Table Of ContentFundamental Uncertainty
99778800223300559944227722__0011__pprreexxxx..iinndddd ii 1111//2244//22001100 11::0055::3366 PPMM
This page intentionally left blank
Fundamental Uncertainty
Rationality and Plausible Reasoning
Edited by
Silva Marzetti Dall’Aste Brandolini
and
Roberto Scazzieri
99778800223300559944227722__0011__pprreexxxx..iinndddd iiiiii 1111//2244//22001100 11::0055::3366 PPMM
Editorial and selection matter © Silva Marzetti Dall’Aste Brandolini and
Roberto Scazzieri 2011
Individual chapters © Contributors 2011
All rights reserved. No reproduction, copy or transmission of this
publication may be made without written permission.
No portion of this publication may be reproduced, copied or transmitted
save with written permission or in accordance with the provisions of the
Copyright, Designs and Patents Act 1988, or under the terms of any licence
permitting limited copying issued by the Copyright Licensing Agency,
Saffron House, 6–10 Kirby Street, London EC1N 8TS.
Any person who does any unauthorized act in relation to this publication
may be liable to criminal prosecution and civil claims for damages.
The authors have asserted their rights to be identified as the authors of this
work in accordance with the Copyright, Designs and Patents Act 1988.
First published 2011 by
PALGRAVE MACMILLAN
Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited,
registered in England, company number 785998, of Houndmills, Basingstoke,
Hampshire RG21 6XS.
Palgrave Macmillan in the US is a division of St Martin’s Press LLC,
175 Fifth Avenue, New York, NY 10010.
Palgrave Macmillan is the global academic imprint of the above companies
and has companies and representatives throughout the world.
Palgrave® and Macmillan® are registered trademarks in the United States,
the United Kingdom, Europe and other countries.
ISBN 978–0–230–59427–2 hardback
This book is printed on paper suitable for recycling and made from fully
managed and sustained forest sources. Logging, pulping and manufacturing
processes are expected to conform to the environmental regulations of the
country of origin.
A catalogue record for this book is available from the British Library.
A catalog record for this book is available from the Library of Congress.
10 9 8 7 6 5 4 3 2 1
20 19 18 17 16 15 14 13 12 11
Printed and bound in Great Britain by
CPI Antony Rowe, Chippenham and Eastbourne
99778800223300559944227722__0011__pprreexxxx..iinndddd iivv 1111//2244//22001100 11::0055::3366 PPMM
Contents
List of Figures xii
List of Tables xiv
Preface xv
Notes on Contributors xvi
1 Introduction: Fundamental Uncertainty
and Plausible Reasoning 1
Silva Marzetti Dall’Aste Brandolini and Roberto Scazzieri
1.1 Uncertainty, plausible reasoning and the continuum of
inductive methods 1
1.2 A refinement of the distinction between risk and
uncertainty: fundamental uncertainty 5
1.3 Plausible reasoning under fundamental uncertainty 7
1.4 The scope of the volume and its contributions 10
1.5 Epilogue 19
2 Logic, Empiricism and Probability Structures 23
Henry E. Kyburg, Jr.
2.1 Keynes’s Treatise on Probability 23
2.1.1 Logic and empiricism 23
2.1.2 Probability a logical relation 24
2.1.3 Probability values 25
2.2 Ramsey’s criticism 26
2.2.1 A logical relation? 27
2.2.2 Dutch Book 27
2.2.3 Black and white balls 29
2.3 Assessing the philosophical issues 32
3 The Weight of Argument 39
Isaac Levi
3.1 The somewhat novel question 39
3.2 Peirce on balancing reasons 39
v
99778800223300559944227722__0011__pprreexxxx..iinndddd vv 1111//2244//22001100 11::0055::3377 PPMM
vi Contents
3.3 Keynes as a conceptualist 43
3.4 Keynes on frequentism 45
3.5 Is more information always a good thing? 48
3.6 Keynes and Ramsey (and Savage and Good) 49
3.7 Inductive warrant and weight of argument 54
4 A Critical Evaluation of Comparative Probability 59
Vincenzo Fano
4.1 Introduction 59
4.2 Probability as the degree of rational belief 59
4.3 The epistemological dilemma between quantitative
and qualitative probability 62
4.4 Sufficient condition of comparability 64
4.5 Comparative indifference principle 67
4.6 Conclusion: epistemological advantages of
comparative probability 69
5 A Theory of Similarity and Uncertainty 73
Roberto Scazzieri
5.1 Two sides of uncertainty 73
5.2 Variety and types 76
5.2.1 Imprecise circumscriptions and limited
uniformity 76
5.2.2 Polar and intermediate cases 76
5.3 Patterns of similarity 80
5.3.1 Ontological criteria and epistemic criteria 80
5.3.2 Similarity relations and measures of distance 82
5.4 Uncertainty under similarity structures 84
5.4.1 Orders of similarity and similarity judgements 84
5.4.2 Similarity, likelihood and uncertainty 88
5.4.3 Short-term and long-term orders of likelihood 89
5.5 Reasoning with uncertainty 92
5.5.1 Cognitive tension under fundamental
uncertainty 92
5.5.2 Balancing acts 93
5.6 Non-exclusion, uncertainty, and similarity thresholds:
concluding remarks 95
99778800223300559944227722__0011__pprreexxxx..iinndddd vvii 1111//2244//22001100 11::0055::3377 PPMM
Contents vii
6 Generalized Theory of Uncertainty: Principal Concepts
and Ideas 104
Lotfi A. Zadeh
6.1 Introduction 104
6.2 Uncertainty and probability theory 105
6.3 The concept of NL-computation 109
6.4 The concept of precisiation 110
6.5 The concept of cointensive precisiation 113
6.6 A key idea—the meaning postulate 116
6.7 The concept of a generalized constraint 117
6.8 Principal modalities of generalized constraints 118
6.9 The concept of bimodal constraint/distribution 122
6.10 The concept of a group constraint 125
6.11 Primary constraints, composite constraints,
and standard constraints 126
6.12 The generalized constraint language and standard
constraint language 127
6.13 The concept of granular value 128
6.14 The concept of protoform 129
6.15 The concept of generalized constraint-based
computation 132
6.16 Protoformal deduction rules 135
6.17 Examples of computation/deduction 141
6.17.1 The Robert example 141
6.17.2 The tall Swedes problem 142
6.17.3 Tall Swedes and tall Italians 145
6.17.4 Simplified trip planning 146
6.18 Concluding remark 146
7 Weight of Argument and Economic Decisions 151
Alessandro Vercelli
7.1 Introduction 151
7.2 Definitions of ‘weight of argument’ 152
7.3 Weight of argument and modalities of uncertainty 154
7.4 Practical relevance of the weight of argument:
preliminary analysis 158
99778800223300559944227722__0011__pprreexxxx..iinndddd vviiii 1111//2244//22001100 11::0055::3377 PPMM
viii Contents
7.5 The weight of argument in the light of the theory
of decision under uncertainty 160
7.6 The weight of argument in the General Theory 163
7.7 Concluding remarks 167
8 The Relevance Quotient: Keynes and Carnap 171
Domenico Costantini and Ubaldo Garibaldi
8.1 Introduction 171
8.2 From the coefficient of influence to the relevance
quotient 172
8.3 The formal definition of the relevance quotient 173
8.3.1 Regular probabilities 174
8.3.2 Exchangeable probabilities 176
8.3.3 Exchangeability, independence and relevance
quotient 178
8.3.4 Invariant probabilities 180
8.3.5 The main theorem 181
8.3.5.1 A new demonstration 182
8.3.6 Inductive applications 184
8.4 Particle statistics 186
8.5 Closed dynamical systems 190
8.5.1 Destruction and creation probabilities 191
8.5.2 Unary moves 193
8.6 Two economic applications 194
8.6.1 Customers and shops: Ewens’s limit 194
8.6.2 Stock-price dynamics: thermodynamic limit 198
8.7 The role of probability in economics 200
9 Non-Self-Averaging Phenomena in Macroeconomics:
Neglected Sources of Uncertainty and Policy
Ineffectiveness 211
Masanao Aoki
9.1 Introduction 211
9.2 Coefficient of variation and policy ineffectiveness 212
9.2.1 Binary choice: model 1 212
9.2.2 Policy multiplier in binary choice: model 2 214
9.3 Two-parameter Poisson–Dirichlet distribution
of innovation arrival processes 214
99778800223300559944227722__0011__pprreexxxx..iinndddd vviiiiii 1111//2244//22001100 11::0055::3377 PPMM
Contents ix
9.4 GDP is non-self-averaging 216
9.5 Mittag–Leffler function 218
9.6 I neffective policy choices in non-self-averaging models 219
9.6.1 T he non-self-averaging phenomenon in
two blocks of economies 219
9.6.2 T he master equation, probability generating
function and cumulant equations 220
9.6.3 T he dynamics for the mean and the
covariance matrix elements 221
9.6.4 The behaviour of the coefficients of variation 222
9.7 Related simulation studies 223
9.8 Non-self-averaging triangular urn models 224
9.8.1 Balanced triangular urn models 225
9.9 Concluding remarks 228
9.10 Appendix: Mittag–Leffler function and
its distribution 228
10 A Critical Reorientation of Keynes’s Economic
and Philosophical Thoughts 232
Izumi Hishiyama
10.1 Introduction 232
10.2 Induction and the ‘atomic’ universe 233
10.3 A model of ethical behaviour and the atomic
hypothesis 236
10.4 T he Quarterly Journal of Economics article and
the atomic hypothesis 238
10.5 The General Theory and the atomic hypothesis 241
10.6 The ‘theory’ of effective demand 244
10.7 T he ‘principle’ of effective demand: Pasinetti’s view 246
10.8 T he Keynesian world view: deterministic uniformity
in the macro world versus the uncertainty of an
individual unit in the micro world 248
10.9 T he ‘whole-parts’ issue: views of scientists
and scholars in the natural and moral sciences 250
10.9.1 Physicists’ views of natural laws 250
10.9.2 T he sociological view according
to Emile Durkheim 253
10.9.3 Saussure’s view of ‘système’ 255
99778800223300559944227722__0011__pprreexxxx..iinndddd iixx 1111//2244//22001100 11::0055::3377 PPMM
