true in general and thus not just in special cases. His argument, though, contains a minor mistake (which is perhaps merely typographical). Festa (2012, Sect. 2, p. 93) claims that:
$$\begin{aligned} \frac{p(H\,|\,E)-p(H)}{p(H\,|\,E)+p(H)-p(H\,|\,E)p(H)}=\frac{\frac{p(E\,|\,H)}{p(E)}-1}{\frac{p(E\,|\,H)}{p(E)}-p(H)+1} \end{aligned}$$This is wrong. The denominator on the right should be \(\frac{p(E\,|\,H)}{p(E)}\left( {1-p(H)} \right) +1\).
The same is true with respect to “Weak Informativity” and “Strong Informativity” in Sprenger (2016, p. 10). For further discussion of PC, and for references, see Rowbottom (2011).
References
Brössel, P. (2013). The problem of measure sensitivity redux. Philosophy of Science, 80, 378–397.
Crupi, V. (2016). Confirmation. In E. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2016 ed.). http://plato.stanford.edu/archives/fall2016/entries/confirmation/.
Crupi, V., & Tentori, K. (2013). Confirmation as partial entailment: A representation theorem in inductive logic. Journal of Applied Logic, 11, 364–372.
Crupi, V., & Tentori, K. (2014). Erratum to “Confirmation as partial entailment”. Journal of Applied Logic, 12, 230–231.
Díez, J. (2011). On Popper’s strong inductivism (or strongly inconsistent anti-inductivism). Studies in History and Philosophy of Science, 42, 105–116.
Douven, I. (2011). Further results on the intransitivity of evidential support. Review of Symbolic Logic, 4, 487–497.
Festa, R. (2012). “For unto every one that hath shall be given”. Matthew properties for incremental confirmation. Synthese, 184, 89–100.
Festa, R., & Cevolani, G. (forthcoming). Unfolding the grammar of Bayesian confirmation: Likelihood and anti-likelihood principles. Philosophy of Science.
Fitelson, B. (1999). The plurality of Bayesian measures of confirmation and the problem of measure sensitivity. Philosophy of Science, 66, S362–S378.
Fitelson, B. (2008). A decision procedure for probability calculus with applications. Review of Symbolic Logic, 1, 111–125.
Hajek, A., & Joyce, J. (2008). Confirmation. In S. Psillos & M. Curd (Eds.), The Routledge companion to philosophy of science (pp. 115–128). London: Routledge.
Joyce, J. (1999). The foundations of causal decision theory. Cambridge: Cambridge University Press.
Joyce, J. (2008). Bayes’ theorem. In E. Zalta (Ed.), The Stanford encyclopedia of philosophy (Fall 2008 ed.). http://plato.stanford.edu/archives/fall2008/entries/bayes-theorem/.
Kuipers, T. (2000). From instrumentalism to constructive realism: On some relations between confirmation, empirical progress, and truth approximation. Dordrecht: Kluwer.
Milne, P. (1996). log[P(h/eb)/P(h/b)] is the one true measure of confirmation. Philosophy of Science, 63, 21–26.
Popper, K. (1954). Degree of confirmation. British Journal for the Philosophy of Science, 5, 143–149.
Popper, K. (1983). Realism and the aim of science. London: Routledge.
Roche, W. (2012). Transitivity and intransitivity in evidential support: Some further results. Review of Symbolic Logic, 5, 259–268.
Roche, W. (2014). A note on confirmation and Matthew properties. Logic & Philosophy of Science, XII, 91–101.
Roche, W. (2015). Evidential support, transitivity, and screening-off. Review of Symbolic Logic, 8, 785–806.
Roche, W. (2016). Confirmation, increase in probability, and partial discrimination: A reply to Zalabardo. European Journal for Philosophy of Science, 6, 1–7.
Roche, W., & Shogenji, T. (2014). Confirmation, transitivity, and Moore: The screening-off approach. Philosophical Studies, 168, 797–817.
Roush, S. (2005). Tracking truth: Knowledge, evidence, and science. Oxford: Oxford University Press.
Rowbottom, D. (2011). Popper’s critical rationalism: A philosophical investigation. New York: Routledge.
Sober, E. (2015). Ockham’s razors: A user’s manual. Cambridge: Cambridge University Press.
Sprenger, J. (forthcoming). Two impossibility results for measures of corroboration. British Journal for the Philosophy of Science.
Acknowledgements
I am indebted to two anonymous referees for very detailed and helpful comments. Their efforts are very much appreciated and helped to significantly improve the paper on many fronts.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1
Is PDe distinct from IP? Suppose that:
It follows by IP3 that \(c(H_{1},\, E_{1}) < c(H_{2},\, E_{2})\). It follows by PDe3, in contrast, that \(c(H_{1},\, E_{1}) > c(H_{2},\, E_{2})\). Hence PDe is distinct from IP. \(\square \)
Appendix 2
Is PE distinct from IP and PDe? Suppose that:
It follows both by IP3 and by PDe3 that \(c(H_{1},\, E_{1}) \ne c(H_{2},\, E_{2})\). But since \(p(H_{1} \,{\vert }\, E_{1}) = 1> p(H_{1})\) and \(p(H_{2} \,{\vert }\,E_{2}) = 1\,> p(H_{2})\), it follows by PE4, in contrast, that \(c(H_{1},\, E_{1})=c(H_{2},\, E_{2})\). Hence PE is distinct from IP and PDe. \(\square \)
Appendix 3
Is PDi distinct from IP, PDe, and PE? Suppose, first, that:
It follows by IP3 that \(c(H_{1},\, E_{1}) < c(H_{2},\, E_{2})\) and by PE4 that \(c(H_{1},\, E_{1})=c(H_{2},\, E_{2})\). It follows by PDi2, in contrast, that \(c(H_{1},\, E_{1}) > c(H_{2},\, E_{2})\). Suppose, second, that:
It follows by PDe2 that \(c(H_{1},\, E_{1}) < c(H_{2},\, E_{2})\). It follows by PDi3, in contrast, that \(c(H_{1},\, E_{1}) > c(H_{2},\, E_{2})\). Hence PDi is distinct from IP, PDe, and PE. \(\square \)
Appendix 4
Is PC distinct from IP, PDe, PE, and PDi? Suppose, first, that \(E_{1}\) entails \(\lnot H_{1}\), that \(E_{2}\) entails \(\lnot H_{2}\), and that:
It follows by IP3 that \(c(H_{1},\, E_{1}) > c(H_{2},\, E_{2})\). It follows by PC2, in contrast, that \(c(H_{1},\, E_{1}) = -1 = c(H_{2},\, E_{2})\). Suppose, second, that:
It follows by PDe1 that \(c(H_{1},\, E_{1})=c(H_{2},\, E_{2})\), by PE4 that \(c(H_{1},\, E_{1})=c(H_{2},\, E_{2})\), and by PDi1 that \(c(H_{1},\, E_{1})=c(H_{2},\, E_{2})\). It follows by PC2, in contrast, that \(c(H_{1},\, E_{1}) = 0.5 < 0.51 = c(H_{2},\, E_{2})\). Hence PC is distinct from IP, PDe, PE, and PDi. \(\square \)
Rights and permissions
About this article
Cite this article
Roche, W. Is there a place in Bayesian confirmation theory for the Reverse Matthew Effect?. Synthese 195, 1631–1648 (2018). https://doi.org/10.1007/s11229-016-1286-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-016-1286-7