## Abstract

According to a widespread but implicit thesis in Bayesian confirmation theory, two confirmation measures are considered equivalent if they are ordinally equivalent—call this the “ordinal equivalence thesis” (OET). I argue that adopting OET has significant costs. First, adopting OET renders one incapable of determining whether a piece of evidence substantially favors one hypothesis over another. Second, OET must be rejected if merely ordinal conclusions are to be drawn from the expected value of a confirmation measure. Furthermore, several arguments and applications of confirmation measures given in the literature already rely on a rejection of OET. I also contrast OET with stronger equivalence theses and show that they do not have the same costs as OET. On the other hand, adopting a thesis stronger than OET has costs of its own, since a rejection of OET ostensibly implies that people’s epistemic states have a very fine-grained quantitative structure. However, I suggest that the normative upshot of the paper in fact has a conditional form, and that other Bayesian norms can also fruitfully be construed as having a similar conditional form.

## Keywords

Bayesian confirmation Measurement theory Scales of measurement Confirmation measures Ordinal equivalence## References

- Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference.
*Journal of the Royal Statistical Society. Series B (Methodological)*,*41*(2), 113–147.CrossRefGoogle Scholar - Brössel, P., & Huber, F. (2014). Bayesian confirmation: A means with no end.
*The British Journal for the Philosophy of Science*,*66*, 737.CrossRefGoogle Scholar - Carnap, R. (1962).
*Logical foundations of probability*(2nd ed.). Chicago: University of Chicago Press.Google Scholar - Christensen, D. (1999). Measuring confirmation.
*Journal of Philosophy*,*XCVI*, 437–461.CrossRefGoogle Scholar - Crupi, V., & Tentori, K. (2014). Measuring information and confirmation.
*Studies in the History and Philosophy of Science Part A*,*47*, 81–90.CrossRefGoogle Scholar - Easwaran, K. (2016). Dr. Truthlove or: How I learned to stop worrying and love Bayesian probabilities.
*Nous*,*50*(4), 816–853.CrossRefGoogle Scholar - Fitelson, B. (1999). The plurality of Bayesian measures of confirmation and the problem of measure sensitivity.
*Philosophy of Science*,*66*, S362–S378.CrossRefGoogle Scholar - Fitelson, B. (2007). Likelihoodism, Bayesianism, and relational confirmation.
*Synthese*,*156*, 473–489.CrossRefGoogle Scholar - Fitelson, B., & Hawthorne, J. (2010). How Bayesian confirmation theory handles the paradox of the ravens. In E. Eells & J. Hawthorne (Eds.),
*The place of probability in science*. Boston Studies in the Philosophy of Science (Vol. 284, pp. 247–275). Dordrecht: Springer.Google Scholar - Gillies, D. (1986). In defense of the Popper–Miller argument.
*Philosophy of Science*,*53*, 110–113.CrossRefGoogle Scholar - Good, I. J. (1985). Weight of evidence: A brief survey. In J. M. Bernardo, M. H. DeGroot, D. V. Lindley, & A. F. M. Smith (Eds.),
*Bayesian statistics 2*(pp. 249–270). Amsterdam: Elsevier.Google Scholar - Joyce, J. (1999).
*The foundations of causal decision theory*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Kemeny, J. G., & Oppenheim, P. (1952). Degree of factual support.
*Philosophy of Science*,*19*(4), 307–324.CrossRefGoogle Scholar - Kullback, S., & Leibler, R. (1951). On information and sufficiency.
*Annals of Mathematical Statistics*,*22*(1), 79–86.CrossRefGoogle Scholar - Myrvold, W. (2003). A Bayesian account of the virtue of unification.
*Philosophy of Science*,*70*(2), 399–423.CrossRefGoogle Scholar - Myrvold, W. (2016). On the evidential import of unification. Unpublished manuscript.Google Scholar
- Popper, K., & Miller, D. (1983). The impossibility of inductive probability.
*Nature*,*302*, 687–688.CrossRefGoogle Scholar - Redhead, M. (1985). On the impossibility of inductive probability.
*The British Journal for the Philosophy of Science*,*36*(2), 185–191.CrossRefGoogle Scholar - Rinard, S. (2014). A new Bayesian solution to the paradox of the ravens.
*Philosophy of Science*,*81*(1), 81–100.CrossRefGoogle Scholar - Royall, R. (1997).
*Statistical evidence: A likelihood paradigm*. Boca Raton: Chapman and Hall/CRC.Google Scholar - Schlesinger, G. (1995). Measuring degrees of confirmation.
*Analysis*,*55*, 208–212.CrossRefGoogle Scholar - Shogenji, T. (2012). The degree of epistemic justification and the conjunction fallacy.
*Synthese*,*184*(1), 29–48.CrossRefGoogle Scholar - Stevens, S. S. (1946). On the theory of scales of measurement.
*Science*,*103*(2684), 577–580.CrossRefGoogle Scholar - Tentori, K., Crupi, V., & Osherson, D. (2007). Comparison of confirmation measures.
*Cognition*,*103*, 107–119.CrossRefGoogle Scholar - Vassend, O. B. (2015). Confirmation measures and sensitivity.
*Philosophy of Science*,*82*(5), 892–904.CrossRefGoogle Scholar - Vranas, P. (2004). Hempel’s raven paradox: A lacuna in the standard Bayesian solution.
*The British Journal for the Philosophy of Science*,*42*, 393–401.Google Scholar - Zalabardo, J. (2009). An argument for the likelihood ratio measure of confirmation.
*Analysis*,*69*, 630–635.CrossRefGoogle Scholar