# Lewis’ Triviality for Quasi Probabilities

## Abstract

According to Stalnaker’s Thesis (S), the probability of a conditional is the conditional probability. Under some mild conditions, the thesis trivialises probabilities and conditionals, as initially shown by David Lewis. This article asks the following question: does (S) still lead to triviality, if the probability function in (S) is replaced by a probability-like function? The article considers plausibility functions, in the sense of Friedman and Halpern, which additionally mimic probabilistic additivity and conditionalisation. These quasi probabilities comprise Friedman–Halpern’s conditional plausibility spaces, as well as other known representations of conditional doxastic states. The paper proves Lewis’ triviality for quasi probabilities and discusses how this has implications for three other prominent strategies to avoid Lewis’ triviality: (1) Adams’ thesis, where the probability function on the left in (S) is replaced by a probability-like function, (2) abandoning conditionalisation, where probability conditionalisation on the right in (S) is replaced by another propositional update procedure and (3) the approximation thesis, where equality in (S) is replaced by approximation. The paper also shows that Lewis’ triviality result is really about ‘additiveness’ and ‘conditionality’.

## Keywords

Probability of a conditional Conditional probability Triviality Plausibility measures Conditional plausibility space Stalnaker thesis Adams’ thesis Conditional valuation functions## Notes

### Acknowledgements

I would like to thank Alan Hájek for encouraging me to publish these ideas, Arno Göbel for several discussions on this topic, Niels Skovgaard-Olsen for helpful comments, the people present at the European Epistemology Workshop 2016 in Paris and colleagues from the DFG-funded ‘What-if?’ research group in Konstanz for their comments, as well as two anonymous referees who have significantly contributed to improving the quality of the paper.

## References

- Adams, E. (1975).
*The logic of conditionals*. Dordrecht: Reidel.CrossRefGoogle Scholar - Adams, E. (1998).
*A primer of probability logic*. Stanford, CA: CSLI, Stanford University.Google Scholar - Arló-Costa, H. (1999). Belief revision conditionals: Basic iterated systems.
*Annals of Pure and Applied Logic*,*96*, 3–28.CrossRefGoogle Scholar - Bennett, J. (2003).
*A philosophical guide to conditionals*. New York: Oxford.CrossRefGoogle Scholar - Charlow, N. (2016). Triviality for restrictor conditionals.
*Noûs*,*50*(3), 533–564.CrossRefGoogle Scholar - Darwiche, A., & Ginsberg, M. L. (1992). A symbolic generalization of probability theory. In
*Proceedings of the national conference on artificial intelligence AAAI’92*(pp. 622–627). Menlo Park, CA: AAAI Press.Google Scholar - Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping.
*The Annals of Mathematical Statistics*,*38*(2), 325–339.CrossRefGoogle Scholar - Dietz, R., & Douven, I. (2011). A puzzle about Stalnaker’s hypothesis.
*Topoi*,*30*(1), 31–37.CrossRefGoogle Scholar - Douven, I. (2016).
*The epistemology of indicative conditionals: Combining formal and empirical approaches*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Douven, I., & Verbrugge, S. (2010). The Adams family.
*Cognition*,*117*, 302–318.CrossRefGoogle Scholar - Douven, I., & Verbrugge, S. (2013). The probabilities of conditionals revisited.
*Cognitive Science*,*37*, 711–730.CrossRefGoogle Scholar - Dubois, D., & Prade, H. (1988).
*Possibility theory: An approach to computerized processing of uncertainty*. New York: Plenum Press.CrossRefGoogle Scholar - Evans, J., Handley, S., & Over, D. (2003). Conditionals and conditional probability.
*Journal of Experimental Psychology: Learning, Memory and Cognition*,*29*, 321–355.Google Scholar - Evans, J. S. B. T., & Over, D. (2004).
*If*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Franke, M., & de Jager, T. (2010). Now that you mention it: Awareness dynamics in discourse and decisions. In A. Benz, et al. (Eds.),
*Language, games, and evolution. LNAI 6207*(pp. 60–91). Berlin: Springer.Google Scholar - Friedman, N., & Halpern, J. Y. (1995). Plausibility measures: A user’s guide. In
*Proceedings of the eleventh conference on uncertainty in artificial intelligence UAI’95*(pp. 175–184). San Francisco, CA: Morgan Kaufmann.Google Scholar - Fugard, A., Pfeifer, N., Mayerhofer, B., & Kleiter, G. (2011). How people interpret conditionals.
*Journal of Experimental Psychology: Learning, Memory, and Cognition*,*37*, 635–648.Google Scholar - Gärdenfors, P. (1982). Imaging and conditionalization.
*The Journal of Philosophy*,*79*(12), 747–760.CrossRefGoogle Scholar - Gärdenfors, P. (1986). Belief revisions and the Ramsey test for conditionals.
*The Philosophical Review*,*95*, 81–93.CrossRefGoogle Scholar - Gärdenfors, P. (1987). Variations on the Ramsey test: More triviality results.
*Studia Logica*,*46*(4), 321–327.CrossRefGoogle Scholar - Gärdenfors, P. (1988).
*Knowledge in flux. Modeling the dynamics of epistemic states*. Cambridge, MA: MIT Press.Google Scholar - Hájek, A. (1989). Probabilities of conditionals: Revisited.
*Journal of Philosophical Logic*,*18*, 423–428.CrossRefGoogle Scholar - Hájek, A. (1994). Triviality on the cheap? In E. Eells & B. Skyrms (Eds.),
*Probability and conditionals*(pp. 113–140). Cambridge: Cambridge University Press.Google Scholar - Hájek, A. (2012). The fall of ‘Adams’ thesis’?
*Journal of Language, Logic and Information*,*21*(2), 145–161.CrossRefGoogle Scholar - Hájek, A., & Hall, N. (1994). The hypothesis of the conditional construal of conditional probability. In E. Eells & B. Skyrms (Eds.),
*Probability and conditionals*(pp. 75–111). Cambridge: Cambridge University Press.Google Scholar - Jackson, F. (1987).
*Conditionals*. Oxford: Blackwell.Google Scholar - Kaufmann, S. (2009). Conditionals right and left: Probabilities for the whole family.
*Journal of Philosophical Logic*,*38*, 1–53.CrossRefGoogle Scholar - Kaufmann, S. (2015). Conditionals, conditional probabilities, and conditionalization. In H.-C. Schmitz & H. Zeevat (Eds.),
*Bayesian natural language semantics and pragmatics*(pp. 71–94). Berlin: Springer.CrossRefGoogle Scholar - Kern-Isberner, G. (2004). A thorough axiomatization of a principle of conditional preservation in belief revision.
*Annals of Mathematics and Artificial Intelligence*,*40*(1–2), 127–164.CrossRefGoogle Scholar - Khoo, J. (2013). Conditionals, indeterminacy, and triviality.
*Philosophical Perspectives*,*27*(1), 260–287.CrossRefGoogle Scholar - Khoo, J. (2016). Probabilities of conditionals in context.
*Linguistics & Philosophy*,*39*(1), 1–43.CrossRefGoogle Scholar - Khoo, J. (1991). Modality. In A. von Stechow & D. Wunderlich (Eds.),
*Semantics: An international handbook of contemporary research*(pp. 639–650). Berlin: de Gruyter.Google Scholar - Leitgeb, H. (2010). On the Ramsey test without triviality.
*Notre Dame Journal of Formal Logic*,*51*(1), 21–54.CrossRefGoogle Scholar - Levi, I. (1967). Probability kinematics.
*British Journal for the Philosophy of Science*,*18*, 197–209.CrossRefGoogle Scholar - Lewis, D. (1976). Probabilities of conditionals and conditional probabilities.
*Philosophical Review*,*85*, 297–315.CrossRefGoogle Scholar - Lewis, D. (1986). Probabilities of conditionals and conditional probabilities II.
*Philosophical Review*,*95*, 581–589.CrossRefGoogle Scholar - Meacham, C. J. G. (2016). Ur-priors, conditionalization and ur-prior conditionalization.
*Ergo*,*3*(17), 444–492.Google Scholar - Milne, P. (2003). The simplest Lewis-style triviality proof yet?
*Analysis*,*63*(4), 300–303.CrossRefGoogle Scholar - Morgan, C. G. (1999). Conditionals, comparative probability, and triviality: The conditional of conditional probability cannot be represented in the object language.
*Topoi*,*18*, 97–116.CrossRefGoogle Scholar - Morgan, C. G., & Mares, E. D. (1995). Conditionals, probability, and non-triviality.
*Journal of Philosophical Logic*,*24*(5), 455–467.CrossRefGoogle Scholar - Oaksford, M., & Chater, N. (2007).
*Bayesian rationality: The probabilistic approach to human reasoning*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Over, D. (2009). New paradigm psychology of reasoning.
*Thinking and Reasoning*,*15*, 431–438.CrossRefGoogle Scholar - Over, D., Hadjichristidis, C., Evans, J., Handley, S., & Sloman, S. (2007). The probability of causal conditionals.
*Cognitive Psychology*,*54*, 62–97.CrossRefGoogle Scholar - Politzer, G., Over, D., & Baratgin, J. (2010). Betting on conditionals.
*Thinking and Reasoning*,*16*, 172–197.CrossRefGoogle Scholar - Popper, K. R. (1955). Two autonomous axiom systems for the calculus of probabilities.
*British Journal for the Philosophy of Science*,*6*, 51–57.CrossRefGoogle Scholar - Raidl, E. (2018). Open-minded orthodox Bayesianism by Epsilon-conditionalisation.
*The British Journal for the Philosophy of Science*. https://doi.org/10.1093/bjps/axy075. - Ramsey, F. P. (1926). Truth and probability. In R. B. Braithwaite (Ed.),
*The foundations of mathematics and other logical essays*(pp. 156–198). London: Kegan, Paul, Trench, Trubner.Google Scholar - Rehder, W. (1982). Conditions for probabilities of conditionals to be conditional probabilities.
*Synthese*,*53*, 439–443.CrossRefGoogle Scholar - Rott, H. (2011). Reapproaching Ramsey: Conditionals and iterated belief change in the spirit of AGM.
*Journal of Philosophical Logic*,*40*, 155–191.CrossRefGoogle Scholar - Shafer, G. (1976).
*A mathematical theory of evidence*. Princeton: Princeton University Press.Google Scholar - Skovgaard-Olsen, N., Kellen, D., Krahl, H., & Klauer, K. C. (2017). Relevance differently affects the truth, acceptability, and probability evaluations of ‘and’, ‘but’, ‘therefore’, and ‘if then’.
*Thinking and Reasoning*,*23*(4), 449–482.CrossRefGoogle Scholar - Skovgaard-Olsen, N., Singmann, H., & Klauer, K. C. (2016). The relevance effect and conditionals.
*Cognition*,*150*, 26–36.CrossRefGoogle Scholar - Stalnaker, R. (1968).
*A theory of conditionals. Studies in logical theory, American Philosophical Quarterly Monograph series*(Vol. 2). Oxford: Blackwell.Google Scholar - Stalnaker, R. (1970). Probability and conditionals.
*Philosophy of Science*,*37*, 64–80.CrossRefGoogle Scholar - Stalnaker, R. (1975). Indicative conditionals. In W. L. Harper, R. Stalnaker, & G. Pearce (Eds.),
*Ifs: Conditionals, belief, decision, chance and time. The University of Western Ontario series in philosophy of science*(Vol. 15, pp. 193–210). Dordrecht: Springer.Google Scholar - Stalnaker, R. (1976). Letter to van Fraassen. In W. L. Harper & C. A. Hooker (Eds.),
*Foundations of probability theory, statistical inference and statistical theories of science*(Vol. I, pp. 302–306). Dordrecht: Reidel.Google Scholar - Stalnaker, R. (2012).
*The laws of belief: Ranking theory and its philosophical applications*. Oxford: Oxford University Press.Google Scholar - Stalnaker, R. (2014).
*Context*. Oxford: Oxford University Press.CrossRefGoogle Scholar - Stalnaker, R. (2015). Conditionals: A unifying ranking-theoretic perspective.
*Philosopher’s Imprint*,*15*(1), 1–30.Google Scholar - Stalnaker, R., & Jeffrey, R. (1994). Conditionals as random variables. In E. Eells & B. Skyrms (Eds.),
*Probabilities and conditionals: Belief revision and rational decision*(pp. 31–46). Cambridge: Cambridge University Press.Google Scholar - Spohn, W. (1988). Ordinal conditional functions: A dynamic theory of epistemic states. In W. L. Harper & B. Skyrms (Eds.),
*Causation in decision, belief change, and statistics*(Vol. 2, pp. 105–134). Dordrecht: Kluwer.CrossRefGoogle Scholar - Van Fraassen, B. (1976). Probabilities of conditionals. In W. L. Harper & C. A. Hooker (Eds.),
*Foundations of probability theory, statistical inference, and statistical theories of science*(Vol. I, pp. 261–301). Dordrecht: Reidel.CrossRefGoogle Scholar - Wenmackers, S., & Romeijn, J.-W. (2016). New theory about old evidence.
*A framework for open-minded Bayesianism. Synthese*,*193*(4), 1225–1250.Google Scholar - Weydert, E. (1994). General belief measures. In R. López de Mántara & D. Poole (Eds.),
*Proceedings of the tenth conference on uncertainty in artificial intelligence, UAI ’94*(pp. 575–582). San Francisco: Morgan Kaufmann.Google Scholar - Williams, J. R. G. (2012). Counterfactual triviality: A Lewis-impossibility argument for counterfactuals.
*Philosophy and Phenomenological Research*,*85*, 648–670.CrossRefGoogle Scholar - Williamson, T. (1998). Conditionalizing on knowledge.
*British Journal for the Philosophy of Science*,*49*, 89–121.CrossRefGoogle Scholar