Abstract
SOME men disclaim certainty about anything. I am certain that they deceive themselves. Be that as it may, only the arrogant and foolish maintain that they are certain about everything. It is appropriate, therefore, to consider how judgments of uncertainty discriminate between hypotheses with respect to grades of uncertainty, probability, belief, or credence. Discriminations of this sort are relevant to the conduct of deliberations aimed at making choices between rival policies not only in the context of games of chance, but in moral, political, economic, or scientific decision making. If agent X wishes to promote some aim or system of values, he will (ceteris paribus) favor a policy that guarantees him against failure over a policy that does not. Where no guarantee is to be obtained, he will (or should) favor a policy that reduces the probability of failure to the greatest degree feasible. At any rate, this is so when X is engaged in deliberate decision making (as opposed to habitual or routine choice).
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Notes
- 1.
Logical Foundations of Probability (Chicago: University Press, 2nd ed., 1962), pp. 219–241.
- 2.
“Inductive Logic and Rational Decisions,” in Carnap and R. C. Jeffrey, eds., Studies in Inductive Logic and Probability (Berkeley: ucla Press, 1971), p. 27.
- 3.
Confirmational commitments built on the principle of confirmational conditionalization are called “credibilities” by Carnap (ibid., pp. 17–19). The analogy is not quite perfect. According to Carnap, a credibility function represents a permanent disposition of X to modify his credal states in the light of changes in his corpus of knowledge. When credibility is rational, it can be represented by a “confirmation function.” Since I wish to allow for modifications of confirmational commitments as well as bodies of knowledge and credal states, I assign dates to confirmational commitments. Throughout I gloss over Carnap’s distinction between credibility functions and confirmation functions (ibid., pp. 24–27).
- 4.
“A Basic System of Inductive Logic,” in Carnap and Jeffrey, op. cit., pp. 51–52.
- 5.
Logic and Statistical Inference (New York: Cambridge, 1965), p. 135.
- 6.
“Subjective Probability as the Measure of a Non-measurable Set,” in P. Suppes, E. Nagel, and A. Tarski, Logic, Methodology, and the Philosophy of Science (Stanford: University Press, 1962), pp. 319–329.
- 7.
“Consistency in Statistical Inference and Decision” (with discussion), Journal of the Royal Statistical Society, series B, XXIII (1961): 1–25.
- 8.
“Upper and Lower Probabilities Induced by a mutivalued Mapping,” Annals of Mathematical Statistics, XXXVIII (1967): 325–339.
- 9.
“The Bases of Probability,” Bulletin of the American Mathematical Society, XLVI (1940): 763–774.
- 10.
Dempster, op. cit.; Good, op. cit.; Kyburg, Probability and the Logic of Rational Belief (Middletown, Conn.: Wesleyan Univ. Press, 1961): Smith, op. cit.; Schick, Explication and Inductive Logic, doctoral dissertation, Columbia University, 1958.
- 11.
The difference between my approach and Smith’s was drawn to my attention by Howard Stein. To all intents and purposes, both Dempster and Smith represent credal states by the largest convex sets that generate the interval-valued functions characterizing those credal states. Dempster (332/3) is actually more restrictive than Smith. Dempster, by the way, wrongly attributes to Smith the position I adopt. To my knowledge, Dempster is the first to consider this position in print—even if only to misattribute it to Smith.
- 12.
As in section “I”, I am supposing that “states of nature” are “independent” of options in the sense that, for every permissible Q-function, Q(h j ) = Q(o ij ;A i ). I have done this to facilitate the exposition. No question of fundamental importance is, in my opinion, thereby seriously altered.
- 13.
“Rational Decisions,” Journal of the Royal Statistical Society, Ser. B, XIV (1952): 114.
- 14.
The possible consequences of a “mixed act” constructed by choosing between “pure options” A i and A j with the aid of a chance device with known chance probability of selecting one or the other option is the set of possible consequences of either A i or A j . Consequently, the security level of such a mixed option for a given u-function is the lowest of the security levels belonging to A i and A j . Thus, my conception of security levels for mixed acts differs from that employed by von Neumann and Morgenstern and by Wald in formulating maximin (or minimax) principles. For this reason, starting with a set of P-admissible pure options, one cannot increase the security level by forming mixtures of them. In any case, mixtures of E-admissible options are not always themselves E-admissible. I shall leave mixed options out of account in the subsequent discussion. See D. Luce and H. Raiffa, Games and Decisions (New York: Wiley, 1958), pp. 68–71, 74–76, 278–280.
- 15.
Smith, op. cit, pp. 3–5, 6–7. The agreement applies only to pairwise choices where one option is a gamble in which there are two possible payoffs and the other is refusing to gamble with 0 gain and 0 loss. In this kind of situation, it is clear that Smith endorses the principle of E-admissibility, but not its converse. However, in the later sections of his paper where Smith considers decision problems with three or more options or where the possible consequences of an option to be considered are greater than 2, Smith seems (but I am not clear about this) to endorse the converse of the principle of E-admissibility—counter to the analysis on the basis of which he defines lower and upper pignic probabilities. Thus, it seems to me that either Smith has contradicted himself or (as is more likely) he simply does not have a general theory of rational choice. The latter sections of the paper may then be read as interesting explorations of technical matters pertaining to the construction of such a theory, but not as actually advocating the converse of E-admissibility. At any rate, since it is the theory Smith propounds in the first part of his seminal essay which interests me, I shall interpret him in the subsequent discussion as having no general theory of rational choice beyond that governing the simple gambling situations just described.
- 16.
See Luce and Raiffa, op. cit., pp. 288/9. Because the analysis offered by Smith and me for cases 1 and 2 seems perfectly appropriate and the analysis for case 4 also appears impeccable, I conclude that there is something wrong with the principle of independence of irrelevant alternatives.
A hint as to the source of the trouble can be obtained by noting that if ‘E-admissible’ is substituted for ‘optimal’ in the various formulations of the principle cited by Luce and Raiffa, p. 289, the principle of independence of irrelevant alternatives stands. The principle fails because S-admissibility is used to supplement E-admissibility in weeding out options from the admissible set.
Mention should be made in passing that even when ‘E-admissible’ is substituted for ‘optimal’ in Axiom 9 of Luce and Raiffa, p. 292, the axiom is falsified. Thus, when .5 ≤ P/S ≤ .6 in case 4, all three options are E-admissible, yet some mixtures of B 1 and B 2 will not be.
- 17.
I mention this because I. J. Good, whose seminal ideas have been an important influence on the proposals made in this essay, confuses permissible with possible probabilities. As a consequence, he introduces a hierarchy of types of probability (Good, op. cit., p. 327). For criticism of such views, see Savage, The Foundations of Statistics (New York: Wiley, 1954), p. 58. In fairness to Good, it should be mentioned that his possible credal probabilities are interpreted not as possibly true statistical hypotheses but as hypotheses entertained by X about his own unknown strictly bayesian credal state. Good is concerned with situations where strict bayesian agents having precise probability judgments cannot identify their credal states before decision and must make choices on the basis of partial information about themselves. [In Decision and Value (New York: Wiley, 1964), P. G. Fishburn devotes himself to the same question.] My proposals do not deal with this problem. I reject Good’s and Fishburn’s view that every rational agent is at bottom a strict bayesian limited only by his lack of self-knowledge, computational facility, and memory. To the contrary, I claim that, even without such limitations, rational agents should not have precise bayesian credal states. The difference in problem under consideration and presuppositions about rational agents has substantial technical ramifications which cannot be developed here.
Acknowledgements
Work on this essay was partially supported by N.S.F. grant GS 28992. Research was carried out while I was a Visiting Scholar at Leckhampton, Corpus Christi, Cambridge. I wish to thank the Fellows of Corpus Christi College and the Departments of Philosophy and History and Philosophy of Science, Cambridge University, for their kind hospitality. I am indebted to Howard Stein for his help in formulating and establishing some of the results reported here. Sidney Morgenbesser, Ernest Nagel, Teddy Seidenfeld, and Frederic Schick as well as Stein have made helpful suggestions.
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Levi, I. (2016). On Indeterminate Probabilities. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_7
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