# A weak symmetry condition for probabilistic measures of confirmation

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## Abstract

This paper presents a symmetry condition for probabilistic measures of confirmation which is weaker than commutativity symmetry, disconfirmation commutativity symmetry but also antisymmetry. It is based on the idea that for any value a probabilistic measure of confirmation can assign there is a corresponding case where degrees of confirmation are symmetric. It is shown that a number of prominent confirmation measures such as Carnap’s difference function, Rescher’s measure of confirmation, Gaifman’s confirmation rate and Mortimer’s inverted difference function do not satisfy this condition and instead exhibit a previously unnoticed and rather puzzling behavior in certain cases of disconfirmation. This behavior also carries over to probabilistic measures of information change, causal strength, explanatory power and coherence.

## Keywords

Symmetry conditions (Probabilistic measures of) Confirmation (Probabilistic measures of) Information change (Probabilistic measures of) Causal strength (Probabilistic measures of) Explanatory power (Probabilistic measures of) Coherence## Notes

### Acknowledgements

I would like to thank Vincenzo Crupi, Igor Douven and Michael Schippers for helpful comments and discussion. This work was supported by Grant SI 1731/1-1 to Mark Siebel from the DFG as part of the priority program *New Frameworks of Rationality* and by Grant SCHU 3080/3-1 to Moritz Schulz from the DFG as part of the Emmy-Noether-Group *Knowledge and Decisions*.

## References

- Carnap, R. (1950).
*Logical foundations of probability*. Chicago: University of Chicago Press.Google Scholar - Christensen, D. (1999). Measuring confirmation.
*Journal of Philosophy*,*96*, 437–461.CrossRefGoogle Scholar - Crupi, V. (2014). Confirmation. In E. N. Zalta (Ed.),
*The Stanford Encyclopedia of Philosophy*(Fall 2014 edition).Google Scholar - Crupi, V., & Tentori, K. (2014). State of the field: Measuring information and confirmation.
*Studies in History and Philosophy of Science Part A*,*47*, 81–90.CrossRefGoogle Scholar - Crupi, V., Tentori, K., & Gonzales, M. (2007). On Bayesian measures of evidential support: Theoretical and empirical issues.
*Philosophy of Science*,*74*, 229–252.CrossRefGoogle Scholar - Douven, I., & Schupbach, J. N. (2015). Probabilistic alternatives to Bayesianism: The case of explanationism.
*Frontiers in Psychology*,*6*, 459.CrossRefGoogle Scholar - Eells, E., & Fitelson, B. (2002). Symmetries and asymmetries in evidential support.
*Philosophical Studies*,*107*, 129–142.CrossRefGoogle Scholar - Fitelson, B. (2006). Logical foundations of evidential support.
*Philosophy of Science*,*73*(5), 500–512.CrossRefGoogle Scholar - Fitelson, B., & Hitchcock, C. (2011). Probabilistic measures of causal strength. In P. Illari, F. Russo, & J. Williamson (Eds.),
*Causality in the Sciences*(pp. 600–627). Oxford: Oxford University Press.CrossRefGoogle Scholar - Frankel, L. (1986). Mutual causation, simultaneity and event description.
*Philosophical Studies*,*49*(3), 361–372.CrossRefGoogle Scholar - Gaifman, H. (1979). Subjective probability, natural predicates and Hempel’s ravens.
*Erkenntnis*,*21*, 105–147.Google Scholar - Glass, D. H. (2014). Entailment and symmetry in confirmation measures of interestingness.
*Information Sciences*,*279*, 552–559.CrossRefGoogle Scholar - Greco, S., Słowiński, R., & Szczęch, I. (2012). Analysis of symmetry properties for Bayesian confirmation measures. In T. Li, H. S. Nguyen, G. Wang, J. Grzymala-Busse, R. Janicki, A. E. Hassanien & H. Yu (Eds.),
*Rough sets and knowledge technology. RSKT 2012. Lecture notes in computer science*(Vol. 7414, pp. 207–214). Heidelberg: Springer.Google Scholar - Hammersley, J. M., & Handscomb, D. C. (1964).
*Monte Carlo methods*. London: Methuen & Co Ltd.CrossRefGoogle Scholar - Hosiasson-Lindenbaum, J. (1940). On confirmation.
*Journal of Symbolic Logic*,*5*(4), 133–148.CrossRefGoogle Scholar - Kemeny, J., & Oppenheim, P. (1952). Degrees of factual support.
*Philosophy of Science*,*1952*, 307–324.CrossRefGoogle Scholar - Keynes, J. (1921).
*A treatise on probability*. London: Macmillan.Google Scholar - Kolmogorov, A. (1956).
*Foundations of the theory of probability*. New York: AMS Chelsea Publishing.Google Scholar - Matsumoto, M., & Nishimura, T. (1998). Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator.
*ACM Transactions on Modeling and Computer Simulation*,*8*(1), 3–30.CrossRefGoogle Scholar - Milne, P. (2014). Information, confirmation, and conditionals.
*Journal of Applied Logic*,*12*(3), 252–262.CrossRefGoogle Scholar - Mortimer, H. (1988).
*The logic of induction*. Paramus: Prentice Hall.Google Scholar - Nozick, R. (1981).
*Philosophical explanations*. Oxford: Clarendon.Google Scholar - Popper, K. R. (1954). Degree of confirmation.
*British Journal for the Philosophy of Science*,*5*, 143–149.CrossRefGoogle Scholar - Roche, W. (2013). Coherence and probability: A probabilistic account of coherence. In M. Araszkiewicz & J. Savelka (Eds.),
*Coherence: Insights from philosophy, jurisprudence and artificial intelligence*(pp. 59–91). Dordrecht: Springer.CrossRefGoogle Scholar - Schippers, M. (2014). Probabilistic measures of coherence: From adequacy constraints towards pluralism.
*Synthese*,*191*(16), 3821–3845.CrossRefGoogle Scholar - Schupbach, J. N., & Sprenger, J. (2011). The logic of explanatory power.
*Philosophy of Science*,*78*(1), 105–127.CrossRefGoogle Scholar - Suppes, P. (1970).
*A probabilistic theory of causality*. Amsterdam: North Holland.Google Scholar