TRUTHLIKENESS

SYNTHESE LIBRARY

STUDIES IN EPISTEM OLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Managing Editor:

JAAKKO HINTIKKA, Florida State University, Tallahassee

Editors:

DON A L D D A V IDS 0 N, University of California, Berkeley GABRIEL NUCHELMANS, University of Leyden WE S LEY C. SAL M 0 N, University of Pittsburgh

VOLUME 185

ILKKA NIINILUOTO

Department of Philosophy, University of Helsinki

TRUTHLIKENESS

D. REIDEL PUBLISHING COMPANY

A MEMBER OF THE KLUWER ACADEMICPUBUSHERSGROUP

DORDRECHT/BOSTON/LANCASTER/TOKYO

Library of Congress Cataloging-in-Publication Data

Niiniluoto, Ilkka. Truthlikeness.

(Synthese library; v. 185) Bibliography: p. Includes indexes. I. Truth. 2. Logic. I. Title. II. Title: Truth likeness.

BCI7l.N54 1987 121 87-4311

ISBN-13: 978-94-010-8170-2

DOl: 10.1007/978-94-009-3739-0

e-ISBN-13: 978-94-009-3739-0

Published by D. Reidel Publishing Company P.O. Box 17,3300 AA Dordrecht, Holland

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers,

101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group,

P.O. Box 322,3300 AH Dordrecht, Holland

All Rights Reserved © 1987 by D. Reidel Publishing Company, Dordrecht, Holland

Softcover reprint of the hardcover 1st edition 1987

No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical

including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

To Petro, Riikka-Maria,

andAtro

CONTENTS

PREFACE Xl

CHAPTER 1. DISTANCE AND SIMILARITY 1 1.1. Metric Spaces and Distances 1 1.2. Topological Spaces and Unifdrmities 18 1.3. Degrees of Similarity 22 1.4. The Pragmatic Relativity of Similarity

Relations 35

CHAPTER 2. LOGICAL TOOLS 39 2.1. Monadic Languages L~ 39 2.2. Q-Predicates 43 2.3. State Descriptions 47 2.4. Structure Descriptions 50 2.5. Monadic Constituents 51 2.6. Monadic Languages with Identity 58 2.7. Polyadic Constituents 61 2.8. Distributive Normal Forms 72 2.9. First-Order Theories 77 2.10. Inductive Logic 80 2.11. Nomic Constituents 91

CHAPTER 3. QUANTITIES, STATE SPACES, AND LAWS 103 3.1. Quantities and Metrization 103 3.2. From Conceptual Systems to State

Spaces 106 3.3. Laws of Coexistence 109 3.4. Laws of Succession 114 3.5. Probabilistic Laws 118

vii

Vlll CONTENTS

CHAPTER 4. COGNITIVE PROBLEMS, TRUTH, AND

INFORMA TION 122 122 126 134 143 147

4.1. Open and Closed Questions 4.2. Cognitive Problems 4.3. Truth 4.4. Vagueness 4.5. Semantic Information

CHAPTER 5. THE CONCEPT OF TRUTHLIKENESS 156 5.1. Truth, Error, and Fallibilism 156 5.2. Probability and Verisimilitude 160 5.3. Approach to the Truth 164 5.4. Truth: Parts and Degrees 172 5.5. Degrees of Truth: Attempted Definitions 179 5.6. Popper's Qualitative Theory of Truth-

likeness 183 5.7. Quantitative Measures of Verisimilitude 192

CHAPTER 6. THE SIMILARITY APPROACH TO TRUTH-

LIKENESS 198 6.1. Spheres of Similarity 199 6.2. Targets 204 6.3. Distance on Cognitive Problems 209 6.4. Closeness to the Truth 217 6.5. Degrees of Truthlikeness 222 6.6. Comparison with the Tichy-Oddie

Approach 232 6.7. Distance between Statements 242 6.8. Distance from Indefinite Truth 256 6.9. Cognitive Problems with False Presup-

positions 259

CHAPTER 7. ESTIMATION OF TRUTHLIKENESS 263 7.1. The Epistemic Problem of Truthlikeness 263 7.2. Estimated Degrees of Truthlikeness 268 7.3. Probable Verisimilitude 278 7.4. Errors of Observation 280 7.5. Counterfactual Presuppositions and

Approximate Validity 284

CONTENTS ix

CHAPTER 8. SINGULAR STATEMENTS 290 8.1. Simple Qualitative Singular Statements 290 8.2. Distance between State Descriptions 297 8.3. Distance between Structure Descriptions 302 8.4. Quantitative Singular Statements 303

CHAPTER 9. MONADIC GENERALIZATIONS 310 9.1. Distance between Monadic Constituents 310 9.2. Monadic Constituents with Identity 321 9.3. Tichy-Oddie Distances 323 9.4. Existential and Universal Generalizations 335 9.5. Estimation Problem for Generalizations 341

CHAPTER 10. POLYADIC THEORIES 346 10.1. Distance between Polyadic Constituents 346 10.2. Complete Theories 362 10.3. Distance between Possible Worlds 365 10.4. First-Order Theories 368

CHAPTER 11. LEGISIMILITUDE 372 11.1. Verisimilitude vs Lcgisirnilitude 372 11.2. Distance between Nomic Constituents 374 11.3. Distance between Quantitative Laws 382 11.4. Approximation and Idealization 394 11.5. Probabilistic Laws 403

CHAPTER 12. VERISIMILITUDE AS AN EPISTEMIC UTILITY 406 12.1. Cognitive Decision Theory 406 12.2. Epistemic Utilities: Truth, Information,

and Truthlikeness 410 12.3. Comparison with Levi's Theory 416 12.4. Theoretical and Pragmatic Preference 420 12.5. Bayesian Estimation 426

x CONTENTS

CHAPTER 13. OBJECTIONS ANSWERED 442 13.1. Verisimilitude as a Programme 442 13.2. The Problem of Linguistic Variance 446 13.3. Progress and Incommensurability 460 13.4. Truthlikeness and Logical Pragmatics 469

NOTES 474

BIBLIOGRAPHY 490

INDEX OF NAMES 508

INDEX OF SUBJECTS 514

PREFACE

The modern discussion on the concept of truthlikeness was started in 1960. In his influential Word and Object, W. V. O. Quine argued that Charles Peirce's definition of truth as the limit of inquiry is faulty for the reason that the notion 'nearer than' is only "defined for numbers and not for theories". In his contribution to the 1960 International Congress for Logic, Methodology, and Philosophy of Science at Stan­ford, Karl Popper defended the opposite view by defining a compara­tive notion of verisimilitude for theories.

The concept of verisimilitude was originally introduced by the Ancient sceptics to moderate their radical thesis of the inaccessibility of truth. But soon verisimilitudo, indicating likeness to the truth, was confused with probabilitas, which expresses an opiniotative attitude weaker than full certainty. The idea of truthlikeness fell in disrepute also as a result of the careless, often confused and metaphysically loaded way in which many philosophers used - and still use - such concepts as 'degree of truth', 'approximate truth', 'partial truth', and 'approach to the truth'.

Popper's great achievement was his insight that the criticism against truthlikeness - by those who urge that it is meaningless to speak about 'closeness to truth' - is more based on prejudice than argument. Indeed, no one had seriously tried to give a precise definition for this concept. In Popper's view, the realist correspondence conception of truth had already been 'saved' by Alfred Tarski in the 1930s. So he made in 1960 his own attempt to make the notion of truthlikeness respectable - in spite of his general misgivings about the 'scholasticism' of the programme of formal explication within analytical philosophy of science. And he also cherished the hope that the proposal to build a fallibilist theory of science upon the concept of verisimilitude would finally help us to get rid of formal systems of inductive logic, as developed by Rudolf Carnap.

Popper's attempt at explication failed, as papers published in 1974 by David Miller and Pavel Tichy showed. Nevertheless, Sir Karl had succeeded in making an intuitively convincing distinction between truth-

Xl

xii PREFACE

likeness (as some kind of combination of truth and information content) and probability. Further, he had clearly separated the logical or semantical problem of truth likeness (i.e., what does it mean to say that a theory is closer to the truth than another?) from the epistemic problem (Le., on what evidential grounds can one rationally and conjecturally claim that one theory is closer to the truth than another?).

But something essential was missing from Popper's qualitative and quantitative definitions of truthlikeness: perhaps slightly paradoxically, this was the idea of likeness or similarity. Already in 1974, a new programme for explicating verisimilitude was initiated by Pavel Tichy, Risto Hilpinen, and myself, and was soon joined by Raimo Tuomela and Graham Oddie. The basic idea of this 'similarity approach' is that the truthlikeness of a statement h depends on the similarities between the states of affairs allowed by h and the true state of the world.

This book gives a self-contained and comprehensive exposition of the similarity theory of truthlikeness. It summarizes all the main work done in this field since 1974 - with the exception of the use of higher-order logic that Graham Oddie develops in his forthcoming book Likeness to Truth (D. Reidel, 1986). These two works - which have been written simultaneously and independently of each other -are the first book-length treatises on truthlikeness. In spite of some remaining disagreements between Oddie and me, I hope that our books will convince even the prejudiced readers that the concept of veri­similitude is by no means 'meaningless' or 'absurd'. On the contrary, I claim, it is a fascinating and rewarding subject of study within logical pragmatics.

In a companion volume to the present work, Is Science Progressive? (D. Reidel, 1984), I have argued that the concept of truthlikeness is an indispensable ingredient of critical scientific realism. It has important applications as a tool within the history and the philosophy of science - and also a great significance for our understanding of the nature of knowledge-seeking enterprises like science and their role in human culture. This book develops systematically the logical details for a theory of truth likeness that is needed to support the wider philosophi­cal theses of the essays in Is Science Progressive?

In brief, I am confident that the concept of truthlikeness, which has a long but not entirely honourable history, will have a long and bright future. Perhaps all the readers do not share my optimism. But at least I hope that I have done a service for them too. Even if many things still

PREFACE xiii

remain to be done, the frequently voiced complaint that no one has yet given a sufficiently explicit account of truthlikeness, which would make it possible to evaluate the arguments and theses of scientific realists, can now be forgotten. While it cannot be predicted to what extent, if at all, the theory of truthlikeness in the particular form developed here will survive the test of time, at least it is now open to evaluation both by the supporters and the critics of realism.

My own interest in truthlikeness arose from my earlier work on induc­tion, where I attempted to apply inductive logic in an 'anti-inductivist' way to problems involving theories and conceptual change (Niiniluoto and Tuomela, 1973). I had a vague idea that Hintikka's measure of corroboration might have something to do with the epistemic problem of verisimilitude - more than Popper's own measure of corroboration. Moreover, Larry Laudan's (1973) excellent historical survey of the thesis that science "approaches to the truth" had convinced me that the concept of truth likeness is indispensable for a fallibilist and realist theory of scientific progress.

Further stimulus for solving the logical problem of verisimilitude came in the autumn of 1973, when the news about Miller's refutation of Popper's comparative concept of truthlikeness reached Finland. Laudan took the failure of Popper's definition as one of the motives for developing a model of scientific progress which denies that science is a truth-seeking activity. Some of Popper's followers eventually withdrew back to a formulation of critical rationalism with truth and content -but without verisimilitude. But the supporters of the new similarity approach instead wanted to face the challenge of finding a workable account of degrees of truthlikeness.

The first wave of results about truthlikeness, obtained in 1974-79 by Tichy, Miller, myself, Tuomela, and Oddie, relied heavily on a specific tool-box in philosophical logic: Carnap's Q-predicates and state descriptions for monadic languages, and Hintikka's constituents for first-order logic. Already this was disappointing to some critical rationalists who had hoped for a simpler way of saving verisimilitude. What is more, I suggested that a solution to the epistemic problem of verisimilitude can be obtained by calculating expected degrees of truthlikeness relative to a system of inductive probabilities.

In this spirit, I argued in 1 978 that the task of explicating the notion of truthlikeness is "important for all supporters of the 'critical' (as

XlV PREFACE

opposed to 'naive') scientific realism - independently of their relation to the Popperian school". In particular, "the interest in truthlikeness is not incompatible with simultaneous interest in Sir Karl's bete noire, inductive logic".

The systematic use of the Carnap-Hintikka tools from first-order logic had an unintended side effect. I suspect that some philosophers regarded the logical problem of truthlikeness, if not exclusively a Popperian problem, in some sense 'artificial' - not applicable to the relevant and interesting real-life scientific problems involving quantita­tive mathematical theories. Indeed, in the late 1970s there seemed to be a wide gulf separating those philosophers who primarily based their metascientific investigations on qualitative first-order languages and those who employed set-theoretical reconstructions of quantitative theories (among others, Suppes, Sneed, and Stegmiiller). I argued in 1978 that the Sneedian structuralist programme for representing the 'empirical claims' of theories contains as a special case the problem of truthlikeness, but this remark (in spite of its correctness) was not convincing, since I was not then able to show how the treatment of truthlikeness can be translated into cases with quantitative statements.

The situation changed with the second wave of results in 1980-83: Roger Rosenkrantz defined truthlikeness for probabilistic laws; I observed that some standard results about Bayesian statistical estima­tion can be interpreted in terms of the estimated degrees of truthlike­ness; I realized that, as Carnap's qualitative conceptual spaces are simply countable. partitions of the state spaces of quantitative theories, there is a uniform method for analysing approximation relations between lawlike statements relative to such (qualitative or quantitative) spaces.

These observations led me to formulate my theory of truthlikeness in an abstract framework of cognitive problems, which may be interpreted in several alternative ways. At the same time, after ten year's work in this problem area, I somewhat unexpectedly realized that the most reasonable abstract definition of the truhlikeness measure· in a sense contains as a special case Isaac Levi's definition of epistemic utility in his classical Gambling with Truth (1967).

Hence, the solution of the logical and the epistemic problems of truthlikeness given in this book has the nature of a synthesis which merges together two major trends within the theory of scientific inference: Bayesianism (covering the inductive logic of Carnap and

PREFACE xv

Hintikka, and the personalist statistics of Savage) and Popperianism. This synthesis turns out to be also an extension Levi's cognitive decision theory.

The structure of the book is based on the following plan. Chapters 1-4 contain preliminaries which will be used mainly in Chapters 8-13. An impatient reader may start directly from Chapter 5, which is an appetizer for the main course served in Chapters 6-7. A philosopher, who does not have much taste for technical details, may get an idea of the main message of the book by reading the following sections: 4.2, 4.3,5.1-5.4,5.6,6.1-6.6,7.1-7.5,12.1,12.4,13.1-13.4.

To be more specific, Chapter 1 is a general introduction to the concepts of similarity and distance. As these notions constitute the key element of my approach to truth-likeness, it is useful to collect for further reference examples and definitions of similarity relations from various fields - and in this way prepare our intuition for the later chapters (especially Chapters 6-13).

Chapter 2 summarizes the basic tools of logic, including inductive logic and modal logic, that will be needed in Chapters 8-11. Chapter 3 introduces quantitative concepts and laws within a state space frame­work that will be used mainly in Chapters 8.4 and 12.

Chapter 4 gives precise formulations to the concept of cognitive problem, truth, and semantical information. Together with the idea of similarity, these are the basic elements that we need for building up our definition of truthlikeness.

Chapter 5 traces the origin of the concept verisimilitude to the sceptic Carneades and his infallibilist critic St. Augustine. A history of fallibilist dynamic epistemology from Cusanus to Hegel, Peirce, Bradley, Engels, Ewing, and Popper is outlined. This critical survey allows us to distinguish the concept of verisimilitude from other related explicanda like 'partial truth' and 'degree of truth'.

Chapter 6 formulates my general definition for the degree of truth­likeness, Tr(g, h*), of a statement g relative to the 'target' h*. Here g is a disjunction of mutually exclusive hypotheses from set B =

{ hi liE I}, and h* is the most informative true statement in B. Measure Tr is defined in terms of a metric or distance function ~ on B. The proposed definition is compared with a number of rivals. It is also generalized to the cases, where the target h* is disjunctive or only counterfactually true.

XVI PREFACE

Chapter 7 continues the abstract treatment by showing how the estimated degree of truthlikeness of g on evidence e, ver(gl e), can be defined as the expected value of Tr(g, h*), given an epistemic prob­ability distribution on the set B.

The general theory of truthlikeness is then applied to special cases: singular sentences (Chapter 8), monadic generalizations (Chapter 9), polyadic generalizations and first-order theories (Chapter 10), qualita­tive and quantitative laws (Chapter 11). In each case, the distance function ~ is explicitly introduced for the relevant cognitive problem. Distances between sentences are also seen to induce distance measures between structures. This observation leads to a general treatment of approximation and idealization for lawlike quantitative statements (see Chapter 11). Chapter 12 analyses cognitive decision making by taking truthlikeness as the relevant epistemic utility. Maximization of expected verisimilitude as a principle of inference is compared with Levi's acceptance rule, Popper's solution to the problem of pragmatic prefer­ence between theories, and Bayesian statistical estimation of real-valued parameters.

The final Chapter 13 replies to some possible objections to the similarity approach, and compares my metric treatment of 'closeness to the truth' with some alternatives. I argue that Miller's demand for the invariance of comparative truthlikeness relations under all one-to-one translations between conceptual frameworks is definitely too strong. I also show how it is possible to appraise cognitive progress in theory­change with meaning variance. This problem, which depends on logical and philosophical issues about translation and incommensurability, has to be left to some extent open in this work.

Standard notation from logic, set-theory, and elementary mathe­matics is used, but otherwise the book does not presuppose any previous knowledge from the reader. Sections, formulas, figures, examples, and notes are numbered separately for each chapter. In the same chapter they are referred to simply by their number. But when the reference, e.g., to Formula 15 of Chapter 7 occurs in Chapter 8, it has the form 'formula (7.15),.

I have once characterized my treatment of truthlikeness as having "Popper's voice but Carnap's hands": my ambition has been to combine Popper's deep insight about the significance of verisimilitude with Carnap's uncompromising and admirable rigour in working out the

PREFACE XVll

details. Evidently lowe my greatest debt to these two great masters of the philosophy of science in our century.

It is equally clear that my views on the methods and tools of philosophy have been decisively influenced by two great students of scientific inference, laakko Hintikka and Isaac Levi.

In the different periods of my work on truthlikeness, I have profited enormously of the chance of discussing, debating, and disagreeing -through publications, correspondence, and personal contacts - with Risto Hilpinen, David Miller, Pavel Tichy, Raimo Tuomela, Graham addie, Isaac Levi, Roberto Festa, and David Pearce. Other important influences and contacts will be visible in the text that follows the preface.

As always, I have profited from the stimulating intellectual atmos­phere of the Department of Philosophy, University of Helsinki. For practical help in preparing the manuscript, I wish to thank especially Mrs. Auli Kaipainen (for most of the typing) and Mr. Ilpo Halonen (for drawing the figures). And once again I am grateful to my family for encouragement and patience.

I dedicate this book to my children: some day they will be closer to the truth than we ever were.

ILKKA NIINILUOTO