Medieval Formal Logic – Obligations, Insolubles and Consequences

The New Synthese Historical Library Texts and Studies in the History of Philosophy VOLUME49

Managing Editor: SIMO KNUUTIILA,

University of Helsinki

Associate Editors: University of Chicago University of London

DANIEL ELLIOT GARBER, RICHARD SORABJI,

Editorial Consultants: Thomas-Institut, Universitiit zu Koln, Germany Virginia Polytechnic Institute E. JENNIFER ASHWORTH, University of Waterloo MICHAEL AYERS, Wadham College, Oxford GAIL FINE, Cornell University R. J. HANKINSON, University of Texas JAAKKO HINTIKKA, Boston University, Finnish Academy PAUL HoFFMAN, University of California, Riverside DAVID KONSTAN, Brown University RICHARD H. KRAUT, Northwestern University, Evanston ALAIN DE LIBERA, Ecole Pratique des Hautes Etudes, Sorbonne JOHN E. MURDOCH, Harvard University DAVID FATE NoRTON, McGill University LUCA 0BERTELLO, Universita degli Studi di Genova ELEONORE STUMP, St. Louis University ALLEN WooD, Cornell University

JAN A. AERTSEN,

RoGER ARIEW,

The titles published in this series are listed at the end of this volume.

MEDIEVAL FORMAL LOGIC Obligations, Insolubles and Consequences Edited by

MIKKO YRJONSUURI University of Jyviiskylii, Finland and Academy of Finland, Helsinki, Finland

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TABLE OF CONTENTS

PREFACE

Vll

pART I OBLIGATIONS AND INSOLUBLES MIKKO YRJONSUURI I Duties, Rules and Interpretations in Obligational Disputations HENRIK LAGERLUND AND ERIK J. OLSSON I Disputation and Change of Belief-Burley's Theory of Obligationes as a Theory of Belief Revision CHRISTOPHER J. MARTIN I Obligations and Liars FABIENNE PIRONET I The Relations between Insolubles and Obligations in Medieval Disputations PARTIICONSEQUENCES

3

35 63 95 115

PETER KING I Consequence as Inference: Mediaeval Proof Theory 1300-1350 117 IVAN BOH I Consequence and Rules of Consequence in the PostOckham Period 14 7 STPEHEN READ I Self-reference and Validity Revisited 183 PART III 1RANSLATIONS

197

ANONYMOUS I The Emmeran Treatise on False Positio ANONYMOUS I The Emmeran Treatise on Impossible Positio PSEUDO-SCOTUS I Questions on Aristotle's Prior Analytics

199 21 7

Opposite of the Consequent?

225

INDEX OF NAMES

235

v

PREFACE

One of the most important cornerstones of logic is the relation of consequence. This relation is something that is supposed to obtain between the premises and the conclusion of a valid inference. However, spelling out this relation in any further detail has proved to be extremely difficult. In fact, logicians of various times who have tried to provide a comprehensive account of what an inference is have always found themselves in serious difficulties. The purpose of this book is to look more closely at medieval discussions of inference. The authors of the various essays aim at bringing the field of medieval logic closer to the concerns of contemporary philosophers and logicians. Thus, although the papers do represent the peak of present-day scholarship, they are not primarily designed to further specialist research in medieval logic. Instead, the purposes of the book follow from the present situation of medieval scholarship: historical research has advanced quite quickly, but the general philosophical audience still has rather outdated views of the medieval developments of philosophy in general and of logic in particular. At present, there is a need for presentations that bring the results of historical research to a wider audience. This book is intended to serve such a purpose, and accordingly it should also be suited to the needs of courses in the history of logic. The essays are independent, but they are organized in a way that should make their argumentation easy to follow. As the case often is in historical research, one of the major problems in our understanding of medieval logic derives from fundamental conceptual differences. Most modern logicians have understood their subject as something with close connections to mathematics. On the other hand, medieval scholars often thought that the account of an inference is best given against the framework of a disputation. Medieval university life was strongly dependent on dialectical practices. Academic argumentation and consequently, practically all intellectual reasoning was understood to take place in contexts where someone is trying to convince another person by presenting a sequence of sentences. Such a conception of logic

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was of course deeply embedded in the ancient tradition. Aristotle's Topics, for example, put logic in the context of an encounter between an opponent and a respondent. In this context, an inference became a structure by which the opponent can force the respondent to accept something because of what he has already granted to the opponent. The topics covered by the papers in this collection can be defmed with reference to three genres of the so-called logica moderna arising in the thirteenth century: obligationes, insolubilia and consequentiae. Part one of this volume is dedicated to obligationes and insolubilia, while part two concerns consequentiae. The third part provides three medieval texts in translation. The two first ones belong together and provide an early representative of the theory of obligationes. The last one is taken from a commentary on Aristotle's Prior Analytics, but can be classified into the genre of consequentiae because of its subject matter. The paper by Mikko Yrjonsuuri provides a general historical survey of the medieval theories of obligationes. Although the name of the genre of logic comes from the word obligatio (an obligation, or a duty), the issues discussed have little to do with deontic logic. More accurately, the genre can be described as a logical theory of a special kind of dialectical encounter similar to that discussed by Aristotle in his Topics. The name comes from the idea that in a disputation the respondent may be given special duties that he or she must follow during the disputation. The treatises on obligationes discuss the logical issues arising in such special disputations. At the focus of attention, we find the rules that the respondent must follow in his answers during the disputation. In his paper, Yrjonsuuri provides a systematic account of three main medieval versions of such rules (by Walter Burley, Richard Kilvington and Roger Swineshed), and gives some guidelines for the variety of interpretations that seem possible for disputations following these rules. In their paper, Henrik Lagerlund and Erik J. Olsson compare Walter Burley's theory of obligations with certain modem techniques of beliefrevision. This is not to say that Burley would have been aiming at the systems that were successfully construed by modem logicians. Rather, the comparison provides the modem reader with an intelligent way of looking at the logical structures employed in Burley's procedures. In essence, the problems encountered and tackled both by Burley's theory of obligationes and modem theories of belief revision concern the ways in which formal inferential techniques can be applied to epistemic contexts with the inherent aim of consistency.

PREFACE

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Lagerlund and Olsson have used Walter Burley's Treatise on obligations from 1302. Modem scholars have often taken it as the paradigm example of an obligational treatise. It indeed seems that the set of rules and practical tricks presented in Burley's text were rather widely taken as the starting point in the fourteenth century. Further, Burley's theory differs little in its essential features from the system presented in the early thirteenth century anonymous text translated in this volume. As Yrjonsuuri shows in his paper, Duns Scotus can be credited for a central generally accepted revision of the standard approach, and Richard Kilvington and Roger Swineshed provided two alternative approaches to obligations. Nevertheless, it seems that for the most part the central philosophical problems discussed in treatises on obligationes can be tackled with reference to Burley's text. The basic structure of obligational disputations resembled closely but not completely the way in which Aristotle described dialectical encounters in his Topics. This is of course no accident: Topics had a strong effect on the formation of medieval logic. Nevertheless, it seems equally clear that treatises on obligations developed certain themes of Aristotle's Topics in an original way not intended or thought about by Aristotle. These themes are further discussed by Yrjonsuuri in his paper, but let us here pay some attention to one specific development that seems to have taken logicians actually outside the theory of obligationes. It was connected to the Aristotelian idea that in all disputations the opponent aims at forcing the respondent to grant a contradiction. This may, of course, result from either of two mistakes. Either the respondent has taken an incoherent position from the beginning, or he defends his position badly. It seems that quite early in the development of the theory of obligationes, a third and even more problematic mistake was recognized. This was that the position from which the respondent starts might be paradoxical. If, for example, the respondent has as the positum "the positum is false," he will be led into rather similar inconsistencies as those encountered in the so-called liar's paradox. When the respondent is asked whether the positum is true or false, he cannot give either answer. Nonetheless, he may have to answer because of the general requirements of the game. In medieval parlance, these paradoxes were called insolubilia. Not all medieval solutions devised for them were dependent on the obligational or even disputational context. Nevertheless, even in such cases it pays to recognize the dialectical setting in which medieval logicians worked.

X

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In her paper, Fabienne Pironet looks at William Heytesbmy's ways of dealing with insolubilia. His solutions are strongly dependent on disputational and obligational techniques, and thus they provide a good vantage point from which to survey the ways in which the disputational setting is relevant to the paradox. The relation between ob/igationes and insolubi/ia is perhaps at its clearest in Heytesbury's text. Christopher J. Martin's paper takes the reader further down to the early stages of the medieval traditions of ob/igationes and insolubi/ia. The primary aim of his paper is to reconstruct the early histories of these two logical genres in a more comprehensive way. As Martin shows, the origins of the medieval discussions of the Liar may be found within the theory of ob/igationes. This, in tum, seems to come down from late ancient discussions located at the borderlines of possibility and conceptual imaginability. Thus, the theory of obligations seems to have been developed in order to treat problems connected with imaginability within disputational contexts. As Martin shows, early medieval authors developed many of their central logical concepts within such contexts. From his discussion of ob/igationes, we achieve a better grasp of how early medieval logicians dealt with concepts that have to do with how two or more statements stand together-that is, concepts like consistency, cotenability and compossibility. On the other hand, in Martin's discussion of insolubilia we can see many interesting ways in which the medieval conceptions of assertion (as distinct from mere utterance) were developed against a disputational background, and in a technical sense within the context of an obligational disputation. The general aim of the papers of the second part, dedicated to consequences, is to give the reader a grasp of the ways in which medieval logicians explicitly tackled problems arising from the theory of inference. On the one hand, the papers give a picture of the historical development in logic in the fourteenth century, which was the time when medieval logic was at its peak. On the other hand, the papers cover the field in a systematic sense: What is an inference? How is it related to conditionals? What makes an inference valid? What is the role played by logical form in inferences? Why did the medieval authors look at inferences especially from an epistemic perspective? Peter King takes up the distinction between conditionals and inferences. It has been claimed that medieval logicians confused the two, and thus their central concept of consequentia may be variously translated into English as conditional or as inference. King has looked at

PREFACE

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all the available texts from the crucial period 13 00-13 50, and argues that in these texts the confusion is very rare and always insignificant from the logical point of view. The important thing to come out of this discussion is an interesting picture of the proof theories in the period considered. According to King's conclusion, far from being confused with conditionals, inferences were seen as the heart of logic in the fourteenth century. Furthermore, King also rejects the idea that logic was exclusively understood as a discipline concerned with formal validity. As King sees it, as far as formal validity was considered, it was generally taken as one specific kind of validity, and medieval logicians thought that they must consider validity in general. Some recent studies have suggested that epistemic or psychological considerations were developed in the late Middle Ages to substitute for attention to the formal properties of inferences when evaluating their validity. If this is so, late medieval logic paved the way for Descartes' criticism of scholastic logic and his idea of deduction as a chain of clear and distinct intuitions. Ivan Bob's paper tackles this problem in a systematic fashion. His idea is to look closely at the epistemic, doxastic and disputational rules given in treatises on consequences in the post-Ockham period. While confirming the thesis that there was an interesting historical change in the ways of describing the idea of validity, Boh also challenges the main formulations of the thesis. Boh opposes the idea that there are psychological overtones in the ways in which late fourteenth-century authors defined the validity of inferences. As he sees it, the development went into a more mentalistic direction without being straightforwardly naturalistic in the psychologistic sense. According to Boh, such a mentalistic approach can already be seen in John Buridan, who was perhaps the most important logician of the early fourteenth century. He was looking at inference from a mentalistic viewpoint although it is clear that he was not in any interesting sense psychologistic in his discussion of the validity of an inference. Indeed, he relied quite heavily on the concept of logical form in his account of validity. Thus, the fourteenthcentury "mentalistic tum" ought not to be understood as something opposed to an approach based on formal considerations. Bob's investigations make it clear that the main representatives of medieval logic did not understand inference as obtaining between formulas, but rather between conceptual representations of what is the case. Stephen Read's paper tests an interesting hypothesis adopted by an anonymous author from the early fourteenth century, who is usually

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called Pseudo-Scotus. According to the hypothesis, the inferential analogue of the Liar paradox (an argument inferring from a single necessary premise that it itself is invalid) proves paradoxical to the socalled classical account of validity. Pseudo-Scotus thought that the paradox forced him to qualify his account of inferential validity: in his discussion we can, in fact, see many central features of his conception of validity. Thus, Read's discussion also provides a look at how PseudoScotus treated the concept of validity. This seems especially interesting if the reader keeps in mind that Pseudo-Scotus was one of the most elaborate late medieval logicians to lean on considerations of logical form in the definition of validity. In this sense, Read's discussion also sheds light on the debate treated by Boh. The text used by Read is included in Question 10 of Pseudo-Scotus' commentary on the Prior Analytics (Super librum primum et secundum Priorum Analyticorum Aristotelis quaestiones), and it is provided here as the third text of the Appendix. The text has traditionally been printed in collections of Duns Scotus' works, but it is now well known that he is not its author. For want of a better name, the author has been called Pseudo-Scotus. In his paper, Read discusses who this Pseudo-Scotus might have been and when he most probably wrote his commentary. He concludes by dating the treatise into approximately two decades after 1331, which provides a rather definite post quem. As for finding out the author's name, Read is more pessimistic than some other scholars: he rejects the view that Pseudo-Scotus would have been John of Cornwall and thus leaves us with no other name than Pseudo-Scotus. In any case, several modem commentators have discussed his questions of the Prior Analytics, and therefore they qualify as one central source for students of medieval logic. The two first texts of the Appendix occur together in the manuscript from which they originate. We know little about their author, and even the dating of them in the early thirteenth century is considerably less exact than is the case for Pseudo-Scotus. They have been known as the 'Emmeran' treatises because of their geographical origin since L. M. de Rijk edited them in Vivarium (vol. 12/1974 and vol. 13/1975). Together, these early treatises provide a simple but philosophically elaborated picture of the rules and practices of different obligational disputations. Yrjonsuuri and Martin discuss in their respective papers these texts in further detail. All three texts have been translated by Mikko Yrjonsuuri.

PART I OBLIGATIONS AND INSOLUBLES

MIKKO YRJONSUURI

DUTIES, RULES AND INTERPRETATIONS IN OBLIGATIONAL DISPUTATIONS

An obligational disputation, as it was known in the Middle Ages, consisted basically of a sequence of propositions put forward by one person, called the opponent, and evaluated by another person, called the respondent. In the most typical variations of the technique, the sequence would begin with a special proposition, called the positum. It was taken as the starting point, which the rest of the sequence would develop. The respondent had to accept the positum, if it was free from contradictions. Then he had to take into account in his later evaluations of the other propositions that he must at any time during the disputation grant the positum and anything following from it. The disputations were called obligational precisely because the respondent would admit to such a special duty or obligation to follow this procedure in his answers. 1 In this paper, my main aim is to give a concise account of the main versions of the rules given by medieval authors for these disputations, and to discuss some general issues concerning the interpretations that can be given to the philosophical content and idea of these disputations. However, in order to be able to look at the obligational disputations from a historically appropriate perspective, we must start with some remarks about how medieval authors located obligational disputations within the theory of disputations in general. 1. ARISTOTELIAN DISPUTATIONS The medieval authors cite two passages from Aristotle as giving the background of the obligational theory. One of them comes from the Topics, and one from Prior Analytics. Book VIII of the Topics describes a dialectical game, where some thesis is being questioned following a dialectical process. In Topics VIII, 4 (159a15-24) Aristotle summarizes the roles of the two players: 2

3

M. Yrjonsuuri (ed.), Medieval Formal Logic 3-34. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

4

MIKKO YRJONSUURI

The business of the questioner is so to develop the argument as to make the answerer utter the most implausible of the necessary consequences of his thesis; while that of the answerer is to make it appear that it is not he who is responsible for the impossibility or paradox, but only his thesis.

As this text shows, in the Aristotelian game the questioner tries to lead the answerer into embarrassment by developing arguments in opposition to the defended thesis. In the standard medieval Latin translations, Aristotle's questioner is called the opponens, and the answerer is called the respondens. For the thesis, the Latin word is positum. These words are also used in the technical texts on the obligational theory. In Prior Analytics I, 13 (32a 18-20) on the other hand, Aristotle defines his terms as follows: 3 I use the terms 'to be possible' and 'the possible' of that which is not necessary but, being assumed, results in nothing impossible.

Aristotle's idea that from the possible nothing impossible follows has played a major role in later discussions of possibility, even if his other condition, that the possible is not necessary, has often been dismissed. Here too, the terminological connection of the early Latin translations to obligational theory is evident. 4 As I see it, Aristotle's discussion of dialectical encounters in the Topics provides an important background for the understanding of disputations assumed in the obligational theory. On the other hand, the test of possibility suggested in the Prior Analytics seems to provide an interesting way of looking at the import of the modal concepts. From this viewpoint, the specific modal discussions that we find in the obligational theory seem to have a basis in Aristotle. For an Aristotelien background of the dialectical context, we have to turn to the Topics. Aristotle's aim in Topics VIII was to give detailed advice on how to behave in dialectical encounters having the form standard in Plato's academy. These dialectical games proceeded through yes/no -questions, which were selected and put forward by the opponent and answered by the respondent. The idea of the game was that the answers were to be used in an inferential manner. The respondent was defending a thesis, and the opponent aimed at building an argument to refute the thesis. 5 In Topics VIII, 5 Aristotle claims that no one has previously given any articulate rules on how to proceed in co-operative disputations for the purposes of inquiry. It is clear that Aristotle does not mean that he is the first to give rules for dialectical disputations in general; his point in the subsequent discussions is to develop a specific version of a standard

DUTIES, RULES AND INTERPRETATIONS

5

technique. The rules for dialectical disputations with the purpose of inquiry are mainly aimed at the respondent, and only hints are given about how the opponent should proceed. The basic idea of these rules is to follow and evaluate the steps of a process of building an argument against the thesis from premises which are more readily acceptable than the conclusion aiumed at. Aristotle employs the idea that dialectics proceeds from that which is better known towards that which is less well known. It must always be the case that the respondent grants only what is more acceptable than the conclusion aimed at; otherwise the disputation could not provide real support for its conclusion. Aristotle also enunciates the strange principle that if the opponent asks for something irrelevant to the argument being constructed, the respondent should grant it whether it is acceptable or not. He should just point out the status of the question in order to avoid appearing foolish. The idea behind such a rule seems to lie in the co-operative character of the game: the respondent should grant the opponent whatever he is asked, if it does not lead to difficulties in defense of the thesis. Anything external to the argumentation can therefore always be granted. Aristotle's rules show how the disputational game can be characterized rather as argument-seeking than directly truth-seeking. A co-operative game cannot be aimed at deciding whether the opponent can beat the respondent's defense, and therefore the aim cannot straightforwardly be to decide the truth-value of the thesis. Rather, the point of the game is in the search for the most interesting refutation of the disputed thesis. Nevertheless, the game is closely bound to actual reality, and the search for interesting arguments does not move freely in the logical space. Granting individual steps in the argumentation is dependent on acceptability in relation to the actual reality, and the idea of the argument is to support the actual acceptability of the conclusion. 2. DIALECTICAL DUTIES IN COMMENTARIES ON THE TOPICS The obligational technique as described in the beginning of my essay is presented in a relatively clear form in some early thirteenth century texts. For example, the Emmeran treatises translated in the appendix to this volume contain rather full-fledged discussions of the rules of the obligational disputations. However, the theory seems to be already at this stage quite close to the refined form that we find in, for example, Walter Burley's treatise from 1302. The early thirteenth century authors must have been working with an already exisiting tradition.

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Drawing a picture of the origins of the technique have proved difficult for the modems scholars. In his contribution below Christopher Martin concentrates on this issue (see also Martin 1990). Martin does not pay attention to Aristotle's Topics, although it may have influenced the obligational technique already at the origins. He refers to other ancient texts, such as Boethius's De hypotheticis syllogismis and Quomodo substantiae. Partly this is because Martin concentrates on the features of obligational disputations that are connected to issues in modal theory. It seems that early medieval theory of disputation developed with a strong connection to the Topics. 6 When we tum to the relatively developed treatises of the obligational technique from the early thirteenth century, such a disputational background is straightforwardly assumed rather than discussed. 7 The relation between obligations and the Topics seems not to have raised much interest. However, in an anonymous Parisian treatise on obligations from the second quarter of the thirteenth century (possibly by Nicholas of Paris) we find a relatively clear statement to the effect that obligations lean on the picture of disputations developed in the Topics but contain further elements not considered by Aristotle (Braakhuis 1998). It is not clear to modem scholars what the purpose of obligations disputations originally was. It seems that they were used at least for the purposes of exercise in the skills of logcial reasoning. Furthermore, modem scholars have recognized that one of the most interesting contexts where obligational technique occurs in the thirteenth century is theological. Already in the early thirteenth century "Emmeran treatise on impossible positio" (included in the Appendix) we find a theological motivation for the discussion of an impossible assumption. The anonymous author argues that because we can imagine God and man becoming one-which has actually happened in Christ-we can also imagine a man and a donkey becoming one, since humanity and donkeyhood are closer to each other than divinity and humanity are to each other. The author seems to think that the obligational technique is useful in discussing the logical import of theological doctrines (De Rijk 1974, 117-118). Towards the end of the Thirteenth Century, obligations are used, among other contexts, also in a straightforwardly theological discussion about the Trinitarian relations. In this context, obligational principles are referred to as an aid of understanding how we are to understand the divine persons. 8 Here we need not go into these theological discussions. For our interests here, it is enough to recognize the presence of the

DUTIES, RULES AND INTERPRETATIONS

7

obligational technique as a philosophical tool that was used by several authors in their search for a theologico-metaphysical problem. Obligational disputations cannot be taken as straight descendants of the Aristotelian dialectical technique as presented in the Topics. Nevertheless, it seems useful to look at their relations more closely. How was the Aristotelian theory of disputation used in the theory of obligations? In this respect, medieval commentaries on Topics might seem to be the most interesting source. However, it turns out that few of the earliest commentaries discuss book VIII of the Topics at all. Also, there is very little modem work on the tradition, and few texts have been critically edited. For these reasons, I have looked only at two easily accessible late thirteenth century commentaries, by Albert the Great and by Boethius de Dacia. Both connect ars obligatoria to some specific traits in Aristotle's work. 9 Let us look briefly at these two texts. Boethius de Dacia's Quaestiones super librum Topicorum introduces the obligational theory as a special technique to be used within the context of dialectical disputations proceeding along the lines discussed in Topics VIII. In general, Boethius de Dacia accepts Aristotle's model of dialectical disputation. However, he makes a clear distinction between sophistical or competitive disputations and dialectical or co-operative disputations. He assumes that the purpose of Aristotle's On Sophistical Refutations is to discuss the sophistical kind of disputation, while that of the Topics is to discuss only the co-operative kind. This makes much of Aristotle's advice in the first three books of the Topics problematic, because much of it is based on techniques of concealment and misguidance. How could such contentious means be applied in cooperative disputations? Again and again, Boethius de Dacia stresses that such means can indeed be used in some contexts. To explain their applicability he refers to disputations used as an exercise, where the opponent may use contentious methods in order to give the respondent exercise in the quick recognition of inferential relations even in less easily recognizable situations. (See esp. Boethius de Dacia 1976, 31 0-321.) It may be of some interest to notice that Aristotle's methods of concealment are discussed in a genuinely contentious sense in a family of thirteenth century treatises discussed and edited by L. M. De Rijk in Die mittelalterlichen Tractate De modo opponendi et respondendi (De Rijk 1980). In these treatises the advice given is clearly read as advice on how to win a disputation, by fair means or foul. While Boethius de Dacia's main problem is to show how co-operative disputants may use such

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contentious methods, these treatises develop further Aristotle's ideas on how to conceal the argumentation and how to mislead the respondent. Aristotle mentions in Topics (VIII, 4, 159a23-24) two kinds of mistakes that the respondent can make: For one may, no doubt, distinguish between the mistake of taking up a wrong thesis to start with, and that of not maintaining it properly, when once taken up.

Boethius de Dacia understands the poor maintenance of the thesis as a defense that allows the argument refuting the thesis to be built too easily. In such a case, the argument does not achieve full credibility, either because a problematic premise is granted or because a questionable step of inference is allowed. Here Boethius follows Aristotle quite faithfully. On the other hand, taking up a wrong thesis to be defended seems to be treated quite differently by Boethius de Dacia and Aristotle. Aristotle seems to have had in mind the competitive game and the problems of trying to defend an incredible thesis. Boethius de Dacia, for his part, seems to have been ready to admit even an impossible thesis (provided that it is not logically impossible), probably simply because Aristotle himself allows the possibility of an implausible thesis in the next chapter (VIII, 5, 159a38-159b2). According to Boethius de Dacia, anything that can give grounds for a good exercise, or some kind of truth-seeking, ought not to be called a wrong thesis. In many cases, it is clear that the issue in the disputation is not the truth of the thesis, but rather just the construction of the argumentation. Boethius de Dacia calls wrong only a thesis that does not provide any basis for an interesting disputation. Boethius de Dacia's example is the parity of the stars: there can be no interesting arguments to show either that the number of stars is even or that it is odd. (Boethius de Dacia 1976, 323-325.) In general, Boethius de Dacia seems to take a step away from reality in his discussion of Aristotle's rules. This step seems to show a crucial difference between Aristotelian dialectical encounters and obligational disputations. In an obligational disputation, the participants are not interested in real facts in such a straightf01ward way as in the Aristotelian dialectical encounters. However, before going on to the obligational disputations proper, let us take a slightly closer look at Boethius de Dacia's discussion of the duties of the respondent. In order to clarify further what is to be understood as poor defense of the thesis, Boethius de Dacia discusses how the respondent should act in the disputation. He gives a list of three requirements: 10

DUTIES, RULES AND INTERPRETATIONS [ 1]

[2]

[3]

9

A good respondent ought to be such that he grants to the opponent all that he would grant for himself thinking by himself, and [such that he] denies in the same way. He ought to be inclined from his inborn nature or from acquired habit to grant truths and deny falsities and he ought to love truth for its own sake. Third he ought to be aware that he should not be impudent, that is, to hold to some thesis for which he has no reasons and from which he cannot be turned away by any reason. Such a person, namely, cannot come to understand the truth.

There seems to be no doubt that a useful truth-seeking disputation is possible only if the respondent meets these requirements. The respondent must try to tell the truth, as is implied by [1] and [2]. If arguments do not affect the views of the respondent, as is required by [3], the dispute can make no progress in any interesting way. With obligational theory in mind, it is useful to see [1]-[3] as basic dialectical duties, to be prima facie followed in any dialectical encounter. In order to handle these duties more conveniently, let us formulate formal analogues for them. In this respect it seems suitable to join [1] and [2] into the analogous duty according to which if the respondent knows something to be true, he ought to grant it to his opponent, if he is asked to. Similar duties concerning denial and doubt can also be formulated. Formally these duties can be stated as follows: Ta

(p)((Krp & Rp) ~ OCp)

Tb

(p)((KrJJ&Rp)~ONp)

Tc

(p)(( _,K,p & _,KrJJ & Rp)

~

ODp)

(Ta is read: For any proposition p, if the respondent r knows (' K') it, and it is put forward ('R'), it must be granted ('OC'). 'N' stands for denying and 'D' for expressing doubt.) These duties can be characterized as the general duty to follow truth. However, it is important to recognize that they must be characterized as prima facie duties, which can be overridden by other duties in certain disputational settings. As it turns out, the whole theory of obligations is concentrated on situations where these duties are overridden to some extent by other special duties. Boethius de Dacia's requirement [3] is connected to the idea of supplying reasons for one's beliefs. It can technically be expressed as the idea that accepting p and the entailment o(p ~ _,q) should have an adverse effect on the defense of q. Accepting reasons for the opposite of

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the thesis ought to make the respondent grant the opposite of the thesis. With the theory of obligations in mind, it seems appropriate to generalize: the respondent ought to grant, if asked to, anything that he knows to be entailed by anything that he has already granted. Such a duty can be formalized as follows: E

(p)(q)((Cp & K,I:J(p

~

q) & Rq)

~ OCq

(Without the deontic operator 'Cp' is comfortably read in the perfect tense as 'p has been granted.') It is noteworthy that in these dialectical duties the uncertainty essentially connected with Aristotelian dialectical reasoning is almost lost. Aristotle's concept of acceptability is replaced by the concept of knowledge, which implies truth and allows no uncertainty. Generally, the respondent following Boethius de Dacia's rules for dialectical disputations is, nevertheless, less bound by truth than the one following Aristotle's rules. Boethius de Dacia places very clear emphasis on the duty to be consistent and gives less weight to the duty to follow truth (E has primacy over Ta-T c). Especially in disputations designed as exercises, the duty to seek the truth is almost completely overridden by the duty to defend the thesis as well as possible. But a parallel overriding of the duty to be consistent seems not to be allowed. As Boethius de Dacia points out, the respondent defending an impossible thesis may grant falsities and impossibilities, but he may not grant anything that is inconsistent (cf. Boethius de Dacia 1976, 328-329). According to Boethius de Dacia, the truth is especially to be forgotten in an obligatio, which he introduces as a special technique to be employed in dialectical disputations: 11 And with this you must know that in dialectical disputations, which are [undertaken] for inquiry into truth, or for exercise in easy invention of arguments for whatever proposition or in defense of the thesis, the art of obligations is often used.

Within the obligational technique developed by Boethius de Dacia after this introductory clause, the respondent is given the duty to grant false, even impossible propositions, provided only that they are consistent (compossibilia). These propositions are called the posita. The respondent is to grant whatever follows from the posita and deny whatever is repugnant to them, regardless of truth value. The respondent must also grant all propositions which neither follow from nor are repugnant to the posita, again regardless of truth value. Boethius's rules for obligational disputations were in certain respects different from those

DUTIES, RULES AND INTERPRETATIONS

11

of the majority of authors, but it seems unnecessary to go into details here. Rather, it suffices to point out that according to him all propositions in an obligational disputation are to be evaluated regardless of their truth values, solely on the basis of considerations of consistency. In addition to Boethius of Dacia, Albert the Great's commentary on Aristotle's Topics provides an interesting picture of the relation of the obligational technique to Aristotelian dialectical encounters. Generally, Albert's view seems to be similar to Boethius's, although as far as we can derive obligational rules from his remarks, they seem to be closer to what became the standard set in the fourteenth century than to Boethius's rules. The most interesting passage is that in which Albert gives a short description of the way in which the respondent should serve his thesis. He writes that 12 the way [of good defense] is to grant the consequences of the thesis (positum) and deny incompatible [sentences]; from this arises the technique ofpositio fa/sa.

Albert understands the thesis to be something to which the respondent is primarily committed. Consequently, he must grant whatever follows from it and deny whatever is incompatible with it. This is one of the general dialectical duties of the respondent, the duty of answering logically. Furthermore, Albert connects this art to positio fa/sa, which is a standard species of ars obligatoria. Soon after this remark, Albert goes on to point out that in some kinds of disputations the respondent always has to grant what seems to be true (ea quae videntur esse vera; Albertus Magnus 1890, 506). Although this general duty of answering truthfully is according to Albert important in all disputations where the primary aim is to find out the truth, it is not as widely applicable as the duty of answering logically, which is to be followed in all disputations of whatever kind. Albert's discussion is an interesting statement of the primacy of logic over truth in terms of what can be defended in a disputation. Violations of logic are much worse than violations of apparent truth. Albert introduces exercises as the most obvious case of a situation in which one needs to be logical but not truthful. The example is still valid, for textbooks of logic rarely stick to true sentences in their examples and exercises. The point of these exercises is not to discuss facts, but, as Albert says, to improve the students' skills (ut per[s]picaciores fiant; Albertus Magnus 1890, 506). I think that Albert's remark that the duty of answering logically is the origin of the technique of positio fa/sa is not accidental. Furthermore, it seems quite clear that Albert would not give much weight to the duty to

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answer truthfully in an obligational disputation. Ars obligatoria is a technique where the duty of answering logically is of foremost importance, and the duty of answering truthfully is pushed into a corner. This makes it a technique where the semantic interpretation of sentences is relatively unimportant, because the main issue is to study inferential and syntactic relations between sentences. 3. WALTER BURLEY'S RULES OF OBLIGATIONS Modem scholars have generally accepted that Walter Burley's Treatise on obligations (1302) can be treated as spelling out the standard form of the medieval theory of obligations. 13 This is not to say that the majority of medieval authors would have agreed with all the details of Burley's presentation. Neither can we say that Burley was very original in the composition of his treatise. 14 Rather, it seems that the basic structure of Burley's theory can serve as the paradigm against which different versions of the technique can be looked at. In the following I follow this practice. I first give a short presentation of Burley's theory. After this, I sketch the two most discussed alternatives to Burley's approach, which were presented by Richard Kilvington in his Sophismata (between 1321-1326) and Roger Swyneshed in his Obligationes (between 1330-1335). In Burley's presentation, there are altogether six classes of obligation: petitio, institutio, positio, depositio, dubitatio, and 'sit verum. ' 15 Of these six classes, Burley gives substantial attention only to institutio and positio. Petitio ('demand') is treated as a general way of imposing any obligation whatsoever on the respondent. Thus, in some cases the obligations given are not even dialectical in any interesting sense. (Green 1963, 41-45; for translation see Burley 1988, 373-378). Also, Burley's discussion of the class of obligations called 'sit verum' ('let it be true') shows that he did not think that any technical rules could be given for it. The idea in this class was that the respondent should behave is if he knew some sentence to be true (Green 1963, 94-96). As a class of obligations, it is interesting mainly in comparison to the paradigmatic class, which is positio. In this class, the respondent is given the duty to treat a certain proposition, called the positum, as true-that is, to grant it whenever put forward during the disputation. Nevertheless, in positio the respondent does not pretend to know that his positum is true. Rather, it is often made clear in the disputation that the positum is not in fact true, but only must be granted. The respondent may usually even grant the

DUTIES, RULES AND INTERPRETATIONS

13

sentence 'the positum is false.' He is not committed to the truth of the pas itum, but only to granting it in the disputation. 16 The central role of the class called positio is emphasized by the fact that two other species, depositio and dubitatio, can be derived from it. In depositio, the respondent must deny a certain sentence, called the depositum, and the resulting disputation can in its central features be treated as a mirror image of a disputation of the class of positio. 17 In dubitatio, the respondent has duty to doubt-that is, refrain from evaluating-a certain proposition, the dubitatum. The rules aim at guaranteeing that the respondent does not make any evaluation that would logically force him to make an evaluation of his dubitatum at a later stage of the disputation. 18 In Burley's treatment, institutio is a class of obligational disputation where some linguistic expression is given a new meaning. Then propositions containing this linguistic expression are evaluated in accordance with the new meaning. Thus, if 'A' is given the meaning 'a man is a donkey,' the respondent must deny 'A' if it is put forward as a proposition in the disputation. Burley uses the technique for the construction of semantic paradoxes. His main interest in the class seems to be in its usefulness for exploring how linguistic expressions can be used to signifY something. 19 The basic idea of the most typical obligational technique (positio) is that a certain sentence, which is given as the positum, must be maintained as true in the disputation. In essence, this idea amounts to constituting a limited exception to the general prima facie dialectical duty of following the truth. The positum must be granted in the disputation regardless of its truth value and it is indeed taken to be typically false. Burley explains separately whether it makes sense to give a true sentence as the positum. 20 Burley's formulation of the very basic first essential rule of obligational disputations in the class of positio is the following: 21 everything that is posited and put forward in the form of the positum during the time of the positio must be granted.

Burley's rule contains two crucial qualifications. The first qualification is that the positum must be put forward in the specific form in which it was originally given. Burley's examples of the rule show that this qualification is connected to the idea that the positum is the very sentence mentioned in the actual speech act giving the obligation for the respondent. If the positum is 'Marcus runs,' it does not follow that

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'Tullius runs' should be granted, even if Tullius is Marcus. 22 In addition to being posited, the sentence must be put forward during the time when the positing is valid, that is, during the obligational disputation proper. The rule can conveniently be formalized as follows: 23 R1

(p) ((Pp & Rp)

--t

OCp)

(To be read: for any sentence p, if it is the positum ('P') and it is put forward ('R'), it must be granted ('OC').) The quantifier (p) ranges over sentences, as Burley's first qualification requires. The sentential operator 'P' standing for "is the positum" must be understood so that it includes a reference to the technical time of the disputation. Being a positum is bound to a specific disputational exchange, outside of which the sentence cannot be treated as a positum. The sentential operator '0' is used as a deontic operator. The sentence 'OCp' states that there is a norm to the effect that p should be granted. In the next rules Burley spells out the obligation to remain consistent after accepting the usually false positum. Thus, while the first essential rule overrides the general duty of following the truth, these rules spell out the import of the general duty of answering logically without allowing any exceptions. The kind of consistency that Burley has in his mind in these rules requires them to guarantee that the accumulating set of answers remains consistent. Thus, at each step of the disputation, sentences possibly put forward at that step fall into three classes: those that follow, those that are repugnant, and others, technically called irrelevant. At each step, the respondent naturally has to grant any one of those sentences that follow. Burley decrees: 24 Everything that follows from the positum must be granted. Everything that follows from the positum either together with an already granted proposition (or propositions), or together with the opposite of a proposition (or the opposites of propositions) already correctly denied and known to be such, must be granted.

The two parts of the rule can be formalized as follows: R2a

R2 b

(p)(q) ((Pp & o(p --t q) & Rq) --t OCq) (p)(q)(r) ((Pp & Gq & o((p & q) --t r) & Rr)

--t

OCr)

(R 2a is to be read: For any sentences p and q, if p is the positum, and p entails q, and q is put forward, then q must be granted.) R2b introduces the sentential operator G, which is quite complicated in order to avoid even worse complications. 'Gq' states that q is a conjunction of sentences, which have either been granted or whose opposites have been denied earlier in the same disputation.

DUTIES, RULES AND INTERPRETATIONS

15

The rule for denying repugnant sentences is analogous: 25 Everything incompatible with the positum must be denied. Likewise, everything incompatible with the positum together with an already granted proposition (or propositions) or together with the opposite of a proposition (or the opposites of propositions) already correctly denied and known to be such, must be denied.

The formalization is as well analogous: R3a

R3 b

(p)(q) ((Pp & o(p---* _,q) & Rq)---* ONq) (p)(q)(r) ((Pp & Gq & o((p & q)---* _,r) & Rr)---* ONr

Here 'ON' stands for 'must be denied.' At each step of disputation, when a sentence is put forward by the opponent, it can be answered on the basis of rules R 1-R3 , if it is logically dependent on what has been maintained earlier in the disputation. For other sentences (called irrelevant), Burley gives the following rules: 26 If it is irrelevant, it must be responded to on the basis of its own quality; and this [means] on the basis of the quality it has relative to us. For example, if it is true [and] known to be true, it should be granted. If it is false [and] known to be false, it should be denied. If it is uncertain, one should respond by saying that one is in doubt.

Since irrelevant sentences cannot be evaluated by the previous rules, by the principle of keeping consistent, they are evaluated according to their actual truth value as far as it is known-this is the quality Burley has in mind. Thus in the case of irrelevant sentences the general principle of following the truth is followed. Formally the rule for irrelevant sentences can thus be represented as follows: R4a R4b R4c

(p) ((/p & Krp & Rp)---* OCp) (p) ((/p & Kr-p & Rp)---* ONp) (p) ((/p & _,Krp & --.Kr-p & Rp)---* ODp)

Here 'OD' stands for 'it must be doubted whether,' and 'Kr' for 'the respondent r knows that'. The epistemic conditions included in these rules for irrelevant sentences are less interesting than they may seem to be. It may seem that these rules allow a way in which the results of the disputation may depend on what the respondent knows about the world. However, the majority of the examples in Burley's treatise, as in other treatises on obligations, assume that knowledge of actual facts does not vary. The opponent can easily predict the correct answers, since the only examples where the respondent may show ignorance are cases where ignorance IS indubitable in the context (whether the king is seated, for example).

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Another feature of the interpretation Burley gives to the epistemic conditions which undermines their importance, is the idea that a doubtful answer has no consequences for the disputation. The respondent may not grant what he has previously denied, and he may not deny what he has previously granted. However, he may grant or deny what he has doubted previously. First doubting and then denying does not count as giving different answers to the same proposition. Thus a doubtful answer looks like refraining from a response where sufficient basis for an evaluation is missing. (See, e.g. Green 1963, 62; translation in Burley 1988, 397.) In the formalization of rules R2 and R3 I have consciously omitted the epistemic conditions given by Burley, although these conditions may be found interesting, if studied systematically. In his formulations of the rules Burley says that a sentence must be granted, if it is known to follow, and denied, if it is known to be repugnant. My reason for omitting these epistemic conditions is that it does not seem clear that Burley himself respects them in his applications of the rules. Often he shows that a sentence is relevant by showing that it follows from earlier granted sentences-without any explanation of whether the respondent knows it to follow. In later treatments of obligational theory these epistemic conditions are often simply omitted. 4. RICHARD KILVINGTON'S REVISION OF THE THEORY It is not clear whether Richard Kilvington's discussion of the obligational technique should be considered as a full-scale revision of the theory. Kilvington's remarks are included in his Sophismata (written somewhere between 13 21-1326); and in the context it is clear that his main aim is not to consider obligational theory .27 The main aim of sophisma 4 7, where the discussion is located, includes certain problems best characterized as issues within the field of epistemic logic. It seems that Kilvington thought that his method of solving the sophisma must conform to the principles of obligations, but the theory as he found it could not allow his sophisma to be solved. Thus, obligational principles should be altered. This seems to be Kilvington' s reason for going into a substantial discussion of certain rules of obligations. To begin his explicit discussion of obligations, Kilvington asks us to consider the following example (I have included answers according to Burley's rules):

DUTIES, RULES AND INTERPRETATIONS Dl Po Pr 1

You are in Rome 'You are in Rome' and 'you are a bishop' are similar in truth-value You are a bishop

17

Accepted, possible

Granted, true and irrelevant Granted, follows

Kilvington's example is connected to an idea that can be found from many obligations treatises. According to a principle frequently discussed it is possible to prove any falsehood compatible with the positum, if the positum is false. In this disputation the proposition Pr2 ("You are a bishop") is proved although it lacks any connection to the positum. Burley, for example, accepted the principle explicitly. The disputation is also connected to another principle, according to which the order of presentation may effect the evaluations of sentences put forward. Burley's rules of obligations would demand the respondent to deny Pr2 if it was put forward straight after the positum. At that point, it would not follow, and thus it should be judged irrelevant and false. Kilvington rejects this principle explicitly, and thus it can be assumed that he does not accept the reasoning. According to Kilvington, the answers of D 1 contain three mistakes; two of them interest us here (see S47, (q)-(bb)). First, Kilvington bluntly points out that the respondent should not grant Prl if he is not a bishop, since positing that he is in Rome should not bind him to accepting anything more than if he actually were in Rome (see S47, (q)). Second, if it were the case that 'you are a bishop' should be granted as Pr2 , the same should, according to Kilvington, already be granted at the first step, if it were put forward. Just as it follows at the step Pr2 from the positum Po and the granted sentence Pr~> at the first step it "follows from the positum and from something else that is true and irrelevant."28 This remark is based on a technical distinction between granted sentences and true sentences that have not been evaluated and are logically irrelevant to the positum. Kilvington states that according to those who hold the criticized view, a sentence following from something true and irrelevant together with the positum should be granted just as a sentence following from the positum and a granted sentence. At this point, Kilvington seems to be either misrepresenting the criticized theory, or criticizing a theory that is unknown to modem scholars. 29 Burley, for example, was always careful to maintain a distinction between answered and unanswered sentences.

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As I have already mentioned, Kilvington states, without providing good reasons for it, that the respondent should deny Pr 1 in D 1. Kilvington's explanation is, unfortunately, confusing. To support his view, he distinguishes two ways of using the word 'irrelevant.' The 'commonly assumed' way is that of Burley: a sentence is irrelevant if it neither follows from nor is repugnant to the positum together with previously granted sentences and opposites of previously denied sentences. The other way of using the term 'irrelevant' is such that it also refers to a sentence, which is true now and that would not be true in virtue of its being in fact as is signified by the positum. 30 By this phrase, Kilvington seems to mean sentences which neither follow from the positum nor are repugnant to it, but whose truth-values should be evaluated differently if the positum is taken as a counterfactual assumption. Kilvington thinks that such sentences should be denied. For example, Pr 1 in Dl is such a sentence, and according to standard obligational rules, it is irrelevant and true, but it must, as he claims, be denied. His point is that it does not follow that if a sentence is irrelevant and true in the common sense, it must be granted. Kilvington thus rejects the standard rule for irrelevant sentences. As I see it, Kilvington is not trying to give a new definition of irrelevant propositions. He is satisfied with the standard idea that a proposition is irrelevant if it neither follows from or is incompatible with the set of sentences determining relevance. Kilvington's concept of an irrelevant sentence covers the same class of sentences as Burley's concept. Both authors can use the same definition. Instead, it seems that Kilvington wants to change the way in which the truth values of irrelevant sentences are evaluated: their truth values are not to be read from actual reality, but with respect to the situation that would obtain if the positum were true. This would not change the concept, but the way in which the role of irrelevant propositions in obligational disputations is conceived. In fact, Kilvington's theory makes the answers of an obligational disputation reflect the counterfactual state of affairs that would obtain if the positum were true. Therefore, the alternative way of speaking about irrelevant propositions suggested by Kilvington concerns their evaluation in the disputation, not the extension of the term 'irrelevant'. In terms of the disputation Dl discussed above, Kilvington's revision would thus amount to the following. The sentence Pr 1 in D 1 is still irrelevant, as it lacks any logically necessary connection to the positum,

19

DUTIES, RULES AND INTERPRETATIONS

but we should not consider it as true, as it would not be true if the positum were true. This kind of interpretation of Kilvington's obscure remarks receives support from the fact that it makes rather easy to formulate Kilvingtonian rules for obligational disputations so that they are not altogether different from the standard rules. On my reading, Kilvington's remarks can be interpreted simply as a slight but significant revision of the traditional theory of obligations. From this viewpoint, it seems that he accepts the standard rules for the positum and for the sentences following straight from the positum or repugnant to the positum alone. Thus the following rules remain at the core of obligational theory: R1 R 2a

R3 a

(p) ((Pp & Rp) ---t OCp) (p)(q) ((Pp & o(p ---t q) & Rq) ---t OCq) (p)(q) ((Pp & o(p ---t _,q) & Rq) ---t ONq)

Kilvington rejects Burley's rules R2b and R 3 b. This rejection implies that a proposition will remain irrelevant at any step of the disputation if it is irrelevant at the first step. Kilvington's rules do not contain any reference to earlier answers in the disputation, and consequently there is no basis for giving a rule requiring attention to the order of propositions. Kilvington's rules for irrelevant propositions are central. As Burley, so also Kilvington evaluates irrelevant sentences according to their truth values. However, he uses different truth values. His idea seems to be that an irrelevant proposition is to be evaluated in accordance with its counterfactual truth value, as far as this is known, and not according to its actual truth value. Incorporating this idea, the rules for irrelevant sentences become the following: R4aK R4bK

~cK

(p)(q) ((Pp & lq & Kr(p o---tq) & Rq) ---t OCq) (p)(q) ((PaP & Iq & Kr(p o---t_,q) & Rq) ---t ONq) (p Xq) ((Pap & fq & -,K,.(p o---tq) & -,KrCF ~ -q) & Rq)

---t

ODq)

(Where 'p D---t q' is to be read as a subjunctive counterfactual conditional.) From the systematic viewpoint it is interesting to notice that, since any entailment is true as a subjunctive counterfactual conditional, the rules for relevant sentences are redundant (some assembly required, especially with the epistemic conditions). The whole of Kilvington's theory can thus be compressed to the rule that the counterfactual truth-values are to be followed, as far as they are known. Rule ~K is especially interesting. It is through this rule that Kilvington is saved from the inconveniences attributed to him by Paul

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Spade in his reconstruction of Kilvington's theory (Spade 1982, 27). Intuitively, it seems acceptable that Kilvington's rules are exhaustive and consistent, even if proof of these features is impossible as long as subjunctive counterfactual conditionals are not satisfactorily described. The consistency of these rules follows from the idea that all answers are related to one situation imagined on the ground of the positum. A description of one situation ought to be consistent. The exhaustiveness is achieved through R4cK• which allows a doubtful answer, if some details of the situation cannot be decided. On the basis of these rules Kilvington's treatment of the disputation D 1 becomes clear. Pr 1 must in that disputation be denied as irrelevant and false on the basis of R4hK. It is irrelevant since it neither necessarily follows from nor is repugnant with the positum. Furthermore, it is judged false because its negation follows with a subjunctive counterfactual from the positum. If the respondent were in Rome, 'you are in Rome' and 'you are a bishop' would not be similar in truth-value (unless the respondent actually is a bishop). Pr 2 is clearly irrelevant and false, and thus it too is denied on the basis of R4h K_ It may be remarked that the claim that Pr2 is sequentially relevant is based on Burley's rule R2b, which is rejected by Kilvington. 5. ROGER SWINESHED'S RESPONSIO NOVA In his treatise on obligations, written probably between 13 3 0-13 3 5, Roger Swineshed formulated a remarkably loose rule about conjunctive and disjunctive propositions: 31 Because the parts of a conjunction have been granted, the conjunction is not to be granted, nor because a disjunction has been granted is any part of the disjunction to be granted.

This rule has been much discussed both by medieval authors and by modem commentators. 32 The primary comment has been that this rule allows the respondent to grant inconsistent sets of sentences. It is indeed the case that through this rule Swineshed allows the respondent to grant both parts of a conjunction and actually deny the conjunction. According to Burley's rules, the only inconsistency allowed was among propositions, which potentially were to be granted at a single step of the disputation. As only one proposition can be answered at each step, inconsistencies within the set of actually given answers could not occur. 33

DUTIES, RULES AND INTERPRETATIONS

21

Swineshed himself accepts the conclusion that his rule makes the respondent grant inconsistencies. As he says: 34 The conclusion is to be granted that three repugnant propositions must be granted, and four and so forth.

However, some lines later he points oue 5 This is true, but, however, no contradictory repugnant to the positum is granted during the time of the obligation.

Swineshed's point seems to be that although the respondent's answers may include inconsistencies, the status of the positum is not to be questioned. Nothing inconsistent with it may be granted. This raises the question: how are the inconsistencies limited? It seems that many modern commentators have discontinued their work of interpretation as soon as they have identified the source of the inconsistencies in Swineshed's theory. But such an approach seems to underestimate the merits of the theory (especially as Swineshed himself recognizes the inconsistencies, and thinks that they can be limited). Let us therefore spell out the rules of this theory in detail, and try to defend it. Swineshed's point of departure is the standard conception of the positum as something that must be maintained during the disputation despite its falsity. He also agrees with other authors on the principle that anything following from the positum is also to be granted. It seems that the originalities in his theory derive from the following principle, implied by Topics, VIII, 5, but not (to my knowledge) pointed out by any other author writing on obligational theory: 36 Because of a lesser inconvenience (inconveniens) a major inconvenience is not to be granted.

It seems that Swineshed attacks

a principle accepted by Burley. According to the principle a false positum may lead the respondent into granting almost anything. This feature is due to rules R2 b and R3b, which dictate that anything following from the positum together with what has already been granted must be granted, and that anything repugnant to the positum, together with what has been granted must be denied. These rules allow those propositions, which have previously been judged irrelevant, to have an effect on what becomes relevant. This leads to the feature of Burley's theory that the order and selection of irrelevant propositions put forward has an effect to the answers. Furthermore, this leads to a situation in which, with a suitable selection of irrelevant propositions, anything compatible with the positum has to be granted.

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It seems that Swineshed's point in not allowing "a major inconvenience" to be granted is that the assumption given as the positum should not be unnecessarily widened. Swineshed simply rejects rules R2 b and R3b, and redefines the concept of an irrelevant proposition accordingly, to include all those propositions which neither follow from nor are repugnant to the positum alone. The set of rules thus becomes the following:

R1 R2a R3a ~a R4b

~c

(p) ((Pp & Rp) - t OCp) (p)(q) ((Pp & o(p - t q) & Rq) - t OCq) (p)(q) ((Pp & o(p - t _,q) & Rq) - t ONq) (p) ((Ip & Krp & Rp) - t OCp) (p) ((Ip & KrfJ & Rp) - t ONp) (p) ((Ip & -.Krp & -.KrfJ & Rp) - t ODp)

(Here 'Ip' applies to all propositions p, which are not covered by the rules RI> R2a or R3a.) Swineshed's rules are merely a simplification of Burley's rules. From the formal viewpoint it is noteworthy that rejecting R2 b and R 3b makes unnecessary the clumsy sentential operator 'G' (true of any conjunction of sentences which have been granted or whose opposites have been denied). All answers can in principle be determined without reference to earlier answers. Only the positum and the actual reality must be taken into account. In essence, these rules are the simplest and the most straightforward in the tradition of obligations. The respondent has a relatively easy task in answering: he has to keep in mind, in addition to logical principles, only the positum. No complicated connections between the positum and earlier answers need to be remembered, as they do in Burley's theory. Irrelevant and relevant sentences need not even retrospectively be connected to each other. Further, there is no need to consider counterfactual states of affairs in a subjunctive way as in Kilvington's theory. Irrelevant sentences are evaluated as the state of affairs actually is. The price of this simplicity in answering is that the intelligibility of the set of answers is achieved only through two-column bookkeeping. An imagined bookkeeper attending an obligational disputation following Swineshed's rules must separate relevant and irrelevant propositions into different columns. Considered as one set, the answers may easily turn out to be inconsistent, if the positum is false. While contradictions may occur between answers to relevant and irrelevant sentences, in both sets consistency must be maintained in standard cases. Thus one may point out that as anything following from the positum must be granted and anything repugnant to it must be denied, similar rules of reasoning can be

DUTIES, RULES AND INTERPRETATIONS

23

employed in the set of irrelevant sentences, as far as no change in the actual world is reflected in the answers. Historically it seems that Swineshed developed to a systematic end an important motive present in many texts related to obligational theory. In fact, Swineshed's rules for conjunctive and disjunctive propositions are not completely novel, but have their predecessors in several texts, including Walter Burley's treatise. It was already earlier quite explicitly recognized that an obligational disputation combines the distinct domains of assumption and fact. However, while the earlier rules of conjunctive propositions presented by, for example, Burley, concerned sentences that would have to be granted at a certain stage of the disputation, if put forward, only Swineshed allows the respondent to actually grant the parts of a conjunction but not the conjunction itself. In Burley's model, the domains of assumption and fact are combined into one coherent set of answers, but in Swineshed's model, they are kept explicitly separate. This makes Swineshed's rules so different that they deserve to be called the "responsio nova," as they indeed were in treatises of obligations written after Swineshed. 37 Swineshed's two-columnar model of obligational disputations did receive some support in the fourteenth century. At about the time of Swineshed's Obligationes, 38 Roger Rosetus attacks the standard rules of obligations along the same lines as Swineshed. The discussion is in an epistemic context in his commentary on the Sentences. 39 Robert Fland, writing sometime between 1335 and 1370, presents Swineshed's rules of positio as a responsio nova, as an alternative to the responsio antiqua, which is basically the model presented by Walter Burley (Spade 1980). Richard Lavenham seems to have accepted Swineshed's model unconditionally (Spade 1978). It is noteworthy that while the early fourteenth century treatises on obligations seem to have been related to Oxford, we do not know of any obligations treatise that would originate from Paris between approximately 1250 and 1350. From the mid-fourteenth century Paris we have the treatises by Albert of Saxony (Albertus de Saxonia 1975), William Buser of Heusden and Marsilius of Inghen. All three treatises reject Swineshed's responsio nova and favor rules more like Burley's rules. 40 Towards the end of the century, Paul of Venice (see Paul of Venice 1988), Paul of Pergula (see Paul of Pergula 1961 ), and John of Holland (see John of Holland 1985) advance a theory basically like Burley's. This seems to be true also for the fifteenth and sixteenth centuries.41 It appears that the authors took it to be important that all answers form

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one consistent set, which can be understood as a description of a situation. Swineshed's acceptance of inconsistencies between answers to relevant and irrelevant sentences was seen to be more problematic than the feature of Burley's theory that had provoked it; namely, that the respondent may be led into granting almost anything. 6. PROBLEMS OF INTERPRETATION In the Prior Analytics I, 13 (32al8-20), as quoted above, Aristotle puts forward the principle that from the possible nothing impossible follows as a short definition of possibility. He does not dwell on the idea. Nevertheless, the principle was well known in ancient and medieval theories of modality. It seems that it also provides a good vantage point from which to see many general issues in the interpretation of the obligational disputations. It is quite clear that obligational disputations provide a methodology by which propositions can be assumed in order to see whether something impossible follows. Through such a procedure, one can test in an obligational disputation whether a sentence is possible. When a positum has been laid down, the opponent aims at forcing the respondent to grant a contradiction. If he succeeds, either the positum has been shown to be inconsistent, or the respondent's defense inadequate. This seems, indeed, to be one of the main motivations behind the development of the theory of obligations, at least in the thirteenth century. There is one very interesting qualification that has to be made to this picture. While Aristotle's principle denies that anything impossible may follow, medieval scholars studying the theory of obligations soon noticed that certain kinds of impossibilities typically do follow from possible assumptions. This was, of course, due to the ways in which modal concepts were usually understood in the thirteenth century. Among the modal principles most often discussed in obligations treatises we find the so-called 'necessity of the present' -principle: omne quod est, necesse est, quando est. Within the theory of obligations, the idea has the consequence that whenever the positum is false, the respondent must connect it to some future instant of time. As a rule, this means that he must deny the presence of the present instant, if it is referred to by a proper name instead of the standard indexical expression (e.g., denying 'A est,' where A names the present instant). Authors writing on obligations soon noticed that the respondent has no convenient way of developing his answers, but can always be led into making answers that

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run counter to the 'necessity of the present' -principle. In effect, the respondent may have to grant sentences that are in this sense impossible whenever his positum is false, even when the actually false positum is clearly possible. This means, furthermore, a violation of Aristotle's principle in the Prior Analytics that from the possible nothing impossible follows. As is well known, Duns Scotus rejects the necessity of the present. In his discussion of modality, he also refers to obligations, and recommends leaving out the peculiar rule about instants of time. As Scotus points out, no other alterations follow if this rule is omitted. His suggestion seems to have been well received, since the rule in fact disappeared from obligations treatises. 42 On a more general view, the peculiar rule about instants of time seems to be but one example of the predominance of consistency over possibility as the aim that the respondent of an obligational disputation strives for. It seems that authors writing on obligations were already in the thirteenth century quite conscious of such a distinction between two kinds of possibility. The Latin word compossibile was used to refer to a more or less syntactic kind of modality based on non-contradiction. The simpler concept possibile seems to pick out a somewhat different kind of modality based on considerations of the powers of agents or on realization at some instant of time. 43 A particularly interesting group of texts in which this distinction seems to be especially visible are the thirteenth century treatises on the type of positio in which the positum is impossible. Within these texts, the authors readily admit that something impossible is admitted, so that the respondent will still have to refrain from granting a contradiction. For example, the early thirteenth century Emmeran treatise on impossible positio allows the respondent to accept anything imaginable as an impossible positum. He introduces the idea in a way that shows close affinity to the Aristotelian idea of assuming a possibility in order to see whether anything impossible follows. 44 Just as we say that something possible must be conceded in order to see what follows from it (quid inde sequitur), similarly we have it from Aristotle that something impossible must be conceded in order to see what happens then (quid inde accidat).

As the anonymous author develops the theory of the impossible positio, it becomes very clear that the impossibility of the positum does not allow the respondent to grant contradictory opposites. The author even states this idea as a general principle: 45

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Thus, one must note that no obligation should be accepted which forces the respondent to concede two contradictory opposites.

It seems that this principle can be taken as guiding all the different versions of the theory of obligations. Whatever else is taken to happen or to follow because of a false positum or some other kind of obligation, contradictories may not be accepted. Given that in all disputations some kind of deductive closure is intended, it appears that obligational disputations can in general be characterized as procedures for 46 constructing consistent sets of propositions. It seems impossible to spell out semantic interpretations for the consistent sets of sentences built up in obligational disputations. In an important sense, the set of sentences is constructed regardless of any interpretation, merely as a consistent set. However, if we look at the revisions of obligational theory proposed by Kilvington and Swineshed, it appears that fourteenth-century scholars wanted to develop the technique in a direction where the answers would form a more easily intelligible whole. This aim also seems to underlie the revision suggested by Duns Scotus to the peculiar rule about instants of time. After Scotus, there was no reason to connect the various answers to different instants of time. Such multiplicity would be quite impossible if the set of the answers were to be taken as forming a single semantic unity. In an obligational disputation that follows Walter Burley's rules, any semantic interpretation can be given only retrospectively. An imagined situation in which all the answers will be true cannot be identified during an ongoing disputation. It would always be possible to select further propositions, and to order them sequentially, in such a way as to ensure that some of the correct answers would be false for the imagined situation. This possibility is due to the feature of Burley's rules attacked by Richard Kilvington. According to Burley's rules, the order in which the propositions are put forward may make a difference to their evaluation. Both Richard Kilvington and Roger Swineshed seem to intentionally provide systems which provide for the possibility of a semantic interpretation right from the beginning. Kilvington's way of achieving this aim is to lean on subjunctive counterfactuals, while Swineshed leans on the idea of keeping evaluations of irrelevant propositions explicitly outside the part of the disputation based on the positum. In both cases, the positum determines a possible situation, so that its determination does not require the evaluation of later propositions. The subsequent disputation can then be taken as merely describing the assumed situation further.

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As I see it, the fact that the revisions of Kilvington and Swineshed were not successful has a specific implication for the semantic interpretation of obligational disputations. It seems that the medieval authors did not think that it would be a problem that an interpretation could be given only retrospectively. It thus seems that any interpretation of the answers of an obligational disputation must comply with such an approach. This seems to exclude the possibility of interpreting obligations in terms of subjunctive counterfactual reasoning, but leaves open many possible interpretations, from thought experiments to belief revision models. 47 University of Jyviiskylii

NOTES 1

There is no direct evidence that real obligational disputations would have been conducted strictly following the rules. However, the medieval university life contained many different kinds of disputations, and it seems reasonable to suppose that the rules of obligations regulated at least some of them. See Perreiah 1984 for a pragmatic interpretation of obligations and Weisheipl 1964 and Weisheipl 1966 for the role of obligations in medieval university curriculum at Oxford. 2 Translation is from Aristotle 1984, 268. 3 Boethius's translation (in Aristoteles latinus): "Dico autem contingere et contingens quod, cum non sit necessarium, ponatur autem esse, non erit propter hoc impossibile." English translation is from Aristotle 1984. 4 Cf. also 34a25: "falso posito et non impossibili et quod accidit propter positionem falsum erit et non impossibile." For a modern discussion of the dialectical method presented in the Topics, see, e.g., Momux 1968, Brunschwig 1985, 31-40, and Kakkuri-Knuuttila 1990. Ryle 1965 gives a general picture of dialectic in the Academy. 6 See, e.g., texts edited in De Rijk 1967, esp. pp. 148; 556-558. Further, see pp. 611-612 for a discussion that connects some elements of the theory of obligations rather straightforwardly to material found in the Topics. 7 De Rijk 1974, De Rijk, 1975. See also De Rijk 1976. 8 See esp. Henry of Ghent 1953, f. 92v; Godefroid de Fontaines 1914, 295 (Quodlibet VII, q. 4); Duns Scotus 1963, 135-138 (Lect. I, d. 11, q. 2, n. 23-28). For discussion, see Knuuttila 1997 and Yrjonsuuri 2000. 9 Boethius de Dacia 1976, Albertus Magnus, 1890. N. J. Green-Pedersen has looked at a large number of commentaries on the Topics and says that ars obligatoria is seldom referred to. See Green-Pedersen 1984, 388. I have discussed Boethius de Dacia in Yrjonsuuri 1993a and Albertus Magnus in Yrjonsuuri 1998. 10 " ... bene respondents debet esse talis, quod concedat opponenti omnia, quae concederet sibi ipsi secum cogitanti, et eodem modo negare. Debet ex naturali suo

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ingenio vel ex habitu acquisito esse aptus ad concedendwn verum et ad negandum falsum et debet diligere propter se verum. Debet tertia cavere, ne sit protervus, id est velle aliquam positionem, pro qua non habet rationem et a qua per nullam rationem potest removeri. Talis enim ad cognitionem veritatis non potest Bervenire." Boethius de Dacia 1976, 321. 1 "Et cum hoc debes scire, quod in disputatione dialectica, quae est ad inquisitionem veritatis vel ad exercitium in argumentis ad quodlibet propositum de facili inveniendis sive ad sustinendum positionem, saepe attenditur ars obligatoria . .. " Boethius de Dacia 1976, 329. 12 "Modus autem iste est, ut consequentia ad positionem concedat, et repugnantia neget: quod ex hoc oritur scientia falsae positionis." Albertus Magnus 1890, 505. 13 Critical edition in Green 1963; partial translation in Burley 1988. For discussion, see D'Ors 1990; Spade 1982a; Spade 1982b; Stump 1982; Stump 1989, 195-213. My presentation here leans on Yrjonsuuri 1994, 36-63. 14 There is an anonymous treatise that has sometimes been attributed to William of SheiWood (edition in Green 1963 ), and which has been argued to be written by Burley himself(see Spade and Stump 1983). Whether the treatise has been written by Burley or not, it seems clear that Burley's theory is not very much different from the early thirteenth century treatises (see De Rijk 1974, De Rijk 1975, Braakhuis 1998). It is also interesting to note that Ockham's discussion of obligations is rather similar to that of Burley (see Ockham 1974, 731-744. Richter 1990 argues that the author of the discussion (whether it is Ockham or not) has compiled the treatise straightfoiWardly from Burley's text. As I see it, there are some important doctrinal differences. See also Stump 1989, 251-269. 15 For a more detailed discussion of the division, see Yrjonsuuri 1994, 38-43. 16 For further discussion, see Yrjonsuuri 1993b. See Knuuttila and Yrjonsuuri 1988; Yrjonsuuri 1994, 152-158 and D'Ors 1990 for arguments on this issue directed at Stump 1982, 323-327 and Stump 1989, 382-383. 17 Green 1963, 84-89; translation Burley 1988, 404-408. 18 Green 1963, 89-94; translation Burley 1988, 409-412. 19 Green 1963, 35-41; translation Burley 1988, 371-373. 20 " ••• ut contra protervientes, qui aliquando verum negant scitum esse verum; non enim semper verum scitum esse verum habetur pro vero." Green 1963, 45; for translation, see Burley 1988, 378. 21 "Ornne positum, sub forma positi propositum, in tempore positionis, est concedendum." Green 1963,46. Translation in Burley 1988, 379. 22 "Et ponitur haec particula: sub forma positi propositum, quia si proponatur sub alia forma quam sub forma positi, non oportet quod concedatur. Ut si Marcus et Tullius sit nomina eiusdem, et ponatur Marcum currere, non oportet concedere Tullium currere." Green 1963, 46. Translation in Burley 1988, 379. 23 Formalizations of this kind were first developed in Knuuttila and Yrjonsuuri 1988. For the logical background of the formalizations see also von Wright 1963. 24 "Omne sequens ex posito est concedendum. Omne sequens ex posito cum concesso vel concessis, vel cum opposito bene negati vel oppositis bene negatorum, scitum esse tale, est concedendum." Green 1963, 48. Translation in Burley 1988, 381.

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2s "Omne repugnans posito est negandum. Similiter Omne repugnans posito cum concesso vel concessis, vel opposito bene negati vel oppositis bene negatorum, scitum esse tale, est negandum." Green 1963, 48. Translation in Burley 1988, 381. 26 "Si sit impertinens, respondendum est secundum sui qualitatem, ct hoc, secundum qualitatem quam habet ad nos. Ut, si sit verum, scitum esse verum, debet concedi. Si sit falsum, scitum esse falsum, debet negari. Si sit dubium, respondendum est dubie." Green 1963, 48. Translation in Burley 1988, 381. 27 Critical edition in Kilvington 1990a and an English translation in Kilvington 1990b. Instead of page numbers, I use the passage codes to be found both in the edition and in the translation. For the dating see the introduction in Kilvington 1990b. For discussion see D'Ors 1991a; Spade 1982a; Stump 1982. My discussion here leans on Yijonsuuri 1994, 102-144 and Yijonsuuri 1996. 28 " ••• sequitur ex posito et alio vero impertinenti." S47, (r). 29 The anonymous treatise edited and translated in Kretzmann and Stump 1985 contains the rule that Kilvington has in mind here. In general, the treatise seems to me, nevertheless, to come rather close to Kilvington's own approach. See Yijonsuuri 1994, 76-89; Spade 1993, 239-241; Ashworth 1993. 30 " •.• nunc est vera et quae non foret vera ex hoc quod ita foret a parte rei sicut significatur per positum." S47, (cc). 31 "Propter concessionem partium copulativae non est copulativa concedenda nee propter concessionem disjunctivae est ali qua pars ejus concedenda" Spade 1977, 257. 32 See Ashworth 1981; Ashworth 1996; D'Ors 1991b; Spade 1982a, Stump 1989, 215-249. My discussion here leans mainly on Yijonsuuri 1994, 89-101. 33 For Burley's principles to this effect, see Green 1963, 58: "Ad primum dicitur quod disiunctiva est concedenda ubi neutra pars est concedenda", and p. 59: "Ideo dico aliter quod copulativa est neganda, et non solum ratione positi nee solum ratione veri impertinentis, sed est neganda quia falsa et non sequens, et ideo est neganda ratione utriusque; neutra tamen pars est neganda primo loco." Note also that the issue comes up in Heytesbury 1988, 447-448. Stump has mistakenly read there a view like Swineshed's model. Ashworth 1993, 385-386 and Yrjonsuuri 1994, 138-142 have corrected the mistake and shown that Heytesbury's idea of obligational disputations is closer to Burley's approach than to Swineshed's model. 34 "Concedenda est conclusio quod tria repugnantia sunt concedenda et quattuor et sic deinceps." Spade 1977, 274. 3 s "Et hoc est verum dum tamen nullum contradictorium repugnans posito concedatur infra tempus obligationis." Spade 1977, 274. 36 "Propter minus inconveniens non est maius inconveniens concedendum." Spade 1977, 253. 37 Swineshed's two-columnar model can also be compared to the idea of elaborating the relations of two parallel disputations going on simultaneously. References to such situations can be found in earlier treatises. See, e.g., Green 1963, 49; translation Burley 1988, 382; Kilvington 1990a and 1990b, sophisma 47, passages G)--(n). A short systematic treatment of such a theme can be found in Richard of Campsall 1968, 227-229 and 237-238; see also Knuuttila 1993b. Swineshed's two-columnar model dividing relevant and irrelevant sentences into separate domains, if not disputations, is a rather natural development of this theme.

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38

Swineshed's Obligationes was written around 1330-1335 (Spade 1977, 246); Rosetus's commentary on the Sentences around 1332-1337 (Courtenay 1987, 109). 39 Rosetus concludes: "Et ideo ista regula est neganda: sequitur ex posito et bene concesso, ergo est concedendum, et multe alie regule que conceduntur ab aliquibus in obligationibus." Rosetus manuscript, 36v (q. 1, a. 3, a. 3). I am thankful to Olli Hallamaa for allowing me to see his edition in preparation. 40 See Braakhuis 1993, Kneepkens 1993 and Pozzi 1990. 41 For discussion of the later obligations treatises, see Ashworth 1985, Ashworth 1986, Ashworth 1992 and Ashworth 1993. 42 Duns Scotus 1963, 417-425; Ord. I, d. 38, pars 2. For discussion, see Knuuttila 1993a and Ytjonsuuri 1994, 64-75. 43 I have discussed this theme more thoroughly in Ytjonsuuri 1998. 44 "Sicuti enim nos dicimus quod possibile est concedendum ut videatur quid inde sequitur, similiter habemus ab Aristotile quod impossibile est concedendum ut videtur quid inde accidat." De Rijk 1974, 117; translation below, p. 000. Cf. also Aristotle's Prior Analytics, 34a25 as quoted above in footnote 3. 45 "Unde notandum quod nulla obligatio est recipienda que cogit respondentem concedere duo contradictorie opposita." De Rijk 1974, 118; translation below, p. 000. 46 Swineshed's theory can be characterized in this manner, if irrelevant propositions are understood as literally irrelevant and thus not included in the constructed consistent set of propositions. 47 For discussion of how to interpret obligations, see also Angelelli 1970; Ashworth 1981; Ashworth 1984; Brown 1966; King 1991; Knuuttila 1989; Knuuttila 1997; Perreiah 1984; Spade 1982a; Spade 1982b; Spade 1992, Spade 1993; Stump 1982; Stump 1989. Ashworth 1994 is a rather complete bibliography of both medieval obligations treatises and modern discussion of them.

REFERENCES Albertus de Saxonia 1975. Tractatus de obligationibus, (in Sophismata, Parisii, Denis Roce, 1502), Facsimile edition Hildesheim, New York, Georg Olms. Albertus Magnus 1890. Commentarii in Aristotelis Topiciis, (ed.) Borgnet, Opera Omnia, vol. II, Paris, 233-524. Angelelli, Ignacio, 1970. "The Techniques of Disputation in the History of Logic," Journal of Philosophy 67, 800-815. Aristotle 1984. The Complete Works of Aristotle, (Bollingen series 71(2)), Princeton, Princeton University Press. Ashworth, E. J., 1981. "The problems of Relevance and Order in Obligational Disputations: Some Late Fourteenth Century Views," Medioevo 7, 175-193. Ashworth, E. J., 1984. "Inconsistency and Paradox in Medieval Disputations: A Development of Some Hints in Ockham," Franciscan Studies 44, 129-139. Ashworth, E. J., 1985. "English Obligationes Texts after Roger Swyneshed: The Tracts beginning 'Obligatio est quaedam ars,"' in P. Osmund Lewry (ed.), The Rise of British Logic, (Papers in Medieval Studies 7), Toronto, Pontificial Institute of Medieval Studies, 309-333.

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Ashworth, E. J., 1986. "Renaissance Man as Logician: Josse Clichtove (1472-1543) on Disputations," History and Philosophy of Logic 7, 15-29. Ashworth, E. J., 1992. "The Ob/igationes of John Tarteys: Edition and Introduction," Documenti e studi sulla tradizione filosofica medievale, Ill, 2, 653-703. Ashworth, E. J., 1993. "Ralph Strode on Inconsistency in Obligational Disputations," in K. Jacobi (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns, (Studien und Texte zur Geisteggeschichte des Mittelaterns, Bd. 38), Leiden, E. J. Brill, 363-384. Ashworth, E. J., 1994. "Obligationes Treatises: A Catalogue of Manuscripts, Editions and Studies," Bulletin de philosophie medievale 36, 118-147. Ashworth, E. J., 1996. "Autour des Ob/igationes de Roger Swynneshed: Ia nova responsio," Les Etudes philosophiques 3, 341-360. Boethius de Dacia 1976. Quaestiones super librum Topicorum, (ed.) N.J. GreenPedersen and J. Pinborg, (Corpus Philosophorum Danicorum Medii Aevi, vol. 6), Copenhagen, Gad. Braakhuis, H. A. G., 1993. "Albert of Saxony's De obligationibus. Its place in the Development of Fourteenth Century Obligational Theory," in K. Jacobi (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns, (Studien und Texte zur Geisteggeschichte des Mittelaterns, Bd. 38), Leiden, E. J. Brill, 323-341. Braakhuis, H. A. G., 1998. "Obligations in Early Thirteenth Century Paris: The Ob/igationes ofNicholas of Paris(?)" Vivarium 36. Brown, Mary Anthony, 1966. "The Role of the Tractatus de obligationibus in Mediaeval Logic," Franciscan Studies 26, 26-55. Brunschwig, J., 1985. "Aristotle on Arguments without Winners or Losers," Wissenschaftskolleg, Jahrbuch 1984/1985, 31-40. Burley, Walter, 1988. Obligations (selections), trans!. N. Kretzmann and E. Stump, in The Cambridge Translations of Medieval Philosophical Texts: Volume One: Logic and the Philosophy of Language, Cambridge, Cambridge University Press, 369-412. Courtenay, William J., 1987. Schools & Scholars in Fourteenth Century England, Princeton, N. J., Princeton University Press. De Rijk, Lambertus Marie, 1967. Logica Modernorum: A contribution to the history of early terminist logic, vol. II, part 2, (Wijsgerire teksten en studies, 16), Assen, Van Gorcum. De Rijk, Lambertus Marie, 1974. "Some Thirteenth Century Tracts on the Game of Obligation I," Vivarium 12, 94-123. De Rijk, Lambertus Marie, 1975. "Some Thirteenth Century Tracts on the Game of Obligation II," Vivarium 13, 22-54. De Rijk, Lambertus Marie, 1976. "Some Thirteenth Century Tracts on the Game of Obligation III," Vivarium 14, 26-49. De Rijk, Lambertus Marie, 1980. Die mittelalterlichen Tractate De modo opponendi et respondendi, (Beitriige zur Geschichte der Philosophie und Theologie des Mittelalters, N. F., Bd. 17), Munster, Aschendorff. D'Ors, Angel, 1990. "On Stump's Interpretation of Burley's De ob/igationibus," in S. Knuuttila, R. Tyorinoja and S. Ebbesen (eds.), Knowledge and the Sciences

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in Medieval Philosophy, (Publications of the Luther-Agricola Society, B: 19), Helsinki, vol. II, 468-478. D'Ors, Angel, 1991a. "Tu scis regem sedere (Kilvington, S47[ 48])," Anuario Filos6fico 24, 49-74. D'Ors, Angel, 1991b. "Sobre las Obligationes de Richard Lavenham," Archives d'histoire doctrinale et litteraire du moyen age 58, 253-278. Duns Scotus, Johannes, 1963. Lectura in librum primum Sententiarum, in Opera Omnia, vol. 17, (ed.) C. Balic et al., Vatican, Vatican Scotistic Commission. Godefroid de Fontaines 1914. Quodlibets, vol. IV, (Les Philosophes Belges, Textes & Etudes), Louvain, Universite de Louvain. Green, Romuald, 1963. The Logical Treatise 'De obligationibus': An Introduction with Critical Texts of William of Sherwood (?) and Walter Burley, Ph. D. Thesis, Louvain. Green-Pedersen, Niels J0rgen, 1984. The Tradition of the Topics in the Middle Ages. The Commentaries on Aristotle's and Boethius' 'Topics, ' (Analytica), Miinchen, Philosophia Verlag. Henry of Ghent 1953. Summae questionum ordinariarum II (reprint of the 1520 edition), (Franciscan Institute Publications, Text Series no. 5), St Bonaventure, N.Y., The Franciscan Institute. Heytesbwy, William, 1988. The Verbs 'Know' and 'Doubt,' transl. N. Kretzmann and E. Stump, in The Cambridge Translations of Medieval Philosophical Texts: Volume One: Logic and the Philosophy of Language, Cambridge, Cambridge University Press, 435-475. John of Holland 1985. Four Tracts on Logic, (ed.) E. P. Bos, (Artistarium, vol. 5), Nijmegen, Ingenium Publishers. Kakkuri-Knuuttila, Matja-Liisa, 1990. "Dialogue Games in Aristotle," in M. Kusch and H. Schroder (eds.), Text-Interpretation-Argumentation, Hamburg, Buske, 221-272. Kilvington, Richard, 1990a. The Sophismata of Richard Kilvington, ed. N. Kretzmann and B. E. Kretzmann, (Auctores Britannici Medii Aevi, vol. XII), Oxford, British Academy, Oxford University Press. Kilvington, Richard, 1990b. The Sophismata of Richard Kilvington, introduction, translation and commentary by N. Kretzmann and B. E. Kretzmann, Cambridge, Cambridge University Press. King, Peter, 1991. "Mediaeval Thought-Experiments," in T. Horowitz and G. J. Massey (eds.), Thought-Experiments in Science and Philosophy, Savage, MD, Rowman and Littlefield, 43-64. Kneepkens, C. H., 1993. "Willem Buser of Heusden's Obligationes-Treatise 'Ob rogatum': Aressourcement in the Doctrine ofLogical Obligation?," inK. Jacobi (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln lwrrekten Folgerns, (Studien und Texte zur Geisteggeschichte des Mittelaterns, Bd. 38), Leiden, E. J. Brill, 343-362. Knuuttila, Simo, 1989. "Modalities in Obligational Disputations," in Atti del Convegno Internationale di Storia della Logica, Le teorie delle Modalita, Bologna, Clueb, 79-92. Knuuttila, S., 1993a. Modalities in Medieval Philosophy, London, Routledge.

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Knuuttila, Simo, 1993b. "Trinitarian Sophisms in Robert Holkot's Theology," in S. Read (ed.), Sophisms in Medieval Logic and Grammar, (Nijhoff International Philosophy Series 48), Dordrecht, Kluwer, 348-356. Knuuttila, Simo, 1997. "Positio impossibilis in Medieval Discussions of the Trinity," in C. Marmo (ed.), Vestigia, Imagines, Verba. Semiotics and Logic in Medieval Theological Texts, Turnhout, Brepols, 277-288. Knuuttila, S. and Yrjonsuuri, M., 1988. ''Norms and Action in Obligational Disputations," in 0. Pluta (ed.), Die Philosophie im 14. und 15. Jahrhundert, (Bochumer Studien zur Philosophie 10), Amsterdam, Griiner, 191-202. Kretzmann, Norman, and Stump, Eleonore, 1985. "The Anonymous De arte obligatoria in Merton College MS 306," in E. P. Bos (ed.), Medieval Semantics and Metaphysics. Studies dedicated to L. M. de Rijk on the occasion of his 60th birthday, (Artistarium Supplementa 2), Nijmegen, Ingenium Publishers, 239-280. Martin, Christopher John, 1990. "The Logic of the Nominates, or, The Rise and Fall oflmpossible Positio," Vivarium 28, 110-126. Moraux, P., 1968. "Lajoute dialectique d'apn!s le huitieme livre des Topiques," in G. E. L. Owen (ed.), Aristotle on Dialectic, Proceedings of the Third Symposium Aristotelicum, Oxford, Clarendon Press, 277-311. Ockham, William, 1974. Summa Logicae, (eds.) P. Boehner, G. Gal, S. Brown, Opera Philosophica, vol. I, St. Bonaventure, N.Y., The Franciscan Institute. Paul of Pergula 1961. Logica and Tractatus de Sensu Composito et Diviso, (ed.) Sister Mary Anthony Brown, St. Bonaventure, N.Y., The Franciscan Institute. 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), Oxford, Oxford University Press. Perreiah, Alan R., 1984. "Logic Examinations in Padua circa 1400," History of Education 13, 85-103. Pozzi, Lorenzo, 1990. La coerenza logica nella teoria medioevale delle obbligazzioni: Con l'edizione del trattato "Obligationes" di Guglielmo Buser, Parma, Edizioni Zara. Richard ofCampsall 1968. The Works of Richard ofCampsall, vol. I: Quaestiones super librum Priorum Analeticorum, (ed.) E. A. Synan, Toronto, Pontificial Institute of Medieval Studies. Richter, Vladimir, 1990. "Zu 'De obligationibus' in der Summa logicae," in W. Vossenkuhl & R. SchOnberger (eds.), Die Gegenwart Ockhams, VCH Verlagsgesellschaft. Ryle, Gilbert, 1965, "Dialectic in the Academy," in R. Bambrough (ed.), New Essays on Plato and Aristotle, London, Routledge & Kegan Paul. Spade, Paul V., 1977. "Roger Swyneshed's Obligationes: Edition and Comments," Archives d'histoire doctrinale et litteraire du moyen dge 44, 243-85. Spade, Paul V., 1978. "Richard Lavenham's Obligationes," Rivista critica di Storia della Filosofia 33, 225-242. Spade, Paul V., 1980. "Robert Fland's Obligationes: An Edition," Mediaeval Studies 42, 41-60.

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Spade, Paul V., 1982a. "1bree Theories of Obligationes: Burley, Kilvington and Swyneshed on Counterfactual Reasoning," History and Philosophy of Logic 3, 1-32. Spade, Paul V., 1982b. "Obligations: Developments in the fourteenth century," in N. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 335-341. Spade, Paul V., 1992. "If Obligationes were Counterfactuals," in Philosophical Topics20, 171-188. Spade, Paul V., 1993. "Opposing and Responding: a New Look at 'positio,"' Medioevo 19, 233-270. Spade, Paul V., and Stump, Eleonore, 1983. "Walter Burley and the Obligationes Attributed to William of Sherwood," History and Philosophy of Logic 4, 9-26. Stump, Eleonore, 1982. "Obligations: From the beginning to the Early Fourteenth Century," in N. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 315-334. Stump, Eleonore, 1989. Dialectic and its Place in the Development of Medieval Logic, London, Cornell University Press. Von Wright, Georg Henrik, 1963. Norm and Action, London, Routledge. Weisheipl, James A., 1964. "Curriculum of the Faculty of Arts at Oxford in the Early Fourteenth Century," Mediaeval Studies 26, 143-185. Weisheipl, James A., 1966. "Developments in the Arts Curriculum at Oxford in the Early Fourteenth Century," Mediaeval Studies 28, 151-175. Yrjonsuuri, Mikko, 1993a. "Aristotle's Topics and Medieval Obligational Disputations," Synthese 96, 59-82. Yrjonsuuri, Mikko, 1993b. "The Role of Casus in some Fourteenth Century Treatises on Sophismata and Obligations," in K. Jacobi (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns, (Studien und Texte zur Geisteggeschichte des Mittelaterns, Bd. 38), Leiden/NewYork/Koln, E. J. Brill, 301-321. Yrjonsuuri, Mikko, 1994. Obligationes: 14th Century Logic of Disputational Duties, (Acta Philosophica Fennica 55), Helsinki, Societas Philosophica Fennica. Yrjonsuuri, Mikko, 1996. "Obligations as Thought Experiments," in I. Angelelli and M. Cerezo (eds.), Studies in the History of Logic, Berlin, Walter de Gruyter, 79-96. Ytjonsuuri, Mikko, 1998. "The Compossibility of Impossibilities and Ars Obligatoria," History and Philosophy of Logic 19, 235-248. Yrjonsuuri, Mikko, 2000. "The trinity and positio impossibilis: Some remarks on inconsistence," in G. Holmstrom-Hintikka (ed.), Medieval Philosophy and Modern Times, (Synthese Library 288), Dordrecht, Kluwer, 59-68.

HENRIK LAGERLUND & ERIK J. OLSSON

DISPUTATION AND CHANGE OF BELIEF BURLEY'S THEORY OF OBLIGATIONES AS A THEORY OF BELIEF REVISION I. IN1RODUCTION As Paul V. Spade remarks "[t]here are many puzzles for historians of medieval logic" and "[o]ne of them concerns the peculiar form of disputation described in treatises de obligationibus ." (Spade 1992, 171.) In the present paper, we claim that the theory of obligationes as presented by Walter Burley in the section de positione of his Treatise on obligations can be seen as a theory of belief change. On the surface there are many structural parallels between Burley's theory of obligationes and the modern theory of belief revision. First of all, a disputation is a dynamic process driven by 'epistemic input' in the form of incoming sentences to be incorporated into a larger body of sentences. Secondly, that the first sentence (the positum) in a disputation should always be accepted corresponds to the so-called success postulate in belief revision theory. Third, the central goal in both frameworks is to avoid inconsistency, and, fourth, there is also an inherent conservativity or minimal change principle at work in both cases. The paper starts in Section 2 by introducing Burley's theory of obligational disputation. In that section we also present a complete formalization of this theory, a formalization which is faithful to the dynamic nature of a disputation. In Section 3 we deal with the problem of how to interpret Burley's theory. The standard formal theory of belief revision, the so-called AGM theory, is outlined in Section 4. In Section 5 we show how to construct a belief revision operation from a disputation. The belief revision interpretation is closely connected with Paul V. Spade's counterfactual interpretation. The exact nature of the connection is the topic of Section 6, where we consider the relation between revision and conditionals and where our account is compared to that of Spade.

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2. BURLEY ON OBLIGATIONAL DISPUTATION In the English philosopher of the thirteenth century Walter Burley's Treatise on obligation (Burley 1963) we find what was to become the standard view on the subject know as the study of obligationes or obligational disputation. 1 An obligational disputation involves two persons: an opponent and a respondent. The role of the opponent is to put forward sentences to the respondent, whereas the respondent should, for each sentence put forward by the opponent, decide upon the acceptability of that sentence in a way which guarantees that no inconsistency is introduced into her gradually increasing set of accepted sentences. What actually constitutes the core of Burley's theory of obligational disputation is a set of rules for how to start, proceed with and end the disputation. An obligational disputation starts with the opponent putting forward an initial sentence, the positum: Everything that is posited and put fotWard in the form of the positum during the 2 time of the positio must be granted.

According to this rule, the respondent has an obligation to accept the posited first sentence and consider it true throughout the disputation (hence the name obligationes of this form of disputation). In short: (B 1)

The first proposition put forward by the opponent, the positum, must be granted.

Let a be the positum. According to Burley's first rule, the initial set of sentences in the disputation should be D 0 = {a}. The sentences put forward by the opponent following the positum will be denoted

f3o,

f3~>

....

The next rule is formulated as follows: Everything that follows from the positum must be granted. Everything that follows from the positum either together with an already granted proposition (or propositions), or together with the opposite of a proposition (or the opposites of 3 propositions) already correctly denied and known to be such, must be granted.

This rule stipulates what sentences should be granted given an existing set of already granted sentences. Since the positum is among the granted sentences (by Burley's first rule), Burley could have simplified his rule as follows:

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Everything that follows from an already granted proposition (or propositions) or from the opposite of a proposition (or the opposites of propositions) already correctly denied and known to be such must be granted.

In fact, Burley's rule admits of further simplification, if we are allowed to make the reasonable assumption that to deny a proposition is the same as to grant its negation. 4 Furthermore, a useful idealization is to assume that the respondent is logically omniscient, i.e. capable of recognizing the logical consequences of what she believes. 5 Given these assumptions, Burley's second rule reduces to the following rule: (B2)

Everything proposed that follows from propositions must be granted.

already granted

Corresponding to this rule, there is a rule specifying what should be denied. Everything incompatible with the positum must be denied. Likewise, everything incompatible with the positum together with an already granted proposition (or propositions), or together with the opposite of a proposition (or the opposites of propositions) already correctly denied and known to be such, must be denied. 6

As before it seems safe to assume that to deny a sentence is the same as to grant its opposite. By the same kind of reasoning that led us to (B2), we can justify a simplification of Burley's third rule: (B3)

Every sentence proposed whose negation already granted propositions must be denied.

follows from

An equivalent formulation of (B3) is: the negation of every proposed sentence whose negation follows from already granted sentences must be granted. Given a sequence /30 , f3I> ... , the rules (B2) and (B3) are used to update the disputation set in the following way:

Di+l =Diu {f3i} if f3i follows from Di, and =Diu {-,f3J if -,f3i follows from Di.

Di+I

The second and third rules concern the case where the sentence proposed by the opponent is relevant to the disputation set in the sense that either the sentence itself or its negation follows from that set. In the remaining case of an irrelevant sentence, the opponent should respond in accordance with the following rule: Everything proposed is either relevant or irrelevant. If it is irrelevant, it must be responded to on the basis of its own quality; and this [means] on the basis of the quality it has relative to us. For example, if it is true [and] known to be true, it

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should be granted. If it is false [and] known to be false, it should be denied. If it is 7 uncertain, one should respond by saying that one is in doubt.

The rule for irrelevant sentences is more complicated than the other rules since it involves the respondent's background knowledge. An irrelevant proposition should, as Burley puts it, be evaluated "on the basis of the quality it has relative to us," i.e. relative to what the respondent thinks is true outside of the disputation. This is a clear statement that irrelevant sentences should be evaluated epistemically, as is the reference to "doubt" in the last sentence which also lends strong support to that interpretation. It is therefore slightly confusing that Burley also uses the expression "true [and] known to be true" indicating an objective criterion (truth) against which irrelevant sentences should be measured. But since the latter sentence is explicitly marked as an exemplification of the main subjective idea, it is reasonable to conjecture that Burley by "true [and] known to be true" meant just "believed and that for (epistemically) good reasons." On that interpretation, the first clause of the rule says that if a sentence is justifiably believed, then it should be granted, and the remaining two rules should be interpreted accordingly. 8 Hence, we end up with this interpretation of Burley's rule for irrelevant sentences: (B4)

If a sentence is irrelevant to the disputation but the respondent believes that it is true, then it should be granted; if, under the same condition, it is believed to be false it should be denied, and in the remaining case it should be doubted, i.e. neither accepted nor denied.

Our purely epistemic interpretation of the irrelevance condition is not uniquely supported by how Burley explains that rule, but it makes the theory we ascribe to Burley more coherent than any other alternative interpretation we are aware of, a claim to be substantiated as we proceed. We can now add the following rules for how to update the disputation set at stage i of the disputation: D; u {{3;} if neither {3; nor its negation follows from D;, but {3; follows from K, i.e. the respondent's background beliefs, D;+ 1 = D; u {-.{3;} if neither /3; nor its negation follows form D;, but -,{3; follows from K, and D;+I = D; if neither /3; nor its negation follows from D;, nor from K.

D;+I =

Let us combine the rules we have arrived at so far:

DISPUTATION AND CHANGE OF BELIEF (i) (ii)

(iii)

(iv)

39

Do= {a}, Di+l =Diu {f3i} if (a) f3i follows from Di> or (b) neither f3i nor its negation follows from Di, but f3i follows from K, Di+l =Diu {-.f3i} if (a) -.f3i follows from Di> or (b) neither f3i nor its negation follows form Di, but -,f3i follows from K, and Di+I = Di if neither f3i nor its negation follows from Di, nor fromK.

Unfortunately, these rules are, as they stand, not formally satisfactory from the point of view of modem sentential logic, since Di+l is, in fact, not well-defined: if f3i is irrelevant to Di but K is inconsistent, then both (ii) and (iii) apply, since everything follows from a contradiction, including f3i and -.f3i· To our knowledge, Burley did not explicitly comment on the case of the respondent entertaining inconsistent background beliefs, and it is not evident what he would have said about this case and its consequences for the course of the disputation. However, if we take rule (B4) at face value, it entails that both f3i and -.f3i should be added at step i+ 1 in this case, since these sentences both follow from K. Following this line, the amended version below represents one reasonable solution to this problem of interpretation: (i) (ii)

(iii)

(iv) (v)

D 0 ={a}, Di+I = Di u {f3i} if either (a) f3i follows from Di, or (b) neither f3i nor its negation follows from Di, but f3i follows from a consistent K, Di+I =Diu {-.f3J if K is consistent and either (a) -.f3i follows from Di> or (b) neither f3i nor its negation follows form Di, but -,f3i follows from a consistent K, and Di+I = Di u {f3i> -.f3J, if neither f3i nor its negation follows from Di, and K is inconsistent. Di+I = Di otherwise.

Clause (iv) handles the case of an inconsistent background set of belief. The only logical possibility not covered by the above rules for how to update the disputation set, Di, is the case when Di is inconsistent. Since

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everything follows from an inconsistent set, we may define D;+l to be the set of all sentences in this limit case. A very compact representation of a (Burley) disputation can be obtained if we enclose all relevant information about the disputation in a set-theoretical structure. This is done in the following final definition. Here we use a standard propositional language L and a corresponding derivability relation to obtain a complete formalization. 9 Definition: A Burley-disputation D is a quadruple D that (1 ) (2)

(3) ( 4)

=

such

a is a sentence (the pos itum), ::;A is a linear ordering on A = {{30 , f3t. ... , f3n, ... }

~ L (the sentences put forward by the opponent after the positum, in order of appearance)/ 0 K is a set of sentences (the background beliefs of the respondent), and D is a set of sentences such that D = uD;, where D; is defmed inductively as follows: (i) D 0 = {a}; (ii) if D; is inconsistent, then D;+l = L; else D;+l is defined by (iii)-(v): (iii) D;+ 1 = D; u {{3;} if either (a) D; ~ {3;, or (b) D; ~ {3;, D; ~ __,{3;, K ~ {3; and K is consistent; (iv) D;+ 1 = D; u {_,{3;} if either (a) D; ~ __,{3;, or (b) D; ~ {3;, D; ~ __,{3;, K ~ _,{3; and K is consistent; (v) D;+l = D; u {{3;, _,{3;} if D; ~A, D; ~_,{3; and K is inconsistent; (vi) D;+ 1 = D; otherwise.

The final disputation set, D above, which we shall call the outcome of the disputation, is the union of all disputation sets that are formed at some point in the disputation. 11 An example of a Burley-disputation might serve to make the idea behind Burley's construction more accessible: Example 1: Let a= 'Ronald Reagan (RR) is president of the US,' f3 = 'Nancy Reagan (NR) is married to Ronald Reagan,' y= 'Nancy Reagan is the first lady.' Let K = {-.a, {3, a&{3-7y, -,y} containing some expected beliefs about the presidency of the US at the time when this paper was

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written ( 1997). Consider the following disputation: D 1 = where A= {/311 /32 , /33 } and /3 1 = /3, /32 = a&f3--7yand /33 = y. Since a is the positum, it should be accepted, i.e. D 0 = {a}. The next sentence to be considered is /3. This sentence is independent of D0 , but it follows from K, so D 1 = {a, /3}. The next sentence is a&/3--?Y, which says that if RR is the president and NR is married to RR, then NR is the first lady, which is independent of D 1 but follows from the background beliefs represented by K. Consequently, D2 = {a, /3, a&f3--7y}. The last sentence to be considered is y, which follows from D 2 and is added for that reason. Since there are no more sentences to be evaluated, D = D3 = {a, /3, a&/3--?Y, y}. According to this set of propositions, Reagan is the president, Nancy his wife and, being married to the president, also the first lady. We have arrived at a precise formalization of Burley's abstract rules. But the most important question remains to be answered: what purpose were these rules designed to serve? This is the question to which we now tum. 3. THE PROBLEM OF INTERPRETING OBLIGATIONAL DISPUTATION There are two fundamentally different ways to interpret the theory of obligational disputation. One might hold that the theory really describes some kind of disputation involving two actual participants, but it is not easy to see what purpose such a disputation could possibly have served, and, moreover, there is no historical evidence of real disputations of this kind having ever been conducted. For these reasons, several authors have argued that the disputational setting is just a 'convenient fiction,' i.e. that the disputation form is used to illuminate some other phenomenon/ 2 although the opinions diverge as regards the exact nature of this phenomenon. We shall in the following adopt a version of the convenient fiction interpretation and argue that the 'opponent' is indeed merely a fictitious entity. The convenient fiction interpretation is not without problems of its own, one having to do with the order of the sentences put forward by the fictitious opponent. According to Burley, there are rules that do not constitute the practice of the art of obligational disputation but are merely useful. One such rule is that "[ o]ne must pay special attention to the order [of the propositions ]" 13 • It is exemplified as follows by Burley:

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HENRIK LAGERLUND & ERIK J. OLSSON

[S]uppose it is the case that Socrates and Plato are black, and let it be posited that Socrates is white. Then if 'Socrates and Plato are alike' is proposed in first place, it must be granted, because it is true and irrelevant. And if 'Plato is white' were proposed after this, it would have to be granted, because it follows. If, however, 'Plato is white' were proposed in ftrst place, it would have to be denied, because it is false and irrelevant. If 'Socrates and Plato are alike' were proposed after this, it 14 would have to be denied because it is incompatible.

Burley here describes two disputations differing only in the order in which the sentences were proposed, yet leading to radically different outcomes. Curiously, as Spade notices, Burley seems to be giving advice rather than merely stating a fact about the behaviour of his rules. 15 One is under the impression that there is, according to Burley, one correct ordering and that one has to be careful not to deviate from that ordering, which raises the question which ordering Burley has in mind. In our view, a reasonable interpretation of Burley's theory has to provide some kind of answer to that question. Note also how difficult it is to make sense of Burley's advice that one must pay particular attention to the order if one adopts a non-fictional interpretation of Burley's theory. On such an interpretation the respondent just has to face the ordering selected by the opponent, and Burley's advice would seem pointless. For another example of the effects of varying the order, an example that makes use of sentential logic only, compare the following disputation to that of Example 1 above: 16

Example 2: Let a, /3, yand K be as in Example 1. Let D 2 = where A = {f3J. /32 , /33 } and /3 1 = y, /32 = f3 and /33 = a&f3~y. This disputation differs from that of Example 1 only in that here the sentence y is considered first, not last. Since a is the positum, it should be accepted, i.e. D 0 = {a}. The next sentence to be considered is now y which says that Nancy Reagan is the first lady. This sentence does not follow from D 0 , but its negation follows from K, so D 1 = {a, -,y}. The next sentence is /3, which is accepted since it follows from K so that D 2 = {a, -,y, /3}. Finally, the negation of a&f3~y is accepted since it follows from D 2 • In this case, the outcome of the disputation is D 3 = {a, f3, -,y, -,( a&f3~y)}. Notice that this set is quite different from the outcome of the disputation in Example 1. The outcome is that Ronald Reagan is the president and Nancy, though not the first lady, is Ronald's wife.

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Notice the slightly odd outcome of Example 2, a feature we will return to in Section 5. The problem with the ordering, its nature and origin, is a genuine problem for the convenient fiction interpretation, including our own interpretation. 17 We will argue that Burley is really trying to provide a theory of how an agent, the respondent in the disputation, should revise her beliefs in the light of the new information represented by the positum. The idea is quite evident from Example 1 and 2; the two different results of these disputations can be interpreted as two possible cognitive responses to the new information that, contrary to what we believed, Reagan, and not Clinton, is the president of the United States. It will prove useful to refer. to the modern discussion on the subject of belief revision, and in the next section we outline the relevant aspects of contemporary theories of belief revision, concentrating on what has become the standard theory: the AGM theory of Alchourr6n, Gardenfors and Makinson. 4. THE AGM THEORY OF BELIEF REVISION In the AGM theory a belief state is represented as a logically closed sets of sentences (called a belief set). 18 There are three principal types of belief change: expansion, revision and contraction. In expansion, a new belief is added without any old belief being given up. In revision, the new information is added in a way that preserves consistency. Even if the new information is inconsistent with the original belief set, revision guarantees that the new belief set is consistent (provided that the new information is non-contradictory). Finally, to contract a belief means to remove it from the belief set. Expansion of a belief set K by a sentence a, denoted K +a, is the simplest of the three operations and is defined as the logical closure of the union of K and {a}. Hence, K +a = Cn(K u {a}). Closing under logical consequence ensures that the result of expansion is a new belief set. According to the AGM trio, a reasonable revision operation should at least satisfy the following so-called basic revision postulates: (K*l) (K*2) (K*3) (K*4) (K*5) (K*6)

K*a= Cn(K*a). aE K*a. K*a~ Cn(Ku {a}). If -.a~ K, then Cn(K u {a}) ~ K* a. K* a= Kj_ if and only if 1- -,a. If 1- a H [3, then K* a = K* [3.

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The first postulate says that the result of revision should be a logically closed set. According to the second postulate, the sentence a should be believed after the revision of K by a. This postulate is known as the success postulate. The postulates (K*3) and (K *4) together express, essentially, that the revision of K by a be identified with the expansion of K by a if a is consistent with K. The meaning of (K*5) is that K*a is consistent unless a is contradictory. Finally, (K*6) says that logically equivalent input sentences should give rise to identical revised states. According to Gardenfors, the purpose of the revision postulates is to capture the intuition that revisions should be, in a sense, minimal changes so that information is not lost, or gained, without compelling reasons. Gardenfors is here appealing to a principle of informational economy. As he puts it, "the main trust of the criterion of informational economy is that the revision of a belief set not be greater than what is necessary in order to accept the epistemic input." 19 Of the six basic postulates, only (K*3) and (K*4) seem directly related to the principle of minimal change. According to (K*3), K*a must not contain more information than what is included in Cn(K u {a}), whereas (K *4) stipulates that, in the case when -,a is not an element of K, K*a must not contain less information than that found in Cn(K u {a}). Clearly, these postulates place but very weak constraints on the principle of minimal change, constraints that far from exhaust the full presystematic meaning of that principle. This holds in particular when the new information contradicts the background beliefs, in which case only (K*3) is applicable of these two postulates. The revision postulates impose constraints on reasonable operations of revision, but they do not suggest how specific examples of such operations can be constructed. An interesting problem is how to construct operations that satisfy these constraints. It is commonly assumed that the revision of K by a can be reduced to first removing the negation of a from K (i.e. contracting by -.a) and then adding a (i.e. expanding by a). This procedure is given a precise representation in the Levi-identity (here + and * denote contraction and revision, respectively): K*a = (K +-.a)+ a (Levi identity).

Given the Levi identity, we can define revision in terms of contraction and expansion. Since expansion is trivial, the Levi identity reduces the problem how to construct a revision operation to the problem how to construct a

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contraction operation. Gardenfors suggested that a contraction operation be constructed on the basis of an ordering ::;; of epistemic entrenchment between sentences. That a sentence a is at least as entrenched in the agent's belief system as the sentence f3 is expressed by writing f3::;; a. According to Gardenfors, "[t]he fundamental criterion for determining the epistemic entrenchment of a sentence is how useful it is in inquiry and deliberation." 20 Moreover, "certain pieces of our knowledge and beliefs about the world are more important than others when planning future actions, conducting scientific investigations, or reasoning in general." As an example, Gardenfors notes that the combining weights is more important in today's chemistry than facts about the color or taste of substances. The idea here is that we should not represent a state of belief simply as a belief set but as an ordered pair of a belief set together with an ordering of epistemic entrenchment. He goes on to show that it is possible to construct a well-behaved contraction functions from a relation of epistemic entrenchment, provided that the relation satisfies some structural requirements. For instance, one of these requirements says that non-beliefs should be less entrenched than beliefs. The contraction function so constructed can then be used to define a revision operation via the Levi identity. Currently, the field of belief revision is growing rapidly, and it has attracted attention from computer scientist and logicians as well as philosophers. Although AGM is the standard theory in the area, several alternative approaches have emerged. One of the main options is to give up the requirement of logical closure and to concentrate instead on the revision ofnon-closed sets, or beliefbases. 21 5. FROM DISPUTATION TO REVISION Given a set K of background beliefs and a linear ordering ::;A on the set A of sentences put forward by the opponent, we can use the following definitorial idea (to be refined below): * is an operation of disputational revision for K and ::;A if and only if is a Burleydisputation. The definition says that we can construct the revision of K by a, for any sentence a, by carrying out an (imaginary) obligational disputation in which a is the positum and K the background beliefs of the respondent. The outcome of the disputation represents the new set of beliefs. A central principle behind the AGM theory is the principle of minimal change, which prescribes that changes of belief be maximally

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conservative. For Burley's theory to be a theory of belief revision, it must adhere to the principle of minimal change. However, our preliminary definition of disputational revision is too uncommited when it comes to the set of sentences to be evaluated (the opponent sentences); as it stands it represent too liberal an approach in the light of the principle of minimal change. It is compatible with that approach that only sentences not logically related to the background beliefs are considered, something which normally is sufficient to ensure that the outcome is non-conservative. What sentences, then, should be considered? It seems reasonable to give the respondent the opportunity to reconsider all previous beliefs in the light of the new information represented by the positum. Indeed, once we adopt an idealised picture, nothing prevents us from considering not only all previous beliefs but all sentences of the whole language. By considering all sentences we can build a new revised state of belief that is as complete as possible. In our formalism this means that the variable set A used in the preliminary definition of disputational revision should be replaced by the constant L, the complete background language, leading us to the final definition of disputational revision: Definition: Let K be a set and ~ = ~L a linear ordering of L. * is an operation of disputational revision for K and ~ if and only if
The AGM postulates were designed to capture the meaning of "minimal change," and it is therefore an interesting question what basic AGM postulates hold true under the disputational interpretation of revision. We have verified the following formal properties (see the appendix for proofs): (K*l.1) K*a= Cn(K*a), if K r a. (K*2) aE K*a. (K*3) K*ac;; Cn(Ku {a}). (K *4.1) If K Y-.a, then Cn(K) u Cn( {a}) c;; K*a. (K *5.1) If K is consistent, then K* a is inconsistent if and only iff- -,a. (K*6) Iff- a~ f3, thenK*a=K*f3. We notice that (K*2), (K*3) and (K*6) can be directly transferred from AGM revision to disputational revision, while, in the case of the other AGM postulates, some modification is necessary. (K*l.1) says that the outcome of disputational revision is a logically closed set in the special

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case that the new information can be derived from the background beliefs. The property (K *4.1) amounts to a weakening of the fourth AGM-postulate: in the case in which -.a does not follow from the background beliefs K* a must not contain less information than what can be derived either from K or from a. The statement (K *5 .1) expresses the sense in which disputational revision is 'consistent'; it says that, provided that the background beliefs are consistent, the consistency of the outcome depends entirely on the consistency of the input. This property is related to the issue of logical omniscience. If the respondent is logically omniscient in the sense of being a perfect deducer, capable of drawing all the correct conclusions (and no incorrect ones) of any given set of premises, then consistency is guaranteed, provided that the input is consistent and there is no contradiction among the respondent's prior beliefs. Note that, unlike AGM, we do not assume that the respondent/agent is logically omniscient in the sense that the beliefs she entertains at any given time are closed under logical consequence. The set K in (K * 1.1 )-(K *6) may, but need not, be logically closed. The fact that disputational revision satisfies (K*3) and (K*4.1) means that this form of revision adheres to a weak form of conservatism. We will however argue that in general disputational revision does not guarantee full conservatism in the absence of further constraints imposed on the ordering. The crucial observation in this connection was made already by Spade who noted that, "[i]n general, a given disputation is more 'conservative' about the sentences proposed earlier than it is about the sentences proposed later; the former stand a greater chance of being irrelevant and so judged according to the respondent's knowledge of the facts." 22 In fact, a mere difference in order can lead to a radical difference in perceived degree of conservatism, as is well illustrated by Example 1 and 2. Recall that in both examples the disputation was used to construct a revised state of belief given the new, belief-contravening, information that Reagan is the president of the US. The disputation in Example 1, in which beliefs were considered before non-beliefs, led to the revised state that Reagan is the president and Nancy his wife and, being the wife of the president, also the first lady. The second disputation (Example 2), in which a non-belief ( y = 'NR is the first lady') was considered before beliefs, gave rise to the more surprising or, equivalently, less conservative result that Reagan is the president and that Nancy is his wife but, surprisingly, not the first lady. The awkwardness of the outcome of the second disputation is due to the fact

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that it, unlike the outcome of the first disputation, is incompatible with the firmly rooted definitorial principle that anyone married to the (male) president of the US is the first lady_2 3 The effect of Burley's rule for irrelevant sentences on the outcome of the disputation lends considerable plausibility to the hypothesis that the purpose of that rule was to introduce a degree of conservatism into the model, but it is also important to note that the rule does not by itself guarantee a conservative result. What is also crucial is that the sentences put forward arrive in the right order. As Space observes, the sooner a sentence is evaluated in a given disputation, the more likely it is that it will be evaluated relative to the background beliefs rather than relative to the set of sentence established so far in the disputation. A consequence is that for as much as possible of the body of background beliefs to be preserved, in accordance with the principle of minimal change, it is necessary that beliefs be evaluated before non-beliefs. Not only must beliefs be considered before non-beliefs in order to guarantee conservatism; more important ('entrenched') beliefs must be evaluated before less important beliefs for the same reason. 24 Whether or not a change of belief is minimal or not is not simply a matter of counting how many beliefs were preserved; it is also imperative that as many as possible of the important, deeply entrenched beliefs are not lost. To sum up, full-blown epistemic conservatism requires the following constraint on the ordering: (P 1)

The ordering :::; = :s;L used in a disputation D = must order all sentences of L according to their relative entrenchment in K.

Surprisingly, our belief revision interpretation led us to a rather plausible solution to two of the most puzzling facts about Burley's theory: 1) what sentences to evaluate, and 2) in what order. The solution is so simple that it is no wonder Burley did not think it necessary to expand on the subject: all sentences should be evaluated (or if time is an issue, as many as possible), and they should be considered in order of epistemic importance. We think that the far-reaching structural agreements between Burley's theory and the AGM theory lends considerable support to our belief revision interpretation of the former. A remaining question is to what extent Burely's theory is also a theory of counterfactuals. As we shall try to show in the next section, that question too can be given a satisfactory answer.

DISPUTATION AND CHANGE OF BELIEF

49

6. FROM REVISION TO COUNTERFACTUALS There is a close connection between, on the one hand, belief revision and, on the other hand, acceptability criteria for counterfactual conditionals. The connection is provided by the so-called Ramsey test, which is described as follows by Gardenfors: In order to find out whether a conditional sentence is acceptable in a given state of belief, one first adds the antecedent of the conditional hypothetically to the given stock of beliefs. Second, if the antecedent together with the formerly accepted sentences leads to a contradiction, then one makes some adjustments, which are as small as possible without modifying the hypothetical belief in the antecedent, such that consistency is maintained. Finally, one considers whether or not the 25 consequent of the conditional is then accepted in this adjusted state ofbelief.

A more compact formulation of the idea is: Accept a sentence of the form 'If a, then /3' in a state of belief K if and only if the minimal change of K needed to accept a also requires accepting /3. It is clear that the Ramsey test presupposes an account of belief revision. The phrase "the minimal change of K needed to accept ci' denotes exactly the revision of K by a. If we use 'o--/ for the 'would' -conditional, then we can express the Ramsey test as follows: (RT)

a o--+ f3 is acceptable relative to Kif and only if K*a f-

f3

Hence, given a belief revision operation, we can formulate the conditions under which a counterfactual conditional is acceptable relative to a given state of belief. Note that the Ramsey test provides acceptability conditions rather than truth-conditions for conditional sentences. The most striking difference between acceptability conditions and truth-conditions is that the former, but not the latter, are person-relative; they refer to the background beliefs of a person. Since the standard account of conditionals, represented by David Lewis' theory, is concerned with the conditions under which a conditional sentence is true, the theory of conditionals associated with the Ramsey test is fundamentally different from that account. In the previous section we saw how to go from an obligational disputation to an operation of belief revision. The Ramsey test takes us even further: in the form of (RT), it provides us with a route from disputation, via belief revlSlon, to acceptability criteria for counterfactuals. In that sense, our belief revision interpretation of obligational disputation is also a 'counterfactual interpretation.' 26

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In his influential paper from 1982, Paul V. Spade argues that the medieval theory of obligational disputation is a theory of counterfactual reasoning. Spade's interpretation is one of the most systematic interpretations in the literature and has served as a major source of inspiration for us. Still there are some differences between Spade's and our account that are worth pointing out. In his interpretation of Burley's theory, Spade makes use of the notion of a possible world, his main idea being that an obligational disputation specifies a class of possible worlds in which the positum is true but which otherwise differ as little as possible from the actual world. It is, according to Spade, the irrelevant sentences that guarantee that the specified possible worlds are as close as possible to the actual world, since these sentences are evaluated on the basis of how things are, or on the basis of how the respondent knows things to be. 27 In contrast, our interpretation does not refer to possible worlds, but this difference between our theory and Spade's is clearly superficial, although one might want to add that our sentence-based account is closer to Burley's own manner of presentation. In our formalization we have borrowed and further developed ideas from Spade. The main difference is that while Spade settles with a rather unwieldy semi-formalization, our formalization is complete (regarding the language, the consequence relation et cetera) and, in our view, more transparent. The difference in degree of precision reflects, to some extent, a difference in purpose; what mainly motivated us to go a step further in direction was the prospect of being able to compare the theory we ascribe to Burley with the AGM theory. Another difference is that Spade does not consider the case of an impossible positum, nor the case of a respondent starting out with inconsistent beliefs/ 8 whereas our model is more general in that it handles both these limit cases. Our use of the Ramsey test corresponds to Spade's use of two principles that he ascribes to the medieval authors. 29 (Sl)

(S2)

For every true (or assertable) counterfactual 'Were a, then f3' there is an obligational disputation whose positum is a in which f3 is accepted or __,/3 is denied. For every obligational disputation with positum a in which f3 is accepted or __,/3 denied, there is a true (or assertable) counterfactual 'Were a, then /3,' and for every such disputation in which f3 is denied or __,/3 accepted, there is a counterfactual 'Were a, then __,/3.'

DISPUTATION AND CHANGE OF BELIEF

51

On our interpretation of these principles, (S I) gives a necessary condition for the validity of a counterfactual conditional, and the first conjunct of (S2) gives a sufficient condition. But, as far as we can see, it is not possible to extract one combined condition that is both sufficient and necessary from (Sl) and (S2). 30 In particular, the following truth (acceptability) criterion does not follow: "The counterfactual 'Were a, then /3' is true (assertable) if and only if: f3 is accepted or -,f3 denied in every obligational disputation in which a is the positum." The Ramsey test, on the other hand, provides us with a condition which is necessary as well as sufficient for the acceptability of a counterfactual. 31 Perhaps the most important difference between our and Spade's account is that we advance a pure epistemic interpretation of Burley, focusing on belief rather than on truth. In contrast, consider this passage from Spade: The epistemic clauses in the rules of obligations make it unclear whether we are dealing with the conditions for the truth of the counterfactual or for the respondent's being epistemically in a position to assert the counterfactual. The rules for relevant sentences favour the former; those for irrelevant sentences favour the 32 latter. It is Spade's view that "we have in obligational disputation a kind of

counterfactual reasoning, a kind that mixes in certain epistemological factors" (Spade I982, II). As pointed out in Section 2, it is admittedly very difficult to see exactly what role, if any, the notion of truth plays in Burley's theory. Nonetheless, the reason Spade gives for his mixed interpretation in the above quotation is not very convincing and seems to rest on the misconception that the relevant sentences are evaluated entirely without (direct or indirect) reference to the respondent's background beliefs. Recall that a relevant sentence is such that either the sentence itself or its negation follows from the set of sentences accepted so far in the disputation. The point is that the latter set is (normally) not as epistemically innocent as Spade seems to think since nothing prevents it from containing sentences that were previously found irrelevant (to the disputation) but were accepted because they follow from the respondent's background beliefs. Hence, also the relevant sentences (normally) depend on background beliefs for their acceptance, although the dependence is indirect in this case. Finally, it is important for Spade that his interpretation can really explain the treatment of the irrelevant sentences in an obligational disputation. As he writes in his I992 paper, "[i]t is the strikingly odd treatment of irrelevant propositions that gives the counterfactual

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interpretation much of its plausibility." (Spade 1992, 173.) According to Spade, the purpose of the rule for the irrelevant sentences is to guarantee that the disputation describes "what would happen if the positum were true but everything else stayed as much as possible the same." (Spade 1982, 12.) We would like to add to Spade's observation that Burley's rule for irrelevant sentences does not in itself ensure a conservative outcome; what is also needed is a correct ordering of the proposed sentences, an ordering which reflects the entrenchment of the sentences relative to the respondent's system of belief. 7. CONCLUDING DISCUSSION

The medieval treatises of ob/igationes, their nature and purpose, present a challenging puzzle to modem scholars of medieval logic, and many of the pieces are still missing and others yet to be assembled. Since this initial study has shown so many direct similarities between Burley's theory of obligationes and belief revision, it would be of interest to continue this line of investigation and look at, for example, the theories of Kilvington and Swyneshed. Another area of future inquiry would be to consider other types of obligational disputations, not just the positio. Yet a possible area of research centres around the question what the medieval theory of obligationes might contribute to the contemporary discussion of belief revision. The idea of focusing on one sentence at a time and in this way construct an operation of revision has not received much attention in the contemporary discussion, but for an attempt in this general direction see Alchourr6n and Makinson (1981 ). A peculiarity of the modem theory of belief revision is that it is commonly believed to lack predecessors in the history of logic, very much in contrast to the situation in modal logic (see Lagerlund 2000). With this paper we hope to have contributed to a revision of this belief; if we are correct, belief revision indeed has a history, and that history, though discontinuous, goes back at least to the medieval treatises de ob/igationibus. 33 APPENDIX: PROOFS OF OBSERVATIONS

We want to prove the following for an operation revision:

*

of disputational

DISPUTATION AND CHANGE OF BELIEF

53

(K * 1.1) K* a= Cn(K* a), if K f- a. (K*2) aE K*a. (K*3) K*a~ Cn(Ku {a}). (K*4.2) If K ~-.a, then K u {a}~ K*a. (K *4.1) If K ~-.a, then Cn(K) u Cn( {a}) ~ K* a. (K * 5. 1) If K is consistent, then K* a is inconsistent if and only iff- -,a. (K*6) Iff- aH {3, thenK*a=K*f3. Since proofs of later principles depend on the proofs of previous principles, the order of the proofs does not correspond to the order of the principles above (which follows the order of the AGM postulates). In all proofs below K is a set of sentences and ~ is a linear ordering of all sentences of L = {Xo, X~> ... , Xm ... }. All clauses referred to are clauses in the definition of a Burely-disputation. Lemma 1: Let D I> D 2 , .•• , D m •.. be a sequence of sets of sentences. UD; is consistent if the following two conditions are satisfied: (i) (ii)

D; ~ D;+h

i E N (non-decreasing sequence), and D; is consistent for i E N.

Proof Standard result which follows using compactness. Assume for contradiction that uD; is inconsistent so that uD; f- .l (where .l is arbitrary contradiction). By the definition of f- there is a derivation of .l with hypothesis in uD;. By compactness there is a derivation of .l from a finite subset {/31> {32 , ••• , f3d of UD;. For each i ~ k, {3; E Dm for some m. Let n be the maximum of all m such that {3; E Dm for some i ~ k. Then f3I> {32 , ••• , f3k E Dn and hence Dn f- .l. But Dn is consistent. We have a contradiction and we may conclude that Lemma 1 holds. o Observation 1: Let (K*2)

* re an

operation of disputational revision for K

and~.

aE K*a.

Proof By clause (i) of the definition of a Burley-disputation, a Hence, aE uD; = K*a. o Observation 2: Let

* re an

E

D0•

operation of disputational revision for K and~.

(K * 5.1) If K is consistent, then K* a is inconsistent if and only iff- -,a. Proof Assume that K is consistent. The if-part: follows from (K*2).

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HENRIK LAGERLUND & ERIK J. OLSSON

The only if-part: Assume ¥-,a is non-contradictory. We will show that K*a= uD; is consistent. It is immediate from the definition of a Burleydisputation that {D;} is a non-decreasing sequence. We will now show that D; is consistent for each i E N and then use Lemma 1. We use induction on i. Base case: i = 0. D 0 ={a}. Since a is non-contraditory, D 0 is consistent. Induction hypothesis: D; is consistent. We will show that D;+t is consistent. Since K and D; are both assumed consistent, we need to prove two cases, corresponding to clause (iii) and (iv) of the definition of a Burley-disputation: Case I: Di+ 1 = D; u {X;}. Case 1.1: D; f- Xi· Since D; is consistent, D; Y By classical logic, D; u {X;} is consistent. Case 1.2: D; Y X;, D; ¥ K f- X; and K is consistent. Since D; ¥ it follows by classical logic that D; u {X;} is consistent. Case 2: Di+ 1 = D; u {-x;}. Analogous to Case I. We see that in all cases it holds that D;+t is also consistent. Hence D; is consistent for each i E N. We can now use Lemma I and conclude that uD; = K*a is consistent. o Observation 3: Let * be an operation of disputational revision forK and:::;.

-x;.

-x;,

(K*3)

-x;

K*ak Cn(Ku {a}).

Proof We must show that K* a = uD; k Cn(K u {a}). It suffices to show that D; k Cn(K u {a}), fori E N. We prove this by induction on i. Base step: i = 0. D 0 = {a} k Cn(K u {a}). Induction hypothesis: D; k Cn(K u {a}). It suffices to show that D;+t k Cn(Ku {a}). Let f3 E D;+t· If f3 E D;, then by the induction hypothesis, f3 E Cn(K u {a}). Assume that f3 ~ D;. There are two principal cases: either f3 = X; or

f3=...,x;.

Case 1: f3 = X;· Since f3 was added at step i+ I it must have been added according to either clause (ii), (iii) or (v) of the definition of a Burleydisputation. Case 1.1: f3 was added according to clause (ii) so that D i+ 1 = L. In this case uD; = L = K* a so that K* a is inconsistent. By (K *5.I), either a is contradictory or K is inconsistent. In either case, Cn(K u{ a}) = L which trivially entails that f3 E Cn(K u {a}).

DISPUTATION AND CHANGE OF BELIEF

55

Case 1.2: f3 was added according to clause (iii). Then f3 must have been added with reference to either condition (a) or condition (b) of that clause. Case 1.2.1: condition (a) of clause (iii) holds so that D; 1-/3. Since D; ~ Cn(K u {a}) it follows from the monotony of Cn that Cn(D;) ~ Cn(Cn(K u {a})). It follows using the iteration property of Cn that Cn(D;) ~ Cn(K u {a}). Since f3 E Cn(D;) it follows that f3 E Cn(K u {a}). Case 1.2.2: condition (b) of clause (iii) holds so that D; ¥/3, D;¥..,/3, K lf3. Since K ~ K u {a} it follows using the monotony of Cn that Cn(K) ~ Cn(K u {a}). By hypothesis, f3 E Cn(K) and hence f3 E Cn(K u {a}). Case 1.3: f3 was instead added in accordance with clause (v). Since in that case -,X; = -,f3 was also added, it follows that D;+I is inconsistent. Since K is inconsistent, Cn(K u {a})= L (the language) so that f3 E Cn(K u {a}) as desired. Case 2: f3 =-,Xi· This case is symmetrical to case 1, but we will carry out the proof anyway. Since f3 was added at step i+ 1 it must have been added according to either clause (ii), (iv) or (v). Case 2.1: f3 was added according to clause (ii). This case is analogous to Case 1.1 above. Case 2.2: f3 was added according to clause (iv). Then f3 was so added with reference to either condition (a) or condition (b) of that clause. Case 2.2.1 condition (a) of clause (iv) holds so that D; 1- -,Xi· Since D; ~ Cn(K u {a}) it follows from the monotony of Cn that Cn(D;) ~ Cn(Cn(K u {a})). It follows using the iteration property of Cn that Cn(D;) ~ Cn(Ku {a}). By hypothesis, -,Xi E Cn(D;). Hence, /]E Cn(Ku {a}). Case 2.2.2: condition (b) of clause (iv) holds so that D; ¥-,X,., D; Y ••X;, K 1- -,Xi· Since K ~ K u {a} it follows using the monotony of Cn that Cn(K) ~ Cn(K u {a}). By hypothesis, -,X; E Cn(K) and hence f3 E Cn(Ku {a}). Case 2.3: f3 = -,Xi was instead added in accordance with clause (v). Then X; was also added so that D;+I is inconsistent. Since in this case K is inconsistent, Cn(K u {a}) = L (the language) it follows trivially that f3 E Cn(Ku {a}). We have shown that D; ~ Cn(K u {a}) for i E N. It follows that uD; ~ Cn(Ku {a}). o

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Observation 4: Let * te an operation of disputational revision for K and :S:. (K *4.1) If K Y~a, then Cn(K) u Cn( a) k K*a. Proof Let f3 E Cn(K) u Cn( {a}). Clearly, f3 =Xi for some i E N. We will show that f3 E uDi. There are two principal cases: either a 1- f3 or K 1- {3. The clauses referred to are those in the definition of a Burleydisputation. Case I: a 1- {3. Since D 0 = {a} it follows from the monotony of Cn and the fact that {DJ is non-decreasing that Di 1- {3. Case 1.1: Di is consistent. Since condition (a) of clause (iii) is satisfied, it follows that f3 E Di+I· Hence, f3 E uDi. Case 1.2: Di is inconsistent. By clause (ii), Di+l = L. It follows trivially that f3 E Di+I· Hence, f3 E uDi. Case 2: K 1- {3. Case 2.1: D i is consistent. Case 2.1.1: Di 1- {3. Since condition (a) of clause (iii) is satisfied it follows that f3 E Di+I· Hence, f3 E uDi. Case 2.1.2: Di Y {3. Assume for contradiction that Di 1- ...,f3. By (K*3), uDi k Cn(K u {a}). Since Di 1- ...,f3, it follows by the monotony of Cn that uDi 1- ...,f3. From monotony and iteration of Cn it now follows that ...,f3 E Cn(K u {a}). By hypothesis, f3 E Cn(K) k Cn(K u {a}). Hence, K u {a} is inconsistent. Using classical logic we conclude that K 1- ~a which contradicts the hypothesis of this theorem. We may conclude that Di Y ~{3. By hypothesis, K Y~a. Hence K is consistent. Since K 1- {3, condition (b) of clause (iii) is now satisfied. It follows that {3 E Di+I· Hence, f3 E uDj.

Case 2.2: Di is inconsistent. By clause (ii), Di+I = L. It follows trivially that f3 E Di+I· Hence, f3 E uDi. Finally, we may now conclude that Cn(K) u Cn( {a}) k K* a, provided that K Y~a. o Observation 5: Let

* te an

operation of disputational revision for K and :S:.

(K*l.l) If K 1- a, then K*a = Cn(K*a), Proof Assume that K 1- a. It is trivial that K* a k Cn(K* a). It remains to show that Cn(K* a) k K* a. We first show that K* a = Cn(K). For the inclusion K*a k Cn(K) it suffices to prove that Di k Cn(K) fori E N, where uDi = K*a.

DISPUTATION AND CHANGE OF BELIEF

57

Base case: i = 0. Since D 0 = {a} and since by assumption K f- a, it follows that D 0 k Cn(K) so that the inclusion holds in this case. Induction hypothesis: D; k Cn(K). We will show that D;+I k Cn(K). Let {3 E D;+I· If {3 E D; it follows directly by the induction hypothesis that {3 E Cn(K). Assume therefore that {3 e D;. Hence, {3 =X; or {3 = -,z;. Since {3 was added at step i+ 1, it must have been added because of one of the rules (ii), (iii), (iv) or (v) in the definition of a Burley-disputation since only these rules add something. Case 1: {3 was added in accordance with clause (ii). Then D;+I = L. Hence, K* a = uD; = L. By (K *5.1) this means that either K or a is inconsistent. Since K f- a it follows that in either case K is inconsistent so that {3 E D;+I = L = Cn(K) as desired. Case 2: {3 =X; was added in accordance with clause (iii). Case 2.1: {3 was added in accordance with clause (iii) (a) and consequently D; f- {3. Hence, {3 E Cn(D;). By the induction hypothesis and the monotonicity of Cn, Cn(D;) k Cn(Cn(K) and, furthermore, using the iteration and inclusion property of Cn, Cn(Cn(K)) = Cn(K) so that Cn(D;) k Cn(K). Consequently, {3 E Cn(K). Case 2.2: {3 was added in accordance with clause (iii) (b). It follows directly that {3 E Cn(K). Case 3: {3 = -,z; was added in accordance with clause (iv). This is parallel to case 2 above. Case 4: {3 was added in accordance with clause (v). In this case K is inconsistent and therefore {3 E Cn(K).

We now tum to the other direction of the inclusion: Cn(K) k K*a. Let {3 E Cn(K). Clearly {3 =X; for some i;:::: 0. We must show that {3 E D;+I· Case 1: K is inconsistent. Then {3 is added by clause (iv) in the definition of a Burley-disputation. Case 2: K is consistent. Since K f- a it follows that a is noncontradictory. Case 2.1: D; f- {3. In that case it follows by clause (iii) (a) that {3 E D;+I· Case 2.2: D; Y {3. We want to show that D; Y -,{3. We first note that since K f- a and K is consistent, it follows that K Y -,a. It now follows from (K*4.1) that K k K*a. We also note that since K is consistent and a is non-contradictory it follows from (K *5.1) that K* a is consistent. Now, assume that D; f- -.{3. It follows using clause (iv) (a) that -,{3 E D;+I and hence that -,{3 E uD;= K*a entailing that -,{3 E Cn(K*a). But since K k K*a it follows using the monotonicity of Cn that Cn(K) k Cn(K*a) and, since {3 E Cn(K) by assumption, that {3 E Cn(K*a), which is

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HENRIK LAGERLUND & ERIK J. OLSSON

contradicted by -,{3 E Cn(K*a) and the consistency of K*a. Consequently, Di ¥-.{3. Since the four conditions in clause (iii) (b) are all satisfied we conclude, using that clause, that f3 E Di+l· This concludes the proof of Cn(K) ~ K*a. We have shown that K*a = Cn(K). What we really wanted to show was that Cn(K*a) ~ K*a. Note that it follows from the monotonicity and iteration property of Cn that Cn(Cn(K)) ~ Cn(K). Substituting K*a for Cn(K) we get the desired inclusion Cn(K*a) ~ K*a. This concludes the proof. o

Observation 6: Let * (K*6)

Iff. a

H

re an

operation of disputational revision for K and::;.

{3, then K*a = K*f3.

Proof sketch: Assume that a H f3 and consider the two Burleydisputations D = and D' = <{3, ::;, K, D'>. It sufficies to show that D = D.' That this indeed holds is clear considering three facts: (i) that the sentences are considered in the same order in the two disputations, (ii) that all sentences are considered in both cases, and (iii) that the rules for the computation of the outcome of a disputation in no way depends on the syntactic form of the positum. The details are left to the reader. o Uppsala University and Konstanz University NOTES 1

For the historical development of the theory see Stump 1982 and Yrjonsuuri 1994. 2 "Omne positum, sub forma positi propositum, in tempore positionis, est concedendum." (Burley 1963, 46.) See also Kretzmann and Stump 1988, 379. 3 "Omne sequens ex posito est concedendum. Omne sequens ex posito cum concesso vel concessis, vel cum opposito bene negati vel oppositis bene negatorum, scitum esse tale, est concedendum." (Burley 1963, 48.) See also Kretzmann and Stump 1988, 381. 4 In this we follow Spade 1982. See clause (2) (b) in his semi-formalization of Burley's theory on p. 6. 5 A similar idealisation is made by Spade 1982, 6. See also his footnote 22. 6 "Omne repugnans posito est negandum. Similiter omne repugnans posito cum concesso vel concessis, vel opposito bene negati vel oppositis bene negatorum, scitum esse tale, est negandum." (Burley 1963, 48.) See also Kretzmann and Stump 1988, 381.

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7 "Omne propositum aut est pertinens aut impertinens. Si sit impertinens, respondendum est secundum sui qualitatem, et hoc, secundum qualitatem quam habet ad nos. Ut, si sit verum, scitum esse verum, debet concedi. Si sit falsum, scitum esse falsum, debet negari. Si sit dubium, respondendum est dubie." (Burley 1963, 48.) See also Kretzmann and Stump 1988, 381. 8 For our purposes, the distinction between what is believed and what is justifiably believed can be disregarded. We may assume that our respondent is epistemically rational in the sense that all her beliefs are justified. 9 The language L is assumed to be a standard propositional language built up from the usual connectives. We use lower case Greek letters as variables over elements of L and normal upper case letters as variables over subsets of L. By a consequence operation on the language L, we mean a function Cn such that the following conditions hold for all subsets A and B of L: (i) A k Cn(A) (inclusion), (ii) If A k B, then Cn(A) k Cn(B) (monotonicity) and (iii) Cn(Cn(A)) k Cn(A) (iteration). We will also assume that the following properties are satisfied: (iv) if a can be derived from A by classical truth-functional logic, then a E Cn(A) (supraclassicality), (v) if f3 E Cn(A u {a}), then a-'? f3 E Cn(A) (deduction), and (vi) if a E Cn(A), then there is a fmite set A' k A such that a E Cn(A') (compactness). We use 'A f- a' and 'a E Cn(A)' interchangably. A set A is logically closed iff A = Cn(A). A set A is consistent if and only if there is no sentence a such that A f- a&-, a. A set which is not consistent is inconsistent. 10 This means, here and in the following, that the linearly (simply) ordered set is isomorphic with an initial segment of the natural numbers for A k L. 11 Note the similarity between Burley's construction and the standard construction used in proofs of the completeness of classical propositional logic using the Lindenbaum method, of constructing out of a set A of formulae and a formula X such that A~ X a maximal set A' ;;::2 A of formulae with A' ~X· In the Lindenbaum construction all formulae are ordered into a sequence [3], {32, ••• and

Ao=A At+ I= A 1 u {{31}, if A; u {[3} is consistent, and At+I =AtU {-,{3;} else. Finally, A' = uA;. See also Martin 1993, 366. 12

See Spade 1982, 4, and Yrjonsuuri 1994, 8. "[O]rdo est maxime attendendus." (Burley 1963, 52.) See also Kretzmann and Stump 1988, 385. 14 "[S]it ita quod Socrates et Plato sint nigri, et ponatur Socratem esse album. Tunc, si primo loco proponatur haec: 'Soctates et Plato sunt similes,' esset concedenda, quia vera et impertinens. Et si postea proponeretur, haec: 'Plato est albus,' esset concedenda, quia sequens. Si, tamen, ista: 'Plato est albus,' primo loco proponeretur, esset negenda, quia falsa et impertinens. Si postea proponeretur: 'Socrates et Plato sunt similes,' esset negenda, quia repugnans." (Burley 1963, 52.) See also Kretzmann and Stump 1988, 385. 15 See Spade 1982, 17, for this observation. 16 Burley's example is most naturally formalized using features of predicate logic not found in sentential logic. We leave it as an exercise to carry out that formalization. 13

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Superficially, an obligational disputation shares many of the characteristics of a game: it has two participants that appear to compete in a process evolving in accordance with formally specified rules, yet admitting some degree of creativity, at least on the opponent's part. For someone defending the game-interpretation of obligational disputation, it is crucial to explain who wins the game in the different outcomes. The issue loses its interest if one instead adopts a convenient fictioninterpretation, since, on that interpretation, obligational disputation is not a game in the first place (unless, of course, one thinks of obligational disputation as form of logical solitaire, a rather remote possibility considering the amount of energy devoted to its study in the Middle Ages). Spade opposes the game-interpretation in Spade 1977, 244. 18 See Alchourr6n, Gardenfors and Makinson 1985. 19 See Gardenfors 1988, 53. 20 Gardenfors 1988, 87. 21 For the belief base approach, see Hansson 1994. 22 See Spade 1982, 16. 23 The revision operations constructed in Examples 1 and 2 are, strictly speaking, not operations of disputational revision in the technical sense of the above definition, since the opponent sequences do not include the whole language. But the sequences could clearly be continued without affecting the point made here. 24 It might be held that the entrenchment and epistemic importance are two separate things. In not making that distinction we essentially follow Gardenfors 1988. 25 Gardenfors 1988, 147. See also Ramsey 1931, 248, and Stalnaker 1968, 102. 26 Martin 1993, 366 writes that the procedure of positio "does recall the use of the so-called Ramsey test". 27 See Ytjonsuuri 1994 for a more thorough discussion on this. 28 "Since we are not considering that special kind of obligatio in which the positum is impossible, JG will be a consistent set" (Spade, 1982, 7). (Here Xt is the set of sentences granted at step i of the disputation.) Spade's statement is unproblematic only under the assumption that the background beliefs of the respondent form a consistent set. If that assumption is not made we fail to see why the inconsistency of Xt could not be traced back to the inconsistency of the respondent's beliefs. There are examples in the medieval literature where liar-type paradoxes are considered, and the contradictions arising in these examples are sometimes best classified as belonging to the sphere of background knowledge. See Martin 1993 for the medieval treatment of liar-paradoxes in the disputation setting. 29 See Spade 1982, 13-14. The principles (S1) and (S2) correspond to (6) and (7) in s;ade, s paper. 3 In his 1982 paper, footnote 41, Spade refers to the account of counterfactual conditionals developed by Sobel 1970. Spade claims (without proof) that his account of Burley's theory validates many of the inference patterns (V1-V9 and V12-V13) considered by Sobel, given (S1) and (S2) and omniscience on the part of the respondent. This is a surprising claim since the theory Spade ascribes to Burley is not precise enough to admit that kind of exact validation. A related problem is that some of Sobel's inference patterns make use of iterated counterfactuals, a feature that would presumably require the presence of counterfactual conditionals in the object language.

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31 In both the 1982 paper (pp. 9-14) and the 1992 paper (pp. 178-180) Spade stresses the fact that he is talking about, so called, would-conditionals. On the other hand, (Sl) seems more plausible as an account of might-conditionals than of would-conditionals. A similar observation is made by Martin 1993, 366. 32 Spade 1982, 13, footnote 42. 33 We would like to thank Kaj Borge Hansen, Sven Ove Hansson, David Makinson and Mikko Yrji:insuuri for their helpful comments on earlier drafts of this paper. The material contained in the paper was discussed at the philosophy seminar in Uppsala, and we are indebted to Sten Lindstrom, Krister Segerberg, Lennart Aqvist and the other participants for their feedback.

REFERENCES Alchourr6n, C. E., Gii.rdenfors, P., and Makinson, D., 1985. "On the Logic of Theory Change: Partial Meet Functions for Contraction and Revision," Journal of Symbolic Logic 50, 510-530. Alchourr6n, C. E., and Makinson, D., 1981. "Hierarchies of regulations and their logic," in R. Hilpinen (ed.), New Studies in Deontic Logic, Dordrecht, D. Reidel, 125-148. Ashworth, E. J., 1981. "The Problems of Relevance and Order in Obligational Disputations: Some Later Fourtheenth Century Views," Medioevo 7, 175-193. 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. Gii.rdenfors, P., 1988. Knowledge in Flux: Modeling the Dynamics of Epistemic States, Cambridge, MA, MIT Press. Hansson, S. 0., 1994. "Taking Belief Bases Seriously," in D. Prawitz and D. Westerstahl (eds. ), Logic and Philosophy of Science in Uppsala, Dordrecht, Kluwer, 13-28. Knuuttila, Simo, 1993. Modalities in Medieval Philosophy, London, Routledge. Knuuttila, Simo, and Yrji:insuuri, Mikko, 1988. ''Norms and Action in Obligational Disputations," in 0. Pluta (ed.), Die Philosophie im 14. und 15. Jahrhundert, (Bochumer Studien zur Philosophie 10), Amsterdam, Griiner, 191-202. Kretzmann, Norman, and Stump, Eleonore, 1988. The Cambridge Translations of Medieval Philosophical Texts: Volume One: Logic and the Philosophy of Language, Cambridge, Cambridge University Press. Lagerlund, Henrik, 2000. Modal Syllogistics in the Middle Ages, Leiden, Brill. Martin, Christopher, 1993. "Obligations and Liars," in S. Read (ed.), Sophisms in Medieval Logic and Grammar, Dordrecht, Kluwer, 357-381. Revised version in the present collection, pp. 65-96. Paul of Venice 1988. Logica Magna, Part II, Fascicule 8, [Tractatus De obligationibus ], (ed. with trans I. and notes) E. J. Ashworth, (Classical and Medieval Logic Texts, vol. 5), Oxford, British Academy, Oxford University Press.

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Ramsey, F. P., 1931. "General propositions and causality," in his Foundations of Mathematics and Other Logical Essays, New York, Routledge & Kegan Paul, 237-257. Sobel, J. H., 1970. "Utilitarianisms: Simple and General," Inquiry 13, 394-449. Spade, Paul Vincent, 1977. "Roger Swyneshed's Obligationes: Edition and Comments," Archives d'histoire doctrinale et litteraire du moyen tige 44, 243-285. Spade, Paul Vincent, 1982. "Three Theories of Obligationes: Burley, Kilvington and Swyneshed on Counterfactual Reasoning," History and Philosophy of Logic 3, 1-32. Spade, Paul Vincent, 1992. "If Obligationes Were Counterfactuals," Philosophical Topics20, 171-188. Stalnaker, R., 1968. "A theory of conditionals," in N. Rescher (ed.), Studies in Logical Theory, (American Philosophical Quarterly Monograph Series, no. 2), Oxford, Blackwell, 98-112. Stump, Eleonore, 1982. "Obligations: A. From The Beginning to The Early Fourteenth Century," in N. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 315-334. Yrjonsuuri, Mikko, 1994. Obligationes: 14th Century Logic of Disputational Duties, (Acta Philosophica Fennica 55), Helsinki, Societas Philosophica Fennica. YIjonsuuri, Mikko, 1996. "Obligations as Thought Experiments," in I. Angelelli and M. Cerezo (eds.), Studies in the History of Logic, Berlin, Walter de Gruyter, 79-96.

CHRISTOPHER J. MARTIN

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The best efforts of some of the best historians of logic have failed to find a clear source in material inherited from the ancient world for the earliest mediaeval discussions of the Liar paradox. 1 The obvious candidates, and most importantly Aristotle's reference to the puzzles of oath-breakers and liars in the Sophistical Refutations, seem to have been of little importance in the first theorising about insolubilia. 2 Despite this I would like to suggest that there was at least a very distant cause in late antique logic for the appearance of the Liar and its relatives in the twelfth century. It cannot be emphasised enough, however, that twelfth century logicians devised the paradoxes for themselves and that their solutions were all their own work. Three things are important in the construction of the Liar and variations upon it. As Saul Kripke has emphasised, in the appropriate circumstances just about any statement involving truth or falsity may tum out to be paradoxical. 3 Once this is pointed out it is easy to see that there can be no syntactic or semantic sieve in which to trap all instances of ungroundedness. Whether or not, for example, utterances of the sentences 'everything that Socrates says is false' and 'everything that Plato says is false' are paradoxical depends upon who says them, what else they say, when they are said, and the way the world happens to be then. No doubt circularity sometimes occurs by chance but in general we need to arrange the circumstances of utterance, inscription, or thought so as to guarantee it. This stage-setting was done in the middle ages by means of a hypothesis or positio. The theory of such hypotheses was regimented in treatises on the obligatio of the same name and it is to these, I suggest, that we should look if we. want to understand how the mediaeval concern with insolubilia originated and the character of the earliest solution. Secondly, in order to achieve paradoxical circularity or self-reference a certain amount of semantical and syntactical equipment is necessary. In particular truth and falsity must be available as predicates in the language and must behave in the appropriate way in conjunction with a suitable referential apparatus. The paradoxes arise because of a conflict 63

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between our intuitions about truth and falsity and our intuitions about the power of natural languages to deploy these predicates. Furthermore the choice between sentences, statements, and propositions as bearers of truth will lead to different formulations of and different solutions to the Liar. An inscriptionalist like Buridan can, for example, make moves which would be entirely unacceptable to someone who maintains a realistic account of propositional contents. Thirdly, the acceptance or rejection of particular logical principles may facilitate or prevent the derivation of disturbing consequences from the Liar sentence. I have in mind for example the Parvipontanian adoption of strict implication and the associated paradoxes, Abaelard's qualifications of the principles of quotation and disquotation, and the requirement of the Melidunenses and Porretani that for a conditional to be true it is necessary for the antecedent and the consequent to be about the same thing. In this paper I will limit myself to a discussion of the first two points. 4 1. OBLIGATIONS The writers of the mediaeval treatises on obligations are irritatingly vague about both the history and the purposes of their works. There are some clues, however, to the development of the obligation known as positio and these suggest an account of its intended application.

1.1 Eudemian Hypotheses and Eudemian Procedures The treatises published by de Rijk as TE and OP are certainly the oldest accounts of obligations discovered so far. Each justifies impossible positio with an appeal to Aristotle who, as OP claims, requires that 'an impossibility be posited in order to see what follows from it. ' 5 There seems to be no such injunction in Aristotle's writings but something rather like it is found in Boethius' De Hypotheticis Syllogismis. 6 There Eudemus is credited with distinguishing between a hypothesis, or positio, which is the condition of a consequence, i.e. the antecedent of a conditional, and a hypothesis by means of which 'something which can in no way come about is agreed to ... in order that reason may be pursued to its limit.' As an example Boethius cites the separation in thought of the form of a body from its matter so that 'for a little while that is

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understood to be so which can in no way come about' in order that we may consider what follows. Let us call these conditions of agreement Eudemian hypotheses and the arguments carried out under them Eudemian procedures. Garlandus Compotista characterises Eudemian hypotheses as being per consensum and offers the example of conceding 'that a man is a stone, in order to see how far reason can go. ' 7 Nothing like this hypothesis is found in the authority which Garland defers to as The Book, which consists of the works of Boethius, but the use of positio impossibilis in the exploration of the logic of impossible unions is the main topic of TEl. An account of such unions is also of great importance to Abaelard in developing his theory of the conditional. Abaelard characterises a Eudemian hypothesis as by 'consent and concession ... a proposition which is not taken as true in itself, but is rather conceded for the sake of argument, so that we may see what can be extracted from it. ' 8 Notice that he does not insist upon the impossibility of the hypothesis. Boethius' example is not particularly helpful but one thing is clear, he supposes that Eudemus had in mind something quite different from the method of reduction to an impossibility employed in the theory of the syllogism. There the aim is to show that a hypothesis is impossible by showing that an impossibility follows from it and appealing of Aristotle's rule that the impossible cannot follow from the possible. In a Eudemian procedure on the other hand we start with something agreed to be impossible and set out to explore its logical structure. As we will see, Aristotle's rule construed as requiring actualisability justifies possible rather than impossible positio. It is indeed cited to this end in some later accounts of obligations though it does not appear in TE or OP. Boethius employs Eudemian procedures on a number of occasions but most explicitly in his treatment in Quomodo Substantiae of the problem of how created things can be good in so far as they exist but not be substantial goods. If we look at some twelfth century commentaries on Quomodo we find an interesting development in the descriptions of Boethius' request there that we do in thought what is impossible in act. Gilbert of Poitiers does not have a special term for it but Thierry of Chartres refers to the exercise as positio per impossibi/e per consensum. Finally Clarembald of Arras calls Boethius' hypothesis a positio impossibilis and finds in his argument a positio possibilis. 9 None of these writers mentions obligatio but it is clear that the kinds of thought-experiment that Boethius proposes can be well conducted only if there are some rules about what counts as an acceptable

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inference. Philosophers spend much of their time reasoning under hypotheses and in the middle ages they registered this activity with the word 'ponatur.' One of the functions of the positio was certainly to regiment the varieties of hypothetical reasoning and to provide practice in the rules of inference applicable in them. Unfortunately Boethius tells us nothing about the formalities of the kind of 'counterpossible' reasoning employed in Eudemian procedures. As I have shown elsewhere, however, from various of his works it is possible to extract two notions of inseparability and corresponding to them two accounts of consequence. In the weaker sense two things are inseparable if they cannot actually be separated but they may nevertheless be conceived apart. I call this strict inseparability. The corresponding conditional is true just in case it is impossible for the antecedent to be true and the consequent false at the same time. In the stronger sense two things are inseparable if they cannot even be conceived apart. This I call conceptual inseparability. The corresponding conditional holds just in case one cannot conceive the antecedent without conceiving the consequent. 10 Though poorly conducted, the thought-experiment of Quomodo gives some content to the distinction between strict and conceptual inseparability. If a situation, albeit an impossible one, can be constructed in which two things exist in some sense independently of one another, then they are not conceptually connected and so neither belongs in the definition of the other. Boethius' authority makes this sort of reasoning respectable and important but he offers no hint of how we are to avoid begging the question of what is conceptually separable from what. 1.2 Impossible Positio

In the twelfth century the most important work on this problem was done by Abaelard and its results can be seen in the TEl. AI though insolubilia are not introduced in this kind of positio, I would like to look at the work briefly since I think it can provide some useful clues to the early history of their treatment and also help to answer a singularly vexed question in the history of logic. TEl in accordance with the principle attributed to Aristotle will allow us to posit any impossibility which may be understood. It claims that since we can understand the claim that that God is man is true and that this is harder to understand than that that a man is an ass is true we may posit the latter. The comparison and the example turn out not to be

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accidental. TEl is for the most part devoted to a discussion of the various kinds of impossible union and has a logic to offer for reasoning about them. In impossible positio the positum is of course impossible but since the aim is to avoid conceding contradictory proposita it cannot be such that contradictory opposites may be derived from it by means of the principles of inference appropriate for this sort of positio. Thus in agreeing to take part in an impossible positio we must forgo the pleasure of appealing to the thesis, attributed to the Adamatae in TEl, that whatever you like follows from an impossibility. That is not all we must give up, however, since inconsistency may be derived from an impossibility independently of the rule. Suppose, for example, that we posit as TEl suggests that a man is an ass. We may then appeal to the locus 'from opposites' to prove that a man is not a man. This move is exactly what Abaelard's theory of the conditional is devised to avoid. TEl embraces at least part of that theory by insisting that consequences with affirmative antecedents and negative consequents must not be accepted in impossible positio and in allowing no appeal to the locus 'from opposites.' More interestingly TEl rehearses but confuses Abaelard's arguments against the truth of 'if its a man, then it's not an ass.' Abaelard maintains that conditionals of mixed quality are false since the sense of the antecedent cannot contain the sense of the consequent and so they do not express natural consequences. TEl, on the other hand, claims that such conditionals may express natural consequences but that since we can understand the combination of the forms of, say, a man and an ass in a single subject, the understanding of one does not exclude that of the other. Since it is understanding rather than nature with which impossible positio is concerned we are not allowed to appeal to such consequences there. TEl thus relies on Abaelardian logic in reasoning from an impossibility and this is also the logic associated with the Nominales. There is, however, at least one curious thesis identified with that school which, pace Calvin Normore, Abaelard apparently would not have accepted. 11 Jacques of Vitry tells us that, in opposition to the Adamatae, the Nominales maintained that nothing grows. 12 It is striking that just this conclusion follows immediately from the principles of predication set down in TEl. It seems, then, that in TEF and TEl we have works connected with the Nominales and interesting information on their beliefs about essential and personal predication.

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Having mentioned one outrageous positio Nominalium let me note that another, obviously related to it, is found in the Obligationes attributed to William of Sherwood. In what is known as vicarious positio we are asked to respond to questions as someone else would. Walter Burley gives the example of being asked to respond as Zeno would to the proposition that nothing moves. In the parallel passage in Sherwood the proposal that something moves is put to someone required to answer as a Nomina/is. The proper response is to deny it. It seems to me that the evidence of this positio on its own outweighs all the similarities pointed out by Spade and Stump in arguing that Burley 13 was the author of the work attributed to Sherwood. It is surely much more likely that Burley was updating a old, perhaps standard, text and in this case providing an example that would make sense to his contemporaries. The only other argument offered by Spade and Stump for the late dating of 'pseudo-Sherwood' is the sophistication that they find in its distinction of different kinds of consequence. They are quite wrong, however, in supposing that such sophistication is a late development in mediaeval logic. It is all available and much more in Abaelard's Dialectica. 14 Though I think that reflection upon Eudemian procedures and their regimentation was quite probably an important factor in the early development of positio it cannot be denied that in the texts that have survived positio impossibilis appears as very much a poor relative of positio possibilis. This is, I suppose, how it should be. It is not often that we are required to indulge in counterpossible inference and nowadays, 15 indeed, the prevailing view seems to be that it can't be done at all. Counterfactual reasoning where the fact is contingent is, however, something that we do all the time and something which needs to be codified. 1.3 The Purpose of Possible Positio That much being obvious, there has been considerable disagreement over exactly what purpose is served by the rules for positio. Recently the debate has turned upon whether or not they were intended to provide a means for evaluating counterfactual conditionals. Conditionals, that is, such as 'if you were to promise me a penny, I would be very suspicious of your motives.' Paul Spade has argued that positio was indeed constructed for the evaluation of such sentences but that it was not very good at its job. 16 Eleonore Stump on the other hand thinks that this was not its job

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at all. 17 As far as I know Stump does not address Spade's positive arguments but rather takes the failure of positio to give the right results as a proof that it was not supposed to give them. Stump's view is, I am sure, correct. Whatever motivated the introduction of the so called nova responsio in about 1330 there is no suggestion in treatments of positio fa/sa current before that date that the device was to be employed in the evaluation or generation of counterfactuals. 18 The discussion has been confused by a failure to distinguish between 'would' and 'might' conditionals and between the corresponding forms of counterfactual reasoning. With these distinctions made we can give a much more plausible account of the workings and purposes of positio. Accounts of positio provide two things at once. Firstly a procedure for putting claims of possibility to the test of actualisation suggested by Aristotle's criterion of possibility. Secondly, and in consequence, the rules for an exercise in which one may gain practice in recognising the truth of conditionals and the validity of arguments. The latter is made more complicated by the cumulative character of positio. At each stage the respondent has to decide the relationship of the propositum to the set of sentences all of which he is required by his earlier answers to hold true. What follows from or is repugnant to this set is more than what follows from or is repugnant to the positum alone since at any point arbitrary truths, logically independent, of what has gone before may be added to the set. This feature of positio seems to serve an essentially pedagogical function. It complicates the antecedent that the respondent has to consider in deciding how to reply and can make delightfully difficult the exercise of detecting consequence and repugnance. The more so, since there is no requirement that the principle of compositionality be observed in proposing compound proposita. That is to say a compound may be posited or proposed without all or any of its components having already been posited or proposed. Thus, notoriously, starting from the same positum one may be required to concede a given propositum in one positio and deny it in another. Spade finds accounts of positio reminiscent of treatments of the semantics of counterfactual conditionals in recent philosophical logic by means of possible worlds. 19 In such accounts, however, possible worlds are taken as primitive and a given counterfactual conditional is evaluated by considering the properties of the the worlds in which the antecedent is true. In the various treatises on positio on the other hand there is absolutely no suggestion that the problem is to decide whether a given counterfactual is true or not. Nor is there any hint that at some point in

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a positio one may stop and conditionalise to generate a true counterfactual with the pas itum for its antecedent and the last conceded propositum for its consequent. Indeed, at least according twelfth and thirteenth century logic, the result of such a conditionalisation would often be false. In the theory of consequences there was no place for a contingently true conditional with a false antecedent. A necessary condition for the truth of a conditional was always that it be more or less strictly necessary and on the rare occasions when subjunctive conditionals were noticed it was not in virtue of any semantical peculiarity but rather because they do not satisfy Boethius' requirement that conditionals have propositions as their parts. What one obtains in a positio starting from a possible positum is not a sequence of propositions which would be true if the positum were true but rather a sequence of propositions each compossible, or better cotenable, with the positum. The rules for handling irrelevant proposita guarantee that the relationship between the positum and any given concessum can be no stronger than that. Thus the most that can be 'read off a properly conducted possible positio is that the positum is cotenable with a given concessum or the opposite of a given negatum. Spade is impressed by the relationship which exists between the positum P and a propositum Q properly conceded in a properly conducted positio. It is (1) intransitive and such that both (2) strengthening the antecedent and (3) contraposition fail. These are certainly all properties of 'would' counterfactuals but that shouldn't impress us too much since they are so in virtue of the necessity that the antecedent and consequent of a true counterfactual be compossible. Unlike, say, the antecedent and consequent of a true material implication. The important point is that the relationship between P and Q has the properties of cotenability which 'would' counterfactuals do not have. It is (4) symmetric and (5) such that if P and Q are contingent and logically independent, then both Q and ~Q may be properly conceded in positiones with positum P. This latter because cotenability follows from logical independence plus contingency. Far from regarding it as an embarrassing difficulty to be explained away accounts of positio dignify this last feature as the rule that: With a possible falsehood as positum any contingent proposition maybe proved and conceded. OP20 proves the case for a contingent falsehood as follows. Let 'P' stand for 'You are in Rome' and 'Q' for 'You are a bishop':

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the positum, does not follow from Po, and it is false so it is to be denied, follows from the denial of Pr 1 so it is to be conceded, follows from Po and Pr2 so it is to be conceded.

Having stated the rule for possible posita and contingent falsehoods, however, OP goes on to claim that the 'rule will not hold for the consequentia Nominalium 'since if one takes for Q an impossibility, the Nominales would still have to deny Pr 1• Given OP's account of the tests of relevance in terms of the truth of appropriately constructed conditionals, the alternative must have been to believe that the addition of a necessity to the consequent of a true conditional makes no difference. That is to say the Nomina/es rejected oQ F P - (P & Q) Note that this a weaker claim than the second Parvipontanian thesis that a necessity follows from anything and in rejecting it the Nominales are rejecting more. 21 Assuming that what they required was containment of sense then the rejection of this principle seems more than reasonable. It is not clear quite where OP stands in the debate since in the discussion of depositio, for obvious reasons, it instructs those who uphold the rule that anything follows from an impossibility not to accept a necessity as a depositum. Rather than talking of cotenability here we could use the so-called 'might' counterfactual for example: 'if you were to promise me a horse, then I might ask to see it.' The only true counterfactual that can be read off the positio in the example is the relatively uninteresting one: 'if you were in Rome, then you might be a bishop.' If P is false but might be true, then a respondent accepting it as positum and obeying the rules of positio will never be led to contradict himself. If at some point he concedes Q, then the relation between P and Q is such that we can say that if P were so, then Q might be so. Conversely if this is so then, there is a positio starting from P in which Q is conceded. That is what we might say but mediaevals would not have done so since the positio already says it all? 2 Though the procedure of positio cannot be construed in terms of the most recent possible worlds accounts of counterfactual conditionals it

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does recall the use of the so-called Ramsey test in earlier modern attempts to deal with such propositions? 3 The 'metalogical' accounts of counterfactuals proposed by Chisholm, Goodman and Mackie, each in their simplest form propose that a counterfactual 'P o--+ R' is true just in case there are propositions Qb ... , Qn which are true of the actual world, compatible with P, and from which, in conjunction with P, R follows in virtue of some general principle L. The case of the man who is a bishop when he is in Rome illustrates why such accounts must fail. If P and R are logically unrelated and R is contingent then not (P & -.R) is cotenable with P and true. By modus ponens R follows from P and -.(P & -.R). This result trivialises such 'metalogical' accounts of counterfactuals and positio interpreted as one of them. Since the result is known and not opposed in any way in the early treatises we surely cannot regard them as attempts at an account of 'would' counterfactuals but at best as exploring the structure of cotenability. I am not claiming that the procedure of positio cannot be used in the construction of possible worlds. Indeed, my suggestion about the origins of insolubilia will require just the opposite. A positio in which the respondent can succeed in avoiding inconsistency despite the best efforts of a logically omniscient opponent provides a partial valuation of the sentences of their language. This valuation may be extended by Lindenbaum's lemma to yield a possible world. The mediaevals did not know Lindenbaum's lemma but the development of a positio exactly parallels its proof. The basic intuition is expressed in the rule that an impossibility cannot follow from a possibility. What positio provides is a an interpretation of that rule which has it that a proposition is possibly true only if it may consistently be supposed to be true. That is to say there is a consistent valuation in which it is assigned the value true or, equivalently, some possible world in which the state of affairs that it describes is actual. Positio used in this way is especially common in fourteenth century discussions of future contingency. Notice that the arguments here for the 'cotenability interpretation' cannot apply to impossible positio. None of our sources provide rules for concession and denial in such positiones but it would hardly make sense to allow the introduction of arbitrary contingent truths in reasoning there. From an impossible positio we may thus read off, that peculiar being, a true conditional with an impossible antecedent. And this indeed is what Clarembald takes Boethius to derive from the impossible positio set out in Quomodo. The resulting counterfactual will not be contingent,

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however, but rather true in the strongest sense of all, the antecedent contains the consequent. 2. POSITIO AND PARADOX The aim of the respondent in possible, or false, positio is to avoid being forced to concede a per se impossibility. That is to say both a proposition and its contradictory opposite or a proposition like 'a man is a an ass' in which 'the form of the predicate is naturally repugnant to the subject thing. ' 24 The respondent is bound to fail if he starts off by allowing a positum which is inconsistent. Consider for example a twelfth century version of Russell's barber. Suppose that as a matter of fact Socrates is not looking at himself and I posit and it is admitted that Socrates is looking at all and only men who are not looking at themselves. I next propose that Socrates is not looking at himself. This does not follow from the positum and it is true. It should therefore be conceded. My respondent is now required to concede both a proposition and its contradictory opposite. Consequently the positum should not be accepted. Propositions and sets of propositions such as this are the topic of the little treatise De Petitionibus Contrarium. 25 None of the sophisms dealt with there tum upon the application of semantical terms. They correspond rather to what in the earlier part of last century were called the logical paradoxes. The solution to them is perfectly straightforward. The situation described by the propositions is impossible and so imponible. The solution offends no intuition and seems to threaten no theoretical claims. It is quite otherwise with two other classes of propositions. If I posit that Socrates utters only the sentence 'what Plato says is false' and that Plato utters only the sentence 'what Socrates says is true' and I then propose 'what Plato says is false,' it seems that no consistent response is possible. 26 Yet the positum is perfectly well formed and could easily enough be realised in practice. Our first intuition is surely that the propositum must have a determinate truth-value but a moment's reflection seems to show that if it is either true or false, then if is both true and false. Something deeper is wrong here and Buridan and the rest of the fourteenth century theoreticians of insolubi/ia try to save the intuition at the expense of naive Aristotelian semantics. Neither TE nor OP notices insolubilia generated in this way but TE warns of and OP explores an even more pathological phenomenon. In

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order to generate a Liar paradox we need to find a predicate 'P' uniquely true of the sentence 'if a sentence is P, then it is false.' Today we might use the demonstrative expression 'this sentence' or the Godel code of the sentence for 'P.' The second alternative obviously was not available in the twelfth century and curiously the first seems not to have been employed. A suitable predicate was available, however, from positio. The sentence 'the positum is false' is perfectly well formed and true when uttered by a witness to a false positio. TEF, however, insists that neither it nor anything convertible with it is acceptable as a positum. The hypothesis that I would like to propose is that the earliest mediaeval interest in insolubilia developed out of a concern with avoiding just this sort of positum. The best evidence for this, I think, is the form taken by the earliest known solution. 27 TEF describes the problem created by the enuntiabile that a falsehood is posited as follows: 28 For suppose that the enuntiabile that a falsehood is posited could be posited, there would follow a contradiction in this way if that a falsehood is posited were posited, and afterwards it were said 'let the time cease' and asked 'was the positum false or was it true?.' If it was true, therefore that a falsehood is posited was true, therefore a falsehood was posited. But nothing except that. Therefore that was false. And that was the positum. Therefore the positum was false. If it was false, therefore that a falsehood is posited was false. Therefore a falsehood was not posited. But something was posited. Therefore a truth. But nothing except that. Therefore it was true and it was said that it was false. On account of this it should be said that this may not be posited since from its positing there would follow a contradiction.

To dmw a parallel with modem work on the Liar let us say that an enuntiabile is ungrounded with respect to positio just in case it cannot be consistently assigned a truth value as the positum of a positio. I take it that the problem with admitting that the positum is false as a positum is the implication that the positum expresses an enuntiabile. Since an enuntiabile must be either true or false the proof shows that if the sentence is admitted it cannot express an enuntiabile. We may also say that a sentence is ungrounded in a positio just in case it cannot be consistently assigned a truth value in that positio. Buridan, reinterpreting ungroundedness in a positio as what George Hughes has called 'contextual inconsistency,' will argue that such sentences are false. 29 The first problem for a respondent in positio is thus to decide whether he is confronted with a claim of possibility which when demodalised yields an enuntiabile ungrounded with respect to positio. OP and TE assume without argument that such enuntiabilia are

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equivalent to that a falsehood is posited. 30 Some simple examples from TEF are that the positum has the same truth-value as a falsehood and that the positum has the opposite truth-value to a truth. Let us say that these enuntiabilia are simply ungrounded with respect to positio. They cannot be posited by anyone or at any time without becoming circular. In contrast there are enuntiabilia which are only accidentally ungrounded with respect to positio. For example that the positum has the same truth-value as that you are a bishop. This enuntiabile will be ungrounded with respect to positio just in case you are not a bishop, which is to say just in case the world happens to be a certain way. The same holds mutatis mutandis of the disjunction or conjunction of the enuntiabile that the positum is false with a contingent enuntiabile. Insisting that the respondent not accept as a positum an enuntiabile which has 'positum' for it's subject term will not save him. It will exclude perfectly acceptable enuntiabilia while failing to trap all insoluble posita. I will justify the qualification 'insoluble' in a moment. One example mentioned by TE is that Socrates posits a falsehood and another that a falsehood is conceded. The first is accidentally ungrounded with respect to a positio in which Socrates is the opponens. The second certainly describes a possibility since it is true of most well conducted positiones. Furthermore no inconsistency follows if we suppose simply that it is posited. It is ungrounded, however, with respect to any positio in which it is proposed since as positum it must not be denied but if it is conceded, it can no longer be consistently assigned a truth value. Though TEF does not prove it, it is easy to see that on being conceded that a falsehood is conceded becomes equivalent to that the positum is false. TEF offers no instructions to a respondent for dealing with cases like this but an appropriate answer can I think be recovered from the treatment of two other enuntiabilia. The first of them is decidedly peculiar. The enuntiabile that the positum and the propositum have the same truth-value can be maintained, according to TEF, if the casus falls out in one way but not if it falls out in another. If the first propositum is true then the positio is perfectly safe. This, incidentally, shows us, I take it, how TEF would deal with the truth positing enuntiabile that the positum is true. If we start of by agreeing to maintain its truth, no inconsistency can result. On the other hand, that the positum and the propositum have the same truth-value cannot be posited if the first propositum is false. In that case, according to TEF, the positio is destroyed (interimitur) by the proposal.

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But what exactly is it that is destroyed. Throughout TEF the distinction between sentences and enuntiabi/ia, or dicta, is carefully observed with the latter picked by means of accusative plus infinitive or quod constructions. The aim in positio is to answer as if the posited enuntiabile were true. Unlike Sherwood, TEF allows that the positum may be proposed in a different form of words to that in which it was accepted. If the positio is destroyed, it is because the positum can no longer be treated as true and this because it can be neither be true nor false. But in that case the positum sentence can no longer express an enuntiabi/e since all there is to being an enuntiabile is being either true or false. Putting this in more modem terms the problem is that of responding to utterances of sentences in situations in which they fail to express propositions. TEF does not say how such proposals should be greeted but I think that it would require that the respondent say 'nugaris' or 'nil dicis,' 'you aren't saying anything.' This at any rate is what we are told to say to the propositum 'mulier a/bus est' in a positio with positum 'hec vox mulier sit masculini generis.' The point is that the metalinguistic positum says nothing about the gender of a/bus and we should not give a reply to the propositum which presupposes its congruity. 31 If this is the appropriate response for an insoluble positum according to TE, then it agrees with the OP. Both works maintain that that the positum is false and anything convertible with it is unacceptable as a positum but the sophisms that appear in OP are very much more complicated than those presented in TEF. For example, prior to Duns Scotus' criticism of the account of time and modality presupposed by the theory of positio a standard rule was what we may call 'the Present Instant Principle' (PIP): 'If A is the name of the present instant, then, if it is proposed in a false positio, "A exists" must be denied.' OP thus sophistically proposes that it is possible that I should concede that A does not exist. This indeed seems eminently plausible since it is just what I must do in a false positio. Thus, unsuspecting, I allow it to be posited. Though the dialectic is a little complicated my opponent can relatively quickly convict me of an improper response no matter how I reply to his proposals. I attach the details in an Appendix. What has gone wrong of course is that I have admitted a proposition simply ungrounded with respect to positio. Here is the proof: 'if you should concede that A does not exist, you should concede a falsehood, and if you should concede a falsehood, then the

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positum is false' Conversely 'if the positum is false, that A exists is to be denied, (and) from this you should concede that A does not exist. ' 32 OP offers the following advice: 33 ... one should consider whether in the time of the positio, on account of one propositum or another the <positum> might become convertible with that a falsehood is posited or not. And if it does so so, it should be said that the <positum> falls at the time at which it is convertible with that a falsehood is posited. Whence if after that time 'stop the time' should be said, or something be proposed, one should say 'nugaris,' and respond to the proposita as if there were no positum. For indeed it has then departed (cessit). If it does not become convertible with that a falsehood is posited in the time of the positio, one should consider whether an insoluble arises from it and if so then the casus is terminated for you.

In the exercises which follow OP sophistically proposes that it is possible that you are a man is repugnant to the positum and what has been conceded. 34 Again it certainly looks possible but if we accept it as our positum we are by a rather complicated series of proposita led to embarrassment. The solution is to notice that at a certain point the positum became convertible with that a falsehood is posited and thereupon departed (cessit). At that point we should have replied 'you are talking nonsense' ('nugaris'). Again I attach the details in an Appendix. What has happened, of course, is that the positum sentence has become ungrounded with respect to positio. The opponent may utter it but if he does, he succeeds in saying nothing at all. Exactly the same may happen in depositio, that form of obligatio in which one has to respond to propos ita as if the enuntiabile accepted as the depositum were false. The skill in depositio is essentially that of applying modus to/lens, of grasping at a glance the antecedents of the depositum. The hazard is the same as in positio and it will only be avoided if one is able to detect propositions that are or will become ungrounded with respect to depositio by becoming convertible with the enuntiabile that the depositum is false. OP's description of what happens here is, however, interestingly different from that which it gives for positio. In depositio just as in positio the Parvipontanian machine is a threat to those who take it seriously. They must not accept as the depositum a disjunction of two contradictory opposites: 35 A necessary (enuntiabile) is not to be accepted in depositio by those upholding the rule that from an impossibility anything would follow. From this it is clear that a

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disjunction consisting of contradictory opposites is not deponible. Thus this has to be disambiguated: let it be deposited that there will be a naval battle or there will not. If it is intended to deposit the dictum as a disjunction, then the depositio should be cassed. For, as Aristotle says in the Perihermenias, it is necessary. And thus the intention is that there should be a determinate depositio. If the depositum is disjoined, the depositio is to be admitted, whether it comes about determinately or indeterminately. And a like judgment is to be made of other disjunctions consisting of contradictory opposites. Those which have for one of the contradictory opposites a necessary (enuntiabile) and for the other an impossible (enuntiabile) should be accepted in deposition only for the impossible, unless an insoluble should prevent this. On account of this the disjunction: that a falsehood is deposited or not deposited is not deponible, neither for the whole nor for the parts because that a falsehood is not deposited is per se necessary, that a falsehood is deposited is impossible and is as well insoluble if it is deposited.

The proof of its insolubility is the standard one. The consequence is that: 36 Its depositio, therefore, should be cassed as should that of everything convertible with it either ut nunc or ut simpliciter.... Furthermore. Other circumstances which cause insolubles in false positio are also to be cassed in depositio.

In Scotland 'cassed' is still available as a translation of cassatur. It is used in the law in exactly the right sense of cancellation or annulment. The solution to the problem of propositions ungrounded with respect to positio or depositio found in OP and TE is thus cassatio, apparently the earliest of the solutions to the Liar. The word is certainly cassatio rather than quassatio. 37 What is proposed is cancellation and annulment not shaking and breaking. The terminology of obligationes indeed reeks of the law and it would certainly be worthwhile to investigate late twelfth century thinking about legal disputation as a source for the structure adopted in regimenting the logical procedure. It seems to me that it makes much more sense to speak of cancellation or annulment with respect to a positio than it does apart from that context. I would thus suggest not only that twelfth century logicians first became aware of insolubles via obligations but also that they developed their first solution in terms of them. To see what that solution really amounts to, however, we must now tum finally to what seems to the oldest surviving treatise entirely devoted to insolubility, the Insolubilia Monacensis (!M), which is preserved in the same manuscript as TEF and TEl.

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3. INSOLUBILIA, CASSATIO, AND TRUTH VALUE GAPS Like both TE and OP, IM is careful to maintain a clear distinction between the propositional content expressed in the assertive utterance of an indicative sentence, that is the enuntiabile, or dictum, and the sentence itself, the propositio. Strictly speaking, according to IM, the enuntiabile that what I assert is false is not insoluble. It would only be so if no appropriate answer could be given to someone proposing it. IM does not explicitly appeal to positio here but clearly needs it to guarantee that this is the only thing that I say. 38 In such a positio that what I assert is false is ungrounded. However, far from there being no suitable response to someone who proposes only this, IM claims that it, and any questions that may be asked about it, can be answered. The questions suggested are: 'is what I assert false?', 'is what I assert true?', 'do I assert anything?'. We are not told the answers but we will see that they must be 'no,' 'no' and 'no.' Like later treatments of insolubilia IM appeals to the Aristotelian distinction between being X without qualification and being X in some qualified way. The later treatments will have the Liar sentence false simpliciter but true secundum quid. IM on the other hand has it insoluble simpliciter but soluble secundum quid. What is meant, I think, is that neither 'verum est' nor 'fa/sum est' is an appropriate response but nevertheless there is one. This, I suppose, is arguably evidence that the solutions which apply Aristotle's distinction to truth and falsity had not been developed when IM was written. IM goes on to claim that enuntiabilia insoluble secundum quid are all characterised by circularity: 'what is insoluble is a circular and necessary deduction to both sides of a contradiction. ' 39 It follows from its account of the significant use of sentences that there are three ways in which an enuntiabile might force itself to go round in circles. IM distinguishes between enuntiabilia, the physical and mental acts involved in presenting enuntiabilia to ourselves and others, and the results of those acts, the token physical, or mental, embodiment of the enuntiabilia. For example, in an act of assertion I assert the enuntiabile that snow is white by uttering a token of the sentence 'snow is white' assertively. That snow is white is of course perfectly straight. For there to be the possibility of a circular deduction the enuntiabile must in some way refer to one of the three elements required to make it manifest-the enuntiabile itself, the act in which it is made manifest, and the sentence which manifests it. For example in asserting that I assert a falsehood I

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refer, or try to refer, to the act of assertion and so long as the only thing that I utter is the corresponding sentence the circular deduction of both parts of a contradiction is immediate. IM proposes in effect that an enuntiabile of the form that what I X is false is simply insoluble with respect to the act of Xing. Thus that what I concede is false is simply insoluble with respect to conceding and that what I think is false with respect to the mental act of thinking. According to IM, the solution secundum quid to the problem of responding to speech acts of the form 'what I X is false' is to realise that with such an utterance I cannot X on pain of circularity. With it posited, of course, that this is the only thing that I say out loud, if X is a physical speech act, or say to myself, if it is a mental speech act. The appropriate response is thus to say 'you aren't Xing anything.' In general we must cass the act of Xing just as we cass the act of positing when the proposed positum is ungrounded with respect to positio. While cassing seemed to make good sense in the context of a positio it is rather less clear what it amounts to here. IM goes on to offer an argument in favour of the objection that in asserting that what I assert is false I must assert something: 40 To assert something is none other than first to judge and then to utter. But the judgement is not to be cassed, since from it no embarrassment would follow. For suppose that someone judges but does not utter and one may with impunity maintain that it (i.e. the enuntiabile) is false. Similarly the utterance is not to be cassed nor is it usual to cass it. And so since neither the judgement nor the utterance is to be cassed and these two are assertion, the assertion is not to be cassed.

1M's solution parallels the account of the negation of implications found in the treatise de Implicationibus in the same manuscript. Neither assertion nor utterance alone is to be cassed but rather the one in respect of the other. That is to say, I take it, that when I respond to your words with 'nil dicis' my claim is that you failed to achieve the combination of judgement and utterance necessary for assertion. As it stands the solution is entirely schematic and ad hoc-assertion succeeds except when it cannot. Furthermore the enuntiabile itself still seems to be insoluble secundum quid. Since, surely, if I fail to assert anything with an assertive utterance of 'what I assert is false,' then the enuntiabile that what I assert is false is false etc. Whether IM has an answer will tum upon what it supposes an enuntiabile to be. If it can show that somehow there is no enuntiabile in this case, then perhaps it does have an answer. Here we need to know more about 1M's theory of

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meaning. The question of the nature of enuntiabilia is raised directly but it is hard to see whether the answer helps with our problem: 41 ... the appellation 'that a human being runs' stands for an enuntiabile. Suppose it is asked what that is, whether a thing or an expression or an understanding. In so far as we are dealing with insolubilia, we should maintain that an enuntiabile is an understanding or a conjunction of things or of understandings.

Later, addressing the problem of the insolubility that arises when an enuntiabile is in some way about the enuntiabile itself, IM develops this point a little in a version of Grelling's paradox almost identical with that presented recently by Tyler Burge.42 Unlike Burge, however, IM is happy to solve the paradox by forcing a gap between truth and falsity even here: 43 ... the third species of insoluble which comes from the existence of the insoluble. And for this reason the enuntiabile should be cassed. But in cassing the enuntiabile we do not cass the substantial but rather the accidental. Granted that an enuntiabile is a thing or an understanding, nevertheless it is not a thing nor an understanding which is cassed but rather a conjunction of things or understandings. For since the conjunction of things or understandings is enuntiabile and that is accidental, the enuntiabile may indeed be cassed.

To cass an enuntiabile it seems that we must say that it is not an enuntiabile ('Propter hoc debet dici quod non est enuntiabile').44 Is there a confusion here between the enuntiabile as an understanding and what it is that may be asserted, a combination of things or understandings and presumably of things and understandings? Perhaps the intended answer is a development of the familiar notion of semantic congruity. An enuntiabile is an understanding consisting of a combination of understandings of things or, presumably, of understandings. A combination which cannot occur cannot be understood and in consequence cannot constitute an enuntiabile. Whatever the details of their epistemology and semantics the cassers have in effect provided a truth-gap solution to the problem of insoluble sentences. Assertive utterances may be either true or false or say nothing at all. Let me conclude by considering briefly how well they understood the demands of this kind of solution. An account the Liar should do its best not to offend too many of our intuitions about meaningfulness and truth. While the claim that if I say only 'what I say is false,' I really say nothing at all perhaps conflicts with a general intuition about the meaningfulness of well-formed sentences, it seems to accord well with our intuitions about this particular instance. If the author of the treatise on insolubles in B. N. Lat. 11.412

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is to be believed it also agrees with the intuitions of the rusticus whose opinion is canvased there. Our intuitions about a sentence like 'this sentence is in English' are quite different and it would be highly offensive to them to propose that in uttering it I am merely babbling. IM does not deal with such sentences but presumably would have to assign this one to the truths. More problematic is the 'truth-teller,' the enuntiabile that what I assert is true, which locates itself exactly at the point at which our intuitions about the connection between meaningfulness and truth-value become unsure. It is certainly not paradoxical but if we assign it a truth-value we are led immediately back to that assignment in an infinite circle. IM agrees that the sentence is not insoluble in the sense that a contradiction follows from it but against indifference it offers an argument from symmetry. Suppose that of two people hearing the sentence one remarks that it is false and the other that it is true. They cannot both be right so they must both be wrong. Some grant the argument, IM tells us, and consign the enuntiabile to the insolubilia and the utterance, presumably, to those with which nothing is said. That is perhaps the best thing to do with it since there can be no matter of fact to settle its truth-value. IM does not agree. Prior to the utterance of the corresponding sentence the enuntiabile is surely false. When it is uttered it either remains false or it becomes true. The argument presumably is that it certainly doesn't become both true and false and unlike the Liar there is no reason to suppose that it becomes neither. It seems that the enuntiabile is either true or false but we cannot say, or even know, which. Though solutions like cassatio can reasonably claim to have dispatched the Liar they can never rest content in their victory. Rather they must be perpetually on guard against an army of its fortified descendents threatening to pour through the gap. One such descendent arrives to take up the challenge in IM. In an objection to the rule that if an insoluble is disjunctively combined with a true enuntiabile the disjunction is true but if with a false enuntiabile the whole is insoluble we are introduced to the Son of the Liar: 45 The enuntiabile that I assert something false or that I assert nothing is either true or it is false. If it is tme, but it is not tme on account of the part: that I assert nothing, therefore it is tme on account of this part: that I assert something false; therefore it is tme that I assert something false, therefore I assert something false, and nothing except this, therefore this is false, and it was said to be true. If it is false, therefore in virtue of both parts, therefore in virtue of this one: that I assert something false, therefore it is false that I assert something false, therefore I do not

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assert something false, and I assert something, therefore something true, and nothing but this, therefore it is true and it was said to be false.

But what about the possibility that I assert nothing with an utterance of the sentence? JM notes that that is just what is usually said but seems to tire of difficulty here. It follows, it claims, that if I assert nothing, then the disjunction that I assert is true. But then, of course, if the disjunction is true, it is not true and if it is not true, it is true. All IM has to say is that its rule was not intended to cover disjunctions like this in which the same act appears in both disjuncts. Though it runs through the proof, it seems not to notice that in agreeing that I say nothing with this disjunction it has handed victory over to the Son of the Liar. What would an appropriate resolution be? Recent gappy accounts of the Liar have suggested a number but all seem to be objectionable in one way or another. In any event the critics of the cassers did not trumpet the return of the Liar. Rather, conflating assertion and utterance, they contented themselves with the facile request that we read their lips for a refutation of cassatio46 • This, of course, would not have impressed the cassers but exactly what the details of their semantical theory were we cannot yet say. Until that problem is solved we will not be able to fully evaluate their analysis of the Liar. Let us hope that it can be solved, and soon, since cassatio is arguably the most interesting and certainly the least contrived of the mediaeval treatments of the test case for semantical theories. APPENDIX: TWO SOPHISMS FROM THE OBLIGATIONES PARISIENSES

I give these two examples to illustrate not only the points made in the text but also the complexity of obligational exchanges. As respondens your aim is to avoid conceding a set of proposita from which a contradiction follows. These two sophisms show how difficult this may be to do in practice. Sophism 1: It is possible that you should concede that A does not exist. 1. It is possible that a possible falsehood is posited. 2. If a possible falsehood is posited, then you should deny that A exists. 3. If you should deny that A exists, you should concede that A does not exist.

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4. Therefore it is possible that you should concede that A does not exist. Let the possibility be posited: Let Q be 'A exists.' The Positum (Po), is 'you should concede that not Q.' Let the time of obligation cease. Po was true~ Po was false 0. [Proof:] 1. Po was true Q was true 2. Q was not repugnant to Po 3. You should not have 4. denied Q You should not have 5. conceded that not Q 'You should concede that 6. not Q' was false Po was false 7. Po was false~Po was true 8. [Proof:] 9. Po was a possible falsehood I 0. If a possible falsehood is posited you should deny Q II. You should have denied Q 12. You should have conceded not Q 13. 'You should have conceded not Q' was true 14. Po was true

Claim. Hypothesis. Positio took place at A. A truth is not repugnant to a truth. 3, rules for positio. 4, rules for positio. 5, quotation. 6, substitution. Claim. Hypothesis. PIP. 9, I 0. 11, rule for positio. 12, quotation. 13, substitution.

Sophism I 0: It is possible that you are a man is repugnant to the positum and what has been conceded. Let it be posited. Let Q be 'you are a man.' Let Cat step n be the conjmction of everything conceded before step n. Let Pri be the ith propositum. Let R(P, Q) be 'P is repugnant to Q.' The positum, Po, is 'R(Q, (Po & C).'

OBLIGATIONS AND LIARS

0. Q is repugnant Po & C [Proof:] 1. Only this, indicating the sentence displayed at 0, has been posited. 2. Not Pr 1 3. 4.

5. 6.

7.

8.

9.

85

Po (the positum). Pr 1 (the first propositum ).

Suppose that the respondens denies Pr 1•

Let the time of obligation cease. You have denied something Pr 1 is true. true which is not repugnant to the positum so you have responded badly. Pr 1 Suppose that Pr 1 is conceded by the respondens. No truth has been denied Pr2 • True and not repugnant to what has already been conceded so it should be conceded by the respondens. No falsehood has been Pr3 . True and not repugnant to conceded what has already been conceded so it should be conceded by the respondens Let the time of obligation At step 8: C = Pr 1& Pr2 & Pr3 • cease. At step 8 Po was either true or false. Po was true~ Po was false Claim.

10. [Proof:] Po was true Hypothesis. 11. 12. Cwas true 5, 6, 7. The respondens is a man. 13. Q is true Truth is not Q was not repugnant to 14. (Po & C) repugnant to truth. 14, Quotation. 15. 'Q is repugnant to (Po & C)' was false 16. Po was false. Substitution. 17. Po was false~ Po was true Claim. [Proof:] Hypothesis. 1 8. Po was false

CHRISTOPHER J. MARTIN

86 19. 20.

(Po & C) - not Q Q is repugnant to Po & C

21.

Po was true

(i) To be proved. 19 instantiates the definition of repugnance. 20, quotation.

It remains to prove (i): (R(Q, (Po & C)) & C) -

SO.

(R(Q, (Po & C)) & C)not Q

[Proof:] Sl. R(Q, (Po & C)) & C S2. 'R(Q, (Po & C))' is true S3. Only 'R(Q, (Po & C))' is posited S4. Po is true Pr 1 is true. S5. S6.

Pr2 is true.

S7.

Pr3 is true.

S8. S9.

Q is repugnant to truth Q is false

S10.

not Q

not Q

Claim.

Hypothesis. S 1, Simplification and quotation. PrJ. conceded at stage 1 of the main argument. S2, S3, substitution. Stage 5 of the main argument, quotation. Stage 6 of the main argument, quotation. Stage 7 of the main argument, quotation. S2, S4, S5, S6, S7. Whatever is repugnant to truth is false. S9, disquotation.

Solution: 'The positio (Po) is to be accepted and the proposita (PrJ. Pr 2 , Pr 3) are to be conceded. But when (the opponens) says "let the time of obligation cease" after the last concession (i.e. at stage 8 of the main argument) you should say "nugaris" because (the positio) ceases in the time of the last concession and in "let the time etc" (the positum) is convertible with "the positum is false."' Proof that the positum is convertible with 'the positum is false':

NO.

Po is false~ R(Q, (Po & C))

[Proof:] N 1. R(Q, (Po & C)) Po is false [Proof:] N2. R(Q, (Po & C))

Claim.

Claim.

Hypothesis.

OBLIGATIONS AND LIARS N3. N4.

R(a truth, (Po & C)) Po is false or Cis false

N5.

Po is false

N6.

Po is false~ R(Q, (Po & C))

[Proof:] N7. Po is false s~ not Q N8.

N9. NlO. Nil.

R(Q, S) S= (Po & C) R(Q, (Po & C))

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Q is true. N2, definition of repugnance, De Morgan. C is not false, disjunctive syllogism. Claim.

Hypothesis. N7, modus ponens, since the conditional is true and necessary (ii) and a necessity follows from anything. Where S is the conjunction at any stage of the positum and concessa. N8, definition of repugnance. As soon as Pr3 is conceded. N9, N I 0, substitution.

(ii) is not proved in OP but the following is presumably what is intended. ('S' is at any stage '(Po & C)'): CO. (Po & C) ~ not Q [Proof:] Cl. Po C2. C C3. R(Q, (Po, C)) C4. (Po & C) ~ not Q C5. not Q C6. (Po & C) ~ not Q

Claim. Hypothesis. Hypothesis. Cl, stating Po. C3, definition of repugnance. Cl, C2, C4, modus ponens. C I, C2, conditionalisation.

Note that while C6 is equivalent to C3, C3 holds only under the two hypotheses which have been discharged to obtain C6. Since S becomes equal to (Po & C) at the moment Pr 3 is conceded, 'up until then the positum may remain, but in the moment of last concession it falls.' For the Latin text see de Rijk 1975; sophism I, p. 33; sophism I 0, pp. 40-41. University of Auckland

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CHRISTOPHER J. MARTIN NOTES

1

See de Rijk 1966; Spade 1973, 292-309; Spade 1987, 24-46. See in particular Spade 1987, which discusses St Paul's letter to Titus, Gellius' Noctes Atticae, Cicero's Academica, and Augustine's Contra Academicos. The earliest use of the distinction between being X simpliciter and being X secundum quid in the solution of the Liar seems to date from the beginning of the thirteenth century. There is much disagreement over the appropriate X 3 Kripke 1975, 690-716. 4 For the beginnings of an account of the different logics invented in the twelfth century see Martin 1988. 5 OP, 52.10-11 :" ... impossibile ponendum est ut videatur quid inde sequatur"; cf. TEl, 117.28-31: "Quod impossibilis positio habeat sustineri sic probatur. Sicuti enim nos dicimus quod possibile est concedendum ut videatur quid inde sequitur, similiter habemus ab Aristotile quod impossibile est concedendum ut videtur quid inde accidat." TEF, 103.21-23 gives the justification for postio possibilis, which it calls positio falsi, or positio fa/sa. 6 Boethius 1969, I, ii, 5--6 (PL 64, 833D): "Sed quoniam de hypotheticis loquimur, quid significet hypothesis praedicendum est. Hypothesis namque, unde hypothetici syllogismi accepere vocabulum, duobus (ut Eudemo placet) dicitur modis: aut enim tale adquiescitur aliquid per quamdam inter se consentientium conditionem, quod fieri nullo modo possit, ut ad suum terminum ratio perducatur; aut in conditione posita consequentia vi coniunctionis vel disiunctionis ostenditur. Ac prioris quidem propositionis exemplum est, veluti cum res omnes corporales materiae formaeque concursu subsistere demonstramus. Tunc enim quod per rerum naturam fieri non potest ponimus, id est omnem formae naturam a subiecta materia, si non in re, saltern in cogitatione separamus; et quoniam nihil ex rebus corporeis reliquum fit, demonstratum atque ostensum putamus eisdem convenientibus corporalium rerum substantiam confici, quibus a se disiunctis ac discedentibus interimatur. In hoc igitur exemplo posita consentiendi conditione, ut id paulisper fieri intelligatur quod fieri non potest, id est ut formae a materia separentur, quid consequatur intendimus, perire scilicet corpora, ut eadem ex iisdem consistere comprobemus." 7 Garlandus Compotista 1959, 127.7-8:" ... quando aliquid impossibile conceditur ut "homo est lapis", gratia videndi a quem finem ratio perveniat ... " 8 Abaelard 1970, Dialectica IV, 1, p. 471.28-30: "Per consensum autem et concessionem hypothesis propositio ilia dicitur quae non in se vera recipitur, sed ~ratia argumentandi conceditur, ut quid ex ea possit extrahi videatur ... " Thierry of Chartres 1971, 421.24; 423.76, and Clarembald of Arras 1965, 212.4sq. 10 See Martin 1988. 11 See Normore 1987. Abaelard considers the claim that nothing grows and agrees that it is true if you identify a thing with its integral parts and by growth you mean that it undergoes a multiplication of these parts. His response is that this is not the proper way to understand growth. To mark growth we must locate something that remains fixed as extra parts are added. What remains fixed, Abaelard claims, is at some level of generality the kind of composite. Thus if to a heap of three stones I 2

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add a fourth, the heap of stones grows but the heap of three stones does not. If I add a horse to a collection of three men the size of the collection of animals increases. If God adds matter to the flesh of a child, the child becomes larger. Abaelard' s view seems to me to be exactly the opposite of that constructed for him by Normore. He holds, to use Normore' s terminology, but not I think his own, that things certainly grow but that growth only occurs where some status remains the same. 12 Jacques of Vi try 1914; cf. exemplum 105, p. 62. 13 Spade and Stump 1983. 14 See Martin 1988. Spade and Stump 1983, n. 22, also make a curious mistake in supposing that 'dubie' is the answer to be given when one is in doubt. It is rather the form of the answer. One responds doubtfully by giving the answer 'proba!.' What we have here is not, as they suggest, a new and an old terminology but simply the standard response and a description of it. See for example OP, 46.24-7: "Si dicat ad hanc 'Antichristus est coloratus,' 'proba!': cedat tempus. Petitum erat ut non responderes dubie ad aliquid nisi ad quod directe obligatus esses ad dubitandum." 15 But see Yagisawa 1988. 16 Spade 1982. 17 Stump 1981. 18 Curiously, something more like 'would' counterfactual reasoning is found in OP's very brief account, p. 28.3-8, of the way in which one resonds in the species of obligation known as 'rei veritas.' If it has been established in rei veritate that the Antichrist exists and it is then proposed that the Antichrist is white, the proper response is 'proba!.' The Antichrist would certainly have a colour but we cannot say which colour. In a positio with the corresponding positum the appropriate response would be 'falsum est' since, according to OP, in positio a propositum which is impertinens is to be denied. Rei veritas thus perhaps anticipates Kilvington's notorious response to Sophism 47. 19 Spade1982, 11. 20 OP, 31.13-27. 21 Pace Normore 1987, 204. The principles are equivalent, however, if the Nominates accept the equivalence of 'P ---t (P & Q)' and 'P ---t Q.' In the the example given the Nominates might perhaps have cited the principle that a conjunction of an affirmative and a negative is negative and their general rule that a negative does not follow from an affirmative. OP, however, states the objection as a general one and so as holding presumably for afTmnative P and Q with Po = P and Pr 1 = (P & Q). For the Nominates' account of the quality of compounds see Martin 1988. 22 I should emphasise that I am not claiming that positio was intended to provide a model of 'might' counterfactual reasoning. The treatment of irrelevant falsehoods means that it is no more useful for that purpose than for 'would' counterfactual reasoning. 23 See for a start the introduction to Harper, Stalnaker and Pearce 1981. 24 TEF, 113.12-21. 25 De Rijk 1976.

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If you concede the propositum, I argue: If it is true that what Plato says is false, then what Socrates says is true, since he says only that what Plato says is false, but if what Socrates says is true, then what Plato says is true since he says only that what Socrates says is true. If on the other hand you deny the propositum, I argue: If it is false that what Plato says is false, then what Socrates says is true, since Plato says only that what Socrates says is true, but if what Socrates says is true, then what Plato says is false. 27 Eleonore Stump has suggested on a number of occasions that there is a connection between the sophisms found in texts on obligations and insolubilia. She is obviously right about TE and OP but in the examples she gives from Sherwood the reference to the respondent in the posita is quite accidental. The opponent needs to find a positum or propositum which will change its truth value while the positio is in progress. He could watch the clock but its a lot easier to fix on the utterances of the respondent. See Stump 1980, 249-264. 28 TEF, 104.7-15: "Dato enim quod hoc enuntiabile 'falsum poni' possit poni, inde sequitur contradictio, si ponatur falsum poni, et postea dicatur 'cedat tempus.' Et queratur. Positum aut fuit falsum aut fuit verum. Si verum ergo verum fuit falsum poni. Ergo falsum ponebatur. Et nichil nisi hoc. Ergo hoc fuit falsum. Et hoc fuit positum. Ergo positum fuit falsum. Si falsum, ergo falsum fuit falsum poni. Ergo falsum non ponebatur. Et ponebatur aliquid. Ergo verum. Sed nichil nisi hoc. Ergo fuit verum. Et dictum <est> quod falsum. Propter hoc debet dici ~uod hoc non potest poni cum ex eius positione sequatur contradictio." Hughes 1982, esp. 18 sq. 30 Neither TE nor OP considers the yet more pathological enuntiabile: that the f.,ropositum is false which can neither be posited nor proposed without circularity. 1 TEF, 110, is here dealing with the problem of how one should answer a quaestio disciplinalis such as quid? or quare? in a positio. It describes the respondens under such circumstances as being in an apparentia-something that seems like the truth but isn't so. This unusual word is used as an alternative to 'fantasia' by the Summa Sophisticontm Elenchorum and in the Fallaciae Parvipontanae in giving the division of sophisms according to 'Alexander.' Both texts are included in De Rijk 1962. Apparently alluding to Top., VIII, 156a7, but giving Aristotle' s advice to the respondent rather than the opponent, TEF maintains that in such cases the respondent should refuse to answer the question 'quia debet celari rei veritas in fa/sa positione.' 'Nugaris' as used by TE does not seem to be connected with refutation by nugatio as described in Soph. El. 3, 165bl5. 32 OP, 33.19-26. 33 OP, 36.16-32: My suggestions for emendations are marked *<>* a suggested addition, or change, *[]* a suggested deletion: "Praeterea. Quoddam enuntiabile potest poni et permanere positum et *<non>* cadere a positione, ut 'Sortes est albus'; quoddam potest poni et permanere positum sed *[non]* potest cadere a positione, ut 'falsum positum **' [edn. 'cadere'] et quodlibet convertibile cum illo. Unde quotiens intendit aliquis ponere, diligenter considerandum est an sit convertibile cum falsum poni annon. Et si sic, non recipiatur. Vel si fiat, positione facta, simile vel convertibile, similiter non recipitur. Si vero non sit nee fiat, recipitur et tum consideretur an in tempore positionis propter propositum a1iquod vel aliud fiat convertibile cum

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'falsum poni.' Et si sic, dicendum quod cadit in eodem tempore positum in quo fuit convertibile cum 'falsum poni.' Quare si post illud tempus dicat: 'cedat tempus' vel proponat aliqua, dicendum: 'nugaris' et ad proposita respondenum est ac si positum non fuerit. lam enim cessit. Si vero non fiat convertibile cum falsum poni in tempore positionis, considerandum est an ex ea fit insolubile et si sic, casus tibi terminatur." 34 OP, 40, reading 'repugnare' for De Rijk's 'respondere.' 35 OP, 48.17-32: "Praeterea. Sustinentibus bane regulam quod ex impossibili sequatur quidlibet, necessarium non est recipiendum in depositione. Et ex eodem patet quod disiuncta constans ex contradictorie oppositis non est deponibilis. Unde distinguenda est bee: deponatur navale bellum fore vel non fore. Et si dictum disiuncte intendat deponere, cassetur depositio. Est enim necessarium, ut dicit Aristotiles in Perihermenias. Et sic fit determinata depositio secundum intentionem. Si depositum disiungatur, admittenda est depositio, sive fiat determinate sive indeterminate. Simile iudicium est de aliis disiunctis constantibus ex contradictorie oppositis. Que autem habent unum contradictorie oppositorum necessarium, alterum impossibile, pro impossibili tantum recipiatur in depositione, nisi impediat insolubile. Et propter hoc bee disiuncta: 'falsum deponi vel non deponi,' non est deponibilis nee pro toto nee pro partibus, quia bee per se est necessaria: fa1sum non deponi, bee autem est impossibilis: fa1sum deponi; et cum hoc est insolubile si deponatur." 36 OP, 49.7-12: "Cassetur ergo eius depositio et omnis convertibilis cum ipsa, vel ut nunc vel ut simpliciter. Ut nunc, ut deponatur depositum esse dissimile te sedere et contingit te sedere in rei veritate. Simpliciter, ut deponatur depositum esse dissimile *dictum esse. Praeterea. Alie circumstantie que causant insolubile in falsa positione, cassande sunt etiam in depositione." (*Should this be 'Deum esse '-something always true? 'Dictum esse' will be true if there is a depositum but whether it is true simpliciter will depend on the theory of dicta at work.) 37 As it is appears in the treatise on Insolubilia in B. N. Lat 11.412 excerpted in De Rijk 1966. 'Cassatio' is a very uncommon term. 'Cassare' is more common and interestingly is used by Abaelard 1970, III.1, p. 255.31-34, in just the right sense to describe the way in which a 'formally' good consecution 'can in no way be cancelled (cassari)' by a uniform substitution of terms. 38 The description of insolubilia in the tract in B. N. Lat. 11.412 is much more explicitly formulated in terms of positio, De Rijk 1966, 94: "Et ut melius pateat quod querendum est, faciamus deductiones ad hanc propositionem: 'ego dico falsum.' Si dicatur quod verum est, sequitur: hoc est verum, et ego dico hoc; ergo ego dico verum; ergo ego non dico falsum; ergo hec est falsa: 'ego dico falsum'; et concessisti ** verum, ergo male;-si dicatur quod falsum, hoc est falsum, et ego dico hoc; ergo ego dico falsum; ergo bee est vera: 'ego dico falsum'; et concessisti ** ergo male." 39 IM, 105.12-13: " ... insolubile est ad utrumque partem contradictionis circularis et necessaria deductio." 40 IM 106.3-8: "Sed contra. Dicere nil aliud est quam prius asserere et postea proferre. Sed assertio non est cassanda, cum ex ipsa non sequatur inconveniens. Dato enim quod asseratur et non proferatur, inpune potest sustineri quod sit falsum.

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Similiter prolatio non est cassanda nee solet cassari. Et ita cum assertio non sit cassanda nee prolatio et ista duo sint dicere, dictio non est cassanda." 41 IM, I 06.18-22: "Et cum tractandum sit de enuntiabilibus insolubilibus, potest queri quid sit enuntiabile. Verbi gratia, hec appellatio hominem currere supponit enuntiabile. Queratur quid sit id, sive res sive voces, sive intellectus. Secundum quod tractandum est de insolubilibus, sustineamus quod enuntiabile sit intellectus sive coniunctio rerum sive intellectuum." 42 Compare: Burge 1979, 169-98: "Suppose I conduct you into a room in which the open sentence type 'it is not true of itself is written on the blackboard. Pointing at the expression, I present the following reasoning: Let us consider it as an argument for its own variable or pronoun. Suppose it is true of itself. Then since it is the negation of the self-predication of the notion of being true of, it is not true of itself. Now suppose it is not true of itself. Then since it is the negation of the self predication of the notion of being true of, it is true of itself." And IM, 115.9-17: "Dato enim quod hoc esset enuntiabi1e: a1iquid non es*<se>* [edn. 'est' ] verum pro se, inde sequitur contradictio sic: hoc aut est verum pro se aut non est verum pro se; si est verum pro se *[non]* ergo ei convenit suus predicatus; sed suus predicatus est non es*<se>* [edn. 'est' ] verum pro se; ergo ei convenit non esse verum pro se; ergo non est verum pro se; et dictum est quod verum; --- si non est verum pro se, sed est verum pro omni eo cui convenit suus predicatus; sed suus predicatus convenit ei; ergo est verum pro se; et dictum est quod non est verum pro se. Propter hoc debet dici quod non est enuntiabile." 43 IM, 115.4-9: "Sequitur de tertia specie insolubilium. Que provenit ex essentia insolubilis. Et propter hoc debet cassari enuntiabile. Sed cassando enuntiabile non cassatur substantiate, sed accidental e. Licet enim enuntiabile sit res vel intellectus, tamen non cassantur res vel intellectus, sed coniunctio rerum vel intellectuum. Cum enim coniunctio rerum sit enuntiabile et ilia sit accidentalis, enuntiabile bene p4otest cassari." IM, 115.17. 45 IM, 111.9-19: "Hoc enuntiabile, me dicere falsum vel nil dicere, aut est verum aut est falsum. Si verum, sed non est verum ratione istius partis, me nil dicere; ergo est verum ratione istius partis, me dicere falsum; ergo verum est me dicere falsum; ergo ego dico falsum; et nil nisi hoc; ergo hoc est falsum; et dictum est quod verum; --- si falsum, ergo ratione utriusque partis; ergo ratione istius, me dicere falsum; ergo falsum est me dicere falsum; ergo ego non dico falsum, et dico aliquid; ergo verum; et nil nisi hoc; ergo hoc est verum; et dictum est quod falsum. Et ita cum hoc sit quoddam verum, me nil dicere secundum commune iudicium respectu huius enuntiabilis, verum est me dicere falsum **. Hie verum enuntiabile disiungitur ab insolubile et tamen ex responsione illius sequitur contradictio." The point of the penultimate sentence-and the emendation-is that the the common view is that me nil dicere is true with respect to the disjunction of which it is one of the disjuncts. The whole disjunction is thus true. 46 SeeP. V. Spade 1973, 307-8.

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ABBREVIATIONS TEF

=

Tractatus emmeranus de falsi positione, published in De Rijk 1974;

translated below in the appendix.

TEl= Tractactus emmeranus de impossibili positione, published in De Rijk 1974; translated below in the appendix.

TE= TEF+ TEl OP = Obligationes parisienses, published in De Rijk 1975. JM = lnsolubilia monacensis published in De Rijk 1966.

REFERENCES Abaelard, Peter, 1970. Dialectica, (ed.) L. M. de Rijk, Assen, Van Gorcum. Boethius, A. M. S., 1969. De Hypotheticis Syllogismis, (ed.) L. Obertello, Brescia, Paedia. Burge, Tyler, 1979. "Semantical Paradox," Journal of Philosophy 76, 169-98 Clarembald of Arras 1965. "Expositio Super Librum Boetii 'De Hebdomadibus,"' in Nikolaus Haring (ed.), Life and Works of Clarembald of Arras, Toronto, Pontifical Institute of Mediaeval Studies. De Rijk, Lambertus Marie, 1962. Logica Modernorum, vol. 1, Assen, Van Gorcum. De Rijk, Lambertus Marie, 1966. "Some notes on the Mediaeval tract De insolubilibus, with the edition of a tract dating from the end of the twelfth century," Vivarium 4, 83-115. De Rijk, Lambertus Marie, 1974. "Some Thirteenth Century Tracts on the Game of Obligation I," Vivarium 12, 94-123. De Rijk, Lambertus Marie, 1975. "Some Thirteenth Century Tracts on the Game of Obligation II," Vivarium 13, 22-54. De Rijk, Lambertus Marie, 1976. "Some Thirteenth Century Tracts on the Game of Obligation III," Vivarium 14, 26-49. Garlandus Compotista 1959. Dialectica, (ed.) L. M. de Rijk, Assen, Van Gorcum. Harper, W. L., Stalnaker, R., and Pearce, G., (eds.) 1981. Jfs, Dordrecht, Reidel. Hughes, George, 1982. John Buridan on Self-Reference: Chapter Eight of Buridan 's Sophismata, (An Edition and a Translation with an Introduction and a Philosophical Commentary), Cambridge, Cambridge University Press. Jacques of Vitry 1914. Die Exempla des Jalwb von Vitry, (ed.) Joseph Graven, Heidelberg, Carl Winter. Kripke, Saul, 1975. "Outline of a Theory ofTruth," The Journal of Philosophy 72, 690-716. Martin, Christopher J., 1988. "Embarrassing Arguments and Surprising Conclusions in the Development of the Theory of the Conditional in the Twelfth Century," in J. Jolivet & A. de Libera (eds.), Gilbert de Poitiers et ses Contemporains, Naples, Bibliopolis, 377-401. Normore, Calvin, 1987, "Medieval Nominalism," in J. Wippel (ed.), Studies in Medieval Philosophy, Washington D. C., Catholic University of America Press.

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Spade, Paul Vincent, 1973. "The Origins of the Mediaeval Insolubilia Literature," Franciscan Studies 33, 292-309. (Reprinted in Spade 1988.) Spade, Paul Vincent, 1982. "Three Theories of Obligationes: Burley, Kilvington and Swyneshed on Counterfactual Reasoning," History and Philosophy of Logic 3, 1-32. Spade, Paul Vincent, 1987. "Five Early Theories in the Mediaeval lnsolubilia Literature," Vivarium 25, 24-46. Spade, Paul Vincent, 1988. Lies, Language and Logic in the late Middle Ages, London, Variorum Reprints. Spade, Paul Vincent, and Stump, Eleonore, 1983. "Walter Burley and the Obligationes Attributed to William Sherwood," History and Philosophy of Logic 4, 9-26. Stump, Eleonore, 1980 "William of Sherwood's Treatise On Obligations," Historiographia Linguistica 7: 1/2, 249-264. Stump, Eleonore, 1981. "Roger Swyneshed's Theory Of Obligations," Medioevo 7, 135-174. Thierry of Chartres 1971. "Commentum Super Ebdomadas Boetii," in Nikolaus Haring (ed. ), Commentaries on Boethius by Thierry of Chartres and his School, Toronto, Pontifical Institute of Mediaeval Studies. Yagisawa, Takashi, 1988. "Beyond Possible Worlds," Philosophical Studies 53, 171-204.

FABIENNE PIRONET

THE RELATIONS BETWEEN INSOLUBLES AND OBLIGATIONS IN MEDIEVAL DISPUTATIONS

I. IN1RODUCTION Almost as ancient as philosophy, the paradox of the Liar in all its forms has always disturbed logicians because it questions several basic principles of logic and hence undermines its own foundations. The paradigmatic insoluble, 'What I say is false' is a proposition that, made in certain circumstances, puts forward paradoxical evidence due to its self-reflexive form and to the nature of its alethic predicate. That is why it leads to a vicious circle: a)

b)

if the proposition 'What I say is false' is true, it is true that what I say is false, therefore what I say is false, and what I say is the proposition 'What I say is false,' therefore the proposition 'What I say is false' is false; if the proposition 'What I say is false' isfalse, it is false that what I say is false, therefore what I say is true, and what I say is the proposition 'What I say is false,' therefore the proposition 'What I say is false' is true.

Does this mean that, the same proposition being at the same time true and false or, what is no better, true and false by turns, our principles of bivalence and non-contradiction are not guaranteed anymore? Any logician would say no. Let us see how medieval logicians tried to save us from this threat. One of the oldest medieval solutions consisted in denying any meaning to such propositions. The cassantes (cassers) argued that I can actually utter a proposition such as 'What I say is false,' but if the predicate 'false' refers to the proposition it is a part of, the proposition has no meaning (the link with its meaning is "cassed" or broken), no more meaning, in fact, than there would be if I were to say 'I am deeply asleep' or 'I am dead' 1• 95

M. Yrjonsuuri (ed.), Medieval Fonnal Logic 95-114. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Another solution, more widely adopted than the previous one, consisted in denying the self-reference of such propositions. This means that the predicate 'false' can refer to any proposition but the one it is a part of. Some logicians considered that the scope of this restrictive rule should be wide2 ; i.e. the rule should be applied to any self-reflexive proposition, others that it should be weak, i.e. only applied to insoluble propositions. But it seems that each of these positions easily invites criticism: it is too large in the first case (why should we say that the predicate 'in English' should refer to any proposition but the one it is a part of, e.g. 'What I say is in English'?); it is ad hoc in the second case (why should we create special rules for insolubles?). In spite of the differences between these two solutions (cassation and restriction), we see that they adopt the same strategy: the aim is somehow or other to expel insoluble propositions from all (meaningful) discourse. Another kind of strategy first developed, as far as we know, by Thomas Bradwardine (ca. 129 5-1349) consisted in analysing the notions of truth and signification so as to find a solution inside language itself. Insoluble propositions are not rejected, but, as they signify themselves to be false and, at the same time, themselves to be true (because every proposition signifies or implies 3 itself to be true4 ), it is claimed that they are simply false (because a proposition is true if and only if howsoever it signifies the case to be, so it is; and it is never the case that one and the same proposition is true and false at once). So insolubles are not expelled from all discourse, they have only one truth-value and no paradox results from that. Compared to the solution by weak restriction, this solution has the advantage of not being ad hoc (because the principle applies to eve1y proposition and not only to insoluble ones), and, although widely applied to every proposition, it does not lead to the same inconveniences as the wide-restriction solution does. If you indeed accept the principle that every proposition signifies or implies itself to be true, you do not expel any proposition from all discourse and it does not produce any undesired effect on not-insoluble propositions. This is not the place to catalogue every solution to this problem medieval logicians put forward. There are so many of these, and some of them are so subtle that no one who is acquainted with them could ever doubt the great theoretical interest the question had at that period; when truth was not just a matter of logic, but also of theology. As an extreme example of the logical difficulties which propositions intended to express the truth could face, the discussion of insolubles was integrated into every subject: treatises on reference (suppositio ), on meaning

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(signiflcatio), on inference (consequentia), on ambiguous propositions (sophismata) and on obligation (obligatio). With regard to reference and meaning, the connexion is clear because those notions are closely related to the truth-conditions of propositions. With regard to sophisms, the connexion is also clear since insolubles are a special kind of proposition presenting a problem of interpretation. With regard to inference, S. Read tackles the question in his article "Self-reference and Validity Revisited" below. 2. INSOLUBLES AND OBLIGATIONS Treatises on obligations are, from a theoretical point of view, treatises in which the rules of disputation are established and discussed. The theories which such treatises expound are illustrated and put into practice in obligational disputations, i.e. those "games" in which two interlocutors are opposed; one of whom (the opponent) has to present propositions to which the other (the respondent) has to answer in certain ways, depending on the special kind of obligation chosen5 • The opponent tries to lead his adversary into maintaining contradictory propositions, while the respondent tries to escape from this. In general, propositions proposed by the opponent are set in a casus, a case, a situation t:1at will be accepted by both interlocutors as true, e.g. "Let it be posited that every man is in Rome, I propose to you 'You are in Rome."' Here, the case has to be admitted because it is not logically impossible, and the proposition has to be granted because it follows from the casus. We note here that insolubles and obligations have something in common, the recourse to a casus. Indeed, self-reference in an insoluble proposition only takes place under very precisely-defined circumstances, a casus, outside of which the problem vanishes: the proposition 'What I say is false' will be insoluble if and only if it is the only thing I say; the proposition 'Every proposition is false' will be insoluble if and only if only false propositions exist or if this proposition 'Every proposition is false' is the only one that exists. The case is exactly the same for pragmatic insolubles: the proposition 'You will throw me in the water' is not in itself insoluble; it will only be so in a context such as this: Plato is the guard of a bridge that no one can cross without his assent. Socrates comes asking Plato if he will allow him to cross. The answer is: if in the first proposition you utter you speak the truth, I will allow you to cross the bridge, but if you speak falsely, I shall throw you in the water. Then Socrates says: You will throw me in the water.

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The link between insolubles and obligations was established especially by authors who, like William Heytesbury (before 1313-1372/3), solved the problem of insolubles in terms of obligations. It is underlined in his definition of an insoluble sentence: An insoluble sentence is one of which mention is made in an insoluble casus, such that if in the same casus it signifies precisely as its words commonly pretend, 6 from its being true it follows that it is false, and vice versa •

And the solution, given in the form of five rules, seems quite simple:

Rule 1: If someone constructs a casus of an insoluble, either he posits how that insoluble should signify, or he does not. If not, then when the insoluble is proposed, one should respond to it exactly as one would respond when the casus is not assumed. Rule 2: Second, notice that if a casus of an insoluble is posited, and together with that it is assumed that the insoluble precisely signifies just as its terms commonly pretend, the casus may in no way be admitted. Rule 3: Third, if someone constructs a casus of an insoluble, and together with that it is assumed that the insoluble signifies as its terms pretend, but not precisely, then when this casus is admitted, the insoluble has to be conceded as following, in whatever position it is proposed, but that it is true has to be denied as being inconsistent. Rule 4: If someone constructs a casus of an insoluble, and together with that it is assumed that the insoluble signifies conjunctively precisely as its words pretend-call that A-and that B, or something else whatever may be, exists if the opposite of that conjunct is inconsistent with the whole casus, the casus has to be denied as formally including contradictories. Rule 5: But if someone constructs a casus of an insoluble, and together with that it is assumed that the insoluble signifies precisely disjunctively as its words pretend or that A is B, or something of this sort, , unless what is disjoined with the insoluble is consistent with the whole casus, the casus is not to be admitted. 7 At first sight, this way of solving insolubles seems to adopt a strict strategy of rejection: insoluble cases are in no way admitted because they lead to a formal contradiction. But we see too that several kinds of

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answer are distinguished, depending on the way the terms signify. As I have already mentioned, a proposition becomes insoluble only when it is inserted into a casus, a part of which consists, in Heytesbury's terms, in positing that the proposition precisely signify just as its terms commonly pretend. Before commenting on Heytesbury's rules, and showing how the formal contradiction that arises depends on the precise signification of the terms, it will be necessary to discuss some general points concerning the theory of signification. It was well known in medieval logic that spoken and written languages are purely conventional and arbitrary while mental language is natural, i.e. signifies the same things for every man. An immediate consequence of this is that we can play with language and modify the signification of spoken and written terms at pleasure. The operation by which we give a signification to a term is called 'imposition' or 'institution'. In general, every categorematic term 8 has a signification which is commonly adopted by all the speakers of a language9 , e.g. the word 'man' commonly signifies men, the verb 'to run' commonly signifies the action of running, etc. This common signification is what a term 'firstly signifies' (primarie significat). This convention is important in order to maintain a minimum of intelligibility in discussions. In a tract on insolubles inspired by Heytesbury's, a certain Magister Iohannes Eclif(?) 10 considers it necessary to recall this convention: To respond to insolubles I first suppose that every proposition in Latin in a disputation precisely signifies according to the common institution of grammar unless the opponent and the respondent agree upon a new imposition. This supposition is evident because in every school and in every faculty where disputations are held one has to presuppose the grammar that expresses the significate of words according to the idiom in which disputation is held. Otherwise propositions such as 'God exists' or 'something exists' or any other proposition submitted would have to be discussed, because the way in which it signifies would be unknown, which is inconsistent and absurd. This is confirmed: it happens that one and the same proposition has different meanings in different idioms; therefore either the respondent has to know every language or he has to doubt every proposition or he just has to understand what is proposed to him according to the common institution of the idiom in which the disputation is held where no new imposition has been applied to the term. The first two parts [of the disjunction] are ~enerally false; therefore the thrid is true, that is 1 equivalent to the first supposition.

To change the signification of a term, we have to operate a new imposition and this, again according to Magister Iohannes Eclif(?), can be done in any of three ways:

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A new imposition can be made in three ways. 12 <1> A term or a proposition can be imposed to signify something more than it does according to its common institution, and what this signification will be is not specified to the respondent, e.g. let this proposition 'a falsehood exists' signify just as its terms commonly pretend, but not precisely. <2> The new signification is specified to the respondent, e.g. let this proposition 'a falsehood exists' signify precisely that a falsehood exists and that God exists, or precisely signify disjunctively that a falsehood exists or that you run. <3> A term or a proposition receives a completely different signification, e.g. let this proposition 'a man exists' precisely signify that an ass is running, or let this term 'A' (or any other term that did not first precisely signify something) signify a man, or be convertible with the term 'man.' 13

In case <1>, the respondent should ask the opponent what the new signification of the terms is and then answer according to Heytesbury's rules 2-5 discussed below. If the opponent refuses to specify what the new signification of the terms is, the respondent should reject14 the case because he cannot answer to a proposition he does not understand. 15 In case <2>, the respondent has to answer according to Heytesbury's rules 4 and 5 discussed below. In case <3>, we see that a proposition can have, at least for the time of the disputation, a completely new signification; the respondent then has to adapt his answer to this new signification: if the opponent posits the case that the proposition 'a man exists' precisely signifies that a stone exists, the respondent should admit that case-because it is possible-and deny the consequence (inference) 'a man exists, therefore, an animal exists,' because it is invalid according to the new signification of the term 'man'-even if it is valid according to the common signification of terms. In case <1>, something will be added to the common signification. The proposition where such a change is made will then have two significations: its first signification (prim aria significatio ), that depends on the common meaning of terms, and a secondary one (secundaria significatio ), that depends on the new imposition. In case <2>, the common signification is modified but the modified signification is not unrelated to the original one. It is only in case <3> that the common signification will be totally replaced (sequestrata 16 ) by a new one, that will become its primary signification for the time of the disputation. These remarks now lead us to a better understanding of Heytesbury's rules on insolubles, as well as of the link that can be made between his solution and others.' I will here repeat earlier quoted rules in order to analyze them.

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Rule I: if someone constructs a casus of an insoluble, either he posits how that insoluble should signify, or he does not. If not, then when the insoluble is proposed, one should respond to it exactly as one would respond when the casus is not assumed. Let us take the proposition "Socrates is saying what is false" (Socrates dicit fa/sum). It is not in itself insoluble. To become insoluble, two conditions must be fulfilled: 1° it must be the case that it is the only proposition Socrates utters, 2° the terms of that proposition have to signify precisely, i.e. exclusively, as its words commonly pretend. If condition I is not fulfilled, the proposition is not insoluble, because the predicate 'what is false' could refer to any other proposition uttered by Socrates. It will then be easy to determine the truth-value of the proposition "Socrates is saying what is false": it will be true if the proposition it refers to is false, and false if the proposition it refers to is true. If condition 2° is not fulfilled, the proposition is not insoluble, because it could signify that a man is an animal, or anything else. Conditions I and 2° are in fact two casus that are superposed one on the other. One should not be amazed at this, because imposition is one of the several kinds of obligation, i.e. that kind of obligation in which the casus is that one term or another is imposed to signify something else than what it usually does. In general, any new imposition has to be admitted, because spoken and written languages are completely arbitrary. Therefore, if someone posits the casus that Socrates says the proposition "Socrates is saying what is false" and no other, without making any assumption about how the terms signify (so that condition 2° is not fulfilled), the respondent must admit the casus, because, as he does not know which is the precise signification of the proposition, it is possible that the casus implies no contradiction. In the absence of specification, the proposition "Socrates is saying what is false" can signify anything and anyhow. Indeed, as at least one of its possible significations would not imply any contradiction, e.g. if it signifies precisely that a man is an animal, the casus is possible and must be admitted. Then, when the proposition "Socrates is saying what is false" is proposed for the first time, the respondent should answer as if the casus was not posited, i.e. as if it was not the only proposition Socrates utters (i.e. as if condition 1o was not fulfilled). To explain this apparently strange rule, we should mention some other principles of obligation games. An obligation is a kind of contract concluded by two adversaries, from which results a kind of link between 0

0

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them for a time: they cannot do anything they like, they must compel themselves to respect certain rules. Inside an obligational disputation, we see that there is also a kind of link between propositions: some propositions are relevant (pertinens) to others, because they follow from or are inconsistent with them, some propositions are irrelevant (impertinens) to others, because they neither follow from nor are inconsistent with them: 'You are an animal' is relevant-following to 'You are a man,' 'You are not an animal' is relevant-inconsistent to 'You are a man,' 'You are a musician' is irrelevant to 'You are a man.' This logical link that exists or does not exist between propositions has some effect on the answers the respondent must give. The general rules are: every relevant-following proposition has to be granted, every relevant-inconsistent proposition has to be denied, every irrelevant proposition has to be granted or denied or doubted according to its own truth-value, i.e. according to the truth-value it has outside the casus: it will be granted if it is true outside the casus, denied if it is false outside the casus and doubted if it is impossible to determine its truth-value 17 • If we now go back to Heytesbury's first rule, we can better understand why he says that the respondent should answer as if the casus were not posited: it is because the proposition "Socrates is saying what is false" is irrelevant. And this proposition is irrelevant because, as the opponent refuses to say how its terms signify, it is not linked in any way to anything. If it is not linked, it is as if no casus were posited ... Now, how should the respondent answer to this irrelevant proposition? Is it true, is it false? He cannot know, as he does not know how it signifies. So he has to doubt it or to ask for its signification 18 • If the opponent then specifies the signification, the respondent will answer according to rules 2-5, but, in this case, we can wonder why the opponent did not specify it at the beginning of the disputation. If the opponent again refuses to specify the signification, the disputation will lead to no result, and we can wonder why he should play in such a foolish way, as his aim is to lead his adversary to a contradiction. We should notice here that Heytesbury does not agree with Eclif(?) on the answer to make in such a case. If the opponent refuses to specify the signification of terms, the respondent should either I

admit the casus, because it is possible and then answer to the proposition as if the casus had not been posited (Heytesbury), or

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reject the casus because he cannot answer to a proposition he does not understand (Eclif(?) 19 ).

Eclif(?)'s position seems to make much more sense than Heytesbury's, especially as Heytesbury's solution leads either to the application of other rules, if the opponent specify the signification of terms, or, if not, to the impossibility of continuing the disputation, i.e. to the same result as Eclif(?)'s solution. So why this seemingly useless detour? Perhaps because he wanted his account to be comprehensive; in which case he felt he should specify at some point in his treatise, how one should respond to an irrelevant proposition? Or perhaps because he thinks that the respondent could answer in other ways than by doubting. The respondent would then answer according to the signification he himself gave to the terms, and grant the proposition if he gives it a signification according to which it is true or deny it if he gives it a signification according to which it is false. But in this case it is clear that the opponent will have considerable difficulty in leading his adversary to a contradiction20 • Rule 2: if a casus of an insoluble is posited, and together with that it is assumed that the insoluble precisely signifies just as its terms commonly pretend, the casus may in no way be admitted. If we continue with the example mentioned above, we see that here the proposition "Socrates is saying what is false" is insoluble, because conditions 1° and 2° are fulfilled: the casus is constructed so that this proposition is self-reflexive, and according to that casus, the only possible common meaning of the terms is that the predicate 'what is false' refers to the proposition it is a part of. According to condition 2 ° (the terms precisely, i.e. exclusively, signify as they commonly pretend), the only proposition the predicate 'what is false' can refer to is the proposition it is a part of and no other. Why does Heytesbury say that a casus like this has to be rejected? One of the basic rules of obligations is that a casus has to be rejected if and only if it is impossible, i.e. if it leads to a formal contradiction. So, while some logicians admit that it is possible for Socrates to say "Socrates is saying what is false," and nothing else, which precisely signifies that Socrates is saying what is false, and consider that what Socrates says is not a proposition or that this proposition is false or that it has no truth-value2 1, Heytesbury considers that the second condition of this casus is impossible and that, as such, the casus has to be rejected. It is impossible, because it implies

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that one and the same proposition is true and false at the same time, which is logically impossible22 • Rule 3: if someone constructs a casus of an insoluble, and together with that it is assumed that the insoluble signifies as its terms pretend, but not precisely, then when this casus is admitted, the insoluble has to be conceded as following, in whatever position it is proposed, but that it is true has to be denied as being inconsistent. According to Heytesbury, a proposition is true if and only if it precisely, i.e. exclusively, signifies in accordance with what is actually the case (praecise significat sicut est). We have seen, above, that when a term or a proposition receives a new imposition, it then has two significations, its first signification and its secondary signification-unless of course its common signification would be completely replaced or suspended for the time of obligation. Let us go back now to our insoluble casus "Socrates is saying what is false" accompanied by the condition that Socrates says nothing else and that the terms signify just as they commonly pretend, but not precisely so. We note that condition I· is fulfilled and that, according to that casus, the only common signification, i.e. the first signification, of the terms is that the predicate 'what is false' refers to the proposition it is a part of. But, as condition 2· is not fulfilled, the proposition could also signify something else according to which it could be true, e.g. if its secondary signification is that God exists. In that case, the proposition "Socrates is saying what is false" would be false according to its first signification and true according to its secondary signification. As it could be true in a way, the casus has to be admitted, because it is possible; the proposition "Socrates is saying what is false" has to be granted as following from the casus, because it is true that Socrates is saying what is false; but it must be denied that the proposition "Socrates is saying what is false" is true, because to be true it should signify only as is the case, i.e. signify as is the case according to both of its significations, and this is not so. The proposition "Socrates is saying what is false" is false according to its first signification and thus simply false, because its first signification is that Socrates is saying what is false, where 'what is false' refers to the proposition "Socrates is saying what is false" that, according to the casus admitted, signifies that Socrates is saying what is false and that God exists. So to be true according to its first signification, it should be the case that the proposition "Socrates is saying what is false" be false

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according to both of its significations; which is not the case, since it is true according to its secondary signification at least. Immediately after rule 3, Heytesbury notes that: If someone ask under this casus what the sentence uttered in this way by Socrates signified other than that Socrates is saying what is false, I say to him that the 23 respondent does not have to solve or to give his determination for that question.

It is thus enough for the respondent to know that the proposition could

possibly be true to admit the casus. And if the opponent later proposes that the proposition is true or false, the respondent does not even have to ask what the proposition secondarily signifies before answering, because it is clear that the proposition is false whatever its secondary signification may be. We have seen, above, why the proposition "Socrates is saying what is false" is false even if its secondary signification is true. It is clear, for similar reasons, that the falsity of its secondary signification is sufficient to make the proposition false. After having conceded that the proposition is false, the respondent will not be forced into a contradiction by the following argument: [I] [2] [3] [ 4]

[5]

the proposition uttered by Socrates, "Socrates is saying what is false" is false; therefore it signifies otherwise than is the case, and it signifies that Socrates is saying what is false; therefore it is not the case that Socrates is saying what is false; therefore if Socrates is saying what is false, Socrates is saying what is true, which is a contradiction

because to validly conclude [4] from [2] and [3], it should be added in [3] that the proposition that Socrates is saying what is false is precisely what-is the only thing that-the proposition "Socrates is saying what is false" signifies, which is contrary to the casus admitted. In fact, [3] should become [3 ']

and it signifies that Socrates is saying what is false and that God exists

[3"]

and it signifies that Socrates is saying what is false and that God does not exist.

or

And [4] should become

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therefore it is not the case that (Socrates is saying what is false and God exists),

which is equivalent to 'Socrates is not saying what is false or God does not exist' and thus to 'Socrates is saying what is true or God does not exist'; or [4"]

therefore it is not the case that (Socrates is saying what is false and God does not exist),

which is equivalent to 'Socrates is not saying what is false or God exists' and thus to 'Socrates is saying what is true or God exists.' To be true, [4'] should be true according to its first part, i.e. 'Socrates is saying what is true.' But this is impossible because, according to the casus admitted, it refers to the proposition "Socrates is saying what is false" that signifies that Socrates is saying what is false and God exists. As this last proposition is true only according to its secondary signification (cf. supra), it cannot be precisely true. With [4"] we go back to the situation outlined above: this proposition is true, but only according to its second part, thus it is not precisely true. The first part 'Socrates is saying what is true' cannot be true, because, according to the casus admitted, it refers to the proposition "Socrates is saying what is false" that signifies that Socrates is saying what is false and God does not exist. As this last proposition is false, as regards its secondary signification at least, it cannot be precisely true. As it is impossible to conclude that the proposition is precisely true, the respondent is not forced into a contradiction. The opponent could also modify the significate of terms or propositions in the second way described, i.e. by adding something to the common signification, not as a secondary signification, but as constituting a part of the new first signification. He could for example posit the casus that the proposition "Socrates is saying what is false" precisely signifies that Socrates is saying what is false and that God exists, or that Socrates is saying what is false and that no God exists. To give an appropriate answer to such a casus, Heytesbury proposes rules 4 and 5. Rule 4: if someone construct a casus of an insoluble, and together with that it is assumed that the insoluble signifies conjunctively precisely as its word pretend-call that A-and that B, or something else whatever may be, exists if the opposite of that conjunct is inconsistent with the whole

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casus, the casus has to be denied as formally including contradictories.

This rule is to be applied in the first casus mentioned just above, because it is clear that it would imply a formal contradiction. Indeed, if the proposition "Socrates is saying what is false" precisely signifies that Socrates is saying what is false and that God exists, the respondent should reject the casus, otherwise he would be forced into a contradiction by the following arguments: [I] [2]

if the casus is admitted, the proposition is possible so signifying, therefore it is true or false. if it is true, it must be true for each of its conjuncts; therefore it is true for its first conjunct that precisely signifies that Socrates is saying what is false; therefore it is the case that Socrates is saying what is false; therefore what Socrates says is false, and Socrates says "Socrates says what is false"; therefore the proposition "Socrates says what is false" is false; therefore if the proposition "Socrates says what is false" is true, it is false. CONTRADICTION

[3]

if it is false, it must be false for at least one of its conjuncts, and the second conjunct (God exists) is not false; therefore the first conjunct is false that precisely signifies that Socrates is saying what is false; therefore it is not the case that Socrates is saying what is false, and Socrates says "Socrates is saying what is false"; therefore the proposition "Socrates is saying what is false" is true; therefore if the proposition "Socrates is saying what is false" is false, it is true. CONTRADICTION

Note that no contradiction would arise if the opposite of the second conjunct was necessary. If the casus was that the proposition "Socrates says what is false" precisely signifies that Socrates says what is false and that God does not exist, this proposition would be simply false because the falsity of its second conjunct. Rule 5: if someone constructs a casus of an insoluble, and together with that it is assumed that the insoluble signifies precisely

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This rule is to be applied in the second casus mentioned just above, because it is clear that it would imply a formal contradiction. Indeed, if the proposition "Socrates is saying what is false" precisely signifies that Socrates is saying what is false or that no God exist, the respondent should reject the casus, otherwise he would be forced into a contradiction by the following arguments: [ 1] [2]

if the casus is admitted, the proposition is possible so signifying, therefore it is true or false. if it is true, it must be true for at least one of its disjuncts, and the second disjunct (no God exists) is not true; therefore the first conjunct is true that precisely signifies that Socrates is saying what is false; therefore it is the case that Socrates is saying what is false, and Socrates says "Socrates is saying what is false"; therefore the proposition "Socrates is saying what is false" is false; therefore if the proposition "Socrates is saying what is false" is true, it is false. CONTRADICTION

[3]

if it is false, it must be false for each of its disjuncts; therefore it is false for its first disjunct that precisely signifies that Socrates is saying what is false; therefore it is not the case that Socrates is saying what is false; therefore what Socrates says is true, and Socrates says "Socrates says what is false"; therefore the proposition "Socrates says what is false" is true; therefore if the proposition "Socrates says what is false" is false, it is true. CONTRADICTION

Note that no contradiction would arise if the opposite of the second disjunct was impossible. If the casus was that the proposition "Socrates says what is false" precisely signifies that Socrates says what is false or that God exists, this proposition would be simply true because of the truth of its second disjunct24 .

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If we now compare Heytesbury's solution to others, we note that they are not totally unrelated. In sophisms 17 and 18 of his Sophismata asinina, Heytesbury tackles the problem of insolubles. In both places, he mentions three possible answers: restriction (in insoluble matters a term cannot refer to the proposition it is a part of), cassation (an insoluble proposition is unintelligible) and his own solution which is qualified as "according to the common way" (secundum modum communem). These three solutions have something in common: they adopt a strategy of rejection, but each one of them lays emphasis on different aspects. Restrictors insist on the fact that self-reference is forbidden (condition 1°), while Heytesbury insists on the way terms signify (condition 2°); but it is clear that the result is the same: if the terms of an insoluble proposition precisely signify as they commonly pretend, the proposition is self-reflexive. 'Cassers' insist on the unintelligibility of insoluble propositions while Heytesbury insists on the fact that they lead to a formal contradiction, but it is clear that a formal contradiction is something unintelligible. Besides, it is noteworthy that he does not particularly criticize restriction and cassation; while, as I have already said, he strongly criticizes authors who admit that an insoluble proposition can signify precisely as its terms commonly pretend. 3. INSOLUBLES AND DISPUTATIONS

Up to this point we have mainly set out what is theorically at stake in the many discussions on insolubles. Let us now take a look at the related practical aspects. Heytesbury recognizes himself that his solution is not totally satisfying. In fact it does not solve the theorical problem, but merely provides a practical way of escaping from contradictions in disputations. As a borderline case in logic, insolubles were introduced in order to provide practise material for young students: Therefore, let these things said about insolubles serve for the introduction and drill of the young, so that when it is seen that insolubles, as their name implies, cannot be solved without evident objection, everyone max quickly go beyond them to a 5 more useful study, pursuing it the more diligently.

In medieval universities, exercises and exams were held in the form of disputations between students or between a student and a master. With the development of the theory and practice of obligations, and because of the link I mentioned above between insolubles and obligations, obligational disputations, or texts written on the model of obligational

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disputations became the place par excellence to introduce insoluble matters, at least in what we could call the Oxford tradition of the fourteenth century. This was so natural that, as we saw, Magister Johannes Eclif(?) included a chapter on imposition inside his treatise on insolubles, even though this traditionally belongs to treatises on obligations. As an example of disputation in which insolubles could appear, I will mention here the sophism 25 of Heytesbury's Sophismata asinina. The text entitled Sophismata asinina is a collection of 37 sophisms in which the aim is to try to prove the proposition 'you are an ass.' Anything is allowed in order to attain this goal... Here are the arguments proposed by the opponent in sophism 25 (some comments are given after each argument). The respondent should answer to each of them trying not to contradict himself. [ 1]

You are an ass.

This proposition should be denied, because it is impossible. The opponent should now try to prove it. [2]

I prove: you are one of these, and any one of these is an ass; therefore you are an ass, indicating by the word 'these' two asses. The consequence is evident.

This consequence has to be granted, because it is valid, but the antecedent has to be denied, because its first part 'you are one of these' is false if the word 'these' indicates two asses. The opponent now has to prove the antecedent. [3]

I prove the antecedent, and I posit the case that whenever you answer to this proposition 'you are one of these' otherwise than affirmatively, you know that you are one of these, and that whenever you answer affirmatively, you know that you are not one of these. If this is admitted, I propose this proposition 'you are one of these.'

The case posited here should be rejected, because, as will be shown in arguments [4] and [5], if the respondent admits it, he will be inevitably led to a contradiction. [4]

If this is rejected, I argue so: whenever you answer to that proposition otherwise than affirmatively, you know that you are one of these, but you answer to that proposition otherwise than affirmatively; therefore you know that you

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III

are one of these, still indicating by the word 'these' two asses, as has been said before. If this proposition 'you are one of these' is granted, I argue so: whenever you answer affirmatively, you know that you are not one of these, but you answer affirmatively; therefore you know that you are not one of these; therefore you should not have granted that you are one of these.

So, if you admit the casus, you are an ass ... University of Montreal NOTES 1

C. J. Martin analyses this solution in his "Obligations and Liars," see pp. 80-85. These notions of wide or strong restriction and weak restriction are discussed by Spade 1987,38-42. See also Roure 1970,219-225. 3 Cf. John Buridan, Sophismata, ch. 8, soph. 7, translated in Hughes 1982, 48-49. 4 Just as every affirmative proposition implies the existence of the significates to which its terms refer, every proposition we assert implies itself to be true, otherwise we would not assert it. This claim rests on a psychological analysis of what a lie is: when we assert something (affirmatively or negatively) we are supposed to assent to it; lying is asserting (i.e. presenting as true) something we know or believe to be false, something we do not give our assent to. As every proposition signifies or implies itself to be true, this solution is not ad hoc. On the several kinds of obligations, cf. Stump, 1982, 315-334, and Spade 1982c, 335-341. 6 Cf. Spade 1979, 46, § 45. 7 Cf. Spade 1979, 47-51, § 48-53. 8 Syncategorematic terms, like 'if,' 'and,' 'then,' 'every,' etc. do not signify by themselves, they 'consignify,' they modify the signification of the terms they are ioined to. Cf. Spade 1982a, 190-191. There is another exception to that general rule: terms like 'A,' 'B,' 'buba,' etc. are considered by some people to have a signification in themselves; while others consider that they should be assimilated to demonstrative pronouns that have no signification as long as we do not know what they exactly refer to. Cf. Pironet 1994, 148-153. 10 This text is found in ms. Oxford, Bodleian Library, Lat. misc. e. 79, ff. 43ra-45vb ('0' in the critical apparatus). I am preparing an edition of it. It is difficult to determine who is its author. De Rijk suggests that it could be John Wyclif, whose name is quite near to the name we read in the explicit, i.e. "Eclif' or "Ecaf', cf. de Rijk 1977, 153. However, in 1986, P.V. Spade published Wyclifs Summa insolubilium, and it seems clear to me that Eclif(?)'s and Wyclifs positions on insolubles are not the same. 2

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Note that the mention "impositione [corr.: propositione 0]" means that I corrected the word 'propositione' in the manuscript 0 to 'impositione.' "Unde per responsionem ad insolubilia primo suppono quod quaelibet propositio in latina lingua disputationis praecise significet secundum communem institutionem grammaticae nisi [corr.: ubi 0] in aliqua nova impositione [corr.: propositione 0] conveniant opponens et respondens. Haec suppositio patet eo quod in omni secta et in omni alia facultate disputabili praesupponitur [corr.: praesupponi 0] grammatica exprimens significatum vocum [corr.: vocalem 0] illius idiomatis in quo fit disputatio. Aliter enim talis propositio 'deus est' vel 'aliquid est' vel quaecumque alia vera proposita semper foret disputanda: quia nesciretur quo modo significat, quod est inconveniens et absurdum. Confirmatur sic: aliquando eadem propositio diversimode significat in diversis idiomatibus; ergo vel oportet respondentem quemlibet scire omnes linguas vel dubitare quamlibet propositionem sibi propositam [corr.: proposita 0] vel solum intelligere propositum juxta communem institutionem idiomatis in quo disputat ubi non fit sibi nova impositio termini. Sed duae primae partes communiter sunt falsae, ut patet; ideo tertia pars est vera, ~uae convertitur cum prima suppositione." f. 43rb. 1 A new imposition can be applied to a single term or to a whole proposition, but it is clear that when it is applied to a term only, the signification of the whole proposition it is a part of will be also affected. 3 Note that "<est>" means that I added the word 'est' to the text while "[ et]" means that I removed the word 'et' from the text. "Nova impositio solet fieri tribus modis. Aut enim terminus sive propositio imponitur ad significandum aliquam ultra communem institutionem et non certificatur respondens qualis ista significatio erit, sicut hie [corr.: hoc 0]: significet ista 'falsum <est>' sicut communiter verba praetendunt, non tam en praecise. Aut de tali significatione certificatur [et 0] respondens sicut hie [corr.: hoc 0]: significet ista 'falsum est' praecise quod falsum est et quod deus est, vel significet disjunctive praecise quod falsum est vel quod tu curris. Aut mutatur terminus sive propositio totaliter in aliam significationem, sicut hie [corr.: hoc 0]: significet quaelibet talis 'homo est' praecise quod asinus currit, vel significet talis terminus 'A'-sive aliquis alius qui non prius significavit ali quid praecise-hominem, sive convertatur cum tali termino 'homo,"' f.44va-b. 14 It should be noted here that in treatises on obligations, the verb 'admittere' is used for admitting a case (because it is possible), and the verb 'concedere' for granting a proposition (because it is true), but the verb 'negare' is used for denying a case (because it is impossible or unintelligible) as well as for denying a proposition (because it is false). To preserve this distinction between cases and propositions, I will translate 'nego casum' by 'I reject the case' and 'nego p.ropositionem' by 'I deny the proposition.' 5 We shall see below that Johannes Eclif(?) and Heytesbury do not agree on this point; the first says that the respondent should not answer, while Heytesbury says that the casus has to be admitted. Cf. infra pp. I 03-105 where I discuss Heytesbury's rule 1. 16 The notion of impounded signification is discussed in a tract on obligations which is to be found in the Logica printed in Oxford in 1483: "Nota etiam quod primarium significatum propositionis non sequestratur per novam impositionem nisi addita dictione exclusiva, unde si ponatur ista 'deus est' ad significandum

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hominem esse asinum, non adhuc sequestratur primarium significatum: quia stat satis bene quod ista 'deus est' significet utrumque significatum, unum ex prima, et alterum ex secunda impositione, sed oportet addere 'solum' vel 'solum modo,' ut impono quod 'deus est' solum significat hominem esse asinum, et tunc sequestratur prima significatio," cf. Pironet 1994, 595-596. Note that the term 'praecise, ' which is the exclusive term that Heytesbmy usually uses, plays exactly the same role as the term 'solum' in this text. 17 These rules are very common. See for example the treatise De obligationibus in Pironet 1994, 582-584. 18 These are the answers suggested in similar cases. Cf. Sophisms 26 and 27 in the Sophismata asinina where Heytesbury is ironic about this kind of attitude: "Pro isto respondeo, et dico quod non intelligo consequentiam (... ) Aliter posset responderi ad primum, quando dicitur quod 'lateat te quid demonstratur,' dicendo quod lateat te responsio mea, et haec responsio satis convenit demonstrationi." Cf. Pironet 1994, 424. 19 Cf. supra, p. 102. 20 These are just suggestions. This point remains quite unclear to me. 21 Heytesbury criticizes these opinions at large in § 4-43. 22 It is quite common to distinguish natural or physical impossibility from logical impossibility. The first can be admitted, because the absolute power of God could see to it that things are contrary to the usual natural rules. The second may in no way be admitted, because even God cannot see to it that two contradictories should be true at the same time. This is discussed in, among others, a treatise called Tractatus Emmeranus de impossibili positione (De Rijk 1974, 117-123; for English translation, cf. below pp. 000-000). 23 Cf. Spade 1979, 49, §51. 24 Rules of this kind are discussed in the Tractatus Emmeranus de falsi positione, (De Rijk 1974, 117-123; for English translation, cf. below pp. 000-000). 25 Cf. P.V. Spade, Guillelmus of Heytesbury On Insolubles Sentences, p. 58, § 64.

REFERENCES Ashworth, E. J., 1994. "Obligationes Treatises: A Catalogue of Manuscripts, Editions and Studies," in Bulletin de Philosophie MMievale 36, 118-147. De Rijk, Lambertus Marie, 1974. "Some Thirteenth Century Tracts on the Game of Obligation I," Vivarium 12, 94-123. De Rijk, Lambertus Marie, 1975. "Some Thirteenth Century Tracts on the Game of Obligation II," Vivarium 13, 22-54. De Rijk, Lambertus Marie, 1977. "Logica Oxoniensis. An Attempt to Reconstruct a Fifteenth Century Oxford Manual ofLogic," Medioevo 3, 121-164. Hughes, George, 1982. John Buridan on Self-Reference: Chapter Eight of Buridan 's Sophismata, (An Edition and a Translation with an Introduction and a Philosophical Commentary), Cambridge, Cambridge University Press. Knuuttila, Simo, 1993b. "Trinitarian Sophisms in Robert Holkot's Theology," in S. Read (ed.), Sophisms in Medieval Logic and Grammar, (Nijhoff International Philosophy Series 48), Dordrecht, Kluwer, 348-356.

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Martin, Christopher .J., 1993 "Obligations and Liars," in S. Read (ed.), Sophisms in Medieval Logic and Grammar, (Nijhoff International Philosophy Series 48), Dordrecht, Kluwer, 357-38 I. Revised version in the present collection, pp. 65-96. Pironet, Fabienne, 1994. Guillaume Heytesbury, Sophismata asmma. Une introduction aux disputes medievales. Presentation, edition critique et analyse, (Collection Sic et Non), Paris, Vrin. Pironet, Fabienne, (forthcoming). Iohanni Buridani Summularum Tractatus nonus: De practica sophismatum (Sophismata), Critical Edition and Introduction, Nimegue, Ingenium Publishers. Read, Stephen, 1979. "Self-Reference and Validity," Synthese 42, 265-274. Roure, M. L., 1970. "La problematique des propositions insolubles au XIIIe siecle et au debut du XIVe, suivie de !'edition des traites de W. Shyreswood, W. Burley et Th. Bradwardine," Archives d'Histoire Doctrinale et Litteraire du Moyen Age 37, 205-326. Scott, T. K., Johannes Buridanus. 'Sophismata, ' Critical Edition with an Introduction, (Grammatica Speculativa I), Stuttgart-Bad Cannstatt, Frommann -Holzboog. Spade, Paul Vincent, 1975. The Medieval Liar, Toronto, Pontifical Institute of Mediaeval Studies. Spade, Paul Vincent, 1979. William Heytesbury. 'On Insoluble Sentences': Chapter one of his 'Rules for Solving Sophisms, ' Translated with an Introduction and Study, Toronto, Pontifical Institute of Mediaeval Studies. Spade, Paul Vincent, "Insolubilia and Bradwardine's Theory of Signification," Medioevo 1, 115-134. Spade, Paul Vincent, 1982a. "The Semantics of Terms," in N. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 188-196. Spade, Paul Vincent, 1982b. "Insolubilia," inN. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 246-253. Spade, Paul Vincent, 1982c. "Obligations: Developments in the Fourteenth Century," in N. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 335-341. Spade, Paul Vincent, 1987. "Five Early Theories in the Medieval InsolubiliaLiterature," Vivarium 25, 24-46. Spade, P. V., and Wilson, G.A., 1986. Johannis Wyclif, 'Summa insolubilium,' (Medieval and Renaissance Texts and Studies 41), Binghamton, New York. Stump, Eleonore, 1982. "Obligations: From the Beginning to the early Fourteenth Century," inN. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 315-334.

PART II CONSEQUENCES

PETER KING

CONSEQUENCE AS INFERENCE: MEDIAEVAL PROOF THEORY 1300-1350

The first half of the fourteenth century saw a remarkable flowering in accounts of consequences (consequentiae). Logicians began to write independent treatises on consequences, the most well-known being those by Walter Burley (De consequentiis) and Jean Buridan (Tractatus de consequentiis). Consequences also came to be treated systematically in comprehensive works on logic, such as those of Walter Burley (both versions of the De puritate artis logicae), William of Ockham (Summa logicae), and, to a lesser extent, Jean Buridan (Summulae de dialectica)-as well as in works written in their wake. 1 The philosophical achievement realized in these various writings was no less than a formulation of a theory of inference: the rules for consequences given by these mediaeval authors spell out a natural deduction system in the sense of Jaskowski and Gentzen. 2 Recognition that mediaeval logicians are dealing with inference in the theory of consequences, rather than with implication, is sporadic at best and nonexistent at worst. 3 This may be due to the emphasis many modem logicians put on presenting logical systems axiomatically, since axiomatic formulations typically have only a single rule of inference (detachment) and focus on logical truth instead of logical consequence.4 But whatever the cause, the point that consequences are inferences has not been appreciated, which in tum has made it hard to see how consequences fit into the mediaeval conception of argument. The discussion will proceed as follows: § 1 argues that consequences are not the same as conditionals; §2 considers two objections to this distinction; §3 argues that consequences are inferences and were understood by mediaeval logicians to be so; §4 examines accounts of formal validity; §5 looks at the place of consequences-the theory of inference-in their general account of argumentation. I'll draw some morals about the mediaeval logical enterprise by way of conclusion.

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A categorical sentence (say) is used to make a statement, that is, "to say something of something" in Aristotle's phrase. 5 Conditional sentences also make statements, that is, they declare that a certain relation obtains (namely that the consequent is conditional upon the antecedent). The statement that a conditional sentence makes is not the same as the statement made by any of its parts taken in isolation, of course; conditionals neither say what their antecedents or their consequents say, nor are they about the subjects of their antecedents or consequents. For all that, conditional sentences do succeed in making statements. Inferences, however, do not "say something of something." They do not make statements. An inference is a performance: it is something we do, perhaps with linguistic items, but in itself it is no more linguistic than juggling is one of the balls the juggler juggles. Furthermore, even the statement of an inference (its linguistic representation) is not a statement-making expression. It has parts that in isolation could be used to make statements-namely any of the premisses or the conclusion-but itself does not make a statement. (One sign of this is that neither an inference nor the statement of an inference is assessed as true or false.) In a slogan: conditionals make statements whereas inferences do things with statements. 6 Modem logicians regiment this distinction between conditionals and inferences by presenting them as categorically different parts of the logical landscape: the former through a primitive or defmed sentential connective appearing in well-formed formulae, for which truth is appropriate, and the latter through rules for transforming well-formed formulae, for which validity is appropriate. Thus conditionals and inferences differ in kind, one belonging to the object-language and the other to the metalanguage. They are not unrelated, however; a Deduction Theorem can be established for many axiomatic systems, so that if A ----+ B then A ~ B, 7 and natural deduction systems typically use conservative introduction and elimination rules to define the conditional connective. Mediaeval logicians, like their modem counterparts, treat implication and inference as logically distinct notions, along the lines sketched above. To begin at the beginning: a conditional sentence is a particular kind of statement-making utterance, but different in kind from the paradigm case of the (simple) categorical sentence. 8 It is instead lumped together with conjunctive and disjunctive expressions under the generic

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heading of "compound sentence" (propositio hypothetica), on the grounds that these three kinds of utterance all have parts that would qualify as sentences taken by themselves, although they are not simply reducible to their parts-a mediaeval version of our notion of the connective of widest scope. 9 So much is commonplace, derived from Boethius and ultimately from Aristotle (De interpretatione 5 17a9-1 0 and 20-22). Another mediaeval commonplace is that logic is divided into three parts, namely into words, statements, and arguments, ordered by composition: statements are made out of words, and arguments out of statements. These parts are not reducible to one another, for we use words to make statements and we use statements to make arguments. 10 Each part of logic thus constitutes its own level of analysis and carves out a distinct part of the logical landscape. Conditional sentences, as statement-making utterances, must therefore differ in kind from arguments, since they belong to different parts of the landscape. Hence consequences will be distinct from conditionals-at least, to the extent that mediaeval logicians classify consequences with arguments. The strength of this line of reasoning lies precisely in its premisses being commonplace. It does not depend on any particular feature of the doctrine of consequences. We can reason our way to the categorical distinction between implication and inference from entrenched mediaeval views about logic and language. The only question that remains is whether consequences and arguments do belong together. In comprehensive works on logic, where systematic concerns readily come to the fore, consequences are classified with arguments and not with sentences. William of Ockham provides a clear example. The Summa logicae is organized into three parts based on the division of logic recounted above. Conditional sentences are treated in Summa logicae 2.31 (devoted to sentences) as a species of compound sentence, whereas consequences are the subject of the third treatise of Summa logicae 3 (devoted to arguments)." Ockham even refers to the later discussion of consequences in his brief chapter on conditionals, so he is aware of the distinction at precisely the point at which it matters. 12 The fragmentary nature of Walter Burley's De puritate artis logicae in both versions makes it less useful as evidence, but he does describe consequences as rules (60.12-14) and not as sentences. Jean Buridan doesn't have a separate discussion of consequences in his Summulae de dialectica, 13 but at the beginning of his Tractatus de consequentiis 3 he classifies all arguments as species of consequence.

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These architectonic considerations give us some presumptive evidence that consequences are logically grouped with arguments rather than sentences. They cannot do more than that, since such considerations do not rule out the possibility that conditionals are a species of consequence-that the term consequentia was used to describe arguments and to describe conditionals. But there is both negative and positive textual evidence against this objection, in support of the claim that mediaeval logicians not only recognized a difference between implication and inference but found them not to overlap at all. The negative evidence is as follows. In all the available literature of this period, which runs to hundreds of pages, I have found no instance of any author treating 'conditional' and 'consequence' as synonymous. Nowhere does the expression consequentia seu condicionalis or the like occur. 14 Of course, the architectonic considerations given above suggest that these terms would not be everywhere interchangeable. But they might well be interchangeable in certain contexts. For example, when speaking of conditionals proper, some feature that they have in common with consequences generally might be under investigation. It is striking, though not conclusive, that such expressions are never employed even in such contexts. The positive evidence comes in two varieties. First, the authors under consideration not only resist treating the terminology as interchangeable, they also use it to mark a logical distinction: conditionals do not merely appear along with consequences; they are actively contrasted with consequences. Second, conditionals have different properties, since they are true or false whereas consequences are not. 15 We'll take each in turn. The evidence for the first claim is as follows. Walter Burley mentions conditionals and consequences together in his De consequentiis §8, where he is talking about the legitimacy of inferring a conditional composed of the consequent of the last of a string of conditionals from the antecedent of the first of the string, that is: A --t B, B --t C f- A --t C. This classic example of cut-elimination, which Burley calls "the start-tofinish inference" (consequentia a prima ad ultimum ), also appears in both versiOns of his De puritate artis logicae (70.1-23 and 200.20-201.3 ). In each instance Burley explicitly contrasts the conditional sentences that enter into such reasoning with the consequence made out of them. Furthermore, Burley devotes De consequentiis §§66-72 to consequences that hold among conditionals, clearly assigning different properties to each. For example, he gives

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truth-conditions for conditional sentences (§68), contrasting them with consequences, which hold in virtue of topics (§§71-72). 16 The parallel section in the longer version of the De puritate artis logicae (66.9-79 .I 0) reiterates these claims. Other logicians also contrast conditionals with consequences. William of Ockham, as noted above, refers us in his chapter on conditional sentences (Summa logicae 2.31) to his later discussion of consequences. The commentary De consequentiis on Ockham, possibly written by Bradwardine, declares in §7 that we can move from a consequence A 1- B to a conditional A ~ B; Rule 15 of the Liber consequentiarum says that we can move in either direction (a claim to be explored more fully in §5 below). Richard Lavenham in his late work Consequentiae §§41-47 gives seven rules describing consequences that hold among conditionals. The claims put forward by these philosophers would not make sense unless consequences were something other than conditionals. The evidence for the second claim is as follows. Burley and Ockham, for example, explicitly call conditional sentences true or false: see the De consequentiis §68 and Summa logicae 2.31 respectively. Likewise for the Logica ad rudium 2.76, the Tractatus minor logicae 2.2, and the Elementarium logicae 2.16. Even Buridan calls conditionals true in his Summulae de dialectica I. 7. 3. 17 Of course, the fact that conditionals may be true or false follows from the fact that conditional sentences are statement-making utterances, since what it is to be a statement is, at least in part, to have a truth-value (De interpretatione 4 17a3-4)-putting aside for now worries about future contingents and other puzzling cases. Consequences, on the other hand, are neither true nor false. Here the negative evidence has quite a bit of weight. In the hundreds of pages of the available literature there are countless opportunities to say of consequences that they are true or false, opportunities that are all the more pressing since the writers are usually grappling with the question which consequences are to be approved and which not. Yet in all these pages I know only three passages in which consequences are called true or false. 18 One occurs in Pseudo-Scotus and is a mere slip. 19 But the other two are found in the writings of Jean Buridan, a logician of the first rank. Now Buridan's view of consequences might simply be idiosyncratic; we could set his testimony aside, given that there is no similar evidence in any other author of the period. But we do not have to do so. The context of each passage shows that we are not to take seriously Buridan's mention of truth-values in connection with consequences.

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The first passage is found in Buridan's Tractatus de consequentiis 1.3.4-6 (21.16-25), where he points out that some people (aliqui) say that any hypothetical sentence formulated with 'if or 'therefore' is a consequence and that there are thus two kinds, namely true and false consequences; he replies: In this treatise-whether or not it be true-words signify by convention; I mean to understand by 'consequence' a true consequence, and by 'antecedent' and 'consequent' sentences one of which follows from the other by a true or legitimate consequence (uera seu bona consequentia).

Buridan concedes the terminology to the unnamed thinkers whose view is under discussion. He stipulates what he will call a consequence after reminding us of the conventionality of language, characterizing consequences not only as true but as "true or legitimate"-and then never calling them 'true' again in the rest of his treatise (a treatise devoted to consequences, mind you!). 20 This passage therefore cannot serve as evidence that consequences, like conditionals, have truth-value. The second passage is found in Buridan's Summulae de dialectica 1.7.6: 21 It seems to me that a hypothetical sentence joining together two categorical

sentences by 'therefore' should likewise be counted as false if the consequence is not necessary (which is denoted by the word 'therefore'!), and also that it is false simply speaking if it were to have some false premiss.

Buridan proposes that we count an argument as false (reputari) if it fails to establish its conclusion by being either invalid or unsound. Yet there is no suggestion here that a consequence is literally true or false the way a statement must be. On the contrary, Buridan's plain meaning is that consequences can be invalid or unsound, and that these are defects in consequences just as falsity is a defect in a statement. To sum up: in neither passage does Buridan seriously propose that conseqences have truth-value; even if he were to do so, we can oppose to this the negative testimony of the rest of his writings, wherein consequences are not called true or false. And, as remarked above, even if Buridan were to allow consequences to have truth-value, no other logician in this period does. Instead, they say that consequences are "legitimate (bona)" and that they "hold (tenet)" or "are valid (ua/et)"-properties explored further in §4. 22 So much for the positive evidence that mediaeval logicians recognized the distinction between consequences and conditionals. The story is not complete, of course; to say that consequences aren't conditionals does

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not establish what they in fact are. Before presenting the positive case that consequences are inferences, though, we need to look at two objections to the thesis that conditionals and consequences are logically distinct notions. 2. TWO OBJECTIONS The first objection runs as follows. Conditional sentences are made up of parts, namely the antecedent and the consequent. Similarly, arguments are made up of parts, statements that we call the premiss(es) and the conclusion. But the parts of consequences are uniformly called the 'antecedent' and the 'consequent' throughout the available literature. Hence consequences must be a form of conditional sentence rather than of argument. The factual claims in this objection are correct, but the conclusion that consequences must be a form of conditional sentence does not follow. The mistake here is easy to make. The Latin terminology is antecedens and consequens, the ancestors and cognates of the English words 'antecedent' and 'consequent.' Modem logicians regiment their use so that they properly apply only to conditional sentences. 23 Well, they do apply to conditionals in Latin, but they are not tied to them the way the Greek grammatical terms i} rrp6Tams and i} arr68oaLs are-that is, unlike the Greek terms, the Latin terms are not simply defmed relative to one another by their occurrence in conditional sentences. Instead, antecedens and consequens carry the broader senses of 'what comes before' and 'what comes after. ' 24 Hence they are equally applicable to the parts of conditional sentences and to consequences. The De consequentiis possibly written by Bradwardine says so explicitly (§2):25 Note that a consequence is an argumentation made up of an antecedent and a consequent.

Consequences are arguments, and, as arguments, they have two logically distinct parts: one that comes before (the antecedent) and one that comes after (the consequent). The terminology is more general than that of premisses and conclusion, but no less legitimate. The pull of the cognate word and its restricted English sense is hard to resist. But resist it we should. The terminology used to talk about consequences doesn't give us any reason to interpret them as conditionals, although it tempts us to.

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The second objection is this. Buridan explicitly says that consequences are compound sentences-Tractatus de consequentiis 1.3.2 (21.9-10): 26 There are two types of sentence, namely categorical and compound, and a consequence is a compound sentence made up of many sentences joined together by 'if or 'therefore' or their equivalent.

Yet we have seen above that there are only three kinds of compound sentence, namely disjunctive, conjunctive, and conditional sentences. Consequences are surely neither of the first two, and hence must be identified as a kind of conditional sentence. Thus consequences belong to the same part of the logical landscape as conditionals, and so we can reject the presumptive evidence explored in § 1 in favor of distinguishing them. There are two replies available to this second objection. 27 First, while it is true that Buridan calls consequences compound sentences, he also calls the syllogism-the paradigm case of an argument-a compound sentence, and in fact reducible to a conditional sentence. 28 If syllogisms are reducible to conditional compound sentences, consequences can still be identified with arguments, although we may have to redraw the line between (compound) sentences and arguments in some fashion. Given that Buridan classifies all arguments as species of consequence, as noted in § 1, it would follow that the distinction between arguments and nonarguments would have to be made among kinds of conditional sentences. The drawback to this first reply is that it would require us to admit that consequences (and arguments generally) are in fact conditionals, which was the problem the reply was supposed to avoid. However, it does suggest that the way to approach the second problem is by considering what might have led Buridan to think that consequences were sentences in the first place. Recall from the start of § 1 the slogan that conditionals make statements whereas inferences do things with statements. True enough, but we can also describe the inferences that we make, and we do so with sentences describing how we manipulate statements. Here is one: "All swans are white objects; therefore, some white objects are swans." What kind of sentence is this? Modem logicians would say that this sentence does not belong to the object-language, despite its similarity in surface grammar to, say, the conditional sentence "If all swans are white objects, then some white objects are swans." The inference has the logical form A f- B (rather than A ----+ B); the turnstile 'f-' acts as a kind of metalinguistic connective. Hence 'A f- B' and 'A ----+ B' are not on a par.

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Mediaeval logicians do not have our distinction between objectlanguage and metalanguage. A sentence representing an inference is on all fours with a conditional sentence, or any other sentence for that matter. Yet Buridan captures the spirit of the modem reply. He admits that the statement of an inference is a compound sentence. But this admissison is compatible with the claim that the statement of an inference (its linguistic representation) is not a statement-making expression, whereas a conditional sentence is a statement-making expression? 9 Significantly, Buridan only says that consequences are reducible to conditional sentences, not that they are conditional sentences; Buridan, unlike Ockham and his followers, does not think we can pass from one to the other in any direct fashion (as we shall see in §5). Hence Buridan can reject the trichotomy of choices among compound sentences, on the reasonable grounds that it taxonomizes the kinds of statements that different sentences can make, whereas the sentence describing a consequence does not make a statement at all, but instead describes something done with statements. 3. CONSEQUENCEAND INFERENCE In the course of disentangling consequences from conditionals we have run across evidence that consequences are arguments, or at least closely related to arguments. From the position they occupy in comprehensive logical treatises to Buridan's classification of argument as a species of consequence to the bald statement in De consequentiis §2 cited above, mediaeval logicians take pains to underline the inferential force of consequences. Ralph Strode, perhaps in the 1360s, explicitly says that "a consequence is a deduction (il/atio) of the consequent from the antecedent" (1.1.02). 30 Buridan contrasts conditionals, which are not arguments at all, with consequences in the proper sense, which are indeed arguments (Summulae de dialectica 7.4.5). There is also a wealth of secondary evidence that consequences are inferences in the terminology and the proof-procedures employed by logicians and philosophers alike during this period. Let's have a look. Consequences underwrite arguments. 31 They argue for (arguitur) or permit us to draw (concluditur) a conclusion from the premiss or premisses, and to say in general what follows (sequitur) from what. They can be established (probatur) by supporting grounds. Furthermore, they are said either to be valid (ualet) or hold (tenet), or, if not, to be fallacious (fallit). If a consequence is appropriately truth-preserving-a

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feature to be investigated in §4-then it is said to be legitimate (bona). 32 Burley is especially clear about these features of consequences in his De consequentiis: in § 11 he says that when a consequence is legitimate a given conclusion ought to be inferred (debet inferri) through it; in § 12 he says that a test to see whether a consequence is valid determines whether it is legitimate; in § 13 he refers to legitimacy (bonitas) as a property suited to consequences, as truth is to sentences, whose presence depends on the inference drawn (quod inferatur) in a given case. Mediaeval philosophers, not just mediaeval logicians, recognized in practice that the consequence provides the inferential force of an argument. Typically, after stating an argument, a proof will be offered of each of the premisses, followed by a proof of the consequence (probatio consequentiae) to ensure that the conclusion does in fact follow from the premisses. Often the consequence is established by showing that it conforms to an accepted rule, or that its violation would conflict with such a rule. Given true premisses and a valid inference, of course, the result is a sound argument; nothing but the consequence can play the role of the latter. The rules for consequences found in the treatises of this period spell out the admissible sequents of a natural deduction system. Consider, for example, the first rule for consequences offered by Ockham in his Summa logicae 3-3.2 (591.9-11): There is a legitimate consequence from the superior distributed term to the inferior distributed term. For example, "Every animal is running; hence every man is running."

Such rules are typically given in metalogical or schematic terms (often in both ways), and they clearly refer to inferences that hold in virtue of the logical form of their constituents. One of the earliest independent works on consequences assimilates the legitimacy of a consequence to its formal validity: 33 This rule may be employed for seeing which consequences are legitimate and which not. We should see whether the opposite of the consequent can obtain with the antecedent. If not, the consequence is legitimate. If the opposite of the consequent can obtain with the antecedent, the consequent is not formally valid (non ualet de forma).

Hence the rules for consequences determine what inferential moves can be made; at least some rules require that the inferences hold in virtue of the logical form of the statements involved.

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Ordinary principles of natural deduction are readily found in the mediaeval literature, as one might expect from the 'naturalness' of natural deduction. Walter Burley gives a concise formulation of detachment in the longer version of his De puritate artis logicae that can be virtually transcribed from the Latin: A ~ B, A 1- B (66.13: Si A est, B est; sed A est; ergo B est). 34 Most of the treatises in this period give cut-elimination among their very first rules: A ~ B, B ~ C 1- A ~ C (cited for Burley in §I above). Examples could easily be multiplied. Some mediaeval rules for consequences have no modem parallel, since they depend on the details of mediaeval term-logic and syntactic analysis; Ockham's first rule, cited above, is a handy instance. 35 Likewise, some modem rules of natural deduction have no mediaeval parallel, such as those depending on mathematical features of the formulae (recursiveness, arbitrary depth, normal form). Then again, at certain points mediaeval logic and modem logic arguably diverge, as perhaps they do over existential import. For the most part, however, there is a remarkable degree of consensus between mediaeval rules for consequences and modem natural deduction principles of first-order logic. There is even some agreement between mediaeval and modem logic on higher-order deductive principles, namely on the proof-procedure for establishing the validity of syllogisms other than the first four moods of the first figure: Barbara, Celarent, Darii, Feria. Following Aristotle's lead, mediaeval logicians adopted a general reductio-method to validate at least some syllogisms (traditionally only Baroco and Bocardo); Buridan offers a clear statement of it in the third theorem of his Tractatus de consequentiis 3.4 (87.99-103). 36 In modem systems Buridan's theorem can be restated as a metalogical rule for classical reductio. Naturally, there is no mediaeval parallel to other techniques of modem proof theory, many of which are artifacts of the mathematical nature of modem logic (such as induction on proof-length). But when idiosyncratic features of mediaeval logic or of modem logic do not intrude, the deduction-rules provided by each system are largely the same. The rules for consequences, then, spell out a natural deduction system. The elements of this system are inferences-that is, consequences -which can be used to license arguments. 37 Hence the rules for consequences state legitimate inference-schemata. But what makes any inference-scheme legitimate, or even preferable to another? What, if anything, makes inferences valid?

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4. FORMAL VALIDITY The mediaeval consensus on legitimate inference-schemata does not extend to the explanation of legitimacy itself. In the first half of the fourteenth century we find three competing accounts of what makes a consequence legitimate. The first is little more than a suggestion, similar to a modem informal characterization of deductive validity. The second explains the legitimacy of A f- B modally, such that it is impossible for A to be true and B false; this account, then as now, is the favored view. But it is not without its problems. Hence a third account, based on substitutivity, was specifically designed to capture formal validity. We'll 38 consider each in tum. First, Robert Fland opens his Consequentiae by giving rules for knowing when consequences are formal, which is the case "when the consequent is understood in the antecedent formally" (§ 1). This psychological or epistemic account seems resistant to logical treatment, and, on the face of it, more appropriate to characterizing implication-relevant implication at that-than inference. (To say nothing of its circularity!) However, around 1370 Richard Lavenham took up the same train of thought in his Consequentiae, and his remarks, though equally brief, give us a clue how to interpret Fland (§2): A consequence is formal when the consequent necessarily belongs to the understanding of the antecedent (necessaria est de intellectu antecedentis), as it is in the case of syllogistic consequence, and in many enthymematic consequences. The tip-off that we are dealing with inference is seen in Lavenham's mention of syllogisms and enthymemes, which are types of argument. Lavenham is thus claiming that in an argument the understanding of the conclusion (consequent) necessarily belongs to the premisses (antecedent), which is a reasonable way to gloss Fland's criterion. The Lavenham-Fland account, then, is recognizably the same as our informal characterization of a valid argument as one in which the conclusion is "contained" in the premisses, and a cousin of the view that deductive inference is not ampliative-unlike, say, inductive inference-since the conclusion contains no more information than the premisses. Whether such an account can be made sufficiently precise for logical treatment is another matter. (Modem information-theoretic accounts of deducibility have not met with general acceptance.) Fland is alone among logicians in the first half of the fourteenth century in mentioning it, and so we shall set it aside for now.

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The second and most common account of inferential validity among our authors is modal: the consequence A 1- B is legitimate when it is impossible for A to be true and B false. More precisely, the modal criterion spells out at least a necessary condition for consequences in general to be legitimate. 39 The intuition at work here is familiar. Modal accounts of logical consequence date back to Aristotle, and live on today in Tarskian model-theoretic explanations of logical consequence that take possible worlds to be the models in which an interpretation is evaluated. 40 Well, with Aristotle on one side and Tarski on the other, how did the mediaeval logicians of the first half of the fourteenth century explicate the modal account of consequence? A battery of distinctions were available that would allow them to construct a fairly precise nonmathematical analogue of Tarskian satisfaction. Jean Buridan offers a clear and lucid presentation of the material, so I'll concentrate on his exposition. Roughly, a sentence is true for Buridan when what it says is the case. (This claim has to be tweaked for tense and quality, of course, but we can ignore such niceties for now.) Thus a consequence A 1- B is legitimate when it is impossible for what A says to be the case and for what B says not to be the case. More precisely, it is impossible for the situation that B describes not to hold in the situation that A describes. These situations may be Bltemative possibilities. Buridan distinguishes between situations that a sentence may describe and also belong to, and those situations which it may describe but not belong to. This is his well-known distinction between sentences that are possibly-true and those that are (merely) possible. 41 For instance, the sentence "No sentence is negative" is possible but not possibly-true, because it describes a possible situation but cannot belong to it. Hence we can clearly distinguish a sentence from the situation it describes and also from its truth-value with respect to that situation. In modem terminology, a possible situation functions as a model, and sentences are assigned truth-values relative to the model. Such an assignment of truth-values is a nonmathematical version of Tarskian satisfaction. Hence the consequence A 1- B is legitimate when it is impossible for A to be true and B false, that is, when there is no situation in which A is assigned truth and B falsity. Inferential legitimacy is a function of the truth-value of sentences with respect to situations. The situations are possibilities-possible worlds, if you like. They can be constructed to evaluate sentences, and were extensively used to do exactly that, particularly in the case of sophisms, where they supplied a technique for both modelling and countermodelling: the description of a

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situation (casus) was the starting-point of these investigations. Hence even if the fine points of Buridan's account were not available to or accepted by everyone, the common use of possible situations in sophisms and obligations shows that the philosophical machinery for explicating the modal account of validity was widely available. The widespread agreement among mediaeval logicians on the modal account didn't settle all the philosophical questions, however. Is quantification over such possibilities, as the modal account seems to demand, itself a legitimate procedure? What about inference from the impossible, where by definition there is no possible situation to start with? But put these difficulties, as challenging as they are, aside for the moment. There is a deeper worry about the proto-Tarskian theory sketched here, one recognized in the first half of the fourteenth century. As it stands, the account of truth (as a satisfaction-relation relative to a model) incorporated in the modal account has no clear connection with formal validity-or even with semantics at all. 42 How does inferential legitimacy depend on the formal features of sentences or on their meanings? Consider the three proposals that A 1- B is legitimate when: 43

(1) (2) (3)

the truth of A guarantees the truth of B in virtue of the meanings of the terms in each the truth of A guarantees the truth of Bin virtue of the forms of A and B there is no uniform substitution of nonlogical terminology that renders A true and B false.

Now to the extent that the meanings of the terms in A and B determine the situations-the range of possibilities or models-we evaluate our sentences against, (1) may provide a semantic dimension to the modal account. Yet (1) will fail to capture formal validity to the extent that meaning is not a formal feature. 44 Inferences such as "Socrates is human; therefore, Socrates is an animal" are legitimate by (1) but are not formally valid: they do not hold in virtue of their form but only hold in virtue of some extrinsic feature, such as the meanings of their terms or the way the world is. (Thus even metaphysical necessity does not entail formal validity.) Several mediaeval logicians turned to the theory of topics to explain such materially valid inferences, sometimes reducing them to formal ones, sometimes the converse. 45 In contrast, the account of legitimacy proposed in (2) tries to explain it by connecting truth and formality. It can even be seen as a special case of (1), wherein the meanings of a special set of terms-called nowadays

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the 'logical vocabulary'-constitute the form of a given sentence. Of course, it isn't clear whether logical vocabulary has meaning at all, even when taken in combination with other terms (as the mediaeval account of 'syncategorematic' terms presupposes). But even so there are three further problems with (2) as an explication of legitimacy. First, the very move from (1) to (2) is suspect. Why should we be interested in logical form in the first place? Why not be content with guaranteed truth, for which (1) is sufficient? It should be an open question whether all validity is formal validity, but (2) closes the subject. Second, mediaeval and modern logicians alike have yet to come up with a criterion to identify the "form" of a sentence that doesn't simply beg the question. What is the logical form of a definite description? Of a paradoxical liar-like sentence? Of sentences involving the word "begins"? Third, even if we could specify the form of a sentence without begging any questions, we need to know how formal features determine possibilities.46 For example, suppose that the (logical) form of the sentence "Socrates is older than Plato" is "x is older than y." Surely not all situations in which one thing is, or is claimed to be, older than another count as possibilities relevant to evaluating the original sentence. What bearing does the situation in which my piano is older than my violin have on Socrates's being older than Plato? It is not that the net of possibility is cast too widely-instead, it seems to be miscast. The age of my musical instruments is simply irrelevant to the respective ages of Socrates and Plato. Insofar as such possibilities are prescribed by (2), the intuitive punch of the modal account is lost. Rather than taking possibilities to be spelled out by the meanings of terms of a sentence or by the structure of a sentence, we could instead look directly at the truth-value of sentences generated by altering a given sentences's (nonlogical) terminology. This is the key intuition behind the substitutional account of legitimacy presented in (3). We can best judge legitimacy by seeing whether an inference holds in terms other than those in which it is originally couched. (Our ability to judge the truth-value of the candidate sentences is assumed.) Furthermore, to the extent that we can identify some terms as part of the logical vocabulary and so as elements of the form of the sentence, (3) will be a formal account as well. Hence Buridan, for example, endorses (3) as the correct account of legitimacy, specifically linking it to formality. 47 Uniform subsitutivity, of the sort proposed by Buridan, is the third account of inferential validity. He is clear that (3) goes beyond the modal account in at least two ways. First, it applies equally to material (non-formal)

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consequences; thus the formality of an inference is a feature that goes beyond its necessity, neither explained by nor reducible to it. Second, it takes legitimacy to be a function of truth-value relative to a set of terms, namely the nonlogical vocabulary, rather than appealing to possibilities. The third account of inferential validity therefore takes a decidedly linguistic and non-metaphysical approach. The mediaeval consensus on a proto-Tarskian account of satisfaction, then, conceals deep divergences in the attempt to explain legitimacy. It may be worthwhile to change directions in pursuing this problem. Rather than looking more closely at the nature of formal and material inference (a topic on which there doesn't seem to be much agreement among mediaeval logicians), we can try to make some headway by understanding how mediaeval logicians reasoned about possible situations and alternatives. Since consequences license arguments, where such possibilities are found, we can start there; after examining the nature of arguments I'll conclude with some reflections on formality and the logical enterprise. 5. ARGUMENT AND ARGUMENTATION Let me pick up a thread from §2 and return to the relation between conditionals and consequences. They are logically distinct notions. Are they correlated in any way? At the beginning of our period the question seems to be ignored, but by the end two schools of thought have emerged. On the one hand, Ockham and his followers hold that conditionals and consequences are logically interchangeable. In Summa logicae 2.31, Ockham declares that since a conditional is equivalent (aequiualet) to a consequence he'll just talk about the latter (347.2-5). So too the Logica ad rudium 2.76. The treatise De consequentiis §7 gives us the other direction: every legitimate consequence is equivalent to a true conditional (Green-Pedersen 1982, 93). The moral is eventually drawn in the fifteenth rule of the Liber consequentiarum 2: 48 Every consequence is equivalent to a conditional composed of the antecedent and consequent of the given consequence with 'if put in front of the antecedent, and conversely every conditional is also equivalent to a consequence composed of the antecedent and consequent of the given conditional with 'therefore' put in front of the consequent.

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This 'mediaeval deduction theorem' permits the logician to pass between conditional and inferential formulations of the same claim, without any logical baggage getting lost in the transfer, as it were. On the other hand, conditionals and consequences might be thought to differ in ways that prevent them from being simply exchanged for one another. This is the position of Jean Buridan (and Albert of Saxony, as usual, in his wake). He sketches the difference in his Summulae de dia/ectica 7.4.5, when he disentangles the sloppy use of 'consequence' in place of 'conditional' :49 Note that 'consequence' is twofold: (1) a conditional sentence, which asserts neither its antecedent nor its consequent (e.g. "if an ass flies an ass has wings") but only asserts that the one follows from the other, and such a consequence is not an argument since it doesn't prove anything; (2) an argument wherein the antecedent is known and better-known than the consequent, which asserts the antecedent and on that basis implies the consequent as an assertion. Furthermore, in a conditional we use 'if and in an argument 'therefore.'

Conditionals do not involve the assertion of their parts, whereas consequences do. Buridan makes the same point earlier as regards syllogisms (Summulae de dialectica 1.7.3): 50 The syllogism differs from the conditional sentence too, because in the conditional sentence its categorical parts aren't put forward in the manner of an assertion (i.e. affirmatively), whereas they are put forward in the manner of an assertion in syllogisms-e.g. that every B is A and every C is B, and the conclusion that every C is A is drawn in the manner of an assertion. Thus we say that a syllogism with false premisses is materially defective, which shouldn't be said of the conditional "If an ass is flying, an ass has wings."

An argument, as noted in § 1, does not make a statement. Yet it does license the making of a statement by anyone who accepts its premisses, namely the detachable statement of its conclusion. Implication does not work like this: conditionals do make statements, namely statements about the relation between the antecedent and the consequent, but unlike arguments they do not license further statements. Hence, for Buridan, consequences and conditionals are not interchangeable. 51 Buridan's view is that accepting or rejecting premisses, committing oneself to an inference, warranting further freestanding statements, and other activities that we might broadly call "dialectical" are partially constitutive of the sense of an argument. 52 However, we don't have to believe that giving an argument will automatically commit us to asserting its conclusion in order to take Buridan's point. Even Buridan didn't think so-otherwise, we wouldn't be able to draw conclusions from an

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opponent's views to refute him. Instead, the point well-taken from Buridan's discussion is that making an argument necessarily involves taking a dialectical stance. What an agent is doing dialectically in stringing together statements will depend, at least in part, on whether the agent is (say) accepting the statements, or rejecting them, or granting them temporarily, or is in doubt about what to do. Which commitments the agent has will depend on which dialectical stance he adopts. The factors listed here as making up an agent's dialectical stance are, of course, precisely those that enter into obligationes. They enable arguments to be what they are in the first place, namely a kind of activity in which we do things with statements. And, as such, they are ways of doing things with statements. 53 For arguments are not independent objects that can be analyzed apart from the contexts in which they occur. Part of their sense-or at least part of what it is to string statements together in an inference-making performance-is arguably constituted by these obligational attitudes (for want of a better description). 54 Obligational treatises are, among other things explored at length in this volume, conscious attempts to work out how certain obligational attitudes are related to inferences. They are efforts to explore the logical features of arguments-dialectical performances-found in the wild. To be sure, we can domesticate obligational attitudes to some extent. If we consider arguments solely from the point of view of accepted (or perhaps even conceded) premisses, a theory of valid consequence that makes only tacit reference to its dialectical origins can be constructed. This is, in essence, the theory of the syllogism. But the task of the mediaeval logician is to examine arguments wherever they may be found, including their natural habitats, and on that reading obligations are part of logical theory proper. Yet obligational attitudes are not, or not in any obvious way, formal features of arguments-that is, they aren't part of the logical form of an argument as such; we seem to be able to talk about arguments without referring to their dialectical contexts. To get straight on how obligations are part of the logical enterprise, we need to look a bit more closely at formality and its connection with logic, both mediaeval and modem. In so doing we'll get a better idea of the logical enterprise generally.

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CONCLUSION Modem logicians, who spend much of their time either devising logical systems that are mathematically-defined objects or investigating the properties of such systems (metatheory), are engaged in a fundamentally modem enterprise. Mediaeval logicians were in no position to do either of these tasks. Yet mediaeval logic is still logic, after all; its relation to modem logic is not like the relation of alchemy to chemistry. The glory of modem logic is rather that it succeeds in treating logic mathematically. But logic is not intrinsically mathematical; it would have little past before Principia mathematica if it were. Yet the influence of mathematics on logic has undeniably changed its character: mediaeval and modem logic are overlapping but distinct enterprises. Each is concerned with logic as in some sense the study of correct reasoning, but without more content this slogan doesn't get us very far. What more can be said? Well, mediaeval and modem logic both attempt to be rigorous and systematic. And, more importantly, each is a formal discipline. That is, they are concerned with studying properties of formal features, e.g. determining which inferences hold in virtue of the logical form of the premisses and of the conclusion (truth-preserving formal inferences). Modem logic is formal and formalized (symbolic); mediaeval logic is formal but not formalized. To this extent Ockham and Tarski are engaged in a common endeavor and the history of logic stretches back to Aristotle. Mediaeval logic is also nonformal. 55 That is, mediaeval logic deals with inferences and assertions that do not hold in virtue of their formal features as well as those that do. Here Ockham and Tarski part ways: modem logic concentrates exclusively on formal properties whereas mediaeval logic is more inclusive. Some sense of the scope of mediaeval logic can be gotten by looking at the variety of subjects falling within its scope: semantics, reference, syncategoremata, syllogistic, consequences, topics, sophisms, paradoxes, obligations, and fallacies. Yet I think there is a single conception of logic here, with consequences at its heart. It is this. Mediaeval logic is the enterprise of devising theories about inference. Inferences may be formal or material, legitimate or illegitimate, and are found in different dialectical circumstances. The unity of mediaeval logic is grounded in its conception of inference (consequence), the key to nonformal logic. Now mediaeval logic is recognizably related to modem mathematical logic, since it studies formal legitimate inferences, the sole subject of modem logic. But it also

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studies much else besides, such as illegitimate inferences (the theory of fallacies). Whether the mediaeval conception of logic as the nonformal study of inference is a worthy competitor to the modem mathematical conception of logic is another question. We cannot make a start on answering it until we recognize the centrality of the notion of inference in mediaeval logic. An obvious first step in that process is clarifying the notion of inference itself. As I have argued here, this was accomplished in the first half of the fourteenth century through a natural deduction system and articulated in discussions of consequences, which are the heart of argument and, by extension, the very heart of (mediaeval) logic 56 itself.

()hio State

[Jniversi~

NOTES 1

All translations are mine. I what follows I cite the Latin text only when it is not readily available (e.g. for much of Buridan's Summulae de dialectica), when there is a textual difficulty, or when a point depends on its original phrasing. The texts on which this study is based are all listed in Part (A) of the Bibliography; when I speak of "the available literature" these are the works I have in mind. GreenPedersen 1983 catalogues several other texts about consequences that exist only in manuscript. The available literature seems to fall roughly into four groups. [ 1] The anonymous two earliest treatises on consequences, along with Walter Burley's De consequentiis and his De puritate artis logicae-the longer version being influenced by Ockham. [2] William of Ockham's Summa logicae, whose influence can be seen in the Elementarium logicae and the Tractatus minor logicae (formerly ascribed to Ockham himself), the anonymous treatises Liber consequentiarum and Logica "Ad Rudium ", and the unusual commentary De consequentiis possibly written by Bradwardine. [3] Jean Buridan's Tractatus de consequentiis and Summulae de dialectica, whose influence can be seen in Albert of Saxony's Perutilis logica or Marsilius of Inghen's De consequentiis. [4] The Consequentiae of Robert Fland and of Richard Ferrybridge, dating from the close of the first half of the fourteenth century, which have many affinities with the later works of (for example) Richard Billingham, Richard Lavenham, and Ralph Strode. 2 See Jaskowski 1934 and Gentzen 1935; Prawitz 1965 gives a modern presentation of natural deduction systems. The claim defended here is the mediaeval counterpart of the case put forward for Aristotle's logic initially by Smiley 1973 and Corcoran 1974, since developed in Lear 1980, Thorn 1981, and Smith 1989. Note the limited scope of my thesis: whatever consequences may have been before 1300 and whatever they may have become after 1350, in the first half of the fourteenth century they constituted a natural deduction system. An admirably clear statement of this position is given in Moody 1953, 15: "The theory of consequence, taken as

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a whole, constituted a formal specification of inference-conditions for the formulated language." Kneale and Kneale 1962, 4.5, describe "a change of fashion" in writings on consequences around 1300, "something like that from Aristotle's presentation of syllogistic theory by means of conditional statements to Boethius' presentation by means of inference schemata" (277). In these historians natural deduction has been glimpsed, but only as in a mirror darkly; I intend to show it to the reader face-to-face. 3 Three recent examples, each a near miss. Boh 1982, 300 writes: "Implication, entailment, and inference are all distinct from one another. .. Nevertheless, medieval logicians disconcertingly use the single notion of consequence to cover all three of these relationships between propositions." (They did no such thing, as we shall see in §§1-3 below.) In King 1985, 59-60 I argued that consequences have features of conditionals as well as inferences, and hence are neither fish nor fowl. Adams 1987, 458-490, who quite properly renders consequentiae as 'inferences,' discusses at length whether Ockham's rules define a version of strict implication. 4 Most modem interpretations of mediaeval rules for consequences take them at best to present axioms, or perhaps theorems, of a connexive logic (as in MacCall 1966). 5 Aristotle, De interpretatione 5 17a21-22 (Boethius's translation): "Harum autem haec quidem simplex est enuntiatio, ut aliquid de aliquo uel aliquid ab aliquo ... " See also De interpretatione 6 17a25-26: "Affrrmatio uero est enuntiatio alicuius de aliquo, negatio uero enuntiatio alicuius ab aliquo." 6 This account oversimplifies the complex nature of conditionals, even the "ordinary'' present-tense indicative conditional. See Woods 1997 or the articles in Jackson 1991 for an account of some of the difficulties. There are other reasons for distinguishing conditionals from inferences; the argument in Carroll I 895 shows that axioms need to be supplemented by rules of inference. Haack 1976 argues that the need for a justification of deduction outlined in Dummett 1973 generalizes Carroll's argument into a dilemma, so that there is either an infinite regress or circularity. But these arguments were unknown in the fourteenth century, so I will not treat them here. (This is not to say that mediaeval logicians did not recognize the need to justify particular inference-rules; they surely did, but just as surely didn't see the enterprise of doing so as deeply problematic.) 7 The Deduction Theorem can be proved from the axioms: A~ (B~A)

(A ~ (B ~ C)) ~ ((A ~ B) ~ (A ~ C)) along with detachment, by induction on proof length. Modem logical systems sharply distinguish syntactic consequence (1-) from semantic consequence (~); the mediaeval analogue is discussed in §4. 8 A sentence may be categorical whether it be affrrmative or negative (the quality of a sentence is part of its logical form); universal, particular, or indefinite (so too the quantity); assertoric or modal; even-within limits-internally complex. 9 Negation is not a connective: sentential negation is accomplished by a categorically distinct copula, so that 'is' and 'is not' are two different functors (mutatis mutandis through all the tenses and modes). The mediaeval account of compound sentences doesn't precisely match the modem notion of widest scope, since the latter has mathematical properties the former lacks, e.g. embedding of

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formulae at arbitrary depth. The precise kind of statement made by compound sentences is a matter of some complexity. We need not explore it here. It is not the details but the bare fact that conditional sentences make statements (and thereby are true or false) that is significant. 10 More exactly: sentences consist in words but are not simply sequences of words; the combination of words into a sentence used to make a statement goes beyond anything in the words themselves-sentences are a way of doing something with words (namely making a statement). Likewise, arguments are a way of doing something with statements, as noted above. 11 The third part analyzes arguments in general, as Ockham tells us in Summa logicae 3-1.1 (359.2-3). 12 The anonymous Logica ad Rudium, structured in the same fashion as the Summa logicae, likewise treats conditionals as a kind of compound sentence (2. 76-78) and consequences as a form of argument (3.64-84). So too the Tractatus minor logicae 2.2 for conditionals and Book 5 for consequences, as well as the Elementarium logicae 2.16 and Book 6. Now Ockham and other logicians sometimes restrict consequences to nonsyllogistic inferences, but this is a matter of terminology and not doctrine: Ockham expressly says that it is a terminological convenience. 13 Buridan's Summulae de dialectica is divided into the following treatises: [I] introductory material and sentences; [2] predicables; [3] categories or categorematic expressions; [4] supposition; [5] syllogisms; [6] dialectical topics; [7] fallacies; [8] demonstrations; and sometimes [9] sophisms. Although there is no treatise devoted to consequences, Buridan does discuss them in [5]-[6], whereas he describes and defines conditionals in [ 1]. 14 The only possible exception: in Summulae de dialectica 7.4.5 (discussed in §5), Buridan does say that 'consequence' can mean either a conditional sentence or an inference. But his entire discussion of consequences uses the second sense, not the first, which he never mentions again. 15 Consequences in fact have a distinct property: they can be legitimate, and thereby they may hold or be valid. This will be discussed in §§3-4. 16 In §69, Green-Pedersen renders the text "Exemplum primae: situ es Romae, ergo illud quod est falsum est uerum ... " adding the ergo with 0 (rather than omitting it with CL). But Burley is giving here an example of a conditional, not a consequence; the ergo should be suppressed. 17 "Notandum est quod haec est una condicionalis uera et necessaria: si homo est asinus, homo est animal brutum." 18 A fourth passage can be set aside as merely terminological. In the second Anonymi de consequentiis §19 (Green-Pedersen 1980), a mention is made of a 'false consequence'-an instance of asserting the consequent-but this is plainly an extension of 'false' to inferences that are fallacious, not meant to ascribe a truthvalue; it is no more to be taken seriously than Burley's willingness to speak of the same fallacy as being a "false rule" in both versions of his De puritate artis logicae (200.16-17 [shorter] and 62.14-15 [longer]). 19 In speaking of material consequences, Pseudo-Scotus says that some are true simpliciter and some ut nunc-the former can be reduced to formal consequences by the assumption of a necessary proposition, whereas the latter refer to consequences that hold contingently, not at all times. Yet by the time he gets to ut

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nunc consequences he switches back to speaking of legitimacy, never returning to truth again. (See the translation included in this volume.) It seems clear that this is no more than a slip of the pen, since nothing in his discussion turns on whether legitimacy or truth is at stake and the usage is completely isolated. 20 Elizabeth Karger has proposed that Buridan is here stipulating that he will use 'consequence' to pick out only true conditionals, and hence that consequences do have truth-value. This reading is possible, but, I think, not borne out by other evidence: nowhere else in the Tractatus de consequentiis does Buridan ever rely on consequences having truth-value. The passage is surely anomalous. 21 "Et uidetur mihi quod talis hypothetica coniungens categoricas per 'ergo' debet similiter reputari falsa si non sit necessaria consequentia, quae designatur per istam dictionem 'ergo,' et quod etiam sit falsa simpliciter Ioquendo si habeat aliquam ~raemissam falsam." 2 The Elementarium logicae 2.16 contrasts consequences and conditionals by their possession of different properties (94.3-8): "Just as a consequence can be legitimate even though neither of its sentences is true, and even though the antecedent is false and the consequent true-but is never legitimate if its antecedent is true and its consequent false-so too a compound conditional sentence can be true even if neither of the categorical sentences of which it is composed is true, and even if the first is false and the second true, but not if the first is true and the second false." 23 Modern logicians have been largely successful with 'consequent' (although it sometimes carries the sense of 'important' outside logical circles) but not at all with 'antecedent,' which still has a broad range of uses not tied to either conditionals or consequents-for instance, in speaking about one's background or f,enealogy. 4 See The Oxford Latin Dictionary 1982 at 138AB s.v. antecedo and also at 413BC s.v. consequens I consequor. Note that the constituents of a Gentzen sequent in a natural deduction system are called the 'antecedent' and the 'succedent'-acceptable translations of antecedens and consequens! 25 Burley refers explicitly to the "syllogistic antecedent" (antecedens syllogisticum) of a consequence in the longer version of his De puritate art is logicae ( 65. 7). 26 Buridan repeats the point at Tractatus de consequentiis 1.3.12 (22.61). The same claim is made in passing by the Pseudo-Scotus at the start of q.l 0; see also Pinborg 1972, 170. 27 A third reply-that Buridan can be discounted as a single voice against many others-will not do for two reasons. First, Buridan was a superb logicians, and voices must be weighed rather than counted. Second, he may not be a lone voice; the longer version of Burley's De puritate artis logicae seems to classify consequences under the generic heading of conditional compound sentences (the first part of the second treatise), although the incomplete nature of the text makes it hard to put much weight on its structural divisions. 28 Summulae de dialectica 5.1.3: "Respondeo quod licet syllogismus sit compositus ex pluribus orationibus, tamen est una propositio hypothetica, coniungens conclusionem praemissis per bane coniunctionem 'ergo.' Et potest reduci ad speciem propositionum condicionalium, quia sicut condicionalis est una consequentia, ita et syllogismus; unde syllogismus posset formari per modum

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unius condicionalis sic 'si omne animal est substantia et omnis homo est animal, omnis homo est substantia.'" 29 There are special challenges for Buridan, as a nominalist, to distinguish sentences (particular mental tokenings perhaps correlated with individual utterances or inscriptions) from the statements they make. We need not pursue this point here. 3 Consequences are identified either (quasi)-syntactically as sentences connected by an illative particle, or as the relation obtaining among such sentences-for example, in §I of the first anonymous treatise on consequences in Green-Pedersen 1980, a consequence is defined as a "relationship (habitudo) between an antecedent and a consequent". (Note that Green-Pedersen 1983 calls the second definition 'semantic' and says that one or the other is given in all the writings on consequences, that is, in both the published and unpublished manuscript texts.) 31 See §2 of the De consequentiis possibly written by Bradwardine: "Every consequence is taken to underwrite* some argument" (*probandum L; GreenPedersen adopts producendum from the badly defective V). 32 Mediaeval logicians, like modem logicians, vacillated about whether to say that a fallacious inference was an inference, and hence whether 'legitimate inference' was pleonastic. The sense is usually clear from context. I'll follow the mediaeval lead here. 33 The first Anonymi de consequentiis § 18 (Green-Pedersen 1980, 7.12-15). 34 Detachment is a rule of inference and not to be confused with the law of p.ropositional logic (A & (A ---7 B)) ---7 B. 5 Note that Ockham's first rule treats the relation between the terms 'animal' and 'man' as a formal feature. Modem first-order logic does not normally respect such relationships, but could do so in a number of ways: indexing or sorting the termvariables; adding semantic rules along the lines of meaning-postulates; and the like. Are such consequences formal? How would we decide? What difference does it make? See §4 below. 36 I badly mangled the analysis of Buridan's account of the reductio-method in King 1985, 73-74 (not least by using conditional form in my account). 37 There is some looseness here: do we identify the consequence as the inferential force of the argument, or as the argument constituted by the inferential force? (Is the inference the whole formula A ~ B or just the open formula ... ~ --- ?) Different mediaeval authors answered the question differently. 38 The three accounts canvassed in this section have usually been identified as truthconditions for implication. As such, they seem to spell out intensional (psychological), modal (strict entailment), and formal conceptions of implication. However, they are accounts of validity rather than truth-conditions, as we shall see. 39 Mediaeval logicians drew several distinctions among kinds of consequences, such as the distinction between consequence simpliciter and consequence ut nunc. Does the modal account of validity range over times or just possibilities at a time? I'm inclined to the latter, and that the common mediaeval view was that "all consequences are necessary'' (as the De consequentiis possibly written by Bradwardine asserts in §7). I will proceed as though the question were settled, but it deserves more attention than I can give it here. 40 The success of the model-theoretic notion of logical consequence, derived from (but not identical to) the version presented in Tarski 193 5, has been challenged

°

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recently in Etchemendy 1990. The discussion of these matters in Shapiro 1998 is extremely helpful, and I make use of his distinctions among accounts of logical consequence in what follows. 41 Prior 1969 is the locus classicus. The distinction is drawn from the ftrst two chapters ofBuridan's Sophismata 8. 42 This point can be pressed against Tarski, and is vigorously argued in Etchemendy 1990. 43 Taken from Shapiro 1997, 132: his (6), (9), and (8) respectively. 44 Modem logicians have made various attempts to treat meaning as a formal feature: see n.35 above. Mediaeval logicians made no such comparable attempt, although they were divided on how to treat certain kinds of structured meaningrelations (notably between subordinate and superordinate elements in a categorial line). One technique was to use the theory of topics-see the following note. 45 See Green-Pedersen 1984 and Stump 1989 for discussion of the use of topics in this period. (Interestingly, Tarski also speaks of "material consequences": Shapiro 1998, 148.) Burley, for example, says in the longer version of his De puritate artis logicae that every consequence holds in virtue of a logical topic (75.35-76.10). Ockham's awkward doctrine of intrinsic and extrinsic middles may be seen as addressing some of these worries. 46 See Shapiro 1998, 143 on interpretational and representational semantics. 47 Tractatus de consequentiis 22.5-9. See also Summulae de dialectica 1.6.1: "Et quia nunc locutum est de consequentia formali et materiali, uidendum est quo modo conueniant et differant: conueniunt enim in hoc quod impossibile est antecedens esse uerum consequente exsistente falso; sed differunt quia consequentia 'formalis' uocatur quae si ex quibuscumque terminis formarentur, propositiones similis formae ualeret similiter consequentia." Buridan's account of substitutivity is similar to Bolzano 1837. 48 Schupp 1988, 123.198-203: "Decima quinta regula est haec quod omnis consequentia aequiualet condicionali compositae ex antecedente et consequente illius consequentiae cum nota condicionalis praeposita antecedenti, et econuerso omnis condicionalis etiam consequentiae compositae ex antecedente et consequente illius condicionalis cum nota consequentiae praeposita consequenti." 49 "Deinde notat duplicem esse consequentiam, scilicet unam quae est propositio condicionalis, et ilia nee asserit antecedens nee asserit consequens (ut 'si asinus uolat, asinus habet alas'), sed solum asserit quod hoc sequitur ad illud. Et ideo talis consequentia non est argumentum; nihil enim concludit. Alia consequentia est argumentum si antecedens sit notum et notius consequente, quae asserit antecedens et ob hoc infect assertiue consequens. In condicionali autem utimur hac coniunctione 'si' et in argumento hac coniunctione 'ergo."' 50 "Et etiam syllogismus differt a propositione condicionali, quia in propositione condicionali nullo modo categoricae proponuntur modo assertiuo, id est affrrrnatiuo, sed in syllogismis proponuntur modo assertiuo, ut quod omne B est A et omne C est B, et concluditur assertiue quod omne C est A. Et ideo dicimus syllogismus ex falsis praemissis peccare in materia, quod non sic est dicendum de ista condicionali 'si asinus uolat, asinus habet pennas. "' 51 Modem logicians are divided over whether arguments do in fact license freestanding occurrences of their conclusions. For instance, if we think of logical

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consequence along purely syntactic lines, the formula A f- B says only that B can be deduced from A-a claim that seems to carry no commitment to B (or presuppose any endorsement of A). This is evidence that Buridan and other mediaeval logicians did not think of consequence as simple deducibility. 52 Perhaps a more fmc-grained approach would be useful here. David Kaplan's distinction of propositional context, character, and content allows us to state Buridan 's claim more exactly: dialectical activities fix the character of propositions as they occur in arguments; they do not enter into their content. 53 Some of the dialectical features described here have been explored at length in Brandom 1994. But his project of"inferential semantics" is not the mediaeval one: to insist that arguments have some irreducible social features is a far cry from maintaining that (all) meaning is constituted by inferential roles and permissible moves of our language-games. 54 If arguments depend on obligational attitudes for their sense, it is misleading to represent them as operators that extend or enrich an independent logical s?'stem-as, for example, in Bob 1993 (for epistemic operators). 5 This is not the same as our modem conception of informal logic, which is at best the general study of deductive and inductive reasoning, the latter based on probability and statistics. Unfortunately, "informal logic" is usually taken to be synonymous with "critical thinking": equal parts of rhetoric, traditional fallacies, and epistemic good sense. There is nothing particularly logical about informal logic taken in this sense. 56 I'd like to thank Anna Greco, Elizabeth Karger, Stewart Shapiro, William Taschek, and Mikko Yrji:insuuri for helpful comments and conversations.

REFERENCES

A. Mediaeval texts 1300-1350 Anonymous. Anonymi de consequentiis (ca. 1300), edited in Green-Pedersen 1980, 4-11. Anonymous. Anonymi de consequentiis (ca. 1300), edited in Green-Pedersen 1980, 12-28. Anonymous. Liber consequentiarum (1330-1340), edited in Schupp 1988, 104-171. Anonymous. De consequentiis (1325-1340), (ed.) Niels Jorgen Green-Pedersen in "Bradwardine(?) On Ockham's Doctrine of Consequences: An Edition," Cahiers de l'institut du moyen-dge grec et latin 42 (1982), 85-150. Anonymous. Logica "Ad Rudium" (1335), (ed.) L. M. De Rijk, (Artistarium 1), Nijmegen, Ingenium 1981. Buridan, Jean. Tractatus de consequentiis (1335), (ed.) Hubert Hubien in /ohannis Buridani tractatus de consequentiis, (Philosophes medievaux 16), Louvain, Publications Universitaires 1976. Translated in King 1985. Buridan, Jean. Summulae de dialectica (1340s?). References are to book, chapter, and section number. The second book has been edited by L M. De Rijk as the Summulae de praedicabilibus (Nijmegen, Ingenium 1995). The third book has

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been edited by E. P. Bos as the Summulae in Praedicamenta (Nijmegen, Ingenium 1994). For other parts of the text I have used the unpublished transcription of Hubert Hubien, employed in the critical editions, and provide the Latin in the notes. Burley, Walter. De consequentiis (1300), (ed.) Niels J0rgen Green-Pedersen in "Walter Burley's 'De Consequentiis': An Edition," Franciscan Studies n.s. 40 (1980), 102-166. Burley, Walter. De puritate artis logicae (tractatus brevior) (early 1320s), (ed.) Philotheus Boehner O.F.M. in Walter Burley: De puritate artis logicae, Tractatus Longior. With a Revised Edition of the Tractatus Brevior, St. Bonaventure: The Franciscan Institute 1955, 199-260. Burley, Walter. De puritate artis logicae (tractatus longior) (later 1320s), (ed.) Philotheus Boehner O.F.M. in Walter Burley: De puritate artis logicae, Tractatus Longior. With a Revised Edition of the Tractatus Brevior, St. Bonaventure, The Franciscan Institute 1955, 1-197. Ferrybridge, Richard. Consequentiae (ca. 1350), printed in Consequentiae Strodi cum commento Alexandri Sermonetae. Declarationes Gaetani in easdem consequentias. Dubia magistri Pauli Pergulensis. Obligationes eiusdem Strodi. Consequentiae Ricardi de Ferabrich. Expositio Gaetani super easdem, Venetiis 1507. Fland, Robert. Consequentiae (1350s?), (ed.) Paul Spade in "Robert Fland's Consequentiae: An Edition," Mediaeval Studies 38 (1976), 54-84. Pseudo-Ockham. Tractatus minor logicae (ca. 1340-1347), originally edited by Eligius M. Buytaert; revised by Gedeon Gal and Joachim Giermek, printed in William of Ockham, Opera philosophica VII (opera dubia et spuria), St. Bonaventure, The Franciscan Institute 1988, 1-57. Pseudo-Ockham. Elementarium logicae (ca. 1340-1347), originally edited by Eligius M. Buytaert; revised by Gedeon Gal and Joachim Giermek, printed in William of Ockham, Opera philosophica VII (opera dubia et spuria), St. Bonaventure, The Franciscan Institute 1988, 58-304. Pseudo-Scotus (John of Cornwall?). Super librum primum et secundum Priorum Analyticorum Aristotelis quaestiones (around 1350), printed and wrongly ascribed to Scotus in Joannis Duns Scoti Doctoris Subtilis Ordinis Minorum opera omnia, ed. Luke Wadding, Lyon 1639; republished, with only slight alterations, by L. Vives, Paris 1891-1895. Question 10 is translated in the present volume. William of Ockham. Summa logicae (1323). Edited by Philotheus Boehner O.F.M., Gedeon Gal OF.M., Stephanus Brown, Opera philosophica I, St. Bonaventure, The Franciscan Institute 1974.

B. Selected mediaeval texts 1350-1400 Albert of Saxony. Perutilis logica (1370s), Venetiis 1522. Reprinted in Documenta Semiotica 6. Hildesheim, Georg Olms 1974. Lavenham, Richard. Consequentiae (1370), (ed.) Paul Spade in "Five Logical Tracts by Richard Lavenham," in Essays in Honor of Anton Charles Pegis, ed.

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J. R. O'Donnell, Toronto, University of Toronto Press 1974, 70-124 (text 99-112). Strode, Ralph. Tractatus de consequentiis (1360?), text and translation in W. K. Seaton, An Edition and Translation of the Tractatus de cnnsequentiis of Ralph Strode, Ph.D. Dissertation, University of California at Berkeley 1973. Ann Arbor, University Microfilms 1974.

C. Modern works Adams, Marilyn, 1987. William Ockham, 2 volumes, Notre Dame, Indiana, University ofNotre Dame Press. Boh, Ivan, 1962. "A Study in Burley: Tractatus de regulis generalibus consequentiarum," in The Notre Dame Journal of Formal Logic 3, 83-101. Boh, Ivan, 1982. "Consequences," in The Cambridge History of Later Medieval Philosophy, Norman Kretzmann, Anthony Kenny, Jan Pinborg, eds. Cambridge, Cambridge University Press. Boh, Ivan, 1993. Epistemic Logic in the Later Middle Ages, London, Routledge & Kegan Paul. Bolzano, Bernard 1837. Wissenschaftslehre (4 vols.), Leipzig, Felix Meiner. Bos, E. P., 1976. "John Buridan and Marsilius of Inghen on Consequences," in Jan Pinborg (ed.), The Logic of John Buridan, Acts of the Third European Symposium on Medieval Logic and Semantics, Copenhagen, Museum Tusculanum. Brandom, Robert B., 1994. Making It Explicit. Reasoning, Representing, and Discursive Commitment, Cambridge, MA., Harvard University Press. Carroll, Lewis, 1985. "What the Tortoise Said to Achilles," Mind 4, 278-280. Corcoran, John, 1973. "Aristotle's Natural Deduction System," in John Corcoran (ed.), Ancient Logic and Its Modern Interpretations, Dordrecht, D. Reidel 1973. Dummett, Michael, 1973. "The Justification of Deduction," Proceedings of the British Academy 1973, 201-232. Etchemendy, John 1990. The Concept of Logical Consequence, Cambridge, MA., Harvard University Press. Gentzen, Gerhardt, 1935. "Untersuchungen iiber das logische Schliessen," Mathematische Zeitschrift 39, 176-210 and 405-431. Green-Pedersen, Niels J0rgen, 1980. "Two Early Anonymous Tracts on Consequences," Cahiers de l'institut du moyen-age grec et latin 35, 1-28. Green-Pedersen, Niels J0rgen, 1983. "Early British Treatises on Consequences," in P. Osmund Lewry (ed.), The Rise of British Logic, Acts of the Sixth European Symposium on Medieval Logic and Semantics, (Papers in Mediaeval Studies 7), Toronto, Pontifical Institute of Mediaeval Studies, 285-307. Green-Pedersen, Niels J0rgen, 1984. The Tradition of the Topics in the Middle Ages, Miinchen, Philosophia Verlag. Haack, Susan, 1982. "The Justification of Deduction," Mind 95, 216-239. Jackson, Frank, (ed.) 1991. Conditionals, Oxford, Oxford University Press 1991. Jaskowski, Stanislaw, 1934. "On the Rules of Suppositions in Formal Logic," Studia logica 1, 5-32.

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King, Peter, 1985. Jean Buridan 's Logic: The Treatise on Supposition and the Treatise on Consequences, Dordrecht, D. Reidel. Kneale, William, and Kneale, Martha, 1962. The Development of Logic, Oxford, Oxford University Press. Lear, Jonathan, 1980. Aristotle and Logical Theory, Cambrdige, Cambridge University Press. McCall, Storrs, 1966. "Connexive Implication," The Journal of Symbolic Logic 31, 415-433. Moody, Ernest A., 1953. Truth and Consequence in Medieval Logic, Amsterdam, North-Holland. Nute, Donald, 1980. Topics in Conditional Logic, Dordrecht, D. Reidel. Pinborg, Jan, 1972. Logik und Semantik im Mittelalter. Ein Oberblick, Stuttgart, Fromann-Holzboog. Pozzi, Lorenzo, 1978. Le consequentiae nella logica medievale, Padova, Liviana editrice. Prawitz, Dag, 1965. Natural Deduction. A ProofTheoretical Study, Stockholm, Almqvist & Wiksell. Prior, Arthur N., 1953. "On Some Consequentiae in Walter Burley," New Scholasticism 27, 433-446. Prior, Arthur N., 1969. "The Possibly-True and the Possible," Mind 78, 481-492. Schupp, Franz, 1988. Logical Problems of the Medieval Theory of Consequences, (History of Logic 6), Napoli, Bibliopolis. Shapiro, Stewart, 1998. "Logical Consequence: Models and Modality," in Mathias Schiro (ed.), Philosophy of Mathematics Today, Oxford, Oxford University Press, 131-156. Smiley, Timothy, 1973. "What is a Syllogism?," The Journal of Philosophical Logic 2, 136-154. Smith, Robin, 1989. Aristotle: Prior Analytics, Indianapolis, Hackett Publishing Company. Stump, Eleonore, 1989. Dialectic and Its Place in the Development of Medieval Logic, Ithaca, NY, Cornell University Press. Tarski, Alfred, 1935. "Der Wahrheitsbegriffin den formalisierten Sprachen," Studia philosophica 1, 261-405. (Derived from an earlier Polish paper published in 1933.) Thorn, Paul, 1981. The Syllogism, Miinchen, Philosophia Verlag. Woods, Michael, 1997. Conditionals, (ed.) David Wiggins, with a commentary by Dorothy Edgington, Oxford, Clarendon Press.

IVAN BOH

CONSEQUENCE AND RULES OF CONSEQUENCE IN THE POST -OCKHAM PERIOD

Late medieval logicians after 1350 continued to be concerned with a cluster of topics connected with the idea of logical inference. Their efforts revolved around the following topics: (a) the nature of consequence in general and of sub-types of consequence; (b) the criteria of valid or sound consequence; (c) the most general propositional rules including those governing modal, epistemic, and obligational realms; (d) the problem with the so-called Ex impossibili -rule: 'From any impossible proposition any other proposition follows'. The number of logicians and philosophers who were engaged in these fields of investigation was quite large' and we can only concentrate on a few major philosophers who seem to be most interesting in developing the cluster of themes just mentioned. We chose John Buridan, Ralph Strode, Peter of Mantua, and Domingo de Soto. Although some symbolic apparatus and some concepts of our times have been employed in our analyses of medieval texts, this was done with the hope to clarify ideas and not as a claim about "modernity" of the medieval logicians. Our project is not so much one of reconstructing their view in our own terms as it is one of understanding them in our own terms 2 • I. THE NATURE OF CONSEQUENCE IN GENERAL Following the practice of the day John Buridan, writing in the midfourteenth century, reminds the reader that in every scientific investigation a preliminary nominal delineation of the subject matter is necessary. He states that in logic the relevant parts of consequence, the ideas of the 'antecedent' and the 'consequent' and the relationship between the two will have to be elucidated. He proceeds to describe consequence as a hypothetical proposition, constituted of several propositions conjoined by the particle 'if' or the particle 'therefore' or one equivalent to it. The particles mentioned signify that of propositions conjoined by them one follows from the other. (Buridan 1976, 21.) 147 M. Yrjonsuuri (ed.), Medieval Formal Logic 147-181. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Buridan observes at this point that some people hold that any such hypothetical proposition joining several propositions by 'if' or by 'therefore' is consequence as such (una consequentia) which can be divided into true and false consequence. He also notes that some others say that if a consequence is false, it should not be called consequence, but only if it is true. He considers this to be a matter of terminology. He, in any case, opts for the latter position: In this tractate I want to understand (vola intelligere) by this term consequentia a true consequence and by antecedent and consequent I want to understand propositions of which one follows from another in a true or sound consequence. (Buridan 1976, 21.)

Since he subscribes to the view that only true or sound consequences are genuine, he examines a common charaterization of the components of consequence, i.e. the antecedent and the consequent, as well as of the relation of consequential dependence. Many say that of two propositions that one is antecedent to the other one which cannot be true without the other one being true also; and that proposition is consequent which cannot not-be-true while the other [the antecedent] is true. (Buridan 1976, 21.)

Taking this characterization at its face value it would seem that it states at least the sufficient condition for consequence: It is not possible that 'p' is true and 'q' not true (definitionally) entails 'If p, then q ': Substituting F'q' for T'q' we get

....,<>(T'p' & F'q'

-----t

(p

-----t

q).

We might even understand it as claiming that defining antecedent and consequent amounts to defining consequence itself. In any case Buridan's treatment of the first counterexample to the initial characterization suggests that we are entitled to the statement of necessary condition for a consequence:

(p

-----t

q)

-----t

....,<>(T'p' & F'q).

The counterexample arises because this is a sound consequence, 'Every man is running, therefore some man is running, and it is possible for the first sentence to be true while the second one is not. (Buridan 1976, 21.)

How do we know this is a sound consequence? Buridan does not say. He seems to assume that we know what a consequence or inference in general is when we encounter its instances in various kinds of cases. And

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inasmuch as he equates genuine consequence with sound consequence, he must also be assuming here that we know what a sound consequence is. 3 As for this first counterexample, he thinks that it would be dissolved if we added the requirement that the antecedent and the consequent be formed at the same time (illis simul formatis). The possibility of 'Every man is running' being true and 'Some man is running' being false is based on the simple fact that there could be a time when 'Every S is P' is satisfied and 'Some S is P' not be satisfied simply on the ground that it does not even exist. We again encounter the negative results: the antecedent is true (or being the case) while the consequent is not true (or not being the case). But even the addition of the simul formatis requirement would not save us from the next counterexample: This consequence is not sound, 'No proposition is negative, therefore no donkey is running'; and yet according to the description mentioned it should be sound, since it is not possible that the antecedent be true and the consequent false. To prove the claim that the consequence is not sound Buridan invokes the principle that 'if a consequence is sound, then from the opposite of consequent there follows the opposite of the antecedent': (p

---t

q)

---t ( ~q ---t

-p).

Applying it to the consequence in question, he confidently asserts that 'Some donkey is not running, therefore some proposition is negative' is not a valid consequence. Again, he must be appealing to a general and unproblematic notion of consequence somehow at our disposal. That the consequence should be proclaimed valid according to the description of consequence and its parts is obvious, since it is impossible for the antecedent to be true, therefore it is impossible that it be true without the other proposition being true also. The definition of consequence and its parts has thus proved to be unsatisfactory even with additions of various qualifYing clauses. Buridan next proposes a final definition of consequence and of its parts: That proposition is antecedent to another which is so related to it that it is impossible howsoever the one signifies, to be the case that it is not howsoever the other proposition signifies to be so, both being proposed at the same time. (Buridan 1976, 32; King 1985, 182.)

It was assumed that this revised definition may stand the test of all kinds

of counterexamples that have been brought forth from various quarters.

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2. BURIDAN'S DIVISION OF CONSEQUENCE

Buridan divided consequence into formal and material. He defined formal consequence as one which is acceptable in any terms, keeping the same form; or, a consequence is formal if it is such that every proposition of the form, if stated, would be a sound consequence; for example, 'What is A is B, therefore what is B is A.' (Buridan 1976, 23.)

He defined material consequence as one which does not hold in all terms, the form remaining the same; for example, 'A man is running, therefore an animal is running'; for it does not hold in these terms, 'Ahorse walks, therefore wood walks.' (Buridan 1976, 23.)

In Buridan's view, no material consequence is evident in inferring the consequent from the antecedent unless it is reduced to a formal consequence by addition of some further premise which turns an enthymematic structure into a full-fledged valid inference. A subdivision of material consequence is that into simple and as-ofnow: Some material consequences are called 'simple, because they are, simply speaking, sound consequences, since it is not possible that the antecedent be true with the consequent being false ... (Buridan 1976, 23).

These consequences become evident in inferendo by addition of a necessary premise; for example, 'A man is running, therefore an animal is running' can be turned into a formal consequence by addition of the necessary proposition (i.e. a proposition in necessary matter) 'A man is an animal.' The simple consequence is still fully acceptable as it stands since one could always remind oneself of the principle that necessary premises may be omitted: that is, the claim expressed by (((p & q)

-t

r) & oq)

-t

(p

-t

r)

is logically true. The second kind of material consequences are called 'as-of-now' consequences; [these] are, simply speaking, not sound, because it is possible for the antecedent to be true without the consequent being true; but they hold as-of-now, because it is impossible, things being related as they are now, that the antecedent be true without the consequent being true. (Buridan 1976, 23)

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Take, for example, 'Socrates is running, therefore a student is running.' This consequence can be reduced to a formal consequence by addition of a true, but not necessary, premise, 'Socrates is a student.' While addition of factually true premisses secures the reduction to formal inference, the principle analogous to the one found in the realm of simple material consequences which allowed the omission of necessary premisses is not available for the realm of as-of-now consequences: there is no legitimate principle which would allow omission of factually true premisses; that Is, assuming that 'q' is a factually true statement, the claim expressed by (((p & q)

~

r) & T'q ')

~

(p

~

r)

is not logically true. At this point we should raise a question about associating consequence with conditional sentences. It was a prevalent view among medieval logicians that all true conditionals are necessary, all false conditionals are impossible and none are contingent. It seems also that truth of a conditional depended, not on the truth-values of the antecedent and the consequent, but on the soundness of the consequence corresponding to the conditional, and not conversely. However, various kinds of conditionals were recognized and, in particular, the illative and the promisory. One would be inclined to think that only illational conditionals, e.g., 'A horse is running; therefore an animal is running,' could be held to be necessary, and be supported by the corresponding sound simple consequences. But Buridan also recognized promisory consequences. These are a subclass of as-of-now consequences. For example, suppose Plato says to Socrates 'If you come to me, I shall give you a horse; this proposition might be a true and sound consequence, or it might be false and thus not a consequence. For if the antecedent is impossible, namely that Socrates cannot come to Plato, the consequence is simply true, because from the impossible anything follows . . . and if the antecedent is false, but not impossible, then the consequent is sound as-of-now, because from the false anything follows in an as-ofnow consequence ... (Buridan 1976, 24.)

Buridan can thus preserve his general thesis that corresponding to any true conditional there is a sound consequence; but he cannot maintain that any conditional, if true at all, is necessary; only those which are supported by sound simple material consequences (and, a fortiori, by strictly formal consequences) are necessary. But those conditionals which, such as the promisory ones, are supported by sound as-of-now consequences, are, if true, only contingently true.

IVAN BOH

152

3. BURIDAN'S ENCOUNTER WITH THE PARADOXICAL RULES OF SIMPLE AND OF AS-OF-NOW CONSEQUENCES

Given this division of consequence and the definitions of the sub-types, we can now appreciate the occurrence of two pairs of "conclusions" which are not readily seen as obviously sound consequences but which are implied by our definitions of consequence and its parts. The first pair comes from the realm of simple material consequences: 'From any impossible proposition any other proposition follows' and, 'A necessary proposition follows from any other. ' (Buridan 1976, 31.) Buridan sees clearly that these "conclusions" are based on our nominal definitions of 'antecedent' and of 'consequent.' For it is impossible that some impossible proposition be true or that howsoever it signifies be the case; therefore it is impossible that howsoever it signifies to be without being the case howsoever any other proposition signifies. (Buridan 1976, p. 32.)

In symbols:

_,op

---t

_,o(p & _,q)

or

_,op- (p- q). Further: It is likewise impossible that it not be howsoever any necessary proposthon signifies; therefore it is impossible that howsoever it signifies to be does not obtain regardless of howsoever the other proposition signifies to be. (Buridan 1976, 32.)

That is:

or

_,o_,q- (p- q). Not only does Buridan recognize the validity of the paradoxes of simple consequence; he also recognizes the parallel paradoxes of the as-of-now consequence: From any false proposition any other proposition follows as an as-of-now consequence, and also, any true proposition follows from any other in an as-of-now consequence. (Buridan 1976, 31.)

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It would be too hasty to represent these as paradoxes of material implication, i.e. as -p:::) (p:::) q)

and q :::) (p:::) q);

for Buridan still thinks in terms of possibility, stating that the reason why the two principles hold is that it is impossible as-of-now that a proposition which is true be not true as things are now related. Hence neither is it possible that it not be true with any other true proposition obtaining. (Buridan 1976, 32.)

The discovery of these strange principles had already been made by Abelard and the recognition of them as legitimate albeit somewhat strange logical principles extended throughout the Middle Ages. Buridan was one of the great defenders of these principles. There appeared in the mid-fourteenth century a more general reaction to the old definitions of 'consequence,' of 'sound consequence' etc., formulated in terms of truth values and alethic modalities, such as this: 'A consequence is sound iff it is not possible that the antecedent be true and the consequent false.' Various counterexamples showed that it did not succeed in stating either the necessary or the sufficient conditions of validity. Even with adjustments designed to take care of all kinds of difficulties, the results seemed unacceptable to many. Buridan's replacement of the above definition with one which defines (sound) consequence in terms of signification was not wholly successfull, either; in particular, it still leaves us with the paradoxical Ex impossibili and Ex fa/so rules which follow from the description of (sound) consequence and its parts. Philosophers in the midfourteenth century and later, often voiced serious misgivings about these rules and about the definitions on which they ultimately rest. They were led to examine the concise proofs which were offered for them, and one detects in their treatments concerns similar to those found in our times by relevant logic.4 These include discussions on the nature of disjunction, on the possible restrictions on the use of disjunctive syllogism, on the principles of simplification of conjunction and the subsequent principle of addition, etc. We shall trace these developments in a later section, but we must first observe the rise of new thinking on the nature of consequence, especially among the English.

IVAN BOH

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4. "EPISTEMIC/PSYCHOLOGICAL" DEFINITIONS OF CONSEQUENT/A

It has been noted and stressed by many medievalists that in the English tradition the characterization of consequence, or at least of formal consequence, often involved a reference to "epistemic" or "psychological" modalities and other mental vocabulary such as understanding, imagining, mental containment, and the like. E. A. Moody, for example, writes: Some of the English logicians, such as Ralph Strodus, introduced epistemic considerations into their definition of formal consequence, saying that a consequence is formal if the consequent is known through the antecedent-if it is de intellectu antecedentis. (Moody 1953, 71.)

He notices that Paul of Venice also gives a psychological definition, saying that a valid formal consequence is one whose antecedent 'cannot be imagined' to be true without the consequent being true. (Moody 1953, 71.)

Moody further contends that in contrast to the English or Englishinspired authors this psychological emphasis is not found in most of the authors, such as Ockham, Buridan, Albert of Saxony, or the Pseudo-Scotus. (Moody 1953, 71.)

Actually, it seems that Buridan could not be considered to be a clear-cut case of logicians juxtaposed to the philosophers in the English tradition with respect to the definition or characterization of consequence. For in certain respects his views may be seen as structurally related to those of Thomas Bradwardine ( 1295-1349) and Richard Kilvington (1302/5-1361 ). In fact, it is the Parisian logician Buridan who is usually associated with the definition of valid inference in terms of signification rather than of truth (An inference is sound iff it is impossible for it to be as signified by the antecedent without its being as signified by the consequent), rather than Kilvington whose own definition became common among English logicians. (Cf Ashworth 1992, 521.) In their joint article, "Logic in Medieval Oxford" E. J. Ashworth and P. V. Spade present some specific illustrative material for the 'psychological' tum in the English tradition. They cite, for example, Bradwardine's thesis that every proposition signifies or denotes as-ofnow or simply whatever follows from it as-of-now or simply. (Ashworth & Spade 1992, 38, n. 12.) The authors also suggest that Bradwardine

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may have gone even further, holding not only that whatever is signified by a name or a proposition follows from it, but also the converse, that whatever follows from a proposition is signified by it. (Ashworth & Spade 1992, 39.) They then suggest a link between the idea of signification and the definition of consequence in mentalistic terms: Since signification was always a psychological-causal notion in the middle ages, Bradwardine's thesis had the effect of linking the logical relation of consequence to psychological considerations. Perhaps it was at least in part because of this that later English authors often defined a 'good and formal' consequence as one in which the consequent was 'contained in the understanding' of the antecedent. (Ashworth & Spade 1992, 38f.)

Ashworth and Spade also collected a number of examples of 'psychological' coloring of logic as it is found in descriptions and definitions of consequence by various late medieval authors. Their findings show how widely spread the new, 'epistemic' definition of consequence and related concepts was; for their research shows that not only the well-known figures, such as Ralph Strode (d. 1387), Richard Billingham (fl. ca. 1344-61), Peter of Mantua (d. 1399), Paul of Venice (d. 1429), Paul of Pergula (d. 1451), Gaetanus of Thiene (1387), Alexander Sermoneta (d. 1486), etc., but also the less known or even anonymous ones adopted epistemically charged definitions in place of, or in addition to, the old definitions couched in terms of truth-value and alethic modes. The following definitions and names of their authors (where available) are given (Ashworth and Spade 1992, 39, n. 15): (a)

(b)

(c)

(d)

Henry Hopton: "In every sound and formal consequence it is so that the consequent is in the understanding of the antecedent"; Robert Fland (c. 1335-c. 1370): "To recognize when a consequence is formal there are rules. The first one is this: When the consequent is understood in the antecedent formally"; Richard Billingham(?) (fl. mid-14th century): "A material consequence is one in which the antecedent is an impossible proposition or in which the consequent is necessary and is not understood in the antecedent"; Ralph Strode (d. 1387): "A consequence is called sound by form when, if the way in which facts are adequately signified by the antecedent is understood, the way in which they are adequately signified by the consequent is also understood; for

156

(e)

IVAN BOH instance, if anyone understands that you are a man, he will understand also that you are an animal." (Seaton 1973); Richard Lavenham (d. 1390), Consequentie: "A formal consequence obtains when the consequent is necessarily in the understanding of the antecedent.

They add two definitions by unknown authors preserved in manuscript form in Italian libraries (Ashworth and Spade 1992, 39, n. 15): (f)

(g)

"Material consequence is one whose antecedent is an impossible proposition and the consequent is not understood in it, or one whose consequent is necessary and is not understood in the antecedent" ( Consequentia materia/is est ubi antecedens est propositio impossibilis et consequens non intelligitur in illud [sic] vel cuius consequens est necessarium et non intelligitur in antecedente); "Any consequence is sound and formal when the consequent is formally understood in the antecedent" (Quelibet consequentia est bona et forma/is quando consequens forma/iter intelligitur in antecedente).

The last item on the list is the definition of consequence by William of Ware: (h)

"Every consequence is sound and formal in which the consequent is formally understood in the antecedent, or the significate of the consequent in the significate of the antecedent."

To this we could add a definition from the Logica Oxoniensis: (i)

"A consequence is sound and formal when the consequent is formally understood in the antecedent; for instance 'A man is running, therefore an animal is running."' Log. Oxon., s. xv, f. 4vb.)

The characterization 'psychological' may be misleading. It may suggest that the acts of inferring a consequent from the antecedent was interpreted as a factual matter of our associating images or ideas, possibly governed by some laws of psychology. Or it may suggest that the mind has a certain structure in virtue of which a normal reasoning subject cannot but make inferences in certain ways which we judge correct or sound. However, if we look at the examples of definitions given above, we will find that it is most likely the conceptual foundation

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that is insisted upon and not the contingent acts of imagination, apprehension and judgement which are, as such, not a matter of logic at all. The characterization 'epistemic' is less misleading. It may be viewed as pointing out the fact that (most) consequences (conditionals) were interpreted as intentional, in the sense that the connection between the antecedent and the consequent was not based on truth-values of components but on objective conceptual relationship apprehended by the mind. This last clause should be taken seriously, for there seems to be little doubt that the 'psychological' epistemological phase in Western logic between around 1330 and 1550 paved the way for Cartesian inference and the stress on mental certitude. 5 Observing the above definitions, we must first note that all but (c) and (f) purport to define soundness offormal consequence. It is in them that conceptual involution of consequent in the antecedent is claimed to hold. It is most likely that earlier in the middle ages, before the transformation of topics and the absorption of them into systems of consequences, conceptual relations were taken care of by the principles of topics, but without reference to mental acts or the epistemic subject. During the sway of modal and truth-value systems of the late thirteenth and the early fourteenth centuries topical principles were invoked to 'ground' inferences, and not only enthymems but sometimes even (categorical) syllogisms which were normally considered as perfect. Some definitions (g, h, i) call for the consequent to be formally understood in the antecedent and (e) calls for the consequent to be understood necessarily in the antecedent. It seems that this difference can be disregarded as one of locution. However, the insistence on 'formally understood' suggests that here there is a matter of the relationship of concepts and not a matter of associating images or contingent facts. Only two definitions (d, h) make reference to significates of propositions or to "the way in which what is signified" is understood, which seems to give importance to modalities and complexities of mental acts. Finally, two definitions (c, f) are not concerned with formal but with material consequences. It is important to note the proviso, i.e. that the consequent not be understood in the antecedent; for these 'epistemic' logicians it is not sufficient for a consequence to be material that its antecedent be impossible or its consequent necessary. For example, 'Man is a stone,' is an impossible proposition, but the consequence, 'Man is a stone, therefore man is lifeless' is, on this view, not a material consequence because the consequent is understood in the antecedent, and this suffices for its being a formal consequence.

158

NANBOH

One sense, then, in which consequences of this tradition are "epistemic" is the presence of epistemic constitutive notes in the characterization of consequence which we have just witnessed. The second sense is the presence of the most general principles or rules for the epistemic realm analogous to the most general principles for the assertoric, alethic, and obligational realms. These are presented in the next section. 5. S1RODE'S RULES OF CONSEQUENCES The sets of rules of consequences collected and organized by the early fourteenth century authors such as Burley, Ockham, Buridan, Albert of Saxony and others invariably included not only rules governing assertoric propositions but also rules for alethic modalities. The sense of quasideductive organization of the rules, distinguishing the derived rules from the principal ones, in their logical systems is impressive. In Strode's Consequentie we encounter a set of rules of consequences which is not only quantitatively much larger (24, as opposed to 10 in Burley) but also quantitatively much richer in the sense that it includes rules which are envisioned to serve as basic principles not only of assertoric and modal, but also of epistemic and obligational realms. Strode first defines consequence as "an inferring of a consequent from the antecedent." It is not clear whether he means by 'inferring' (i) an act of inferring or (ii) the conditions for inferring. He does allow that a consequence may be sound (bona), or else unsound (mala); for a consequent may be inferred properly (debite) or improperly (indebite). Thus, Strode treats consequence as sequence of two (or more) propositions, the antecedent and the consequent, joined with the inferential sign 'therefore': p; therefore q; and since the pattern may or may not hold (syntactically for formal consequences, semantically for material consequences), we can envision particular or concrete inferences to be instances of good or of bad inference patterns. Of course we do not start with patterns; we may come to them after applying the definition of sound consequence to any concrete case, expressed in natural language. A consequence is called sound when the facts cannot be as they are adequately signified by the antecedent, without being as they are adequately signified by its consequent. (Strode 1973, 137)

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For example, 'You are a man; therefore you are an animal,' meets the conditions and is therefore a sound consequence. On the other hand, 'You are a man, therefore you sit' does not meet the conditions and is therefore unsound. We should note that here the question is not of truthvalues signifying propositions but of facts (or states of affairs) signified by the antecedent and the consequent. Next a division is made of sound consequences into two groups: those which are sound by form and those which are sound materially only. The definition of a consequence sound by form (bona de forma), as we saw above, makes reference to "the way in which facts are signified is understood." For example, if anyone understands that you are a man, he will also understand that you are an animal. It is said that in such a consequence the consequent is derived from a formal understanding of the antecedent. A consequence sound by matter only (bona de materia tan tum) is one in which the consequent is not derived from a formal understanding of the antecedent, while, nevertheless, the conditions required for sound consequence as described are preserved; for example, 'Man is a donkey, therefore a stick stands in the comer.' Strode adds the reason why this is a consequence sound by matter only, viz. that you can understand a man to be a donkey, even though you may not understand, nor even think about, whether a stick stands in the comer. The personal pronoun 'you' in the last sentence can of course be replaced by the indefinite pronoun 'one.' However, the presence of an epistemic subject cannot be eliminated. That same sentence may open up a serious problem if we interpret it simply as an empirical claim. For even in the case of consequences which are supposed to be sound de forma it may happen the consequent is not understood, or that we are not thinking of it at all; e.g., 'This figure is Euclidean triangle, therefore the sum of the interior angles of this figure equals 180 degrees.' Yet, if understanding or at least thinking about or considering the consequent in relation to the antecedent is essential for a consequence to be de forma and sound, this geometrical truth would not be sound de forma. In my opinion, Strode does not intend to make definitions or descriptions or epistemic rules containing psychological or attitudinal elements as if they were of empirical-descriptive nature. Strode, as well as other logicians of his tradition, encountered the Ex impossibili and Necessarium ad quodlibet rules, but he argued that they are merely material consequences. For they do not meet the requirement that the consequent be understood in the antecedent. They are the only

160

NANBOH

rules of material consequences stated by Strode, and neither of them appears among the 24 principal rules. The set of general rules can conveniently be divided into several subgroups. First, we have six rules of 'classical' propositional logic. Using the arrow ·~· as contextually ambiguous sign of 'therefore' or 'if,' we might render these rules in a schematic way (we will identify the rules by ordinal numbers on Strode's list; moreover, the italicized main hypothesis holds for all the subsequent rules): Rl

R2 R 17

R 18

R21

R22

If a consequence is sound and formal and its antecedent is true, then its consequent is also true. (p ~ q, p F q) If a consequent is false, then its antecedent is also false. (p ~ q, _,q F fJ) If something is antecedent to the antecedent, the same thing is antecedent to the consequent. (p ~q F(r~ p) ~ (r~ q)) If something follows from the consequent, the same thing follows from the antecedent. (p ~ q F (q ~ r) ~ (p ~ r)) Arguing from the contradictory of the consequent to the contradictory of the antecedent makes a sound inference. (p ~ q F _,q ~ JJ) From the opposite of the consequent and one of its premises there follows the opposite of the other premise. ((p& q) ~ d(-,r&p) ~_,qand(p& q)~ d(-,r& q) ~JJ)

Secondly, Strode's list contains a sub-group of seven rules governing a/ethic modal propositions: R7 R8

R9 R 10

If the antecedent is possible, the consequent is also possible. (p ~ q F <>p ~ <>q) If the consequent is impossible, the antecedent is also impossible. (p ~ q F _,<>q ~ _,<>p) If the antecedent is necessary, the consequent is necessary. (p ~ q F Op ~ oq) If the consequent is contingent, then the antecedent is also contingent or else impossible. (p ~ q F <>_,q ~ <>JJ v _,<>p)

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Other rules involving aletic modal concepts of consistency and repugnance are: R 19

R20

R24

If something is consistent with the antecedent, that same thing is consistent with the consequent. (p ~ q I= (r o p) ~ (r o q)) If something is repugnant to the consequent, that same thing is repugnant to the antecedent. (p ~ q I= -'(r o q) ~ -.(r o p)) If a consequence is sound, then the opposite of the consequent is not consistent with the antecedent. (p ~ q I= -,(p 0 -,q))

Thirdly, there is a group of three epistemic rules: R 13 R 14

R23

If the antecedent is known, the consequent is also known. (p ~ q I= K(a, p) ~ K(a, q)) If the consequent is doubtful, then the antecedent is also doubtful or else known to be false. (p ~ q I= D(a, q) ~ (D(a, p) v K(a, -p)) If the antecedent is understood by you, then the consequent is also understood by you. (p ~ q, U(a, p) I= U(a, q))

As is evident from Strode's elaboration on each rule, the application of the epistemic rules presupposes not only that 'p ~ q' is sound but also that the epistemic subject knows it to be such. The full statement of Rl3, for example, would be p

~

q, K(a, p

~

q) I= K(a, p)

~

K(a, q)

However, in virtue of the generally accepted principle that nothing but what is true (or what is the case) can be known (Nihil scitur nisi verum), i.e. 'K(a, p) ~ p, 'we can omit 'p ~ q' as unnecessary and accordingly shorten the rule to K(a, p ~ q) I= K(a, p) ~ K(a, q)

Strode points to another reason. In discussing his R 14 he writes: It should be noted that the phrase 'if it is known to be sound' is added to the rule

because it could be maintained that a consequentia is sound of which the antecedent is known by you and the consequent is doubtful to you or believed to be impossible. For instance suppose that you believed that Socrates does not exist, and yet suppose that Socrates runs in front of you and that you see him. Then this consequentia is sound, 'That man runs; therefore Socrates runs,' and the antecedent

NANBOH

162

of it is known by you, and the consequent is doubtful or believed to be impossible. (Strode 1973, 160.)

Likewise for R14 as well as for R23. Examining the 'proofs' of these rules we encounter the use of 'understanding' as crucial. In fact R23 itself deals explicitly only with this concept. R23 is clarified as follows: If the consequent is not understood by you, then neither is the consequence understood by you. This consequence holds because understanding of the composite presupposes understanding of the simple components ... and if a consequence is not understood by you, then you do not know whether it is sound-which is opposed to first part. (Strode 1973, 175f.)

Schematically: 1. 2.

3.

4. 5.

6. 7.

K(a, p ---t q)

Assumption that you know that the consequence is sound --.u(a, q) Assumption that you do not understand the consequent for indirect proof --.u(a, q) ---t --.u(a, p ---t q) from (2), failure to understand simple components entails failure to understand compounds --.u(a, p---tq) from (3), (2) l::o/ Rl (modus ponens) --.u(a,p---tq )---t--.K( a,p---tq) from (4), in virtue of the analytic connection between understanding and knowing: K(a, p) ---t U(a, p) or --.u(a, p) ---t --.K(a, p) --.K(a, p---tq) from (5) and (4) by Rl K(a, p---tq) & --.K(a, p---tq) (1 ), (6), conjunction-an explicit contradiction

It should be remarked that Strode holds that 'K( a, p )' entails 'U( a, p )' but not the converse: 'U(a, p)' does not entail 'K(a, p).'

The fourth major sub-group consists of four rules governing the realm of logical obligationes. To represent these rules schematically we use 'N*(a, p)' for 'pis to be denied by a,' 'D*(a, p)' for 'pis to be doubted by a,' and 'G*(a, p)' for 'pis to be granted or conceded by a': R5

If the antecedent is to be granted by someone, the consequent is also to be granted by that same person (K(a, p ---t q) t= G*(a, p) ---t G*(a, q))

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R6

R 15

R 16

163

If the consequent is to be denied, then its antecedent is to be denied (K(a, p - q) t= N*(a, q)- N*(a, p)) If the antecedent is to be doubted by someone, the consequent is not to be denied by the same person. (K(a, p - q) t= D*(a, p)- -.N*(a, q)) If the consequent is to be denied, then the antecedent is not to be doubted. (K(a, p - q) t= N*(a, q)- -.D*(a, p))

Strode stresses the fact that the basic concepts involved in these obligational rules are not descriptive but normative in character: 'to be granted' (concedendum) means 'worthy to be granted,' 'to be denied' (negandum) means 'worthy to be denied,' 'to be doubted' (dubitandum) means 'worthy to be doubted,' and 'to have a distinction made' (distinguendum) means 'to be such that it must have its senses distinguished.' Again, his discussion of these rules brings out the necessity to strengthen the basic assumption from 'if a consequence is sound and formal' to 'if a consequence is sound and formal and known to be such.' The general rules are not constitutive rules of the game of obligatio, i.e. rules which express the nature of the game. They already presuppose those rules that determine various systems of obligatio, such as positio, depositio, etc. and they are not tied to the 'old response' or to the 'new response.' The 'proofs' or dialectical defense offered for them, however, do involve reference to rules such as Everything possible put forward to you and known by you to be such is to be admitted by you. (Pergula 1961, 102.)

That is:

(P(a, p) & K(a, p))

t=

G*(a, p)

This is in fact identical with one of the general duties [yet, understood as prima facie rather than absolute duties] to follow truth. It seems very appropriate to list here three sets of basic rules which determine the moves of the participants (the respondent and the opponent) in an obligational disputation. The sets are those of M. Yijonsuuri (1993) and the symbolic language that of Knuuttila & Yijonsuuri (1988). The general duties to follow truth (Boethius of Dacia):

164 Ta Tb Tc

NAN BOH (p) ((K,p & Rp) - t OCp) (p) ((K,}J & Rp) - t ONp) (p) ((-.K,p & -.K,}J & Rp)

-t

ODp)

Read Ta: For any proposition p, if it is known ( 'K') by the respondent r and it is put fmward ('R '), it must be granted. 'N' stands for denying and 'D' for doubt. '0' ('ought') comes from deontic logic, (cf YIjonsuuri 1993, 23) Rules of how the respondent should evaluate propositions put forward to him (Boethius of Dacia):

R1 R2 R3

(p) ((Pp & Rp) - t OCp) (p)(q) ((Pp & (p - t q) & Rp) - t OCq) (p)(q) ((Pp & (p - t _,q) & Rp) - t ONq)

Here 'Pp' stands for 'p is the positum.' R 1 is thus to be read: For any proposition p, if it is the positum and it is put forward, it must be granted. (Yijonsuuri 1993, 34) Rules for irrelevant sentences ( 'Jp ') (Burley): ~a R4b

~

(p) ((Ip & K,p & Rp) - t OCp) (p)((Ip & K,}J &-.Rp) - t ONp) (p)((Ip & -.K,p & -.K,}J & Rp) - t ODp)

Now 'OD' stands for 'it must be doub1ed whether.' (Yijonsuuri 1993, 52.) Returning to the rules of consequences involving obligational concepts, we should scrutinize Strode's proof of R5

K(a, p

-t

q) I= G*(a, p)

-t

G*(a, q)

Strode writes: If it [the rule] is not true, let the opposite be granted, namely, that the antecedent is to be granted and the consequent not.

That is: 1. 2. 3.

K(a, p - t q) G*(a, p) -.G*(a, q)

Assumption for Indirect Proof

Then, either the consequent is to-be-denied, to-be-doubted, or have a distinction made:

4.

N*(a, q) v D*(a, q) v X

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If it is to-be-denied, therefore it is either known to be false, or conflicting 5.

N*(a, q)

~

(K(a,

~q)

v ~(q o q))

If it is false, then the antecedent is false by R2. Strode applies 'K(a, p' (truth-condition for knowing) to (I), to get

6.

p~q

and to K(a,

7.

p)~

~q)

to get

~q

from (6) and (7) by R2 (modus to/lens)

8. Strode goes on observing that

since the antecedent is to be granted, therefore the antecedent follows from something rightly (debite) admitted. But whatever that admitted thing may be, the consequent just as rightly follows from it. Therefore the consequent should be granted for the same reason that the antecedent is. (Strode 1973, 147.)

Since it makes no sense to say that someone must concede or grant a proposition unless that proposition is put forward and admitted, we should take the next steps to be something like this:

9. I 0.

G*(a, p) A(a, p)

~

A(a, p)

from (2) and (9) by RI

for'~'

and given that everything that has been properly admitted is to be granted, there is a sense in which 'G*(a, p)' follows from 'A(a, p).' But since p formally implies q, and the epistemic subject a knows that it does so imply it, then 'A(a, p )' is sufficent for 'G*(a, q )' also. But inferring 'G*(a, q)' contradicts the assumption for indirect proof at step (3), and thus the rule has been vindicated: The supposition that the rule is "false" led to its own denial. 6. SOUND CONSEQUENCEAND THE RULES OF CONSEQUENCE IN PETER OF MANTUA

In place of definition of 'consequence' Peter of Mantua provides a description of it as a conditional (If p, then q) or a rational (p; therefore q) proposition. We do get a definition of sound (bona) consequence as

166

IVAN BOH

a necessary relationship (habitudo) between two propositions, the contradictory of the second of which is inconsistent with (non potest stare cum) the first one without the new imposition." (Logica, hlvA).

He insists on the existence requirement and rejects the following inference as erroneous: 'This consequence is sound, therefore the contradictory of the consequent is repugnant to the antecedent': for perhaps the contradictory of the consequent does not exist at all, either because this consequence has neither an antecedent nor a consequent; or perhaps it has an antecedent, but does not have a consequent, or conversely. (Logica, hlvA.)

He does not stipulate some mysterious non-existing possible 'structures' but rather points to linguistic capabilities; e.g., we can use the absolute ablative form and interpret it consequentially: 'Socrate currente ipse movetur' and understand it as 'Si Socrates currit, ipse movetur, ' i.e. 'If Socrates is running, he is moving.' He is also concerned with subjunctive conditionals such as 'If Socrates were running, he would be moving,' maintaining that as they stand, they are neither true nor false, since the components are not propositions, i.e. things which are true or false, things which could be incompatible, etc. Peter also seems to depart from Strode in his apparent rejection of the role of 'understanding' in his definition of sound consequence which he gives in terms of the impossibility of the conjunction 'p & _,q.' He rejects this inference: 'This consequence is sound and formal, therefore, if it is understood to be as is signified by the antecedent, it is understood to be so as is signified by the consequent.' His argument for this rejection is as follows: For this consequence is sound and formal: 'A man is running, therefore something capable of laughing is running'; but it is not the case that if you understand that a man is running, you understand that something capable of laughing is running, therefore, etc. (Logica, h2rA).

Yet, this is only a part of the story. While his characterizations of consequence and sound consequence, etc. are made without invoking intellectio, one of the difficulties brought against his definition of formal consequence suggests that the definition neglects the relevance between the antecedent and the consequent. It makes the following inference sound: 'A man is not a man, therefore a goat is disputing,' and Yet there is no relation (habitudo) between the antecedent and the consequent. For every relation which is a consequence is a following-upon (consecutio). But there is here no following-upon of the second proposition from the other one, 'Man is not

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man,' since it is composed of logically independent terms (terminis impertinentibus). (Logica, h2rB.)

Unfortunately there is no specific reply to this difficulty, but only a general one, i.e. that formality of consequence depends on the mutual relevance of terms of the consequent and the antecedent, and on propriety and order. (Logica, h2rB.)

Peter also adds that there are degrees of formality. Moreover, as we will see shortly, Peter attaches the condition of 'understanding' of 'antecedent' and 'consequent' to all obligational and epistemic general rules of consequences. Formulating his principal rules of consequences, Peter tries to preempt as many counter examples as possible by packing into the very statements of rules several conditions. In Rule One, we find five conditions demanded in all principal rules: Rl

If a consequence is (a) sound (bona), (b) affirmative, (c) determined by 'if' or 'therefore,' (d) having signification through composition of its terms, (e) with its antecedent and consequent fully expressed and neither of them having multiple senses, and its antecedent is true, then its consequent is also true.

In stating other rules, further conditions were added. Thus, all six obligational rules of consequence (R3-R8) stipulate the epistemic condition (f) that the epistemic subject know that the consequence is formal. A further epistemic condition (g) was added to each of these obligational rules: the respondent was supposed to know that from something-to-be-granted only what is to-be-granted follows: K(a,(G*(a, p) & -,G*(a, q))

--t

_,(p

--t

q));

that from something true only what is true follows: K(a, ((T'p' & F'q')

--t

_,(p

--t

q)));

that from something to-be-denied only what is to-be-denied follows: K(a, (N*(a, p) & _,N*(a, q))

--t

_,(p

--t

q)).

Still other conditions are placed specifically on the antecedents and on consequents of obligational and also on the two epistemic rules (R9, RIO). The four alethic modal rules (Rll-R14) are stated elliptically, but

168

IVAN BOH

presumably require that the consequences governed by them be formal. R 15-R22 all belong to non-modal propositional logic. The rules selected as the most general rules of consequence by Peter of Mantua cover the same logical areas as those by Strode, but the way he states them exhibits much more clearly the interconnections of the epistemic and obligational areas with one another. Our schematic representation above the inferential line includes the bare essential conditions (a)-(e) plus the characteristic conditions imposed upon consequents and antecedents in obligational and epistemic rules; it does not include the variable condition (g) for the sake of simplicity. R3:

If a consequence is sound ... known to be formal, and the antecedent is to-be-granted by a person, the consequent is proposed, and in addition it is well known [to the same person] that from what is to-be-granted nothing but what is to-be-granted follows, then the consequent is also to-begranted. (Logica, h2rA) K(a, p----* q), G*(a, p), P(a, q) t= G*(a, q)

It should be kept in mind that to-be-granted presupposes other acts. As we saw, for Strode the act selected was the acceptance of a proposition by the epistemic subject before he can grant it. (Of course, the acceptance may itself presuppose the act of proposing by some other agent, i.e. by the opponent). One cannot simply grant a proposition anymore than one can accept an invitation unless it has been extended to him. Signs of obligation are, as it were, speech-acts within a formal disputation governed by rules. Although the variable condition (g) is not listed, we should not undermine its importance. We should at least be aware that our rules may be stated loosely or incompletely. As Peter points out: It is invalid to argue: 'This consequence is sound, known to be such, and its

antecedent is to-be-granted by a person and the consequent is understood, therefore the consequent is to-be-granted.' For perhaps it is believed that the antecedent is false or perhaps it is believed that from what is to-be-granted something which is not to-be-granted follows. (Logica, h3rB.)

In the next five rules as well as in the two epistemic rules Peter employs the concept of understanding (intelligere) as one of the multiple conditions jointly sufficient for the consequents in question. In fact, he makes the mental act of understanding a precondition for any response in obligational disputation:

CONSEQUENCE IN THE POST-OCKHAM PERIOD

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One ought not to respond to any proposition unless he ftrst understood it.

(Logica, i4vb.)

Thus, a background principle, _,U(a, p)- _,R*(a, p)

(where 'R*(a, p)' is short for 'a is obligated to respond to p') is fully endorsed and through it the connection between 'understanding' and specific acts of responding is established: R*(a, p) F G*(a, p) v N*(a, p) v D*(a, p) v X*(a, p)

Although Peter found no use for 'intellectum' in his definition of consequence, or for definition of sound consequence, he now does take recourse to it, provided that 'understanding antecedent' or 'understanding consequent' means not merely understanding the sentences but understanding them precisely as such, that is, as terms of the relation (habitudo) of consequence. R4

R5

R6

If a consequence is sound ... known to be formal, and the antecedent is to-be-granted [by you] and the consequent is understood and proposed, and in addition it is well known [by you] that from what is to-be-granted nothing but what is tobe-granted follows, and that from truth falsehood does not follow, then the consequent is neither to-be-doubted nor tobe-denied. (Logica. h3rB.) K(a,p- q), G*(a,p), U{a, q), P(a, q) F _,D*(a, q) & _,N*(a, q) If a consequence is sound ... known to be formal, and the consequent is to-be-denied by someone, and the antecedent is understood and proposed, and it is also known that from what is to-be-denied only what is to-be-denied follows, then the antecedent is, by the same person, to-be-denied. (Logica, h3rB.) K(a, p - q), N*(a, q), U(a, p), P(a, p) t= N*(a, p) If a consequence is sound ... and known to be formal, and the consequent is understood and proposed to someone, and it is also well known that from truth falsehood does not follow, and the consequent is to-be-denied, then the antecedent is neither to-be-doubted nor to-be-granted by the same person. (Logica, h3rB/h3vA.) K(a,p- q), U(a, q), P(a, q), N*(a, q) t= _,D*(a,p) & _,G*(a,p)

170 R7

IVAN BOH If a consequence is sound ... known to be formal, and the antecedent is to-be-doubted and the consequent is understood and proposed, and it is also known that from truth falsehood does not follow, then the consequent is not to-be-denied. (Logica, h3rB/h3vA.) K(a, p---+ q), D*(a, p), U(a, q), P(a, q)

R8

t=

--.N*(a,q)

If a consequence is sound ... and known to be formal, and the consequent is to-be-doubted, and the antecedent is understood and proposed, and it is also known that from truth falsehood does not follow, then the antecedent is to-be-denied or to-bedoubted. (Logica, h3rB/h3vA.) K(a, p---+ q), D*(a, q), U(a, p), P(a, p)

t=

N*(a, p) v D*(a, p)

The two epistemic rules are: R9

If there is a sound consequence ... known [by a person] to be sound and its antecedent is known (scitum) and its consequent understood (mentally grasped, intellectum) and it is not repugnant (non repugnat) for the consequent to be known, and it is well known [by the same person] that from truth nothing but truth follows, and he sufficiently considers (pays attention to, considerat) the consequent, then the consequent also is known. (Logica, h3vB) K(a, p---+ q), K(a, p), U(a, q), C(a, q)

R 10

t=

K(a, q)

If a consequence is sound ... known [by you] to be sound, and the consequent is understood and not known (nescitum), and it is not repugnant for it to be known . . . and you have sufficiently considered the antecedent and the consequent, then the antecedent is also not known. (Logica, h4rA.) K(a, p---+ q), U(a, q), --.K(a, q), C(a, p), C(a, q) t= --.K(a, p)

An important condition added in the case of epistemic rules is that of consideratio of the antecedent and the consequent. One must sufficiently consider these propositions to see how any response to them (granting, denying, doubting, distinguishing senses) would fit into the context of an obligational disputation, keeping in mind all the rules of the game. 7. ON LOGICAL STATUS OF GENERAL RULES OF CONSEQUENCES What is the logical status of the general rules of consequence? In raising this question we should, of course, keep in mind that we are not dealing

CONSEQUENCE IN THE POST-OCKHAM PERIOD

I 71

here with formalized systems, and we might fmd even Moody's observation rather anachronistic when he writes: The rules of consequence are themselves formal, so that if they are expressed as theorems or formulas of the object language, they constitute logically true sentences of conditional form. (Moody 1953, 77.)

The idea that logical truth (tautologous or analytic character of sentences) is the foundation of medieval rules of consequences does not have any support in historical texts. On the contrary, as we saw above, medievals held that truth of a conditional was determined by the soundness or validity of the corresponding consequence; i.e. T'P ---t Q' if 'P.!:. Q' is sound. But soundness of a consequence depended (a) on the impossibility of deriving a false consequent from a true antecedent or/and (b) on the understanding of the consequent through the antecedent. There is, however, a systematic correlation between (true, necessarily true) conditionals and sound consequences which could be invoked for various purposes. The rules themselves are such that if we suppose that a given rule does not hold, we are led to the reassertion of the rule. How did men come to possession of principles of logic, such as rules of consequences? Buridan says that others had treated the consequences in a posteriori manner but that he was investigating the "causes" of the validity of laws of inference. (Cf. Moody 1953, 8.) In fact, his Tractatus de Consequentiis is a quasi-deductive system of rules with a clear distinction between principal rules and those derived from them. His system of principles of deduction, however, is restricted to assertoric (propositional) and alethic modal areas. A very interesting view about the place of the treatise entitled Consequentie in the logical corpus and about the nature of its subjectmatter is expressed by Alexander Sermoneta who in the "Prologue" of his commentary on Strode's Consequentie in the 15th century writes: I say that this book [Consequentie] is the most universal part of the Prior Analytics, or else is introductory to it; and therefore it should be placed immediately after the De Interpretatione, and before the Topics, Sophistical Refutations, and Posterior Analytics. This order is evident, because this book is concerned with consequence as its subject, and this is more universal than any special kind of argumentation, or than the syllogism, with which the Prior Analytics is concerned. (Sermoneta 1493, "Prologue"; quoted by Moody 1953, 10)

Here the claim of universality of consequences covers the assertoric and modal consequences, but it also extends to the obligational, epistemic and other general rules included among the 24 rules found in Strode. The

172

IVAN BOH

obligational and epistemic rules are related to assertoric or bare propositional logic in a way analogous to the way the alethic modal principles are related to it: the latter are not simply a case of substitution of modal sentences to the logic of propositions but are the most general principles in the field of alethic modalities. Likewise, obligational structures such as G*(a, p), K(a, p), etc. are not simply substitution instances in theorems of propositional logic; rather they are primitive elements entering the principal rules of their own respective fields. Even though Strode and others often produced "proofs" for these principles, these proofs are basically dialectical defenses, indirect proofs starting with the assumption that a given rule does not hold. Such an assumption should lead us, ideally, to its own denial, if the rule was in fact a fully stated correct rule. We might, for example, not be able to dialectically defend the rule which did not take into account the fact that the respondent knows that the consequence is sound. For the consequence may in fact be sound and the respondent may have legitimately granted the antecedent, but if he does not know that the consequence is sound or if he believes that it is not sound, he may not grant the consequent. 8. A SIXTEENTH CENTURY PERSPECTIVE ON CONSEQUENCE: DOMINGO DE SOTO In the second edition of his Summulae (Salamanca, 1554; Olms Repr. 1980) the respected Spanish philosopher Domingo de Soto presents his basic system of consequences wholly within the chapters on hypothetical proposition. Soto says that the name 'hypothetical' in its etymological sense applies only to conditional, causal, and rational propositions which are consequentiae in which if the antecedent is assumed, the consequent follows. (Summulae, 81rA)

He adds a little later that rational and causal propositions are subsumed under conditionals (81 rB). Conditional proposition is one in which two categorical propositions are conjoined by the conjunction 'if,' although sometimes what is conjoined is not propositions but propositional complexes such as 'if a man should dispute, he should act diligently,' in which it is not propositions but propositional complexes (complexa propositionalia) which are conjoined (82rA). For truth of an affirmative illative conditional such as 'If a man is, an animal is'

CONSEQUENCE IN THE POST-OCKHAM PERIOD

173

it is sufficient and necessary that it be a sound consequence, that is, that it not be possible for the antecedent to be true and consequent false (82rB).

On the other hand, for truth of a promisory conditional (conditionalis promissiva), it is not required that the consequence be sound; e.g., 'If you serve me, I will reward you,' may be true even if it is not a sound consequence. Soto sees no problem with the traditional view that every true conditional is necessary and every false one impossible because every consequence once it is sound is always sound, and once it is unsound (mala), it is always unsound. (82rB.)

He concludes that since it suffices and is required for truth of a conditional that it be a sound consequence it follows that the true conditional be necessary and a false one impossible. (82rB.)

The other two types of hypothetical propositions which he considered to be consequences (8lrA), the rational and the causal propositions, need not, if true, be necessary and, if false, be impossible, because for their truth it is not sufficient that they be sound consequences. For truth of affmnative rational proposition it is sufficient and necessary (a) that it be a sound consequence and (b) that its antecedent be true. And for truth of a causal proposition it is sufficient and necessary (a) that it be a sound consequence and (b) its antecedent be true and (c) that it be the cause of the consequent. (8lrA.)

In terms of examples, the conditional 'If a man flies, he has wings,' is true, but the rational proposition 'A man flies, therefore he has wings' is false. Likewise, the rational proposition 'Man is capable of laughing, therefore he is rational' is true, but the causal proposition, 'Because man is capable of laughing, he is rational, is false; on the contrary the converse is true, 'Because man is rational he is capable of laughing' is true. From these stipulations Soto concludes that the definition of true conditional' and of 'sound consequence' is the same. And, while true rational and causal propositions are sound consequences, more is required of them than being sound consequences. Having defined 'true conditional' or 'sound consequence' Soto deduced the principle that in every sound consequence the opposite of the consequent is repugnant to the antecedent: (p- q)- -,(p

0

-,q).

He seems to consider this as a principle in some sense preliminary to the statement of the ten rules of sound consequence.

IVANBOH

174

There are ten principal rules, mostly modal, in his system of consequences. (82vB I 83rA.) Three of those rules are of special interest to us; the two "paradoxical" rules *R7 *R8

...,<>p F p ~ q oq t=p ~ q

(with qualification) (with qualification)

and the rule "from first to last": R9

p

~

q F (q

~

r)

~

(p

~

r)

Soto goes on to state: There are other rules, namely, that if a consequence is sound, and its antecedent is known, the consequent is also known; and if the consequent is not known (nescitum), the antecedent is not known; and likewise other rules with 'it is believed' (credito), 'it is conjectured' (opinato), and 'it is doubted' (dubitato).

However, it seems that the reason why he did not list these with principal rules is that "they belong to the Posterior Analytics." (83rB.) The text just cited contains, in essence R9 and Rl 0 of Peter of Mantua and R13 of Strode; it also suggests that an equivalent of Strode's R14 as well as other epistemic/doxastic rules could be formulated. In a chapter on disjunction he recognizes fides (faith) as an epistemic mode as well. In spirit of the new age of humanism Soto turns in the very next section (still in Ch. 8) to "topical" arguments (loci) involving conditional, rational, and causal propositions which he wanted to gather from the last chapter of Prior Analytics. The first locus is the formal inference modus ponens: He proclaims p

~

q, p F q

to be a formal inference. Example is "suggesting" ancient sources: 'If the sun shines, it is day, but the sun shines, therefore it is day' (83rB). Whereas the first rule of consequence gives in metalinguistic terms an explication of one immediate feature of consequence Rl

From truth nothing but truth; i.e. if a consequence is sound and the antecedent is true, the consequent is true. This rule is from Aristotle's Topics 6, and it follows directly from the definition of sound consequence. (82vNB.)

The topical argument, on the other hand, offers, as it were, a commonly used formal schema of reasoning: From a complete conditional, and positing the antecedent, to positing the consequent, there is a formal consequence.

CONSEQUENCE IN THE POST-OCKHAM PERIOD

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The second locus is the formal consequence modus to/lens:

p

---t

q, _,q F -p

The third locus,

p---t qFqv-p states that "there is a sound formal consequence from a conditional to a disjunction formed from the consequent and the contradictory of the antecedent" The fourth locus, p

---t

q F -,q ---t -p

endorses the Principle of Transposition. The fifth and final locus has to do with rational and causal propositions. Adopting Angel d'Ors's symbolic depiction and definition of rational and causal propositions (D'Ors 1981, 245) as p

~

q =dr((p- q) &p)

p

~

q =df ( (p

and ~

q) & (p

0

q)) = (((p -

q) & p) & (p

0

q))

we can see that causal proposition entails rational and conditional proposition, that rational proposition entails a conditional one, but that the converse entailments do not hold. Soto recognizes promisory conditionals as well. He states that they, unlike the illative conditionals, are not, if true, necessary, but contingent (85vA). There are many promisory conditionals which only God knows whether they are true or false (85vB). Suppose I made this promise to you: If you win in the disputation, I will give you a book. Now, if you will never win in the disputation, only God knows whether, if you had won, I would have given you a book.

He rejects the view of those who hold that any promisory conditional whose antecedent is impossible is true. He only admits that 'If you came first, I would have given you a book' could be true, but that it is not alloted to us to know whether it is true or false (85vB). On the other hand, he says that every promisory conditional whose consequent is necessary is true (85vB), but does not explain why. Finally, contrary to the ("moderni") summulists, 'for a conditional to be true' and 'to obligate' are not the same; thus, if in an ugly business I make a promise, 'If you will give me I 00 gold coins, I will kill your enemy,' it may be that this is a true promise, yet it is nevertheless not obligatory. And

176

IVAN BOH

conversely it is possible that the promise be obligatory, although it is not true; as when anyone does not observe what was promised in relation to possessions. Indeed it happens that a conditional promise is obligatory even ifthe condition will never be fulfilled, e.g.: if Peter promises Mary to take her for a wife on condition of something unsavory which is held to be unacceptable, at least by the positive law. (85vB.)

9. QUESTIONING THE VIABILITY OF THE EX IMPOSSIBILI -RULE As indicated earlier, the later medieval philosophers expressed two major concerns with the "classical" early fourteenth century theories of consequences. One was the definition of consequence and the stipulation of conditions of its soundness-which was probably the main reason for the rise of "epistemicized" definitions of consequence. The second concern was with certain unconvincing rules, namely, ex impossibili (ex fa/so) quodlibet sequitur and Necessarium (verum) sequitur ex quolibet rules which tum up, apparently as "conclusions" from old ("preepistemic") definitions of 'consequence.' Domingo de Soto posed the problem as follows: The conditional 'If God does not exist, then God exists' is a consequence in which it is impossible to be so as is signified by the antecedent without being so as is signified by the consequent. And yet it is not sound; and therefore the definition does not hold (nihil valet). (85vB.)

The claim that the consequence is not sound is based on a principle from Aristotle's second book of Prior Analytics to the effect that it is impossible for one of the contradictories to infer the other. Soto is aware that "all moderns" (omnes moderni) grant that the strange consequence is sound without qualification because the line of reasoning from definition to -.<>p- -.o(p & q)

where q is any particular proposition you want is hard to resist. As he had proposed rule R7, he interpreted quodlibet ('anything') as distributing "for genera of singulars," so that the meaning of R7 was that a consequence could be sound ('p- q') and the antecedent be impossible (-.<>p) and the consequent be either impossible, contingent or necessary. Soto holds that R7 and R8, if understood as sound without qualification (absolute) are both against reason and against the authority of Aristotle:

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Who in his sane mind would concede that if you were a stone, you would therefore be God, since there is no relationship between you being a stone and you being God. For nothing is understood to follow from another thing except what has some relation to it, for example, of an effect to cause, of a genus to species, or something like that. (83vB.)

Sometimes these "modem" summulists invoked a distinction, saying that a sound consequence is of two sorts. One sort is sound in an intrinsic manner (per modum intrinsecum ); in it the consequent does have a relationship to its antecedent. The other kind is sound in an extrinsic manner in that its antecedent is impossible or its consequent is necessary. But Soto finds the distinction a mere pronouncement without reason or authority. (83vB.) Soto's own position is that no consequence is sound unless it holds in the intrinsic manner. Therefore it does not follow that the necessary follows from anything; nor that from the impossible anything follows (in their sense).

He adds: I deny that this consequence holds: 'God does not exist, therefore God exists.' I deny that the antecedent could not be true without the consequent; for indeed it could be true without the consequent because 'could be' makes here a composite sense, namely, that if truth of the antecedent is posited as possible or as impossible, the truth of the consequent is not thereby posited. (83vB.)

If arguing from a semantically-determined impossible proposition, such as 'Man is a stone,' to any proposition whatever, is not sound, must we at least admit that from a syntactically determined impossible proposition, for example, from any pair of contradictories, any other proposition follows? Soto scrutinizes the following example: 'Peter is and Peter is not, therefore man is a stone or whatever else you wish.' He takes the following steps: 1.

2. 3. 4. 5. 6.

p&-p -p p

pvq q (p & -p)

~

q

1, Simplification of conjunction 1, Simplification of conjunction 3, Principle of Addition 4, 2, Disjunctive Syllogism 1-5, Conditional Proof

Actually, while justifications of the first five steps are fully recognized as they stand by Soto's text, step 6 is seen by him as an application of the principle of transitivity of entailment, expressed by his

178 R9

IVAN BOH

Whatever follows intrinsically from the consequent of a sound consequence also follows from its antecedent. (p ----+ q) F (q ----+ r) ----+ (p ----+ r)

Applying this rule to justify step (6) we should think of the proof as stating that if p = (1) entails q = ((2) & (3) & (4)), then if q = ((2) & (3) &(4)) entails r = (5), then p = (1) entails r = (5). Soto admits that this argument has a sophistical appearance of a good argument, but goes on to say: In truth to me the argument does not prove its point. For in order that a consequence from first to last (i.e. R9) hold, it is necessary that in the intermediate consequences only that which is the consequent of the preceding consequence be affirmed as antecedent of the next consequence in the chain." (84rB).

Soto does not think that this condition has been met in the above proof. He argues: In the consequence from a disjunction with denying one of its parts to positing the other part, namely, 'Peter is or man is a stone and Peter is not, therefore man is a stone,' this proposition, 'Peter is not' is assumed, although it was not a part of the consequent of the preceding consequence, but had been the consequent of another distinct and separate consequence; nor was it at any time granted as true. And yet when it is assumed to negate one part of the disjunctive proposition, it is accepted as true. Therefore, I believe that there is no impossible proposition from which anything you wish would follow. (84rB.)

To unravel Soto's remarks about the misapplication of R9 to the effect that the consequent of the earlier consequence and the antecedent of the next consequence in line must be the same we reconstruct the proof in another way:

1. 2. 3. 4. 5.

(p & -p) ----+ -p (p & ----+ p) ----+ p p----+(pvq) ((p v q) & -p)----+ q (p&-p)----+q

The culprit designated by Soto is at step (4 ); whereas in the consequent of the earlier consequence is (p v q)

the antecedent of the next consequence in the chain is (p v q) & -p

CONSEQUENCE IN THE POST-OCKHAM PERIOD

179

With p, smuggled into step (4), the "middle term" of the presumed chain of reasoning has thus been vitiated. This seems to be one point that Soto is making. There is a second point of critique, viz. that 'p' was a result of a different, separate consequence and in any case was never asserted as true. Thus, it is not eligible for a modus tollendo ponens disjunctive reasoning to help obtain the remaining disjunct. Surprisingly, Soto does not scrutinize here the fact that a proposition such as p & -p itself is merely supposed and could not be asserted as true. 6 If the philosophers selected for this study are fairly representative as to the conception of logical inference and related matters, it can be concluded that in the two centuries after the death of Ockham "epistemically I psychologically" colored logic held its sway. Even Buridan was already half-way into the new understanding of logic by moving from talk in terms of truth of antecedents and consequents to the talk about what is signified, or the manner of being signified, being the case or not being the case. A more drastic move towards mentalistic characterization of consequence and a development of epistemic/doxastic rules of consequence was not far away and it made its mark on the history of logic. The problems discussed by these late medieval authors are interesting in themselves as well as historically. It will take many researchers to go through details of extant literature before a viable comprehensive picture of the period could be constructed.

Ohio State University NOTES 1

Cf. Ashworth 1973; Ashworth and Spade 1992. Cf. Normore 1990 for an examination of doxology in relation to history of philosophy. In a recent article, A. D'Ors argued that "it is not possible, within logic, to establish an effective criterion which would permit the determination of the soundness or unsoundness of a consequence, for adequate response to such a question can be no other than logic in its entirety... " (D'Ors 1993, 196) "[Moreover] ... a 'proprium' of soundness of consequence does not exist ... ; there are at best 'necessary accidents,' properties which apply to all sound consequences but not only to them. These properties, therefore cannot serve as a criterion for the soundness of consequence, but at best, negatively, as a criterion of of unsoundness." (Ibid.) The proposed criterion of soundness of consequence, "It is impossible for the antecedent to be true and consequent false" is viewed by d'Ors as a "necessary accident", and he considers it to be a mistake treating it as if it were the proprium of sound consequence (Ibid., pp. 196f.)

2

180

NANBOH

4

Cf. Read 1993; D'Ors 1983, 1984, 1993. Cf. Normore 1993 for a more comprehensive picture of the structural connections between late medieval logic and Descartes. 6 Cf. D'Ors 1993, Read 1993 and other selections in Jacobi 1993. 5

REFERENCES Ashworth, E. J., 1973. "The Theory of Consequence in the Late Fifteenth Century," Notre Dame Journal of Formal Logic 1413, 289-315. Ashworth, E. J., 1979. "The Libelli Sophistarum and the Use of Medieval Logic Texts at Oxford and Cambridge in the Early Sixteenth Century," Vivarium 17, 134-157. Ashworth, E.J., 1985. "English Obligationes Texts after Roger Swyneshed: The Tracts beginning 'Obligatio est qua edam ars, "' in 0. Lewry, (ed. ), The Rise of British Logic, (Papers in Mediaeval Studies 7), Toronto, Pontifical Institute of Mediaeval Studies, 309-333. Ashworth, E. J., 1992. ''New Light on Medieval Philosophy: The Sophismata of Richard Kilvington," Dialogue 31, 517-21. Ashworth, E. J., and Spade, P. V., 1992. "Logic in Medieval Oxford," in J. I. Catto and Ralph Evans, (eds.), The History of the University of Oxford, vol. II: Late Medieval Oxford, Oxford, Clarendon Press, 35-64. Boh, Ivan, 1993. Epistemic Logic in the Later Middle Ages, London I New York, Routledge. Broadie, Alexander, 1993. "Assent in Inference Theory," in Jacobi 1993,637-652. Buridan, John, 197 6. Joannis Buridan: Tractatus de Consequentiis, (ed.) Hubert Hubien, Louvain, Publications Universitaries. D'Ors, Angel, 1981. "En Torno a Ia Una Figura de Oposici6n de Proposiciones Hipoteticas Condicional y Consecuencia Intrinseca," in L6gica, Epistemologia y Teorla de Ia Ciencia, (Estudios de Educaci6n, no. 9), Madrid, Ministerio de Educacion y Ciencia. D'Ors, Angel, 1983. "Las Summulae de Domingo de Soto," Anuario Filos6jico 1611, 209-217. D'Ors, Angel, 1993. "Ex impossibili Quodlibet Sequitur (John Buridan)," in Jacobi 1993, 195-212. D'Ors, Angel, 1984. "Los limites de Ia regia 'tollendo ponens': Juan Versor y Lamberto del Monte," Anuario Filos6jico 17/1, 9-26. D'Ors, Angel, 1985. "La Doctrina de las Proposiciones Hipoteticas en Ia L6gica de Pedro de Castrovol," Antonianum 60, 120-159. Jacobi, Klaus, (ed.) 1993. Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns, (Studien und Texte zur Geisteggeschichte des Mittelatems, Bd. 3 8), Lei den I New York I Koln, E. J. Brill. King, Peter, 1985. Jean Buridan 's Logic: The Treatise on Supposition and the Treatise on Consequences, Dordrecht, D. Reidel.

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Knuuttila, Simo, 1981. "The Emergence of Deontic Logic in the Fourteenth Century," in R. Hilpinen (ed.), New Essays on Deontic Logic, Dordrecht, Kluwer, 225-248. Knuuttila, Simo, 1993. "Ober praktische Argumentation und Logik des Wollens im Mitte1alter," in Jacobi 1993, 607-619. Knuuttila, S. and Yrjonsuuri, M., 1988. ''Norms and Action in Obligational Disputations," in 0. Pluta (ed.), Die Philosophie im 14. und 15. Jahrhundert, (Bochumer Studien zur Philosophie 10), Amsterdam, Griiner, 191-202. Maieru, Alfonso, 1983. English Logic in Italy in the 14th and 15th Centuries, Atlantic Highland, Humanities Press. Moody, Ernst A., 1953. Truth and Consequence in Mediaeval Logic, Amsterdam, North-Holland. Normore, Calvin, 1990. "Doxology and the History of Philosophy," Canadian Journal of Philosophy, Supplementary Volume 16, 203-226. Normore, Calvin, 1993. "The Necessity in Deduction: Cartesian Inference and its Medieval Background," Synthese 96, 437-454. Paul ofPergula, 1961. Logica and Tractatus de Sensu Composito et Diviso, (ed.) M.A. Brown, St. Bonaventure, N.Y., The Franciscan Institute. Peter of Mantua, 1492. Logica Petri Mantuani, Venice, Simon Berilaqua Pluta, Olaf, ( ed.) 1988. Die Philosophie im 14. und 15. Jahrhundert, (Bochumer Studien zur Philosophie 10), Amsterdam, Gruner. Pozzi, Lorenzo, 1978. Le Consequentiae nella Logica Medievale, Padova, Liviana Editrice. Read, Stephen, 1993. "Formal and Material Consequence, Disjunctive Syllogism and Gamma," in Jacobi 1993, 233-259. Seaton, Wallace K., 1973. An Edition and Translation of the ''Tractatus de Consequentiis" by Ralph Strode, Fourteenth Century Logician and Friend of Geoffrey Chaucer (Ph. D. Dissertation, Univ. of California, Berkeley 1973), Ann Arbor, Mich., University Microfilms. Soto, Domingo de, 1554. Summulae (2nd.ed.), Salamanca, 1554. (Reprint: Hildesheim, Georg Olms 1980). Strode, Ralph, 1484, Consequentie Strodi, Venice.

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1. An argument is valid if its conclusion follows from its premises; it is invalid if it is possible for its premises to be true while its conclusion is false. How can we be certain of these claims? ( 1) That the conclusion follows from the premises is a sufficient condition of validity because an argument is a piece of discourse which purports to deduce a conclusion from certain premises. Its success (its validity) is measured by its succeeding in that derivation. Of course, this condition is pretty vacuous until we give some account of the methods of deduction: that is the central task of logic. But (1) is nonetheless right. (2) That the premises cannot be true while the conclusion is false is a necessary condition of validity because it is essential to the notion of validity of an argument that it guarantee to take one from truth to truth. It is for this reason that sustaining modus ponens is required of any connective expressing entailment which corresponds to valid argument. But might there not appear a gap between (1) and (2)? Perhaps to give different necessary and sufficient conditions for validity will permit an argument which can neither be shown to be valid, for its conclusion cannot be deduced from its premises, nor shown to be invalid, for its conclusion could not be false while its premises were true. A natural way to prevent this situation arising is to take just one condition to be both necessary and sufficient for validity. One such account of validity takes (2) to express both a necessary and a sufficient condition. I shall call it 'the Classical Account of Validity.' It states that an argument is valid if and only if it is impossible for its premises to be true while its conclusion is false. What is distinctive of the classical account is that it takes the impossibility of true premises and a false conclusion to be sufficient for validity. But can this be accepted? The paradoxes of strict implication are often put forward as a counterexample to this claim. But perhaps they just show that any argument whose premises cannot be true together or whose conclusion must be true is, often despite appearances, valid. To support the objection we need to produce an argument which is 183 M. Yrjonsuuri (ed.), Medieval Fonnal Logic 183-196. © 200 I Kluwer Academic Publishers. Printed in the Netherlands.

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clearly invalid and yet has, for example, a necessarily true conclusion. For if its conclusion must be true, then it would indeed be impossible for its conclusion to be false jointly with the truth of its premises. So by the classical account a clearly invalid argument would be valid. 2. Such an argument was put forward by Pseudo-Scotus. 1 The epithet 'Pseudo-Scotus' derives from the fact that the treatise by which we know the author, though not by Scotus, was included in the collected works of John Duns Scotus in the seventeenth century? This treatise is a commentary on the Prior Analytics of Aristotle. The collected works also contain a commentary on Aristotle's Posterior Analytics. Neither commentary is by Scotus himself. So both are attributed to 'PseudoScotus.' The following argument comes from the commentary on the Prior Analytics. It dates after 1331 (Scotus died in 1308), since it discusses the notion of the complexly signifiable (complexe significabi/e: PseudoScotus 1891-5, question 8, 98b-10lb), a notion introduced in that year by Adam Wodeham (see Gal 1977, 70-71) in his Sentences commentary. Bendiek (1952, 206) used this fact to argue for a date after 1344, since at the time he was writing, it was thought that the notion of the complexly signifiable was due to Gregory of Rimini. Gal ( 1977), in editing a question from W odeham, showed that he had anticipated Gregory by some years. Nonetheless, it may well be that our author took the notion from Gregory, and that the correct date is indeed in the decade or so after 1344. Boh (1982) dates it around 1350, but gives no reason. In an Oxford manuscript, the commentary on the Posterior Analytics is attributed to John of St Germain of Cornwall. Emden, in his list of Oxford scholars, identifies John of St Germain as studying at Oxford from 1298-1302 and teaching at Paris from 1310-15 (Emden 1959, col. 1626). Some modern commentators have chosen to cite the author of the questions on the Prior Analytics as John of Cornwall. 3 However, there is no reason to suppose the two treatises have the same author. Indeed, the late date of the questions on the Prior Ana/ytics shows that the work cannot be by the John of St Germain listed by Emden, who also reveals no connection with Cornwall. Perhaps the Cornish St Germain is a scholar of the next generation; or our Pseudo-Scotus is not him at all. We do not know; we must continue, therefore, to call its author 'PseudoScotus.'

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3. Let A be the argument: God exists Hence this argument is invalid Pseudo-Scotus took the premise to be necessarily true. (Atheists may substitute 'I =I.') We can then reason as follows. If the argument A is valid, A has a true premise and a false conclusion. But every argument with a true premise and a false conclusion is undoubtedly invalid. (This follows from the necessity of the condition offered earlier. What is under attack is its sufficiency.) So A is invalid. That is, if A is valid, then it is invalid. So A is invalid, by reductio ad absurdum. Our only assumption (if we may call it that) m demonstrating the invalidity of A was that God exists (or that 1 = I). And that is necessarily true. By a plausible thesis concerning modal terms, what is deduced from a necessarily true proposition is itself necessarily true.4 So it is necessarily true that A is invalid. But that shows that A has a necessarily true conclusion. If the classical account of validity were correct, and the necessary truth of the conclusion of an argument were sufficient for the argument's validity, it would follow that A was valid. Hence, if the classical account is correct, A is both valid and invalid. The classical account leads to contradiction, and so must be wrong. A is an argument which is clearly invalid, yet which the classical account maintains is valid. So the classical account is incorrect. 4. One may, however, have reservations about the self-reference present in argument A. On taking the classical account of validity, we find that A leads to contradiction. But certain sentences exhibiting self-reference lead to contradiction anyway. Various ways of dealing with those paradoxes were suggested both in the Middle Ages and in more recent times. Two kinds of solution popular in the twelfth and thirteenth centuries were cassatio and restrictio. The idea of the former was that such selfreferential utterances as the Liar paradox, 'What I am saying is false,' simply say nothing at all; the latter went further, in claiming that strictly speaking, self-reference is impossible, and that if such an utterance as the Liar means anything at all, it is that, e.g., one's previous utterance was false. 5 There is considerable similarity between both these

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ideas and Tarski's familiar claim that semantic closure (a language's containing its own truth-predicate) leads to incoherence, and his recommendation of a hierarchy of object language and metalanguage. 6 Cassatio and restrictio became less popular in the fourteenth century, and had few proponents in the later fourteenth, fifteenth and sixteenth centuries. An extremely influential alternative theory seems first to have been proposed by Thomas Bradwardine in his treatise on insolubles (inso/ubi/ia-problems which are not strictly insoluble but only "solved with difficulty," as William Ockham put it). Bradwardine's treatise was written in the 1320s, many years before he became Archbishop of Canterbury. Forty years later, Ralph Strode called Bradwardine that "prince of modem natural philosophers who first came upon something of value concerning insolubles." This idea was that every insoluble not only has a primary standard meaning, but also means that it itself is true. Bradwardine proved this by an extended argument hinging on the postulate that every proposition includes in its meaning whatever logically follows from it (see Roure 1970, 297). Albert of Saxony and others went further. Every proposition, they said, signifies itself to be true. Once again, this was proved from a set of postulates analysing what exactly it is to be true (Albert of Saxony 1988, 339-40). John Buridan, Albert's teacher at Paris in the 1340s and '50s, whose development of these ideas has been most frequently commented on and analysed (e.g., Prior 1962, Scott 1966, Hughes 1982), and who at one time adopted Albert's expressed view, qualified this claim. We can't say that every proposition means that it itself is true, for then we would either have a use/mention confusion (not every proposition refers to itself) or at least a reference to the proposition's own truth, and so no adequate account of falsehood, for there would be nothing to refer to when it was false (Hughes 1982, §7.7.1 and commentary). Buridan preferred to say that every proposition "virtually implies" its own truth. Whatever the exact detail of Bradwardine's and his followers' theses about the meaning or implication of each insoluble, they agree on this: that each insoluble is false. For it means, or implies, something which is not the case, namely, that it itself is true. It can't be, for it also (primarily) means that it's not true. Whatever entails a contradiction is false, and the insolubles entail the contradiction that they are both true and false. So they are all false. 5. It is time to return to argument A. If A should be found to lead to contradiction, independently of acceptance of the classical account of

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validity, then whatever solution one takes to the paradoxes of selfreference will undercut the demonstration in §3 that the Classical Account of Validity is mistaken. Whatever solution removes the contradiction resulting from A independently of the classical account will certainly remove that resulting from it in conjunction with that account. So does A result in contradiction independently of the classical account? Let B be the argument: This argument is valid Hence this argument is invalid. If the argument B is valid, then it has a true premise and a false conclusion. Therefore B is invalid. That is, if B is valid, then it is invalid. So B is invalid, by reductio. But what have we shown? Look again at that sentence: 'if B is valid, then it is invalid.' We have deduced the invalidity of B from the premise that B is valid. That is precisely what B says we can do. So B is valid. And that is a contradiction. B is both valid and invalid. Any solution to the paradoxes of self-reference must deal with argument B as well. When we deduced a contradiction from A in §3, we had made an assumption, namely that the classical account of validity was correct. Hence we were able to evade the contradiction (we thought) by denying the correctness of the classical account. With B, however, we have deduced a contradiction-by unquestionable facts about validity: 1) that any argument whose conclusion follows from its premises is valid; and 2) that any argument whose premises might be true and conclusion false is invalid. If the establishment above of the validity of B is to be faulted that can only be done by faulting some step in the derivation of B's invalidity from the hypothesis of its validity. If that deduction is sound, as I claim it is, then it is immediate, from 1) that B is valid. 6. We can now diagnose the fault in Pseudo-Scotus' example. The classical account of validity is not needed to establish that argument A is valid, and so to derive a contradiction from it. 7 In establishing that A was valid, Pseudo-Scotus reasoned as follows. We take as our premise that God exists. Then suppose A is valid. In that case, A has a true premise and a false conclusion. Therefore it is invalid. That is, given that God exists, then if A is valid, it is invalid.

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So, if God exists, A is invalid. (From this, by modus ponens for 'if,' Pseudo-Scotus deduced that A was indeed invalid, since God exists.) What we have done is precisely to deduce A's conclusion (that A is invalid) from its premise (that God exists). So A is valid. Regardless of our acceptance of the classical account of validity, A is both valid and invalid. Nor does A's .premise need to be necessarily true. If A's premise is true, then if A is valid, it is invalid. Hence, by reductio ad absurdum, if A's premise is true, A is invalid. Since we have deduced A's conclusion from its premise, A is valid. And since its premise is true, A is invalid. 7. The paradoxical nature of argument A was in fact recognised in the medieval period by Albert of Saxony ( 1988, 360-1 ), also writing in about 1350. Clearly A is invalid, Albert says, by an argument similar to that we gave in §3. But then, suppose A is invalid. It follows that its consequent is true. So the antecedent cannot be true without the consequent's also being true. Hence (see below) A is valid, i.e. if we suppose A is invalid, it follows that it's valid. So it's valid. Albert has in fact used the Classical Account of Validity here-which, we have just seen, he didn't need to do. He wrote: "if argument A is not valid, it is possible for [its antecedent] to be true while [its consequent] is false" (ibid., p. 361 ). That is, condition (2) in § 1 is taken to be sufficient for validity (because necessary for invalidity) in the penultimate step (marked 'see below'), where he concludes that A is valid on hypothesis that it is invalid. Nonetheless, Albert does not proceed to reject (or revise) the Classical Account, as did Pseudo-Scotus. Instead, he applies his analysis of insolubles, rejecting the move from the supposition that A is invalid to the conclusion that the consequent is true. For supposing the consequent is true, it follows that A is valid, and so it means that A is valid (as well as meaning, primarily, that A is invalid). But we have supposed A wasn't valid, so "things are not however [the consequent] signifies them to be." So A is invalid, but the consequent is not true. One may not agree with Albert's solution. 8 It appears to achieve consistency at the expense of preventing us saying that what is the case is true. What is important is the recognition of the connection between A and the other insolubles such as the Liar. Indeed, Albert considers some propositional variants of the paradox. The propositional form of A is the conditional a:

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If God exists then this (conditional) sentence is false. 9 If a is true, then it is a true conditional with true antecedent, so its consequent is true, and so if a is true, it's false. So a is false. Albert proceeds to show that the supposition that a is false leads to contradiction, relying on the principle that a conditional is true only if it is impossible for the antecedent to be true and the consequent false-the analogue for conditionals of the Classical Account of Validity. But we can simplify his argument as we did in §6. For in showing that a was false, we relied on the fact that 'God exists' was true. That is, what we showed was that if God exists then a is false. But that is what a says. So a is true (as well as false). A further variation, which both Albert (1988, 357-9) and Buridan (Hughes, 60-1) consider, gives the paradox a conjunctive or disjunctive form. Let X be the conjunction: God exists and this (conjunctive) sentence is false and

othe disjunction: God does not exist or this (disjunctive) sentence is false. 10

o

o

Suppose is true. Then either God does not exist or is false. But God does exist. So o is false, i.e. if o is true it's false. So o is false. Hence either God does not exist or o is false, which is what o says. So o is true too. A similar argument shows that X is also paradoxical. Albert and Buridan both use their theories of insolubles to diagnose the error, claiming that a, X and o are all false. Paradox is rife here, and Pseudo-Scotus and others (e.g., Priest and Routley 1982) should hesitate to use argument A to question any account of validity. If the Classical Account is wrong, a different proof of that fact must be found. 8. An argument which could (superficially) be called the contrapositive of A was considered by a number of fifteenth and sixteenth century authors. 11 Let C be the argument: This argument is valid Hence God does not exist. (C is not strictly the contrapositive of A since 'this argument' now refers to C, not to A.) If C is valid, then since God exists, C has a true premise and a false conclusion. Therefore C is invalid. That is, if C is valid, then C is invalid. So C is invalid, by reductio.

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On the other hand, if C is valid, then its conclusion can be deduced from its premise, and its premise is true. So its conclusion is true. That is, if C is valid, then God does not exist. So C is valid, since its conclusion has been deduced from its premise. Pseudo-Scotus' own solution to his objection (argument A) to the classical account of validity was to add to that account an exceptive clause. He said that an argument was valid if and only if it is impossible for the premises to be true and the conclusion false together, except when the conclusion explicitly denies the connecting particle (here 'hence'), that is, when it denies that the argument is valid. Argument C serves to show this clause insufficient. Pseudo-Scotus' revision covers argument A, but not its 'contrapositive,' C. On the other hand, of course, neither A nor C in fact need be allowed for by an account of validity. They will already have been excluded from consideration by the account of self-referential paradox. 9. Argument C corresponds to Curry's paradox. For C is the inferential version of a conditional, y: 12 If this (conditional) sentence is true, then God does not exist. (Of course, the reference changes, from arguments to conditionals -sentences-and truth of conditionals replaces validity of arguments.) Sentences of the form 'if this sentence is true, then p' can be used to show that any sentence is true. For suppose y is true. Then if its antecedent is true, so is its consequent, and its antecedent is true. So its consequent is true, that is, God does not exist. That is, if y is true, then God does not exist. But that is what y says. So y is true. Hence, if its antecedent is true, so is its consequent, and its antecedent is true. So its consequent is true, that is, God does not exist. We can use C in the same way to establish its conclusion. (y and C are, we might say, the ultimate ontological argument.) Indeed, we can use A to show, to Pseudo-Scotus' dismay, that its premise is false. For A is valid. So its conclusion is false. So its premise must be false too! And in the manner in which we showed that C is both valid and invalid we can show that y is both true and false. We have already seen the conditional form, a of A. There will also be a conditional form, ~ of B. Indeed we can treat B (and A) as we did C: if B is valid, then its conclusion can be deduced from its premise and its premise is true.

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Therefore its conclusion, that is, 'B is invalid,' is true, and so B is invalid. (I'll remark on this final step in § 11.) That is, if B is valid, then it is invalid. That conditional contains all we need to conclude both that B is valid, since we have deduced B's conclusion from its premise, and that B is invalid, by reductio. I 0. The conditional form ~ of the argument B lies behind Rosser's proof of Godel's theorem. 13 For~ is: If this sentence is true, then this sentence is false. Suppose ~ is true. Then it is false. That is, if~ is true, then it is false. So ~ is true, since that is what ~ says, and ~ is false, by reductio. Godel used the Liar paradox, 'This sentence is false,' to construct an undecidable sentence G of arithmetic. G says informally: G is not provable. Neither G nor its negation is provable in arithmetic. However, the demonstration that -G is not provable requires the assumption that arithmetic is m-consistent. 14 To reduce this assumption to the assumption only of simple consistency, Rosser considered instead of G a sentence H which says informally: if H is provable, then there is a simpler proof of ..H. (What I am here calling 'simpler' was defined precisely in terms of one proof's having a smaller Godel number than the other.) It was the condition that the proof of ...,H be simpler than that of H which allowed the assumption of m-consistency to be dropped. Without this restriction on size of proof, we obtain a sentence J which says informally: if J is provable, then ...,J is provable. J corresponds to J3 as G corresponds to the Liar sentence. It is straightforward to show that both the assumption that J is provable and the assumption that ...,J is provable lead to contradiction, given the ro-consistency of arithmetic. Hence, following Godel, we can conclude that J is an undecidable formula. That is, neither J nor its negation is provable. If the demonstration that if J is provable then ...,J is provable were formalizable in arithmetic, it would constitute a proof of J, by the deduction theorem, and a proof of ...,J, by reductio, just as we showed B to be both valid and invalid. But although we have that iHJ then 1- ...,J, we do not have J 1- ...,J; if arithmetic is consistent. On that assumption, the derivation of ...,J from J cannot be performed in arithmetic.

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11. Does Rosser's use of H rather than J have any significance for us? I think not. From the assumption that J is provable, it follows that '....,J is provable' is provable. The problem is to extract a proof of ....,J from this. Lob (1955, 116) showed that (what informally expresses) 'if S is provable then S' is provable only if S is provable. To obtain the proof of ....,J we need to assume ro-consistency, to ensure that the Godel number which '-.J is provable' asserts to exist is in fact one of 0,1 , .... In the case of H however, we are given a bound on the Godel number which indexes the proof of ...,H. This yields a proof of ....,H with the assumption only of simple consistency. But the bound on size of proof has no analogue in the natural language context of B and ~· What might have significance is the analogue of Lob's result, namely that 'if Sis true then S' should be true only when S is true. Recall that in §9 we inferred immediately from the fact that B's conclusion, that is, 'B is invalid,' was true, that B was invalid. (We used 'if Sis true then S' when dealing with B in §5 also, but there it took the form of concluding that 'B is invalid' was false from the hypothesis that B was valid, and so was not invalid.) We did this at precisely the point where in the corresponding arithmetical proof we need to use ro-consistency. We need to make the move in order to show that B is valid (respectively, that if J is provable then -.J is provable). But we would be allowed to make it only if B was valid (J was provable). So we would never get started. Suppose we try to treat truth in the natural language examples as we treat provability in arithmetic. Immediately we need to deny the law of excluded middle. Arithmetic is consistent only if it is not negationcomplete. Godel showed that some sentences of (a consistent) arithmetic are neither provable nor refutable. But such a lead from arithmetic would not end with claiming certain self-referential sentences to be neither true nor false. We would have also to reject that half of the truth-equivalence corresponding to Lob's result, permitting the inference of S from 'S is true.' With the Liar sentence we can avoid the establishment by Dilemma that L is both true and false by denying that it is either. With the Curry paradox and arguments A, B and C, excluded middle is not used. It appears that one can prove that, for example, "(is true, or B is valid. To refuse to move from, say, '"B is invalid' is true" to "B is invalid" would block the demonstrations both that B is valid and that it is invalid. Yet to do this we have now to deny not excluded middle but 'if S is true then S.' Otherwise any claim that B was neither valid nor invalid would not only be totally unsupported but contrary to fact.

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The proposal is the converse of the solution proposed by Bradwardine and others. They rejected the move from S to 'S is true' (at least for insolubles, and for all propositions for Albert and Buridan). Their proposal promises to solve the paradoxes, but at the price of compromising one's theory of truth altogether. 15 The present proposal suggests instead that we block the move from the denial of S to the denial that S is true, at least when S is not true. Again, the paradoxes may be blocked, but at the price of being unable to deny what is plainly not true. So to deny the truth-equivalence is, for me, too high a price to pay. It is constitutive of the notion of truth that if S is true then things are as S states them to be. (Of course, it is constitutive of the notion of proof that if S is provable then there is a proof of S. Lob's result shows not that the arithmetical predicate which informally expresses 'provable' does not have this property, but that only for provable sentences can one show in arithmetic that it has the property.). Formal arithmetic is (we hope) consistent. That is why we can conclude that it is negation-incomplete. But natural language is at first blush inconsistent. Its deductive power seems unlimited; we cannot easily constrain it in the way we can choose to constrain formal theories. f3 is both true and false and B is both valid and invalid. 12. Argument B and the others remind us that self-reference can be indirect. B's premise contains an expression referring to a piece of discourse of which that premise is a part. Further, whether a sentence leads to paradox may depend on how the world is, on whether certain other sentences are true (as Epimenides showed). A is contradictory only if God might exist. Moreover, the semantic paradoxes cannot be evaded simply by denying excluded middle (for truth and for validity). A, B and C can be proven to be valid, just as a, f3 and y can be proven to be true. Lastly, semantic closure does not mean simply 'contains its own truthpredicate,' though that is a useful shorthand for it. There are other semantic concepts besides truth and falsity, and paradox can arise through them too. Validity is one. That is what Pseudo-Scotus, and others, should have seen. A leads to paradox independently of any account of validity. The proper account of validity has no more to deny the validity of A (and B and C) than the proper account of truth has to deny that the Liar sentence is true. (Though perhaps the proper accounts do do this.) Unfortunately, the

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classical account of validity emerges unscathed from Pseudo-Scotus' attack. 16 University of St. Andrews

NOTES 1

See Pseudo-Scotus 1891-5, question 10 (pp. 103a-l 08a), p. 104b. The relevant text is reproduced in Spade 1975, 44-5 and the whole question in Pozzi 1978, 150-60. An English translation is given in the Appendix to the present volume (pp. 000-000). The objection was discussed in Mates 1965a, and also retold, with little or no comment, in Moody 1953, 69, Kneale and Kneale 1962, 287-8, Mates 1965b, 213, McDermott 1972, 288-90, Ashworth 1974, 184, and Bob 1982, 308-9. 2 Indeed, all the printed texts of the treatise, from 1500 onwards, attribute it to Scotus. 3 See Read 1993, 236 n. 10. Indeed, McDermott (1972, 274) and Back (1996, 205-7, 274-8) take the questions on the Sophisticis Elenchis (also printed in volume 2 of Wadding's Opera Omnia of Scotus, 1-80), on the Prior Analytics (ibid., 81-197) and on the Posterior Analytics (ibid., 199-347) all to have the same author, viz John of Cornwall or Cornubia. Note that both extant mss. of the questions on the Sophisticis Elenchis, unlike those of the other two commentaries, explicitly ascribe them to Scotus. 4 The principle alluded to is K: L(p ~ q) ~ (Lp ~ Lq). See, e.g., Hughes and Cresswell 1996, 25. 5 See, e.g., Martin 1993 and Panaccio 1993. 6 See Tarski 1956, § 1. 7 One has also of course to ensure that the deduction of the contradiction does not use methods of argument only justifiable classically. The deduction which follows is acceptable in a relevance logic such as FE of Anderson and Belnap 1975. 8 I have expressed by own reservations about this manner of solution in Read 1984, 425. 9 Albert's example is 'If God exists, some conditional is false' on hypothe!.is that this is the only conditional. The effect is the same. Similarly with X and obelow. 10 Albert and Buridan use 'A man is a jackass' in place of 'God does not exist.' 11 Ashworth 1974, 125 and Roure 1962, 275-6. Again, 'A man is a jackass' is commonly used in place of 'God does not exist.' 12 It is often claimed that the medievals, following Aristotle's lead, did not clearly distinguish arguments from conditionals (e.g., Mates 1965b, 133, Boh 1982, 306). This is certainly not true of Jean of Celaya, who used C and y to observe that a true conditional could correspond to an invalid argument and a false conditional to a valid one: see Ashworth 1974, 125 and Roure 1962, 262. 13 See Rosser 1936, 89 (Theorem II), and Kleene 1952/71, 204-13 (Theorem 29). 14 Iff- ~A(O), 1- ~A(l), ... , then not 1- 3xA(x). 15 See Read 1984 and Spade 1982, 249.

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16

This paper is a revision and expansion of my 'Self-Reference and Validity,' Synthese, 42, 1979, 265-74. The major changes consist in the addition of new §§2, 4 and 7. The reader may be interested to see subsequent discussion of the issues in Priest and Routley 1982, Sorensen 1988, 299-310 and Drange 1990.

REFERENCES Albert of Saxony, 1988. "Insolubles," transl. N. Kretzmann and E. Stump, in The Cambridge Translations of Medieval Philosophical Texts: Volume One: Logic and the Philosophy of Language, Cambridge, Cambridge University Press, 337-68. Anderson, A.R. and Belnap, N.D., 1975. Entailment: the logic of relevance and necessity, vol. I, Princeton, Princeton University Press. Ashworth, E. J., 1974. Language and Logic in the Post-Medieval Period, Dordrecht, D. Reidel. Back, A., 1996. Reduplicatives, Leiden, Brill. Bendiek, J., 1952. "Die Lehre von den Konsequenzen bei Pseudo-Scotus," Franzikanischer Studien 34, 205-34. Boh, Ivan, 1982. "Consequences," inN. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, Cambridge, Cambridge University Press, 300-14. Drange, W., 1990. "Liar Syllogisms," Analysis 50, 1-7. Emden, A. B., 1959. A Biographical Register of the University of Oxford to A.D. 1500, vol. 3, Oxford, Oxford University Press. Gal, G., 1977. "Adam of Wodeham's Question on the 'Comp1exe significabile' as the Immediate Object of Scientific Knowledge," Franciscan Studies 37, 66-102. Geach, P. T., 1954-5. "On Insolubilia," Analysis 15, 71-2. Hughes, G. E., 1982. John Buridan on Self-Reference, Cambridge, Cambridge University Press. Hughes, G. E., and Cresswell, M., 1996. A New Introduction to Modal Logic, Routledge, London. Kleene, S.C., 1952/71. Introduction to Metamathematics, North-Holland, Amsterdam. Kneale, William, and Kneale, Martha, 1962. The Development of Logic, Oxford, Oxford University Press. Lob, M. H., 1955. "Solution of a Problem of Leon Henkin," Journal of Symbolic Logic 20, 115-8. Martin, Christopher .J., 1993 "Obligations and Liars," in S. Read (ed.), Sophisms in Medieval Logic and Grammar, (Nijhoff International Philosophy Series 48), Dordrecht, Kluwer, 357-381. Revised version in the present collection, pp. 65-96. Mates, B., 1965a. "Pseudo-Scotus on the Soundness of Consequentiae," in Contributions to Logic and Methodology in Honor of J. M. Bochenski, edited by A.-T. Tymieniecka, Amsterdam, North-Holland, 132-41. Mates, Benson, 1965b. Elementary Logic, New York, Oxford University Press.

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McDermott, A. C. S., 1972. ''Notes on the Assertoric and Modal Propositional Logic of the Pseudo-Scotus," Journal of the History of Philosophy 10, 273-306. Moody, E.A., 1953. Truth and Consequence in Medieval Logic, Amsterdam, North-Holland. Panaccio, C., 1993. "Solving the Insolubles: hints from Ockham and Burley in S. Read (ed.), Sophisms in Medieval Logic and Grammar, (Nijhoff International Philosophy Series 48), Dordrecht, K1uwer, 398-410. Pozzi, Lorenzo, 1978. Le consequentiae nella logica medievale, Padova, Liviana editrice. Priest, G., and Routley, R., 1982. "Lessons from Pseudo-Scotus," Philosophical Studies 42, 189-199. Prior, A., 1962. "Some problems of self-reference in John Buridan," Proceedings of the British Academy 48, 281-96. Pseudo-Scotus, 1891-5. In Librum Primum Priorum Analyticorum Aristotelis Quaestiones, in Joannis Duns Scoti Opera Omnia, edited by L. Wadding, Paris, Vives, vol. 2, 81-177. Read, S., 1984. Review of Hughes 1982, Australasian Journal of Philosophy 62, 423-6. Read, S., 1993. "Formal and Material Consequence, Disjunctive Syllogism and Gamma," inK. Jacobi (ed.), Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns, (Studien und Texte zur Geisteggeschichte des Mittelaterns, Bd. 38), Leiden, Brill, 233-59. Rosser, J. B., 1936. "Extensions of some Theorems of Godel and Church," Journal of Symbolic Logic 1, 87-91. Roure, M. L., 1962. "Le traite 'Des Propositions Insolubles' de Jean de Celaya." Archives d'Histoire Doctrinale et Litteraire du Moyen Age 29, 235-336. Roure, M. L., 1970. "La problematique des propositions insolubles au x:me siecle et au debut du xrve, suivie de !'edition des traites de W. Shyreswood, W. Burley et Th. Bradwardine," Archives d'Histoire Doctrinale et Litteraire du Moyen Age 37, 205-326. Scott, T. K., 1966. John Buridan: Sophisms on Meaning and Truth, New York, Appleton-Century-Crofts. Sorensen, R., 1988. Blindspots, Oxford, Clarendon Press. Spade, Paul Vincent, 1975. The Medieval Liar, Toronto, Pontifical Institute of Mediaeval Studies. Spade, Paul Vincent, 1982. "Insolubilia," in The Cambridge History of Later Medieval Philosophy from the Rediscovery of Aristotle to the Disintegration of Scholasticism, 1100-1600, N. Kretzmann, A. Kenny, J. Pinborg and E. Stump (eds.), Cambridge, Cambridge University Press, 246-53. Tarski, A., 1956. "The Concept of Truth in Formalized Languages," trans!. in his Logic, Semantics, Metamathematics, (ed.) J.H. Woodger, Oxford, Oxford University Press.

PART III TRANSLATIONS

ANONYMOUS 13TH CENTURY AUTHOR

THE EMMERAN 1REATISE ON FALSE 1 POSITIO

Since there are various ways in which a respondent can be obligated in a disputation, we must now discuss that obligation which is called false positio. Thus, we must see what it is to posit and what positio is. But first we should note that every obligation consists of two [parts], namely of the opponent's positio and the respondent's consent. For the respondent is not obligated if he does not consent [to it]. 1. WHAT POSIT/0 IS

To posit is to prefix the verb[al phrase] it is posited that to some statement in some disputation in order for it to be upheld as a truth. I say "the verb[al phrase] it is posited that," because sometimes obligation occurs with the verb to demand, and such an obligation is called a petitio. From this it is obvious that positio is giving a prefix to some statement so that it will be upheld as true for the purpose of seeing what follows from it. And we must note that this obligation is not called false positio because only falsehoods would be posited, but because they are posited more often than truths. Since every positio occurs for the sake of concession, and one must concede truths because of their truth, they need no positio. But since falsehoods do not have a cause for concession in themselves, they need a positio so that they can be conceded and we can see what follows. 2. HOW APOSITIO HAS TO BE RECEIVED

And we must note that since [it is] statements [that] can be posited, we need to know that there are some statements which can be posited-that is: such that from their positio no contradiction follows-and others which cannot be posited-that is: such that from their positio a contradiction follows. The following are of the [latter] kind: a falsehood is posited, something dissimilar from truth is posited, something similar 199 M. Yrjonsuuri (ed.), Medieval Fonnal Logic 199-215. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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to falsehood is posited and anything convertible [with these]. For given that one could posit the statement a falsehood is posited, then a contradiction would follow, if one posits that a falsehood is posited and then says that the time is finished. And this is questioned [in the following way]: The positum was either false or true. If true, then a falsehood is posited was true. Therefore, a falsehood was posited. And nothing but this. Therefore, this was afalsehood and this was the positum. Thus the positum was false. If false, then a falsehood is posited was false. Therefore, no falsehood was posited. And something had been posited. Therefore, a truth. But nothing but this. Therefore, it was a truth. And it was said that it was false. Because of this [argument], one must say that this [statement] cannot be posited, since a contradiction follows from its positio. 2.1 Some rules Moreover, the following rules are given about statements which cannot be posited: [I]

If a statement which cannot be posited is conjoined to a true statement by a disjunctive connection, that whole can be posited correctly.

Thus the following can be posited correctly: a falsehood is posited or God exists, because when it is said that the time is finished, one can hold that it is true because of the part God exists so without any contradiction following. [II]

If a statement which cannot be posited is conjoined to a false statement by a disjunctive connection, that whole cannot be posited.

Thus the following whole cannot be posited: a falsehood is posited or Socrates is a donkey, because when it is said that the time is finished, then a contradiction follows. Moreover: [III]

If a statement which cannot be posited is conjoined to a true statement by a conjunctive connection, that whole cannot be posited.

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Thus the following whole cannot be posited: a falsehood is posited and God exists, because given that it was posited, then a contradiction follows when it is said that the time is finished. Moreover: [IV]

If a statement which cannot be posited is conjoined to a false statement with a conjunctive connection, the whole can be posited correctly.

Thus the following whole can be posited correctly: a falsehood is posited and Socrates is a donkey, because when it is said that the time is finished, one can hold that the positum was false so that a contradiction does not follow. By this [the following] holds, if it is posited that a falsity is posited or Marcus is called Tullius if Marcus and Tullius are names of the same [person], Marcus is called Tullius is a truth, and thus due to this part the whole can be posited correctly, without a contradiction following. But if they are names of different [persons], Marcus is called Tullius is a falsehood, and thus [the whole] cannot be posited. Likewise, we must note that this and the positum are similar cannot be posited, if it is conjoined to something false. Thus the following can in no way be posited: 'It is posited that you are a bishop and the positum are similar.' For given that one could posit this, then a contradiction would follow in the following way: The time is finished. The positum was either true or false. If true, then it was true that you are a bishop and the positum are similar. But it was false that you are a bishop. Therefore, the positum was false, and it was said that it was true. If false, then it was false that you are a bishop and the positum are similar. Therefore, you are a bishop and the positum were not similar, and they were of some kind. Therefore, they were dissimilar. But it was false that you are a bishop. Therefore, the positum was true. Because of this [argument] one must say that it cannot be posited. And in similar cases the same judgement applies. Likewise, it must be recognized that a positio is sometimes excluded by [that which is] put forward or by the response. Thus, when it is said: "it is posited that the positum and [that which is] put forward are similar," one must say that in one contingent case [the statement] can be posited, in another not. If indeed a truth is posited/ it can be correctly posited. If a falsehood it cannot. Similarly, when it is said: it is posited that the positum and [that which is} put forward are dissimilar, one must say that in one contingent case it can be posited, in another not. If indeed a

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truth is put forward, it cannot be posited. But if a falsehood, it can be posited correctly. 3. HOW ONE MUST RESPOND TO THE POSITUM Now that we have seen what a positio is and how it has to be received, we must see how one should respond to the positum. 3.1 Some rules About this the following rules are given: [V]

If the respondent knows that the positum is put forward to him, he must concede it if it can be conceded.

I say "if the respondent knows that the positum is put forward to him" because if it is posited that Socrates is white and Marcus is white is posited, the respondent need not concede it unless he knows that Socrates is white is signified by it. I say "if it can be conceded," since there are some statements which cannot be conceded although they can be posited, such as: a falsity is conceded. Indeed, given that this a falsity is conceded is posited, no contradiction follows from the positio, and thus it can be posited correctly. But given that it is conceded, a contradiction follows from the concession. Thus this statement can be correctly posited, but nevertheless it cannot be conceded. Likewise, we must note that [VI]

If the respondent doubts whether the positum is put forward to him, he must not simply concede or deny it, but he must say: "prove it."

For example: it is posited that Socrates is white and Marcus is either a name of Socrates or of Plato, but you do not know of which. If Marcus is white is posited, since the respondent doubts whether the name Marcus is a name of Socrates or of Plato, he must respond: "prove it." Moreover: [VII]

Everything that follows 3 from the positum must be conceded, if it can be conceded.

Thus, if it is posited that Socrates is white and Socrates is colored is put forward, one must concede it during the positio, because it follows from

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the positum. For it follows correctly that if Socrates is white, Socrates is colored. [X]

Everything that is incompatible with the positum must be denied, if it can be denied.

And we must note that something is said to follow from the positum when the positum cannot be true without it; that is, if the positum is true, it is true. Something is said to be incompatible, when its contradictory opposite follows from the positum. Thus, if it is posited that Socrates is white and during the positio [the statement] Socrates is not colored is posited, one must deny it, because its contradictory opposite follows from the positum, namely that Socrates is colored. Moreover: 'If it can be conceded' or '[if it can be} denied' are posited in the preceding rules because there are some statements, which cannot be conceded or denied although they either are posited, follow from the positum or are incompatible with the positum. The following are like this: falsehood is conceded, falsehood is denied. For given that the statement falsehood is conceded could be conceded, a contradiction would follow as has been said above. Similarly, we must note that falsehood is denied can in no way be denied. Thus, although these statements can be correctly posited, they nevertheless cannot be conceded or denied. Likewise, we must note that [IX]

Everything that follows from a statement or statements which have been conceded, together with the positum, must be conceded, if it can be conceded.

Moreover: [X]

Everything that is incompatible with a statement or statements which have been conceded, together with the positum, must be denied, if it can be denied.

Likewise, we must note that during a false positio, not only those statements to which one correctly responds "it is true" are said to be conceded, but also contradictory opposites of those one has correctly denied. Thus, if I deny Socrates is white, I am said to concede Socrates is not white. Similarly, not only is that to which one responds "it is false" said to be denied, but also the contradictory opposite of that which one concedes. It is also a rule that:

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[XI]

ANONYMOUS 13m CENTURY AUTHOR When something false and possible has been posited, something false and possible must sometimes be conceded, not when put forward in any order, but in some [order].

As in the following case: The truth of the matter is that Socrates is black. It is posited that he is white. Next this is put forward: Socrates is white and you are not a bishop. This is something false that does not follow from the positum. Therefore, one must deny it. Therefore, its contradictory opposite must be conceded, namely this: [it is} not [the case that] Socrates is white and you are not a bishop. But Socrates is white; this is the positum. Therefore, it must be conceded. Therefore, [it is] not [the case that] you are not a bishop. Therefore, you are a bishop. It is obvious therefore that one must concede this falsehood in this order, since it follows from the positum and those statements which have been correctly conceded with the positum. But if it were put forward in the first place, one would have to deny it, since it would be something false that does not follow. And in similar cases the same judgement applies. 3.2 Some sophisms Moreover, the following objections are raised against the preceding rules. The truth of the matter is that Socrates is necessarily white and Plato is contingently black. Then it is said "it is possible that Plato is white." It is posited. Then the following is put forward: Plato is of some kind. This is something true which is not incompatible with the positum. Therefore, one must concede it. Similarly, the following is put forward: Socrates is not such. This is something true and not repugnant. Therefore, one must concede it. (That it is true and not incompatible is proved as follows: I want to take these two adjectives of some kind and such so that they join whiteness and blackness . But the adjective of some kind is used to join the colour that Plato has . But Socrates does not have that kind of color.) And Socrates is white. Therefore, Plato is not white. The time is finished. You have conceded the contradictory opposite of the positum. Therefore, badly. Likewise, the truth of the matter is that Socrates sees some man necessarily, and it is impossible for Socrates and Plato to see the same [person]. Nevertheless Plato can certainly see that which Socrates sees. Then it is said: "Socrates sees some man; it is possible for Plato to see

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him." It is posited. Socrates sees some man; Plato sees him. If one denies [this], the time is finished. You have denied the positum retained in the same verbal form. Therefore, badly. If one responds it is true, it is inferred that therefore Socrates and Plato see the same [person]. The time is finished. You have conceded something [which is] impossible in itself on the basis of a possible positio. Therefore, badly. Likewise, two contingent contradictorily opposite sentences are pointed out to you. "One of these is true, it is possible that the other is true." This is posited. Then this is put forward: one of these is true. This is something necessary. Therefore, one must concede it. Then: the other is true. If one denies it, the time is finished. You have denied the positum in the same verbal form in which it was posited. Therefore, badly. If one responds: "it is true," it is inferred: therefore both of these are true. The time is finished. You have conceded something [which is] impossible in itself on the basis of a possible positio. Therefore, badly. Likewise, the truth of the matter is that Socrates and Plato are white. "It is possible that only one of them is white." It is posited. One of them is white. This is something true and not repugnant. Therefore, one must concede it. Then this is put forward: the other is white. If it is conceded, the time is finished. You have conceded something that was incompatible with the positum. Therefore, you have responded badly. For it follows correctly that if only one of these is white, the other is not white. And you have conceded the opposite. Therefore, badly. If it is denied, [one can argue] against: the other is white is either Socrates is white or Plato [is white]. But each of these is something true and not incompatible. And you have denied it. Therefore, [you have responded] badly. 3.3 Solutions To the first sophism some [people] give the solution that one must deny this: Socrates is not such, because it is incompatible in this verbal form. But if it was put forward in this [form]: Socrates is not black, one would have to concede it, since it would not be incompatible. Others say that this is something false that does not follow. Thus one must deny it. For it does not follow that if Plato is white, Plato is of some kind and Socrates is not such. And thus it is something false that

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does not follow, because this adjective of some kind is restricted to joining whiteness by the adjective white posited in the positio. To the second and the third sophism some [people] say that the positio must be accepted and [those sentences which] are put forward must be conceded, but the arguments made are not valid, since in them one proceeds from disjuncts to something conjoined. Others say that a relative [pronoun] can refer to the signified accident as belonging to the substance. Thus, when it is said: "Socrates sees some man; it is possible for Plato to see him," one must say: "this is false, rather: it is impossible," since its sense is that it is possible for Plato to see that [person] whom Socrates sees. Similarly, when it is said: "one of these is true, [and] it is possible that the other is true," one must say "this is false, indeed: it is impossible," because its sense is that it is possible that what is other than true is true. Those who uphold this solution, say to the last sophism that the other is white must be denied. For when it is said "the other is white means either Socrates is white or Plato [is white]," one must say "this is false, for it rather means Socrates, who is other than the white, is white or Plato, who is other than the white, is white." But whichever of these is signified, it is something false that does not follow, and thus one must deny it. 3.4 Rules continued Moreover, we must note that because a false positio sometimes binds the respondent to conceding something false, so that the respondent appears to be involved in fallacy, it is usually said that in this problem the truth of the matter must be hidden and, as a result, one must not answer [questions about] which or why or when, or any substantial question, during a false positio. This is obvious in the following example: It is posited that this word mulier is masculine. 4 Then this is put forward: mulier a/bus est. If one answers "it is true" or "it is false" or "prove it," on the contrary: You have replied to mulier a/bus est, which is ungrammatical, as if it was grammatical; and it does not follow that it would be grammatical. Therefore, [you answered] badly. In fact, it does not follow that if this word mulier is masculine, then this mulier5 a/bus est would be grammatical, since the word album could change its gender. From this is is clear that one must say to mulier a/bus est either "you speak nonsense" or "you are not saying anything." But if it is said that these words together signify something and there is no discord of gender,

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number or case, then one must say "it is true," but when it is said "nor of any other [grammatical] property," one must say "it is false." But if it is asked "which one?", one must not answer, because the truth of the matter must be hidden during a false positio. Moreover, we must note that although a false positio binds one to conceding something false, the aim [of the opponent] is [to provoke a] bad response. But sometimes it happens that when the truth of the matter is hidden, the respondent must concede that he answers badly. This is obvious in the following example: it is posited that you concede that Socrates is a donkey. Then this is put forward: you concede Socrates to be a donkey. This is the positum and it is put forward in the same verbal form in which it was posited. Therefore, one must concede it. If one concedes it, against: You concede that Socrates is a donkey. But that Socrates is a donkey is impossible. Therefore, you have conceded something impossible on the basis of a possible positio. Therefore, [you have responded] badly. Therefore, it is obvious that sometimes it happens that the respondent must concede that he is responding badly during the positio, because it follows from the pas itum. But if it is said: "The time is finished. You conceded that you were responding badly, therefore, you responded badly," this does not follow, but rather "therefore, you responded correctly." Furthermore, keeping the same positio one can make the following objection. You concede that Socrates is a donkey. But that Socrates is a donkey is a falsehood. Therefore, you concede a falsehood. The time is finished. When you have conceded this: you concede a falsehood, this was either true or false. If true, then it was true that you concede a falsehood. Therefore, you have conceded a falsehood. And nothing but this. Therefore, it is false. And it was said that [it is] true. If false, then it was false that you concede a falsehood. Therefore, you have not conceded a falsehood. And you have conceded something. Therefore, a truth. And nothing but this. Therefore, this is true. And it was said that it was false. Because of this one must note that when two acts determine the application of this verb it is posited, the respondent is not obliged to concede only one of them, but both together. Thus, when it is said it is posited that you concede that Socrates is a donkey, I am not obliged by this to concede only Socrates is a donkey, but the [following] whole together: I concede that Socrates is a donkey. Thus, if this is put forward: Socrates is a donkey, one must deny it. But if it is said you are obliged to concede that Socrates is a donkey, one must say "it is false."

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That is, I am not obliged to concede that Socrates is a donkey but the following: that I concede that Socrates is a donkey. But if it is said, keeping the same positio, "a possible positio has been put to you," this is a truth which is not incompatible. Therefore, it must be conceded. Then the following is put forward: No other obligation is put to you. Therefore you should not concede [anything] impossible, but you must concede that Socrates is a donkey, and that Socrates is a donkey is impossible. Therefore, you concede [something] impossible. The time is finished. You have conceded two contradictory opposites in the same disputation. Therefore, [you have responded] badly. In this case, one can say that to the following: you must concede that Socrates is a donkey one must say "it is false, indeed, [I am rather bound] to this whole: I have to concede that Socrates is a donkey." Alternatively, one can say that to the following: a possible positio has been put to you one must say "it is true." But when it is said "but no other obligation is put to you," one must say "it is false," even if it is something true, because it is incompatible. For it follows correctly that if you must concede [something] impossible on the basis of a possible positio, some other obligation has been put to you. Thus the following is incompatible: no other obligation is put to you. Moreover, even if it is given as a rule that everything true that is not incompatible with the positum must be conceded, nevertheless it sometimes happens that a truth that is not incompatible with the positum must be denied, because the answer to it is given for an instant at which it was false. As in the following example: As things are, Socrates is black at the moment of the positio, and at the moment of the first statement after the positum he will be white. Then it is said: "it is posited that Socrates is white." Then the following is put forward: Socrates is white and you are not a bishop. This is something true which is not incompatible with the positum. And it was false at the moment of the positio. And one must answer in relation to that [instant]. Therefore, you must deny it. We concede this because of this [above mentioned] reason. And the following argument is not valid: This is something true which is not incompatible with the positum, and you have denied it. Therefore, you have responded badly. Indeed, one must add: and you have not denied it for the moment at which it was false, and this is false. 3.5 About a rule which is usually given Moreover, it is usually given as a rule that

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When anything false and possible is posited concerning the present instant, it must be denied that [the present instant exists].

Thus, if A is the present instant and as a matter of fact Socrates is black, and it is posited that Socrates is white, then if the following is put forward: A exists, one must deny it, since then it would follow that Socrates is white at A, which is impossible. But if the following is said: "A exists is something true which is not incompatible with the positum and you have denied it, therefore, you have answered badly," one must say: this is not valid, because although it is not incompatible with what has been put forward, it is incompatible with the positum and [something that is] held to be a truth. For it follows correctly that if Socrates is white and he is not white at A, A does not exist, given that one holds that Socrates is not white in A is true. 3.6 An objection But the following objection is raised against [this argument]: When it is said it is posited that Socrates is white, the verb is either conjoins whiteness [to the subject] with respect to this instant discretely, or with respect to instants in general, or with respect to the instant at which Socrates will be white. If with respect to this instant discretely, something impossible is posited, and thus it is no wonder if something impossible is conceded. If for instants in general, either it is in general with respect to this instant or with respect to some other [instant]. If with respect to this instant, that is impossible, if with respect to another [instant], that is possible. And when it is inferred therefore, Socrates is white at A, this does not follow, because the verb is does not conjoin whiteness [to the subject] with respect to this instant, but with respect to another [instant]. Thus, the proposition Socrates is white is equivalent to this: Socrates will be white, according to this [explanation]. And if this verb is conjoins whiteness [to the subject] with respect to the instant at which Socrates will be white, let that instant be B. Then it is posited that Socrates is white at B. But this is either posited with respect to this [present] instant or with respect to that [instant] at which Socrates is white. If with respect to this [present] instant, that is impossible, and thus it is no wonder if something impossible follows. If with respect to that [instant] at which Socrates will be white, it is possible and the following argument is not valid for the above mentioned reason.

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Others solve [the problem] otherwise. That is, they distinguish between the impossible per se and the impossible per accidens. The impossible per se is that which cannot in any way be true, namely when the form which is predicated is naturally incompatible with the thing which is the subject, as in the following: a man is a donkey. And this kind of impossible [statement] must not be conceded on the basis of a possible positio. The impossible per accidens is that which is not impossible by itself (per se) but by virtue of something else (per aliud). That is: with respect to some determination, namely when the form which is predicated is not naturally incompatible with the thing which is the subject, as in the following: Socrates is white. For whiteness is not naturally incompatible with Socrates, but [only] with respect to this [present] instant. And this kind of impossible [statement] can be correctly conceded on the basis of a possible positio. 3. 7 Continued Moreover, we must note that it is the same [thing] to posit the whole conjunctive sentence and to posit both of its parts. For in both cases the respondent is bound to concede both parts. But one must note that it is not the same [thing] to posit the whole disjunctive sentence and to posit one of the parts not knowing which one. For if the whole disjunctive sentence is posited, then to the part which is first put forward one must answer in accordance with its own quality. 6 To the part that is put forward in the second place, one must answer "it is true"-even if it is something false-because it follows from the positum and [the opposites] of correctly denied statements together with the positum. Thus, if the following whole sentence is posited: Socrates is white or Plato is white, and both [parts] are false, then to this Socrates is white, if it is put forward first, one must answer "it is false," since it is a falsehood that does not follow from the positum. But to this: Plato is white, if it is put forward later, one must answer "it is true," even if it is something false, because it follows from the positum and [the opposites] of correctly denied statements together with the positum. For it follows correctly that if Socrates [is white] or Plato is white and [it is] not [the case that] Socrates is white, then Plato is white. And if one of the parts is posited and it is not known which one, then one must answer to both parts: "prove it," for about each part one is uncertain whether it is the positum.

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3. 8 A sophism From this the solution of the following sophism is obvious. As a matter of fact Socrates is black. And it is posited that Socrates is white or you have to concede that Socrates is white. And the whole disjunctive sentence is posited. Then the following is put forward: Socrates is white. This is something false which does not follow from the positum. Therefore, one must deny it. Then it is put forward that you have to concede that Socrates is white. This follows from the positum and [the opposites] of correctly denied statements together with the positum. Therefore, one must concede it. For from Socrates is white or you have to concede that Socrates is white and [it is] not [the case that] Socrates is white, it follows correctly that you have to concede that Socrates is white. If one concedes it, against this: That Socrates is white does not follow from the positum, or from [the opposite] of a correctly denied statement, or from correctly conceded statements together with the positum, or from any obligated [proposition]. And you have to concede it. Therefore, it is true. The time is finished. You have conceded two contradictory opposites. Therefore, you have answered badly. 3. 9 Solution Solution. To this Socrates is white, which it is put forward in the first place, one must answer "it is false," since it is a falsehood which does not follow from the positum. But to this: you have to concede that Socrates is white, which is put forward in the second place, one must answer "it is tme"-even if it is something false-because it follows from the positum and [the opposites] of correctly denied statements together with the positum, as has been proved above. But when it is said "that Socrates is white does not follow from the positum etc.," then if all the causes of conceding [a proposition] are listed, one must answer "it is false." But if some [causes] are listed and not all, one must say "it is tme." But one must deny the statement in which [all] those causes are listed, even if it is true, because it is repugnant to the positum and [something] correctly conceded with the positum.

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3.10 Whether a positio containing a relative [term] must be upheld Now we must see how one should sustain a positio containing a relative [term]. But that such a positio need not be upheld is proved as follows: Every concession exists with respect to some specific thing. Thus, since positio exists with respect to concession, every positio exists with respect to some specific thing. But since a positio which contains a relative [term] does not exist with respect to some specific thing, it seems that it need not be sustained. But this reason is not sufficient. For I can correctly bind myself to something specifically or non-specifically. Assuming therefore that such a positio should be sustained, we must see how it should be sustained.

3.11 How it should be sustained We must note that when a positio contains a relative [term] related [to a term] with diS.rirutive supposition, one mUS. concede all the singulars of that universal. Thus, when it is said: "every man exists; it is posited that he is running" one mrnt concere all these: Socrates is running, Plato is rwzning and so on for other cases. Similarly, when it is said: "Socrates exists; it is posited that he is ruming," one mu;t concere this singular: Socrates is running.

3.12 Concerning a doubt But when a positio is made with a relative [term] related to a term with determinate supposition, it is uncertain how one should judge. Some judge that when it is said "a man exists; it is posited that he is running," (if there are only three men, Socrates, Plato and Cicero), then if the three singulars are put forward, one must answer to the two first put forward "it is false since I am not bound to these." But to the last one put forward one must say "it is true, not because I am bound to that one but because I am bound to at least one." Others say that one must answer to each according to its own quality. And the following argument is not valid: you are bound to at least one, and you have conceded none; therefore, you have answered badly. This [claim of invalidity] is clear from the [following] explanation: I was bound to at least one indeterminately, and I have conceded none determinately; therefore I have responded badly, which is not valid. But

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if the following were put forward: a man is running, he is running, one indeed must concede them. Furthermore, one can say that this is like the natural case when, if someone promises to rescribe something indeterminately, [and he describes it determinately], then he does fulfil the promise. Similarly, if I oblige myself to concede something indeterminately, and I concede something determinately, then I fulfil the promise. Thu;;, if one of these [statements] is put forward to me and I concede it, then I have fulfilled the promise. 3.13 Continued Moreover, we must note that it sometimes happens that impossible statements are conceded in a dependent form, as some [people] say. But this impossible statement is conceded on the basis of a possibility. 3.14 Sophism As in the following example: As a matter of fact Socrates is black. It is posited that he is white. Then the following is put forward: A color is in Socrates. It is whiteness. If one answers "it is false" or "prove it," it is proved as follows: Whiteness is in Socrates. And it is a color. Therefore, a color is in Socrates. And it is whiteness. If one concedes it, against this: You have conceded that it is whiteness. But it is whiteness was blackness is whiteness. But this is impossible. Therefore, you have conceded something impossible on the basis of a possible positio etc. Therefore, you have answered badly. 3.15 Solution Solution. Some [people] say that the last argument is not valid, namely the following: You have conceded something impossible on the basis of a possible positio etc. Therefore, you have answered badly. For I have conceded this impossible statement on the basis of a possible [statement]. But it can be solved otherwise. If you were to sustain that Socrates is white, you would deny Socrates is black, as if you knew that Socrates was in fact white. But if you knew that Socrates was in fact white, then behind the verbal form it is whiteness there was whiteness is whiteness. 7 But this is

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not impossible. And so you did not concede [anything] impossible, for if you knew the truth of the matter you were bound to give this answer. 3.16 Continued Moreover, we must note that just as this statement I posit a falsehood cannot be posited by anyone, similarly this statement Socrates posits a falsehood cannot be posited by Socrates, but it can be correctly posited by others. And in similar cases the same judgement applies. About those statements to which one cannot answer correctly, it is usually said that they can be correctly posited, in accordance with the following rule: [XIII]

Only those statements cannot be posited from whose positio a contradiction follows immediately.

Thus since from their positio a contradiction does not follow immediately, they can be correctly posited. But against this: Every positum must be conceded in order to see what happens then. But this seeing consists of opposition and response. Therefore, if there cannot be a response, there cannot be a positio, since [the positio] takes place forthe sake of that [response]. Therefore, since such statements cannot be answered, such statements cannot be posited. Furthermore, every positio takes place for the sake of disputation. A disputation consists of opposition and response. Therefore, if a response cannot be based on such [statements], there cannot be a positio. But a response cannot be based on such [statements]. Therefore, they cannot be posited. This is conceded by some [people] for the [above mentioned] reason. And according to this [opinion] the following argument is not valid: From the positio a contradiction does not follow. Therefore, such statements can be posited. For [the argument] proceeds from insufficient [premises], 8 for one must add: and when these are posited there can be a disputation and a response and it can be seen what follows then. But this is false. Or one can say that it can be correctly posited. And according to this [opinion] the following argument is not valid: Every positio takes place for the sake of the response. But there cannot be a response. Therefore, there cannot be a positio. An example [of similar form]: Every man exists for happiness. But that man is not happy. Therefore, he is not a man. This is not valid. And this is enough about false positio. 9

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NOTES 1

Reading throughout the treatise fa/sa instead of De Rijk's emendation falsi, when combined with positio. Cf. De Rijk's footnote on p. 98. 2 The author uses the verb pono (to posit) in a role in which later authors consistently use propono (to put forward). This causes some ambiguity here (and below in some cases). 3 Omitting the second sequens. 4 Mulier= woman. In Latin the adjective a/bus should agree with the noun, so that the sentence mulier a/bus est (a woman is white; a/bus (white) is masculine) would be grammatical if mulier really was masculine. As a feminine, mulier requires the feminine form of the adjective (mulier alba est). 5 Reading mulier instead of homo, since homo est a/bus is in fact grammatical, and thus the claim makes no sense about this sentence. 6 The author tacitly assumes that the part which is first put forward is false, and it must be denied. Without this assumption, the following argument makes no sense. 7 Reading albedinem esse albedinem. Cf. De Rijk's apparatus. 8 Reading fit enim ab insufficienti for sit enim insufficiencti, which is unintelligible, as already noted by De Rijk. 9 I am grateful to Jenny Ashworth for help in many of the problems of translation.

ANONYMOUS

13TH

CENTURY AUTHOR

THE EMMERAN 1REATISE ON IMPOSSIBLE POSJTJO

I. THAT AN IMPOSSIBLE POSIT/0 MUST BE UPHELD That an impossible positio must be upheld is proved as follows. Just as we say that something possible must be conceded in order to see what follows from it, similarly we have it from Aristotle that something impossible must be conceded in order to see what happens then. Furthermore, we say that God is a man, and we say that correctly. But deity and humanity differ more than humanity and donkeyhood in the course of nature. Therefore, just as we can understand it to be true that God is a man, so can we understand it to be true that Socrates is a donkey. And when we can understand, we can posit, and thus concede. And thus it is clear 1 that an impossible positio must be admitted. Furthermore, we have it from Aristotle that something impossible can be understood, for he talks about taking a fish from the water so that nothing assumes its place-which is impossible. Hence it is possible to understand something impossible. Therefore, since we can posit that which we can understand, it is clear that an impossible positio must be accepted and something impossible must be conceded. Assuming therefore that an impossible positio must be upheld, we proceed accordingly. 2. HOW AN IMPOSSIBLE POSIT/0 HAS TO BE CONS1RUCTED Therefore, we must note that one must not concede two contradictory opposites in this question. For this is the aim [of the opponent] in every disputation or question. Thus neither in this question nor in [any] other must one concede two contradictory opposites. Thus, one must note that no obligation should be accepted which forces the respondent to concede two contradictory opposites. Furthermore, one must note that the art of a false positio is the same as that of an impossible positio. Thus, we must note that just as one must concede during a false positio all that follows from the positum, so during 217 M. Yrjonsuuri (ed.), Medieval Formal Logic 217-223. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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an impossible positio one must concede everything that follows from the positum. [The word] "follows" means "in accordance with a direct (recta) consequence. And a consequence is direct (recta) when the understanding of the consequent is contained in the understanding of the antecedent. And we should note that in this question everything does not follow from an impossible obligation. Thus, in this question one must not concede the consequence of the Adamites-namely that from the impossible anything follows. Instead one must concede in this question only those consequences in which the understanding of the consequent is contained in the understanding of the antecedent. Thus, since one must admit only that kind of onsequence in this question, one must note that one must not admit in this question those consequences in which negation follows from an affirmation. Thus one must not concede the following kind of consequence: if it is a man, it is not a donkey. This is clear, if a man is united with a donkey according to every kind of identity. 3. OBJECTIONS

But against this: It is not in the nature of humanity to be compatible with donkeyhood in the same subject. Therefore, humanity and donkeyhood cannot be in the same subject. Therefore, it follows naturally that if it is a man, it is not a donkey. Furthermore, substantial difference makes a species and divides it from others. Therefore, rational, since it is a substantial difference, produces no species other than man. Therefore, it follows correctly: if something is a man, it is different from [any} other [species]. Therefore, it follows correctly: if it is a man, it is not a donkey. 4. SOLUTIONS

Solution. Since an impossible positio need not be constructed with respect to the nature of things, but at the level of understanding, and since those two forms cannot naturally be in the same subject, it follows correctly at the level of nature if it is a man, it is not a donkey. But because one can understand these two forms to be in the [same] subject, it does not follow at the level of understanding. Thus, when an impossible positio is taken at the level of understanding, it is clear that

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during an impossible positio one should not concede a consequence in which a negation follows from an affirmation. To the other [objection] we say that the intention of a substantial form properly and in itself (per se) is to produce a species, and it divides [that species] from others per accidens, namely by the contrary relation which it has to others. And since this contrariety is not in the thing, but in the understanding, it is clear at the level of the understanding that this does not follow: if it is a man, it is not a donkey. 5. CONTINUED

And we should note that although something impossible can be posited, we should nevertheless note that something impossible from which two contradictory opposites follow cannot be posited. Thus, if one makes the assumption that mortal is included in the definition of man, then this impossible statement-namely: [a man] exists by necessity-cannot be posited in any way. Since if it was posited, then two contradictory opposites would follow like this: if Socrates is a man, Socrates is a rational and mortal animal, and if he is rational [and} mortal animal, he can die; if he exists by necessity, he cannot die. Therefore, if he can die, he cannot die. From this it is obvious that this kind of impossible statement cannot be posited in any way. Moreover, one must note that this impossibility: Socrates ceases to know that there is nothing he ceases to know cannot be posited in any way, because then two contradictory opposites would follow-that is: Socrates ceases to know that there is nothing he ceases to know and2 if he ceases to know that there is nothing he ceases to know, he knows that there is nothing he ceases to know. And if he knows that there is nothing he ceases to know, it is true, because everything that is known is true. And ifit is true that there is nothing he ceases to know, there is nothing Socrates ceases to know. And thus, if Socrates ceases to know, there is nothing he ceases to know. And thus two contradictory opposites follow. And thus [this sentence] cannot be posited in any way.

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And we must note that an impossible positio is sometimes brought about by a union, and sometimes without one. And a union, as it is taken here, is a predication of one on the basis of two. And one must note that sometimes only a union of essence is made, and sometimes only of person, and sometimes both of essence and person. And even if essence and person are the same and one cannot exist without the other, nevertheless one can be properly understood without the other. And since it can be understood, it can be correctly posited, since an impossible positio is made at the level of the understanding. And essence is the suppositum undeerstood without the form, and person is the suppositum understood with the form. And we must note that those terms are called essential which are predicated in the same way of a whole as of each of its parts, for instance stone, wood and the like. Those terms are called personal which are not predicated in the same way of a whole as of each of the parts, for instance, this term man and this term animal. Furthermore, we must note that an adjective with a neuter ending is an essential term. Now that we have seen what essential and personal terms are, and what essence, person and union are, we must see how one must answer to a positio made by a union. Thus, one must note that when the union is only of essence, if an essential predication is predicated of one, one must [also] concede it of the other. But if a personal predication is predicated [of one], one must deny it of the other. Thus, if Socrates is united with Brunellus with a union only of essence, one must concede this: Socrates is the same as Brunellus, but deny this: Socrates is Brunellus. But if the union is made only of person, the opposite occurs, since if personal predication is conceded of one, one must concede it of the other. Thus, if Socrates is united with Brunellus with a union of person alone, one has to concede this: Socrates is Brunellus, but deny this: Socrates is the same as Brunellus. From this it is clear that in this question an argument from an adjective with a neuter ending to an adjective with a masculine or a feminine ending is not valid.

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7. A QUESTION But it is possible to ask why an adjective with a neuter ending is any more an essential term than an adjective with a masculine or a feminine ending. To this one should answer that masculine and feminine are imposed by the form of the thing, and because of this they are called formal terms, and hence [also] personal. Neuter gender is not imposed by any form that is in the thing, but rather by pure privation. And so [by it] a thing is understood without form, and [it is] an essential term. 8. CONTINUED Moreover, one must note that there are some forms which agree with [both] essence and person, for instance, whiteness and blackness. Thus, whatever kind of union is posited: if one is conceded of one, it must be conceded of the other. Thus, if the case is that Socrates is white and Plato is black and any kind of union is made, one must concede this: whiteness is in Socrates, and this also follows: blackness is in Socrates. And this argumentation is not valid: there is whiteness, therefore [it is} not: [the case that] there is blackness. Thus one must note that the topic of opposites does not hold in this question. Furthermore, one must note that there are some forms which agree only with person, like growth and decrease. Thus, if Socrates grows and Brunellus decreases, and a union of essence alone is made between Socrates and Brunellus, one should concede this: Socrates grows, but deny this: Brunellus grows. And one must note that there are some forms which agree only with essence. Thus one must note that if a union of person alone is made, then if one has conceded an essential predication of one, one must deny it of the other. Hence one should note that there are some essential forms, those which are meant by essential terms, like carnality and corporeality, and similarly that form which is meant by the term something.

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Moreover, there is a common doubt. If a union is made to unite this man and this donkey, these two [statements] are true: this man is this donkey and this donkey is this man. But this man will rise again. Therefore, this donkey will rise again. Some [people] concede this argument. But it is better to say that there is a fallacy secundum accidens. I 0. AN OBJECTION Moreover, one must note that just as in a false positio the truth of the matter is concealed, and the order of the things put forward is taken into account, so it is in an impossible positio. From this the solution of the following objection is clear. As a matter of fact, Socrates is a grammarian, Plato a grammarian and a musician, and they are united in every kind of union. Then the following is proved: Socrates is just a grammarian. This is something true which is not incompatible with the positum. Therefore it should be conceded. Similarly Plato is a grammarian and a musician. This is something true which is not incompatible etc. If it is conceded, against this: Socrates is Plato. But Socrates is just a grammarian. Therefore, Plato is just a grammarian. The time is finished. You have conceded two contradictory opposites in the same disputation. Therefore [you have responded] badly. An example: This is a count. This is a bishop. And this bishop celebrates a mass. Therefore count [celebrate a mass]. [This] is not valid. And one must note that just as a union is made between living things, so one can be made between inanimate things and between statements. 11. A QUESTION Furthermore, it is asked whether from a union of statements a union of things follows. This is proved as follows. If these two statements God exists and Caesar exists are united, it follows correctly that if that Caesar exists is true, Caesar exists; but that Caesar exists is true; therefore, Caesar exists. That Caesar exists is true is proved as follows. That Caesar exists is the same as that God exists; but that God exists is true; therefore

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that Caesar exists is true. Therefore, Caesar exists. And thus from a union of statements follows a union of things. 12. SOLUTION To this it is said that a union can be made in two ways. For a union can be made [both] at the level of things and at the level of the understanding. And in this way, from a union of statements follows the union of things. Moreover, a union can be made with respect to the level of understanding alone. And in this case a union of things does not follow from a union of statements. From this it is clear that the following argument is not valid: That Caesar exists is the same as that God exists; but that God exists is true; therefore that Caesar exists is true. For when it is said that Caesar exists is the same as that God exists, this is understood at the level of the understanding. But when it is said that Caesar exists is true, this cannot be in any way understood except at the level of both thing and understanding. 13. A QUESTION Furthermore, it is asked whether a union of truth and falsity has to be held. That [it should] not is proved as follows. For a statement to be true is nothing other than for it to signify as things are in reality. For a statement to be false is nothing other than for it to signify as things are not in reality. And thus in the whole union [at issue] it is implied that things are as they are in reality and that things are not as they are in reality. But no obligation in which two contradictory opposites are implied should be upheld. Therefore, since this [union] implies two contradictory opposites, it should not be upheld. 3 NOTES 1 2

3

Reading patet for pater. Omitting De Rijk's addition. I am grateful to Jenny Ashworth for help in many of the problems of translation.

PSEUDO-SCOTUS

QUESTIONS ON ARISTOTLE'S PRIOR ANALYTICS QUESTION X WHETHER IN EVERY VALID CONSEQUENCE THE OPPOSITE OF THE ANTECEDENT CAN BE INFERRED FROM THE OPPOSITE OF THE CONSEQUENT?

1. ARGUMENTS

1.1 Arguments for the negative answer It is argued that not, for then it would follow that one universal affirmative [proposition] could be simply converted into [another] universal affirmative. The consequent is false, as Aristotle says in the text, c~apter 2. The consequence is proved by arguing as follows: every A is B, therefore, every B is A, since the opposite of the antecedent, that is: no A is B, follows from the opposite of the consequent, that is: noB is A. Secondly, for if the opposite of the antecedent can be inferred from the opposite of the consequent, this is the case only in as much as the opposite of the consequent is incompatible with the antecedent, but [it is] not [the case] for this reason, for the following is a valid consequence: only the Father exists, therefore, not only the Father exists, and nevertheless the opposite of the consequent is not incompatible with the antecedent, but is rather the same as the antecedent. Thirdly, for the following is a valid consequence: no man runs, therefore someone does not run, yet nevertheless the opposite of the antecedent cannot be inferred from the opposite of the consequent, since it does not follow [from the premise that] everyone runs, [that] therefore some man runs, since when one posits the case that there is no-one except running women, then the antecedent is true and the consequent false. 1 Fourthly, if this would suffice for the validity of a consequence, it would follow that the following consequence would be valid: you are [a 225 M. Yrjonsuuri (ed.), Medieval Formal Logic 225-234. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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donkey} or you are not a donkey, therefore, you are a donkey. The conclusion is false, since the antecedent of this consequence is true and [its] consequent is false. The consequence is proved: for if the opposite of the consequent is given, that is, [it's} not [that] you are a donkey, then one can infer, therefore, [it's} not [that] you are a donkey, or you are not a donkey, by this rule "a disjunction of any proposition with any other proposition can be inferred from itself." In this way the last consequent is in contradiction with the first antecedent, for a contradiction cannot be given in a truer way than by positing a negation in front of the whole proposition. 1.2 Argument for the positive answer Aristotle argues for the opposite in the text, chapter 2, where he proves the conversion of the universal negative [proposition] by this [principle]. 1.3 Division of the question In [our discussion of] the question we shall first see what is required for the validity of a consequence. Secondly, we shall see what a consequence [is] and how its kinds are divided. Thirdly, [we shall answer] the question and fourthly we shall add certain other rules. 2. WHAT IS REQUIRED FOR THE VALIDITY OF A CONSEQUENCE

About the first [article], one must know that there are three ways of describing [the requirements for the validity of a consequence]. The first way is [to say] that for the validity of a consequence it is necessary and sufficient that it be impossible that the antecedent is true and the consequent false. The second way is to say that for the validity of a consequence it is necessary and sufficient that it be impossible that the case is as is signified by the antecedent without the case being as is signified by the consequent. The third way is [to say] that for the validity of a consequence it is necessary and sufficient that it be impossible that when the antecedent and the consequent are formed simultaneously, the antecedent is true and the consequent false.

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Against the first [way], one argues that the following consequence is valid: every proposition is affirmative, therefore, no proposition is negative, and nevertheless it is possible that the antecedent is true, and it is impossible that the consequent is true, and so, that [principle] is not sufficient for [explaining] the validity of a consequence. That this [consequence] is valid is proved: for the opposite of the antecedent can be inferred from the opposite of the consequent, since if some proposition is negative, it follows that not every proposition is affirmative, which is the opposite of the antecedent. And that the antecedent can be true is obvious through [considering the case of] the destruction of all negative propositions. But that the consequent cannot be true, is proved: for the consequent is a negative proposition, therefore, whenever it exists, it is false, yet nevertheless, it cannot be true except when it exists, therefore it follows that it can never be true. Against the second way, one argues by assuming that for the truth of a negative proposition it is not required that something is the case, but it suffices that it is not the case as would be signified by the affirmative [proposition] contradictory to it, if it existed. Then if that [principle] sufficed for [explaining] the validity of a consequence, the following consequence would be valid: no chimera is a goat-stag, therefore, a man is a donkey, which is false, for the antecedent is true and the consequent false. For it is impossible that [things] be as is signified by the antecedent without their being as is signified by the consequent. And this suffices in itself (per se) for the validity of a consequence, therefore, the said consequence was valid. Against the third way, one argues by proving that this does not suffice for the validity of a consequence, because some consequences can be formed whose antecedent and consequent are both necessary, yet nevertheless the consequence is not valid, and so, this [principle] does not suffice for [explaining] the validity of a consequence. The consequence holds, for where both the antecedent and the consequent are necessary, it is impossible that when the antecedent and the consequent are formed simultaneously, the antecedent is true and the consequent false, for the consequent cannot be false when it is necessary. One proves the antecedent by [considering] this [consequence]: God exists, therefore, this consequence is not valid (with reference to [the consequence] itself). It is certain that this consequence is not valid, for [if] it were possible for it to be valid, then the consequent would be false and the antecedent true in a valid consequence. And it is known that the

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antecedent is necessary. But I prove that the consequent is necessary: For it is impossible for the said consequence to be valid, therefore, it is necessary that [things] are howsoever is signified by the consequent, because the consequent does not signify anything except that this consequence is not valid. And this is by the material signification of the terms, and by the formal signification it signifies that it is true that this consequence is not valid. So according to both significations the consequent is necessary. Therefore, I describe in a final way what is necessary and sufficient for the validity of a consequence. That is: it be impossible that when the antecedent and the consequent are formed simultaneously, the antecedent is true and the consequent false, except in one case, namely when the signification of the consequent is incompatible with the signification of the sign of consequence, for example, of a conjunction which means that the consequence obtains, as [was the case] in the preceding argument. Thus, in the above mentioned consequence this word 'therefore' means that the consequence is valid. This reference or signification is incompatible with what is signified by the consequent. Therefore it does not follow that in this case the consequence is valid. And this is [enough] about the first [article]. 2. WHAT A CONSEQUENCE IS AND HOW ITS KINDS ARE DIVIDED

About the second [article] one must note that a consequence is a hypothetical proposition composed of an antecedent and a consequent connected by a conditional or argumentative conjunction, which means that it is impossible that when these (that is: the antecedent and the consequent) are formed simultaneously, the antecedent is true and the consequent false, and thus if [things] are as this conjunction means, then the consequence is valid, and if not, then the consequence does not hold. And note that I say: "the antecedent is true," and one must not say that the antecedent can be true, for in the above mentioned consequence: 2 every proposition is affirmative, therefore, no [proposition] is negative, when the antecedent and the consequent are formed simultaneously, the antecedent can be true and the consequent cannot be true, but nevertheless it is impossible that when the antecedent and the consequent are formed simultaneously, the antecedent is true and the consequent false, since the consequent is always incompatible with the

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antecedent, and therefore, when the consequent exists, the antecedent is always false. 3 Secondly, one must note that consequences are divided as follows: some are material, some formal. A formal consequence is one which holds for all terms, when the disposition and form of the terms remains the same. And in the foregoing, we call 'terms' the subjects and the predicates of propositions, or parts of the subject and the predicate. Relevant to the form of the consequence are all the syncategorematic [elements] posited in the consequence, like conjunctions, signs of universality or particularity, negations and suchlike. Secondly, relevant to the form of the consequence is the copula of the proposition, and thus the form of a consequence [composed] of propositions whose copula is non-modal is not the same as that of one [composed] of propositions whose copula is modal. Thirdly, relevant to the form is the number of premises, and affirmation and negation of propositions, and suchlike, and thus the form of arguing from affirmative [propositions] is not the same that of arguing from negative [propositions], and so also for other cases. Formal consequences are further divided, for there are some whose antecedent is a categorical proposition, for example, conversion, equivalence, and so on. Others are such that their antecedent is a compound proposition. And each of these kinds can be further divided into several other kinds. Material consequences are those which do not hold for all terms, when kept in the same disposition and form so that there is no change except for the terms. And there are two kinds of them, for some are true unconditionally and some are true for now. Consequences are true unconditionally if they can be reduced to a formal [consequence] by assuming a necessary proposition. And so the following is an unconditionally valid material consequence: a man runs, therefore, an animal runs, for it can be reduced to a formal [consequence] by [adding] the following necessary [proposition]: every man is an animal. And these are further divided into many kinds according to the variety of dialectical topics. 4 But a material consequence is valid for now if it can be reduced to a formal consequence by assuming some contingently true proposition. And thus given that Socrates is white, the following consequence is valid for now: Socrates runs, therefore a white [thing] runs, for it can be reduced to a formal [consequence] by [adding] the following [true] contingent [proposition]: Socrates is white. It is clear, therefore, what a

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consequence is and what are its kinds, and this [suffices] about the second [article]. 3. ANSWER TO THE QUESTION As for the third [article], we present [some] conclusions. The first [conclusion] is that when a consequent follows from an antecedent, the contrary opposite of the antecedent need not follow from the contrary opposite of the consequent, and if it does follow, this nevertheless does not suffice for the validity of the consequence. Proof of the first part: for it follows [from the premise that] no animal is a man, [that] therefore, no man is an animal, and nevertheless in the argument from the contrary of the consequent to the contrary of the antecedent, the consequence is not valid, for it does not follow [from the premise that] every man is an animal, [that] therefore, every animal is a man. And by the same [argument] we prove the second part, since it does not follow from the fact that the contrary of the antecedent follows from the contrary of the consequent, that this consequence would be valid. The second conclusion is that in every valid consequence the contradictory opposite of the antecedent follows from the contradictory opposite of the consequent. Proof: by the definition of valid consequence. Let A be the antecedent and B the consequent. Then by the definition it follows that when A and B are formed simultaneously, it is impossible that A is true and the consequent false, and so, it is impossible that the contradictory of the consequent is true and the contradictory of the antecedent is false. Therefore, by the definition of consequence, the contradictory of the consequent will be antecedent to the contradictory of the consequent. And [so] we have the conclusion. The third conclusion is that when a consequent follows from an antecedent, the opposite of the consequent is incompatible with the antecedent. Proof: since the opposite of the consequent cannot stand as true with the antecedent, therefore, it is incompatible with it. The consequence holds by the nominal definition of incompatibility, and the antecedent is obvious, for the opposite of the consequent implies the opposite of the antecedent by the previous rule. Therefore, since the opposite of the antecedent is incompatible with [the antecedent], it follows also that the opposite of the consequent is incompatible with it, although not in the same way.

QUESTION X

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It is clear, therefore, how the opposite of the antecedent is inferred from the opposite of the consequent, and this [suffices] about the third [article]. 4. CERTAIN OTHER RULES As for the fourth [article], the first conclusion is that from any proposition which formally implies a contradiction, any other proposition follows by a formal consequence. For example, a man is a donkey, a stick is broken, or anything at all, follows from this: Socrates exists and Socrates does not exist, which formally implies a contradiction. Proof: for it follows [from the proposition that] Socrates exists and Socrates does not exist, [that] therefore Socrates does not exist, for the consequence from a conjunction to one of its parts is formal. Then this consequent is retained, and further, it follows [from the proposition that] Socrates exists and Socrates does not exist, [that] therefore, Socrates exists, by the same rule. And [from the proposition that] Socrates exists, it follows [that] therefore, Socrates exists or a man is a donkey. For from any proposition we can formally infer itself together with any other [proposition] in a disjunctive [proposition]. Then it is argued from the consequent: Socrates exists or a man is a donkey, but Socrates does not exist, as was [inferred and] retained earlier, [that] therefore, a man is a donkey. And just as has been argued about this [proposition], so we can argue about any other [proposition], for all these consequences are formal. The second conclusion is that from any impossible proposition follows any other proposition, not by a formal consequence, but by an unconditionally valid material consequence. Proof: for such a consequence is unconditionally valid, and can be reduced to a formal consequence by assuming only a necessary proposition. But the consequence by which one infers another proposition, whatever it is, from an impossible [proposition] can be reduced to a formal consequence by assuming a necessary proposition. Therefore, such a consequence is unconditionally valid. The major [premise] is clear by the definition of an unconditionally5 valid [consequence]. The minor [premise] is proved by reducing such a consequence to a formal consequence by assuming the contradictory of the impossible proposition, for its contradictory is necessary. For example, we say that from this: a man is a donkey, any other proposition follows by an unconditionalll valid consequence, as, for example, from [the premise that] a man is a donkey, it follows that

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therefore, you are in Rome. For if we take the contradictory of the antecedent, then it follows [from the premises that] no man is a donkey and a man is a donkey, [that] therefore, no man is a donkey. Similarly, it follows [that] therefore, a man is a donkey (from a conjunction to each of its parts). Similarly, it follows [from the proposition that] a man is a donkey, [that] therefore, a man is a donkey or you are in Rome. But no man is a donkey (as has already been shown), therefore, you are in Rome, and we have what we aimed at. And just as was argued about this [proposition], so it is possible to argue about any other [proposition]. The third conclusion [is] that a necessary proposition follows from any proposition by an unconditionally7 valid consequence, with the exception of the case mentioned earlier, where the signification of the consequent is incompatible with the signification of the sign of consequence. The conclusion is proved [by noting that when] the consequent follows from the antecedent, the opposite of the antecedent follows from the opposite of consequent; but since any other proposition follows from an impossible proposition, therefore, the contradictory of an impossible proposition follows from the contradictory of any proposition. And since any proposition is the contradictory of some [other], it follows that the contradictory of an impossible proposition follows from any proposition, and since the former is necessary, it follows that a necessary [proposition] follows from any proposition, and we have what we aimed at. The fourth conclusion is that from any false proposition any other proposition follows by a material consequence valid for now. Proof: [recall that] a material consequence is valid for now if it can be reduced to a formal [consequence] by assuming a [true] contingent proposition; but that consequence by which from a false proposition another [proposition] follows, whatever it is, can be reduced to a formal [consequence] by assuming a true contingent proposition, therefore, etc. The major [premise] is clear by the definition of a material consequence valid for now, and the minor [premise] is proved by an example. Given that Socrates is seated, I say that from the [proposition]: Socrates moves, any other proposition follows by a material consequence valid for now, for by the contradictory of this [proposition]: Socrates moves, which is true, the consequence at issue can be reduced to a formal [consequence] by taking the following conjunction: Socrates moves and Socrates does not move, from which each of its parts follows formally, and proceeding as we did above.

QUESTION X

233

The fifth conclusion [is] that every true proposition follows from any other proposition in a material consequence valid for now. Proof: for when a consequent follows from an antecedent, the opposite of the antecedent follows from the opposite of the consequent; but from any false proposition any other proposition follows by a consequence valid for now; therefore, from the contradictory of any proposition there follows the contradictory of a false proposition. But the contradictory of a false proposition is true, therefore, 8 any true [proposition] follows from any other proposition, and we have what we aimed at. It is clear, therefore, how from an impossible proposition anything follows formally in some cases, and in other [cases] only materially, namely when the impossible [proposition] does not formally imply a contradiction. 5. ANSWERS TO THE ARGUMENTS FORT HE NEGATNE ANSWER 9 Now to the arguments. To the first I say that [the rule] must be understood [to apply] to contradictory opposites, and not to contrary opposites. To the second [argument]: I concede that "[this is the case only in as much] as the opposite of the consequent is incompatible with the antecedent," and say that the following: only the Father exists, is incompatible with itself, for it implies a contradiction, as it follows [from the premise that] not only the Father exists, [that] therefore, the Father exists, and nothing else than the Father exists, and from [the proposition that] the Father exists, it follows that the Son exists. Therefore, the said proposition is incompatible with itself. To the third [argument]: I concede the first consequence and say that the opposite of the antecedent can be inferred from the opposite of the consequent. But because the term one is common to both genders, one ought not to infer [from the premise that] everyone runs, [that] therefore some man 10 runs, but one ought to infer an indefinite [proposition] without the sign [of quantity], as follows: everyone runs, therefore, [some}one runs. And if the sign [of quantity] is added, then it must be taken in disjunction, with the masculine ending and with the feminine ending as follows: everyone runs, therefore, some man (quidam) or some woman (quaedam) 11 runs. To the fourth [argument]: the consequence is denied. To the first [step], I admit that from the opposite of the consequent one can infer the following disjunction: [It's} not [that] you are a donkey, or you are

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not a donkey, but then this disjunction must be distinguished in accordance with the fallacy of composition and division, for this negation not, which precedes the whole disjunction, may affect only the first part of the disjunction, and this is the divided sense, [in which case the proposition] is a disjunction, for it follows from the opposite of the above mentioned consequent. But if it is taken in the composite sense, then the negation not affects the [whole] disjunction and not [only] one part of the disjunction. And in this way [the proposition] is not a disjunction, but rather a conjuction, since it is equivalent to a compound conjunction, and in this way it does not follow from the opposite of the said consequent. And so we have answered the [original] question. 12 NOTES 1

The argument is based on playing with feminine and masculine endings. In Latin, omnis ('every') is a shared form of both masculine and feminine, while quidam ('some') is a specifically masculine determiner. The author takes it that the word homo ('man') can be used for both men and women, but with a masculine determiner it will refer only to men, and with a feminine one it will refer only to women. In attempting to render the effect in English, 'one' has been used for the fender-neutral sense of homo, 'man' for its use with quidam. Cf. below, p. 11. Reading consequentia for propositione. 3 Reading/a/sum for verum. 4 Cf., e.g. Aristotle's Topics. 5 Reading simpliciter for simplicis, et. 6 Reading simpliciter for simplici. 7 Reading simpliciter for simplici. 8 Here we omit a repetitive and redundant phrase: "the contradictory of any false proposition follows from any proposition. And since any true proposition is the contradictory of some false [proposition], it follows that". 9 That is, to the initial arguments on pp. 1-2. 10 Quidam ('some') is masculine. Cf. footnote 1. 11 Quidam ('some') is masculine and quaedam ('some') is feminine. Cf. footnote 1. 12 I am grateful to Stephen Read for help in many of the problems of translation.

INDEX OF NAMES Abaelard 64-68, 88, 89, 91, 93, 153 Adams, Marilyn 137, 144 Albert the Great 7, II, 27, 28, 30 Albert of Saxony 23, 30, 133, 136, 144, 154, 158, 186, 188-189, 193-195 Alchourr6n, C. E. 43, 52, 60, 61 Andersson, A. R. 194-195 Angelelli, Ignacio 30 Aristotle viii-ix, 3-8, I 0-11, 21, 24-25, 27, 30, 63, 64-66, 73, 88, 90, 118-119, 127, 129, 135, 136, 137, 176, 184, 217,225,234 Ashworth, Jennifer 29, 30-31, 61, 113, 154-156, 179-180, 194-195, 215, 223 Augustine 88 Back, Alan 194-19 5 Belnap, N.D. 194-195 Bendiek, J. 184, 195 Billingham, Richard 136, 155 Boethius, A. M. S. 6, 27, 64-66, 72, 88, 93, 118, 137 Boethius de Dacia 7-11, 27, 28, 31, 163-164 Boh, Ivan xi, xii, 137, 142, 144, 180, 184, 194-195 Bolzano, Bernard 141, 144 Bos, E. P. 144 Braakhuis, H. A. G. 6, 28, 30, 31 Bradwardine, Thomas 96, 121-122, 136, 140, 154-155 186 Brandom, Robert 142, 144 Broadie, Alexander 180 Brown, M. A. 30, 31 Brunschwig, J. 27, 31 Burge, Tyler 81, 92, 93 Buridan, John xi, 64, 73-74, Ill, 117, 119, 121-122, 124-125, 127, 129-131, 133-134, 136, 138-143, 147-154, 158, 171, 179-180, 186, 189, 193-194

Burley, Walter viii-ix, 5, 12-24, 26, 28-29, 31, 35-43, 45-46, 48, 50-54,57-61,68, 117, 119-121, 126-127, 136, 138-139, 141, 143, 158, 164 Carroll, Lewis 137, 144 Chisholm, R. M. 72 Cicero, Marcus Tullius 88 Clarembald of Arras 65, 72, 88, 93 Corcoran, John 136, 144 Courtenay, William 30, 31 Cresswell, M. 194 Curry H. B. 190, 192 De Rijk, L. M. xii, 6, 7, 27, 28, 30, 31, 64, 87, 88-91, 93, 111, 113, 215 Descartes, Rene xi, 157, 180 Domingo de Soto 147, 172-179, 181 D'Ors, Angel 28, 29, 31-32, 175, 179-180 Drange, W. 195 Dummett, Michael 137, 144 Duns Scotus, John ix, xii, 25, 27, 30, 32, 76, 184, 194 Eclif(?), Johannes 99, 102-103, 110, Ill, 112 Emden, A. B. 184, 195 Etchemendy, John 141, 144 Eudemus 64-66, 68 Ferrybridge, Richard 136, 143 Fland, Robert 23, 128, 136, 143, 155 Gaetanus ofThiene 155 Gal, Gedeon 184, 195 Garlandus Compotista 65, 88, 93 Geach, P. T. 195 Gellius, Aulus 88 Genzen, Gerhardt 117, 136, 139, 144 Gilbert of Poi tiers 65 Godefroid de Fontaines 27, 32 Goodman, Nelson 72 235

M Yrjonsuuri (ed.), Medieval Fonnal Logic 235-237. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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INDEX OF NAMES

Greco, Anna 142 Green, Romuald 12, 16, 28, 29, 32 Green-Pedersen, N. J. 27, 32, 132, 136, 138, 140-141, 144-145 Gregory of Rimini 184 Grelling 81 Giirdenfors 43-45, 49, 60, 61 Gode1, Kurt 74, 191-192 Haack, Susan 137, 145 Hallamaa, Olli 30 Hansen, Kaj Borge 61 Hansson, S. 0. 60, 61 Harper, W. L. 89, 93 Henry of Ghent 27, 32 Heytesbury, William x, 29, 32, 98-100, 102-106, 109-110, 112, 113 Hopton, Henry 155 Hughes, George 74, 90, 93, 111, 113, 186, 189, 194-195 Jackson, Frank 137, 145 Jacobi, Klaus 180 Jacques ofVitry 67-89, 93 Jaskowski, Stanislaw 117, 136, 145 Jean of Celaya 194 John of Cornwall xii, 184, 194 John of Holland 23, 32 John of StGermain 184 Kakkuri-Knuuttila, M.-L. 27, 32 Kaplan, David 142 Karger, Elizabeth 139, 142 Kilvington, Richard viii-ix, 12, 16-20, 22, 26-27, 29, 32, 52, 89, 154 King, Peter x-xi, 30, 32, 137, 140, 145, 149,180 Kleene, S. C. 194-195 Kneale, Martha 137, 145, 194-195 Kneale, William 137, 145, 194-195 Kneepkens, C. H. 30, 32 Knuuttila, Simo 27, 28, 29, 30, 32-33,61, 113, 163, 181 Kretzmann, N. 29, 33, 58, 59, 61 Kripke, Saul 63, 88, 93 Lagerlund, Henrik viii-ix, 52, 61 Lavenham, Richard 23, 121, 128, 136, 144, 156

Lear, Jonathan 136, 145 Lewis, David 49 Lindenbaum 72 Lindstrom, Sten 61 Lob, M. H. 192, 195 MacCall, Storrs 137, 145 Mackie, J. L. 72 Maieru, Alfonso 181 Makinson, David 43, 52, 60, 61 Marsilius of Inghen 23, 136 Martin, Christopher J. x, 6, 33, 59, 60, 61, 88, 93, 111, 114, 194-195 Mates, Benson 194-195 McDermott, A. C. S. 194, 196 Moody, Ernest A. 136, 145, 154, 171, 181, 194, 196 Moraux, P. 27, 33 Nicholas of Paris 6 Normore, Calvin 67, 88, 89, 93,179-181 Nute, Donald 145 Ockham, William xi, 28, 30, 33, 117, 119, 121, 125-127, 132, 135-138, 140-141, 143, 147, 154, 158, 179, 186 Olsson, Erik J. viii-ix Panaccio, Claude 194, 196 Paul of Pergula 23, 33, 155, 163, 181 Paul ofVenice 23, 33, 61, 154-155 Pearce, G. 89, 93 Perreiah, Alan 27, 30, 33 Peter of Mantua 147, 155, 165-169, 174, 181 Pinborg, Jan 139 Pironet, Fabienne x, 111, 113, 114 Pluta, Olaf 181 Pozzi, Lorenzo 30, 33, 145, 181, 194, 196 Prawitz, Dag 136, 145 Priest, G. 189, 195-196 Prior,Arthur N. 141, 145, 186, 196 Pseudo-Ockham 143 Pseudo-Scotus xii, 121, 138-139, 143, 154, 184-185, 187-190, 193-194, 196

INDEX OF NAMES Ramsey, F. P. 49-51, 60, 62, 72 Read, Stephen xii, 97, 114, 180-181, 194, 196, 234 Richard of Campsall 29, 33 Richter, Vladimir 28, 33 Rosetus, Roger 23, 30 Rosser, J. B. 191-192, 194, 196 Roure, M. L. Ill, 114, 186, 194, 196 Routley, R. 189, 195-196 Russell, Bertrand 73 Ryle, Gilbert 27, 33 Seaton, Wallace 181 Segerberg, Krister 61 Schupp, Franz 141, 145 Scott, T. K. 114, 186, 196 Sermoneta, Alexander 155, 171 Shapiro, Stewart 141-142, 145 Smiley, Timothy 136, 145 Smith, Robin 136, 145 Sobel, J. H. 60, 62 Sorensen, R. 195-196 Spade, Paul, 19-20, 23, 28-30, 33, 35, 42, 47-48, 50-52, 58-62, 68-70, 88, 89, 92, 93, 111, 113, 114, 154-156, 17~ 194, 196 Stalnaker, R. 60, 62, 89, 93 Strode, Ralph 125, 136, 144, 147, 154-155, 158....:166, 168, 171--172, 174, 181, 186 Stump, Eleonore 28, 29, 30, 33, 34, 58, 59, 61, 62, 68-69, 89, 90, 93, 111, 114, 141, 145 Swineshed, Roger vm-tx, 12, 20-24, 26-27, 29, 30, 52 Tarski, Alfred 129-130, 132, 135, 140-141, 145, 186, 194, 196 Taschek, William 142 Thierry of Chartres 65, 88, 93 Thorn, Paul 136, 145 Von Wright, G. H. 28, 34 Weisheip1, James A. 27, 34 William Buser ofHeusden 23 William of Sherwood 28, 68, 76, 90 William of Ware 156 Wodeham, Adam 184 Woods, Michael 137, 145

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Wyclif, John 111 Yagisawa, Takashi 89, 94 Ytjonsuuri, Mikko vm-ix, xii, 27-30, 34, 58-62, 142, 163-164, 181 Zeno ofElea 68 Aqvist, Lennart 61