Skip to main content
Log in

Is there a place in Bayesian confirmation theory for the Reverse Matthew Effect?

  • Published:
Synthese Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 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).

  2. This approach is on display in Hajek and Joyce (2008) and Joyce (1999, Chap. 6, Sect. 6.4; 2008). A very different approach—a monistic approach—is on display in Milne (1996).

  3. 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).

  4. 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 \(c^*(H_{1},\, E_{1}) > / = / < c^*(H_{2},\,E_{2})\).

  5. 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.

  6. 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.

  7. This is also true with respect to (a) RME and conditions (P0), (P1), and (P3) in Crupi (2016, Sect. 3), (b) RME and conditions (IFPD), (FPI), (IPI), and (E) in Festa (2012, Sect. 2), and (c) RME and conditions P1, P2, and P3 in Festa and Cevolani (forthcoming, Sect. 2).

  8. 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 \(H_{2}\).

  9. It is not essential that \(p(H_{1} \,{\vert }\, E)\) equals unity on Distribution D. Any value very close to unity would suffice.

  10. 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.

  11. The same is true with respect to MI.

  12. The expression “partial dependence”, though, is mine. Joyce (2008, Sect. 3) speaks in terms of “effective evidence”. Hajek and Joyce (2008, p. 122) speak in terms of “probative evidence”.

  13. MI entails that \(c(H_{1},\, E)=c(H_{2},\, E)\). So PDe also runs counter to MI.

  14. The title of Crupi and Tentori (2013) is “Confirmation as partial entailment: A representation theorem in inductive logic”.

  15. MI entails that \(c(H_{1},\, E)=c(H_{2},\,E)\). So PE also runs counter to MI.

  16. See Roche (2016) for an argument to the effect that \(c_{8}\) (the so-called “likelihood measure”) is perhaps best understood as a measure of confirmation in the sense of partial discrimination. See also Roush (2005, Chap. 5).

  17. MI entails that \(c(H_{1},\, E)=c(H_{2},\, E)\). So PDi also runs counter to MI.

  18. 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 \(H_{2}\).

  19. 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.

  20. 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.

  21. MI entails that \(c(H_{1},\, E)=c(H_{2},\, E)\). So PC also runs counter to MI.

  22. 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 \(\frac{p(E\,|\,H)}{p(E)}\left( {1-p(H)} \right) +1\).

  23. 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Crupi, V., & Tentori, K. (2014). Erratum to “Confirmation as partial entailment”. Journal of Applied Logic, 12, 230–231.

    Article  Google Scholar 

  • Díez, J. (2011). On Popper’s strong inductivism (or strongly inconsistent anti-inductivism). Studies in History and Philosophy of Science, 42, 105–116.

    Article  Google Scholar 

  • Douven, I. (2011). Further results on the intransitivity of evidential support. Review of Symbolic Logic, 4, 487–497.

    Article  Google Scholar 

  • Festa, R. (2012). “For unto every one that hath shall be given”. Matthew properties for incremental confirmation. Synthese, 184, 89–100.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Fitelson, B. (2008). A decision procedure for probability calculus with applications. Review of Symbolic Logic, 1, 111–125.

    Article  Google Scholar 

  • Hajek, A., & Joyce, J. (2008). Confirmation. In S. Psillos & M. Curd (Eds.), The Routledge companion to philosophy of science (pp. 115–128). London: Routledge.

    Google Scholar 

  • Joyce, J. (1999). The foundations of causal decision theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • 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.

    Book  Google Scholar 

  • Milne, P. (1996). log[P(h/eb)/P(h/b)] is the one true measure of confirmation. Philosophy of Science, 63, 21–26.

    Article  Google Scholar 

  • Popper, K. (1954). Degree of confirmation. British Journal for the Philosophy of Science, 5, 143–149.

    Article  Google Scholar 

  • Popper, K. (1983). Realism and the aim of science. London: Routledge.

    Google Scholar 

  • Roche, W. (2012). Transitivity and intransitivity in evidential support: Some further results. Review of Symbolic Logic, 5, 259–268.

    Article  Google Scholar 

  • Roche, W. (2014). A note on confirmation and Matthew properties. Logic & Philosophy of Science, XII, 91–101.

    Google Scholar 

  • Roche, W. (2015). Evidential support, transitivity, and screening-off. Review of Symbolic Logic, 8, 785–806.

    Article  Google Scholar 

  • Roche, W. (2016). Confirmation, increase in probability, and partial discrimination: A reply to Zalabardo. European Journal for Philosophy of Science, 6, 1–7.

    Article  Google Scholar 

  • Roche, W., & Shogenji, T. (2014). Confirmation, transitivity, and Moore: The screening-off approach. Philosophical Studies, 168, 797–817.

    Article  Google Scholar 

  • Roush, S. (2005). Tracking truth: Knowledge, evidence, and science. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Rowbottom, D. (2011). Popper’s critical rationalism: A philosophical investigation. New York: Routledge.

    Google Scholar 

  • Sober, E. (2015). Ockham’s razors: A user’s manual. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Sprenger, J. (forthcoming). Two impossibility results for measures of corroboration. British Journal for the Philosophy of Science.

Download references

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

Authors

Corresponding author

Correspondence to William Roche.

Appendices

Appendix 1

Is PDe distinct from IP? Suppose that:

$$\begin{aligned} p\left( {H_1 \,|\,E_1 } \right)= & {} 0.99>p\left( {H_1 } \right) =0.98>p(H_1 \,|\,\lnot E_1 )=0.01\\ p\left( {H_2 \,|\,E_2 } \right)= & {} 0.99>p\left( {H_2 } \right) =0.03>p(H_2 \,|\,\lnot E_2 )=0.02 \end{aligned}$$

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:

$$\begin{aligned} p\left( {H_1 \,|\,E_1 } \right)= & {} 1>p\left( {H_1 } \right) =0.98>p(H_1 \,|\,\lnot E_1 )=0.01\\ p\left( {H_2 \,|\,E_2 } \right)= & {} 1>p\left( {H_2 } \right) =0.03>p(H_2 \,|\,\lnot E_2 )=0.02 \end{aligned}$$

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:

$$\begin{aligned}&\displaystyle p\left( {H_1 \,|\,E_1 } \right) =1=p\left( {H_2 \,|\,E_2 } \right)&\\&\displaystyle p\left( {H_1 } \right) =0.5>0.49=p\left( {H_2 } \right)&\\&\displaystyle p\left( {E_1 \,|\,H_1 } \right) =1>0.25=p\left( {E_2 \,|\,H_2 } \right)&\\&\displaystyle p(E_1 \,|\,\lnot H_1 )=0=p(E_2 \,|\,\lnot H_2 )&\end{aligned}$$

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:

$$\begin{aligned}&\displaystyle p\left( {H_1 \,|\,E_1 } \right) =0.01<0.99=p\left( {H_2 \,|\,E_2 } \right)&\\&\displaystyle p(H_1 \,|\,\lnot E_1 )=0=p(H_2 \,|\,\lnot E_2 )&\\&\displaystyle p\left( {E_1 \,|\,H_1 } \right) =1=p\left( {E_2 \,|\,H_2 } \right)&\\&\displaystyle p(E_1 \,|\,\lnot H_1 )=0.001<0.002=p(E_2 \,|\,\lnot H_2 )&\end{aligned}$$

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:

$$\begin{aligned} p\left( {H_1 } \right) =1/101<49/2599=p\left( {H_2 } \right) \end{aligned}$$

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:

$$\begin{aligned}&\displaystyle p\left( {H_1 \,|\,E_1 } \right) =1=p\left( {H_2 \,|\,E_2 } \right)&\\&\displaystyle p\left( {H_1 } \right) =0.5>0.49=p\left( {H_2 } \right)&\\&\displaystyle p(H_1 \,|\,\lnot E_1 )=0=p(H_2 \,|\,\lnot E_2 )&\\&\displaystyle p\left( {E_1 \,|\,H_1 } \right) =1=p\left( {E_2 \,|\,H_2 } \right)&\\&\displaystyle p(E_1 \,|\,\lnot H_1 )=0=p(E_2 \,|\,\lnot H_2 )&\end{aligned}$$

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-016-1286-7

Keywords

Navigation