Abstract
In his Bayesian Nets and Causality, Jon Williamson presents an ‘Objective Bayesian’ interpretation of probability, which he endeavours to distance from the logical interpretation yet associate with the subjective interpretation. In doing so, he suggests that the logical interpretation suffers from severe epistemological problems that do not affect his alternative. In this paper, I present a challenge to his analysis. First, I closely examine the relationship between the logical and ‘Objective Bayesian’ views, and show how, and why, they are highly similar. Second, I argue that the logical interpretation is not manifestly inferior, at least for the reasons that Williamson offers. I suggest that the key difference between the logical and ‘Objective Bayesian’ views is in the domain of the philosophy of logic; and that the genuine disagreement appears to be over Platonism versus nominalism (within weak psychologism).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This passage is from the call for papers for PROGIC 2005, at which I presented a much earlier version of this paper. URL: http://www.kent.ac.uk/secl/philosophy/jw/2005/progic/ (Accessed 30/06/08).
Note that Popper also endorsed a logical interpretation, despite his anti-inductivism. He accepted ‘the probability calculus as a generalisation of ordinary logic’ (Popper 1983, p. 293), yet argued that degrees of confirmation should not be confused with probabilities—p(a,b) ≠ C(a,b). For more on this, see Rowbottom 2008.
Note also that Keynes (1921, p. 14) earlier says, in another passage which his critics appear to overlook, ‘Of which logical ideas and relations we have direct acquaintance…it is not possible to give a clear answer.’
By way of contrast, an event could be impossible in an aleatory sense, yet nevertheless certain on the basis of one’s evidence.
My thanks to Jon Williamson for explaining to me (in personal correspondence) that he is a pluralist. It appears we would both agree with Popper (1938) that: ‘From the formal point of view of “axiomatics”, probability can be described as a two-termed functor…whose arguments are variable or constant names (which can be interpreted, e.g., as names of predicates or as names of statements according to the interpretation chosen).’
Note that even Popper defines rational degrees of belief (understood as corroboration values) in terms of probabilities. But rational degrees of belief do not satisfy the probability calculus according to Popper (e.g. because he alleges that universal laws have zero probability).
Notice that Keynes (1921, p. 7) admits something similar: ‘We may fix our attention on our own knowledge and, treating this as our origin, consider the probabilities of all other suppositions—according to the usual practice which leads to the elliptical form of common speech; or we may, equally well, fix it on a proposed conclusion and consider what degree of probability this would derive from various sets of assumptions, which might constitute the corpus of knowledge of ourselves or others, or which are merely hypotheses.’ Williamson has also told me (in personal correspondence) that he has recently started to use ‘whatever the agent takes for granted’ in place of ‘background knowledge’.
Williamson has told me (in personal correspondence) that he intended ‘a total absence of differentiating empirical information’. I retain my original discussion, in the main body, because (a) one of the anonymous referees agreed with a literal reading of Williamson’s original text, and (b) an argument is still required that Keynes did not disregard empirical constraints. I also reiterate, as earlier mentioned, that Jaynes (1957, p. 623) incorrectly suggests that the maximum entropy principle reduces to the principle of indifference only when ‘no information is given except enumeration of the possibilities x i ’.
It is true that Keynes (1921, p. 41) allowed for non-numerical probabilities, holding that ‘In order that numerical measurement may be possible, we must be given a number of equally probable alternatives’. However, Keynes does not say that ‘we must be given a number of equally probable alternatives’ at any particular point in time in order to (correctly) acquire a probability relation. With reference to the example in the text, there will have been a time at which the probability of a roll of the die giving a value of six was equiprobable with that of it giving a five.
Note also that probability is not the only measure which Keynes (1921, p. 72) employs, since he also suggests ‘Starting… with minimum weight, corresponding to a priori probability, the evidential weight of an argument rises, though its probability may either rise or fall, with every accession of relevant evidence.’
Williamson (Forthcoming B) discusses such critiques.
References
Bernoulli, J. (1713). The art of conjecturing (trans.: Sylla, E. D.). Baltimore: Johns Hopkins University Press, 2006.
Bertrand, J. (1889). Calcul des Probabilités (3rd ed.). New York: Chelsea (c.1960).
Carnap, R. (1962). Logical foundations of probability. Chicago: University of Chicago Press.
De Finetti, B. (1974). Theory of probability: A critical introductory treatment (Vol. 1). London: Wiley.
Franklin, J. (2001). Resurrecting logical probability. Erkenntnis, 55, 277–305. doi:10.1023/A:1012918016159.
Gillies, D. A. (1998). Confirmation theory. In D. M. Gabbay & P. Smets (Eds.), Handbook of defeasible reasoning and uncertainty management systems (Vol. 1, pp. 135–167). Dordrecht: Kluwer.
Gillies, D. A. (2000). Philosophical theories of probability. London: Routledge.
Haack, S. (1978). Philosophy of logics. Cambridge: Cambridge University Press.
Hacking, I. (1975). The emergence of probability. Cambridge: Cambridge University Press.
Hájek, A. (2005). Scotching Dutch books? Philosophical Perspectives, 19, 139–151. doi:10.1111/j.1520-8583.2005.00057.x.
Jaynes, E. T. (1957). Information theory and statistical mechanics. Physical Review, 106, 620–630. doi:10.1103/PhysRev.106.620.
Jaynes, E. T. (1973). The well posed problem. Foundations of Physics, 3, 477–493. doi:10.1007/BF00709116.
Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge: Cambridge University Press.
Keynes, J. M. (1921). A treatise on probability. London: Macmillan.
Laplace, P. S. (1814). A philosophical essay on probabilities. New York: Dover, 1951.
Locke, J. (1689). An essay concerning human understanding. In P. H. Nidditch (Ed.) Oxford: Clarendon Press, 1979.
Marinoff, L. (1994). A resolution of Bertrand’s paradox. Philosophy of Science, 61, 1–24. doi:10.1086/289777.
Mikkelson, J. M. (2004). Dissolving the wine/water paradox. The British Journal for the Philosophy of Science, 55, 137–145. doi:10.1093/bjps/55.1.137.
O’Donnell, R. (1990). The epistemology of J. M. Keynes. The British Journal for the Philosophy of Science, 41, 333–350. doi:10.1093/bjps/41.3.333.
Popper, K. R. (1938). A set of independent axioms for probability. Mind, 47, 275–277. doi:10.1093/mind/XLVII.186.275.
Popper, K. R. (1976). Unended quest. La Salle: Open Court.
Popper, K. R. (1983). Realism and the aim of science. London: Routledge.
Ramsey, F. P. (1926). Truth and probability. In H. E. Kyburg & H. E. Smokler (Eds.), Studies in subjective probability (pp. 61–92). New York: Wiley.
Rowbottom, D. P. (2007). The insufficiency of the Dutch book argument. Studia Logica, 87, 65–71. doi:10.1007/s11225-007-9077-2.
Rowbottom, D. P. (2008). Intersubjective corroboration. Studies in History and Philosophy of Science, 39, 124–132. doi:10.1016/j.shpsa.2007.11.010.
Russell, B. (1912). The problems of philosophy. London: Oxford University Press.
Shackel, N. (2007). Bertrand’s paradox and the principle of indifference. Philosophy of Science, 74, 150–175. doi:10.1086/519028.
Van Fraassen, B. C. (1989). Laws and symmetry. Oxford: Clarendon Press.
Van Fraassen, B. C. (2002). The empirical stance. New Haven: Yale University Press.
Williamson, J. (2005). Bayesian nets and causality: Philosophical and computational foundations. Oxford: Oxford University Press.
Williamson, J. (Forthcoming A). Philosophies of probability: Objective Bayesianism and its challenges. In A. Irvine (Ed.), Handbook of the philosophy of mathematics Volume 4 of the Handbook of the Philosophy of Science. Elsevier.
Williamson, J. (Forthcoming B). Objective Bayesianism, Bayesian conditionalisation and voluntarism. Synthese.
Acknowledgements
I am grateful to several audience members at PROGIC 2005 for comments on an embryonic version of this paper, and have also benefitted from the comments of two anonymous reviewers for Erkenntnis. I am especially grateful to Jon Williamson for his swift and thorough comments on two earlier drafts.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rowbottom, D.P. On the Proximity of the Logical and ‘Objective Bayesian’ Interpretations of Probability. Erkenn 69, 335–349 (2008). https://doi.org/10.1007/s10670-008-9117-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10670-008-9117-y