The Logic of Bayesian Probability
 Colin Howson
 … show all 1 hide
Abstract
For the last eighty or so years it has been generally accepted that the theory of Bayesian probability is a theory of partial belief subject to rationality constraints. There is also a virtual consensus that both the measure of belief and the constraints to which it is subject can only be provided via utility theory. It is easy to see why this should be so. The underlying idea, accepted initially by both de Finetti and Ramsey in their seminal papers ([1964] and [1931] respectively, though the paper 1964, first published in 1937, built on earlier work), but going back at least as far as Bayes’ Memoir [1763], is that an agent’s degree of belief in or uncertainty about a proposition A can be assessed by their rate of substitution of a quantity of value for a conditional benefit [S if A is true, 0 if not]. The natural medium of value is, of course, money, but the obvious difficulties with sensitivity to loss and the consequent diminishing marginal value of money seem to lead, apparently inexorably, to the need to develop this idea within an explicit theory of utility. This was first done only in this century, by Ramsey [1931]; today it is customary to follow Savage [1954] and show that suitable axioms for preference determine a reflexive and transitive ordering ‘at least as probable as’ and thence, given a further assumption about how finely the state space can be partitioned, a unique probability function.
 Aczel, 1994] P. Aczel. Schematic consequence. In [Gabbay, 1994, pp. 261–273].
 E. W. Adams. The Logic of Conditionals, Reidel, Dordrecht, 1975. CrossRef
 E. W. Adams. A Primer of Probability Logic, CSLI, Stanford, 1998.
 Albert, 2001] M. Albert. Bayesian learning and expectations formation: anything goes. This volume, pp. 347–368.
 F. J. Anscombe and R. J. Aumann. A definition of subjective probability, Annals of Mathematical Statistics, 34, 199–205, 1963. CrossRef
 T. Bayes. An essay towards solving a problem in the doctrine of chances, Philosophical Transactions of the Royal Society of London, 1763.
 J. Bernoulli. Ars Conjectandi, Basel, 1715.
 B. Bolzano. Theory of Science, 1850
 R. Carnap. A basic system of inductive logic. In Studies in Inductive Logic and Probability, Volume I, R. Carnap and R. C. Jeffrey, eds. pp. 33–167. University of California Press, 1971.
 R. Carnap. The Logical Foundations of Probability, Chicago: University of Chicago Press, 1950.
 L. Couturat. La Logique de Leibniz, Paris, 1901.
 de Finetti, 1964] B. de Finetti. Foresight; its Logical Laws, its Subjective Sources’, Studies in Subjective Probability,H. Kyburg and H. Smokier, eds. pp. 93–159. Wiley, 1964. (de Finetti’s paper was published originally in 1937 in French.)
 R. Fagin and J. Y. Halpern. Reasoning about Knowledge and Probability: Preliminary Report. In Proceedings of the Second Conference on Theoretical Aspects of Reasoning about Knowledge, M. Y. Vardi, ed. pp. 277–293. Morgan Kaufmann, 1988.
 D. M. Gabbay, ed. What is a Logical System?, Oxford: Oxford University Press, 1994
 Gabbay, 1994a] D. M. Gabbay. What is a logical system? In [Gabbay, 1994, pp. 179–217].
 H. Gaifman. Concerning measures in first order calculi. Israel Journal of Mathematics, 2, 1–18, 1964. CrossRef
 Hacking, 1994] I. Hacking. What is logic? In [Gabbay, 1994, pp. 1–35].
 P. Halmos. Lectures on Boolean Algebras, Van Nostrand, Princeton„ 1963.
 Heifetz and Mongin, forthcoming] A. Heifetz and P. Mongin. The Modal Logic of Probability, forthcoming.
 G. Hellman. Bayes and beyond. Philosophy of Science, 64, 190–205, 1997.
 W. Hodges. Logic, Harmondsworth: Penguin Books, 1974.
 C. Howson. Logic and probability’, British Journal for the Philosophy of Science, 48, 517–531, 1997.
 C. Howson. Logic With Trees, London: Routledge, 1997.
 C. Howson. Hume’s Problem: Induction and the Justification of Belief, Oxford: Oxford University Press, 2000.
 C. Howson and P. Urbach. Scientific Reasoning: the Bayesian Approach, 2nd edition, Chicago: Open Court, 1993.
 R. I. G. Hughes. Rationality and intransitive preferences, Analysis, 40, 132–134, 1980.
 M. Kac and S. Ulam. Mathematics and Logic, New York: Dover, 1968. [Kolmogorov, 1956 ] A. N. Kolmogorov. Foundations of the Theory of Probability, New York: Chelsea, 1956.
 I. Lakatos. Falsification and the methodology of scientific research programmes. In Criticism and the Growth of Knowledge, I. Lakatos and A. Musgrave, eds. pp. 91–197. Cambridge: Cambridge University Press, 1970.
 D. Lewis. Probabilities of conditionals and conditional probabilities. Philosophical Review, vol. LXXXV, 297–315, 1973.
 P. M. Milne. Bruno de Finetti and the logic of conditional events. British Journal for the Philosophy of Science, 48, 195–233, 1997. CrossRef
 J. Paris. The Uncertain Reasoner’s Companion. A Mathematical Perspective, Cambridge: Cambridge University Press, 1994.
 S.D. Poisson. Recherches sur la probabilité des jugements en matière civile et en matière criminelle, Paris, 1823.
 F. P. Ramsey. Truth and probability. In The Foundations of Mathematics, R.B. Braithwaite, ed. London: Kegan Paul, 1931.
 L. J. Savage. The Foundations of Statistics, New York: Wiley, 1954.
 D. Scott and P. Krauss. Assigning probabilities to logical formulas. In Aspects of Inductive Logic, J. Hintikka and P. Suppes, eds. pp. 219–264, 1970.
 A. Shimony. Coherence and the axioms of confirmation. Journal of Symbolic Logic, 20, 1–28, 1955. CrossRef
 R. Smullyan. First Order Logic, New York: Dover, 1968. CrossRef
 P. Teller. Conditionalisation and Observation, Synthese, 26, 218–58, 1973. CrossRef
 I. Todhunter. A History of the Mathematical Theory of Probability, Cambridge and London, 1865.
 P. Walley. Statistical Reasoning with Imprecise Probabilities, London: Chapman and Hall, 1991.
 J. Williamson. Probability logic. In ‘Handbook of the Logic of Inference and Argument: The Turn Toward the Practical’, D. Gabbay, R. Johnson, H.J. Ohlbach and J. Woods, eds. Elsevier, 2001.
 Title
 The Logic of Bayesian Probability
 Book Title
 Foundations of Bayesianism
 Book Part
 Part II
 Pages
 pp 137159
 Copyright
 2001
 DOI
 10.1007/9789401715867_6
 Print ISBN
 9789048159208
 Online ISBN
 9789401715867
 Series Title
 Applied Logic Series
 Series Volume
 24
 Series ISSN
 13862790
 Publisher
 Springer Netherlands
 Copyright Holder
 Springer Science+Business Media B.V.
 Additional Links
 Topics
 eBook Packages
 Editors

 David Corfield ^{(2)}
 Jon Williamson ^{(2)}
 Editor Affiliations

 2. Department of Philosophy, King’s College
 Authors

 Colin Howson ^{(3)}
 Author Affiliations

 3. Department of Philosophy, Logic and Scientific Method, London School of Economics, London, UK
Continue reading...
To view the rest of this content please follow the download PDF link above.