Studies on the History of Logic: Proceedings of the III. Symposium on the History of Logic [Reprint 2019 ed.] 9783110903829, 9783110148299

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Studies on the History of Logic

Perspektiven der Analytischen Philosophie Perspectives in Analytical Philosophy Herausgegeben von Georg Meggle und Julian Nida-Rümelin Band 8

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G Walter de Gruyter • Berlin • New York 1996

Studies on the History of Logic Proceedings of the III. Symposium on the History of Logic Edited by Ignacio Angelelli and Maria Cerezo

w DE

G

Walter de Gruyter • Berlin • New York 1996

® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication Data Symposium on the History of Logic (3rd ; 1993 : University of Navarra) Studies on the history of logic ; proceedings of the III. Symposium on the History of Logic / edited by Ignacio Angelelli and Maria Cerezo. p. cm. — (Perspektiven der analytischen Philosophic ; Bd. 8 = Perspectives in analytical philosophy) ISBN 3-11-014829-3 1. Logic — History — Congresses. I. Angelelli, Ignacio. II. Cerezo, Maria, 1964III. Tide. IV. Series, Perspectives in analytical philosophy ; Bd. 8. BC15.S96 1993 160'.9-dc20 96-5201 CIP

Die Deutsche Bibliothek - Cataloging-in-Publication Data Studies on the history of logic : proceedings of the III. Symposium on the History of Logic / ed. by Ignacio Angelelli and Maria Cerezo. - Berlin ; New York : de Gruyter, 1996 (Perspektiven der analytischen Philosophie ; Bd. 8) ISBN 3-11-014829-3 NE: Angelelli, Ignacio [Hrsg.]; Symposium on the History of Logic ; GT

© Copyright 1996 by Walter de Gruyter & Co., D-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany Typesetting and Printing: Arthur Collignon GmbH, Berlin Binding: Liideritz & Bauer, Berlin Cover design: Rudolf Hiibler, Berlin

Preface For the third time it has been possible to organize a meeting on the History of Logic at the University of Navarra, in Pamplona (Spain). It seems appropriate to pause, and to take a retrospective look at what has been accomplished. The first Symposium of History of Logic took place on May 14 and 15 of 1981. The proceedings were published in a special issue of Anuario Filosófico (University of Navarra), vol. XVI, 1, 1983. Here is the list of papers read at the conference and published in that volume: I. Angelelli: "Presentación del Simposio"; M. Mignucci: "La teoria della quantificazione del predicato nell'antichità classica"; C. Imbert: "Histoire et formalisation de la logique"; K. Jacobi: "Aussagen über Ereignisse. Modal- und Zeitlogische Analysen in der mittelalterlichen Logik"; V. Muñoz Delgado: "Pedro de Espinosa (m. 1536) y la lògica en Salamanca hasta 1550"; A. d'Ors: "Las Summulae de Domingo de Soto. Los límites de la regla bollendo tollens' "; J. L. Fuertes Herreros: "La lógica de Sebastián Izquierdo (1601-81): Un intento precursor de la lógica moderna en el siglo XVII"; L. Hickman: "The Logica Magna of Juan Sanchez Sedeño (1600). A sixteenth century addition to the Aristotelian categories"; H. Burkhardt: "Modaltheorie und Modallogik in der Scholastik und bei Leibniz"; Ch. Thiel: "Die Revisionsbedürftigkeit der logischen Semantik Freges"; I. Angelelli: "Sobre una clase especial de proposiciones reduplicativas"; A. García Suárez: "Fatalismo, trivalencia y verdad: un análisis del problema de los futuros contingentes"; G. Kalinowski: "La logique juridique et son histoire". In the Preface to that first gathering it was observed that "The recent decades have seen an extraordinary development of the research in the history of logic. Naturally, this has been possible because of the increase of logical studies occurred since the second half of the 19th century, after the relative silence of the 'modern philosophy' period. To today's scholars an enormous and diverse set of sources is available. Aside from the Greek-Latin-European tradition, we have the arabic and oriental sources.... it is necessary to insist on the unity and harmony that really exists among these various forms of logical theory." The concept of 'forms', or 'varieties' of logic in its history, was first introduced into logical historiography by H. Scholz (Abriss der Geschichte der Logik, Berlin, 1931, §1: Die Gestalten der Logik)1 and thereafter quite systematically applied by Bochenski in his epoch-making Formale Logik of 1956, where four historical forms were distinguished: Ancient, Scholastic, Mathematical, and Indian.2

VI

Preface

Such was the historiographical conception inspiring the organization of the I Symposium, in 1981. As pointed out in the Preface, "the intention was to somehow cover the principal forms of the logical tradition, with the exception of the oriental and islamic ones." The same emphasis on the unity of logic under its diverse historical forms presided the organization of the II Symposium, in May 1987, but then three rather than just two days of sessions were necessary to accomodate the increasing number of participants. The proceedings of the II Symposium were published in a 591 pages volume titled Estudios de Historia de la Lógica, ed. by I. Angelelli and Angel d'Ors (Pamplona: Ediciones Eunate, 1990). Here is the list of the contributions: I. Angelelli: "Presentación"; E. J. Ashworth: "The doctrine of signs in some early sixteenth-century spanish logicians"; I. Boh: "On medieval rules of obligation and rules of consequence"; A. Broadie: "Act and object in latescholastic logic"; H. Burkhardt: "Contingency and probability: a contribution to the Aristotelian theory of science"; J. Coombs: "John Mair and Domingo de Soto on the reduction of iterated modalities"; D. Felipe: "Johannes Felwinger (1659) and Johannes Schneider (1718) on syllogistic disputation"; N. Hinske: "Kant by computer. Applications of electronic data processing in the humanities"; H. Hochberg: "Predication, relations, classes and judgment in Russell's philosophical logic"; J. Hruschka: "The hexagonal system of deontic concepts according to Achenwall and Kant"; S. Knuuttila: "The varieties of natural necessity in medieval thought"; W. Lenzen: "Precis of the history of logic from the point of view of the Leibnizian calculus"; J. C. León and A. Burrieza: "Identity and necessity from the Fregean perspective"; A. C. Lewis: "An introduction to the Bertrand Russell editorial project: axiomatics in Russell"; Ch. Martin: "Significatio nominis in Aquinas"; M. Mignucci: "Alexander of Aphrodisias on inference and syllogism"; V. Muñoz Delgado: "El análisis de los enunciados de 'incipit et desinif en la lógica de Juan de Oria (1518) y en la de otros españoles hasta 1540"; N. Óffenberger: "Die Oppositionstheorie strikt partikulárer Urteilsarten aus der Sicht der Vierwertigkeit"; A. d'Ors: "La doctrina de las proposiciones insolubles en las Dialecticae introductiones de Agustín de Sbarroya"; J. Sánchez Sánchez: "Quine y Kripke sobre el análisis objetual de los enunciados de identidad"; Ch. Thiel: "Must Frege's role in the history and philosophy of logic be rewritten?" The III Symposium was again structured with the leading principle of the various forms of logic and their unity in mind. No more days than three were scheduled for the meetings, but the number of participants continued to increase, which has made our third gathering more "intense" than ever.

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Preface

Some papers are included from scholars who have not been able to attend the III Symposium (Cates), or from scholars who did attend but the time constraints made it impossible for them to present orally their contributions (McMahon, Jimenez Cataño, Dufour, Legris). Also, a number of authors 'Who read a paper in the III Symposium chose not to submit a written version of it for publication in the proceedings (Burkhardt, Burton, Sousedik). A special feature of this III Symposium was the organization of a bibliographical exhibition entitled 'Manuscritos y Libros de Lógica en las Bibliotecas de Pamplona' ('Manuscripts and Books of Logic in the Libraries of Pamplona'). Thirty logical works from the 14th to the 18th centuries were showed: three manuscripts, six incunabula, eighteen printed books from the 16th to the 18th centuries and three manuscript class notes from the 18th century. Exhibits were selected from five libraries of Pamplona: Biblioteca Capitular, Biblioteca de los P.P.Capuchinos Extramuros, Biblioteca General de Navarra, Biblioteca del Seminario Diocesano and Biblioteca de la Universidad de Navarra. The exhibition took place at the University of Navarra during the three days of the Symposium and was partially supported by the Government of Navarra. We are indebted to Paloma Pérez-Ilzarbe who organized the exhibition and to all the Libraries which let us show some of their treasures. The arithmetical question of whether a fourth symposium is being planned must remain unanswered here. As the principle of plenitude demands, if it is possible, it will happen. We want to express our gratitude to all the Institutions and persons who helped in the organization of the III Symposium, especially to the University of Navarra, and to its President, Prof. Alejandro Llano. Special thanks are due to Jaime Nubiola and Angel d'Ors for their suggestions in organizing the Symposium and editing this volume. The preparation of the manuscript was coordinated by María Cerezo. We are indebted to many people who were involved in the work; especially to Ruth Breeze who translated into English some of the contributions, and to Juan Cruz, Manuel García-Clavel and Idoya Zorroza for their assistance in editing several papers. Finally, but most importantly, we are grateful to the contributors themselves for allowing their work to appear in this volume. Ignacio Angelelli Austin, Texas

María Cerezo Pamplona, Spain

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Preface

Notes 1

2

Instead of Gestalten der Logik, Scholz has considered also the phrase Bedeutungsmannigfaltigkeit des Namens 'Logik'; he preferred the former phrase because of its being shorther (p. 1 fn). The title Formale Logik of Bocheriski's famous volume is surely uninformative, since it hides the historiographical central significance of it, better highlighted by the title of the series in which Bocheriski's work was published: Problemgeschichten der Wissenschaft in Dokumenten und Darstellungen. Thus, it was wise on Ivo Thomas' part to title his translation of Bocheriski's book as A history of formal logic. Hopefully, this defect in the title is corrected in an eventual third, German edition.

Contents Preface List of Contributors

V XI

MARIO MIGNUCCI

Aristotle's Theory of Predication

1

ROBIN SMITH

Aristotle's Regress Argument

21

HERMANN WEIDEMANN

Alexander of Aphrodisias, Cicero, and Aristotle's Definition of Possibility

33

DONALD FELIPE

Fonseca on Topics

43

ALAN PERREIAH

Modes of Skepticism in Medieval Philosophy

65

MIKKO YRJÖNSUURI

Obligations as Thought Experiments

79

ANGEL d ' O R S

Utrum propositio de futuro sit determinate vera vel falsa (Antonio Andrés and John Duns Scotus)

97

E . J. ASHWORTH

Domingo de Soto (1494-1560) on Analogy and Equivocation

117

ALLAN BÄCK

The Triplex Status Naturae and its Justification

133

WILLIAM E . MCMAHON

The Semantics of Ramon Llull

155

PALOMA PÉREZ-ILZARBE

The Doctrine of Descent in Jerónimo Pardo: Meaning, Inference, Truth... 173 JEFFREY COOMBS

What's the Matter with Matter: Materia Propositionum in the PostMedieval Period

187

RAFAEL JIMÉNEZ CATAÑO

Copulatio in Peter of Capua (12th Century) and the Nature of the Proposition

197

X

Contents

L Y N N CATES

Wyclif on sensus compositus et divisus

209

MAURICIO BEUCHOT

Some Examples of Logic in New Spain (Sixteenth-Eighteenth Century).. 215 ADRIAN DUFOUR

Necessity and the Galilean Revolution

229

G U Y DEBROCK

Peirce's Concept of Truth within the Context of his Conception of Logic. 241 PIERRE T H I B A U D

Peirce's Concept of Proposition

257

JAIME NUBIOLA

Scholarship on the Relations between Ludwig Wittgenstein and Charles S. Peirce

281

JOSÉ M I G U E L GAMBRA

Arithmetical Abstraction in Aristotle and Frege

295

HERBERT HOCHBERG

The Role of Subsistent Propositions and Logical Forms in Russell's 1913 Philosophical Logic and in the Russell-Wittgenstein Dispute 317 ALFONSO GARCÍA SUÁREZ

Are the Objects of the Tractatus Phenomenological Objects?

343

M A R Í A CEREZO

Does a Proposition Affirm every Proposition that Follows from it?

357

JAVIER LEGRIS

Carnap's Reconstruction of Intuitionistic Logic in The Logical Syntax of Language

369

ALBERT C . LEWIS

Some Influences of Hermann Graßmann's Program on Modern Logic.... 377 J U A N CARLOS LEÓN

Indeterminism and Future Contingency in non-Classical Logics

383

CHRISTIAN THIEL

Research on the History of Logic at Erlangen

397

Index

403

Contributors Ignacio Angelelli, The University of Texas at Austin, Department of Philosophy, Waggener Hall, 316, Austin, Texas 78712-1180, USA E. J. Ashworth, University of Waterloo, Department of Philosophy, Waterloo, Ontario N2L 3G1, CANADA Allan Bäck, Kutztown University, Department of Philosophy, Kutztown, Pennsylvania 19530, USA Mauricio Beuchot, Apartado Postal 23-161, Xochimilco, 16000 México D.F., MEXICO Lynn Cates, 1401 E. Rundberg 357, Austin Tx 78753, USA María Cerezo, Departamento de Filosofía, Universidad de Navarra, 31080 Pamplona, SPAIN Jeffrey Coombs, Our Lady of the Lake University, 411 S.W. 24th Street, San Antonio, Texas 78207-4689, USA Guy Debrock, Catholic University, Faculty of Science, Department of Philosophy, 6525 ED Nijmegen, THE NETHERLANDS Adrian Dufour, rue Franijois-GuilLimann, 3, 1700 Fribourg, SWITZERLAND Donald Felipe, Department of Philosophy, Iowa State University, Ames, Iowa 50010, USA José Miguel Gambra, Departamento de Lógica y Filosofía de la Ciencia, Facultad de Filosofía, Universidad Complutense, 28040 Madrid, SPAIN Alfonso García Suárez, Dept. de Filosofía, Teniente Alfonso Martínez, 1, 33001 Oviedo, SPAIN Herbert Hochberg, The University of Texas at Austin, Department of Philosophy, Waggener Hall, 316, Austin, Texas 78712-1180, USA Rafael Jiménez Cataño, Via S. Girolamo della Caritá, 64, 00186 Rome, ITALY Javier Legris, Instituto de Filosofía, Universidad de Buenos Aires, Puán 470, RA-1406 Buenos Aires, ARGENTINA Juan Carlos León, Departamento de Filosofía y Lógica, Universidad de Murcia, Edificio Luis Vives, 2.53, 30071 Murcia, SPAIN Albert C. Lewis, Bertrand Russell Editorial Project, McMaster University, Hamilton, Ontario L8S 4M2, CANADA William E. McMahon, Department of Philosophy, University of Akron, Akron, Ohio 44325, USA Mario Mignucci, Kings College, London, Department of Philosophy, Strand, London WC2R 2LS, UNITED KINGDOM Jaime Nubiola, Departamento de Filosofía, Universidad de Navarra, 31080 Pamplona, SPAIN Angel d'Ors, Dpto. de Lógica y Filosofía de la Ciencia, Facultad de Filosofía, Universidad Complutense, 28040 Madrid, SPAIN

xn

List of Contributors

Paloma Pérez-Ilzarbe, Departamento de Filosofía, Universidad de Navarra, 31080 Pamplona, SPAIN Alan R. Perreiah, Department of Philosophy, University of Kentucky, 1425 Patterson Office Tower, Lexington, Kentucky 40506-0027, USA Robin Smith, Department of Philosophy, Texas A&M University, College Station, Texas 77843-4237, USA Pierre Thibaud, Université de Provence, 29, avenue Robert Schuman, 13621 Aix en Provence Cedex 1, FRANCE Christian Thiel, Institut für Philosophie, Universität Erlangen-Nürnberg, Bismarckstr. 1, D-91054 Erlangen, GERMANY Hermann Weidemann, Philosophisches Seminar A, Universität Bonn, Am Hof, D-53113 Bonn, GERMANY Mikko Yijönsuuri, Dept. of Social Policy and Philosophy, University of Joensuu, PoB 111, 80101 Joensuu, FINLAND

Aristotle's Theory of Predication MARIO MIGNUCCI

I Consider the following two propositions: (1)

John is pale

(2)

Man is mortal

and

Both have a common grammatical structure which in traditional terms can be analysed as constituted by two terms linked by a copula.1 In (1) 'John' and 'pale' are the two terms involved and 'is' expresses the copula. In the same way in (2) 'Man' and 'mortal' are linked by the copula 'is'. Notwithstanding this similarity most logicians would be prepared to deny that the logical structure of (1) and (2) is the same. It is a common view that the so-called 'copula' in the two statements refers to two different logical relations which are expressed by different symbols. As a rule, the form of (1) is expressed by (3)

F(a)

and a possible way to formalise (2) is as (4)

VX(F(X)

->

G(x))

According to the different ways of representing (1) and (2) we have different truth conditions for them. In general, we say that (1) (or (3)) is true if the individual to which 'John' Ça') refers belongs to the set to which 'pale' ( l F(x)') is supposed to correspond. In other words, the 'is' of (1) has the membership relation of set theory as its semantic counterpart. The same holds neither for (2) nor for its formal representation (4). The more natural way of stating the truth conditions for (2) is by saying that (2) is true if and only if the set to which 'man' refers is included into the set to which 'mortal' relates. Thus, in this case the 'is' of (2) has the inclusion relation as its semantic correspondent, and it would be a regrettable mistake to muddle set-theoretical membership and the inclusion relation which are normally distinguished by

2

Mario Mignucci

two different symbols, the usual epsilon, ' e ' , for the membership relation and the horse-shoe, ' c ' , for inclusion. By adopting this interpretation we face a reassuring symmetry. On the one hand, it is easy to get convinced that the 'is' of (1) is much more fundamental than the 'is' of (2). It expresses the basic predicative relation out of which the meaning of the 'is' of (2) is built up with the help of some logical notions such as quantifiers and material implication. On the other hand, the membership relation, which is the semantic counterpart of the 'is' of (1), is the primary relation by means of which the semantic counterpart of the 'is' of (2) is characterised, since inclusion can easily be defined in terms of membership belonging, as is well known. II This approach to syntax and semantics of propositions is so widespread and usual that we are inclined to think that it is a distinctive feature of any logical theory of propositions. We start from singular simple propositions and from them we build up more complex predicative relations, which involve logical operators and functors. In a parallel way, we interpret simple singular propositions through the set-theoretical membership relation and from this basic operation we proceed to define the truth conditions of more complicated propositions by introducing new set-theoretical relations defined in terms of the membership relation. Historians of the logic have not been able to escape the appealing and architectonic simplicity of this model. To become convinced of this it is sufficient to think of the way in which the following passage of Aristotle's Prior Analytics is interpreted: (A) "We say that one term is predicated of all of another when nothing [of the subject] can be taken of which the other is not said."2 The Greek is not sure since Ross has deleted toi) ûtcokeiiiévoi) at 24^29 on the basis of Alexander's commentary to the passage,3 although the expression is present in most MSS.4 However, even if we decide to follow him in omitting the word, the sense clearly requires that xoû {moKeiiaivou is tacitly understood. The text contains a well-known definition of universal predication and it is normally taken to mean that asserting that a term G is predicated of every F amounts to stating that there is no individual of which F is true and G is not true. According to this interpretation 'mortal' is predicated of every man if

Aristotle's Theory of Predication

3

there is no individual man who is not mortal, and in general 'G is predicated of every F ' means that there is no individual which is F and not G. A familiar pattern comes to the mind and we are immediately tempted to formalise the Aristotelian definition as (5)

-.3JC(F(JC) A - i G ( X ) )

Thus, since (5) is formally equivalent to (4) it is reasonable to claim that (4) is the adequate formal representation of (2) and in general of universally quantified affirmative propositions such as: (6)

every man is mortal

Therefore, we are inclined to think that (6), even from the Aristotelian point of view, is dependent for its form on the form of (1), as well as (4) depends on (3). Because of this way of putting things, one is driven to attributing to Aristotelian propositions the same sort of semantic interpretation one finds attached to formulae such as (3) and (4). By systematising Aristotle's intuitions along these lines we may think that a set of individuals is associated with every general term that plays the role of a predicate or a subject in a proposition, while just one individual is connected to each singular term. Let us call what is associated with a term in a proposition the 'extension' of this term. The extension of a general term will be a set of individuals, while the extension of a singular term is constituted by just one individual. Then, in the Aristotelian spirit, one might say that a singular proposition such as (3) is true if the individual constituting the extension of a belongs to the extension connected to F, and that (4) is true if the extension of F is included in the extension of G. Something like the membership relation of set theory comes spontaneously to the mind as the formal representation of the belonging of an individual to the extension of a predicate. In the same vein, (6) (or ( 4 ) ) is true if every individual in the extension of the subject is also an element in the extension of the predicate. The formal counterpart of this kind of situation is expressed by means of the inclusion relation.

Ill I do not claim that this interpretation is locally false. If we keep to the text of the Prior Analytics alone there is no reason to reject it. However, doubts can be cast on it if we consider the logic in which Aristotle's definition is embedded and the way in which universal propositions are used by him. In my argument I will proceed this way. First, I will propose an alternative reading of text ( A ) , which is as plausible as the standard interpretation. Secondly, I will

4

Mario Mignucci

try to justify my position by referring to some general features of Aristotelian logic which do not fit the traditional understanding of (A), and rather lead to the alternative interpretation. Let us consider again text (A). In characterising universal predication Aristotle does not say that there is no individual of which the subject is said such that the predicate is not true of it, or, equivalently that every individual in the extension of the subject is an individual in the extension of the predicate. In a much more elliptical and vague way he claims that in a universal predication there is nothing of the subject of which the predicate is not said. It is by our partisan view about the nature of predication that we are persuaded to interpret this "nothing" (|ir)8ev) in the sense of 'no individual'. If we stay away from the influence of modern semantics we could make a different suggestion and suppose that "nothing" (|ar|8ev) means simply 'no part'. According to this view the characterisation of universal predication could be paraphrased in the following way: A is predicated of every B iff there is no part of B of which A is not predicated.

IV Before presenting some evidence for this move let us try to clarify the meaning and the consequences of this proposal by comparing it with the traditional interpretation. Consider again proposition (6) and imagine that to its subject, 'man', we associate the collection of its individual instances, Peter, John, James and so on. This is the same move we make in the perspective of the standard semantics. But the two approaches depart from each other at this point. In the traditional view the collection of instances of 'man' is taken to be a set, while here it is not a set. To accept this we must abandon the view according to which the obvious way of thinking of a collection is by taking it as a set. Collections are by no means sets nor is there any reason to claim that taking a collection as a set is the only way to make precise our confused intuitions about a multiplicity of objects. To stricture collections into sets we must give up some natural intuitions we have about collections and more than one philosopher has stressed this point. 5 To consider a collection as a set we must introduce in it a basic relation, the set-theoretical membership relation, our ' e ' . As everybody knows, this relation is not transitive, since (7)

aeb,

bec\-aec

Aristotle's Theory of Predication

5

does not hold in general, and this means, for instance, that we cannot claim that a collection of art collections is a collection of art objects if we take art collections as sets of art objects. The situation is different if we take a collection as defined by the partwhole relation. First of all, it is reasonable to claim that the part-whole relation is transitive, i.e. that (8)

a q' is the philosophical theory that is criticized. II. A token of the antecedent p of this consequence is put forward as the positum.

III. The corresponding consequent q of this consequence is put forward to be evaluated. (The point is that it should be denied, but this means providing a counterexample to the discussed philosophical theory. Thus, it can be denied only if the theory is rejected.)

Obligations as Thought Experiments

93

The example is simple, and thus shows very clearly how to construct in an obligational disputation a counterexample for an inferential principle. Basically, the idea is to start with the antecendent and force the respondent to deny the consequent or to grant its opposite. Walter Burley gives as a useful rule for obligational disputations that it is possible to prove anything that is compossible with the positum.20 In terms of criticism of inferential principles this is exactly what we would wish. If an inferential principle is valid, the opposite of the consequent is not compossible with the antecedent. Some modern scholars have interpreted obligations in terms of some kinds of subjunctive conditionals. Burley's rule allowing anything compossible to be proved comes as a problem to such an interpretation. There hardly is any ordinary language conditional which would allow as consequent any sentence compossible with the antecedent regardless of whether there is any (real or putative) logical connection, even if there are some conditionals designed to deny the existence of such connection.21 It has been clear for some time to most modern scholars that Paul Spade's suggestion22 that obligations are a theory of 'would'-conditionals is untenable as such. There simply is no guarantee that if the positum were true, the case would be as the sentences granted in the disputation tell. Often the opposite is true. Christopher Martin has modified the interpretation into an interesting direction. His idea is that obligations give a theory of 'might'-conditionals.23 Burley's useful rule telling that anything compossible can be proved is interesting in respect to this interpretation. It seems acceptable that anything compossible with the positum might be the case, if the positum is true. My interpretation of obligations as thought experiments is to some extent consistent with Martin's idea. Especially if we look at obligational disputations as constructing counterexamples to putatively valid principles of inference. Basically, it is rather natural to attack an entailment 'if p, then q' by pointing out that 'if p, it might not be the case that q.' Thus an obligational disputation might be seen as aiming at a valid 'might'-conditional, if it is interpreted as a thought experiment. However, it is clear that generally the positum and granted sentences do not form valid 'might'-conditionals. For directly relevant sentences this is clear. It does not make sense to claim that if Socrates were standing, he might not be sitting. Equally for irrelevant sentences, it is rather pointless to remark that if Socrates were standing, Bill Clinton might be the president of the USA, unless one wants to hide ones views into mysterious sentences.

94

Mikko Yrjonsuuri

There is a theoretical reason why evaluations of sentences in obligational disputations are not connected to 'might'-conditionals, or to any other kind of conditionals. In an obligational disputation relevant sentences are evaluated on the basis of inferential considerations. Their interpretation comes only retrospectively. They do not describe but construct a possible situation. Irrelevant sentences, on the other hand, are evaluated on the basis of semantical considerations based on the actual situation. The semantical background of the positum and irrelevant sentences is consequently different. On the other hand, no conditional makes sense with different semantics for the antecedent and for the consequent. Burley offers a beautiful example. In an obligational disputation where the positum is any false sentence, it must in the absence of special reasons be granted as irrelevant and true that the positum is false. On the other hand, no true conditional can have as the consequent that the antecedent is false.

Notes 1

Burley 1963, pp. 34-35. 2 Paul of Venice 1988, pp. 2-21, esp. p. 11. This text also discusses different ways of defining obligation. In her footnotes Ashworth gives ample references to other authors. 3 Burley 1963, p. 81. 4 Burley 1963, pp. 76-81. Cf. also Knuuttila 1993, pp. 190-196. 5 Burley 1963, p. 35. 6 See Yrjonsuuri 1993 for discussion and references. 7 Swyneshed's Obligationes is edited in Spade 1977. 8 Spade 1977, pp. 254-255. 9 Yrjonsuuri 1990, p. 654. 10 Yrjonsuuri 1993, pp. 319-321. 11 Edited in Kretzmann and Stump 1985. For the quoted text, cf. pp. 251-252. 12 Kretzmann and Stump 1985, pp. 252-253. 13 See Perreiah 1984. 14 Kilvington 1990a and 1990b. For modern discussion, see e. g. Spade 1982; Stump 1989, pp. 222-231; Yrjonsuuri 1994, pp. 108-127, and Kretzmann's commentary in Kilvington 1990b. Heytesbury 1494 and 1988. I discuss Heytesbury more widely in Yrjonsuuri 1994, pp. 138-144. Heytesbury's De sensu composito et diviso is printed in Heytesbury 1494. See Ashworth 1993, pp. 385-386, and Stump 1989, pp. 237-239, for alternative discussions of Heytesbury's theory. 17 Kilvington 1990a and 1990b, S47, passages (t)-(z).

Obligations as Thought Experiments

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See Heytesbury's introduction to the work. Spade's translation is available in HeyteSbury 1979. For a particularly illuminating discussion of how Heytesbury's 'rules' work, see Murdoch 1979. 19 Paul of Venice 1988, pp. 32-33. 20 Burley 1963, p. 57; 1988, p. 391. I am thankful to Calvin Normore for pointing out 'even if-conditionals to me. 22 See Spade 1982. Spade 1992 modifies the interpretation very much to the same direction as I am defending. 23 Martin 1993.

References Ashworth, E. Jennifer 1993 "Ralph Strode on Inconsistency in Obligational Disputations" in Jacobi 1993, pp. 363-386. Bos, Egbert P. (ed.) 1985 Medieval Semantics and Metaphysics, Artistarium Supplementa, vol. 2. Nijmegen: Ingenium Publ. Burley, Walter 1963 Tractatus de obligationibus, ed. R. Green, in The Logical Treatise 'De obligationibus': An Introduction with Critical Texts of William of Sherwood (?) and Walter Burley, Ph. D. Thesis, Louvain. 1988 Obligations (Selections), trans. N. Kretzmann and E. Stump, in Kretzmann and Stump 1988, pp. 369-412. Heytesbury, William 1494 Tractatus Gulielmi Hentisberi de sensu composito et diviso, Regualae eiusdem cum Sophismatibus... Venice. 1979 On 'Insoluble' Sentences. Chapter One of his Rules for Solving Sophisms, trans, with an introduction and study by Paul Spade, Medieval Sources in Translation, vol. 21. Toronto: Pontificial Institute of Medieval Studies. 1988 The Verbs 'Know' and 'Doubt', trans. N. Kretzmann and E. Stump, in Kretzmann and Stump 1988, pp. 435-475. Jacobi, Klaus (ed.) 1993 Argumentationstheorie. Leiden: Brill. Kilvington, Richard 1990a The Sophismata of Richard Kilvington, ed. N. Kretzmann and B. E. Kretzmann, Auctores Britannici Medii Aevi, vol. XII, British Academy. Oxford: Oxford University Press. 1990b The Sophismata of Richard Kilvington, introduction, translation and commentary by N. Kretzmann and B. E. Kretzmann. Cambridge: Cambridge University Press.

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Knuuttila, Simo 1993 Modalities in Medieval Philosophy. London: Routledge. Knuuttila, Simo; Työrinoja, Reijo and Ebbesen, Sten (eds.) 1990 Knowledge and the Sciences in Medieval Philosophy. Helsinki: Publications of Luther-Agricola Society (Series B 19). Kretzmann, Norman and Stump, Eleanore 1985 "The Anonymous De arte obligatorìa in Merton College MS. 306" in Bos 1985, pp. 239-280. 1988 (eds.) Cambridge Translations of Medieval Philosophical Texts, vol. 1, Logic and the Philosophy of Language. Cambridge: Cambridge University Press. Martin, Christopher J. 1993 "Obligations and Liars" in Read 1993, pp. 357-381. Murdoch, John 1979 "Prepositional Analysis in Fourteenth-Century Natural Philosophy" in Synthese 40, pp. 117-146. Paul of Venice 1988 Logica Magna, Part II, Fascicule 8 [Tractatus De obligationibus], ed. with trans, and notes E. J. Ashworth, Classical and Medieval Logic Texts, vol. 5, British Academy. Oxford: Oxford University Press. Perreiah, Alan R. 1984 "Logic Examinations in Padua circa 1400" in History of Education 13, pp. 85-103. Read, Stephen (ed.) 1993 Sophisms in Medieval Logic and Grammar. Dordrecht: Kluwer. Spade, Paul 1977 "Roger Swyneshed's Obligationes: Edition and Comments" in Archives d'histoire doctrinale et littéraire du moyen age 44, pp. 243-285. 1982 "Three Theories of Obligationes: Burley, Kilvington and Swyneshed on Counterfactual Reasoning" in History and Philosophy of Logic 3, pp. 1-32. 1992 "If Obligationes were Counterfactuals" in Philosophical Topics 20, pp. 171-194. Stump, Eleonore 1989 Dialectic and Its Place in the Development of Medieval Logic. Ithaca: Cornell University Press. Yrjönsuuri, Mikko 1990 "Obligationes, Sophismata and Oxford Calculators" in Knuuttila, Työrinoja and Ebbesen 1990, vol. II, pp. 645-654. 1993 "The Role of Casus in some Fourteenth Century Treatises on Sophismata and Obligations" in Jacobi 1993, pp. 301-321. 1994 Obligationes: 14th Century Logic of Disputational Duties, Acta Philosophica Fennica, vol. 55. Helsinki: Societas Philosophica Fennica.

Utrum propositio de futuro sit determinate vera vel falsa (Antonio Andrés and John Duns Scotus) ANGEL d'ORS

In a previous paper 1 1 have already dealt with Antonio Andrés's Scriptum super librum Perihermeneias, focusing on the question Utrum nomen signified rem vel passionem in anima. There, I devoted some attention to the figure and work of Antonio Andrés; to the problems posed by the Scriptum primi et secundi libri Peryhermeneias cum notabilibus et dubiis et quaestionibus, which is to be found in manuscript 6 of Pamplona Cathedral (attributed to Antonio Andrés); and to the relations between the Scriptum super librum Perihermeneias of Antonio Andrés and the In librum Perihermeneias quaestiones2 of John Duns Scotus. By examining this question I tried to show how, under the guise of literal fidelity to Scotus' teaching, Antonio Andrés is defending a radically different view. In this paper I will deal with another of the questions contained in Antonio Andrés's Scriptum super librum Perihermeneias, question 11, entitled Utrum propositio de futuro sit determinate vera vel falsa. This question is the result of the fusion of questions VII and VIII in In duos libros Perihermeneias, operis secundi, attributed to Scotus, the titles of which are, respectively, An propositio de futuro sit determinate vera vel falsa and An possibile sit neutram partem contradictionis esse veram.3 In this case, this is merely a reorganization of Scotus' questions, in slightly simplified form, in which faithfulness to Scotus' thinking is allied to literal fidelity to Scotus' texts. Consequently, the doctrines which I propose to scrutinize can be attributed without discrimination to Antonio Andrés and John Duns Scotus. I shall here deal primarily with the work of Antonio Andrés, on the one hand because of a larger project in hand to study the logical work of this outstanding figure of Spanish Scotism, of which my previous study also formed part—and on which Paloma Pérez-Ilzarbe has also collaborated—; 4 and on the other, because there exists a literal commentary by Antonio Andrés on the passage in Aristotle's Perihermeneias which is the historical origin of this question, whereas no equivalent by Scotus is available. 5 This commentary should provide us with some passages which help towards a better understanding of the question under examination.

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My concern with this quaestio, apart from its intrinsic logical and philosophical interest, springs not from a preoccupation with the Scotist tradition (whether or not it is faithful to Scotus' own theses in the mainstream of this tradition), but rather from an interest in the contemporary interpretation of Scotus' teaching. In particular, my underlying concern is with the contemporary interpretation of his modal doctrines, to which Simo Knuuttila has devoted so much attention.6 Knuuttila draws our attention to the historical importance of Scotus' modal doctrines, and proposes an interesting interpretation of them. Knuuttila bases his interpretation on Scotus' theological writings (Tractatus de Primo Principio, Ordinatio, Lectura in Sententiarum). In his logical writings, however, we find doctrines which seem to be in conflict with this interpretation, and which therefore appear to call it into question. It is not my present aim to take on the task of re-examining this issue, as this would require reinterpretation of the theological texts on which Knuuttila builds his interpretation; I intend only to draw attention to some difficulties which are raised in this respect by Scotus' logical texts. 1. Introduction My purpose, as stated above, is to analyze the Scotist solution to the question Utrum propositio de futuro sit determinate vera vel falsa. None the less, before proceeding to an analysis of the texts of Antonio Andrés (or Scotus), it would be appropriate to deal with some general aspects of this question, which provide the frame within which the interpretation of such texts is to be carried out. I shall divide this introduction into two sections: in the first I shall address the philosophical problems raised by contingent futures; and in the second I shall confront the logical problems raised by the propositions which speak of such contingent futures. a) Contingent futures Contingent futures constitute a subject of primary philosophical importance, around which there have arisen many questions of a very diverse nature which must be distinguished. 7 Metaphysical questions were raised concerning the structure of contingent being (in as far as it is contrasted to necessary being 8 ), and the structure of time; physical questions concerning causality and change; logical questions concerning the truth of the propositions which speak of contingent futures and the formal conditions of modal discourse about the modes of being, whether necessary or contingent, of such entities; theological questions concerning the Creation and Divine Omnipotence, Prescience and Providence were also posed.

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Central to all these are the metaphysical and physical questions concerning the structure of contingent being and time, causality and change. The most drastic and immediate solution is determinism, which denies contingency and thus nips all the other problems in the bud. From a determinist point of view, there are neither logical nor theological problems about contingent futures. 9 The logical and theological problems raised with regard to contingent futures are only problems for those who adopt a non-determinist solution which recognizes the effective existence of contingent futures. Such problems, however, are no longer properly speaking problems about contingency, but about truth or God. These problems, in both the logical and the theological order, present a triple dimension. On the one hand, there is a central problem: that of how to understand the way in which truth exists in propositions which tell of contingent futures, or how to understand the Omnipotence, Prescience and Providence of God, the creator of a world open to contingency. On the other hand, there are two complementary problems of a metatheoretical nature: that of whether the existence or not of contingency in any way affects our understanding of the truth of propositions, or of God's Omnipotence, Prescience and Providence; and that of whether a given doctrine regarding truth, or Divine Omnipotence, Prescience and Providence, is compatible or not with the previous determinist or non-determinist positions. In the sphere of logic it is still necessary to distinguish between the questions concerning the truth of propositions which speak of contingent futures, which forms a chapter in the theory of truth, and the questions concerning the propositions which speak not of contingent futures, but of their contingency, which constitutes a chapter in modal logic. Both questions mingle in the discussions centring on the issue of contingency, but they are different in nature and should be distinguished and separated. In question 11, Antonio Andrés does not tackle the metaphysical or physical questions raised by contingent futures, but presupposes the nondeterminist theses from the outset. In his commentary on chapter 9 of the Perihermeneias he states expressly that the proposition 'aliquod ens est contingens' is an undemonstrable principle, the truth of which is made manifest by experience. 10 His argument against those who deny contingency leaves no room for doubt: "Et ideo negantes talia manifesta, indigent pena vel sensu, quia secundum Avicennam, primo Metaphisicorum, negantes primum principium sunt vapulandi et expoliandi quousque concedant quod non est idem comburi et non comburi, vapulare et non vapulare; et isti qui negant aliquod ens esse contingens, exponendi sunt tormentis quousque concedant quod possibile est eos torqueri sicut non torqueri."

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Nor does Antonio Andrés (or Scotus) address the theological questions here. His attention focuses on the logical questions, and in particular on the questions concerning the problem of truth. The title of the question is a faithful reflection of the problems examined: Utrum propositio defuturo sit determinate vera vel falsa. These will therefore be the only questions which I shall examine in the present study. b) The logical problem of the truth of propositions of contingent future In all propositions, three times (at least) should be distinguished: the time of the proposition, that is, the time in which the speaker utters the proposition; the time of the copula, that is, the time of the belonging (or not belonging) of the predicate to the subject, which is being spoken of; and the time of truth, that is, the time at which it belongs to the proposition (as subject) to be true or false (this belonging could in turn be spoken of, and it can therefore become the subject of a further proposition). In the propositions of present time, these three times are the same, and therefore do not raise any problem in this respect: the proposition that it is uttered now, states that the predicate belongs now to the subject, and is now true or false (according to whether or not the predicate belongs now to the subject); likewise, according to whether this proposition is now true or false, the proposition which speaks of its truth will now be true or false. In propositions of extrinsic time, of the past or future, the times of proposition and copula are separated (today we talk of the belonging, yesterday or tomorrow, of the predicate to the subject). This separation is precisely the root of the problem. In what way does this separation affect the time of truth? Is it separated as a third time, or does it follow one of the other two? If so, which? This is in my opinion the nub of the problem raised by the propositions of extrinsic time, but it is obviously not the only problem. A further four problems of a general nature, if not more, must be taken into account. In the first place, it is necessary to deal with the (triple) problem of the way in which its temporal being affects the very being of each one of the three terms of the problem: the being of the proposition, the being of the belonging of the predicate to the subject, and the being of the truth. Is the being of the proposition restricted to its being uttered now, is the being of the belonging of predicate to subject limited to the predicate's belonging to the subject now, and is the being of truth confined to the proposition being true now? Or, on the contrary, do they somehow transcend this 'now'? Do these three terms of the problem all behave in the same way in this respect? And if any of these three terms does transcend this 'now', does it do so within the limits of time, or does it even transcend the limits of time?

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Secondly, it is necessary to deal with the way in which the time of the proposition affects its individuality. Can the same proposition (the same single prepositional individual) be uttered at different times? If so, how does this difference in time affect its meaning and truth? Thirdly, it is necessary to deal with the way in which the time of the proposition, the time at which the speaker utters it, affects the prepositional being of the proposition, that is, what the proposition says, its meaning. Is the time of the proposition an extrinsic determiner of the proposition, or, on the contrary, is it in fact one of the ingredients of its meaning? Lastly, it is necessary to deal with the way in which language affects the being of propositions. Is the proposition essentially linguistic, that is, is its being reduced to the being of language (or at least, is language an essential ingredient of its being); or, on the contrary, is it merely the trappings, an instrument used to utter it which is radically extrinsic to its being? Can the same proposition be uttered using different linguistic instruments (in the same language, or in different ones), or different propositions using identical linguistic instruments; or, on the contrary, is the individuality of the proposition reducible to the individuality of its linguistic expression? All these are problems involved in the logical treatment of the propositions of extrinsic time, but problems which affect propositions of past and future time equally, and which therefore do not serve to define in its full specificity the problem raised by propositions of future time, still less that raised by propositions of contingent future. We might well ask, however, whether there is indeed a specific problem in the case of propositions of future time, and, more particularly, a specific problem in the case of propositions of contingent future. But this is part of the problem. In order to admit the specificity of these problems it is necessary to have solved, in a certain sense, many of the problems outlined above. The recognition of the specificity of such problems is, in my opinion, the characteristic feature of Aristotelian teaching. It presupposes the recognition of the possibility that the being of truth, like the being of the belonging of the predicate to the subject, and the being of the proposition, transcends its own respective now, and that they all do so not in a single way but in different ways. In this respect, necessary being does not behave in the same way as contingent being; the way in which the belonging of the predicate to the subject transcends its particular now through the cause (a parte ante) or through matter (a parte post) is not the same; nor does the proposition transcend its particular now through the faculty (a parte ante) or through habit (a parte post) in the same way. These are the reasons why Aristotle holds that propositions of future time, with regard to propositions of

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past time, and propositions of contingent future with regard to propositions of necessary future, constitute specifically distinct logical problems. However, it is not Aristotelian thought which is in question here, but Scotist teaching. Whether or not the above represents genuine Aristotelian teaching, in my opinion it defines the framework of questions on which the medieval debate on propositions of contingent future hung, a framework which is of service for describing the specific features of Scotist thinking. 2. Utrum propositio de futuro sit determinate vera vel falsa Under this title, as I have indicated, Antonio Andrés examines three complementary problems, which he formulates in the following terms: i) an absolute loquendo propositio de futuro sit determinate vera velfalsa\ ii) (supposito quod illudfuturum 'a' eveniat postea), an haec sit determinate vera 'a erit'; and iii) an possibile sit neutrampartem contradictionis esse veram. The differences between the treatment of these three problems in the work of Antonio Andrés and that of Duns Scotus seem to be merely formal. Whereas in the work of Scotus they are presented as a series of three interlinked questions of the same rank (which are first formulated, one by one, each with its respective difficulties, and then resolved, again one at a time, though on the basis of the same body of ideas expounded in the context of the first question),11 in Antonio Andrés's work they are presented as three parts of a single question. The first question assumes an authoritative role, and the two other questions are subordinated to it. This is why the difficulties raised by Scotus about the first question are presented in Antonio Andrés's work as difficulties of a general nature, and why they are not answered at the point which would have been appropriate, but at the end of the question, once the two complementary questions have been solved. This also explains why Antonio Andrés considerably simplifies the treatment of the last two questions. It would seem, however, that no change in thinking underlies such alterations. The formulation of the first two questions presents some peculiar features which it is worth pointing out with a view to gaining a better understanding of these questions' scope and of the sense of the difference between them. In their literal formulation, these questions refer to the truth or falsehood of propositions of future time in general, and not only to propositions of contingent future.12 This is the root of the main difficulties we come up against when trying to interpret Scotus' views. The difficulties which are examined, of a traditional nature, seem to refer specifically to propositions of contingent future, but the background ideas which are drawn on in order to solve such difficulties apply equally to all propositions of future time (in fact, both Antonio

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Andrés and Scotus extend their main conclusion to all types of propositions of future time). This apparent disagreement between the basis of ideas and the difficulties scrutinized reveals that neither Antonio Andrés nor Scotus recognized the specificity of the problems raised by propositions of contingent future. A s far as the difference between the sense of each individual question is concerned, this is found in the establishment of a supplementary hypothesis: the hypothesis that what is now being stated as future will in fact happen in the future. Both questions presuppose the reality of contingent being—what is contingent taken as that which, in respect of the same time, can be and can not be—, which can be expressed in the form: [ae tnPJ is contingent: M[aet n P] a M[a£ tnPL and the principles of contradiction and of excluded third, which can be expressed respectively in the form: (a)(P)(t„) - n M [ a € t n P A a g t n P ] , and (a)(P)(tn) L [ a e t n P v a e t n P ] . Expressed in the form (ae tnP) the proposition which states the contingent being [aet n P], and expressed in the form (aet n P)t m the proposition which states this same contingent being as future—t m prior to t n — , what Antonio Andrés (and Scotus) asks (ask) himself (themselves) in the first question is whether 'absolute loqueado'—that is, without any hypothesis about what will happen in future—, the proposition (ae tnP)tm is true o r false at the time t m at which the proposition is uttered. In the second question, what he (they) is (are) asking is whether the hypothesis that [ae tnP] will happen in the future ('suppositio quod illudfuturum 'a' eveniatpostea') in any way affects the question being posed. Antonio Andrés (and Scotus) hold that the hypothesis does not introduce significant differences regarding the question under scrutiny. Their analysis, however, provides an opportunity to clear up some interesting difficulties. To examine these three questions, I will first deal with the background thinking, then go on to expound the way in which Antonio Andrés (and Scotus) answers (answer) these questions. Lastly, I shall examine the consequences which can be drawn from these ideas with respect to Scotus' modal doctrines. a) Background thinking The basis from which Antonio Andrés (and Scotus) approaches (approach) the task of solving the three questions raised consists of four observations: the first, which has three facets, refers to the order of being; the second to the order of truth; and the last two to the order of the meaning of propositions of future time.

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The first observation introduces a contraposition between 'esse in seipso' and 'esse in causa', from which, in turn, derive further contrapositions, on the one hand between 'esse quod est' (or 'verum') and 'esse quod non est' (or 'falsum'), and between 'esse determinatum' and 'esse indeterminatum' on the other. The contraposition between 'esse in seipso' and 'esse in causa' springs from the consideration of the being [ae tnP] in its own time t n ('esse in seipso' or 'esse actualiter in praesenti') or at some previous time13 ('esse in causa'). Antonio Andrés (and Scotus) admits (admit) that this being can transcend its own present and assume a certain being before it comes to be the case ('esse in causa')-,14 moreover, it can do so in three different ways: as determination to be the case, as inclination to be the case, or as possibility of being the case or not being the case. The presence which this being has in the cause as a determination to be the case (or not to be the case) can be expressed in the form [e t m c a e t„P]> tmca£ t„P]; the presence which this being has in the cause as a mere possibility of being the case or not being the case can be expressed as a joint negation of the latter expressions: [£ t m c a G tnP a £t m ca £t n P]> or as the affirmation of their contingency: M[aet n P] A M[a£ tnP]-1 shall forbear to enter into an analysis of the Scotist basis for this contraposition at this juncture. As far as the contraposition between 'esse quod est' and 'esse quod non est' is concerned, this belongs to the sphere of 'esse in seipso', and can be regarded as being derived immediately from the principle of contradiction: (a)(P)(tn) —iM[ae tnP A ae tnP]- Given that at any time t n only one of the parts of this conjunction can be the case, at this time t n a contrast is established between the part which is the case ('esse quod est' or 'verum') and the part which is not the case ('esse quod non est' or 'falsum'). This contraposition also extends to the 'esse' of the cause, in as much as it too is an 'esse in seipso', with regard to which the principle of contradiction is also valid: (tm)(a)(P)(tn) -iM[e tm cae t„P A £ tm cae tnP], (tm)(a)(P)(tn) -nM[e tm cag tn P a É^caÉ^P]. 'Verum' and 'falsum', considered in respect of the being of which they are predicated ('esse in seipso'), constitute an 'esse determinatum'. In the case of the cause, however, it is appropriate to consider not only the being of the cause—[e t m cae tnP]—, but also the being of that which is said to be in the cause—[aetnP]> the 'esse in causa' properly said—, in which case, if there is the cause by which it has to be or it has not to be—that is, if the 'verum' is [e t m cae tnP] o [e t m ca£ t„P]—, we can say that [ae tnP] and [a£ tnP] have an 'esse determinatum in causa', while if there is not the cause by which it has to be or it has not to be—that is, if the 'verum' is M[ae t„P] a M[ag tnP]—, we can say that [ae tnP] and [a£ tnP] have an 'esse indeterminatum in causa'. The 'esse determinatum in causa', however, is not properly speaking an 'esse determinatum' but one which is 'indeterminatum', as it is not an 'esse in

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seipso' but an 'esse in causa'; only at time t n , once it has been determined as 'verum' or 'falsum', will it be constituted properly as an 'esse determinaturri. Antonio Andrés (and Scotus) introduces (introduce) the contrapositions 'esse quod esf (or 'verum')!'esse quod non est' (or ' f a l s u m ' ) , 'esse determinatum'I 'esse indeterminatum' as contrapositions subordinated to the contraposition 'esse in seipso'I'esse in causa', which in turn is subordinated to the contraposition present/future. The 'esse determinatum' {'verum' or 'falsum'), however, is not restricted to the present, but also extends to the past. The past, in as much as it has already been present, also has an 'esse determinatum' and it is also 'verum' or 'falsum'. This extension, however, in as far as the 'esse in seipso' and derivatedly the 'esse determinatum', have been linked to the 'esse actualiter in praesenti', does not seem sufficiently justified. The second observation refers to the order of truth. Truth and falsehood are not constituents of the being of propositions, but accidents which arise out of it: its 'proprium' (potential). The being of the proposition is made up by the 'esse' of the asserting, and the being which is asserted. The being which is asserted has in the proposition an 'esse' which is not its own, but that of the asserting (which is an 'esse' in the mind). Its own 'esse', if considered with respect to its asserted being, appears as an to be the case or not to be the case. The behaviour of propositions regarding their truth and falsehood is parallel to the behaviour of the asserted being with regard to its to be the case or not to be the case. Now given that the being which is asserted is a temporal being, relative to a certain time t n , and that only at this exact time t n is it determined with regard to its to be the case or not to be the case, then only at this exact time t n propositions will be determined with regard to their truth or falsehood. The proposition (ae tnP)tm will be determined with regard to their truth or falsehood when, and only when, the being [ae tnP] asserted by it will be determined with regard to its to be the case or not to be the case; that is: (t) {[(ae tnP)tm e t Vera] o [[ae tnP] e t Verum]}, (t){[(ae tnP)tm e t Falsa] o [[ae tnP] e t Falsum]}; the consequents are only fulfilled when t > t n , and therefore the antecedents will be fulfilled only when t > t n . Obviously, at any time t m , [ae tmP] is determined with regard to its to be the case or not to be the case, but it is not this 'esse determinatum' the being which the proposition asserts, and on which its truth or falsehood depends. The last two observations refer to the order of the meaning of propositions of future time. By means of the third, it is shown that whether the being which the proposition asserts be the case or not in no way affects the proposition's meaning. The proposition means what it means, irrespective of whether it is true or false ('re existente et non existente'). The proposition of future time attributes a certain predicate to a certain subject at a certain time, and

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therefore, from the very moment at which it is uttered it has a certain meaning. 15 In as far as it signifies propositionally, the proposition has a claim to truth, but a claim to truth which, in as much as it arises not from the being which is asserted 16 (from its meaning) but from the 'esse' of the asserting, must be referred to the time to which the asserting refers, the time at which the proposition says that the asserted being will be the case, and not the time of the asserting. By means of the fourth and final observation, propositions of future time are shown to be ambiguous, as they can be understood in two different senses: it is possible to understand that a proposition of future time is now asserting the future 'esse in seipsoor that it is now asserting the present'esse in causa' of this future being; that is, we might understand that the proposition of future time means "in advance", by virtue of the autonomy of meaning, the 'esse in seipso' which the thing will have in the future, or that it signifies the being which the thing has "in advance" in its cause. What justifies the grammatical form of the proposition of future time is, in any case, the "advance", but this advance can be understood either via the independence of meaning or through the presence in the cause. That is, the proposition of future time ltu curres in a' can be understood in the form (ae t P)tl or in the form v(e l t cae t t P)tl . In v L n ' m

m

n

' m

accordance with this second sense, the present t m of the utterance is not reduced to the mere time of the proposition, but forms part of the meaning, the proposition does not only say that 'you will run tomorrow' (first sense), but also that this is already in its cause today (or that its cause exists). b) Answers to the questions The four observations set out above clearly anticipate the responses to the questions posed. To the first question, "an absolute loquendo propositio de futuro sit determinate vera vel falsa", Antonio Andrés (and Scotus) answers (answer) by distinguishing the two meanings of propositions of future time: if 'tu curres in a' means the same as (e t m cae tnP)tm> a t time tm this proposition is determinately false,' as LL [[e ltm c a e LtnP J £ tl m Falsum], and, in accord with the second 7

observation, (t){[(et m caet n P)t m e t Falsa] [[et m caet n P] e t Falsum]}. On the contrary, if it means the same as (ae tnP)tm> a t time t m it is neither true nor false, but indeterminate, as 'tu curres in a'—[ae t n P]—, is only determined as 'verum' or falsum' at time 'a'—t n —, and therefore the proposition (ae tnP)tm is not determined to be true or false at time t m but only when time t n has been reached. This is the solution which Scotus and Antonio Andrés give to the first question. Its range, however, can perhaps only be displayed through its

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immediate corollary, which may well serve to justify his title of 'Doctor Subtilis'—in fact, Scotus here shows himself to be 'subtilissimo'—: propositions referring to the necessary future, according to Antonio Andrés (and Scotus), in as far as it is understood that they assert not the present'esse in causa' but the future 'esse in seipso', are not determinately true or false either; there is no difference in this regard between propositions of contingent future and propositions of necessary future, because the problem of truth is not a problem relative to necessity or contingency, but to time. At first sight, it might seem that a conflict exists between this thesis and the previous one, in which an 'esse indeterminatum in causa' was assigned to contingent being, and an 'esse determinatum in causa' to necessary being. For Antonio Andrés (and Scotus) there is no conflict. In effect, he recognizes (they recognize) a major difference in the order of being between necessary being and contingent being, between the former's 'esse determinatum in causa' and the latter's 'esse indeterminatum in causa', but the fact is that propositions of future time do not speak of such a difference, and it therefore in no way affects their truth or falseness. Propositions of future time speak of necessary beings and contingent beings, but they speak of their future being, not of their necessity or contingency. Propositions of future time are not modal propositions. The problem of the truth of propositions of future time is a problem only of time, not of modality. The necessity of the being which we are speaking of is not a sufficient cause of truth; it can enable us to know the necessity of the future truth of the proposition, but it cannot yet make it be true. To the second question "(supposito quod illud futurum 'a' eveniat postea), an haec sit determinate vera 'a erit' ", Antonio Andrés (and Scotus) answers (answer) in an analogous manner: 'a erit' is neither determinately true, nor determinately false. The hypothesis is not at all relevant, as it in no way affects the real presence required for truth. The third question, "an possibile sit neutram partem contradictionis esse veram", is answered by Antonio Andrés (and Scotus) by admitting the possibility that neither of the two parts of a contradiction is true, but for different reasons, according to the two possible meanings of the propositions of future time. If the proposition of contingent future is understood as meaning the same as (e t m c a e tnP)tm> then neither of the two parts of the contradiction (e i c a íLnt P)t N lt cae L 7 Lt 7 tnP) m m' x(elm 411 is true, ' because there is neither the cause of its having to be, nor the cause of its having not to be, and both are therefore false. If, on the contrary, it is understood as meaning the same as (ae tnP)tm> then neither of he two parts of the contradiction (ae tnP)tm> t„P)tm is true, because both are still indeterminate regarding truth and falsehood. Unfortunately, Antonio Andrés (and Scotus) tells (tell) us nothing about the truth or falsehood of the disjunction of these propositions.

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The solution to this third question brings to light one of the weakest points in Scotus' doctrine:17 the propositions (e t m cae t„P)tm, (e t m c a£ tnP)tm are not mutually contradictory propositions: the contradictory of (e t m cae tnP)tm is not (e t m cag t„P)tm but (g t m c a e t„P)tm- The contradictory of 'tu curres in a', should therefore not be 'tu non curres in a' but 'non tu curres in a\ which will be true whenever 'tu curres in a' is false. In this sense, then, it is impossible that neither of the two parts of the contradiction is true. Assigning to propositions of future time this second meaning relative to the cause thus turns out to be dubious. It seems to be no more than a technical contrivance. Yet a technical artifice of this kind appears unnecessary, as the modal proposition which this temporal proposition was set up against could, it seems, have been able to carry out the task for which Scotus requires this new sense of propositions of future time. To consolidate the doctrines defended here, and confirm the solutions put forward to the three questions set, Antonio Andrés (and Scotus) has (have) to confront various objections and difficulties. The two most relevant ones are those referring to the redundancy of the predication of truth, by which propositions of future time are reduced to a certain type of propositions of present time, and to the analogy with propositions of past time. The first objection—which was formulated by Scotus in the context of the first question and repeated when discussing the third—, rests on the equivalence between the propositions 'tu curres in a' and lte esse cursurum in a est verum'; it states that the latter, in as much as it is a proposition of present time, is determinately true or false; and concludes that the former, too, must therefore be determinately true or false. 18 Antonio Andrés (and Scotus) admits (admit) the equivalence between the two propositions, but denies (deny) that the second is 'simpliciter' a proposition of present time. Antonio Andrés (and Scotus) draws (draw) on the distinction 'sensu composite'/1 sensu diviso'. 'In sensu composito' the proposition is properly a proposition of present time, meaning the same as ([e t m c a e t„P] e t m Verum)tm, and is false—as [fe t m c a e tnP] £ tm Falsum]. 'In sensu diviso', however, the proposition is no longer truly a proposition of present time, as it means the same as ([ae tnP] £ tn Verum)tm—that is, the present of the main copula by means of which the 'verum' is attributed to [ae tnP] should not be understood as the present of the speaker but as the present of the copula by which the predicate 'P' is joined to the subject 'a'. For this reason, like (ae tnP), it is neither determinately true nor determinately false: 'te esse cursurum in a est verum' becomes synonymous with 'te esse cursurum in a erit verum'. The second objection—formulated in the context of the second question—, relies on analogy with propositions of past time. The proposition of past time (ae tnP)tm—tm after t n —is judged to be true or false at time t m of

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the proposition, not with regard to the state of matters at the present time, but with regard to their state in the past: [(aet n P)t m e tm Vera] because [[ae tnP] et n Verum]. For the same reason, the proposition of future time (aet n P)t m —t m prior to t n —, must not be judged at time t m with regard to the state of matters at the present time, but with regard to their state in the future: [(ae t n P)t m e tm Vera] if [[aet n P] G tn Verum]. Consequently, at least in the hypothesis which characterizes the second question, the proposition of future time must also be, like the proposition of past time, determinately true or false at time t m . Although the response to this question is a fragmentary one, in which we are told nothing about the problem of the truth or falsehood of propositions of past time, Antonio Andrés (and Scotus) seems (seem) to reject the validity of this analogy. If (ae tnP)tm is a proposition of past time, the being [ae tnP] which is asserted—and the same can be said with regard to [[ae tnP] e tn Verum] and [(aet n P)t n e t n Vera]—has an 'esse determination', 'verum' or falsum', and therefore, even though the assertion is made at time t m , (ae t n P)t m is determinately true or false because it is asserted in respect of this past time t n . On the contrary, if (ae tnP)tm is a proposition of future time, the being [ae tnP] which is asserted—and the same can be said with regard to [[ae t n P] e tn Verum] and [(aet n P)t n et n Vera]—still has no 'esse determinatum', 'verum' or 'falsum', and therefore, even though it is asserted in regard of time t n , (ae tnP)tm is not yet true or false at time t m . It is not a question here of the 'esse determinatum1 or 'indeterminatum' which past and future may have at time t m —as this is not the being which propositions of past or future time speak of—, 19 but of the 'esse determinatum' or 'indeterminatum' which they have at that time t n , past or future, which they are speaking of. If t n is a future time, the being [ae t n P] not only is not determinate as 'verum' or ' f a l s u m ' with respect to t m , but it is also not determinate with respect to t n , since M[aet n P] A M[ag t„P]- Past things already have a past disposition, but future things do not yet have a future disposition. In the opinion of Antonio Andrés (and Scotus) this would seem to be the root of the difference between propositions of future and past time. c) On Scotus' modal conceptions Once the two objections outlined above have been overcome, Antonio Andrés (and Scotus) has (have) to confront a further objection, according to which for the truth of the proposition of future time (ae t n P)t m at time t m , it is not necessary that the future being [ae tnP] which is being spoken of to have an 'esse determinatum' at time t m . The proposition of future time (ae t n P)t m means that [aet n P], without excluding the possibility that [ag t„P], and therefore for it to be true at t m its 'esse determinatum' with respect to t n is not required, but it is enough that, at the future time tn> [ae tnP] to be determined as 'verum'. The

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truth of the proposition of future time (ae t n P)t m is not incompatible with the contingency of the being [aet n P] which it asserts; [(aet„P)t m e t m Vera] is said to be compatible with M[ag t„P]Antonio Andrés (and Scotus) explicitly and emphatically reject this suggestion of compatibility. The truth of (ae tnP)tm is not compatible with the possibility of [ag tn P]- If [(ae tn P)t m e tm Vera], then [(ae tnP)t„ e tn Vera], and just as [(aet n P)t n e t n Vera] is incompatible with M[a£ tnP], so also [(aet n P)t m e t m Vera] is incompatible with M[ap)(Bel{m*, Bel, m, a, Redness, fx, 0(0, x, g, L)} corresponds to p)' fails in that it is a permutative context containing'm*' a n d ' m ' as terms for particular minds, forgetting that a is also a particular. With a text like 'a is human' ('a is a mind'?), the permuting of 'a' and'm' is only ruled out by fiat. Interestingly, Russell himself raised the problem: "One special objection is t h a t . . . we have to regard its atomic constituents, xjCiy, X2C2Y, etc. as really its constituents, and what is more, we have to regard the corresponding propositions as constituents of the proposition 'there is a complex y in which xiQy, X2C2Y, etc.' This seems to demand a mode of analyzing molecular propositions which requires the admission that they may contain false atomic propositions as constituents, and therefore to demand the admission of false propositions in an objective sense. This is a real difficulty, b u t . . . we will not consider it further at present." 37 Russell did not get to the section on molecular propositions, so one can only speculate about how he sought to handle it, but it is worth noting his use

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of the phrase 'in which' above, indicating the connection between the complex and its constituents.38 Russell was quite explicit about construing contexts involving nonsymmetric relations in terms of existential claims: "When we assert 'A is before B', we are asserting 'there is a complex in which A is earlier and B is later'. It is impossible to find a complex name which shall name this complex directly, because no direct name will distinguish it from 'B-before-A'. Complex names, in fact, are only directly applicable to non-permutative complexes, where the mere enumeration of simple names determines the complex meant. Thus the only propositions that can be directly asserted or believed are non-permutative, and are covered by our original simple definition."39 This also shows that for a non-permutative complex, he thinks of an atomic sentence as a "complex name" and it points to a familiar ambiguity of 'corresponds to'. In one sense 'Fa' corresponds to a complex irrespective of its truth; in another sense it can correspond only to an existent fact. This ambiguity points to the problem posed by non-existent or possible facts. Russell avoids such entities by using 'There is a p consisting of F and a and with 0 y as form', not '(EpX'Fa' corresponds to p)', to specify the truth ground for 'Fa'. Russell gives truth grounds for permutative contexts, like 'aRb', in the same way, by 'E!(tp)(aR 1 p & bR 2 p)' or '(Ep)(aR'p & bR 2 p)', not by asserting that a specific judgment fact corresponds to a further fact. Likewise, he analyzes 'S judges that aRb' by ' J ( S , E K i p X a R 1 ? & bR 2 p))' or 'J(S, (Ep)(aR'p & bR 2 p))'. This is clear from passages cited earlier and: "Owing to the above construction of associated non-permutative complexes, it is possible to have a belief which is true if there is a certain permutative complex, and is false otherwise; but the permutative complex is not itself the one directly 'corresponding' to the belief, but is one whose existence is asserted, by description, in the belief, and is the condition for the existence of the complex which corresponds directly to the belief."40 Thus, Russell did not employ the pattern he used for nonsymmetrical relational judgment facts in the case of symmetrical relational judgments and monadic judgments. Since these latter contexts are non-permutative, he has no need to. But, since he does not use such existential statements in the analysis of judgments with non-permutative contents, as he does in the case of permutative contexts, his analysis of them faces Wittgenstein's argument.41 He must rely on the intentional relation, J, U, etc., to unify S, F, a, and the form 0 y into a judgment fact, J(S, F, a, 0y), as well as unify the "objects", F, a and 0y, of the judgment. In the case of a permutative content, the definite

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description (or existential claim) connects the "objects" by describing the relevant truth maker. Such definite descriptions do two things. First, they provide for the unity of content, and thus answer Wittgenstein's objection, and, second, they resolve the problems posed by specifying truth grounds for atomic sentences (judgments) without recognizing nonexistent facts. But they do not appear able to resolve a further, familiar problem posed by intentional contexts. To use 'J(S, E!(ip)(aR J p & bR 2 p))', or 'J(S, (Ep)(aR'p & bR 2 p))\ requires specifying what S stands in the relation J to when there is no fact denoted by the description, i. e. when 'aRb' is false. Russell cannot take the existential sentence to represent either a propositional entity or a possible but non-existent state of affairs with J, as a dyadic relation, relating S to such an entity. His theories of truth and judgment are designed to avoid doing that. As we noted earlier, Russell mentioned the problem posed by constituent clauses like 'aR'p', when *E!(ip) (aR'p & bR 2 p)' is false, but he did not go into the problem. His Principia use of functional abstracts will not help in the present case. We can construe 'S judges that something has R to b' in terms o f ' J ( S , R, b, (Ex)0(x,y))', since the context is non-permutative. But we cannot reconstrue 'J(S, ( E p ) ( a R ' p & bR 2 p))' in terms of 'J(S, R 1 , R 2 , a, b, ( E p ) ( x 0 p & y * p ) ) \ with ( E p ) ( x 0 p & y ^ p ) as an appropriate logical form, as 'a' and 'b', as well as R1 and R2, can be permuted. Russell appears to have no solution. Landini merely mentions in passing that he "would presumably require... 'position relations' " determined by the relation judges (believes). But, in describing such a complex, Landini reverts to using 'R' as a subject term—not the position predicates 'R 1 ' and 'R 2 '. This neither resolves the problem nor fits with what Russell says is really believed.42 Russell was concerned about the truth ground of ' ( E 0 ) ( E y ) 0 y \ He held that such a ground was not complex, due to his concern about the constituents of such a fact. What he did, as we noted earlier, was take the monadic logical form ( E 0 ) ( E y ) 0 y to be its truth ground, as he took the dyadic form (E0)(Ex)(Ey)0(x,y) as the truth ground for ' ( E 0 ) ( E x ) ( E y ) 0 ( x , y ) \ and so on for n-adic relational forms. This helps explain his identifying the logical forms of monadic predication, dyadic predication, etc. with purportedly simple existentially general facts. As neither forms nor such existentially general facts had constituents, both were simples. Thus such existential facts cannot contain forms but can be identified with them. Yet, while Russell identifies the predicative forms with existentially general facts, he sees problems with the identification. First, for each such general fact or form to exist, a basic relation and atomic fact of the appropriate logical kind must exist, but it cannot be a matter of logic that they exist. A form must be the form of some fact, but it

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seems evident to him that a form could be grasped even if there should be no such fact. Thus, the form of an n-term relational complex can be understood or be an object of acquaintance whether or not there are n terms standing in an nterm relation and whether or not there is such a relation. Second, to say that there are no n-term relations, for a certain n, would require understanding the appropriate form. Thus he faces the same problem he noted as the chief "demerit" of his definition of 'proposition'. 43 Wittgenstein's criticisms aside, one might speculate that problems about logical forms, construed as facts, and the failure of his own theory of descriptions to provide a key to the analysis of intentional contexts with nonsymmetric relations were among the reasons Russell abandoned the later chapters of the 1913 manuscript. Pears' claim that Wittgenstein's criticism of Russell's identification of logical forms with existentially general facts played a role could well be right.44 As we noted, Russell raised problems about that identification in the manuscript, and this would fit with his reporting, in the 1913 letter to Ottoline Morrell, that Wittgenstein had said that he had tried Russell's analysis and that it did not work. Pears argues that identifying forms with existentially general propositions, and taking these to be simple, makes "their truth unintelligible". 45 But his argument presupposes that a ground of truth for a logical truth must be a complex, as is a ground of truth for a non-logical truth. On Russell's account there is nothing unintelligible about taking the fact (E0)(Ey)0y to be a simple and a ground of truth. The problem stems from Pears' focusing on Russell's use of the term 'proposition'. This is misleading, since, for Russell, a logical form is a fact: "if a logical form, i.e. a fact containing no constants.... " 4 6 The point is that if we state Russell's view more carefully than Pears does, or than Russell himself does in the passage Pears quotes, the proposition would be represented by the expression ' U ( S , ( E 0 ) ( E y ) 0 y ) ' , read as 'There is a U and an S such that U(S, (E0)(Ey)0y)'. The proposition is a complex, and it is true, since the object, the logical form, (E0)(Ey)0y, which is its ground of truth, is an object of acquaintance and, hence, exists. Unlike the proposition, which is complex, since it contains the logical form (E0)(Ey)0y as a constituent, the logical form is simple. The difference between a proposition, whose ground of truth is a logical form (fact), and a proposition like 'U(S, F, a, 0y)' is two-fold. First, in the former case, a logical form or fact is the' only constituent of the proposition, and, second, its being a constituent means that the proposition contains its own ground of truth. One may take that to show that it is a necessary truth, as the logical form is a logical or necessary fact. Pears thus fails to show that Russell's view is unintelligible. But there is an ambiguity in Russell's view. He speaks of understanding the form (E0)(Ey)0y, as well as

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being acquainted with it, and that in such a case understanding, like acquaintance, is a dyadic relation. Such an understanding fact would be represented by 'U(S, (E0)(Ey)0y)\ for example. But, by abstracting from such a fact, we obtain the proposition (EU)(ES)U(S, (E0)(Ey)0y), which is true since (E0)(Ey) 0 y is a fact. Just as the sentence 'a is F' expresses a proposition—(EU)(ES)U(S, (E0)(Ey)0y, F, a)—so '(E0)(Ey)0y' should express the true proposition: (EU)(ES)U(S, (E0)(Ey)0y). As (i) the object is here represented by a sentence and is a fact, (ii) Russell speaks of such facts as "truths", and (iii) U is dyadic here, one easily fuses the fact with the true proposition.47 Russell may have come to believe that there was a problem with his use of definite descriptions of facts to state truth grounds. He sought to use such descriptions not only to specify the order in a fact and to avoid non-existent facts in connection with false atomic sentences, but also to avoid negative facts as truth makers for true negations of atomic sentences. But it seems that Wittgenstein's criticisms led him to believe that specifying the truth ground for '-iR(a, b)' by '—iE!(ip)(aR'p & bR 2 p)' was to recognize the fact that (Ep)(aR'p & bR 2 p) does not exist when '-iR(a, b)' is true. He not only argued for negative facts in the logical atomism lectures, but a page, attached as Appendix B.l Folio 2 to the 1913 manuscript by the editors, shows him thinking along such lines earlier. " 'xRy' has different uses: (1) as the name of the positive fact xRy. (2) as the name of the neutral proposition or of the neutral fact. (3) as the expression of a judgment. These must be distinguished. (1) Call the positive fact +(xRy), and the negative fact -(xRy). (2) Call the neutral fact ±(xRy), and the proposition xRy. . . . . Judgment involves the neutral fact, not the positive or negative fact. The neutral fact has a relation to a positive fact, or to a negative fact. Judgment asserts one of these. It will still be a multiple relation, but its terms will not be the same as in my old theory. The neutral fact replaces the form."48 The passage not only reveals Russell's concern with negative facts, but that such a concern was a motive in his reconsideration and modification of his theory of judgment. With the neutral fact replacing the form, he has a complex sentential form as a linguistic term in a sentence expressing a judgment. This may show that he was concerned about his theory of descriptions failing to allow him to analyze judgment facts, in addition to believing that it did not enable him to avoid negative facts. His considering a neutral fact as a

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constituent of a judgment fact and his recognition of negative facts makes it reasonable to think that he saw the problems in the light of Wittgenstein's criticism. The use of descriptions in 1913 continued a theme of "On Denoting", where Russell took his theory to avoid acknowledging non-existent complexes in the case of false statements.49 By introducing negative and neutral facts, Russell abandoned a basic theme from his 1905 paper that he elaborated in 1913: the use of descriptions to denote complexes and state truth grounds. This may have been due to Wittgenstein's "ab-functions", taking a proposition to have "poles", in the 1913 "Notes on Logic".50 Wittgenstein may even have been referring to Russell's use of definite descriptions, to specify truth grounds and analyze judgments, when he wrote: "But it is easy to see that every attempt to replace functions with sense (ab-functions) by descriptions, must fail." 51 A passage in the 1921 The Analysis of Mind is relevant: "We may say, metaphorically, that when today is Tuesday, your belief that it is Tuesday points towards the fact, whereas when today is not Tuesday your belief points away from the fact. Thus the objective reference of a belief is not determined by the fact alone, but by the direction of the belief towards or away from the fact." 52 Here Russell appears to think along Wittgenstein's lines. The passage ends with a footnote that contains the only reference to Wittgenstein in the book: "I owe this way of looking at the matter to my friend Ludwig Wittgenstein."53 While rejecting propositions at places in the 1913 manuscript, as in earlier versions of his theory, Russell also took propositions, not judgments or statements, to be the basic bearers of truth and falsity: "But it is fairly obvious that the truth or falsehood which is attributed to a judgment or statement is derivative from the truth or falsehood of the associated proposition.... Thus the opposition of truth and falsehood, in its primary and fundamental sense, is applicable only to propositions, not to particular thoughts or statements."54 But Russell then reiterates his view that propositions are incomplete symbols and that false propositions are unacceptable. 55 However, he here argues against propositions by denying (1) that beliefs affirm the reality of propositions and (2) that propositions are truth makers of beliefs.56 Such grounds for rejecting propositions do not apply to propositions taken as forms that are bearers of truth, not truth makers. Russell even appears to give a further reason for accepting propositions. He argues that to distinguish 'A precedes B' from 'B precedes A', we must understand what it is for a complex

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to be logically possible, and: " . . . the notion of what is 'logically possible' is not an ultimate one, and must be reduced to something that is actual before our analysis can be complete." 57 He then states: "Now although we do not yet know what a proposition is, it is obvious that what we had in mind, when we said that a complex was 'logically possible', may be expressed by saying that there is a proposition having the same verbal form." 58 But it is still in doubt "how propositions are to be explained."59 A page and a half later he considers the subsistence of propositions. Thus, he might have thought propositions helped explicate logical possibility. In the 1914 Monist papers Russell rejected propositions, and in 1918 he discussed the earlier form of the multiple relation theory, crediting Wittgenstein with the discovery that belief facts are of a different "form" from any fact that "occurs in space". This suggests that he had not grasped that fact earlier but had taken properties and relations (verbs) to function as mere terms in such facts. 60 He again rejected propositions as nothing and considered logical forms in terms of expressions, not entities, that contain free variables.61 Not able to resolve the problem of the embedded verb and rejecting both judging subjects and mental acts, he abandoned the multiple relation theory in 1919 and sketched a new analysis, influenced by Wittgenstein. He developed the view in The Analysis of Mind62 and in appendix C to the second edition of Principia63 in a way that is explicitly derived from his construal of Wittgenstein's 5.542 in the Tractatus.M

Notes 1 The second type of view, appealing to properties rather than propositional entities, was implicitly held by Moore (cfr. Moore 1953) and many years later by Gustav Bergmann. On Moore's view see Hochberg 1978, chapters I and III. 2 Russell's use of propositional functions in the years 1907-1912 enabled him to handle nonatomic contexts. Consider 'S judges that something is F' and 'S judges that a is not F'. They are easily transcribed by 'J(S, (Ey)0y, F)' and 'J(S, —iFy, a)', respectively, with '(Ey)0y' and '—iFy' as signs for propositional functions. For such a treatment of Russell's early view see Hochberg 1979 and Cocchiarella 1987. Russell considered non-atomic judgments in Principia, but there he was not concerned with the analysis of the judgment facts, but of their contents. Thus he spoke of the judgment "all men are mortal" and "any judgment (x).0x" (not of the judgment facts) and was concerned to point out that such judgments did not correspond to one complex but to complexes as "numerous as the possible values of x" (Whitehead and Russell 1950, vol. 1, p. 46). His basic concern in Principia was with the different "types" of "truth" involved in various kinds of judgment

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in connection with his consideration of the liar paradox. On this question of truth-types in Principia, see Hochberg 1992b, pp. 104-105. 3 Russell 1984, p. 115. 4 Russell 1984, p. 112. 5 Russell 1984, p. 112. 6 Russell 1984, p. 113. 7 Russell recognized propositional entities in 1905 in "On Denoting" (Russell 1971). This is clear from the fact that the scope differences he employs there for different readings of 'George IV wished to know whether Scott was the author of Waverley' require a proposition as a term of the intentional verb 'wished to know whether'. 8 Russell 1984, p. 109. Earlier, he says: "But for reasons which I have set forth elsewhere, it would appear (1) that no reality is a proposition.... (3) that the unreal is simply nothing . . . . " (p. 25). This passage belongs to one of the articles printed in The Monist, while the passage quoted in note 6 above and the present passage both occur later in Part II, Chapter I. 9 Russell 1984, pp. 114-115. The use of the numeral '2' shows he is responding to the question of that number. Russell uses the letters 'x', 'R', and 'y' in the manuscript as both variables and indefinite constants. Russell 1984, pp. 115-116. Russell's objection to his definition reveals a concern about the identification of logical forms with existentially general logical facts, since that requires that there be instances of relations for there to be logical forms. Thus there must be an exemplified n-term relation for each n-term relational form. This consequence might have provoked Wittgenstein's remarks from 5.55 through 5.5571 in the Tractatus regarding the possible forms of elementary propositions. 11 Russell 1984, p. 114. 12 Russell 1984, p. 117. 1 3 In the Principia notation one would here use circumflexes over the free variables to distinguish the function sign from the open sentence. But as the notation is not necessary and as Russell does not use it in the manuscript, I simply use free variables. I will follow Russell's tendency to speak of both signs and purported entities they represent as incomplete symbols. As we will see, in passages where Russell rejects propositional entities he characteristically speaks of them as "incomplete symbols". 15 We will later see further reasons for his taking logical forms like (E0)(Ey)0y to be facts rather than functions, in particular his taking them as terms of a dyadic relation of acquaintance and as truth grounds for logical truths. 16 Russell 1984, p. 153. One must keep in mind the diverse uses of the term 'proposition', by Russell and others—as a synonym for 'statement', as a term indicating what is expressed by a sentence or statement, and as a somewhat neutral term, where it may be the one or the other. 17 Russell 1984, p. 153. 18 Russell 1984, p. 116. Accepting propositions under such a condition is like accepting only instantiated universals, a familiar form of realism about universals.

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Russell 1984, p. 153. Russell 1984, p. 153. In some cases of "understanding", the intentional relation is a dual relation, as, for example, in the understanding of a logical form like "something has some relation to something" (Russell 1984, p. 131). But the second term is a logical form and not a "proposition", as an existential fact or form with constituents. Russell 1984, p. 109. He goes on to consider the grounds of such falsehood and its bearing on propositional entities, which we will take up below. 22 Russell 1984, p. 114. 23 Russell 1984, p. 131. 24 Russell 1984, p. 177. Note that the form—proposition, fact—(EU)(ES)U(S, F, a, 0 y ) will itself be of the form (Ef)(Ex)(E0)(EU)(ES)U(S, 0 , x, f), which is the form "of an understanding"—for example, of the understanding fact, J(o, F, a, 0y). Russell also writes as if such a proposition would have the form of something having some property, determined by the form of its truth maker or content. Since it is a judgment that a is F, what is judged is of the form of something having some property. . Russell considers appealing to classes in the case of logical forms like 0 y but remarks: " . . . we might of course define the form of a complex as the class of all complexes having the same form. Or, if we wish to avoid classes in so fundamental a question, we can s a y . . . . It is, however, obvious that such an explanation will land us in endless regress.... the form must be something exceedingly simple" (Russell 1984, pp. 113114). For simplicity, I will follow Russell and sometimes use open sentences, rather than existential quantifications, as signs for logical forms and propositions. 26 Russell 1956, p. 128. Pears 1989, p. 174 and p. 180. When Russell adopts Wittgenstein's Tractarian analysis, as he interprets it, in appendix C to the second edition of Principia, he faces exactly the same problem. For he takes the coordination of the elements of a "thought" to the elements of a fact to suffice for taking the one complex to represent the other. Whitehead and Russell 1950, p. 662. 28 Russell clearly recognized the problem in 1913, as well as in the later logical atomism lectures, when he writes: "Thus a first symbol for the complex will be U{S, A, B, similarity, R(x, y)}. This symbol, however, by no means exhausts the analysis of the form of the understanding-complex. There are many kinds of five-term complexes, and we have to decide what the kind is. It is obvious, in the first place, that S is related to the four other terms in a way different from that in which any of the four other terms are related to each other." (Russell 1984, p. 117) He then draws a diagram showing how the various relations involved are connected to their terms. Russell 1984, p. 118. 29 Russell 1984, p. 128. 30 Pears, 1989, p. 179. 31 Russell 1984, p. 116.

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32

Russell 1984, p. 148. Landini 1991, p. 55. Russell 1984, pp. 144-145. Russell mentions the need to explicate 'correspondence' more than once. 35 Russell 1984, p. 154. 36 Landini 1991, p. 58. His truth condition is '(E'p)(Bel{m*, Bel, m, a, Redness, fx, 0(0, x, g, L)} corresponds to p)', where m* is "the mind of the person attributing the belief' to a person, m is the latter's mind, fx is the logical form of predication, 0(0, x, g, L) is the logical form "which accounts for our understanding of the simplest kind of belief-relation", " '0' represents a mental state such as belief', and " ' 0 ' represents a mind." It can be argued that the statement of a truth condition that Landini gives, involving 'corresponds to', does not enable a theory embodying it to satisfy Tarski's Convention-T. Alternatively, if one takes a sentence like 'aRb' to be transcribed by the existential claim '(Ep)(aR'p & bR 2 p)' which also states its truth condition, Convention-T is satisfied. For arguments to this effect see Hochberg 1992b. 37 Russell 1984, p. 154. Russell also uses a conjunction of such positional clauses in descriptions of the complex. Russell 1984, pp. 146-147. 39 Russell 1984, p. 148.

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Russell 1984, p. 148. Symmetric relations do not "permute" as there is only one fact. He could have done so by using a relation like consisting of or introduce relations like is a particular in and is the property (relation) in, obtaining between complexes and their constituents. This would not have been foreign to Russell for in an appendix to the second edition of Principia, he spoke of particulars and properties being construed as classes of facts having, respectively, particular-resemblance and predicate-resemblance to a given fact. Landini 1991, p. 59. Landini does consider a context like 'For anything if it is an F then if it is a G then it is an F'. But, as that is a permutative context, since 'F' and 'G' are subjects in his analysis 'Bel{m, F, G, ( E 0 ) ( E 0 ) ( a ) ( 0 a z> ( 0 a z> 0a))}', that will not do, as the different variables '0' and ' 0 ' cannot carry the burden of order (Landini 1991, p. 61). Russell seems to have thought no problem arose in such a case. Russell 1984, p. 147. Russell 1984, p. 134. Pears 1989, pp. 176-178. Russell might have been influenced by Wittgenstein's views, as recorded in the latter's notebooks. Though the entries cited below date from later than Russell's manuscript, they had been discussing and communicating about such matters in the period prior to and during Russell's work on the manuscript. Thus, in a letter of January, 1913, Wittgenstein writes: "I now think that qualities, relations (like love) etc. are all copulae! That means I for instance analyze a subject-predicate proposition, say, 'Socrates is human' into

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'Socrates' and 'something is human', (which I think is not complex)." (Wittgenstein 1969, appendix iii, 120-121) In his Notebooks Wittgenstein writes that he "had thought that the possibility of the truth of the proposition 0 a was tied up with the fact '(Ex, 0 ) . 0 x ' "(Wittgenstein 1969, p. 17), and "(Similarly ( E x ) 0 x would be the form of 0 a , as I actually thought)" (Wittgenstein 1969, p. 22). 45 Pears 1989, p. 178. There is a problem with "forms" like (0)(x)0x, that neither Russell nor Pears discusses. Such a form cannot be a logical fact, since the corresponding sentence is a logical falsehood, but it must be taken as a simple by Russell's reasoning in the existential case. The problem seems to obvious for Russell not to have noticed. 46

Russell 1984, p. 131. it is crucial to keep in mind that, while propositions are not terms of dyadic intentional relations, forms like (E0)(Ey)0y are terms of intentional relations, like U and J, that are dyadic in cases like U(S, (E0)(Ey)0y) and in cases of acquaintance. 48 Russell 1984, p. 197. 49 Russell 1971, p. 48. On this point see Hochberg 1989. 50 Wittgenstein 1969, p. 97. Wittgenstein 1969, p. 97. The "Notes on Logic" are printed as an appendix to the Notebooks and are dated September, 1913, when he dictated some of them to Russell and supplemented them with a typescript sent to Russell a "few days later" (Monk 1990, p. 93). 52 Russell 1921, p. 272. 53 Russell 1921, p. 272. 54 Russell 1984, pp. 108-109. 55 Russell 1984, pp. 109-110. 56 Russell 1984, p. 109. 57 Russell 1984, p. 111. 58 Russell 1984, p. 111. 59 Russell 1984, p. 111. Russell's use of the phrase 'same verbal form' may go back to Moore's holding that the only way of referring to the fact that was the truth ground for a proposition, the proposition that-p, was by an expression like 'the fact that-p'. On Moore's view see Hochberg 1992a. Russell's contrast of belief facts with spatial facts clearly derives from Wittgenstein's discussion of "A judges (that) p" in the 1913 "Notes on Logic" (Wittgenstein 1969, p. 96). Russell 1971, pp. 237-238. Russell does not say that the variables are real but there is no indication that they are apparent. 62 Russell 1921, pp. 231-278; see especially section IV, pp. 271-278. 63 Whitehead and Russell 1950, vol. 1, pp. 659-666. 64 5.542 reads: "It is clear, however, that 'A believes that p', 'A has the thought p', 'A says p' are of the form ' "p" says p': and this does not involve a correlation of a fact with an object, 47

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but rather the correlation of facts by means of the correlation of their objects." (Wittgenstein 1961, p. 109) For Russell's construal of 5.542 see his "Introduction" ( Wittgenstein 1961, pp. xix-xx).

References Cocchiarella, Nino 1987a "Russell's Theory of Logical Types and the Atomistic Hierarchy of Sentences" in Cocchiarella 1987b, pp. 193-221. 1987b Logical Studies in Early Analytic Philosophy. Columbus: Ohio State University Press. Hochberg, Herbert 1978 Thought, Fact, and Reference: The Origins and Ontology of Logical Atomism. Minneapolis: University of Minnesota Press. 1979 "Belief and Intention" in Philosophy and Phenomenological Research XL, 1, pp. 74-91. 1989 "Descriptions, Situations, and Russell's Extensional Analysis of Intentionality" in Philosophy and Phenomenological Research XLIX, 4, pp. 555-581. 1992a "Moore's Anticipation of Tarski's Convention-T and his Refutation of Truth as Coherence" in History of Philosophy Quarterly 9, 1, pp. 97-117. 1992b "Truth Makers, Truth Predicates, and Truth Types" in Mulligan 1992, pp. 87-117. Landini, Gregory 1991 "A New Interpretation of Russell's Multiple Relation Theory of Judgment" in History and Philosophy of Logic 12, pp. 37-69. Marsh, Robert (ed.) 1971 Logic and Knowledge. New York: Allen & Unwin. Monk, Ray 1990 Wittgenstein: The Duty of Genius. New York: Free Press. Moore, George E. 1953 Some Main Problems of Philosophy. London: Allen & Unwin. Mulligan, Kevin (ed.) 1992 Language, Truth and Ontology. Amsterdam: Kluwer Academic Publishers. Pears, David F. 1989 "Russell's 1913 Theory of Knowledge Manuscript" in Savage and Anderson 1989, pp. 169-182. Russell, Bertrand A. W. 1921 The Analysis of Mind. London: Allen & Unwin. 1956 The Problems of Philosophy. London: Williams and Norgate. 1971 "On Denoting" in Marsh 1971, pp. 41-56.

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1984 Theory of Knowledge. The Collected Papers of Bertrand Russell, vol. 7. London: Allen & Unwin. Savage, C. Wade and Anderson, C. Anthony (eds.) 1989 Rereading Russell: Essays on Bertrand Russell's Metaphysics and Epistemology. Minneapolis: University of Minnesota Press. Whitehead, Alfred N. and Russell, Bertrand A. W. 1950 Principia Mathematica, 2nd ed. Cambridge: Cambridge University Press. Wittgenstein, Ludwig 1961 Tractatus Logico-Philosophicus, trans. D. F. Pears & B. F. McGuinness. London: Routledge & Kegan Paul. 1969 Notebooks 1914-1916, ed. by Georg H. von Wright and G. E. M. Anscombe. New York: Harper and Row.

Are the Objects of the Tractatus Phenomenological Objects? ALFONSO GARCÍA SUÁREZ

1. The Empiricist Interpretation of the Tractatus It is widely known that Viennese positivists considered the author of the Tractatus as an anti-metaphysical fellow-traveler and as a consistent empiricist. This approach to the Tractatus not only overlooks the profound disparities existing between the positivist attitude and the Wittgensteinian one towards that which falls beyond the limits of what can be expressed through language, but also it distorts Wittgenstein's position regarding meaningful language. In fact, logical positivists read certain cardinal notions in the Tractatus from an empiricist point of view. They identified atomic or elementary propositions with empirically verifiable observational propositions and assimilated the simple objects designated by their names to entities susceptible of acquaintance, in the Russellian sense. Thus Alfred Ayer stated, in one of his last works, that the reading of the Tractatus in the early 30's had convinced him of the validity of the verification principle and that "in the last resort all empirical facts were atomic, identifiable with items of sensory experience." Ayer comments upon this doubtful interpretation: "If I was not in doubt about it, the reason is that I wished to harmonize my interpretation of Wittgenstein with the approach to the theory of sense-perception shared by Russell and Moore."1 Under this approach, sense data are immediately given in perception and the external world is a logical construction from those primary materials. Even as late as 1967 it is possible to find a phenomenalist reading of the Tractatus in David Favrholdt's book.2 The positivist interpretation of the Tractatus started falling into disrepute in the 60's. Two of the authors that have most contributed to this much deserved relegation were Elisabeth Anscombe and Michael Dummett. In her book published in 1959,3 Miss Anscombe established a relationship between the Tractatus and the Fregean semantic tradition and she fixed its limits with the reductivist epistemological aims of Russell and the logical empiricists. As regards Dummett, 4 he taught us to bear in mind the differences between a realist theory of meaning, based on truth-conditions, like Frege's or the one in the Tractatus, and an anti-realist theory of meaning, based on verification- or

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assertability-conditions, like that of the logical positivists, Quine's or Wittgenstein's last theory. Nevertheless, in recent years a revival of the empiricist interpretation of the Tractatus has taken place due to the work of Merrill and Jaakko Hintikka5 and, more cautiously, to David Pears.6 The Hintikkas and Pears see in the Tractatus not an ancestor of the Viennese verificationism of the 30's, but a descendant of the Russellian empiricism of the 10's, with its insistence on the notion of acquaintance or knowledge by direct awareness. In this paper I will try to prove that this new empiricist version of the Tractatus is going astray and to do so I am going to concentrate on the problem of the nature of simple objects. The new empiricist version presents Tractarian objects as objects susceptible of immediate experience, without necessarily prejudging whether they are mental subjective phenomenic entities (sense data) or objective entities (phenomena in the Kantian sense). Using Wittgenstein's terminology from his transition period we will refer to this kind of entities as "phenomenological objects". I will start out by rejecting three arguments in favour of the thesis that Tractarian objects are phenomenological objects: the argument based on Wittgenstein's testimonies and texts (section 2), the reasoning based on solipsism (section 3), and the argument based on the Russellian background of the language theory of the Tractatus (section 4). I will then expound the main objection to the identification of Tractarian objects with phenomenological entities: the problem of colour incompatibility (section 5). Finally, I will reject the way in which the Hintikkas are trying to save this objection (section 6).

2. Wittgenstein's Testimonies and Texts Several interpreters have based their theories on texts and testimonies by Wittgenstein himself, from which, at first sight, the goodness of the empiricist interpretation seems to derive. We will pay attention here to texts coming from the Notebooks 1914-1916, the Tractatus and the transition period. In the Notebooks that Wittgenstein kept during the Great War there are numerous texts in which he plays with the idea that the simple objects required by logical analysis might be sense data.7 However, he did not adopt this position definitively and he counterbalanced it by contemplating the opposite possibilities that the objects could be ordinary things—like a clock 8 —or unobservable physical entities—like material points.9 Therefore, Wittgenstein's position at this moment is indecision. He himself regrets it in his Notebooks:

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"Our difficulty was that we kept on speaking of simple objects and were unable to mention a single one."10 Wittgenstein is even more enigmatic in the Tractatus with regard to the nature of objects. There are certain passages that have slanted in favour of the phenomenological interpretation, such as 2.0131, in which he uses as illustrations "a spatial object", "a speck in the visual field", "musical notes", and "objects of the sense of touch". However, as he clarifies in 4.123, it is in this case a "shifting use" (schwankende Gebrauch) of the word "object". I understand that in these cases the word is only used in order to illustrate, without pretending to be strict. It would be something similar to Russell's usage of the word "name" when he presents the expression "Scott" as an example of a relative name, even when his official doctrine states that only demonstratives, if the speaker uses them to refer to his present sense data, are genuine proper names. While Wittgenstein's position in the Notebooks was one of indecision, his position in the Tractatus is one of calculated detachment, as is illustrated by an anecdote told by Norman Malcolm in his Memoir. Malcolm asked Wittgenstein, while he visited America, if when he wrote the Tractatus any example of simple object had occurred to him: "His reply was that at that time his thought had been that he was a logician: and that it was not his business, as a logician, to try to decide whether this thing or that was a simple thing or a complex thing, that being a purely empirical matter!"11 The writings of his transition period seem more favourable towards a phenomenological interpretation of objects. Thus in his article "Some Remarks on Logical Form" Wittgenstein seems to assume that at least some attributions of degrees to an experienced quality are atomic propositions. But we must take into account that, consequently, he gave out from that time the requirement of mutual independence for atomic propositions, a requirement that plays a basic role in the logical atomism of the Tractatus. Therefore, this is not the position that he maintained in the Tractatus with regard to the nature of objects. The Hintikkas also quote a text from the Philosophische I, 1, which states:

Bemerkungen,

"I do not now have a phenomenological language, or 'primary language' as I used to call it, in mind as my goal." According to their interpretation, Wittgenstein would be contrasting his aim at the time of writing that text and that one he had planned in the Tractatus: a phenomenological language in which we might report on what was primarily given in experience. Nevertheless, they cannot directly support this reading. It is very probable that Wittgenstein was contrasting his aim in the Bemerkungen

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with the aim that he had proposed a little earlier. In fact, the editor of the Bemerkungen, Rush Rhees, believes that he was referring to the goal that had been set in 1929, in "Some Remarks on Logical Form". Perhaps the most impressive testimony in favour of a phenomenological reading is the one found in the Cambridge lectures in 1930-1932, edited by Desmond Lee. During 1930-31 Wittgenstein explained the first propositions in his book to Lee, and told him that objects were things such as colours, spots in visual space, etc. However, this testimony by Wittgenstein himself cannot be seen without a critical eye either. Firstly, it seems to contradict the testimony that Malcolm recorded in his Memoir and that we have cited above. What Wittgenstein told Malcolm is not that he had thought that we could not say whether any given entity is a specific example of an object. After all, many of the things said in the Tractatus are officially considered as something that cannot be said. His answer to Malcolm was that in those days he considered the question of ascertaining whether something was simple or complex was an empirical matter, not a logico-philosophical task. Secondly, as strange as it may seem, this testimony is not decisive because, as the Hintikkas recognize elsewhere, "Wittgenstein is not always a completely reliable witness concerning his own earlier views." 12 Peter Carruthers gives an excellent explanation of why this is so: "Eleven years elapsed between the completion of Tractatus in 1918 and the first of the recorded remarks in 1929, during which time Wittgenstein, not only did very little philosophy, but found thinking about his own work extremely slow and painful. Notice also that the writing of Tractatus seems to have been highly intuitive, with much apparently going unsaid, even in Wittgenstein's own thoughts. He may therefore, in later years, have had difficulty in thinking his way back into the full complexity of his earlier text—specially given the restless and forward-looking nature of his intellect."13 Thirdly, and as we will see later on, in his Notebooks as well as in the Tractatus colours are rejected as candidates for objects. Taking Wittgenstein's texts and testimonies as a whole, we can distinguish three stages in his points of view regarding the nature of simple objects: in the first stage, established with documentary evidence in his Notebooks, he observes the possibility that the notion of object may be satisfied by sense data, by everyday objects or by punctual masses, without deciding on any of the three alternatives; in the Tractatus indecision is replaced by official silence: the necessary existence of the simples is transcendentally deduced from requirements for the working of a meaningful language, but the determination of what kind of entities are considered as simple is thought of as

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an empirical matter, not as a semantic question; finally, in the transition period, and coinciding with a time in which Wittgenstein kept very close positions to those of the Viennese positivists, he considers a phenomenalist interpretation, but he is aware that it makes him scrap the thesis of the mutual independence of the atomic propositions and, therefore, give up the system of the Tractatus. 3. The Argument Based on Solipsism It has been frequently thought that the fact that Wittgenstein endorsed solipsism in the Tractatus involved him in phenomenalism. David Favrholdt, for instance, defended this interpretation. But this association of solipsism with phenomenalism overlooks the peculiar character of Wittgenstein's solipsism, which is a "critical" or "transcendental" one. As I have tried to prove in my 1976 book, 14 but apparently with only limited success, the solipsism in the Tractatus is closely connected to the projection doctrine. According to Wittgenstein, the key to the inexpressible truth of solipsism stems from the fact that the limits of my language mean the limits of my world. But why my language and not simply the language? It is here where we have to resort to the projection doctrine. Wittgenstein claimed that language is made up of propositions that are pictures of reality. Now, for something to be a picture it is not enough for it to be a sequence of signs. Wittgenstein calls this the prepositional sign: the sign, perceptible by the senses, of the proposition. The prepositional sign is not the senseful proposition yet; it is only a mere utterance or inscription. In his Blue Book he expressed this idea by saying that the prepositional sign is dead in itself: "Without a sense, or without the thought, a proposition would be an utterly dead and trivial thing. And further it seems clear that no adding of inorganic signs can make the proposition live. And the conclusion which one draws from this is that what must be added to the dead signs in order to make a live proposition is something immaterial, with properties different from all mere signs."15 What gives life to the prepositional sign in the Tractatus is precisely the thought. Through thought the perceptible sign of the proposition turns into a picture of reality, since "the proposition is the prepositional sign in its projective relation to the world" 16 and "the method of projection is thinking about the prepositional sense." 17 In the proposition thought is expressed in such a way that the elements of the prepositional sign correspond with the objects of thought. That is why it is necessary for thought to be articulated: it must be composed of psychical constituents which correspond to the elements of the prepositional sign. But, as Wittgenstein states in a letter to Russell, what

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those elements are and what kind of relation they have with words is an empirical question of psychology, not a philosophical matter. Thus, by virtue of the projection doctrine, the sense of the proposition depends on the thought. Now, Wittgenstein does not understand "thought" as something abstract which is a common property of anyone who grasps it, but as something subjective and private. That is why the only language that I understand is my language and why its limits are the limits of the world. One could wonder after all if this reading would not strengthen the Hintikkas' position according to which the objects of the Tractatus, although not necessarily sense data, must be at least objects open to acquaintance, phenomenological objects given to me in my experience. Hence, the Hintikkas argue, the important thing is not that the objects are subjective sense data, the important thing is that they are directly given to me. Otherwise, how could they be relevant to my language and my thought? I think the mistake underlying this reasoning consists of not realizing that in order to use language meaningfully—in order to mean and to understand— we do not need to be able to reach the last analysis of the propositions. As Wittgenstein himself points out in 4.002: "Man possesses the ability to construct languages capable of expressing every sense, without having any idea how each word has meaning or what its meaning is—just as people speak without knowing how the individual sounds are produced." Everyone who understands the propositions in their unanalyzed form must already know that there are elementary propositions in which they are decomposed. The elements of the "completely analyzed" proposition are simple signs or names that stand for objects. But it is not stated anywhere that we need direct awareness of the objects in order to know this. We know it a priori, as the result of a logical necessity.

4. The Argument from the Russellian Background of the Tractatus In a fine display of scholarship, David Pears has explained how Wittgenstein criticized and modified the theory of judgement that Russell had designed in his failed Theory of Knowledge reaching this way his own theory of language.18 While Russell posited two kind of objects of acquaintance— concrete objects and logical forms—, Wittgenstein disregarded logical forms and, a fortiori, the necessity of acquaintance with those Platonic entities. The forms of simple objects passed over to perform all the functions that Platonic logical forms used to have in the Russellian theory. Hence the elementary

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propositions "absorb", if you will, the forms of the objects. Well then, the Hintikkas hold that this developmental story told by Pears: "makes little sense... unless we assume that [Wittgenstein] retained the idea that simples—the building blocks of forms—are still objects of acquaintance."19 This argumentation looks to me like a patent non sequitur. From the fact that Wittgenstein disregarded logical forms and kept objects we cannot derive that his objects were to have all the characteristics of Russellian objects. It would be enough if they kept all the characteristics required by the explanation of the process of "absorption" of logical forms: in the explanation of how complex logical forms are determined by the logical forms of the elementary propositions, which at the same time are determined by the logical forms of objects. Now, the Hintikkas would have to prove that the task that objects perform in this double process of transmission of the form is performed qua objects of acquaintance and not merely qua entities possessing certain combinatorial powers. In other words, they would have to prove that objects would lack in forms if they were not objects of acquaintance. The Hintikkas also referred in this respect to 5.552, that runs: "The 'experience' that we need in order to understand logic is not that something or other is the state of things, but that something is: that, however, is not an experience." In their opinion, Wittgenstein would be saying here that we must have acquaintance with objects in order to grasp logical forms and that the reason why this is not a genuine experience is that it is an inexpressible experience. I think they are wrong in both points. As the context of the 5.55 makes clear, Wittgenstein's position is that we must know on a purely logical basis that there are elementary propositions and therefore that there are simples denoted by the names that they contain. But nothing is said about a supposed need of achieving immediate experience of those objects. The issue at stake is a kind of knowing-that and not knowing-of: we must know that there are simple objects, but that does not mean that we must have a direct knowledge of those objects.

5. The Problem of Colour Incompatibility So far we have discarded several arguments that have been used in favour of the thesis that the objects in the Tractatus are phenomenological entities. But there is also a powerful argument against that thesis. In my opinion it is a decisive argument.

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In 6.375 it is claimed that the only necessity and the only impossibility existing are logical need and impossibility, i.e. the need and impossibility that is respectively characteristic of truth-functional tautologies and contradictions. Since the negation of an impossible proposition is a necessary proposition, we can refer shortly to this claim by calling it the Tautologicity Thesis. The Tautologicity Thesis is paired with the principle according to which the elementary propositions are logically independent of each other. Let us call it the Independence Thesis. If there were not elementary propositions independent of each other, the truth-functional explanation of necessity would collapse. Let us suppose that two elementary propositions 'p' and 'q' were not independent, but rather that 'no-*?' would follow from '/>'. Then the conjunction 'p & q' would be a non-truth-functional contradiction. In other words, in the truth-table of that conjunction there would be an extra line, a line that represents an impossible situation. Well then, a suitable example of this kind of non-truth-functional contradiction is the conjunctive proposition 'A is red and A is green'. This proposition looks like a counterexample to the Tautologicity Thesis. Don't we have here a case of an impossibility that cannot be reduced to a non-truth-functional contradiction? We must think that exactly this kind of impossibilities were presented by Husserl as a testimony of the existence of synthetic a priori propositions. Wittgenstein's solution in 6.3751 consists of declaring that this kind of impossibilities are in the final analysis logical impossibilities: "For example, the simultaneous presence of two colours at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour." This reference to the logical structure of colour implies that colours are not simple objects and that the propositions 'A is red' and 'A is green' are not elementary propositions, for the mark of elementary propositions is, as we have seen, their mutual logical independence. In fact, Wittgenstein implies this in the final paragraph of 6.3751 in which he states: "(It is clear that the logical product of two elementary propositions can neither be a tautology nor a contradiction. The statement that a point in the visual field has two different colours at the same time is a contradiction)." And in the Notebooks he clearly affirm it: "If the logical product of two propositions is a contradiction, and the propositions appear to be elementary propositions, we can see that in this case the appearance is deceptive. (E.g.: A is red and A is green)."20 Up to here the solution that Wittgenstein gives to the problem rests upon a promissory note: that an analysis of the structure of colour will be found so

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that when colour ascriptions are analyzed in their most basic components it will be seen by the symbol alone that a proposition like ' A is red' entails that A is not green. Let us call that desideratum the Requirement of Perspicuity. However, at the time of writing the Tractatus, Wittgenstein did not undertake the task of trying to find an analysis of colour that would satisfy this requirement. Later, when in the transition period he tries to develop this analysis he will realize that it is impossible to harmonize it with the Independence Thesis and he will drop the mutual independence of atomic propositions, starting thus the dismantling of the Tractatus. But this is another story. What we want to highlight here is that if we admit the Tractarian doctrine of the logical independence of elementary propositions, then our only option is to reject that the attributions of colour are elementary propositions and that words such as 'red' and 'green' are names of simple objects. And that is the conclusion which Wittgenstein comes to.

6. The Hintikkas' Way Out and its Difficulties The Hintikkas think that arguments such as the one we have just presented depend on a presupposition not accepted by Wittgenstein: that colour ascriptions are subject-predicate propositions, so that ' A is red' and ' A is green' would translate into the logical notation as 'R(a)' and 'Gfa)'. The problem is that, although those two propositions are incompatible, their incompatibility is not logical in the sense that it is not shown in the notation used. The solution would be, according to the Hintikkas, to reject that colour attributions are of the subject-predicate form. If we interpret the concept of colour as a mapping of points into a colour space, colour incompatibilities would not create non-logical necessities. Let us suppose that the concept of colour is not represented by a class of colour-predicates, but by a function c that maps points in visual space into a colour space. Then the form of ' A is red' and ' A is green' would respectively be c(a)=r and c(a)=g, where r and g are the objects red and green. The incompatibility would be reflected by the fact that the colours red and green are represented by different names. In such a circumstance, both propositions are logically incompatible and their incompatibility is perspicuously shown by their logical representation, since a function cannot have two different values for the same argument due to their "logical form". Although the way out is clever, it does not solve the problem that colour exclusion presented to the thesis that the objects of the Tractatus are phenomenological entities. Firstly, there is no evidence that the functional analysis proposed by the Hintikkas was the analysis that Wittgenstein was bearing in mind. The most that the Hintikkas dare to affirm is that the analysis

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"is in keeping with the spirit of Wittgenstein's thinking."21 However, the texts seem to suggest that Wittgenstein was considering a rather different kind of analysis. In the Notebooks as well as in the Tractatus Wittgenstein reduces the impossibility of simultaneous coinstantiation of two colours in the same place to a kinetic impossibility. The second paragraph of 6.3751 affirms: "Let us think how this contradiction appears in physics: more or less as follows—a particle cannot have two velocities at the same time; that is to say, it cannot be in two places at the same time; that is to say, particles that are in different places at the same time cannot be identical." Pace the Hintikkas, Wittgenstein does not limit himself here to present a solvable analogue to the problem in the field of the mechanics of particles, but he offers a reductive explanation, as he states in the parallel paragraph of the Notebooks which runs: "A point cannot be red and green at the same time: at first sight there seems no need for this to be a logical impossibility. But the very language of physics reduces it to a kinetic impossibility. We see that there is a difference in structure between red and green. And then physics arranges them in a series. And then we see how here the true structure of the object is brought to light. The fact that a particle cannot be in two places at the same time does look more like a logical impossibility."22 I think that the fact that the "reduction" of a phenomenological impossibility to a kinetic impossibility is mentioned in this text as well as the fact that "the true structure of objects" is referred to, shows that we clearly have here a reductive explanation. Nevertheless, this cannot be the final analysis yet, as the expression "does look more like" shows. That is to say, we have not yet found an analysis which reduces the phenomenological impossibility to a full logical impossibility, to a truth-functional contradiction. We only have something that "does look more like" what we are looking for. Judging by the texts of his intermediate period, it seems that Wittgenstein had in mind an analysis of the logical structure of colour in terms of logical products. Thus, in "Some Remarks on Logical Form" he writes: "One might think—and I thought so not long ago—that a statement expressing the degree of a quality could be analyzed into a logical product of single statements of quantity and a completing supplementary statement."23 It was a question of analyzing the degrees of a quality by means of a conjunction of quantity statements plus a closure clause of the kind "and nothing else", just as we can

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describe the contents of my pocket by saying: "It contains three coins, a handkerchief, some keys, and nothing else." But even if this proposal could satisfy the Requirement of Perspicuity, it would not solve the problem. For if the degrees of brightness that take part in the analysis are identical, the statement "E(2b)"—"The entity E has two degrees of brightness"—would not be equivalent to the logical product "E(b) & E(b)". (For idempotency this last statement would be equivalent to the mere "E(b)".) But if we distinguish between units of brightness and we write "E(2b)=E(b') & E(b")'\ then the original problem of exclusion is reproduced again at the level of the different units. We have already anticipated that, as a consequence of the difficulties which he came across when trying to solve this problem, Wittgenstein gave up the Independence Thesis. In "Some Remarks on Logical Form" he writes: "The mutual exclusion of unanalyzable statements of degree contradicts an opinion which was published by me several years ago and which necessitated that atomic propositions could not exclude one another." 24 The solution that he begins to see, and later will give up, consists of modifying the rules that govern logical connectives. Secondly, even if we were to accept the functional analysis that the Hintikkas propose, the problem would still remain unsolved. Although they show how it is possible to comply with the Requirement of Perspicuity, they do not show how it is possible to satisfy at the same time the Independence Thesis. That is to say, they offer an analysis of colour ascriptions that reduces the phenomenological impossibility to a logical impossibility, but they do not show how it is possible to make compatible that analysis with the doctrine that atomic propositions must be mutually independent. In the notation proposed by the Hintikkas the incompatibility of the propositions " c ( a ) - r " and "c(a)=g" is shown by the symbol alone. But neither r nor g are simple objects nor are the two propositions in question elementary propositions, r and g are not simple objects because their nature generates logical connections between the apparently atomic sentences that contain their names; and the propositions are not elementary because they exclude each other. In the Notebooks as well as in the Tractatus the conclusion that Wittgenstein reached is that it is impossible to maintain at the same time the Independence Thesis and the status of colours as simple objects. From that follows his affirmation that 'A is red' and 'A is green' are not really elementary propositions although they might seem to be. (It is highly meaningful that the Hintikkas never cite this paragraph.) Finally, the functional analysis that the Hintikkas propose as a way out to the problem of colour exclusion complies

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with the Requirement of Perspicuity by sacrificing the Independence Thesis. Certainly, Wittgenstein reached the point of sacrificing the Independence Thesis in his intermediate period. But that meant undermining the basis of the logical atomism of the Tractatus. Whatever virtues could be attributed to phenomenological objects, there is something that they could never be: Tractarian objects. Therefore, it is an intellectual perversion to present an analysis that does not comply with the requirement of the mutual independence of atomic propositions as an argument in favour of the phenomenological nature of objects. The conclusion is rather the inverse one: the objects in the Tractatus cannot be phenomenological entities because, if they were, the Independence Thesis would have to be given up.25

Notes 1 Ayer 1987, p. 25. 2 Favrholdt 1967. 3 Anscombe 1959. 4 Dummett 1973, 1978. 5 Hintikka 1986. 6 Pears 1987, vol. 1. 7 Cf. Notebooks 1914-1916, 3.9.14, 6.5.15, 7.5.15, 24.5.15, 25.5.15, etc. 8 Notebooks 1914-1916, 20.5.15, 14.6.15, 15.6.15, 16.6.15, etc. 9 Notebooks 1914-1916, 20.6.15. 10 Notebooks 1914-1916, 21.6.15. 11 Malcolm 1958, p. 86. 12 Hintikka 1986, p. 129. 13 Carruthers 1990, p. XIII. 14 Garcia Suarez 1976, pp. 49-51. 15 Wittgenstein 1964, p. 4. 16 Tractatus 3.12. 17 Tractatus 3.11. 18 Pears 1977; 1987, chapter 6. 19 Hintikka 1986, p. 55. 20 Notebooks 1914-1916, 8.1.17. 21 Hintikka 1986, p. 124. 22 Notebooks 1914-1916, 16.8.16 (The second italics are mine). 23 Wittgenstein 1929, p. 167. 24 Wittgenstein 1929, p. 168. 25

This work was supported by the DGICYT, Project PS 92-0121.

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References Anscombe, Elizabeth 1959 An Introduction to Wittgenstein's Tractatus. London: Hutchinson. 1977 Introducción al Tractatus de Wittgenstein. Buenos Aires: El Ateneo. Ayer, Alfred 1987 "Reflections on Language, Truth and Logic" in Gower 1987, pp. 23-34. Carruthers, Peter 1990 The Metaphysics of the Tractatus. Cambridge: Cambridge University Press. Dummett, Michael 1973 Frege: Philosophy of Language. London: Duckworth. 1978 Truth and Other Enigmas. London: Duckworth. Favrholdt, David 1967 An Interpretation and Critique of Wittgenstein's Tractatus. Copenhagen: Munksgaard. Ferrater Mora, José et al. 1966 Lasfilosofias de Ludwig Wittgenstein. Barcelona: Oikos-Tau. Garcia Suärez, Alfonso 1976 La lògica de la experiencia: Wittgenstein y el problema del lenguaje privado. Madrid: Tecnos. Gower, Barry (ed.) 1987 Logical Positivism in Perspective: Essays on Language, Truth and Logic. London: Croom Helm. Hintikka, Merril and Jaakko 1986 Investigating Wittgenstein. Oxford: Blackwell. Malcolm, Norman 1958 Ludwig Wittgenstein: A Memoir. Oxford: Clarendon Press. 1966 "Recuerdo de Ludwig Wittgenstein" in Ferrater Mora et al. 1966, pp. 39-95. Pears, David 1977 "The Relation Between Wittgenstein's Picture Theory of Proposition and Russell's Theories of Judgment" in The Philosophical Review 86, pp. 177-96; repr. in Shanker 1986, vol. 1. 1987 The False Prison: A Study of the Development of Wittgenstein's Philosophy. Oxford: Clarendon Press. Shanker, Stuart G. (ed.) 1986 Ludwig Wittgenstein: Critical Assessments. 1. London: Croom Helm. Wittgenstein, Ludwig 1929 "Some Remarks on Logical Form" in Proceedings of the Aristotelian Society, supplementary, vol. 9, pp. 162-171. 1961 Tractatus Logico-Philosophicus, trans. D. F. Pears and B. F. McGuinness. London: Routledge & Kegan Paul.

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1964 The Blue and Brown Books. Oxford: Blackwell. 1968 Los cuadernos azuly marron. Madrid: Tecnos. 1969 Notebooks, ed. by Georg H. von Wright and G. Elizabeth M. Anscombe. New York: Harper and Row.

Does a Proposition affirm every Proposition that Follows from it?1 MARÍA CEREZO

The problem of the relations between the orders of signification, truth and logical consequence is central to logic. The way in which these relationships are established goes some way towards determining the logical theory to which the writer suscribes. This problem has been a major issue in the history of logic. In this essay, I propose to show that the problem of these relations is present in the course of the historical development of logic, and that there is no one solution to it. My aim is thus not to offer an answer to the question posed, but to highlight the contrast between different solutions, approaching this issue from the standpoint defended by Ludwig Wittgenstein (1889-1951) in his Tractatus Logico-Philosophicus.2 Nor do I intend to provide a detailed explanation of Wittgenstein's view, but only to outline it and bring out the contrasts with other possible solutions, solutions which predate it by some considerable time. 1. The Solution in the Tractatus Logico-Philosophicus The question as to whether a proposition affirms everything that follows from it meets with a clear enough response in Wittgenstein's Tractatus: "A proposition affirms every proposition that follows from it" (TLP 5.124).3 That the proposition affirms (bejaht) something means that it says that this is the case; it is true if it does in fact happen that things behave as is expressed by the proposition. The correlate in Frege is judgement (Urtheil) or assertion (Bejahung), to affirm a proposition is to state a judgeable content, recognising it to be true. Propositions state facts, and for Wittgenstein the condition which enables this to be so is the peculiar relationship between language and reality, proposition and state of affairs. The explanation for this relationship is provided by the picture theory. The Tractatus offers the picture theory as a key to explain the meaningfulness of language. The proposition is meaningful to the extent that it is a picture of reality, and for this reason, the pictorial character of the

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proposition establishes itself as a condition for the proposition to have sense,4 for the proposition can be true or false.5 The proposition is a picture of reality because it represents a possible state of affairs, which may be the case or may not be the case, and consequently the proposition can be true or false. The next question is, what does depiction consist of, what explanation of it does Wittgenstein offer us? The genuinely pictorial character is proper to elementary propositions which can represent possible states of affairs in an immediate manner. Elementary propositions are concatenations of names which stand for simple objects, and represent the structure or mode in which such objects relate to each other or are arranged with regard to each other. The elementary proposition therefore immediately reflects the possible state of affairs which it represents, because it is a reproduction of it. The elementary proposition represents a possible state of affairs, and asserts the actual existence of this state of affairs.6 The application of the picture theory to the non elementary proposition is somewhat more difficult. 7 The question, then, is how the non elementary proposition depicts if the condition for any proposition to be meaningful is that it should be a picture of reality, and this is only immediate in the case of elementary propositions. Wittgenstein's solution is to consider that "a proposition is a truth-function of elementary propositions" (TLP 5),8 and it depicts reality in a mediated way, through its truth-arguments, that is, by means of elementary propositions. This is so because "truth-possibilities of elementary propositions mean possibilities of existence and non-existence of states of affairs" (TLP 4.3).9 The non elementary proposition therefore depicts through its truth-conditions.10 The notion of sense in the Tractatus is thus established in accord with the pictorial nature of language, as has been explained above. The sense (Sinn) of the proposition is its directedness towards a state of affairs which may be the case or may not be the case,11 and for this reason, the sense is also to be determined by way of the truth-conditions.12 The sense of the elementary proposition is determined by the being the case or not being the case of the state of affairs which it represents; that of the non elementary statement is determined by its truth-grounds, that is, by the truth-possibilities of its arguments which make it true.13 The relation "it follows from" is presented in the Tractatus as a relationship between truth-conditions14 and therefore between sense-conditions. The proposition affirms everything that follows from it because what is being asserted is the proposition with sense, and the sense of the first proposition contains that of the one which follows because there are no truth-grounds of the first which do not also belong to the second.15 Thus Wittgenstein affirms that

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"the truth-grounds of the one are contained in those of the other: p follows from q" (TLP 5.121),16 and therefore it is also true that "if p follows from q, the sense of 'p' is contained in the sense of 'q' " (TLP 5.122).17 The relation of consequence is a necessary connection between propositions, because it is a necessary connection between their truth-values. The theory of logical consequence in the Tractatus is eminently semantic. It is not that one proposition follows from another because we can deduce it by means of some rule, but rather that we can deduce it because it follows from it, because there is a link between the two, in concrete, because the truth-grounds of the one are included in those of the other. There are no rules of deduction which justify the chain of inferences, but rather there is a direct link between propositions through their sense, through the truth-conditions. All inferences are a priori, and the axiomatic method is trivialised. The reduction of the relation of consequence and sense to conditions of truth makes the approach in the Tractatus into an extensional semantics: the semantic relationships are not properly established in terms of sense, as is the case in intensional semantics, but in terms of truth-conditions. The picture theory of the proposition leads to the reduction of the meaning of the proposition to its causes of truth. Analysis of the proposition is the means of determining its sense, and precisely because the proposition signifies pictorially, figuratively, analysis is conceived of as being a reduction to what is basic in the reference. The process of determining the sense is carried out in absolutely referential terms.18 The simple nature of the object and its lack of intelligible form determine a concrete conception of sense: sense is not what has been understood, it is not an intelligible form of the object, but the external correlate of the arrangement of names which constitutes the proposition. Meaning has been dissolved into truth-conditions. Gottlob Frege and Rudolf Carnap also maintain a certain tendency towards an extensional view of meaning. Despite recognising the sphere of sense, Frege's theory of sense and reference entails a weakening of the intensionality of meaning: the demand that concept-words should be referential, and the approach of the reference of concept-words to conceptual extension, also tend to result in a reduction of meaning to the extensional order.19 Frege's notion of "judgeable content" in his Conceptual Notation is a further example of this phenomenon. The judgeable content is that part of the proposition's content which is relevant for inference, and so the judgeable contents which have the same consequences will be the same, and will have the same conceptographical expression. 20 On the other hand, this trend culminates in Carnap, who in Meaning and Necessity (1947) reduced the meaning of a sentence to the class of sentences which are equivalent to it.21

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2. "Contradiction is the Outer Limit of Propositions" (Tractatus 5.143) In the Tractatus, contradiction is one of the limiting cases of the proposition, and study of it brings out the scope and implications of the thesis expounded in this work. The contradiction is the proposition in which all the truth-possibilities are negative, that is, the proposition entirely without truthconditions, and therefore the proposition whose sense cannot be determined. The application of Wittgenstein's notion of logical consequence to contradiction is of great interest. On the one hand, as we have noted, in Wittgestein's view the proposition affirms every proposition which follows from it; yet on the other, as the result of a notion of consequence established in terms of truth-conditions, it is recognised that anything can be inferred from a contradiction.22 But then the contradiction affirms anything or, in other words, the supposed sense of the contradiction includes the sense of all possible propositions. The relation of consequence, as seen by Wittgenstein, makes it possible to assign total meaning to the contradiction: the contradiction is the proposition which says most,23 the outer limit of propositions.24 In this respect, Wittgenstein's note in his Notebooks for 3 June 1915 is interesting. Here the scope of the application of the theory of consequence to the case of the contradiction is shown.25 On the one hand, the note shows an awareness of the difficulty underlying the very notion of logical impossibility: nothing should be inferred from a contradiction, for the very reason that it is a contradiction. On the other, the sinnlos character of the contradiction is explained precisely because, as it contains the sense of all propositions, the possibility of determining the sense is cancelled out. By saying this, Ludwig Wittgenstein ventured into the long and important debate on the subject of the principle Ex impossibili quodlibet sequitur.26 The underlying problem is not new, it is the same problem as was debated by medieval logicians, who put forward various solutions.27 An analogous solution to that of Wittgenstein is the one adopted by Thomas Bradwardine in his Treatise on the Insolubilia, which seems to be the earliest work containing a defence of the affirmation that the proposition signifies everything that follows from it.28 This thesis is postulated, and it is a necessary presupposition of the particular solution to the problem of the insolubilia which Bradwardine offers. 29 However, this thesis is mentioned and repudiated by the Summa Dialecticae Oxoniensis (Sophisteria Angliae), in which a different relationship between these notions is suggested: "ilia non est generaliter vera, scilicet, omnis propositio significat omne illud quod sequitur ad illam, quia tunc sequeretur quod omnis propositio impossibilis significaret omne imaginabile, et sic quodlibet verum et

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necessarium et impossibile, quia quodlibet tale sequitur ad istam, eo quod ex impossibili sequitur quodlibet." The interest of this text lies precisely in the fact that it defends the opposite point of view to that noted in TLP 5.124, even though, like Wittgenstein, it accepts the principle Ex impossibili quodlibet sequitur. In the Summa Dialecticae Oxoniensis, the idea that the proposition signifies everything that follows from it is rejected, and the justification for this lies in the commitment both to a notion of consequence established in the extensional order, in terms of truth-conditions,30 and to an intensional notion of meaning which is not reduced to truth-conditions. Meaning, in this case, is not reduced to external reference or the state of affairs which is denoted, but lies chiefly in what has been understood. It is for this reason that the impossible proposition cannot signify everything, as it would be unintelligible. The text of the Summa Dialecticae Oxoniensis thus brings to light the maladjustment between the two orders, and avoids the difficulty by denying that the proposition affirms everything that follows from it. Both Wittgenstein and the author of the Summa Dialecticae Oxoniensis defend the principle Ex impossibili quodlibet sequitur, but they differ in the notion of meaning, as the former regards it as belonging to the extensional and the latter to the intensional order. Other solutions to the problem are also possible, and scrutiny of the discussion on the principle Ex impossibili quodlibet sequitur in the medieval period sheds some light on this issue. Those who do not accept the principle give a third possible relationship. For them, as for Wittgenstein, the consequent must in some way be included or presupposed in the antecedent; but the form of inclusion is unlike that suggested in the Tractatus. It is not truth-conditions that are included, but meanings. The antecedent has in some way to contain the consequent, so that the latter can be understood in the former. The relation of consequence is thus understood as a relationship between meanings, not between truth-conditions. In this case, meaning and consequence are placed in the intensional order. The answer to the question as to whether the proposition affirms all propositions that follow from it is in the affirmative, as in Wittgenstein, but in a completely different sense. The proposition asserts everything that follows from it because intelligens antecedens intelligit consequens. To illustrate this view, it might be useful to look at the fifth and sixth objections to the question Utrum ad impossibile sequatur quodlibet in the Tractatus Consequentiarum of Martin Lemaistre,31 who was among those who defended this principle, and who sets out the opinions against it in the form of objections.

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The fifth objection32 embodies the opinion of those who require for the bona consequentia relation that the consequent should be included in the antecedent. For this reason, the Ex impossibili quodlibet sequitur cannot be accepted, because it would mean that the impossible antecedent had to be infinite. Precisely because the proposition includes in its meaning what follows from it—a relation of consequence viewed as a relationship between meanings—and because the meaning is what is understood, be this actual or virtual—the intensional notion of meaning—the principle Ex impossibili quodlibet sequitur cannot be accepted, because then the antecedent would be infinite, and therefore unintelligible. The answer to this objection33 falls into a circular argument when it reduces includere to inferre: if the objection turns about a notion of inferentia which depends on that of includere—quia nihil sequitur ad aliud nisi quod virtualiter includatur in illo—, the answer cannot lie in indicating that the difficulty is overcome if we take includere to mean inferre. The sixth objection34 alludes to the requirement that intelligens antecedens intelligit consequens, that the consequent is understood in the antecedent. If the consequent must be understood from the antecedent (bonae consequentiae consequens debet esse de intellectu antecedentis), it is difficult to think how can every proposition be understood from any impossible antecedent. In this case, the solution is reduced to the rejection of such a notion of consequence, but no reason is given for doing so.35 3. Conclusion The various answers outlined in this essay show that Wittgenstein suggests one among several possible solutions: the proposition affirms every proposition that follows from it because the relation of consequence is understood as a relationship between truth-conditions, and because the meaning similarly dissolves into truth-conditions. But this is not the only possible solution: the Summa Dialecticae Oxoniensis offers another, and the detractors of the principle Ex impossibili quodlibet sequitur a third. We must therefore conclude first that the solution given in the Tractatus is not necessary; and second, that this solution can be found to contain a reduction of signification to truth-conditions, through which the function of understanding in signification and logical inference is practically eliminated. The picture theory of the proposition imposes excessively stringent requirements on the underlying vision of language and the world. Signification consists of the direct correlation between the picture and the state of affairs of which it is the model, not of the intellection or possession of the intelligible form of the object.

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Finally, I also hope I have s h o w n that the relations b e t w e e n the orders of signification, truth and logical consequence is a problem o f prime importance. It w a s m y aim to s h o w that study of the debate over the principle Ex impossibili quodlibet sequitur o f f e r s a g o o d perspective f r o m w h i c h to approach this question, inasmuch as the acceptance of this principle g o e s s o m e w a y towards conditioning the relationship b e t w e e n these three notions. It is m y belief that this d i s c u s s i o n serves as further confirmation o f the l o g i c a l interest w h i c h arises f r o m examination o f extreme cases, such as that of the principle Ex impossibili quodlibet sequitur.

Notes 1

2

3 4

5

6

7

This paper is the result of research carried out in the Department of Logic and Philosophy of Language at the University of Navarre on the medieval doctrine of consequence, and my preliminary work on the theory of logical inference in Ludwig Wittgenstein's Tractatus Logico-Philosophicus. I therefore owe a debt of gratitude to the members of this Department, particularly to former member Angel d'Ors, to Paloma P6rez-Ilzarbe and Jaime Nubiola, for all the help they provided, both through their written work and their suggestions. I should also particularly like to extend my thanks to the participants at the III Symposium on the History of Logic for their extremely useful observations. I quote the German text from the bilingual edition published in 1933 (Wittgenstein 1933). For the English version of the text, I follow Wittgenstein 1961a. To refer to the propositions in the Tractatus I shall henceforth use the letters TLP followed by the corresponding number. "Der Satz bejaht jeden Satz der aus ihm folgt." Cf. TLP 4.064: "Every proposition must already have a sense; it cannot be given a sense by affirmation. Indeed its sense is just what is affirmed. And the same applies to negation, etc." ("Jeder Satz muß schon einen Sinn haben; die Bejahung kann ihn ihm nicht geben, denn sie bejaht ja gerade den Sinn. Und dasselbe gilt von der Verneinung, etc.") Cf. TLP 4.06: "A proposition can be true or false only in virtue of being a picture of reality." ("Nur dadurch kann der Satz wahr oder falsch sein, indem er ein Bild der Wirklichkeit ist.") Cf. TLP 4.21: "The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs." ("Der einfachste Satz, der Elementarsatz, behauptet das Bestehen eines Sachverhaltes.") It should be noted that the relation of consequence, or "it follows from" requires the consideration of non elementary propositions, as for Wittgenstein, since the elementary propositions are independent from each other, no relation of consequence can exist between them; from one elementary proposition we cannot infer another elementary proposition. The fact that from one elementary proposition there follows the disjunction of it and any other is not an obstacle: in this case, the truth-grounds in the second include those of the first, and therefore follow from the former. Cf. Wittgenstein 1961b, p. 55: "In the real sign for p there is already contained the sign

. (For it is then possible to form this sign WITHOUT FURTHER ADO.)"

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8 "Der Satz ist eine Wahrheitsfunktion der Elementarsätze." 9 "Die Wahrheitsmöglichkeiten der Elementarsätze bedeuten die Möglichkeiten des Bestehens und Nichtbestehens der Sachverhalte." Cf. TLP 4.4: "A proposition is an expression of agreement and disagreement with truthpossibilities of elementary propositions." ("Der Satz ist der Ausdruck der Übereinstimmung und Nichtübereinstimmung mit den Wahrheitsmöglichkeiten der Elementarsätze.") TLP 4.41: "Truth-possibilities of elementary propositions are the conditions of the truth and falsity of propositions." ("Die Wahrheitsmöglichkeiten der Elementarsätze sind die Bedingungen der Wahrheit und Falschheit der Sätze.") 11 Cf. TLP 4.2, 4.3, 4.4, 4.41, 4.431. The analysis of the notion of sense in Wittgenstein falls beyond the scope of this essay; Wittgenstein scholars have devoted ample attention to this subject, see especially Maury 1977, Black 1964 and Shwayder 1966. 12 Cf. TLP 4.063: "in order to be able to say, ' "p" is true (or false)', I must have determined in what circumstances I call 'p' true, and in so doing I determine the sense of the proposition." ("um sagen zu können: 'p' ist wahr (oder falsch), muß ich bestimmt haben, unter welchen Umständen ich 'p' wahr nenne, und damit bestimme ich den Sinn des Satzes.") Cf. TLP 5.101: "I will give the name truth-grounds of a proposition to those truth-possibilities of its truth-arguments that make it true." ("Diejenigen Wahrheitsmöglichkeiten seiner Wahrheitsargumente, welche den Satz bewahrheiten, will ich seine Wahrheitsgründe nennen.") Cf. TLP 5.11. "If all the truth-grounds that are common to a number of propositions are at the same time truth-grounds of a certain proposition, then we say that the truth of that proposition follows from the truth of the others." ("Sind die Wahrheitsgründe, die einer Anzahl von Sätzen gemeinsam sind, sämtlich auch Wahrheitsgründe eines bestimmten Satzes, so sagen wir, die Wahrheit dieses Satzes folge aus der Wahrheit jener Sätze.") At this point I do not share the view of Black (1964, p. 247), when he alludes to the expression "contained" in the Tractatus as being unclear and perhaps unfortunate. To my mind, in the context of the relations between the notions of signification, truth and consequence, this word in no way obfuscates the thesis which Wittgenstein is putting forward. "Die Wahrheitsgründe des einen sind in denen des anderen enthalten; p folgt aus q." 17 "Folgt p aus q, so ist der Sinn von 'p' im Sinne von 'q' enthalten." 18 Cf. Griffin 1964, pp. 149ff. 19 Frege establishes conceptual extension as a criterion for the identity of concepts: "what two concept-words mean is the same if and only if the extensions of the corresponding concepts coincide" (Frege 1979a, p. 122). The concept-word, according to Frege, has reference when for any object it can be rigorously established whether it falls under that concept, or not. The reference of the concept-word is determined by the conceptual extension. "Thus, its chief purpose should be to test in the most reliable manner the validity of a chain of reasoning and expose each presupposition which tends to creep in unnoticed, so that its source can be investigated. For this reason, I have omitted the expression of everything which is without importance for the chain of inference" (Frege 1972, p. 104).

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21 "We defined earlier the L-equivalence class of a designator in S as the class of all designators in S L-equivalent to it. It is easily seen that there is one-one correlation between the L-equivalence classes in S and the intensions expressible in S. Therefore, the L-equivalence class of a designator may be taken as its intension or at least as a representative for its intension" (Carnap 1947, p. 152). 22 The classic version of the principle is Ex impossibili quodlibet sequitur. Wittgenstein understands impossibility as a logical contradiction. The impossible is understood as logically impossible or formally contradictory. 23 Cf. TLP 5.14: "If one proposition follows from another, then the latter says more than the former, and the former less than the latter." ("Folgt ein Satz aus einem anderen, so sagt dieser mehr als jener, jener weniger als dieser.") 24 Cf. TLP 5.143: "Contradiction is the outer limit of propositions: tautology is the unsubstantial point at their centre." ("Die Kontradiktion ist die äußere Grenze der Sätze, die Tautologie ihr substanzloser Mittelpunkt.") 25 "One could certainly say: That proposition says the most from which the most follows. Could one say: 'From which the most mutually independent propositions follow'? But doesn't it work like this: If p follows from q but not q from p, then q says more than p? But now nothing at all follows from a tautology. It however follows from every proposition. The analogous thing hold of its opposite. But then! Won't contradiction now be the proposition that says the most? From 'p.~p' there follows not merely 'p' but also '~p' ! Every proposition follows from them and they follow from none!? But I surely can't infer anything from a contradiction, just because it is a contradiction. But if contradiction is the class of all propositions, then tautology becomes what is common to any classes of propositions that have nothing in common and vanishes completely." (Wittgenstein 1961b, p. 54.) 26 The principle Ex impossibili quodlibet sequitur is the most striking among the principles or rules of good consequence which have generally been discussed within the framework of the paradoxes of strict implication. This issue was hotly disputed in the medieval period, and interest has been reawakened among our contemporaries, as can be seen in developments in logic of relevance and minimal logic, to take two examples. 27 An interesting discussion is to be found in Ashworth 1974, pp. 133-136 and Martin 1987. 28 Cf. Spade 1981. 29 "Suppositiones sunt:... secunda est ista: quelibet propositio significai sive denotai vel omne quod sequitur ad illam vel vel " (Roure 1970, p. 297). 30 The basis for the principle Ex impossibili quodlibet sequitur in medieval writers is linked to the notion of consequence understood as a relationship between truth-conditions. To gain some idea of the scope and difficulties of the bases on which this principle rests, see d'Ors 1990 and 1993, Martin 1987, Iwakuma 1993 and Spruyt 1993. Martin Lemaistre (1432-1483) worked under the influence of the nominalist tendency which was very widespread at the end of the fifteenth century. Bom in Tours, he studied and taught at the Collège de Navarre, and afterwards in Santa Barbara. He was Rector of the University of Paris in 1460. He was Louis XI's confessor, and his intervention was decisive in bringing about the repeal of the 1473 decree against the nominalists. Two

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works on Logic are known: Tractatus Consequentiarum (1489) and Expositio perutilis et necessaria super libro Praedicabilium Porphyrii (1490). See Fraile 1966, vol. 3, pp. 384385. "Quinto sic: sequeretur quod impossibile includerei quodlibet (et non solum contradictoria), immo infinita, et per consequens quodlibet impossibile esset virtualiter infinitum; et patet consequentia: quia nihil sequitur ad aliud nisi quod virtualiter includatur in ilio." "Ad quintam conceditur consequens si 'includere' accipiatur pro 'inferre', ita quod antecedens includat omnem propositionem quam inferi (qua inclusione unum oppositorum includit aliquando reliquum), et in hoc sensu conceditur quod impossibile includit infinita, id est, infinitorum illativum." "Sexto sic: cuiuslibet bonae consequentiae consequens debet esse de intellectu antecedentis; sed in multis consequentiis ubi antecedens est impossibile, consequens non est de intellectu antecedentis; ergo non ad impossibile sequitur quodlibet. Maior patet: quia ex hoc multi assignant causam bonae consequentiae. " "Ad sextam negatur maior, et ad probationem dicitur quod illud non est verum quid nominis bonae consequentiae (neque ex eo adaequate quaelibet consequentia dicitur bona)."

References Anonymous 1503 Summa utilissima Dialecticae Oxoniensis, quae communiter Sophisteria dicitur Angliae. Exacta nobili in civitate Hispalensi, impensis largissimis Lazari de Gazanis viri optimi, arte quoque praecellenti Ioannis Pegnicer de Nuremberga alemani, anno salutis christianae millesimo quingentesimotertio, decimoquarto kalendas octobris. f.erb. Hispalis. (Biblioteca Universitaria "Marqués de Valdecilla", Universidad Complutense, Madrid.) Ashworth, E. Jennifer 1974 Language and Logic in the post-medieval period. Dordrecht: Reidei Publishing Company. Black, Max 1964 A Companion to Wittgenstein's 'Tractatus'. New York: Cornell University Press. Carnap, Rudolf 1957 Meaning and Necessity. 2nd ed. Chicago: The University of Chicago Press. Copi, Irving M. and Beard, Robert W. ( eds.) 1966 Essays on Wittgenstein's Tractatus. London: Routledge & Kegan Paul. d'Ors, Angel 1990 "Ex impossibili quodlibet sequitur (Walter Burley)" in Archives d'Histoire Doctrinale et Littéraire du Moyen Age LVII, pp. 121-154. 1993 "Ex impossibile quodlibet sequitur (John Buridan)" in Jacobi 1993, pp. 195-212. Fraile, Guillermo 1966 Historia de la Filosofia. Madrid: BAC.

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Frege, Gottlob 1972 Conceptual Notation and related articles, trans, and ed. by T. W. Bynum. Oxford: Clarendon Press. 1979a "Comments on Sense and Meaning" in Frege 1979b, pp. 118-125. 1979b Posthumous Writings, ed. by H. Hermes, F. Kambartel and F. Kaulbach, trans, by P. Long and R .White. Oxford: Blackwell. Griffin, James 1964 Wittgenstein's Logical Atomism. Oxford: Clarendon Press. Iwakuma, Yukio 1993 "Parvipontani's Thesis ex impossibili quidlibet sequitur: comments on the sources of the thesis from the twelfth century" in Jacobi 1993, pp. 123-151. Jacobi, Klaus (ed.) 1993 Argumentationstheorie. Leiden: E. J. Brill. Joli vet, Jean and Libera, Alain de (eds.) 1987 Gilbert de Poitiers et ses contemporains aux origines de la Logica Modernorum. Napoli: Bibliopolis. Lemaistre, Martin 1489 Tractatus Consequentiarum. Editions: 1494, Parisiis, 4 (Bibliothèque Royale, Bruxelles); 1497, Parisiis, 4 (Bibliothèque Nationale, Paris); 1501, Parisiis, 4 (Universitätsbibliothek, Freiburg; Stadtbibliothek, Trier). Martin, Christopher J. 1987 "Embarrasing arguments and surprising conclusions in the development of theories of the conditional in the twelfth century" in Jolivet and Libera 1987, pp. 377-400. Maury, André 1977 The Concepts of Sinn and Gegenstand in Wittgenstein's Tractatus in Acta Philosophica Fennica XXIX, 4. Roure, Marie-Louise 1970 "La problématique des propositions insolubles du XlIIe siècle et du début du XlVe, suivie de l'édition des traités de William Shyreswood, Walter Burleigh et Thomas Bradwardine" in Archives d'Histoire Doctrinale et Littéraire du Moyen Age XXXVII, pp. 205-326. Shwayder, David S. 1966 "On the Picture Theory of Language: Excerpts from a Review" in Copi and Beard 1966, pp. 305-312. Spade, Paul Vincent 1981 "Insolubilia and Bradwardine's Theory of Signification" in Medioevo VII, pp. 115134. Spruyt, Joke 1993 "Thirteenth Century Positions on the rule 'ex impossibili sequitur quidlibet' " in Jacobi 1993, pp. 161-193.

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Wittgenstein, Ludwig 1933 Tractatus Logico-Philosophicus, trans, by C. K. Ogden and F. P. Ramsey. London: Routledge & Kegan Paul. 1961a Tractatus Logico-Philosophicus, trans, by David F. Pears and Brian F. McGuinness. London: Routledge & Kegan Paul, London, 1961. 1961b Notebooks 1914-1916, ed. by G. H. von Wright and G. E. M. Anscombe. New York: Harper & Brothers.

Carnap's Reconstruction of Intuitionistic Logic in The Logical Syntax of Language JAVIER LEGRIS

Carnap's The Logical Syntax of Language can be regarded as an attempt to solve the controversy in the foundations of logic and mathematics at the end of the 1920's and the beginning of the 1930's. In this context Carnap undertook a reconstruction of intuitionism to which not much attention had yet been paid. This paper aims to show the motivations and limitations of this reconstruction. In the argumentation both historical and purely systematical aspects will be considerated. I The main goals of Carnap's Logical Syntax can be summed up as follows: firstly, to give a solution to the controversy in the foundations of mathematics at that time, and secondly, to develop a conception of logic and mathematics consistent with his own ideas about empirical knowledge. According to Carnap the foundations of logic and mathematics were the same as an explanation of how logic and mathematics were possible. This implied setting up a criterion for the analytic character of logic and mathematical truths, and this criterion was to be given in relation with sentences of a given language. The fulfillment of these goals led Carnap to construct a formal theory about the linguistic forms of the language in which the rules determining the analytic sentences of the system should be formulated. Carnap called this theory the syntax of the language (Carnap 1937, p. 1). Now, following his own "principle of tolerance"—by which there is no privileged logic or language—Carnap decided to present the syntax of two different formal languages, viz. language I conceived to formalize finitary and intuitionistic mathematics, and language II, used to express classical mathematics and physical theories. As it will be noted later, this clear distinction between the two languages corresponds to a certain extent to the distinction between real (or finitary) and ideal (or infinite) mathematics introduced by Hilbert in order to accomplish his own program (v.g. Hilbert 1926).

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Language I (LI) was a first order language with identity. Its individual constants are numerals, intuitively conceived to denote natural numbers or 'positions' of empirical objects. Hence, predicates and function symbols of LI refer to numerical properties, relations and functions. Besides, they are what Carnap called definite (1937, p. 45), in the sense that it is always resoluble whether a predicate is satisfied by a specific number or whether a function symbol has a definite value. Logical connectives are characterized in LI as the classical truth functions. Quantifiers are not the usual standard ones but they are finitely limited, that is, the scope of the quantifiers is restricted to a finite number of individuals. So, quantified formulas of LI have the forms '(V;c)n A[x]' or '(3JC)/I A[JC]' where V designates a natural number. (Carnap used a quite different notation including German letters for metalinguistic variables.) It is by virtue of these limited quantifiers that every definite predicate is definite. Unlimited universality can be expressed in LI only by free variables, so that sentences with free variables are to be understood as unlimited universal quantifications. Unlimited existential propositions are not expressible in the language. These features of the logical symbols of LI are reflected in a system of axioms and rules. The axioms for logical connectives together with the rule of modus ponens build a system for classical sentential logic. From the three axioms for limited quantifiers it follows that universal and existential quantifiers can be reduced to finite conjunction and disjunction respectively. Because of the lack in the language of unlimited quantifiers nothing will be proved in the system about unlimited generality, i.e. about an unlimited (possibly infinite) domain of individuals. Besides the rules of substitution and modus ponens, the principle of induction is added to the system as a rule. This principle reveals itself as necessary to prove some logical theorems, playing in part the role universal generalization (absent in the system) has in standard formalizations. According to this account of LI, it turns out clearly that it was a decidable system; in a finite number of steps it could be decided for every sentence of the language whether it was derivable in LI or not. Hence, every derivation might be seen as 'effective' or 'constructive'. In this way, LI seemed to be an accurate version of finitary mathematics in Hilbert's sense, i.e. primitive recursive arithmetic. Moreover, Carnap originally intended to construct only LI, and only later Language II was developed in order to formalize classical mathematics (Carnap 1963, pp. 48ff.). Furthermore, Carnap conceived LI as the most elementary mathematical language, being something like a 'minimal' language common to all possible richer mathematical systems (including classical mathematics).

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Now, there existed in LI unlimited universal sentences (that is, with free variables) of the form A[*] not derivable in LI, even if every instance A [a] was (1937, p. 37). For there were no theorems in LI with free variables. From an intuitive point of view, however, these universal sentences should be regarded as logically true, i.e. 'analytic'. Hence, LI was 'incomplete' in relation to what it is normally understood as 'analytic sentences'. Carnap's syntactism of that time required the notion of analyticity to be an extension of that of derivability. More accurately, logical consequence was defined as a generalization of derivability for infinite sets of premises (1937, p. 38), similar to the w-rule formulated by Hilbert a short time before, consisting of an universal generalization from infinite premises (Hilbert 1931). Thus, there resulted 'two different methods of deduction': derivation, where all steps were definite, and logical consequence, consisting of a finite series of not necessarily finite sets of sentences, and being, therefore, not definite (1937, p. 39). Analytic sentences were defined as logical consequences from the empty set. This way of extending the notion of derivability allowed Carnap to reach a complete criterion of analyticity according to which there are no undecidable sentences with respect to their analyticity. II Carnap considered LI as an appropriate language to reconstruct 'in a certain sense' intuitionistic logic and mathematics (1937, p. 46). When Carnap began to write The Logical Syntax—at the beginning of the 1930s—a formally accurate and generally accepted characterization of intuitionism was not yet available, even though A. Heyting had developed an axiom system for intuitionistic sentential logic. Hence, Carnap used to refer with more generality (and vagueness) to 'constructivism' or 'finitism', subsuming under these notions—among others—Brouwer's intuitionism, Hilbert's finitism and Wittgenstein's new ideas on mathematics. The Vienna Circle—Carnap in particular—had a deep—at first sight even surprising—interest in mathematical intuitionism. It can be said that this interest was due to certain resemblances between intuitionism and the logical empiricist program. Thirty years later, in his "Intellectual Autobiography", Carnap remembered his own sympathy with intuitionism (Carnap 1963, pp. 48ff.). These resemblances had different reasons. First, constructive proofs seemed to be epistemologically more certain and accurate than the classical ones. Second, a strict empiricism should require every meaningful sentence to be decidable according to its verification—and, consequently, according to its truth value. Note that in the Circle it was doubted on several opportunities whether

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universal empirical sentences, like scientific hypothesis, should be regarded as meaningful because of the impossibility of verification. Furthermore, the fact that, for instance, from an intuitionistic point of view an existentially quantified sentence had to be proved by showing a concrete existing example was fully compatible with the epistemological ideas of the Circle. Hence, if intuitionistic ontological commitments—like assuming mathematical objects as mental constructions—are left aside, its conception of mathematical proof became a counterpart in formal science of the early verificationism in empirical sciences sometimes sustained by members of the Vienna Circle. These resemblances provoked Carnap to construct first language I and only later, guided by his principle of tolerance, language II (Carnap 1963, pp. 55ff.). Anyhow, the reconstruction of intuitionism undertaken by Carnap was to be also rooted in the tradition in logic and semantics he expressly followed. This tradition was represented firstly in modern logic by Frege and had been conveyed in Wittgenstein's Tractatus through his conception of logical truths as tautologies. Its influence, explicitly acknowledged by Carnap himself in his "Intellectual Autobiography", prevented him on the one hand from questioning or discussing the intuitive semantical basis for classical logic, so that principles like the principle of excluded middle belonged to logical laws in the system. (In fact, it is theorem 13.2 of LI.) On the other hand, they led him also to match— in this respect—theoremhood and analyticity. Both facts led to a situation in which Brouwer's dictum could be applied paradigmatically: The validity of the principle of excluded middle raised from an unjustified extension from finite domains to infinite ones (Brouwer 1929, p. 160). Now, keeping in mind the whole situation, how was it possible to reconstruct intuitionistic logic on this basis? The answer is in Carnap's assumption that the problem intuitionists faced was not to be found in logical connectives but in quantifiers, and these problems could be avoided by means of the limited quantifiers of LI. However, intuitionists considered quantification over infinite domains— under certain conditions—as constructive and, hence, meaningful. For the notion of infinite should be understood now as 'potential infinite', that is, as a not well defined totality. In particular, they were able to provide an interpretation for the existential quantifier: A proof of a sentence '(3*) A[JC]' is a proof of the sentence 'A[n]' for some number denoted by ln'. Carnap argued in the following way (1937, p. 47). Provided that from the sentence ( 1 ) (VJC)

Px

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a contradiction is inferred, the following sentence can be classically derived (2)

( 3 J C ) -,PX

.

In its derivation the principle of excluded middle (or an equivalent proposition) must be used, and hence it is not admissible from an intuitionistic point of view. (More accurately, although (1) is refuted, the existence of a constructible number a for which we have a proof that the sentence '—¡Pa' is derivable, does not follow intuitionistically. That is, in reality, one of the reasons why the principle of excluded middle is not accepted.) On the other hand, such a derivation was not possible in LI. For (2) was not expressible in this language while (1) was. Unlimited universality was expressed in LI via free variables, but there existed no way to express an unlimited existential quantification. This procedure had the same effect as the reduction of logic to finite and decidable domains. As it is known, in this case intuitionistic and classical logic coincide. More precisely, every language—even an intuitionistic one—supplied with decidable primitive predicates and a finite domain of quantification will make valid the principles of classical logic. In this sense, Carnap's reconstruction worked. However, intuitionistic logic attempted to be valid also in infinite domains. (At this point the divergences of intuitionism with classical mathematics originated. See, v.g. Brouwer 1929, p. 160.) Furthermore, this reconstruction was carried out from a finitist conception of what should be intuitive or contentual mathematics. Hence, Carnap's proposal should seem very doubtful to intuitionistic minds. Ill The whole problem becomes more evident if Carnap's interpretation of logical symbols is analyzed. On the one hand, as it was already pointed out, intuitionistic logical constants are interpreted in terms of constructive proofs and not in terms of truth values. As a consequence, the semantical principle of bivalence is no longer valid and the principle of excluded middle cannot be a theorem. On the other hand, in spite of his syntactism, by virtue of which no meaning is assigned to logical constants but are introduced as mere syntactical symbols characterized by the rules governing them, Carnap had in mind an underlying interpretation for them and this interpretation had to be the classical one. For the sentential connectives were characterized intuitively as bivalent truth functions, and this characterization could be extended to limited

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quantifiers. So the principle of bivalence should be (implicitly) valid and the principle of excluded middle was a theorem of the system. Now—assuming the truth tables to be a mere mechanical device without any semantical import—the principle of excluded middle does not necessarily imply the principle of bivalence. For the former is always valid with respect to decidable domains—notwithstanding the interpretation of the connectives— while the latter is more an assumption than a consequence of the system. Moreover, it was true that proofs in LI of sentences containing quantifiers were constructive (in any sense of constructivity) because of their decidable character. Nevertheless reducing classical logical constants to their application to decidable domains is not equivalent to obtaining intuitionistic logic. In general, the situation presented here can be summarized as follows. In developing LI, Camap had a double purpose: first, to construct a logic whose proofs were all constructive and second to gain for this language a complete criterion of analyticity. In a certain sense, both purposes obeyed different— mutually inconsistent—conceptions of logic. The first purpose was motivated by Camap's tendency to constructivism and finitist methods and his ideas about empirical knowledge. The second purpose was motivated, as already suggested, by the need for a clear-cut distinction between formal and factual truth, and here the principle of bivalence was in some way involved. This situation had undesirable consequences. For Carnap pretended to use LI as a constructive (finitary) basic logic for his further metalogical tasks in relation with richer languages, but analyticity in LI required a non constructive (non finitary) extension of the system. It is obvious that a constructive and at the same time complete notion of analyticity is not possible—insofar as logic and mathematics are not restricted to finite domains. In developing the logical syntax, Carnap wanted to be independent from any semantic conception, but the very idea of analyticity implied specific semantical assumptions. They would be explicit later as Carnap adopted Tarski's semantics in the late 1930s.

References Brouwer, Luitzen Egbertus Jan 1929 "Mathematik, Wissenschaft und Sprache" in Monatsheft pp. 153-164. Carnap, Rudolf 1937 The Logical Syntax of Language, Kegan Paul.

f . Math. u. Physik

36,

trans, by Amethe Smeaton. London: Routledge &

1963 "Intellectual Autobiography" in Schilpp 1963, pp. 3-83.

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Hilbert, David 1926 "Über das Unendliche" in Mathematische Annalen 95, pp. 161-190. 1931 "Die Grundlegung der elementaren Zahlenlehre" in Mathematische Annalen 104, pp. 485-494. Schilpp, Paul (ed.) 1963 The Philosophy of Rudolf Carnap. La Salle: Open Court.

Some Influences of Hermann Graßmann's Program on Modern Logic ALBERT C . L E W I S

1. Graßmann 's Program Hermann Graßmann, the "universal genius" as E. T. Bell called him, was born in Stettin, Pommerania, (now in Poland) in 1809 and died there in 1877. His schoolmaster and pastor father, Justus, was perhaps the single greatest intellectual influence on his life since it was from him that Hermann not only learned mathematics but an approach to science which combined a theological-philosophical standpoint with research and teaching. Like his father, Hermann Graßmann's entire career was as a schoolmaster in Stettin. He attended Berlin University in 1834-35 and attended lectures in philology and philosophy. One of the lecturers, the theologian Friedrich Schleiermacher, was later singled out by Graßmann as a particular influence on his approach to science (Lewis, A. C. 1977). The Ausdehnungslehre of 1844 was to be a new branch of mathematics and Graßmann introduces it through it's place in a scheme for Wissenschaften that I summarize here in diagrammatic form: Wissenschaften Real Formal Dialectic (Logic) - general principles of thought Philosophy Formal Logic (Begriffslehre) Theory of Forms (Pure Mathematics) - the particular generated by thought Graßmann can, I believe, stand with figures like Josiah Willard Gibbs, Oliver Heaviside, and C. S. Peirce—to name a few in related areas—whose importance has not been recognized during their lifetimes as much as after. Their importance today is not just that they anticipated later work or were

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unrecognized co-discoverers, but that their lives, works and methods continue to be a source of ideas and inspiration. Not until after his death was Graßmann's work used in any substantial way. The first major recognition given to him was by the twenty-eight-year-old Hermann Hankel (1839-1873) in his theory of complex number systems (Hankel 1867). Hankel, in extensive historical notes, gave Graßmann credit for forming the most general "theory of forms" (Formenlehre) up to that time in his Ausdehnungslehre. In this presentation I would like to list a number of influences and connections between Graßmann and the development of modern logic. 2. A Comparison with C. S. Peirce 's Logical System Graßmann is not usually compared with C. S. Peirce but there are a number of significant parallels between these two polymaths. One is that each was motivated by an overall philosophy of science that, though quite different for each of them, and having roots in essentially different cultures and different generations, still played a similar role. Attending Friedrich Schleiermacher's lectures in 1839 and reading his Dialektik in 1840 was a major influence on Graßmann; following the debate between John Stuart Mill and William Hamilton on experience versus consciousness formed an analogous experience in 1865 for Peirce. Of course, as well they both put mathematics at the center of their work and formed new ways of doing mathematics and of relating logic to mathematics. In 1877 Peirce read one of the papers that Graßmann published late in his life to demonstrate the usefulness of his calculus of extension— usefulness to mechanics in this case. Peirce reacted by publishing a short paper on the similarities between Graßmann's work and his own (Peirce 1877-78). But Graßmann died in September 1877 without, as far as I know, ever hearing of Peirce. 3. Robert Graßmann and the Formenlehre Though they are sometimes confused with each other by philosophers of science or logicians, Robert Graßmann (1815-1901), Hermann's younger brother, is associated more with logic than mathematics. Mathematicians, since they have little to do with logic and have probably never heard of Robert, do not confuse them. Robert's best known work is probably Die Formenlehre oder Mathematik (Graßmann, R. 1872). As is clear from the title this is a continuation of the program started by Hermann's Ausdehnungslehre in 1844. It is usually described in histories, such as (Lewis, C. I. 1918), as an

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interesting, novel, and idiosyncratic development in the history of the algebra of logic between Boole and Schröder. One of the novel features of Robert's system is in that part of the Formenlehre entitled Die Begriffschrift oder Logik. Here he derives the laws of concepts, Begriffe, or classes as C. I. Lewis calls them, from the laws for Stifte or Elemente, or individuals as Lewis calls them. Thus it follows from the "special principles" that e +e =e

e.e = e

ere2-0

i.e., the class-sum and the class-product of two equal elements is again the same element, as Robert expresses it. C. I. Lewis is probably typical in not recognizing how Robert's approach ties in with the Graßmann brothers' approach to mathematics. After all, this connection is not explicit in their works and only stands out when looking at the whole corpus. Here I will just point out that in the Lehrbuch der Arithmetik Hermann develops numbers out of the same sort of general series of elements that Robert has for Begriffe.

4. Lehrbuch der Arithmetik Graßmann's Lehrbuch is divided into numbered definitions, notations, and theorems. (Graßmann does not admit "axioms" though we see the equivalent to Peano's axioms in Graßmann's "definitions".) The first, introductory, section includes definitions of mathematics, equality and inequality, introduces the terminology of "connection" between "quantities", and gives examples of how the notation for connections (such as minus and plus) are written along with letters for quantities and parentheses for grouping. Finally arithmetic is defined as that branch of mathematics which is concerned with those quantities which can be produced from a single quantity (denoted by e) through connection. The next sections concern properties of addition and subtraction. Addition is the connection which generates a fundamental series from a unit e which has an inverse, the "negative unit", -e. Successor and predecessor are defined and properties are proven that are used in proving the identity (on the right) property of 0. Associativity of addition and commutativity of the unit with an arbitrary member of the series are proven. This last proof also introduces the inductive method for the first time in the book. This sketch may provide enough of the spirit of Graßmann's method.

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Note that up to now no mention has been made of numbers. What we have in Graßmann's fundamental series is, in modern terminology, an infinite cyclic group. The treatment up to this point corresponds approximately to the development of logic that Robert Graßmann presented in Die Begriffschrift oder Logik (mentioned above). Numbers are introduced as a special fundamental series in the next section, on multiplication. An identity element, 1, is defined with respect to multiplication and a number series is defined as a fundamental series whose unit is 1. Multiplication of these elements by elements not in the same series is on the right. Thus what is being described is close to a right module in modern terms. Hao Wang remarked that Graßmann's system is not perfect but he admired the non-set theory approach and recast Graßmann's work in the Lehrbuch to show how close it comes to what he called a "formalization of logic"—the sort of thing we see later in Frege, Whitehead-Russell, and Quine, "except that their works are entangled with their particular theories of sets" (Wang 1957, p. 149). 5. Peano 's Compliment Peano was one of those influenced by Graßmann: in 1888 he published Calcolo geometrico secondo 1'Ausdehnungslehre di H. Graßmann preceduto dalle operazioni della logica deduttiva. But Peano's first attempt at an axiomatization of arithmetic was his Arithmetices principia, nova methodo exposita (1889) in which he applies his notation to the mathematics, including the proofs, in Graßmann's Lehrbuch. In it Peano stated: "With these notations, every proposition assumes the form and the precision that equations have in algebra; from the propositions thus written other propositions are deduced, and in fact by procedures that are similar to those used in solving equations. This is the main point of the whole paper" (Peano 1967, p. 85).

6. Frege's Critique Graßmann did not escape Frege's sharp criticisms of treatments of arithmetic in Die Grundlagen der Arithmetik (1884) where Frege noted that in the Lehrbuch Graßmann defined the sum (a+b) by means of an expression which presupposed such a definition. Frege allowed that this could be gotten around if one takes sum to be addition. But then the possibility could not be ruled out that a+b would be an empty symbol if there were no term, or more than one, of the fundamental series satisfying the condition. "That this does not

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in fact ever happen, Grassmann simply assumes without proof, so that the rigour of his procedure is only apparent" (Frege 1960, p. 8). 7. Russell's Consideration In the 1890s Graßmann's reputation began to rise and over the next several decades his algebras were, for example, a part of the mathematical curriculum in American universities. Whitehead's "universal algebra" was based largely on Graßmann's work which may not have helped to further spread Graßmann's work in England since this universal algebra had rather limited appeal to both mathematicians and logicians. Russell, however, as a student of Whitehead, read the Ausdehnungslehre and wrote out a passage from it in his notebook of 1896. GraBmann's algebras became manifolds in Whitehead's Universal Algebra (1898) and as such were Russell's initial "fundamental notions" that were in turn precursors of classes and propositional functions. This particular shift, away from Graßmannian notions, is evident in the altered table of contents for an early version of The Principles of Mathematics (1903) (Russell 1990, p. 154).

References Frege, Gottlob 1960 The Foundations of Arithmetic: of Number. New York: Harper. Graßmann, Hermann G. 1844 Die lineale Ausdehnungslehre,

A Logico-Mathematical

Enquiry into the

ein neuer Zweig der Mathematik.

Concept

Leipzig: Otto

Wigand. Also in Graßmann 1910-11, II.l, pp. 1-319. 1860 Lehrbuch der Arithmetikßr höhere Lehranstalten. Stettin; Enslin, Stettin. 1910-11 Hermann Graßmanns gesammelte mathematische und physikalische Werke, ed. Friedrich Engel. Leipzig: B. G. Teubner. Graßmann, Robert 1872 Die Formenlehre oder Mathematik, Stettin. (Reprinted with an introduction by J. E. Hofman 1966 Hildesheim: Olms). Hankel, Hermann 1867 Vorlesungen über die complexen Zahlen und ihre Funktionen. 1 Theil. Leipzig. Heijenoort, Jean van (ed.) 1967 From Frege to Gödel: A Source Book in Mathematical Cambridge, Massachusetts: Harvard University Press.

Logic,

1879-1931.

382

Albert C. Lewis

Lewis, Albert C. 1977 "H. Grassmann's 1844 Ausdehnungslehre and Schleiermacher's Dialektik" in Annals of Science 34, pp. 103-162. Lewis, Clarence Irving 1918 A Survey of Symbolic Logic. California: University of California Press. Peano, Giuseppe 1967 "Principles of arithmetic, presented by a new method" in Heijenoort 1967, pp. 8 5 97. Peirce, Charles Sanders 1877-78 "Grassmann's Calculus of Extension" in Proceedings of the American Academy of Arts and Science 13 (May 1877-May 1878), pp. 115-116. Also in 1986 Writings of Charles S. Peirce, vol. 3. Bloomington: Indiana University Press, pp. 238-239. Russell, Bertrand 1990 Philosophical Papers, 1896-99, ed. Nicholas Griffin and Albert C. Lewis. London: Unwin Hyman. (The Collected Papers of Bertrand Russell, vol. 2.) Wang, Hao 1957 "The axiomatization of arithmetic" in Journal of Symbolic Logic 22, pp. 145-158.

Indeterminism and Future Contingency in non-Classical Logics JUAN CARLOS LEÓN

Few problems have stirred so much interest throughout the history of logic as the one raised by Aristotle about future contingency in chapter 9 of his Peri Hermeneias. In our century, the question has had a special impact on one of the most outstanding features of the recent history of logic: the development of the so-called "non-classical logics". My aim here is to carry out a critical examination of the fundamental moments of that development, in which the goal of solving the logical problems posed by the question of the indeterminacy of the future has always had a central character. My interest, however, is not merely historic, but also systematic, because, although these problems have been the direct cause of the mentioned logical developments in recent history, it is my opinion that also due to them can we achieve a better understanding of the question itself. As it is widely known, the problem originally formulated by Aristotle consisted of designing a logical analysis of the propositions expressing temporal events that does not entail a determinist or fatalist position with respect to the future. The difficulty of escaping determinism is clearly seen in the following argumentation: 1 if a proposition "p" is true in the present moment, then it seems that " F p " must have been true in any previous moment. That is, we have the thesis p

HFp.

But, let us choose any of those past moments, and let us suppose that it should then have been possible that "p" could have failed to come about; in that case, " F p " could not have been true in that moment. It seems then that we are committed to the truth of the determinist principle p H> HLFp. Aristotle himself seems to express this difficulty when he says that "if it was always true to say that it was so, or would be so, it could not not be so, or not be going to be so."2 In the setting of modern logic, the first attempt to develop a logical system escaping these difficulties is due, as it is known, to Jan Lukasiewicz. 3

384

Juan Carlos León

The idea was, in principle, fairly simple. The determinist argument is blocked in its first step just by rejecting that every proposition about the future should be true or false. This idea, that as Prior notes,4 has historical precedents in Peter de Rivo,5 leads almost naturally to think that such propositions will have a third (different from true or false) truth-value: a truth-value that could be called "neuter" or of "mere possibility". Thus, the path was cleared for the development of a three-valued logic, first, and then, many-valued logic. Many-valued logic underwent a strong development during the first half of XXth century.6 Nowadays, however, the expectations created around it have stayed relatively frustrated, and interest on it is almost reduced to the mere historical one.7 But I do not intend to focus here on such overall estimations, but just on the aspects directly related to our topic. And not even these aspects will be extended to a great point, since I think that it constitutes a subject that is generally well known and sufficiently treated.8 These treatments normally ground their criticisms of the many-valued view of future contingency on the high price that entails the rejection of classical logical principles as fundamental as the excluded middle or the non-contradiction principle. As Prior points out,9 it does not seem reasonable to reject those principles, not even as a consequence of the acceptation of neuter propositions: it is true that if "Fp" is neuter (since it is a future contingency), so it will be "—¡Fp"; but "Fp v —¡Fp" seems plainly true and "Fp A —¡Fp" plainly false.10 I endorse such criticisms. But, in my opinion, the main reproach that could be made to the three-valued solution does not lie there: the price to be paid is even greater. As Richmond Thomason has noted,11 the problem of three-valued truth-functional semantics is that it also forces us to accept as valid propositions that do not match in any case with the intended interpretation. Such is the case of (.MFp

A

M-,Fp

) A

( M F q A M-Fq )

(Fp

Fq ).

That is, a contingent future event will take place if and only if any other contingent future event also takes place. In conclusion, then, it seems clear that we must agree with Prior when he says that "the truth-functional technique seems simply out of place here."12 Towards mid-century, this conviction lead Arthur Prior to raise the question of our concern in the framework of modal logic. Of course, Lukasiewicz was rather conscious that he was dealing with modal notions, but he mistakenly thought that it was possible to formulate their satisfaction conditions using many-valued semantics. It must be recognized, though, that ordinary modal logic is not well suited for our purposes either. In spite of the multiple conceptions of modalities covered by the different systems of modal logic that have been developed, the one we are studying now could not (indeed,

Indeterminism and Future Contingency in non-Classical Logics

385

cannot) be captured in this scheme. If we focus, for example, on the notion of necessity, we would wish to say that every true proposition about the past or the present is necessary, not in the sense of logical necessity characteristic of ordinary modal logic, but in the historical sense: past or present events cannot not have been. Some future events, on the other hand, will be unpreventable from the present perspective of evaluation, and are therefore historically necessary; but that will not be the case of many others, whose occurrence is not absolutely determined from the present point of view, and that will be, therefore, future contingencies. We can say, along with John Burgess, "that it is now necessary which is a physical or metaphysical consequence of the way the world is now and has been in the past: what is now unpreventable."13 This seems to be the sense in which Aristotle says, in the famous sea battle passage, that "what is, necessarily is, when it is; and what is not, necessarily is not, when it is not."14 It is evident, then, that the concept of necessity that we are considering involves temporal notions along with the strictly modal ones. Prior had widely investigated the relations between modal and temporal logic15 and thus he was in an excellent position to approach successfully the question of how to analyze logically those notions escaping from the fatalist thesis. As a matter of fact, in chapter VII of Past, Present and Future, entitled "Time and Determinism", he supplied two different ways of doing this, although both coincide in using the same type of semantic structure on which to define the satisfaction conditions: the semantic structure which is typical of branching tense logic. Prior is not too precise in the description of semantic models, and in the object language he also uses metric tense operators, which unnecessarily complicates things. Following the reformulation made by Thomason 16 and by Burgess.17 I will present the question ignoring such complications, and will consider only languages whose modal operators will be "L" and the non-metric tenses "F' and "P". The semantic structure that Prior seems to have in mind is that which we would call today "tree frame", and that could be defined in the following way: DEFINITION 1: A tree frame is a pair (T,

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