Abstract
We examine some old and new paradoxes of probability, give a rough account of probabilistic conditionals, and prove a new result about non-monotonicity in probabilistic conditionals. It is well known that such conditionals are not monotonic – a conditional which is true can become false when additional hypotheses are added. We show that nonetheless, the conditionals are usually monotonic, or roughly speaking that we do not actually have to worry about non-monotonicity in practice.
Dedicated to Anil Nerode on his 75th birthday. Research supported in part by a grant from PSC-CUNY grants program. Versions of this paper were given in colloquia at Boston University, CUNY Graduate Center, and at IHPST in Paris.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, E.: Probability and the logic of conditionals. In: Suppes, P., Hintikka, J. (eds.) Aspects of Inductive Logic, pp. 265–316. North-Holland, Amsterdam (1968)
Costa, H.A., Parikh, R.: Conditional probability and defeasible inference. Journal of Philosophical Logic 34, 97–119 (2005)
Bradley, D., Leitgeb, H.: When betting odds and credences come apart: more worries for the Dutch book arguments. Analysis 66(2), 119–127 (2006)
The binomial theorem, http://www.stat.yale.edu/Courses/1997-98/101/binom.htm
Edgington, D.: On Conditionals. Mind 104, 235–329 (1995)
Elga, A.: Self-locating belief and the Sleeping Beauty problem. Analysis 60, 143–147 (2000)
de Finetti, B.: Foresight: its logical laws, its subjective sources (translation by Kyburg, original French version, 1937). In: Kyburg Jr., H.E., Smokler, H.E. (eds.) Studies in Subjective Probability, pp. 53–118. Krieger, Malabar (1980)
Gabbay, D.: Theoretical foundations for nonmonotonic reasoning in expert systems. In: Apt, K.R. (ed.) Proceedings NATO Advanced Study Institute on Logics and Models of Concurrent Systems, pp. 439–457 (1985)
Gnedenko, B.V.: Theory of Probability (translated from the Russian by Seckler, B.D.). Chelsea, New York (1962)
Hawthorne, J.: Nonmonotonic Conditionals that Behave Like Conditional Probabilities Above a Threshold. Journal of Applied Logic (May 2006)
Halpern, J.: Sleeping Beauty reconsidered. In: Gendler, T.S., Hawthorne, J. (eds.) Oxford Studies in Epistemology, pp. 111–142 (2005)
Jeffreys, H.: Theory of Probability, 3rd edn. Oxford University Press, New York (1961)
Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44, 167–207 (1990)
Levi, I.: On indetrminate probabilities. Journal of Philosophy 71, 391–418 (1974)
Lewis, D.: Probabilities of conditionals and conditional probabilities. Philosophical Review 85, 297–315 (1976)
Parikh, R.: Logical omniscience and common knowledge: WHAT do we know and what do WE know? In: Meyden, R. (ed.) Proceedings of the Tenth Conference on Theoretical Aspects of Rationality and Knowledge - 2005, pp. 62–78. National U. Singapore Press, Singapore (2005)
Parikh, R.: Review of: Sanford, D.: If P then Q (2003). Essays in Philosophy 7(1) (2006)
Ramsey, F.P.: Truth and probability (1926). Reprinted in: Mellor, D.H. (ed.) F.P. Ramsey, Philosophical Papers, Cambridge University Press, Cambridge (1990)
Schwarz, C.: Cumulative probability for the standard normal distribution, http://www.stat.sfu.ca/~cschwarz/Stat-301/Handouts/nod122.html
Sanford, D.: If P then Q, 2nd edn. Routledge, New York (2003)
Stroock, D.: Probability Theory: An analytic view. Cambridge U. Press, Cambridge (1993)
Weirich, P.: The St. Petersburg Gamble and Risk. Theory and Decision 17, 193–202 (1984)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Parikh, R. (2007). Some Puzzles About Probability and Probabilistic Conditionals . In: Artemov, S.N., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2007. Lecture Notes in Computer Science, vol 4514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72734-7_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-72734-7_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72732-3
Online ISBN: 978-3-540-72734-7
eBook Packages: Computer ScienceComputer Science (R0)