PART II METHODOLOGY: PROBABILITY AND UTILITY

Transcription

1 PART II METHODOLOGY: PROBABILITY AND UTILITY

2 The six articles in this part represent over a decade of work on subjective probability and utility, primarily in the context of investigations that fall within the general area of decision theory. Articles 6 and 8 are closely related to the theory of measurement. Because of doubts about the possibility of measuring either subjective probability or utility, much of the theory of these subjects has been devoted to an explicit working out of the theory of measurement. Article 9 on the behavioristic foundations of utility is related closely to the articles in Part IV on the foundations of psychology. The discussion in this article of learning theory overlaps the more detailed analyses given in Articles 16 and 23. To those readers who want a quick survey of decision theory without confronting the technical problems, I would recommend Article 7 on the philosophical relevance of decision theory. Duncan Luce and I (1965) have attempted a much more substantial and technical survey in an article not reprinted here. The last article in this part, Article 11 on probabilistic inference, makes the closest connection of any of the articles with much of the recent philosophicalliterature on induction. I think the line of attack begun in this article can be considerably extended, particularly in areas of experience and those parts of science not yet weil organized from a theoretical standpoint. Above all, however, the problems raised about rational behavior at the end of this article seem to me the most important open problems that I have raised in any of the six articles in this part, and in this respect, the article is closely related to the tradition of analysis in decision theory exemplified by the first article of this part, Article 6. From a general philosophical standpoint, the central theme of these six articles is the problem of characterizing and analyzing the elusive concept of rationality. I suppose it is clear to everyone who thinks about the matter very much that we are still only in the beginning stages of a satisfactory analysis, and there are many people who are skeptical of ever giving a systematic characterization that is intuitively satisfactory. I do not think we should yet aim or hope for anything that is complete, but,

3 84 PART 11. METHODOLOGY: PROBABILITY AND UTILITY as in the case of work in the foundations of mathematics over the past century, there is now some ground at least for believing that progress of adefinite and objectively agreed upon sort is possible. The work that originates with the theory of games is turning out to be one of the most useful general lines of approach, even though the classical article by Milnor (1954) shows how treacherous and difficuit it is to give an intuitively complete, but consistent list of attributes of a rational strategy in an uncertain situation, even when that situation is highly restricted. In many respects, the great classical tradition in economics, going back to Adam Smith, can be viewed as an attempt to work out a normative theory of rational behavior in economic contexts. The recent literature in normative economics has generalized the relatively narrow economic context to a wider context of decision or action, as exemplified, for example, in Arrow's classical book (1951). I turn now to a more detailed consideration of the last two articles in this part. In an as yet unpublished book on welfare economics, Dr. Amartya K. Sen of the Delhi School of Economics, University of Delhi, India has made a number of acute comments on the grading principle of justice introduced in Article 10. The fundamental point he makes is that some possible relations J i of more just than can violate Pareto optimality. The relation J i is person i's preference ordering of the possible consequences accruing to him and the possible consequences accruing to the other person as weil (in Article 10 I restricted the number of persons to two, but the generalization to n is straightforward and has been carried out by Dr. Sen). Here is a simple instance of Sen's demonstration of incompatibility with Pareto optimality. Consider two vectors of consequences X=(Xl' X2) and Y=(Yl' Y2)' Let person 1 order these four consequences X2 PI Yl' Yl P Xl and let person 2 order them thus and Xl P Y2' Xl P2 Y2' Y2 P X2, and X2 P Yl. Then according to Definition 5 (p. 159), we have X J l Y and X J 2 y, but on the other hand, for person 1, Y! PI Xl and for person 2, Yz P X2' whence by Pareta optimality, the appropriate social choice is Y over x. This undesirable result follows whenever each man presumes to know his

4 PART H. METHODOLOGY: PROBABILITY AND UTILITY 85 neighbor's preferences better than the neighbor does hirnself. Thus in Sen's example, person 1 thinks that for 2, X2 is better than J2, even though 2 thinks the opposite. Person 2 judges similarly the ranking of x, and Yl for 1. I accept Dr. Sen's criticism and believe that it calls for a change. Fortunately, one is already implicit in his analysis. This is to require that in ordering the set C 2 of consequences for person 2, person 1, with ordering relation R l, agree on C 2 with R 2, i.e., with person 2's own ordering on C 2 ; a similar constraint is placed on R 2 with respect to R l on Cl' Formally, we need to add to Definition 4 (p. 158) the condition that on the subsets Cl and C 2 of Cl u C 2 the ordering relations R l and R 2 agree. Dr. Sen's criticism leads to an emendation in the right direction, because it forces more structure on the concept of justice being set forth. I am, however, still far from satisfied with matters as they now stand. Far stronger structural principles are required to rule out other counterintuitive examples, such as the one given at the end of the article. The issues concerning probabilistic inference, its nature and its justification, have received extensive discussion in recent philosophical literature. I originally intended to relate what I had to say about these matters in Article 11 to what other people have said in the past couple of years. A wide-ranging and informative discussion of many of the central issues in inductive logic is to be found in the volume edited by Lakatos (1968), and the 1968 volume of Philosophy 01 Science contains useful papers by Hempel and others. When I attempted a preliminary review of the rapidly increasing literature, however, it soon became apparent that it would not be possible to deal with it briefly and in a way that was Iimited to trying to extend my own work to meet it. For example, a good part of the Lakatos volume is taken up by discussions by Salmon and others of rules of acceptance. In my judgment, the issues raised need to be analyzed in the context of modern statistical decision theory, not in terms of extending the theory of inference and the theory of explanation. In other words, to take the idea of acceptance seriously, we must proceed to an analysis of behavior and a theory of decisions. The lottery paradox, which has been so much discussed in relation to rules of acceptance, seems to me an example of the sort of artificial puzzle generated by considering rules of acceptance apart from a theory of

5 86 PART 11. METHODOLOGY: PROBABILITY AND UTILITY decisions. In a way, perhaps, the St. Petersburg paradox of utility theory is similar in spirit to the lottery paradox, but in terms of the concepts of acceptance and certainty, there is a total lack of similarity. From still another standpoint, the law of large numbers, the centrallimit theorem, and other asymptotic results in probability theory are related both to probabilistic inference and rules of acceptance, because they describe what, under rather general assumptions, may be predicted to happen with near certainty in the long run. But to examine these relations is not possible here. lan Hacking's criticisms of Salmon and Reichenbach in the Lakatos volume are also pertinent. Hacking presses his remarks from the standpoint of de Finetti's ideas on the foundations of probability and induction. Hacking argues weh for the Bayesian conception of learning by experience, especially in criticizing relative-frequency theories of induction. I share his skepticism of the ability of Bayesian ideas to deal with large parts of our cognitive experience. In another article (Suppes, 1966) published at the same time as Article 11, I tried to show in some detail why Bayesian ideas are not adequate to that part of learning by experience which requires the learning of a new concept. Some brief remarks about these matters are made at the end of Article 11. The learningtheoretic account of finite automata in the very last article of the present volume says as much as I can sharply formulate at the present time about the manner in which a learning mechanism might operate in learning a new concept. In a detailed critique of Article 11 given in Levi's review (1967), I am accused, probably rightly, of adopting a radical psychologism toward the problems of induction. I am increasingly prepared to defend this general way of looking at both deductive and inductive logic. I suppose I feel the real test of a theory of concept formation or a theory of induction is its ability to generate the drawings for a machine, or more specifically, a computer that can form concepts and make inductions. Theories of this kind will not answer many sorts of Humean puzzles about predicting the future from knowledge of the past. Nevertheless, the contribution of such theories, once developed, to the philosophy of induction should be as substantial as have been the contributions of explicitly formulated set theories to the philosophy of mathematics.