Collected Papers of Stig Kanger with Essays on his Life and Work, Vol II (Synthese Library)

COLLECTED PAPERS OF STIG KANGER

WITH ESSAYS ON HIS LIFE AND WORK Vol. II

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

Managing Editor:

JAAKKO HINTIKKA, Boston University, U.S.A.

Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley, U.S.A. THEO A.F. KUIPERS, University of Groningen, The Netherlands PATRICK SUPPES, Stanford University, California , U.S.A. JAN WOLENSKI, Jagiellonian University. Krakow, Poland

VOLUME 304

COLLECTED PAPERS OF STIG KANGER WITH ESSAYS ON HIS LIFE AND WORK Vol. II Edited by

GHITA HOLMSTROM-HINTIKKA Boston University, Boston, U,S.A.

STEN LINDSTROM Umea University, Umea, Sweden

and RYSIEK SLIWINSKI Uppsala University, Uppsala, Sweden

KLUWER ACADEMIC PUBLISHERS DORDRECHT /BO STON / LONDON

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

ISBN 1-4020-0111-8

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht , The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers , 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands .

Printed on acid-free paper

All Rights Reserved © 2001 Kluwer Academic Publishers 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. Printed in the Netherlands .

TABLE OF CONTENTS

Introduction

VB

Stig Kanger as We Remember Him: Nine Biographical Sketches edited by Krister Segerberg with contributions by Krister Segerberg Soren Hallden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan Berg Soren Stenlund Ingmar Porn Lars Gustafsson Brian F. Chellas Lennart Aqvist Kaj Berge Hansen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

3 11 13 17 19 21 23 25 29

Goran Sundholm: The Proof Theory of Stig Kanger : A Personal Recollection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Dag Prawitz: A Note on Kanger's Work on Efficient Proof Procedures Anatoli Degtyarev and Andrei Voronkov : Kanger 's Choices in Automated Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Kaj Berge Hansen : Kanger's Ideas on Non-Well-Founded Sets: Some Remarks Jaakko Hintikka: The Proper Treatment of Quantifiers in Ordinary Logic Sten Lindstrom: An Exposition and Development of Kanger's Early Semantics for Modal Logic Risto Hilpinen: Stig Kanger on Deontic Logic Lars Lindahl: Stig Kanger's Theory of Rights Lennart Aqvist : Stig Kanger's Theory of Rights : Bearers and Counterparties, Sources-Of-Law, and the Hansson Petaluma Example Ghita Holmstrom-Hintikka: Stig Kanger 's Actions and Influence Sven Ove Hansson : Kanger's Theory of Preference and Choice

31 43 53 69 87 97 131 151 173 185 205

vi Wlodek Rabinowicz : Preference Logic and Radical Interpretation: Kanger Meets Davidson 221 Amartya Sen: Non-Binary Choice and Preference: A Tribute to Stig Kanger 243 Notes on the Contributors 255 Index of Names 261 Subject Index .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

INTRODUCTION

This is the second volume of Collected Papers of Stig Kanger with Essa ys 011 his Life and Work. The first volume contains Kanger's own published papers, most of which have become virtually inaccessible even in his own country, together with one previously unpublished manuscript: 'Choice based on preference'. In this second volume we have collected critical essays on the various aspects of Kanger' s work as well as some biographical sketches. Stig Kanger made groundbreaking contributions to a broad range of areas within both mathematical and philosophical logic: (i) General proof theory: In 1955-57, several logicians - Beth, Hintikka, Kanger and SchUtte, independently of each other - brought about a kind of synthesis between the proof-theoretic methods of Gentzen and the modeltheoretic ones of Tarski. Exploiting the close correspondence between the rules of inference of Gentzen' s calculus of sequents and the semantic clauses of Tarski' s definition of truth, they obtained very natural and simple proofs of Godel's completeness theorem for classical first-order predicate logic . The fundamental idea was to view a proof of a logically valid formula as an unsuccessful attempt to find a counter-model to it. Kanger's completeness proof in his 1957 dissertation Provability in Logic - perhaps the most elegant - established in a direct way the connection between Gentzen' s sequent calculus and Tarski's model theory. As an immediate corollary, Kanger's completeness proof yielded a simple (but non-constructive) proof of Gentzen' s Hauptsatz: Kanger's work in general proof theory is described in Goran Sundholm's contribution to this volume : 'The proof theory of Stig Kanger: a personal recollection'. Sundholm also describes how the Beth- Hintikka- KangerSchUtte proof method has been extended beyond elementary logic after Kanger. In addition, Sundholm's article contains information about Kanger's early work in mathematical logic . Additional light is thrown on Kanger's proof theory and semantics by Kaj Berge Hansen in his 'Kanger's ideas on non-well-founded sets'. Hansen describes how, at one point in his dissertation, Kanger outlines a theory of non-wellfounded sets, and makes use of it in a proof of a version of his completeness theorem for predicate logic. Hansen gives a careful and thorough analysis of Kanger's proof and points out that the particular version of the vii

G. Holmstrbm-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. II, vii-xii. © 200! All Rights Res erved. Printed by Klu.....er Academic Publishers, the Netherlands .

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INTRODUCTION

completeness theorem does not hold without the assumption of non-wellfounded sets. Nowadays, non-wellfounded sets have of course become a topical research subject due especially to the work of Peter Aczel. This was far from the case when Kanger wrote his book. (ii) Efficient proofprocedures and automated deduction: As a by-product, Kanger's completeness proof yields a proofprocedure that is effective in the sense of providing an algorithm for finding a proof of any given logically valid sequent. To construct a proof of a valid sequent r => ~, we start from below with the given sequent and construct a tree of sequents above it by means of repeated backwards applications of the rules of the cut-free sequent calculus . We continue until the process terminates and we have reached an axiom at the top of each branch in the tree. The resulting tree is then a proof of the valid sequent that we started with. Kanger's completeness proof guarantees that the process terminates after finitely many applications of the rules, provided, of course , that the sequent we started with was indeed valid. In the paper 'A simple proof procedure for elementary logic' , Kanger describes how the proof procedure can be extended to predicate logic with identity and how it can be made more efficient for actual implementation on a computer. Kanger's work on efficient proof procedures was carried further by Dag Prawitz. In 'A note on Kanger's work on efficient proof procedures' Prawitz gives a personal account of Kanger's and his own work to make proof procedures more efficient. He describes how in the late 1950s Kanger's first proof procedure was implemented on a computer and the difficulties that then arose. Prawitz gives a fascinating account of the genesis of the so-called "dummy method" at one of Kanger's seminars. The subsequent fate of the Kanger- Prawitz method of "dummies" for making proofs more efficient is also described in Prawitz's article. In 'Kanger's choices in automated reasoning' Anatoli Degtyarev and Andrei Voronkov discuss how Kanger's classical 1963 paper 'A simple proof procedure for elementary logic' fares compared to modem work on automated deduction. They write: "Now, when we are equipped with the impressive amount of techniques developed in this are, we are amazed by the incredible intuition of Kanger that allowed him to choose elegant, interesting (and correct) solutions among many possible choices. This article explains these choices and their place in modem automated deduction". (iii) Algebraic logic. In the 1960s, Kanger came into contact with the group of logicians around Tarski at Berkeley and the work that they pursued on the algebraic study of first-order predicate logic by means of so-called cylindric algebras. Intuitively , cylindric algebras playa role in the study of predicate logic that is analogous to that played by Boolean algebras in the study of

INTRODUCTION

ix

sentential logic . Kanger was very impressed by this work and it inspired him to develop an algebraic logic calculus, where the ordinary sentences of predicate logic are treated as terms, the statements are equations between terms, and the only rule of inference is substitution of equals by equals. In his contribution to this volume, 'The proper treatment of quantifiers in ordinary logic', Jaakko Hintikka reviews Kanger's algebraic approach to standard first-order logic and discusses whether it can be applied more generally, in particular to Hintikka' s own generalization of standard first-order logic, so-called independence-friendly first-order logic (IF-logic). There are several reasons why Kanger's equational approach does not seem well suited for the study of IF-logic. First of all, the set of valid formulas of IFlogic is not recursively enumerable. Hence, IF-logic does not admit of a complete proofprocedure. On the other hand, there exists a complete disproof procedure: If A is an unsatisfiable formula of IF-logic, then there is a taubleaux-type (i.e., Gentzen-type) demonstration of this fact. In ordinary twovalued logic, the existence of a complete disproof procedure is tantamount to the existence of a complete proof procedure. Ordinarily, if a formula is irrefutable, i.e., lacks a counter-model, then it is valid. Due to the failure of the law of excluded middle, this implication does not hold in IF-logic. But couldn't Kanger's equational approach still be applied to IF-logic simply by formulating the rules of logical disproof as an equational calculus? This is not a simple matter either, due to the apparent failure in IF-logic of the principle of compositionality. Intuitively speaking, the semantic interpretation of a formula of IF-logic depends, not only on the semantic interpretations of its subformulas, but also on the context in which the formula occurs. Consequently, substitution of equals by equals (applied to formulas) does not, in general, preserve satisfiability. Hintikka points to a way around this problem. Kanger's algebraic methods can still be used, once predicate symbols and quantifiers have been eliminated in favor of so-called Skolem functions. By means of this technique, the problem of testing a finite set of formulas of IF-logic for satisfiability can be reduced to the problem of testing whether a certain Boolean combination of equations is derivable in an equational calculus ala Kanger. Hintikka remarks: "Kanger's calculus of functional equations can handle more than he himself pointed out". (iv) Semanticsfor modal logic. In Kanger's dissertation from 1957, appears, for the first time in print, a detailed exposition of a Tarski-style modeltheoretic semantics for quantified modal logic. A crucial innovation was the use of accessibility relations in the semantic evaluation clauses for modal operators. Kanger points out that by imposing various formal requirements on

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the accessibility relation one can make the operator satisfy corresponding wellknown axioms of modal logic. In this way, the introduction of accessibility relations made it possible to apply semantic and model-theoretic methods to the study of a variety of modal notions . Kanger's early semantics for modal logic differs in interesting ways from the semantic frameworks developed, at about the same time, by Hintikka, Kripke and Montague . Kanger 's work on modal logic is discussed in Sten Lindstrom's paper 'An exposition and development of Kanger's early semantics for modal logic'. (v) Deontic logic . In 'New Foundations for Ethical Theory' from 1957, Kanger developed a model-theoretic semantics also for normative concepts , the so-called deontic modalities "It ought to be that ...", "It is right that ...", and to imperatives: "Let it be the case that ... !" . Kanger's formal language contains quantifiers as well and he discusses the interplay between these and deontic operators. It is noteworthy that Kanger already in this early paper discusses the notion of agency. In terms of the deontic operators and the notion of agency, Kanger, already in 1957, takes the first steps in developing a theory of rights. Kanger's contributions to deontic logic are discussed in Hilpinen's paper 'Stig Kanger on deontic logic'. (vi) Theory ofrights and actions. Kanger's work in deontic logic led him to develop a typology ofrights , inspired by the work of the American jurist W. N. Hohfeld, within the framework of a formal language containing among its primitive concepts , in addition to deontic operators, the action operator "X sees to it that ...". Kanger's theory of rights is arguably his most substantial and influential contribution outside of the field of pure logic. In this volume, Kanger's theory of rights is dealt with in Lars Lindahl's 'Stig Kanger's theory of rights' and in Lennart Aqvist's 'Stig Kanger's theory of rights: bearers and counterparties, sources-of-law, and the Hansson Petaluma example'. Kanger's contributions to the theory of action are described in Ghita Holmstrom-Hintikka's 'Stig Kanger's actions and influence'. Holmstrom-Hintikka also discusses Kanger's attempts at developing a typology of different kinds of influence that is analogous to his typology of rights. (vii) Theory ofpreference and choice. The theory of preference and rational choice occupied Kanger intermittently during the last 20 years of his life. A comprehensive overview of Kanger's contributions to this area is given by Sven Ove Hansson in 'Kanger' s theory of preference and choice' . Hansson discusses Kanger's attempts to develop a preference logic in the tradition of Hallden, his so-called paradox of exclusive disjunction (more extensively

INTRODUCTION

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treated by Rabinowicz, see below), as well as Kanger's contribution to the theory of rational choice (more extensively treated by Sen, see below) . In his contribution 'Preference logic and radical interpretation: Kanger meets Davidson' , Wlodek Rabinowicz discusses a paradox in preference logic (referred to by Hansson as 'the paradox of exclusive disjunction') that was formulated by Kanger and that led Donald Davidson to modify his theory of radical interpretation. Rabinowicz argues that although Kanger's paradox can be dissolved, Davidson's theory of radical interpretation still confronts serious difficulties. Finally, Amartya Sen in 'Non-binary choice and preference: a tribute to Stig Kanger' discusses Kanger's contribution to the theory of rational choice in 'Choice based on preference'. In this paper Kanger generalizes the standard theory of preference and choice to choice functions that select a set of alternatives from a "menu" of available alternatives against a "background set" of alternatives. As the background set varies, the selected set may vary as well, even if the menu of available alternatives is kept fixed. Sen compares Kanger's approach to the standard theory of rational choice and discusses the reasons that Kanger might have had for adopting his alternative approach. The process of publishing these two volumes dedicated to the work of Stig Kanger has been a genuinely joint venture in which many people have contributed in essential ways. We are grateful to them all, including the contributors of essays on Kanger's life and work. We owe special thanks to Jaakko Hintikka and Krister Segerberg for their enthusiastic and steadfast support of the project as well as for their inspiration and good advice. We also wish to thank Jaakko Hintikka for including the two volumes in the Synthese Library Series and Krister Segerberg for editing the section of biographical sketches. We are grateful to Ms. Annie Kuipers and Mr. Rudolf Rijgersberg at Kluwer for their patience and cooperation, Ms. Kaipainen at the Department of Philosophy of the University of Helsinki for doing an excellent work in transforming Kanger 's typographically difficult texts into camera-ready copy, Sharon Rider and Kaj Berge Hansen for translating some of Kanger' s Swedish texts into English, and Sven Ove Hansson and Lars Lindahl for valuable editorial assistance. Kaj Berge Hansen helped us prepare the indexes for the two volumes and Anders Berglund assisted us with proof reading. Thanks are due to the Department of Philosophy at Uppsala University for arranging the colloquium In memory of Stig Kanger: A Symposium on Stig Kanger's Contributions to Logic and Philosophy, March 13-15, 1998, thereby giving the contributors to Volume II an opportunity of trying out their ideas. We thank Elsevier and Kluwer and the Swedish philosophy journal Theoria for permission to reprint some of Kanger' s papers and some of the papers in

xii

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Volume II. Helle Karman kindly gave her permission to publish the paper that Stig Kanger wrote jointly with her. Finally, we are grateful to Rune Kanger for providing us with pictures of his brother, to Thomas Kanger, Kim Kanger and the estate of Li Kanger for giving us permission to publish Stig Kanger's papers, and to Dagmar Soderberg for her assistance and encouragement. We gratefully acknowledge generous financial support from the Vilhelm Ekman University Fund and the Anders Karitz Foundation without which this project would not have been possible. Boston & Helsinki, Umea, Uppsala, August 2000 GHITA HOLMSTROM-HINTIKKA

STEN LINDSTROM

RYSIEK SLIWINSKI

STIG KANGER AS WE REMEMBER HIM: NINE BIOGRAPHICAL SKETCHES

1. 2. 3. 4. 5. 6. 7. 8. 9.

Krister Segerberg: "Stig Kanger (1924-1988)" Soren Hallden: "Stig Kanger: The Young Philosopher" Jan Berg: "Stig Kanger: Teacher and Friend" Soren Stenlund: "Stig Kanger: Thesis Supervisor and Friend" Ingmar Porn: "Stig Kanger: Games, Research, Friendship" Lars Gustafsson: "Stig Kanger: The Case of the Lost Model Language" Brian F. Chellas: "Stig Kanger: The Summer of 1966" Lennart Aqvist: "Stig Kanger' s Influence on my Philosophical Development" Kaj Berge Hansen: "Stig Kanger as an Educator and as a Thinker"

KRISTER SEGERBERG

STIG KANGER (1924-1988)

The Mission Covenant Church of Sweden - Svenska Missionsfiirbundet, a free church not part of the Swedish State Church - performed missionary work in China 1890-1951. Its effort was concentrated in Hupei, a province in central China through which flows the great Yangtse. In 1919 two young missionaries joined the mission, Gustav Karlsson, a farm labourer from the south of Sweden, and Sally Svensson, a nurse from Stockholm. Gustav's only education beyond elementary school (jolkskolan) was four years of mission school; nevertheless he later gained, by correspondence, two academic degrees in theology from Webster University, Atlanta, Georgia: a bachelor's degree in 1936 and a doctorate in 1948. Sally had obtained a midwife's certificate before her two years of mission school. Gustav and Sally were engaged to be married already before their departure from Sweden, but only three years later were they actually married. Their marriage was blessed with two sons of whom the elder, Stig Gustav, was born on 10 July 1924 in Kuling (short for Ku Niu -ling , the Mountain of the Wild Ox) in the province ofKiangsi, a popular summer resort in the mountains. Stig began school when he was six, and he had six years of schooling before the family returned to Sweden. The first year he was taught by his mother, then by two missionaries: for four years by Ida Pettersson and then one year by Lisa Bjorkdahl, During Ida Pettersson's time the class consisted first of two (Stig and another boy) and later of four (when their brothers had joined). But during the final year, Stig and his brother, Rune, were the only students. In 1930 Gustav and Sally Karlsson decided to adopt Kanger as their family name. The most common traditional Swedish surnames are of the Karlsson type - literally, "Karl's son". So common have these names been that, especially at the end of the nineteenth century and the beginning of the twentieth, many have preferred to change to a more distinctive, often made-up, name . Kanger is such a name, a combination of Karlsson and Hanger. (The latter was the name of Gustav Karlsson 's birth place, a village in Smaland in the south of Sweden, where his forebears had been peasants for generations and his father still operated a small farm. Later Stig inherited and used as a

3 G. Holmstrom-Hintikka, S. Linstrom and R. Slivinski (eds.}, Collected Papers ofStig Kanger with Essays on his Life and Work. Vol. II. 3- 9. © 2(0) Kluwer Academic Publishers. Printed in the Netherlands.

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KRISTER SEGERBERG

holiday home his grandfather's cottage at Erikslund in a forest a few miles from Hanger , usually referred to by Stig as "the Middle of Nowhere".) Except for a visit to Sweden in 1927-29, the family remained in China until 1936. In 1939 the parents left for a final sojourn in China, which was to last through World War II until 1946; the sons were left at a home for missionaries' children in Stockholm. (The parents, both born in 1893, both died in 1954.) Stig attended Palmgrenska Samskolan and then Tekniska Laroverket, passing teknisk studentexamen in May 1942 and studentexamen pa reallinjen in December 1944. The latter examination was the formal prerequisite for university entrance, but by the time he sat it, he had already begun informal studies at the University of Stockholm. In 1945 there was a brief interlude of military service: Stig was called up but was discharged after only a short time. At the university, Kanger followed a normal path, gaining the degree of filosofie kandidat in 1949. In those days the requirement was at least seven units of courses in at least three subjects. Kanger's degree consisted of three units of theoretical philosophy, three units of practical philosophy and one unit of statistics. Two years later, in 1951, he received the degree of filosofie licentiat, a higher degree for which a thesis was required. In his thesis, entitled "En studie i modallogik, med sarskild hansyn till 'bora' -satser" ["A Study in Modal Logic, with Special Attention to 'Ought' -sentences "], Kanger showed how, in a certain sense, deontic logic is reducible to modal logic plus a new primitive constant. After the thesis was accepted - it recei ved the highest grade - Kanger asked his professor, Anders Wedberg, whether he thought that publication was warranted. Wedberg thought not, and the thesis was never published. But when a few years later an idea equivalent to Kanger's was published by Alan Ross Anderson, it attracted a good deal of attention from philosophical logicians. Unfortunately, no copy of Kanger's thesis seems to have survived. In 1957 he defended his doctoral dissertation, Provability in Logic. At that time dissertations were graded : Kanger's was given the second highest grade. The dissertation earned Kanger a position as docent in theoretical philosophy 1957-1963 at the University of Stockholm. The docentur was a much coveted research position of a kind unfortunately no longer existing; just about the only obligation was to lecture seventy-five hours a year - thus between two or three hours a week - on subjects freely chosen. The idea was of course to leave the docent ample time to develop as a researcher. Kanger made good use of this freedom (even though he spent several terms acting in place of professors on leave, something that was better paid). As a formal logician he may have been limited in his methods , but the applications of his work spanned an impressive array of subjects: meaning theory, measurement theory ,

STIG KANGER (1924-1988)

5

ethics, theory of action, theory of rights, theory of preference, phonematics and even (unpublished) ~sthetics. The academic year 1965-66 he spent as a visiting associate professor at the University of Michigan, Ann Arbor, and in the summer of 1966 he taught at Stanford. In 1968-69 he was a visiting professor at the University of California, Berkeley. The drawback of the position of docent was that it was for a limited period only. The future was uncertain, a fact that became even more pressing when the years as docent were up. By luck, this coincided with the appointment in 1963 of Erik Stenius to the Swedish language chair of philosophy at the University of Helsinki. Stenius left vacant the chair of philosophy at the Swedish language university in Finland, Abo Academy in Turku (Abo), a chair originally created for Edward Westermarck. Here Kanger became acting professor for several years before finally being appointed professor ordinarius on 9 February 1968. But by that time his appointment to the chair of theoretical philosophy at the University ofUppsala from 1 July 1968 was well under way. When after his Berkeley year Kanger took up his duties as the new professor at Uppsala in the fall of 1969, he inaugurated a new era. This was at a time when there was money around, and Kanger was good at getting hold of it. For some years, the Uppsala philo sophy department became a thriving hive of activity with visitors coming and going in numbers unprecedented in Sweden. Student numbers, too, rose at all levels. The list of PhDs who wrote their dissertations under Kanger is long by Uppsala standards. Soren Stenlund, Lennart Nordenfelt, Paul Needham, Lars Lindahl, Ingmar Porn, Lars Gustafsson, Craig Dilworth and Bengt Molander received their degree s during Kanger's lifetime, while Jan Odelstad, Patrick Sibelius, Ghita HolmstromHintikka, Sven Ove Hansson and Kaj Berge Hansen finished later. Yet another dissertation influenced by Kanger was one written in political science by Helle Kanger. Furthermore, Kanger dispatched several students to Stanford, notably Ingrid Lindstrom, Sten Lindstrom and Patrick Sibelius, who received their PhDs there. (Thus Sibelius holds two doctorates, as does Porn who earned his first doctoral degree from Birmingham.) Of the many initiatives that Kanger took during his two decades as professor and head of the department, some are worth mentioning here. One was the Hagerstrom Lectures to be given annually by a philosopher of international repute; the lecturer would spend a week in Uppsala, giving one lecture on each of five days but also being available to meet faculty and graduate students. The first Hagerstrom lecturer, in 1970, was Konrad MarcWogau, Kanger's immediate predecessor as professor in Uppsala, followed by von Wright in 1972 and by Quine in 1973; the number of distinguished

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philosophers following in their footsteps is still growing. A second initiative was the Adolf Phalen Annual Memorial Picnic, an informal, three-day affair involving at times considerable numbers of staff and students in the philosophy departments of Uppsala and the two universities of Turku: Abo Academy and University of Turku (in the early years) as well as Helsinki (later). (Phalen and Hagerstrom were nationally famous philosophy professors in Uppsala in the first part of the twentieth century .) A third initiative was the Scandinavian Logic Symposium; today it is languishing, but the first few meetings - Turku (Abo) 1968, Oslo 1970, Uppsala 1973 - were remarkably successful. A fourth initiative , still flourishing, was the revival ofthe Uppsala Philosophical Studies, an in-house monograph series that was considerably enlarged. As the years passed, money became less plentiful. Operations at Villavagen 7, later Villavagen 5, lost some of their momentum . Personal problems began to develop. Kanger's output , never massive, dwindled. Even though his personal situation improved during the last few years, he produced little. However, his enthusiasm for philosophy and logic never ceased. He had a repertoire of pet problems that he would bring up in conversation, as a challenge to himself as much as to his listeners. When he died on 13 March 1988, he still had not solved them all. Kanger was married three times: to Neita Petrini 1949-1960, to Helle Komerup 1961-1978 and to Dagmar SOderberg from 1980. He had two children with Neita (Elisabeth (Li) and Thomas, born 1950 and 1951, respectively) and one son with Helle (Kim, born 1963). He is buried in the cemetery at Hanger.

* Stig Kanger was a hard man to figure out, a mixture of many contrary qualities: gregarious but a loner; sensitive under a crust of insensitivity; unconventional in some ways, conventional in others. He could be caring, yet was not seldom brusque . He could joke about anything , yet be offended when others did so. He gave an impression of being boisterous, yet he said little; he was one of those people it is awkward to talk to on the telephone. In some ways he changed over the years. In his youth he had a lean and hungry look ; later he became substantial. His older friends remember him as devoted to discussion. But in later years he was not very open to the ideas of others , and discussion became one-sided : he was willing to give, not to take. Among human qualities he admired intelligence the most. Becoming a friend of his, one had a feeling of having been admitted to an ordered set, each member ranked according to intelligence; to be lacking in intelligence was a flaw of character. He was certainly himself intelligent, if the word is used in

STIG KANGER (1924-1988)

7

the traditional sense with its emphasis on formal or mathematical ability. Yet it is not clear how much Stig had of the other "intelligences" we hear about today, for example, emotional intelligence, knowledge of self and knowledge of others. Apart from occasional remarks, sometimes very perceptive, he did not like to talk about personal matters. Like all academics, Stig wanted recognition for his work and, like most, felt that he had not got enough of it. In his case the feeling of frustration may have been justified, for Kanger' s work has not had the impact it could have had , had it been better known . Yet the fault was to a certain extent his own. First, he published little ; this he saw as a virtue and used to boast that no other Swedish philosophy professor had ever been appointed on so slim a corpus as he had. Second, his publications all appeared in local or at least peripheral venues. Third, his style of writ ing is off-putting to many readers. Stig loved games, and perhaps he saw writing logic as a game : to give readers as little explanation as possible - but always, in a strict sense, enough - and then challenge them to understand. It is clear that the years of childhood and adolescence were extremely important in forming the adult Kanger. We know little about this part of his life, but I am certain that it holds the key to understanding this complex man. (One Swedish psychologist - himself a child of missionaries who spent an important part of his childhood in the same missionaries' children's home as Stig and his brother - has written about missionaries that "they should not have children".) The early years at the Univers ity of Stockholm must also have been important. Anders Wedberg (1913-1978), professor of theoretical philosophy at the University of Stockholm from 1949 till his death, was an eminent philosopher; he will be remembered, among other things, for doing history of philosophy in a new way and for being instrumental in bringing formal logic into Swedish philosophy. Wedberg was very gifted but also very critical. There used to be a saying that Wedberg had been able to prove a new theorem: "Almost everything is trivial". Then (the saying went on) Kanger came along and succeeded in strengthening this result, establishing the definitive Wedberg/Kanger Theorem: "Everything is trivial ". Wedberg was a perfect example of an analytical philosopher - one good at analysis. For his part, Kanger used to deny that he was one , maintaining that it is not clear what analytic philosophy is, if anything, and that at any rate he, Kanger, was a synthetic philosopher. One wonders what it was like for the young Kanger intense and vulnerable, probably then as later given to occasional coarseness - to try to find his way under the refined, patrician, ever critical Wedberg. Kanger admired Wedberg's intellect, perhaps greatly, but his overall attitude

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to Wedberg was one of ambivalence. Later in life Kanger would sometimes ask younger colleagues what help they had received from their supervisors while writing their dissertations. This question was merely to set up an opportunity for Kanger to make his own cynical comment: that all a student had a right to expect from a supervisor was that the latter not work actively against the student. And that the student would be lucky to be granted that right. If the WedberglKanger Theorem is a myth, there is another thesis that Kanger could actually be heard to propound from time to time: "Philosophy that cannot be mathematized is not philosophy". This is an extreme position, and no doubt Kanger advanced it partly for the sake of provocation. But only partly! Mathematizing philosophy was a method that worked well for him, and he is not the only philosopher in history who has felt that what was good enough for him should be good enough for all. In this spirit he would ask people to name the greatest philosophers of the twentieth century. His own answer was, 1. Godel, 2. Tarski , 3. Church (on occasion he would wonder whether or not to replace Church by Kleene). Needless to say, this attitude amused some, outraged a few and alienated many. One thing Kanger did as professor in Uppsala was to get hold of as many pictures of previous Uppsala professors of philosophy as he could , which he then put in the departmental common room. He himself was the forty-eighth (known) holder of the chair which was (and still is) named the Chair of Theoretical Philosophy. Now a photo of Kanger (taken at his beloved Erikslund) has been added to this collection. However, there is no picture of him in the philosophy department at Abo Academy (where he was the fourth to hold the chair) . There is a big oil painting ofWestermarck and a smaller one of Lagerborg; and after Stenius' s death a water colour of him has been added as well. Once Stig and I talked about this matter. I proposed that Stig donate a photo of himself. Stig did not totally reject the idea of a representation of himself, but of course it would have to be on his terms. He at once enumerated three possibilities. The first: a life size statue in the nude. The second: a picture of the back of his head. The third: a miniature portrait painted on the head of a pin with an attached magnifying glass of sufficient power. I tried to think of some equally extravagant idea to join in the spirit of playful absurdity and suggested that he have a bust made of frozen mercury to be kept in a deep freeze in the department. He told me that was not funny. I am reminded of the famous passage in Malcolm's memoir of Wittgenstein where the latter explains that Cassiopeia is a Wand means Wittgenstein, the former pointing out that it might equally well be thought to be an upside down M and mean Malcolm: "He gravely assured me that I was wrong."

STIG KANGER (1924-1988)

9

After 1965, Stig and I never lived in the same place or even in the same country. When we met, the occasion was always special. Perhaps that is why I remember Stig as surrounded by a nimbus of festivity. He was a generous host but an uneasy guest. I think this was because it was so important for him always to be in control. Power was the subject of some of his research, but it was also something he wanted for himself. He was at his best at Erikslund, "in the Middle of Nowhere"; there he was King. He would often talk, perhaps jokingly, perhaps seriously, of one day erecting a flagpole from which to fly the old Dragon Flag of Imperial China. One of my most vivid memories ofStig is from Erikslund in May 1968. By this time his appointment to Uppsala had been announced, and Stig was full of plans for the future; there were twenty years ahead of him and no end to all the things he wanted to do. Then I asked him what he would do when he retired . His face fell, perhaps the only time I (unwittingly) caught him off guard. "Then it is all over", he said gruffly. "Then one might as well be dead. "

SÖREN HALLDÉN

STIG KANGER: THE YOUNG PHILOSOPHER

Pale. thin . intense and fluently communicative...• this was my first impression of Stig Kanger. It was a cold evening in the late 1940s, and we were standing outside the not very impressive building in which Stockholm College resided. probably after a meeting in the Philosophy Club . Fellow students were usually stem and taciturn, so he surprised me in a nice way. He had a lot to say about Carnap; his comments were sharp and penetrating and also distinctively his own. It immediately became clear that there was something exceptional about this new acquaintance. There was nothing problematic about our encounters these first years . He radiated friendliness. His later way of treating guest lecturers was foreshadowed one evening at the Philosophy Club in Stockholm. I had come from Uppsala to present what, years afterward, was to become my "logic of better", and he immediately invited me to spend the night in his home so that I would be spared the fifty minute trip back to Uppsala. A couple of years later , in the 'fifties, he moved to Uppsala with his family . and it was pleasant to witness the warmth with which he treated his children and the inventiveness with which he played with them. The ideal ofliberty was very much present. and it was clear that he was in principle willing to accept the hardships connected with the corresponding life style. I do not think I was fully aware of his deeper sensitivity in those days, but it became noticeable when we. two junior teachers , together invited Harald Ofstad, full Professor of Practical Philosophy in Stockholm (and a prophet of freedom) to a dinner party in Uppsala. My wife Ruth and I prepared the food, Stig and his wife Neita came with bottles of red wine. It was a festive occasion. but Ofstad was very, very late. Moreover, he was not at all apologetic - he just said that he did not like to be chained down by any binding duties. The pleasantness of the evening was not disturbed. but I discovered the next day that Stig was furious. So his allegiance to the ideal of liberty was not complete! At the end of the 'fifties our relationship became more strained. The one reason was. I think. that the problem of academic survival became acute for both of us. Each wanted a steady job. and nothing of the sort was in sight. I remember a series of seminars in Stockholm which we conducted jointly. The II

G. Holmstrbm-Hintikka , S. Linstrom and R. Slivinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. II, 11- 12. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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two leaders of the seminar got into violent disputes, so violent that the students began to look embarrassed. But between seminars we got on well. We were tense and somewhat unhappy, but I believe that we both found that this kind of relationship has its charm.

JAN BERG

STIG KANGER : TEACHER AND FRIEND

Stig Kanger was born in 1924 and died in 1988. He married thrice and had three children. He received his Ph.D. from the University of Stockholm in 1957 and was Assistant Professor (docent) of Theoretical Philosophy there until 1963, when he was appointed (first acting, later ordinary) Professor of Philosophy at Abo Academy in Finland. In 1968 he was appointed to the Chair of Theoretical Philosophy at the University of Uppsala. Throughout his professional career he stayed in touch with colleagues in the English-speaking world; he was a Visiting Professor of Philosophy at the University of Michigan, Ann Arbor, in 1965/66, and the University of California, Berkeley, in 1968/69. My first encounter with Stig occurred during the Christmas vacation in 1948 at the guest house of an institution often frequented by authors and other Swedish intellectuals. It was in Sigtuna, a small town with roots in the Middle Ages, situated between Stockholm and Uppsala, the two main centres of Stig' s life-work. In this as in all discussion, Stig made a strong impression on me by his way of ripping up arguments of his opponents - occasionally interspersing sarcastic grunts and drawing geometric figures in the air. It emerged that he was studying for afilosofie licentiat, a higher degree at the University of Stockholm. It was a pleasant surprise to find that he took a special interest as well in Dylan Thomas , whose poetry I was just beginning to translate into Swedish. When Stig heard that I had completed my first term in Psychology at the University of Stockholm and was disturbed by the obscurities in the relevant concept formation , he advised me to switch as soon as possible to Theoretical Philosophy, embracing Logic, Semantics and the Theory of Knowledge and Science. Stig had a personal researcher's desk at the Liberal Arts Library, and there one could go and find out which modem works in Philosophy would be worth studying. One obstacle to overcome was that Stig often - grunting and drawing in the air - pronounced world-renowned books he had recently read to be completely unnecessary . But I still possess a yellowed slip of paper from 1952 on which Stig has scribbled the names of some authors of such publications and corresponding articles as a suggestion for my Master's course:

13 G. Holmstrom -Hintikka, S. Linstrom and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work, Vol. II, 13-15 . © 2001 Kluwer Academic Publishers. Printed in the Netherlands .

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Morris, Stem, Richards, Hyman, Carnap (4 titles) , Tarski (2), Church, Quine (2).

By and by I learned that Stig was born in China to a family of Swedish missionaries and that he had lost his parents early on. The only vestige of Christianity in his mind emerged, however, when he declared, with an abrupt grunt and flushed cheeks, "But there is no God, you know!" I used to call him "Jenghis Kanger". During the spring terms of 1955 and 1956 Stig gave two courses in logic to a minimal audience ; these remain the best organized and best executed series of university lectures I have ever attended. After a strict development of the classical sentence calculus and its metatheory there followed an exposition of Boolean algebra culminating in theorems about nonnormal interpretations of the sentential calculus. Then came modal logic, mainly Lewis's S5, and Gentzen's formalisation of sentential logic. All these sections contained a huge variety of well-chosen exercises to be solved for the following week. The second series of lectures began with a formalisation of set theory . In this context Stig presented, for the first time, his heuristic net model, which was to become the butt of sarcastic smiles in Stockholm but forty years later got its recognition in connection with the theory of nonwellfounded sets. Subsequently Stig proceeded to logical semantics, concatenation theory, semantic antinomies and a generalised theory of modalities leading to an analysis of the notion of Ought and a proposal for a logic of imperatives. The whole matter was concluded by a new theory of truth in which even the Antinomy of the Liar and Grelling's Paradox were treated. This was a solid foundation, indeed, for my future studies in philosophy and its history, with mathematical logic functioning as a sort of lingua franca. In the early 1960s Stig delved deeper and deeper into algebraic logic. He would amuse himself by playing the maverick to infuriated senior Swedish professors by proclaiming philosophy to be a mere subdiscipline of abstract algebra . By that time I had decided on choosing the logic of Bernard Bolzano as the topic of my Ph.D. dissertation. During Stig's tutorials I had to exert a discreet counterpressure against his attempts to cram Bolzano into the Procrustean bed of algebraic logic. After the public defence of my Ph.D. thesis in 1962 my wife and I emigrated to positions in the United States, and Stig began commuting to Finland. During my short stay in Sweden in the latter half of the 1960s we had only occasional contact. This changed in 1969 after Stig had taken up the Chair of Theoretical Philosophy at Uppsala and I had accepted an offer from the Technical University in Munich and settled in Germany. My family and I always spend the summers in Sweden, and I was frequently invited, before returning to Munich in September, to deliver lectures to qualified audiences in Stig 's graduate seminar. Stig was a very hospitable person , fond of children. On the way through

STIG KANGER: TEACHER AND FRIEND

15

Sweden heading for Stockholm my family and I were often invited to his inherited cottage deep in the dark woods of southern Sweden. There Stig and I could sit till late at night and discuss a wide range of philosophical problems. In spite of his expertise in mathematical logic, Stig maintained a keen interest in classics like Plato, Aristotle, Kant and Bolzano . Inter alia, I remember a dispute concerning the notion of analyticity in Kant; Stig advocated an interpretation based on Beth 's conception which I found absurd at that time but have later been able to incorporate into a more detailed presentation of Kant's philosophy of mathematics. In 1982 I obtained a position for Stig, financed by the German Research Council, as a Visiting Professor at my university . From February to July 1983 Stig pursued research and gave lectures and seminars about his latest results in algebraic logic and decision theory . Stig's ideas on the former topics are of importance for the philosophy of language through the application of cylindric algebra to problems and concept formations in logical semantics. In the treatment of decision theory he started from a modified version of Sen's theory of preference relations and choice functions. The exposition was carried out in an exact manner with proofs of theorems and detailed explanations of concrete examples and counterexamples. The discussion culminated in a critical appraisal of the notion of rationality in voting procedures, which is of fundamental importance, of course, for our western democratic systems. In particular, the question was examined as to which conditions a preference relation must satisfy in order that the corresponding choice would qualify as rational. In 1987, on the initiative of Professor Essler in Frankfurt and me, Stig received the international Research Award of the Alexander von Humboldt Foundation for his "important contributions to logic and ethics". This award entitles the recipient to carry out his research at learned institutions in the German Federal Republic. On his way to Frankfurt in March 1988, Stig and his wife stayed overnight at their house in southern Sweden. In that dark wood he suddenly took a deep breath and passed away.

SOREN STENLUND

STIG KANGER: THESIS SUPERVISOR AND FRIEND

My first meeting with Stig Kanger was arranged by Professor Konrad MarcWogau, Stig's predecessor in the chair of theoretical philosophy in Uppsala. It was about a year before Stig entered upon his duties. As a result of that meeting I became Stig's first graduate student. We met at Stig's apartment at Brantingstorg, a place I was to visit many times during the following years and enjoy Stig's and Helle's hospitality. Stig's straightforward and unacademic manner of responding to my questions about the subject for my dissertation appealed to me. I was impressed by the great number of ideas he had at this time. The kind of determined attitude with which he presented them made me feel as though I ought to deal with all of them, and I was somewhat bewildered when I left him after four or five hours . One thing I remember Stig saying as I left is this: "Just don't become too philosophical!" Some might be inclined to say that I did not follow his advice . But I am not so sure. As the new professor, Stig was full of great ideas and plans for the Philosophy Department and for the new logical orientation he was determined to carry through. Since that orientation suited me well at that time, Stig often wanted me to be with him in his office discussing and working on his plans, maybe to hear someone agree who was not afraid of his bold ideas. These working sessions often ended in his apartment at Brantingstorg late at night, with several empty bottles of wine. I am sure that Stig felt that these discussions sometimes became "too philosophical". On these occasions we sometimes discussed philosophical issues that Stig refused to talk about in public or allowed anyone else to talk about in his seminars. It soon became clear to both of us that our philosophical temperaments were different. Stig was often satisfied when he could reduce a problem to some simple formal or mechanical scheme or procedure. I was impressed by his skill in finding formal patterns, but that was not enough for me. What I wanted from philosophy was something more. We had many discussions about Wittgenstein in which we could never quite agree. Stig was repelled by the air of pretentious abstruseness that surrounded, not so much Wittgenstein himself, as many of his followers and commentators and the picture of Wittgenstein they had created. Still, there is a feature of Stig's way of working that has some similarity to the feature ofWittgenstein's 17

G. Holmstrom -Hintikka, S. Linstrom and R. Slivinski (eds.}, Collected Papers ofStig Kanger with Essays on his Life and Work, Vol. II, 17-18. © 2(0) Kluwer Academic Publishers. Printed in the Netherlands .

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philosophy that Wittgenstein described as "business-like". They both detested general philosophical jargon. When a discussion became too general and gaseous, Stig, like Wittgenstein, came up with some new, concrete example to test the cash value of the claims made. They both wanted their work to be business-like, although they were engaged in quite different sorts of business. In the first years of our acquaintance we developed a friendship that never really ended. It appears to me now that our friendship was based, not so much on common philosophical interests, as on our common experience of not being quite at home in academic life, and Stig helped me quite a lot in dealing with such difficulties. Our relationship became more tense and our conversations more rare when my interests changed and I decided to go my own way in philosophy. Stig was of course disappointed that I no longer wanted to take part in his projects, but he showed respect for my work in a manner that was of some importance to me. A few years before his death I visited Stig in his new apartment. We drank together and talked "just like old times", as I remember him saying.

INGMAR PORN

STIG KANGER: GAMES, RESEARCH, FRIENDSHIP

At a meeting in Uppsala, about a quarter of a century ago, Stig Kanger invited me to give the logical form of the sentence "Walking is nicer than running". Gleefully he watched me tryout some not very likely alternative structures in elementary logic. Almost at my wit's end I said that there is an elegant solution to the problem if "walking" and "running" are accepted as singular terms, for in that case the sentence may be parsed as a two-place predication. Kanger's response was, "Of course!", and more laughter, and quickly we agreed that the infinitives in "To err is human, to forgive , divine" can be understood in the same way. In this there was a boyish will of a sort, not to playa prank on me but to engage me in a battle he had arranged and knew he would win. The boyish attitude, but without preset design, was expressed in races lasting two or three days, be it in the execution of some practical task, sometimes at his summer cottage; or injunketing in the company of students who received and delivered booster and injection rockets in a colourful disp lay of fireworks ; or in the playful pursuit in discussion of points and paths afforded by quick wit and imagination. In study and research he maintained continuity and control, producing lasting results worthy of his impressive mind. As an academic supervisor he was extremely demanding. Candidates with ill-defined projects or gaps in their philosophical education were quickly made aware of their limitations under his close scrutiny. Students willing to accept his discipline received his unswerving support. I know from experience how stimulating and exciting it was to grasp the solutions he proposed and to follow him in attempts to construct and analyse them. He was somewhat weary of being called an analytical philosopher; he preferred the label constructive philosopher. After many years of friendship we said goodbye in Uppsala two days before his death. Immediately prior to that meeting he had been released from intensive care. His face was ash-grey , his hand was warm. Both of us knew that his end was near. In friendship, generous loyalty was his abiding passion.

19 G. Holmstrom -Hintikka , S. Linstrom and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work, Vol. ll, 19-20. © 2001 Kluwer A cademic Publishers. Printed in the Netherlands.

LARS GUSTAFSSON

STIG KANGER: THE CASE OF THE LOST MODEL LANGUAGE

In the autumn of 1978 I was writing one of my few philosophical books, one entitled Language and Lie. Stig Kanger, who had read an early version, was of the opinion (why, I have never understood) that this book ought to be adorned by a modal logical model language. There is nothing stranger than the advice one receives on showing one's manuscripts to other people! I tried to explain that modal logical model languages, although no doubt very interesting, were not really for me. Furthermore, a model theoretical approach to the concept ' linguistic meaning' was rather alien to my enterprise, which actually contained a critique of Tarski' s theory of meaning. But the cheerful Kanger, with his typical enthusiasm, at once offered to show me how to manufacture a satisfactory model language. He invited me to visit him in the countryside. His summer place was somewhere in the vicinity of Varnamo in the south of Sweden, and as I was on my way to Berlin it would be easy for me to leave the train and spend a week-end at his place. I travelled through a snow storm of increasing viciousness. The evening papers boded ill; more snow was expected . A woman had just been brutally slain not far from Vamarno; the police were looking for the murderer, with the help of road blocks among other things . I had just returned from a month in China and was still feeling the time difference, in the mood of unreality that is characteristic of this condition. When, in darkness and whirling snow, the train finally came to a stop at the Varnamo railway station there was only one person on the platform, namely, Professor Stig Kanger, the great logician, snugly bundled up in a great coat, woollen cap and scarves. He received me with a big smile and an exhortation to hurry up. "We must get to the liquor store before they close!" After several miles over badly ploughed roads we arrived at his small, wood-heated forest cottage. I realised that a couple of bottles of potent beverages was exactly what would be needed in this place. When my modest luggage was transferred to the trunk of Professor Kanger's car, I discovered, to my consternation, two colossal, black plastic bags, which evidently contained a couple of heavy, frozen bodies. "Oh", Stig said, "you wonder what those are?" He removed a bit of the plastic, and a magnificent pig's head became visible. "You see, I have sold a small parcel of forest to a farmer. And he pays me a little now and then. When he is able to. In 21

G. Holmstrbm-Hintikka. S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. II, 21-22. © 2001 Kluwer Academic Publishers. Printed in the Netherlands . Originally published in L. Gustafsson Ett Minnespalats, Bokforlaget Natur och Kultur, Sweden, 1995

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pigs. You will help me to cut them up, won't you, while we talk about model theory?" Such business was fairly common, then as now, in Sweden, the land of high taxes. The next two days were memorable. Enveloped in smoke that made our eyes smart at times, equipped with axes and long knives, we cut, ground, preserved and froze Stig's two pigs. It was a formidable job and, paradoxically, very cold. We partitioned and dissected, in complete parallel, two pigs and one formalised language. When our fingers became too cold and our toes tended to lose all feeling we imbibed a glass or two. Our discussions were richly rewarding. I don't think I have ever learned so much logic in so little time. And a good deal about the art of butchery into the bargain . When a few years later the German edition of Language and Lie was being prepared, one respectful but somewhat curious editor asked me what, in the name of the Lord , model theory had to do with my argument. To this I was not able to give a rational answer. This is how it is: There is no real place for a model theoretic language in Language and Lie. One might say that it appears in the way imaginary numbers can appear in a complicated equation only to disappear in time for the solution. It is there only because I was so fond of Kanger.

BRIAN F. CHELL AS

STIG KANGER: THE SUMMER OF 1966

I first met Stig Kanger in the summer of 1966, when I was assigned as his teaching assistant for an introductory logic course he'd been engaged to teach at Stanford University, where I was working toward my doctorate. Stig and his work were unknown to me, but I quickly came to know the force of his intellect, both in a graduate seminar he conducted and through many conversations. To this day, I think of Stig' s New Foundat ions f or Ethical Theory as an exempl ar of philosoph y. It was Stig who first stimulated my interest in imperatives, and it was also Stig who urged me to ask Dana Scott to supervise my work (which resulted in my dissertati on, The Log ical Form of Imperatives) - two things for which I shall be ever grateful. Shortly after he began teaching at Stanford, Stig invited me and my wife, Merry , to a party. We arrived to find a considera ble gathering of Stig' s friends and colleagues from colleges and universities in the San Francis co Bay area. Stig and his wife Helle greeted us and introduc ed us to their son, Kim, a sturdy two-year-old clad only in a diaper and topped by a mop of blazing blond curl s. With manic energy, Kimmi e, to everyone's delight , was runn ing madl y about the garden. Late in the afte rnoon he discovered the hose, managed to tum it on, and began watering those assembled. Consternation! Pandemonium! Guests ran amok trying to esca pe the spray, tables and chairs overturned and tumbled about the yard . Then our champion, Stig, appeared. Respectively roaring and squealing with laughter , father chased son around the garden, wrested the hose from him, and squirt ed him up a tree with it - a style of parenting I'd never before and I've never since observed. We spent many evenin gs that summe r at the Kan gers' . Stig would produce a pitcher of iced tea and rum for Merry and Helle , and a bowl of ice cubes and a bottle of bourbon (always Jim Beam) for himself and me. Hour s later, usuall y between 2:00 and 3:00 a.m., when there was nothin g left for Stig and me to drink, Merry and I would find our way home . T hose eve nings were harde st on Merry, a reporter for the Palo Alto Times, who had to beat work at 6.00 a.m. Learning that Merry and I were going to Europe that summer , Stig and Helle invited us to visit them at their summer place in the Middle of Nowhere , and help put a new roof on the bam. This we did, though we do n' t reca ll much 23 G. Holmstriim-Hintikka, S. Linstrom and R. Slivinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. Il , 23-24. © 2001 Kluwer A cademic Publishers. Printed in the Netherlands .

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about the bam (except that the carpenter, who walked the bare beams in wooden clogs, fell off and broke his arm). The weeks of our visit were a haze of continuous partying . The days began with champagne and scrambled eggs, followed by sunbathing (we were burned on parts that had never seen the sun, and haven't since). Wine-bibbing continued throughout each day's meals, and every night Stig placed bottles of Danish elephant beer at our bedside against our early rising. Along with energy and intellect, Stig had a magnificent sense of humor, sometimes wry. In 1970, he invited me to give six weeks of lectures on imperatives at the University of Uppsala, where he had taken up the Chair of Theoretical Philosophy. I accepted and with high anxiety asked him what the level of the audience would be for the lectures. He replied - this was in writing and without elucidation - "Unbelievable!" Other occasions with Stig and Helle and Kim crowd my memory - a Valborg 's Night in Uppsala, traveling in Norway (our daughter, Semi, celebrated her first birthday in Oslo), the party for Stig's fiftieth birthday in the Middle of Nowhere - times filled with laughter and friendship. Stig Kanger did more to combine vigor, wit and genius than anyone else I've ever known. He died too soon, and I miss him still.

LENNART AQVIST

STIG KANGER 'S INFLUENCE ON MY PHILOSOPHICAL DEVELOPMENT

Although I was never "officially" a student of Stig Kanger, he never ceased to fascinate me, either person ally or professionally. We met on several occasions already in the 1950s at meetings with the Philosophical Society in Uppsala and at Ingemar Hedenius 's seminars. What particularly impressed me was Kanger's firm belief in logic as a tool - or even the main tool - in philo sophy, a belief that remained unshaken throughout his life. Also, his high opinion of logic in philo sophical methodology extended to all branches of what is nowadays known as philosophical logic, not ju st to the foundat ions of mathematics (although that is the area where it all started). An unforgett able episode took place at the Nordic Summer University in Hillered 1959, where Stig gave a talk with essentially the following provocative content: we have all these famous philosophic al proble ms (he gives a long list of them); insofar as any of these problems have been solved, they have been solved by mathematicians or mathematical logicians but never (Stig insisted), neve r by philosophers. There were quite a few distinguish ed Scand inavian philosophers around. Listening to this message of Kanger ' s, their faces became greener and greener. . . Howe ver, there is an episode via hearsa y (also from the 1950s) that made an even stronger impre ssion upon me. Kanger had given a talk on formal semantics a la Tarski on an occasion when I had been absent. Later Thorild Dahlquist reported its content to me in his usual lucid , exemplary way. I am still unable to decide what impressed me most, Thorild's report or its KangerTarski contents. But I got a real kick out of it, and that kick has never been forgotten.

* Now, I would like to call attention to two problem areas where Kanger's influence on my own work is especially apparent, viz., (1) deontic logic and the emotive theory of ethics, and (2) logic of action and agency and the theory of rights and responsibility.

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1. DEONTIC LOGIC AND THE EMOTIVE THEORY OF ETHICS

In the preface to the volume Deontic Logic: Introductory and Systematic Readings (Dordrecht: Reidel, 1971), the editor, Risto Hilpinen, makes the following pertinent statement: In the study of deontic logic, as well as other branches of what is now commonly called 'philosophical logic' , we can distinguish between two main aspects. On the on hand , deontic logicians have constructed formal theories of various normative concepts, and on the other, they have tested the adequacy of these theories by applying them to analysis of ethical discussion. These applications may also involve criticism of ethical or meta-ethical views. In this book , the former aspect is emphasized in Kanger's paper, and the latter, e.g., in Hintikka's essay. A characteristic feature of this book is the prevalence of semantical methods. These methods are most likely to lead to significant further developments, not only in deontic logic, but in philosophi cal logic and analysis in general.

The Kanger paper mentioned in this quotation is the well known "New Foundations for Ethical Theory", an earlier version of which was published as a mimeographed, privately distributed pamphlet already in 1957; I got acquainted with it in its original form. Kanger's attitude in that essay towards the so-called emotive theory in meta-ethics or moral philosophy came as a surprise to many ethical theorists in Uppsala, where the emotive theory had been, since Hagerstrom and in spite of its controversial nature, a sort of "received view" among moral and legal philosophers. Kanger 's assessment of it made a strong impression on me, and still, in 1999, it continues to do so: The importance of the emotive theory lies particularly in its emphasis on the phenomenon of emotive meaning. Thus, the emotive theory cannot be properly characterized as the theory which denies truth and falsehood to ethical judgments. The phenomenon of emotive meaning cannot be analyzed unless we consider the concrete instances of expressions in the context of human communication, but that is not the concern of this paper.

As everybody knows, or should know, Kanger's concern in this paper was rather to provide semantical foundations for deontic logic precisely by treating ethical judgments as true or false and by giving truth-conditions for them. Kanger's approach then appeared to me to possess two definitive virtues : (i) it elegantly avoided the problem whether deontic sentences could stand in logical relations to one another, a problem known to Swedish emotivists as de s.k. praktiska slutledningamas problem [the problem of the so-called practical inference]; and (ii) it was perfectly consistent with C. L. Stevenson's so-called first pattern of analysis to the effect, roughly, that ethical judgments have at least two meaning components in ordinary discourse, viz., a descriptive, trueor-false meaning component which is of an indexical, egocentric character, as well as an emotive meaning component. Now, virtue (i) , as it were, opened the door to further research into deontic logic as a main tool in meta-ethics research in which I am still engaged. And virtue (ii) led me into the logical study of indexical languages because of the egocentric character of Stevenson's descriptive meaning component. See my paper "Om indexikala spraks

STIG KANGER'S INFLUENCE ON MY PHILOSOPHICAL DEVELOPMENT

27

logik" ["On the logic of indexical languages]" in Sanning, dikt, tro: till Ingemar Hedenius (Uppsala, 1968), where the methodological inspiration from Kanger is conspicuous. The same is also true (although to a smaller degree, perhaps) of my contributions to the logical theory of interrogatives and perforrnatives, e.g., A New Approach to the Logical Theory of Interrogatives (Uppsala 1965, Tubingen 1975), "Revised Foundations for Imperativeepistemic and Interrogative Logic", Theoria 37 (1971) and Performatives and Verifiability by the Use of Language (Uppsala 1972). Finally, my paper "The Emotive Theory of Ethics in the Light of Recent Developments in Formal Semantics and Pragmatics" in Modality , Morality and Other Problems of Sense and Nonsense: to Soren Hallden (Lund 1973) should be mentioned in the present context. 2. LOGIC OF ACTION AND AGENCY AND THE THEORY OF RIGHTS AND RESPONSIBILITY Stig Kanger certainly shared Hilpinen' s optimism concerning the significance of semantical methods, not only in deontic logic but in philosophical logic and analysis in general (cf. our initial quotation from Hilpinen supra). I, for one, inherited that optimism from Stig and was greatly encouraged by it. However, our general agreement in philosophical outlook did not always amount to agreement as to the proper treatment of central issues in specific branches of philosophical logic, such as the logic of actions and agency (causation by agents) and the theory of rights and responsibility; both areas are all-important in the philosophy of law. This appears clearly enough, I think, from a detailed comparison of Kanger's "Law and Logic" (Theoria 38 (1972» to my own contributions "A New Approach to the Logical Theory of Actions and Causality" in S. Stenlund (ed.), Logical Theory and Semantic Analysis: to Stig Kanger (Dordrecht: Reidel, 1974), the book co-authored with Ph. Mullock, Causing Harm: A Logico-legal Study (Berlin: de Gruyter , 1989), and "Stig Kanger's Theory of Rights: Bearers and Counterparties, Sources-of-law, and the Hansson Petaluma Example" (vol. 2 of the present work). In my opinion, two important aspects were missing or inadequately handled in Kanger's treatment of the central issues mentioned above, viz., the aspects of temporality and conditionality. He never tried to base his analysis of agent-causation on conditionallogic, and he seems to have lacked an appropriate temporal framework for working out the analysis of unavoidability or historical necessity, although he was fully aware of the need for this important notion already in 1957. However, all things considered, Kanger 's approach to the theory of rights remains a major achievement in the analytic philosophy of law: his general idea to look for its foundations in a combination of action-logic and deontic logic always struck me as basically sound, convincing and ingenious.

*

28

LENNART AQVIST

To some of his friends, e.g., Ingmar Porn, Lars Lindahl and myself, Stig was, philosophically, a catalyst. He often characterized himself as a constructive rather than a merely analytic philosopher. This "engineering" attitude gives a good clue to both his strength and his weakness as a scholar. One of the last times I met Stig was at a conference in Baden-Baden in 1986, where he was his friendly, charming self, enthusiastic about certain fantastic ideas that were around at that meeting. I did not know at the time that he had just another year and a half to go.

KAJ B0RGE HANSEN

STIG KANGER AS AN EDUCATOR AND AS A THINKER

I started my studies in philosophy at the University of Aarhus towards the end of the 1960s. Because of the stress laid by the philosophy department there on the importance of a solid training in history of philosophy and classical problems, I was for some years prevented from doing what I really wanted to do: logic and the foundations of mathematics . Sometime during 1973 I found in the library of the philosophy department in Aarhus an information folder about the education in theoretical philosophy at Uppsala university, at the time headed by Stig Kanger. I found that some of the books I wanted to study were used as textbooks in courses at Uppsala, for example, Stoll 's Set Theory and Logic and Shoenfield's Mathematical Logic. 'After some months of pondering, weighing risks against possibilities, I decided to move to Uppsala and arrived there in February 1974. Finally I was able to study what I wanted to study! By the time I had finished my undergraduate education during the summer of 1975, I decided to stay in Uppsala and enroll as a graduate student in theoretical philosophy. It was not difficult for me to choose between a department where logic was not tolerated and one where logic was not only tolerated but encouraged by the professor. The first time I read about Bell's theorem was during the autumn semester 1986. I quickly got some new ideas on the problems in the foundations of quantum mechanics connected with the EPR argument and Bell's theorem. The essence of these ideas, later elaborated in my books Ideas on Bell 's Theorem (1989) and Logical Physics: Quantum Reality Theory (1996), was that with a suitable change in the underlying logic, quantum theory could be compatible with both realism and locality . Early on during the spring semester 1988, shortly before Kanger 's untimely death , I presented my ideas in a seminar led by him. During my presentation and in the ensuing discussion he was silent. I then turned to him and asked, "What do you think about my ideas?" He answered with considerable empha sis, "1 don't like them! If you change the logic, everything [that is, quantum mechanics] becomes even more opaque than it already is." As I understand it, this statement gives an excellent characterisation of Kanger's own way of thinking. For him, classical logic, possibly extended with the axioms of set theory or modal operators, was a

29 G. Holmstrom-Hintikka , S. Linstrom and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work, Vol. I/, 29-30. © 200] Kluwer Academic Publish ers. Printed in the Netherlands.

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KAJ B0RGE HANSEN

form of the understanding. A theory or a phenomenon could only be made intelligible if it could be formalised or represented in classical logic . For me, Kanger was primarily important as an educational organiser. During the 1970s and part of the 1980s, the department of philosophy in Uppsala under Stig Kanger was the only one in the Nordic countries where an ambitious student could acquire an adequate education in logic, methodology and foundational issues. Had I not come to Uppsala, I would hardly have been able to get the kind of intellectual training I wanted. The debt of gratitude I feel lowe to Stig Kanger is based mainly on this circumstance. Kanger published a number of remarkable ideas. He is, possibly with one exception, the only individual I have ever met personally for whose ideas and creativity I have had any respect. I had no private social contacts with Kanger. Though he was formally my supervisor, I had little academic contact with him. I nevertheless felt that there was considerable mutual sympathy and respect between us. With his creativity, independent-mindedness and goodness of heart, he is one of the very few great personalities I have met in my life.

GÖRAN SUNDHOLM

THE PROOF THEORY OF STIG KANGER: A PERSONAL RECOLLECTION. *

I.

HOW TO TURN GERHARD GENTZEN ON HIS HEAD : THE SEMANTIC COMPLETENESS OF CUT-FREE SYSTEMS 1. Semantics versus Proof Theory

The term Proof Theory shows a certain ambiguity. In the fifties when Stig Kanger carried out his logical work it stood for a cluster of topics pertaining to the syntactic turnstile 1-, that is, the syntactic counterpart to the semantical notion of (logical) consequence 1=. On the other hand, and more narrowly, it also stood for investigations of the properties of the syntactic turnstile by means of systematic transformations of derivation trees. Stig Kanger was a proof theorist only in the former sense. For him, model-theoretic semantics, couched in a rich set-theoretic framework, held pride of place , and in this he was very close to the then main European school of logic, namely the MUnster School, under the leadership of Heinrich Scholz . There are indeed many questions to be asked with respect to the mere 26 (!!) non-modal pages of Provability in Logic,' Not the least of these is the question : where did Stig Kanger find his semantics? He admired Alfred Tarski above all other logicians . By the side of Finnegan's Wake, Tarski - Mostowski- Robinson, Undecidable Theories.' and, of course, Der Wahrh eitsbegriffin den formalisierten Sprachen.' would have been with him on the Desert Island . The rare off-print copy of the German (1935) version of Tarski's masterpiece from 1933, formerly in Stockholms Hogskolas Humanistiska Bibliotek, now in the University library at Stockholm, bears the mark of careful study, but it does contain the model-theoretic semantics in question only derivatively at pp. 36162: Tarski's official definition of truth in §3, for the general calculus of classes, is not relativized to a domain of individuals, but quantifies over a universe of everything. The key-concept of model-theoretic semantics, as we now know it, is the three-place relation R(U,
=dcf S

satisfies

G. Holmstriim-Hintikka, S. Linstrom and R. Slivinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. fl, 31-42. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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GORANSUNDHOLM

As far as I know Tarski only gave the explicit definition of R, by a metamathematical recursion over the complexity of the wff
2. Conflu ence of Ideas in 1955: Backwards Application of Gentzen Rules My Oxford supervisor, the late Robin Gandy, wrote a splendid paper called The Confluence of Ideas in 1936, in which he dealt with the origins of recursive function theory and the many different ways of defining the same class of functions by Church, Kleene, Herbrand, Godel, Hilbert-Bernays, Turing and POSt. 12 The early history of contemporary mathematical logic is replete with such confluence of ideas. Post, Lukasiewicz and Wittgenstein provide decision-methods for the propositional calculus. Skolem and Fraenkel emend Zermelo's set theory in similar fashion . Tarski and Herbrand prove the Deduction Theorem. Godel , Gentzen and Bernays give the intuitionistic Double-negation interpretation in 1932. Stig Kanger played a central role in two major instances of such confluence. One of these is becoming well-known and concerns the semantics of modal 10gic.13 The other concerns the new method for proving completeness of the classical predicate calculus that was independently developed around 1955 by

THE PROOF THEORY OF STIG KANGER

33

Ewert Willem Beth, 14 J aakko Hintikka, 15 Stig Kanger and Kurt Schutte, 16 Beth and Hintikka took the search for a counter-model, where frustration of the search is held to be a proof, as their starting point and worked with, respectively, semantic tableaux and model sets. [The latter are now also known as 'semi-val uations' (Schutte), 17 '(analytic) consistency properties' , and, perhaps even more appropriately, 'Hintikka sets' (SmulIyan).] Schutte invented a new book-keeping device in terms of positive and negative parts, but it has never caught on outside his immediate circle. The Beth- Hintikka approach was streamlined by Raymond Smullyan, magician friend of Kanger' s, in the 1960's, and codified in his text First-Order Logic in terms of 'Consistency properties' .18 This approach has now become part of standard teaching, above all through the successful Penguin-paperback textbook Logic by Wilfrid Hodges. 19 Of our four inventors only Kanger stayed close to the original Gentzen format." The key observation is the folIowing. Consider the sequent

or r => /)., for short. It holds (logicalIy) when

A] & ... & An

~

B] V ... V B,

is (logically) true . However, and this is the main discovery, the sequent Scan also be read as a task to be resolved, namely: make every antecedent formula in r true and every succedent formula in /). false. This is what Kanger does. For instance, in order to resolve the task A

~

B,

r

=> L1

(that is, the task: make A ~B and all of r true and make alI of /). false), it is necessary either to make alI of r true and to make A and alI of /). false or to make B and all of r true and alI of L1 false . Set out in schematic form this becomes: B,

r

=> /).

A~,r=>/).

This, however, is nothing but an instance of ~ =>, Gentzen's antecedent introduction-rule for ~ . Similar backwards applications of the Gentzen rules yield a systematic search-tree, along whose open branches generate certain semi-valuations - Hintikka sets. The search is frustrated when the systematic decomposition of the sequent-tasks issues in an impossible task. These take the form

34

GORANSUNDHOLM (*)

<1>, C, S =

e, C, 'P,

where one would have to make the wff C both true and false, which is clearly impossible. From the other perspective, though, the sequent (*) is nothing but an axiom of the Gentzen sequent calculus, so if each branch of the search-tree is thus truncated it is converted into a sequent-calculus proof, when read from top to bottom. An open branch in the search-tree, on the other hand, allows one to read off a semi -valuation which - in the case of first-order logic - can be extended immediately to a total valuation: an atomic wff without a value in the semi-valuation is assigned the value true. We have found the desired countermodel to the original sequent. This procedure is perfectly deterministic: at each stage it can be explicitly laid down which wff to attack and, in the case of a backwards application of the quantifier-rules \/= and :3=, what new free variable to choose as a witness. In this fashion we get a primitive recursive two-place function 'I'(x, u) such that when S is a sequent and u = -ck., ..., k.> (with n20), that is, a finite sequence of natural numbers considered as a node in the universal spread, T(fSl,u) gives the (Godel-number [S"] of the) sequent S' (if any) that has to be placed at the node u in our derivation. (For nodes v such that no sequent is placed there , T(fSl,v) is trivially put == 0.) This information can be squeezed out, in some version or other, from all four approaches to the backwards proof-methods. The treatment becomes particularly smooth when one stays close to the Gentzen format, though, and a beautiful exposition of (essentially) the Kanger treatment can be found in Kleene's second text-book Mathematical Logic (as Kleene himself came to realise upon completion of his work)." Dag Prawitz's contribution to the SchUtte-Festschrift from 1974 is another useful exposition." 3. Kanger's Background when Working Backwards

How did Kanger reach this point? Who, if any, were his precursors? First, and foremost, Godel and Gentzen. Godel' s original completeness proof can, in retrospect, be seen as working with semi-valuations, that is, minimal countermodels." rather than with the total valuations that are obtained through the Lindenbaum maximalization technique, now very well-known from the Henkin completeness-proof." Gentzen himself, in his dissertation from 1932 (published 1934-35), used the cut-free formalism to prove that intuitionistic propositional logic is decidable, by means of applying the rules backwards in a hypothetical derivation (which, in virtue of the Hauptsatz; may be assumed cut-free ).25 Kanger knew Oiva Ketonen' s dissertation from 1944,26 which gives a Gentzen-like refinement of the Kriterien der Widerlegbarkeit des reinen

THE PROOFTHEORY OF STIG KANGER

35

Priidikatkalkids from Hilbert-Bemays". He also knew Erik Stenius' bookDas lnterpretationsproblem der fo rmalis ierten Zahl enreihe und ihre formale Widerspruchsfreiheit,28 which gives a Hilbert-Bernays inspired Herbrandtreatment of the proof-theory of classical predicate logic, and of arithmetic with the omega-rule. 4. How Did the Backwards Method Fare After Kanger ? i) Beth shortly afterwards designed also a version of his semantic tableaux that was complete for an intuitionistic system," but , as was shown by work of Godel, Kreisel and Dyson, the completeness proof in question was not constructive. After two decades, in the mid-seventies, W . Veldman" and H . de Swart" at Nijmegen were able to circumvent the God el - Krei sel obstacle by considering con structive Beth-models in which 1- was allowed to be true at certain "exploding" nodes. Later refinements were give n by Friedman and

Dummett." ii) A decade after the Beth - Hintikka- Kanger-Schtitte pro of, the method was extended to the then emerging infinitary logics. E. G. K. Lopez-Escobar, in particular, gave a Kan ger-like completeness-proof for cut- free L"'I"" in his dissertation from 1963, with applications to Craig's Interpolation Theorem and Beth's Theorem ." Also Jon Barwise originally developed his version of the theory of admissible sets on a proof-theoretic basis." but Makkai later eliminated the proof theo ry in favour of an approach in term s of infinitary consistency properties ," now con veniently accessible in Keisler's Model Theory for lnflnitary Logic37 and Barwise's Admissible Sets and Structures/" iii) On a more mode st level, it was realised, e. g. by Kent" and LopezEscobar,'? that the Shoenfield completeness-theorem for the recursive omegarule in arithmetic" was readily provable using the backwards method of proof, and then it yields even Kalmar-elementary proof-trees. I have no information as to whether Kanger knew of the work under i)-iii). Lopez-Escobar's thesis was written under the supervision of Dana Scott, a good friend of Kan ger' s, and it is not improbable that it was known to him . iv) In the mid -sixties Takeutis conjecture con cerning cut-elimination in second-order predicate calculu s" was established by Tait,43Prawitz(2 X),44 and Takahashi. 45 Kang er knew and appreci ated Prawitz's extremely elegant Theoria-proof: a two-sort ed semi-valuation is obt ained by running the backwards method. The predic ate univers es of this sem i-valuation need not be closed under definability. The required total second-order counter- valuation is obtained by closing the ramifi ed analytical hierarchy based on the predic ate

36

GORANSUNDHOLM

universe(s) of the given semi-valuation. These proofs were streamlined a decade later by J.-Y. Girard and put into place in a larger framework." iv) Similar themes were also explored by Kreisel in joint work with Mints and Simpson around 1975.47 By considering search-trees where every branch of recursion-theoretic complexity M closes in an axiom , one obtains "restricted completeness theorems ": Sequent S is true in all M-models if and only if it has an M-founded generalised proof-tree, that is, a not-necessarily well-founded derivationtree (possibly using the rule of repetition: from S conclude S) in which all branches of complexity M are finite. v) Herman Jervell's (1979) proof of Girard's ~-completeness theorem represents a culmination of the development from Kanger's Gentzen-related work via ii)-iv): the semantical, constructive and infinitary aspects are blended into a most harmonious whole." Here optimal use is made of the strong uniformity of the back-wards search -tree. For each countable ordinal a consider a-logic, which in place of numerical constants and the co-rule, uses constants for all ordinals L1,A[Wx]>llL1,(VxL1,A[a/x], guarantees that VX
THE PROOF THEORY OF STIG KANGER

37

II. TRICKS OF THE TRADE: STIG KANGER AS A LOGICAL CHILD OF HIS TIME. In the remainder of my talk I wish to describe a couple of typical efforts of Kanger's that give a good picture of him as a typical child of his logical time. Kanger was very much a logician when he set out on his work in the early fifties, that is, a scholar who contributes to the theory, or conceptual foundations, of logic. When I knew him well - for an academi c year and a half at Uppsala 1975-76, I had virtually daily contact with him - he was no longer a logician in this sense, but a moral and political philosopher who used the tools and concepts of logic for formulating his ideas." The technical development of elementary logic had, by 1950, not yet reached its present mature stage. Matters were not always dealt with in as smooth a way as we now know. It is highly instructive to compare the expository style of, say, Church's Introduction to Mathematical Logic, with Dirk van Dalen's modem Logic and Structure. 50 As I remarked even standard Tarski-semantics had yet to be given. There were no decent expositions of Godel's theory of constructible sets. Kleene ' s theory of hyperarithmetical sets and their connection to well-founded recursive orderings could only be learned from his own heavy-going publications. The Natural Deduct ion revolution was a decade away: finding a derivation in a formal system could still be a very painful experience, demanding considerable ingenuity . The tedium of actually deriving theorems is reduced there to an absolute minimum, using attractive natural deduction methods. Owing to these circumstances it was still possible to give worthwhile contributions to logic that consisted of no more than a couple of pages in the JSL. Also Stig Kanger contributed to this genre, but then often in the form of a letter, or a throw-away remark. Here I wish to mention three such gems of Kanger 's prooftheory. i) In Handbok i Logik (p. 41) he considers classical second order logic with Ackermann's axiom of choice V... Vx 3 y kA

=

V ... 3y k+ IVxA [AXI

. ..

Xk yk+l(X,X\,...,Xk)/y k],

which later gets an important role in the proof-method for 'den utvidgade icke elementara logiken' - a phrase that perfectly captures the flavour and style of the logical works of Stig Kanger - in chapter 6, p. 86. His actual execution of the method is not so impressive : what is impressive is that so early on he had fixed on this principle and considered its relation to semantical cut-elimination. The matter became notorious: classical second-order logic with the

38

GORANSUNDHOLM

axiom of choice proved to be one of the very hardest nuts to crack prooftheoretically whereas in the intuitionistic case you get it virtually for free. For a long time it was not known if the Gentzen rules with and without cut generated the same provable sequents. Only recently Gregori Mints, now at Stanford, was able to give a proof that this was indeed the case, using a second-order formulation of Hilbert's epsilon-calculus (where you get the Ackermann choice-axiom for free) and a suitable technique for extending a semi-valuation to a total valuation along the familiar pattern. ii) The separation problem for classical implicationallogic. Kanger remarked on this in a letter to Hugues Leblanc. (I never saw this letter, but I did see a fair number of other letters to and from Leblanc, who was courting Kanger in the late fifties and wanted to coax him into co-operation. It was clear from the correspondence that Kanger was not keen on the idea and only kept his end up sporadically, when he happened to have had his logical imagination triggered.) What he did was the following, using sequential natural deduction methods in place of Kanger' s Hilbert- Frege style calculus: (~Ec)

r

=>

A

~

B

A

~

C, A => A

I', A=> B. The only difference with standard natural deduction is that we are allowed to cancel certain assumptions on which the minor premise of modus ponens depends. The standard (~E) is a special case of the classical (~Ed, just cancelling no assumptions. We derive Peirce's law as follows: [AP

A A

2

The assumptions I and 3 are here discharged according to ( ~I) and the assumption 2 according to the classical (~Ed. But (~I), (~E) and Peirce's law suffice to derive all implicational tautologies." Furthermore, the rule ( ~Ec) is sound in the sense that when the premises hold logically so does the conclusion. Accordingly, we have a very elegant complete axiomatization ofCIC, the classical implicational calculus. Kanger wrote this up in a letter. Leblanc, on

THE PROOF THEORY OF STIG KANGER

39

the other hand, wrote 9 (nine!) papers in which this idea played a, or even the major role." iii) The omega-rule in predicate logic . This was Kanger at his vintage best. Consider 'elementary logic with identity ' in which we have a set C = {co' c. , c 2, ••• } of constants. Assume that for every k Then also

1- 1A[c/x] . I-

\/xA.

Thus, PC=is closed under an omega-rule. The proof of this observation, when cast in a natural deduction mould, could not be simpler. In the wff A only a finite number of C-constants occur. Let c be a C-constant that does not occur in A. Then, by assumption, A[c/x] is derivable using , say, the derivation D. Since c does not occur in A and there are no open assumptions in D, c functions as an eigenparameter of D. Let a be a parameter that does not occur in D. Then D[a/c] is a derivation of A[a/x] (= (A[c/xJ)[a/c]). An application of (\/1) provides the desired derivation of \/xA. The ubiquitous Leblanc used this remark as well: Paul Bernays noted (my emphasis) in his very readable review (JSL 32 (1967), pp. 256-258) of a French paper of Leblanc' S:53 Here Leblanc points to the astonishing circumstance (observed also by L. Henkin, St. Kanger, and Th . Hailperin) that ... the [omega-rul e] is dispensable .

I was preparing my thesis on the omega-rule at the time and was systematically checking the literature: when I drew Kanger's attention to Bernays' way of abbreviating Stig, he looked up at me, gave a wolf-grin and said: "He-he! Han gar handel serna nagot i forvag. " 54

With this my tour of Kanger' s work in proof theory has come to an end. It only remains for me to put on the record that for Stigprooftheory was second best . He cared little for natural deduction, preferring always to use a symmetric sequent calculus. Constructive methods held no premium and he was troubled by the Swedish constructivist upsurge in the mid-seventies. For him, model theory and semantics always came first. Tarski was the greatest logician and cylindric algebra the culmination in the development of logic from Aristotle onwards. In spite of this, and, perhaps, with a fitting touch of irony that Stig might have been the first to appreciate, as long as proof theory is done after the

40

GORANSUNDHOLM

fashion of Gentzen, the name of Stig Kanger will live on, through his splendid completeness-proof for cut-free systems.

Leyden University NOTES Revised text of an invited lecture read at the Kanger Memorial Symposium, Uppsala University, March, 1993. \ Almqvist & Wiksell, Stockholm, 1957. North-Holland, Amsterdam, 1953. Studia Philosophica Vol. 1 (1936), pp. 261 -405. Here U is a set-theoretic structure, with do(U)=Ai0, s:N-A, and q> is a wff. Enzyklopddle der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (II :e Auflage), Band I , Heft 1, TeilI , H. G. Teubner Verlag, Leipzig , 1952. 6 Is this the price that semantically inclined logicians have to pay ? Owing to their semantic proclivities their touch is less sure when it comes to matters syntactic and then they prefer to play it rigorously by the book . Alnozo Church, Introduction to Mathemat ical Logic, Vol. I, Princeton U. P., 1956 , is the foremost example of this phenomenon. 7 llandbok i Logik. Dell: Logisk Konsekvens, Stocholms Hogskola, 1959. Bonniers, Stockholm, 1958, 1961, 1966. Heinrich Scholz, Mathesis Universalis, Birkhauser, Basel, 1961. 11 10 Springer, Berlin, 1918 (1925 ) . This dual background to Wedberg's work suggests a stimulating task for an historian of Swedish philosophy: trace the influences behind Wedberg's conception of philosophy and its history. II Whether they left Wedberg completely untroubled, I cannot say. 12 In Rolf Herken (ed.) , The Universal Turing Machine , Oxford University Pre ss, 1988, pp . 55- Il2. 13 See Sten Lindstrom's contribution to the present volume. \4 ' Semantic entailment and formal derivability' , Mededelingen der Kon. Nederlandse Akademie van Wetenschappen, Afd . letterkunde, n. S., 18, pp. 309- 342, Amsterdam, 1955. \5 ' Form and content in quantification theory', in: Two Papers in Symbolic Logic, Acta Philosophica Fennica 8, Helsinki, 1955, pp. 7-55. \6 'Ein System des verkniipfenden Schliessens' , Archiv fiir mathematische Logik und Grundlagensorschung,2 (1956), pp. 55-67. 17 'Syntactical and seman tical properties of simple type theory ', Journal ofSymbolic Logic, 25 (1960), pp . 305 -326. 18 Springer, Berlin , 1968. 19 Penguin, Harmondsworth, 1977. 20 An unusually perceptive move of Kanger 's which is still not part of the common logical fare - is the use of "quasi-deductions" and "assumption sequ ent" (p. 19). Usually, the antecedent formulae , that is, the antecedents of consequence relations, are the only assumptions considered, but Kanger clearly perceived that one can also make use of assumption at one level above, so to speak, and assume that a consequence, that is, a sequent, holds, or holds logically. 2\ John Wiley & Sons , N.Y., 1967, at p. 285 . 22 'Comments on Gentzen-type procedures and the classical notion of truth ', in: Justus Diller and Gert Muller (eds .), ProofTheory Symposion KielI974, Lecture Notes in Mathematics, 500 , Springer, Berlin, 1975, pp. 290-319.

THE PROOF THEORY OF STIG KANGER

41

23 Kurt Godel, ' Die Voll standigkeit der Axiome des logischen Funktionenkalktils', Monatshefte fur Mathematik und Physik, 37 (1930), pp . 349 - 360 . 24 Leon Henkin, 'The completeness of the first-order funct ional calculus' , Journal ofSymbolic Logic, 14 (1949). pp. 159 -166. 25 ' Untersuchungen tiber das logi sche Schliessen ' , Mathematische Zeitschrift, 39 (1934). pp. 176-210,405-431 . 26 Oiva Ketonen, Untersuchungen zum Prddikatenkalkid , Annales Academiae Scientarium Fennicae, Ser. A. 1, Mathematica-Physica 23. Helsinki , 1944 . 27 Grundlagen der Mathematik, I (1934), II (1939), Springer, Berlin, in Band Il, § 3.4. 28 Acta Academiae Aboensis, Math. et Phys.• 17:3 . Abo, 1952 . 29 E.W. Beth, 'Semantic construction of intuitionistic logic'. Mededelingen der Kon. Nederlandse Akademie van Wetenschapp en, Afd. letterkunde, n. s., 18, pp . 357-388, Amsterdam, 1956. 30 Verena Dyson and Georg Kreisel, 'Analysis of Beth's Semantic construction of intuitionistic logic', Technical Report No.3, Applied Math ematics and Statistical Laboratories, Stanford University, 1961. 31 'An intuitionistic completeness theorem for intuitionistic predicate logic' , Journal of Symbolic Logic, 41 (1976). pp . 159 -166. 32 'Another intuitionistic completeness proo f, Journal of Symbolic Logic, 41 (1976), pp . 644- 662 . (De Swart used a Smullyan notational variant of the Gentzen sequent calculus that was popularized by Melvin Fitting: the sequent Aj....,An => B" ...,B. is written

TA" ...,TA n, FB" ...,FBn, using so called signed formulae, thereby making explicit the operational instruction embodied in the sequent.) 33 Michael Dummett, Elements ofIntuitionism , Oxford U.P.• 1977, p. 288 . Friedman's work is unpublished. 34 ' An interpolation theorem for denumerably long formulas ', Fundam enta Mathematicae , 58 (1965), pp . 190-205. 35 Infinitary Logic and Admissible Sets, Ph.D. Thesis, Stanford, 1967 . 36 'Preservation theorems for logic with denumerable conjunctions and disjunctions' , Journal ofSymbolic Logic, 34 (1969), pp . 437 -459. 37 North-Holland, Amsterdam. 1971. 38 Springer, Berlin, 1975. 39 c.P. Kent , 'Further restricted ro-rulc' , Part I, Preprint, Case Western Reserve University, c:a 1967 . 40 'Remarks on an infinitary language with constructive formulas' , Journal ofSymbolic Logic, 32 (1967), pp . 305 -318 . 41 J. Shoenfield, ' On a restricted to-rule' . Bulletin de I'Academie Polonaise des Sciences, Ser. Sc. Math., Phys . et Astr. , 7 (1959 ). pp . 405 -407. 42 Gaisi Takeuti, ' On a gene ralized logical calculus ' , Japanese Journal of Mathemati cs, 23 (1953), pp. 39 -96. 43 W.W. Tait, 'A non-constructive proof of Gent zen's Hauptsatz for second-order logic', Bulletin ofthe American Mathematical Society, 72 ( 1966). pp . 980 - 983 . 44 Dag Prawitz, 'Completene ss and Hauptsatz for second order logic' , Theoria, 33 (1967), pp. 246-258, and ' Hauptsatz for higher orde r logic' , Journal of Symbolic Logic, 33 (1968), pp. 452 -457. 45 M. Takahasi , 'A proof of cut-elimination theorem in simple type theory', Journal of the Mathematical Society ofJapan , 19 (1967) , pp. 399 -410.

42

GORANSUNDHOLM

Three-valued Logic and Cut-elimination: the Actual Mean ing of Takeut i' s Conjecture, Dissertationes Mathematicae, 136, Warsaw, 1976. 47 'The use of abstract language in elementary mathematics : some pedagogic examples', in: R. Parikh (ed.), Boston Logic Colloquium, Lecture Notes in Mathematics, 453, Springer, Berlin, 1976, pp. 38- 131. 48 Herman Ruge Jervell, ' Reasoning in trees' , in: D.G. Skordev , Mathematical Logic and its Applications, Plenum Press, N.Y., 1987, pp. 125-135 . 49 The description is his own, applied approvingly, not to himself, but to Lennart Aqvist in a sakkunnigutliuande (expert evaluation of the candidates in the competition for an academic post). Kanger wrote a number of such reports and they give considerable insight into his mature views of his subject: a complete file ought to be assembled . 50 Springer, Berlin , 1980. 51 See Church 's Introduction to Mathematical Logic, op. cit., fn. 6, exercises 12.7, 12.8, and 18.3 for the proof. 52 All but one of which are easily available in his Existence, Truth and Provability, State University of New York Press, Albany, 1982. 53 The one that was not reprinted out of the nine referred to above . 54 "He-he! He is anticipating events ." 46

DAG PRAWTIZ

A NOTE ON KANGER'S WORK ON EFFICIENT PROOF PROCEDURES

Three of Stig Kanger's works belong to proof theory taken in a wide sense: the monograph Provability in Logic of 1957, the paper "A simplified proof method for elementary logic" of 1963, and, in between these two , the mimeographed Handbok i logik written in 1959. I concur in Goran Sundholm's remark in his paper of the present volume that Kanger's main interest in this connection was not proofs themselves but provability and derivability and in particular the relation of these notions to semantical ones. A case in point is Kanger's variant of Gentzen' s calculus of sequents for classical logic , LK, which Kanger develops in Provability in Logic . The purpose is there to give a new demonstration of Godel' s completeness result that every valid formula is provable, i.e. has some proof, no matter which. This picture of Kanger's proof theoretical interests is in need of some supplementations and qualifications, however. Kanger also gave a proof of Gentzen's Hauptsatz, a corner-stone in proof theory, which takes up most of Gentzen's classical paper. The theorem is now obtained as an easy corollary of Kanger's completeness result for a cut-free version of LK: If the sequents r => ~, A and A, r => ~ are both provable, then in view of the soundness of the calculus they are valid and so is the sequent r =>~ (by the semantical validity of the cut rule), and hence by the completeness theorem, the sequent r => ~ is provable without use of the cut rule. Kanger was very fond of his semantical proof of the Hauptsatz and in particular of the ease with which he obtained it. He devotes a section of eight lines to it in his otherwise very condensed monograph. Modestly he remarks that Gentzen' s proof is superior to his own since it is finitary. But Kanger did not really care about a result being established in a finitary way. Therefore, his real attitude, as I remember it, was that the Hauptsatz, having been obtained for free, could hardly be a deep theorem. This indifference to the significance of the Hauptsatz seems to confirm the initial impression that proofs themselves on the object level and their properties were not the kind of things that interested Kanger. But this is not the whole truth. Kanger had an interest in the art of engineering, in how things are made , and he saw that his way of establishing the completeness result for a 43

G. Holmstrom-Hintikka, S. Linstrom and R. Slivinski (eds.}, Collected Papers ofStig Kanger with Essays on his Life and Work, Vol. 1/. 43-52. © ZOO} Kluwer Academic Publishers. Printed in the Netherlands.

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DAGPRAWITZ

cut-free formalism gave rise to a specific proof procedure. Although proof theory did not please his philosophical interest, model theory being his favourite, more practical questions concerning how to find proofs in an efficient way appealed to him. Kanger's monograph Provability in Logic was his doctoral dissertation, presented in theoretical philosophy in Stockholm in the academic year 195657, at which time I was a beginner in philosophy. When my teacher Anders Wedberg, who had also been the teacher of Kanger, was to describe the content of Kanger's thesis to us beginners, he described it as essentially amounting to a new proof procedure for predicate logic, which in principle could be implemented on a computer.' This stimulated me to try to automatize Kanger' s procedure, and this in tum led Stig and me to some new insights in that field. I shall here give some glimpses of this early phase of automatic deduction. 1. KANGER'S FIRST PROOF PROCEDURE

The proof procedure that Kanger developed in Provability in Logic, also described in Sundholm's paper in this volume, consists essentially in applying the rules of a cut-free version of Gentzen's calculus of sequents backwards until one reaches an axiom at the top of each branch of the resulting tree. What did such a proof procedure amount to? We may separate the sentential and the quantification part of the procedure. As for the first part, it is instructive to make a comparison. Every beginning student of logic learns after a while that to verify that a formula in sentential logic is a tautology one does not need to go through all the possible truth value assignments to the atomic formulas. Given for instance the formula [A - (B - C)] - [(A & B) - C]

it is sufficient to reason as follows : A falsifying assignment must make the antecedent true and the succedent (A & B) - C false ; to achieve the latter A and B must be assigned truth and C falsity; but to achieve the first, i.e. to make the antecedent A - (B - C) true, either A must be assigned falsity, which possibility has already been excluded, or B - C must get the value truth, requiring in tum that either B is assigned falsity or that C is assigned truth, which two possibilities have also been excluded. Hence there is no falsifying assignment. This way of reasoning has exactly the same structure as applying the sequent rules as formulated by Kanger backwards: to make a formula true corresponds to putting it in the antecedent of the sequent, i.e. before the arrow,

A NOTE ON KANGER 'S WORK ON EFFICIENT PROOF PROCEDURES

45

and to make it false to putting it in the succedent of the sequent, i.e. after the arrow; that a formula cannot be both true and false corresponds to the fact that a sequent in which a formula occurs both before and after the arrow is an axiom. To use sequent rules in this way thus seemed to be a very appropriate way of finding proofs in sentential logic corresponding to known short cuts in the handling of truth tables. As for the quantificational part, backward applications of the quantification rules amounted to the generation of instances of quantified formulas. In the case of an existential formula in the antecedent or a universal formula in the succedent a new constant was introduced to replace the quantified variable (which corresponds semantically to introducing in the counter model sought for the name of an individual that satisfies an instance of the existential formula or falsifies an instance of the universal formula, respectively). In the case of a universal formula in the antecedent or an existential formula in the sue cedent, instances were instead formed by systematically substituting for the quantified variable all constants that had been introduced in operations of the first kind. The method was quite straightforward and not very sophisticated. But it was a complete proof search that took advantage of the sub-formula property of cut-free proofs, and the achievement was to formulate it precisely. After Kanger's public defence of his dissertation in the spring of 1957, I sat down in the summer to automatize the procedure. I was then also inspired by Beth's semantical tableaux used in his proof of the completeness result, which parallels Kanger's method in many respects. At that time there was a computer in Stockholm known as BESK. It was a huge machine occupying several rooms of what had been the premises of the Royal Technical University. Just then it had the record as the fastest computer of the world. But this was a time when there were no programming languages. One had to use the machine code, which I did not know and did not want to learn. Instead I invented a programming language of my own suitable for the particular task in question, and it was translated to the machine code by my father, who had sometimes been using BESK for certain mathematical calculations. The program was run on the machine by Neri Voghera in 1958, and the whole project was presented at the First International Conference on Information Processing, which was arranged in Paris in 1959 by UNESCO. It was published in 19602 and was one of the very first automated proof procedures. Some roughly equivalent procedures were implemented on other computers and presented in journals at more or less the same time. Our program proved simple theorems of logical textbooks, e.g. the one saying that a transitive and irreflexive relation is asymmetric was proved in 12

46

DAGPRAWITZ

seconds. But as soon as it came to more advanced theorems, it was hopelessly inadequate, in spite of the record speed of the computer. 2. THE DUMMY METHOD The reason for the inefficiency of the method was very obvious. After having introduced a few constants because of existential formulas occurring in the antecedent or universal formulas occurring in the succedent, the number of possible substitutions becomes very large when one is to apply the other two quantification rules backwards to universal formulas in the antecedent or existential formulas in the succedent . As an illustration suppose that we have a sequent of the form 't/xI't/x3't/xz't/x4A( x.,xZ,X3 'X4 )

=>

v»: \'t/xz't/x3't/x4A(x 1,xz,x 3 'x 4 )

where the antecedent and succedent are the same except for a permutation of the quantifiers. A human recognizes the validity of the sequent as soon as she sees that the only difference between the two formulas is the permutation of 't/X Z't/x3• However, to prove the sequent by our procedure, one has to try out different ways of breaking down the antecedent formula by substituting the four constants C j generated by the succedent formula , i.e., after having first obtained the sequent

vx,'t/x3't/xz't/x4A(x l ,xZ,X3'X4 )

=>

A(c,,cZ, C 3 , C 4 ) ,

one has to test at random substitutions of c., Cz, c-, and C4 for the four universally quantified variables in the formula in the antecedent. There are thus 4 4 = 256 possible formulas of the form A(t l,tZ,t3,t4 ) to generate, and we may assume that only one of them, A(c l,cZ' C 3, C4 ) , yields a proof of the original sequent. There was no method for seeking out the relevant substitution instance, and hence one could expect that, on an average, one had to generate a sequent containing 128 substitution instances in the antecedent - quite a long sequent in view of the simplicity of the original sequent. Imagine a similar example but with 10 quantifiers instead of 4 - then there are 1010 possible formulas of the form A(t\,tz,...,t lO) to generate , and the right one will be found after 5 000000 000 steps on an average! Clearly the speed of the computer did not matter much. At a seminar where my automatization of Kanger's and Beth's proof procedures was presented, I draw attention to this inefficiency of the method and suggested that one should try to find the right substitutions by some kind of calculation. I compared our method to solving equations, say 8x + 37 =445, by systematically trying different values for x until the right one was found. Just

A NOTE ON KANGER'S WORK ON EFFICIE NT PROOF PROCEDURES

47

as we find the roots of equations by performing certain calculations and not by running through possible values 1, 2, 3, ..., we should find the right substitution instance directly by some kind of calculation. At this seminar Kanger suggested as a solution simply to replace the variable by a dummy, which could also be called a meta-variable, and to let it stand while continuing as before until one sees what substitutions are appropriate to make for the dummy . He provokingly pronounced that this dummy-method solved the problem. I remember that I was slightly irritated by this proposal. Just to replace the variables by something called dummies did not solve the problem; in the essay discussed at the seminar I had also suggested that one should drop quantifiers, replace the variables by meta-variables ranging over the constants to be substituted later , and go on with applyin g the sentential operations to the formulas - the problem was to "see", as Kanger expressed it, what values should be assigned to the dummies, or how to "put together different proof branches to a proof tree" as I had expressed it. Nevertheless, it did turn out that to replace the variables by dummies or to treat them as meta-variables was a very good strategy, or even the right strategy. The crucial idea was to postpone substitutions for the variables in question until there was a better opportunity for making a suitable choice . When attention was focused on this idea, it was not difficult to point out situations in which it was fairly easy to see what the appropriate substitutions should be. Somewhat later it turned out that one could make different selections of such situations. Kanger described his idea in his Handbok i logik and again in the paper "A simplified proof method for elementary logic". This paper is note-worthy also for other features to be mentioned below. The dummy method is described as follows. New constants are introduced by backward applications of quantification rules as before . At applications ofthe other quantification rules (that do not generate new constants), we substitute not a constant but instead a new dummy d and list at the same time what values the dummies can assume, i.e. we make a note d/t., t2, .. , tn'called a substitution list, containing all constants and dummies that occur in the sequent to which the rule is applied. At some stages we stop and "check whether we can choose values for the dummies from the substitution lists in such a way that all top sequents will be directly demonstrable when we replace the dummies by their values". "Directly demonstrable" means here to be either an axiom or derivable from an axiom by inference rules for identity. . Applied to the sequent occurring in the example above, the method works as follows: The formula in the succedent is replaced by A(c\ ,c2,c3,c4 ) , where c., c2 ' c3 , and C 4 are four new constants, while the variables of the formula in the

48

DAGPRAWITZ

antecedent are replaced by dummies d., which gives rise to the sequent A(d l.d3,dz,d4) => A(c l,cz,c3,c4) and the substitution lists: d/cl,c z,c3,c4; dz/cl,cz,c3,c4,d\; d/c\ .cz,c3,c4 ,d\,d z; d/c\.cz,c3,c4,dl,dz,d3' Now we check whether we can choose values for the dummies in such a way that the sequent becomes an axiom. If A is a predicate symbol in the given example it is easy to see that there is exactly one way to choose values for the dummies so as to make the sequent an axiom. 3. THE SUBSEQUENT FATE OF THE DUMMY METHOD For my part I handled the problem of finding values of the dummies such that top-sequents became axioms by working with systems of equalities and inequalities. It was presented in the paper "An improved proof procedure" in 1960. The example above is of course an easy case: a comparison of what stands before and after the arrow yields the following sufficient and necessary condition for the sequent to be an axiom: d,

=C 1 & dz =C3 & d, =Cz & d, =c4 ·

In general the conditions are more complicated. There may be several ways of turning a sequent into an axiom, and the conditions will then be disjunctive. Furthermore, since some rules of sentential logic have two premisses, the backward applications of inference rules will in general give rise to a tree with several branches, each of which is to have an axiom at the top, and one has thus to form a conjunction of the disjunctive conditions. Furthermore, some conditions may be contradictory, e.g. they may demand that two constants that have to stay different are made identical. To illustrate this more general situation, consider the sequent VxVyVz(Rxy & Rxz - Rxz), vx-Rxx

=>

VxVz(Rxy - -Ryx).

Introducing dummies when applying the quantification rules backwards, we get a sequent

Rd.d, & Rd.d, - Rd.d., ..,Rd4d4 => Rc,c, - ..,Rczc\, which in tum , when applying the rules of sentential rules backwards, gives rise to a tree with the three top-sequents Rd 1d3, Rc.c., Rcc, => Rd4d4 Rc.c., Rc.c, => Rd4d4, Rd.d, Rc\c z, Rczc\ => Rd4d4, Rd zd3·

A NOTE ON KANGER'S WORK ON EFFICIENT PROOF PROCEDURES

49

A sufficient and necessary condition for turning the first sequent into an axiom is (d,

=d, & d, =d.) V (c. =d, & Cz =d 4 ) V (C z =d, & c, =d.),

However, in order to turn the tree into a proof, two different constants must never be identified (since otherwise the correctness of the applications of the quantification rules is not preserved, the restriction of the rules being violated). Hence, the second clause, implying c] =cz, is inconsistent, and the same holds for the third clause. Leaving out such inconsistent clauses, we can form the following sufficient and necessary condition for turning all the three sequents into axioms and the whole tree into a proof: [d, =d, & d, =d 4] & [(c, = d. & Cz =d z) V (c z =d, & & [(c] = dz & C z =d3 ) V (c z = dz & c] =d3 ) ] .

C1

=d z)] &

Rewriting this in disjunctive normal form , we find that two of the four disjuncts are inconsistent, and we are left with the following two disjuncts (after dropping repetitions): [(d 1 = d, & d, =d 4) & (c, =d, & Cz = dz) & c, = d 3] V [(d 1 =d, & d 3 =d.) & (c z =d, & c, = dz) & Cz =d.] . These correspond to the two ways in which the tree can be turned into a proof. This illustrates a general method for finding all sets of values of the dummies such that the tree obtained at a given stage of the application of the inference rules backward becomes a proof. When there is no such assignment to the dummies, we must repeat the backward applications of the quantification rules so that further instances are formed with new dummies and constants. We must also respect not only the restriction that two different constants cannot be identified but also certainordering relations : a dummy d cannot be identified with a constant c if d replaces a variable bound by a quantifier that is in the scope of the quantifier which binds the variable replaced by c. Formally, we may express this by d < c, counting conditions that imply that c < c as inconsistent. This has to suffice as an indication of the method of constructing a sufficient and necessary condition in the form of Boolean combination of equalities and inequalities which the dummies have to satisfy in order for the tree constructed at a certain stage to be a proof. All assignments of values (if any) to the dummies that make the tree a proof can thus be found from the Boolean combination in question by bringing it into conjunctive normal form. The method described was never implemented on a computer.' It was soon surpassed by the resolution method proposed by Robinson [1965] . Robinson,

50

DAGPRAWITZ

who states that he was inspired by the method described in Prawitz [1960], took over the technique of dummies , but instead of postponing the choice of values for the dummies until one has a complete assignment that gives the desired proof as in Prawitz [1962]. the dummies are determined stepwise so as to make possible applications of the resolution principle, the rule of sentential logic that Robinson is using instead of the Gentzen rules. To facilitate comparison with what has been said above, we can formulate the resolution principle as an inference rule in a calculus of sequents as follows: - (B & C) - (A & B), (.A & C)

Given a sequent - (Ptl"t 2••··.tn & B), (.PU 1.,U 2, ••• .u, & C)• ... one determines, if possible, the dummies so as to make t, identical to u, (i = 1, 2, ..., n) (using an algorithm called unification, if the terms are compound), thus permitting a backward application of the resolution principle. Robinson is actually treating the dual problem of showing a formula to be inconsistent, but this difference is insignificant and is ignored here. The use of dummies is thus essential in the resolution method. but the dummies are now used to generate applications of the resolution principal, regardless of whether a specific application of this principle is contributing to finding a proof. In other words. one is not any longer looking for the shortest possible proof. This may help the speed since it is quite possible that fewer operations are needed to find a long proof than to find the shortest one. The resolution method soon came to dominate the field of automatic deduction. It was in turn developed in many directions. but the technique of dummies was always kept. In this way the dummy method lives on as an essential ingredient of automatic deduction today (whereas, contrary to the first impression. the way of handling sentential logic inspired by Gentzen's calculus of sequents has turned out to be a very inefficient method). 4. SOME REMARKS ON THE FIELD OF AUTOMATIC DEDUCTION Kanger's paper "A simplified proof method for elementary logic" deals with first order logic with identity and functional symbols and, besides the dummy method, it contains several interesting ideas for how to handle identities between individual terms. The use of function symbols is quite essential in many actual applications of automatic deduction. I refer the reader to the paper

A NOTE ON KANGER 'S WORK ON EFFICIENT PROOF PROCEDURES

51

by Degtyarev and Voronkov in this volume, which describes these ideas of Kanger's in more detail. In conclusion I shall only offer a short comment on the present status of automatic deduction. When Kanger made his pioneering contributions to the field, some people (but not Kanger, I believe) had the hope that automatic deduction would be used in the future to solve difficult mathematical problems. I think it is fair to say that this hope has been frustrated. Nevertheless, automatic deduction is today a thriving subject. One main reason for this is that automatic deduction is important in allowing us to prove very long formulas that express requirements that actually existing physical systems have to satisfy . When one is designing a security system, e.g. for railways or atomic energy plants, it is important to prove that certain situations cannot occur when the system works as designed. Such a task can often be recast as a problem of proving that certain first order formulas hold logically or follow logically form given premisses. Since there often are many parameters involved, the formulas can become very long. Although the formulas do not express anything of theoretical interest and although their proofs are not sophisticated, they are not easily managed by hand because of their size. It thus becomes mandatory to process them by efficient automatic proof procedures. We have here a field where automatic deduction, in spite of its shortcomings when there is a question of solving hard theoretical problems, is playing a role of increasing importance. Stockholm University NOTES I For the most part Kanger and Wedberg had quite different preferences. For instance, Wedberg appreciated Kanger's semantics for modal logic because of its instrumental value and did not attach any philosophical interest to it. He saw it as a purely meta-theoretical device for getting proof theoretical results about modal logic, not as something that throws any light upon the meaning of modalities, which certainly was an interest of Kanger's. But his proof procedure for predicate logie was something that Wedberg appreciated. 2 D. Prawitz, H. Prawitz and N. Voghera [1960] . 3 A further improvement and more condensed description of the method appeared in a later paper of mine [Prawitz 1969], which more conveniently uses Skolem functions .

REFERENCES Kanger, Stig, 1957, Provability in Logic, Acta Universitatis Stockholmensis, Stockholm studies in philosophy 1, University of Stockholm.

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Kanger, Stig , 1963, "Handbok i logik" (A simplified proof method for elementary logic) , in P. Braffort & D. Hirschberg (eds .), Computer Programming and Formal Systems . Studies in logic and the foundations of mathematics , North-Holland, pp. 87-94. Prawitz, Dag, 1960, An improved proof procedure, Theoria, 26, pp. 102-139. Prawitz, Dag, 1969, Advances and problems in mechanical proof procedures, in: Machine Intelligence 4, ed. Meltzer, B. and Michie, Edinburgh University Press, pp. 59-71. Prawitz, D., Prawitz, H., and Voghera, N., 1960, A mechanical proof procedure and its realization in an electronic computer, Journal ofthe Associationfor Computing Mach inery, 7, pp. 102-128. Robinson, 1. A. 1965, A machine oriented logic based on the resolution principle, Journal ofthe Association for Computing Machinery, 12, pp. 23-44.

ANATOLI DEGTYAREV AND ANDREI VORONKOV

KANGER 'S CHOICES IN AUTOMATED REASONING

Automated deduction, or automated theorem proving is a branch of science that deals with automatic search for a proof. The contribution of Kanger to automated deduction is well-recognized. His monograph [1957] introduced a calculus Le, which was one of the first calculi intended for automated proofsearch. His article [1963] was later republished as [Kanger 1983] in the collection of "classical papers on computational logic". Kanger's [1963] (and also [1959]) calculi used some interesting features that have not been noted for a number of years, and the importance of which in the area of automated deduction has been recognized only much later. Kanger [1963] gives no proofs and uses very succinct presentation. Automated deduction is an area in which very subtle chang es in definitions and assertions may lead to inconsistent conclu sions. Kanger ' s [1963] area was theorem proving in sequent calcul i with equality and funct ion symbols. Most papers published in this area before 1995 contained serious mistakes, except for Kangers. Now, when we are equipped with the impressive amount of techniques developed in this area , we are amazed by the incredibl e intuition of Kanger that allowed him to choose elegant, interesting (and correct) solutions among many possible choice s. Thi s article explains these choices and their place in modem automated deduction. 1.1==f-

The title of this section 1= :; f- is the logo of the Association for Logic Programming : truth is equivalent to provability. The equivalence of validity and provability for classical logic was proved by Godel [1930] and is known as Godel 's completeness theorem. The notion s of truth and validity in logic are formulated as semantical properties, while the notion of provability is defined in a purely syntacti cal way, so there seems to be a gap between the two notions . In 1955-1957 several new proofs of Gode ls completene ss theorem appeared [Beth 1955, Hintikka 1955, Schtltte 1956, Kanger 1957] in which model theory and proof theory were connected in a very natural manner. The y 53 G. Holmstrom-Hintikka, S. Linstrom and R. Slivinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work . Vol. fl. 53-67. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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ANATOLI DEGTYAREV AND ANDREIVORONKOV

are based on the idea of searching for countermodels of a given formula Fby applying a proof-search procedure to F (i.e. trying to establish s-F). Kanger proposed to search for a proof in a sequent calculus named LC [Kanger 19571. Cut-free sequent calculi for first-order logic have been introduced by Gentzen [1934]. They turned out to be an important tool for investigating basic proof-theoretic problems [e.g . Gentzen 1936, Girard 1987] . It has also been realized that sequent systems give a convenient tool for designing proof-search algorithms by using the rules of a calculus backwards (i.e, from the conclusion to the premise). To prove a sequent S "we start from below with S and proceed upwards from level to level in the tree form. At each level the sequent of the next level above are uniquely and effectively determined - if there is such a level. lfthere is no such level, this fact is effectively determined, so that the process may brought to an end. " [Kanger 1957, page 31]. Consider some choices that arise when one formalizes sequent calculi. Choice 1 (structure rules) In the original Gentzen's LK a sequent was an expression r - 11, where I', 11 are sequences of formulas . Since rand 11 play the role of a conjunction and a disjunction, respectively, the logical semantics of a sequent is independent of the order of formulas in I', 11. Neither does it depend on duplicate occurrences of formulas in r or 11. Therefore, Gentzen had to introduce several structure rules that allow one to interchange and duplicate formulas in I', 11, and also add new formulas :

rr-

11" B, A, 11 2 11" A, B, 112

r 1,B,A ,r2 - 11

r., A, B, r, - 11 r, A, A - 11 r, A - 11

r-11

r-11

r - 11, A

I', A - 11

These rules are called exchange , contraction and weakening. The use of these rules introduced unnecessary technical details in proofs of [Gentzen 1934] . In order to avoid complications, other structures than sequences should be adopted. One obvious choice is the use sets instead of sequents. This again makes the formalization of sequent calculi quite complex. Suppose that I', 11 are sets and consider the following rule of sequent calculi:

KANGER 'S CHOICES IN AUTOMATED REASONING

55

r-~ u{A}

r -~u

{A V B} (- V)

Let r be empty and consider four different instantiations for ~: {}, {A}, {A V B}, and {A , A V B}. We obtain the following four instances of this rule: -. {A}

- {A} _ {A V B } (- V)

- {A , A V B } (- V)

- {A, A V B } _ {A V B } (- V)

- {A, A V B} - {A , AV B } (- V)

The last one is absurd, among all four instances only the first one is enough to preserve completeness. Therefore, if we choose sets, we have to impose several restrictions on the inference rules. If we prohibit A and A V B occur in ~, we may eventually loose completeness. Even if we impose no restrictions we might still be in need of the weakening rule . So what is the right choice for sequent and structure rules in sequent calculi?

Kanger's Choice lOne distinctive feature of the calculi used in [Kanger 1957, Kanger 1963] is the full absence of structure rules. In order to achieve this, sequent are made of multisets of formulas and some rules are modified. The use of multi sets eliminates the exchange rule . The use of contraction rule is replaced by the explicit duplication of formulas in some (but not all!) rules and changes in some other rule s. For example, the (- :3) rule in Kanger's system is

r - ~, ::Jx
(- :3)

(the formula ::Jx
rT -

~ ,A,

~,A

B

V B (- V).

Finally, to get rid of weakening axioms I', A traditional A-A.

~,A

are used instead of more

Completeness can be proved for virtually any variant of sequent calculi, but even completeness proofs meet small technical problems when it comes to

56

ANATOLI DEGTYAREV AND ANDREIVORONKOV

structure rules. The choice made by Kanger to design a system without structure rules at all has now become de facto standard. Kleene [1952] also described the sequent system G3 with invertible rules, but this property was realized straightforwardly by retaining the principal formula in the premise(s). Later Kleene's G3 was transformed to the system G4 [Kleene 1967]1, which was essentially the system LC. Choice 2 (variants of rules) For some logical connectives, we have a choice among various sequent calculus rules. For example, for the proof disjunction one can use either the following two rules :

r - D., A r - D., A V B

(- V)

and

r - D., B r - Do, A V B

(- V)

or just one rule

r - D., A, B r - D., A V B

(- V)

The first choice seems to reflect the semantics of disjunction in a more intuitive way. Nevertheless, in Kanger's system the choice ofthe second rule is made. Why? Kanger's Choice 2 The main answer is: all inference rules of Kanger' s system are invertible. A rule is called invertible if the derivability of the conclusion implies the derivability of the premises. For automatic proof-search invertibility of rules is really a remarkable property. If a sequent S is unprovable, then any derivation tree for S has a branch containing a countermodel for S. It allowed Kanger to prove completeness "by means ofarguments which are new in some respect and which involve a new turn to the notion of validity " [Kanger 1957, page 7]. It also allows one to search for a proof in a "don't care" matter : after we have selected a rule to apply, there is no need to undo the selection. Kleene [1967] notes that the use of his system G4 for proving the completeness theorem"is quite close to Beth [1955J which gave the present writer the idea for it. 1n some respect it more resembles Kanger [1957J, as the author learn ed after working it out. "

57

KANGER 'S CHOICES IN AUTOMATED REASONING

2. USE OF A SEQUENT CALCULUS AS A DECISION PROCEDURE FOR PREDICATE LOGIC Kanger was one of the first who used a particular logical calculus as a decision procedure in the backward directiorr', The point was to guarantee termination of the procedure on target classes of formulas. As examples Kanger considered the class of quantifier-free formulas and the class of \I*~* formulas (without functional symbols). Later, Wang [1960] described and implemented a procedure solving this class of formulas, also using backward proof-search in sequent calculi. The possibility to obtain decision procedures for propositional logics, classical and intuitionistic, using backward proof-search in cut-free Gentzen type calculi was also noted by Kleene [1952] . Later, the use of derivations in machine-oriented calculi to decide some classes of predicate logic has become a generally accepted area of research [Maslov 1964, Kallick 1968, Maslov 1968, Joyner jr 1976, Fermtiller, Leitsch, Tammet & Zamov 1993 , Leitsch, Fermtiller & Tammet 1999]. 3. PROOF-SEARCH VIA LOGICAL CALCULI As soon as the first programs for proving theorems in predicate logic appeared [Prawitz, Prawitz & Voghera 1960, Wang 1960, Gilmore 1960, Davis & Putnam 1960] it has become clear that the main problem consists in instantiating variables in the application of (~ ~ ) and (\I ~ » rules (also called y-rules due to [Smullyan 1968, Fitting 1996]).

r

~

L\,
r

~

~x
L\, ~x
and

r,


r, \lx

~

L\

L\ (~

\I )

Here sequent are represented by expressions of the form I" ~ L\, where rand L\ are multisets of formulas . Choice 3 (variable instantiation in y-rules) How to instantiate variables by terms in y-rules? The early methods of automated reasoning used the so-called level saturation. The set of all variable-free terms was enumerated (usually respecting depending on the term depth) and terms have been substituted one by one in that order. However, it was clear that such a solution is far from the best. Kanger's Choice 3 The system of Kanger used a new strategy for instantiating variables in the applications of y-rules. His strategy of instantiating variables

58

ANATOLI DEGTYAREV AND ANDREIVORONKOV

is based on two ideas: the use of fre e variables and the subterm instantiation explained below. Free variables used by Kanger have been originally introduced by Prawitz [1960J for logic without equality. The idea was to introduce a new kind of variables, called "dummies" , in both Prawitz [1960] and Kanger [1963], and to delay instantiation of these variables until necessary information for it has been obtained. Comparing this approach to an earlier work of Beth [1959], Prawitz [1960] noted: "the solution proposed here is quite different but wellsuited for mechanical use". This method was independently proposed in Russia by N. Shanin in 1962 [see Maslov 1964] and characterized as the "rnetavariable method" by Maslov , Mints & Orevkov [1983]. Dummies or metavariables have later been called "free variables" by Fitting [1996] . Information for instantiation in Kanger [1963] is provided by constructing an uninstantiated proof, and checking from time to time whether one can find values for dummies which make it a valid proof. Kanger [1963] reduces this check to verifying that the top sequent are "directly demonstrable", i.e. can be obtained from axioms by applications of equality rules. Now we know a generally accepted solution to the problem of variable instantiation. When we apply y-rules, we temporarily replace the variables by dummi es or metavariables, and delay the instantiation until we try to close one branch of a derivation, i.e. to make it into an axiom. To close the branch one uses the notion of unification due to Robinson [1965] that allows to instantiate dummies by most general terms. However, at the early stage of research in automated reasoning, this solution has not yet been found . We shall explain the Kanger's idea of subterm instantiation below , when we consider logic with equality. 4. LOGIC WITH EQUALITY AND VARIABLE INSTANTIA nON INy-RULES For free-variable sequent-based calculi more practical way of instantiating variables is the introduction of unification in inference rules, that has been considered for calculi without equality in ([Voronkov 1988, Fitting 1996, Voronkov 1992]). For the case with equality the problem of variable instantiation turned out much more difficult' . The Kanger's idea of subterm instantiation discussed below was combined with another nice idea of considering a normal form of derivations in a sequent calculus with equality. All equality inferences, i.e. applications of equality rules, have been moved on top of the proof so that to precede all other steps in

KANGER ' S CHOICES IN AUTOMATED REASO NING

59

the proof. Later, the deri vation s of this form have been named regular [e.g. Lifschitz 1967] , and used in the generalization of the inver se method to predicate calculus with equ alit y in Maslo v [1971] . A typical branch of a regular derivation in logic with equality has the following form: f"~/),. ",t =t

f'

: derivation equality rules only /),. ' : deri vation with no equality rules

~

f ~/)"

The derivation with no equality rules can use instantiation of variables by dummies. In order to make the deri vation a valid one, one should substitute dummies by terms so that the subderivation of I" ~ /),. ' becomes a valid one. Let us call a skeleton of this deri vation the subderivation obtained by removing the equality rule s and using dummies instead of terms. To es tablish provability of a sequent, one has to solve the skeleton instan tiation problem: find a replacement of dummies by terms so that every top sequ ent of a deri vation is provable using onl y equality rules.

Choice 4 (skeleton instantiation) Can we find any rea sonable strategy of skeleton instantiation? For several years, a number of techniques have been de veloped with the ai m of solving skeleton instantiation. The first time this problem has been described as s{,nultaneolls rigid Esunification in [Gallier, Raatz & Snyder 1987] . Severa l fault y algorithms solv ing simultaneous rigid Eunification have been proposed in e.g. [Gallier, Narendran , Plaisted & Snyder 1988, Gallier, Narendran, Raatz & Snyder 1992, Goubault 1994], until the problem was proved undecid able in [Degtyarev & Voronkov 1996b]. Later, it was shown that even very small fragments of the problem are still undecidable [Plaisted 1995, Veanes 1997a, Veane s 1997b, Degtyare v, Gurevich, Narendran, Veanes & Voronkov 1998]. The exact connections of simultaneous rigid E-unifica tion problem with other decision problems, includ ing skeleton instantiation are described in [Degtyarev, Gurevich & Voroukov 1996, Voronkov 1998b]. In view of the undecidabil ity, there is no computational way of solving skel eton instantiati on. The unification-based methods to handl e variable instantiation were found only recentl y [Degtyare v & Voronkov 1996 a, Degtyarev & Voronkov 1998b]. One naive solution would be to blindly substitute variables by all possible term s, thus obtaining at least a semi-

60

ANATOLI DEGTYAREV AND ANDREIVORONKOV

decision procedure. All other proposals published before 1996 were essentially incomplete (or had mistakes in the completeness proofs), except for Kanger's.

Kanger's Choice 4 Kanger [1963] proposed a method for instantiating free variables that can be characterized as subterm instantiation. This method has been referred to by the name of minus -normalization in [Matulis 1962, Norgela 1974, Maslov & Mints 1983] . According to this method, instantiation of variables in the backwards application of y-rule is made only by ground terms, i.e, by terms without variables, explicitly occurring in its conclusion. This method is complete for first-order classical logic and incomplete for intuitionistic logic. Unfortunately even in simplest cases, minus-normalization can require a huge number of instantiation. Some results on subterm instantiation are proved by Norgela [1974]. For logic without equality and function symbols a similar restriction on yrules was used in proof procedures developed in [Quine 1955, Hintikka 1955 , Beth 1955 , Schutte 1956]. Kanger's intuition was that instantiations admitted by subterm instantiation contains the set that could be generated by applying the most general unification technique to proof search in sequent calculi. Consider the following example. Suppose we want to find a proof of the formula ::Jx(P(f(x), x) ::> Pix, g(c))). We can start from computation of the most general unifier of literals P(f(x), x) and P(y, g(c)), because they have opposite occurrences (variables in the second literal are renamed). This computation consists of two steps. Firstly, we replace x by g(c), and secondly, y by fig(c)). The same steps would be done by applying Kanger's subterm instantiation. ... P(f(g(c)), g(c)) ... - ... P(f(g(c)), g(c)) ... - ... P(f(g(c)), g(c)) - P(f(c), c)

::> P(g(c),g(c)), ::> P(c,

- P(f(c), c)

P(f(f(g(c))),fig(c)))

g(c)), P(f(g(c)), g(c))

::> P(c,

::> P(g(c),

g(c)), ::Jx(P(f(x), x)

- ::Jx(P(f(x), x)

::> P(x,

::> P(f(g(c)),

::> P(x,

g(c)) ...

g(c)) ...

g(c)))

g(c)))

Let us now consider the connection of (simultaneous) rigid E-unification with Kanger' s subterm instantiation. Rigid E-unification can be formulated as follows . Given a (finite) set of equations E = {s] = t l , . .. , s, = tn } and the equation s = t, does there exist a substitution 8 such that I- V(s)8 = t 18 /\.../\ snS = tnS ::> sS = 18)4.

KANGER'S CHOICES IN AUTOMATED REASONING

61

A rigid E-unification problem (E, s = t) corresponds to the sequent E - s = t. A simultaneous rigid E-unification problem corresponds to a finite number of such sequent situated in the leaves of a sequent derivation. Undecidability of simultaneous rigid E-unification was proved in 1995 [see Degtyarev & Voronkov 1996b], its comprehensive investigation can be found in [Degtyarev et al. 1996, Voronkov 1998b, Voronkov 1998a]. After that Degtyarev & Voronkov [1998b] proposed a complete proof procedure for sequent-type calculi with equality based on incomplete but terminating procedure for rigid E-unification. We can say that the subterm instantiation of Kanger was the first incomplete but terminating algorithm for simultaneous rigid E-unification giving a complete proof search procedure. Kanger's method cannot be called a truly free-variable method, but it avoids exhaustive search in the set of all terms. In addition, Kanger's method is complete unlike several other methods proposed between 1987 and 1994. A detailed survey of proof-search methods in sequent based calculi with equality can be found in [Degtyarev & Voronkov 1998a, Degtyarev & Voronkov 1999]. 5. MORE ON EQUATIONAL LOGIC Handling equality in automated theorem proving is one of the central topics in automated reasoning. Kanger has anticipated many tendencies used in the modem methods of reasoning with equality. A sequent-based proof procedure for logic with functional symbols and equality was described in [Kanger 1959] and [Kanger 1963]. It was the first extensive analysis of a sequent-style system for equational logic. A more easily accessible publication is [Kanger 1983] . Choice 5 (equality rules in sequent calculi) Essentially all formalizations of sequent calculi with equality use variants of substitution of equals by equals:

s = t,

r~

s = t, r

- ~~ -

~

The number of variations is huge. One can allow to replace one occurrence of s by t or several occurrences, put restrictions on the form of r and ~, replace s by t only in ~, only replace s by t but not t by s for some terms sand tete. Kanger's Choice 5 As for the formalization of equality rules, the contribution of Kanger to automated deduction is really remarkable. We shall consider four novelties introduced by Kanger and discuss their connections with the state-ofthe-art techniques in automated reasoning with equality. One novelty intro-

62

ANATOLI DEGTYAREV AND ANDREIVORONKOV

duced by Kanger on the application of equality rules has been mentioned above: to use regular derivations. The other three are discussed below. Two other novelties introduced by Kanger are the use of simultaneous replacement of all occurrences of the same subterm (we consider simultaneous and replacement as two novelties): s = t, r~

- ~~

s =t, T -

~

where r~ denotes the result of the simultaneous replacement of all occurrences of the term s by t in I', It has appeared that this rule is enough to establish that a sequent is "directly demonstrable". In other words, this rule decides the uniform word problem: whether Sl = t 1,. .. , s; = tn I- S = t, where s.; t.; s, t are ground terms. Th is problem is also equivalent to the decidability problem of the quantifierfree theory of equality. Its decidability has been proved in [Ackermann 1954] but there no practical algorithm had been proposed. Now the uniform word problem is solved by the so called congruence closure algorithm [Shostak 1978, Nelson & Oppen 1980]. The termination of Kanger's method of establishing of direct demonstrability is based on very important restriction put on the above equality rule which we consider to be the fourth novelty. Kanger only allowed its non-increasing applications of equality rules, i.e. those in which the depth of t is not greater than the depth of S (we do not distinguish here the formula S = t from t = S and consider non-increasing constraint in the above rule as orientation of equality). From the viewpoint of the current knowledge of the area Kanger's rule is interesting in the following. Firstly, it can be considered as the rule of demodulation , or simplification, introduced to automated deduction by [Wos, Robinson, Carson & Shalla 1967] as a heuristic tool for discarding apparently irrelevant clauses. It is essential that r and ~ are replaced but not retained, and this calculus does not contain the contraction rule", It has long ago been acknowledged that simplification and other techniques for elimination of redundancy are indispensable for an acceptable behavior of any practical theorem prover [see Bachmair & Ganzinger 1999]. However, the first complete procedure for solving the word problem (with free variables) combining orientation of equations with simplification technique was represented only much later in the famous paper of Knuth & Bendix [1970]. Secondly, Kanger's rule is close to the rule of simultaneous paramodulation of Benanav [1990] . Simultaneous paramodulation is a refinement of paramodulation - the main rule for handling equality introduced by Robinson

KANGER'S CHOICES IN AUTOMATED REASONING

63

& Wos [1969]. Paramodulation is defined on clauses with free variables, and uses most general unifiers. Completeness of paramodulation was an open problem for many years. A great obstacle was that paramodulation does not have the lifting property. It means that in the case of paramodulation standard techniques of proving completeness could not be applied. This technique is to first prove the existence of a ground derivation and then to lift it to the nonground case. It is remarkable that the system with simultaneous paramodulation has the lifting property, and it was sufficient to prove completeness of simultaneous paramodulation only on the ground level. In the same way Kanger's rule could also be lifted from the ground to the non-ground level which would guarantee completeness.

Computing Science Department Uppsala University NOTES The propositional part of G4 coincided with the propositional rules of Ketonen [1944] . Gentzen [1934] was the first to describe a decision procedure for propositional intuitionistic logic as proof-search in his cut-free calculus. The method was based on upwards applications of the rules. Now such approach is known as the inverse method [see Mints, Degtyarev, Tammet & Voronkov 1999]. 3 A significant step in the spirit of subterm instantiation was made by Shostak [1978] who proved that for any set M of ground clauses satisfiability of M in the first-order logic with equality is equivalent to Boolean satisfiability of M together with all ground instances of equality axioms obtained by substituting terms from M for variables. 4 The word "rigid" is introduced to distinguish rigid E-unification from E-unification: given a (finite) set of equations E = (Sl =t" ..., s; =tnl and the equation s =t, find a substitution 0 such that 'If(SI =t l ) , •• • , 'If(sn =tn) f- sO =to. S In opposite to Lifschitz [1967] and Orevkov [1969] who considered more extravagant ways of orientation of equations but in calculi with contraction. 2

REFERENCES Ackermann, W. [1954], Solvable Cases of the Decision Problem , North-Holland. Bachmair, L. & Ganzinger, H. [1999], 'A theory of resolution ' , in A. Robinson & A. Voronkov (eds .), Handbook ofAutomated Reasoning , Elsevier Science and MIT Press . To appear. Benanav, D. [1990] , 'Simultaneous paramodulation ' , in M. Stickel (cd.) , Proc. 10th Int. Con! on Automated Deduction, Vo!. 449 of Lecture Notes in Artificial Intelligence, pp. 442-455 . Beth, E. [1955], 'Semantic entailment and formal derivability' , Mededelinger der Koninklijke Nederlandse Akademie van Wetenschappen, Afd. Letterkunde, Nieuwe Reeks 18 (13). Beth , E. [1959], The Foundation of Mathematics, North -Holland Pub!. Co .. Amsterdam. Davis, M. & Putnam, H. [1960] , 'A computing proc edure for quantification theory ' , Journal of the Associationfor Computing Machinery 7 (3). Reprinted as Davis & Putnam [1983] .

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Davis , M . & Putnam, H. [1983] , 'A computing procedure for quantification theory', in J. Siekmann & G. Wrightson (eds .), Automation ofReasoning. Classical Papers on Computational Logic, Vol. 1, Springer Verlag, pp . 125-150. Originally appeared as Davis & Putnam [1960] . Degtyarev, A., Gurevich, Y., Narendran, P. , Veanes, M. & Voronkov, A. [1998] , 'The decidability of simultaneous rigid E-unification with one variable' , in T. Nipkow (ed .), Rewriting Techniques and Applications. RTA '98, Vol. 1379 of Lecture Notes in Computer Science, Springer Verlag, pp . 181-195 . Degtyarev, A., Gurevich, Y. & Voronkov, A. [1996], ' Herbrand ' s theorem and equational reasoning: Problems and solutions' , in Bulletin ofthe European Associationfor Theoretical Computer Science, Vol. 60, pp . 78 -95. The "Logic in Computer Science" column. Degtyarev, A. & Voronkov, A. [1996a], 'Equality elimination for the tableau method', in J. Calmet & C. Limongelli (eds .), Design and Implementation of Symbolic Computation Systems. international Symposium , DiSCO'96, Vol. 1128 of Lecture Notes in Computer Science , Karlsruhe, Germany, pp . 46-60. Degtyarev, A. & Voronkov, A. [1996b], 'The undecidability of simultaneous rigid E-unification ' , Theoretical Computer Science 166 ( 1- 2), 291- 300 . Degtyarev, A. & Voronkov, A. [1998a], Equality reasoning in sequent-based calculi, Technical Report MPI-I-98-2-011 , Max-Planck Institut fur Informatik, Saarbriicken. Degtyarev, A. & Voronkov, A. [l998b], 'What you always wanted to know about rigid Eunification', Journal ofAutomated Reasoning 20 (1), 47-80. Degtyarev, A. & Voronkov, A. [1999], ' Equality reasoning in sequent-based calculi', in A. Robinson & A. Voronkov (eds .), Handbook ofAutomated Reasoning, Elsevier Science and MIT Press . To appear. Fermuller, D., Leitsch, A., Tarnmet, T. & Zamov, N. [1993], Resolution Methods for the Decision Problem, Vol. 679 of Lecture Notes in Computer Science , Springer Verlag. Fitting, M. [1996], First Order Logic and Automated Theorem Proving, 2nd ed ., Springer Verlag, New York . 1st edition appeared in 1990 . Gallier, 1., Narendran, P., Plaisted, D. & Snyder, W. [1988], 'Rigid E-unification in NPcomplete' , in Proc. iEEE Conference on Logic in Computer Science (LlCS) , IEEE Computer Society Press, pp . 338- 346. Gallier, J., Narendran, P., Raatz, S. & Snyder, W. [1992], 'Theorem proving using equational matings and rigid E-unification', Journal of the Association for Computing Machinery 39 (2),377-429. Gallier, J., Raatz, S. & Snyder, W . [1987], 'Theorem proving using rigid E-unification: Equational matings '. in Proc. IEEE Conference on Logic in Computer Science (LlCS), IEEE Computer Society Press, pp . 338- 346 . Gentzen, G. [1934], 'Untersuchungen tiber das logische SchlieJ3en', Mathematical Zeitschrift 39, 176 - 210,405-431. Translated as Gentzen [1969]. Gentzen, G. [1936] , ' Die Wiederspruchsfreiheit der reinen Zahlentheorie' , Mathematische Annalen 112,493 - 565 . Gentzen, G. [1969], ' Investigations into logi cal deduction ' , in M. Szabo (ed.) , The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, pp . 68-131. Originally appeared as Gentzen [1934]. Gilmore, P. [1960], 'A proof method for quantification theory: its justification and realization ', IBM J. of Research and Development 4, 28- 35 . Reprinted as Gilmore [1983]. Gilmore, P. [1983], .A proof method for quantification theory: its justification and realization', in J . Siekmann & G. Wrightson (eds.), Automation of Reasoning . Classical Papers on

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Computational Logic, Vol. I, Springer Verlag , pp. 151 -158. Originally published as Gilmore [1960J . Girard, J.-Y. [1987] , Proof Theory and Logical Complexity, Studies in Proof Theory, Bibliopolis , Napoly. Godel , K. [1930] , 'Die vollstandigkeit der axiome des logischen funktionenkalktils', Monatshefte fur Mathematik und Physik 37, 349 - 360 . Goubault, J. [1994], ' Rigid if -unifiability is DEXPTIME-complete', in Proc. IEEE Conference on Logic in Computer Science (LICS) , IEEE Computer Society Press. Hintikka, K. [1955], ' Form and content in quantification theory' , Acta Philosophica Fennica 8, 7-55. Joyner jr, W. [1976], 'Resolution strategies as decis ion procedures' , Journal ofthe Association for Computing Machinery 23,398-417. Kallick, B. [1968] , ' A decision procedure based on the resolution method' , in IFlP '68, NorthHolland, pp . 365- 377. Kanger, S. [1957], Provability in Logic, Vol. I of Studie s in Philosophy, Almqvist and Wicksell, Stockholm. Kanger, S. [1959], Handbook in logic, Stockholm. Kanger, S. [1963], 'A simplified proo f method for elementary logic ' , in P. Braffort & D. Hirschberg (eds.), Computer Programming and Formal Systems, North-Holland, pp . 87 -94. Reprinted as Kanger [1983] . Kanger, S. [1983], 'A simplified proof method for elementary logic' , in J. Siekmann & G. Wrightson (eds.), Automation ofReasoning . Classical Papers on Computational Logic, Vol. 1, Springer Verlag , pp. 364- 371. Originally published as Kanger [1983]. Ketonen, O. [1944], 'Untersuchungen zum Pradikatcnkalkul', Annales Academia Scientiarum Fennicae 23. Ser. A, 1 Mathematixa-physica. Kleene, S. [1952], Introduction to Metamathematics, Van Nostrand P.c., Amsterdam. Kleene , S. [1967], Mathematical Logic, John Wiley and Sons . Knuth, D. & Bendix , P. [1970], 'Simple word problems in universal algebras', in J. Leech (ed .), Computational Problems in Abstract Algebra , Pergamon Press , Oxford , pp. 263 -297. Leitsch, A., Fermuller, C. & Tammet, T. [1999] , 'Resolution decision procedures', in A. Robinson & A. Voronkov (eds .), Handbook ofAutomated Reasoning, Elsevier Science and MIT Pres s. To appear. Lifschitz, V. [1967], 'A normal form of deriv ations in predicate calculus with equality and function symbols (in Russian)' , Zapiski Nauchnyh Seminarov LOMf 4, 58-64. English Translation in: Seminars in Mathematics: Steklov Math. Inst. 4, Consultants Bureau , NYLondon, 1969. Maslov, S. [1964] , 'The inverse method of establishing deducibility in the classical predicate calculus', Soviet Mathematical Doklady 5, 1420 -1424. Maslov, S. [1968] , 'The inverse method of estab lishing deducibility of logical calculi (in Russian)' , in Collected Works of MIAN, Vol. 98, Nauka, Moscow, pp . 26-87. Maslov, S. [1971] , 'The generalization of the inverse method to predicate calculus with equality (in Russian) ', Zapiski Nauchnyh Seminarov LOM f20, 80- 96 . English translation in: Journal of Soviet Mathematics 1, no. 1. Maslov, S. & Mints, G. [1983] , 'The proof-search theory and the inverse method (in Russian)' , in M.G. (ed .), Mathematical Logic and Automatic Theorem Proving , Nauka, Moscow, pp . 291-314. Maslov, S., Mints, G. & Orevkov, V. [1983] , 'Mechanical proof-search and the theory oflogical deduction in the USSR', in 1. Siekmann & G. Wrightson (eds.), Automation of Reasoning (Classical papers on Computational Logic) , Vol. I , Springer Verlag , pp . 29 - 38.

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Matulis, V. [1962], 'Two variants of classical predicate calculus without structure rules (in Russian)', Soviet Mathemat ical Doklady 147 (5), 1029-1031. Mints, G., Degtyarev, A., Tammet, T. & Voronkov, A. [1999], 'The inverse method', in A. Robinson & A. Voronkov (eds.), Handbook ofAutomated Reason ing, Elsevier Science and MIT Press . To appear. Nelson , G. & Oppen, D. [1980], 'Fast decision procedures based on congruence closure', Journal ofthe Association for Computing Machinery 27 (2), 356- 364. Norgela, S. [1974], 'On the size of derivations under minus-normalization (in Russian)' , in V. Smimov (ed.), The Theory of Logical Inference, Institute of Philosophy, Moscow. Orevkov, V. [1969] , 'On nonlengthening applications of equality rules (in Russ ian)', Zapiski Nauchnyh Seminarov LOMI16, 152-156. English translation in: Seminars in Mathematics: Steklov Math . Inst. 16, Consultants Bureau, NY-London, 1971, pp. 77-79 . Plaisted, D. [1995] , Special cases and substitutes for rigid E-unification, Technical Report MPI1-95-2-010, Max-Planck-Institut fur Informatik. Prawitz, D. [1960] , 'An improved proof procedure' , Theoria 26 , 102-139. Reprinted as Prawitz [1983] . Prawitz, D. [1983], 'An improved proof procedure', in J. Siekmann & G. Wrightson (eds.), Automation of Reasoning. Classical Papers on Computational Logic, Vol. I, Springer Verlag, pp. 162-201. Originally appeared as Prawitz [1960]. Prawitz, D., Prawitz, H. & Voghera , N. [1960], 'A mechanical proof procedure and its realization in an electronic computer' , Journal ofthe Associationfor Computing Machinery 7 (2). Reprinted as Prawitz, Prawitz & Voghera [1983] . Prawitz, D., Prawitz , H. & Voghera, N. [1983], 'A mechanical proof procedure and its realization in an electronic computer', in J. Siekmann & G. Wright son (eds.), Automation of Reasoning. Classical Papers on Computational Logic, Vol. I , Springer Verlag , pp. 162- 20 1. Originally appeared as Prawitz et al. [1960] . Quine , W. [1955] , 'A proof procedure for quantification theory ', Journal ofSymbolic Logic 20, 141-149. Robinson, G. & Wos, L. [1969] , 'Pararnodulation and theorem-proving in first order theories with equality', in B. Meltzer & D. Michie (eds.) , Machine Intelligence , Vol. 4, Edinburgh University Press, pp. 135-150. Robinson, J. [1965], 'A machine-oriented logic based on the resolution principle' , Journal ofthe Associationfor Computing Machinery 12 (I), 23-41. Schutte, K. [1956], 'Ein System des verknupfenden Schliessens', Archiv fur Mathematische Logik und Grundlagenforschung 2, 34-67. Shostak, R. [1978] , 'An algorithm for reasoning about equality', Communications of the ACM 21,583- 585 . Smullyan, R. [1968], First-Order Logic, Springer Verlag . Veanes , M. [1997a] , On Simultaneous Rigid E-Unification, PhD thesis , Uppsala University. Veanes , M. [1997b], 'The undecidability of simultaneous rigid E-unification with two variables' , in G. Gottlob, A. Leitsch & D. Mundici (eds.), Computat ional Logic and Proof Theory. 5th Kurt Godel Colloquium, KGC-97, Vol. 1289 of Lecture Notes in Computer Science, Vienna, Austria, pp. 305- 318. Voronkov, A. [1988], 'A proof search method for first orderlogic', in Preliminary Proceedings of COLOG-88 , Vol. 2, Tallinn , pp. 104- 118. Voronkov, A. [1992], 'Theorem proving in non-standard logics based on the inverse method', in D. Kapur (ed.), J lth International Conference on Automated Deduction, Vol. 607 of Lecture Notes in Artificial Intelligence, Springer Verlag, Saratoga Springs, NY, USA, pp. 648-662.

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Voronkov, A. [I998a], ' Herbrand' s theorem, automated reasoning and semantic tableaux ' , in Proc. IEEE Conference on Logic in Computer Science (LICS), IEEE Computer Society Press, pp. 252-263 . Voronkov, A. [1998b] , Simultaneous rigid E-unit1cation and other decision problems related to Herbrand's theorem, UPMAIL Technical Report 152, Uppsala University, Computing Science Department. To appear in Theoretical Computer Science . Wang , H. [1960], 'Towards mechanical mathematics ', IBM J. of Research and Development 4, 2-22. Reprinted as Wang [1983] . Wang , H. [1983] , 'Towards mechanical mathematics ' , in J. Siekmann & G. Wrightson (eds.) , Automation of Reasoning . Classical Papers on Computational Logic , Vol. I, Springer Verlag, pp. 244-264. Originally published as Wang [1960] . Wos, L., Robinson, G., Carson , D. & Shalla, L. [1967] , 'The concept of demodulation in theorem proving' , Journal of the Association fo r Computing Machinery 14, 698-709.

KAJ B0RGE HANSEN

KANGER'S IDEAS ON NON-WELL-FOUNDED SETS : SOME REMARKS

1. INTRODUCTION 1.1 Provability in Logic. Stig Kanger 's small book from 1957, Provability in Logic, contains eight chapters. The last two chapters are concerned with modal logic. This part has received considerable attention and recognition. Chapters 2-6 treat elementary extensional logic. This part has drawn scantier attention. The present essay contains an exposition and comments on chapters 2- 5, with an emphasis on the contributions to set theory and model theory . I take for granted that the reader has access to Provability in Logic, either the original edition from 1957 or the reprint in Collected Papers of Stig Kanger, Vol. I. 1.2 Content. Kanger develops a calculus LC for one fixed predicate logical language L. His intention seems to be to show that LC is all that is needed for general predicate logic. This forces him to develop new ideas on non-wellfounded sets which are of great interest in their own right. These ideas are the main subject of the present essay. I expose the language L, the calculus LC, the ideas on non-well-founded sets, and the use Kanger makes of them in the model theory for LC. 2. THE CALCULUS LC 2.1 Language. We first indicate the language L. 2.2 Primitive Symbols . The language L is built from the following symbols. (1)

(2) (3)

(4) (5)

Parentheses. Propositional constants: p, q, PI' ql' Pa- qz, ... Set symbols: For t = 1,2, , variables of type t: xi, xi, . constants of type t: c], ci, . Notation for ordered sets: < > A two-place predicate: E 69

G. Holmstrom-Hintikka, S. Linstrom and R. Slivinski (eds.) , Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. ll, 69-86. © 2001 Kluwer Academic Publishers. Printed in the Netherlands .

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(6) (7)

(8)

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Connectives» (material implication), & (conjunction), V (disjunction), - (negation) Quantifier symbols: V (universal quantification), ::J (existential quantification) A symbol for Gentzen entailment: -

2.3 Formulas. The atomic formulas of L are all expressions of the form (a E b) or (ca, ...an> E b)

where a, b, a.,....a, are set symbols. The set of formulas are obtained in the usual way by closing the set of atomic formulas under the connectives and quantifiers. 2.4 Sequents. Let I' and e be sequences of closed formulas of LC. A quasisequent is an expression of the form f-

e

A quasi-sequent S is a sequent if there are infinitely many set constants of each type which do not occur in S. This distinction is relevant for the formulation of the deduction rules *10 and *13 of LC, and we shall not need it in the sequel. 2.5 Intended Interpretation. Kanger considers two interpretations of the atomic formula «al' ''~> E b). 'E' may be interpreted as denoting an arbitrary 2-place relation, or it may be interpreted as representing the membership relation for sets. 2.6 Remark. The language L with the calculus LC is predicate logic without identity . This is important. One of Kanger's completeness theorems for LC cannot be extended to predicate calculus with identity as will be shown in Section 7. 2.7 Remark. All variables and constants in L are typed. This complicates somewhat the semantics ofL, the deduction rules ofLC, and the constructions in the proofs of the completeness theorems . It is not easy to see any justification for having several types rather than just untyped variables and untyped constants. A typing of symbols may be justified when the domain is a type structure of sets. The set universe Kanger eventually chooses for his model

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theory contains non-well-founded sets and allows loops like a E a and a E b E a. This makes a typing inappropriate. In the comments later in the essay, I will sometimes make reformulations where types are neglected.

2.8 Remark. The usual way to do general predicate logic is to consider the family {La} of all predicate logical languages and define a predicate calculus for each La. Kanger's intention is clearly to do general predicate logic ; but he considers only one fixed language, the language L with one predicate E and individual constants c., c2 , .... To motivate this approach, consider, e.g ., a language L * which contains a one-place predicate P and a 2-place predicate R and also constants c and d. Let 5t be a model for L *. Then (2-1) (2-2)

5t 1= P( c) R(c , d)

51 1=

We see that atomic formulas in the semantics always are interpreted in terms of the s-relation. This suggests the possibility of doing general predicate logic by having only one predicate, namely E. Individual terms like c, d and predicates like P, R should then be represented by constants intended to denote atoms or sets . This is exactly how Kanger's language L is built up. The difficulties with such an approach are considerable. The e -relation on the right -hand side of equivalences (2-1) and (2-2) is governed by the axioms of ZF. A logical calculus for L should presumably be incomplete if the' E ' of L were interpreted in this way since for completeness, the calculus should have to include at least one of the non-logical ZFaxioms. Kanger therefore needs to invent another set universe which contains sets not occurring in Zermelo's cumulative type structure. It should be pointed out that the considerations stated in the present remark are my own and do not occur in Kanger's work. I nevertheless feel that they must have motivated Kanger in his work. They also make it intelligible that he attaches so great importance to norma l models and completeness with respect to normal models (see paragraphs 2.11 - 2.14 and Section 3 below). 2.9 Semantics. Kanger defines a semantics for L. It consists of a frame together

with a valuation. Aframe for L is an infinite sequence r = of classes where r' ;/; 0 and r' c r"! for t = 1,2,... .

2.10 Remark. The frame r is the domain of the model. r' is the class of entities of type at most t. Since r' s:;; r'", the type structure is cumulative.

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2.11 Valuations. A primary valuation is a 2-place function V. The first argument is a frame ; the second argument is either a propositional constant, the predicate ' E' , or a set symbol. V satisfies: (1)

(2) (3)

VCr, P) = 1 or VCr, P) =0 if P is a propositional constant. 1 and 0 are the truth-values true and false, respectively; VCr, 'E') is a class of finite non-unitary ordered sets of elements of r; VCr, s) is an element of r' if s is a set symbol of type t.

A primary valuation is normal if it holds for each frame r that any ordered set of elements of r belongs to V(r,'E') if and only if is a member of w, for n E Z+. The primary valuation gives rise to a secondary valuation T(r, V, S) which, given a frame r and a primary valuation V, assigns a truth-value, 0 or 1, to each formula or sequent S. The extension of V to T is done in the natural way.

2.12 Remark. If we use modem notation and disregard types, the semantics can be reformulated as follows. A structure (sometimes called an arbitrary structure) is a sequence (2-3)

.9l

= (A, E

J \ • •• ,

c",...)

where A::/; 0 is a set, E-" k A+ X A with A+ = Un>o An, and c" E A for each constant c in L. Thus in an arbitrary structure, 'E' is interpreted as any relation over A+ x A. A normal structure is a structure .9lsuch that E-" is a set theoretical membership relation. Thus if A is a class of atoms and sets, then (2-4)

.9ll= E d <=>
,C? is an element of d"

Note that the definition of a normal structure is vague and ambiguous as is Kanger's concept of a normal primary valuation. This is due to the fact that the exact meaning of "element of' is left open so far. The meaning given to "element of' by the ZFaxioms is not adequate for Kanger's purposes. A main task for him is to find a more suitable concept.

2.13 Validity and Logical Truth. Let S be a sequent or sentence of L. S is valid if S is true in every structure. S is logically true if S is true in every normal structure.

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2.14 Remark. Thus validity is defined with reference to structures .9l. where E 51 is any relation. Logical truth is defined with reference to such structures .9l. where EJl is a set theoretical membership relation . 2.15 The Calculus LC. Kanger defines a sequent calculus LC for the language L. The rules of the calculus are essentially the usual ones for a sequent calculus, though with some minor changes. The purpose of one of these is to ensure that the proof procedure will be effective. Another one allows the exclusion of the Structural Rule from LC. The purpose of a third change is to cope with the types in the language L. 2.16 Proofs. Let Il = <S, T I' T 2" "> be a sequence of sequents. Il is a quasideduction in LC of S from the class 8 of assumption sequents if the following conditions are satisfied :

(1) (2)

Il begins with an occurrence of S; each component U of Il is either - an instance of Postulate * 1 (the identity axiom), or an element of 8, or - inferable in one step by a rule of inference from one or two succeeding components of Il,

A proofin LC of a sequent S is a finite quasi-deduction of S from the empty class of assumptions. A theorem of LC is a sequent which has a proof in LC. 2.17 Remark. (I) A curiosity about the definition is that Kanger defines a deduction as a linear sequence though what he needs and actually uses is the picture of a deduction as a labeled tree.

(II) If S is a provable sequent , then a proof for S is a finite tree with S at the root and with instances of the Identity Postulate (2-5)

r -, B, r -

8 ', B, 8

labeling the leaves . If S is not provable, then the proof tree for S contains a branch ~ which does not begin with an Identity Postulate and is extended as far backward as the rules ofLC allow. (III) Note that Kanger writes a deduction in the opposite order of the usual one, i.e., he writes the conclusion first and not last. When one works in ordinary formulations of the sequent calculus, one soon discovers that the only reasonable way to construct a deduction is to construct it backward starting

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with the sequent S to be derived. Kanger has made this heuristic procedure part of the official definition of a deduction. 3. SOUNDNESS AND COMPLETENESS 3.1 Theorem (Soundness). (I) Every theorem of LC is a valid sequent. (II) Every theorem of LC is logically true . Proof' (I) Induction on the length of proofs in LC. (II) Follows immediately from (I). If we cannot find a counterexample to a sequent S, which is not provable in LC, by considering all relations on A + x A, then we cannot find a counterexample to S by considering only such relations on A+ x A which are membership relations. No preciser characterisation of "element of' is needed. 3.2 Theorem (CompLeteness with Respect to Validity). Every valid sequent is a theorem of LC. Proof' Kanger gives a new proof of this theorem by what is now called the BethKanger- Schtitte- Hintikka method. The theorem is, however, a trivial corollary to the usual Godel- Henkin completeness theorem. This follows since LC essentially is the usual predicate calculus for the language L, and the class of all structures 51 =(A, E'!\ ... , c",...) includes all models for L because E>l is an arbitrary relation on A + x A. 3.3 Theorem (Completeness with Respect to LogicaL Truth). Every logically true sequent is a theorem of LC. 3.4 Remark. There is a major difficulty about proving this theorem. Kanger needs to make the concept "element of' in the definitions of normal structure and logical truth sufficiently exact to allow a proof. "Element of' cannot be understood in the sense of ZF since in this case LC should certainly not be complete. Instead we need a concept of set and membership which implies the existence of the sets of Zermelo's cumulative type structure as well as an abundance of other sets. These sets must supply us with counterexamples to all sequents which are not theorems of LC. Moreover, the defined concept must agree in a convincing way with our intuitive idea of set membership.

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4. NETS

4.1 Introduction. A quite common way of visualing a set is as a container or bag. A set contains its elements in essentially the same way a bag contains the things which are in it. Kanger's idea is to visualise a set as a cord. The set contains whatever the cord is tied on. It turns out that this idea of a set gives rise to a considerably richer membership relation than does the idea of sets as bags utilised in the cumulative type structure. Cords allow all the membership relations in the cumulative type structure; but in addition, they allow an abundance of other membership relations. E.g., a cord can be tied on itself, thus allowing self-membership. Consequently, in Kanger's set universe there is a set x satisfying x = {x}. 4.2 Example. We imagine two men on the sea in a row-boat on a hot summers day. They have brought two bottles of beer; but because of the hot weather, the beer has become lukewarm. The two men resolve the problem. They find a piece of twine . (Such can almost always be found on board a boat.) They tie the string around the neck of first one and then the other bottle. They fasten one end of the string to the gunwale and let the rest of the string with the bottles hang down in the sea water. After c. half an hour, they draw the piece of twine up, untie the knots, open the bottles, and drink the beer, now at an acceptable temperature. In an intuitively very natural sense, the piece of twine has contained the two bottles. 4.3 Remark. For Kanger's purposes, it is essential that the interpretation he gives to 'E' agrees in a convincing way with an intuitive idea of membership, since only then can his idea of replacing predication with statements involving only set symbols and ' E', as explained in Remark 2.8, be accepted. The observations in paragraphs 4.1 and 4.2 show that this is indeed the case. 4.4 Remark. A characteristic feature of a cord is that it can be tied on itself. Therefore sets based on nets can have themselves as members. This makes Kanger sets well suited to deal with circular statements, e.g., self-referential sentences, and autological predicates, i.e., predicates which apparently apply to themselves. Consider, e.g., the predicate 'predicate'. It applies to all predicates. Since 'predicate' is itself a predicate, one is inclined to think that it should apply to itself, i.e., it should belong to its own extension. This is excluded in set theories, like ZF, based on the cumulative type structure. The solution in logics based on well-founded sets is to say that 'predicate' is a second-order predicate and introduce an infinite hierarchy of types of

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predicates. In LC, a sentence like (4-1)

'Predicate' is a predicate

can be expressed at the first-order level. Let c be the constant representing 'predicate'. Then 'c E c' expresses (4-1). The corresponding phenomenon among nets is the following. Consider the universe 'l1 of all cords representing sets. Select the class of all cords which represent first-order predicates. Choose a new cord k and tie it on all these cords. Then tie k on itself. Now k represents the autological predicate 'predicate'. No infinite hierarchy of predicates is needed. I now indicate the main features of Kanger 's ideas on nets and sets . 4.5 Nets. A net consists of a nonempty collection of cords . A cord may, but need not, be tied on one or more other cords in the collection, and even on itself. The main structure of a net n is a pair such that cis the set of cords in nand e is a 2-place relation in Csatisfying

(4-2)

keI

ee-

I is tied on k

The net n is elementary if Cand E in the main structure of n are both countable. A cord k in n is an empty cord in n if k is not tied on any cord in n. n is an extensional net if there are no two distinct nonempty cords k and I in n such that k and I are tied on the same cords in n. 4.6 Remark. Kanger uses extensional nets to represent classes of sets. The impact of the condition of extensionality is that the associated class of sets should satisfy the Weak Axiom of Extensionality:

(4-3)

Vx Vy «3z Z E x 1\ Vz (z E X- Z E y» - x = y)

The usual Axiom of Extensionality (4-4)

Vx Vy (VZ(ZE X - z E y) - X= y)

is not satisfied in Kanger's universe of sets. The reason is that Kanger allows atoms in his universe. It is, of course, possible to build a set theory based on nets without atoms . In that case, the stronger Axiom of Extensionality (4-4) will be satisfied. Note that the Weak Axiom of Extensionality (4-3) cannot be expressed in L because Land LC do not have identity.

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4.7 Sets. is a set structure if Z is a nonempty class of sets and EO is a 2place relation in Z satisfying (4-5)

v EO w

<=>

v is an element of w

The set structure is based on the class Z' if Z' is the class of elements of Z that do not contain any member of Z as an element. Kanger formulates the following set theoretical principle.

4.8 Kanger's Set Theoretical Principle. Let n be an extensional elementary net and Z' a set of positive integers in one-to-one correspondence with the class of empty cords in n. Then there is a set structure based on Z' and isomorphic with the main structure of n. 4.9 Remark. The limitation to elementary nets is not essential ; but by the Lowenheim-Skolem feature of first-order elementary logic , this limited principle is all Kanger needs for the completeness proof. Principle 4.8 supplies all the counterexamples Kanger needs for his completeness proof. Unfortunately, the principle is somewhat opaque and not as transparent as an axiom should be. We consider two examples. We define a net n. Consider a countably infinite set of cords labeled 1, k., k2, · .. . Let 1 be empty and tie k, on I. Then tie k,,+] on k, for n = 1,2,... . n is clearly an extensional elementary net. By Principle 4.8 , there is a set structure based on Z' = {I} and isomorphic with the main structure of n. The set structure must be defined as follows.

(4-6)

Z = {l, k., k2, .. . }

and EO contains all pairs (l, k.), (k., k2 ) , (k 2 , k.), ; .. In this example, one connected net n has given rise to an infinity of sets. A simple example with a finite net can also be given. Consider, e.g ., a net consisting of the cords a, b, c, d, e. a, b. c are empty. d is tied on a and b while e is tied on band c. This is thus a net where all cords are connected with each other. It also satisfies the hypothesis of Kanger' s principle. The corresponding set structure contains five entities, namely the atoms a, b, c together with the sets d = {a, b} and e = {b, c}. Thus one connected net may give rise to several distinct sets. In some cases, this decay of a net into sets may take place in highly complicated and opaque ways. The principle should be simpler and more transparent if every connected net corresponded to only one set.

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4.10 Remark. Kanger's proof of the Completeness Theorem 3.3 is based on the intuitive idea of a set universe based on nets rather than on an exact axiomatisation of a theory of non-well-founded sets. This corresponds to giving a proof of the usual Godel- Henkin completeness theorem based only on an intuitive understanding of the cumulative type structure. Though this is possible, it is a less satisfactory approach than basing it on an axiomatisation of a suitable set theory. 4.11 Remark. Unfortunately an even stronger objection can be directed towards Kanger's Set Theoretical Principle 4.8. The principle is false and implies a contradiction. Consider a net n containing two cords a and b. a is tied on band b is tied on a. Let a* and b* be the sets corresponding to the cords a and b, respectively. Then a* and b* satisfy the equations (4-7)

a* = {b*} b*={a*}

The system (4-7) has a unique solution (4-8)

a*=b*= {{{{ . . . . }}}}

Now we apply Principle 4.8 to n. n is an extensional elementary net. Since empty cord , Z' = 0. By Principle 4.8, there is a set structure based on 0 and isomorphic with the main structure of n. Let a* and b* be the elements of Z corresponding to a and b under the isomorphism. Then a* and b* must satisfy the equations (4-7), and therefore, by (4-8), a* = b*. But then and cannot be isomorphic since C has two elements, namely a and b, while Z has only one element , namely a* = b*.

n contains no

5. A THEORY OF NON-WELL-FOUNDED SETS 5.1 Introduction. In this section, I formulate a modification of Kanger's Set Theoretical Principle 4.8 which will, I hope, avoid the objections in remarks 4.9-4.11. The exposition is based on chapters 1 and 2 in Aczel (1988). 5.2 Graphs. A graph is a pair (G, A) where G is a nonempty set whose elements are called nodes. A (;;; G x G is a set of pairs (n, m) of nodes . Each pair in A is called an edge. It is common to represent nodes in G by discrete points and an edge (n, m) in A by an arrow from point n to point m, n - m.

KANGER'S IDEAS ON NON -WELL-FOUNDED SETS

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If (n, m) E A, then we say that m is a child ofn. A path is a sequence, finite or infinite, of nodes no, n., nz, connected by edges: (5-1)

no- n 1 -

nz -

.

5.3 Remark. Nets, graphs, and 2-place relations on a non-empty set are essentially the same. This claim for graphs and relations follows immediately from Definition 5.2. Now consider graphs and nets . Let n be a net with main structure
(k, l)

E

A

<=>

l ek

Then nand G are isomorphic. Conversely, let G = (G, A) be a graph. Let n be a set of cords in one-to-one correspondence with G. If n EG, let n* denote the associated cord . Define (5-3)

n* is tied on m*

<=>

(n, m) E A

Then nand G are isomorphic. An advantage about nets as representations of sets is that they visualise in a more concrete way than graphs the element relation . Otherwise it is mostly more convenient to use graphs as pictures of sets. I will do that from now on. But every statement about graphs can easily be translated into a statement about nets via the isomorphism above .

5.4 Pictures of Sets. A pointed graph is a graph G together with a distinguished node no in G, called the point of G. Let (G, no) be a pointed graph. (G, no)is an accessible pointed graph (APG) if for every node n in G there is a path no- n 1 -

• •.

-n

from the point no to n. If for every n there is a unique path, then (G, no) is a tree with noas root. Let G be a graph . A decoration of G is an assignment of a set to each node in G in such a way that the elements of the set assigned to node n are exactly the sets assigned to the children of n. In other words, (5-4)

d(n)

= {d(m) In -

m}

where d(n) is the set assigned to n by the decoration. Thus a node having no children must be assigned the empty set 0.

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Let (G, no) be an APG which has a decoration. Then G is a picture of the set assigned to the point 00. (Note that so far it is neither excluded that an APG (G, no) has no decoration nor that it has several decorations.) 5.5 Well-Founded Graphs. A graph is well-founded if it has no infinite path . For well-founded graphs, the following propositions can easily be proved:

(I) Every well-founded graph has a unique decoration (Mostowski 's Collapsing Lemma). (II) Every well-founded APG is a picture of a unique set. (III) Every set has a picture. Mostowski's Collapsing Lemma is proved by assigning 0 to all the childless nodes of the graph and then inductively working backward along the paths against the direction of the arrows. A set is well-founded if it has a picture which is well-founded. Otherwise it is non-well-founded. (Note that the proof of Proposition (III) applies to nonwell-founded sets as well.) A set, well-founded or not, is heriditarily finite if it has a finite picture. 5.6 The Foundation Axiom (FA) . One of the axioms of ZFC is Zermelo's Foundation Axiom (FA): (5-5)

\Ix C:Jy YE x - ::Jy (y E x 1\ \lz ""(z E X 1\ z E y)))

This axiom, which is true in the cumulative type structure, excludes all nonwell-founded sets from the models ofZFC. E.g., it excludes sets with infinitely descending membership chains (5-6)

... E a2 E a, E ao

and therefore also all loops (5-7)

b JEb 2E .. . EbnEb J

There is a way to get a much richer set universe with all the well-founded sets of ZFC as well as an abundance of non-well-founded sets. This is by letting all APGs, well-founded or not, be pictures of sets . 5.7 The Anti-Foundation Axiom, AFA.

(5-8)

Every graph has a unique decoration.

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81

5.8 Weak Set Theoretical Principle. Every APG is a picture of a unique set. Proof' Immediate from AFA and the definitions. 5.9 Remark. An axiomatised set theory which implies the existence of both well-founded and non-well-founded sets can be obtained. Let ZFe- be ZFC with the Foundation Axiom FA deleted. Then add the Anti -Foundation Axiom AFA to obtain ZFC- + AFA . The Weak Set Theoretical Principle 5.8 is a simple theorem in this theory . ZFe- + AFA is a theory for a universe of pure sets. If we, as Kanger, want atoms in the set universe , then some of the axioms must be suitably modified. E.g. , the Axiom of Extensionality (4-4) must be replaced by the Weak Axiom of Extensionality (4-3). In the proof of the completeness theorem in the next section, I use the Weak Principle 5.8 rather than Kanger ' s Principle 4.8. Since Principle 5.8 is a theorem in the axiomatic theory ZFe- + AFA, the objection in Remark 4.10 is avoided. Since we use APGs instead of nets, every conne cted graph/net we consider will give rise to exactly one set, namely the one assigned to its point. We therefore also avoid the objectio n in Remark 4.9. The main trouble with Kanger's Set Theoretical Principle 4.8 detected in Rema rk 4.11 is that the principle demand s a one-to-one mapping from the set of cords in the net into the class of sets. The Weak Set Theoret ical Principle 5.8 only demands a many-to-one mappin g from the set of nodes in the APG into the class of sets and therefore avoids the objec tion in Remark 4.11. 6. COMPLETENESS WITH RESPE CT TO LOGICAL TRUTH

6.1 Introduction. In this section , I outline a proof of the compl eteness of LC with respect to normal models. It is essentially Kanger' s original proof; but some minor changes will be introduced. The typing of set symbols will be discarded. The proof will be carried out in ZFC- + AFA, and it will be based on the Weak Set Theoretical Principle 5.8 rather than on Kanger 's Set Theoretical Principle 4.8. 6.2 Theorem (Completeness with Respect to Logical Truth). Every logically true sequent is a theorem of LC. Proof' Let S be a sequent which is not a theorem ofLC. Consider an attempt to prove S by Kanger's proof procedure. It must contain a branch ~ with no instance of

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KAI B0RGE HANSEN

Postulate *I of LC: (6-1)

r-, B, r -

0 ', B, 0

We define a normal structure 5l= (A, E.'1, ... , c",...) such that every sequent in E, including S, is false in .9l. We assume that the constants of L are just c., c2 ,. •• • We first define a function U which to each constant c assigns a set U(c): (1) If Cn is a constant occurring in ~, then let

(6-2)

Utc.)

= In} u Y,

where Y, is the class of ordered sets E c, is an antecedent sentence in some member of ~ ; (2) if c, is a constant not occurring in ~, let

(6-3)

U(c n )

= In}

We thus get a system consisting of an infinite sequence of equations (6-4)

U(c 1) U(c 2)

= {I} Y = {2} u Y 2 U

I

We can use this equation system to define a tree with Ufc.) labeling the root, for a given c., Consider any labeled node in the tree. (l) If the node is labeled by an integer m, let it have no children. (2) If the node is labeled by a (k+l)-tuple

(6-5)

X

={{
-ca.,...,ak,ak+1> rewrite it as

{, ak+d}

Let the node have two children and assign them the two elements of X as labels . (3) If the node is labeled U(c k), let it have a child for each element in U(c k)

according to the right-hand side of the equation (6-6)

U(c k)

= {k} u Y,

(4) If the node is labeled {w.; ...w.}, let it as children have k nodes labeled WI'···'Wk•

The tree we obtain this way is an APG. By the Weak Set Theoretical Prin ciple 5.8, it is a picture of a unique set. This is the set which the decoration assigns to the root Utc.). Thus all the sets Utc .), U(c 2) , •.• exist. We can now define the model 5l = (A, E.'1, ..., c", ...).

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KANGER 'S IDEAS ON NON-WELL-FOUNDED SETS

(6-7) (6-8) (6-9)

A = {U(c) I c is a constant ofL} c 51 = U(c) EO 51 = {
EO

U(d), k

EO

Z+}

The definition of E0 51 ensures that Jl is a normal structure. Two earlier lemmas in Kanger's essay ensure that the construction of Jl is possible and that Jl makes all antecedent statements in ~ true and all succedent statements in ~ false. In particular, S will be false in JL

6.3 Example. The crucial step in the proof is the definition of the sets U(c) and the claim that they exist. An example may be clarifying. Suppose, e.g ., that 'c EO d' and 'd EO c' occur as antecedents in ~ and that 'c' and 'd' occur nowhere else in ~. Let c = c, and d =c m • Then by (6-2) and the definition of the Yk' (6-10)

U(c) = {n} u {U(d)} U(d) = {m} u {U(c)}

We can draw a tree with U(c) as root as suggested in the proof. A simpler APG G can be obtained as follows . Let U(c) be the point of G. Draw one arrow from U(c) to a childless node n. Draw another arrow from U(c) to a node U(d). Draw one arrow from U(d) to a childless node m. Draw another arrow from U(d) back to U(c). This is the APG G. By Principle 5.8 , the existence of G implies that there are unique sets U(c) and U(d) which are solutions of the equation system (6-10). If we unfold the equations (6-10), we get the solution

(6-11)

U(c) = [n, {m, {n, {rn, U(d) = {m, {n, {m, {n,

}}}} }}}}

6.4 Remark. Aczel (1988) gives another proof of the Completeness Theorem 6.2. He proves the following result in ZFC- + AFA. Lemma. Every arbitrary structure is isomorphic to some normal structure. The Completeness Theorem 6.2 with respect to normal structures is then an immediate corollary to the lemma and the Completeness Theorem 3.2 with respect to arbitrary structures.

6.5 Remark. We can draw the following conclusions. (I) The calculus LC is a sound system with respect to arbitrary models and with respect to normal models.

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(II) LC is complete with respect to arbitrary models. Kanger has found a new proof of this theorem; but it also follows immediately from the classical Godel- Henkin completeness proof. (III) LC is complete with respect to normal models when 'E ' is interpreted as allowing non-well-founded sets as in ZPC- + AFA. Kanger's own proof of this theorem is based on the contradictory Set Theoretical Principle 4.8 . However, a slightly different proof can be given based on the sounder Weak Set Theoretical Principle 5.8. This result implies that all problems of elementary first-order predicate logic without identity can be formulated and solved in one simple and fixed language, Kanger's L.

(IV) LC is not complete if we limit the interpretation of E to membership in the sense of the cumulative type structure, e.g., as governed by ZP. Every model 51= (A, E}1, ... , c", ...) where E}1 is membership in the sense of ZP would satisfy the Foundation Axiom (5-5). Since FA can be expressed in L without identity and FA is not a logical theorem, LC is incomplete with respect to such model s.

7. IDENTITY

7.1 Predi cate Logic with Identity. The calculus LC is a predicate logical system without identity. We now add identity. Let L* be the first-order language obtained from L by adding the 2-place predicate '='. Extend LC to LC * by adding the usual axioms for identity: the reflection principle and the relevant substitution principles. Modify the notion of an (arbitrary) structure by adding an interpretation = }1 of '=' and letting = }1 be identity. The subclass consisting of normal structures is obtained as before by demanding that E}1 is the membership relation. 7.2 Completeness. LC* is sound with respect to arbitrary models and with respect to normal models. LC* is complete with respect to arbitrary models. This follows from the Godel- Henkin completeness theorem for predicate logic with identity. It can also be proved directly by suitably modifying Kanger's proof of completeness with respect to validity for LC. LC * is not complete with respect to normal models. To see this , take any non-logical axiomofZPC- + AFA involving identity, e.g ., the Weak Axiom of Extensionality. It can be expressed in L*. It is true in all normal models; but it cannot be proved in LC* since it is not true in all non-normal models for LC *.

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8. CONCLUSIONS

8.1 Conclusion 1. An advantage about Kanger ' s picture of sets as nets compared to the graph model outlined in Sectio n 5 is that it gives a natural, simple, and concrete visuali sation of the membership relation. An argument for the classical conception of sets as well-founded is that heriditarily finite well-founded sets can be constructed in a natural and concrete stepwise process leadin g from lower to higher types of sets. This process can by judicious generalisation be extended into the tran sfinite realm. This is, it is claimed, how sets are brought about. It leads to the cumulative type structure. Kanger shows that there is another process, ju st as simple, natural, and concrete, by which sets, well-founded or non-well-founded , can be constructed: use cords tied on cord s. Thi s process can also by judicious generalisation be extended into the tran sfinite realm . This is another and more general way of bringing about sets. Kanger nets can play an equally important role in the foundations of set theor y as Zermelo' s cumulative type structure does. Kanger's ideas on sets as nets should be of interest to anybody thinking about the foundation s of set theory. 8.2 Conclusion 2. Kanger had the remarkable insight that all problems of firstorder elementary predicate logic without identity can be formul ated and solved in one simple formal langu age, the language L. This claim is sustained by the completeness theorem with respect to normal models. The theorem cannot, as shown in Section 7, be generalised to pred icate logic with identity. At first the limitation to first-order logic without identity seems to diminish severely the importance of Kanger ' s result because most applications of logic involve identity. However, the advent of computers and logic programming has changed the situation. Normally logic programs are based on predicate logic without identity. The reason is that the introduction of identi ty makes the programs so slow that they become almost useless. Kanger's result implies that most logic programs can be written in just one simpl e language, namely L. Kanger's insights can possibly be useful in the theory and practice of logic programming. As far as I know, his work has hitherto not been studied from this point of view . 8.3 Impact and Recognition. For three decades, Kanger 's idea s on nets and non-well-founded sets receive d virtually no attention outside his native

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Sweden. Only with Peter Aczel's book Non-Well-Founded Sets from 1988 did his pioneering work begin to win international recognition. Uppsala Univers ity

REFERENCES Aczel , P. (1988), Non-Well-Founded Sets . Center for the Study of Language and Information, Stanford. Barwise, J., and J. Etchemendy (1987), The Liar: An Essay on Truth and Circular Propositions. Oxford UP, Oxford. Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals . North-Holland Publishing Company, Amsterdam. Fischer. F. H. (1958), "Review of Provability in Logic" . Journal ofSymbolic Logic , 23 , 37 - 38. Fitting, M. (1996), First-Order Logic and Automated Theorem Proving . Second edition. Springer-Verlag, Berlin. Kanger, S. (1957), Provab ility in Logic . Acta Universitatis Stockholmiensis. Almqvist & Wiksell , Stockholm. Lindstrom, Ingrid (1986), "A Construction of Non -Well -Founded Sets Within Martin-Lars Type Theory." Department of Mathematics, Uppsala University. Mirimanoff, D. (1917a), "Les antinomies de Russell et de Burali-Forti et Ie probleme fondamental de la theorie des ensembles." L 'enseignement mathematique, 19,37-52. Mirimanoff, D. (1917b), "Remarques sur la theorie des ensembles." L'enseignement mathematique, 19,209-217. Nerode, A., and R. A. Shore (1997), Logic for Applications. Second edition. Springer-Verlag, Berlin. Zermelo, E. (1908), "Untersuchungen tiber die Grundlagen der Mengenlehre, I." Mathematische Annalen, 65, 261 - 281. Zermelo, E. (1930), " Uber Grenzzahlen und Mengenbereiche ." Fundamenta mathematicae, 16, 29-47.

JAAKKO HINTIKKA

THE PROPER TREATMENT OF QUANTIFIERS IN ORDINARY LOGIC

The best known work in logic by Stig Kanger is undoubtedly his Provability in Logic. This monograph is a detailed analysis of the foundations of modal logic. However, Stig Kanger published also a few smaller studies which reveal a different facet of his personality as a logician. This aspect is his keen sense of how logic can actually be done in practice, in contradistinction to the systematizations of logic serving the study of its theoretical foundations. This facet of Stig Kanger' s logical work is represented e.g. by the two small papers "En algebraisk logikkalkyl" (1966) and "Equational calculi and automatic demonstration" (1970) . An important component in the motivation of symbolic logic all the way from Leibniz to John von Neumann is the idea of replacing ordinary verbal argumentation by calculation. But it is not obvious that this idea is implemented very efficiently in the usual logical systems . Admittedly, it is often said that contemporary symbolic logic treats logic as a calculus. Unfortunately, this is not true in practice in the strictest sense of the word . Typically, a branch of logic is not in fact dealt with like a calculus, but like a deductive axiom system. Logical truths (valid formulas) are obtained by deriving them as theorems from suitably chosen axioms or axiom schemata by means of fixed rules of inference. This is in certain respects a clumsy method . Since the usual rules of inference depend for their application to a formula F only on the principal quantifier or connective in F, we have to wait for an opportunity to deal with an occurrence of a quantifier or a connective in F until this quantifier or connective surfaces so as to become the principal operator of a formula. This can be very clumsy from a purely calculatory point of view. This clumsiness is familiar to everybody who teaches introductory logic from the usual textbooks. On the one hand, one has to insist that the rules of inference are applicable only to the principal (outmost) quantifier or connective. On the other hand, one has to alert the student, for the purpose of planning one's deductive arguments, to logical relations between subformulas of the given one. The usual axiomatic approach may even hide certain features of the logic of quantifiers, viz. those that are lost sight of in the course of the process of 87 G. Holmstriim-Hintikka , S. Linstrom and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work , Vol. II, 87-95 . © 200! Kluwer Academi c Publishers . Printed in the Netherlands.

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paring away of the outer layers of quantifiers and propositional connectives within the scope of which the original given quantifier occurs. Other logicians have apparently been aware of the same tension between the calculatory ideal and the axiomatic practice. Tarski's first preference was a categorial logic, which would have been a nice implementation ofthe calculatory ideal. But he realized that at least for the time being his best bet was the usual first-order (or higher-order) logic which was compositional but not as easy to deal with computationally. Later (1948- 52) , however, he went back to his calculatory idea and developed the generalization of first-order logic that is known as cylindrical algebra. Of course, what Tarski was doing was not only and not primarily to enhance our methods to actually manipulate logical formulas. He was interested in the algebraic structure of those formulas , the possibilities of dealing with this structure by means of well-established algebraic techniques, and generalizing this algebraic structure. To return to ordinary first-order logic, it is therefore very natural to try to formulate rules which enable us to manipulate subformulas inside a larger formula without having to dredge them first to its surface, i.e. to the position of a principal operator. Stig Kanger realized the tremendous calculatory advantages of such inside rules. These advantages are especially conspicuous when the only rule of inference is the replacement of equals by equals. Both of the nonstandard calculi that he devised are of this kind. In the former (see Kanger 1966), Kanger presents a formulation of firstorder logic where all that are being cons idered are equations between different well-formed formulas . The only rule of inference is the substitution of equivalent expressions from each other in an equation. The crucial quantifier and identity equations are the following (in a notation different from Kanger's): (1) (2)

(F[x] & (~)F[x]) ... F[x] (~)(F[x] & (~)G [x]) ... «~)F[x] & (~)G [x])

(3)

(~)(::Jy)F[x. y]

(4 ) (5)

(x= x)=T

(x

=y ) =

(::Jz)(z

= (::Jy)(~)F[x.y]

=x & z =y )

In (4) Tis the symbol for truth. (These are Kanger's equations (9)- ( 13) .) The choice of these basic equations depends on the choice of the logical primitives of the system. A different choice might result in using instead, or in addition to, (1)- (3) their duals. Kanger's little paper turns out to lead to interesting comparisons. First, Kanger's basic equations (1)-(5) are to all practical (or theoretical) purposes

THE PRO PER TREATMENT OF QUANTIFIERS IN ORDINARY LOGIC

89

the same as the correspondin g postulates in a cylindrica l algebra. (Cf. Henkin et al. 1971, p. 162.) They may be formulated (in a notation different from the one cylindrical algebraists employ) as follo ws: (6) (7) (8) (9) (10)

(3x)(S [x] & (3x)T[x]) = «3x)S [x] & (3x)T[x]) (S [x] V (3x)S [x])'" (3x)S [x ] (3x)(3 y )S[x,y] = (3y)(3x)S [x,y] (x x) = T (x= y ) = (3 z)(x= z& z= y)

=

Here (6) is tantamount to (2), (8) to (3), (7) is a kind of dual to ( 1), and (9)(10) are essentially (4) -(5). What this means is that Kanger discovered, apparently independently of Tarski, the structure that Tarski captured and generalized in his notion of a cylindrical algebra. Moreover, as the title of Kanger' s paper shows, he was aware of the algebraic significance of the structure he was dealing with. However, unlike Tarski, he did not generalize this structure and study the algebraic prop erties of the genera lized structure. The cons ilience of the two algebraic treatments of first-order logic is not accid ental. Why it is not can be illustrated by pointing out that Tarski, Henkin et al. naturally did not use the logical notation I have employed but an algebraic one in which each quantifier (with a different variable found to it) corresponds to a different operator Cl.' In such a notation, (6) -(1 0) can be written as (1 I) (12 ) ( 13)

(14) (15)

cix ' c,y) = c,x . c,y (x + c, x) = c.x c,cl.x Cl.C,x d KK 1

=

If K

=

*A,f.! ' then

d). ~ =

c-C dl.' .

d,~)

Here d,l. is intuitively speaking the constant proposition asserting the identity of the variables bound to c, and Cl.' A comparison between (6)-(1 0) and ( 11)- (15) shows the restrictions that a Tarski-style algebrai zation imposes on an equational treatment of logic. Since the algebraization does not use any (explicit) variables, one cannot use any logical rule s (equations) which involve the substitution or the renaming of bound variables. These restrictions are of course satisfied by (6) -(1 0), and surprise, surpri se - they are also satisfied by Kanger ' s equations (1) -(5). It is fairl y clearly the possibility of a Tarski-type algebraization that motivated Kanger here. For if one gives up the requi rements of such algebraization, one can formulate an eve n simpl er equational approach to first-

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order logic. Apparently unbeknownst to Kanger, I had presented such a nonalgebraic equational treatment of first-order logic in my dissertation of 1953. Its basic equations might seem to resemble Kanger's, but they involve the substitution of variables for each other and are therefore not suitable as a basis of an algebraic treatment. A merely technical difference is that I used both quantifiers as primitives, thus requiring the use of a dual for each rule . One half of my set of equations can be written as follows (omitting propositional rules): (16)

If G does not contain free occurrences of x,

(17)

«=u)F & G) ... (=u)(F & G) F[y,y]'" (F[y,y] & (=u)F[x,y]) where F[y,y] is the result of replacing x by yin F[x,y]

For identity, the onl y rule needed is the following (18)

F[y,y] ... (=u)(F(x,y) & (x = y))

These rules are more economical than Kanger's, but they are not suitable for the same kind of algebraization as his. But what is there to be said of a Tarski-type algebraization of first-order logic? What does it accomplish? Are its merits only those of algebraic theft over honest logical toil? Let us see. One test case is the possibility or impossibility of generalizing the treatment in question. It has turned out meanwhile that ordinary first-order logic is not our true elementary ("ordinary") logic . That role belongs to what I have called independence-friendly (IF) first-order logic. Can an equational treatment be extended to it? (For the phenomenon of informational independence, see Hintikka 1996, chapters 3-4.) It has been shown that IF first-order logic does not admit a complete axiomatization of rules of logical proof. However, it does admit a complete set of rules of logical disproof. Can they be treated equationally? In my non-algebraic rules, there is little problem. The main change needed is in the rule dual to (16): (19)

If G does not contain free occurrences of x, then «Vx)F V G) ... (Vx)(F V G)

If we admit independent quantifiers, we must change (19) by replacing the second G by a formula G in which each existential quantifier (::ly) of G (assuming that it is in a negation normal form) is replaced by the quantifier (::lyNx) independent of (Vx). In this sense, an equational treatment of IF first-order logic is perfectly possible. However, an algebraic treatment of IF first-order logic does not seem

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to be feasible without radical complications. This is connected with the fact that the advantages of an algebraic treatment lie in the idea that algebraic equations justify the substitution of its two sides for each other independently of the context. It follows that context-dependences (failures of compositionality) are hard to accommodate in Tarski -type algebraization. The difficulty is therefore due to the fact that the most natural formulations of IF first-order logic (including its semantics) are not compositional. In so far as this observation reflects on Kanger's algebraic treatment of logic, it also reflects on the very idea of a cylindrical algebra. But what else can be done here? To manipulate formulas by means of equations is not the only way of improving the calculatory properties of firstorder logic . Logical arguments are usually made much easier and more perspicuous when they are carried out by means of identities between functional terms . Anyone who has ever had to deal with group-theoretical arguments as logical arguments knows what I mean. If such arguments are formulated in terms of a usual axiom system of first-order logic, they are intolerably long and complicated. It is much easier to carry them out as symbolic calculations in terms of the group-theoretical operation, which is a function, plus the inverse function. Hence it would be most desirable if we could in general replace in a similar way arguments carried out in the usual formulations of first-order logic by manipulations of symbolic equations. In the papers I am discussing here, Kanger develops a way of doing just that. The way he does this is in three steps. First , he develops the equational treatment of logical formulas along the lines I have indicated. Then he proposes, very much in the spirit of an algebraic approach a la Tarski, to treat the logical operations involved as if they were ordinary functions. Then he sketches an equational treatment of such functional equations. It is this functional calculus that is calculated (no pun intended) to serve the symbolic calculations needed in logic . The formalism uses only equations (identities) between functional terms. There are no propositional connectives or quantifiers in the system. The main question is to decide when a number of such equations logically imply a disjunction of other such equations. Kanger outlines a sequent -type proof procedure for such implications and proves that it is complete. My task here is not to examine the nitty-gritty details of Kanger' s equational calculus but study its general nature and to ask what it can accomplish. Since this overall procedure involves a Tarsk i-type algebraization, it might seem that it is subject to the limitations pointed out above. For instance, it might be doubted whether it can be extend ed to IF first-order logic . It therefore puts an interesting perspective on Kanger's work here to point out that

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essentially the same reduction to all questions of satisfiability and hence validity in ordinary first-order logic can be accomplished without going by way of an equational treatment of first-order deduction. Indeed, Kanger's equational calculus can be taken to be a fragment of first-order logic with functions as non-logical primitives and identity as a logical primitive. It is a remarkable observation that this fragment can do the same job as the entire first-order logic. This observation is relevant to computer-based implementations of logic, as Kanger was very well aware. But it can be accomplished more directly, without an algebraic treatment of deductive logic . Here is a sketch of how this can be done: Assume that we are asking whether a finite set o of first-order formulas is satisfiable. In ordinary first-order logic, questions of validity can be reduced to such questions. Transform now o as follows: (i) Turn all the members of o into negation normal form. (ii) Replace every atomic sentence with a nonlogical predicate, say Ptx-, x z,...) by (20)

I(x l , xz' ...) = 0

where I is a new function symbol, different for different predicates P. (iii) Replace every existentially quantified subformula, say (21)

(:lx)S[x]

occurring in some member of o by (22)

S[g(y\ , Yz , ...)]

where g is a new function symbol, different for different subformula occurrences, and where (Vy\),(VY2)" " are all the quantifiers within the scope of which (21) occurs. These functions g are known as the Skolem-functions of the formula in question. (iv)

Replace every disjunction

(23)

(S\ V Sz)

occurring as a subformula of the given formula by (24)

=

(SI & h(y\,yz, ) 0) V (Sz & h(y\,yz, ) =I: 0)

where h is a new function symbol, different for different disjunctions (and

THE PROPER TREATMENT OF QUANTIFIERS IN ORDINARY LOGIC

93

different from the functions g) and where (VYl),(VY2)"" are all the universal quantifiers within the scope of which the disjunction (23) occurs. I will call functions h also Skol em functions (in an extended sen se). (v)

Omit all universal quantifiers.

This eliminates all quantifiers , leav ing us with truth-functions of equations of functional terms t j • (vi) Tum the conjunction of the resulting formulas into disjunctive distributive normal form. It is of the form (25)

(C 1 V C2 V ...)

where each C, is of the form (26)

(til = t*il) & (tiZ = t*i2) & 00. & (tij -:/= t *y) & (t j(j+l ) -:/= t *j(j+l») & 00.

The original set of formulas satisfiable.

(J

is satisfiable if and only if one of the C, is

(vii) Now it suffices to point out that (13) is satisfiable if and only if the following sequent is not valid : (27)

(til

= t *il)' (tiZ = t *i2)'

00'

~ (tij

= t *ij)' (ti(j+l) = t *i(j+I»)' 00 .

Kanger's calculus of functional equations gives a complete logic for sequents like (vii) . This way of reducing the usual axiomatic treatment of first-order logic to an equational calculus it la Kanger has certain theoretical advantages. Among them there are the following: (a) By the same procedure, we also obtain a calculatory treatment of the problem of recognizing (recursively enumerating) inconsistent formulas of independence-friendly first-order logic . All we have to do is to change step (iii) to read as follows : Replace every subformula of the form (28)

(3xNz 1, Vz 2 ,

... )

S[x]

occurring in one of the members of (J by (29)

S[g(Yt, Yz, 00 ')]

where g is a new function symbol, different for different subformula occurrences, and where VYl,VY2'oo . are all the universal quantifiers other than Vz\,Vz 2 ,oo . within the scope of which (28) occurs.

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JAAKKO HINTIKKA

Actually, we must have also a similar rule for transforming independent disj unctions: Replace any subformula of the form (30)

(5\

(VNYI' VYz, ... ) 52)

occurring in any member of (31)

(51 & (h(zl' Z2' (52 & (h(zl' Z2'

(J

by

) =0)) V ) i= 0))

where h is a new function symbol, different from all the g' s and different from different subformula occurrences (30), and (Vz 1),(VZ2) , ... are all the universal quantifiers other than (VYI),(VY2)'''' within the scope of which (30) occurs. In this way, not only all quantifiers but also all slashes (independence indicators) are eliminated. (b) There is a further twist that can be given to this reduction. We can replace (ii) by (or, if you prefer, insert between (ii) and (iii)) the following step: (ii)' For any nonlogical predicate P and every function symbolfin a member of (J replace it by a variable of the same number of arguments say X or g, and bind each of these variables to a sentence-initial existential quantifier. Then we also need a new concluding step: (viii) Translate the result into independence-friendly first-order logic. (For such translatability, see Hintikka 1996, pp. 61-63.) These transformations obviously preserve satisfiability. It leaves us with a set of formulas with no nonlogical primitives, i.e. with = as the only predicate, and without any function symbols . Hence in the equations that result from the reduction the only functions that occur are the Skolem functions of the different formulas . This shows strikingly that first-order logic is at bottom the logic of the Skolem functions of different formulas. Kanger 's calculus of functional equations can thus handle more than he himself pointed out. It can also be given a twist that enhances further its theoretical interests. For me, the main insight available here is not only the applicability of algebraic techniques in logical theory but the idea of quantificational logic as the theory of Skolem functions . Boston University

THE PROPER TREATMENT OF QUANTIFIERS IN ORDINARY LOGIC

95

REFERENCES Henkin, Leon, J.D. Monk and Alfred Tarski (1971), Cylindrical Alg ebra s, Part 1. Amsterdam , North-Holland. Hintikka , Jaakko (1953) , Distributive Normal Forms in the Calculus of Predicates (Acta Philosophica Fennica vol. 6). Helsinki , Societas Philosophica Fennica. Hintikka, Jaakko (1996), The Prin ciples of Mathema tics Revisited. Cambridge University Press. Kanger, Stig (1959) , Handbok i logik . Stockholm. Kanger, Stig (1963), "A simplified proof method for elementary logic" in P. Braffort and D. Hirchberg (cds.), Computer Programming and Formal Syst ems . Amsterdam , North-Holland, pp.87-94. Kanger, Stig (1966) , "En algebraisk logikkalkyl" in Analyser och argument: Filosofiska uppsatser tilldgnade Andries MacLeod (Filosofiska Studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala Univcrsitct, vol. 4). Uppsala, pp. 117125. Kanger, Stig (1970) , "Equational calculus and automatic demonst ration" in Tom Pauli (ed.), Logic and Value: Essays Dedicated to Thorild Dahlquist (Filoso fiska Studier utgivna av Filosofiska foreningcn och Filosofiska institutionen vid Uppsala Universitet , vol. 9). Uppsala , pp. 220 - 226.

STEN LINDSTROM

AN EXPOSITION AND DEVELOPMENT OF KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

1. INTRODUCTION Stig Kanger - born of Swedish parents in China in 1924 - was professor of Theoretical Philosophy at Uppsala University from 1968 until his death in 1988. He received his Ph. D. from Stockholm University in 1957 under the supervision of Anders Wedberg . Kanger's dissertation, Provability in Logic, was remarkably short, only 47 pages, but also very rich in new ideas and results. By combining Gentzen-style techniques with a model theory a la Tarski, Kanger obtained new and simplified proofs of central metalogical results of classical predicate logic: Godel's completeness theorem, Lowenheim -Skolem's theorem and Gentzen's Hauptsatz. The part that had the greatest impact, however, was the 15 pages devoted to modal logic. There Kanger developed a new semantic interpretation for quantified modal logic which had a close family resemblance to semantic theories that were developed around the same time by Jaakko Hintikka, Richard Montague and Saul Kripke (independently of each other and independently of Kanger) . Although his work did not receive much attention at the time, it is generally accepted nowadays that Kanger played a crucial role in the development of model-theoretic semantics for modal logic. The precise nature of his contribution is, however, not so widely appreciated. It is sometimes said that Kanger was the true originator of the so-called possible worlds or Kripke semantics for modal logic . Thus, Dagfinn Fellesdal, in his address to the Stig Kanger Memorial Symposium in Uppsala August 1991, proposed the label "Kanger- Kripkesemantics" for the possible worlds approach.' Such an assimilation of Kanger-type semantics to standard possible worlds semantics could, however, be misleading . Kanger 's formal semantics for modal logic does not, at least explicitly, utilize the notion of a possible world. Nor is there in his early works from 1957 any mention or discussion of possibilia (possible worlds, counterfactual states of affairs, possible individuals). 1 do not think this is an accident. Kanger's goal was to generalize and extend the standard Tarski-style definition of truth in a model for first-order 97

G. Holmstrtim-Hintikka . S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. 1I. 97-130. © 2001 Kluwer Academic Publishers. Printed in the Netherlands. Originally published in P.w. Humphrey s and J .H. Fetzer (ed s.), The New Theory of Reference - Kripke, Marcus. and Its Origins, Kluwer Academic Publishers , 1998, pp. 203 -233 .

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predicate languages to languages of quantified modal logic. He wanted to do so without introducing any new primitives in the metalanguage. In (1957b), he explicitly mentions it as an advantage of his approach that it does not presuppose any "intensional" entities like Fregean senses, meanings or intensions. Although he does not discuss the matter , one gets the decided impression that Kanger's ontology is no more hospitable toward possibilia than it is toward intensional entities. There are, I shall argue, important differences, both of a conceptual and a technical nature, between Kanger's approach to modal semantics and the possible worlds approach. In this connection, one should distinguish between: (i) possible worlds semantics proper: a particular type of model-theoretic semantics for modal logic; and (ii) the possible worlds interpretation of modal concepts. Accordingly , we may pose the two questions: "Is Kanger' s semantics a kind of possible worlds semantics?" and "Is Kanger 's interpretation of modal notions a 'possible worlds interpretation'?". In this paper I shall argue that both questions should be answered in the negative . Kanger semantics differs from standard possible worlds semantics in many ways. As we shall see, the notions of logical truth and logical consequence are defined differently for the two kinds of semantics. Moreover, the underlying intuitions about modality are different. 2. SEMANTICS FOR QUANTIFIED MODAL LOGIC: FROM CARNAP TO KRIPKE The proof theoretic study of quantified modal logic was pioneered by Ruth Barcan Marcus (1946a , 1946b, 1947) and Rudolf Camap (1946, 1947) who were the first to formulate axiomatic systems that combined quantification theory with (S5- and S4-type) modal logic. The attempts to interpret quantified modal logic by means of formal semantic methods also began with Camap (1946, 1947), where he presented a semantics for logical necessity based on Leibniz's old idea that a proposition is necessarily true if and only if it is true in all possible worlds. In his formal semantics, Camap used syntactic entities - state-descriptions - as representatives of possible worlds. Suppose that we are considering a first-order predicate language L with a countably infinite set of individual constants, an infinite sequence of n-ary predicate letters for each n ~ 1, but without function symbols or the identity symbol. In addition to Boolean connectives and quantifiers, the language L contains the modal operator 0 for logical necessity. A state-description for L is simply a set of (closed) atomic sentences of £,2 Camap (1947) , p. 9, writes "...the statedescriptions represent Leibniz's possible worlds or Wittgenstein's possible

KANGER' S EARLY SEMANTICS FOR MODAL LOGIC

99

states of affairs". Relative to a state-description S, the notion of truth for sentences

S F P(a l , .. . , an) iff P(a l , . .. , an) EO S. S F -'

Here,
SF O

That is, the modal formula "it is (logically) necessary that
S F O

It is easy to verify that 0 satisfies the usual laws of the system S5, together with the so-called Barcan formula and its converse, and the rule of necessitation, (K) (T) (S5) (Ba) (CBa) (Nee)

F O(

Let QC(L) be the set of all logically true sentences, according to the above semantics, in the language L.

Theorem. The set QC(L) is not recursively enumerable, so there is no formal system with this set as its theorems.

Proof Suppose the set QC(L) is recursively enumerable. Let

Either F D

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STEN LINDSTROM

Since, by assumption, there is an effective enumeration of the logically true sentences of L, we can effectively decide which of ~ D

~

D

~

<po

That is, D

S

~

[!J

I

~ <po

Thus, R[Q is the universal relation between systems. Kanger points out that by imposing certain formal requirements on the accessibility relation, like reflexivity, symmetry, transitivity, etc., one can make the operator satisfy corresponding well-known axioms of modal logic. In this way, the introduction of accessibility relations made it possible to apply semantic and model-theoretic methods to the study of a variety of modal notions other than logical necessity . Although Kanger was the first to publish,

KANGER 'S EARLY SEMANTICS FOR MODAL LOGIC

101

other researchers, among them Hintikka and Montague, also came up with the idea of utilizing accessibility relations in the semantics of modal notions. One source of inspiration for Kanger 's use of accessibility relations in modal logic was no doubt the work of Jonsson and Tarski (1951) on representation theorems for Boolean algebras with operators.' Jonsson and Tarski define operators 0 on arbitrary subsets X of a set U in terms of binary relations R c U x U in the following way : OX = {x

EO

U: ::Jy

EO

X(yRx)},

that is OX is the image of X under R. They also point to correspondences between properties of 0 and properties of R. Among other things, they prove a representation theorem for so-called closure algebras that , via the TarskiLindenbaum construction, yields the completeness theorem for propositional S4 with respect to Kripke models with a reflexive and transitive accessibility relation. However, Jonsson and Tarski do not say anything about the relevance of their work to modal logic. Perhaps they considered the connection too obvious or of too little importance to mention it. A semantic approach to first order modal predicate logic that has a close resemblance to Kanger's was developed by Montague (1960).4 Like Kanger, Montague starts out from the standard model-theoretic semantics for nonmodal first-order languages and extends it to languages with modal operators. He defines an interpretation for an ordinary first -order predicate language L to be a triple S =, where (i) D is a non-empty set (the domain); (ii) I is a function that assigns appropriate denotations in D to the non -logical con stants (predicate symbols and individual constants) of £ ;5 and (iii) a function g that assigns values in D to the individual variables of L. For each non-logical constant or variable X, let SeX) be the semantic value (i.e., denotation for nonlogical constants and value for variables) of X in the interpretation S. Then the notion of truth relative S is defined as follows: (1)

(2) (3) (4) (5)

S S S S S

~ ~ ~ ~

~

P(tt,···, tn) iff <S(t l ) , . . . , S(t n» EO S(P). (t, = t z) iff S(t l ) = S(t z). -'

~

<po

Here, Sta/x) is the interpretation which is exactly like S, except for assigning the object a to the variable x as its value. Montague now asks the same question as Kanger: How can this definition of the truth-relation ~ be generalized to first-order languages with modal operators? As we recall, Kanger solved the problem by modifying the notion

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of an interpretation: a Kanger-type interpretation (what he called 'a system') assigns denotations to the non-logical constants and values to the variables not only for one single domain (the ' actual' one) but for all domains. Montague's approach is simpler than Kanger's: he keeps the notion of an interpretation S of first-order logic intact, and just adds semantic evaluation clauses for the modal operators. As in the Kanger semantics, each modal operator D is associated with an accessibility relation Ro. Now, however accessibility relations are relations between interpretations S = of the underlying nonmodal first-order language. The semantic clause corresponding to the operator D, with associated accessibility relation Ro' is: (6)

S

F

D

F


Montague observes that for each variable x the universal quantifier Vx, that binds the variable x, can be viewed as a modal operator with the accessibility relation R'v'xdefined by the condition: R'v'x iff D = D', I every variable y different from x.

= l'

and g(y)

= g'(y) for

Given this definition, (5) becomes a special case of (6). Montague associates with the operator [I] of logical necessity the accessibility relation R[I] defined by : R[I] iff D = D' and g = g ' . Thus, his semantic clause for [I] becomes: (1)



F

[I]

F


This semantic clause should be compared with Kanger's stricter condition: (2)

S

F

[I]

F


The difference between (1) and (2) corresponds to a difference between two different conceptions of logical truth: Tarski's (1936) conception and the modem model-theoretic one . According to Tarski (1936), truth and logical truth are properties that primarily apply to interpreted formal languages. An interpreted first-order language Lcomes with a domain ofdiscourse D and an interpretation function I that gives the denotations in D of the non-logical constants of L Hence, the (absolute) notion of truth is well-defined for such a language. A formula

F .6 Now, according to Tarski (1936), a sentence (closed formula)

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103

if it is true and its truth is invariant with respect to all reinterpretations of its non-logical constants relative to the given domain of discourse D. That is,

with the given domain D, .. <poIf

.. tp,

where D is the domain of discourse of Land g is any assignment of values in D to the variables. Hence, for closed formulas of the underlying first -order language, Montague's truth-clause (1) above coincides with Tarski's (1936) definition of logical truth. It also accords well with Camap's definition of logical truth as truth in all state descriptions, provided that the language contains names for all the objects in a fixed domain of discourse. Tarski's (1936) definition yields the result that all true sentences of an interpreted language that do not contain any non-logical symbols are logically true. So, if the domain of discourse is infinite (and provided that = is taken as a logical constant), then all the sentences of the form: (n) n

~

::Jxl ...::JXn(x l =f. x2 /\ ... /\ XI =f. x, /\ X2 =f. x3 1\.../\ X2 =f. x, 1\.../\ Xn_ 1 =f. xn)

2, are logically true . So according to Montague (1960), the sentences:

([gn)

[g ::Jx! ...::Jxn(x\ =f. x2 1\.../\ XI =f. x, /\ x 2 =f. x3 1\.../\ X2 =f. x, /\.../\ 1\.../\ Xn _1 =f. x.)

are true for an interpreted language with an infinite domain . Notice, however, that according to Montague (1960), it is not the case that : .. [g ::Jx1...::JXn(x l =f. X2 1\.../\

/\ .../\ x n_1 =f. x.)

XI

=f. x, /\ X2 =f. x, 1\.../\ X2 =f. x, /\.../\

that is the sentences ([g n) are not logically true. This is so, since the definition of logical truth that Montague adopts in the metalanguage is the standard model-theoretic one of truth in all interpretations, not just truth in all interpretations with a given domain. In conclusion, Montague's and Kanger's respective truth clauses for [g yield, for modality-free sentences
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STEN LINDSTROM

There are other possibilities for interpreting logical necessity within Montague's framework, one interesting alternative being:

(3)

1=

[]J

such that D c 0' , <0 ' ,1', g> 1= <po

Notice that on Montague's interpretation (1) of OJ , both the Barcanfonnula: (Ba)

VxOJ
and its converse: (CBa)

OJ'v'x
are logically true. However, on the interpretation (3), (CBa) still holds, but the Barcan formula fails. Kanger's interpretation (2), finally, makes both (Ba) and (CBa) fail. As a matter of fact, on the interpretation (2) the following is a logical truth: OJ
'"

x.),

the initial quantifier does not really "bind" the free occurrences of the variable XI in the formula <po In other words, Kanger's interpretation does not allow us to "quantify in" past the modal operator OJ . The interpretations (I) and (3 ), on the other hand , give coherent sense to "quantifying in". For example, given interpretation (3), the following sentence is clearly meaningful. As a matter of fact, it is easily seen to be false in any domain with two elements: 3x3y(x

z = yj).

-:/=

y 1\ VzOJ(z

= x V z = y»

- 3x3y(x -:/= y 1\ OJ'v'z(z = x V

The semantical frameworks ofCarnap (1947), Kanger (1957a, 1957b) and Montague (1960) - although important developments in themselves, were still not what we nowadays call possible worlds semantics. Something was still missing: an ingredient that was provided by Jaakko Hintikka and Saul Kripke.' In order to see what was missing, let us just make a seemingly small change in Carnap's semantics. Instead of considering the set of all state-descriptions, we shall only consider those state-descriptions that belong to some arbitrary non-empty set C. Let us call such a set C of state-descriptions a Carnap frame. The notion of truth in a state-description now becomes relativized to a Carnap frame. Thus for any S EC,

KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

(1)

(2) (3) (4) (5)

S S S S S

105

""c P(al,···, an) iffP(a 1,. ··, an) EO S. ""c ""

Observe how the relativization of the definition to C changes the meaning of O. Instead of O

(2) (3) (4) (5)

w w w w w

"=cP(al , · .. , an) iffP(a1 ,·oo, an) EO Sw' "=c ""

Finally, we get a full-fledged possible worlds semantics if we replace the use of state-descriptions by an indexed family of domains {D w} WEW together with an interpretation function I that assigns, for each possible world w EO W , an appropriate semantic value to the non-logical symbols of LIn Carnap's, Kanger's and Montague's early theories, the space of possibilities is represented by one comprehensive collection containing all state descriptions, systems or models, respectively. Hence, every state description, system, or model is thought of as representing a genuine possibility. Hintikka, Kripke and modern possible worlds semantics are instead working with semantic interpretations in which the space of possibilities is represented by an arbitrary non-empty set K of model sets (in the case of Hintikka) or possible worlds (Kripke). Following Hintikka' s (1980, 1989) terminology, one may say

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STEN LINDSTROM

that the early theories of Carnap, Kanger, and Montague were considering standard interpretations only, where one quantifies over what is, in some formal sense, all the possibilities. In the possible worlds approach, one also considers non-standard interpretations, where arbitrary non-empty sets of possibilities are considered." The consideration of interpretations (model structures) that are non-standard in this sense - in combination with the use of accessibility relations between worlds in each interpretation - made it possible for Kripke (1959, 1963b and 1965) to prove completeness theorems for various systems of propositional and quantified modal logic (T, B, S4, etc.). Kripke's major innovation was perhaps his use - within each model structure - of a set of abstract points (indices, "possible worlds") to represent the space of possibilities. This innovation made it possible for Montague - building on ideas from Camap - to represent intensional entities (senses, intensions) by set-theoretic functions from points (representing possible worlds) to extensions." These Camap- Montague intensions could then be used to give model-theoretic interpretations of comprehensive fragments of natural language. Montague grammar combined categorial grammar with the idea of representing meanings of lexical items with functions from points (indices, "possible worlds") to extensions. We end this section by presenting a version of Kripke's (l963a) semantics for modal predicate logic with identity, where the notion of a possible world is an explicit ingredient of the semantic theory. A (Kripke) model structure for a language L of first-order modal predicate logic is a quintuple S = <W , D, R, E, wo> where, (i) W is a non-empty set (of possible worlds); (ii) D is a non-empty set (of possible objects); (iii) R c W x W is the accessibility relation of the structure; (iv) E is a function which assigns to each WE W a subset E, ofD (intuitively E, is the set of objects that exist in w); and (v) W o is a designated element of W (the actual world) . It is required that D = U WEWE W' i. e., that every possible object exists in at least one world. A Kripke model is an ordered pair VJl =<S, 1>, where S =<W, D, R, E, wo> is a model structure and I is a function satisfying: (vi) for each w E W, I assigns to each n-ary predicate constant P of La subset I(w, P) of D"; and (vii) I assigns to each individual constant c of L an element I(c) E D . A model VJl of the form <S, I> is said to be based on the model structure S. Observe that I(w, P) is not necessarily a subset of (E w ) " , i. e., the extension of P in w may involve objects that do not exist in w. Moreover, individual constants are treated as rigid designators, i. e., they are assigned denotations in a world-independent way. An assignment in is a function g which as-

m

KANGER'S EARLYSEMANTICS FOR MODAL LOGIC

107

signs to each variable x an element g(x) in D. For any term t in L, we define 9)1 (t, g) to be g(t) if t is a variable; and I(t) if t is an individual constant. With these notions in place, we can define what it means for a formula

w w w w w w

"" m P(t l , . .. , tn)[g] iff <9)1(t l , g),..., 9)1(tn, g) E I(w, P). ""m (t, = tn)[g] iff 9)1(t[, g) = 9)1(t2 , g). ""m ""
We say that

w "" m O

Let QS5= be the class of such structures. As Kripke (l963a) showed, neither the Barcan formula nor its converse is (QS5=)-valid. It should also be pointed out that Kripke's semantics validates the Law of Identity, (L=)

'lix(x = x),

as well as the principle of Indiscernibility of Identicals, (1=)

vxvy]x

= Y ~ (
applicable without restrictions also to modal contexts
'lix'liy(x = Y ~ O(x = y)) (Necessity of Identity) 'lix'liy(x 4: Y ~ O(x 4: y)) . (Necessity of Distinctness)

The latter principles are controversial. Hintikka has repeatedly stressed that they are unacceptable for propositional att itude constructions like "John knows that ...", "George the IV believes that .;.", etc." On the other hand, Kripke's formal semantics fits well with his conception of metaphysical

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STEN LINDSTROM

necessity - as it is expounded in Kripke (1980) - and the possible worlds metaphysics that goes with it. In section 6.1, I shall argue for the thesis that (QS5=) is the (first-order) quantified modal logic corresponding to Kripke's conception of metaphysical necessity. And I shall write [Hj rather than D when the necessity operator is interpreted as metaphysical necessity. The dual operator, metaphysical possibility, is written as ~, that is, ~q> =df. -,[Hj-'q>.

3. METALINGUISTIC VERSUS OBJECT-LEVEL INTERPRETATION OF MODALITIES According to the dominant intuition underlying Kanger's work, modal operators of the object language, "It is logically necessary that", "It is settheoretically necessary that", etc., are obtained "from above" as it were, by "reflecting" corresponding predicates of the metalanguage: "truth in all interpretations", "truth in every interpretation that is normal with respect to the settheoretic vocabulary", etc. According to this intuition, it is (analytically) necessary that all bachelors are unmarried, just because the sentence "All bachelors are unmarried" is an analytic truth. Writing [E]q> for "it is analytically necessary that q>", where q> is a sentence of the object language, we have: (1)

[E]q> is true iff q> is an analytic truth.

By relativizing this schema to a semantic interpretation S, we get:

(2)

[E]q> is true w. r. t. S iff q> is an analytic truth w. r. 1. S.

The basic idea can be expressed in the form of a reflection principle: (R)

S

>=

Oq> iff P( q>, S),

where 0 is an operator of the object language and P is a metalinguistic predicate that may be satisfied by a formula q> and an interpretation S.lI By itself, the schema (R) only gives "truth-conditions" for formulas containing O. But if we think of (R) as explaining the meaning of the operator 0 in terms of the metalinguistic predicate P, we have what may be called a metalinguistic interpretation of O. Kanger's interpretation of modal operators is usually metalinguistic in this sense. Consider, for example, his introduction of the operator OJ of logical necessity. The underlying intuition is expressed by the schema: (3)

OJq> is true iff the sentence q> is logically true .

In virtue of the standard model-theoretic definition of "logical truth" as "truth in every interpretation of the object language", we get:

KANGER 'S EARLY SEMANTICS FOR MODAL LOGIC

(4)

109

lIJ

(4) corresponds to the following clause of Kanger semantics:

[g

This corresponds to the following semantic clau se for the operator [E]: (6)

[E]

v. The possible worlds approach, by contrast, is based on the intuition that there is a multitude of ways in which the world could have been different. On this conception, nece ssity is not primaril y a metalin guistic notion: the necessary truths are rather the truths that would have remained true had any of these possibilities been realized. It is necessarily true that 2 + 2 =4 just because, for any of the ways (" poss ible worlds") in which the world could have been different, had the world been different in that way, the proposition that 2 + 2 =4 would still have been true. These two intuitions about the source of necessity are clearly distinct and indeed, as we shall see, give rise to very different formal semantic theori es. To further clarify the relation ship between Kangers semantics for modal logic and possible worlds semantics, we may make use of a terminology due to Etchemendy (1990 ) and distingui sh between interpretational and representational semantics. Intuitively, the truth-value of a sentence is determined by two factors: (i) the interpr etation of the symbols in L; and (ii) the fa cts (the way the world is). We may keep the world fixed but let the interpretation of the symbols vary. Thi s is the basic idea behind the standard definitions of logical truth and logical consequence in the model-theoretic tradition after Tarski (1936 ). Etchemendy calls it interpretational semantics. On this approach, settheoretic models represent ways of reinterpretin g the language. Or, we may keep the interpretation of the linguistic units fixed, but consider what would have happened to the truth- value of the sentence had the facts been different. A semantics based on the latter idea Etchemendy calls representational . On this approach set-theoretic models rep resent ways in which the world could have been different (possible worlds) . The clas sical exa mple of representa-

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STEN LINDSTROM

tional semantics is, of course , Carnap' s semantics in terms of state descriptions . We can use either of the two ideas, the representational or the interpretational, to interpret modal operators. For example, (i) and (ii) below are alterna tive ways of interpreting a necessity operator: (i)

(ii)

[gq> is true (in the actual domain relative to the intended interpretation) iff q> is true in every domain relative to every interpretation of the (non-logical) symbols of q>. [Hjq> is true (in the actual world relative to the intended interpretation) iff q> is true and would have remained true (relative to the intended interpretation of the language) even if the world had been different in any of the ways in which it (metaphysically) could have been different.

The definition of [g is an example of the interpretational approach: it corresponds to Kanger's explication of logical necessity. The definition of [Hj, corresponding to Kripke's notion of metaphysical necessity, is an example of the representational approach. Both ideas playa role in Kanger's semantics, but the first one is clearly the one that dominates his thinking . The second idea turns up, in a relatively mild form, in his semantics for analytic necessity: it is analytically true that q> (in the actual domain and on the intended interpretation of object language) iff q> is true in every domain. In order for this definition to make intuitive sense, we must allow domains that contain not only actually existing objects but also merely possible objects, i.e., objects that do not exist but might have existedor at least representatives of such objects. Otherwise, all true universal generalizations of the form Vx(Fx ~ Ox) would on the proposed definition be analytically true. We could not distinguish between the analytically true sentence "All bachelors are unmarried" and the presumably synthetic truth "All bachelors are less than 150 years old". Kanger expresses the idea behind his definition of analyticity: "Our definition of analyticity may be regarded as an explication ... of the idea that an analytic proposition is a proposition that is true in every possible universe.?" Actual truth is defined with respect to the domain that Kanger describes as "the class of all 'real' individuals"." We see that alternative domains of individuals play somewhat the same role in Kanger semantics as possible worlds in standard possible worlds semantics. Kripke's notion of metaphysical necessity - truth in every metaphysically possible world - is , however, foreign to Kanger's way of thinking . Modal operators are viewed as projections of metalinguistic concepts rather than as expressing genuine properties of propositions.

III

KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

4. KANGER'S SEMANTICS FOR FIRST-ORDER QUANTIFIED MODAL LOGIC We shall now take a closer look at Kanger's semantics for quantified modal logic. We consider a first -order predicate language L with identity and a family of unary modal operators [ L]: i EO I}. A domain is a non-empty set D. A (primary) valuation (for L) is a function v, which, given any domain D, assigns: (i) an element ofD to each term t of L ; (ii) a set of ordered n-tuples of elements in D to each n-place predicate constant P in L. We may, whenever convenient, think of a valuation as consisting of two components: (i) an interpretation I that assigns denotations I(D, a) to the nonlogical constants of L relative to domains D; and (ii) an assignment g that assigns, for each domain D and each variable x, a value g(D, x) to x in D. Of course, v = lug. For any I and D, we define ID to be the unary function defined by letting ID(a) =I(D, a). VDand gD are defined analogously. A system is an ordered pair S = , where D is a domain and v is a valuation. When we want to emphasize the distinction between the interpretation I and assignment g, we also write a system as: S = . Kanger defines the notion of a formula

(in symbols, S 1=
(2) (3) (4) (5) (6) (7)

S 1= (t, = t2) iff v(D, t\) = v(D, t2) . S 1= P(t\ ,..., tn ) iff 1= Vxrp iff 1= 1= ~x

1=

Explanation: g ' is like g except possibly at x (also written, g' =xg) if and only if, for each domain D and each variable yother than x, g(D, y) = g(D, y). In the above definition, Ro is a binary relation between systems that is associated with the modal operator D . Ro is the accessibility relation associated with the operator D. Among the modal operators in L , there are two designated ones [E] ("analytic necessity") and [I] ("logical necessity") with the semantic clauses:

1= 1=

[E]

1IJ

We define a Kanger model for L to be an ordered pair Wi

1=

<po

= of a

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STEN LINDSTROM

domain and an interpretation. Notice that for any D and any I, the structure is a model for the non-modal fragment I.v of L. Hence, an interpretation I can also be represented as a family { } DE2> of first-order models inde xed by the collection 'D of all domains. Notice also that this family contains exactly one model for each domain D. A Kanger model can then be written as an ordered pair of the form 9Jt = }DE-z?' A system, finally, can be represented as an ordered pair <9Jt , g> of a Kanger model 9Jt and an assignment g. We say that formula

. If g and g' are assignments that agree for each domain on all the variables that occur free in F

F tp, Hence, a sentence (closed formula) is true in a Kanger model 9Jt = { }DE2> iff it is true in some system <9Jt , g>. A formula

of a sentence (closed formula)

, where P is an n-ary predicate constant and I is an interpretation. A name is a pair , of a constant c and an interpretation I. The proposition < is said to be true in the domain D if F <po-op, I> is called analytic if it is true in every domain D. Notice that for any domain D, ( I) (2)

<[E] is true in D iff < is analytic. <[I] is true in D iff

The notion of (non-relativi zed) truth is explained as follows : 16 Let Do be the set of all "real" individuals. Let 10 be the interpretation which, for every domain D, assigns to the non-logical symbols the denotations that they have according to the intended interpretation of L. 17 A proposition -op, I> is true if it is true in the domain Do. A sentence

is true.

is analytic. Notice that [E]

F

D

F

<po

In (1970) Kanger presents a version of his semantics where he considers ontological operators only. That is, every operator 0 is associated with a relation Ro between domains rather than between systems. In (1972) the ontological approach has become his official semantics for modal logic .

KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

113

An important difference between Kanger's semantics for modal operators and that of Kripke is that Kanger assigns to each modal operator a fixed accessibility relation, once and for all, while Kripke ("standard possible worlds semantics") lets the accessibility relations vary from one model structure to another. One could say that Kanger treats the modal operators as logical symbols - having fixed interpretations - while Kripke treats them as non-logical symbols - i.e., allowing their interpretations to vary. 5. THE TREATMENT OF "QUANTIFYING IN" IN KANGER SEMANTICS In Provability in Logic (1957a) and in "The morning star paradox"(l957b) Kanger gives the following evaluation clause for the universal quantifier: (i)

<0, I, g> ~ v xtp iff <0, I, g'> that g' = x g.

~

According to this clause, the formula Vx

if and only if

which is exactly like <0, I, g> except, possibly, for the values it assigns to the variable x in the various domains. In (l957c) and (l957d) he considers the following alternative clause: (ii)

<0, I, g> ~ Vx

>=

That is, the formula Vx

if and only if

which is exactly like <0, I, g> except possibly, for the value it assigns to the variable x in the domain 0. Kanger (l957c) suggests the following informal readings of these two alternatives: (xup is true iff

114

STEN LINDSTROM

Consider, namely, a system S = such that Fx is true in some domain D' different from D, and Fy is false in every domain. Suppose also that the extension ofF in D is the empty set. Then, ~Fx and "~Fy are both true in S. By the semantic clause for (Ux), (Ux)~Fx will also be true in S. But, .,~Fy is true for every assignment g' satisfying the conditions (i) g' =y g; and (ii) for all domains D' that are distinct from D, g'(D', y) =g(D', y). Hence, (Uy)~Fy is false in S. That is, substitutivity of alphabetic variants fails for the operator (Ux). For this reason, 1shall not consider it further. So, when 1speak of Kanger' s interpretation of the quantifiers, 1 shall understand the treatment in (l957a) and (1957b), according to which the universal quantifier "i has the semantic clause (i) and the existential quantifier has the dual clause : (iii)



1=

~xq>

iff

1=

q> for some g' such that g'

= x g.

In order to get a clearer understanding of Kanger' s treatment of quantification, 1shall speak of selection functions that pick out from each domain an element of that domain as individual concepts. To be more precise, an individual concept, in this sense, is a function f, with the collection of all domains as its range, such that for every domain D, f(D) E D. We can think of a system S = as assigning to each individual constant c the individual concept {: D is a domain}. The formula P(t l , ... , t,,) is true in S = <0, I, g> if and only if the individual concepts designated by t l , ... , t" pick out objects in the domain D that stand in the relation I(D, P) to each other . The identity symbol designates the relation of coincidence between individual concepts (at the "actual" domain D). That is. t( =t2 is true in a system S = if and only if the individual concepts designated by t, and t2, respectively, pick out one and the same object in the domain D of S. Kanger's quantifier "ix, with the semantic clause (i), can now be thought of as an objectual quantifier that ranges not over the "individuals" in the "actual" domain D, but over the (constant) domain of all individual concepts. That is, "ixq> is true in a system if and only if q> is true in every system which is exactly like except, possibly, for the individual concept that it assigns to the variable x. Note , that interpreted in this way, the range of the quantifiers "ix and ~x is independent not only of the domain D but also of the system S: the range of the quantifiers "ix and ~x is fixed, once and for all, to be the collection of absolutely all individual concepts. While formulas of the form r, = t2 express coincidence, identity between individual concepts is expressed by formulas of the form [H](t} =t2) . Writing x = y for [H](x =y), the following principles are valid:

KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

(L=)

\ix(x = x)

(1=)

\ix\iy(X

IlS

=Y ~ (
that is, = satisfies the formal laws of the identity relation . None of the sentences: (n)

~Xl ... ~Xn(xl

;/; X2 1\...1\ Xl;/; x, 1\ X2 ;/; X3 1\...1\ X2 ;/; x, 1\...1\ Xn-1 ;/; xn),

saying that the "actual" domain has at least n elements (for n <: 2), is logically true according to Kanger's semantics. In contrast all sentences of the form : ~x l", ~ xn(-' (x i

1\...1\ -'(x n_1

= x2 ) 1\. ..1\ -'(x 2 = x3 ) 1\...1\ -,(x 2 = x.) 1\...1\

= x n) ) ,

are logically true. Intuitively, these sentenc es say that there are, for each n, at least n individual concepts. Consider now Kanger's (l957b) discussion of the so-called Morn ing Star paradox. The paradox arises from the following premises: (1)

(2) (3)

[E](Hesperus =Hesperus) Phosphorus =Hesperus -,[E](Phosphorus = Hesperus) ,

where "Phosphorus" and "Hesperus" are two proper names (individual constants) and [E] is to be read "it is analytically necessary that". We assume that "Phosphorus" is used by the language community as a name for a certain bright heavenly object visible in the morning and that "Hesperus" is used for some bright heavenly object visible in the evening. Unbeknown to the community, however, these objects are one and the same, namely, the planet Venus. "Hesperus =Hesperus" being an instance of the Law of Identity is clearly an analytic truth . It follows that the premise (1) is true. (2) is true, as a matter of fact. "Phosphorus = Hesperus" is obviously not an analytic truth, "Phosphorus" and "Hesperus" being two different names with quite distinct uses. So, (3) is true. However, using sentential logic together with the following laws of predicate logic: (UI) (1=)

\ix

we can infer from (2) and (3): (4)

-,[E](Hesperus = Hesperus).

But (1) and (4) contradict each other, so something must have gone wrong in this argument.

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STEN LINDSTROM

According to Kanger's diagnosis of the Morning Star paradox it is (1=) that is at fault. Given Kanger's semantic clause (i) for the universal quantifier and his semantic treatment of individual terms, (UI) is valid but (1=) is not. Only the following restricted version of (1=) is valid: (1=')

= y - (q>(x/z) - q>(y/z»), provided that no free occurrence of z in q> is within the scope of a modal operator.

\lx\ly(x

With this change in the underlying logic, (4) can no longer be inferred from (2) and (3). As Kanger himself points out, there are still some difficulties left. From (UI) and the equivalence :::Jxq> - -Nx-'q>, we get: (EG)

q>(t/x) - :::Jxq>.

(Existential Generalization)

But, in view of (1)- (3) and the Law of Identity, the following sentences are true: (5) (6)

Phosphorus =Hesperus A -,ffi](Phosphorus = Hesperus). Hesperus =Hesperus A ffi](Hesperus =Hesperus).

So, it follows that: (7) (8)

:::Jx(x = Hesperus A -'ffi](x = Hesperus» . :::Jx(x = Hesperus A ffi](x = Hesperus».

Although unintuitive, this result is perfectly compatible with the interpretation of the quantifiers as ranging over individual concepts and of the identity symbol as designating coincidence between individual concepts. According to this interpretation, (7) and (8) mean: (7')

(8')

There is an individual concept x which actually coincides with the individual concept Hesperus but does not do so by analytical necessity. There is an individual concept x which not only happens to coincide with the individual concept Hesperus but does so by analytic necessity.

As Quine (1947) was the first to point out, however, (7) and (8) are incompatible with interpreting \Ix and :::Jx as objectual quantifiers meaning "for all objects x (in the domain D)" and "for at least one object x (in D)" and letting the identity sign stand for genuine identity between objects (in D) . Because, under this interpretation, (7) and (8) have the readings :

KANGER 'S EARLY SEMANTICS FOR MODAL LOGIC

(7")

(8/1)

There is an object with Hesperus and rus. There is an object with Hesperus and

117

x (in the actual domain D) which is identical which is not necessarily identical with Hespex (in the actual domain D) which is identical which is necessarily identical with Hesperus.

meaning that one and the same object, Hesperus, both is and is not necessarily identical with Hesperus, which is absurd. So Kanger' s semantics for quantified modal logic is incompatible with interpreting the quantifiers as ranging over actually existing individuals (as opposed to individual concepts) and at the same time interpreting = as identity between individuals. In Kanger's semantics there are no means of identifying individuals from one domain to another. In particular, the truth-values of formulas will not be affected if we make all the domains disjoint, by systematically replacing every domain D by the set: {: a ED}. In other words , set-theoretic relations between domains like inclusion, overlap and disjointness, have no semantic significance. Suppose we make the cla im: (9)

Something is such that it is the number of planets but might not have been so.

It seems reasonable to formalize this claim in quantified modal logic as : (10)

::Jx(Px f\ ..,OPx).

We cannot use any of the Kanger's quantifiers for this purpose, however. Suppose, namely, that: g(D, x)

E

I(D, P), DRoD', D =/:. D ', g(D', x) if I(D ', P).

Intuitively this means that one thing is the number of planets in the domain D and one thing or another is not the number of planets in the modal alternative D' to D. From this , we should not be able to conclude (10) . But on any of Kanger's interpretations of the universal quantifier, (10) follows . So his approach does not allow us to express the claim that one and the same object has a given property in one domain and lacks that property in another domain. Now, we might ask how we could repair Kanger's semantics in order to allow for genuine quantification over individuals. There are many possibilities. One that is particularly straightforward technically is to adapt Kripke's (1963a) treatment of quantification to Kanger's approach. This means that we modify the notion of an assignment g in such a way that an individual variable

lIS

STEN LINDSTROM

x is assigned an object g(x) in a domain-independent way. That is, we make two changes with respect to Kanger's notion of an assignment: (i) the value g(D, x) of an individual variable x in a domain D is no longer required to be a member of D; (ii) for all domains D and D', we require that g(D, x) = g(D' , x). After these changes are made, an assignment simply becomes a function g that assigns to each variable x an object g(x). We then adopt the following evaluation clauses for the universal and existential quantifies: 0= Vx

0= tp, for every g' such that (i) g' is like g except possibly at x; and (ii) g'(x) E D. 0= :3xq> iff 0= q>, for some g ' such that (i) g ' is like g except possibly at x; and (ii) g '(x) E D. With these clauses our previous objection to Kanger's approach that it could not make correct sense of sentences like :3x(Px /\ ..,DPx) , seems to have been met. For this sentence to be true at , there needs to be an element a E D such that: a

E

I(D, P) and for some D', DRoO' and a

(f

I(D', P).

We have been able to express the claim that one and the same individual has the property P in domain D and lacks that property in some domain D ' that is possible relative to D. Let us now see how the modified semantics might handle the Morning Star paradox. In this semantics, = is interpreted as genuine identity between objects. Accordingly the logical principles for = are the expected ones : (L=) (1=)

Vx(x = x). VxVy(x = y - (q>(x/z) - q>(y/z))).

(Law of Identity) (lndiscernibility of ldenticals)

However, instead of (UI) we have: (UI ' )

Vxq> /\ :3x(x = y) - q>(y/x).

Now, how should we handle individual constants within the modified Kanger semantics? An intuitively appealing approach is to assign denotations to constants in a domain-dependent way as before, but not require the denotation I(D, c) of a constant c relative to a domain D to be a member ofD. With this treatment of indi vidual constants, we cannot infer from (1=) to: (11)

Phosphorus = Hesperus ([E](Phosphorus = Phosphorus) -[E](Phosphorus = Hesperus)),

KANGER 'S EARLY SEMANTICS FOR MODAL LOGIC

119

unless the following requirements are met: ::lxlEJ(x = Phosphorus),

::lxlEJ(x = Hesperus).

But these conditions hold , only if: ::lxlEJ(x = Phosphorus),

IEJ(Phosphorus

=Hesperus).

The last of these conditions contradicts (3), so it cannot be assumed. It would, presumably, hold only if "Phosphorus" and "Hesperus" were synonymous. Hence, we cannot infer (11) from (1=). We can also verify, directly, that the modified semantics does not allow the inference from (2) and (3) to (4) . So the Morning Star paradox, in the form that Kanger presented it, is resolved. Let us say that a modal operator 0 is a constant assignment operator, if there exists a binary relation Ro between Kanger models , such that for every assignment g, 0= O

such that Ro, 0= <po Intuitively, constant assignment operators do not affect the values that are assigned to free variables within their scopes. In contrast, Kanger's operator of logical necessity, which is not a constant assignment operator, binds all free variables within its scope. Hence, quantifying in, past this quantifier, does not make sense. As Kaplan (1986) has shown , however, we can interpret logical necessity in a way that adm its of quantifying in. Within the present framework, Kaplan's treatment of logical necessity amounts to the following semantic clause: 0= [g

, 0= rp, Although, we do have for this operator, Barcan Marcus's (1947) Necessity of Identity principle: (0=)

VxVy(x = y

~[g

(x = y)),

we, of course, do not have : (12)

Phosphorus

=Hesperus ~[g (Phosphorus =Hesperus),

and this, we think, is as it should be. It is, of course, very doubtful whether Kanger would have approved of these changes to his semantics.

120

STEN LINDSTROM

6. THE SET-DOMAIN VERSUS THE CLASS-DOMAIN SEMANTICS Kanger presents his semantic theory within an informal set-theoretic framework, where all sets are treated on a par, as genuine objects that can be members of other sets." This theory - what we might call naive Kanger semantics - is, however, threatened by paradoxes : given normal assumptions about sets the theory is inconsistent. For example, primary valuations are themselves non-empty sets, that is, domains . It follows that for any valuation v, v itself belongs to the first argument domain of v. In consequence, for an individual constant c, «v, c>, v(v, c) E v. This is contrary to the ordinary assumption of set theory that sets are well-founded. This consequence may not seem especially serious from Kanger's point of view, since in section 5.6 of (l957a) he briefly discusses the introduction of non-well-founded sets in set theory (cf. K. B. Hansen's contribution to the present volume). However, there are more serious problems . Consider any valuation v. Its first argument domain is the collection U of all domains. But in order for v to be a set, the collection U must also be a set. Let t> +(U) be the set of all nonempty subsets of U. Then, by Cantor's theorem, cardtp'(U) > card(U). But this contradicts the fact that t>+(U) s U. Thus, given normal set-theoretic assumptions the collections U and v cannot be sets. But this is in apparent contradiction to Kanger's free use of the valuations v in various set-theoretic constructions, for instance, when he defines the notions of a system, a proposition, truth in a system, logical truth, etc., or when he introduces accessibility relations between systems. We are going to consider two alternative ways of modifying Kanger's approach in such a way that the threat of paradox is avoided: the set-domain approach and the class-domain approach. 6.1. Kanger Set-domain Semantics

A natural way of modifying naive Kanger semantics is to assume, as before, that the domains are arbitrary non-empty sets but that the valuation functions are proper classes . The elements of a domain are, of course, either individuals (urelements) or sets. A (primary) valuation v is now defined in the same way as before, except that it is taken to be a functional (proper) class rather than a functional set. The same holds for the notions of an interpretation I and an assignment g. The models for the non-modal base language L 0 are sets. An interpretation is a proper class { }OEV of such models (relational structures) indexed by the class 'f) of all non-empty sets. A Kanger (set-domain) model consists of a domain D and an interpretation

KANGER ' S EARLY SEMA NTICS FOR MOD AL LOGIC

121

1 = { l DW Such a model (we use the notation (D, 1) for it) cannot be a set nor, in fact , a cla ss, since it is intuitively an ordered pair, one of the components of which is a proper cla ss. Although, we cannot define it in the usual systems of set-theory (Zerrnelo- Fraenkel, God el - Bern ays- von Neumann, Kelley - Morse), it make s intuiti ve sense to speak of the ordered pair (D, 1). An assignment g is now a functional class which for any domain D and any variable x assigns a value g(D, x) in 0 to x. A system is an ordered pair S = (wt , g) con sisting of a Kanger model wt and ari assignment g. The accessibility relation Ro that is associated with a modal operator 0 of L is a collection of ordered pair s (S, S ' ) of systems. Or, in the case of ontological operators: a collection of ordered pair s of domains. The following notions are defined exactly as before: (i) S 1= cp and wt 1= cp; (ii) a formula cp being valid or logically true (written as 1= qi) ; (iii) logical consequence, r 1= cp, where r is a set of formul as and cp a formula. The Kanger set-domain semantics presupposes a strong meta theory in which one can spea k not only of the usual cumulative hierarch y of sets over a set of indi viduals, but also of classes that may cont ain sets and indi vidu als, collections that may cont ain such cla sses, collections that may contain collections, and so on. Onl y the indi viduals and sets are regarded as genuine objects , while cla sses and collecti ons are thou ght of as essentially predicati ve in nature. As a proper formal language for formalizing the metatheory, we think of a language of simple-type theory, where the individual variables (i.e., variables of type '1') range over the elements of the cumulative hierarchy and the predicate variables of type (a .,..., an) range over arbitrary n-ary relations whose i'th domain for 1 ~ i ~ n consist of the cla ss of all entities of type a i . So for instance, the variables of type (t) range over arbitrary classes of objects (urelements and sets). Vari ables of type i), (t)) range over arbitrary relations between classes of objects, etc. A metalanguage of this kind should be appropriate also to formalize the next version of Kan ger semantics.

«

6.2. Kanger Class-doma in Semanti cs This semantics differs from the pre vious one in the following respect s: A domain is now defined to be a non-empty class of elements of the cumulative set hierarchy, i.e., we also allow domains to be proper classes . A valuation is a function v, which given any domain D, ass igns appropriate denotations over D to the non-logical con stant s (proposi tiona l con stants, indi vidual constant s and predicate constants) and indi vidual variables of L. We now allow the denotation of an n-plac e predic ate con stant P of L to be a class of n-tuples of

122

STEN LINDSTROM

elements in D. The notions of an interpretation and an assignment are adjusted accordingly. Hence , we allow models (D, ID) for the base language 4, where D is a class and ID assigns appropriate classes to the non-logical predicate constants of L Thus , Kanger models now have the form 9Jt = (D, {(D, I D)} Dc1»)' 6.3. Comparing the Two Approaches We might now ask how validity with respect to the class-domain semantics is related to validity with respect to the set-domain semantics. If the language £ has sufficient expressive capacity, neither implies the other. Let, for example, £ ' be the language which is obtained from our language I: by adding the generalized quantifier (=:JabS inf X) as a new logical constant." The intuitive reading of (=:Jabs inf x)

~=:Jabs inf x)(x

= x)

is true in every domain (and for every valuation v). Hence, it is logically true according to the Kanger c1ass-domain semantics. On the other hand, according to the Kanger set-domain semantics the sentence (::Jabs inf X)(X = X) is false in every domain. Hence, according to this semantics, it is instead (2)

"' ~=:Jabs inf x) (x

= x)

that is a logical truth . Here we have a dramatic difference between the two semantic theories. Clearly, it is the class-domain semantics that yields the intuitively correct result in this case. Instead of adding the "artificial" quantifier (=:Jabs inf x) to L, we could instead have assumed that I: contained a modal operator 0 with the semantic clause: (D, v) 1= D

1=

tp.

123

KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

Then, we would have for the dual operator 0 , (D, v)

t=

o

t=

The class-domain semantics would then pronounce: (3)

O\fx(x = x)

logically true, but according to the set-domain semantics it would instead be its negation: (4)

D::Jx(x 4: x)

that is logically true . Once again, the set-domain semantics gets the wrong result by arbitrarily excluding interpretations that are intuitively legitimate. 7. LOGICAL VERSUS METAPHYSICAL NECESSITY

7.1. On the Adequacy of Kripke's Logic (QS5=) as the Logic of Metaphysical Necessity According to the metaphysical picture of modal reality inspired by Kripke 's Naming and Necessity (1980), there is a space W of possible worlds in which the actual world W o is just one of the worlds. There is also a collection D of possible objects and there are prop erties that the possible objects can have and relationships that they can have to each other (we call these properties and relationships attributes). For each possible world w, n-place attribute A, and possible objects a., ..., an' it is determinately either true or false that A(a 1, .. · , an) holds in w. For each world w there is also the collection E, of all the objects that exist in that world. Presumably, every possible object exists in at least one possible world. Also, certain objects like pure sets exist in all possible worlds. A reasonable existence principle for sets is that a set exists in a world w if and only if all its elements exist in that world. Given that sets are objects and that the pure sets exist in all possible worlds, it follows that the collection E, of all individuals that exist in a world w is alwa ys a proper class rather than a set. Consider now the first-order modal language 1." with identity, and with [MJ as its only modal operator. In terms of the metaphysical picture described above, we can describe the intended interpretation of L and define the notion: "truth in a world w relative to an assignment g", for formulas of L. The intended interpretation consists of the following ingredients: (i) the class W of all metaphysically possible worlds; (ii) the class D of all possible objects; (iii) for each w E: W, a class E, ~ D of objects existing in the world w; (iv) the n-ary predicate symbols of L designate n-ary attributes and the individual

124

STEN LINDSTROM

constants of £ designate possible objects; (v) the attributes have extensions relative to possible worlds (the extension of an n-ary attribute relative to a world w is a (possibly proper) class of n-tuples of possible objects). A sentence q> of the interpreted language £ is true if it is true at the actual world (w. r. t. the intended interpretation) relative to every assignment. Now, it may very well be that the collections of possible worlds , possible individuals, and individuals existing in particular worlds do not form sets. In that case, the intended interpretation of £ does not correspond to a Kripke model structure. We cannot, then, directly conclude from a sentence q> being true to it being true in some model structure. As long as we are considering the language of first-order modal logic, however, we can still make this inference in an indirect way." By the completeness theorem for Kripke 's system of quantified S5, there exists a formal system QS5= which is (strongly) complete with respect to the set of all (QS5=)-structuresY Hence, for any sentence q> of

c

q> is true => q> is (QS5=)-eonsistent (by the intuitive soundness of QS5= with respect to the intended interpretation) => q> is (QS5=)satisfiable (by the completeness theorem for QS5=).

It follows that q> is (QS5=)-valid => q> is true.

=>

""q> is not (QS5=)-satisfiable

=>

""q> is not true

Now, consider the language £ with the intended interpretation given above. We have a notion of truth for £, but what could it mean for a sentence of L to be logically true? The notion of the intended interpretation for £ suggests a notion of an interpretation for L An interpretation for £ is just like a Kripke model 9J1 = <S, I> based on a model structure S = <W, D, R, E, wo> except that the collections W, D, R, E, for wE W, are not required to be sets but are allowed to be proper classes. (We are of course only considering interpretations where R = D x D.) Hence, the intended interpretation becomes one of the interpretations. Let us call a sentence q> of £ supervalid if it is true in every interpretation of £ and valid if it is true in every (QS5=)-model (that is, if it is (QS5=)-valid).23 We now make the following conjecture: The logical truths of the language £ of metaphysical necessity are precisely those sentences of I: that are supervalid. Adapting an argument due to Kreisel (1969) , we can prove that supervalidity coincides with validity for the language L The argument goes as follows :

125

KANGER'S EARLY SEMANTICS FOR MODAL LOGIC

Since Kripke (QS5=)-models are interpretations, we have: (1)

if

The completeness theorem for the system (QS5=) yields: (2)

if

However, the system (QS5=) is intuitively sound with respect to supervalidity. That is the axioms are easily seen to be supervalid and the only rule of inference, modus ponens, preserves supervalidity. Hence : (3)

if

(2) together with (3) yield: (4)

if

Hence, the notions of validity and supervalidity are coextensional for the language L. From this together with the conjecture, we conclude that (QS5=) is the first-order logic of metaphysical necessity."

7.2. Logical Necessity An (interpreted) sentence

:3x\...:3x n(x 1 f. x2 1\.../\

X\

f. x, /\ X 2 f.

X 3 /\ ... /\

x2 f. xn l\. ../\ xn_\ f. x.),

Each of these sentences is presumably metaphysically necessary. So for each positive n, the following is a truth of metaphysics: ([HIn)

[HI :3x\ ...:3x n(x\ f.

/\.../\ xn_\ f. xu),

X 2 /\ . .. /\

x, f. x, /\

X2

f. X 3 /\ ... /\

X2

f.

Xu /\,. ./\

126

STEN LINDSTROM

It is, of course, not a logical truth. We do not have for any n ~ ""(QSS=>

1,25

[Hj::Jxl···::JXn(X1 -:/:. x 2 1\.· ·A x, -:/:. x, A X2 -:/:. x 3 1\.··A X2 -:/:. x, I\. ..A I\. ..A Xn _1 -:/:. x.).

Nor do we have for any n, ""(Qss=)

~:3xl ...::JXn(Xl -:/:. x 2 A...A x, -:/:. x, A x 2 -:/:. x 3 A...A x 2 -:/:. x, I\. ..A A...A xn • l -:/:. x.),

In sharp contrast to this, Kanger's semantics for logical necessity validates every instance of "" ~ ::Jx( ... :3xn(x I

-:/:.

X2 A...A XI

-:/:. x n A x 2 -:/:. x 3

A...A x 2 -:/:. x, A...A x n_l -:/:. x.).

This is as I think it should be. It is a logical truth that it is logically possible that there are at least n objects . When comparing Kanger's semantics for modal logic with Kripke's we come to the conclusion that the former (at least in its class-domain version) is adequate for the notion of logical necessity, while the latter adequately captures a form of metaphysical necessity. Neither semantics can handle adequately the notion that is captured by the other. To devise a semantics that can treat both notions is a challenge that still remains to be met. As we have seen, Kanger's model-theoretic semantics for quantified modal logic differs in many respects from modem possible worlds semantics. However, it raises sufficiently many questions both of a technical and of a philosophical kind to motivate an interest that is not merely historical. Umea University

NOTES

* The present paper is a very slightly revised version of Lindstrom (1998) . A shorter version was published as Lindstrom (1996). Various versions have been presented at seminars in Umea, Uppsala, Boston. Pittsburgh and Los Angeles. I am especially indebted to Joseph Almog, Lennart Aqvist, Nuel Belnap, Thorild Dahlquist , Kit Fine, Bengt Hansson , Risto Hilpinen , Jaakko Hintikka, David Kaplan, Paul Needham, Peter Pagin, Wlodek Rabinowicz, Krister Segerberg and Rysiek Sliwinski for their very helpful comments and advice. A fellowship at the Swedish Colleg ium for Advanced Studies in the Social Sciences (SCASSS) during the Autumn of 1996 provided an excellent research environment for working on the paper. I At the 9th International Congress of Logic, Methodology and Philosophy of Science . See Fellesdal (1994 ). 2 Actually Camap's state descriptions are sets of literals (i.e., either atomic sentences or negated atomic sentences) that contain for each atomic sentence either it or its negation. However, for our purposes we may identify a state descript ion with the set of atomic sentences that it

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contains. Also, in order to make things simple, I am not discussing here Camap's treatment of identity statements. 3 On p. 39 in (1957a) Kanger makes an explicit reference to Jonsson and Tarski (1951) . Montague (1960) writes : "The present paper was delivered before the Annual Spring Conference in Philosophy at the University of California, Los Angeles, in May, 1955. It contains no results of any great technical interest; I therefore did not initially plan to publish it. But some closely analogous, though not identical, ideas have recently been announced by Kanger [(1957b)], [(1957c)] and by Kripke in [(1959)]. In view of this fact, together with the possibility of stimulating further research, it now seems not wholly inappropriate to publish my early contribution." 5 We are not going to consider languages that contain function symbols . Here we ignore the possibility of D not being a set but a proper class and I not assigning sets but proper classes as extensions to the predicate symbols of L. If this were the case, then the intended interpretation of L would not be a model in the formal sense of model theory. Of course, there are interpreted first-order languages whose intended interpretations are not models in the formal sense, e.g., the first-order language of set theory, with the proper class V of all sets as its domain and the proper class [ex, y>: x is a member of y) as the interpretation of ' E' . This opens up the possibility for a sentence q> of an interpreted formal language of being true although it may be false in all models in the sense of model theory. This possibility is precluded for firstorder languages, by the Lowenheim- Skolem theorem : the truth of lp implies its consistency (by the intuitive soundness of first-order logic), which in tum, by the Lowenheim-Skolem theorem, implies lp having a (countable) model. But for formal languages that are able to express such notions as 'there is a proper class of x's such that lp' the implication : (1) if lp is true, then lp is true in some model (which is built up from sets). fails. Let, for example, lp be the sentence 'there is a proper class of x's such that x = x' . This example is due to McGee (1992) . 7 Cf. Kripke (1959,1963) and Hintikka (1957a, 1957h, 1961). For the standard -non-standard distinction , see also Cocchiarella (1975). Cf. Montague (1974) and the papers reprinted therein . 10 See, for example, Hintikka (1969). II We leave it open, for the time being, exactly what is meant by a (semantic) interpretation. 12 Kanger (1970) , p. 49. 13 Kanger (1957b) , p. 4. Cf. also Kanger (1970), p. 50. 14 Kanger uses the notation T(D , v, lp) = I instead of our '" lp[g] and he speaks of the operation T which, for every domain D, every primary valuation v and every sentence cp, assigns one of the truth-values 0 or I to lpas the secondary valuation for L. 15 Actually he uses valuations here, but since the assignment of values to the variables is immaterial in this context it is more natural to work with interpretations. 16 Cf. Kanger (I 957b), p. 4. 17 Kanger (1957b) uses the terminology "standard usage of f)' instead of "intended interpretation of L". 18 Here our terminology differs slightly from Kanger's . Our ontological operators correspond to what Kanger calls purely ontological operators. Cf. Kanger (1957a) p. 34. 19 In his definitions Kanger speaks of "classes" rather than "sets", but this terminological difference is inessential, since he does not make any distinctions within the category of all classes but rather treats all classes that he speaks of as genuine objects that can be members of other classes. 20 I have taken the quantifier (:Jab'inrX) from McGee (1992) , where he uses it to show that there are interpreted formal languages for which the equivalence:

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(M)

REFERENCES Barcan (Marcus) , R., 1946a, "A functional calculus of first order based on strict implication", The Journal ofSymbolic Logic 11, 1-16. Barcan (Marcus), R., 1946b, "The deduction theorem in a functional calculus of first order based on strict implication", The Journal ofSymbolic Logic 11, 115-118. Barcan (Marcus) , R., 1947, "The identity of individuals in a strict functional calculus of second order", The Journal of Symbolic Logic 12,12-15. Boolos , G., 1985, "Nominalist platonism" , The Philosophical Review, XCIV, No.3, 327 -344. Carnap, R., 1946, "Modalities and quantification", The Journal ofSymbolic Logic 11, 33-64. Carnap, R., 1947, Meaning and Necessity: A Study in Semantics and Modal Logic , University of Chicago Press, Chicago. Second edition with supplements, 1956. Cocchiarella, N., 1975, "On the primary and secondary semantics oflogical necessity", Journal of Philosophical Logic 4, 13-27. Etchemendy, J., 1990, The Concept of Logical Consequence, Harvard University Press , Cambridge, Massachusetts. Fellesdal, D., 1994, "Stig Kanger in memoriam " in D. Prawitz and Westerstahl, D. (eds.), Logic, Methodology and Philosophy of Science IX, pp. 885 -888. Elsevier, Amsterdam. Garson , J. W., 1984, "Quantification in Modal Logic", in D. Gabbay and Guenthner, F. (eds.), Handbook of Philosophical Logic, Vol II, Ch. 5, pp. 249 - 307, D. Reidel , Dordrecht.

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HintikkaL, 1957a, "Quantifiers in deontic logic" , Societas Scientiarum Fennica, Commentationes humanarium litterarum, vol. 23, no. 4. Hintikka , J., 1957b, "Modality as referential multiplicity", Ajatus 20,49-64. Hintikka, 1.,1961 , "Modality and quantification", Theoria 27,110-128. Hintikka , J., 1969, "Semantics for propositional attitudes", in J. W. Davies et al. (eds.), Philosophical Logic, pp. 21-45, D. Reidel, Dordrecht. Reprinted in J. Hintikka , Models for Modalities, D. Reidel, 1969; and in L. Linsky (ed.), Reference and Modality, Oxford University Press, London , 1971. Hintikka.J, 1980, "Standard vs. nonstandard logic : higher-order, modal, and first-order logics" , in E. Agazzi (ed.), Modern Logic - A Survey, pp. 283 -296. D. Reidel, Dordrecht. Hintikka, J. 1989, "Is alethic modal logic possible?" , in J. Hintikka & M. B. Hintikka: The Logic of Epistemology and the Epistemology of Logic . Kluwer, Dordrecht. Hughes , G. E. and Cresswell, M. J., 1996, A New Introdu ction to Modal Logic, Routledge, London and New York. Jonsson, B. and Tarski, A., 1951, "Boolean algebras with operators" , American Journal of Mathematics , vol. 73 (1951) , 891-939, vol. 74 (1952) ,127 -162. Kanger, S., 1957a, Provability in Logic , Acta Universitatis Stockholmiensis, Stockholm Studies in Philosophy 1, Almqvist & Wiksell, Stockhol m. Kanger, S., 1957b, "The morning star paradox" , Theoria 23, 1-11 . Kanger, S., 1957c, "A note on quantification and modalities", Theoria 23, 133-134. Kanger, S., 1957d, "On the characterization of modalities", Theoria 23, 152-155. Kanger, S., 1970, New Foundations for Ethical Theory , in Hilpinen (ed.), Deontic Log ic: Introductory and Systematic Read ings, Reidel, pp. 36 -58. Earlier mimeographed version : New Foundations for Ethical Theory, Part 1, Stockholm 1957. Kanger, S., 1972, "Law and logic", Theoria 38, 105-132. Kaplan, D., 1986, "Opacity", in Hahn and Schilpp (eds.), The Philosophy ofW. V. Quine, The Library of Living Philosophers, Volume XVIII, Open Court , La Salle, Illinois. Kleene, S. C., 1967, Mathematical Logi c, John Wiley & Sons, inc., New York. Kreisel, G., 1969, "Informal rigour and completeness proofs", in 1. Lakatos (ed.), Problems in the philosophy of mathematics, North-Holland, Amsterdam . Kripke, S., 1959 "A completeness theorem in modal logic", The Journal ofSymbolic Logi c 24 , 1-14. Kripke, S., 1963a, "Semantical considerations on modal logic", Acta Philo sophica Fennica, fasc. 16, Helsinki, pp. 83 -94. Kripke, S., 1963b, "Semantical analysis of modal logic 1, normal propositional calculi ", Zeitschrift fur math ematische Logic und Grundlagen der Mathematik 9 (1963) , 67 - 96. Kripke, S., 1965, "Semantical analysis of modal logic II, non-normal propositional calculi", in 1. W. Addison , L. Henkin , A. Tarski (eds.), The Theory of Model s, pp. 206 -220, NorthHolland, Amsterdam . Kripke, S., 1980, Naming and Necessity , Basil Blackwell, Oxford. Lewis, D., 1985, On the Plurality of Worlds, Basil Blackwell, Oxford. Lindstrom, S., 1996, "Modality without worlds : Kanger's early semantics for modal logic", in Odds and Ends, Philosophical Essays Dedicated to Wlodek Rabinowicz on the Occasion of His Fiftieth Birthday, Uppsala Philosophical Studies, 45, Department of Philosophy, Uppsala University. Lindstrom, S., 1998, "An Exposition and Development of Kanger's Early Semantics for Modal Logic", in P. W. Humphr eys and J. H. Fetzer (eds.), The New Theory ofReference - Kripke, Marcus, and Its Origins. Kluwer, 1998.

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McGee, V., 1992, "Two problems with Tarski's theory of consequence", Proceedings of the Aristotelian Society , new series, vol. 92, pp. 273-292. Montague, R., 1960, "Logical necessity, physical necessity , ethics and quantifiers", Inquiry 4, 259-269. Reprinted in Montague (1974) . Montague, R., 1974, Formal Philosophy: Selected Papers of Richard Montague. Edited and with an Introduction by Richmond H. Thomason . Yale University Press, New Haven. Quine, W. V, 1947, "The problem of interpreting modal logic", The Journal ofSymbolic Logic 12,43 -48. Tarski , A., 1936, "Uber den Begriff der logischen Folgerung", Actes du Congres International de Philosophie Scientijique, vol. 7, Paris, pp. 1- II (English translation : "On the concept of logical consequence", pp. 409-420 in Logic, Semantics, Metamathematics, second edition, Hackett Indianapolis, 1983).

RISTO HILPINEN

STIG KANGER ON DEONTIC LOGIC

The year 1957 saw the publication of two ground-breaking contributions to deontic logic, Stig Kanger's 'New Foundations for Ethical Theory' (Kanger 1957171)1 and Jaakko Hintikka's 'Quantifiers in Deontic Logic' (Hintikka 1957). 2 Superficially the papers look very different, but they contain many significant similarities: 1. Both papers contain a semantic analysis of the basic normative concepts the obligatory, the permitted, and the prohibited (or the concepts of ought, may, and may not). The semantical theories proposed by Kanger and Hintikka are closely related to, or variants of, the possible worlds semantics for modal concepts. 2. Kanger and Hintikka discu ss the appl icability of the concepts of truth and consequence to normative discourse, that is, to imperatives, norms, and normative statements. 3. Both authors analyze the role of quantifiers in deontic logic and the interplay of quantifiers and deontic operators. 4. The concepts and results of deontic logic are applied to conceptual problems in ethics and the philosophy of law. 5. In these papers, Kanger and Hintikka discuss the importance of the representation of actions and agency in deontic logic and , more generally, the role of actions and action descriptions in normative discourse. In the present paper I shall briefly discuss Kanger's contributions to these topics. II

It is interesting to note that even though Kanger and Hintikka were among the principal architects of the possible worlds semantics of modal logic , neither used the expression 'possible world ' in their semantical discussions. Hintikka used the concept of model set (downward saturated set), nowadays often called a Hintikka set, as a representation of a possible world, whereas Kanger (1957/ 1971, 44-45) represented possible worlds by the concept of a range of 131

G. Holmstrom -Hintikka, S. Linstrom and R. Slivinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Yol.Tl , 131-1 50. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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individuals (a domain of individuals), coupled with what he called a primary valuation, that is, an interpretation which assigns denotations to non-logical constants (predicates and individual constants) and values to individual variables for each range (non-empty set) of individuals. In (1972) Kanger replaced the expression 'range' by the more common expression 'domain of individuals' . Kanger called a pair consisting of a domain and a primary valuation a system, and analyzed the truth-eonditions of modal sentences (including deontic sentences) in tenus of systems. Thus systems are the possible worlds of Kanger's semantics. The fact that Kanger and Hintikka did not use the expression 'possible world ' is of minor importance. Their most important semantical insight was the use of an alternativeness relation between model sets or domains (or systems) for the explication of modal concepts. Different modal concepts can be represented by different alternativeness relations, and by giving the alternativeness relations different structural properties one can analyze the logical differences between different modalities. For the concept of ought, Kanger's analysis looks as follows: Let Vbe a domain of individuals, let Vbe a primary valuation (an assignment of an interpretation to each non-logical constant and variable in each domain of individuals), and let 0 be a modal operator for the concept of Ought.' An ought-sentence Op is true relative to V and V if and only if p is true (relative to V) at every deontic alternative to V (Kanger 1957/ 1971, 45 ; 1972, 114):

(T.O)

T(Op,V, V) = 1 if and only if T(p,V ~ V) = 1 for each V I such that Ro(V ~U).

'Ro(V I,U)' means that VI is a deontic alternative to U. This is Kanger's relativized version of Leibniz's old idea that the concept of obligatoriness (or ought) can be regarded as species of necessity, and defined in tenus of truth in possible worlds . If a primary valuation and a domain are thought of as corresponding to a "possible world", possible worlds are denoted by u, v, W, oo, and the truth of a statement p at a possible world u is expressed by 'u~ p', Kanger's condition (T.O) assumes the familiar form

(T.O*) u~ Op if and only if w~ p for all w such that Ro(w,u). Kanger accepts the semantic principle that R o is a serial relation, in other words , that every domain (every possible world) has some deontic alternative: (RD)

For every u, there is some w such that Ro(w,u).

According to (T.O*) and (RD), the propositional part of Kanger's deontic logic corresponds to what is often called the "standard system of deontic

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logic", or the modal system KD (Fellesdal and Hilpinen 1971, 13-15, Chellas 1980, 131).

III The application of the concepts of logical consequence and truth to prescriptive discourse, for example, to norms and imperatives, has been widely discussed among philosophers since the 1930's and 1940's. The concept of logical consequence is usually defined in terms of the concept of truth, but commands and other prescriptions cannot literally be said to be true or false. How should the concept of logical consequence be defined for normative sentences?' Some philosophers have tried to solve this problem by regarding the satisfaction of an imperative (direc tive) by an action as a semantic counterpart of the truth of a declarative statement. This is a plausible idea, because both concepts characterize the relationship between an utterance (an assertion or a command) and the world; their main difference lies in the "direction of fit", to use John Searle's terminology: in the case of an assertion, the direction of fit is "from the words to the world" (a satisfactory assertion should fit the world), but in the case of commands the direction of fit is from the world to the words, that is, a "correct" action ought to fit the command (Searle 1975, 346 -347). However, as Kanger pointed out, this approach to the semantics of normative discourse is unsatisfactory: it fails to do justice to the modal (or intensional) character of normative discourse. Let us assume that the satisfaction of ought-norms is defined as follows : (Ol.sat)

OA is satisfied if A is performed, otherwise OA is dissatisfied,

and let the consequence relation among norms be defined by (Nl.cons) Nz follows from (is a consequence of) N 1 if and only if the satisfaction of N, follows from the satisfaction of N 1• The proposition that a norm is satisfied is an ordinary (indicative) statement, thus the logical relationships among statements of this kind can be understood in the standard way . To make this definition applicable to permissive as well or mandatory norms, the definition of satisfaction must be extended to permissive sentences. It is clear that the "satisfaction" of a permissive norm cannot be defined in term s of the performance, or omission, of the permitted action, because this would undermine the distinction between mandatory and permissive norms. In fact , it does not seem reasonable to apply the concept of satisfaction to permissive norms at all: a person can "avail herself' or "make

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use" of a permissive norm, but a permission cannot be "dissatisfied" or violated, and the satisfaction semantics is therefore insufficient for normativeprescriptive discourse in general - at least in the simple form considered above. In (1957/1971) Kanger pointed out the counter-intuitive consequences of G. H. von Wright's early (1955) attempt to define the consequence relation in terms of the concept of norm-satisfaction.5 The proposal considered by Kanger was slightly more complicated than that discussed above, but the difficulty is the same. According to the proposal, the satisfaction of ought-norms is defined by (01.sat), and the satisfaction of may-norms by (P2.sat) PA is satisfied if A is performed; PA cannot be dissatisfied, but it is not satisfied if A is not performed, where 'P ' represents the concept of permission. According to this definition, the concept of dissatisfaction is not the negation of satisfaction: permissive norms cannot be dissatisfied, but they can fail to be satisfied. Kanger (1957/ 1971, 49) considers the followings definition of the concept of consequence: (N2.cons) N2 follows from N, and a statement p if and only if (i) the satisfaction of N 2 follows from the satisfaction of N) and the truth of p, and (ii) the dissatisfaction of N) or the falsity of p follows from the dissatisfaction of N2 , provided that N2 can be dissatisfied. This definition is unsatisfactory in the same way as the simpler definition considered above. If an action A which is permitted is performed, then, according to (0 Lsat), OA is satisfied. On the other hand, if OA is dissatisfied, the proposition that A is performed is false; thus, as Kanger points out, according to (N2.cons), OA follows from PA and the statement that A is performed. Again, the proposed definition cannot account for the distinction between mandatory and permissive norms. The fundamental problem with the satisfaction semantics of norms is that it does not do justice to the modal (or intensional) character of normative discourse, a feature which was observed by medieval logicians as well as G. W. Leibniz and his contemporaries. The satisfaction semantics is an attempt to characterize normative discourse in terms of a single action-world, and this leads to the loss of the modal distinctions between different kinds of norms and normative propositions.

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IV Kanger's formal language for deontic logic contains an imperative operator and modal operators for deontic and alethic modalities . He divides the sentences of the language (formulas without free variables) into imperative sentences and statements: the former contain occurrences of the imperative operator; all other sentences are called statements. Thus the statements of the language include deontic statements, e.g., ought-sentences; imperative sentences and deontic statements form the class of deontic sentences. A deontic statement together with an interpretation (in Kanger's terminology, a primary valuation) forms a deontic proposition, and an interpretation of an imperative sentence produces an imperative (Kanger 1957/1971, 48). In Kanger's semantics, imperatives and deontic ought-propositions correspond to each other: even though imperatives are not called true or false, but, for example, correct or incorrect , the correctness of an imperative !A relative to a system is equivalent to the truth of the corresponding deontic sentence OA relative to . (In this context we may assume that the relevant concept of ought is an "imperative ought'") Thus imperatives are synonymous with the corresponding deontic statements. Deontic statements are statements, and express propositions; thus the logical relationships among deontic statements can be explained in the usual way in terms of the concept of truth. The difference between imperative sentences and deontic statements, and thus between deontic propositions and imperatives , is a merely linguistic difference: it is clear that the same command can be expressed by using the imperative mood, e.g., "John, open the door!", or by a deontic statement, e.g., "John must open the door" . Some philosophers have suggested that the problem of defining the concept oflogical consequence in deontic logic can be solved by adopting a descriptive interpretation of deontic sentences: according to this interpretation, deontic sentences (represented by the formulas of deontic logic) describe what is regarded as obligatory, permitted , or prohibited, etc., in some unspecified norm system, and the principles of deontic logic are then regarded as conditions of consistency or rationality for norm systems." The descriptive use of deontic sentences may also be termed their assertoric use: according to the descriptive interpretation, deontic sentences are assertions about a norm system. However, the applicability of the concept of truth to deontic sentences does not require a descriptive interpretation in the sense that deontic statements should be regarded as representations of some independently or antecedently existing "normative reality". The truth of deontic sentences can also be understood performatively: we may assume that in the genuinely

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prescriptive (i.e., nonnative) use of deontic sentences, the utterance of a deontic statement in the right circumstances, or an appropriate communication of a nonn, makes it true . Kanger was on the right track when he observed (1957/1971, p. 55): The argument that deontic propositions of the form do not state anything about reality ... involves nothing that might refute the valuation clause or the idea that deontic propositions are truth or false.

Hans Kamp (1979, p. 263) has observed that the perfonnative use of deontic sentences should be regarded as their primary use: the truth-conditions of deontic assertions should reflect (or correspond to) the pragmatic effects of their perforrnative use. It is not always clear whether a deontic sentence is used perfonnatively or assertorically - but both kinds of utterances can direct an agent's actions in the same way. For example, a performative utterance of a permission sentence makes certain options available (permitted) to an addressee (an agent); the corresponding assertion states that the same options are already open to the agent. Thus a true assertoric utterance and a perfonnative , i.e., genuinely nonnative, utterance of a permission sentence have the same practical consequences for the agent (Kamp 1979,264).

v Kanger's treatment of quantifiers in modal logic has been discussed in detail by Sten Lindstrom (2000), and I shall make here only a few observations on Kanger's analysis of the interplay between quantifiers and the deontic operators. Kanger (1957/1971, 47) observes that the formula (:3x)PAx, where Ax is a non-modal formula, is intuitively stronger than P(:3x)Ax; thus, according to Kanger, a correct semantics for deontic modalities ought to validate the principle (1)

(:3x)PAx ~ P(:3x)Ax,

but not its converse (2)

P(:3x)Ax ~ (:3x)PAx.

(2) is equivalent to the deontic version of the Barcan formula (3)

(\fx)OAx ~ O(\fx)Ax,

and (1) can be written as the converse Barcan formula (4)

O(\fx)Ax ~ (\fx)OAx.

Jaakko Hintikka (1957 , 21) has given to (2) a counter-example in which an act

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of kind A (for example, a demonstration) may be performed on a certain occasion only on the basis of a special permit obtained for that occasion: even though it is permitted to perform act s of kind A, there is no particular occasion x on which A would be permitted if the required permission has not been applied for and obtained. (In this example, the domains of quantification consist of occasions or individual acts.) In the same way, even though it may be obligatory for every actual indi vidual to pay traffic fines (if everyone has been caught for violating the traffic regulations), it need not be obligatory, i.e., true in all "deontically perfect" worlds, that everyone pay traffic fines . On the other hand, as Kanger observes, (1) and (4) seem intuitively valid. Kanger's semantical rules for quantifiers are cho sen in such a way that they fit these

intuitions.' Kanger (1957/1971,53) also regards (5)

(3x)OAx::> O ( 3x)Ax

as a valid deontic principle. Th e converse of (5), (6)

O (3x)Ax ::> (3x)OAx

is obviously not v alid. In fact, the non -validity of (6) is one of the few uncontroversial features of quantified deontic logic and other areas of quantified modal logic," Even though someone ought to jump into the river to save a drowning ch ild (in the sense that it ought to be the case that someone jumps), no particular person need have that duty . But it has been argued that (5) is not valid either. Jaakko Hintikka (1971, 102) has given the following counter-example to (5) : Let 'Rxy' mean that y is a puni shment for x, where x and yare individual act s or events. The following formula has the form (5): (7)

(~x)O( ~y)R.xy ::> O ( ~x) ( ~y)R.xy .

The antecedent of (7) says that there is an act which ought to be followed by a punishment, that is, that a puni shable act has been performed; the consequent says that there ought to exist a punishable act , i.e., that there ought to be an act which ought to be followed by a punishment. It is clear, according to Hintikka, that the latter does not foll ow from the former; thus (5) cannot be regarded as a valid principle. Kanger' s treatment of quantification in deontic logic was in man y respects inconclusive and unsatisfactory, and he did not discu ss it at any length , but by making one of the first attempts to develop a form al semantics for first-order deontic logic he opened a door for the discu ssion of the top ic. The interpre tation of quantifie rs in modal logic, especi ally in deontic logic , continues to be a problem-ridden area. "

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VI Different assumptions about the properties of the altemativeness relation used in the truth-definitions of modal sentences (for example, deontic statements) lead to different modal and deontic logics. Kanger also thought that the altemativeness relation is helpful for expressing various philosophical assumptions about the nature and justification of norms even when such assumptions do not affect the logic of normative concepts. For example, he observes that the condition

can be regarded as an expression of "moral relativism". The deontic alternatives to a given world (domain of individuals) can be regarded as "normatively perfect worlds". According to (8), what is normatively perfect relative to one domain need not be so relative to another - perhaps this can be regarded as a form of moral or normative relativism. Kanger also makes the following assumptions about the Ro-relation: (9)

(VU) (3UJRo(U ',U)

and (10)

(3 U)...,Ro( U,U).

(9) is the seriality assumption (RD) mentioned earlier; it guarantees the truth of the principle of deontic consistency, (D)

OA:J PA,

(10) expresses the assumption that Ro is non-reflexive, which means that OA does not entail A; (10) distinguishes the concept of ought (i.e., normative necessity) from alethic necessity. Kanger observes that the denial of principle (8) means that the concept of Ought can be defined by ( 11)

OA

=df.

N(Q :J A),

where N represents the concept of analytic necessity, and Q is a propositional constant; it may be thought of as stating "what morality prescribes" (Kanger 1957/1971,53). According to Kanger (1957171) , schema (11) was first put forward in his University of Stockholm thesis for the Lie. Phil. degree in 1950. 10 The idea that normative concepts can be defined in this way by means of alethic modalities goes back to G. W. Leibniz, who suggested that the obligatory (debitum) can be defined as "that which is necessary for a good man to do", and the permitted (licitum) is "what is possible for a good man to do"

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(Leibniz 167111930, 465). According to (11), deontic modalities can be defined in terms of a1ethic modalities and a "normative constant" Q; an equivalent proposal was made by Alan Ross Anderson (1956), who suggested that deontic logic can be reduced to alethic modal logic by means of the schema (12)

OA

"'df.

N(A

::l

S),

where S can be taken to mean that the requirements of morality have been violated or as the threat of a sanction associated to such violations. In (1957/ 1971, 54) Kanger observed that he was inclined to reject definition (11), "because some deontic propositions seem to be synthetic". But if N is not regarded as an expression for analytic or universal necessity, but as a "contingent" necessity so that N(Q ::l A) can be true at some possible worlds but not at others, this philosophical objection to (11) does not hold. VII

Kanger seemed to be interested in deontic logic mainly for the sake of its possible applications to ethical theory, social philosophy, and the philosophy of law. In the applications studied by Kanger, the concept of action and the logic of action play an important role and are intimately connected to deontic logic . Kanger represented the concept of action by a modal operator for agency, ' Do(a,p )' , where a is an agent and p is a state of affairs or an event, the result or the outcome of the action. 'Do(a ,p)' is read "a sees to it that p", In some early publications, for example, in the original 1957 version of (1957/ 1971), in Kanger (1963) and in Kanger and Kanger (1966), Kanger used the locution "a causes p" but he adopted the expression "a sees to it that p" in (1957/1971) and in later publications (1972 , 1985). The Do -operator makes it possible to distinguish the following "modes of action" with respect to a result (state of affairs or condition) p:11 (13) (i) (ii) (iii) (iv)

Do(a,p) : .,Do(a,p): Do(a,"p): .,Do(a,"p) :

a sees to it thatp, a does not see to it that p, a sees to it that »p, and a does not see to it that "p.

The combination of different modes of action with deontic concepts makes it possible to represent several types of obligation and permission and different legal or deontic relations between individuals. Consider a state of affairs involving two parties, F(a,b). According to Kanger (1957/1971, 42; Kanger and Kanger 1966, 86- 89), the Do-operator can be combined with deontic

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operators to distinguish four basic types of right, corresponding to different senses of the expression 'right' :

(R l ) (R2) (R3) (R4)

ODo(b,F(a,b» : a has the claim in relation to b that F(a,b), -,ODo(a,-,F(a,b» :; P-,Do(a ,-,F(a,b»: a has the freedom (or liberty or privilege) in relation to b that F(a,b) , -,O-,Do(a,F(a,b» :; PDo(a ,F(a,b»: a has the power in relation to b that F(a,b), O-,Do(b,-,F(a,b»: a has the immunity in relation to b that F(a,b).

(R1)-(R4) define relational concepts of right. The replacement of the state of affairs F(a,b) by -,F(a,b) yields four additional concepts of right which Kanger and Kanger (1966, 86 -87) call counter-claim (R l ' ), counter-freedom (R2'), counter-power (R3'), and counter-immunity (R4 '). Kanger and Kanger call the relations (R1)-(R4) and (Rl ')-(R4') simple types of right. The normative relationship between any two individuals a and b with respect to a state of affairs p can be characterized completely by means of the conjunctions of the eight simple types of right or their negations. There are 256 such distinct conjunctions, but according to the standard principles for 0 and certain plausible assumptions about the logic of the Do-operator," only 26 combinations of the simple types of right or their negations are logically consistent; Kanger and Kanger call these 26 relations the atomic types ofright (1966,9394). It is perhaps misleading to call these 26 relations "types of right ", because they include as their constituents negations of rights (e.g., duties and "disabilities") as well as rights (e.g., freedoms and powers): the 26 atomic types give a complete characterization of the possible legal relationships between two persons with respect to a single state of affairs . Lars Lindahl (1994, 894-895) has suggested that Kanger's theory of normative relations represents "an improvement in the theory of duties", but suffers from a number of difficulties as a theory of rights (Lindahl 1994, 896- 909). Kanger's concepts (Rl- R4) correspond to W. N. Hohfeld's (1919) account of the four ways using the word 'right' (or four concepts of a right) , and he adopted the expressions 'privilege', 'power' and 'immunity' from Hohfeld. Hohfeld (1919, 35-36) called the counterpart of Kanger's "claim" simply "right". Kanger apparently intended (R1)- (R4) as approximate explications of Hohfeld's notions. However, Kanger' s concepts of power and immunity differ from Hohfeld's concepts ." According to Kanger, the concepts of power and freedom are closely related. Both are permissions: a power consists in the permissibility of actively seeing to it that something is the case, whereas freedom means that there is no obligation to see to it that the opposite state of affairs should be the case. Lindahl (1977, 51) and many others have argued

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that Hohfeld's concept of power should be analyzed as a legal ability rather than a permission (i.e., a can rather than may). (See Lindahl 1994,898-899, Bulygin 1992, Makinson 1986,408-409.) However, it is clear that Kanger's way of combining deontic operators with operators for agency provided useful conceptual tools for the analysis of legal relations . It enabled him and Helle Kanger to develop a rich theory of normative relations which found interesting applications in the study of governmental position structures, different forms of parliamentarism, and human rights (Kanger and Kanger 1966, Helle Kanger 1984), and formed a good basis for further research in this area (Lindahl 1977, 1994).

VIII In his 1957 paper (1957/1971) Kanger did not say much about the interpretation of the Do-operator, but in (1972) he presented an interesting analysis of the concept of seeing to it that p: A statement of the form Do(a,p) is regarded as a conjunction (CDO) Do(a,p)

'=

Dn(a,p) & Ds(a,p),

where 'Ds' may be said to represent the sufficient condition aspect of agency and 'Dn' the necessary condition aspect of agency. Kanger used the expression 'D6' for the sufficient condition aspect of agency (Ds), and 'Do' for the necessary condition aspect of agency (Dn); he read 'Ds(a,p)' as p is necessary for something a does,

and 'Dn(a,p)' as p is sufficient for something a does.

These readings are equivalent to

(14)

Ds(a,p): Something a does is sufficient for p,

and

(15)

Dn(a,p): Something a does is necessary for p.

Kanger interpreted Ds and Dn in terms of two alternativeness relations on possible worlds : (CDS)

u ~ Ds(a,p) if and only if w~ p for every w such that Sos(w,u),

and (CDN) u ~ Dn(a,p) if and only if w~"'p for every w such that SON(W,U).

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To simplify this presentation, I am using here the standard possible worlds notation, not Kanger' s notation of primary valuations and domains. The words W such that SDS(W,U) can be regarded as worlds in which the agent a performs the same actions as in u. Kanger takes 'SDN(W,V)' to mean that "the opposite" of everything a does in u is the case in W (Kanger 1972, 121). It is not quite clear what this means. One possible interpretation is that a does not do any of the things she does in u, but for example is completely passive (insofar as this is possible) or, for any action B that a performs at u, she does something else (i.e., some action alternative to B) at w. Kanger's analysis of the concept of agency has a form which has become widely accepted in the recent work on the logic of agency. The first condition of agency, the Ds-condition, may be termed the positive condition, and the second condition, the On-condition, may be termed the negative condition of agency." The negative condition may also be termed the counterfactual condition of agency: it states that if the agent had not acted the way he did, p would not have been the case . G. H. von Wright presented an analysis of this kind in the his work Norm and Action (1963), but his logic of action and agency was otherwise quite different from Kanger's. (For discussions of von Wright's theory, see Segerberg 1992, 351- 359, and Hilpinen 1997a, 5-10; 1997b, 84-91.) Other versions of the analysis of agency by means of a positive and a negative condition have been presented by Ingmar Porn (1974, 1977), who was directly influenced by Kanger's work, Lennart Aqvist (1974) , Aqvist and Mullock (1989), and more recently by Nuel Belnap and his associates (Belnap 1991, Belnap and Perloff 1990,1992, Horty and Belnap 1995, Perloff 1991; for a review of these proposals, see Hilpinen 1997a, 1997b). Philosophers have disagreed about the formulation of the negative condition. Ingmar Porn (1974, 1977) has argued that we should accept instead of Kanger's Oncondition only a weaker negative requirement, viz. '--,Dn(a,-.p)' , abbreviated here 'C n(a,p )' : (ACN) u pCn(a,p) if and only if W P--'p for some w such that SDN(W,U). This condition can be read: but for a's action it might not have been the case that p (Porn 1974, 96; 1977, 7). This means that it is not unavoidable for a that p. Lennart Aqvist (1974, 86) has accepted a similar weak form of the counterfactual condition. According to Porn and Aqvist, the negative condition should be formulated as a might-statement or a might-conditional, not as a would-eonditional. Porn has defended the weak negative cond ition on the ground that if the concept of agency is defined in terms of (COO), then the following conjunction is logically inconsistent:

STIG KANGER ON DEONTIC LOGIC

(16)

143

Do(P) & Do(p :::> q);

according to Kanger's conditions (CDS) and (CDN), Do(p :::> q) entails ...,Do(p). (Porn gives credit for this observation to Andrew J. I. Jones ; cf. Porn 1977,7.) In other words, 'a sees to it that p :::> q' is inconsistent with'a sees to it that p' . According to Porn, there are many action situations which can be adequately described by means of consistent conjunctions of the form (16). However, the usual sine qua non test of causal dependence as a condition of agency is usually expressed by a would-conditional rather than by a might-conditional (Hart and Honore 1959, 104-108), that is, by means of a strong negative condition. There are strong presystematic grounds for favoring a wouldconditional.

IX As I mentioned above, Kanger's characterization of the negative condition suggests that it refers to situations in which the agent is passive" or in which the agent performs none of the actions she performs in the actual situation. It is not clear how the second characterization could be satisfied, and it is easy to see that the interpretation of the negative condition as passivity does not always give the right results . Let us consider a situation u in which a person - call her Elsie - is in a room which is too warm; the door is closed and there is a cat in the room. Assume Elsie wants to cool the room . She can do this by opening the door; however, if she opens the door, the cat will run out unless she restrains the cat. Elsie wants the cat to remain inside . Let us adopt the following abbreviations: r =The room is cooled. s =The cat remains inside. D = (the action of) opening the door, C = (the action of) restraining the cat.

Elsie wishes to see to it that the room is cooled and that the cat remains inside. She performs an action - opening the door (D) - which guarantees or necessitates the result r and anothe r action, restraining the cat (C), which ensures the result s. Thus we can say that Elsie sees to it that the room is cooled and sees to it that the cat remains inside. It is also natural to say that by restraining the cat while opening the door , Elsie sees to it that if the door is opened, the cat will remain inside. Thus the following sentences seem true in this situation:

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(17) (i) Do(e, r), (ii) Do(e, s), and (iii) Do(e, If r . then s). However, according to Kanger's version of the negative condition, (17.ii) is not true, because in the situations in which "the opposite of everything" Elsie does at u is the case, the cat will remain inside: in such situations Elsie neither opens the door nor restrains the cat. In this respect Porn's weak negative condition does not work much better, because we may assume that the cat remains inside in all situations in which Elsie performs neither D nor C, that is, "does not do any of the things she does in a given situation"; Porn (1977, 5). It is nevertheless clear that C is in circumstances u a sine qua non condition and in that sense a necessary condition of the result s: the following counterfactu al is true: (18)

If Elsie had not restrained the cat, he would not have remained inside.

(18) cannot be understood as a strict conditional ; it is a variably strict conditional: it should be taken to mean that the cat would not remain inside but would escape in the situations in which Elsie does not restrain him, but which are otherwise maximally similar to the actual situation; in such situations Elsie opens the door and cools the room just as in the actual situation . The negative condit ion of agency should be formulated as a counterfactual conditional like the following where the corresponding concept of agency is expressed by 'Do*' : (19)

up Do*(a,p) only if a performs at usome action B such that if a had not done B, p would not have been the case .

In fact, (19) corresponds reasonably well to Kanger's intuitive reading of the Do-operator. 16 This version of the negative condition agrees with the ordinary sine qua non interpretation of agent causation. As was stated above, the conditional used in (19) should be regarded as a variably strict conditional whose truth-conditions can be analyzed for example in accordance with the possible worlds semantics of counterfactuals developed by David Lewis (1973 ). It must be recognized, of course , that the semantical theories of counterfactuals were only being developed when Kanger was working on the semantics of his Do-operator. What about the conditional 'if r, then s' in the example given above - can we say that Elsie brings it about that this conditional is true? Let us assume that it can be construed as a material conditional ' -y V s' , Does Elsie do something which is necessary for the truth ofthis disjunction? Given that Elsie

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restrains the cat, can we also say that she performs the disjunctive action of not opening the door or restraining the cat ? If this is regarded as an action performed by Elsie, then there is an action (performed by Elsie) which is necessary for or V s, because the following counterfactual is true: (20)

If Elsie had not performed the action »D V C, it would not have been the case that or V s.

Insofar as (20) makes sense and can be regarded as true, the Jones-Porn objection to the modified (counterfactual) formulation of the strong negative condition of agency fails. It seems possible that the performance of a certain action in a given situation is necessary for p (e.g. , for that there is a fire) , and the performance of another action in the same situation is necessary for 0p V q (e.g., for that if there is a fire, a fire brigade will arrive), but the same action cannot be the conditio sine qua non of both states of affairs. There is no doubt that in some contexts the concept of agency (or agent causation) requires a strong negative (counterfactual) condition. However, in the analysis of the concept of seeing to it that p such a condition is questionable. For example, Lars Lindahl (1977 , 70) has observed that the expression 'x sees to it that p' can characterize merely an intention or preparedness to act in order to sustain the state of affairs p; in such circumstances the counterfactual condition as formulated above, need not hold .'? Brian Chellas (1992,515) has argued that the negative condition "does not form a proper part of the meaning of sees to it that" ; according to Chellas, assertions of agency using the expression "sees to it that" may carry an implication of "seeing to it really", but this does not justify making "a negative stipulation intrinsic to the meaning of this idiom" (ibid.). In this respect 'x sees to it that p' differs from 'x brings it about that p' or the expression used by Kanger in his early writings, 'x causes p '; the latter two usually indicate a causal dependence of the result p on the agent's actions, and thus their meaning should include a strong negative (i.e ., counterfactual) condition. The concept of seeing to it that p suggests a strong positi ve (or "necessitating") condition, whereas the concept of bringing it about involves a strong negative condition but only a weak positive condition, since one can bring things about (e.g., cause events) by accident or through coincidences." It may be observed that the applications of de ontic logic and the logic of action in which Kanger was interested, for example, the analysis of different concepts of right, are not sensitive to slight shifts in the interpretation of the Do-operator. In the logic of action, as in many other areas of philosophical logic, Kanger's pioneering work gave rise to interesting conceptual questions and

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stimulated a great deal of new research in the field. 19 His way of combining the logic of action with deontic logic extended the applicability of deontic logic far beyond the traditional questions about the logic and interpretation of mandatory and permissive norms. University ofMiami (Coral Gables)

NOTES The paper appeared first as a privately distributed booklet which circulated widely among the philosophers interested in deontic logic . A revised version was published in 1971 as Kanger (1957171). 2 Part of the content of Hintikka (1957) is included in Hintikka (1971). Kanger uses 'Rightp ' as the abbreviation of '-.Ought-.p'; I shall use here the customary notation of 0 for the concept of 'ought' (obligation) and P for the concept of ' may' (permissibility). The latter concept is one of the four senses of 'right' distinguished by Kanger (195711971, 42) . 4 Jergen Jergensen (1937-1938, 290) formulated this problem as follows : "According to a generally accepted definition of logical inference only sentences which are capable of being true or false can function as premisses or conclusions in an inference; nevertheless it seems evident that a conclusion in the imperative mood may be drawn from two premisses one or both of which are in the imperative mood ." Alf Ross (1941, 55) has called this difficulty "Jergensen's dilemma". 5 von Wright has analyzed the semantics of norm sentences by means of the concept of 'satisfaction' (or satisfiability) also in his more recent publications; in von Wright (1983, 130) he calls deontic logic "a logic of norm-satisfaction". 6 The distinction between these two uses of deontic sentences has sometimes been formulated as the distinction between norms (norm-formulations) and normat ive propositions (normative statements); see G. H. von Wright 1963, 93-106, and E. Bulygin 1982, 128-130. 1 In (1957) Kanger distinguishes two ways of interpreting quantifiers in modal contexts, and expresses the corresponding two universal quantifiers by '(x)' and '(Ux)'; the quantifier rules adopted in Kanger (1957/1971) are those of the (Ux)-quantifier and its existential counterpart. Sten Lindstrom (2000) has shown that these quantifiers have some very odd and unusual properties. 8 The intuitive non-validity of (V'x)PAx :::> P(V'x)Ax, a formula equivalent to (6), is equally obvious. Unlike many problematic operator switch principles of modal logic, these formulas fail to be valid even if the domains of individuals under consideration are assumed to remain the same across possible worlds . 9 See, for example, Makinson (1981) , for the interpretational problems that arise in this area. For a discussion of the problems in Kanger's interpretation of quantifiers in modal logic, see Lindstrom (2000) . 10 Unfortunately I have been unable to find a copy of this work. II According to Krister Segerberg (1992, 348-350), this way of analyzing the concept of 'action' may go back to St. Anselm (1033-1109), who distinguished the constructionsfacere esse (to do p),facere non esse (to do not-p), nonfacere esse (not to do p), and nonfacere non esse (not to do not-p) .

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12 These principles include the principle that if p and q are logically equivalent, so are Do(a ,p) and Do(a,q), and the principle that Do(a ,p) entails p; see Kanger and Kanger (1966 ,89). 13 Kanger and Kanger (1966, 101-102) observe some differences between their concepts of 'claim' and 'immunity' and those of Hohfeld' s system. 14 Cf. Aqvist and Mullock (1989, 37, 93); Belnap (1991 , 792). 15 G. H. von Wright (1963 , 28 - 36) understands the negative condition as referring to situations in which the agent is passive, i.e., docs not "interfere with the course of nature". For discussions of von Wright's theory, see Segerberg (1992, 351 - 354), Hilpinen (l997a, 5- 8). 16 In view of (19), it is clear that a Kanger-type semantics of action sentences does not give an analysis of the concept of ' action' , but rather makes it possible to distinguish several "modes of action". 17 However, some more complex form of the counte rfactual condition may apply to such cases . 18 Cf. Hilpinen (l997a, 18). For modes of agency with a weak positive condition, see Hilpinen (1997a, 11-12, 18). 19 For some recent developments and applications of Kanger's (or Kanger-type) logic of action in deontic logic and computer science , see Jones and Sergot (1993, 292 - 301) , Santos and Carmo (1996), and Sergot (1999) .

REFERENCES Anderson, Alan Ross, 1956, The Formal Analysis of Normative Systems (Technical Report No. 2, Contract No. SARlNonr-609 (16), Office of Naval Research, Group Psychology Branch), New Haven . Reprinted in N. Rescher (ed.), The Logic of Decision and Action, University of Pittsburgh Press, Pittsburgh, 1967, pp. 147-213. Aqvist, Lennart, 1974, "A New Approach to the Logical Theory of Actions and Causality", in S. Stenlund (ed.), Logical Theory and Semantic Analysis, D. Reidel, Dordrecht, pp. 73 -91. Aqvist, Lennart, and Philip Mullock , 1989, Causing Harm, Walter de Gruyter, Berlin -New York. Belnap, Nuel, 1991, "Backwards and Forwards in the Modal Logic of Agency", Philosophy and Phenomenolog ical Research 51, 777 - 807. Belnap, Nuel, and Michael Perloff, 1990, "Seeing to It That: A Canonical Form for Agentives", in H. Kyburg et al. (eds.), Knowledge Representation and Defeasible Reasoning , Kluwer Academic Publishers, Dordrecht- Boston, pp. 167- 190. Belnap, Nuel, and Michael Perloff, 1992, "The Way of the Agent", Studia Logica 51, 463 -484. Bulygin, Eugenio, 1982, " Norms, Normative Propositions, and Legal Statements", in G. Fleistad (ed.), Contemporary Philosophy. A New Survey . Vol. 3: Philosophy of Action, Martinus Nijhoff, The Hague, pp. 127-152. Bulygin, Eugenio , 1992, "On Norms of Competence", Law and Philosophy 11, 201 - 216 . Chellas, Brian F., 1980, Modal Logic : An Introdu ction, Cambridge University Press, Cambridge . Chellas, Brian F., 1992, "Time and Modality in the Logic of Agency" , Studia Logica 51, 485 517 . Fellesdal. Dagfinn, and Risto Hilpinen, 1971, "Deontic Logic: An Introduction", in R. Hilpinen (ed.) , Deontic Logic: Introductory and Systematic Readings, D. Reidel , Dordrecht, pp. 1- 35. Hart, Herbert, and Anthony Honore , 1959, Causation in the Law, Clarendon Press, Oxford . Hilpinen, Risto, 1997a, "On Action and Agency ", in Sten Lindstrom and Eva Ejerhed (eds.), Log ic, Action and Cognition, Kluwer Academic Publishers, Dordrecht and Boston, pp. 3- 27.

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Hilpinen, Risto , 1997b, "On States , Actions , Omissions and Norms ", in G. Holmstrom-Hintikka and R. Tuomela (eds.), Contemporary Action Theory, Vol. 1, Kluwer Academic Publishers, Dordrecht and Boston, pp. 83- 107. Hintikka, Jaakko, 1957, "Quantifiers in Deontic Logic", Societas Scientiarum Fennica, Commentationes Humanarum Litterarum 23:4, Helsinki. Hintikka, Jaakko, 1971, "Some Main Problems of Deontic Logic" , in R. Hilpinen (ed.), Deontic Logic: Introductory and Systematic Readings, D. Reidel , Dordrecht, pp. 59 -104. Hohfeld , Wesley Newcomb , 1919, Fundamental Legal Conceptions as Applied in Judicial Reasoning, ed . by W. W. Cook , Yale University Press, New Haven . Horty, John , and Nuel Belnap, 1995, "The Deliberative Slit : A Study of Action, Omission, Ability, and Obligation", The Journal of Philosophical Logic 24,583-644. Jones, Andrew J. I., and Marek Sergot, 1993, "On the Characterization of Law and Computer Systems : The Normative Systems Perspective", in 1. J. Ch. Meyer and R. 1. Wieringa (eds.), Deontic Logic in Computer Science: Normative System Specification, John Wiley & Sons, Chichester - New York, pp. 275-307. Jergensen, Jergen, 1937-1938, "Imperatives and Logic", Erkenntnis 7, 288 -296. Kamp, Hans , 1979, "Semantic versus Pragmatics", in F. Gunther and S. J. Schmidt (eds.), Formal Semantics and Pragmatics for Natural Languages, D. Reidel, Dordrecht, pp. 255 - 287 . Kanger, Helle, 1964, Human Rights in the U'N. Declaration . Acta Universitatis Upsaliensis , Uppsala University, Uppsala. Kanger , Stig, 1957, "A Note on Quantification and Modalities", Theoria 23, 133-134. Kanger, Stig , 1957/1971, "New Foundations for Ethical Theory ", in R. Hilpinen (ed.), Deontic Logic: Introductory and Systematic Readings . D. Reidel, Dordrecht, pp. 36- 58. Originally published as a privately distributed booklet New Foundations for Ethical Theory, Part I. Stockholm 1957. Kanger, Stig , 1963, "Rattighetsbegreppet", in Sjufilosofiska studier tilldgnade Anders Wedberg den 30 mars 1963 . Philosophical Studies published by the Department of Philosophy, University of Stockholm, No.9. Stockholm. Kanger, Stig , 1972, "Law and Logic", Theoria 38, 105-132. Kanger , Stig, 1985, "On Realization of Human Rights ", in G. Holm strom and A. J. I. Jones (eds.), Action, Logic and Social Theory. Acta Philosophica Fennica 38, Societas Philosophica Fennica, Helsinki . Kanger , Stig, and Helle Kanger, 1966 , "Rights and Parliamentarism", Theoria 32 , 85-115 . Reprinted (with changes) in R. E. Olson and A. Paul (eds.), Contemporary Philosophy in Scandinavia, The Johns Hopkins Press, Baltimore and London, pp. 213 -236. Leibniz, Gottfried Wilhelm, 167111930, "Elementa iuris naturalis ", in G. W. Leibniz, Siimtliche Schriften und Briefe. Sechste Reihe: Philosophische Schriften, Bd, 1, Otto Reichl Verlag, Darmstadt, pp. 431-485. Lindahl, Lars, 1977, Position and Change, D. Reidel, Dordrecht and Boston . Lindahl, Lars, 1994, "Stig Kangcr's Theory of Rights" , in D. Prawitz et al. (eds .), Logic, Methodology and Philosophy of Science IX, Elsevier Science B. V., Amsterdam, pp. 889 -911. Lindstrom, Sten, 2000 , "An Exposition and Development of Kanger 's Early Semantics for Modal Logic" , in G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds .), Collected Papers of Stig Kanger with Essays on His Life and Work, Vol. II, Kluwer Academic Publi shers, Dordrecht and Boston. Makinson, David , 1981, "Quantificational Reefs in Deontic Waters ", in R. Hilpinen (ed.), New Studies in Deontic Logic, D. Reidel, Dordrecht, pp. 87 -91.

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Makinson , David, 1986, "On the Formal Representation of Rights Relations. Remarks on the Work of Stig Kanger and Lars Lindahl ", Journal of Philosophical Logic IS , 403- 425. Perloff, Mich ael, 1991, "Stit and the Language of Agency", Synthese 86, 379- 408. Porn .Ingmar, 1974, "Some Basic Concepts of Action ", in Soren Stenlund (ed.), Logical Theory and Semalllic Analysis.D.Reidel. Dordrecht, pp. 93-101. Porn, Ingrnar, 1977, Action Theory and Socia l Science, D. Reidel, Dord recht. Ross, Alf, 1941, "Imperati ves and Logic", Theoria 7, 53-7 1. Santos, Filipe, and l ose Carrno, 1996, "Indirect Action , Influence and Respon sibility", in M. A. Brown and 1. Carmo (eds.), Deontic Logic, Agency and Normative Systems, Sprin ger Verl ag, New York and Berlin , pp. 194- 215. Searle,lohn, 1975, "A Taxonomy of Illocutionary Acts", in K. Gunderson (ed.), Languag e, Mind and Knowledge. Minnesota Stud ies in the Philosophy of Science 7, Univer sity of Minnesota Press, Minne apolis, pp. 344- 369. Segerberg, Kriste r, 1992, "Getting Started: Beginnings in the Logic of Action", Studia Logica: An Intemational Journal for Symbolic Logic 51:3-4 (Special issue: Logic of Action), 347378. Sergot, Marek, 1999, "Normative Positions", in P. McNamara and H. Prakken (eds.), Norms, Logic and Information Systems, lOS Press, Amsterdam and B erlin , pp. 289- 308. von Wright , Geor g I-Ienrik, 1955, "a m s.k. praktiska slutledningar", Tidsskriftf or Rettsvitenskap 68,465-495. von Wright, Georg Henrik, 1963, Norm and Action, Rout ledge & Kegan Paul, London. von Wright, Georg Henrik, 1983, "Norms, Truth and Logic", in G. H. von Wright, Practical Reason. Philosoph ical Papers, I. Cornell University Press, Ithaca, pp. 130- 209.

LARS LINDAHL

STIG KANGER'S THEORY OF RIGRTS'

1. INTRODUCTION Stig Kanger regarded his theory of rights as one of his substantial contributions to philosophy; he worked on it, intermittently, for nearly thirty years. A starting-point was Kanger's interest in the classification of "fundamental jural relations" proposed by the American jurist W. N. Hohfeld, in the second decade of this century. Hohfeld's theory concerns an area which is mainly legal, and it belongs to the tradition of juri sts such as Jeremy Bentham and John Austin. Hohfeld distinguished the relations right, privilege, power, immunity, and their "correlatives" duty, no-right, liability, disability; one of Hohfeld's tenets was that each of these relations is a relation between two parties with regard to an action by one of them. 1 In his little book New Foundations for Ethical Theory, from 1957, Kanger presented his first explication of Hohfeld, He suggested that standard deontic logic, with only a deontic operator applied to sentences, is not adequate for expressing the Hohfeldian distinctions. The improvement he proposed was to combine a standard deontic operator with an action operator and to exploit the possibilities of external and internal negation of sentences where these operators are combined. In Kanger's 1963 paper "The Concept of a Right", his explication of Hohfeld was restated as a system of so-called simple types of rights . In this paper, however, the simple types are the basis of a theory of atomic types of rights, which is more of a genuine typology. In the explication of atomic types, the combinatory method of "maxi-conjunctions" is used for providing an elegant logical typology of normative relations . During the last two decades of his life, Kanger was interested in the application of his theory of rights in connection with human rights and social justice; in particular, he turned to the problem of what, in the U.N . Declaration on Ruman Rights , is meant by having a right. In this connection, Kanger became aware of the distinction between a person's having a right and this right 's being realized for the person. And so, in his last paper on rights, from 1985, Kanger dealt with the notion of realization of rights . lSI

G. Holmstrbm-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. II. / 5/-/7/. © 2001 Kluwer Academi c Publishers. Printed in the Netherlands. Originally published in D. Prawitz, B. Skyrms and D. Westerstahl (eds.) Logic. Methodology and Philosophy of Science IX Elsevier Science B.Y., 1994, pp. 889-9 11. Reprinted here with so me minor changes

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The first part of my paper contains a brief presentation of Kanger's typologies. After this, there follows a discussion of problematic points. The final part offers some suggestions for a positive solution to the most central problems. Kanger's ideas about realization make use of a much enlarged logical framework, the treatment of which would lead too far in the present essay. His basic theory of rights, however, is independent of these ideas.' II. STIG KANGER'S THEORY OF RIGHTS: A PRESENTATION

1. The Language used by Kanger The sentences on rights that Kanger tries to explicate are taken from juristic usage or plain ordinary language. Moreover, Kanger's explications are not stated within a strictly formal language but only semi-formally. Only two kinds of entities are explicitly referred to, namely agents, on the one hand, and states ofaffairs or conditions, on the other. Agents are either persons, like Mr. Smith, or so-called collective agents, such as the Swedish Government or Smith & Co, Ltd. As illustrations of the second group of entities we have, for instance, the state of affairs (or condition) that Mr. Smith gets back the money lent by him to Mr. Black, or that Mr. Smith walks outside Mr. Black's shop . In Kanger's view, negation, conjunction, disjunction etc. can be applied to states of affairs (or conditions) in the same way as they are applied to sentences, and the notion of logical consequence is applicable to them by analogy as well.' In order to state his explications in a general way, Kanger introduces letters for referring to agents or states of affairs that are chosen arbitrarily. He assumes that x.y.z,... are agents and that F,G,H,... are states of affairs. Moreover, Ftx, y), G(x, y), ... are assumed to be states of affairs "involving" (as Kanger says) agents x and y. To the Boolean connectives of negation, conjunction etc., Kanger adds the modal expressions "Shall" and "Do". Shall F is to be read "It shall be that P' and Do(x, F) should be read "r sees to it that P'. In his explication of rights, Kanger exploits the possibilities of combining the deontic operator Shall with the action operator Do. One example is Shall Do(x, F) which means that it shall be that x sees to it that F; another is ...,Shall Do(y, ""F) which means that it is not the case that it shall be that y sees to it that not F.5 The logical postulates for Shall and Do assumed by Kanger are as follows (where - is a relation of logical consequence, satisfying some reasonable postulates"):

STIG KANGER 'S THEORY OF RIGHTS

1. 2. 3. 4. 5.

If F - G, then Shall F - Shall G. (Shall F & Shall G) - Shall(F & G). Shall F - -Shall ..,F. If F - G and G - F, then Do(x, F) Do(x, F) - F.

153

Do(x, G).

2. The Simple Types of Rights?

In Kanger's theory, there are several types of rights . A type of rights is always a relation between two agents with respect to a state of affairs or a condition. For instance, ifMr. Smith has lent 100 dollars to Mr. Black , and, therefore, has a right to get back the money lent, then, according to Kanger, Smith has a right of the simple type Claim against Black with regard to the state of affairs (or condition) that Smith gets back the money he has lent to Black. In this example, Claim is the type , Smith is the bearer, Black is the counter-party, and the state of affairs (or condition) that Smith gets back the money lent is (what may be called) the "object-matter". In the history of the analysis of rights, there is a traditional distinction between, on the one hand, "passive rights", or rights to have something done, and on the other hand, "active rights" or rights to do something. In Kanger's theory of simple types of rights , the first group , henceforth called Oirights, is explicated by "counter-party obligatives", while the second, called P-rights, are explicated by "bearer perrnissives", In the first group, we have four simple types , explicated as follows: Explicandum: O-right

Expli cans: Counter-party obligative

Claim(x, y, F(x, y)) Counter-claim(x, y, F(x, y )) Immunity(x, y, F(x, y )) Counter-immunity(x, y, F(x, y ))

ShallDo(y, F(x, y)) ; ShaIlDo(y, ..,F(x, y)) Shall .., Do(y, -n». y)) Shall .., Do(y , F(x, y))

For example, if Mr. Smith has an immunity against Mr. Black with regard to the condition that Mr. Smith walks outside Mr. Black's shop, this is explicated by: It shall be that Mr. Black does not see to it that Mr. Smith does not walk outside Mr. Black's shop. Each explicans satisfies the scheme, Shall ± Do(y, ± F(x, y)) , where ± stands for the two alternatives of affirmation or negation. The four bearer permissive types are explicated in this way:

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Explicans: Bearer permissive

Explicandum: P-right Power(x, y, F(x , y» Counter-power(x, y, F(x, y)) Freedom(x, y, F(x, y)) Counter-freedom(x, y , F(x, y»

'.., Shall .., Do(x, Fix , y» .., Shall > Do(x, ..,F(x, y» .., Shall Do(x, y» .., Shall Do(x, F(x, y»

-n».

Here, each explicans satisfies the scheme, .., Shall ± Do(x, ± F(x , y)). As an example, consider Mr. Smith's counter-freedom versus the police with regard to the condition that the police are informed about Mr. Smith's private life. In Kanger's explication, this would amount to: It is not the case that it shall be that Mr. Smith sees to it that the police are informed about Mr. Smith 's private life. Between the types of O-rights and the types of P-rights there exists a correspondence f such that if T is a type of O-right and T ' is a type of P-right, then T ' =f(T) in case for any x, y, F it holds that x has a right of type T versus y with regard to F(x, y) if and only if y has not a right of type T ' versus x with regard to F(x, y) . For example, Claim is the counter-part of counter-freedom, in the sense , that x has a claim versus y with regard to F(x, y ) if and only if y has not a counter-freedom versus x with regard to Fix , y ). According to the logical postulates , for some types it holds that membership of one type implies membership of another. For example, since Do(y, Fix, y» , y», Do(x, ..,F(x, y)), are inconsistent, Shall Do(y, F(x, y», Shall Do(x, are inconsistent as well; therefore, according to Kanger's explication, Claim(x, y, F(x, y», not Freedom(x, y, F(x, y)), are inconsistent.

-n».

3. The Atomic Types of Rights

The construction of atomic types is as follows . We begin with the list, Claim(x, y, F(x, y» , Counter-claim(x, y, F(x, y» , Imrnunity(x, y, F(x , y» , Counter-imrnunity(x, y, Ftx, y» , Power(x, y, F(x, y» , Counter-power(x, y, F(x, y» , Freedom(x, y, F(x , y», Counter-freedom(x, y , Fix, y)'),

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Starting from this list, we form every new list that can be obtained by negating either 0, 1,2,. 00 ' up to all 8 members of the list, while keeping the other members unnegated. Obviously, the number of all such lists will be 2 8 , i.e ., 256 . (Each choice of negated members of the list corresponds to the choice of a subset of the original list ; since the list has eight members, the number of its subsets is 28 . ) Of the 256 lists , however, all but 26 are inconsistent according to the logic of Shall and Do. Each of the remaining 26 lists, when regarded as a conjunction of its members, specifies an atomic type of rights. As an example, we consider atomic type No .5.

Name "Power, immunity, counter-power, counter-immunity". Explicans {<x, y, F> I..., Shall ..., Do(x, F(x , y» & Shall ..., Do(y, -Ftx, y» & ..., Shall ..., Do(x, ...,F(x, y» & Shall ..., Do(y, F(x, y» }.

We see that each conjunct in the explicans satisfies the scheme, (*)

± Shall ± Do G, ± F(x, y» ,

where ± and ; represent choices, as befo re. As suggested by David Makinson," we can say that each atomic type is explicated by a "maxi-conjunction", i.e., a maximal and consistent conjunction such that each conjunct satisfies scheme (*) . Maximality means that if we add any further conjunct, satisfying (*), then this new conjunct either is inconsistent with the original conjunction or redundant. Given the underlying logic , the atomic types are mutually disjoint and their union is exhaustive. Not all of Kanger's types of atomic rights are types of rights in any rea sonable sense. Consider Kangers atomic typ e No. 23. According to Kanger, x has a right of atomic type No. 23 versus y with regard to F(x, y) if the following is true: Not freedom(x, y, F(x, y» , Not immunity(x, y, F(x , y», Not counter-claim(x, y, F(x, y». (Type 23 is specified by the list we obtain if all the lines of the original list of bearer permissives and counter-party obligatives are negated, and redundant members of the list have been dropped.) Since all members of the list are negated, x's relationship versus y with regard to F(x, y) is one of not having a right of any kind, rather than one of having a right of a certain type. To say, in this case, that x has a right of a particular kind is like saying that poverty is a

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particular kind of opulence. Kanger 's atomic typology , therefore, is a typology of normative relations from the "rights perspective" rather than a typology of rights. III. SOME ASPECTS OF KANGER'S THEORY In this section I will argue that Kanger 's typology represents an improvement in the theory of duties ; as a theory of rights, it suffers from a number of difficulties. 1. Kanger's Theory as a Theory of Duties Kanger's typologies are primarily typologies of duties and non-duties; x's 0rights versus yare explicated in terms of y' s duties, i.e., in terms of counterparty obligati ves; correspondingly, x' s P-rights versus yare explicated in terms of x' s non-duties, i.e., in terms of bearer permissives. Thus, the counter-party obligative Shall Do(y, Ftx, y» is an explication of " y has a duty to the effect that Do(y, F(x, yj)", Correspondingly, the bearer permissive .,Shall Do(x, F(x, y» is an explication of "x has no duty to the effect that Do(x, F(x, y»". Other types of duty/non-duty are explicated if a negation sign is inserted before Do, before F(x, y) or before both. It follows that the atomic types are intersections of different types of duty/ non-duty for two agents with regard to one and the same state of affairs . If conceived of as typologies of duties/non-duties, Kanger 's typologies represent a considerable improvement on earlier representations. In deontic logic, statements of duties are sometimes reproduced with the help of deontic operators carrying an index, like 0 ;, OJ'' '' where i.j are parameters or variables for agents; an expression of the form OF is read "F is obligatory for i".9 Compared with this construction, Kanger's combinations of Shall and Do have greater expressive power ; for example, instead of staying with "not-F is obligatory for x", as expressed by Ox -.F, a distinction can be made between the cases Shall -.Do(x, F), Shall Do(x, -.F) . The idea of combining a non-relativized deontic operator with an agentrelative action operator has another advantage as well (though this was not exploited by Kanger himself) . This advantage consists in the possibility of iterating operators in a meaningful way. It is controversial whether iterations of the kind OOF, O.,OF etc., make sense; in any case it is not clear what is meant by statements of this form. 10 If we combine Shall and Do, however, new possibilities of iterations are opened. For example, in an organization, the boss is the superior of the clerk who is the superior of the errand-boy; it may well

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be the case that the boss is permitted to impose a duty on the errand-boy to work over-time, while the clerk is not permitted to impose such a duty on him. This distinction can be expressed by the two sentences .., Shall .., Do(x, Shall Do(z, F»); Shall .., Do(y, Shall Do(z, F»); where x is the boss, y is the clerk and z is the errand-boy. II In the sentences just illustrated there is an instance of the Do operator between the two instances of the deontic operator. 2. Problems for Kanger's Theory of Rights

There are well-known problems connected with Kanger's theory conceived of as a theory of rights. (i) IDENTlACATlON OF BEARER AND COUNTER-P ARTY . As remarked by 1.S. Mill , the notion of a claim-right is connected with the idea that particular actions or omissions constitute cases of injustice committed against an assignable person (the bearer of the right); the injustice may consist in "depriving a person of a possession, or in breaking faith with him, or treating him worse than he deserves, or worse than other people who have no greater claims". The assumption that an injustice is committed, in turn, implies that the bearer of the right is wronged: "in each case the supposition implies two things - a wrong done, and some assignable person who is wronged"." In accordance with this suggestion, a criterion of appropriateness for the explication of a claim-right is as follows :

(1)

x has a claim-right versus y to the effect that F(x, y)

only if it is true that, (2)

If F(x , y) is not the case, then x is wronged ,

(or x has a legitimate complaint). There are many interpretations of x, y, F such that Kanger's explicans for (1), i.e., (3)

Shall Do(y, F(x , y)),

holds, while (2) is false . The policeman has a duty to seize the murderer, who tries to get away. If we set x = the murderer, y = the policeman, and F(x, y) for "x is seized by y" , (3) is true. On the other hand, (2) is false in this case ; the murderer is not wronged, and has no legitimate complaint, if the policeman does not succeed to seize him. The murderer has no right to the effect that he be seized .

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Assume, on the other hand, that Creditor has lent 100 dollars to Debtor, and that, as a consequence, Debtor has a duty to pay this amount back. If we set x = Creditor, y = Debtor, and F(x, y) for "x receives 100 dollars from y", the same counter-obligative formula (3) is true for this interpretation of the variables as well. In this case, however, (2) is true, and Creditor has a right to get his money back. Kanger's explicative formula (3) does not suffice to distinguish the two cases. 13 One might try to defend Kanger's theory by going to the theory of atomic types of rights. But this does not help much since the same atomic type, viz. No 6 (claim, power, counter-freedom) seems to be appropriate in both of the two examples illustrated. As applied to x versus y with regard to F(x, y) , type No 6 is explicated as follows : Shall Do(y, F(x, y )) , ..., Shall ..., Do(x, F(x, y )), ..., Shall Do(x, F(x, y )). The three sentences are true in the murderer case as well as in the Creditor case . (Observe that the third formula is true for the murderer, since he has no duty to see to it that he is seized by the particular policeman in view.) The problem just illustrated for Claim-rights is that the explicandum is not entailed by the explicans. This problem can be shown to exist as well for the other types of a-rights, i.e., counter-claim, immunity, counter-immunity. If this objection is correct for a-rights, there will be a problem for P-rights as well. This time , however, the problem is that the explicans is not entailed by the explicandum. Let us remember that, in Kanger 's construction, if T is a type of P-right, there is a type T* of O-right such that T(x, y , F(x, y ) ) if and only if not T*(y , x, F(x, y)). Furthermore, the types are constructed in such a way that q> is the explicans of Ttx, y, F(x, y)) if and only if'-xp is the explicans of T*(y, x, F(x , y)). By contraposition, therefore, if q> does not entail T*(y, x, F(x, y)) , then Ttx, y, Fix, y )) does not entail ""q> . Let us illustrate the technical argument with an example. Suppose that y has a house in a suburban area. We may plausibly assume : y has no right that x does not walk around in the garden of y' s neighbor (x's walking in that garden is no concern of y ' s). In Kanger's language, this means that (1)

not Counter-immunity(y, x, F(x, y))

where Ftx, y) expresses that x walks in the garden of y's neighbor. (1) is equivalent to (2)

Power(x, y , F(x, y )).

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159

However, from (1) and (2) it ought not to follow, as in Kanger's theory , (3)

.., Shall .., Do (x , F(x , y)),

i.e., it should not follow that it is permitted that x walks in the garden of y ' s neighbor. For example, we may well suppose that x is a mortal enemy of y' s neighbor, and that this neighbor has expressly forbidden x to walk in his garden ; if so, the negation of (3) is true. RI GHTS OF RECIPIENCE WmIOUT A COUNTER- PARTY. There are statements about "rights to receive", which do not imply statements about duties and which are not tractable in terms of Kanger' s typologie s. An example is as follows:

(ii)

(1)

Children have a right to be nurtured.

If x is a child, nothin g follows from (1) about who has a duty to nurture it. Rather , it has been suggested, the acceptance of ( I) is a first and basic point of departure from which further considerations can be made concerning duties for others (parents , guardians , authorities and so on)." Indeed, from (1) it does not even follow that for each child there is some y such that y has a duty to nurture it; i.e., if x is a child it does not follow that (2)

C:Jy)(Shall Do(y, F(x , y))

where Ftx, y) means that x is nurtured by y. It may be suggested that (1) entail s that if x is a child , then, (3)

Shall (::Jy) [D o(y , F(x , y ))].

(2), however, does not satisfy the Kanger scheme for counter-party obligatives since a quantifier is embedded between Shall and Do. Since the quantifier is located after Shall, not before it, (2) does not say that anyone has a duty; rather (2) prescribes that there be someone who nurture s x. (iii) LE GAL POWER. It is often maintained that so-called legal power is a type of right not tractable in terms of duties or non-duties. Suppo se that F is a legal condition; F(x , y ) signifies, for example, that the ownership of the Glenro y estate is transferred from x to y . Then (it is argued), the statement (4)

x has the legal power to see to it that F(x , y) ,

cannot be analyzed as (5)

.., Shall .., Do(x, F(x, y)) ,

which is Kanger's general explication scheme for the simple type of right

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called "power": (5) expresses permission, while (4), it is usually held, expresses a capacitive dimension. IS On this point, I think that Kanger's analysis can be defended. Admittedly, as "legal power" is usually understood, (4) and (5) are not synonymous, and Kanger's use of the term "power" is misleading. What Kanger wants to assert, however, is rather that (5) is an explication of a general notion of a right-to-do (what in German would be called Befugnis), i.e., of (6)

x has a right to see to it that F(x, y).

(Apparently, Kanger did not find a suitable word in English corresponding to Befugnis.) Admittedly, in some circumstances, a thief is able to transfer the ownership of stolen goods to a purchaser who is in good faith (the sale will be legally valid). But, obviously, the thief has no right to do this . Perhaps (4) is true for this interpretation of F, x , y, but since (5) is false , (6) is false as well. In one sense of "legal power", the thief has the legal power to sell the stolen goods. But if so, "legal power" is not a type of right. (iv) RELEVANCEOF CLAIM -HOLDER'SWILL. Suppose that Mr. Smith has a claim versus

the community to receive medical care. If x = Mr. Smith , y = the community, and F(x, y) is the condition that x receives medical care from y , then (1)

Claim(x, y, F(x, y))

is explicated by (2)

Shall Do(y, F(x, y)).

According to (2), the laws are disobeyed if y does not see to it that x receives medical care, even if this is due to x's refusing to receive it. However, all duties can be fulfilled even if x does not receive medical care, namely in the case that he does not want to have it. However, we might say that the "object-matter" of Smith's claim, expressed by F(x, y), should appropriately be constructed in a different way, namely as the condition that medical care is made available to him by the community. The latter is another way of saying that Smith receives medical care, if he wants to have it. Of course, the expression F(x, y) does not make it explicit that a conditional is involved, and it will be a problem how such a conditional should be expressed within the simple language presupposed by Kanger. However, this is a difficulty about expressing the "object-matter" of rights rather than an objection to the typology of rights itself. A possible way out, in the specific example, is to replace F(x , y ) in (2) by the material equivalence G(x, y) ~ Hix, y ), i.e., to substitute (2) by

STIG KANGER'S THEORY OF RIGHTS

(2')

161

Shall Do(y, G(x, y) - H(x, y)),

where G(x , y) expresses that x (informs y that he) wants medical care and Hix, y) that x receives medical care from y . This would keep the analysis within Kanger's basic framework; however, it remains an open question whether the construction is a good one. As regards bearer-permissive rights, the problem is somewhat different. Mr. Brown has a right to walk in the municipal park, if he wants to, but need not walk there if he does not want to. In Kanger's typology, the relevance of Mr. Brown's will in this case can be expressed by the conjunction -'Shall -,Do(x, Ftx, y)) & -'Shall -,Do(x, -,F(x, y )), where F(x, y) expresses that x walks in y's park; the sentence says that x has both power (=:: Befugnis) and counter-power, as regards his walking in the park. Since, in this case, the power is "two-sided" (power and counter-power), it is sometimes described as bilateral. Among theories of rights the so-called will theory, making relevance of the right-holder's will a conceptual characteristic of rights, has a respectable ancestry. A modern version of this theory has been developed by the Oxford legal philosopher Sir Herbert Hart. In Swedish philosophy, views similar to Hart's have been proposed by Sven Danie lsson." However, there are claim-rights where the claim-holder's will is irrelevant, and there can be powers (in Kanger's sense) which are not bilateral. The statement that all children have a right to be given elementary education is compatible with the proposition that such education is compulsory, i.e., that refusal to partake in the education is inoperative. This shows that the objectmatter of a claim-right should not always be construed by a conditional ofthe kind illustrated, where the claim-holders will is made relevant: relevance of the claim-holder's will is not a general characteristic of claim-rights. Similarly, the statement that the policeman has a right to try to seize the thief is compatible with the statement that trying to do so is compulsory. The policeman's power is not bilateral, and it is not relevant what the policeman wants to do. As is well-known, the notion of a right plays, and has played, an important part in many moral and political theories . Various theories emphasize different features of the notion of a right, or even define the notion in different ways, using it as a tool for an ideological message. This fact can be described in various ways: we might say that the notion of a right is "theory-dependent", or, that it is a "contested concept", or with Charles Stevenson, that there exist various persuasive definitions of the notion . I? Those modem theories emphasizing relevance of the right-holder's will can be called liberal theories, in a

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wide sense. Since liberal theory occupies an important place in political thought, it is only to be expected that we are apt to regard cases where the right -holder's will is relevant as the central cases of rights. On the other hand, a general philosophical analysis of rights ought to avoid incorporating as definitional characteristics such features that are asserted by a specific moral or political theory. (v) THE HETEROGENEITY OR HOMOGENEITY OF RIGfITS. Kanger never addresses the question whether the various types have anything in common which justifies calling all of them types of rights. He seems to hold that this problem is not worth pursuing, since the term "a right" is ambiguous; in fact, in the opening of his 1963 paper on rights, he says: "It is almost a commonplace that the idea of a right is vague and ambiguous...". The problem is whether there is any predicate

(1)

x has a right to (the effect that) A if and only if
where A is any condition, and
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IV. A POSITIVE PROPOSAL I. "Being Wronged" and a New Start After the foregoing survey of problems, some positive proposals will now be made. A central subject is the identification of bearers and counter-parties of rights within a minimalistic theory. In the suggestions that follow, the notion of being wronged, introduced in the previous sect ion, features prominently. As is well-known, in 1956 A.R. Anderson suggested an interpretation of the deontic operator 0 (for "obligatory") in terms of alethic modal logic," Applied to Kanger's expression Shall, this interpretation amounts to the following: Shall F - N(,F - S). In the expression to the right, N stands for "necessary", - , as usual, is the symbol for material implication, and S is a propositional constant. S can be understood as "deontic", expressing that the Code is violated or that the Bad Thing occurs . N is supposed to satisfy what Anderson calls the minimal requirements of a normal alethic modal logic. For S, Anderson assumes the axiom 'N(S). The so-called system T for alethic propositional modal logic has the following rule of inference and axioms ." If A is a theorem, then N(A) is a theorem. N(A - B) & N(A) - N(B). N(A) - A . With system T for N, and with ,N(S), the Anderson interpretation of Shall yields the theorems of standard deontic logic, as formulated by: If A is a theorem, then Shall(A) is a theorem. Shall(A - B) & Shall(A) - Shall(B). Shall(A) - ,Shall(,A). Therefore, this interpretation remains basically within Kanger's framework, which is also the framework accepted in this essay. F A useful tool for the explication of rights is obtained if we substitute Anderson's propositional constant S by a two-place predicate constant W for "is wronged by". The notion of an agent's being wronged, introduced above with reference to J. S. Mill, is important in criminal, private, and procedural law; moreover, it plays a prominent part in moral theory . (See, for instance, G. E. M. Anscombe's essay "Who is wronged'i? ") In what follows, W(x, y) is to be read "x is wronged by y".

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In order to define a relativized deontic operator, the Anderson construction will be applied to the notion of "being wronged by". We start with system T for N, the constant S, and Anderson 's axiom (I)

-.N(S) .

We add the predicate constant Wand the axiom (II)

W(x, y) - S,

expressing that if x is wronged by y, then the Code is violated etc." Next, using Wand the Anderson construction, we introduce a relativized "obligative" operator R. If A is a condition, R(x, y, A) is defined by: R(x, y, A) - N(-.A - W(x, y)) .

Accordingly, R(x , y, A) expresses that, by necessity, x is wronged by y if A is not realized. For a fixed pair <x, y> of agents, the logic of R will be standard deontic logic: If A is a theorem, then R(x, y, A) is a theorem; R(x, y, A) & R(x, y, A - B) - R(x, y, B); R(x, y, A) - -.R(x, y, -.A) .25 To simplify the exposition, we introduce a corresponding "permi ssive" operator R * by the definition R*(x, y, A) - -.R(y, x, -.A),

where the right hand side expresses that -.N(A - W(y, x)).

Thus R*(x, y, A) expresses that y is not necessarily wronged by x if A is realized.

2. Simple and Atomic Types of Rights As will be remembered from section II. Kanger's explicans-formulae for simple types of O-rights (x versus y with regard to F(x, y)) all satisfy the scheme, Shall ± Do(y, ± F(x , y)).

In this scheme, let us substitute Shall( ...) by R(x, y, ...) and F(x, y) by F (where F can be any condition, involving x, y, or not) . We obtain,

STIG KANGER 'S THEORY OF RIGHTS

R(x , y, ± Do(y, ±

165

F».

Using this latter scheme, we can reconstruct all of Kanger's simple types of O-rights: claim, counter-claim, immunity, and counter-immunity. Due to the introduction of the notion of being wronged, however, their explication will differ from Kanger' s, and the problem of identifying the bearer does not occur. For example, Claim(x, y, F(x, y» is explicated by, R[x, y, Do(y, F)],

i.e .,

Nl-Doty, F)

~

W (x, y»).

We will no longer have to say , as in the example discussed in section III, that the murderer ha s a claim versus the policeman to the effect that he is arrested by the policeman. In a similar wa y, all of Kanger' s simple types of P-rights can be reconstructed within the new system. Kanger's ex plicans- sentences for P-rights (x versus y with regard to F(x , y» all satisfy the scheme,

-sien ± Do (x, ± F(x , y» . If we substitute ",Shall(...) by R* (x , y,...), and F(x, y) by F, we get the scheme, R*(x, y, ± Do (x , ±

F»,

i.e. , ..,N[± Do (x , ± F)

~

W(y, x )] .

Using this scheme, all of Kanger' s types of P-rights can be reconstructed: power (= Bejugnis), counter-power, freedom, counter-freedom. For example, Power(x, y, F) becomes R* (x, y, Do(x, F» , and is explicated by ..,N[Do(x, F)

~

W(y, x )].

We avoid the problem about counter-parties that is connected with Kanger' s explication. In the example from section III, of x' s walking in the garden of y' s neighbor z, we can make the two statements, Power(x, y, F) , not Power(x, z, F) i.e., x has a power (= Befu gnis) versus y with regard to walking in z' s garden, but x does not ha ve this power vers us z (the owner) himself. The distinction is accomplished, since we ha ve the res pe ctive explicatio ns,

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..,N[Do(x, F) - W(y, x)] , N[Do(x, F) - W(z, x)].

Thus it appears that, using R, we can reconstruct the complete lists of four types of O-rights and four types of P-rights. Given the list of the eight simple types, we can, of course, reconstruct a theory of atomic types of rights by the method of "maxi-conjunctions". The number of atomic types, however, will be greater than Kanger admitted. This is due to the fact that while (1)

Shall Do(y, F) - Shall ..,Do(x,"'F)

is a theorem in Kanger's theory (since Do(y, F), Do(x,"'F) are inconsistent), the corresponding reconstructed formula (2)

R(x, y, Do(y,

F») -

R(y, x, -Doo,

..,F»)

i.e., [N(..,Do(y, F) - W(x, y))] - [N(Do(x, "'F) - W(y, x))],

does not follow from the axioms hitherto assumed in the reconstructed theory. (If (2) were a theorem, we would get 26 atomic types , as does Kanger.) It would lead too far afield to discuss in any detail the merits of (2). If, however, we want to have (2) as a theorem, while keeping the former basis of the reconstructed theory untouched, the question arises which further axiom or axioms should be added. There may be various possibilities. Among these is the following pair of additions: III. IV.

N(F - G) - N[Do(x, F) - Do(x, G)]; N[Do(x, W(x, y)) - W(y, x)].

(If these are added, (2) can be derived.") III is easily understood; but IV needs some comment. It says that, necessarily, if x himself sees to it that he is wronged by y, then it follows that y is wronged by x. (This seems, in fact, to be the rationale behind the Kanger theorem (1).) For example, suppose that a child, by escaping from school, sees itself to it that it is wronged insofar as it does not receive the education that is due to it. Then it follows that those who have the duty to give the child its education (teachers, schoolmasters etc.) are wronged by the child 's escaping, which prevents them from full filling their duty . The acceptability of III and IV as logically valid may well be questioned. But if so, the Kanger theorem (1) can be questioned with as much justification.

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3. Rights without a Counter-party

We often use statements of the kind "x has a right to..." without mentioning any counter-party. Is it possible to explicate such statements using our twoplace predicate W? Three examples will be discussed. The first one concerns the colloquial use of "having a right", emphasized by Alan White. Suppose we say to x: "You have the right to feel proud." Such a statement is somewhat ambiguous. One plausible interpretation, however, might go as follows . If x does not see to it that he feels proud, then he is wronged by himself; furthermore, for any y other than x, if y sees to it that x does not feel proud , then x is wronged by y. This way, counter-parties are seen as implicitly referred to, and the statement can be explicated in terms of the reconstructed notion. The next example is adapted after one proposed by Bengt Hansson." Petaluma is an area of private property, where different parts are owned by different people; we assume that for each land-owner y, y is wronged if x walks on his land. If F(x) expresses that x walks on Petaluma land, we have -,(3y)(N[F(x) - W(y , x)]),

since no land -owner is wronged if x walks on Petaluma land belonging to another land-owner (cf. the example, above , concerning x walking in the garden of y' s neighbor). On the other hand, in the example, N[F(x) - (3y)(W(y, x)] .

This sentence expresses, simpliciter, that x has no right to walk on Petaluma land. The third example is the one referred to in section III, that all children have the right to receive nutrition. We suppose that x is a child and that F(x) expresses that x receives nutrition ; we want to express that x has the right to receive nutrition. This sentence is compatible with -,(3y)N[-,Do(y, F(x)) - W(x, y )],

i.e. there need not be any particular agent by whom the child is wronged if that agent does not see to it that the child receives nutrition . On the other hand, we might suggest the following as an improved interpretation: N[-,(3y)(Do(y, F(x))) - ( 3y)(W(x, y ))].

That is: if no-one sees to it that x receives nutrit ion, then there is someone by whom x is wronged. The last two examples illustrate how predicate W can be used in a flexible way to explicate sentences that cannot be well interpreted even in terms of the reconstructed notions of rights against a counter-party. In the last of the three

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examples, however, the explication given may be questionable. Indeed, the example may suggest that, in addition to the two-place predicate W, we can be in need of a one-place predicate W for "is wronged", such that W(x) expresses that x is wronged, simpliciter. If we introduce such a notion, we should assume that W(x, y) implies W(x) but not that W(x) implies (3y)(W(x, y». The purpose of introducing a one-place predicate W would be to use it for interpreting a notion R(x,...), where there is no counter-party, according to the formula: R(x, A) - N(-,A - W(x». W ith axiom (I), as well as W(x, y ) - W(x) and W (x) - S, we would get standard deontic logic for R(x,...), as well as further theorems like R(x, A) - -,R(y, ...,A); R(x, y, A) - R(x, A); and so on . The question whether there is a need for introducing the one-place predicate W, however, is left open here .

4. The Impersonal Operator Shall and the Reconstructed Notion ofa Right A typology of rights, based on the notion of "being wronged by", as developed in the foregoing, is more akin to Hohfeld's original idea of jural relations between parties than is the Kanger typology, based on the impersonal operator Shall." By the axiom W(x, y ) - S, we established a connection since, from our assumptions, it follows that R(x, y, F) - Shall F. The suitability of establishing this connection may be questioned. In any case, however, we ought not to assume any of

S - (3.x)(::Jy)(W(x, y»; N(-,F - S) - (3.x)(:3y)N(-,F - W(x, y» . That is, we should not assume that if the Code is violated, then someone is wronged by someone, or that if something is prescribed, then someone has a right versus someone as regards the fulfillment of what is prescribed. There are many prescriptions (administrative regulations, traffic prescriptions etc .) which do not imply rights for particular agents; the contrary assumption would lead to an inflation of rights where the group of right-holders is very diffuse. This shows that there is room for the reconstructed typologies of rights that are genuine relations of rights between parties, alongside with typologies of normative positions expressed in terms of the operator Shall. For the latter kind of typologies, Stig Kanger's idea of combining Shall and Do is very

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useful. As suggested in the foregoing, typologies satisfying the Kanger schemes can be seen as typologies of posit ions of duty or non-duty.

University of Lund NOTES The present essay is part of a project which was supported by The Soderberg Foundations. On the theories of Bentham, Austin and Hohfeld , see Lindahl (1977), Chapters I and 6, with further references. 2 See Kanger (1985). The enlarged logical apparatus is developed in Kanger (1977) and (1986). 3 Kanger (1963), Kanger & Kanger (1966), and Kanger (1972). 4 During the last years of his life, Kanger planned to develop a general theory of conditions based on cylindric algebra; unfortunately, however, the plan was never realized. 5 The systematical use of "sees to it that" in combination with other operators is a characteristic feature in the work of Kanger' s pupils within the Fenno-Scandian school of legal theory and social science. It is used in Porn (1970), (1971), (1974), (1977), in Lindahl (1977), in H. Kanger (1984), in S.O. Hansson (1986), (1990-91), and in Holmstrom-Hintikka (1991) . For some early suggestions, resembling Kanger's idea of combining Shall and Do, see Anderson (1962) and Fitch (1967). 6 The principles assumed by Kanger for the relation of logical consequence are as follows : (i) (ii) (iii)

If I' and I' - G, then G; If F - G, then .,G - .,1'; If I' - G and G - 1J, then F -

H,

See , Kanger & Kanger (1966), at p. 88, note 3. See , concerning Kanger's typologies, Kanger (1963), Kanger & Kanger (1966), Lindahl (1977), and Makinson (1986). 8 See Makinson (1986), at pp . 405 f. 9 See , for example, B. Hansson (1970). 10 For a discussion of this problem, see Barcan Marcus (1966), v. Wright (1968), Szewak (1974) and Opfermann (1977). 11 For a theory exploiting these possibilities. see Lindahl (1977), Part II (the theory of "ranges of legal action" or Spielraum). For a comment, see Talja (1980), where the tools of lattice theory are used. 12 Mill (1910), p. 46 . 13 Cf. Lindahl (1977), pp. 45 f., and Makinson (1986). 14 See N. MacCormick's essay "Children ' s rights : A Test-case for the Theories of Right", in MacCormick (1982). 15 See Lindahl (1977), p. 51 and pp. 194- 211 , with further references. 16 See Hart (1972), and S. Danielsson's essay "Fri- och rattigheter" in Danielsson (1986). 17 See Stevenson (1944). 18 Tuck (1979). Cf. M. Golding (1990), at p. 55 . 19 White (1984), especially at p. 114. Of course, the idea of unambiguity is compatible with holding that there are, nevertheless, several types of rights . To make an analogy, the unambiguity of the term "bird" in zoology is perfectly compatible with assuming that there are various kinds of birds. 7

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See Anderson (1956) . reprinted in Rescher (1967) . See. for example . Hughes & Cresswell (1968). 22 The Anderson construction is. of course. connected with the problem of how to express "If .... then ---" in a satisfactory way within a logically well-written language . Our reason for not discussing this problem is that. even if N(... - ---) is questionable in the context at hand. it will keep us close to the Kanger typologies and logical framework. 23 Anscombe (1967 ). 24 In passing, we observe that a weaker logic is obtained if we drop S together with (I) and (II). and rather stay with the axiom, 20 21

(1)

-'N[W(x. y»).

expressing that it is not necessary that x is wronged by y . Thus . from (I) and (II) we can derive (2 )

-,N[W(x. y) V W (z, -n

but (2) cannot be derived from (1). As will appear, the stronger logic resulting from (I ) and (II ) will yield typologies closer to those proposed by Kanger. 25 Since we have (I ) and (II) among the axioms. we obtain , as well, further theorems for cases where x, y are not kept fixed; in particular . we have. R(x. y. A) - -,R(z, w, -,A ). 26 The antecedent of (2) is equivalent to N(-,Do(y. F) - W(x . y» which implies N(.,F - W(x. y». From this formula and III we get N[Do(x. -'F) - Do(x, W(x . y»); using IV we get N( Dotx, "F) - W(y, x». which is equivalent to the consequent of (2). 27 B. Hansson (1970 ). at pp. 245 f. 28 For an approach closerto Hohfeld 's than Kanger 's, see. as well, B. Hansson (1970 ); cf. also Makinson (1986 ). at pp. 48 ff.

REFERENCES Anderson , A. R.. The Formal Analysis of Nonnative Systems. Technical report N:o 2. contract N:o SARInonr-609 (16), Office of Naval Research. Group Psychology Branch. New Haven, Conn. 1956. (Reprinted in Rescher (1967). at pp. 147-213.) Anderson , A. R.• "Logic. Norms and Roles". Ratio 4 (1962). pp. 36 -49. Anscombe, G. E. M.• "Who is Wronged?" The Oxford Review (1967) . Barcan Marcus, R.,"lterated deontic modalities ". Mind 75 (1966), pp. 580 - 582 . Danielsson, S.. Filosofiska inviindningar. Stockholm 1986. Fitch , F. B.. "A Revision of Hohfeld's Theory of Legal Concepts", Logique et Analyse 10 (1967 ), pp. 269- 276. Golding, M. P., "The Significance of Rights Language" , Philosoph ical Topics 18 (1990 ), pp. 53 -64. Hansson . B.. "Deontic Logic and Different Levels of Generality". Theoria 36 (1970). pp. 241 248. Hansson, S. 0 ., "Individuals and collective actions", Theoria 52 (1986 ). pp. 87 -97. Hansson, S. 0. , "A formal representation of declaration-related legal relations". Law and Philosophy 9 (1990 -91). pp. 399 -416.

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Hart, H ., "Bentham on Legal Rights ", in Oxford Essays in Jurisprudence , Second Series, ed . A. W . B. Simpson, Oxford 1973, pp . 171-201. Hilpinen, R., ed ., Deontic Logic: Introdu ctory and Systematic Readings, Dordrecht 197 I. Hohfeld, W . N., Fundamental Legal Conceptions As Applied in Jud icial Reasoning, and Other Legal Essays, ed . W. W . Cook, New Haven 1923. Holrnstrom-Hintikka, G. , Action, Purpose, and Will: A Formal Theory. Helsinki 1991. Hughes, G. E. and Cresswell, M . J., An Introduction to Modal Logic. London 1968. Kanger, H., Human Rights in the U. N. Declaration . Acta Universitatis Upsaliensis, Uppsala

1984. Kanger, S ., New Foundation s for Eth ical Theory . Part I. Stockholm 1957. (Reprinted, with minor changes, in Hilpinen (1971).) Kanger, S ., "Rattighetsbegreppet'' ("The Concept of a Right") , in Sju Filosofiska Studier tilldgnade Anders Wedberg den 30 mars 1963. Philosophical Studies published by the Department of Philosophy, University of Stockholm; N:o 9, Stockholm 1963. (Reprinted, in English translation, as the first part of Kanger & Kanger (1966». Kanger, S ., "Law and Logic", Theoria 38 (1972), pp. 105-132. Kanger, S ., " Nagra synpunkter pa beg reppet inflytande" ("Some aspects of the concept of influence") , in Filosofiska Smulor tilliignade Konrad Marc-Wogau . Filosofiska Studier utgivna av Filosofiska Foreningen och Filosofiska Institutionen vid Uppsala Un iversitet, Uppsala 1977. Kanger, S ., "On Realization of Human Rights", in Action, Logic, and Social Theory, cd . by G. Holmstrom and A. J. I. Jones, 38 Acta Philosophica Fenni ca (1985), pp . 71- 78. Kanger, S., "Unavoidability" , in Logic and Abstraction. Essays Dedicated to Per Lindstrom on His Fiftieth Birthday, cd. M . Furberg et aI., Goteborg 1986. Kanger, S. & Kanger, H. "Rights and Parliamentarism", Theoria 32 (1966), pp . 85-116. Lindahl, L., Position and Change: A Study in Law and Logi c, Dordrecht 1977. MacCormick, N., Legal Right and Social Democracy . Oxford 1982. Makinson, D ., "On the formal representation of rights relations. Remarks on the Work of Stig Kanger and Lars Lindahl" , Journal of Philosophy 15 (1986), pp. 403-425. Mill , J. S., Utilitarianism, Liberty, Representative Government. Everyman' s Library. London 19IO (repr. 1964). Opfermann, W ., "Zur Deutung normlogisch er Metaoperatoren " , In Deontische Logik und Semantik, cd . by A. G . Conte et al. Wiesbaden 1977. Porn, I., The Logic of Pow er . Oxford 1970. Porn, I., Elements ofSocial Analysis . Filosofiska Studier utgivna av Filo sofiska Foreningen och Filosofiska Institutionen vid Uppsala Universitet. Uppsala 1971. Porn, I., "Some basic concepts of action" , in Logical Theory and Semantic Analysis, ed . by S . Stenlund, Dordrecht 1974, pp . 93-1OI. Porn, I., Action Theory and Social Science, Dordrecht 1977. Rescher, N., ed ., The Logic of Decis ion and Action , Pittsburgh 1967. Stevenson, Ch . L., Ethics and Language. New Haven 1944. Szewak, E. 1., "Iterated modalities and the parallel between deontic and modal logic", Logique et Analyse 67-68 (1974), pp . 323- 333. Tuck, R., Natural Rights Theories: Their Origin and Development. Cambridge 1979. Talja, J. "A technical note on Lars Lindahl's Po sition and Change", Journal of Philosophical Logic 9 (1980), pp . 167-183 . White, A. R., Rights. Oxford 1984. Wright, G.H. v; An Essay in Deontic Logic and the General Theory of A ction . Amsterdam

1968.

LENNART AQVIST

STIG KANGER'S THEORY OF RIGHTS : BEARERS AND COUNTERPARTIES, SOURCES-OF-LAW, AND THE HANSSON PETALUMA EXAMPLE

1. INTRODUCTION In spite of the many conspicuous virtues of Stig Kanger's well-known theory of rights, as presented e.g. in Kanger (1957), Kanger & Kanger(1966), Kanger (1972), and also in Lindahl (1977, Chapter 1), there are quite a few intriguing problems connected with that theory, especially when it is considered from a legal or juristic point of view. Some of these problems have been very ably discussed in two fairly recent important contributions, viz. Makinson (1986) and Lindahl (1994). For instance, they are both concerned with the difficulty, on Kanger's approach, of capturing the "full Hohfeldian relationality" involved in rights relationships (Hohfeld (1919) was concerned not just with claims and duties simpliciter, but with claims held by yon x, and duties borne by x towards y, with regard to specified states of affairs). Already Hansson (1970) dealt with this type of difficulty as a problem for so-called deontic logic, although without explicitly relating it to Kanger's theory of rights; but Hohfeld (1919) forms the starting point of his paper. Following Makinson's terminology, we shall refer to this problem as the task of finding a suitable double indexation for bearer and counterparty in the formal representation of a rights relationship. The main purpose of the present paper is to outline a new way of handling the problem. The plan of the paper is as follows . In Section 2 infra we present the afore-mentioned difficulty for Kanger's theory, and in Section 3 we diagnose it as arising from failure of the theory to pay explicit attention to socalled sources-of-law, which are, of course , all-important to lawyers and legal scientists alike. On the basis of this (alleged) insight we then propose, in Section 4, a new formalism for representing rights relations, which will be an extension of Dyadic Deontic Logic, i.e. a logic for conditional obligation and permission; the formalism will also have to contain notation both for sourcesof-law and their application to various agents, or persons, e.g. parties to a contract. Section 5 supplies a series of definitions, which will be put to work

173 G. Holmstrom-Hintikka , S. Linstrom and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work. Vol. II. 173-183. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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in the concluding Section 6, where we deal in some detail with an interesting example due to Hansson (1970 ). Our source-of-Iaw approach to the double indexation problem can be seen to differ from the approaches of Makinson (1986) and Lindahl (1994) in the following important respect: their informal explications of such locutions as x bears a duty towards y that F under the code N x has a claim-right versus y to the effect that F(x,y)

(Makinson) (Lindahl)

are both in terms of what happens in the case that F is not true / if F(x,y) is not the case, viz. for Makinson: that y has a power under N to initiate legal action against x for non-fulfillment of F, and for Lindahl: that x is wronged by y. Thus, their explicata both refer to certain legal consequences of breaking the duty / violating the claim-right at issue. On the other hand , my own explications below of those locutions rather refer to certain legally relevant (conditioning , operative, ultimate) facts, viz. that there exist valid sources-of-Iaw, which apply to the parties x and y , and according to which x has that duty to y/ x has that claim-right against y. The distinction legally relevant fact vs. legal consequence is a well-known theme in Scandinavian jurisprudence of this century; see e.g. Wedberg (1951) . It corresponds closely to the Alchourron & Bulygin (1971 ) distinction Case vs. Solution. My emphasis in this paper on sources-of-Iaw as legally relevant facts on which rights and duties are based, so to speak, is not intended as a criticism of the Makinson - Lindahl emphasis on legal consequences. Both aspects are clearly important; but the importance of the source-of-law aspect must not be underrated. 2. A CURIOUS DIFFICULTY IN KANGER'S THEORY OF RIGHTS Kanger's explication of the notion of a claim, or a claim-right, results from the addition to his formal language of a definitional schema of the form : Dl.

Claim(x,y,A)

~

ShallDOyA

where x, yare variables over agents, or parties, A is any formula denoting a state of affairs, Shall is a one-place impersonal deontic operator for oughtness (obligation), and where DO is Kanger's operator for "sees to it that". Following Lindahl (1977 , p. 45 f.), we may take as an instance of the definiens here the statement (1)

Smith shall see to it that Jones receives White's bankbook

where we set y

= Smith and A = "Jones receives White's bankbook". Again ,

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setting x =Jones, we obtain as an instance of the defin iendum in D1 the statement (2)

Jones has versus Smith a claim that Jones receives White 's bankbook

And, if we set x = White, we obtain as an instance of that definiendum the statement (3)

White has versus Smith a claim that Jones receives White's bankbook

Now, as observed by Lindahl (1977, p. 46), since (1) is the explication both of (2) and of (3) according to Dl, (2) and (3) tum out to be equivalent in Kanger's theory. On the other hand, according to common legal usage, they are not equivalent: (3) might well be true, whilst (2) is not (see e.g. Hart (1973), p. 195). Clearly, this is a strange consequence ofDl. The present objection to the adequacy of the Kanger explication of Claim can be sharpened as follows . Suppose that Kanger's theory of rights is formulated in an extension of standard first-order quantificational predicate logic, as it obviously appears to be in Kanger (1957) and Kanger (1972) . (Such a formulation is clearly required, if one is to do justice e.g. to the distinction right in personam / right in rem, which important distinction is elaborately and extensively discussed in the later part of Hohfeld (1919) .) Then, we easily obtain as a consequence of D1: Tl.

Claim(x,y,A) ~ 'izClaim(Z,y,A) ;

where the variable z is distinct from both x and y and is not free in A.

If the equivalence in Kanger of (2) and (3) was a strange consequence of D1, the present result T1 is an even stranger one: it enables us to infer from (3) that anybody has versus Smith a claim that Jones receive White's bankbooknot only the "favoured party" Jones , but even anyone having no dealings whatsoever with White, Smith, Jones or the bank involved. What has gone wrong here with Kanger 's theory in relation to current legal usage of "claim" and "claim-rights"? 3. DIAGNOSIS : THE NEED FOR TAXING SOURCES-OF-LAW INTO EXPLICIT ACCOUNT Let us reflect for a moment on the reason why the statements (2) and (3) fail to be equivalent in many jurisdictions (as observed by Hart (1973), p. 195).

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The reason could be that, whereas there exists a valid source-of-law which applies to the parties White and Smith (say , an agency agreement with Smith the agent and White the principal) and according to which Smith has a duty to White to transfer White's bankbook to Jones, there does not exist any valid source-of-law applying to Jones and Smith according to which Smith has that duty to Jones. So (3) is true by virtue of the valid source-of-law applying to White and Smith, while (2) is false because of the non-existence of any valid source-of-law applying to Jones and Smith that could make (2) true. On the basis of this explanation, which we take to be basically sound, the following amendment of Kanger' s formulations suggests itself: append to the statement (3) the clause "which is based on the valid source-of-law SL that applies to White and Smith"; and append to the statement (2) the clause "which is based on the valid source-of-law SL that applies to Jones and Smith". (In these appendages, the relative pronoun "which" refers back to "claim", not to "White' s bankbook", of course.) Note that these amended formulations immediately provide an identification of bearer and counterparty in a rights relationship. We recall that such an identification is an important concern of e.g. Makinson (1986) and Lindahl (1994). Moreover, our above explanation why it may be that (3) is true, whilst (2) is false, can now be articulated more clearly as follows : it is easy to find and point out a value of SL which makes the amended formulation (3) true, say, the supposed agency agreement between White and Smith, whereas it might be impossible to find a value of SL making the amended (2) true .

=

=

4. A NEW FORMALISM FOR REPRESENTING RIGHTS RELATIONS: AN EXTENSION OF DYADIC DEONTIC LOGIC How is Kanger's D 1 to be modified, if we adopt the "source-of-law" approach just outlined? To begin with, I suggest that his three-place relation Claim, relating two parties to a state of affairs , be replaced by a four-place one, which relates two parties and a state of affairs to a source-of-law. Our definiendum would then be, instead of the one in D I: Claim-BasedOn(x,y,A; SL) to be read as "x has versus y a claim that A, which is based on the valid sourceof-law SL that applies to x and y" . The "amended" formulations of (2) and (3) considered above would then instantiate this new definiendum. The question how to reformulate the definiens of D 1 is more tricky. We can let SL to a binary relation symbol such that Slxy

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is a well-formed formula which asserts that the valid source-of-law SL applies to the parties x and y . SLxy should then go into the new deflniens as a conjunct; but how do we relate it to the old deflniens ShallDOyA? For a number of reasons that will not be discussed here , I do not think that taking it to strictly imply the latter will yield the best analysis of source-of-law-based claimrights. A more promising course is, in my opinion, to let the new deflniens contain a clause to the effect that the state of affairs SLxy "requires" that DOyA in the sense of Chisholm (1964). In tum, as argued in my Aqvist (1998), the logic of Chisholm's notion of requirement is nothing but a dyadic deontic logic for conditional obligation. In the present context we take its basic formal locution to be Shall(A / B) to be read as: "it shall (ought to) be the case that A, if (given that) B". We can now re-formulate Kanger's Dl in this way : Dl *. Claim-BasedOn(x,y,A ; SL) - SLxy & Shall(DOyA / SLxy) the deflniens of which is read as: "the valid source-of-law SL applies to x and y; and it shall be that y sees to it that A , if (given that) SL applies to x and y" . Upshot: in what follows we are going to use as our formalism for representing rights/duties relations a language of temporally relative modal and deontic predicate logic with, at least, quantifiers and variables over agents, times and sources-of-law as well as matching names (constant symbols), plus the action operator DO (for agent-causation), the dyadic deontic operators Shall( /) (for conditional obligation) and May( / ) (for conditional permission), together with appropriate alethic modal operators (e.g. for historical necessity or, as Kanger calls it, unavoidability). Without entering on a detailed study of the logic I have in mind, we just observe that it will combine features and ideas from Aqvist & Hoepelman (1981), van Eck (1981), Bailhache (1993) and Aqvist (1998) . 5. A SERIES OF DEFINITIONS Given a formal language of the sort just outlined, we now propose a ser ies of definitions, or definitional schemata, which are to be added to an axiomatic formulation of the logic mentioned in the preceding section. In them, x, y will be variables over agents (parties), A, B will be any formulas (thought of as referring to states of affairs, as in Kanger), and SL will be a variable ranging over sources-of-law (assumed to be valid in whichever jurisdiction is under consideration).

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CondDuty-BasedOn(x,y,A,B; SL) SLxy & Shall(DOxA/B & SLxy)

~

Here, the definiendum is to be read as: "x has versus y a conditional duty that x sees to it that A. given that B. based on the valid source-of-Iaw SL that applies to x and y"; and the definien s as "SL applies to x and y; and it shall be that x sees to it that A, given that B and that SL applies to x and y". D3 .

CategDuty-BasedOn(x,y,A; SL) SLxy & Shall(DOxAIT & SLxy)

~

where the definiendum has this reading: "x has versus y a categorical (or unconditional) duty that x sees to it that A. based on the valid source-of-Iaw SL that applies to x and y" . The defini ens of D3 is read as in the case of D2 , except that we have a "designated" tautology T (verum) in the place of B. Thus, we obtain D3 from D2 by exploiting the familiar device of defining categorical (unconditional, "peremptory") duties in terms of conditional ones by means of the propositional constant verum . Note that the relation CategDuty-BasedOn is a "merely" four-place one. whereas CondDutyBasedOn is a five-place relation. On the basis of D2 and D3. two obvious notions of a claim (claim-right) readily suggest themselves: D4 . D5.

CondClaim-BasedOn(x,y,A,B ; SL) ~ SLxy & Shall(DOyA/B & SLxy) CategClaim-BasedOn(x,y,A; SL) ~ SLxy & Shall(DOyA/T & SLxy)

The readings of these definienda are straightforward: in the case of D4: "x has versus y a conditional claim that y sees to it that A, given that B, based on etc .", and in the case of D5: "x has versus y a categorical claim that y sees to it that A. based on etc.". We then have the following results. asserting correlativity in the Hohfeld sense : T2. T3.

CondClaim-BasedOn(x,y,A ,B ; SL) ~ CondDuty-BasedOn(y,x ,A,B; SL) CategClaim-BasedOn(x,y,A; SL) ~ CategDuty-BasedOn(y,x,A; SL)

Apart from the definitions D2- D5, the only additional assumption needed in the proofs of T2 and T3 is one to the effect that the involved source-of-Iaw relation SL is symmetric, i.e.• that for any agents x and y,

STIG KANGER'S THEORY OF RIGHTS

SLxy

~

179

SLyx

which assumption seems uncontr oversial and can be adopted as an axiom. A familiar manoeu vre, enabling us to reduce the number of free variables in definitions D2 - D5 by one, is the following: prefi x each defini ens with an existential quantifier :JSL ("there is a valid source-of-law SL such that "), and delete the variable SL in each definiendum as well as the rider BasedOn; call the resulting series of definitions D2*-D5 *. The first two members of that series will then look like this: D2*. D3 *.

CondDuty(x,y,A,B) ~ :JSL(SLxy & Shall(DOxA/B & SLxy» CategDuty(x,y,A) ~ :JSL(SLxy & Shall (DOxAIT & SLxy»

and so on, in like manner, for the remaining two resultin g definitions. Analogues of T2 and T3 are immediate for the new notions of conditional! categorical duties/claims. Various further definitions might be added to our form al framework. For the time being, let us cont ent ourselves with ju st two new items: D6. D7.

CondPriv-BasedOn(x,y,A,B; SL) ~ SLxy & May(DOxA/B & SLxy) CategPriv-B asedOn(x,y,A; SL) ~ SLxy & Ma y(DOxAIT & SLxy)

the readings of which should be obvious , except possibly that Priv suggests "privilege" in the Hohfeld sense of "permission" . 6. HOW TO HANDLE BENGT HANSSON'S PETALUMA EXAMPLE IN OUR FRAMEWORK The Hansson (1970) Petaluma example runs as follo ws. Petaluma, Calif., is an area of private property , where different parts are owned by different people. Now, every lando wner in Petaluma has forbidden x to walk on his (the landowner's) land ; on the other hand , no lando wner in Petaluma has the power to (' can') forbid x to walk on land owned by others in Petalum a. For simplicity, assume that there are exactly three landowners in Petaluma, viz. a, b, and c, which are all distin ct from x (and from one another). We shall deal with the que stion how to characte rize x's position with regard to these lando wners , using the conc ept s defined in the last section. First of all, let us introduc e a bit of formal notation and write:

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W.xy /Wxa, Wxb, Wxcl for "x walks on y's la's, b's, c's/land",

and Wx

for

"x walks on private land in Petaluma".

Moreover, we shall write SLl for SL2 for

"the source-of-law constituted by a having prohibited x from walking on a's land, but not on b's or c's"; "the source-of-law constituted by b having prohibited x from walking on b's land, but not on a's or c's";

and SL3 for

"the source-of-law constituted by c having prohibited x from walking on c's land, but not on a's or b's".

We can now state three unproblematic assumptions, or axioms, governing the Petaluma example (where the variable y ranges over the landowners): Al , A2.

A3.

Vy(y=aVy=b Vy=c) Wx ~ (Wxa V Wxb V Wxc) SLlxa & SL2xb & SL3xc

~

:3yW.xy

Consider next the following four statements:

(4) (5) (6) (7)

Vy :3SL CategOuty-BasedOn(x,y,-'W.xy; SL) :3SL Vy CategOuty-BasedOn(x,y,-,W.xy; SL) Vy :3SL CategOuty-BasedOn(x,y,-'Wx; SL) :3SL Vy Categljuty-BasedOrnxw-wx: SL)

We then consider each of these statements in tum, with a view to finding out its truth-value. Ad (4). As every landowner in Petaluma has forbidden x to walk on his land, we have the following results: SLlxa & Shall(OOx -,Wxa/ SLlxa) SL2xb & Shall(OOx -rt Wxb/ SL2xb) SL3xc & Shall(OOx -'Wxcl SDxc)

From these assumptions, true ex hypothesi in the example, we easily derive (4), which must then be true as well (for the derivation, use definition 03, axioms Al and A3, predicate logic, and our logic for dyadic deontic operators). Again, appealing to 03*, we may conclude from (4) that x has a categorical

STIG KANGER'S THEORY OF RIGHTS

duty versus every landowner not to walk on his land, formally:

CategDuty(x,y,""Wxy).

181

vv

Ad (5). Familiarly, (5) is a stronger statement than (4), so the truth of (4) does not preclude (5) from being false . For what does (5) assert? It asserts that there is a single, valid source-of-law SL such that:

SLxa & Shall(DOx ...,Wxal SLxa) SLxb & Shall(DOx ...,Wxbl SLxb) SLxe & Shall(DOx ..., Wxc/ SLxe) Clearly, none of SLl , SL2 or SL3 could serve as the desired SL here: SLl, for instance, satisfies the first clause but fails to satisfy the second and the third one, and similarly for SL2 and SL3. In general, even if there were an SL of the desired sort (applying to all of x.a.b,c and prohibiting x from walking on any of the different parts of Petaluma land), its existence and validity certainly does not follow from the truth of (4) in our example. So we might well deem (5) false.

Ad (6). First of all, we must observe here that the relevant clause in (6), ...,Wx ("x does not walk on private land in Petaluma"), is stronger than ...,Wxy ("x does not walk on y's land in Petaluma"), which clause figured in (4) and (5). In effect, by the axiom A2 , ...,Wx is equivalent to the conjunction (...,Wxa & ...,Wxb & ""Wxe). Moreover, according to the premisses of the example, no landowner in Petaluma had the power to forbid x to walk on land owned by others in the area: x's walking on such a land is no concern of any given landowner, who worries only about his own land . We can express this assumption as follows in our formalism: A4.

SLlxa & May(DOxWxbISLlxa) & May(DOxWxc/SLlxa) SL2xb & May(DOxWxaISL2xb) & May(DOxWxc/SL2xb) { SL3xe & May(DOxWxaISL3xe) & May(DOxWxbISL3xe)

A more compact way of expressing A4 is the following:

'tIy'tlz(u y -

3SL CategPriv-BasedOn(x,y, Wxz; SL»

where the four-place relation in the scope of the existential quantifier is defined by D7 supra. Let us now go back to the statement (6) . Since ...,Wx is intuitively stronger than ...,Wxy, the truth of (4) should not preclude (6) from being false. So the situation of (6) vis-a-vis (4) is similar to that of (5) vis-a-vis (4) just dealt with. However, we may ask: is it possible, using the assumption A4 in particular, to

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prove the negation, or falsity, of the statement (6)? We shall argue that this is indeed the case, provided only that we make the following additional assumption concerning the sources-of-law which are applicable in the context of the present example: A3+.

VSL«SLxa ~ SL=SLl) & (SLxb (SLxc - SL=SL3»

~

SL=SL2) &

Clearly, taken together with A3 supra, A3+ asserts that SLi is the only source-of-law that applies to x and a (in the context at hand), that SL2 is the only source-of-law that applies to x and b, and similarly for SL3 with respect to x and c. An attempted refutation of (6) may run as follows. We start by assuming (6) for reductio ad absurdum, and then go on to derive e.g. SLixa & Shall(.,DOxWxbISLlxa)

which result contradicts the first clause in A4 by virtue of the principle May(AIB)

~

.,Shall(.,AIB)

of our dyadic deontic logic. In the derivation of that result we appeal (i) to the axiom A3+ (in order to get SLlxa ), (ii) to the axiom A2 together with the Kanger- Lindahl rule RI in the logic of DO (in order to be able to replace the clause .,Wx in (6) by the conjunction (.,Wxa & .,Wxb & .,Wxc); see Lindahl (1977 ), p. 68), (iii) to the principle (iii) on p. 76 of Lindahl (1977) asserting distributivity of DOx over conjunction (in one direction; note that this principle is not forthcoming in the Kanger- Lindahl basic , or minimal, logic of DO) and (iv) to their axiom Al (see again Lindahl (1977), p. 68) from which we easily derive the principle: DOx.,A -

.,DOxA

which is needed in our proof. The remaining details can be left to the reader. Before leaving the statement (6), let us just observe that the additional axiom A3 + is also useful in enabling us to refute the statement (5): the latter will then be seen to imply the absurd conclusion that the three sources-of-law SLl , SL2 and SL3 are identical to each other! Ad (7) . Since (7) is logically stronger than (6), and (6) is false , (7) must be false as well.

Upshot : the high ly interesting distinctions, which Hansson (1970), Makinson (1986) and Lindahl (1994) try to bring out in different formal ways, amount in our source-of-law framework to one as between the true statement

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183

(4) and the false stronger statements (5) -(7). Among the latter three, (6) seems to be the one that best fits the argument of our colleagues; but (5) is an interesting candidate, too. Uppsala University

REFERENCES Alchourr6n, C.E . & E. Bulygin (1971), Normative Systems . WienlNew York, Springer. Bailhache, P. (1993) , "The Deontic Branching Time : Two Related Conceptions," Logique et Analyse 36, 159-175. Chisholm, R.M. (1964), "The Ethics of Requirement," American Philosophical Quarterly 1, 147-153. van Eck, lA. (1981) , A System of Temporally Relative Modal and Deontic Predicate Logic and its Philosophical Applications. University of Groningen, Department of Philosophy. Also in Logique et Analyse 25 (1982), 249 - 290 and 339- 381 . Hansson, B. (1970), "Deontic Logic and Different Levels of Generality," Theoria 36, 241- 248 . Hart, H.L.A . (1973), "Bentham on Legal Rights" in A.W.B. Simpson (ed .), Oxford Essays in Jurisprudence (2nd series) . Oxford, Clarendon Press, pp. 171- 20 1. Hohfeld, W.N . (1919), Fundamental Legal Conceptions as Applied in Judicial Reasoning and Other Legal Essays (edited by W.W. Cook) . New Haven, Yale University Press , 1919 , 1923, 1964. Kanger, S. (1957), New Foundations for Ethical Theory . University of Stockholm, Department of Philosophy. Also in R. Hilpinen (ed.), Deontic Logic: Introductory and Systematic Readings. Dordrecht, Reidel, 1971. Kanger, S. (1972), "Law and Logic," Theoria 38,105 -132. Kanger, S. & H. Kanger (1966) , "Rights and Parliamentarism," Theoria 32,85-115 . Lindahl, L. (1977), Position and Change: A Study in Law and Logic . Dordrecht, Reidel. Lindahl, L. (1994), "Stig Kanger's Theory of Rights" in D. Prawitz, B. Skyrms and D. Westerstahl (eds.), Logic, Methodology and Philosophy of Science IX. Elsevier Science B.Y., pp . 889-911. Makinson, D. (1986), "On the Formal Representation of Rights Relations. Remarks on the Work of Stig Kanger and Lars Lindahl ," Journal of Philosophical Logic 15,403-425. Wedberg, A. (1951) , "Some Problems in the Logical Analysis of Legal Science," Theoria 17, 246-275 . Aqvist, L. (1998), "Prima Facie Obligations in Deontic Logic : A Chisholmian Analysis Based on Normative Preference Structures" in C. Fehige & U. Wessels (eds .), Preferences. Berlinl New York, W. de Gruyter, pp. 135-155. Aqvist, L. & J. Hoepelman (1981), "Some Theorems About a 'Tree' System of Deontic Tense Logic" in R. Hilpinen (cd.) , New Studies in Deontic Logic . Dordrecht, Reidel, pp. 187-221.

GHITA HOLMSTROM-HINTIKKA

STIG KANGER'S ACTIONS AND INFLUENCE

1. INTRODUCTION This essay is mainly historical in character. Thus, a great part of it is devoted to Kanger' s own development of his concepts and theory. For a brief comparison, some aspects of his theory are put in perspective in order to show his originality and influence. This is the case for instance concerning his Dopredicate which is mirrored in Porn's Ii-predicatc for action (1977). One section is also devoted to extensions and further developments of Kanger's action theory . Although undertaken by mysel f and in many respects diverging from his thoughts , these extensions arc to a great extent triggered by my intensive discussions with Kanger in the years of 1980-1987. His influence is also plain in publications by Lindahl (1977) and Porn (1970, 1972, 1977) and Helle Kanger (1966, 1981) as well as by myself (as late as 1997) and others as for instance A. J. I. Jones . This Fenno-Scandian school of Act ion Theory is also being taught in our universities till this very day. 2. BACKGROUND Stig Kanger's Action Theory was gradually developed along with his ethical theory and theory of human rights. Thus, in his early writings such as New Foundations for Ethical Theory (1957) and his essay "Rattighetsbegreppet" (1963) he introduces agent causation through the concept 'that Y causes that S(X,Y)'. Still in the reprinted and extended version of this essay renamed "Rights and Parliamentarism" and written together with Helle Kanger and published in Theoria (1966) we can read among other things that a party Z breaks a rule of rights if it holds either that (1) according to the rule it shall be that Z causes ..., but actually it is not so that Z causes ..., etc. (1966 , pp. 9899). When New Foundations for Ethical Theory is reprinted 1971 in Deontic Logic ed. by R. Hilpinen, as we will see, we then find the action terms familiar from Kanger 's later writings 'Y sees to it that' . Thus the clauses above reads as (1) ... that Z sees to it that ... but actually it is not the case that Z sees to it that ... etc.

185

G. Holmstrom-Hintikka, S. Linstrom and R. Slivinski (eds .), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. ll, 185-204. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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GHITA HOLMSTROM-HINTIKKA

Also, another action concept which plays a key role in several of his writings, the concept of 'unavoidability', enters his thought at an early stage. In New Foundations (1957) he says that 'OughtA' always implies that' -A is avoidable', where avoidability is taken in the wide sense: only such facts are unavoidable which would be or would have been outside the range of reasonable and foreseeingly planned joint human efforts .

As we all know, Stig' s last paper was named "Unavoidability" (Kanger, 1986). In New Foundationsfor Ethical Theory (1957) Kanger discusses the deontic notions 'Ought', 'Right' and 'Wrong' and their relations within a framework of ethical theory . In this context he spells out the content of the schema (Kanger, 1971, p. 42) (0)

X has a right in relation to Y to the effect that F(X,Y)

where X and Yare 'moral personalities' and F(X,Y) is a relation between X and Y. The vagueness of 'X has a right in relation to Y to the effect that' is clearly shown by the four idiomatic instances (1)-(4) of (0) (1) (2) (3) (4)

X has a right to get back the money she loaned to Y. X has a right to walk into Y's shop when it is open. X has a right to give all her money to Y. X has a right to walk on the street outside Y's shop.

The ambiguity in (0) as seen in (1)-(4) reflects the different meanings of 'right' in that "in (1) 'right' means claim, in (2) 'right' means liberty or privilege, in (3) 'right' means power and in (4) 'right' means immunity". When these four senses of 'right' are explicated, in the very explication we see in the first version of 1957 the term cause as in 'Y causes that' whereas in the reprinted version we find action terms familiar from Kanger's later writings 'Y sees to it that'. Thus the alternative meanings of 'right' are explicated as (1 ') (2') (3') (4')

Ought(Y causes that F(X,Y)) Right-(X causes that -F(X,Y)) Right(X causes that F(X,Y)) Ought-t Y causes that -F(X,Y))

In the reprinted version it reads (1/1) (2/1) (3/1) (4/1)

Ought(Y sees to it that F(X,Y)) Right-(X sees to it that -F(X,Y)) Right(X sees to it that F(X,Y)) Ought-t Y sees to it that -F(X,Y))

STIG KANGER'S ACTIONS AND INFLUENCE

187

In this same connection Kanger recognizes that the different senses of Right as well as some other moral notions "cannot be analysed unless such concepts as: It is avoidable for X that, X sees to it that and X can know that, are available" (1971 , p. 43) . Then he simply states that since they are not available in the language L, "we postpone all further troubles to a planned second part of this paper" (Kanger, 1972, p. 43) . Unfortunately this second part never appeared but Kanger did continue to slowly develop a concept of action which can be extracted from his later writings. Let me, however, first go back in time to "Rattighetsbegreppet'' (1963) and "Rights and Parliamentarism" (1966). 3. CAUSING

Being elements of the simple types ofrights agent causation 'that Y causes that S(X,Y)' gets its own explication. The eight types of rights (a) (b) (c) (d)

(a ') (b') (c') (d')

claim freedom power immunity

counter-claim counter-freedom counter-power counter-immunity

(p. 86f.) are given the explication (1a) (1b) (1c) (1d)

X X X X

has has has has

versus versus versus versus

Y a claim that S(X,Y) Y a freedom that S(X,Y) Y a power that S(X,Y) Y an immunity that S(X,Y)

(1966, p. 88). The types (a')-(d ') are expressed in an analogous way. It is in the process of interpreting these explications that we first meet the cause relation in a semi-formalised language: (2a) (2b) (2c) (2d)

It shall be that Y causes that S(X,Y) Not : it shall be that X causes that not-S(X,Y) Not: it shall be that not: X causes that SeX,Y) It shall be that not: Y causes that not-SiX, Y)

For the expression: Not: it shall be that not ... its dual expression is introduced: It may be that ...

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GHITAHOLMSTROM-HINTlKKA

Kanger's familiar examples of the concept of a right may also be used as an example of this relation of causation: It shall be that Y causes that X receives from Y what X has lent to y. It may be that X causes that a manuscript written by X is published in Sweden. I shall in this paper concentrate on the development of Kanger's notion of agent causation, from his first vague expression of 'Y causes that ...' to his action concept as expressed by the 'Do' predicate for 'seeing to it that' and beyond. At an early stage of his concept of 'cause' Kanger himself recognizes its vagueness commenting that it admits "different specifications in different contexts" and that we "shall assume only that they are interpreted in a reasonable way" and that it satisfies certain logical principles (ibid., p. 88). These logical principles are then simply introduced by means of the long arrow ' - ' which denotes the relation of logical consequence: F - G if G follows from F by ordinary logic extended in a suitable way by logical principles for the concepts 'shall' and 'cause' . Note that the relation - is assumed to fulfill principles like (i) if F and if F - G, then G; (ii) if F - G, then not-G - not-F ; (iii) ifF - G and G - H, then F - H. (p. 88, footnote) . As the concepts 'shall' and 'cause' are linked together it may be appropriate in this context to present all five logical principles which they are assumed to satisfy: I. II. III. IV. V.

If F - G, then shall-F - shall-G (shall-F and shall-G) - shall-(F and G) shall-F - not shall-(not-F) IfF - G and G - F, then X causes F X causes F - F

X causes G

(1966 , p. 89). As will be shown later in this paper, when Kanger's essay was translated and republished it had undergone some striking changes and developments. One point concerns agent causation which in the later version occurs in its new form ' seeing to it' . In Kanger's writings one important notion is the one of 'state of affairs'

STIG KANGER'S ACTIONS AND INFLUENCE

189

(later to be replaced by 'condition'). About this notion he says in 1966 that it "in this essay always means a relation between parties, so that if X and Yare parties S(X,Y) means that the party X stands in the relation S to the party Y". For instance, S may be specified "as the relation between any two parties PI and P2 such that Pi receives from P2 what PI has lent to P 2' Then, of course, S(X,Y) means: X receives from Y what X has lent to Y" . We can note that X or Y need not always occur in S(X,Y) when S is specified (Kanger, 1966, p.89). In the development of the 26 atomic types of rights, the notion of action is imbedded but nowhere actually spelled out. Yet, in discussing the rules of rights and in particular breaking such rules agent causation is immanent. According to an example of a rule of rights it shall be that Y causes that Y does not run into X and, by the strength diagram for rights, it shall be that X does not cause that Y runs into X. Then again, an agent breaks a rule of rights if (1) -(2) hold: (1) (2)

according to the rule it shall be that Z causes ..., but actually it is not so that Z causes ..., according to the rule it shall be that Z does not cause ..., but actually it is so that Z causes ....

(1966, pp. 98-99.) As has been plain , agent causation in the form of causing plays a significant role in the development of Kanger' s types of rights. Nevertheless, so far he has not paid attention to the very conceptual, i.e., semantical aspects of 'causing'. 'Cause' and 'causing' appears as an unanalysed primitive. 4. LINGUISTIC CONSIDERATIONS

In "Law and Logic" we find Kanger' s first extensive analyses of actions. In the very beginning of this article he states that a system of law is "any system of rules which has the purpose of regulating human action under certain conditions" (Kanger, 1972, p. 105). The linguistic framework, later called an Llanguage, he says , should be narrow but should contain sentences by means of which one can (1)

(2) (3)

describe states of affairs or conditions, including numerical conditions state that a state of affairs is unavoidable prescribe that something shall be, or ought to be, the case

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(4) (5)

GHITA HOLMSTROM-HINTIKKA

state that an agent , i.e., a person or an ordered set of people, does something or sees to it that something is the case state that an agent decides upon a certain state of affairs

(1972 , p. 106). A few more points are listed on the wish list but are more related to systems of law and can be neglected in this context. As to the codification, Kanger assumes that "sentences of type (1) can be formulated within the framework of the language of many-sorted elementary logic extended by elementary algebra. "The language of many-sorted elementary logic", he says, "differs from the usual one-sorted type due to the fact that it has several kinds of individual variables". He assumes that there are at least four kinds : variables

x, y, z,... r, s, t,... a, b, c,... p, q,...

for for for for

things time numbers people (or agents regarded as units).

The sentences of type (2) are formed by means of the modal operator it is unavoidable that. Type (3) sentences in tum are formed by means of the deontic operators ought and shall. Greek letters, a,~, ..., stand for a sequence of person variables. For instance, a can be p, or pq, or qpq etc. Thus sentences of type (4) are formed by means of the operators a sees to it that and a sees to it at t that. In an analogous way sentences of type (5) are formed with the operators a decides that and a decides at t that (1972, p. 107). Some of the operators introduced can be analysed in terms of more basic operators. This is the case with the operator seeing to it: (Def)

a sees to it that (---) is per definition the conjunction: (---) is necessary for something which a does and (---) is sufficient for something which a does.

(1972, p. 108). 5. DEDUCTIVE DEVELOPMENT Kanger introduces a set of abbreviations and notions among those directly connected to actions which are from our perspective the important ones (F, G, H are letters for formulas in the L-language) (1972 , pp. 108-109):

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STIG KANGER 'S ACTIONS AND INFLUENCE

Do(«r, Do( a,t, F) Do(«r, Do(a,t,F)

for for for for

F is necessary for something a. does F is necessary fo r something a. does at t F is suffic ient for something a. does F is sufficient for something a. does at t.

The previously stated definition can now be written as (Defl) Do(a,F) = df Do(a ,F) & Do(a,F) (Def2) Do(a,t,F) =dfDo(a,t,F) & Do( a,t,F) The more frequently used variants are the ones where the time component is ignored. Two rules of inference (I -II) are con sidered along with a set of formulas (1)-(5) where 0 is a general letter for any of the operators above: I II

If I-F , then 1-0 F. If I-(F '= G), then I-(OF '= OG) .

1. 2. 3. 4. 5.

OF & O(F ::> G) ::> OG. OF & OG ::> O(F & G). O(F & G) ::> OF & OG. OF ::>F. OF ::> -O-F

(1972, p. 109) . As we can read from Kanger' s tabl e these rules and formulas are taken to be valid for Do and Do to the following extent

I

Do Do

II

2

II

2

3

4

5 5

Thus, if Do(a,F) expresses what the agent do es we can easily infer that the rules and formul as are val id only as to Do(a,F)

II

1

2

This is to say II

If I-(F '= G), then I-Do(a,F) '= Do(a,G).

1. 2. 5.

Do( a,F) & Do(a,F ::> G) ::> Do(a,G) Do(a,F) & Do(a,G) ::> Do(a,F & G) Do(a,F) ::> -Do(a,-F)

It is also believed that the following formulas are valid Do(a,F) & DOCa,G)

::> Do(a,

(F V G))

5

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GHITA HOLMSTROM-HINTIKKA

Do(a,r,F) & r < t

::> Unav-Do(/U,-Do(a,r,F)

.

(For a complete list of formulas assumed valid see Kanger, 1972, p. l l Of.) The concept of unavoidability also gets structured in this paper in that all the rules and formulas (1- II) and (1)-(5) are considered valid for UnavF (it is unavoidable that F). Moreover, Unavf' > -Do(a,F) UnavDo(a,F) ::> -Ought-Do(a,F) OughtDo(a,F) ::> -Unav-Do(a,F) Unav(F = G)::> OughtF = OughtG Unav(F = G)::> Do(a,F) = Do(a,G). This concept is analysed in detail in his last paper, "Unavoidability" (1986) and I shall return to this in that connection. In a comparison to other attempts in the direction of structuring the operator seeing to it Kanger takes us to Chellas (1969) and Porn (1970, 1971). The difference between his operator and the one of Porn's he says is that Porn 's theory "contains an inference rule of type I for the operator seeing to it, thus identifying it with our 06 rather than with Do". A comparison with Porn (1977) is in place here. The basis for Porn 's concept of action is a possible world semantics in terms of which he defines his concept E(a,p) as a conjunction of two other concepts, Dta.p) and C'(a.p) which are two of his basic concepts (Porn uses the notation EaP, DaP, CaP, etc., where a is the name of the agent). Translated into the terminology used in this work, Porn 's basic concepts may be written as follows: D(a,p) = it is necessary for something which a does that p. (p. 4) D'(a.p) = but for a's action , it would not be the case that p. (p. 5) C(a,p) = it is compatible with everything that a does that p. (p. 7) C'(a.p) = but for a's activity, it might not be the case that p. (p. 7) An alternative reading for D'(a.p) is: p is dependent on a's action, whereas C'(a.p) may be read as: p is not independent of a's action . The connection between these concepts Porn defines (p. 6) as follows: (Dfl) (Df2)

C(a,p) =-Dta.-p) C'(a,p) = -D/(a,-p)

After this, Porn gives the following definition of a's action : (DO)

E(a,p)

=(D(a,p) & C/(a,p»

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STIG KANGER 'S ACTIONS AND INFLUENCE

The reading of (Dfl) is the same as for C(a,p) above, and the reading of (Df2) as for C'(a.p). E(a,p) stands for the sentence "a brings it about that p" (p.7). 6. SEMANTICS

In his semantical discussion Kanger first gives a brief account of well-known semantic theories (1972, p. I 12ff.). This account , however, is strongly limited to the language of classical two-valued elementary logic. Not only does he rule out, and rightly so, intuitionistic logic and many-valued logic, he also limits himself to one-sorted logic without symbols for operations in the domain of individuals. What we find, then, is an expansion of Tarski type theory "with the purpose of obtaining semantics for modal formulas" (p. 114). I shall not go into this in detail here. (The interested reader is advised to see Kanger, 1972, pp. 114-115 also reprinted in Vol. I of this edition.)

6.1 Semantics for Some Operators in Language L The semantics for D6(p, F) and Do(p,F) can be constructed in Kanger's theory as follows: T(D6(P, F) , (U, W; V)) RDoCV(p, U), U ~ U),

=tiff T(F, (U ~ W, V)) =t for all U

I

such that

where RD6 is a 3-place relation such that RD6( V(p,U), U ~ U) means that everything the person V(p,U) does in U is the case in U ~ We should note that "the assignment V of values to the variables applies to person variables as well as to individual variables of other sorts" (1972, p. 121). In a similar way Kanger gives the semantics for sufficiency, Do: T(Do(P,F), (U, W, V)) RDi>(V(P,U), U ~U),

=tiff T(-F, (U ~ W; V)) =t for all U

I

such that

where RDoCV(p, U), U ~ U) means that the opposite of everything V(p, U) does in U is the case in U ~ RDoCV(p, U), U ~ U) and (::JU JRDi>(V(p, U), U ~ U) are assumed to always hold true . Action at time t, Do(p,t,F), has an analogous semantics (1972, p. 121).

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GHITA HOLMSTROM-HINTIKKA

7. ACTIONS IN GENERAL In his general remarks about actions Kanger starts with pointing out three distinctions which need to be made - but which are not always so made (1972, p. 122): (1) Act-Acting. An act is an action expressed by a noun phrase ; for instance, moon-walking, murder, ice-dancing. Acting is action expressed by a sentence; for instance, P walks on the moon, P kills someone, p and q dance on ice. (2) Acting - Pseudo-acting. Acting involves some sort of activity performed by an agent; for instance, p walks on the moon. Pseudo-acting does not have to involve activity; for instance, p and q, p did not kill anybody . (3) Acting - Instances of acting. A type of acting is acting regardless of time; for instance, p and q dance on ice. An instance of acting is acting at a certain time; for instance, p and q are dancing on the ice at time t. Since there are all sorts of borderline cases, the distinction needed is the one between acting (in general) and instances of acting (at a particular time t); for instance p takes his morning walk. In addition to these three distinctions Kanger points out three other main problems which need more attention than what usually has been given : the characterization problem for acting, the elimination problem for acts and the identity problem for acts. The characterization is given in the theoretical framework of language L. Kanger says that "a formula F without occurrences of person names expresses an n-person acting if and only if there is a choice of n person variables p\, ....p; such that (P1)"'(Pn)(F = Do(p],..·,Pn,F» & (~PI) · · ·(~Pn)(PI,· .. -P« are distinct & F)

is true" (p. 123). (For further qualifications and distinctions also see p. 123.) The elimination problem for acts concerns noun phrases denoting acts. The phrase denoting an act is put in a standard form. For instance 'ice-dancing' will be rephrased in standard form as 'the act done by every P such that P dances on ice' (see further 1972, p. 124). The elimination problem is easily solved also in modal contexts such as 'sailing is necessary'. This is reduced to either (P)N(P does sail) or N(P)(P does sail). The identity problem for acts concerns the question "when are acts identical?" At this stage Kanger simply says that this problem does not arise in the

STIG KANGER'S ACTIONS AND INFLUENCE

195

language L, unless it is extended with for instance variables and quantifiers for acts and predicates and relations of acts . This is, however, not being done in this context. (Later on, in "Unavoidability" the identity for actions is defined. ) In this thorough article "Law and Logic" Kang er returns to earlier explications for rights and takes the full move into the language L. In this framework the explications can be formulated as in for instance the following (p. 125): (1)

(2)

a has a claim again st ~ with respect to F iff Shall Do(~,F) a has a power against ~ with respect to F if and only if May Do(O"F).

The action operator, Do for see to it has a structure not present in Kanger's previous presentations but now it has come to stay. The rights relations shall be bypassed in this paper. 8. INFLUENCE

8.1 'Infl uence' introduced The first signs of an analysis for ' influence' are visible in "Law and Logic". Kanger says that this notion seems similar to that of a right. Thus, we may distinguish simple types of influence which a part y may have in relation to another party with respect to a state of affairs or condition. By replacing Shall by Unav, and May by Can = -Unav- we can express the different influence types , he explains: (1 )

(2) (3)

being forc ed: a is said to be forced in relation to ~ to F if Unav Do(o',F) having power: a is said to have power in relation to ~ with respect to F if Can Do(o',.F) irresistibl e power: a has irresistible power in relat ion to ~ with respect to F if and only if Can Do(O"F) & Unav-Do(~ ,-F)

The last clau se, (3), corresponds to the combination of power and immunity. (For further details see 1972, p. 127.) Kanger does not proce ed with a fullblown anal ysis of influence in this paper. The first signs of second-order action can, nevertheless, be seen very briefly as a distinction connected to "the overlapping of the Do-operator" (p. 127). The distinction concerns 'power in relation' to a party and 'power over' a party.

196

Def

GHITA HOLMSTR0M-HINTIKKA

The party a is said to have power over the party F if Can Do(a,Do(~,F))

~

with respect to

Yet another matter is that of exercising power: Def

a is said to exercise power over

~

with respect to F if Do(a,

Do(~,F)).

This in turn may be distinguished from the influence a exercises over ~ with respect to F when Do(a,Can Do(~,F)) etc.

8.2 Revision of 'Influence' In a later paper, "Nagra synpunkter pa begreppet inflytande" ("Some Aspects on the Concept of Influence") (1977), Kanger returns to his thoughts in "Law and Logic". But now this concept of 'influence' undergoes an extensive revision as does the concept of action . The starting point is now the notion Possible and See to it that. By means of these operators types of influence may be analysed, analogously to types of rights . Thus the influence type Capacity tFormaga) may be interpreted as follows: X has in relation to Y an influence of type capacity with respect to S(X,Y) is synonymous to It is possible that X sees to it that SeX,Y) (Kanger, 1977, p. 12). The possibility in this context means practical possibility, says Kanger. In what then follows, the earlier approach to ' influence' (Kanger, 1972) is radically revised due to his further development of the action operator see to it that but more importantly to a further analysis of 'practical possibility' . For a comparison let me mention that the concept of 'practical possibility' was later by myself developed far beyond Stig 's analysis , yet starting from the modal notion of possibility. As this modal operator is then relativized to an agent I consider the notion ::JmM(x,E(x,m,r)) the equivalent of practical possibility. The interpretation for the formula is: there is some means m such that it is possible for x to see to it that r by means of m. (Holmstrorn-Hintikka, 1991)

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197

9. SEE TO IT THAT

See to it that is an action operator which is now (in Kanger, 1977) developed into a three-place relation. When X sees to it that the state of affairs S is the case, then it is reasonable to assume that X sees to it that S by some means (or with the help of) which in the frame of the order of nature and society leads to it that S becomes the case. I The interpretation of the statement X sees to it that S is synonymous to There is some means A such that X sees to it that S by means of A. 2 For instance if X sees to it that the lamp is lit then X sees to this by turning the switch. The means can, but need not, be an active action. If the lamp is already lit then X can see to it that it remains lit by e.g. keeping somebody from turning it off. But if nobody tries to switch it off then X's means consists in remaining prepared to intervene - should somebody try.' In this paper on influence Kanger introduces the notation Sf-eX,S) and Sf-(X,S,A) for see to it and Mojl for practical possibility. I shall nevertheless stay with his Do-operator in particular since this is what he returns to in later papers (1985 and 1986). Influence is interpreted as follows (1977 , p. 14): X has in relation to Y an influence of the type capacity with respect to SeX,Y) (PI)

Poss( ::JA) Do(X,S(X,Y),A)

Being too weak a relation, allowing everybody to have such a capacity, Kanger strengthens this notion to (P2)

(::JA)Poss Do(X ,S(X ,Y),A) by some means it is possible for X to see to it that SeX,Y)

which he finds adequate for the concept of 'influence' (1977 , p. 15).

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GHITA HOLMSTROM-HINTIKKA

10. TYPES OF INFLUENCE 10.1 Atomic Types of Influence In analogy with types of rights Kanger defines types of influence by means of non-contradictory, maximal conjunctions of sentences of the form : (~A)Poss

Do(X,S(X,Y),A)

(~A)poss

Do(Y,S(X,Y),A)

or or by sentences we arrive at by the denial of any of the components "(~A)", "Poss", "Do" or "S(X,Y)". Such sentences are named simple sentences of influence (1977, pp. 15-16). Due to the acceptance of the two principles I-II below the atomic types received at are reduced to 26. I II

not Poss Do (X,S,A) Poss not Do (X,S,A) (~A)

The first one states the trivial truth that there is always some means by means of which it is impossible to see to it that S. The second principle states that it is always possible that X abstains from seeing to it that S by means of A. X can always remain passive. All simple sentences of influence containing either form I or II should be excluded from the non-contradictory conjunctions of simple types of influence. Such sentences are either always true and thereby redundant in the conjunction or else always false and thereby excluded from the non-contradictory conjunctions . Having made this observation we shall go one step further and introduce the following notation. We shall write Can Do (X,S) for

(~A)

Poss Do (X,S,A)

thereby utilizing one of Stig 's variants (albeit not yet in Kanger, 1977). The reading of Can Do (X,S) is: X can see to it that S becomes the case. Note that Can Do (X,S) and Poss Do (X,S) are distinct (cf. HolmstromHintikka, 1997, p. 108). The four simple types ofinfluence are called Capacity (Formag a), Security (Sakerhet), Counter-capacity and Counter-security. As the explication of these types resemble those of simple types of rights there is no need for a complete list here . Suffice it to mention only one (for the complete list see Kanger, 1977)

STIG KANGER'S ACTIONS AND INFLUENCE

199

X has in relation to Y an influence of type capacity with respect to S(X,Y) This is interpreted in the symbolic language just introduced: Can Do (X,S(X,Y» In addition we have Not: Can Do (Y, not-S(X,Y» Can Do (X, not-SeX,Y» Not: Can Do (Y, S(X,Y» The atomic types of influence are constructed as combinations of simple types and their denials. (For further developments of 'practical possibility' initially inspired by discussions with Kanger see Holmstrorn-Hintikka, 1991, pp. 96 -123.)

10.2 Higher Order of Influence Sometimes we need a capacity to have a capacity but more interestingly we may also practice our influence. It is easy to see that if Y executes his influence of the type capacity in relation to X concerning S(X,Y) this simply means that Y sees to it that S(X,Y). Kanger accepts a set of principles (not necessarily exhaustive) govering an agent's influence: III IV V VI VII

If Do (X,S) then S If Do (X,S) then Can Do (X,S) If Can Do (X, Can Do (X,S» then Can Do (X,S) If Can Do (X,S) then Can Do (X, Can Do (Y,S»4 If Can Do (X,S) then Can Do (X, not-Can Do (Y, not-Sj)

10.3 Unavoidability As was mentioned earlier, the concept of ' unavoidability' enters Kanger's writings at an early stage (New Foundations, 1957). In the beginning Kanger saw 'unavoidability' as a modal concept and says that 'OughtA' always implies that '-A is avoidable' . 'Unavoidable' is as we noticed to be understood in the following way: only such facts are unavoidable which would be or would have been outside the range of reasonable and foreseeingly planned joint human efforts .

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GHITA HOLMSTROM-HINTIKKA

Kanger's essay named "Unavoidability" is in many respects an interesting one . For one thing, this is an attempt to develop "a non-modal explication of the notion: P is unavoidable for agent A" (Kanger, 1986, p. 227). Moreover, the concept of 'condition' is here systematically used for the previously employed 'state of affairs' and here Kanger also returns to and further develops his thoughts about 'judgement' first introduced in "Law and Logic".5 Furthermore, one of the principles accepted (principle (3», makes a major difference in the later Kanger inspired developments of Action Theory, for instance in my own work (Holmstrom-Hintikka, 1991). In addition, in "Unavoidabili ty" Kanger modifies the equivalence rule of action to become an on-a-par relation defined for three-place action predicates (cf. Holmstrom, 1985, p. 60) . Conditions may be exemplified by "Agent A turns on the electric switch" (1986, p. 227) . In what follows the letters M,P,Q,R are used as variables for conditions. Thus, the Do-predicate Do(A,P,Q) is also considered a condition. But since "all conditions are judgements", Do(A,P,Q) is also a judgement. As a reading of this predicate Kanger suggests among others: "By means of the fact that P, the agent A brings about the result Q", or as: "With P at hand, A sees to it that Q", or: "By means of P agent A causes it to be the case that Q" (cf. Holmstrom, 1985). A central concept in this study is the equivalence relation P-AQ, for for A P is on-a -par with Q defined as" Def

(P-AQ)

= «R)(Do(A,P,R)

- Do(A,Q,R))) & «M)(Do(A,M ,P) - Do(A,M,Q»)

In Kanger (1972), as we have seen, when actions were still expressed as two-place relations Kanger thought of identity in terms of the principle II

If f-(F = G), then f-Do(u,F)

= Do(u,G)

A still stricter equivalence than the on-a-par relation is a congruence relation: Def

(P "" AQ) = (R)«P - R) -A (Q - R» As far as A's activities are concerned P and Q are equal. (1986, p.229)

By means of the parity condition Kanger introduces the concepts of being 'avoided' and more importantly 'unavoidability'. As 'avoidability' the way it is introduced may feel less intuitive for somebody - including myself -let me here give a full presentation of Kanger' s thoughts:

STIG KANGE R'S ACT IONS AND INFLUENC E

201

For most agents, many conditions are, so to speak , out of reach, or avoided, in the sense of not being involved in any of the agent' s activities as means or as results . Th us, for example, for me all false co nditions as well as true condi tions such as: Kiwis breed on Kapiti Island , are clearcut insta nces of avoi ded conditions. Obviously, the more passive an agent is, the more conditio ns would be avoided. This notion of being avoided is defined in the straightforward way: Def

Avd(A ,P) =« R) ..., Do(A,P,R) & (M) ..., DO(A,M,P»

( 1986, p. 229) Kanger' s action theory conce ntrates on an aspec t of action where intentions or purposes have no place. Had these mental aspect s or ac ts been included I belie ve that the concept of avoiding coul d have been expressed as follow s: agent A see s to it for the purp ose that P that. he does not see to it that R by mean s of M i.e., Int Do(A,-'Do(A,M,R),P) where HInt Do" is the predi cate for purp osive action. (Cf. Holrnstrom-Hintikka, 199 1. In part icular see p. 127.)7 Unav oida bility is defined simply by mean s of the equivalence Def

Unav(A,P) = (Q)(( P & Q)

- A

Q)

'T rivially true cond itions and co nditions that always are at hand for the agent are often unavoid able" Kanger says .

10.4 Assumptions and Consequences Among the assumpti ons made for the Do-predicate, in his paper (1986), Kanger mentions three : (1 ) (2) (3)

Do (A,P,Q) - (P & Q) -,Do(A,P,T) (::JQ)Do(A,T,Q)

Fro m my point of view, the cruc ial principle is the third one according to whic h it is ass umed that the age nt sees to a result by means of a tautology at hand , doing nothing with respect to the result, leaving thin gs as they are . Myself, in developin g my own theory (Holmstrom-Hintikka, 199 1) I realized that actions need to be divided into three, not only two as I had thought (see e.g. Holm strom , 1985): 1.

2.

mere causation : C(A,M,R) for agent A, M suffi ces to make sure that R instrumental action : E(A,M,R) by means of M age nt A sees to it that R

202

3.

GHITA HOLMSTROM-HINTIKKA

purposive action: A(A,R,P) agent A sees to it that R for the purpose that P.

Although Kanger's action concept resembles intrumental action the on-apar definition and principle (3) turns it into a mere causation i.e., causing in my terms. For instance when a bus comes to a quick stop and A thereby pushes B (unintentionally), this is an example of a mere causation, agent A causes it that B is pushed. Kanger's readings of his Do-predicate from "sees to it", "brings it about" to "causes" signals that his emphasis is sometimes on the mere causation side whereas seeing to it turns on the intentional aspect. By separating the three action concepts we can grasp the, as it were, more physical doing of an agent. For this kind of doing we can construct a logic where the on-a-par relation holds . We can even show that 'causing' is a lattice (Holmstrom-Hintikka, 1991). Instrumental action, where the C::3R)Do(A,T,R) is not accepted is a partial ordering and the on-a-par relation does not apply either. Had this distinction between 'mere causation' and 'instrumental action' not been made we would have had the following problems: (i) (ii)

Accepting (::JR)Do(A,T,R) is counterintuitive Refuting this principle, i.e., accepting ..,(::JR)Do(A,T,R) leads to a logical inconvenience. We can easily prove that (T -A ..L)

My solution to this problem was to make this tripartition." FURTHER INFLUENCE Higher-order influence can be applied to other agents as to their actions. But it is conceivable that an agent can influence other people's mental acts such as thoughts, intentions and wills as well (see for instance Holmstrom-Hintikka, 1997). It is also easy to understand that one agent, a person, can influence a computer to perform certain moves . Kanger did not explicitly develop his concept of 'influence' to the fields of artificial intelligence nor even to influencing other agent's purposes or wills . But Kanger did by his own philosophy influence other scholars to continue along his lines inspired by his ideas .

Boston University and University of Helsinki

STIG KANGER'S ACTIONS AND INFL UENCE

203

NOTES Cf. Lars Lindahl' s 'instrumental action' in Lindahl (1977). Ingmar Porn also refers to a three-place action oper ator E,(p,q) altho ugh his emphas is is on the two-pla ce relation E,p. 3 Compare Lars Lindahl ' s null actions (Lindahl, 1977 ). It is easy to find counterexa mples to principle VI. Kanger was for a lon g time work ing on various aspects of the co nce pt of 'j udgement' . In oral presentati on s and pri vate discussions I came to be familiari zed with prelimin ary outlines and thou ghts of a logi c of judgement as well as of other expli cation s thereof. Un fortunately all possible further notes on thi s subjec t seem to be lost. 6 A similar definition for congruence was used in Holmstrom (1985, p. 60 ). Althou gh modal operators are at usc here (H-H, p. I27 f.). it is conceivable that Kanger' s method could be empl oyed inter alia. 8 For further and deeper understandin g of this probl em and its sol ution sec HolmstromHintikka (199 1, pp , 29 -52).

REFERENCES Chellas, Brian F. (1969), The Logical Form of Imperatives, Perry Lane Press, Stanford. Holm strom , Ghit a ( 1985), "Wills, Purp oses and Actions" in Ghita Holm strom and Andrew J.I. Jones (eds.), Actio ll, Logic and Social Theory , Acta Philosoph ica Fennica Vo l. 38, Societas Philosoph ica Fennica, Helsinki, pp. 49 -62. Holmstrom-Hintikka, Ghita (199 1). Action, Purpose and Will. A Formal Theory, Acta Philosoph ica Fenn ica Vol. 50, Socie tas Philosophi ca Fenn ica , Helsink i. Holmstrom-H inti kka . Ghita (1997 ), "Actions in Action" in Ghita Holm strom -H intikka and Raimo Tuomela (eds.), Contemporary Action Theory, Vol. I, Kluwer Academic Publ isher s, Dordrecht , Holland/Boston , U.S.A., pp . 109-1 34. Jones, A.J.I. and M.J . Sergot (199 1), "On the Role of Deont ic Logic in the Characterization of Normati ve System s". Proc. First Internat ional Workshop on Deonti c Log ic in Computer Science (DEON '9/), Amsterdam, Decembe r 1991. Jon es, A.J.I. and M.J. Sergot ( 1992), "Deontic Logic in the Represent ation of Law: Towards a Methodology". Artificial Iruelligence and Law, Kluwer. To appea r 1992. Jones, Andrew 1.1. and Mare k Sergot (1992), "Formal Specification of Security Requirements usin g the Theory of Normative Positions" in Y. Deswarte , G. Eizcnbe rg and J.-J. Quisquater (eds .), Computer Security - Esorics 92 (Proceedings of the Second Eur opean Symposium on Research in Computer Securit y), Sp rin ger-V erl ag (Lecture Notes in Computer Science 648) , Berlin, pp. 103- 121. Kanger, Helle (198 1), Human Rights and their Realization, Dep artment of Philosophy, University of Uppsala, Uppsala. Kanger, Stig (195 7), New Founda tions fo r Ethical Theory , Stockho lm. Reprinted in R. Hilp inen (cd.) ( 197 1), Deontic Log ic: Introductory and Systemat ic Readings, D. Reidel Publi shin g Co mpany, Dordre cht , Holl and, pp. 36-58. Kanger, Stig (1963), " Rattighe tsbegreppet " (The Concept of Right) in Sju fi losofis ka studier tilliignade Anders Wedberg , Stockh olm . Repr inted and extended in St ig Kanger and Helle Kanger (196 6), "Rights and Parliamentarism," Theo ria 32, 85- 115. Kanger, Stig ( 1972 ), "Law and Logic ," Theo ria 38, 105- 132. Kanger , Stig ( 1977), "Nagra synpunkter pa begrepp et int1ytande" (Some Aspec ts on the Con cept ofi nfluence) , Filosofi ska smulor tilliignade Konrad Marc-Wogau, Philosoph ical Studies 27,

204

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Department of Philosophy, Uppsala University, Uppsala, pp. 12-23. In translation 2000 by Sharon Rider in G. Holrnstrorn-Hintikka, S. Lindstrom and R. Sliwinski (eds .), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. I, Kluwer Academic Publishers, Dordrecht, Holland/Boston, U.S.A. Kanger, Stig (1985) , "On Realization of Human Rights" in Ghita Holmstrom and Andrew J.1. Jones (eds.), Action, Logic and Social Theory, Acta Philosophica Fennica Vol. 38, Societas Philosophica Fennica , Helsinki , pp. 71- 78. Kanger, Stig (1986) , "Unavoidability" in M. Furberg et al. (eds.), Logic and Abstraction. Essays Dedicated to Per Lindstrom on his Fiftieth Birthday. (Acta Philosophica Gothoburgensia, No.1), Gothenburg, pp. 227 - 236. Lindahl , Lars (1977), Position and Change, D. Reidel Publishing Company , Dordrecht, Holland/Boston, U.S.A. Porn, Ingmar (1970) , The Logic of Power, Basil Blackwell, Oxford. Porn, Ingmar (1971), Elements of Social Analysis, Department of Philosophy, University of Uppsala , Uppsala . Porn , Ingmar (1977), Action Theory and Social Sciences, D. Reidel Publishing Company, Dordrecht , Holland/Boston, U.S.A.

SvEN OvE HANSSON

KANGER'S THEORY OF PREFERENCE AND CHOICE

1. INTRODUCTION

The logic of preference and choice preoccupied Stig Kanger at least from the late 1960's until just before his death in 1988. He devoted three publications to this subject: "Preferenslogik" [Preference logic], 1968. (pp. 199- 208 )1 "Choice and modality", 1976. (pp. 211 -213) "A note on preference logic" , 1980. (pp. 209 -210) In addition, he left behind an unfinished manuscript on choice functions , which I believe that he still worked on in the months before his death: "Choice based on preference". (pp. 214-230) I will attempt to assess these papers in Kanger's own spirit, which means that I will focus on whatever problems and weaknesses I believe myself to have found in them . Needless to say, in doing so I run the risk - or is it certainty? - of exposing my own lack of understanding rather than that of Stig Kanger. The papers under review are 20 years apart , and it is only to be expected that they differ in notational conventions. In what follows, I will use a notation that is close to Kanger's own, but modified to avoid the differences in notation between his different papers. 2. PREFERENCE LOGIC "Preference logic" from 1968 is a discussion of principles for the three relations "at least as good as" (z), "better than" (» , and "equal in value to" ("'), as applied to sentences representing states of affairs .2 Without further ado Kanger accepted as trivial that ~ is reflexive and transitive (a quasi-ordering) and that > and > are definable in the conventional way from ~ (P > q iff p ~ q & q i p, p '" q iff p ~ q & q ~ p).

205

G. Holm strom-Hintikka, S. Linstr om and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work . Vol. /I. 205-219. © 200 1 Kluwer A cademic Publishers. Printed in the Netherlands.

206

SVEN OVE HANSSON

2.1 Intuitive Counterexamples

He devoted a couple of pages to a critical discussion of some further conditions on preferences that had been proposed by Soren Hallden, G.H. von Wright, and Bengt Hansson. Hallden's postulate p zq iff (p & -'q)

z

(q & -yJ) ("conjunctive expansion'")

was said by Kanger to be "not entirely reasonable". It has , he said, "an unnatural consequence: every condition that has a neutral supplement is as good as a contradiction. (By a neutral supplement to a condition p is meant a condition q such that p z (p & q) and p z (p & -'q).)" (p. 199) It may be worth the trouble to write out this argument in detail:

(4)

p z (p & q) p z (P & -'q) p & -,(p & q) (P & -'q) z J.

(5)

p ZJ.

(1) (2) (3)

z

(p & q) &

-yJ

premise premise (1) , conjunctive expansion (3), intersubstitutivity (2), (4), transitivity of z

Kanger's treatment of von Wright's preference principles is strongly critical. For instance, he shows that von Wright's principle p > q iff (p & r) > (q & r) and (p & -,r) > (q & -'r)

implies that p j q for all p and q. The proof is simple: just let p > q and substitute J. for r. Then we obtain p & J. > q & J. , hence by intersubstitutivity J. > J., contrary to the irreflexivity of> (that follows from the reflexivity of ~ and the definition of> in terms of z). Kanger's substitution of J. for r does not comply with von Wright's intensions. von Wright seems to have had in mind some kind of atomic or logically independent sentences, but he did not explain this clearly. In a note, Kanger quoted von Wright as saying "Let r be some state which is different fromp and q and which is not, in its turn, a truth-function of any other states.:" Kanger rightly noted that every r is a truth -function of other states (such as r & u and r & -'u), and in another note (note 3) he declared that he chose to disregard the unclear condition imposed by von Wright on r. A more charitable option would of course have been to try to reconstruct it in a manner compatible with logical consistency. Three principles put forward by Bengt Hansson seem to have interested Kanger the most: BHl

p ~ q or q ~ p

KANGER' S THEORY OF PREFERENCE AND CHOICE

BH2 BH3

207

If p <: q and p <: r then p <: (q V r). If p <: r and q <: r then (p V q) <: r.

Kanger provided a striking counterexample to BH2: " It is at least as good to get ten crow ns as to get a ten-crown book , and it is at least as good to get ten crowns as to get a ten-crown box of choco late. But it is not clear that it is at least as good to get ten crowns as to get a ten-crown book or a ten-crown box of choco lates. The disjun ction can entail that one gets both ." (p. 20 I)

Intere stingly enou gh, Kanger did not let this counterexample deter him from developing preference logics that satisfy the two principles. He did not explain why, but he said: "In what follows , I shall nonetheless give a reasonable interpretat ion of the preference relation that satisfies BH2 and BH3, but not BH I. I shall also give a reasonab le interpretation that satisfies BH I, but not BH2 or BH3. I must leave unsolved the problem of finding a reaso nable interpretation that satisfies all three Hansson principles." (p. 20 I)

I will return in Section 2.3 to this probl em left open by Stig Kanger. Let us first have a look at the main part of his paper. It is devoted to a representation, or as he says, "interpretation" of preference relati ons. 2.2 The Two-Dim ensional Int erp retation of Pref erences

As a basis for his interpretation of preferenc es (I will leave open whether or not it can be called a semantics of preference logic) Kanger introduced a twodimensional value plane with two ortho gonal axes, one of which represents utilit y and the other disutility.' Every maximally specified condition (we may think of them as possible world s) has a position on this value plane. More than one of them may have the same position in the plan e. Every condition can be represented as a set of such maximally specified conditions (corresponding in the finite case, I assume, to their disjunction). Therefore, to each condition is associated a set of points on the plane . For each condition p , Dp is the cylindrification along the utility axis of p . It therefore repre sents the disutilit y of p. Similarly, Up is the cylindrification of the di sutility axis, and hence repre sents the utilit y of p . (See Kanger's diagram s, pp. 202 - 203 .)6Based on this interpretation, Kanger introduced what he called an "elementary preferenc e logic", such that: p <: q iff Up <: Uq and Dq <: Dp .7

BHl (completeness) is not satisfied in this interpretation, whereas BH2 and BH3 are satisfied. Next, Kanger introduced a metric into his value plane. For any two condi-

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SVEN OVE HANSSON

tions p and q, the real number A(p,q) represents the distance between p and q. Letting T (tautology) represent the intersection of the axes", each condition p has the coordinates A( T,Up ) and A(T,Dp). Assuming that utility and disutility can outweigh each other, the value of p is equal to A(T,Up) - A(T,Dp), and Kanger's second constructed preference relation can be defined as follows: p z ' q iff A( T,Up) - A(T,Dp) ~ A(T,Uq) - A(T ,Dq)

where ~ represents arithmetical "at least as great as". This preference relation, he says somewhat cryptically, "may be suitable if one wants to combine preference logic with a theory for value-measurement". (p. 206) The relation z ' clearly satisfies BHI (completeness), Kanger mentions that it does not satisfy BH2 or BH3, but he does not show why. The following example illustrates that BH2 does not follow even if we require that U(p V q) be interposed between Up and Uq, and correspondingly for D(p V q). More precisely, BH2 does not follow even if we require that

If A(T,Up) ~ A(T,Uq) then A(T,Up) ~ A(T,U(P V q» ~ A( T,Uq) If A( T,Dp) ~ A( T,Dq) then A( T,Dp) ~ A(T,D(P V q» ~ A(T,Dq) The example is as follows A( T,Up ) = 7 A(T,Uq) = 10 A(T,Ur) = 5 A( T,U(q V r»

=9

A( T,Dp) =0 A( T,Dq) = 5 A( T,Dr) = 0 A( T,D(q V r»

=1

Kanger also introduced a third type of preference relation, ~ It is an extension of the second metric that allows for (probabilistic) mixtures of conditions. As he noted, ~ n is "as meagre as" z '. In particular, it does not satisfy BH2 or BH3 . /I .

2.3 7'tvO A4ysteries This paper ends with "A pessimistic closing word" in which Kanger deplored that he had not been able to find "a reasonable and clear interpretation of the preference relation that offers an interesting preference logic and which (at least for essential classes of conditions) satisfies the principle ofcomparability and brings preference logic closer to the mainstream of preference theory". The latter, he said , "is largely concerned with decisions and the measurement of value". (p. 207 ) Based on the emphasis that he put in the paper on BH2 and BH3 we can safely assume that he wished them to be satisfied in "a reasonable and clear

KANGER 'S THEORY OF PREFERENCE AND CHOICE

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interpretation". As we saw above, he explicitly left it as an open question to find an interpretation of preferences that satisfies all ofBHl, BH2, and BH3 . This leaves us with two mysteries, the first of which has already been indicated: First, why did he want BH2 and BH3 satisfied, in spite of the striking counterexample to BH2 that he had himself constructed? Se condly, why did he not construct a model that satisfies all of BHl, BH2, and BH3, in spite of the fact that this can easily be done? To substantiate the second question, note the following: OBSERVATION: If ~ satisfies BHl (completeness), transitivity, and

p

~

q

~

p

~

(p V q)

~

q (disjun ctive interpolation),

then it satisfies BH2 and BH3.

r. Due to completeness, either q ~ r or r ~ q. In the first case, disjunctive interpolation yields q ~ q V r. Applying transitivity to p ~ q and q ~ q V r, we obtain p ~ q V r. The second case is proved analogously. Q.E.D. PROOF: Only the proof of BH2 will be given. Let p ~ q and p ~

Disjunctive interpolation is satisfied by a wide range of value-based preference relations, such as - maximax preferences: p ~ q if and only if the best alternative in which p holds is at least as good as the best alternative in which q holds. - maximin preferences: p ~ q if and only if the worst alternative in which p holds is at least as good as the worst alternative in which q holds." It is also possible to construct a preference relation based on Kanger's utility and disutility functions in which disjunctive interpolation (and thus BH2 and BH3) will be satisfied. All you have to do is to perform the subtraction of disutility values from utility values separately for each point in the plane, before the aggregation. 2.4 The Paradox of Exclu siv e Disjunction

In "A Note on Preference-Logic" , Kanger introduced a new, intriguing preference paradox. Given the basic properties of weak preference ( z ) and strict preference (» , as introduced in his previous paper, he showed that the following three postulates are logically incompatible: (I retain his numbering, and follow him in using to denote exclusive disjunction.)

+

(III)

There are two states of affairs p and q such that no two elements of {p ,q,--'p,--'q} are equal in value.

210

(IVa) (IVb)

SVEN OVE HANSSON

If p > q, then p z (p f q) If p > q, then (p f q) 2 q

(Note that the combination of IVa and IVb amounts to a variant of the interpolation principle discussed above, but now for exclusive disjunction.) This paradox is subjected to a penetrating discussion by Wlodek Rabinowicz in another contribution to this volume. My own reaction to it is to reject (IVa) and (IVb) (and with them disjunctive interpolation, that I believe to be plausible for inclusive but not for exclusive disjunction). All that Kanger says about (IVa) and (IVb) is that they "seem to be evident". However, it is not difficult to construct counter-examples. Let p and q be two logically independent states of affairs such that the four composite states of affairs are ordered in terms of value as follows :

p&"'q

V

-.p & q

V

p&q

V

"'p&"'q For a simple example, suppose that my wife and I are both on our way home from work, but cannot communicate with each other before we come home. (This was in the distant past, before the advent of the mobile phone.) We need a loaf of bread, and we both pass a shop in which we can buy bread. The best outcome is that only one of us buys bread, but it is better that both do it than that neither of us does, since in the latter case we will be out of bread. On my way home I pass a bakery with excellent bread, whereas she only passes a grocery store. Therefore, it is marginally better that I buy bread (P) than that she buys bread (q) . My (and her) preferences over complete alternatives can then be as shown above. Let us evaluate this example both with ceteris paribus and weighting methodology. To begin with the first-mentioned of the two methodologies, there is only one total state, p & »q. in which p holds but not q. There is only one total state, "'p & q, in which q holds but not p . The former is better than the latter. We then clearly have p > q. Similarly, there is only one total state , -.p & q, in which P f q holds but not p. There is only one state, p & q, in which p holds but not p f q. The former is better than the latter; hence (p f q) > p, contrary to Kanger's (IVa). Next, let us use probability-weighted utility. Suppose that each of the four

KANGER'S THEORY OF PREFERENCE AND CHOICE

211

complete alternatives has the same probability, and that their values are as follows: p Sc rq -'p&q p&q -'p & -'q

12 10

6 0

Then the value of pis 9, that of q is 8, and that of p f q is 11, again contrary to Kanger's (IVa) . Would Kanger have been convinced by this example that his principles (IVa) and (IVb) are not as "evident" as he said? I am not so sure, given the way he treated his own counterexample to the related disjunctive properties BH2 and BH3 proposed by Bengt Hansson. 3. THEORY OF CHOICE Kanger developed his ideas on choice in the unfinished "Choice based on preference" and in the short note "Choice and modality", published in 1976. 3.1 Binary Choice Functions

In "Choice based on preference", Kanger introduced an unconventional type of choice functions, namely functions with a pair of sets, rather than a single set, as arguments. He did not explain why he did this. One possible explanation may be that he took seriously the dependence of preference relations on alternative sets. It is not self-evident thatx is preferred to y among {x,y} if and only if x is preferred to y among {x ,y,z} . More generally speaking, the preference relation best suited for guiding choices among a certain set of alternatives need not be a suitable guide for choosing among a particular subset of that set - not even if comparison-costs are negligable. Two types of examples of this are well-known from the literature on choice functions. First, the alternative set may carry information, as in Amartya Sen's example: "[G]iven the choice between having tea at a distant acquaintance's home (x), and not going there (y), a person who chooses to have tea (x) may nevertheless choose to go away (y) , if offered - by that acquaintance - a choice over having tea (x), going away (y), and having some cocaine (Z)."lO Hence, in formal terms, letting Ch stand for a conventional choice function, Ch( {x,y}) = {x} and Ch({x,y,z}) = Ivl" Secondly, choice may be positional. In a choice between a small apple (x), a big apple (y), and an orange (z), you may choose the big apple, but in a

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SVEN OVE HANSSON

choice only between the two apples you may nevertheless opt for the smaller one. !2 In formal terms, we then again have have Ch( {x, y }) = {x} and Ch( {x, y, z l) = {y}. There does not seem to be any sensible way to base a choice function such as this on a conventional preference relation. From Ch( {x,y}) = {x} it seems to follow that x is strictly preferred to y, x > y. Similarly, from Ch( {x,y,z}) = {y} it seems to follow that y > x . This is an impossible combination under the weak assumption that strict preference is asymmetric. A rather obvious way to deal with this problem is to distinguish between different preference relations, one for each alternative set. We can then write a >y b to denote that a is preferred to b among the elements of V. In the above examples, we have x>lx,ylY but y>(x.y.z1x, which does not contradict asymmetry since >(x,yj and >(x.y.z) are distinct preference relations. But Kanger goes further than this. He replaces the indexed> by a choice function. Taking only one step at a time in the direction of his formalism, this corresponds to introducing an indexed choice function, ChV<X) with the following properties in our example:

Chlx.yj({x,yD = {x} Ch{x.y.z)({x,y}) = {y} What makes this notation more general than the ~v notation is of course that the argument (X in Chy(X)) need not have exactly two elements. I have difficulties in finding a reasonable intuitive interpretation of this type of choice functions. We can interpret x>yy as saying that x is preferred to y in a comparison among the elements of V. But what does it mean to say, for instance, that X

E

Ch{x.y.z.wj({x,y,z})?

Here, we have two alternative sets, one smaller among which the choice is made ({ x,y,z}) and one larger which provides a sort of background or general context for the choice ({x,y,z,w D. (Note that V is not the set of all potential alternatives, since Kanger also has a larger background set U which is interpreted in that way .)!' Formally, we can define ChV<X) in terms of an underlying, indexed preference relation. This can be done as follows: If, X c V, then ChV<X) = {x E X I (y E X) ""(y>~)} If, X s: V, then ChV<X) is undefined. The reason for the last clause is of course that >y is defined only for arguments that are elements of V. However, Kanger did not wish to introduce such a

KANGER 'S THEORY OF PREFERENCE AND CHOICE

213

restriction. Instead, he had his choice function defined for all backgrounds (V) and all arguments (X) . In "Choice based on preference", he introduced his binary choice function C as follows: "I intend to interpret C(V,X) as the set of those alternatives of (VnX) which , compared with alternatives of V, are regarded as not being worse than any alternatives of (VnX) . In other words: x E C(V,X) iffx E (Vn X) and (y)(y>yX - y E ( V \ X» " (p. 214)

where y>yX has the interpretation "y is better than x in V". We can rewrite Kanger's definition as follows : C(V,X)

= {x EO V n X I (y EO V n X) -'(y >yx)}

We can say that C(V,X) selects those elements of vnX that are preferable" according to the standards for elements of V. Kanger's C and our Cli are interdefinable in a fairly obvious way: C(V,X) = Chy(VnX) If X c V, then Chy(X) = C(V,X). If X g;, V, then Chy(X) is undefined.

Hence , Kanger's binary choice functions are extensions of the indexed choice functions Ch to the case when the argument is not a subset of the index. To see the intuitive meaning of C, let V denote the set of all violin sonatas and X the set of all compositions by Xenakis. Then C(V,X) is the set consisting of those violin sonatas by Xenakis which are, in a comparison including all violin sonatas, not worse than any other violin sonatas by Xenakis. 3.2 The Dual Functions

However, Kanger uses neither C nor >v as primitive notions . Instead, his primitive function is a function D that is dyadic just like C, and with the interpretation: "D(V,X) is the set of those alternatives of V which, compared with alternatives of V, are regarded as not being worse than any alternative of V\X. . In formulas: x E D (V,X) iff x E V & (y)(y> yX - y E ( VnX» " (p. 214)

We can rewrite this as follows : D(V,X)

= {x EO V I(y EO V\ X)-,(y> yx)}

Using the same example as above, D(V,X) is the set of all violin sonatas that are not worse than any violin sonata by anybody else than Xenakis. As was observed by Rabinowicz and Sliwinski (1990), D is a somewhat "artificial concept" in the sense of not immediately suggesting itself, from an

214

SvEN OvE HANSSON

intuitive point of view, as a primitive notion in the theory of choice. It seems as if Kanger had formal rather than intuitive reasons for choosing it as a primitive. For expository convenience, I will introduce an expressively equivalent function E, such that E(V,X) = D(V,V\ X), or more explicitly:

= {x E V I (y E VnX)-'(Y>yx)} Clearly D(V,X) =D(V,vnX) =D(V, V\ (V\X)) =E(V,v\X) . The advantage of E(V,X)

E over D is that it is somewhat more intuitive. We can interpret it as follows : E(V,X) is the set of V-elements that are not >y-worse than any Xelement.

In our example, E(V,X) is the set of all violin sonatas that are not worse than any violin sonata by Xenakis. If weak preference is transitive and complete, then E(V,X) is the set of violin sonatas that are at least as good as Xenakis 's best violin sonatas. We can regard E as an extended choice function, that extends the choice in X to the whole of V. If ~ is transitive and complete, then E(V,X) is the set of V-elements that are at least as good as the best X-elements. Kanger did not name his D function. Since it is the dual of E, it can be called the dual extended choice function. Kanger himself provided a definition of C in terms of D: C( V,X)

=XnD( V,V \ X)

For E, the corresponding definition is: C( V,X)

=XnE(V,X)

Kanger did not provide a definition of D in terms of C. Under the assumption that C and D are based on the same transitive and complete weak preference relation, E and D can be defined in terms of C as follows: E( V,X) = U {C( V,Y) I X c Y} 15 D(V,X) = U{ C(V,Y) I V c XUY}16

3.3 Choic e as Modal Logic In the introductory section of "Choice Based on Preference", Kanger announced that the concluding section - which was never to be written - was intended to point out the close connections "between choice functions - the D-function, in particular - and some kinds of operators studied in certain extensions of Boolean algebras and in modal logic". A preview of what he

KANGER'S THEORY OF PREFERENCE AND CHOICE

215

intended can be found in his brief "Choice and Modality" from 1976. There he discussed a monadic operator, which is essentially the D of the longer paper but with a fixed background set V. I ? He pointed out that if D satisfies the following three axioms, then it "determines" a normal modal logic in Segerberg's sense: (I) (II) (1lI)

If X c Y, then D(X) \:: D( Y) n XEF D (X) \:: D( nXEFX) if (/) f. F \:: &:>( \1) . D(\I) = V

In what way does Kanger's D function correspond to the modal operator of necessity? Let X denote a set of alternatives. Then the corresponding sentence X can denote that the actual alternative is an element of X. We then get the following nice correspondences: X ~Y

~

x-x 1..

(a contradictory sentence)

T (a tautology) J:: X nY X&X

etc.

Now for the difficult part: How can D be interpreted? In the last sentence of "Choice and modality", Kanger says that "there is a close and direct connection between choice theory and modal logic which might be worth some further exploration". He did not, however, specify the nature of this connection. Sten Lindstrom has provided an interpretation of the D operator that seems to capture what Kanger must have had in mind." Let us consider Kripke structures of the form a binary relation on U. Subsets of U are denoted X, Y,... and its elements are denoted x, y ,... We can use x 0= Yas an alternative notation for x E Y. Next, let D be an operation from and to &:>( U), such that for all X c U and x, y E U: x E D(X) if and only if (y)(y>x - y E X)

(This coincides with the definition of D given above, provided that V is fixed and coincides with u.) Equivalently:

x

0=

D(X) if and only if (y)(y>x - y

0=

X)

Hence, D is a necessity operator in a modal logic in which> has taken the place of the accessibility relation . A tentative reading of D(X), also proposed by Lindstrom, is "If things were better, then X would be the case".

216

SVEN aVE HANSSON

A large part of "Choice based on preference" (Sections 3- 8) is devoted to demonstrations of what properties of D are needed to ensure that it can be based on a preference relation in the intended way, and what further conditions on D are necessary to obtain additional properties of the preference relation. These sections are conceptually fairly straight-forward. They are expressed in terms of the binary D function, but V is kept constant so that they could also have been expressed in terms of the unary D function of his 1976 paper "Choice and Modality". These results do not differ drastically from what are now standard results on the relations beween choice and preference. 3.4 Shifts in the Background Set

Still more interesting problems are discussed in Section 9 and the final unfinished Section 10 of "Choice based on preference". According to the plan announced in the introduction, these sections and an unwritten Section 11 were meant to provide "rationality principles of another kind in which the background V is no longer kept fixed" . Kanger rejected as implausible two extreme views. One of these is that choices are completely stable, or more precisely: determined by the restriction to V of a preference order among the "grand domain" U which includes all possible background sets V. The other extreme view is that there are no rulebound connections between preference orders for different background sets. His own proposal, he says, "goes midpoint between the two extremes". (p.216) His proposal is expressed in terms of preference orders determined by the binary choice function, as follows :

X>~Y iffy f£ C(V,{x,y}) He assumed that there is a set of basic preference orders that corresponds to "simple aspects" with respect to which alternatives can be compared. Furthermore, he assumed that these aspects are finite in number . Each of these aspects corresponds to a subset of U, and for each of these subsets VI"" Vm there is a preference relation >f that is obtained by applying the binary choice function C with Vk as a background set. Each >f is required to be a strict weak ordering k (but he mentioned the possibility of relaxing that condition and only requiring them to be semiorders). In my view, the (tacitly made) assumption that each aspect can be represented by a set Vk is far from unproblematic. From a formal point of view it has the obvious consequence of limiting the number of aspects to 2m where m is

217

KA NGER'S TH EORY OF PREFERENCE AN D CHOI CE

the number of alternatives. More importantly, in practical examples of preferences combining several aspects, it doe s not seem intuitively sound to require that each aspect be bound to a set of altern ative s to which no other aspect is bound. It would not have been difficult for Kanger to generalize his approach and introduce a set of basic preference relations that are unrel ated to subsets of U. He did not explain why he did not do so. The reason may have been either metaphysical or connected with his strict standards of economy in terms of formal primitives. Kanger formed the tran sitive closure of the union of all basic preference orders, T= (>~ u ...u >~ )* 1

f1I

A

A

Beside s T he also had use for its converse T. In particular, he used T\ T which is "w hat remains of T after removal of all cycles in T' or, as he also says, "the non-controversial part" of T (p. 228). Kanger ' s stability ax iom requires, for any subset W of U (not necessarily among the sets that give rise to basic preference orders) that; (T \

i , n (W x W) c

>~ ~ T

The right of the two inclu sions in this axiom only requires that >~ be a subset of T. The left-hand conditi on states that the preference relation for the alternative set W contains (the W-part of) the non-controversial part of T. As Kanger himself preferred to express it, this means that >~ is obtainable by resolving cyclical patterns in T. He regarded this as a fairly weak condition, "if there are several conflicting basic preferences, then the axiom says very little about the structure of >~" . A further condition which he wished to impose is that this resolution be based on a preference ordering of the basic preference orders. He introduced a relation ?: such that for any sets V and W, >~ <: >~ means that >~ is a "stronger or more important preference order" than >~ (p. 229). Interestingl y enough, he did not introduce <: as a primitive notion . Instead , he based this second-order preference relation - ju st like the first-order preference relations such as >~ - on the binary choice functi on C. The definition is:

>~ <: >~ iff >~n w ~ >~ c >~uw The first inclu sion is intended to express that the preferenc e order >~nw conforms with >~ and the second that the preferenc e order >~u w conforms

218

SVEN OVE HANSSON

with >~ . He called this "a very natural relation of importance" (p. 216). The last thing he wrote in this paper was an axiom saying that ~ is transitive. It is not easy to guess how he intended to continue. In the introductory section of the paper he announced that "the consistency condition we have in mind reduces to the requirement that this relation be a quasi-ordering" (i.e., reflexive and transitive). 3.5 Kanger's Achievements in the Theory of Choice One of my strongest memories of my supervisor Stig Kanger is an occasion when I had constructed an operator for legal power. 19 He was not as enthusiastic as I had hoped. Instead , he complained that I had packed too much into a single formal notion . He was right, of course, and I later modified the construction in accordance with his advice . However , I must confess that when rereading his unfinished "Choice based on preference" I am close to uttering a tu quoque . This paper contains two important innovations in the theory of choice: binary choice functions and an exciting but elusive connection between choice and modality . Each of them gives rise to substantial difficulties, and they also seem to be independent of each other. Therefore it might have been more appropriate to develop each of them in isolation from the other. Or am I wrong? Perhaps he was on the track of a connection that it now remains for us to discover. Stockholm University NOTES I would like to thank the participants of the memorial symposium on Stig Kanger's Contribution to Logic and Philosophy, Uppsala, March 13-15 1998, for valuable comments on an earlier version of this paper. Special thanks are due to Sten Lindstrom and Sven Danielsson. All page references are to Volume 1of the present edition. In the Swedish text, the arguments of the preference relations are called "villkor" (conditions) . "What is expressed by a formula will be called a condition" (An Algebraic Logic Calculus , p. 71). 3 This term seems to have been introduced by Jennings (1967). For a discuss ion of this and related conditions, see Hansson 1996 and 1998. 4 This sentence appears twice, on pp. 29- 30 and 31, in von Wright's book . Kanger 's quotation has a couple of typos . 5 In Swedish, "nytta" and "skada" . 6 Kanger also associated to each condition a set of advantages and a set of disadvantages. However, these are not necessary for the present outline of his ideas. 7 He wrote Dp " Dq here, but that must be a misprint (p. 204) . 8 -c in Kanger's notation . 2

KANGER ' S THEORY O F PREFERENCE AN D CHOICE

219

See Hansson 1998 for further details on disjunctive interpo lation. Sen 1993, p. 502. See also Kirchste igcr and Puppe 1996. II I use Ch for conventio nal choice functions and follow Kanger in using C for those that he intro duced. 12 Anand 1993, p. 344. On posit ional choice, see Gardenfors 1973. 13 The best exp lanation tha t I am aware of was proposed in conversation by Sven Danielsson: x and y may be imposs ible to distinguish when you look at x, y. and z. However, when IV is availa ble, then you see that x is better than IV, where as y is not better than IV . It may then be reason able for a cho ice among [x.y.z} to depend on whether or not IV is includ ed in the background that can be used to facilitate com parisons. 14 Mor e precisely: unbeaten . 15 The pro of is left to the reader, with the hint that for the left-to -right direction it is sufficient to show that E(V,X) ~ u {C( V,Xu {z}) I z E V }. 16 D(V.X) =E(V, V\ X) = u {C(V,Y) I V\ X c Y} = u {C(V,Y) I Vc Xu Y} . 17 The background set was denoted V in "Choice Based on Preference" and A in "Choice and Modality". Vwill be used here. The monadic (dual extended choice) operator was denoted 0 in "Choice and Modalit y". D will be used here, 18 In a letter to the author, August 26, 1998. 19 Han sson 1986. 10

REFERENCES Anand , Paul (1993), "The Philosoph y of Intransitive Preference", Economic Journal 103, 337 346. Gardenfors, Peter (1973), "Positionalist Vot ing Functions", Theory and Decision 4, 1- 24. Hansson , Sven Ove (1986), "A Note on the Typo logy of Rights", in Paul Needham and Jan Odclstad, Changing Positions. Essays Dedicated to Lars Lindah l on the Occasion of His Fiftieth Birthday, Philosophical Studies no. 38, Dep artment of Ph ilosoph y, Uppsala University, Uppsala, pp. 47- 57. Hansson , Sven Ove ( 1996), "Wh at is ceteris parib us preference ?", Journal of Philosophical Logic 25,307-332. Hansson , Sven Ove (1998), Structures of Value, An Investigation of the Statics and Dynamics of Values and Norms. Lund Philoso phy Report s 1998: I. Jennings, R.E. (1967), "Preference and Choice as Logical Corre lates", Mind 76, 556-567. Kirchst eiger, Georg and Clemens Puppe ( 1996), " Intransitive Choic es Based on Tran sitive Preferen ces: The Case of Menu-Depend ent Information", Theory and Decision 41,37 - 58. Rabin owicz, Wlod ek (1999), " Preference Logic and Radical Interpr etation . Kanger Meets Davidson", this vo lume. Rabinowicz, Wlodck and Ryszard Sliwinski (1990), Some Scandi navian Contributions to Decision Theory, mimeographed, Department of Ph ilosophy, Uppsa la University. Sen , Amartya (1993), " Internal Co nsisten cy of Cho ice", Econometrica 61, 495- 521. von Wright , Georg Henrik (1963), The Logic of Pref erence, Edinburgh University Press, Edi nburg h.

WLODEK RABINOWICZ

PREFERENCE LOGIC AND RADICAL INTERPRETAnON KANGER MEETS DA VIDSON 1

The primary purpose of this paper is to trace the intellectual effects of an encounter between two very different philosophers working in two seemingly unconnected areas . As will be seen, Stig Kanger's meeting with Donald Davidson led the latter to modify his influential theory of radical interpretation and gave the former an inspiration to set up a rather striking paradox in preference logic . While the paradox can be dissolved, radical interpretation confronts some serious difficulties. A PARADOX IN PREFERENCE LOGIC Stig Kanger's paradox can be found in "A Note on Preference Logic". This characteristically short two-page note was his contribution to a Festschrift for Thorild Dahlquist, published in Uppsala in March 1980. To introduce the paradox, let us suppose that 2: is a preference relation on a set of states of affairs (propositions) that is assumed to be clo sed under Boolean operations. I.e., 2: is the type of relation that is studied in preference logic. We may read "A ?:: B" either evaluatively, as "State A is at least as good as state B", or descriptively, as "State A is at least as preferred as state B" . Now, let us consider two conditions that one might want to impose on 2: : Interpolation of Exclusive Disjunction (lED): For all states A, B, if A 2: B, then A ?:: (A.;.-B ) ?:: B. Here , A.;.-B stands for the symmetrical difference of A and B, respectively, i.e., for the Boolean analogue of exclusive disjunction (either A or B, but not both). Four Levels (4L ): There are some states A, B such that A, B, -A and -B all occupy different levels in 2: . That is, either A » B or B >- A, and similarly for all the other pairwise comparisons between the four states. -A and -B stand for the complements of A and B, respectively. Strict preference >- is immediately definable in terms of weak preference 2: : 22\

G. Holmstriim -Hintikka ; S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. 1/. 22 1- 242. © 200 \ All Rights Reserved. Printed by Kluwer Academic Publish ers. the Netherlands.

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A >- B =df A .c B and it is not the case that B

.c A.

Similarly. we can define indifference as follows: A ::: B =df A .c Band B

.c A.

The two conditions above "seem evident", says Kanger, but they cannot be upheld in tandem:

Kanger 's impossibility result: There is no weak (= transitive & complete) ordering .c that satisfies both the Four Levels and the Interpolation of Exclusive Disjunction. The assumption of completeness for .c is very strong . In many cases, we would like to allow for gaps in the preference ordering. If states A and B are significantly different from each other, we may well insist that neither A .c B nor B .c A are the case . Can we avoid assuming completeness and still prove that Kanger's two conditions are mutually incompatible? Fortunately, the answer is: yes, we can . It is quite enough to assume that .c is transitive. ' In fact , as may be seen from the proof below, it would be enough just to assume that > is a transitive relation. While the transitivity of indifference is relatively uncontroversial when indifference is interpreted evaluatively, the transitivity requirement is more problematic given the descriptive interpretation. As is well known, if preferences of a subject are determined by pairwise comparisons, then intransitivities of indifference are to be expected, due to such psychological mechanisms as insufficient discrimination or attention shifts . Still , these phenomena might also be interpreted in a different way as showing that a subject's preference ordering should be determined holistically rather than by a series of independently conducted pairwise comparisons. Transitivity will then function as an apriori constraint on any adequate holistic determination of preference. Here, then , is the paradox in its final version:

Strength ened impossibility result: There is no transitive .c that satisfies both the Four Levels and the Interpolation of Exclusive Disjunction. The proof that follows slightly simplifies Kanger's original version. By the transitivity of .c, >- is transitive.' Consequently, (4L) implies the existence of states A, B, -A and -B that are linearly ordered by strict preference, in one of the twenty four possible ways. In Kanger's somewhat sadistic version of the proof, we are supposed to go through each of these possible cases on its own , one by one, and show that each such case is incompatible with (lED). Here , we shall be more economical.

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Proof Assume (lE D). We first show that, given (lE D), the excl usive disjunction of unequally valued states must be equal in value to one of its disjuncts: Lemma:

If A >- B, then A '" (A-:-B) or B '" (A-:-B) .

Proof of Lemma: Assume that A >- B. By (lE D), (i)

A 2: (A-:-B) 2: B.

Thus, in particular, A 2: (A-:-B). We consider two cas es: Case I : (A-:-B) 2: A. Then A '" (A-:-B), by the definition of "', and we are home. Case 2: It is not the cas e that (A-:-B) 2: A. Th en A >- (A-:-B), by the definition of >- . Thus, appl ying (lE D) once aga in, this time to A and A-:-B , (ii)

A 2: (A-:-(A-:-B)) 2: (A-:-B).

Sinc e A-:-(A-:-B ) = (An -(A-:-B)) u ((A-:-B)n -A)

=(An B) u (Bn -A) =B,

it foll ows from (ii) that B 2: (A-:-B). Since we already know from (i) that (A-:-B) 2: B, it fo llows that B '" (A-:-B), by the defin ition of "'. A nd so we are home aga in. We next show that : Lemma + the transitivity o f '" = (4L) is violated. Pro of by reductio: Suppose (4L) holds for some A and B. We then have either A >- B or B >- A. In each case, Lemma implies: (i)

A '" (A-:-B) or B '" (A-:-B) .

We also have either -A >- -B, or -B >- -A . In each case, Lemma implies: (ii) Since A-:-B (iii)

-A '" (-A -:- -B) or -B '" (-A -:- -B) .

=-A -:- -B, (ii) is logicall y equivalent to: -A '" A-:-B or -B '" A-:-B.

(i) and (iii) imply that two of the four states A, B, -A, -B are on the same level as A-:-B. But gi ven the transiti vity of "', this mean s that these two states occupy the sa me level , contrary to the hypoth esis. 0 How are we to deal with this paradox? To begin with, one might argue that the two conditions are not as ev ide nt as Kan ger sugges ts. Th at (4L) is not quite uncontrover sial will be seen below. But what is es pec ially important is that the see mingly strong intuiti ve appea l of (lED) is decepti ve. In the first place, the

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value of the exclusive disjunction may simply be incomparable with the values of the disjuncts. The evaluation of disjunctive states is notoriously contested. But even if we ignore this possibility, we might still want to reject (lED). If the disjuncts are mutually compatible with each other, the value of their exclusive distinction need not lie somewhere in-between the values of the disjuncts. After all, in the exclusive disjunction of A and B, the alternatives envisaged are not simply A and B but A-and-not-B and B-and-not-A. Thus, the value of such a disjunction should lie somewhere in-between the values of the mutually incompatible alternatives. These alternatives coincide with A and B only if A and B themselves are incompatible with each other. This suggests that (lED) is intuitive only when its weakened: Interpolation of the Exclusive Disjunction of Incompatibles (lED/): For all mutually incompatible states A and B, if A >- B, then A ~ (A-;.B) ~ B.4 Unlike (lED), (1E0I ) is fully consistent with the Four Levels, as will be seen below. Does it mean, then, that (lED) is a condition without any appeal whatsoever? I do not think so. Exclusive Disjunction Interpolation does have some independent plausibility. Even when two unequally valued states are mutually compatible, it is not easy to see how their exclusive disjunction could be preferred or dispreferred to both of them at the same time. In order to finally dissolve the paradox, therefore, we should explain how it is possible that the Four Levels and the Exclusive Disjunction Interpolation, taken separately, appear to be rather plausible (if not quite "evident"), even though , as we have seen, they cannot be upheld together. Such an explanation is not hard to come by. (4L) and (lED) are both plausible, because each of them is separately satisfied by some relatively plausible interpretations of ,::. Examples of such interpretations will be presented below. Case I: The Four Levels is satisfied while the Interpolation of Exclusive Disjunction is violated. Example: the expected utility interpretation. On this interpretation, A ~ B iffU(A) ~ D(B), where U measures the expected utility of a state. Following Jeffrey (1983), the expected utility of a state is here taken to be the weighted sum of the expected utilities of its different possible realisations, with weights being the conditional probabilities of the realisations in question. Thus, if P is the underlying probability function, we assume that the following holds for all states A and B:

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(EU)

225

If P(AnB ) = 0 and P(AuB ) > 0, U(AuB ) = P(NAuB )U(A) + P(B/AuB )U(B)

It is easy to see that (EU) immediately implies (lEDI): If A and B are incompatibl e, then A-;.B = AuB. Consequently, U(A-;.B ) = U(Au B) equals the weighted sum of the utilities of the disjuncts. Thu s, the utility of A-;.B must lie somewhere between the utilities of A and B. 5 (lED), on the other hand, may well be violated by this expect ed utility interpretation, as is shown by the following example. Let C, D and E be three equi-probable states that are mutually exclusive and jointly exhaustive. Thus , P(C) = P(D) = p eE) = 1/3, and P(CnD) = P(DnE) = P(CnE) = O. Let the Uvalues for C, D and E be 0, 2 and 3, respecti vely. Consider A = CuD and B = Cu E. Note that A-;.B = DuE . Using (EU) , we can calculate the utilities of A, B and As-B: U(A) = P(C/A)U(C) + P(D/A )U(D) = 1/2 x 0 + 1/2 x 2 = 1. Similarly, U(B) = 1/2 x 0 + 1/2 x 3 = 1,5, while U(A-;.B) = 1/2 x 2 + 1/2 x 3 = 2,5. T hus (lE D) is violated but the Four Levels holds; states A, B and their both complements have all different utilities: U(-A) = U(E)

= 3, U(-B) = U(D) = 2.6

Case 2: Th e Interpolation of Exclu sive Disjunction is satisfied while the 4Level Condition is violated. Examples: extremal preference (maximin or maximax), or ceteris paribus preference . Suppose that preferences between states are derived from preferenc es between possible worlds. The preference between states is defin ed as the preference between selected world-representatives of the states in que stion . To be more preci se, assume a weak preference ordering ~ on the set U of possible worlds . We identify states with sets of worlds : a state is identified with the subset of U that consists of all worlds in which that state obtains. Let c be a choice fun ction from states to world s, such that c(A) is defined iff A =/:. 0, and for each such non-empt y A, c(A) E A. Then, define A ~ B as c(A) ~ c(B). (The relation ~ does not obtain between A or B, if either of them is empty.) Supp ose that the choice function c is based on some underlying linear ordering 3> of U: c(A) picks out that world in A that comes highest in 3>. (If such a world is to exist, for any non-empt y A, whether fini te or not, it is not enough that 3> is linear; every subset of U, and not ju st the finite one s, must contain the maximal »-element. In other words, U is well-ordered by the converse of 3>.)

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Possible interpretations of»: (i) » is a "linearisation" of ~ on U (with equally good worlds coming in an arbitrary order). That is, » is any linear ordering such that for all A and B, if A comes above B in ~, then A:> B. Then ~ is maximax. (ii) » is a "linearisation" of :; on U (with equally good worlds again coming in an arbitrary order). The ~ is maximin.' (iii) :> is the ordering of worlds with respect to their similarity to the actual world (= the status quo, the reference world). For any worlds x and y, x » y iff x is more similar to the actual world than y is. Then A ~ B stands for: Ceteris paribus, it would be better that A than that B. In other words: It would be better that A rather than that B, other things being equal (to what they actually are)."

It can now be shown that if ~ is based on a preference relation ~ between world-representatives of states, selected by means of a choice function c that is based on a linear ordering » of U, then (lED) is satisfied and (4L) is violated. Proof' Clearly, since ~ is transitive on worlds, the derived ordering ~ is transitive on states. Therefore, in view of the impossibility result, it is enough to prove (lED). Suppose that A rB, i.e., c(A) r c(B). Let x be that element of {c(A), c(B)} that comes highest in the s-ordering of AuB. Clearly, x (f AnB, since otherwise we would have c(A) = c(B) and so it would not be the case that c(A) r c(B). Since x E AuB, it follows that x E A-:-B. Consequently, c(A~B) = x. But then c(A-:-B) = c(A) or c(A-:-B) = c(B). In each case, A ~ (A-:-B) ~ B.9 WHAT'S

rms GOT TO DO WITH DAVIDSON?

Here is Davidson's problem of radical interpretation: What does it take to understand another person, more or less from scratch? Davidson's goal is to elicit what a subject means by what he says, what he believes and what he prefers. The elicitation should be based on some relatively unproblematic set of empirical data. While in his earlier work, collected in Truth and Interpretation (Davidson 1986), the goal of radical interpretation was just to disentangle meaning from belief," the new project involves an additional task: determination of the subject's desires. In this way, understanding a person allows us to understand not just what he thinks but also what he does, given that what he does is a function of what he believes and desires.

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This project of a three-way elicitation - of meaning, belief and preference - has been pursued by Davidson over two decennia in a series of publications starting with his 1978 lecture "Towards A Unified Theory of Meaning and Action " (published two years later, see Davidson, 1980) and continuing with such papers as "Expressing Evaluations" (Davidson, 1982), "A New Basis for Decision Theory" (Davidson, 1985), "The Structure and Content of Truth" (Davidson 1990), and "The Folly of Trying to Define Truth" (Davidson 1996). In order to understand Davidson's project, it is instructive to compare it with Quine's well-known conception of radical translation (cf. Quine, 1960). Here is how Davidson himself describes Quine's undertaking: Noting that , while there is no direct way to observe what speakers mean , all the evidence required to implement communication must be pub licly available, Quine surveys the relevant available evidence, and asks how it could be used to elicit meanings. [...] For Quine, the key observables are acts of assent and dissent , as caused by events within the ambit of the speaker. From such acts it is possible to infer that the speaker is caused by certain kinds of events to hold a sentence true . [Added in footnote : The step from observed assents to the inferred attitude of holding true is not. I think , exp licit . in Quine.] Just here a basic challenge arises. A speaker holds a sentence true as a result of two considerations: what he takes the sentence to mean, and what he belie ves to be the case. [...] How can the roles of these two explanatory factors be distinguished and extracted from the evidence ? [...j Quine 's key idea is that the correct interpretation of an agent by anothe r cannot intelligibly admit certain kinds and degrees of difference between interpreter and interpreted with respect to belief. As a constraint on interpretation, this is often called by the name Neil Wilson gave it [Wilson (1959)], the principle of charity. (Davidson, 1990, pp. 31St)

Quine's problem is then that different hypotheses about the speaker's meaning can be defended by adjustments in the hypotheses about his beliefs. Quine's solution of this problem is that the speaker's beliefs must obey constraints imposed by the principle of charity that require s a far-reaching consensus in beliefs between us and the speaker. Given these constraints on beliefs, the available data about the speaker's assents to and dissents from sentences (including the external circumstances of such assents and dissents), we can fix the speaker's meaning (up to remaining indeterminacies). The differences between Quine and Davidson are at least sixfold: (i) While Quine pursues a project of (radical) translation, Davidson is interested in interpretation. While Quine is concerned with the conditions of successful translation from a speaker's language into the interpreter's, I emphasise that the speaker needs to know of the semantics of the speaker's language, that is, what is conveyed by the T-sentences entailed by a theory of truth [for the speaker' s language] . (Davidson, 1990, p. 3 I9).

(ii) Quine takes the circumstances that prompt assents to (or dissents from)

observation sentences to be patterns of stimulation of nerve endings rather than

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external objects or events. These proximal stimuli are taken to determine the empirical content of such sentences in the speaker's language, which implies that their translation is relatively unproblematic: they are translated into the sentences in the interpreter's language that are correlated with the similar stimulation patterns. (Relativ ely unproblematic, that is. For some second thoughts on this issue, cf. Quine, 1990, sections 15 and 16.) Davidson's view on this issue is more in line with common sense: ... interpretation depends (in the simplest and most basic situations) on the external objects and events salient to both speaker and interpreter, the very objects and events the speaker's words are then taken by the interpreter to have as subject matter. (Davidson, 1990, p. 321)

(iii) For Quine, charity is much less inevitable than it is for Davidson. While we normally assume that the speaker shares our beliefs to a large extent, this assumption of consensus is not inescapable: it might be overturned by considerations of simplicity. The linguist assumes that the native 's attitudes and ways of think ing are like his own , up to the point where there is contrary evidence . He accord ingly imposes his own ontology and linguistic pattern s on the native wherever compatible with the native's speech and other behavior, unless a contrary course offers striking simplifications. (Quine, 1990, pp. 48f., my italics.)

(iv) Furthermore, Quine 's version of charity requires us to assign to the speaker not so much the beliefs we actually hold but rather the beliefs we imagine we would have held in the speaker's shoes. [The translator] will favor translations that ascribe beliefs to the native that stand to reason or are consonant with the native ' s observed form of life. [...] Practical psychology is what sustain s our radical translator all along the way, the method of his psychology is empathy : he imagine s himself in the native 's situation as best he can . (Quine, 1990, p. 46.)

Davidson appears to be less prepared to make such allowances for expected divergences and, when he does make them, he takes empathy to uncover divergences in needs and valuations rather than divergences in beliefs. The interpreter is counselled to interpret agents he would understand as having, in important respects, beliefs that are mostly true and needs and values the interpreter shares or imagines himself sharing ifhe had the history of the agent and were in compatible circumstances. (Davidson, 1985, p. 93)

(v) While Quine's project is strictly behaviourist - the basic data concern an outward assent and dissent behaviour of the speaker - Davidson is fully prepared to allow as empirical data the speaker's mental attitudes to sentences, such as holding a sentence to be true. To be sure, such a mental attitude is manifested in an assent behaviour, but the attitude and the behaviour are not the same thing.

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(vi) However, Davidson thinks that knowing which sentences are assented to or held true by a subject under which external circumstances would be insufficient as the data basis for elicitation. To fix the meaning, and especially to fix the meaning of theoretical sentences, we need to determine evidentiary relations between the different sentences in the speaker's language: what he counts as evidence for what, what sorts of evidence would make him consider a given sentence as more probable . We need probability assignments rather than simple yes-or-no attitudes of holding true. As Davidson illustrates this point: ... a sentence [tentative ly] interpreted as meaning that there is a patter on the roof, if held true (given a high probability), ought to increase the probability of the senten ce [tentatively] interpreted as meaning that it is raining. In this way, by marking what the speaker takes as evidence for the truth of a sentence, it is possible to interpret sentences and words of an increasingly abstract and theoretic al nature . (Davidson , 1982, p. 15)

But how are we to access the subject's probabilities? Here it is time to introduce another important source of inspiration for Davidson: Frank Ramsey's program for decision theory . The task Ramsey put himself was to simultaneously determine a subject's probability assignments (quantitative degrees of belief) and his quantitative degrees of preference for different outcomes . (Cf. his paper "Truth and Probability", posthumously published in Ramsey, 1931.) The data for this elicitation were to be the subject's ordinal preferences over various gambles, as revealed by his (actual and hypothetical) choices . Being prepared to gamble on an event shows something about an agent's probability for the event in question and about his valuation of the possible outcomes. As Quine , Ramsey encountered the problem of compensatory adjustments: changes in the hypotheses about the subject's probabilities could be compensated for by adjustments in the hypotheses about his degrees of preference for various outcomes . He solved that problem by imposing a number of constraints on the subject's preferences over gambles (including such constraints as transit ivity, completeness, etc.). Given the constraints, the preferences could be seen as going by the expected utility of gambles and the constraints made it possible to uniquely determine the probability assignments that underlied these expected utilities . When event probabilities were determined, it was then easy to determine the degrees of preference for outcomes (up to the positive linear transformations). Davidson accepts Ramsey's idea of taking ordinal preferences as basic data, but rejects using preferences over gambl es as base. As he points out, when the subject is given a choice between gambles, each gamble is presented to a subject as a proposition of the form:

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A ifC, B if not C. Thus, preference over gambles is an intensional attitude - such preferences are attitudes towards propositional objects . As such, they are unfit to function as empirical data for radical interpretation. The same objection immediately applies to Ethan Bolker 's and Richard Jeffrey's approach to expected utility (cf. Jeffrey, 1983): their base for the elicitation of probabilities and degrees of preference are a subject's ordinal preferences over propositions: that A is more desirable than that B. Again, such a base is unfit to function as an empirical point of departure for radical interpretation. But from Davidson's point of view, Bolker-Jeffrey constraints on preferences over propositions have one important advantage as compared with Ramsey's: they are formulated in such a way as to allow the objects of preference to be any set of entities whatsoever, as long as the Boolean operations are definable on that set. In view of the close connection between Boolean operations and truth-logical sentential connectives, it thus becomes possible to replace propositions as objects of preference with linguistic entities. This leads to Davidson's own proposal. Provided we can identify the truthfunctional (Boolean) connectives in a subject's language, why not replace Bolker-Jeffrey preferences over propositions with preferences over (otherwise) uninterpreted sentences as basic data for radical interpretation? The subject prefers the truth of p to the truth of q (in symbols, p

~

q).

This is a mental attitude towards linguistic objects . As such, it still is an intentional attitude , but no longer an intensional one. The objects of the attitude are not propositions, but sentences that obey well-defined identity conditions. As Davidson puts it: ... the objective was not to avoid intentional states; it was to avoid individuative intentional states, intensional states, states with (as one says) a propositional object. A preference for the truth of one sentence over another is an extensional relation that relates an agent and two sentences (and a time). Because it can be detected without knowing what the sentences mean, a theory of interpretation based on it can hope to make the crucial step from the nonpropositional to the propositional. (Davidson 1990, p. 323)

Constraints on the preferences among sentences are the same as Bolker-Jeffrey conditions on preferences among propositions, with one extra constraint added :

PREFERENCE LOGIC AND RADICAL INTERPRETATION

If P

H

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q is a truth-functional tautology, then p '" q.

For the subject, his sentences are meaningful. But their meaning is originally not given to the interpreter. To replicate Bolker- Jeffrey project of elicitation, (i) we first need to identify the truth-functional connectives in the subject's language. Otherwise, simple sentences may be treated as black boxes. Then, using Bolker- Jeffrey methods, (ii) we elicit the subject's degrees of preference and probabilities for sentences. Unlike in Ramsey 's case and in the case of other standard expected utility theories, utilities in Bolker- Jeffrey approach are elicitable only up to so-called fractional linear transformations (with four free parameters instead of the usual two), and the probability assignment is only determined within certain limits II . But this remaining indeterminacy is a cost that Davidson is quite willing to pay. In the last step, given the subject's probabilities for uninterpreted sentences (plus our knowledge of the external circumstances), and making use of the Principle of Charity, (iii) we determine the subject's intension al attitudes : his degrees of (intensional) belief and the meaning of the sentences in his language. With the meaning in place, the subject's degrees of preference for sentences allow us to determine the third factor in the subject's mental life, his degrees of intensional preference . Which completes the task ofradical interpretation. But how are we to achieve task (i)? How can we identify truth-functional connectives, in an otherwise uninterpreted language, just on the basis of its user's preferences over sentences? This presupposes, of course, that we have already managed to identify the sentences of the language in question, and that we have found some way to test, for each pair of such sentences, whether the subject prefers one sentence to the other , or vice versa, or is indifferent between them. Davidson does not discuss how this can be done. Nor does he discuss the obvious objection that Bolker- Jeffrey constraints on preferences may be much too exacting for any subject to be able to obey them in full. But let us suppose that we can ignore these problems. What then? How are the truth-functional connectives to be identified? The first try was made by Davidson in the summer of 1978, at the Wittgenstein seminar in Kirchberg. In the published version of this lecture (Davidson, 1980), he suggests that we first should find the connective? that stands for neither ... nor in the subject's language. We can do it, he claims , by examining

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the subject's preferences for sentences. Since , as is well known, all the other truth-functional connectives are definable in terms of that one, the rest is easy. For example: A connective n stands for negation iff for all sentences p, np '" p?p.12 And then, A connective c stands for conjunction, iff for all sentences p and q, pcq '" np?nq. And so on. But how can we determine that? is what we have been looking for? Well, Davidson suggests the following test: ? stands for neither... nor iff: For all sentences p and q, p >- q iff q?q >- p?p. Note that, if the hypothesised interpretation of? is correct, q?q and p?p stand for ""q and ""p, respectively. Thus , what Davidson relies on here is the preference principle according to which p is preferred to q iff r-q is preferred to ""p. As it stands , this proposal is unsatisfactory, for at least two reasons, one of which is partly recognised by Davidson himself: (1) The principle p >- q iff oq >- ""p does not generally hold for the intended expected utility interpretation of the preference relation. On this interpretation, the equivalence only holds for probabilistically independent p and q. This means, that "we must devise a way of telling, from preferences among sentences , that two sentences are independent" (ibid., p. 11). While Davidson thinks it likely that such a test for probabilistic independence can be developed, he does not provide it in the paper . In fact, it can be shown that the needed test may be impossible to obtain. As we have seen, in Bolker-Jeffrey theory, the subject's probability assignments are not uniquely determinable. In particular, as Levi (forthcoming) has proved, if p and q are probabilistically independent on one of the probability assignments that are compatible with the evidence, then there will be infinitely many other such admissible assignments on which p and q are not independent. This means that the test of probabilistic independence is unavailable on the BolkerJeffrey approach! Levi (ibid.) suggests that such a test could still be available to Davidson, who takes the interpreter to rely on more information than the mere preference data about the subject. The interpreter can access his own beliefs and values and thus might be able to reduce the indeterminacies in his interpretation by invoking the Principle of Charity. I am not convinced, however, that this solution can be of help at the present early stage of interpretation. Charity

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comes into play at stage (iii), when the interpreter tries to determine the subject's intensional attitudes and in this process is supposed to maximise the consensus between himself and the subject. At present, we are still at the very beginning of the interpretation process, when the interpreter has not yet managed to fully determine the subject's extensional attitudes (= attitudes to sentences). He still has to find out the subject's degrees of preference and probability for uninterpreted sentences. At this preliminary stage, there is no room for charity. (2) Still, I may be too pessimistic." But even if a test of probabilistic independence could be developed, Davidson would not yet be in the clear. There are other truth-functional connectives that? might stand for and p?p will still correspond to negation. An example is the Sheffer stroke (= not both). So even if we found '! such that, for all probabilistically independent p and q, p >- q iff q?q >- p?p , we would still be unable to tell whether? stands for neither-nor or for not both. Davidson made a second try, which he never published, in his lecture in Oslo , in the fall of 1979. This time, instead of neither-nor, he proposed to start with a search for exclusive disjunction in the subject's language. In fact, the latter looks like a right connective to start with as far as preference data are concerned. Note that Ramsey's gambles may be understood as such disjunctions: If it is read truth-functionally, a gamble description "A if C, B if not C" is equivalent to the exclusive disjunction "Either (C and A) or (not-C and B)" . Admittedly, this truth-functional reading of "if' in the gamble description is quite unsatisfactory: it ignores the subjunctive connection that is supposed to obtain between the gamble event (C) and the prizes (A and B). What is even more important, unlike neither-nor, exclusive disjunction does not suffice for the definition of all the remaining truth -functional connectives. However, if the exclusive disjunction could be identified in the subject's language, we would at least make some progress in the process of interpretation. Davidson took it for granted that exclusive disjunction satisfies the interpolation principle (IED), possibly because he did not clearly distinguish it from the (relatively) innocuous condition (IEDI) .' 4 Therefore, he suggested that connective? stands for either ... or iff it satisfies the interpolation con dition: For all p and q, if p >- q, then p .:: (p?q) .:: q.

It is here that Kanger comes in: In Oslo 1979, Kanger showed Davidson that, on the intended expected utility interpretation of .::' (IED) does not generally hold. 15 Kanger's paradox from 1980 is, as far as I can see, an indirect result of this exchange, even though Davidson's name is never mentioned in that paper.

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As Davidson conjectures (personal communication), this polite silence might have been partly motivated by Kanger's feeling that, whatever might be said against it, (lED) is still an internally plausible principle, worth considering in its own right. Kanger had also been helpful in connection with Davidson's last and final try to solve the connective problem." This time, Davidson proposed to start with the Sheffer stroke. He presented this idea in his Hagerstrom lectures in Uppsala, with Kanger as a host, in the Spring of 1980, and he kept to it in all his later work on the subject (cf. Davidson 1982, 1985, 1990 and 1996). Thus, the idea is to first identify ? that stands for not both and then to identify all the remaining truth-functional connectives in terms of? The latter task is easy since all such connectives are definable by means of the Sheffer stroke . (The identification method to be used is thus the same as the one sketched above in connection with neither -nor.i But what about the Sheffer stroke itself? The expected utility interpretation implies that, for all p, (a)

if p >- T, then T .:: ""p,

where T is an arbitrary tautology. We also have, (b)

if T >- p, then ""p .:: T.

We note first that if ? stands for the Sheffer stroke, then p?p stands for ""p, while q?(q?q) stands for a tautology: ""q V ...,-.q. Consequently, Davidson asserts, ? stands for the Sheffer stroke iff, (i)

for all p and q, (u) if p >- q?(q?q), then q?(q?q) .:: p?p, (P) if q?(q?q) >- p, then p?p .:: q?(q?q);

and (ii)

for some p and q, it is not the case that p?p

z

q?q.

According to Davidson, no truth-functional connective apart from the Sheffer stroke satisfies both (i) and (ii). (I have somewhat simplified Davidson's condition (i). Instead of q?(q?q), he makes use of a more complicated expression in which he substitutes q?r for q in q?(q?q). This is unnecessary, since the simpler q?(q?q) already stands for a tautology if? is the Sheffer stroke.) , Condition (ii) is first added in Davidson (1990), probably in order to avoid the obvious objection that (i) by itself would be insufficient to pick out the Sheffer stroke. It is easy to check that (i) is satisfied even if? stands for the material implication. It is also satisfied if ? stands for the "tautological "

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connective that for any two sentences yields truth as value . In both cases, (i) translates into the trivially valid: (u) ifp )- T, then T ::; T, and (P) if T" > p, then T ::; T. In fact, pace Davidson, conditions (i) and (ii) are still insufficient to pick out the Sheffer stroke. Given just (i) and (ii), ? might stand for the "contradietary" connective that for any two senten ces yields falsity as value . Since on the intended interpretation, the expected utility of contradiction is undefined, the contradictory sentences do not belong to the field of ::; . Consequently, if? were such a connective, (i) and (ii) would be vacuously satisfied. As a matter of fact, given just (i) and (ii), ? might also stand for neither... nor. Since on this interpretation q?(q?q) is again contradictory, (i) would be vacuously satisfied and (ii) would be true. To exclude these remaining interpreti ve possibilities, we should add an extra condition, such as: (iii)

for some p and q, p ::; q?(q?q).

But are we now in the clear? Not quite. Some objections, not mentioned by Davidson, still remain: Objection 1: What if ? is not a truth-functional connective at all? For example: What if it stands for it is impossible that both ? Or it is improbable that both ? The Sheffer stroke is the only truth-functional connective that satisfies conditions (l) -(iii) . This much can be shown. But the subject's language may contain several non-truth-functional connectives that also satisfy these conditions. This possibility has not been excluded by Davidson; it is a potentiality he never even considers! The non-truth-functional interpretation could be excluded if Davidson's interpreter had access to some additional information about the subject's sentential attitudes , apart from the evidence about the subject's preference ordering on sentences. Suppose the interpreter also knows which sentences are held to be true and which are held to be false by the subject. (While it is a requirement of rationality that the two sets of sentences be disjoint, they will normally not be jointly exhaustive.) Then the task of identifying the truthfunctional connectives is easy. Thus, ? stands for the Sheffer stroke iff: (u) (P) (y)

Whenever the subject holds p or q to be false, he holds p?q to be true; Whenever the subject holds p?q to be true, he does not hold both p and q to be true; The subject holds p?q to be false iff he holds both p and q to be true. I?

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But, as we have seen, Davidson wants to make do without the information about which sentences are held to be true/false by the subject. In this respect, Davidson's approach to radical interpretation has changed in recent years, as compared with his earlier work collected in Truth and Interpretation. The reasons for this austerity are not quite clear: After all, the attitudes of holding true/false, if directed to sentences, are just as extensional as the attitudes of preferring one sentence being true rather than another. Possibly, Davidson's motivation was aesthetic: it is clearly more elegant to use only one kind of data (preference data) instead of two. But just as possibly, under Jeffrey's influence, he might have come to suspect that the simple yes-or-no attitudes of holding true/false are not easily ascribable to a person. In particular, their relationship to a person's probability assignments is notoriously unclear. As is well known, holding true cannot be identified with assigning high probability: the former, unlike the latter, is supposed to be closed under conjunction.

Objection 2. What is there to guarantee that the preference data reflect the subject's expected utility comparisons? As we have seen in the first section, there are other plausible interpretations of >. To be sure, if the interpreter knew that the subject's preferences satisfied Bolker- Jeffrey constraints, then he would know that they can be interpreted as in expected utility terms. But in order to know that they satisfy the constraints, he must first identify the truth -functional connectives in terms of which these constraints are formulated . At the same time, the interpreter's procedure for the identification of the connectives assumes that the subject'S preferences do obey the relevant constraints, which the interpreter cannot yet know at that stage. I do not think that the two objections mentioned above are unanswerable. Given Davidson's general holistic approach to theorising, they might both be met by a standard recipe: Why not try it out and see how it works? Thus, suppose we find a connective? in the subject's language that obeys conditions (i)-(iii). We can then start the process of interpretation with the hypothesis that? does indeed stand for the Sheffer stroke. If given this hypothesis, the subject's preferences over sentences do tum out to satisfy the Balker-Jeffrey constraints, then we may conclude that? was the Sheffer stroke and that the preference data in fact reflected the subject's expected utility comparisons. On the other hand, if the hypothesis turns out to be unworkable, we might try to look for some other candidate for the Sheffer stroke, and eventually, if all such attempts would fail, for some wholly different set of preference data. So our two objections can be met. But it is somewhat surprising that Davidson never even considers them in his essays.

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A more serious problem arises in connection with the second stage of interpretation - the stage at which the interpreter, after having identified the truth-functional connectives, moves on to the task of elicitation of the subject's degrees of preference and probabilities for sentences. At this stage, the interpreter is supposed to make use of the Jeffrey - Bolker methods of elicitation, as described in Jeffrey (1983). But are these methods applicable for Davidson's purposes? Unlike Davidson, Bolker and Jeffrey start from a preference ordering on a Boolean algebra of propositions. Davidson seems to think that sentences would do just as well, provided the language contains truth-functional connectives. To be sure, truth-functional connectives are not quite Boolean operations, but there is a close correspondence between the two. For each sentence p, we can determine its equivalence class [p] consisting of all the sentences that are truth-functionally equivalent to p. Then , in terms of the truth-functional connectives, it is easy to define the Boolean operations on such equivalence classes . For example, if A = [p] and B = [q] are such equivalence classes, then the complement of A = ["p] and the intersection of A and B = [pAq]. We get in this way the so-called Lindenbaum algebra, which is an example of a Boolean algebra. However, we are not home yet. Bolker-Jeffrey representation theorem, on which Davidson relies , rests on a very strong presupposition. It presupposes that the Boolean algebra on which the preference order ing is defined is both atomless and complete. 18 To explain these notions, let us first define the notion of implication: we shall say that a state A implies a state B iff A = AnB. An algebra is atomless, if for any non-zero state A (i.e., A -J 0 =An-A), A contains a non-zero state B such that B implies A but is not implied by it (we say that such a state is "strictly smaller" than A). If states of the algebra are propositions, then the algebra is atomless if for every consistent proposition there is a stronger consistent proposition. If X is a set of states, a lower (upper) bound of X is any state that implies (is implied by) every member of X. The infimum inf(X) of X is the greatest lower bound of X, i.e., every other lower bound of X is strictly smaller than inf(X). The supremum sup(X) of X is the least upper bound of X, i.e., it is strictly smaller than every other upper bound of X. An algebra is complete if for any set X of states, whether finite or not, it contains the infimum and the supremum of X. As a matter of fact, if a Boolean algebra contains the infima for all its subsets, it will of necessity contain the suprema as well. It can be shown that sup(X)

= -inf({-A: A E X }).

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Consequently, if the states of a Boolean algebra are propositions, then the algebra is complete if for every infinite set of propositions, it contains the infimum, i.e., the proposition that corresponds to the conjunction of the propositions in that set. Now, what about the Lindenbaum algebra? Does it satisfy the two requirements ofBolker- Jeffrey representation theorem? Roughly, this would mean (i) that for any (truth-functionally) consistent sentence p, the language contains another consistent sentence that is stronger than p (atomlessness), and (ii) that for any set of sentences, whether finite or not, the language contains a sentence that is exactly as strong as the conjunction of the sentences in the set (completeness). It is easy to see that these two requirements pull in opposite directions. The number of sentences in any language is countable. The same applies, therefore, to the number of equivalence classes in the corresponding Lindenbaum algebra. If that number is finite , the language will be complete but it obviously will not be atomless. And if the number of such classes is denumerable (= countably infinite), the algebra will be atomless but it will not be complete. In general, no countable Boolean algebra can be both atomless and complete. For a sketch of the proof, due to Sten Lindstrom, see Appendix. Thus, the Bolker- Jeffrey theorem presupposes a non-denumerable algebra of propositions. The number of propositions needed for the representation theorem exceeds by far the sentential resources of any language that we might encounter. This means that Davidson's project of elicitation, with its strong dependence on Bolker-Jeffrey elicitation methods , is doomed from the start . A move from propositions to sentences is necessary if the elicitation is to build on extensional data. But it is precisely this move that makes the BolkerJeffrey elicitation impossible! You cannot both have your cake, and eat it. APPENDIX Theorem: No countable Boolean algebra is both atomless and complete. Proof' Since all countable atomless Boolean algebras are isomorphic (cf. Bell and Slomson, 1969, Corollary 7.7 in ch. 1, p. 30), and since Lindenbaum algebras are Boolean and countable, it is enough to prove that an infinite Lindenbaum algebra is atomless and incomplete. Then it will follow, by isomorphy, that every countable atomless Boolean algebra is incomplete. We first prove that any infinite Lindenbaum algebra LA is atomless. If LA is infinite, the underlying language L must contain infinitely many atomic sentences (= sentences that are not built up from other sentences in L by means of truth -functional connectives). Therefore, for every equivalence class A that belongs to LA, where A = [p] for some consistent sentence p of L. we can

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always find an atomic sentence q that does not appe ar in p. Since p/\q is consistent (as far as truth-functional sentential logic is concerned) and stronger than p, [p/\q] is a non-zero element of LA that is strictly smaller than [pl. Consequ entl y, [p] is not an atom of LA. Given that LA is atomless, it must be incomplete. For let X be any smallest set of states in LA such that for each atomic sentence p either [p] or [,p], but not both , belongs to X. Suppose for reducti o that LA is compl ete. Then LA contain s inf(X). Thi s, however, is impossible, since the infimum of X, if it existed, would have to be an atom of the algebra . To see this, let q be any sentence such that [q] = inf(X). By the definition of X, q must be (truthfunctionally) consistent; and since for every atomic sentence p, q implies either p or -ip, q cannot be extended to a stronger con sistent sentence.

Lund University NOTES This paper is a revised vers ion of Rab inowicz (1998 ). I wish to than k several peop le who have hel ped me with comments. references and sugge stio ns: John Broome, Thori ld Dahlq uist, Sven Danielsson, So ren Hallden, Paul Needham. Jan Odel stad , Rysiek Sliwinski. Howard Sobel. Fredrik Stjernberg, Fredrick Stoutland . Goran Sundholm, and Folke Tersman. I am espec ially gratefu l to Donald Davidson. who has kin dly supplied historical inform ation. and to Sten Lindstrom. who has pro ved a theorem I needed for my argument . apart from being supportive in many other ways. 2 Kanger had to impose completeness because of his choice of primitive: instea d of .>::, he took ~ to be his only primitive and then he simp ly defined j- as the complem ent of <. Such a definition imme diately yields com plete ness for .>: . Which show s. by the way, that j, sho uld not be define d in this way if we want to uphold a distinction between indiffere nce and incomparability. 3 Proof: Suppose . for reductio , that (i) A ~ B. (ii) B ~ C. but (iii) not A ~ C. By the definiti on of ~ and by the transitivity of >, (i) and (ii) imply (iv) A .>: c. Again by the definition of >, (iii) and (iv) impl y (v) C .>: A. Sinc e j, is tran sitive. (v) and (i) imply (vi) C .>: B. which contradic ts (ii). .j Since A and B are inco mpatib le. we could j ust as well replace the excl usive disjunction A.,.B with the inclusive one: Au B. A word of warni ng: Tha t (lE DI) is an intuitive condition does not mean that it is unassailable . A counte r-exampl e is provided in the next footno te. S In fact. a slight strengthe ning of (lE DI) is an axiom in the Bolker-Jeffrey theory of expected utility: (Averag ing)

For all mutu ally inco mpatible states A and B. (i) if A ~ B, then A ~ (Au B) ~ B; and (ii) if A " B. then A " (Au B) " B.

Since Au B =A.,.B. averaging implies that an excl usive disjunctio n is to be strictly interpolated between unequ ally valued disj uncts. if these are mutually inco mpatible . The role of the averaging axiom in Bolker- Jeffrey theory is somewhat simi lar to the function of the axiom of independence in other axiomati sations of expected utility. Even though independence is a much stronger assumption than averaging. the two axio ms express esse ntial ly the same idea . Consequ entl y. the well-know n Allais- type and Ellsbe rg-type objections to

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independence can be re-formulated as objections to averaging and to (lEDI). Thus, (lEOI) is not quite as innocent as it might seem to be! Here is an Allais-type counter-example: Suppose that, being are risk-aversive, you prefer receiving one million dollars to a high chance of five million dollars, even though the chan ce is as high as .9. Let A be the state in which the former but not the latter occurs, while B the state in which the latter occurs but not the former. A and B are thus incompatible and A >- B. Suppose you are offered the pro spect of either-A-or-B, with equal probability for both disjuncts. This state , A.;-B, may be seen as a prospect of getting one million with the probability of .5, five millions with the probability of .45 , and of getting nothing with the remaining probability of .05. It may well happen that you prefer B to A.;-B (if there is some risk of getting nothing, you may be prepared to take an additional risk in order to win a larger rrize), even though you prefer A to B. Which violates (1E0I). In this case , not only the exclusive but also the inclusive disjunction of A and B has a higher expected utility than A and B: V(AuB) = U(C uDuE) = 5/3. But unlike (lED), the Interpolation of (Inclusive) Disjunction : (10 )

For all states A, B, if A >- B, then A

~

(AuB )

~

B,

is a principle that is consistent with (4L) . To see that, con sider the expected utility model just as the one we have described, in which the three equi-probable, mutually exclusive and jointly exhaustive states C, 0 , E are assigned V-values 1/5,2/5,2/5, respectively. (4L) is still satisfied and if C, 0 and E are assumed to be the atoms of this state algebra, it can be checked that the model satisfies (10). 7 Maximin and maximax are two examples of what might be called "extremal preferences". Such preferences are discussed at length in Sven Ove Hansson (1998 ), Ch . 7. 8 For another interpretation of "ceteris paribus preference" cf. Sven Ove Hansson (1998 ), Ch. 6. Following von Wright (1963 ), Hansson interprets such a preference in a very demanding way: A is ceteris paribus preferred to B iff for any possible realisations of these states, the Arealisation is preferred to the B-realisation, provided that other things are equal between them. More precisely, A is ceteris paribus better than B iff every complete alternative x (= Hansson's analogue of a possible world) that instant iates A rather than B is preferable to a complete alternative y that instantiates B rather than A but otherwise is similar to x as much as possible. While I take ceteris paribus preference to be interpretable in terms of the dyadic relation of comparative similarity of alternatives (worlds) to an assumed status quo , Hansson interprets it in terms of degrees of similarity that obtains between pairs of alternatives: we are supposed to look for a pair of alternatives that are as similar to each other as possible. Thus , the underlying comparative similarity relation is four-place rather than two-place: x is more similar to y than x' to y' . It is easy to show that the preference principles validated by the two interpretations differ from each other (cf. ibid ., p.83f). (lED) is a case in point: it is valid on my interpretation but invalid on Hansson's. Both interpretations of ceteris paribus preference are, I think, legitimate, but the proposal I favor is considerably less demanding and therefore more common in everyday use . 9 Note that we also have it that c(AuB) = x. Thus , the present interpretation validates both (lED) and (10). 10 For an excellent overview of this early work , see Stoutland (forthcoming). 11 Th ese limits are del ineated as follow s (cf Jeffrey, 1983, sections 6.1 and 6.6 .): For any probability-utility pair (P, V ) that pro vides an expected utility representation for a preference ordering ~ on the algebra of propositions (with the logically false proposition removed), (P' , V ' ) is also an expected utility representation of ~ iff for some parameters a, b, c, d such that (i) ad - bc > 0, (ii) cV + d is positive for all the arguments of V, and

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ifT is the logically true proposition, cU(T) + d = L (iii) U' is a fractio nal linear transformation of U with respect to the parameters in question, i.e., U' = (aU + b)/(cU +d), while P' = P(cU + d). 12 For simplicity, I am here ignoring the distinction between use and mention. I hope thi s sjstematic ambigui ty will not co nfuse the reader. 1 Even though chari ty consi dera tions directly apply only at the third stage of the interpretation process, they might, when applicable, make us re-consider the interpretive hypotheses we have started with. Thus, if applying charity at the third stage to the mater ial obtained at the earlier stages turn s out to be dif ficult , the interpreter might at that stage co me to suspect that he has made some mistakes at the outse t of the process. Thu s, he might decide to go back to the first stage again and try out a new interpreta tion of the sentential connec tives . In this sense, it could be argued, charity is relevant to all the intepretive stages. But whether such a back-and -forth process of charity-driven interpre tation would allow us to iden tify all pairs of probabilistically independen t sentence s is by no means clear. 14 For this conject ure, I am indebted to Howard Sobe l. Sobe l has also reminded me that Ramsey' s gambles cannot be described by means of truth-functio nal statements. 15 Davidson, ( 1985), fn. 5, and (199 0), fn. 68: "I am indebted to Stig Kanger for showi ng me why an earlier attempt at a solution to this problem [= identi ficatio n of the co nnectives] would not work." The details of their encount er have been confirmed by Davidson, in personal communication . 16 "[Kanger] also added some needed refi nements to the present proposal." (Davidson, 1985, fn. 5, and 1990, fn. 68.) 17 Strictly speaking, since it is a ration ality constraint that a sentence is neve r held to be both true and false at the same time, condition ( ~) is redunda nt given (y). 18 See Axiom 2 in Jeffrey (1983 ), Ch. 9. The preference orderi ng is defined on such an algebra with its zero element removed. As Jeffrey points ou t, the assumptio ns of atom lessness and comp leteness are used in the proo f of the existence part of the representation theorem. They are not needed for the uniqu eness part. For an excellent short presentation of the theorem, cf. Broome ( 1990). That the requ irements of com pleteness and atomles sness might pose a problem for Davidson is suggeste d in Rawling (forthcoming).

REFERENCES Bell, J.L., and A.B. Siomson, 1969, Models and Ultraproducts: An Introduction, North Holland Publ. Co mp., Amsterd am - Lond on. Broome. John . 1990. "Bolker-Jeffrey Expected Utility Theory and Axio matic Utilitarianism", Review of Economic Studies 57, pp. 477 - 502. Davidson, Donald, 1980, "Towards A Unified Theory of Meaning and Actio n", Grazer Philosoph ische Studi en 11, pp. 1- 12. Davidson , Donald, 1982, "Exp ressing Evaluations", Lindley Lectures, Lawrence, Kansas, University of Kansas. Davidson, Donald, 1985, "A New Basis for Decision Theory", Theory and Decision 18, pp . 87 98. Davidson, Donald, 1986, Truth and Interpretation, ed . by E. Lel' ore, Blackwell , New York . David son, Donald, 1990. "The Structure and Cont ent of Truth ", Dewey Lectures, The Journal of Philosophy 87, pp. 279 - 328.

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Davidson, Donald , 1996, "The Folly of Trying to Define Truth", The Journal of Philosophy 93, pp. 263- 278. Hansson , Sven Ove, 1998, Structures of Value. An Investigation ofthe Statics and Dynami c of Values and Norms, Lund Philosoph y Reports, Department of Philosophy, Lund University. Jeffrey, Richard C. 1983, The Logic of Decision, University of Chicago Press, Chicago and London, second revised edition; first edition 1965. Kanger, Stig, 1980, "A Note on Preference Logic", ThD 60 - Philosophical Essays ded icated to Thorild Dahlquist on his sixtieth birthday, Philosoph ical Stud ies publ. by Philosophical Society and Department of Philosophy, University of Uppsala, vol. 32, Uppsala, pp. 37 - 38. Levi, Isaac, forthcoming , "Representing Preferences", to be publi shed in the Schilpp volume on Davidson. Quine , Willard Van, 1960, Word and Object, MIT, Cambridge , Mass. Quine, Willard Van, 1990, Pursuit of Truth, Harvard University Press, Cambridge , Mass., and London . Rabinowicz, Wlodek, 1998, "Preference Logic and Radical Interpretation: Kanger Meets Davidson", in Lars Lindahl, Jan Odelstad and Rysiek Sliwinski (eds.), Not Without Cause Philosophical Essays Dedicated to Paul Needham on the Occasion ofHis Fiftieth Birthday, Uppsala Philosophical Studies 48, Uppsala University, Department of Philosophy. Ramsey, Frank, 1950, "Truth and Probabilit y", in his Foundat ions of Mathemat ics and Other Logical Essays, ed. by R.B. Braithwaite, Routledge & Kegan Paul , London , pp. 156-198. Rawling , Piers, 1996, "Davidson' s Measurement-Theoretic Reduction of the Mind", draft. Stoutland, Fredrick, forthcoming , "Davidson on Truth and Interpretation", a book chapter. Wilson , Neil, 1959, "Sub stances Without Substrata", Review of Metaphysic s 12, pp. 521-539. von Wright, Georg Henrik , 1963, The Logic of Preference, Edinburgh Univer sity Press, Edinburgh.

AMARTYA SEN

NON-BINARY CHOICE AND PREFERENCE: A TRIBUTE TO STIG KANGER*

1. INTRODUCTION

Stig Kanger was a philosopher of extraordinary power and creativity. In logic, in choice theory, in the theory of rights , and in many other fields, Kanger made far-reaching contributions which were profoundly important for the respective subjects. But he was not invariably a person of the greatest perseverance. He would often make an extremely innovative departure from the received tradition, but then move on to something else without staying on to finish the work he had started. This is especially the case with his deep and penetrating contributions to choice theory. His slender paper "Choice Based on Preference" - a thoroughly original contribution - was written some time in the middle 1970s (it will be called here Kanger I). It was seriously incomplete when it was first presented (with two sections of the text and the entire reference list missing), and it remained incomplete even at the time of his death more than a decade later. A subsequent paper "Choice and Modality" (to be called Kanger II) seemed like an attempt at completing the exercise, and it did extend the analysis, but it too needed more work which never carne.' In this paper, I want to talk about some specific aspects of choice theory that emerge forcefully from Kanger's ingenious contributions in this field . But given the incompleteness of the papers, this exercise must involve some speculation on what Kanger was really after. I am helped in this exercise by the discussions I had with him, first, at the London School of Economics in the mid-seventies, and later on, during my two visits to Uppsala in 1978 and 1987, respectively. In the next section, the standard models of binary and non-binary choice theory are briefly discussed, followed - in section 3 - by some reformulations reflecting Stig Kanger's ideas and suggestions. In section 4, the motivation underlying the reformulations are examined, and the importance of these departures is illustrated with particular substantive examples. The essay ends with a concluding remark on the over-all significance of Kanger' s departures. 243

G. Holmstrbm-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. II, 243-254. © 2001 Kluwer Academic Publishers. Printed in the Netherlands . Originally published in D . Prawit z, B. Skynns and D. Westerstahl (eds .) Logic, Methodology and Philosophy cf Science IX, Elsevier Science B.V., 1994, pp. 913-924.

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2. CHOICE FUNCTIONS AND BINARINESS At the risk of some over-simplification, the literature in choice theory can be divided into two categories in terms of what is taken to be "the primitive", viz, (l) some binary relation R (interpreted as "preference", or "value", or "objective", or "the utility relation" - something seen as prior to choice), or (2) the choice function C (.) itself.' These two standard approaches can serve as the background against which we see Kanger's departures. 2.1. Binary relation as the primitive Consider, first, the traditional view of "relational choice", basing choice on the primitive relation R in the standard way. A binary relation R ranks the set of available alternatives X from which a non-empty "menu" S is offered for choice, S ~ X and from this S an "optimal set" C(S, R) is chosen on the basis of the binary relation R. In fact, only one element of the optimal set must ultimately be picked, but the optimal set reflects the set of "chooseable" elements of S. (I)

C(S,R)={xlxES&VyES:xRy}

C(S, R) is sometimes called the "choice set" of S with respect to the binary relation R. The interpretation of C(S, R) depends on the content of the binary relation R. If, for example, R stands for the relation "at least as good as", then C(S, R) is the set of "best" elements in S. Here we move from a binary relation, taken as the primitive, to the derived choices. Within this general structure, the approach can vary with the characteristics of R, which mayor may not be complete, mayor may not be transitive, and so forth. The symmetric and asymmetric factors of R partition the different cases in which xRy holds into xPy and x1y. (2) (3)

xPy <=> [xRy & not yRx] x1y <=> [xRy & yRx]

If R is interpreted as at least as good as, then P can be seen as the relation "better than" and 1 as the relation "indifferent to" . In another variant of this approach of relational choice, the elements to be chosen may be specified as the set of "maximal" elements, rather than as the "optimal elernents".' In the case of choosing from the "maximal element" set, to qualify for choice, and element x has to be undominated by any other element (that is, for no y should it be true that yPx), even though xRy need not hold either.

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(4)

M(S,P)={xl x ES¬ ~Y ES: yPX}

The distinction between the maximal set M(S, P ) and the optimal set C(S, R) is helpful for relational choice for various reasons , but perhaps most of all because the optimal set C(S, R) might well be empty when R is incomplete. While reflexivity (requiring xRx for all x) may be trivial in the context of many cases in choice theory (it is, for example, hard to dispute that x is "at least as good as" itself), completeness certainly can be a really exacting demand. Even with incompleteness, the maximal set can sometimes exist even though the optimal set is empty. For e xample, if neither xRy, nor y Rx , then {x, y}, R) = 0 , whereas M({x, y}, R) = {x , y }. One type of preference relation much studied in choice theory is a "quasiordering", in which R is transitive but not necessarily complete. Kanger too has tended to take that type of relation as a good starting point of his analysis of "choice based on preference". For a quas i-ordering, an "optimal set" may well be empty even whe n a "maximal set" is clearly non-empty. Indeed, over a finite set S, a maximal set M (S, R ) will always exist for a quasi-ordering R (Sen 1970, Lemma 1*b). However, the following theorem holds (for a proof see Sen 1970, Lemma 1*d, pp. 11-12).

ce

(T . 1) For quasi-ordering R, ijC(S, R ) is non-empty, th en M (S, R )

The interest in the maximal set - as opposed to the optimal set arises when the optimal set does not exis t.

= C(S, R ).

particularly

2.2. Choi ce fun ction as th e primitive

In the alternative traditional approach, the primitive is taken to be the choice function itself, which is a functional relationship that specifies for any non-empty subset S of the universal set X, a "choice set" ceS), a subset of S. It is possible to obtain binary relation s of "revealed" or "underlying" preference, from such a choice function (by making some standard assumptions), and indeed there is quite a literature on this . For example x is weakly "revealed preferred" to y if and only if from some set of which y is a member, x is actually chosen (whether or not y is also chosen)". Further, x is weakly "base relation preferred" to y if and only if x is picked precisely from the pair {x, y}."

ce.)

Weak revealed preferen ce: (5 )

x RcY

<=>

[ ~S : x E C(S )

& yE S ]

Weak base relation : (6)

xRcY

<=>

[x E C( {x , y })]

246

AMARTYASEN

The asymmetric and symmetric factors of R, (denoted P; and I, respectively) can be obtained in the usual way, following (2) and (3) applied to R; Similarly with Re . It is, in fact , also possible to define a strong revealed preference relation P" directly, in terms of x being chosen from a set that contains y but from which y is not chosen (that is, x is chosen and y rejected)."

Strong revealed preference: (7)

xp ey <=> [::JS : x

E

C(S) & y

E

(S - C(S»]

2.3. Binary choice A choice function is binary if and only if the revealed preference relation R, generated by that choice function would generate back the same choice function if R, is used as the basis of relational choice. Invoking (1) and (5), binariness is defined thus.

Binariness of a choice function : A choice function is binary if and only if, for all S ~ X: (8)

C(S)

=C(S, Rc )

Various consistency conditions have been proposed for choice functions, such as the weak axiom of revealed preference, path independence, and so on. The following two elementary conditions are central for the binariness of a choice function.

Property a (basic contraction consistency): For all x in X and all S, T (9)

[x E C(X) & x

E

~

X,

T c S] = [x E C(1)]

Property y (basic expansion consistency): For all x in X and any class of sets Sj ~X:

(10)

[x E

nC(Sj)] = [x J

E C(

U Sj)] J

Prop erty a demands that if a chosen element x from a set S belongs to a subset T of S, then x would be chosen from T as well. Property y requires that if some x is chosen from every set S, in a class, then it would be chosen also from the union of all such Sj' The following result is easily established linking Properties a and y to binariness of choice for a complete choice function , that is, for choice

NON-BINARY CHOICE AND PREFERENCE

247

funct ions such that C(S) is non-empty for any non-emp ty S (see Sen 1971 and Herzberger 1973). (T .2) A complete choice fun ction is binary

if and only if it satisfies Properties

a and y.

Binariness can also be defin ed in terms of the base relation Re , rather than the revealed preferenc e relation Re , in exactly the same way, and it can be shown that "basic binariness" thus defined is equivalent to binariness with respect to the revealed preference relation and thu s equivalent to the combination of Properties a and y (on this and related matters, see Herzberger 1973). By varying the required prop erties, the choice functi on can be made less or more demanding than binariness.'

3. KANGER 'S DEPARTURES The basic variation that Kanger introduc es in this standard structure is the possib ility of choos ing according to a binary relation of preferenc e RV that depend s on the "ba ckgrou nd" set Vrather than being independent of the set of alternatives (as ass umed in the case of R considered in the last section). Whil e the choices are seen as being based firmly on bin ary relations, the parti cular binary relation to be used in the Kanger system varies with the background set V. The far-re aching significance of this variation will be considered in the next section. The present section is concerned mainl y with sorting out the formalities in Kanger 's formulation , which is rather complex and in some ways quite hard to follow.' I shall first present the logical sequence in Kanger ' s own presentation, but it will emerge that the main diffe rences introduced by him can be stated in anoth er - rather simpl er - way in terms of the standa rd format of choice theory. So if the reader is disinclined to go through a lot of forma lities, he or she could move strai ght on to equations (15) and (16) below. Kanger proceeds from a "primitive" notion of a dec ision function D, from which a choice function C is obtained. We shall call them D K and CK respectively, in honour of Kan ger. The different concepts can be perh aps more easily understood by invokin g a diagram of intersect ing sets V and X (at the cost of some loss of generalit y, which will not however affect the form al definition s prese nted here). We take S = V n X.

248

AMARTYASEN

Figure 1 DK(V, X) are the elements of V that are no worse than any element of V-x (equivalently, V-S) according to the strict binary relation pV with respect to the

background set V. (1l)

DK(V, X)

= {x I XE V¬:::Jy E V-X :ypVx }

It is easily checked that the following relations hold : (12) (13)

=

DK(V, X) DK(V, S) DK(V, V-X) D\V, V-S)

=

The choice function CK is defined in terms of DK thus: (14)

CK(V, X)

=DK(V, V-X) n x

With the choice function CK thus established, Kanger proceeds to introduce more structure into the background-dependent preference relation pV: first the elementary need for this notationally "strict" pV to be irreflexive; then the requirement that pV be a strict partial ordering with no infinitely ascending chain; then it be also a semi-ordering; and finally that it be a strict weak ordering. He examines their various properties and relates them to the consistency conditions used in the standard literature (such as Properties a and y). The basic idea behind the choice function C K can be understood in more direct terms in the following way. Consider the maximal set M(S, P), defined earlier, in equation (4) . The strict preference relation P invoked there did not depend on any background set V. Now make it dependent on a selected background set V, and call it Define C*(S, V) simply as M(S, pV), exactly like a traditional maximal set, except for using pV rather than P .

r:

(15)

C*(S, V)

=M(S , pV) = {x IX ES & not:::Jy E S: ypVx }

Now bearing in mind that S is the intersection of V and X, it can be easily established that Kanger's Choice function CK relates to C* (and thus to the standard maximal function M) in the following way:

NON-BINARY CHOICE AND PREFERENCE

249

(T .3)

(16) The result is easily checked by comparing (15) with the characterization of CK(V, X) in the Kanger system, given by (17), based on (14) : (17)

CK(V, X)

= {XIX E VnX¬ ::Jy E VnX:yPVx}

Thus, we are essentially in the same territory as the traditional maximal function M(.) , with the added proviso that the strict preference relation Pis now a background dependent And bearing in mind the old result (T. 1) that the traditional maximal set M(S, P) is the same as the traditional choice set C(S, R) whenever the latter is non-empty and R is a quasi-ordering (Sen 1971), we have a clear relationship between Kanger's choice system and the standard system of choice sets and maximal sets. The Kanger system opts for the idea of maximality rather than that of optimality (underlying the traditional binary choice function) , and furthermore makes the binary relation of preference P" (on the basis of which maximality is defined) dependent on the specification of the background set V. The latter is a truly substantial departure, and in the next section the motivation underlying this change and its extensive importance are discussed and exemplified. But as far as formalities are concerned, we lose nothing substantial by using the simpler notion of a background-dependent maximal function M(S, p V ) , rather than CK(V, X), as in the Kanger system. The discussion that follows will be conducted entirely in these less specialized terms, using the older notion of maximality coupled with Kanger' s ideal of a background-dependent preference relation P".

r:

4 . WHY BACKGROUND DEPENDENCE? At the substantive level , the idea behind a background-dependent maximal choice M(S, pV), equivalent to Kanger's differently formulated choice structure, can be seen in terms of two distinct departures from the standard maximal choice M(S, P): (1) the preference relation P is taken to be dependent on a background set V in terms of which it is defined, and (2) the background set V need not be the set S (the menu) from which choice is being made. I shall briefly consider different types of motivations that can justify the broader conception of choice behaviour proposed by Kanger. Since Kanger himself has tended to shy away from motivational discussions in general, I cannot claim that these motivations explain why Kanger made his proposals. But nevertheless these motivational arguments help us understand some of the advantages

250

AMARTYASEN

of the Kanger formulation over more traditional models of choice behaviour. Let us first consider the former departure without the second (i.e., background-dependence of preference when the background is required to be the menu itself). Take the preference relation pS to be dependent on the set S from which choice is being made: M(S, pS). This is already a considerable departure from the standard model of choice, given by C(S, R) or M(S , P), in which the preference relations Rand P are taken to be menu-independent (and of course, more generally, background -independent). This relaxed requirement can deal with cases in which the nature of the menu from which choice is being made can affect the ranking of alternative elements. The reasons for such menu-dependence of rankings can be diverse and they tend to be comprehensively ignored in the traditional models of binary choice. I present here briefly three quite different - and essentially independentreasons for menu-dependence of preference, which I have discussed more extensively elsewhere (Sen 1992).9 Positional choice: The ranking of alternatives may depend on the position of the respective alternatives vis-a-vis the others in the menu. For example, when picking a slice of cake from a set of slices, a cake-loving person who nevertheless does not want to be taken to be greedy may decide not to pick the largest slice, but choose instead one that is as large as possible subject to its not being the largest, to wit, she may choose the second largest slice.'? This type of choice would violate binariness and even the elementary condition of Property a (basic contraction consistency). If, for example, three slices of cakes are ranked in decreasing order of size as a over b and that over c, then from the menu (a, b. c), the person may pick b, and from (b, c) may choose c. There is nothing particularly "irrational" in such behaviour, even though these choices violate Property a and binariness. Similarly, a person may decide not to pick the last apple from an after-dinner fruit basket , having one of the pears instead, even though she may pick an apple from a larger basket containing many apples and many pears. Epistemic value of the menu: A person may accept the invitation to tea from an acquaintance she does not know well, but refuse that invitation to tea if the acquaintance were also to invite this person to have some cocaine with him. The addition of the latter invitation may give her some extra information about him which might make her more skeptical of the idea of having tea with him. The menu offered has informational value in ranking the individual courses of action . Again, we see here a violation of Property a and of binariness, but the reasoning is canny enough.

NON-BINARY CHOICE AND PREFERENCE

251

Valuation of freedom: The freedom a person enjoys depends on the nature of the menu open to her. The choice of courses of action may be influenced by the extent of freedom. For example, a person may choose to read a particular newspaper when she could read anyone she chooses (or none), and yet decide to protest and read none if she is forced to read that particular newspaper and no others. Contraction consistency and binariness are violated in all these cases, but there is no difficulty in explaining and rationalizing the choices in terms of "choice based on preference" when the preference relation pS depends on the menu from which choice is being made . These and other examples have been discussed and scrutinized elsewhere in terms of the particular properties of menu-dependent preference pS, but they are covered inter alia by the more general case of background-dependent preference pVproposed by Stig Kanger. Now we can turn to the case in which the background set V need not coincide with the menu set S. This is particularly Kanger territory. What can be the reason for choosing a background set that is different from the menu from which choice is being made? While Kanger himself has not discussed the motivational issues in his papers, possible reasons for the additional departure are not hard to seek. The menu tells us what we can choose from. The ranking of the alternatives may depend, however, on the role of the chosen alternatives after the choice has been made . For example, consider the problem of selecting tennis players to represent a country in the Davis Cup - an international tournament. What the selectors have to seek are not the best players in the country in terms of playing against each other, but the best players in terms of playing against tennis players from other nations. Consider a case in which players A and B can defeat players C, D, E and F individually and in pairs. That is a good reason for declaring them to be champion players within the nation. But it is still possible - given differences in the style of playing - that players C and D can defeat the Davis Cup team from the United States while the others cannot do that, and players E and F can defeat the Davis Cup players from Sweden, while the others cannot perform that feat. In that case, in picking Davis Cup players, there would be a good argument for picking C and D if it looks that this country will have to play against the United States, and for picking E and F if it appears that the contest will be against Sweden. The ranking relation p v must , thus, take note of the ranking of the domestic players not vis-a-vis each other, but of their abilities to play against the likely international competitors - the appropriate "background" in this case.

252

AMARTYA SEN

Similarly, in selecting a poet laureate , the selectors may be guided not just by the merits of the likely candidates seen in terms of internal comparisons, but by the respective standings and comparative standards of these candidates vis-a-vis other well-known poets - including dead poets and lyricists from other nations . To take another type of example, in making admission decisions, a college may be guided not just by comparisons of the applicants against each other seen in purely internal terms, but also by comparing them to general categories of students whether or not applicants to this particular college. Many other types of examples can be easily presented. The common factor in all this is the need for external reference - external to the menu - in comparing the alternatives in the menu. It is that general possibility that the Kanger formulation of choice can capture in a neat and elegant way by explicitly bringing in the reference to a background set V that mayor may not coincide with the menu S. 5. A FINAL REMARK

In this essay I have briefly presented the special features of Stig Kanger's model of "choice based on preference". By presenting his formulation in a slightly different way, we can see it as an extension of the standard model of binary choice in terms of maximal sets with the binary relation of choice P" made dependent on a background set V which mayor may not coincide with the menu S. The departures, thus , involve three distinct elements: (1) use of maximality rather than optimality, (2) admitting menu dependence of preference , and (3) admitting dependence of preference on a set different from the menu itself. I have discussed the case for each of these departures, of which the last is most specific to Kanger's own work. I end with a final remark that while Kanger's formulation takes choice theory well beyond the limited framework of binary choice as it is standardly defined, the primitive notion that Kanger invokes is still a binary relation p v defined in terms of a specified background set. In this sense, Kanger's model can be seen as a generalized formulation of binary choice (as he calls it, "choice based on preference"). One of the implications of Kanger's analysis is the need to rethink on the requirements of maximization as the basis of decisions and choice. The Kanger framework violates the standard conditions of maximal choice quite robustly, but the differences arise not from rejecting any intrinsic feature of maximization as such, but from dropping the implicit presumption in the standard literature that the preference relation be background independent. In effect, Stig Kanger has shown that maximization is a much more general discipline

NON-BINARY CHOICE AND PREFERENCE

253

than theorists of maximization have tended to assume. That is the key to a different world of choice through maximization.

Harvard University NOTES For helpful discussions on this and related topics, I am most grateful to Nick Baigent , Ben Fine, Dagfinn Fellesdal , Wlodzimierz Rabinowicz, Ryszard Sliwinski, and of course - over many years - to Stig Kangcr himself. I Both the papers contained, in fact, a small error, which was detected and sorted out by Stig Kanger's associates, Wlodzimierz Rabinowicz and Ryszard Sliwinski, in a forthcoming volume of Scandinavian texts on decision theory and ethics, which will include Kanger's unpublished - and unfinished - paper "Choice Based on Preference" ; Porn et al. (1992). The "Introduction" also comments generally and illuminatingly on the nature of Kanger 's contributions to ~ecision theory. - The distinction applies to choice under uncertainty as well as certainty . However, in this ~aper I shall not go into the former: since neither ~f Kanger' s essays deals with uncertainty . . On the distinction between "optimal " and "maximal" see Dehreu (1959) , Chapter I, and Sen (1970) . 4 See Samuelson (1938), Arrow (1959), Hansson (1968), Herzberger (1973) . 5 Sec Uzawa (1956), Herzberger (1973) , Suzumura (1983). 6 See Arrow (1959), Suzumura (1983) . 7 For the main results , see Arrow (1959), Hansson (1968), Sen (1971), Herzberger (1973), Suzumura (1983). 8 Rabinowicz and Sliwinski point out in their introduction in Porn et al. (1992) that Kanger 's "reason for choosing such an artificial concept as D as his primitive" relates to "the close formal connection between D and modal operators studied in modal logic". Rabinowicz and Sliwinski discuss these connections, and they are indeed important for the formal side of Kanger's reformulation of the choice problem (see Kanger I and Kanger II). In this paper, however, I am mainly concerned with the substantive differences pursued by Kanger. See also Danielsson (1974) on related issues. 9 See also Sen (1982 , 1992), Elster (1983), Levi (1986) , Fine (1990), among others, for different types of reasons for menu-independence. 10 Positional valuation has been extensively investigated in the context of social choice by Gardenfors (1973) and Fine and Fine (1974) .

REFERENCES Arrow, K. J. (1959), "Rational Choice Functions and Orderings", Economica 26. Danielsson, S. (1974), Two Papers on Rationality and Group Preference, Philosophy Department, Uppsala University, Uppsala. Debreu, G. (1959), Theory of Value. Wiley, New York. Elster.J, (1983), Sour Grapes. Cambridge University Press, Cambridge. Fine, B. (1990), On the Relationship between True Preference and Actual Choice, mimeographed, Birkbeck College, London . Fine, B., and Fine, K. (1974), "Social Choice and Individual Ranking", Review of Economic Studies 41.

254

AMARTYASEN

Gardenfors, P. (1973) , "Positional Voting Functions", Theory and Decision 4. Hansson, B. (1968), "Choice Structures and Preference Relations" , Synthese 18. Herzberger, H. G. (1973), "Ordinal Preference and Rational Choice ", Econometrica 41. Kanger, Stig (1970s), "Choice Based on Preference ", mimeographed, Uppsala University (cited here as Kanger I). Kanger, Stig (1980s), "Choice and Modality", mimeographed , Uppsala University (cited here as Kanger II). Levi, I. (1986) , Hard Choices, Cambridge University Press, Cambridge . Porn, I. et al (1992), Choices , Actions and Norms. Conceptual Models in Practical Philosophy - Scandinavian Contributions, to appear. Rabinowicz, W., and Sliwinski , R. (1991), Introduction , Porn et al. (1992) . Samuelson, P.A. (1938) , "A Note on the Pure Theory of Consumers' Behaviour", Economica 5. Sen, A. K. (1970), Collective Choice and Social Welfare, Holden -Day, San Francisco; republished, North-Holland, Amsterdam (1979) . Sen, A. K. (1971), "Choice Functions and Revealed Preference", Review of Economic Stud ies 38; reprinted in Sen (1982) . Sen, A. K. (1982) , Choice, Welfare and Measurement, MIT Press, Cambridge, MA and Blackwell, Oxford. Sen, A. K. (1992), "Internal Consistency of Choice", 1984 Presidential Address to the Econometric Society, forthcoming in Econometrica 1993. Suzumura, K. (1983), Rational Choice, Collective Decisions , and Social Welfare. Cambridge University Press, Cambridge . Uzawa, H. (1956), "A Note on Preference and Axioms of Choice" , Annals of the Institute of Statistical Mathematics 8.

NOTES ON THE CONTRIBUTORS

Lennart A.qvist received his Ph.D. in 1960 and has since then been Docent of Practical Philosophy at Uppsala University. He has also taught at Lund University and Abo Academy and has been a Visiting Professor at Brown University and the University of Stuttgart. For the last twenty years he has been working on projects in linguistics and in the logical reconstruction of legal reasoning. Aqvist's areas of interest include philosophical logic , lingu istics , philosophy of language, ethics, philosophy of law and epistemology. Among his publications are A New Approach to the Logical Theory of Interrogatives (Tubingen 1975) and Introduction to Deontic Logic and the Theory of Normative Systems (Napoli 1987), as well as Causing Harm : A Logico-Legal Study (with Philip Mullock, Berlin, 1989). Jan Berg earned a Ph .D. in Theoretical Philosophy at Stockholm University in 1962 during the period when Stig Kanger upheld a Docentship there. He then taught at the University of Minnesota and at Stockholm University until 1969 when he was appointed Professor of Philosophy at the Technische Universitat Munchen. He has published extensively on history of philosophy, logic, philosophy of science, and general philosophy, and he is a leading expert on Bolzano's philosophy. Also high on his list of credits is a Black Belt in Judo. Brian F. Chellas is Professor Emeritus of Philosophy at the University of Calgary. He received his Ph.D. from Stanford and later taught at the Universities of Pennsylvania and Michigan. He is the author of Modal Logic: An Introduction (Cambridge University Pres s 1980) and many papers in philosophical logic as well as a book on chord systems for the guitar. Anatoli Degtyarev is a Docent at Kiev University and also a visiting researcher at Manchester Metropolitan University. His main research area is automated reasoning. He has made a number of contributions to equational theorem proving, including the discovery of basic paramodulation and results on theorem proving with rigid variables.

255 G. Holmstrom-Hintikka , S. Linstrom and R. Slivinski (eds.), Collected Papers ofStig Kanger with Essays on his Life and Work, Vol. 1/, 255-259. © 2001 Kluwer Academic Publi shers. Printed in the Netherlands.

256

NOTES ON THE CONTRIBUTORS

Lars Gustafsson became a Ph.D. in Theoretical Philosophy at Uppsala University in 1978 with Stig Kanger as supervisor. Since 1981 he has been Professor of Germanic Languages and Philosophy at the University of Texas where he is currently Jamail Distinguished Professor. Besides being a scholar, he is a prominent Swedish poet and prose writer, with many of his works translated into other languages. Soren Hallden is Professor Emeritus of Theoretical Philosophy at Lund University. He received his Ph.D. in Theoretical Philosophy from Stockholm University in 1950 and was a Docent at Uppsala University until 1964 when he was appointed to the Chair of Theoretical Philosophy at Lund University. He has published extensively in the fields of philosophical analysis, decision theory, philosophy of science, and philosophical logic. His latest book is Truth Strategy Simplified (Thales 1999). Kaj Barge Hansen earned his Ph.D. in Theoretical Philosophy at Uppsala University in 1996. The same year, he became Docent of Logic at Uppsala. He has taught at the Universities of Uppsala, Stockholm and Lulea, and has been a Visiting Professor at the University of Tartu and at the Universidad Nacional del Sur in Bahia Blanca. Among his publications are Logical Physics: Quantum Reality Theory (Thales 1996), Applied Logic (Uppsala 1996) as well as a textbook of logic and several articles on pure and applied logic and philosophy of science . Sven Ove Hansson, Professor of Philosophy at the Royal Institute of Technology, Stockholm, was one of Stig Kanger's graduate students. He is the author of Setting the Limit (Oxford University Press 1998), A Textbook of Belief Dynamics (Kluwer 1999), The Structure of Values and Norms (Cambridge University Press, in press), and articles on subjects such as preference logic, deontic logic, decision theory, philosophy of risk, and belief revision. He is the editor of Theoria. Risto Hilpinen, previously Professor of Theoretical philosophy at Torun Yliopisto, is Professor of Philosophy at the University of Miami and has held visiting appointments at many other universities. He has written papers on philosophical logic, epistemology, and the philosophy of science, and edited books in these areas, for example, Deontic Logic (Reidel 1971), Rationality in Science (Reidel 1980) and New Studies in Deontic Logic (Reidel 1981). He is an editor of Synthese.

NOTES ON THE CONTRIBUTORS

257

Jaakko Hintikka is Professor of Philosophy at Boston University. He has held professorial appointments at the University of Helsinki, the Academy of Finland and Florida State University. From 1965 to 1982 he was associated with Stanford University. He is the principal founder of epistemic logic and game-theoretical semantics and a pioneer of possible worlds semantics, though he has published widely in other areas such as philosophy of language, epistemology, inductive logic, philosophy of science , philosophy of mathematics and history of philosophy. He has authored or co-authored over 30 books and monographs and more than 300 scholarly papers. He is the editor of Synthese and Synthese Library . Ghita Holmstrom-Hintikka was one of Stig Kanger's graduate students and received her Ph.D. in Theoretical Philosophy from Uppsala University in 1991. She is Adjunct Associate Professor of Philosophy at Boston University and Docent of Ethics and Philosophy of Religion at the University of Helsinki. Her dissertation, Action, Purpose, and Will, was published in 1991. She has also written papers on action theory, legal philosophy, applied ethics and medieval philosophy. She is the editor of Medieval Philosophy and Modern Times (Kluwer 1999) and a co-editor of Contemporary Action Theory (Kluwer 1997) as well as the guest editor of Communication and Cognition, "Legal Argumentation" (1995) . Lars Lindahl is Professor of Jurisprudence at the University of Lund, Sweden . He has a background in law as well as in philosophy. His book, Position and Change: a Study in Law and Logic (Reidel 1977) was his Ph.D. thesis at the University of Uppsala, written with Stig Kanger as supervisor. Sten Lindstrom received his Ph.D. from Stanford University in 1981, and is Professor of Theoretical Philosophy at Umea University. He has held teaching appointments at the universities of Uppsala and Lund and been a Research Fellow at the Swedish Collegium for Advanced Study in the Social Sciences in Uppsala. He has written papers on intensional logic, belief revision, and the philosophy of language, and co-edited the book Logic, Action and Cognition (Kluwer 1997). Ingmar Porn is Professor Emeritus of Philosophy at the University of Helsinki. He obtained his first Ph.D. from University of Birmingham in 1970 and his second from Uppsala University in 1977. Before being appointed to the Swedish Chair of Philosophy at the University of Helsinki, he taught at the University of Birmingham and at Uppsala University. He is the author of The

258

NOTES ON THE CONTRIBUTORS

Logic ofPower (Blackwell 1970) and Action Theory and Social Science: Some Formal Models (Reidel 1977).His research interests are in philosophical logic, social philosophy and philosophy of health.

Dag Prawitz has been Professor of Philosophy at the University of Oslo and is now Professor of Theoretical Philosophy at Stockholm University. As a student at Stockholm University he worked for a while in the field of automatic deduction, which is the topic of his note in this volume. His main work in logic has been in the field of general proof theory. More recently he has been interested in theory of meaning, to which he has applied some ideas from proof theory. Wlodek Rabinowicz started his philosophical studies in Warsaw, then gained his Ph.D. from Uppsala University in 1979, and is currently Professor of Practical Philosophy at Lund University. His research interests are in moral philosophy, decision theory and philosophical logic and he has published extensively in each of these areas. Currently, he is involved in a research project on sequential choice. He is the author of Universalizability: An Essay in Morals and Metaphysics (Reidel 1979) and is a co-editor of the journal Economics and Philosophy. Krister Segerberg was Professor of Philosophy at Abo Academy and then the University of Auckland before succeeding Stig Kanger to the Chair of Theoretical Philosophy at Uppsala University. Most of his publications are in philosophical logic, including two books, An Essay in Classical Modal Logic (Uppsala 1971) and Classical Propositional Operators (Clarendon Press 1982). He is an editor of the Journal of Philosophical Logic . Amartya Sen received his doctorate from the University of Cambridge in 1959 and has been professor in India, the U.K. and the U.S. Before becoming Master of Trinity College , Cambridge, in 1998, he was Lamont University Professor and Professor of Economics and Philosophy at Harvard University. He was awarded the Nobel Prize in Economics in 1998 for his contributions to the theory of social choice and his work on poverty . Rysiek Sliwinski was an assistant to Stig Kanger and also one of his graduate students. Currently he holds a teaching position in Theoretical Philosophy at Uppsala University. His research deals with paradoxes in game theory and doxastic paradoxes. He is an editor of Uppsala Philosophical Studies.

NOTES ON THE CONTRIB UTORS

259

Soren Stenlund is Professor of Theoretical Philosophy at Uppsala University. He is the author of Combinators, A-terms and ProofTheory (Reidel 1972) and of Language and Philosophical Probl ems (Routledge 1990) and has published several other books and articles on various themes in the philosophies of language, logic and mathematics. Problems concerning the nature and history of philosophy are other themes dealt with in his publications. In 1974 he edited Logical Theory and Semantic Analysis (Reidel 1974), the Festschrift dedicated to Stig Kanger on his fiftieth birthday. Goran Sund holm was educated at Lund, Uppsala and Oxford. He is currently the Professor of Logic in Leyden University, having previously held posts at Oxford, Nijmegen and Stockholm. He is the author of numerous articles on philosophy of mathematics, as well as on the history and philosophy of logic , and he has been a visiting professor at Stockholm, Campinas and Siena. In 1992 he edited (jointly with Joachim Schulte) a Festschrift for Brian McGuinness.

Andrei Voronkov recei ved his Ph.D. in mathematical logic in 1987 . Then he worked in the Institute of Math ematics (Novosibirsk), International Laboratory of Intelligent Systems (No vosibirsk), European Computer Indu stry Res earch Centre (Munich) and Uppsala Univer sity. Since 1999 he has been Professor of Formal Methods at the University of Manchester. He has worked in several areas of logic and computer science, including automated reasoning, logic programming and database theory. He is a co-editor of the Handbook of Automated Rea soning , to be publi shed by Elsevier Science and MIT Press.

INDEX OF NAMES

Ackermann, W. 37, 62 Aczel , P. 78 ,83,86 Alchourr6n, C. E. 174 Anand, P. 219 Anderson ,A.R. 4,139,163 -169,170 Anscombe, G. E. M . 163, 170 Anselm 146 Aqvist, L. 25 -28,42,142,147,173 -1 83 Aristotle 15,39 Arrow, K. J. 253 Austin, J. 151,169 Bachmair, L. 62 Baihache, P. 177 Barcan Marcus, R. 98 - 100, 119, 169 Barwise, J. 35 Bell , J. L. 23 8 Belnap, N. 142, 147 Bend ix, P. 62 Bentham , J. ] 5] , ] 69 Berg, J. 13- 15 Bem ays, P. 32, 35, 39, 53 Beth, E. W. 15,32- 33, 35,41,58, 60 Bjorkdahl , L. 3 Bo1ker, E. 230-232,237-238 Bo1zano , B. 14-15 Boolos, G. 128 Broom e, J. 241 Bulygin , E. 141,146,174 Carm o, J. 147 Carn ap, R. 11,98 -1 00 , 103, 126 Carson , D. 62 Chell as, B. 23 -24, 145, 192 Chellas, M . 23 Ch urch, A. 8, 32,37, 40, 42 Cocchiare lla, N . 127 Cresswe ll, M. 1. 128,1 70 Dahlquist, T. 25 ,22 1 Dalen , D. van 37

Daniel sson , S. 169,219,253 Dav idson, D. 221 - 242 Davis, M. 57 Debreu, G. 253 Degtyarev, A. 5] ,53 -67 Dil worth , C. 5 Dumrn ett, M. 35,41 Dyson , V. 35, 4 1 Eck , J. A. van 177 Elster, J. 253 Ess ler, W. K . 15 Etchemendy , J. 109 Fermullcr, D. 57 Fine, B. 253 Fine, K. 253 Fitch , F. B. 169 Fitt ing, M. 57 Fe llesdal , D. 97 Fraenk el , A. 32 Frie dman , \-I. 35 Gallier, J. 59 Gandy, R. 32 Ganzinger, H. 62 Gardcnfors, P. 219 , 253 Garson , 1. W. 128 Gent zen , G. 31 - 35, 40 , 54 , 63 , 97 Gilm ore, P. 57 Girard , J.- Y. 36, 54 Godel , K . 8,32, 34,35, 4 1, 53, 97 Gold ing, M. 169 Gou bault, J. 59 Gur evich, Y. 59 Gu stafsson , L. 5, 2 I - 22 Hagerstro rn, A. 5, 26 Ha ilperin, Th. 39 Halld en, R. 11 Hallden, S. 11-1 2, 206

262

INDEX OF NAMES

Hansen, K. B. 5,29 -30,69 -86,120 Hansson, B . 167, 169, 170, 173-174, 182,206-207, 211, 253 Hansson, S. O. 5,169,205 -219, 240 Hart, H. 143, 169, 175 Hedenius, 1. 25 Henkin, L. 34,39,41 ,89 Herbrand, 1. 32 Hermes, H. 32 Herzberger, H. G. 253 Hilbert, D. 32, 35 Hilpinen, R. 26- 27, 131- 149 Hintikka,1. 26,32- 33, 53, 60, 87- 95, 97, 101. 104-108, 127, 131 -132, 136137,146 Hodges, W. 33 Hoepe1man, J. 177 Hohfeld, W. N. 140,151. 168, 169, 173, 175 Holmstrom-Hintikka, G. 5, 169, 170, 185-204 Honore, A. M. 143 Horty, J. 142 Hughes, G. E. 128,170 Jefrrey,R.230-232,237-238,240-241 Jennings, R. E. 218 Jervell, H. 36, 42 Jones, A. 1. 1. 143, 185, 147 J6nsson, B. 101, 127 Jergensen, J. 146 Joyner, W. 57 Kallick, B. 57 Kamp, H. 136 Kanger, Dagmar 6 Kanger, Elisabeth (Li) 6 Kanger, Gustav 3-4 Kanger, Helle 5,6,17,23 -24, 139, 141, 147, 169, 185 Kanger, Kim 6,23 -24 Kanger, Neita 6 Kanger, Rune 3 Kanger, Sally 3-4 Kanger, Thomas 6 Kant,T, 15 Kaplan, D. 119 Karlsson, Gustav 3-4 Kent, C. F. 35,41

Ketonen, O. 34, 41, 63 Kirchsteiger, G. 219 Kleene, S. K. 8,32,34, 56,57, 100 Knuth, D. 62 Komerup, Helle 6 Kreisel, G. 35,36,41 , 124, 128 Kripke, S. 97, 104-108, 113, 117, 123126, 127, 128 Leblanc, H. 38- 39 Leibniz, G. W. 87,98, 134, 138-139 Leitsch, A. 57 Levi, 1. 232, 253 Lewis, D. 128 Lifschitz, V. 59,63 Lindahl, L. 5,28,140-141 , 145, 151171,173 -176,182,185 ,203 Lindstrom, 1. 5 Lindstrom, S. 5, 40, 97-130, 136, 146, 215 Lopez-Escobar, E. G. K. 35 Lukasiewicz, J. 32 MacCormick, N. 169 McGee, V . 127 Makinson, D. 141,146,169-170,173 174,176,182 Makkai, M. 35 Malcolm, N. 8 Marcus, R. B. 98-100, 119, 169 Marc-Wogau, K. 5,17 MasJov,S. 57-60 Matulis, V. 60 Mill, J. S. 157, 163, 169 Mints, G. E. 36, 38, 58, 60, 63 Molander, B. 5 Monk, J. D. 89 Montague, R. 97, 101 - 104, 127 Mostowski, A. 80 Mullock, Ph. 27,142, 147 Narendran, P. 59 Needham, P. 5 Nelson, G. 62 Neumann, J. von 87 Nordenfelt, L. 5 NorgeJa, S. 60

IND EX OF NA MES

Odelstad, J. 5 O fstad, H . 11

Opferman, W. 169 Oppen, D. 62 Orevkov, V. 58, 63 Perloff, M. 142 Petrini, Neita 6 Pettersson, I. 3 Phalen, A. 5 Plaisted, D. 59 Plato 15 Porn, I. 5, 19, 28, 142-1 43, 169, 185, 192,203 Post, E. 32 Prawitz, D. 34-35, 4 1,43-52,57 Prawitz, H. 45, 51, 57, 58 Puppe, C. 2 19 Putnam, H. 57

Quine, W. V. O. 5,60, 116,227 -229 Raatz, S. 59 Rabinowicz, W. 2 10,2 13,22 1-242,253 Ramsey, F. P. 229-23 1 Rawling, P. 24 1 Resch er, N. 170 Robinson, G. 62 Robinson , J. A. 49 -50 Ross, A. 146 Samuelson , P. A. 253 San tos, F. 147 Schlick , M. 32 Scholz, H. 31 - 32,40 Schutte, K. 32 - 33, 53, 60 Scott , D. 23 , 35 Scarle. J. 133 Segerberg, K. 3 -9, 142, 146, 147 Sen, A. 15, 211 , 243- 254 Sergot, M. 147 Shalla, L. 62 Shanin, N. 58 Sh oen field , J. R. 35, 4 1 Shostak, R. 62, 63 Sibelius, P. 5 Simp son , S. G. 36 Skolem , T. 32 Sliwinski, R. 2 13,253

263

Slomson, A. B. 238 Smu llyan, R. 33 ,57 Snyde r, W. 59 Soderberg, Dagmar 6 Stenius , E. 5, 8, 35 Stenlund, S. 5, 17- 18,27 Stevenson, C. L. 26, 169 Stoutland, F. 240 Sundholm, G . 3 1- 42, 43 Suzumura, K. 253 Svensson, Sally 3-4 Swart, H. de 35,41 Szewak, E. J. 169 Tait, W. W. 35, 4 1 Takahashi, M. 41 Takcuti, G. 35,4 1 Talj a, J. 169 Tammet, T . 57, 63 Tarski, A. 8,25,3 1-32,39,88 -89, 97, 101-103,1 27 Tuck, R. 162, 169 Turin g, A. 32 Uzaw a, H. 253

Vaught, R. 32 Veanes, M. 59 Veldman. W. 35 Voghera, N. 45 ,5 1,57 Voronkov, A . 5 1,53-67 Wang, H. 57 Wed berg, A. 4,7 - 8, 32,40,44,51 , 97, 174 Westermarck, E. 5, 8 Wh ite, A. R. 162, 167, 169 Wilson, N. 227 Wittgenstein, L. 8, 17- 18, 32 Wos , L. 62 Wright, G. H. von 5, 134, 142, 146- 147, 169, 206, 218 Za mov, N. 57 Zermelo, E. 32, 80, 85

SUBJECT INDEX

V*3*-formulas 57 "A note on prefere nce logic" (1980 ) 205 , 209-2 11 "A note on quantification and modality" (1957) 113 "A simplified proof method for elementary logic " ( 1963) 43 ,47,50,53 ,55 ,6 1 Absolute infinity 122 Accessibility relation 100, 106, 111, 121, 132 Ackermann ' s axiom of choice 37 Action 139, 147, 194-195,201 -202,203 Action , logic of 146, 185- 204 Action, modes of 139, 147 Action , theory of 5,27,146,1 85-204 Admissible set 35 Aesthet ics 5 Agency 139,141 -142,152,187 Agent 152, 147, 187 Agent causation 187- 189, 201 -202 Algebraic logic 14, 15,87 -95 Altemativeness relation 132, 138 "An algebraic logic calcu lus" (1966 ) 87, 88,218 Analytic necessity 108, 111, 138 Analytic philosophy 7, 19,28 Analytic/synthetic 15 Analytic truth 112 Anti-foundation axiom (AFA) 80 At least as good as 205 Autologicality 75 - 76 Automat ic deduc tion 43 - 52, 53 - 67 Background set 2 12-213,215,216-218, 219 , 247 -252 Backward method 33 -36,44-46,54,57, 73 -74 Barcan formula 99, 107, 136 Bearer of right 157- 159, 173 Befugnis 161 Beth-Hintikka-Kanger-Schiitte proof 33, 35

Better than 205 Boolean algebr a 237 -239 Capacity 198 Ceteris paribus preference 225 -226,240 Charity , principle of 227 "Choice and modality " (1976) 205 , 211 2 18,243 "Choice based on prefere nce" (1970s ) 205,21 1-218,243,253 Choice functio n 211 -214,244-247 Cho ice, theory of 205 -2 19,243-254 Church 's theorem 100 Claim 140,154, 157-158, 160,174-179, 186, 187 Class domain semantics 120-123 Completeness 74,81 ,84 Comple teness theorem 32, 40 , 43, 53 - 54, 74, 78,81 - 83,97 Condition 152,169, 218 Conditional 170 Constru ctive philosophy 19, 28 Contr action 54 Cord 76 Counter-capacity 198 Cou nter-claim 154, 187 Co unter-freedom 154, 187 Counter-im munity 154 -1 55, 187 Counter-model 34, 54 Counter-party 157-1 59,1 67 - 168,173 Counter-po wer 154- 155, 187 Counter-security 198 Cumulative type structure 7 1,84,85 Cut elimination 34, 35, 43 Cut-free system 40 , 54, 57 Cylindric algebr a 39,88 -89, 169 Decision function 247 - 248 Decision procedur e 57 Decision theory 15,205 -219,229 -23 1, 243 -254 Decoration 79 - 80

266

SUBJECT INDEX

Demodulation 62 Deonticlogic 26-27,131 -149,176-177 Deontic operator 135 Domain 100,111,120,121 ,1 32 Do-operator 139,141 , 144, 152, 191,201 Dummy 47,58 Dummy method 46- 50, 58 Duty 156-157,178 -179 Educator, Kanger as 5-6,8 -9,13 -14,17, 19,21 -22,29-30 Efficient proof procedure 43-52,53 -67 "En studie i modallogik, med sarskild hansyn till 'bora' -satser" (1951) 4, 138 Epsilon-calculus 38 Equal in value to 205 Equality 58- 63, 84 Equality, predicate logic with 58-63, 84, 85 Equality, predicate logic without 58, 60, 69-70,85 "Equational calculi and automatic demonstration" (1970) 87 Ethics 5,26-27, 139 Exchange 54 Extensionality, axiom of 76 Extensionality, weak axiom of 76 Foundation axiom (FA) 80,84 Four levels-assumption (4L) 221- 222 Frame 71 Free variable 58 Freedom 140, 154, 187 Gamble 229-230 Gamma-rule 57- 58 General predicate logic 71 Gentzen' s Hauptsatz 34,43,97 Government 141 Graph 78-79 Graph, accessible pointed (APG) 79 Graph, pointed 79 Graph, well-founded 80 Hagerstrom lectures 5, 234 Handbook ofLogic (1959) 32, 37, 40, 43. 47,53 ,61 Hintikka set 33, 131 Human right 141 ,151

Humboldt award 15 Identity 58-63,84 Identity postulate 73 Identity, predicate logic with 58-63, 84, 85 Identity, predicate logic without 58, 60, 69-70,85 Immunity 140, 154-155, 186, 187 Imperative operator 135 Implicationallogic 38-39 Independence friendly(IF) first-order logic 90-94 Individual concept 114 Infinitary logic 35 Influence 195-196,198-202 Interpolation of exclusive disjunction (lED) 221- 222, 240 Interpolation of exclusive disjunction of incompatibles (lED!) 224, 239 -240 Interpolation of inclusive disjunction (10) 240 Interpretational semantics 109-110 Intuitionistic logic 35 Invertible rule 56 Jergensen 's dilemma 146 Judgement 203 Kanger model I I 1-112 'Kanger', origin of the name 3 Kripke model 101, 106 "Law and logic" (1972) 27, 139, 169, 173,175,189-190,193 ,195,196,200 Law, philosophy of 139 Legal power 159 Level saturation 57 Liberty 186 Lindenbaum algebra 238-239 Logic programming 49- 50, 85 Logical consequence 112, 121, 169 Logical necessity I 11, 123- 126 Logical truth 72-74,81 ,112,121 -123 Lowenheim-Skolem theorem 77.97 Meaning theory 4 Measurement theory 4 Membership relation 71-75, 85

SUBJECT INDEX Metaling uistic interp retation 108 Metaphysical necessity 107- 110, 123126 Metaphysical possibility 107- 1 IO Metavariab le 58 Minus-no rmalisation 60 Modal logic 4, 97-130,2 14-216 Modal operato r 135 Model 31-36 Model set 33, 105, 131 Model theory 31- 32, 69- 86, 97 Morning star paradox 115- 119 Mostowski colla psing lemma 80 Name 112 Net 14, 75- 78, 79, 85 Net, elementary 76 Net, extensiona l 76 Net, main structure of 76 New Foundations for Ethical Theory (1957) 23,26, 131, 134, 136, 137139, 14 1, 146, 151, 173, 175, 185186,199 Non-well- founde d sets 14, 69- 86, 120 Object-level interpre tation 109 Obligation 177, 131, 132 Omega-rule 35, 39 "On the charac terization of modalities" (1957) 113 "On realiza tion of huma n right s" (1985) 139, 151,1 69 Ontological modal operato r 11 2 a -right 153-1 54,1 58, 165-1 66 Ought 132,1 38,1 63 Ought-operator 4, 132, 135, 138, 163, 190 Paramod ulation 62 Parliamentarism 141 Performative 135- 136 Permission 131, 177 Petaluma 167, 179-1 83 Philosoph ising, style of 4- 5, 8, 14-15, 17-1 8, 19, 23, 25, 28, 29- 30 Phonematics 5 Picture of set 80 Possible world 97,98, 105-108, 132 Possible worlds semantics 105- 108 Power 140,1 54- 155,1 86,1 87,1 95

267

Predicate 112 Preference, theory of 5,205 -2 19,221 242,243-254 "Prefere nce logic" (1968) 205 -209 Preference, logic of 205-2 19,22 1-242, 243-254 Preference relation 221,244-247 P-right 153-1 54, 158,1 65- 166 Privilege 140, 179, 186 Proof 73 Proof theory 3 1- 42, 43-52,53 - 67,69 86 Proposition 112 Propositional attitude 107 Provability in Logic (1957) 4,43-44,53 54, 55, 5~ 69, 87,97, 10~ 113, 11 ~ 127,1 46 Quantifiers 57,87- 95, 97- 130, 136- 137 Quantifying in 11 3-119 Quasi-de duction 73 Quasi-sequent 70 Radical interpret ation 226 - 227 Range 131-1 32 Representational semantics 109- 110 Resolution 49- 50 "Rights and parliamentarism" (1966) 139, 169,1 73,1 85,1 87 Rights, theory of 5,27, 139- 141,15 1171,173 -183 Rights, types of 139-1 40,1 53-1 56,1 64166,1 87 Rigid designator 106 Rigid E-unification 59-61 , 63 Sati sfaction sema ntics 133-1 34 Second-order logic 37-38 Security 198 See to it that 139, 145, 152, 169, 185, 190, 192, 197 Self-reference 75- 76 Semantic tableau 33 Semantics 3 1- 36, 7 1- 73, 97- 130, 193 Semi-valuatio n 33 Separation problem 38- 39 Sequent 33, 54, 70 Sequent calculus 33- 34, 43, 50, 54, 57, 58-63, 73

268

SUBJECT INDEX

Seriality 132, 138 Set domain semantics 120-123 Set, heriditarily finite 80 Set, non-well-founded 80 Set structure 77 Set theoretical principle , Kanger's 77-78, 81,84 Set theoretical principle, weak 81, 82, 84 Set theory 69- 86 Set, well-founded 80 Shall-operator 152, 163, 190 Sheffer's stroke 234 Simplification 62 Simultaneous paramodulation 62 Simultaneous replacement 62 Skeleton instantiation 59-60 Social philosophy 139 "Some aspects ofthe concept of influence" (1977) 169, 196, 197 Soundness 74, 84 Source of law 173,175 -176 Stability axiom 217 State of affairs 152 State-description 98-99, 126-127 Structural rules 54- 56 Structure (arbitrary) 72, 83, 84 Structure, normal 72, 83, 84 Subterm instantiation 58,60-61 Supervalidity 128 Synthetic philosophy 7 System 100, Ill , 121, 132

"The morning star paradox " (1957) 98, 113-115 "The notion of a right" (1963) 151, 162, 169,185, 187 Theorem of LC 73 Tree 79 Truth and prescriptions 133-134 Type 70-71 "Unavoidability" (1986) 169, 195,200 Unavoidability 177,186,190,199-201 Unification 50, 58 Uniform word problem 62 Validity 72-73,74,112,121-123 Valuation 72, 100, Ill , 120, 127, 132 Valuation , normal 72 Variable instantiation 57- 58 Weakening 54 Will theory 161

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J. M. Bochenski, A Precis of Mathematical Logic. Translated from French and German by O. Bird. 1959 ISB N 90-277-0073-7 P. Guiraud, Problemes et methodes de la statistique linguistique. 1959 ISB N 90-277 -0025-7 H. Freudenthal (ed.), The Concept and the Role of the Model in Mathematics and Natural and Soc ial Sciences. 1961 ISBN 90-277-0017-6 E. W. Beth , Formal Methods. An Introduction to Symb olic Logic and to the Study of Effe ctive Operations in Ari thmetic and Logic. 1962 ISBN 90-277-0069-9 B. H. Kazemi er and D. Vuysj e (eds .), Logic and Language. Studies dedicated to Profe ssor ISBN 90-277-0019-2 Rudolf Camap on the Occasion of His 70th Birthday. 1962 M . W. Wartofsky (ed .), Proceedings of the Boston Colloquium f or the Philosophy of Science, 1961-1962. [Boston Studi es in the Philosophy of Science, Vol. I] 1963 ISBN 90-277-0021-4 A. A. Zinov'ev, Philosophi cal Problems of Many-valu ed Logic. A revised edition, edited and translated (from Russian ) by G . Kling and D.D. Corne y. 1963 ISBN 90-277-0091-5 G . Gurvitch, The Spect rum of Social Time. Translated from French and edited by M. Korenbaum and P. Bosserman. 1964 ISBN 90-277-0006-0 P. Lorenzen, Formal Logic. Tran slated from German by F.J. Crosso n. 1965 ISBN 90-277-OO80-X R. S. Cohen and M . W. Wartofsky (eds .), Proceeding s ofthe Boston Colloquium for the Philosophy of Science, 1962-1 964. In Honor of Philipp Frank. [Boston Studies in the Philosophy ISBN 90-277-9004-0 of Scien ce, Vol. II] 1965 E. W. Beth, Mathemat ical Thought. An Introduction to the Philos oph y of Mathematics. 1965 ISBN 90-277-0070-2 E. W. Beth and J. Piaget , Mathemat ical Epistemol ogy and Psychology. Tran slated from French by W. Mays. 1966 ISBN 90-277 -0071 -0 G. Kling, Ontology and the Logisti c Analysis of Language. An Enqu iry into the Contemporary Views on Univer sal s. Revised ed., tran slated from German. 1967 ISBN 90-277-0028 -1 R. S. Cohen and M. W. Wartofsky (eds .), Proceedin gs of the Boston Colloquium for the Philosophy ofSciences, 1964-1966. In Memory of Norw ood Russell Han son. [Boston Studies ISBN 90-277-0013-3 in the Philosophy of Science, Vol. III] 1967 C. D. Broad, Induction, Probab ility, and Causation. Sele cted Papers. 1968 ISBN 90-277-0012-5 G. Patzig, Aristotle 's Theory ofthe Syllogi sm . A Logical-philosophical Study of Book A of the Prior Analytics. Translated from German by J. Bames. 1968 ISBN 90-277-0030-3 N. Rcscher, Topics in Philo sophical Logic. 1968 ISBN 90-277-0084-2 R. S. Cohen and M. W. Wartofsky (ed s.), Proceedings of the Boston Colloquium for the Philosoph y of Science, 1966-1968, Part I. [Boston Studies in the Philosophy of Science,

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Philosophy of Science, 1966-1 968, Part II. [Boston Studi es in the Philo soph y of Sc ience, Vol. V] 1969 ISBN 90-277-0015-X J. W. Davis, D. J. Hockn ey and W. K. Wilson (eds.), Philosophical Logic. 1969 ISBN 90-277-0075-3 D. Davidson and J. Hintikka (eds. ), Words and Objections. Essays on the Work of W. V. Quine. 1969 , rev. ed. 1975 ISB N 90-277 -0074- 5; Pb 90-277-060 2-6 P. Suppes, Studies in the Methodology and Foundations ofScience. Selec ted Papers f rom 1951 to 1969. 1969 ISB N 90-277-0020-6 J. Hint ikka, Models fo r Modalities . Selected Essays. 1969 ISB N 90-277-0078-8; Pb 90-277-0598-4

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75.

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78. 79.

80. 81. 82. 83. 84. 85. 86. 87. 88. 89.

A. Grzegorczyk, An Outline ofMathematical Logic. Fundamental Results and Notions explained with all Details. Translated from Polish. 1974 ISBN 90-277-0359-0; Pb 90-277-0447-3 F. von Kutschera, Philosophy of Language. 1975 ISBN 90-277-0591-7 J. Manninen and R. Tuomela (eds.), Essays on Explanation and Understanding . Studies in the Foundations of Humanities and Social Sciences. 1976 ISBN 90-277-0592-5 J. Hintikka (ed.), RudolfCarnap, Logical Empiricist. Materials and Perspectives. 1975 ISBN 90-277-0583-6 M. Capek (ed.) , The Concepts of Space and Time. Their Structure and Their Development. [Boston Studies in the Philosophy of Science, Vol. XXII] 1976 ISBN 90-277-0355-8; Pb 90-277-0375-2 J. Hintikka and U. Remes, The Method of Analysis. Its Geometrical Origin and Its General Significance. [Boston Studies in the Philosophy of Science, Vol. XXV] 1974 ISBN 90-277-0532-1 ; Pb 90-277-0543-7 J. E. Murdoch and E. D. Sylla (eds.), The Cultural Context of Medieval Learning . [Boston Studies in the Philosophy of Science, Vol. XXVI] 1975 ISBN 90-277-0560-7; Pb 90-277-0587-9 S. Amsterdamski, Between Experience and Metaphysics. Philosophical Problems of the Evolution of Science. [Boston Studies in the Philosophy of Science, Vol. XXXV] 1975 ISBN 90-277-0568-2; Pb 90-277-0580-1 P. Suppes (ed.), Logic and Probability in Quantum Mechanics. 1976 ISBN 90-277-0570-4; Ph 90-277-1200-X H. von Helmholtz: Epistemological Writings. The Paul Hertz / Moritz Schlick Centenary Edition of 1921 with Notes and Commentary by the Editors. Newly translated from German by M. F. Lowe. Edited, with an Introduction and Bibliography, by R. S. Cohen and Y. Elkana. [Boston Studies in the Philosophy of Science, Vol. XXXVII] 1975 ISBN 90-277 -0290-X; Pb 90-277-0582-8 J. Agassi, Science in Flux. [Boston Studies in the Philosophy of Science, Vol. XXVIII] 1975 ISBN 90-277-0584-4; Pb 90-277-0612-2 S. G. Harding (ed.), Can Theories Be Refuted? Essays on the Duhern-Quine Thesis. 1976 ISBN 90-277-0629-8; Pb 90-277-0630-1 S. Nowak, Methodology ofSociological Research. General Problems. 1977 ISBN 90-277 -0486-4 J. Piaget , I .-B. Grize, A. Szerninsska and V. Bang, Epistemology and Psychology ofFunctions. Translated from French. 1977 ISBN 90-277-0804-5 M. Grene and E. Mendelsohn (eds.), Topics in the Philosophy of Biology. [Boston Studies in the Philosophy of Science, Vol. XXVII] 1976 ISBN 90-277-0595-X; Ph 90-277-0596-8 E. Fischbein, The lntuitive Sources of Probabilistic Thinking in Children. 1975 ISBN 90-277-0626-3 ; Pb 90-277-1190-9 E. W. Adams , The Logic of Conditionals. An Application of Probability to Deductive Logic . 1975 ISBN 90-277-0631-X M. Przelecki and R. Wojcicki (eds .), Twenty-Five Years of Logical Methodology in Poland. Translated from Polish . 1976 ISBN 90-277-0601-8 I. Topolski, The Methodology of History. Translated from Polish by O. Wojtasiewicz. 1976 ISBN 90-277-0550-X A. Kasher (ed .), Language in Focus: Foundations, Methods and Systems . Essays dedicated to Yehoshua Bar-Hillel. [Boston Studies in the Philosophy of Science, Vol. XLIII] 1976 ISBN 90-277-0644-1 ; Ph 90-277-0645-X

SYNTHESE LIBR ARY 90. 91. 92. 93. 94. 95. 96.

97. 98. 99.

100. 101.

102. 103. 104. 105. 106. 107.

108. 109. 110. II I. 112. 113. 114.

J. Hintikka, The Intentions of Intentional ity and Other New Models fo r Modalities. 1975 ISBN 90-277-0633-6; Pb 90-277-0634-4 W. Stegmul ler, Collected Papers on Episte mology, Philosophy of Science and lJistory of Set ISB N 90-277-0767-7 Philosophy. 2 Volume s. 1977 D. M. Gabbay, Investigations in Modal and Tense Logics with Applications to Problems in ISBN 90-277-0656-5 Philosophy and Linguistics. 1976 R.1. Bogdan, Local Induction. 1976 ISBN 90-277-0649-2 S. Nowak, Understanding and Prediction. Essays in the Methodology of Social and Behavioral Theories. 1976 ISBN 90-277-0558-5; Pb 90-277-1199-2 P. Mittelstaedt, Philosophical Problems of Modem Physics. [Boston Studies in the Philosophy of Science , Vol. XVlII] 1976 ISBN 90-277-0285-3 ; Pb 90-277-0506-2 G. Holton and W. A. Blanpied (eds.), Science and Its Public: The Changing Relation ship. [Boston Stud ies in the Philosophy of Science , Vol. XXXIIl] 1976 ISBN 90-277 -0657-3; Pb 90-277-0658-1 M. Brand and D. Walton (eds.), Action Theory. 1976 ISBN 90-277-0671-9 P. Gochet, Outline ofa Nominalist Theory ofPropositions. An Essay in the Theory of Meaning and in the Philo sophy of Logic. 1980 ISBN 90-277-1031-7 R. S. Cohen , P. K. Feyerabend , and M. W. Wartofsky (eds.), Essays in Memory of lmre Lakatos . [Boston Studies in the Philosophy of Science , Vol. XXXIX] 1976 ISBN 90-277-0654-9; Pb 90-277-0655-7 R. S. Cohen and J. J. Staehel (eds.), Selected Papers of Leon Rosenfield. [Boston Stud ies in the Philosophy of Science, Vol. XXI] 1979 ISBN 90-277 -0651-4; Pb 90-277-0652-2 R. S. Cohen, C. A. Hooker, A. C. Michalos and J. W. van Evra (eds.) , PSA 1974. Proceedings of the 1974 Biennial Meet ing of the Philosophy ofScience Association. [Boston Studies in the Philosophy of Science, Vol. XXXlI] 1976 ISBN 90-277-0647-6; Pb 90-277-0648-4 y. Fried and 1. Agas si, Paranoia. A Study in Diagno sis. [Boston Studies in the Philosophy of Science, Vol. L] 1976 ISBN 90-277-0704-9; Pb 90-277-0705-7 M. Przelecki, K. Szaniawski and R. Wojcicki (eds.), Formal Methods in the Methodolog y of Empirical Sciences. 1976 ISBN 90-277-0698-0 J. M. Vickers, Beliefand Probability. 1976 ISBN 90-277-0744-8 K. H. Wolff, Surrender and Catch. Experience and Inquiry Today. [Boston Studies in the Philosophy of Science , Vol. Lt] 1976 ISBN 90-277-0758-8; Pb 90-277-0765-0 K. Kosik, Dialectics ofthe Concrete. A Study on Problems of Man and World. [Boston Studies in the Philosophy of Science, Vol. LlI] 1976 ISBN 90-277-0761-8; Pb 90-277-0764-2 N. Goodman, The Structure of Appearance. 3rd ed. with an Introduction by G . Hellman. [Boston Studies in the Philo sophy of Science , Vol. LIIl] 1977 ISBN 90-277-0773-1 ; Pb 90-277-0774-X K. Ajdukiewicz, The Scientific World-Perspective and Other Essays, 193/ -1963. Translated from Polish. Edited and with an Introduction by J. Giedymin . 1978 ISBN 90-277-0527-5 R. L. Cau sey, Unity of Science. 1977 ISB N 90-277 -0779-0 R. E. Grandy, Advanced Logic fo r Applications. 1977 ISBN 90-277-0781-2 R. P. McArthur, Tense Logic. 1976 ISBN 90-277-0697-2 L. Lindahl , Position and Change. A Study in Law and Logic. Translated from Sw edish by P. Needham. 1977 ISBN 90-277-0787-1 R. Tuomela, Dispositions. 1978 ISB N 90-277-08 10-X H. A. Simon, Models ofDiscovery and Other Topics in ihe Methods of Science. [Boston Stud ies in the Philosophy of Science, Vol. LtV] 1977 ISBN 90-277-08 12-6; Pb 90-277-0858-4

SYNTHESE LIBRARY 115. R. D. Rosenkrantz, Inference, Method and Decision. Towards a Bayesian Philosophy of Science . 1977 ISBN 90-277-0817-7 ; Pb 90-277-0818-5 116. R. Tuomela, Human Action and Its Explanation. A Study on the Philosophical Foundations of Psycholog y. 1977 ISBN 90-277-0824-X 117. M. Lazerowitz, The Language of Philosoph y. Freud and Wittgenstein. [Boston Studies in the Philosophy of Science , Vol. LVjI977 ISBN 90-277-0826-6; Pb 90-277-0862-2 118. Not published 119. J. Pelc (ed.), Semiotics in Poland, 1894-1969. Translated from Polish . 1979 ISBN 90-277 -0811-8 120. I. Porn, Action Theory and Social Science. Some Formal Models . 1977 ISBN 90-277-0846-0 121. J. Margolis, Persons and Mind . The Prospects of Nonreductive Materialism. [Boston Studies in the Philosophy of Science, Vol. LVIIj 1977 ISBN 90-277 -0854-1; Pb 90-277-0863-0 122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic. 1979 ISBN 90-277-0879-7 123. T. A. F. Kuipers, Studies in Inductive Probability and Rational Expectation . 1978 ISBN 90-277-0882-7 124. E. Saarinen, R. Hilpinen , I. Niiniluoto and M. P. Hintikka (eds.), Essays in Honour of Jaakko ISBN 90-277-0916-5 Hintikka on the Occasion of His 50th Birthday. 1979 125. G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science. [Boston Studies in the Philosophy of Science, Vol. LVIII] 1978 ISBN 90-277-0921-1; Pb 9Q-277-0922-X 126. P. Mittelstaedt, Quantum Logic. 1978 ISBN 90-277-0925-4 127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi . 1979 ISBN 90-277-0929-7 128. H. A. Bursen, Dismantling the Memory Machine. A Philosophical Investigation of Machine Theories of Memory. 1978 ISBN 90-277-0933-5 129. M. W. Wartofsky, Models . Representation and the Scientific Understanding. [Boston Studies ISBN 90-277-0736-7; Pb 90-277-0947-5 in the Philosophy of Science , Vol. XLVIII] 1979 130. D. Ihde, Technics and Praxis. A Philosophy of Technology. [Boston Studies in the Philosophy ISBN 90-277-0953-X; Pb 90-277-0954-8 of Science , Vol. XXIVj1979 131. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science , Vol. XXIX] 1979 ISBN 90-277-0723-5; Pb 90-277-0956-4 ISBN 90-277-0958-0 132. W. C. Salmon (ed.), Hans Reichenbach: Logical Empiricist . 1979 133. P. Bieri, R.-P. Horstmann and L. Kruger (eds.), Transcendental Arguments in Science . Essays in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5 134. M. Markovic and G. Petrovic (eds.), Praxis. Yugoslav Essays in the Philosophy and Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXXVI] 1979 ISBN 90-277 -0727-8 ; Pb 9Q-277-0968-8 135. R. Wojcicki, Topics in the Formal Methodology ofEmpirical Sciences . Translated from Polish. 1979 ISBN 90-277-1004-X 136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science . [Boston Studies in the Philosophy of Science, Vol. LIXj1979 ISBN 90-277-0994-7; Pb 90-277-0995-5 137. J. C. Webb, Mechanism , Mentalism and Metamathematics . An Essay on Finitism. 1980 ISBN 90-277-1046-5 138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mind and Method. Essays in Honor of Virgil C. Aldrich . 1979 ISBN 90-277-1013-9 139. L. Nowak, The Structure of Idealization . Towards a Systematic Interpretation of the Marxian Idea of Science. 1980 ISBN 90-277-1014-7

SYNTHESE LIBRARY 140. C. Perelman, The New Rhetoric and the Humanities. Ess ays on Rhetoric and Its Applications. Translated from French and Germ an. With an Introduction by H. Zyskind . 1979 IS BN 90-277 - 1018-X; Pb 90-277-1019-8 141. W. Rabinowicz, Universalizability. A Study in Moral s and Metaph ysic s. 1979 ISBN 90-277 -1020- 2 142. C. Perelm an, Justice, Law and Argument. Essay s on Moral and Legal Reasoning. Tran slated from French and Germ an. With an Introduction by H.J . Berm an. 1980 IS BN 90-277-1089 -9 ; Pb 90-277-1090-2 143. S. Kanger and S. Ohman (cds.), Philosophy and Grammar. Paper s on the Occas ion of the Quincentennial of Uppsala Univer sity. 198 1 IS BN 90-277-109 1-0 144 . T. Pawlowski, Concept Formatio n in the Humanities and the Social Sciences. 1980 ISBN 90-277-1096-1 145. J. Hint ikka, D. Grucndcr and E. Agazzi (eds.), Theory Change, Ancient Axiomatics and Galileo 's Methodology. Proceedings ofthe 1978 Pisa Conference on the History and Phi losophy of Sci ence, Volume I. 1981 ISBN 90-277-11 26-7 146. J . Hin tikka, D. Gru ender and E. Agazzi (eds.), Probabilistic Thinking, Thermodynami cs, and the Interaction of the History and Philosophy of Science. Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Volume II. 1981 ISBN 90-277-112 7-5 147. U. Monnich (ed.), Aspects of Philosophical Logic. Some Logical Foray s into Cen tral Notions of Lingui stics and Philosophy. 198 1 ISB N 90-2 77- 1201-8 148. D. M. Gabbay, Semantical lnvestigations in Heyting 's lntuitionistic Logic. 1981 ISB N 90-277- 1202-6 149. E. Agazzi (ed.), Modem Logic - A Survey. Historical, Philosophical, and Mathematical Aspects of Modern Logic and Its Applications. 198 1 IS BN 90-277 -1137-2 150. A. F. Parker- Rhode s, The Theory of lndistinguishables. A Search for Expl anatory Principles ISB N 90-277- 1214-X below the Level of Physics. 198 1 151. J. C. Pitt, Pictures, Images, and Conceptual Change. An Analy sis of Wilfrid Sellars' Philosophy of Science. 1981 IS BN 90-277- 1276-X ; Pb 90-277-1277-8 152. R. Hilp inen (ed.), New Studies in Deontic Logic. Norms, Actions, and the Foundations of Ethics. 198 1 ISBN 90-277- 1278-6; Pb 90-277-1346-4 153. C. Dilworth, Scientific Progress . A Study Concerning the Nature of the Relation between Successive Scientific Theories. 3rd rev. ed ., 1994 ISBN 0-7923 -2487-0 ; Pb 0-7923-2488-9 154. D. Woodruff Smith and R. Mcl ntyre, Husserl and Intentionality. A Study of Mind, Meani ng, and Language. 1982 ISBN 90-277- 1392-8; Pb 90-277- 1730-3 155. R. J. Nelson, The Logic of Mind. 2nd. ed ., 1989 ISBN 90-277-28 19-4; Pb 90-277-2822-4 156. J. F. A. K. van Benthem, The Logic of TIme. A Model-Theoretic Investigation into the Varieties of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed. , 1991 ISBN 0-7923 -1081-0 157. R. Swinburne (ed.), Space, TIme and Causality. 1983 ISBN 90-277-1437-1 158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D. Rozenkrantz. 1983 ISBN 90-277-1448-7 ; Pb (1989 ) 0-7923 -0213-3 159. T. Chapman, TIme: A Philosophical Analysis. 1982 ISBN 90-277 -1465-7 160. E. N. Zalta , Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983 ISBN 90-277-1474-6 161. S. Hard ing and M. B. Hintikka (cds.), Discovering Reality. Femin ist Perspectives on Epist emology, Metaphysics, Methodology, and Philosophy of Science. 1983 ISBN 90-277-1496-7 ; Pb 90-277- 1538-6 162. M. A. Stewart (ed.). U1W, Moralit y and Rights. 1983 ISBN 90-277 -1519 -X

SYNTHESE LIBRARY 163. D. Mayr and G. Siissmann (eds.), Space, TIme, and Mechanics. Basic Structures of a Physical Theory. 1983 ISBN 90-277-1525-4 164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I: Elements of Classical Logic. 1983 ISBN 90-277-1542-4 165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. II: Extensions of Classical Logic. 1984 ISBN 90-277-1604-8 166. D. Gabbay and F. Guenthner (eds.), Handbook ofPhilosophi cal Logic. Vol. III: Alternative to Classical Logic. 1986 ISBN 90-277-1605-6 167. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. IV: Topics in the Philosophy of Language. 1989 ISBN 90-277-1606-4 168. A. J. I. Jones, Communication and Meaning . An Essay in Applied Modal Logic. 1983 ISBN 90-277-1543-2 169. M. Fitting, Proof Methods f or Modal and Intuit ionistic Logics. 1983 ISBN 90-277-1573-4 170. J. Margolis, Culture and Cultural Entities. Toward a New Unity of Science. 1984 ISBN 90-277-1574-2 171. R. Tuornela, A Theo ry ofSocial Action. 1984 ISBN 90-277-1703-6 172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical Analysis in Latin America. 1984 ISBN 90-277-1749-4 173. P. Ziff, Epistem ic Analysis. A Coherence Theory of Knowledge. 1984 ISBN 90-277-1751-7 174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984 ISBN 90-277-1773-7 175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure, Examples, Philosophical Problems. 1984 ISBN 90-277-1811-3 176. A. Peczenik, L. Lindahl and B. van Roermund (eds.), Theory ofLegal Science. Proceedings of the Conference on Legal Theory and Philosophy of Science (Lund, Sweden, December 1983). 1984 ISBN 90-277-1834-2 177. I. Niiniluoto, Is Science Progressive ? 1984 ISBN 90-277-1835-0 178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative Perspective. Exploratory Essays in Current Theories and Classical Indian Theories of Meaning and Reference. 1985 ISBN 90-277-1870-9 179. P. Kroes, TIme: Its Structure and Role in Physical Theories. 1985 ISBN 90-277-1894-6 180. J. H. Fetzer, Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1 181. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. 1986 ISBN 90-277-2126-2 182. M. Detlefsen, Hilbert's Program. An Essay on Mathematical Instrumentalism. 1986 ISBN 90-277-2151-3 183. J. L. Golden and J. J. Pilotta (eds.), Practi cal Reasoning in Human Affairs. Studies in Honor ofChaim Perelman. 1986 ISBN 90-277-2255-2 184. H. Zandvoort, Mod els ofScientific Developm ent and the Case of Nucl ear Magnetic Resonance. 1986 ISBN 90-277-2351-6 ISBN 90-277-2354-0 185. I. Niiniluoto, Truthliken ess . 1987 186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic fo r Science. The Structuralist Program. 1987 ISBN 90-277-2403-2 187. D. Pearce, Roads to Commensurability. 1987 ISBN 90-277-2414-8 188. L. M. Vaina (ed.), Matte rs of Intelligence. Conceptual Structures in Cognitive Neuroscience. 1987 ISBN 90-277-2460-1

SYNTHESE LIBRARY 189. H. Siegel, Relativism Refu ted. A Critique of Contemporary Epistemological Relativism. 1987 ISBN 90-277-2469-5 190. W CalJebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm Program, with a Complete Evolutionary Epistemology Bibliograph. 1987 ISBN 90-277-2582-9 ISBN 90-277-2199-8 191. J. Kmita, Problems in Historical Epistemology. 1988 192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon, with an Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2 193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical Studies of Scientific Change. 1988 ISBN 90-277-2608-6 ISBN 90-277-2640-X 194. H.R. Otto and J.A. Tuedio (eds.), Perspectives on Mind. 1988 195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights and New Perspectives on Their Relation. 1988 ISBN 90-277-2645-0 ISBN 90-277-2648-5 196. J. Osterberg, Self and Others. A Study of Ethical Egoism. 1988 197. D.H. Helman (ed.), Anal ogical Reasoning. Perspectives of Artificial Intelligence, Cognitive Science, and Philosophy. 1988 ISBN 90-277-2711-2 198. J. Wolenski, Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X 199. R. Wojcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations. 1988 ISBN 90-277-2785-6 200. J. Hintikka and M.B. Hintikka, The Logic of Epistemology and the Epistemology of Logic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6 ISBN 90-277-2808-9 201. E. Agazzi (ed.), Probabili ty in the Sciences. 1988 ISBN 90-277-2814-3 202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 203. R.L. Tieszen, Mathematical lntuition, Phenomenology and Mathematical Knowledge. 1989 ISBN 0-7923-0131-5 ISBN 0-7923-0135-8 204. A. Melnick, Space. TIme, and Thought in Kant. 1989 205. D.W Smith, The Circle ofAcquaintance. Perception, Consciousness, and Empathy. 1989 ISBN 0-7923-0252-4 206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of Arthur W Burks. With his Responses, and with a Bibliography of Burk's Work. 1990 ISBN 0-7923-0325-3 207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study in Husser!, Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4 208. T.C. Meyering, Historical Roots of Cognitive Science . The Rise of a Cognitive Theory of Perception from Antiquity to the Nineteenth Century. 1989 ISBN 0-7923-0349-0 ISBN 0-7923-0389-X 209. P. Kosso, Observabilit y and Observation in Physical Science. 1989 ISBN 0-7923-0441-1 210. J. Kmita, Essays on the Theory of Scientific Cognition. 1990 211. W Sieg (ed.), Acting and Reflecting. The Interdisciplinary Tum in Philosophy. 1990 ISBN 0-7923-0512-4 ISBN 0-7923-0546-9 212. J. Karpinski, Causality in Sociological Research. 1990 ISBN 0-7923-0823-9 213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 1991 214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein's Philosophy of Psychology. 1990 ISBN 0-7923-0850-6 215. M. Gosselin, Nominalism and Contempo rary Nominalism. Ontological and Epistemological Implications of the Work ofW V.O. Quine and of N. Goodman. 1990 ISBN 0-7923-0904-9 216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definabil ity. Philosophical Perspectives. 1991 ISBN 0-7923-1046-2 217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the Universe. 1991 ISBN 0-7923-1322-4

SYNTHESE LIBRARY 218. M. Kusch, Foucault 's Strata and Fields. An Investigationinto Archaeological and Genealogical Science Studies. 1991 ISBN 0-7923-1462-X 219. C.1. Posy, Kant 's Philosophy of Mathematics. Modem Essays. 1992 ISBN 0-7923-1495-6 220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomy and Connectionism.1992 ISBN 0-7923-1519-7 ISBN 0-7923-1566-9 221. 1.C. Nyfri, Tradition and Individuality . Essays. 1992 222. R. Howell, Kant's Transcendental Deduction . An Analysis of Main Themes in His Critical Philosophy. 1992 ISBN 0-7923-1571-5 223. A. Garcia de la Sienra, The Logical Foundations of the Marxian Theory of Value. 1992 ISBN 0-7923-1778-5 224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of Our Conceptual Order. 1992 ISBN 0-7923-1803-X 225. M. Rosen, Problems of the Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human Reality. 1993 ISBN 0-7923-2047-6 226. P. Suppes, Models and Methods in the Philosoph y ofScience : Selected Essays . 1993 ISBN 0-7923-2211-8 227. R. M. Dancy (ed.), Kant and Critique : New Essays in Honor ofW. H. Werkmeister. 1993 ISBN 0-7923-2244-4 228. 1. Woleriski (ed.), Philosophical Logic in Poland. 1993 ISBN 0-7923-2293-2 229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional Logic. 1993 ISBN 0-7923-2342-4 230. B.K. Matilal andA . Chakrabarti (eds.), Knowing from Words. Westernand Indian Philosophical Analysis of Understanding and Testimony. 1994 ISBN 0-7923-2345-9 231. S.A. Kleiner, The Logic ofDiscovery. A Theory of the Rationality of Scientific Research. 1993 ISBN 0-7923-2371-8 232. R. Festa, Optimum Inductive Methods . A Study in Inductive Probability, Bayesian Statistics, and Verisimilitude. 1993 ISBN 0-7923-2460-9 233. P. Humphreys(ed.), Patrick Suppes: Scientific Philosopher. Vol. 1:Probability and Probabilistic Causality. 1994 ISBN 0-7923-2552-4 234. P. Humphreys (ed.), Patrick Suppes : Scientific Philosopher. Vol. 2: Philosophy of Physics, Theory Structure, and Measurement Theory. 1994 ISBN 0-7923-2553-2 235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and Psychology. 1994 ISBN 0-7923-2862-0 Set ISBN (Vols 233-235) 0-7923-2554-0 236. D. Prawitz and D. Westerstahl (eds.), Logic and Philosophy of Science in Uppsala. Papers from the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994 ISBN 0-7923-2702-0 237. L. Haaparanta (ed.), Mind, Meaning and Mathematics. Essays on the Philosophical Views of Husserl and Frege. 1994 ISBN 0-7923-2703-9 238. 1. Hintikka (ed.), Aspects ofMetaphor. 1994 ISBN 0-7923-2786-1 239. B. McGuinness and G. Oliveri (eds.), The Philosophy ofMichael Dummett . With Replies from Michael Dummett. 1994 ISBN 0-7923-2804-3 240. D. lam ieson (ed.), Language, Mind, and Art. Essays in Appreciation and Analysis, In Honor of Paul Ziff. 1994 ISBN 0-7923-2810-8 241. G. Preyer, F. Siebelt and A. Ulfig (eds.), Language, Mind and Epistemology . On Donald Davidson's Philosophy. 1994 ISBN 0-7923-2811-6 242. P. Ehrlich (ed.), Real Number s, Generalizations ofthe Reals, and Theories of Continua. 1994 ISBN 0-7923-2689-X

SYNTHESE LIBRARY 243 . G. Debrock and M. Hulswit (eds.), Living Doubt . Essays concerning the epistemology of Charl es Sanders Peirce. 1994 ISBN 0-7923-2898-1 244 . 1. Srzednicki , To Know or Not to Know. Beyond Realism and Anti-Realism. 1994 ISBN 0-7923-2909-0 245 . R. Egid i (ed.), Wittgenstein: Mind and Language. 1995 ISBN 0-7923-3171-0 246. A. Hyslop, Other Minds . 1995 ISBN 0-7923-3245-8 247 . L. P610s and M. Masuch (eds.), Applied Logic: How, What and Why. Logical Approaches to Natural Language . 1995 ISBN 0-7923-3432-9 248 . M. Krynicki, M. Mostowski and L.M . Szczerba (cds.), Quantifiers: Logics, Models and Computation. Volume One : Surveys . 1995 ISBN 0-7923-3448-5 249. M. Krynicki , M. Mostowski and L.M . Szczerba (eds.), Quantifiers: Logics, Models and ComISBN 0-7923-3449-3 putation. Volume Two: Contributions. 1995 Set ISBN (Vols 248 + 249) 0-7923-3450-7 250. R.A. Watson, Representational Ideas from Plato to Patricia Churchland. 1995 ISBN 0-7923 -3453 -1 251. 1. Hintikka (ed.) , From Dedekind to Godel . Essays on the Development of the Foundations of Mathematics. 1995 ISBN 0-7923-3484-1 252. A. Wisniewski, The Posing ofQuestions. Logical Foundations of Erotetic Inferences. 1995 ISBN 0-7923-3637-2 253. J. Peregrin, Doing Worlds with Words. Formal Semantics without Formal Metaphysics. 1995 ISBN 0-7923-3742-5 254 . LA. Kieseppa, Truthlikeness for Multidimensional, Quantitative Cognitive Problems. 1996 ISBN 0-7923-4005-1 255. P. Hugly and C. Sayward: Intensionality and Truth. An Essay on the Philosophy of A.N. Prior. 1996 ISBN 0-7923-4119-8 256. L. Hankinson Nelson and J. Nelson (cds .): Feminism, Science, and the Philosophy of Science. 1997 ISBN 0-7923-4162-7 257. P.1. Bystrov and V.N. Sadovsky (eds.): Philosophical Logic and Lagical Philosophy. Essays in Honour of Vladimir A. Smirnov . 1996 ISBN 0-7923-4270-4 258 . A.E. Andersson and N-E . Sahlin (eds.) : The Complexity of Creativity. 1996 ISBN 0-7923-4346-8 259. M.L. Dalla Chiara, K. Doets , D. Mundici and J. van Benthem (eds.): Lagic and Scientific Methods. Volume One of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, Augus t 1995. 1997 ISBN 0-7923-4383-2 260. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.) : Structures and Norms in Science . Volume Two of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995. 1997 ISBN 0-7923-4384-0 Set ISBN (Vols 259 + 260) 0-7923-4385-9 261. A. Chakrabarti : Denying Existence. The Logic , Epistemology and Pragmatics of Negative Existentials and Fictional Discourse. 1997 ISBN 0-7923-4388-3 262. A. Bilet zki: Talking Wolves. Thomas Hobbes on the Language of Politics and the Politics of Language . 1997 ISBN 0-7923-4425-1 263 . D. Nute (cd .): Defeasible Deontic Logic. 1997 ISBN 0-7923-4630-0 264 . U. Meixner: Axiomatic Formal Ontology. 1997 ISBN 0-7923-4747-X 265. I. Brinck : The Indexical T . The First Person in Thought and Language. 1997 ISBN 0-7923-4741-2 266. G. Holmstrom-Hintikka and R. Tuomcla (eds.): Contemporary Action Theory. Volume 1: Individual Action . 1997 ISBN 0-7923-4753-6; Set : 0-7923 -4754-4

SYNTHESE LIBRARY 267. G. Holmstrom-H intikka and R. Tuomela (eds.): Contemporary Action Theory. Volume 2: Social Action. 1997 ISBN 0-7923-475 2-8; Set: 0-7923-4754-4 268. B.-C. Park: Phenomenological Asp ects of wittgenstein's Philosophy. 1998 ISBN 0-7923-48 13-3 269. J. Pasn iczek : The Logic of Intentional Objects. A Meinongian Version of Clas sical Logic. 1998 Hb ISBN 0-7923-4880-X ; Pb ISBN 0-7923- 5578-4 270. P.w. Humphreys and J.H. Fetzer (eds.): The New Theory of Ref erence. Kripke, Marcus, and Its Origins . 1998 ISBN 0-7923-4 898-2 27 1. K. Szaniawski, A. Chmielew ski and J. Wolenski (eds.): On Science, Inference, Information and Decision Making. Selected Essays in the Philosoph y of Scienc e. 1998 ISBN 0-7923 -4922-9 272. G.H. von Wright: In the Shadow of Descartes. Essays in the Philosophy of Mind . 1998 ISBN 0-7923-4992-X 273. K. Kijania-Placek and J. Wolenski (eds.): The Lvov-Warsaw School and Contemporary Philosophy. 1998 ISBN 0-7923 -5105-3 274. D. Dedrick: Nam ing the Rainbow. Colour Language, Colour Science, and Culture. 1998 ISBN 0-7923-5239-4 275. L. Albertaz zi (ed.): Shapes of Forms. From Gestalt Psychology and Phenomenol ogy to Ontology and Mathemati cs. 1999 ISBN 0-7923-5246-7 276. P. Fletcher: Truth, Proofand Infinity. A Theory of Constructions and Constructive Reason ing. 1998 ISBN 0-7923-5262-9 277. M. Fitting and R.L. Mendelsohn (eds.) : First-Order Modal Logic. 1998 Hb ISBN 0-7923-5334-X; Pb ISBN 0-7923-5335 -8 278. J.N. Mohanty: Logic, Truth and the Modalitiesfrom a Phenomenolog ical Perspective. 1999 ISBN 0-7923-5550-4 279. T. Placek : Mathematicallntiutionism and Intersubj ectivi ty. A Critical Exposition of Arguments for Intuitionism . 1999 ISBN 0-7923-5630-6 280. A. Cantini , E. Casari and P. Minari (eds.): Logic and Foundations ofMathematics . 1999 ISBN 0-7923 -5659-4 set ISBN 0-7923 -5867-8 281. M.L. Dalla Chiara, R. Giuntin i and F. Laudisa (eds.): Language, Quantum, Music. 1999 ISBN 0-7923-5727-2 ; set ISBN 0-7923-5867-8 282. R. Egidi (ed.): In Search ofa New Humanism . The Philosophy of Georg Hendrik von Wright. 1999 ISBN 0-792 3-5810-4 283. F. Vollmer: Agent Causality. 1999 ISBN 0-7923-5848-1 ISBN 0-7923 -5865-1 284. J. Peregrin (ed.): Truth and Its Nature (if Any). 1999 285. M. De Caro (ed.): Interpretations and Causes. New Perspectives on Donald Davidson 's Philosophy. 1999 ISBN 0-7923 -5869-4 286. R. Murawski : Recursive Functions and Metamath ematics. Problems of Completeness and Decidabil ity, Godel 's Theorems. 1999 ISBN 0-7923-5904-6 287. T.A.F. Kuipers: From Instrum entalism to Constructive Realism. On Some Relat ions between Con firmation, Empirical Progress. and Truth Approx imation. 2000 ISBN 0-7923-6086-9 288. G. Holmstrom-Hintikka (ed.): Medieval Philosophy and Modem TImes. 2000 ISBN 0-7923-6102-4 289. E. Grosholz and H. Breger (eds.): The Growth of Mathematical Knowledge. 2000 ISBN 0-7923-6151- 2

SYNTHESE LIBRARY 290 . G . Sommaruga: History and Philosophy of Constructive Type Theory . 2000 ISBN 0-7923-6180-6 291. J. Gasser (cd .): A Boole Anthology. Recent and Classical Studies in the Logic of George Boole . 2000 ISBN 0-7923-6380-9 292 . Y.F. Hendricks, S.A. Pedersen and K.F. Jergen sen (eds .): Proof Theory. History and Philosophical Significance. 2000 ISBN 0-7923-6544-5 293 . wt, Craig: The Tensed Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6634-4 294. WL. Craig : The Tenseless Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6635-2 295 . L. Albertazzi (cd .): The Dawn of Cognitive Science . Early European Contributors. 2001 ISBN 0-7923-6799-5 296. G. Forrai : Reference , Truth and Conceptual Schemes . A Defense of Internal Realism. 2001 ISBN 0-7923-6885-1 297. Y.F. Hendricks, S.A . Pede rsen and K.F. Jergensen (cds .): Probability Theory . Philosophy, Recent History and Relations to Science. 2001 ISBN 0-7923-6952-1 298 . M. Esfeld: Holism in Philosophy ofMind and Philosophy of Physics . 2001 ISBN 0-7923-7003-1 299 . E.C. Steinhart: The Logic of Metaphor . Analogous Parts of Poss ible Worlds . 2001 ISBN 0-7923-7004-X 300. To be published. 301. T.A.F. Kuipers: Structures in Science Heuristic Patterns Based on Cognitive Structures. An Advanced Textbook in Nco-Classical Philosophy of Science. 2001 ISBN 0-7923-7117-8 302 . G. Hon and S.S. Rakover (eds .): Explanation . Theoretical Approaches and Applications. 2001 ISBN 1-4020-0017-0 303. G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds .): Collected Papers ofStig Kanger with Essays on his Life and Work. Vol. I. 2001 ISBN 1-4020-0021-9; Pb ISBN 1-4020-0022-7 304. G. Holmstrorn-Hintikka, S. Lindstrom and R. Sliwinski (eds .): Collected Papers ofStig Kanger with Essays on his Life and Work. Vol. II. 2001 ISBN 1-4020-0111 -8; Pb ISBN 1-4020-0112-6

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