The Dynamics of Thought [1 ed.] 9781402033988, 1402033982

Citation preview

THE DYNAMICS OF THOUGHT

SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Editor-in-Chief:

VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.

Honorary Editor: JAAKKO HINTIKKA, Boston University, U.S.A.

Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLEN´SKI, Jagiellonian University, Kraków, Poland

VOLUME 300

THE DYNAMICS OF THOUGHT by

PETER GÄRDENFORS Lund University, Sweden

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-3398-2 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-10 1-4020-3399-0 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3398-8 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3399-5 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

CONTENTS Preface

vii

Acknowledgements

ix

Decision theory Chapter 1: Probabilistic reasoning and evidentiary value

1

Chapter 2: Unreliable probabilities, risk taking, and decision making

11

Chapter 3: Rights, games and social choice

31

Belief revision and nonmonotonic reasoning Chapter 4: The dynamics of belief systems: Foundations vs. coherence theories

47

Chapter 5: The role of expectations in reasoning

67

Chapter 6: How logic emerges from the dynamics of information

83

Induction Chapter 7: Induction, conceptual spaces and AI

109

Chapter 8: Three levels of inductive inference

125

Semantics and pragmatics Chapter 9: Frameworks for properties: Possible worlds vs. conceptual spaces 145 Chapter 10: The pragmatic role of modality in natural language

161

Chapter 11: The social stance

173

Chapter 12: The emergence of meaning

179

Chapter 13: Does semantics need reality?

201

Evolution and cognition Chapter 14: The nature of man – games that genes play?

215

Chapter 15: The detachment of thought

227

Chapter 16: Cognitive science: From computers to anthills as models of human thought 243 References

261

Index

277

v

PREFACE This book is a selection from the articles that I have written over a period of more than twenty years. Since the focus of my research interests has shifted several times during this period, it would be difficult to identify a common theme for all the papers in the volume. Following the Swedish tradition, I therefore present this as a smörgåsbord of philosophical and cognitive issues that I have worked on. To create some order, I have organized the sixteen papers into five general sections: (1) Decision theory; (2) belief revision and nonmonotonic logic; (3) induction; (4) semantics and pragmatics; and (5) cognition and evolution. Having said this, I still think that there is a common theme to my work over the years: The dynamics of thought. My academic interests have all the time dealt with aspects of how different kinds of knowledge should be represented, and, in particular, how changes in knowledge will affect thinking. Hence the title of the book. The papers have been selected to minimize the overlap with my books Knowledge in Flux (1988) and Conceptual Spaces (2000). However, because some of the articles in this volume were parts of the process of writing these books, there are still common themes. The articles have been published in disparate and often hard to access publications, which is a good reason to present them under one cover. However, this also entails that there is some overlap between the articles. In preparing them for this volume, I have therefore tried to eliminate the most blatant repetitions. I have also edited the articles lightly to make their styles more uniform.

vii

ACKNOWLEDGEMENTS Several people have been helpful in the process. Since it was created in 1988, the seminar in cognitive science at Lund University has been a constantly critical but friendly test bed for my ideas. I also want to thank Eva Börjesson who has been very efficient in transforming some of the articles that appear in this book, Nils-Eric Sahlin for allowing me to reuse one of our joint articles, Annie Kuipers who initiated the project at Kluwer, Ingrid Krabbenbos who kept it alive, and Vincent Hendricks who gently pushed me to finalize it. Over the years, my research has been generously financed by the Swedish Council for Research in the Humanities and Social Science and the Swedish Foundation for Strategic Research, which I gratefully acknowledge. The following list presents the original sources of the chapters. I am grateful to the publishers for granting me the permission to reprint them in this volume. Chapter 1:

Chapter 2: Chapter 3: Chapter 4:

Chapter 5:

Chapter 6:

Chapter 7: Chapter 8:

Chapter 9:

“Probabilistic reasoning and evidentiary value,” in P. Gärdenfors, B. Hansson and N.-E. Sahlin, eds., Evidentiary Value: Philosophical, Judicial and Psychological Aspects of a Theory, Gleerups, Lund, 44-57. (1983) “Unreliable probabilities, risk taking, and decision making,” Synthese 51, 361-386. (1982) “Rights, games and social choice,” Noûs 15, 341–356. (¤ Blackwell Publishing 1981) “The dynamics of belief systems: Foundations vs. coherence theories,” ¤ Revue Internationale de Philosophie 44, 24-46. (1990) “The role of expectations in reasoning,” in M. Masuch and L. Polos, eds., Knowledge Representation and Reasoning Under Uncertainty, ¤ Springer-Verlag, Berlin, 1-16. (1994) “How logic emerges from the dynamics of information,” in J. van Eijck and A. Visser, eds., Logic and Information Flow, ¤ MIT Press, Cambridge, MA, 49-77. (1994) “Induction, conceptual spaces and AI,” ¤ Philosophy of Science 57, 78-95. (1990) “Three levels of inductive inference,” in D. Prawitz, B. Skyrms and D. Westerståhl, eds., Logic, Methodology, and Philosophy of Science IX, X ¤ Elsevier Science, Amsterdam, 427-449. (1994) “Frameworks for properties: Possible worlds vs. conceptual spaces,” ¤ Acta Philosophica Fennica 49, 383-407. (1991) ix

x Chapter 10: “The pragmatic role of modality in natural language,” in P. Weingartner, G. Schurz and G. Dorn, eds., The Role of Pragmatics in Contemporary Philosophy, ¤ Hölder-PichlerTempsky, Vienna, 78-91. (1998) Chapter 11: “Social intentions and irreducibly social goods,” in G. Brennan and C. Walsh, eds., Rationality, Individualism, and Public Policy, Centre for Research on Federal Financial Relations, Canberra, 91-96. (1990) Chapter 12: “The emergence of meaning,” Linguistics and Philosophy 16, 285-309. (1993) Chapter 13: “Does semantics need reality?,” in A. Riegler, M. Peschl and A. von Stein, eds., Understanding Representation in the Cognitive Sciences: Does Representation Need Reality? Kluwer Academic Publishers, New York, NY, 209--217 (1999) Chapter 14: “The nature of man – games that genes play?,” in G. Holmström and A. Jones, eds., Action, Logic and Social Theory (¤ Acta Philosophica Fennica 38), Helsinki, 9–24. (1985) Chapter 15: “The detachment of thought,” in C. Erneling and D. Johnson, eds., The Mind as a Scientific Subject: Between Brain and Culture, ¤ Oxford University Press, Oxford, 323–341. (2005) Chapter 16: “Cognitive science: From computers to anthills as models of human thought,” World Social Science Report, ¤ UNESCO Publishing/Elsevier, Paris, 316-327. (1999)

CHAPTER 1 PROBABILISTIC REASONING AND EVIDENTIARY VALUE (At the beginning of this dialogue on evidentiary value the conversation involves two persons: The first is Basie, a statistician, very much interested in subjective probabilities. When working on a problem involving assessments of probabilities, he always thinks to himself: What would Bayes say in a situation like this? The second person is Lazy, an ordinary man, neither a statistician, nor a lawyer, but a representative of what the lay say.) Lazy: Look, Basie. I’ve just come from the Psychology Department where I participated as a subject in an experiment on probabilistic reasoning. One of the problems given to me was the following:1 Two cab companies operate in a given city, the Blue and the Green (according to the color of cab they run). Eighty-five per cent of the cabs in the city are Blue, and the remaining 15% are Green. A cab was involved in a hit-and-run accident at night. A witness later identified the cab as a Green cab. The court tested the witness’s ability to distinguish between Blue and Green cabs under nighttime visibility conditions. It found that the witness was able to identify each color correctly about 80% of the time, but confused it with the other color about 20% of the time. What do you think are the chances that the errant cab was indeed Green, as the witness claimed?

Well, I answered that the chance was 80%, because that is the probability that the witness was correct. But I smell a rat here. I am not so certain that my reasoning is valid. What do you say? Basie: It seems to me that you are the victim of a very common cognitive illusion. You fail to consider that there are many more Blue cabs than Green. In fact, it is more probable that the cab was Blue and the witness incorrectly identified it as Green than that it was Green and the witness made a correct identification. Since 85% of the cabs are Blue and the witness is wrong in 20% of the cases, the first situation should occur in 85% x 20% = 17% of the cases in the long run. The other situation occurs in only 15% x 80% = 12% of the cases. So, the probability that the cab is Green, given that the witness says it is Green, is only 0.12/(0.12 + 0.17) = 0.41. Lazy: I see how you count, Basie, but where did I go wrong? Basie: The problem is a standard case of Bayesian inference. There are 1

This example is from Kahneman and Tversky (1972). Variants of this example are discussed in Tversky and Kahneman (1982) and Bar-Hillel (1980).

1

2

THE DYNAMICS OF THOUGHT

two pieces of information: one is in the form of background data, often called base-rate information. The second, the witness report, may be called indicator information. In your reasoning, when you fail to consider the first piece of information, you commit what has been called the base-rate fallacy. Lazy: After the experiment at the Psychology Department I was informed by the experimenter that most of the subjects gave the same answer as I did.2 Even lawyers seem to be reasoning in the same way. Basie: It does not surprise me. The base-rate fallacy is a persistent illusion. It is the duty of statisticians to show people where their intuitions go wrong. One way of doing this is to point out that the probability that the cab is Green (let us denote this by G) given that the witness says green (denoted WG), i.e. P(G/WG), is not the same as the probability that the witness says green given that the cab is Green, i.e. P(WG/G). The latter is 0.80 (P(WG/G)), as was given in the problem, but the former (P(G/WG)), as I showed earlier, is only 0.41. Lazy: It does seem very plausible that I indeed confused P(G/WG) with P(WG/G) when I gave my answer in the experiment. I see now that P(G/WG) need not be 80%, because this would in some way mean that the witness report caused the cab to be Green. Basie: That’s a sloppy way of expressing the point. But you seem to understand my lesson. Lazy: Nevertheless. I feel uneasy with your solution. If I were a judge thinking about the particular cab involved in the accident, the witness report that relates to this cab would seem to carry much more weight than some statistics about cabs in general. Statistics cannot be the cause of accidents.3 Basie: I don’t see why you keep referring to causes. In the formulation of the problem you were only given two pieces of information about the cab: the base-rate and the witness report. Since you do not know anything else about the cab you should take both figures into account. And both of them are certainly relevant when judging the probability that it was a Green cab that was involved in the accident. If lawyers make the same mistake as you do, so much the worse. It would be interesting to hear what they say about the problem. Perhaps I can help them avoid the base-rate fallacy. Lazy: Yes, what would they say? Speaking of this, let us pay a visit to my friend Daisy who is writing a thesis on jurisprudence. Perhaps she can explain my feeling of uneasiness with your solution. (Lazy and Basie meet Daisy in her office. Lazy explains the problem and his answer, and Basie gives his account.)

2 3

For a summary of the findings, cf. Bar-Hillel (1980), p. 215. Basically, this seems to be the Cohen's position in (1981a,b). Cf. in particular pp. 365-366.

DECISION THEORY

3

Daisy: This is indeed a problem that I am very much interested in. In fact I have been investigating this problem and some related ones. They seem to offer serious problems for lawyers since their solutions do not accord with Bayesian calculations. Unfortunately, the problem does not seem to be very much discussed in the literature. Basie: As I see it, lawyers are simply committing the base-rate fallacy in the same way as lazy did. Daisy: I am not so certain about that. I have recently been working with an interesting theory about evidentiary values that has been proposed by Ekelöf and developed by Halldén and Edman.4 Basie: I have never heard of it. Daisy: If I follow Halldén’s presentation, an evidentiary argument contains three kinds of components. Firstly, an evidentiary theme that is to be ‘proved.’ In the cab example this is the fact G, that the cab responsible for the accident was Green. Secondly, evidentiary facts, in this case WG, the witness’s report that the cab was Green. Finally, evidentiary mechanisms that say that an evidentiary fact is caused by the evidentiary theme. In this case the only evidentiary mechanism is that which relates the color of the cab to the witness’s report. Let us denote by E the hypothesis that this mechanism was ‘working,’ i.e. that the witness’s report is correct. Basie: What about the base-rate information concerning the frequency of the different colors? Daisy: Well, I suppose that this would not represent an evidentiary fact since there is no evidentiary mechanism that connects it with the evidentiary theme. Lazy: Do you mean by this that there is no causal link between the baserate and the actual accident? Daisy: Basically yes, since causal relations are the most important evidentiary mechanisms. But this does not mean that base-rates should be ignored when judging the probabilities involved. Basie: But then you must arrive at the same probability values as I did. Daisy: No. A key feature of the Ekelöf-Halldén-Edman theory is that the probability that the evidentiary mechanism has worked, in this case P(E/WG), is considered more important judicially than the probability of the evidentiary theme given the evidentiary facts, i.e. P(G/WG). This means that the evidentiary value of the argument is P(E/WG), which in this case the subjects take to be 80%, rather than P(G/WG), which is 41%.5 It seems that

4

The basic references are Ekelöf (1982) §23, Halldén (1973) and Edman (1973). Also cf. Goldsmith (1980). 5 It should be noted that the Ekelöf-Halldén-Edman theory does not say that the evidentiary value of the witness's report is 0.8. Rather, this value is much lower since it cannot be higher than the probability of the evidentiary theme, given the evidentiary facts. However, it seems

4

THE DYNAMICS OF THOUGHT

people, both lawyers and lay people, often evaluate the first probability rather than the second when evidentiary mechanisms of the kind that I have outlined are involved. Basie: But concentrating on P(E/WG) rather than P(G/WG) means that lawyers ignore an important item of information. Daisy: Well, base-rates are normally considered only when they are connected with some evidentiary mechanism. Lazy: This is presumably why I was dissatisfied with Basie’s argument, even though I could not rebut his calculations. Daisy: Besides what I have already presented, the theory includes some technical results by Ekelöf, Halldén and Edman that concern the proper calculations when several evidentiary mechanisms are operating together. Three separate cases are considered: When the mechanisms form a causal chain, when they involve concurring evidentiary facts, and the most intricate case of all when the mechanisms concern conflicting evidentiary facts, i.e. when one fact supports the evidentiary theme and another fact contradicts the theme. Basie: I can see that there are some interesting mathematical problems here. Daisy: Well, I do not think we need to discuss the mathematics here. What is more interesting for me is that this theory can explain the results of several of the experiments involving evidentiary mechanisms that have been performed by the psychologists concerning the base-rate fallacy and other similar effects. For example, let us consider a variant of the cab example, where the information about the frequency of the colors of the cabs was replaced by the following sentence:6 “Although the two companies are roughly equal in size. 85% of cab accidents in the city involve Blue cabs and 15% involve Green cabs.” When presented with this modified problem the subjects’ answers to this problem were not as unanimous as before (when most of them gave the answer 80%), but the base-rate was taken into account. The median answer of the subjects was 0.60. Lazy: I see. In this variant there is a causal connection between the color of a cab and its propensity to be involved in accidents. The Blue drivers are more reckless or something like that.

clear that the subjects try to estimate the evidentiary value of the facts instead of the true probability of the theme, even if they make mistakes in their calculations. This view is supported by the results reported by Bar-Hillel (1980), Goldsmith and Sjöberg (1979), Tversky and Kahneman (1982) and Sahlin (1983). The base-rate is not ignored just because it is a base-rate, but because it is judged to have no influence on the probability of a causal relation between the evidentiary facts and the evidentiary theme. Another error in the subjects’calculations is pointed out by Basie at the end of the dialogue. 6 Tversky and Kahneman (1982), p. 157.

DECISION THEORY

5

Daisy: Yes. This variant is an example of two evidentiary mechanisms that are in conflict with each other. In another variant of the problem the base-rate information is the same as in the original problem but now there are two witnesses both identifying the cab as Green. One witness was found to be able to identify the correct color 80% of the time confusing it with the other color 20% of the time; the second witness identified each color correctly 70% of the time and erred about 30% of the time. In this variant 24 out of 29 subjects answered that the chances that a Green cab was involved in the accident was 75%. This is midway between the two witness-based assessments.7 Basie: But if the two witness reports are independent they reinforce each other and the probability should be higher than each of the assessments in this case around 94%. Daisy: Yes, it appears that lay subjects (at least) are not very good at combining the values of concurring pieces of evidence. The results strongly indicate that they ignore the base-rates given in the problem and concentrate on the evidentiary mechanisms in the witnesses’ reports. I do not wish to claim that subjects never make errors in their calculations. My main point is that in psychological experiments involving evidentiary mechanisms we get a much better understanding of the subjects’ performance if we interpret their answers in terms of the Ekelöf-Halldén-Edman theory of evidentiary values than if we use the standard Bayesian account.8 Lazy: Your argument impresses me. Earlier Basie convinced me that I had made a mistake in my answer to the cab problem, but the theory of evidentiary values seems to give me a rationale for my intuitive judgment. Basie: I don’t care if you can explain people’s behavior! What matters to me is which answer is correct. I claim that the correct answer to the cab problem is P(G/WG), i.e. 41%, but you introduce these mysterious evidentiary mechanisms and say that the correct answer is the probability that the evidentiary mechanism has worked, i.e. P(E/WG), which you say is 80%. I want to see a reason why P(E/WG) is a better answer than P(G/WG) before I am willing to give the Ekelöf-Halldén-Edman theory any credit. After all it seems obvious to me that it is the probability that the cab was Green that is crucial here – the reliability of the witness is only an aid in evaluating this probability. Lazy: Yes, this seems to be the heart of the matter. Daisy: I am afraid that I cannot give you any conclusive argument here.

7

Bar-Hillel (1980). p. 227. For a similar experiment with similar results also cf. Sahlin (1983). 8 For further material cf. Goldsmith (1980), Goldsmith and Andersson (1978) and Goldsmith and Sjöberg (1979).

6

THE DYNAMICS OF THOUGHT

One thing is that the theory seems to accord well with actual judicial reasoning, but this does not answer your question. Basie: Not at all! Daisy: As far as I know, Ekelöf and Halldén do not give a detailed argument either. So I will have to reconstruct an answer from their brief remarks. Halldén says that the problem is a matter of the responsibility of the judge. If a strong causal connection between the evidentiary theme and the evidentiary facts supporting the theme can be established, then the judge can take the responsibility for a sentence of guilty. However, even when the probability of the evidentiary theme is very high given the evidentiary facts, the judge cannot take the responsibility for convicting the defendant if the evidentiary value of the facts is low, i.e. if the evidentiary theme is not ‘proved.’9 Lazy: I see. When one says that the evidentiary theme is not ‘proved’ one means that one has not found any evidentiary mechanism that connects the theme with the evidentiary facts, but this does not mean that the probability that the evidentiary theme is true is low. Daisy: Right. This is connected with the principle that in criminal cases the burden of proof is to lie with the prosecution. It is a strong norm of the judicial system that unless the court can establish the appropriate ‘proof’ the defendant cannot be convicted – it is not necessary that the defendant ‘proves’ his innocence. Basie: I am beginning to suspect that there is no unique correct answer to the cab problem since there are different norms involved. The goal of a statistician is to find out how probable it is that the evidentiary theme is true, while the goal of the court is to find out to which extent the theme has been ‘proved,’ which in your account means to find the total evidentiary value of the facts.10 Lazy: I feel like you have removed the scales from my eyes. Of course there are different goals involved! And which answer is ‘correct’ is relative to which norms are presupposed. Basie: Yes, it seems that this distinction solves the main conflict between the Bayesian approach and the Ekelöf-Halldén-Edman theory. It also explains why the base-rates in the cab example are considered irrelevant by the jurists: Since the base-rates are supposed to have no causal connection with the particular accident, they have only an indirect influence on the evidentiary value, which has to do with the witness’s report. Lazy: So the evidentiary value is more important to the court than the probability of the truth of the theme, because of the responsibility of the

9

Halldén (1973), p. 57. Also cf. Ekelöf (1982), §23 and §25. Cf. Goldsmith (1980), pp. 217-218.

10

DECISION THEORY

7

court. But is the burden of proof always with the court? Daisy: No. In civil cases, the situation is quite different. Here the court’s verdict often seems to be guided by a principle of ‘preponderance of evidence,’ which means that the decision favors the side having the stronger support. In such cases it seems that the Bayesian way of estimating probabilities may be superior to the theory of evidentiary value, from a normative point of view. It seems anyway that the responsibility of the judge in criminal cases is quite different from his role in civil cases. However, it remains to be seen whether this choice of theory for probability assessments accords with the norms and intuitions of lawyers.11 Basie: The thought occurred to me that another area of relevance when comparing the Bayesian approach with the theory of evidentiary value is clinical medicine. There physicians are often involved in probabilistic reasoning concerning an ‘evidentiary theme’ in form of results from different diagnostic tests. The norms involved there are based on the physician’s goal being to select the correct therapeutic actions. From my point of view, which may be biased, it seems that in this area of application the correct way of arguing is to use the Bayesian approach. Daisy: I have no normative arguments against your position here. However, there is some experimental evidence that shows that in many cases physicians make major errors in probabilistic reasoning. Let me show you one example of this that is similar to the cab problem:12 You are suffering from a disease that, according to your manifest symptoms, is either A or B. For a variety of demographic reasons disease A happens to be nineteen times as common as B. The two diseases are equally fatal if untreated, but it is dangerous to combine the respectively appropriate treatments. Your physician orders a certain test which, through the operation of a fairly well understood causal process, always gives a unique diagnosis in such cases and this diagnosis has been tried out on equal numbers of A- and B-patients and is known to be correct on 80% of those occasions. The tests report that you are suffering from disease B.

What treatment should you choose? Lazy: Well, if I reason in the same way as in the cab problem, I should choose the treatment appropriate to disease B. Daisy: You’re in good company. Many physicians would do the same. Basie: Let me see now. The initial probability that you are suffering from B is 5% and in 80% of these cases the test says B, i.e. this situation applies to 4% of the population. On the other hand, the initial probability of A is 95% and among those with disease A the test says B in 20% of the cases, which is 19% of the population. So the probability that you have disease B, 11

Cf. Ekelöf (1982), §25 for the ‘burden of proof’ in criminal and civil procedure in Sweden. Cohen (1981a). p. 329. This example is a variant of an experiment performed by Hammerton (1973). See Tversky and Kahneman (1982), Bar-Hillel (1980) and Eddy (1982) for further references to medical applications. 12

8

THE DYNAMICS OF THOUGHT

given that the test says B is 0.04/(0.04+0.19) = 0.17. Well, I for one would ignore the result of the test and choose the treatment appropriate to disease A. Daisy: It is clear that, in the long run, the physician would make fewer mistakes if he dispensed with the test and always opted for the treatment for A than if he let the outcome of the test determine the treatment. The analogy with the judicial situation breaks down here since for the judge there is a strong normative difference between declaring an innocent person guilty and declaring a guilty person not guilty, while in the medical case it is just as bad to treat a person for B when he has A as it is to treat him for A when he has B. Lazy: But why, then, do so many choose the treatment for B in the example? Basie: As in the cab problem, I am inclined to think that they fail to distinguish between the statement that when the test is applied to people who have B, it is correct 80% of the time, which is true, and the statement that when the test says that someone has B, it is correct 80% of the time, which is blatantly false.13 Daisy: That is one possible answer. Another hypothesis is that in the case of a single patient the subjects see a causal connection between the disease and the test result, just as in the witness’s report earlier, but they do not perceive any such connection between the base-rates and the patient’s disease. If this is correct, it then seems as if they are using the theory of evidentiary value to obtain their response. Basie: Which of these answers is the true one? Daisy: Well, I do not think the present experimental evidence is sufficient to determine whether this hypothesis, or Basie’s argument, is the correct one.14 Basie: Anyhow, it is clear to me that the subjects are mistaken. This shows that the theory of evidentiary value is, at best, only applicable in those judicial situations where the burden of proof lies with the court. Let me return to the concept of an evidentiary mechanism, which I find somewhat puzzling. In the cab example the central probability for the theory was P(E/WG). When you introduced the sentence E you said that it expressed the hypothesis that the evidentiary mechanism was ‘working’ which was supposed to mean that the evidentiary fact is caused by the evidentiary theme. But all this makes no particular sense to me. I do not see how the probability P(E/WG) can be determined at all. Daisy: Perhaps the best way to explain the intention behind the notion of

13 14

This is Mackie's position in (1981), p. 346. Cf. Bar-Hillel (1980), pp. 220-221, for a discussion of the two positions.

DECISION THEORY

9

an evidentiary mechanism is to use the analogy with a measuring instrument that was introduced by Edman.15 In the cab problem the witness would be regarded as a measuring instrument that is used to determine the color of the cab involved in the accident. In this analogy, the sentence E means that the measuring instrument was functioning in the appropriate way, which in this case is the same as that the witness’s report was caused by the color of the cab. According to the theory, we are only interested in cases when there is a causal link between the evidentiary theme (namely that the cab was Green) and the evidentiary fact (i.e. the report that the cab was Green). The cases when the measuring instrument is not working are irrelevant, just as the hands of a broken watch does not tell us anything about what time it is. Basie: Well, I see a problem here, though. The investigation of the court only showed that the witness gave the correct answer in 80% of the cases. But if all of these cases are to be caused by the color of the cab, one must assume that the witness, regarded as a measuring instrument, does not give the correct answer when the measuring instrument is not functioning. Daisy: Yes, I suppose this is how the subjects think of the evidentiary mechanisms. Basie: But this assumption is not very realistic. Even a broken watch shows the correct time once each day and once each night. And in forensic contexts, the witnesses’ reports often happen to be correct, even if the reports are not caused by the evidentiary theme. In the cab problem, for example, the witness may be certain of the color in only 60% of the cases, while he only guesses in the remaining 40%. If his guesses are correct half of the time, he would then give the correct answer in 80% of the cases as in the example. But if this is true, the evidentiary mechanism is working in only 60% of the cases, and P(E/WG) should accordingly be only 0.6 rather than 0.8 as you assumed in your description of the theory.16 Daisy: You seem to have a point here. But many subjects may nevertheless interpret the information that the witness was correct in 80% of the cases as saying that the probability that the evidentiary mechanism is functioning is 0.8. Basie: I am still not very much interested in explaining the subjects’ performance. What worries me is that in order to apply the theory of evidentiary value it is necessary to be able to determine how probable it is that there is a causal relation between the evidentiary theme and the evidentiary facts. We statisticians do not talk much of causes and I do not know how such probabilities are to be computed. For example, how do you

15

Cf. Edman (1973), pp. 181-182. This point is captured in the two measures presented by Edman (1973). The experiment performed by Sahlin (1983) also takes this into account.

16

10

THE DYNAMICS OF THOUGHT

know that the base-rates of the cabs are not causally connected with the accident? Daisy: I am sorry, but I cannot help you with any theory here. In my investigations I have relied on the intuitions of the lawyers and the test subjects. Basie: We seem to be coming back to the problem of causality over and over again. Let us visit Dr. Hazy, who is at the Philosophy Department. He may be able to tell us how a causal relation is to be determined.17

17

I wish to thank Aleksander Peczenik for supplying me with some of Daisy's arguments. I have also benefited from discussions with Bertil Rolf and Robert Goldsmith.

CHAPTER 2 UNRELIABLE PROBABILITIES, RISK TAKING, AND DECISION MAKING

(Co-authored with Nils-Eric Sahlin)

2.1 The limitations of strict Bayesianism A central part of Bayesianism is the doctrine that the decision maker’s knowledge in a given situation can be represented by a subjective probability measure defined over the possible states of the world. This measure can be used to determine the expected utility for the agent of the various alternatives open to him. The basic decision rule is then that the alternative that has the maximal expected utility should be chosen. A fundamental assumption for this strict form of Bayesianism is that the decision maker’s knowledge can be represented by a unique probability measure. The adherents of this assumption have produced a variety of arguments in favor of it, the most famous being the so-called Dutch book arguments. A consequence of the assumption, in connection with the rule of maximizing expected utility, is that in two decision situations which are identical with respect to the probabilities assigned to the relevant states and the utilities of the various outcomes the decisions should be the same. It seems to us, however, that it is possible to find decision situations which are identical in all the respects relevant to the strict Bayesian, but which nevertheless motivate different decisions. As an example to illustrate this point, consider Miss Julie who is invited to bet on the outcome of three different tennis matches.18 As regards match A, she is very well-informed about the two players – she knows everything about the results of their earlier matches, she has watched them play several times, she is familiar with their present physical condition and the setting of the match, etc. Given all this information, Miss Julie predicts that it will be a very even match and that a mere chance will determine the winner. In match B, she knows nothing whatsoever about the relative strength of the contestants (she has not even heard their names before) and she has no other information that is relevant for predicting the winner of the match. Match C is similar to match 18 This example was chosen in order to simplify the exposition. We believe, however, that similar examples can be found within many areas of decision making, e.g. medical diagnosis and portfolio selection.

11

12

THE DYNAMICS OF THOUGHT

B except that Miss Julie has happened to hear that one of the contestants is an excellent tennis player, although she does not know anything about which player it is, and that the second player is indeed an amateur so that everybody considers the outcome of the match a foregone conclusion. If pressed to evaluate the probabilities of the various possible outcomes of the matches, Miss Julie would say that in all three matches, given the information she has, each of the players has a 50% chance of winning. In this situation a strict Bayesian would say that Miss Julie should be willing to bet at equal odds on one of the players winning in one of the matches if and only if she is willing to place a similar bet in the two other matches. It seems, however, perfectly rational if Miss Julie decides to bet on match A, but not on B or C, for the reason that a bet on match A is more reliable than a bet on the others. Furthermore she would be very suspicious of anyone offering her a bet at equal odds on match C, even if she could decide for herself which player to back. The main point of this example is to show that the amount and quality of information that the decision maker has concerning the possible states and outcomes of the decision situation in many cases is an important factor when making the decision. In order to describe this aspect of the decision situation, we will say that the information available concerning the possible states and outcomes of a decision situation has different degrees of epistemic reliability. This concept will be further explicated later. We believe that the epistemic reliability of a decision situation is one important factor when assessing the risk of the decision. In our opinion, the major drawback of strict Bayesianism is that it does not account for the variations of the epistemic reliability in different decision situations. The concept of epistemic reliability is useful also in other contexts than direct decision making. In the next section, after presenting the models of decision situations, we will apply this concept in a discussion of Popper’s ‘paradox of ideal evidence.’ In order to determine whether empirical support could be obtained for the thesis that the epistemic reliability of the decision situation affects the decision, Goldsmith and Sahlin (1983) performed a series of experiments. In one of these, test subjects were first presented with descriptions of a number of events and were asked to estimate for each event the probability of its occurrence. Some events were of the well-known parlor game type, e.g. that the next card drawn from an ordinary deck of cards will be a spade; while other events were ones about which the subjects presumably had very limited information, e.g. that there will be a bus strike in Verona, Italy, next week. Directly after estimating the probability of an event, subjects were asked to show, on a scale from 0 to 1, the perceived reliability of their probability estimate. The experiment was constructed so that for each subject several sets of events were formed, such that all the events in a set had

DECISION THEORY

13

received the same probability estimate but the assessed reliability of the various estimates differed. For each set, the subject was then asked to choose between lottery tickets involving the same events, where a ticket was to be conceived as yielding a win of 100 Swedish Crowns if the event occurred but no monetary loss if it did not occur. One hypothesis that obtained support in this experiment was that for probabilities other than fairly low ones, lottery tickets involving more reliable probability estimates tend to be preferred. This, together with the results of similar experiments, suggested the reliability of probability estimates to be an important factor in decision making. The aim of the present chapter is to outline a decision theory which is essentially Bayesian in its approach but which takes epistemic reliability of decision situations into consideration. We first present models of the knowledge relevant in a decision situation. One deviation from strict Bayesianism is that we use a class of probability measures instead of only one to represent the knowledge of an agent in a given decision situation. Another deviation is that we add a new measure that ascribes to each of these probability measures a degree of epistemic reliability. The first step in a decision, according to the decision theory to be presented here, is to select a class of probability measures with acceptable degrees of reliability on which a decision is to be based. Relative to this class, one can then, for each decision alternative, compute the minimal expected utility of the alternative. In the second step the alternative with the largest minimal expected utility is chosen.19 This decision theory is then compared to some other generalized Bayesian decision theories, in particular Levi’s (1974) theory. 2.2 Models of decision situations Our description of a decision situation will have many components in common with the traditional Bayesian way of describing decision problems. A decision is a choice of one of the alternatives available in a given situation. For simplicity, we will assume that in any decision situation there is a finite set A = {a1, a2, ..., an} of alternatives. Though the decision maker presumably has some control over the factors that determine the outcome of the decision, he does not, in general, have complete control. The uncertainty as to what the outcome of a chosen alternative will be is described by referring to different states of nature (or just states, for brevity). We will assume that, in any given decision situation, only a finite number of states are relevant to the decision. These states will be denoted s1, s2, ..., sm. The result or outcome of choosing the alternative ai, if the true state of nature is sj will be denoted oij. An important factor when making a decision 19

This decision rule is a generalization of the rule suggested by Gärdenfors (1979b), p. 169.

14

THE DYNAMICS OF THOUGHT

is that of the values the decision maker attaches to outcomes. We will make the standard assumption that this valuation can be represented by a utility measure.20 The utility of the outcome oij will be denoted uij. It is assumed that all information on how the decision maker values the outcomes is summarized by the utility measure. A final factor in describing a decision situation is that of the beliefs the decision maker has concerning which of the possible states of nature is the true state. Within strict Bayesianism it is assumed that these beliefs can be represented by a single probability measure defined over the states of nature. This assumption is very strong since it amounts to the agent having complete information in the sense that he is certain of the probabilities of the possible states of nature. The assumption is unrealistic, since it is almost only in mathematical games with coins and dice that the agent has such complete information, while in most cases of practical interest the agent has only partial information about the states of nature. In the strict form of Bayesianism that is advocated by de Finetti (1980) and Savage (1972) among others, it is assumed that the agent’s subjective probability of a state of nature can be determined by his inclination to accept bets concerning the state.21 The so-called Dutch book theorem states that if it is not possible to construct a bet where the agent will lose money independently of which state turns out to be the actual one, then the agent’s degrees of beliefs satisfy Kolmogoroff’s axioms, i.e., there is a unique probability measure that describes these degrees of belief. However, a presupposition of this theorem is that the agent be willing to take either side of a bet, i.e., if the agent is not willing to bet on the state sj at odds of a:b, then he should be willing to bet on not-ssj at odds of b:a. But this assumption makes too heavy demands on people’s willingness to make bets. One is often not willing to accept either of the two bets.22 In our opinion, this is explained by the fact that the estimated probabilities of the different states of nature are unreliable and one is not willing to take the risk connected with 20

This measure is assumed to be unique up to a positive linear transformation. Levi (1974, 1980) has generalized another dimension of the traditional Bayesian decision theory by allowing sets of utility functions that are not linear transformations of each other. We believe that this generalization is beneficiary in some contexts, but we will not discuss it further in the present paper. 21 It is interesting to note that de Finetti (1980), p. 62, note (a), has recognized some problem in such an operational definition as a way of representing the agent's beliefs. 22 The central axiom is the so-called coherence criterion which assumes that if the agent is willing to bet on state sj at the least odds of a:b, then he should be willing to bet on not-ssj at odds of b:a. The first of these betting ratios will thus be equal to one minus the second betting ratio, i.e. a/(a + b) = 1- b/(a + b). Smith (1965), among others, points out that this assumption need not be satisfied. An agent may very well be willing to bet on least odds of a:b for sj, but at the same time bet on least odds of c:d against sj, where a/(a + b)  1- c/(c + d), which contradicts the coherence criterion.

DECISION THEORY

15

this uncertainty.23 This criticism is directed against the assumptions behind the Dutch book theorem, but similar criticism can be constructed against other arguments in favor of the assumption of representing beliefs by a unique probability measure. In this chapter we will relax this assumption and, as a first step in the description of the beliefs that are relevant in a decision situation, we instead assume that the beliefs about the states of nature can be represented by a set P of probability measures. The intended interpretation of the set P is that it consists of all epistemically possible probability measures over the states of nature, where we conceive of a probability measure as epistemically possible if it does not contradict the decision maker’s knowledge in the given decision situation.24 In this way, we associate with each state sj a set of probability values P(ssj), where P  P. The values may be called the epistemically possible probabilities of the state sj. For simplicity, we will assume that the probabilities of the outcomes oij are independent of which alternative is chosen, so that P(oij) = P(ssj), for all P  P and all alternatives aj. Since this assumption can be relaxed, the decision theory to be presented can be extended to the more general case.25 The idea of representing a state of belief by a class of probability measures is not new but has been suggested by various authors. (See, e.g., Dempster (1967), Good (1962), Smith (1961) and (1965).) It has been most extensively discussed by Levi (1974) and (1980), but, as will be seen in the sequel, he does not use the class of probability measures in the same way as we do. Levi also assumes that the set of probability measures is convex, i.e., that if P and P’ are two measures in the set, then the measure a·P + (1-a)·P’ is also in the set, for any a between 0 and 1.26 The motivation for this assumption is that if P and P’ both are possible probability distributions over the states, then any mixture of these distributions is also possible. We will 23

This aspect of risk taking will be further discussed in the next section. In the present paper we do not aim at an elaborate analysis of the concept of knowledge, but we take this as a primitive notion. 25 In this paper we use a decision theory similar to Savage's (1972). We thus deliberately exclude problems connected with conditional probabilities and probabilities of conditionals. One reason for this is that it is rather straightforward to generalize a decision theory based on such probabilities in the same way as we have generalized Savage's theory. The second reason is that Luce and Krantz (1971) have shown that in decision situations with only finitely many states and outcomes it is possible to translate a decision situation containing conditional probabilities into a Savage type situation (and vice versa). For a discussion of this result, cf. Jeffrey (1977). As is easily seen, this result also holds for sets P of probability measures. 26 Levi (1980), p. 402, requires that the set of ‘permissible’ probability measures be convex. The interpretation of ‘permissible’ is discussed in section 4. In this connection it is interesting to note that Savage (1972), p. 58, note (+), mentions that “one tempting representation of the unsure is to replace the person’s single probability measure P by a set of such measures, especially a convex set.” 24

16

THE DYNAMICS OF THOUGHT

discuss the requirement of convexity in section 2.5. If P is assumed to be convex, then the set of epistemically possible probabilities associated with a state sj by the elements of P will form an interval from the lowest probability assigned to sj to the highest. Some authors have taken such intervals as basic when describing beliefs about the states of nature – to each state is assigned a probability interval and this assignment is governed by some consistency restrictions.27 The representation by a convex set of probability measures is, however, more general, since from such a set one can always compute a unique set of associated intervals, but starting from an assignment of consistent probability intervals, there will in general be a large number of convex sets of probability measures that will generate the intervals.28 We believe that not all of an agent’s beliefs about the states of nature relevant to a decision situation can be captured by a set P of probability measures. As a second element in describing the beliefs relevant to a decision situation, we introduce a (real-valued) measure U of the epistemic reliability of the probability measures in P. Even if several probability distributions are epistemically possible, some distributions are more reliable – they are backed up by more information than other distributions. The measure U is intended to represent these different degrees of reliability. In the introductory examples, Miss Julie ascribes a much greater epistemic reliability to the probability distribution where each player has an equal chance of winning in match A where she knows a lot about the players than in match B where she knows nothing relevant about the players. In match C, where she knows that one player is superior to the other, but not who, the epistemically most reliable distributions are the two distributions where one player is certain to win. Since there are only two relevant states of nature in these examples, viz. the first player wins (s1) and the second player wins (s2), a probability distribution can be described simply by the probability of one of the states. We can then illustrate the epistemic reliability of the various distributions in the three matches by diagrams as in figure 2.1.

27

See, e.g., Dempster (1967), Edman (1973), Ekelöf (1982), Gärdenfors (1979b), Good (1962), Halldén (1973), Smith (1961, 1965). 28 We say that a set of probability intervals associated with the states of a decision situation is consistent if and only if, for any state si and for any number x within the interval associated with si, there is a combination of numbers, which lie within the intervals associated with the remaining states, such that the sum of x and these numbers equal 1. Levi (1974), p. 416-417, gives an example which shows that there may be two decision situations with the same alternatives, states and outcomes, but with different sets of 'permissible’probability measures, which give different decisions when his decision theory is used, although the intervals that can be associated with the states are identical.

DECISION THEORY

17

Figure 2.1: A representation of the belief states for the three matches.

Even if examples such as these illustrate the use of the measure U, its properties should be specified in greater detail. Technically, the only property of U that will be needed in this chapter is that the probability distributions in P can be orderedd with respect to their degrees of epistemic reliability. However, it seems natural to postulate that U has an upper bound, representing the case when the agent has complete information about a probability distribution, and a lower bound, representing the case when the agent has no information at all about these distributions. However, we will not attempt a full description of the properties of the measure U, since we believe that this can be done only in a more comprehensive decision theory in which the relations between different decision situations are exploited.29 A fundamental feature of the epistemic reliability of the probability distributions possible in a decision situation, as we conceive of the measure, is that the less relevant information the agent has about the states of nature, the less epistemic reliability will be ascribed to the distributions in P. Where little information is available, therefore, all distributions will, consequently, have about the same degree of epistemic reliability. Conversely, in a decision situation where the agent is well informed about the possible states of nature, some distributions will tend to have a considerably higher epistemic reliability than others. A problem that strongly supports our thesis that the measure of epistemic reliability is a necessary ingredient in the description of a decision situation is Popper’s paradox of ideal evidence. Popper asks us to consider the following example (1974), pp. 407-408):

29

An interesting possibility is to take U to be a second order probability measure, i.e. to let U be a probability measure defined over the set P of epistemically possible probability measures. If P is finite there seem to be no problems connected with such a measure. But if P is taken to be a convex set of probability measures and we at the same time want all measures in P to have a non-zero second order probability, we run into problems. However, nothing that we have said excludes the possibility of taking U to be a non-standard probability measure. For a discussion of such measures see Bernstein and Wattenberg (1969).

18

THE DYNAMICS OF THOUGHT Let z be a certain penny, and let a be the statement ‘the n th (as yet unobserved) toss of z will yield heads’. Within the subjective theory, it may be assumed that the absolute (or prior) probability of the statement a is equal to 1/2, that is to say, (l) P(a) = 1/2 Now let e be some statistical evidence; that is to say, a statistical report, based upon the observation of thousands or perhaps millions of tosses of z; and let this evidence e be ideally favorable to the hypothesis that z is strictly symmetrical.... We then have no other option concerning P(a, e) than to assume that (2) P(a,e)= 1/2 This means that the probability of tossing heads remains unchanged in the light of the evidence e; for we now have (3) P(a) = P(a, e). But, according to the subjective theory, (3) means that e is, on the whole (absolutely) irrelevant information with respect to a. Now this is a little startling; for it means, more explicitly, that our so-called ‘degree of rational belief’ in the hypothesis, a, ought to be completely unaffected by the accumulated evidential knowledge, e; that the absence of any statistical evidence concerning z justifies precisely the same ‘degree of rational belief’ as the weighty evidence of millions of observations which, prima facie, support or confirm or strengthen our belief.

The ‘subjective theory’ that Popper is referring to in this example is what we have here called strict Bayesianism. Now, with the aid of the models of decision situations presented above, we can describe Popper’s example in the following way. There is a set P of possible probability measures concerning the states of nature described by a and not-a. If one is forced to state the probability of a, before the evidence e is obtained, the most reliable answer seems to be 1/2. The degree of epistemic reliability of this estimate is, however, low, and there are many other answers which seem almost as reliable. After the evidence e is obtained, the most reasonable probability assessment concerning a is still 1/2, but now the distribution associated with this answer has a much higher degree of epistemic reliability than before, and the other distributions in P have correspondingly lower degrees of reliability. It should be noted that this distinction between the two cases cannot be formulated with the aid of the set P only, but the measure U of epistemic reliability is also necessary.30 We will conclude this section by briefly mentioning some related attempts to extend models of belief by some measure of ‘reliability.’31 30

We can compare this example with the difference between match A and match B in the introductory example. The reliability measure connected with match A which is depicted in figure 2.l, can be seen as corresponding to the reliability measure after the evidence e has been obtained, and the measure connected with match B as corresponding to the reliability measure before e is obtained. Ideas similar to those presented here have been discussed by Bar-Hillel (1979) and Rosenkrantz (1973). 31 Models of belief similar to ours have been presented in terms of fuzzy set theory by Watson

DECISION THEORY

19

Keynes (1921, p. 71) introduced an interesting concept: As the relevant evidence at our disposal increases, the magnitude of the probability of the argument may either decrease or increase, according as the new knowledge strengthens the unfavorable or favorable evidence; but something seems to have increased in either case – we have a more substantial basis upon which to rest our conclusion. I express this by saying that an accession of new evidence increases the weight of an argument. New evidence will sometimes decrease the probability of an argument, but it will always increase its “weight.”

Here Keynes writes about the probability of an argument, while we are concerned with probability distributions over states of nature. Even if the intuitions behind Keynes’s ‘weight of evidence’ and our ‘epistemic reliability’ are related, it is difficult to say how far this parallel can be drawn. Carnap (1950, pp. 554-555) discusses ‘the problem of the reliability of a value of degree of confirmation’ which obviously is the same as Keynes’s problem. Carnap remarks that Keynes’s concept of ‘the weight of evidence’ was forestalled by Peirce (1932, p. 421) who mentioned it in the following way: [T]o express the proper state of belief, not one number but two are requisite, the first depending on the inferred probability, the second on the amount of knowledge on which that probability is based.

The models of the agent’s beliefs about the states of nature in a decision situation that have been presented here contain the two components P and U, i.e., the set of epistemically possible probability distributions, and the measure of epistemic reliability. These two components can be seen as an explication of the two numbers required by Peirce.32 2.3 A decision theory The models of decision situations that were outlined in the previous section will now be used as a basis for a theory of decision. This theory can be seen as a generalization of the Bayesian rule of maximizing expected utility. A decision, i.e. a choice of one of the alternatives in a decision situation, will be arrived at in two steps. The first step consists in restricting the set P to a set of probability measures with a ‘satisfactory’ degree of epistemic et al. (1979) and Freeling (1980). However, we will not consider this theory in the present paper. 32 In the quotation above it is obvious that Popper uses the traditional definition of relevance, i.e., e is relevant to a if and only if P(a)  P(a, e). We believe that this definition is too narrow. Instead we propose the definition that e is relevant to a iff P(a)  P(a, e) or the evidence e changes the degree of epistemic reliability of P. Keynes is also dissatisfied with the traditional definition of ‘relevance.’ He wants to treat ‘weight of evidence’ and ‘relevance’ as correlative terms so that “to say that a new piece of evidence is ‘relevant’ is the same thing as to say that it increases the ‘weight’ of the argument” (1921, p. 72). For a proof that Keynes's definition of ‘relevance’ leads to a trivialization result, and for a discussion of some general requirements on a definition of ‘relevance’ the reader is referred to Gärdenfors (1978a).

20

THE DYNAMICS OF THOUGHT

reliability. The intuition here is that in a given decision situation certain probability distributions over the states of nature, albeit epistemically possible, are not considered as serious possibilities. For example, people do not usually check whether there is too little brake fluid in the car or whether the wheels are loose before starting to drive, although, for all they know, such events are not impossible, and they realize that if any such event occurred they would be in danger. Now examples of this kind seems to show that what the agent does is to disregard certain states of nature rather than probability distributions over such states. But if a certain state of nature is not considered as a serious possibility, then this means that all probability distributions that assign this state a positive probability are left out of consideration. And there may be cases when some probability distributions are left out of account even if all relevant states of nature are considered to be serious possibilities. So, restricting the set P is a more general way of modeling the process than restricting the set of states. Deciding to consider some distributions in P as not being serious possibilities means that one takes a risk. The less inclined one is to take risks, the greater the number of distributions in P will be that are taken into account when making the decision. A fundamental question is how the agent determines which probability distributions in P are ‘satisfactorily reliable’ and which are not. In our view, the answer is that the measure U of epistemic reliability should be used when selecting the appropriate subset P/ P Uo of P. The agent selects a desired level Uo of epistemic reliability and only those probability distributions in P that pass this U-level are included in P/ P Uo, but not the others.33 P/ P Uo can be regarded as the set of probability distributions that the agent takes into consideration in making the decision. An obvious requirement on the chosen level of reliability is, of course, that there be some distribution in P that passes the level. Which “desired level of epistemic reliability” the agent will choose depends on how large the risks are he is willing to take. The more risk aversive the agent is, the lower the chosen level of epistemic reliability will be. It is important to note that two agents in identical epistemic situations, here identified by a set P and a measure U, may indeed choose different values of Uo depending on their different risk taking tendencies. This is the reason why it is assumed that U yields an ordering of P and not merely a dichotomy. If the agent is willing to take a maximal risk as regards which probability distribution to consider as ‘satisfactory’ it may happen that there 33

The ‘desired level of epistemic reliability’ can be interpreted in terms of levels of aspiration so that the Uo chosen by the decision maker is his level of aspiration as regards epistemic reliability

DECISION THEORY

21

will be only one distribution which passes the desired level of reliability. After such a choice his relevant information about the states of nature will be of the same type as for the strict Bayesian, i.e. a unique probability measure, but this situation will arise for quite different reasons. The second step in the decision procedure starts with the restricted set P Uo of probability distributions. For each alternative ai and each probability P/ distribution P in P/ P Uo the expected utility eikk is computed in the ordinary way. The minimal expected utility of an alternative ai, relative to a set P/ P Uo, is then determined, this being defined as the lowest of these expected utilities eik. Finally the decision is made according to the following rule:34 The maximin criterion for expected utilities (MMEU): U The alternative with the largest minimal expected utility ought to be chosen. In order to illustrate how this decision procedure works, we will return to the introductory examples. For simplicity, let us, for all three tennis matches, denote by s1 the event that the first player to serve wins the match, and by s2 the event that the other player wins. Now assume that Miss Julie is offered the following bet for each of the three matches: She wins 30$ if s1 occurs and loses 20$ if s2 occurs. For each match, she must choose between the alternative a1 of accepting the bet and the alternative a2 of declining it. Let us furthermore assume that Miss Julie’s utilities are mirrored by the monetary values of the outcomes. In match A, where Miss Julie is very well informed about the players, she considers that the only probability distribution that she needs to take into consideration is the distribution P1 where P1(s1) = P1(s2) = 0.5. She is willing to take the (small) risk of letting P/ P Uo consist of this distribution only. The only, and hence minimal, expected utility to compute is then 0.5·30 + 0.5·(20) for a1 and 0.5·0 + 0.5·0 for a2. Hence, according to MMEU, she should accept the bet in match A. In match B, where she has no relevant information at all, the epistemic reliability of the epistemically possible distributions is more evenly spread out. Consequently, Miss Julie is not so willing to leave distributions out of account when forming the subset P/ P Uo as in the previous case. For simplicity, let us assume that P/ P Uo = {P1, P2, P3}, where P1 is as before, P2 is defined by P2(s1) = 0.25 and Ps(s2) = 0.75, and P3 is defined by P3(s1) = 0.75 and P3(s2) = 0.25. The expected utilities for the alternative a1 of accepting the bet, determined by these three distributions, are 5, -7.5 and 17.5 respectively; while the expected utilities for the alternative a2 of not accepting the bet are 0 in all three cases. Since the minimal expected utility of a1 is less than that of a2, the MMEU criterion demands that a1 be the chosen alternative, i.e. 34

Cf. Gärdenfors (1979b), p. 169.

22

THE DYNAMICS OF THOUGHT

Miss Julie should decline the bet offered. In match C, it is reasonable to assume that P/ P Uo contains some probability distribution that assigns s1 a very high probability and some distribution that assigns it a very low probability. A similar analysis as above then shows that Miss Julie should not accept the bet in this case either. This example can be heuristically illustrated as in figure 2.2. In this figure the broken horizontal line indicates the desired level of epistemic reliability.

Figure 2.2: The level Uo determines the set P/ P Uo (indicated by thick lines on the horizontal axis).

To give a further illustration of the decision theory, it should be noted that the hypothesis from the Goldsmith-Sahlin experiments, mentioned in the introduction, is well explained by the MMEU criterion. When an agent is asked to choose between tickets in two lotteries which are estimated to have the same primary probability of winning, he should choose the ticket from the lottery with the epistemically most reliable probability estimate, since this alternative will have the highest minimal expected utility.35 Still other applications of the decision theory will be presented in the next two sections. A limiting case of information about the states of nature in a decision situation is to have no information at all. In the decision models presented here, this would mean that all probability distributions over the states are epistemically possible and that they have equal epistemic reliability. In such a case, the minimal expected utility of an alternative is obtained from the distribution that assigns the probability 1 to the worst outcome of the 35

The results of the majority of subjects of the Goldsmith-Sahlin experiments support the thesis that the degree of epistemic reliability of the probability estimates is an important factor when choosing lottery tickets, but, it should be admitted that not all of these results can be explained by the MMEU criterion. The main reason for this is, in our opinion, that the agent's values are not completely described by a utility measure. In this paper we have concentrated on the epistemic aspects of decision making and used the traditional way, i.e. utility measures, to represent the values of the decision maker. We believe that this part of the traditional Bayesian decision theory should be modified as well, perhaps by including a ‘level of aspiration,’ but such an extension lies beyond the scope of this paper.

DECISION THEORY

23

alternative. This is, however, just another way of formulating the classical maximin rule, which has been applied in what traditionally has been called ‘decision making under uncertainty’ (a more appropriate name would be ‘decision making under ignorance’). Hence, the classical maximin turns out to be a special case of the MMEU criterion. At the other extreme, having full information about the states of nature implies that only one probability distribution is epistemically possible.36 In this case the MMEU criterion collapses into the ordinary rule within strict Bayesianism, i.e. the rule of maximizing expected utility, which has been applied to what traditionally, but somewhat misleadingly, has been called ‘decision making under risk.’ The decision theory that has been presented here thus covers the area between the traditional theories of ‘decision making under uncertainty’ and ‘decision making under risk’ and it has these theories as limiting cases. 2.4 Relation to earlier theories Several authors have proposed decision theories that are based on more general ways of representing the decision maker’s beliefs about the states than what is allowed by strict Bayesianism. The most detailed among these is Levi’s (1974, 1980) theory, which will be discussed in a separate section. In this section we will compare the present theory with some earlier statistical decision theories. Wald (1950) formulates a theory of ‘statistical decision functions’ where he considers a set W of probability measures and a ‘risk’ function. He says that “the class W is to be regarded as a datum of the decision problem” (p. 1). He also notes that the class W will generally vary with the decision problem at hand and that in most cases “will be a proper subset of the class of all possible distribution functions” (p. 1). In the examples, W is often taken to be a parametric family of known functional form. A risk function is a function that determines the ‘cost’ of a wrong decision. Such a function can be seen as an inverted utility function restricted to negative outcomes. On this interpretation it is easier to compare Wald’s theory to the present decision theory. Wald suggests two alternative decision rules. The first is the traditional Bayesian method when an ‘a priori’ probability measure in W can be selected, or, as Wald puts it, when “it exists and is known to the experimenter” (p. 16). The second case is when the entire set W is employed to determine the decision. Wald suggests that in such cases one should 36

Such a distribution may, in many cases, assign the probability 1 to one of the states, but we do not assume that it always will. We interpret ‘full information’ in a pragmatic way, meaning something like ‘having as much information as is practically possible,’ so, even if the world is deterministic, having full information does not entail that one knows which is the true state of nature.

24

THE DYNAMICS OF THOUGHT

“minimize the maximum risk.” In our terminology, using utility functions instead of risk functions, this is the same as maximizing the minimal expected utility with respect to the set W. If Wald’s theory is interpreted as above, the difference between his and our theory is mainly of epistemological character. Since Wald does not say anything about how W is to be determined it is difficult to tell whether it corresponds to our set P of epistemically possible probability functions or to the set P/ P Uo of ‘reliable’ functions. In particular, be does not introduce any factor corresponding to the measure U of epistemic reliability, nor does be associate the choice of W with any form of risk taking. Hurwicz (1951) apparently interprets Wald’s set W as corresponding to our set P. He notes that sometimes some of the distributions in W are more ‘likely’ than others (p. 343). For example, assume that W consists of all normal distributions with mean zero and standard deviation V. In a particular decision situation evidence at hand may support the assumption that V is considerably small. It thus seems reasonable to select a proper subset W0 of W which is restricted to those distributions with standard deviation less than or equal to some value V0. Hurwicz assumes that such a subset W0 of W can be selected in any decision situation. He then suggests a ‘generalized Bayes-minimax principle’ that amounts to using W0 as the base when maximizing the minimal expected utility (minimizing the maximal risk). Obviously, the set W0 corresponds closely to our set P/ P Uo. The main difference between Hurwicz’ theory and the present one is that be does not give any account of how the set W0 is to be determined. In particular he, as Wald, does not introduce any factor corresponding to the measure U. Hodges and Lehmann (1952) suggest an alternative to Wald’s minimax solution that they call a ‘restricted Bayes solution.’ It is of interest here since it is adopted by Ellsberg (1961) as a solution to his ‘paradox.’ Let us start by considering Ellsberg’s problem. Ellsberg (1961, pp. 653-654) asks us to consider the following decision problem. Imagine an urn known to contain 30 red balls and 60 black and yellow balls, the latter in unknown proportion. One ball is to be drawn at random from the urn. In the first situation you are asked to choose between two alternatives a1 and a2. If you choose a1 you will receive $100 if a red ball is drawn and nothing if a black or yellow ball is drawn. If you choose a2 you will receive $100 if a black ball is drawn, otherwise nothing. In the second situation you are asked to choose, under the same circumstances, between the two alternatives a3 and a4. If you choose a3 you will receive $100 if a red or a yellow ball is drawn, otherwise nothing and if you choose a4 you will receive $100 if a black or yellow ball is drawn, otherwise nothing. This decision problem is shown in the following decision matrix.

DECISION THEORY

a1 a2 a3 a4

Red $100 $0 $100 $0

Black $0 $100 $0 $100

25

Yel1ow $0 $0 $100 $100

that a1 is preferred to a2 and a4 is preferred to a3. As Ellsberg notes, this preference pattern violates Savage’s (1972) ‘sure thing principle’ (postulate P2), which requires that the preference ordering between a1 and a2 be the same as the ordering between a2 and a4. When applying the present decision theory to this problem, the main step P Uo. The set P should be the same in the two decision is to determine the set P/ situations, since they do not differ with respect to the information about the states. Now, unless the decision maker believes that he is being cheated about the content of the urn, P is most naturally taken as the class of distributions (1/3, x, 2/3-x), where x varies from 0 to 2/3. If the decision maker chooses a low Uo, P/r P o will contain most of the distributions in this class. For simplicity, let us for the moment assume that P Uo = P. With this choice, the minimal expected utilities of the alternatives P/ are 1/3·u($l00), 1·u($0), 1/3·u($l00) and 2/3·u($100), for a1, a2, a3 and a4, respectively. Assuming that 1/3·u($100) is greater than u($0), the MMEU criterion requires that a1 be preferred to a2 and a4 to a3, which accords with Ellsberg’s findings. Intuitively, a2 and a3 involve greater ‘epistemic risks’ than a1 and a4 – thus a2 and a3 are avoided by most subjects. This feature is well captured by the present decision theory. If the decision maker is willing to take an epistemic risk and chooses a P Uo. If the decision maker higher ro, fewer distributions will be included in P/ has no further information about the distribution of black and yellow balls, then, because of symmetry, it is likely that he judges the distribution (1/3, 1/3, 1/3) as being the highest in the P-ordering. If this is the only distribution included in P/ P Uo, he should be indifferent between a1 and a2 and between a3 and a4 according to the MMEU criterion. In this case Savage’s sure thing principle is not violated.37 In order to explain the ‘paradox’ that decision makers do not act 37 However, if the agent has some information that he judges relevant for the distribution of the black and yellow balls, then the epistemic reliability of the distributions in P may be quite different. He may, for example, believe that the distributions in P/ P Uo cluster around e.g. (1/3, 1/2, 1/6). Then the MMEU criterion will recommend that a2 be preferred to al and that a4 be preferred to a3. This recommendation does not conflict with Savage's sure-thing principle either (cf. Gärdenfors and Sahlin (1983)).

26

THE DYNAMICS OF THOUGHT

according to Savage’s sure thing principle, Ellsberg (1961, p. 661) first introduces for a decision maker in a given situation a set Y0 of probability distributions “that still seems “reasonable,” reflecting judgments that he “might almost as well” have made, or “that his information – perceived as scanty, unreliable, ambiguous – does not permit him confidently to rule out.” Ellsberg also considers a particular probability distribution y0, the “estimated” probability distribution, which can be viewed as the distribution a strict Bayesian decision maker would have adopted in the decision situation. Ellsberg also ascribes to y0 a degree Ue (e for Ellsberg) of the decision maker’s ‘confidence’ in the estimate. The decision rule suggested by Ellsberg, which is the restricted Bayesian solution developed by Hodges and Lehmann, can now be described as follows: Compute for each action aj the expected utility according to the distribution y0 and the minimal expected utility relative to the set Y0. Associate with each action an index based on a weighed average of these two factors, where Ue is the weight ascribed to the first factor and 1 - Ue is the weight ascribed to the latter factor. FinalIy, choose that act with the highest index. When comparing this decision theory with the theory presented in this chapter, one notes that there are differences both of epistemological and formal character. Firstly, since Ellsberg, like his predecessors, does not say anything about how the set Y0 is to be determined, and it is therefore difficult to say whether it corresponds to our set P/ P Uo. Secondly, even if we identify Y0 with P/ P Uo, Ellsberg exploits the degree Ue of ‘confidence,’ which is defined for only one distribution y0, in a way that differs considerably from our use of the measure U, which is assumed to be defined for all distributions in P. In particular we need not assume that U gives a numerical value, only that it orders the distributions in P. Thirdly, since the decision rules are different, the theories will recommend different decisions in many situations. The most important disagreement here is that we reject the need P Uo has been of an estimated distribution y0. We believe that once the set P/ selected, the distribution with the highest degree of reliability (corresponding to Ellsberg’s y0) does not play any outstanding role in the decision making. This difference between the two theories is in principle testable, assuming that Ue is not always close to zero. The experiment performed by Becker and Brownson (1964) is relevant here. They offered subjects ambiguous bets differing in the range of probabilities possible but of equal expected value and found that subjects were willing to pay money for obtaining bets with narrower ranges of probability. This finding seems to highlight the importance of a measure of reliability. But they did not, however, obtain support for Ellsberg’s hypothesis that the distribution y0 is relevant for the decision making.

DECISION THEORY

27

2.5 A comparison with Levi’s theory In this section we compare our theory with Levi’s theory that is presented in Levi (1974) and elaborated on in Levi (1980). He starts out from a description of the decision maker X’s information at the time t about the states of nature. This information is contained in a convex set BX,tt of probability distributions. The distributions in BX,t are, according to Levi, the permissible distributions. As to the meaning of “permissible,” he offers only indirect clarification by indicating the connections between permissibility and rational choice. In order to compare the theories, we will here assume P Uo (or its convex hull) as presented that the set BX,tt corresponds to the set P/ in section 2.3. Levi also generalizes the traditional way of representing the utilities of the outcomes by introducing a class G of permissible utility measures, such that not all of these measures need be linear transformations of one another. An alternative ai is said to be E-admissible if and only if there is some probability distribution P in BX,tt and some utility function u in G such that the expected utility of ai relative to P and u is maximal among all the available alternatives. A first requirement on the alternative to be chosen in a given decision situation is then that it should be E-admissible. The second step in Levi’s decision procedure concerns the opportunity to defer decision between two or more E-admissible alternatives. He argues that a rational agent should “keep his options open” whenever possible. An alternative is said to be P-admissible if it is E-admissible and it is best with respect to E-admissible option preservation. Levi does not, however, explicate what he means by “best” here, and we will ignore the effects of this requirement here, since we have not imposed any structure on the set of alternatives. Let us say that a P-admissible alternative ai is security optimal relative to a utility function u if and only if the minimum u-value assigned to some possible outcome oij of ai is at least as great as the minimal u-value as signed to any other P-admissible alternative. Levi then, finally, calls an alternative S-admissible if it is P-admissible and security optimal relative to some utility function in G. Levi (1980, p. 412) states that he “cannot think of additional criteria for admissibility which seems adequate” so he, tentatively, assumes that all Sadmissible alternatives are ‘admissible’ for the final choice, which then, supposedly, is determined by some random device. In order to illustrate the differences between the decision theory presented in the previous section and Levi’s theory, we will consider the following example that contains two states and three alternatives:

28

THE DYNAMICS OF THOUGHT

a1 a2 a3

s1 -10 11 0

s2 12 -9 0

In this matrix the numbers denote the utilities of the outcomes. Assume that the set P/ P Uo, which here is identified with Levi’s set BX,tt consists of the two probability distributions P, defined by P(s1) = 0.4 and P(s2) = 0.6, and P’, defined by P’(s1) = 0.6 and P’(s2) = 0.4, together with all convex combinations of P and P’. The minimal expected utility of a1 is -1.2 and the minimal expected utility of a2 is -1.0. The minimal expected utility of a3 is of course 0, so MMEU requires that a3 be chosen. In contrast to this, only a1 and a2 are E-admissible. P-admissibility has no effect here, but a2 is security optimal relative to the utility measure given in the matrix, so a2 is the only S-admissible alternative. Thus, according to Levi’s theory, a2 should be chosen. When the uncertainty about the states of nature in this decision situation, represented by P/ P Uo, is considered, we believe that a3 is intuitively the best alternative. Against this it may be argued that using a maximin principle is unnecessarily risk aversive. It should be remembered, however, that when P Uo restricting P to P/ P Uo the agent is already taking a risk and his choice of P/ indicates that he is not willing to take any further epistemic risks. On the other hand, Levi’s requirement of E-admissibility has the consequence that, in many cases, the choices made by his theory seems unrealistically optimistic. A strange feature of Levi’s theory is that if the previous decision situation is restricted to a choice between a2 and a3, then his theory recommends choosing a3 instead of a2! In their chapter on individual decision making under uncertainty, Luce and Raiffa (1957, pp. 288-290) introduces the condition of independence of irrelevant alternatives which in its simplest form demands that if an alternative is not optimal in a decision situation it cannot be made optimal by adding new alternatives to the situation. The example presented here shows that Levi’s theory does not satisfy this condition since in the decision situation where a2 and a3 are the only available alternatives and where a3 is optimal according to Levi’s theory, a2 can be made optimal by adding a1. It is easy to show, however, that the MMEU criterion that has been presented here satisfies the condition of independence of irrelevant alternatives. Levi’s theory also seems to have problems in explaining some of the experimental results considered in this chapter. If Levi’s theory is applied to Ellsberg’s two decision situations as presented earlier, it gives the result that

DECISION THEORY

29

both alternatives in the two situations are S-admissible, and hence that a1 is equally good as a2 and a3 is equally good as a4. This contrasts with Ellsberg’s findings that are in accordance with the recommendations of the present theory. Similar considerations apply to the experiments presented in Becker and Brownson (1964). Decision theories of the kind presented in this chapter are based on several idealizations and they will unavoidably be exposed to some refractory empirical material. We believe, however, that the considerations of this section show that the decision theory presented in section 2.3 is a more realistic theory than Levi’s. 2.6 Conclusion The starting-point of this chapter is that in many decision situation the assumption, made in the strict form of Bayesianism, that the beliefs of an agent can be represented by a single probability distribution is unrealistic. We have here presented models of belief which contain firstly, a class of probability distributions, and, secondly, a measure of the epistemic reliability of these probability distributions. Several authors before us have suggested that a class of probability distributions should be exploited when describing the beliefs of the agent. A main thesis of this chapter is that this is not sufficient, but an assessment of the information on which the class of probability distributions is based is also necessary. We have here tried to capture this assessment by the measure U of epistemic reliability. With the aid of this measure we can account for one form of risk taking in decision situations. On the basis of the models of the beliefs that are relevant in a decision situation we have formulated a decision theory. We have argued that this theory has more desirable properties and is better supported than other decision theories that also generalize the traditional Bayesian theory. The MMEU criterion, which has been suggested here as the main rule of the decision theory, is generally applicable to decision situations where the possible outcomes are non-negative from the point of view of the decision maker. However, there are situations where the MMEU criterion seems to be too cautious. For example, the ‘reflection effect’ and the ‘isolation effect’ suggested by Kahneman and Tversky (1979) cannot be explained directly with the decision theory of this chapter. We believe that in order to cover these phenomena a more general and comprehensive decision theory is needed which includes references to the decision maker’s ‘levels of aspiration.’ A special case of the effect of levels of aspiration would be the ‘shifts of reference point’ discussed by Kahneman and Tversky. Introducing ‘levels of aspiration’ means that the part of traditional Bayesian theory that refers to utilities has to be considerably extended and modified.

CHAPTER 3 RIGHTS, GAMES AND SOCIAL CHOICE 3.1 Introduction It has been argued by Amartya Sen that two of the most fundamental principles in evaluating social states – the Pareto principle and the libertarian claim – conflict with each other.38 The Pareto principle demands that if everyone prefers a certain social state to another, then the former state is better for the society than the latter. The libertarian claim is that everyone has a right to determine certain decisions by himself, no matter what others think. Sen’s proof of the “impossibility of a Paretian liberal” has provoked an extensive debate among economists, political scientists and philosophers. A fundamental assumption in Sen’s argument is that, for any set of individual preference orderings, a collective choice rule determines a social preference ordering of the possible social states. Both the Pareto principle and the condition of “minimal liberalism” are formulated as constraints on social preference relations. A consequence of this approach is that an individual right, in its most elementary form, is conceived of as a possibility of determining the social orderingg of a pair of social states. In all technical discussions that have followed Sen’s original result, this view of individual rights has hardly been questioned.39 I believe, however, that it is misleading. An alternative approach to individual rights, suggested by Robert Nozick (1974, pp. 165-66), is that such rights, if exercised, put constraints on the set of alternatives open to social choice. Nozick sees no conflict between social welfare and liberalism – for him rights take precedence over welfare. In this chapter, I will present a simple framework for rights and the exercising of rights in which Nozick’s suggestion is incorporated. A system of individual and collectivistic rights will be put into a game-theoretical setting where the choices of strategies by individuals or coalitions correspond to the exercising of different rights. This approach is fundamentally different from Sen’s and I believe that rights, and thereby also the libertarian claim, are more appropriately represented in my framework. By this new interpretation of the libertarian claim the paradox of the Paretian liberal will be resolved.

38 39

Sen (1970) was the first paper on the topic. A more recent survey article is Sen (1976). An exception is Bernholz (1974).

31

32

THE DYNAMICS OF THOUGHT

3.2 Rights and constraints Nozick argues that Sen’s approach to individual rights is the source of the impossibility result. He suggests the following remedy: Individual rights are co-possible; each person may exercise his rights as he chooses. The exercise of these rights fixes some features of the world. Within the constraints of these fixed features, a choice can be made by a social choice mechanism based upon a social ordering, if there are any choices left to make! Rights do not determine a social ordering but instead set the constraints within which a social choice is to be made, by excluding certain alternatives, fixing others, and so on. (Nozick, 1974, pp. 165-166)

According to Nozick, the choice of a social state proceeds in two steps: In the first, rights are exercised which restrict the available choice alternatives. In the second step a choice mechanism is applied to the remaining alternatives to determine the final social state. In this chapter I will concentrate on the first of these steps, but in the concluding section I will briefly comment upon the second and its connection with the first step. The basic conflict between Sen’s and Nozick’s views seems to be that while Sen tries to combine liberalism and welfarism by putting conditions on social welfare functions which simultaneously restrict the permitted outcome of such functions, Nozick claims that the rights of individuals have priority over welfare considerations such as the Pareto principle. This conviction that justice takes precedence over welfare Nozick shares with many thinkers – not surprisingly we find the same idea in Kant’s writings: The rights of men must be held sacred however great a sacrifice the ruling power may have to make. There can be no half measures here; it is no use devising hybrid solutions such as a pragmatically conditioned right halfway between right and utility. For all politics must bend the knee before right. (Kant, 1971, p. 125)

In the light of Sen’s impossibility theorem, an important question about this requirement of the priority of rights is whether it means that social welfare is not necessarily maximized. An affirmative answer to this question would not bother someone like Kant or Nozick. However, I will argue in a later section in connection with the function of the social choice mechanism that, at least as regards a reasonable interpretation of the Pareto principle, the requirement of the priority of rights does not mean that welfare cannot be maximized. An argument against Sen’s formulation of the libertarian claim that may be raised already here is that if individual rights determine the social ordering of a pair (or a set of pairs) of social states, then this ordering may not have any influence on the final social choice since this pair need not be among the top-ranked states. For this reason, such rights do not guarantee that individuals can determine decisions by their own actions. Such a guarantee is, no doubt, the essential point of the libertarian claim.40 Another way of supporting Nozick’s view of rights is to refer to 40

A similar argument is given by Bernholz (1974).

DECISION THEORY

33

philosophical and legal analyzes of the concept of a right. The type of individual right that Sen and Nozick seem to have in mind can be presented in the form “i may see to it that F” where F is not a particular social state, but a condition on (or a property of) a social state (see, e.g., Kanger and Kanger (1966) and Lindahl (1977)). Rights do not establish an ordering of the social states, but divide them into classes; if a right is exercised, some class of possible social states is excluded from further consideration and the remaining class of social states may be subject to further restrictions by the exercising of other rights. Another aspect of rights that has been swept under the carpet in the discussion around the paradox of the Paretian liberal is that also groups of individuals may be assigned rights that are different from the rights of the individuals in the group.41 The best example is the institution of contract. A contract requires more than one individual to be effective and a contractual right cannot be exhaustively analyzed in terms of individual rights. There is a strong libertarian tradition of free contract, and, as soon as individual rights are admitted, it therefore seems very natural to include also this kind of rights in an analysis of social decision situations. There are several different types of individual and collectivistic rights. In the main part of the chapter I will be concerned only with the simplest types of rights, but later I will devote a section to other kinds of rights and their relevance to the analysis of social choice situations. 3.3 Rights-systems After this discussion of the nature of rights, a simple formal model of a system of rights will be presented. There is a finite and non-empty set of individuals. Single individuals will be denoted i, j ... and non-empty subsets of I will be denoted G, G1, G2, ... . There is a set S of social states, containing at least two elements. Social states will be denoted x, y, z … and subsets of S will be denoted X, Y … . A rightt can now be described as a possibility for a group G of individuals to restrict the set of social states to a subset X of S. Such a right will be denoted by the ordered pair (G,X). It will be clear from the gametheoretical analysis to be developed in what sense a right is a ‘possibility.’ When G consists of a single individual i, the right will be denoted (i,X) instead of ({i},X). A rights-system is defined as a set of pairs (G,X). Not every set of pairs (G,X) qualifies as a reasonable rights-system. It is necessary to impose some conditions on which rights must be included and which may be combined in a coherent rights-system. A first demand is that different groups of individuals not be assigned conflicting rights. More 41 An exception is Batra and Pattaniak (1972). However, this paper does not consider the game-theoretical dynamics that arise when the rights of individual is compared to the rights of the coalitions he may enter.

34

THE DYNAMICS OF THOUGHT

precisely, this can be formulated as follows: Consistency condition: If L is a rights-system and if (G1,X1)  L and (G2,X2)  L where G1 and G2 are disjoint groups of individuals, then Xl ˆ X2  ‡. The interpretation of the condition is that if both groups G1 and G2 exercise their rights to restrict the set of available social states to X1 and X2 respectively, then there will still be some available social state. A consequence of the condition is that, for all (G,X) in a rights-system L, we have X  ‡. There is no reason to demand that the rights of a particular individual or a particular group should all be mutually compatible in the sense that if both (G,X1) and (G,X2) belong to the rights-system, then Xl ˆ X2  ‡. On the contrary, a common type of right is that an individual or a group may choose between restricting the available social states to X and restricting them to not-X, i.e., S-X, which means that both (G,X) and (G,S-X) may be included in a rights-system. For example, a person may have a right to read the Bible, and he may also have a right to abstain from reading it. It is also natural to demand that the rights of a group G, regarded as a coalition, contain all the rights assigned to the individuals in G and subgroups of G. Condition on the rights of groups: If L is a rights-system such that (G1,X)  L and if G1 Ž G2, then (G2,X)  L. On the other hand, the rights of a group G is, normally, not just the union of the rights of the individuals in G, but the group may agree on contracts and have other forms of collectivistic rights which essentially extend the power of the group beyond the individual rights. The libertarian claim that everyone has a right to determine certain decisions by himself can, in the present framework, be formulated simply as follows: Condition of minimal libertarianism: If L is a rights-system, then, for every i in I, there is some X that is a proper subset of S such that (i,X)  L. I believe that this condition covers the intention behind Sen’s condition of “minimal liberalism” which he formulated in terms of restrictions on, social preference relations. In contrast to Sen’s condition, the present formulation guarantees that an individual may freely decide some, features of the social state that is finally chosen by the society.

DECISION THEORY

35

The game-theoretical analysis to be presented in the sequel will be simplified if it is assumed that the rights of an individual or a group are independent in the sense that rights, which do not exclude each other, may be combined without restrictions. This assumption can be made precise in the following way: Condition on combination of rights: If L is a rights-system such that (G,X1)  L and (G,X2)  L and Xl ˆ X2  ‡, then (G, Xl ˆ X2)  L. This condition is not valid in general, even if only the simplest types of rights are considered. I may, for example, at a certain point of time have the right to drink a bottle of whisky and also have the right to drive my car, but I may not exercise both of these rights at the given point of time. The concept of a rights-system will next be illustrated by an example that is borrowed from Allan Gibbard and which will serve as a paradigm example in this chapter. Gibbard presents the example in the following way: Angelina wants to marry Edwin but will settle for the judge, who wants whatever she wants. Edwin wants to remain single, but would rather wed Angelina than see her wed the judge ... First, Angelina has a right to marry the willing judge instead of remaining single. ... Next, Edwin has the right to remain single rather than wed Angelina. (Gibbard, 1974, p. 348)

It is clear from the example, although Gibbard does not state it explicitly, that Angelina also has the right to remain single, both Edwin and Angelina may waive their rights, and that Edwin and Angelina, as a group, have the contractual right to marry each other. In this example, the set S consists of three alternatives. Denoting “Edwin weds Angelina” as x, “the judge weds Angelina and Edwin remains single” as y, and “both Edwin and Angelina remain single” as z, the present rightssystem can be described as the set of the following pairs: (Edwin, {y, z}), (Edwin, S), (Angelina, {y}), (Angelina, {z}), (Angelina, S), ({Edwin, Angelina}, {x}) and for the group {Edwin, Angelina} also the individual rights assigned to Edwin and Angelina (according to the condition on the rights of groups). It is easy to check that this rights-system satisfies all conditions introduced so far. The rights-systems presented here do not account for any temporal aspects of rights. It is assumed that the rights of an individual or a group are not dependent on whether other individuals exercise or waive their rights. This is, of course, a simplification, since the rights of an individual or a group often changes with the situation, and the situation changes as soon as some persons exercise or waive their rights. Everyone may, for example, sit in the left corner on the park-bench that is now empty, but as soon as someone exercises his right to sit there, the others loose their rights. As will be clear from the sequel, this simplification will make the game-theoretical

36

THE DYNAMICS OF THOUGHT

analysis straightforward. 3.4 Rights and rights exercising Sen connects rights and social choice by requiring that if an individual has a right to x over y and he prefers x to y, then x is socially better than y. Gibbard (1974, pp. 398-400) criticizes this criterion by pointing out that one g a right and exercisingg it. He argues must make a distinction between having that it may be irrational for any individual to exercise a right to x over y (in Sen’s sense), even if he prefers x to y. Gibbard’s argument can be carried over to the types of rights presented here. We can illustrate this by returning to the Edwin-Angelina example. Edwin prefers the alternative z to x and these are both better than y. Angelina prefers x to both y and z, but she finds y definitely better than z. It is now not rational for Edwin to exercise his right to remain single and restrict the alternatives to {y, z}, because he can then expect that Angelina exercises her right to marry the judge, i.e. to restrict the set of available social states to y. Instead, since both of them prefer x to y, they should agree to marry each other, i.e. to exercise their contractual right. In order to guarantee that an individual must not exercise a right but may waive it, the following condition on a rights-system may be added to those presented earlier: Condition on waiving of rights: If L is a right-system, then for each individual i, (i,S)  L. Without this condition the rights guaranteed by the condition of minimal libertarianism might not be rights but rather duties, forcing an individual to restrict the set of available social states. Such rights will be discussed in section 3.9. A central problem is now to give rationality criteria for when an individual or a group ought to exercise a right and when the right ought to be waived. I will outline a solution to this problem by putting the choice in a game-theoretical setting. 3.5 Preferences What rights an individual ought to exercise in a given situation are, among other things, dependent on how he values the different social states. These values can be represented in many ways – by various preference relations or utility measures – depending on how strong assumptions one is willing to make. I will here avoid utility measures in order to rely on as weak assumptions as possible. A minimal presumption concerning individual values is that each individual can order the social states in a preference ordering. I will here

DECISION THEORY

37

assume that these individual preferences relations are weak orderings of S, i.e. transitive and complete orderings of the set of possible social states. The preference relation associated with individual i will be denoted Ri and from this relation we define a strict ordering Pi in the usual manner, i.e. xPix iff xRiy and not yRix. It is, however, not sufficient to introduce preference orderings over single alternatives since rights determine sets of social states and not single states, so the consequences of exercising a right must be evaluated in relation to which set of social states it determines. For this reason I will, for each individual i, introduce a preference relation Ri over non-empty sets of social states. It will be assumed that Ri is transitive and reflexive, but it need not be connected. In the same way as X iY iff above, a strict relation Pi can be introduced by the stipulation that XP Y iX. X iY and not YR XR In connection with investigations of the manipulability of social choice functions some criteria for connecting the relations Ri and Ri have been suggested (cf. e.g. Gärdenfors (1979a)). The extent to which the relation Ri can be determined from Ri is mainly dependent on what information i has about how the final choice of a social state will be made. The relevant information is which social choice mechanism is used and what preferences the other individuals have. Here, I will only make some very weak assumptions about the connections between Ri and Ri that can be defended even if i has no information beyond his own preferences. It will be assumed that (i) {x}Ri{y} iff xRiy and (ii) if xRiy for all x  X and y  Y, then XR X iY. 3.6 Games of rights exercising Individuals’ preferences over social states or sets of social states are not the only factors that determine whether it is rational for an individual or a group of individuals to exercise a certain right. The decision to exercise or waive a right is also dependent on what is known about what rights other individuals or groups will exercise. The exercising or waiving of a right assigned to an individual can be regarded as a move in a game, which can be met with countermoves by other individuals. A combination of choices to exercise or waive a set of rights can be seen as a strategy in the game. Since the condition on combinations of rights has been introduced it is possible to disregard the sequential or temporal aspects of a set of moves and the set of strategies available to an individual can simply be identified with the set of rights assigned to the individual. This simplification is the main reason why I have presumed the condition on combinations of rights. In a more detailed analysis, strategies

38

THE DYNAMICS OF THOUGHT

will be much more complex. The combination of the condition on waiving of rights with the condition of minimal libertarianism can be interpreted as saying that the game of rights exercising is non-trivial for each individual. It is natural to assume that the game is cooperative. Individuals may form coalitions or coordinate their choices of strategies. Since it has been assumed that also groups have rights, the set of strategies available to a group can, in the same way as for individuals, be identified with the set of rights assigned to the group. A consequence of the condition on the rights of groups is that the set of strategies available to a group, which in a sense is a measure of the power of the group, is at least as great as the rights of the individuals in the group taken together. And since the group will normally have additional collectivistic rights available as strategies, the formation of coalitions will be an essential part of the game. The outcome of a game of rights exercising can, for the present purposes, be identified with the set of social states that are still available when all coalitions have selected strategies, i.e. decided which rights to exercise, and thus restricted the set of available social states. After this informal presentation, we can now, with the aid of a rightssystem L and, for each individual i. a preference relation Ri over sets of social states, define a game of rights exercising in normal form as consisting of: (i) a set I of individuals; (ii) for each non-empty subset G of I, a set of strategies, consisting of all X such that (G,X)  L; (iii) a set of outcomes O consisting of all sets X of social states such that there is a partitioning of I into disjoint coalitions G1, G2, … Gk and a set of strategies X1, X2, ... Xk where (G Gj,X Xj)  L, for all j, l ” j ” k, and where X = X1 ˆ X2 ˆ ... ˆ Xk; (iv) for each i  I a preference ordering Ri of the sets of social states in O. This definition conforms to the traditional definition of an n-person game in normal form, except that the individual utility functions that measure the payoff of the outcomes have been replaced with the individual preference relations Ri.42 If the empty set were a possible outcome of a game of rights exercising, this would be hard to interpret since it would mean that the rights exercised by the individuals or groups have excluded all possible social states from being available for social choice. However, with the aid of the consistency 42

For a general introduction to the concepts of game theory the reader is referred to Luce and Raiffa (1957). The definition presented here, which does not mention utility functions, is adapted from Peleg (1966).

DECISION THEORY

39

condition on a rights-system and the condition on the rights of groups, it can be shown, by a simple induction on the number of coalitions, that the empty set is never an outcome in any game of rights exercising. 3.7 Criteria for exercising of rights Having presented games of rights exercising in normal form it is now possible to introduce many of the concepts and criteria that have been used in traditional game theory to determine which strategies are rational to select in a given game situation. Here, only two of the most fundamental criteria will be presented.43 A strategy X1 is said to dominate the strategy X2 for the group G iff and only if no matter what strategies the individuals not in G employ, there is no individual in G who prefers, according to the Ri-relations, the outcome when X2 is chosen to the outcome when X1 is chosen, and for some strategy choices of the individuals not in G, there is some individual in G who prefers the outcome when X1 is chosen to the outcome when X2 is chosen. A weak criterion on strategy choices is that a group G should never chose a strategy that is dominated by some other strategy. An example of a dominated strategy: Suppose that (i,X) and (i,S-X) belong to a rights-system L, i.e. i has a right to see to it that X and a right to see to it that not-X. If now, for all x  X and y  S-X, xPiy, then the strategy X dominates the strategy S-X for i, since no matter how the remaining individuals choose their strategies, an outcome which is a subset of X will always be preferred, according to the relation Ri, to an outcome which is a subset of S-X. An n-tuple X1, X2, … Xn of strategies for a partitioning of I into coalitions G1, G2, … Gn, determining the outcome X = X1 ˆ X2 ˆ ... ˆ Xn, is said to be an equilibrium pointt if and only if there is no strategy X+ for some coalition G+ such that, for all i in G+ and for all strategies X- belonging to IG+, X+ˆX-PiX. Or, more informally, no attempt to choose a strategy X+ in a new coalition G+ will be advantageous to all members of G+ against all counterstrategies X- of the counter-coalition I-G+. The concept of an equilibrium point can be illustrated by the EdwinAngelina case. Let us assume that Edwin and Angelina have the same rights and preferences (over single alternatives) as before. Since Angelina, according to Gibbard, is willing to settle for the judge, if Edwin does not want to marry her, it seems natural to assume that not {x,y,z}Pa{y}, where Pa is Angelina’s preference relation over sets of alternatives. With this assumption it can be shown that the only equilibrium point of the EdwinAngelina game in normal form is the strategy {x} for the coalition {Edwin, Angelina}, i.e. they agree to marry each other. This observation is, in my 43

For an outline of some other criteria, see Peleg (1966).

40

THE DYNAMICS OF THOUGHT

opinion, the precise reason why Gibbard (1974, p. 398) found it “plain in this case how the conflict ought to be resolved.” Strategies in an equilibrium point are never dominated by any other strategy. It seems very natural to require that if a game of rights exercising has equilibrium points, then the outcome of the game should be determined by one of these equilibrium points. For the formulation of a game in normal form presented here, it is possible to construct games without equilibrium points, so this criterion on strategy choice is not universally applicable. For games without equilibrium points, it is necessary to rely on other criteria for the solution of n-person cooperative games without side payments. In an attempt to find a cure for the Paretian liberal, Gibbard proposes a criterion that is based on the distinction between having a right and exercising a right. His condition is, like Sen’s Minimal Liberalism, formulated in terms of rights as possible restrictions on social orderings. It is weaker than Sen’s condition since it is compatible with the Pareto principle, but, as Kelly (1976) points out, it “is an extremely cautious risk averse criterion for rights-exercising.”44 If reformulated as close as possible in the present game-theoretical notions, it requires that an individual i waives a right to restrict S to X as soon as it is possible that other individuals, by exercising their rights, can exclude the social states in X which i prefers most. In the game-theoretical setting, this condition may not naturally yield optimal strategies of rights exercising. 3.8 Social choice and the Pareto principle When the Pareto principle is discussed, the distinction between the process of exercising rights and the social choice process that selects a social state after the rights have been exercised must be kept in mind. As regards the second process, the social choice, the Pareto principle is fairly unproblematic and a weak version may be formulated in the following manner: The Pareto condition on social choice: If x and y both belong to the set of available alternatives (available after the rights have been exercised) and all individuals prefer x to y, then y will not belong to the outcome of the social choice process. In my opinion, the Pareto principle is a valid condition on the outcome of a social choice process. I now turn to the question whether it is reasonable to impose some form of Pareto criterion on the process of rights exercising. This is one way of examining whether there is an unavoidable conflict between libertarianism and social welfare maximization. 44

A simple game of this kind has been investigated by Gardner (1980).

DECISION THEORY

41

A condition, which on the surface is similar to the Pareto condition on social choice, is the following: The Pareto condition on single states in games of rights exercising: If all individuals prefer x to y, then y will not belong to the outcome of a game of rights exercising. Whether a person has a certain right is normally independent of what preferences he has. So, in order to realize this kind of condition, there must exist rights that make it possible to exclude any particular social state from the set of possible states. This can be guaranteed by introducing another condition on a rights-system that bestows full decisive power on the society I regarded as a whole. Condition on the sovereignty of the society: If L is a rights-system, then, for any non-empty subset X of S, (I,X)  L. It is, however, difficult to give a valid rationale for the Pareto condition on single states in games of rights exercising. As a counterexample to the validity of the condition, we can consider a situation where three individuals i, j and k have rank ordered the social states x, y and z as follows (in strict descending order): i: x y z j: z x y k: z x y Suppose furthermore that the rights-system L in question contains (i, {y, z}), (j, {x, y}) and (k, {x, y}) as the only individual rights (which thus are of no interest in this situation) and that no group rights, except for what is demanded by the condition on the rights of groups and the condition on the sovereignty of society, are included in L. A final assumption for this example is that the social choice process that will be used on the set of social states which remain after the exercising of rights will be the Borda rule, i.e. the rule which assigns 1 point to the first position in a preference ordering, 2 to the second, 3 to the third, etc., and then determines the social choice as the alternative or alternatives which obtain the smallest sum of rank order points. According to the Pareto condition on single states in a game of rights exercising, the social state y should be excluded from the set of available social states. The only way to attain this, with the present rights-system, is for the group {i, j, k} to agree to exclude y. However, it is not rational for i to agree to this exclusion, since in the resulting outcome {x, z}, z will be the

42

THE DYNAMICS OF THOUGHT

winning alternative in this restricted set according to the Borda rule, while if the set of alternatives available for social choice is {x, y, z}, the Borda outcome will be {x, z}, and i certainly prefers the latter social choice to the former. This kind of example shows that it is difficult, in general, to defend the Pareto condition on single states in a game of rights exercising. It is, however, easy to conceive of another version of the Pareto condition for games of rights exercising: The Pareto condition on sets of social states in games of rights XPiY, then Y will not be the outcome exercising: If, for all individuals i in I, X of a game of rights exercising. In my opinion, this condition ought to be fulfilled for all games of rights exercising. The criticism presented with the aid of the example above which was directed against the Pareto condition on single states does not apply to this version. In support of the condition, one can mention that if there are equilibrium points in a game of rights exercising, then it can be shown, with the aid of the condition of the sovereignty of the group, that the choice of an equilibrium point will always agree with the Pareto condition on sets of social states. I believe that the distinction between the three types of Pareto condition on games of rights exercising, in connection with Nozick’s criticism, sheds some light on why Sen’s presentation of the Pareto condition in terms of restrictions on social preference orderings is misleading. 3.9 Other types of rights The types of rights that I have included in the rights-system are of a simple nature. There also exist other types of rights where the connection between rights and strategies in a game of rights exercising cannot be made as simple as it has been presented here. In a full-blown analysis of games of rights exercising these other types must also be taken into account. Lindahl (1977, chapter 3) distinguishes between seven basic types of individual rights concerning a given “state of affairs” X. I will briefly present these types and comment upon their connection with strategies in a game of rights exercising. (i) i may see to it that X, and he may see to it that not-X, and he may be passive with respect to X. This type of right corresponds to having all of (i,X), (i,S-X) and (i,S) included in a rights-system. (This correspondence is, as will be seen later, not perfect.)

DECISION THEORY

43

(ii) i may see to it that X, but he may not see to it that not-X, and he may be passive with respect to X. This type corresponds to including (i,X) and (i,S) in a rights-system (but not (i,S-X)). (iii) i may see to it that X, and he may see to it that not-X, but he may not remain passive with respect to X. This type of right corresponds to including (i,X) and (i,S-X) in a rightssystem, but not (i,S). This type of right thus conflicts with the condition on waiving of rights. (iv) i may not see to it that X, but he may see to it that not-X, and he may remain passive with respect to X. This type corresponds to having (i,S-X) and (i,S) in a rights-system (but not (i,X)). From a game-theoretical point of view, this type is equivalent to type (ii). (v) i shall see to it that X. This type is normally not considered as a right but rather as a duty. In the game terminology this means that if an individual has a “right” of this type, he may not waive it, but is forced to restrict the set of available set of social states to a set which is included in X. This type of right conflicts with the condition on waiving of rights, as did type (iii). (vi) i may neither see to it that X nor that not-X, but he shall remain passive with respect to X. This type of right corresponds to including (i,S) in a rights-system, but neither (i,X) nor (i,S-X). (vii) i shall see to it that not-X. From a game-theoretical point of view, this type is equivalent to type (v). Rights of the types (iii), (v) and (vii) do not quite fit with the earlier description of a rights-system and the conditions imposed on such a system. In game-theoretical terms, these types of rights are “forced” moves. Rights of types (v) and (vii) introduce a priori limitations on the set of available social states. These types of rights can be included in a more detailed

44

THE DYNAMICS OF THOUGHT

description of a game of right exercising, but in order to keep the notation and analysis as simple as possible, I have chosen to include only some elementary types of rights in my definitions. A similar classification can be made for two-person rights where one person is said to have a right against another person. Lindahl (1977) between two different classifications of such types, with 35 and 127 types respectively. He also gives a typology of the collective rights of two persons in altering their relations by contract. It would take us far too far to present those types here. In order to cover all game-theoretical aspects of these types a more sophisticated apparatus than the one used here is necessary. I believe, though, that the rights-systems and games of rights exercising that have been introduced in this chapter contain the most important constituents. 3.10 Conclusion In this chapter, I have tried to show how Sen’s paradox of the Paretian liberal may be resolved. The main difference from Sen’s analysis is the change of the role of rights in a social decision situation. By distinguishing between having a right and exercising that right I have been able to formulate a rights exercising situation as a game in normal form. With the aid of some traditional concepts from game theory, I have briefly discussed how to determine the rational outcome of a game of rights exercising. In a recent survey article on the paradox of the Paretian liberal, Sen (1976, p. 224) distinguishes between a ‘pragmatic’ interpretation of the problem based on the distinction between the existence of rights and the exercise of rights and an ‘ethical’ interpretation which concerns how conflicts between rights and welfare should d be resolved. Clearly, this chapter deals with the pragmatic interpretation of the use of rights. My main thesis is that the use of game theory, in some form or other, is the best way to model the pragmatic aspects of rights. However, I have also given some arguments for that under the interpretation of rights presented here there need not be any conflict between the libertarian claim and the Pareto principle. In the game-theoretical analysis presented in this chapter, I have only considered strategic aspects of the exercising of rights. The social choice from the remaining set of social states has to a large extent been ignored in this analysis. However, the example which was presented in connection with the Pareto condition on single states in a game of rights exercising shows that the choice of strategy of rights exercising may be dependent on the social choice process. And, as recent results on the manipulability of social choice methods show, the individual choices of which preference relations to reveal to the social choice mechanism are non-trivial from a gametheoretical point of view (cf. e.g. Gibbard (1973) and Gärdenfors (1976)). In order to give a fair picture of the concurrence of the strategic aspects

DECISION THEORY

45

of rights exercising with the strategic aspects of preference revelation, another type of game, which is more complex than those presented here, ought to be introduced. In such a game, the strategies of an individual or a group consist of combinations of the exercising of certain rights and the revelation of certain preferences. This will provide each coalition with a large set of strategies and a detailed description of a game situation may be difficult to survey. In this kind of game, the decision process does not proceed in two steps by first exercising rights and then making a social choice among the remaining social states, as suggested by Nozick, but the final social choice is determined in one step. The normative analysis of such a game will be more complicated and less illuminating than the analysis of the games that have been presented here. For these reasons I have chosen to discuss only the simpler games in this chapter.45

45

I wish to thank Lars Lindahl for, at several points, making me understand what I was saying. He has also provided me with some examples and counterexamples. I am grateful to Amartya Sen for saving me from one way of misrepresenting his position. This does not imply that I have avoided all of them. Finally, a referee for NOÛS made several helpful suggestions.

CHAPTER 4 THE DYNAMICS OF BELIEF SYSTEMS: FOUNDATIONS VS. COHERENCE THEORIES

4.1 The problem of belief revision In this chapter I want to discuss some philosophical problems one encounters when trying to model the dynamics of epistemic states. Apart from being of interest in themselves, I believe that solutions to these problems will be crucial for any attempt to use computers to handle changes of knowledge systems. Problems concerning knowledge representation and the updating of such representations have become the focus of much recent research in artificial intelligence (AI). Human beings perpetually change their states of knowledge and belief in response to various cognitive demands. There are several different kinds of belief changes. The most common type occurs when we learn something new – by perception or by accepting the information provided by other people. This kind of change will be called an expansion of a belief state. Sometimes we also have to revise our beliefs in the light of evidence that contradicts what we had earlier mistakenly accepted, a process which will here be called a revision of a belief state. And sometimes, for example when a measuring instrument is malfunctioning, we discover that the reasons for some of our beliefs are invalid and so we have to give up those beliefs. This kind of change will be called a contraction of an epistemic state. Note that when a state of belief is revised, it is also necessary to give up some of the old beliefs in order to maintain consistency. Some of the changes of states of belief are made in a rationall way, others not. The main problem to be treated here is how to characterize rational changes of belief. But before we can attack the problems related to the dynamics of epistemic states, we must know something about their statics, i.e. the properties of single states of belief. To this purpose, one can formulate criteria for what should count as a rational belief state. There are two main approaches to modeling epistemic states. One is the foundations theory that holds that one needs to keep track of the justifications for one’s beliefs: Propositions that have no justification should not be accepted as beliefs. The other is the coherence theory that holds that one need not consider the

47

48

THE DYNAMICS OF THOUGHT

pedigree of one’s beliefs. The focus is instead on the logicall structure of the beliefs – what matters is how a belief coheres with the other beliefs that are accepted in the present state. Each of these theories will be presented in greater detail in the two following sections. After that I shall also introduce some of the empirical evidence relating to how people in fact do update their epistemic states and discuss its relevance for the two theories. It should be obvious that the foundations and the coherence theories have very different implications for what should count as rational changes of belief systems. According to the foundations theory, belief revision should consist, first, in giving up all beliefs that do no longer have a satisfactory justification and, second, in adding new beliefs that have become justified. On the other hand, according to the coherence theory the objectives are, first, to maintain consistency in the revised epistemic state and, second, to make minimal changes of the old state that guarantee sufficient overall coherence. Thus, the two theories of belief revision are based on conflicting ideas of what constitutes rational changes of belief. The choice of underlying theory is, of course, also crucial for how an AI researcher will attack the problem of implementing a belief revision system on a computer. In order to illustrate the more abstract aspects of the theories, I shall present two modelings of belief revisions. The first is Doyle’s (1979) Truth Maintenance System (TMS), which is a system for keeping track of justifications in belief revisions, and thus it directly follows the foundations theory.46 The second is the theory of belief revision that has been developed by myself in collaboration with Carlos Alchourrón and David Makinson.47 The theory is often referred to as the AGM modell of belief revision. This modeling operates essentially with minimal changes of epistemic states and is, in its simplest form, thus in accordance with the coherence theory. The advantages and drawbacks of each modeling will be discussed. One of the main criticisms directed against coherence models of belief states is that they cannot be used to express that some beliefs are justifications or reasons for other beliefs. This is true, of course, for single states of belief that only contain information about which beliefs are accepted and which are not. However, in section 4.8 I shall argue that if one has further information about how such a state will potentially be revised under various forms of input, then it is possible to formulate criteria for considering one belief to be a reason for another. In particular, the notion of the epistemic entrenchmentt of beliefs, which plays a central role in a development of the AGM model, is useful here. Adding this kind of information about the beliefs in an epistemic state will thus produce a model 46 For more recent developments of the theory see e.g. Goodwin (1987) and Gärdenfors and Rott (1995). 47 See Alchourrón, Gärdenfors and Makinson (1985).

BELIEF REVISION AND NONMONOTONIC REASONING

49

of belief revision that satisfies most of the requirements of the foundations as well as the coherence theory. 4.2 The foundations theory of belief revision The basic principle of the foundations theory is that one must keep track of the reasons for the beliefs that are accepted in an epistemic state. This means that an epistemic state has a justificational structure so that some beliefs serve as justifications for others. Justifications come in two kinds. The standard kind of justification is that a belief is justified by one or several other beliefs, but not justified by itself. However, since all beliefs should be justified and since infinite regresses are disallowed, some beliefs must be justified by themselves. Harman (1986, p. 31) calls such beliefs foundational. One finds ideas like this, for example, in the epistemology of the positivists: Observational statements need no further justification – they are self-evident. Another requirement on the set of justifications is that it be non-circularr so that we do not find a situation where a belief in A justifies B, a belief in B justifies C, while C in turn justifies A. Sosa (1980) describes the justificational structure of an epistemic state in the following way: For the foundationalist, every piece of knowledge stands at the apex of a pyramid that rests on stable and secure foundations whose identity and security does not derive from the upper stories or sections.

Another feature of the justification relation is that a belief A may be justified by several independent beliefs, so that even if some of the justifications for A are removed, the belief may be retained because it is supported by other beliefs. Probably the most common models of epistemic states used in cognitive science are called semantic networks. A semantic network typically consists of a set of nodes representing some objects of belief and, connecting the nodes, a set of links representing relations between the nodes. The networks are then complemented by some implicit or explicit interpretation rules that make it possible to extract beliefs and epistemic attitudes. Changing a semantic network consists in adding or deleting nodes or links. If nodes represent beliefs and links represent justifications, semantic networks seem to be ideal tools for representing epistemic states according to the foundational theory. However, not all nodes in semantic networks represent beliefs and not all links represent justifications. Different networks have different types of objects as nodes and different kinds of relations as links. In fact, the diversity is so large that it is difficult to see what the various networks have in common. It seems that any kind of object can serve as a node in the networks and that any type of relation or connection between nodes can be used as a link between nodes. This diversity seems to undermine the claims

50

THE DYNAMICS OF THOUGHT

that semantic networks represent epistemic states. In his excellent methodological article, Woods (1975, p. 36) admits that “we must begin with the realization that there is currently no `theory´ of semantic networks.” As a preliminary to such a theory, Woods formulates requirements for an adequate notation for semantic networks and explicit interpretation rules for such a notation. My aim here is not a presentation of his ideas, but only to give a brief outline of semantic networks as models of epistemic states suitable for the foundational theory. If we now turn to the implications of the foundational theory for how belief revisions should be performed, the following general principle formulated by Harman (1986, p. 39) is of interest: Principle of Negative Undermining: One should stop believing P whenever one does not associate one’s belief in P with an adequate justification (either intrinsic or extrinsic). This means that if a state of belief containing A is revised so that the negation of A becomes accepted, not only should A be given up in the revised state, but also all beliefs that depend on A for their justification. Consequently, if one believes that B and all the justifications for believing B are given up – continued belief in B is no longer justified, so it should be rejected. A drawback of this principle, from an implementational point of view, is that it may lead to chain reactions and thus to severe bookkeeping problems. A specific example of this type of process, namely TMS, will be presented in section 4.5. 4.3 The coherence theory of belief revision According to the coherence theory, beliefs do not usually require any justification – the beliefs are justified just as they are. A basic criterion for a coherent epistemic state is that it should be logically consistent. Other coherentist criteria for the statics of belief systems are less precise. Harman (1986, p. 32-33) says that coherence “includes not only consistency but also a network of relations among one’s beliefs, especially relations of implication and explanation. ... According to the coherence theory, the assessment of a challenged belief is always holistic. Whether such a belief is justified depends on how well it fits together with everything else one believes. If one’s beliefs are coherent, they are mutually supporting. All one’s beliefs are, in a sense, equally fundamental.” Sosa (1980) compares the coherence theory with a raft: For the coherentist, a body of knowledge is a free-floating raft every plank of which helps directly or indirectly to keep all the others in place, and no plank of which would retain its status with no help from the others.

A more far-reaching criterion on an epistemic state is that it be closed

BELIEF REVISION AND NONMONOTONIC REASONING

51

under logical consequences, i.e. if A is believed in an epistemic state K, and B follows logically from A, then B is believed in K too. This criterion is presumed in the AGM model of belief revision that will be presented in section 4.6. An immediate objection to this criterion is that human beings can only handle a finite number of beliefs and there are infinitely many logical consequences of any belief, so a human epistemic state cannot be closed under logical consequences. This objection will be discussed in sections 4.7 and 4.8. Turning now to the problem of belief revision, the central coherentist criterion is that changes of belief systems should be minimal. This criterion provides the motivation for many of the postulates formulated for expansions, contractions, and revisions of belief systems in the AGM model. The postulates capture some general properties of minimal changes of belief. However, from an implementational point of view, they leave the main problem unsolved: What is a reasonable metric for comparing different epistemic states that can be used in a computer program to update such states? According to the coherence theory, belief revision is a conservative process in the following sense:48 Principle of Conservation: When changing beliefs in response to new evidence, you should continue to believe as many of the old beliefs as possible. It should be noted that, in general, it is not meaningful to count the number of beliefs changed, but other comparisons of minimality must be applied.49 There is an economic side to rationality that is the main motivation for the Principle of Conservation. When we change our beliefs, we want to retain as much as possible of our old beliefs – information is in general not gratuitous, and unnecessary losses of information are therefore to be avoided. We thus have a criterion of informational economy motivating the coherentist approach. The main drawback of coherence models of belief is that they cannot be used to directly express that some beliefs may be justifications for other beliefs. And, intuitively, when we judge the similarity of different epistemic states, we want the structure of justifications to count as well. In such a case we may end up contradicting the Principle of Conservation. The following example, adapted from Tichy (1976), illustrates this point:50 Suppose that I, 48

Gärdenfors (1988a), p. 67, also cf. Harman (1986), p. 46. This problem is discussed in Gärdenfors (1988a), section 3.5. 50 Also cf. Stalnaker (1984), pp. 127-129. 49

52

THE DYNAMICS OF THOUGHT

in my present state K of belief, accept as known that Oscar always wears his hat when it rains, but when it does not rain, he wears his hat completely at random (about 50% of the days). I also believe that it rained today and that he wore his hat. Let A be the proposition “It rains today” and B “Oscar wears his hat.” Thus both A and B are accepted in K. What can we say about the beliefs in the state where K has been revised to include ¬A? Let us denote this state by K*¬A. According to the Conservativity Principle, B should still be accepted in K*¬A because the addition of ¬A does not conflict with B. However, I no longer have any justification for believing B, so intuitively neither B nor ¬B should be accepted in K*¬A. This and similar examples will be discussed in section 4.8, where a way out of the problem, remaining within the coherentist framework, will also be presented. 4.4 Some empirical evidence Levi (1980, p. 1) criticizes the foundations theory of knowledge as follows: Knowledge is widely taken to be a matter of pedigree. To qualify as knowledge, beliefs must be both true and justified. Sometimes justification is alleged to require tracing of the biological, psychological, or social causes of belief to legitimating sources. Another view denies that causal antecedents are crucial. Beliefs become knowledge only if they can be derived from impeccable first principles. But whether pedigree is traced to origins or fundamental reasons, centuries of criticism suggest that our beliefs are born on the wrong side of the blanket. There are no immaculate preconceptions.

If we consider how people in fact change their beliefs, Levi seems to be right. Although the foundations theory gives intuitively satisfying recommendations about what one oughtt to do when revising one’s beliefs, the coherence theory is more in accord with what people actually do. In a survey article, Ross and Anderson (1982) present some relevant empirical evidence.51 Some experiments have been designed to explore “the phenomenon of belief perseverance in the face of evidential discrediting.” For example, in one experiment [s]ubjects first received continuous false feedback as they performed a novel discrimination task (i.e., distinguishing authentic suicide notes from fictitious ones) ... [Each subject then] received a standard debriefing session in which he learned that his putative outcome had been predetermined and that his feedback had been totally unrelated to actual performance. Before dependent variable measures were introduced, in fact, every subject was led to explicitly acknowledge his understanding of the nature and purpose of the experimental deception. Following this total discrediting of the original information, the subjects completed a dependent variable questionnaire dealing with [their] performance and abilities. The evidence for postdebriefing impression perseverance was unmistakable … . On virtually every measure … the totally discredited initial outcome manipulation produced significant “residual” effects ... . 51

Also cf. Harman (1986), pp. 35-37.

BELIEF REVISION AND NONMONOTONIC REASONING

53

A recent series of experiments ... first manipulated and then attempted to undermine subjects’ theories about the functional relationship between two measured variables: the adequacy of firefighters’ professional performances and their prior scores on a paper and pencil test of risk preference. ... [S]uch theories survived the revelations that the cases in question had been totally fictitious and the different subjects had, in fact, received opposite pairings of riskiness scores and job outcomes. (Ross and Anderson 1982, pp. 147-9)

Ross and Anderson conclude (1982, p. 149) from these and other experiments that it is clear that beliefs can survive potent logical or empirical challenges. They can survive and even be bolstered by evidence that most uncommitted observers would agree logically demands some weakening of such beliefs. They can even survive the total destruction of their original evidential basis. Harman (1986, p. 38) has the following comments on the experiments: In fact, what the debriefing studies show is that people simply do not keep track of the justification relations among their beliefs. They continue to believe things after the evidence for them has been discredited because they do not realize what they are doing. They do not understand that the discredited evidence was the sole reason why they believe as they do. They do not see they would not have been justified in forming those beliefs in the absence of the now discredited evidence. They do not realize these beliefs have been undermined. It is this, rather than the difficulty of giving up bad habits, that is responsible for belief perseverance.

Do these empirical findings concerning belief perseverance mean that people are irrational? Shouldn’t they try to keep track of the justifications of their beliefs? I think not. The main reason is that it is intellectually extremely costly to keep track of the sources of beliefs and the benefits are, by far, outweighed by the costs.52 A principle of intellectual economy would entail that it is rational to neglect the pedigree of one’s beliefs. To be sure, we will sometimes hold on to unjustified beliefs, but the erroneous decisions caused by this negligence have to be weighed against the cost of remembering all reasons for one’s beliefs. The balance will certainly be in favor of forgetting reasons. After all, it is not very often that a justification for a belief is actually withdrawn and, as long as we do not introduce new beliefs without justification, the vast majority of our beliefs will hence remain justified.53 4.5 Doyle’s Truth Maintenance System Doyle’s (1979) Truth Maintenance System (TMS) is an attempt to model changes of belief within the setting of the foundational theory. As Doyle remarks (p. 232), the name “truth maintenance system” not only sounds like Orwellian Newspeak, but is also a misnomer, because what is maintained is the consistency of beliefs and reasons for belief. Doyle (1983) later changed 52

The same seems to apply to computer programs based on the foundations theory: As will be seen in next section one of the main drawbacks of the TMS system is that it is inefficient. 53 Harman (1986, pp. 41-42) expresses much the same point when he writes about “clutter avoidance.”

54

THE DYNAMICS OF THOUGHT

the name to “reason maintenance system.” In a broad sense TMS can be said to be a semantic network model, but its belief structure and its techniques for handling changes of belief are more sophisticated than in other semantic network models. There are two basic types of entities in TMS: nodes representing propositional beliefs and justifications representing reasons for beliefs. These justifications may be other beliefs from which the current belief is derived. A node may be in or out, which corresponds to the epistemic attitudes of accepting and not accepting the belief represented by the node. As should be expected, if a certain belief is outt in the system, this does not entail that its negation is in. On the other hand, as a rationality requirement, if both a belief and its negation are in, then the system will start a revision of the sets of nodes and their justifications in order to reestablish consistency. A justification consists of a pair of lists: an inlist and an outlist. A node is in if and only if it has some justification (there may be several for the same node), the inlist of which contains only nodes that are in and the outlist of which contains only nodes that are out. A particular type of justification, called “nonmonotonic justifications,” is used to make tentative guesses within the system. For example, a belief in A can be justified simply by the fact that the belief in ¬A is out. Beliefs that are justified in this way are called assumptions. This technique gives us a way of representing commonsense “default” expectations. It also leads to nonmonotonic reasoning in the following sense: If belief in A is justified only by the absence of any justification for ¬A, then a later addition of a justification for ¬A will lead to a retraction of the belief in A. The basic concepts of TMS are best illustrated by an example: Node

Justification Status Inlist Outlist ________________________________________________________ (N1) Oscar is not guilty of defamation. (N2) (N3) in (N2) The accused should have the benefit of the doubt. in (N3) Oscar called the queen a harlot. (N4) (N5) out (N4) It may be assumed that the witness´ report is correct. in (N5) The witness says he heard Oscar call the queen a harlot. out ________________________________________________________ In this situation (N1) is in because (N2) is in and (N3) is out. Node (N3) is outt because not both of (N4) and (N5) are in. If (N5) changes status to in (this may be assumed to be beyond the control of system), (N3) will become in and consequently assumption (N1) is out. Apart from the representation of nodes and justifications as presented here, TMS contains techniques for handling various problems that arise

BELIEF REVISION AND NONMONOTONIC REASONING

55

when the system of beliefs is adjusted to accommodate the addition of a new node or justification. In particular, when a contradiction is found, the system uses a form of backtracking to find the fundamental assumptions that directly or indirectly give support to the contradiction. One of these assumptions is chosen as the culprit and is given the status out. This process sometimes needs to be iterated, but it is beyond the scope of this chapter to give full description of the mechanics of TMS. However, the TMS representation of beliefs is not without epistemological problems. The example can be used to illustrate some of the drawbacks of TMS. In the handling of beliefs and justifications, TMS takes no notice of what the nodes happen to stand for. The sentences that I have added to the node names are not interpreted in any way by the system. This means that TMS completely lacks a semantic theory. As a consequence, much of the logic of propositions is lost in the TMS representation of beliefs. All forms of logical inferences that are to be used by the system have to be reintroduced as special systems of justifications. Doyle discusses conditional proofs, but the process for handling such inferences seems extremely complex. Furthermore, TMS leaves much of the work to the programmer. The programmer produces the nodes and their justifications; she has to organize the information in levels, and she also has to decide on how contradictions are to be engineered.54 4.6 The AGM model As an example of a model of belief revision that agrees with the coherence theory I shall now outline the AGM model.55 Epistemic states are modeled by belief sets that are sets of sentences from a given language. Belief sets are assumed to be closed under logical consequences (classical logic is generally presumed), which means that if K is a belief set and K logically entails B, then B is an element in K. A belief set can be seen as a partial description of the world – partial because in general there are sentences A such that neither A nor ¬A are in K. Belief sets model the statics of epistemic states. I now turn to their dynamics. What we need are methods for updating belief sets. Three kinds of updates will be discussed here: (i)Expansion: A new sentence together with its logical consequences is added to a belief set K. The belief set that results from expanding K by a 54

A later development of the TMS system is presented in Goodwin (1987). A system with a slightly different methodology, but still clearly within the foundations paradigm, is de Kleer's (1986) Assumption-based TMS (ATMS). For further comparison between ATMS and TMS, see Rao (1988) and Gärdenfors and Rott (1995). 55 The main references are Gärdenfors (1984b), Alchourrón, Gärdenfors, Makinson (1985), Gärdenfors (1988a) and Gärdenfors and Makinson (1988).

56

THE DYNAMICS OF THOUGHT

sentence A will be denoted K+A. (ii) Revision: A new sentence that is inconsistent with a belief set K is added, but in order that the resulting belief set be consistent some of the old sentences of K are deleted. The result of revising K by a sentence A will be denoted K*A. (iii) Contraction: Some sentence in K is retracted without adding any new beliefs. In order that the resulting belief set be closed under logical consequences some other sentences from K must be given up. The result of contracting K with respect to A will be denoted K-A. Expansions of belief sets can be handled comparatively easily. K+A can simply be defined as the logical consequences of K together with A: (Def +) K+A = {B: K ‰ {A} |– B} Here |– is the relation of logical entailment. As is easily shown, K+A defined in this way is closed under logical consequences and will be consistent when A is consistent with K. It is not possible to give a similar explicit definition of revisions and contractions in logical and set-theoretical notions only. To see the problem for revisions, consider a belief set K which contains the sentences A, B, A & B o C and their logical consequences (among which one finds C). Suppose that we want to revise K by adding ¬C. Of course, C must be deleted from K when forming K*¬C, but at least one of the sentences A, B, or A & B o C must also be given up in order to maintain consistency. There is no purely logicall reason for making one choice rather than the other, but we have to rely on additional information about these sentences. Thus, from a logical point of view, there are several ways of specifying the revision of a belief set. What is needed here is a (computationally well defined) method of determining the revision. As should be easily seen, the contraction process faces parallel problems. In fact, the problems of revision and contraction are closely related – being two sides of the same coin. To establish this more explicitly, we note, firstly, that a revision can be seen as a composition of a contraction and an expansion. Formally, in order to construct the revision K*A, one first contracts K with respect to ¬A and then expands K-¬A by A which amounts to the following definition: (Def *) K*A = ( K-¬A)+A Conversely, contractions can be defined in terms of revisions. The idea is that a sentence B is accepted in the contraction K-A if and only if B is accepted in both K and K*A. Formally:

BELIEF REVISION AND NONMONOTONIC REASONING

57

(Def -) K-A = K ˆ K*¬A These definitions indicate that revisions and contractions are interchangable and a method for explicitly constructing one of the processes would automatically yield a construction of the other. There are two methods of attacking the problem of specifying revision and contraction operations. One is to present rationality postulates for the processes. Such postulates are introduced in Gärdenfors (1984a), Alchourrón, Gärdenfors and Makinson (1985) and discussed extensively in Gärdenfors (1988a), and they will not be repeated here. In these works the connections between various postulates are also investigated. The second method of solving the problems of revision and contraction is to adopt a more constructive approach. A central idea here is that the sentences that are accepted in a given belief set K have different degrees of epistemic entrenchmentt – not all sentences that are believed to be true are of equal value for planning or problem-solving purposes, but certain pieces of our knowledge and beliefs about the world are more important than others when planning future actions, conducting scientific investigations, or reasoning in general.56 It should be noted that the ordering of epistemic entrenchment is not prima facie motivated by justificational considerations. However, the connection between entrenchment and justification will be discussed in section 4.8. The degrees of epistemic entrenchment of the sentences in a belief set will have a bearing on what is abandoned and what is retained, when a contraction of a revision is carried out. The guiding idea for the construction is that when a belief set K is revised or contracted, the sentences in K that are given up are those having the lowestt degree of epistemic entrenchment (Fagin, Ullman, and Vardi (1983), pp. 358 ff, introduce the notion of “database priorities” which is closely related to the concept of epistemic entrenchment and is used in a similar way to update belief sets). I do not not assume that degrees of epistemic entrenchment can be quantitatively measured, but work only with qualitative properties of this notion. One reason for this is that I want to emphasize that the problem of uniquely specifying a revision method (or a contraction method) can be solved, assuming very little structure on the belief sets apart from their logical properties. If A and B are sentences, the notation A ” B will be used as a shorthand for “B is at least as epistemically entrenched as A.” The strict relation A < B is defined in the usual way. Note that the relation ” is only defined in relation to a given K – different belief sets may be associated with different 56

The epistemological significance of the notion of epistemic entrenchment is further spelled out in Gärdenfors (1988a), sections 4.6 - 4.7.

58

THE DYNAMICS OF THOUGHT

orderings of epistemic entrenchment. The following postulates for epistemic entrenchment will be assumed:57 (EE1) If A ” B and B ” C, then A ” C (EE2) If A |– B, then A ” B (EE3) For any A and B, A ” A & B or B ” A & B (EE4) If K is consistent, A  K iff A ” B for all B (EE5) B ” A for all B, only if A is logically valid

(transitivity) (dominance) (conjunctiveness) (minimality) (maximality)

The motivation for (EE2) is that if A logically entails B, and either A or B must be retracted from K, then it will be a smaller change to give up A and retain B rather than to give up B, because then A must be retracted too, if we want the revised knowledge set to be closed under logical consequences. The rationale for (EE3) is as follows: If one wants to retract A & B from K, this can only be achieved by giving up either A or B and, consequently, the informational loss incurred by giving up A & B will be the same as the loss incurred by giving up A or that incurred by giving up B. Note that it follows already from (EE2) that A & B ” A and A & B ” B. The postulates (EE4) and (EE5) only take care of limiting cases: (EE4) requires that sentences already not in K have minimal epistemic entrenchment in relation to K; and (EE5) says that only logically valid sentences can be maximal in ”. After these technicalities we can now return to the main problem of constructing a contraction (or revision) method. The central idea is that if we want to contract K with respect to A, then the sentences that should be retained in K-A are those which have a higher degree of epistemic entrenchment than A. For technical reasons (see Gärdenfors and Makinson (1988), pp. 89-90), the comparisons A < B do not always give the desired result, but in general one has to work with comparisons of the form A < A › B. The appropriate definition is the following: (C-) B  K-A if and only if B  K and (i) A is logically valid (in which case K-A = K) or (ii) A < A › B It can be shown (Gärdenfors and Makinson (1988), Theorem 4) that if the ordering ” satisfies (EE1) - (EE5), then the contraction method defined as above satisfies all the appropriate postulates. Via (Def *), the definition (C-) can also be used to construct a revision method with the desired properties. From an epistemological point of view, this result suggests that the problem of constructing appropriate contraction and revision methods can be reducedd to the problem of providing an appropriate ordering of epistemic 57

Cf. Gärdenfors (1988a), section 4.6, and Gärdenfors and Makinson (1988).

BELIEF REVISION AND NONMONOTONIC REASONING

59

entrenchment. Furthermore, condition (C-) gives an explicitt answer to which sentences are included in the contracted belief set, given the initial belief set and an ordering of epistemic entrenchment. Computationally, applying (C-) is trivial, once the ordering ” of the elements of K is given. 4.7 Finite bases for infinite belief sets I now turn to some of the problems of the AGM model. Although the model and the results presented in the previous section provide some insights into the problem of revising epistemic states, it seems doubtful whether the approach could really be used in a computational context, for example in updating databases. First of all, belief sets cannot be handled directly in a database because they contain an infinite number of logical consequences. What is needed is a finite representation of belief sets. Such a representation will in the sequel be called a finite base for a belief set.58 Secondly, when we contract or revise a belief set it can be argued that what we operate on is not the belief set in toto but rather a finite base for the belief set. Nebel (1988, p. 165) writes that “[p]ropositions in finite bases usually represent something like facts, observations, rules, laws, etc., and when we are forced to change the theory we would like to stay as close as possible to the original formulation of the finite base.” And Makinson (1985, p. 357) observes that “in real life when we perform a contraction or a derogation, we never do it to the theory itself (in the sense of a set of propositions closed under consequence) but rather on some finite or recursive or at least recursively enumerable base for the theory. ... In other words, contrary to casual impressions, the intuitive processes of contraction and revision are always applied to more or less clearly identified finite or otherwise manageable bases for theories, which will in general be either irredundant or reasonably close to irredundant.” If B is a finite base, let Cn(B) be the set of all logical consequences of B. A consequence of using finite bases instead of belief sets to represent epistemic states is that there may be two different bases B and B’ such that Cn(B) = Cn(B’) but where revisions or contractions performed on these bases may lead to different new states. Hansson (1989) gives the following illustrative example: [S]uppose that on a public holiday you are standing in the street of a town that has two hamburger restaurants. Let us consider the subset of your belief set that represents your beliefs about whether or not each of these two restaurants is open. When you meet me, eating a hamburger, you draw the conclusion that at least one of the restaurants is open (A › B). Further seeing from a distance that one of the two restaurants has its lights on, you believe that this particular restaurant is open (A). This situation can be represented by the set of beliefs {A, A › B}.

58

For a detailed treatment of belief revision based on finite belief bases, see Hansson (1991).

60

THE DYNAMICS OF THOUGHT When you have reached the restaurant, however, you find a sign saying that it is closed all day. The lights are only turned on for the purpose of cleaning. You now have to include the negation of A, i.e. ¬A into your belief set. The revision of {A, A › B} to include ¬A should still contain A › B, since you still have reasons to believe that one of the two restaurants is open. In contrast, suppose you had not met me or anyone else eating a hamburger. Then your only clue would have been the lights from the restaurant. The original belief system in this case can be represented by the set {A}. After finding out that this restaurant was closed, the resulting set should not contain A › B, since in this case you have no reason to believe that one of the restaurants is open. This example illustrates the need to differentiate in some epistemic contexts between the set {A} and the set {A, A › B}. The closure of {A} is identical to the closure of {A, A › B}. Therefore, if all sets are assumed to be closed under consequence, the distinction between {A} and {A, A › B} cannot be made.

This example will be analyzed in the following section. 4.8 Representing justifications with the aid of epistemic entrenchment It is now, finally, time to return to the challenge from the foundations theory that models of epistemic states based on the coherence theory cannot express that some beliefs are justifications or reasons for other beliefs. I want to show that if we take into account the information provided by the ordering of epistemic entrenchment, the beliefs sets of the AGM model can, at least to some extent, handle this problem. As an illustration of the problems, let us return to the example in section 4.3 where the belief set K was generated from the finite base consisting of the beliefs A “It rains today” and A o B “If it rains, Oscar wears his hat.” A logical consequence of these beliefs is B “Oscar wears his hat.” Now if it turns out that the belief in A was in fact false so that it does not rain after all, then together with A we would like to get rid of B since A presumably is the only reason to believe in B. However, it seems that if we follow the Conservativity Principle, B should be retained in the revised state since it requires that we keep as many of the old beliefs as possible. My reply is that this is far from obvious: The result of the contraction depends on the underlying ordering of epistemic entrenchment. The belief set K is the set of all logical consequences of the beliefs A and A o B, i.e. Cn({A, A o B}), which is identical to Cn({A & B}). The logical relations between the elements of this set can be divided into equivalence classes that can be described as an 8-element Boolean algebra. This algebra can be depicted by a Hesse diagram as in figure 4.1.

BELIEF REVISION AND NONMONOTONIC REASONING

61

Figure 4.1: Hesse diagram of 8-element Boolean algebra.

Here, lines upwards mean logical implication and T is the logical tautology. Now if we want to contract K by A (in order to be able to later add ¬A) we have to give up the elements A, A & B and some of the elements the conjunction of which entail A (for example, we cannot have both A › ¬B and A › B in K-A because A would then also be in K-A). If we require the principle of recovery, i.e. that (K-A)+A = K,59 then we are left with three alternatives for K-A, namely the belief sets Cn({B}), Cn({A l B}), and Cn({¬A › B}). The first two of these are depicted as Hesse diagrams in figure 4.2 (the black dots and heavy lines):

Cn({B})

Cn({A l B})

Figure 4.2: Hesse diagrams of two belief sets.

Which of these possibilities for K-A is the correct one is determined by the underlying ordering of epistemic entrenchment. Basically this amounts to a choice between whether B or A l B should be included in K-A (we cannot have both). According to the criterion (C-) we should include in K-A exactly those elements C for which A < A › C. The proposition A › (A l B) is equivalent to A › ¬B, so what it all boils down to is to decide which of A › 59

For a presentation and discussion of this postulate, see Gärdenfors (1988), p. 62.

62

THE DYNAMICS OF THOUGHT

¬B and A › B has the highest degree of epistemic entrenchment. To do this we should return to the example and look at the meaning of the formulas. A › ¬B is equivalent to ¬A o ¬B which amounts roughly to “if is does not rain, then Oscar does not wear his hat.” A › B, on the other hand, is equivalent to ¬A o B, or in words “if it does not rain, Oscar wears his hat.” (Both formulas are material conditionals and are included in K since they are logical consequences of A.) If rain is the only reason for Oscar to wear his hat, then A › ¬B is epistemically more entrenched than A › B and K-A will then be identical with Cn({A l B}). On the other hand, if Oscar wears his hat even if it does not rain, then A › B is the more entrenched proposition and in this case we will have K-A = Cn({B}). Finally, if they are regarded epistemically equally entrenched, we have the third case where neither of the two disjunctions A › ¬B and A › B will be included, so K-A = Cn({¬A › B}). The point of this fairly technical exercise is to show that if we consider the extra information provided by the ordering of epistemic entrenchment, then this information suffices to account for much of what is required by a foundations theory of belief revision. A belief set in itself does not contain any justifications of the beliefs included. However, if an ordering of epistemic entrenchment is added to such a belief set, then at least some justifications for the beliefs can be reconstructedd from this ordering. A similar analysis can be applied to the hamburger example of the previous section. Let A be the proposition that the restaurant with the lights on is open and let B be the proposition that the other restaurant is open. Hansson claims that if all sets are assumed to be closed under logical consequence, the distinction between {A} and {A, A › B} cannot be made. This is true enough if we consider belief sets without the ordering of epistemic entrenchment. However, the relevant distinction can be made if such an ordering is introduced. If K = Cn({A}) = Cn({A, A › B}) and if we want to form the contraction K-A, we have to decide which of the disjunctions A › B and A › ¬B is epistemically more entrenched. If you have met a man eating hamburgers so that you believe that at least one restaurant is open, this information is captured by the fact that A › B becomes more entrenched than A › ¬B. Consequently, K-A will in this case be identical to Cn({A › B}) as desired. Without this information, however, you have no reason why A › B should be more entrenched than A › ¬B. If they are equally entrenched, K-A will be Cn(Ø), i.e. only contain logically valid formulas. (The third case where A › ¬B is more entrenched than A › B does not occur in Hansson’s example. In this case K-A would have been Cn({A › ¬B}).) In conclusion, the distinction between {A} and {A, A › B} can be made without resorting to finite bases, if one considers the epistemic

BELIEF REVISION AND NONMONOTONIC REASONING

63

entrenchment of the beliefs in a belief set. The upshot of the analyses of the two examples presented here is that the ordering of epistemic entrenchment of propositions, if added to a belief set, contains information about which beliefs are reasons for other beliefs. From a computational point of view, keeping track of the entrenchment of the propositions in a belief set is much simpler than what is the case in justification-based systems like TMS.60 However, I have not given any definition in terms of epistemic entrenchment of what constitutes a reason. The only analysis of reasons and causes in terms of belief revisions that I know of is that of Spohn (1983). I shall now show how Spohn’s analysis relates to the one presented here, but also indicate some problems connected with his proposal. The guiding idea for Spohn is that “my belief in a reason strengthens my belief in what it is a reason for” (1983, p. 372). More technically, he formulates this as follows: A is a reason for B for the person X (being in an epistemic state K) if and only if X’s believing A would raise the epistemic rank of B.61 In the terminology of the present chapter, the notion of ‘raising the epistemic rank’ can be defined in the following way: (R) A is a reason for B in K iff (a) B  K*A, but not B  K*¬A or (b) ¬B  K*¬A, but not ¬B  K*A. If we now apply (Def *) and (Def +) to (R) it can be rewritten as (R’) A is a reason for B in K iff (a) A o B  K-¬A, but not ¬A o B  K-A or (b) ¬A o ¬B  K-A, but not A o ¬B  K-¬A. And if (C-) is applied to this in the case when both A and B are in K, case (a) can in turn be formulated as (R’’) A is a reason for B in K, if¬A < ¬A › B and A › B ” A. The clause that ¬A < ¬A › B is automatically fulfilled according to the postulate (EE4), since in the assumed case, not ¬A  K, but ¬A › B  K. So what remains is the following simple test: (R’’’) If A and B are in K and A › B ” A, then A is a reason for B in K. (A similar analysis can be applied to case (b) of (R’)). Note that in the case assumed in (R’’’) we also have A o B  K. (R’’’) means that if we want to determine whether A is a reason for B, this can be done by comparing the epistemic entrenchment of A › B with 60

Some computational aspects of the entrenchment relation are studied in Gärdenfors and Makinson (1988). Foo and Rao (1988) also show how an ordering of epistemic entrenchment can be used very fruitfully to solve problems in planning programs. Their principle (P