Ascertainability, Applicability, and Admissibility

Is there some decisive argument against a non-reductive theory that I have not yet considered? Perhaps, but I do not think it is to be found among the problems that have traditionally plagued theories of chance. Wesley Salmon famously argued that any interpretation of the probability calculus, including theories of objective chance, must meet three conditions of adequacy: ascertainability, applicability, and admissibility.100 A thorough defense of my non-reductive theory would involve demonstrating that it can meet each of these, and that is more than I will do here. However, I end this dissertation with a brief look at my non-reductive theory’s prospects for satisfying Salmon’s three conditions.

100 _{Salmon (1967)}

Let’s start with ascertainability. Ascertainability requires that there be some method by which we may, at least in principle, come to learn the values of objective chances. On my view, we ascertain chance values from the values of actual relative frequencies via inference to the best explanation. Justifying abduction is a difficult but familiar task, and poses no special problem for a non-reductive theorist. So long as abduction can be justified and there are (non-causal) probabilistic explanations, my non- reductive theory meets the ascertainability condition.

What about applicability? The applicability condition is roughly that a theory of chance be consistent with the myriad of ways in which we use chances in our everyday lives and our best scientific theories. Theories of objective chance typically run afoul of the applicability condition by failing to respect either that chances constrain rational credences or that chances apply to singular events. I have already argued that my non- reductive theory explains the PP, but does it allow that objective chances apply to singular events? The answer seems to be yes, since objective chances are probabilistic explanations of singular events.

Furthermore, my non-reductive theory does not suffer from a reference class problem. Reference class problems arise when the particular chance value a theory assigns to a singular event varies depending on how that event is characterized, but no characterization stands out as privileged. While this is a serious problem, it only plagues reductive theories of chance.101 If the particular chance value of a singular event depends on how it is characterized, then facts about the objective chance value of that event privilege a characterization of that event. Of course, reductive theories cannot appeal to

101 _{Alan Hájek (2007) makes this point, though he doubts that non-reductive theories can explain the} Principal Principle.

such chance facts on threat of vicious circularity. But non-reductive theories are not similarly constrained. Whatever worries my non-reductive theory might face, the reference class problem is not among them.

Lastly, consider admissibility. An objective theory of chance is admissible if and only if it respects all the mathematical relationships between probabilities entailed by the probability calculus. Any theory that explains the Principal Principle satisfies this condition, since rational credences obey the probability calculus. So, to the extent that my non-reductive theory explains the Principal Principle, it satisfies admissibility.

My non-reductive theory, then, seems to satisfy ascertainability, applicability, and admissibility. Of course, it might fail to meet one of these conditions for some reason I have not yet considered. And, once again, my non-reductive theory might face some further objective even if it meets all three conditions of adequacy and explains the Principal Principle. A full defense of my non-reductive theory requires a more thorough investigation into these possibilities. More work must be done, then, to establish that my non-reductive theory is true. However, this dissertation should dispel the myth that non- reductive theories are non-starters. My non-reductive theory provides illuminating answers to questions about the nature of chance and has prima facie advantages over its reductive competitors

Works Cited

Bigelow, J., Collins, J. and Pargetter, R. 1993: ‘The Big Bad Bug: What are the Humean’s Chances?’, The British Journal for the Philosophy of Science, 44, pp. 443-62

Butler, J. 1736: Analogy of Religion, reprinted by Adamant Media Corporation, Boston. Clausius, R. 1857: ‘The Nature of the Motion which we call Heat’, Philosophical

Magazine, Vol. 14: 108-27.

Damodaran, A. 2002: Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, Second Edition. Wiley.

Dretske, F. 1977: ‘Laws of Nature’. Philosophy of Science, Vol. 44, pp 248-268. Fetzer, J.H. 1981: Scientific Knowledge: Causation, Explanation, and Corroboration,

Boston Studies in the Philosophy of Science, Vol. 69, D.Reidel, Dordrecht. Fitelson, B. 2001: Studies in Bayesian Confirmation Theory, unpublished manuscript. Gillies, D. 2000: Philosophical Theories of Probability. Routledge: New York. Glymour, C. 1980: Theory and Evidence, Princeton University Press, Princeton. Godement, R. 2004:. Analysis, Springer Publishing

Hájek, A. 2007: ‘The Reference Class Problem is Your Problem Too’, Synthese 156 (3): pp. 563-585

Hall, N. 2004: ‘Two Mistakes about Credence and Chance’, Australasian Journal of Philosophy 82: 93-111.

Hempel, C. 1965: Aspects of Scientific Explanation and Other Essays in the Philosophy of Science, New York: Free Press.

Jeffrey, D. 1969: ‘Statistical Explanation vs. Statistical Inference’ N. Rescher, ed., Essays in Honor of Carl G. Hempel, Reidel (1969) pp. 104-113

Kolmogorov, A.N. 1933: Foundations of the Theory of Probability, Second English Edition, Chelsea, 1956

Laplace, P. 1814. A Philosophical Essay on Probabilities, reprinted by Cornell University Libraries, Ithaca.

Lewis, D. 1986 (1980): ‘A Subjectivist’s Guide to Objective Chance’, in Philosophical Papers, Vol. II, New York: Oxford University Press, pp. 83-113.

---. 1994: ‘Humean Supervenience Debugged’, Mind 106: 321-34. Lipton, P. 1991: Inference to the Best Explanation, London: Routledge.

Miller, D. W. 1994: Critical Rationalism. A Restatement and Defense, Open Court. Moore, B. 1973: An Introduction to Modern Economic Theory, The Free Press, NY. Popper, K. 1959: ‘The Propensity Interpretation of Probability’, British Journal for the

Philosophy of Science 10: 25-42

Prior, E. 1985. Dispositions, Aberdeen: Aberdeen University Press.

Railton, P. 1978: ‘A Deductive-Nomological Model of Probabilistic Explanation’, Philosophy of Science 45: 206-26.

Reichenbach, H. 1949. The Theory of Probability, University of California Press. Salmon, W. 1965: ‘The Status of Prior Probabilities in Statistical Explanation, ‘

Philosophy of Science, Vol. 32, No.2 pp. 137-146

---. 1967: The Foundations of Scientific Inference. University of Pittsburgh Press. ---. 1979: ‘Propensities: a Discussion Review of D.H. Mellor The Matter of

Chance’, Erkenntnis 14: 183-216.

---. 1990: Four Decades of Scientific Explanation, University of Minnesota Press, Minneapolis.

Scriven, M. 1959: ‘Explanation and Prediction in Evolutionary Theory,’ Science 30: pp. 477–482

Sober, E. 2010: ‘Evolutionary Theory and the Reality of Macro Probabilities’, in E. Eells and J. Fetzer (eds.), Probability in Science, Springer, pp. 133-62

van Fraassen, B. 1989: Laws and Symmetry, Oxford: Clarendon Press.

Von Mises, R. 1928: Probability, Statistics and Truth, 2nd _{revised English edition, Allen} and Unwin, 1961.

Weisberg, J. 2009: ‘Locating IBE in the Bayesian Framework,’ Synthese, 167: 125– 143.

Winkler, R. 1969: ‘Scoring Rules and the Evaluation of Probability Assessors’, Journal of the American Statistical Association, 64: 1073-1078.