Logic and Ontology in the Syllogistic of Robert Kilwardby (Studien Und Texte Zur Geistesgeschichte Des Mittelalters)

Logic and Ontology in the Syllogistic of Robert Kilwardby

Studien und Texte zur Geistesgeschichte des Mittelalters Begründet von

Josef Koch Weitergeführt von

Paul Wilpert, Albert Zimmermann und Jan A. Aertsen Herausgegeben von

Andreas Speer In Zusammenarbeit mit

Tzotcho Boiadjiev, Kent Emery, Jr. und Wouter Goris

BAND XCII

Logic and Ontology in the Syllogistic of Robert Kilwardby By

Paul Thom

LEIDEN • BOSTON 2007

This book is printed on acid-free paper. Library of Congress Cataloging-in-Publication Data Detailed Library of Congress Cataloging-in-Publication Data are available on the Internet at http://catalog.loc.gov

ISSN: 0169-8028 ISBN: 978 90 04 15795 8 Copyright 2007 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill, Hotei Publishing, IDC Publishers, Martinus Nijhoff Publishers and VSP. All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Koninklijke Brill NV provided that the appropriate fees are paid directly to The Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, MA 01923, USA. Fees are subject to change. printed in the netherlands

For Cassandra

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Chapter One. Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter Two. Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter Three. Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Chapter Four. The assertoric syllogistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter Five. Necessity-syllogisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Chapter Six. Contingency-syllogisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Appendix. Kilwardby and modern logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Index of dubia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Index locorum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 General index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

LIST OF FIGURES 1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22

Fallacious argument with two senses of contingency Proof-patterns for syllogisms with negated universal contingencypremises Proof-pattern for syllogisms with negated particular contingencypremises First division of Discourse Second division of Discourse Scientific syllogism Inductive syllogisms Circular syllogisms Syllogism from opposed premises Apparent syllogism with a single term Reduction of particular first Figure moods Three-Figure ontological model Direct reductions of Camestres to Cesare, Celarent to Camestres, and Darii to Disamis Syllogistic proofs of assertoric conversions Direct reduction of Cesare Apparent syllogistic proofs of conversion Expository proof of conversion, with singular terms Reduction of e-conversion to Ferio Conversion by contraposition First proof of Le-conversion Conversion of contingency-propositions, in the sense of possibility Conversion of contingency-propositions, in the sense of necessity Qa-conversion Purported proof of Qa-conversion Conversion of universal Q-propositions to universal M-propositions Conversion of natural and indeterminate contingencies Indirect reduction of the second and third Figures to the first Rules of indirect reduction Rule of Exposition, proving syllogisms with a particular affirmative premise Rules of Exposition, proving syllogisms with a particular negative premise Aristotle’s Upgrading proofs of Barbara XQM Analysis of Aristotle’s first proof of Barbara XQM Purported rule for the proof of XQ syllogisms Apparent Upgrading proof of Barbara XQX

x 3.23 3.24 3.25 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 6.1

list of figures Counter-example to Barbara XQX Counter-example to Barbara XQM, with as-of-now Major Core inference in Barbara XQM The quality of premises and conclusion Invalid first Figure syllogistic form with negative Minor Valid first Figure syllogistic form with negative terms First Figure hypothetical syllogism with negative Minor Fallacy of denying the antecedent First Figure syllogisms with particular Major premises Ghazali’s division of first Figure premise-pairs Alexander’s division of first Figure premise-pairs Aristotle’s direct or indirect reduction of second Figure syllogisms Second Figure syllogism with particular Major Counter-examples to aa-2 moods Aristotle’s direct or indirect reduction of third Figure syllogisms Categorial counter-example to aaa-3 Reduction of Darapti to Celarent Reduction of Darapti to Disamis Syllogisms by diminution Fapesmo and Frisesomorum Subaltern first Figure moods Apparent counter-examples to indirect second and third Figure moods Indirect second and third Figure moods Direct reduction of Fapesmo and Frisesomorum Direct reduction of non-subaltern indirect second and third Figure moods Direct reduction of subaltern indirect second Figure moods Aristotle’s reductions of LL-2 and LL-3 moods Expository proofs of Baroco LLL and Bocardo LLL, with general terms Expository proofs of Baroco LLL and Bocardo LLL, with singular terms Apparent counter-example to Barbara LXL Aristotle’s direct reduction of XL-2 moods Apparent counter-example to Cesare LXL Aristotle’s counter-example to Camestres / Baroco LXL and Cesare/ Festino XLL Apparent counter-example to Cesare LXL Aristotle’s counter-example to Baroco XLL Kilwardby’s counter-example to Baroco XLL Third counter-example to Baroco XLL Direct reductions of LX-3 moods Aristotle’s counter-examples to Felapton XLL, Datisi XLL, Ferison/ Bocardo XLL and Disamis / Bocardo LXL Apparent counter-example to Felapton LXL Aristotle’s counter-example to Ferison XLL Kilwardby’s counter-example to Ferison XLL Two representations of the terms in Barbara QQQ

list of figures 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 A1 A2 A3

xi

Apparent counter-example to Barbara QQQ Direct reduction of QQQ-1 syllogisms Aristotle’s counter-example to QiQaQi-1 Erroneous Indirect Reduction of Cesare QQQ Erroneous Indirect Reduction of Cesare QQM Erroneous indirect reduction of Cesare QQL Demonstration that Cesare QQQ cannot be indirectly reduced to the first Figure Aristotle’s reductions of QQQ-3 moods Apparent counter-example to Darapti QQQ Perfection of XQM-1 moods by Upgrading Direct reduction of non-standard XQ-1 moods Apparent Upgrading proof of Celarent XQX Direct reduction of XQ-2 moods Apparent counter-example to Cesare XQM Kilwardby’s counter-example to Cesare QXM Apparent Upgrading proof of Cesare QXM Aristotle’s direct reduction of XQ-3 moods Indirect reduction of XQ-3 moods Apparent Upgrading proof of Darapti XQX Apparent counter-example to Felapton XQM Apparent Upgrading proof of Bocardo XQM Apparent expository proof of Bocardo XQM Indirect reduction of imperfect LQ-1 moods Apparent counter-examples to Celarent LQX Apparent Upgrading proof of Barbara LQX Direct reduction of LQM-2 and LQX-2 moods Purported indirect reduction of Camestres LQX Aristotle’s counter-example to Camestres LQX Purported indirect reduction of Baroco QLX Indirect reduction of Camestres and Baroco LQM, and of Cesare and Festino QLM Indirect reduction of Baroco QLM Direct or indirect reduction of LQ-3 moods Indirect reduction of Bocardo QLM The medieval L / X / M systems Perfectibility of non-Aristotelian QL-2 inferences Generalized pons asinorum

LIST OF TABLES 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 A1 A2 A3 A4 A5

Useful first Figure assertoric premise-pairs Useful second Figure assertoric premise-pairs Useful third Figure assertoric premise-pairs Useful LX-1 premise-pairs Useful LX-2 premise-pairs LX-2 syllogisms Useful LX-3 premise-pairs LX-3 syllogisms Useful QQ-1 premise-pairs QQQ-1 syllogisms Useful QQ-3 premise-pairs QQQ-3 syllogisms Useful XQ-1 premise-pairs QXQ-1 and XQM-1 syllogisms Useful XQ-2 premise-pairs XQM-2 and QXM-2 syllogisms Useful XQ-3 premise-pairs XQM-3, XQQ-3, QXQ-3 and QXM-3 syllogisms Useful LQ-1 premise-pairs LQ-1 syllogisms Useful LQ-2 premise-pairs LQX-2, LQM-2, QLX-2 and QLM-2 syllogisms Useful LQ-3 premise-pairs QLQ-3, QLM-3, LQX-3, LQM-3 and LQQ-3 syllogisms LQ-2 premise-pairs yielding a conclusion in Kilwardby’s wider system LQ-2 premise-pairs yielding a conclusion in Kilwardby’s narrower system Proof patterns for modal and non-modal propositions (including Kilwardby’s corrected L /Q/M system) Extra proof patterns in Kilwardby’s maximal L /X/M system Extra proof patterns in Kilwardby’s maximal L /Q/M system

PREFACE My interest in Robert Kilwardby was sparked by the reprinting in 1968 of the 1516 edition of his commentary on the Prior Analytics, under the name of Giles of Rome. Intermittently I pondered sundry matters in the commentary, including its distinction between essential and accidental consequences, and its discussion of the nature of conversion. Then in 2000 Henrik Lagerlund published his book Modal Syllogistics in the Middle Ages which devoted a ground-breaking chapter to Kilwardby and his contemporaries. Inspired by Lagerlund’s work, I undertook the survey that was published in 2003 as Medieval Modal Systems, a chapter of which is devoted to Kilwardby’s modal syllogistic. Then in 2004 I was fortunate to win a grant from the Australian Research Council to work more extensively on Kilwardby’s commentary. The present book is the outcome of the research I did under that grant, and I thank the Australian Research Council for making it all possible. Some of this work has been aired at the annual meetings of the Aristotelian Logic East and West 500–1500 symposium at Cambridge University; and I thank John Marenbon and Tony Street for their hospitality on those occasions. My thanks also go to the participants who helped me clarify some difficult questions—particularly Robert Wisnovsky and Simo Knuuttila. As the book progressed, I had some particularly sunny discussions with Henrik Lagerlund in Byron Bay. He and José Filipe Silva subjected the work to very useful scrutiny in a workshop at the University of Uppsala. In Copenhagen Sten Ebbesen kindly read an earlier version of the book, and made numerous helpful comments. He also generously helped with accessing some important out-of-the-way primary material. For supplying copies of the manuscripts my thanks go to the staff of the Biblioteca Comunale Assisi, the Biblioteca Universitaria Bologna, the Bruxelles Bibliothèque Royale Albert, the Library of Peterhouse Cambridge, the Bibliothèque de la ville Carpentras, the Bibliotheca Amploniana Erfurt, the Biblioteca Nazionale Centrale Firenze, the Biblioteca Laurenziano Firenze, the Stiftsbibliothek Klosterneuburg,

xvi

preface

the Biblioteca Jagiellonska Krakow, the Bodleian Library Oxford, the Library of Merton College Oxford, the Bibliothèque nationale Paris and the Biblioteca Marciana Venezia. Special thanks go to my research assistant Dr Berenice Kerr for preparing the indexes and for her painstaking work in marking up photocopies of the manuscripts and collating readings of difficult passages, and to my doctoral student Heine Hansen for his work in transcribing most of the Latin citations. Any remaining faults are my responsibility.

NOTATION

From time to time I shall employ the following notation. Propositions Propositions The contradictory of p A contrary of p Sequences of propositions If-then Terms (subjects and predicates) Negated terms

p, q, r, etc. ~p ¬p Q , R, etc. →

a, b, c, etc. a, b, c, etc.

Quality and quantity The quality and quantity of categorical propositions is indicated by the following superscripts: Name

Superscript

Quality Affirmative Negative Quantity Universal Particular Universal affirmative Universal negative Particular affirmative Particular negative

qual + quant univ part a e i o

xviii

notation

Modality The following symbols are prefixed to a proposition, or are subscripted to a term, to indicate modality. Modality

Mode

Unrestricted assertoric Negation of unrestricted assertoric Unrestricted assertoric (necessary) Unrestricted assertoric (natural) Problematic (one-way possibility) The necessary contingent The non-necessary contingent Contingency (two-way possibility) Natural contingency Indeterminate contingency Negation of contingency Per se necessity De dicto necessity

X X XL XN M QL QQ Q QN QI Q L o

Categorical propositions are symbolized by a sequence of these symbols in the order: modality—predicate—subject—quality/ quantity (as a superscript). Thus, a universal negative possibility-proposition will take the form Mabe, and a proposition with modalized terms will be written aLbM a. Form

Notation

Unrestricted assertoric Contradictory of unrestricted assertoric Apodeictic unrestricted assertoric Contradictory of apodeictic unrestricted assertoric Per se necessity Per se possibility Ampliated contingency Contradictory of ampliated contingency Unampliated contingency Contradictory of unampliated contingency Natural contingency Contradictory of natural contingency

Xaba, Xabe, Xabi, Xabo Xaba, Xabe, Xabi, Xabo X Laba, X Labe, X Labi, X Labo XLaba, XLabe, XLabi, XLabo Laba, Labe, Labi, Labo Maba, Mabe, Mabi, Mabo Q’aba, Q’abe, Q’abi, Q’abo Q’aba, Q’abe, Q’abi, Q’abo Qaba, Qabe, Qabi, Qabo Qaba, Qabe, Qabi, Qabo Q N aba, Q N abe, Q N abi, Q N abo QN aba, QN abe, QN abi, QN abo

notation

xix

Inferences/arguments I do not distinguish between inferences and arguments. An inference / argument with antecedents / premises p and q, and consequent / conclup q sion r, is written: . r Inference-type

Name

Syllogisms

Barbara Celarent Darii Ferio Cesare Camestres Festino Baroco Darapti Felapton Disamis Datisi Bocardo

Inference aba bca aca abe bca ace aba bci aci abe bci aco abe aca bce aba ace bce abe aci bco aba aco bco aca bca abi ace bca abo aci bca abi aca bci abi aco bca abo

xx

notation Ferison

Conversions

e-conv i-conv a-conv Le-conv Li-conv Q-conv

Subalternation

Sub

Identity

Id

ace bci abo abe bae abi bai aba bai Labe Lbae Labi Lbai Qab--- Qab+ Qab+ Qab--aba abi aaa

Rules of inference Name

Abbreviation

Indirect Reduction

C K

Exposition

Rule p q r p q r

→

p ~r ~q

→

p ¬r ~q

Q bz az p

→

Q Lbne Lana p

Q bai p →

Q Lbao p

Q Lbao Q ~Lbz Laz → p p

INTRODUCTION Robert Kilwardby (c. 1215–1279) has been described as “one of the most remarkable thinkers of the thirteenth century”.1 This judgment is borne out by the sense of system, the logical acumen, the profound understanding of Aristotelian philosophy, and the interpretive inventiveness displayed in his theory of the syllogism as it is presented in his commentary on Aristotle’s Prior Analytics.

Kilwardby’s Life Kilwardby studied at the University of Paris from about 1231, and graduated as a Master of Arts around 1237. He taught in the Paris Faculty of Arts until about 1245. Also present in the Arts Faculty at that time (1241–1247) was Albert the Great. After 1245 Robert joined the Dominican Order, and taught at Blackfriars Oxford. He held the Dominican Chair of Theology at Oxford from about 1256 to 1261. In that year he was elected Provincial of the English Dominicans. In 1272 Pope Gregory X appointed him Archbishop of Canterbury. As Archbishop he presided over the coronation of King Edward I and Eleanor on 16 September 1274. In 1277 he issued a condemnation of 30 propositions that were being taught at the University of Oxford. Among these, some have to do with logical theory, and we shall see in the ensuing chapters that even as early as his Prior Analytics commentary some of these propositions were occupying his attention. In 1278 Pope Nicholas III elevated Kilwardby to the College of Cardinals, and he was called to Rome to serve as Cardinal of Porto. He died in papal service at Viterbo on 11 September 1279. He was buried there in the Church of the Dominican Order. His epitaph reads:

1 Alessandro D. Conti, “Kilwardby, Robert (d. 1279)”, in Concise Routledge Encyclopedia of Philosophy (Florence KY: Routledge, 2000), 438.

2

introduction Venerabilis Fr. Robertus de Kilvarbius; Anglus, Theologus ac Philosophus praeclarus. Archiepiscopus Cantuariensis, Primas Angliae, Cardinalis Portuensis, ordinis Praedicatorum hic sepultus jacet 1280.2

Works While in Paris, Kilwardby commented on Porphyry’s Isagoge, the Categories, the Peri Hermeneias, the Six Principles, Boethius’s De Divisione, Boethius’s Topica, the Prior Analytics, the Posterior Analytics, the Topics, and the Sophistical Refutations.3 This body of work, along with his grammatical commentaries, forms “the earliest comprehensive witness to the teaching of the set books at Paris from one master”.4 His commentary on the Prior Analytics was written about 1240. It was printed at Venice in 1499, 1500, 1502, 1504, 1516, 1522 and 1598—ascribed to Giles of Rome.5 At Oxford he produced a quantity of theological and biblical writings. In 1250 he wrote the wide-ranging De Ortu Scientiarum.6 Other works include De Spiritu iminaginativo sive de acceptione specierum, De praedicamento relationis, a treatise De tempore, a treatise De conscientia et de synderesi, and a commentary on the four books of the Sentences. His theological works show an “explicit concern to harmonize traditional Augustinian theology with modern Aristotelian philosophy”.7

2 Ellen M.F. Sommer-Seckendorff, Studies in the Life of Robert Kilwardby O.P. (S. Sabina Roma, Istituto storico domenicano, 1937), 126. 3 Thomas Kaeppeli O.P., Scriptores Ordinis Praedicatorum Medii Aevi vol. 3 (Roma: S. Sabinae, 1980); Thomas Kaeppeli O.P. and Emilio Panella O.P., Scriptores Ordinis Praedicatorum Medii Aevi vol. 4 (Roma: Istituto storico domenicano, 1993). 4 Robert Kilwardby O.P., On Time and Imagination: De Tempore, De Spiritu Fantastico = Auctores Britannici Medii Aevi IX edited by P. Osmund Lewry, O.P. (London: Oxford University Press, 1987), xiii. 5 Charles H. Lohr S.J., “Medieval Latin Aristotle commentaries: Authors: Robertus–Wilgelmus”, Traditio 29 (1973): 93–197, 112. Simon Tugwell, “Kilwardby, Robert”, in Oxford Dictionary of National Biography [accessed 3 Dec 2004: http://www.oxforddnb.com/ view/article/15546] (Oxford University Press, 2004). 6 Robert Kilwardby, De Ortu Scientiarum = Auctores Britannici Medii Aevi IV ed. A.G. Judy (Toronto: PIMS, 1976). 7 Tugwell, “Kilwardby, Robert”.

introduction

3

Kilwardby’s Prior Analytics Commentary Kilwardby’s exposition of the two books of the Prior Analytics follows Aristotle’s text chapter by chapter, and section by section within the chapters. In some places he interpolates a Note, providing further explanation of the text, or observing that Aristotle has omitted consideration of some case or other. In most chapters, the exposition is followed by a series of dubia. In many cases these deal with literal quotes from Aristotle’s text [dubitatur de dictis in littera], pointing out apparent consequences of Aristotle’s words [sed tunc sequitur] or questioning his examples [aliquis dubitabit de exemplo quod ponit]. In other cases, a question is raised about the absence of something from the text [quare non facit mentionem de illa]. In yet other cases, the question concerns the choices that Aristotle made in writing his text [quare magis ponit … quam…]. Sometimes the question concerns the consistency of one passage with another [cum in principio docuerit … hic autem docet…] and sometimes it asks for a reason why the same topic is covered in more than one part of the Aristotelian corpus [quare hic determinat … cum in topicis…]. Sometimes what is in question is the passage’s relevance [pertinentia], its completeness [sufficientia], or the order [ordo] in which matters are discussed. Sometimes the dubium states a theoretical difficulty with Aristotle’s doctrines [sed contra]. He sets out solutions to these puzzles, sometimes giving more than one solution to a single puzzle. His solutions call on other Aristotelian texts, and distinguish the different purposes of different texts. He frequently dissolves a puzzle by pointing out that something can be “considered” under more than one aspect. He calls on other Aristotelian texts—notably from the Posterior Analytics, Physics or Metaphysics—to illuminate passages in the Prior Analytics. Lewry observes, in relation to Kilwardby’s commentaries on the Ars vetus, “The commentaries are marked by a sense of the unity of the arts course: metaphysics, natural philosophy, ethics and grammar are all drawn upon to elucidate the set books in logic”.8 Throughout his Prior Analytics commentary, Robert displays a penetrating understanding of Aristotle’s philosophical doctrines, and an ability to apply those doctrines in areas where their author did not. 8 Osmund Lewry, “Robert Kilwardby’s Writings on the logica vetus studied with regard to their teaching and method” (D.Phil. thesis, Oxford University, 1978), 354– 355.

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introduction

The commentary, however, is no dry repetition of Aristotle’s ideas. It puts forward an interpretation, and as such brings material from Kilwardby’s own historical context to bear on Aristotle’s text. Lewry speaks of an Augustinianism that is “present in germ in Kilwardby’s Parisian arts course”;9 and more generally he notes “a continuity of thought” between Kilwardby’s earlier and later writings on a number of points.10 We do indeed find Augustinian influences in his Prior Analytics commentary. For example (as we shall see in Chapters Two and Four), Kilwardby thinks that there are multiple formal elements in a syllogism, some of which are more remote than others. Syllogistic Figure he takes to be more remote than Mood, which is needed to turn an inference into a complete syllogism. He holds that an inference may possess the more remote formal elements without possessing the completing form that would make it into a syllogism. Again (as we shall see in Chapter Three), he speaks of some inferences as “yet-tobe-syllogized”. In both these instances, he uses language that is influenced by Augustine’s doctrine of “seminal natures”—things “which are to be but have not yet been made”,11 and which lack their completing form.12 Kilwardby’s work is not presented as a systematic treatise on the topics he touches on; but that doesn’t mean that there is not within it a systematic treatment of those topics. And in fact by drawing together material that is presented at disparate points in the commentary, one can construct a set of systematically related views. These views are based on a thorough-going respect for the logical values of consistency 9

Kilwardby, On Time and Imagination, xvii. Lewry, “Robert Kilwardby’s Writings on the logica vetus”, 355. 11 Augustine, De Genesi ad litteram. 6, 5, 8: “Tunc scilicet secundum potentiam per verbum Dei tamquam seminaliter mundo inditam, cum creavit omnia simul, a quibus in die septimo requievit, ex quibus omnia suis quaeque temporibus iam per saeculorum ordinem fierent.” 12 Augustine, De vera religione 18, 35–36: “Nam illud quod in comparatione perfectorum informe dicitur, si habet aliquid formae, quamvis exiguum, quamvis inchoatum, nondum est nihil, ac per hoc id quoque in quantum est, non est nisi ex Deo.” “Nam et quod nondum formatum est, tamen aliquo modo ut formari possit inchoatum est, Dei beneficio formabile est: bonum est enim esse formatum. Nonnullum ergo bonum est et capacitas formae: et ideo bonorum omnium auctor, qui praestitit formam, ipse fecit etiam posse formari. Ita omne quod est, in quantum est; et omne quod nondum est, in quantum esse potest, ex Deo habet. Quod alio modo sic dicitur: omne formatum, in quantum formatum est; et omne quod nondum formatum est, in quantum formari potest, ex Deo habet. Nulla autem res obtinet integritatem naturae suae, nisi in suo genere salva sit.” 10

introduction

5

and completeness; but this is not say that Kilwardby is putting forward a formal system of syllogistic in the modern sense. Readers of my book on medieval modal systems13 may have been led to think otherwise. There I represented Kilwardby as having subscribed to a syllogistic system structured in a number of layers according to the forms of propositions that are taken as premises. Here I favour an approach that stays closer to Kilwardby’s text. I argue that, while we may interpret Kilwardby as putting forward syllogistic systems, we do thereby interpret him, and it is as well to be aware of the assumptions on which such interpretations rest. Moreover, I shall argue that there are actually important elements in Kilwardby’s text that militate against such interpretations.

Albert the Great’s treatises on the Prior Analytics Albert the Great’s Treatises on the Prior Analytics are, as is well known, based closely on Kilwardby’s commentary. Indeed, Albert’s main aim in those treatises is very similar to the aims of the present work—to expound Kilwardby’s ideas on the Prior Analytics in a systematic way. However, Albert never mentions Robert by name. That omission has led critics, both medieval and modern, to question Albert’s competence. The early fourteenth-century Bolognese scholar Gentile da Cingoli viewed the relationship between Albert’s and Robert’s work in these terms: “… Albertus (qui fuit symia eius [sc. Roberti] in libro priorum, ideo commentum eius nihil ualet in libro Priorum)”.14 True it is that Albert’s views on the Prior Analytics have little independent value, and largely ape Robert’s. Sten Ebbesen has written about the relationship.15 He judges Prantl as being “not far off the mark” in describing Albert as a mere compiler, though he thinks Roger Bacon’s accusation of “ineffable falsity” is overstated.16 13 Paul Thom, Medieval Modal Systems: problems and concepts (Aldershot: Ashgate, 2003), ch. 6. 14 Andrea Tabarroni, “Gentile da Cingoli e Angelo d’Arezzo sul Peryermenias e i maestri di logica a Bologna all’inizio del XIV seculo” ([Appendix: Gentile da Cingoli (attrib.), “Utrum verbum formaliter sumptum predicetur”], in Dino Buzzetti, Maurizio Ferriani, Andrea Tabarroni (eds.), L’insegnamento della logica a Bologna nel XIV secolo (Bologna: Presso l’Istituto per la storia dell’Università, 1992), 432. 15 Sten Ebbesen, “Albert (the Great?)’s companion to the Organon”, Miscellanea Mediaevalia 14 (1981): 89–103. 16 Ebbesen, “Albert (the Great?)’s companion to the Organon”, 93.

6

introduction

On the other hand we do well to notice the difference between Albert’s and Robert’s aims. These can be brought out by applying to both authors the analytical framework of the four causes which Robert uses in relation to Aristotle’s text.17 The material cause of Robert’s work (what it is about) is dual—the Prior Analytics and the syllogism in general; and the same is true of Albert’s work. The formal cause (the order and type of treatment) is different. Let us take as an example the two commentators’ treatments of Prior Analytics A9. Robert proceeds, after giving an exposition of Aristotle’s text, to pose and solve 7 dubia, and closes with two notes. Albert gives no exposition of Aristotle’s text, but he does include versions of Robert’s dubia and Notes, albeit in a different order. His version of dub.1 is found at the start of his I.ii.3; and this is followed, in his next chapter, by versions of Note 1, dub.7, dub.2, dub.3, dub.5, dub.6 and Note 2. Some of his versions of Robert’s material expand on the original. In other cases, he omits parts of Robert’s treatment. To a modern eye, Robert’s work reads like a research work whereas Albert reads like a textbook. There is nothing superfluous in Robert’s commentary; indeed, his writing is sometimes so sparse that one has to labour to find his meaning. By contrast, Albert adopts a leisurely pace and often fills in details that are missing from Robert’s presentation. On a number of occasions he succeeds in clarifying obscure points in Kilwardby.

Sources I have consulted the 1516 Venice edition, and most of the manuscripts mentioned by Lohr18 or by De Rijk and Bos.19 Sources consulted Ed

17

Reverendi Magistri Egidii Romani in libros Priorum analeticorum Aristotelis Expositio et interpretatio sum perquam diligenter

Kilwardby, Prologue dub.7 (2vb). Charles H. Lohr S.J., “Medieval Latin Aristotle commentaries: Authors: Robertus–Wilgelmus”, 93–197. 19 L.M. de Rijk and E.P. Bos, Medieval Logical Manuscripts (Research project of L.M. de Rijk, and E.P. Bos: http://www.leidenuniv.nl/philosophy/publicaties/bos/rijk_/ bos.html. 18

introduction

A

Bo Br

Cm1

Cm2

Cr

E1

E2

F1

7

visa recognita erroribus que purgata. Et quantum anniti ars potuit fideliter impressa cum textu (Venice 1516) Assisi Biblioteca Comunale 322 ff. 41r–62r (XIII or XIV) [cum omnis sciencia sit veri inquisitiva … explicit liber priorum analectorum secundum sanctum Thomas de Aquino] is in good condition but lacks A33–B27, and contains 30 lines (50va) missing from Ed (20va), plus several lines (59vb) missing from Ed (38vb). Bologna Biblioteca Universitaria 1626 ff. 1–55 (Fratri 846) (XIV) is in poor condition and was found to be of no practical use. Bruxelles Bibliothèque Royale Albert 1er 1797–1798 ff. 1r–24v (Cat. 2907) (XIV) [notule supra primos priorum … et quid per discrimina demonstramus] is in good condition, but contains Kilwardby’s commentary on Book A only, and that incompletely. It contains a substantial section (13va–b) missing from Ed (20va), and several lines (18vb) missing from Ed (38vb). It lacks A16 dub. 7–16 (13ra) as well as A19 dub. 4–9 (14v–15r). Cambridge Library of Peterhouse 205 (2.0.8) ff. 85ra–135rb (XIII or XIV) [cum omnis sciencia sit veri inquisitiva … et illa non est inconveniens] is in generally good condition, and contains several lines (109ra) missing from Ed (38vb) but lacks A16 dub. 7–16. Cambridge Library of Peterhouse 206 (2.0.9) ff. 81ra–133vb (XIII or XIV) is a different commentary on the Prior Analytics. There is no correspondence between this and Ed. [voces sunt note earum pasiones que sunt in anima…] (See O1.) [Voces sunt signa intellectuum, idest specierum intelligibilium in anima rerum et intellectus sunt singna rerum ut dicit Aristotiles in libro Perihermenias] Carpentras BV 281 (L278) 48 ff. (XIV) is arranged differently from the other sources in that, while ff. 3ra–36vb contain the exposition, ff. 37ra–48vb contain the dubia (but only as far as A10, dub.4). Erfurt Bibliotheca Amploniana Cod. Quar. 276 ff. 63r–97v (1295– 1333) [cum omnis scientia sit inquisitiva … quod non est inconveniens] is in generally good condition, and contains several lines (81ra) missing from Ed (38vb). This manuscript does not contain the exposition of the text. Erfurt Bibliotheca Amploniana Cod. Quar. 328 ff. 94ra–161vb (end XIII) [cum omnis scientia sit inquisitiva … quod non est inconveniens] is in good condition, and contains some lines (110va) missing from Ed (20va), plus several lines (126r) missing from Ed (38vb) but lacks A16 dub. 7–16. Firenze Biblioteca Nazionale Centrale J. X. 48 ff. 21ra to end

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F2

Kl

Kr

O1

O2

O3

20

(XIII) [cum omnis scientia sit veri inquisitiva … explicunt notule analectorum] is in good condition, and contains a section (35rb–va) missing from Ed (20va), plus several lines (47rb) missing from Ed (38vb), while lacking a substantial section from Ed, B8–B22 (64ra–78vb). Firenze Biblioteca Laurenziano Lat. Plut. 71, 29 ff. 1–54 (XIV init.) [cum omnis scientia sit veri inquisitiva … et ita non est inconveniens quod dictum est] is in good condition, contains some lines (14va) missing from Ed (20va) plus several lines (27ra) missing from Ed (38vb) but lacks A16 dub. 7–16. Klosterneuburg Stiftsbibliothek 847 f. 1r–70v (XIII) [cum omnis scientia sit inquisitiva veri … quod non est inconveniens. Explicunt notule super librum priorum codice analectorum gratia Roberto de Killimebrum] is in very good condition, and contains some lines (16va) missing from Ed (20va) plus several lines (32va–b) missing from Ed (38vb) but lacks A16 dub. 7–16. Krakow BJ 1902n ff. 2r–133r (XV / XVI) (Iste … liber priorum analectorum) is included in Lohr’s list of works by Robert Kilwardby.20 In essence this is Aristotle’s text as far as A45, with annotations / marginalia written in more than one different hand. These indicate chapter divisions and include what could be teaching notes for the user of the manuscript. The work is ascribed to Aegidus Romanus, with folio 1v stating that it is a faithful interpretation of the most excellent teacher of the Arts and Sacred Theology, Giles of Rome, an Augustinian Hermit. There is nothing in the manuscript to suggest that it is Kilwardby’s commentary. Oxford Bodleian Library Canon Misc 403 ff. 134ra–181ra (XIV init.) lacks A1–B20 of Kilwardby’s commentary, and in their place has another commentary on those chapters [voces sunt note earum pasiones que sunt in anima…]; thus the only parts of Kilwardby’s commentary that it contains are B21–B27. Oxford Merton College 289 ff. 33r–100v (XIV) [cum omnis scientia sit veri inquisitiva … quod non est inconveniens] is generally clear. It contains several lines (65va) missing from Ed (38vb). Oxford Merton College 280 ff. 38r–99v (XIII) [Cum omnis scientia sit veri inquisitiva … quod non est inconveniens] is in good condition and is clear throughout. It contains several lines (69rb) missing from Ed (38vb).

Lohr 112.

introduction P1

V

9

Paris Bibliothèque nationale 16620 ff. 2r–51v (XIII) [cum omnis scientia sit inquisitiva … expliciunt notule priore] is in generally good condition with some blotchiness, and contains a substantial section (14va) missing from Ed (20va), plus several lines (26rb) missing from Ed (38vb). A few lines present in Ed (64rb) and in most other manuscripts are missing here (40vb). Venezia Biblioteca Marciana Lat V1220 (X40) 55 ff. (XV). [cum omnis scientia sit inquisitiva veri … et illa quod est inconveniens] is in generally good condition, and contains a substantial section (26va) missing from Ed (38vb). It also includes a section (42vb) missing from Ed (64ra–64v) but lacks A16 dub. 7–16.

Manuscript not consulted P2

Paris Bibliothèque nationale 15661 ff. 97r–98v (c. 1300) [3 folios only]

In quoting from Kilwardby’s commentary I have followed Ed except where it is evidently unsatisfactory—in which case I have cited those manuscripts which are legible. In most cases, where Ed is not satisfactory, all the manuscripts agree; and in these cases the English translation follows the manuscripts. Where the manuscripts disagree, the English translation follows the preferred reading (as indicated in the apparatus). In the passages translated, the most reliable manuscripts are P 1 and V, followed by Cm1, E 2 and Kl.

chapter one PROPOSITIONS

Throughout his commentary, Kilwardby pursues a program of supplementing Aristotle’s text in ways designed to enhance its credibility and to display its systematicity. This program is characteristic of one type of interpretation—in its concern to display the object of interpretation in a favourable light, and in its efforts to integrate that object into a larger meaning-giving context. The context into which Kilwardby assimilates Aristotle’s words draws from other parts of Aristotle’s text, as well as from various doctrines that post-date Aristotle. The Aristotelian material that he invokes in the interpretation of passages in the Prior Analytics includes other passages from the same work, along with material from Aristotle’s other logical and metaphysical works. Any account of the syllogism must include an account of the propositions that are its premises and conclusion. Aristotle offers a simple definition of a proposition: “A proposition, then, is discourse [oratio] affirming or denying something of something”.1 Kilwardby’s account goes beyond this simple definition in several ways. He distinguishes propositions and statements, he identifies types of proposition not mentioned by Aristotle, and he provides semantical interpretations of different proposition-types.

Propositions and statements Aristotle’s definition of “proposition” can be seen as comprising a genus (“discourse”) and differentiae (“affirming or denying something of something”). With regard to the genus Kilwardby notes that though in Grammar discourse is first divided into perfect and imperfect discourse, in Logic the assumption is that all discourse is perfect (i.e. complete); and so the definition is to be understood as taking “perfect discourse” as the genus.2 1 2

A1, 24a16–17. Kilwardby (3vb). Kilwardby ad A1 Part 2 dub.1 (4ra).

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With regard to the differentiae, he says that these can be seen as simultaneously defining “proposition” and dividing it into species.3 These differentiae have been chosen because they are the most relevant ways of differentiating propositions within the context of syllogistic theory.4 Syllogistic propositions are also differentiated as universal or particular, but the differentiation by quality is “more essential” than the one by quantity.5 There is a sense in which this is true: according to Kilwardby, a proposition may have a quality while having no quantity—at least no quantity that is appropriate to a well-formed syllogism. Singular propositions, for instance, are either affirmative or negative; but are neither universal nor particular nor indefinite, and therefore lack the type of quantity that they would have to have in order to generate a syllogistic Mood.6 He distinguishes propositions from statements: A statement [enuntiatio] is proffered solely on its own account, and primarily states what is in the soul; so it is defined through truth and falsity, which are in the soul. But a proposition exists on account of something else, namely a conclusion. But this can be in two ways, according to the double being of a syllogism, to which it corresponds. For a syllogism exists either in significative terms (and the proposition corresponding to this is the same as a statement in subject but different in mode, and it signifies the true or the false—and thus Boethius defines it by truth and falsity in the Topics),7 or else the syllogism exists in terms abstracting from matter, which is the way Aristotle takes it here (and the proposition corresponding to this is in abstract terms, and taken this way it differs from a statement, as is clear, and this is what Aristotle defines here).8

Kilwardby ad A1 Part 2 dub.2 (4ra). Kilwardby ad A1 Part 2 dub.3 (4ra). 5 Kilwardby ad A1 Part 2 dub.4 (4ra–b). 6 Kilwardby ad A24 dub.2 (35va–b). 7 Eleonore Stump, Boethius’s De Topicis Differentiis translated with notes and essays on the text (Ithaca: Cornell University Press, 1978), 1173D37–38. 8 Kilwardby ad A1 Part 2 dub.5 (4rb): “Enuntiatio tamen propter se profertur et primo enuntiat quod est in anima; et ideo definitur {definitur BrCm1CrE1E2F1F2KlO2P1V: differtur A: definit Ed} per verum et falsum, quae sunt in anima. Sed propositio est alterius gratia, scilicet conclusionis. Sed hoc potest esse dupliciter secundum duplex esse syllogismi cui respondet. Aut enim est syllogismus in terminis significativis, et eius propositio est eadem in subiecto cum enuntiatione sed differens in modo, et ideo significat verum vel falsum, et sic definitur per verum et falsum in Topicis Boethii. Aut est syllogismus in terminibus abstrahentibus a materia, de quo hic determinat Aristoteles, et huic respondet propositio in terminis abstrahentibus, et hoc modo differt ab enuntiatione, sicut patet, et hanc definit Aristoteles hic.” 3 4

propositions

13

A statement, unlike a proposition, may possess features that are “accidental”9 and relate to “accidental modes of understanding”.10 A statement like “No substance is destroyed by the destruction of a nonsubstance” doesn’t display the “right arrangement” [recta dispositio] of the terms. It wrongly suggests that this is a denial, whereas in reality it is the affirmation that whatever is such that its destruction destroys a substance, is a substance.11 Only when this right arrangement is displayed do we have a proposition, and do we recognize its correct quality. Thus, he goes beyond Aristotle’s simple definition of the proposition in two ways. Firstly, a proposition is thought of relative to one or more conclusions. Aristotle himself thinks of propositions in this way (though he doesn’t spell this out in his definition). As Ross notes, the Aristotelian definition of propositions [protaseis] gives their form but not their function, which is “to serve as starting-points for argument”.12 This function is evident throughout the Prior Analytics, but especially so in the discussion of the power of the syllogism to produce multiple conclusions, as Kilwardby notes: And it is to be said that a syllogism is implicit in every universal, whether affirmative or negative. For the sense of “Every man is an animal” is “All of that which is a man is an animal”. And it signifies a Major proposition by saying “All of that is an animal”, and a Minor by what is implicit, i.e. “which is a man”.13

Thus every proposition (that is, every categorical proposition) must be expressed in such a way as to be eligible as a premise of a perfect syllogism. Even if this is not part of Aristotle’s definition, it is implicit in Aristotle’s usage of the term “proposition” in the text.14 From time to time, Kilwardby invokes this criterion, speaking of the “setting out” [positio] of terms and propositions that is required by the definition of the syllogism. He states: “… a syllogism has to have the due setting 9

Kilwardby ad A36–A37 dub.2 (46rb). Kilwardby ad A36–A37 dub.3 (46va). 11 Kilwardby ad A32–33 dub.4 (44rb–va). 12 W.D. Ross, Aristotle’s Prior and Posterior Analytics: a revised text with introduction and commentary (Oxford: Clarendon Press, 1949), 288. 13 Kilwardby ad B1 Part 2 dub.1 (55rb): “Et dicendum quod in omni universali sive affirmativa sive negativa implicatur syllogismus. Sensus enim huius ‘Omnis homo est animal’ est iste: Omne id est animal quod est homo, et significatur maior propositio per hoc quod dico ‘Omne id est animal’, et minor per implicationem quae est ‘quod est homo’.” 14 Kilwardby ad A1 Part 2 dub.3 (4ra). 10

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out of the terms and propositions”.15 And he takes a hypothetical proposition (such as “If a man exists an animal exists”) to lack this due setting out. Secondly, the formulation of a proposition is contrasted with the accidental modes of understanding that are expressed in statements; and once set out correctly, a proposition expresses an essential mode of understanding. This way of thinking about propositions, even if it is not implicit in Aristotle’s usage, is after all not uncommon among philosophical logicians. Kilwardby raises a question about propositional identity, namely whether a proposition and its equivalent converse are identical, seeing that in B1 Aristotle says that the converse is different from the convertend,16 but later, in discussing circular syllogisms, he says they are the same.17 Characteristically, Kilwardby responds by accepting more than one sense of propositional identity: And it is to be said that the converting proposition and the converse are the same in subject and in terms; and this is what he means later. But they are different in the position of the terms; and this is what he means here, namely that the conclusion is different from its converse. And so there is no contradiction.18

In drawing the distinction between propositions as displaying a reality correctly arranged, and statements as expressing the soul’s possibly accidental ways of understanding, Kilwardby gives Aristotle’s definition of the proposition an ontological inflection that is absent from the original. The contrast between accidental ways of understanding and a correct arrangement that corresponds to the ways things are in reality bespeaks his realist ontology, which he regularly deploys in contrasting the way things are according to grammatical appearance [secundum vocem, secundum sermonem] with the ways they are in reality. However, that reality, as we are about to see, may be either actual or merely potential; and if actual, it may involve natural being or merely being according to reason and understanding.

15 Kilwardby ad A32–33 dub.6 (44va): “… oportet ad syllogismum quod fiat debita positio terminorum et propositionum.” 16 B1, 53a11–12. 17 B5, 58a27–28. 18 Kilwardby ad B1 Part 1 dub.2 (54vb): “Et dicendum quod convertens et conversa eadem propositio est secundum subiectum et secundum terminos, et hoc intendit inferius, sed diversa propositio est secundum positum terminorum et situm,

propositions

15

Quality Kilwardby explains the meaning of negative assertorics as follows: A negative assertoric proposition actually denies the predicate of those things that are actually under the subject (not of those things that are contingently under it), because when I say “Nothing white is black” the meaning of this utterance is that nothing which is white is black (not that nothing which contingently is white is black).19

One of the propositions Kilwardby condemned in 1277 states that from a negative with a finite predicate there follows an affirmative with infinite predicate without the assumed existence [constantia] of the subject. This proposition would have it, for example, that if Socrates is not just then it follows (without assuming that Socrates exists) that Socrates is not-just. Kilwardby is committed to rejecting this inference, because a negative proposition with a finite predicate doesn’t posit the existence of the subject whereas an affirmative with infinite predicate does. He distinguishes two types of infinite predicate, depending on whether the negated term is a substantive term or an adjectival term. A negated substantive term (like “Non-man”) posits nothing. He explains, “I say nothing in reality, namely among natural things, but only something conceptually or in imagination”: … when there is a non-man, it may be that there is something in nature, such as wood or stone or the like; but this is not necessary. On the contrary, there might be absolutely nothing; and so infinite names of this sort are said to posit nothing. If an exposition is sought, I say that it is to be expounded as follows: Non-man, a Being that is not a man. But when I say “Being” in this exposition, I do not mean a Being according to nature, but I take it broadly for Beings either according to nature or according to reason or understanding.20

et hoc intendit hic, scilicet quod diversa est conclusio ipsa propositio convertens et eius conversa; et ita non dicit opposita.” 19 Kilwardby ad A16 dub.5 (27ra): “In propositione negativa de inesse actualiter removetur praedicatum ab his quae actu sunt sub subiecto et non ab his quae contingit esse sub eo, quia cum dico ‘Nullum album est niger’, intellectus huius sermonis est iste: Nihil quod est album est nigrum, et non iste: Nihil quod contingit esse album est nigrum.” 20 Kilwardby ad A46 Part 1 dub.13 (52va): “… cum sit non homo, contingit quod sit aliquid secundum naturam, sicut lignum vel lapis vel huiusmodi, sed hoc necesse non est. Immo potest omnino nihil esse, et ideo huiusmodi infinita nomina dicuntur nihil ponere. Si autem quaeratur expositio, dico quod exponitur sic non homo: ens quod non est homo. Sed hoc quod dico ‘ens’ in hac expositione non dicit aliquid ens secun-

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A negated adjectival term, by contrast, does posit something in reality, since “Non-white” is to be expounded as “Some thing that is not white”. In either case, then, an infinite term posits something, whether it be something in reality or merely something in the understanding or the imagination. By contrast, a denial doesn’t posit anything: The next question is, what is the difference between a privative and an infinite name, and a pure negative? And it is to be said that they all agree in this, that they signify a form by a privative term. But they differ in this, that a negative term of itself simply negates, positing nothing either in reality or conceptually. Privative and infinite terms posit something in reality and conceptually—but in different ways. A privative term privates the Form but posits something as an aptitude for the Form. An infinite term, if it is a substantive or an adjective signifying a Differentia of a Substance, privates the Form and the Potency for the Form, positing something conceptually but nothing in reality. If it is an adjective, it privates both the finite Quality and the aptitude for it, but it posits something in reality as an infinite substance, and a privative term posits the same substance.21

The distinction between affirmation and denial is an absolute one, denial being thought of as the privation of affirmation.22 This, however, doesn’t mean that it is always a straightforward matter to determine a proposition’s quality. We have already noted his analysis of the seemingly negative statement “No substance is destroyed by the destruction of a non-substance”. Again, in discussing A46 Kilwardby distinguishes between the analysis of a proposition secundum rem and secundum vocem (or secundum quid). He applies this distinction to the question of Double Negation (a question that is not addressed by Aristotle). dum naturam, sed accipitur communiter ad ens secundum naturam et secundum rationem vel intellectum.” 21 Kilwardby ad A46 Part 1 dub.18 (53ra): “Consequens est quaestio quae sit differentia inter nomen privativum et infinitum et pure negativum. Et dicendum quod omnia conveniunt in hoc quod formam significant per terminum privativum. Differunt autem in hoc quod terminus negativus quantum de se est omnino negat nihil ponendo nec secundum rem nec secundum rationem, terminus autem privativus et infinitus aliquid ponit secundum rem et rationem, sed differenter. Terminus enim privativus privat formam sed ponit rem aliquam sicut aptitudinem ad formam, terminus autem infinitus si fuerit substantivum vel adiectivum significans differentiam substantiae, privat formam et potentiam ad formam ponendo aliquid secundum rationem sed nihil secundum rem. Si autem fuerit adiectivum, privat tam qualitatem finitam quam aptitudinem ad eam, sed {sed BrCm1E1E2F1F2KlO2O3P1V: si Ed} aliquid secundum rem ponit sicut substantiam infinitam, et eandem enim substantiam ponit terminus privativus {privativus BrCm1E1E2F1F2KlO2O3P1V: infinitus Ed}.” 22 Kilwardby ad B4 Part 1 dub.3 (59rb).

propositions

17

Secundum rem, the negation of a negation is an affirmation and there is no infinite series of negations. On this analysis—unlike the standard analysis of mathematical logic—negation is a toggle-function. Secundum vocem—and according to the standard modern analysis—the negation of a negation is a negation and there is an infinite series of negations.23

Quantity Given that Aristotle’s interest in the Prior Analytics is in propositions as premises and conclusions of syllogisms, and that for this purpose the most essential difference is between affirmative and negative, it is nonetheless important that propositions are also differentiated according to their quantity. In A41 Aristotle considers a case in which it is difficult to determine the question of quantity. This concerns the propositions 1. What b is in, a is in all of and 2. What b is in all of, a is in all of.24 He explains the difference between them by pointing out that whiteness might be in some, but not in all, beautiful things.25 Albert makes a parenthetic comment here, marking a rare addition to Kilwardby’s discussion: On this account I may say that in essential terms it does follow that if it inheres it inheres in all, just as if a man is an animal it follows that every man is an animal, because the one nature is in all. But in accidental [terms] it doesn’t follow, for in these what inheres may not inhere in all.26 Kilwardby ad A46 Part 2 dub.3 (54rb). A41, 49b14–16. 25 A41, 49b17–20. 26 Albertus Magnus. Priorum Analyticorum Libri II, in Beati Alberti Magni Ratisbonensis Episcopi Ordinis Praedicatorum Opera quae hactenus haberi potuerunt. Tomus primus, ed. Thoma Turco, Nicolao Rodulphio, Ioan. Baptista de Marinis (Lugduni: Claudii Prost, Petri & Claudii Rigaud, Hieronymi de la Garde, Ioan. Ant. Huguetan filii, 1651), I.vii.10 (423B): “Propter hoc dico autem fortasse quia in terminis essentialibus sequitur quod si inest, omni inest, ut si homo est animal, sequitur quod omnis homo est animal, quia una ratio est de omnibus. Sed in accidentalibus non sequitur hoc. In his enim potest inesse ita quod non omnibus inest.” 23 24

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Propositions like (1) and (2)—so-called “prosleptic” propositions27—pose some interesting questions about quantity. The subjects of (1) and (2) appear respectively to be terms in which “b” inheres, and terms in which “b” inheres universally. But are (1) and (2) universal or particular propositions? Kilwardby says that (1) “may be understood as a particular proposition” but he also notes that it can also be regarded as a universal, equivalent to “a is in every b”. He distinguishes it from (2), of which he says: “The other one, however, has a certain utility, namely in regard to what is contained under b but not in regard to b”.28 I take this to mean that (2) is a universal proposition whose subject, however, is not “b” but “what is contained under b”; and I take the latter phrase to mean “terms in which b inheres universally”. Kilwardby also states that “In discrete terms, it is the same for this to be in that, and for this to be in all of that”.29 He notices the existence of other propositional forms, similar to (2), mentioning the following ones (none of which occur in A41): 3. 4. 5. 6. 7. 8.

What b is in none of, a is in none of What b is not in [some of], a is in none of What b is in none of, a is in all of What b is not in [some of], a is in all of What b is in all of, a is in none of What b is in some of, a is in none of. He notes that (7) is equivalent to the universal negative “No b is a”.30 Modality

Kilwardby’s most significant contribution to the theory of the proposition is found in his detailed analyses of the senses that modal and non-modal propositions may have in modal syllogisms, and much of this contribution has an ontological inflection. He distinguishes propositions that express per se necessities (such as “Every man is of necesThom, The Syllogism (Munich: Philosophia, 1981), 205–215. Kilwardby ad A41 dub.1 (48rb): “Reliquam autem habet quamdam utilitatem, scilicet respectu contenti sub b, sed non respectu b.” 29 Kilwardby ad A33, 47b15–40 (44ra): “… in terminis discretis idem est hoc huic inesse et hoc huic omni inesse …” 30 Kilwardby ad A41 dub.2 (48rb). 27 28

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19

sity an animal”) from those that express per accidens necessities (such as “Every literate being is of necessity a man”). Per se necessities, when true, can be seen as expressing the theorems of an Aristotelian ontology (a science of being as being); and their terms are considered as such, in isolation from all other terms. Possibility-propositions would then be those propositions that express something consistent with that ontology. Affirmative necessity-propositions apply only to what actually falls under the subject, whereas negatives extend to what does so potentially. In a per se negative necessity-proposition the subject gives an essential and inseparable cause why the predicate is absent from it. Contingency in the generic sense is one-way possibility; two specific senses respectively cover necessities and two-way-possibilities, and the latter are further divided into natural and indeterminate contingencies. Contingency-propositions may or may not ampliate their subjects. Assertoric propositions may be subject to temporal restrictions or may be unrestricted; and the latter may either express essential predications that hold necessarily (such as “Men are animals”) or may express the actualization of natural contingencies (such as “Old people go grey”). The actualization of indeterminate contingencies is expressed in as-ofnow assertorics. Negative unrestricted assertorics (such as “The healthy are not sick”), like propositions expressing the actualization of natural contingencies, are not necessary. Necessity-propositions Kilwardby holds that there is a semantic distinction between the truth of a necessity-proposition and the necessity of the corresponding assertoric. In his questions on A9 he states that denominative terms enter into true (affirmative) necessary propositions only when they are mutually convertible, or when they are related as inferior and superior, as in “Everything white is coloured”.31 This statement is consistent with his view that true necessity-propositions must have per se terms. That “Everything white is coloured” is necessary doesn’t imply that “Everything white is necessarily coloured” is a true per se necessityproposition. Kilwardby distinguishes two types of necessity-proposition—per se necessities such as “Every man is of necessity an animal” and per

31

Kilwardby ad A9 dub.6 Note (17ra).

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accidens necessities such as “Every literate being is of necessity a man” or “Something white is necessarily an animal”.32 He explains: For, a per se necessity-proposition requires the subject to be per se some of the predicate itself. But when it is said “All who are literate are of necessity men”, the subject is not per se some of the predicate itself; but it is granted that it is necessary, because the literate are not separate from what is some of man. But this is a per accidens necessity.33

The key phrase here is “the subject is per se some of the predicate itself ”; and it includes the expression “the a itself ”, which comes ultimately from Plato34 and draws our attention to the term “a” considered in isolation from any other term that may be connected with it. In Aristotelian language, this is to say that the subject of a true syllogistic necessity-proposition is kath’ hauto, or per se: “… it can also be said with fair credibility that those propositions with the name of an Accident as subject are necessity-propositions only per accidens, not per se. …”.35 The formula “the subject is per se some of the predicate itself ” contains a second important element. This is the locution “the subject is some of the predicate”. This locution is found elsewhere in the Prior Analytics, for instance in the expository proof of e-conversion: If a inheres in none of those which are b, neither conversely will b inhere in any of those that are a. For, if it inhered in some, such as c, then it would not be true that no b is a; for, c is some of those that are b [c eorum que sunt b aliquid est].36

Again, in the expository proof of Baroco LLL, Aristotle uses a similar phrase: “Now, if it is necessary of the exposed [term], then also of some of that. For, that which is exposed is some of that itself [ipsum quidem illud aliquid est]”.37 These expressions suggest that the members of the subject-class count as members of the predicate-class, as men 32

Kilwardby ad A11 dub.3 (18vb). Kilwardby ad A3 Part 1 dub.4 (7rb): “Propositio enim per se de necessario exigit subiectum esse per se aliquid ipsius praedicati. Cum autem dicitur ‘Omne grammaticum de necessitate est homo’, ipsum subiectum non est aliquid per se ipsius praedicati, sed quia grammaticum non separatur ab eo quod est aliquid ipsius hominis, ideo conceditur esse necessaria. Sed quae sic est de necessario, per accidens est de necessario.” 34 For example, Plato, Phaedo 65d, 74a, etc. 35 Kilwardby ad A3 Part 1 dub.4 (7rb): “… et dici potest satis probabiliter, scilicet quod huiusmodi propositiones subicientes nomen accidentis non sunt per se de necessario sed per accidens tantum.” 36 A2, 25a15–17. Kilwardby 5va. 37 A8, 30a12–13. Kilwardby 15rb. 33

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count among the animals. I take it that this could be the case only if the predicate, as well as the subject, were a per se term; and indeed this subject / predicate symmetry is required if, as Kilwardby believes, affirmative per se necessities are convertible (universals to particulars). It is characteristic of per se terms that, as a matter of necessity, whatever falls under them does so of necessity: if “a” is per se then it’s necessary that whatever is a must be a.38 (There is an outer as well as an inner necessity-operator here. For ease of reading, let’s use “It’s necessary that” for the outer necessity, and “must be” for the inner one. The two are distinguishable: it could be the case merely as-of-now that whatever is a must be a. For instance, it could happen that just now only snow is white—in which case, given Aristotle’s belief that snow must be white, it would be the case that everything white must be white. But this would not be the case at other times—when some white things are contingently white.) But this is not all. Per se necessity-propositions have to be true not merely as-of-now, as a proposition made up of per se terms could be. It might be true that all animals are men, but this would be true merely as-of-now. A true syllogistic necessity-proposition has to state a truth of Aristotelian ontology, as “All men are animals” does but “All animals are men” does not.39 Thus, assuming the correctness of Aristotelian ontology, affirmative per se necessity-propositions have to express de dicto necessities with per se terms.40 Thus, an affirmative per se necessity-proposition with predicate “b” and subject “a” states that 1. It’s necessary that (all or some) b is a; 2. It’s necessary that all b must be b; 3. It’s necessary that all a must be a. The core proposition here is that (all or some) b is a. That core proposition is subject to two further conditions—that it be governed by an outer modality of necessity, and that its terms be per se. Given this last condition, the core proposition may be stated either in the unmodalized form “What is b is a”, or with a modalized predicate, as in “What is b must be a”. Kilwardby says that affirmative necessity-propositions 38 39

son.

Thom, Medieval Modal Systems, 19. Modern scholars who have arrived at this position include Van Rijen and Patter-

40 As shown in Thom, Medieval Modal Systems, 85, propositions of this type are recognized by Averröes.

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affirm the “actual inherence” of the predicate in the subject;41 and he evidently takes both the unmmodalized and the modalized form to express actual inherence. Similarly, according to Kilwardby, the subject of a necessity-proposition may be modalized or unmodalized, without harmful ambiguity:42 Now, it is said that necessity-propositions, unlike contingency-propositions, do not have a double sense. To the contrary, just as the proposition “Every b is contingently a” has a double sense—either “Everything that is b is contingently a” or “Everything that is contingently b is contingently a”—so “Every b of necessity is a” either says that everything that is b is of necessity a or that everything that is necessarily b is necessarily a. … And it is to be said that a universal necessity-proposition has those two senses, as objected. But they are inseparable from one another, because if every b is of necessity a, then a is inseparable from every b, and so from everything that is of necessity b and from what is b without qualification—because to inhere without qualification and to inhere of necessity do not differ in inherence but only in definition. He calls either one actual and essential inherence. And because this is so, Aristotle does not in the preceding section distinguish between to-be-said-of-all in necessity-propositions by his said senses, since they do not diversify inherence.43

The equivalence of “Every b is of necessity a” and “Everything that is necessarily b is necessarily a” has not thus far been taken into account by modern formal representations of Kilwardby’s modal logic.44 41

Kilwardby ad A16 dub.1 (26vb). Kilwardby ad A15 dub.1 (24ra). 43 Kilwardby ad A13 dub.10 (21ra): “Sed dicitur quod propositionis de necessario non est duplex intentio sicut propositionis de contingenti. Sed contra: sicut haec propositio ‘Omne b contingit esse a’ duplicem habet intentionem, vel quod omne quod est b contingit esse a vel quod omne quod contingit esse b contingit esse a, similiter ista {similiter ista ABrCm1E1E2F1F2KlO2O3V: similiter P 1: sequitur Ed} ‘Omne b de necessitate est a’ vel dicit {dicit AF 1O2O3V: dicet BrF 2: dicitur EdE 1: dicetur E 2Kl: quia P 1} quod omne quod est b de necessitate est a vel omne quod necesse est esse b necesse est esse a. … Et dicendum quod universalis de necessario habet illas duas intentiones sicut opponitur, sed inseparabiles sunt ab invicem, quia si omne b ex necessitate est a, tunc a est inseparabile ab omni b, ita quod ab omni quod de necessitate est b et quod est b simpliciter, eo quod inesse simpliciter et ex necessitate inesse non diversificant inhaerentiam nisi secundum rationem tantum sed utraque dicit actualem inhaerentiam et essentialem. Et quia ita est, ideo non distinxit Aristoteles in praecedentibus dici de omni in necessariis per dictas suas intentiones cum non diversificent inhaerentiam.” 44 The semantics for Kilwardby’s universal affirmative necessity-propositions that is given in Thom, Medieval Modal Systems, 96 requires that the subject be included within 42

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Given that per se necessity-propositions contain an outer modality of necessity, we shouldn’t be led to identify them with compound necessity-propositions. In fact, Kilwardby hardly ever mentions compound modals; and when he does he distinguishes them from the modals that occur in syllogistic reasoning: But it is to be noted that an affirmative necessity-proposition in the sense of composition is taken otherwise. For in that sense its terms supposit simply. But according to that sense necessity-propositions are not used for syllogisms.45

To say that a term has simple supposition is to say that it stands not for its inferiors but for the universal which it signifies.46 To say this of the terms in compound modal propositions is (in modern language) to say that such propositions generate opaque contexts. Such propositions are unsuitable for syllogistic reasoning precisely because the substitution of an inferior for the proposition’s subject doesn’t preserve truth. If it’s necessary that all b are a, and x is an inferior of b, it doesn’t follow that it’s necessary that x is a. Unlike Kilwardby, Vincent of Beauvais (d. 1264) favours a compound analysis of necessity-propositions.47 Vincent says Aristotle doesn’t discuss modals with adverbial modes because these are just assertoric propositions. By contrast, those with nominal modes are statements, as it were, with names and oblique verbs, where being or not-being are subject and modes are appositions—and these are discussed in the De Interpretatione. Nominal modals, he states, are all singulars with a dictum as subject.

the predicate as a matter of necessity, with no requirement that the terms be per se. No matter how successful this representation may be in the context of that work’s aim of formulating a semantics that delivers just the modal syllogisms which Kilwardby regards as valid, it does not match Kilwardby’s own stated requirements for the truth of per se necessity-propositions. 45 Kilwardby ad A16 dub.7 Note (27va): “Notandum tamen quod affirmativa de necessario in sensu compositionis sumpta aliter se habet. Termini enim in ipsa ad hunc sensum supponunt simpliciter. Sed secundum hunc sensum non sumuntur propositiones de necessario secundum quod ex eis fit syllogismus.” 46 Peter of Spain, Tractatus called afterwards Summule logicales, first critical edition from the manuscripts with an introduction by L.M. de Rijk (Assen: Van Gorcum, 1972), 81:12–18. 47 Vincentius Bellovacensis, Speculum Quadruplex sive Speculum Maius Tomus Secundus (Dvaci: Baltazaris Belleri 1624) (Reprinted Graz: Akademische Druck—u. Verlagsanstalt 1965) III.xl, 242.

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For Kilwardby, not all true de dicto necessities are true per se necessities. It is necessary de dicto that everything white is coloured; but “Everything white is necessarily coloured” is not a true per se necessitystatement. Conversely however, it is the case that all true per se necessities are, or imply, true de dicto necessities. Given that it’s necessary per se that all men are animals, the proposition “All men are animals” expresses a necessary truth. The predicate, as Kilwardby puts it, is “inseparable” from the subject. How, then, is it going to be permissible to substitute an inferior for the subject of a necessity-proposition (as is required by Aristotle’s doctrine of necessity-syllogisms)? The answer will be found in Kilwardby’s understanding of the Minor premise, stating that an inferior falls under the subject of the Major—a matter to be discussed later in this chapter. The subject of an affirmative per se necessity-proposition, unlike that of a compound necessity-proposition, applies to its inferiors. However, in Kilwardby’s view, it applies only to what is (not what could be) under the subject: A universal affirmative necessity [-proposition] affirms the predicate only of those things that are actually under the subject, and not of those that contingently are under the subject. The proposition “Every man of necessity is an animal” does not say that whatever can be a man is an animal, but that whatever is a man is an animal.48

As we saw earlier, actual beings may be either beings in nature or merely in imagination and understanding. But what exists merely in the imagination or the understanding should not be confused with what exists contingently. To talk about non-men isn’t necessarily to talk about potential men. Kilwardby’s doctrine of affirmative necessity-propositions doesn’t commit him to saying that such propositions imply that their subjects apply to something in nature. Indeed, he denies that “Man is of necessity an animal” implies that men exist of necessity.49 The occasion for this comment is Aristotle’s remark that for a man to go grey does not have uninterrupted necessity in that there is not always a man.50 If this seems a rather slender excuse for launching into a lengthy discussion 48 Kilwardby ad A16 dub.7 (27rb–va): “Universalis affirmativa de necessario affirmat praedicatum de his quae actu sunt sub subiecto tantum et non de his quae contingit esse sub subiecto. Haec enim propositio ‘Omnis homo de necessitate est animal’ non dicit quod quicquid potest esse homo est animal, sed quicquid est homo est animal.” 49 Kilwardby ad A13, dub.7 (20rb–va). 50 A13, 32b4–10.

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on the question of the implications of modal propositions, it should be borne in mind that the topic is one that is dear to Kilwardby’s heart. One of the propositions he condemned in 1277 is the thesis that truth with necessity only occurs with the constantia of the subject.51 So, not only did he believe that necessity-propositions (at least some of them) can be true when their subjects do not exist; he also believed that the opposite thesis was a dangerous one. In the context of his Prior Analytics commentary, he considers the question whether, given the categorial chain Animal-Substance-Being, it follows that if a man is an animal then a man is a being, and therefore that a man exists. He doesn’t agree: “… it is to be said that first and foremost, ‘being’ signifies actual being; and if it is taken in this way then ‘If a man is a substance, a man is a being’ doesn’t hold”.52 He is not denying that there is some sense in which it’s true that if a man is a substance then a man is a being; what is denied is that if a man is a substance then a man is a being in the primary sense of “being”, namely actual being. Kilwardby’s position on this matter is consistent with Aristotle’s view that “it is not the same thing to-be-something and just to-be”.53 On Aristotle’s view, it would be true that a man is a substance even if there were no men in existence.

51 See Alain de Libera, “Omnis homo de necessitate est animal: reference et modalité selon ‘anonymus erfordensis, Q. 328 (pseudo-Robert Kilwardby)”, Archives d’histoire doctrinale et litteraire du moyen-age 69 (2002) 201–237. De Libera argues that the condemned thesis was espoused by Roger Bacon. A separate question is whether Kilwardby was the author of the Anonymous Erfordensis edited in de Libera’s article. As I point out below, Kilwardby’s treatment of the question in the present work differs in a number of ways from the treatment by the Anonymus Erfordensis. 52 Kilwardby ad A13 dub.7 (20va): “… dicendum quod ‘ens’ primo et principaliter significat ens actuale; et si sic sumatur, tunc non tenet ‘Si homo est substantia, homo est ens’.” The Anonymus Erfordensis (de Libera, 237) replies to the arguments differently, saying simply that the inference ‘If man is a substance, man is a being’ does hold, but that the inference ‘If man is a being, man is’ doesn’t hold, because just as ‘Man is an animal’ is true on account of the agreement of concepts, so is ‘Man is a being’ true on account of the agreement between the concepts ‘Man’ and ‘Being’. But the latter concept does not signify actual being but only conceptual being. 53 Aristotle, Sophistical Refutations, in Sophistical Refutations and Topics translated by W.A. Pickard-Cambridge, in The Works of Aristotle translated into English under the editorship of W.D. Ross vol. 1 (Oxford: Clarendon Press, 1928) 5, 167a1–2.

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Negatives In Kilwardby’s understanding, negative necessity-propositions are like affirmatives in embedding a core proposition within the context of an outer modality of necessity. What that outer modality declares is that the core proposition is a truth of Aristotelian ontology—a truth stating that the subject is “essentially and inseparably a cause of the predicate’s inherence or absence”.54 In respect of their core propositions, however, negatives and affirmatives differ. In his questions on A16 Kilwardby states that a universal negative necessity-proposition “denies the predicate actually of all that is or that contingently is under the subject”55 (whereas an affirmative affirms the predicate of all that is under the subject). Again, in his questions on A19 he provides the following analysis of the proposition “Of necessity nothing sleeping rests”: A universal negative necessity-proposition denies the predicate of those things that are, and of those things that can be, under the subject—as is clear from what has been said—and so the sense of the Minor is: of necessity nothing that is, or that can be, sleeping rests (which is false).56

So negative necessity-propositions, unlike affirmatives, ampliate their subjects. Since the subject of a negative necessity-proposition stands for everything that can fall under that term, it’s evident that such propositions do not imply the actuality of anything under the subject-term. Aristotle states in A34 that the proposition “Of necessity nothing healthy is sick” is false; and Kilwardby asks why it wouldn’t be true, seeing that “Nothing healthy is sick” is true and cannot not be true. Here is his solution: And it can be said that the argument “If a proposition is true, and it cannot not be true, therefore it is necessary” is not valid unless there was in the subject an essential and inseparable cause why the predicate is absent from the subject or inheres in the subject. But in “Nothing healthy is sick” there is in the suppositum no essential and inseparable cause 54

Kilwardby ad A34 dub.2 (45rb). Kilwardby ad A16 dub.5 (27ra). This feature of universal negative necessity-propositions is captured in Thom, Medieval Modal Systems, 96. 56 Kilwardby ad A19 dub.9 (31ra): “… universalis negativa de necessario removet praedicatum ab his quae sunt et quae possunt esse sub subiecto, sicut patet ex dictis. Unde sensus minoris est: De necessitate nihil quod est vel quod potest esse vigilans quiescit, quod falsum est.” Kilwardby ad A16 dub.5 (27ra) gives a similar treatment of the proposition “Of necessity nothing white is black”. 55

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of the predicate’s being absent from the subject, but an accidental and separable cause. And so the proposition is not necessary, but becomes false with the addition of a mode of necessity. However, “No man is an ass”, which is true and cannot not be true, is necessary because there is in the subject an essential and inseparable cause of the predicate’s absence from the subject; and accordingly, it becomes true when a mode of necessity is added. It’s clear from all this in what way “Nothing healthy is sick” and similar propositions are not necessary, and in what way we should understand the rule that if a proposition is true and cannot not be true, therefore it is necessary.57

This passage is important because it lays out in a clear manner what sense Kilwardby attaches to negative necessity-propositions. This sense is a metaphysical one—based as it is on the notion of the subject’s being “essentially and inseparably a cause of the predicate’s inherence or absence”. Kilwardby’s examples suggest that he wishes to draw a distinction among negatives—as he does among affirmatives—between per se and per accidens necessity-propositions; and that the former must have per se terms if they are to be true. If per se negative necessity-propositions are to be convertible (as required by Aristotle and Kilwardby), their core will have to state that nothing that can be under the subject can be under the predicate, not just that nothing that can be under the subject is under the predicate. Kilwardby states a further difference between affirmative and negative necessity-propositions: “… an assertoric follows from a negative, but not from an affirmative”.58 The assertoric affirmative, unlike the assertoric negative, has existential import. If every b is a then there exists a b. Now, the negative necessity-proposition “It’s necessary that no b is a” implies, with regard to every possible (and hence very actual) b, that it is not a. By contrast, 57 Kilwardby ad A34 dub.2 (45rb): “Et dici potest quod illud argumentum non valet: ‘Si propositio est vera et non potest non esse vera, ergo est necessaria’ nisi fuerit in subiecto causa essentialis et inseparabilis quare praedicatum removetur a subiecto vel ei inest. Sed cum dicitur ‘Nullum sanum est aegrum’, in supposito non est causa essentialis et inseparabilis removens praedicatum a subiecto sed causa accidentalis et separabilis, et ideo propositio non est necessaria, immo falsa erit addito modo necessitatis. Haec autem ‘Nullus homo est asinus’, cum sit vera et non possit non esse vera, est necessaria quia in subiecto est causa essentialis et inseparabilis remotionis praedicati a subiecto, et ideo si addatur modus necessitatis, adhuc erit vera. Ex his patet quo modo haec propositio non est necessaria: ‘Nullum sanum est aegrum’ nec alia consimilis, et quo modo intelligenda est regula quae est: ‘Si propositio est vera et non potest non esse vera, ergo est necessaria’.” 58 Kilwardby ad A16 dub.8 (27va).

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the affirmative necessity-proposition “It’s necessary that every b is a” doesn’t imply that there exists a b, and hence doesn’t imply that every b is a. Contingency-propositions Types of contingency Why is it that Aristotle doesn’t define the necessary, but does define the contingent? The reason for this disparity, Kilwardby says, is that “contingent”, unlike “necessary”, is ambiguous. The contingent includes what he calls the necessary contingent as well as the non-necessary contingent (i.e. two-way possibility).59 Together, these make up what he calls generic contingency (i.e. one-way possibility).60 Generic contingency: one-way possibility A proposition expressing a one-way possibility is contradictory to a necessity-proposition: A negative necessity-proposition is convertible with a genuinely negative contingency-proposition: if it’s necessary not to be then it’s not contingent to be, and vice versa. Equally, a negative contingency-proposition (in the sense of the possible) is convertible with a genuinely negative necessity-proposition: if it’s contingent not to be then it’s not necessary to be, and vice versa. And on this account we can syllogize in the second Figure from such negatives.61

The contradictory of a per se necessity-proposition is a per se possibilityproposition; and since a per se necessity-proposition contains an outer modality of necessity, a per se possibility-proposition contains an outer modality of possibility. The core of a negative necessity-proposition denies that its predicate can belong to what can fall under the subject; therefore the core of its 59

Kilwardby ad A13 dub.1 (19vb). Kilwardby ad A3 Part 2 dub.1 (7vb–8ra). See Henrik Lagerlund, Modal Syllogistics in the Middle Ages (Leiden: Brill, 2000), 23. 61 Kilwardby ad A17 dub.3 (28vb): “… negativa de necessario convertitur cum vere negativa de contingenti, ut si necesse est non esse, non contingit esse, et e converso. Et negativa de contingenti pro possibili convertitur cum vere negativa de necessario, ut si contingit non esse, non necesse est esse, et e converso. Et propter hoc ex {ex ABrCm1E1E2F1F2KlO2O3P1: in V : quia Ed} talibus negativis syllogizari potest {syllogizari potest AE 1F1F2O2O3P1: syllogizare potest BrCm1E2KlV: potest fieri Ed} in secunda figura.” 60

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contradictory—an affirmative possibility-proposition—affirms that its predicate can belong to what can fall under the subject. Kilwardby takes “Some musician is possibly a man” and “Something white is possibly black” to be true one-way possibility-propositions.62 The first contains just one per se term; the second contains none. If we wanted an example of a true possibility-proposition containing two per se terms, I suppose “Some animal is possibly a man” might suffice. All three propositions are capable of being read as contradictory to necessitypropositions. It’s possible that something which can be a musician can be a man, it’s possible that something which can be white can be black, and it’s possible that nothing which can be an animal can be a man. These possibilities would be actualized if (respectively) a potentially musical man, a potentially white black thing, and only non-human animals, existed. The contradictory per se necessities—“Of necessity no musician is a man”, “Of necessity nothing white is black” and “Of necessity no animal is a man”—are all false. The falsity of the first two follows from the fact that true per se necessities must have per se terms. The third is false because it isn’t necessary that nothing which can be an animal can be a man. The core of an affirmative necessity-proposition affirms that its predicate must belong to what actually falls under the subject; therefore the core of its contradictory—a negative possibility-proposition—denies that its predicate must belong to what actually falls under the subject. Kilwardby takes “It’s possible that nothing white is a man” to express a true one-way possibility.63 This proposition is capable of being read as contradictory to a per se necessity. It’s possible that no actual white thing must be a man. The possibility would be actualized if there were no white men. The contradictory per se necessity—“Something white is of necessity a man”—is false (and would be false even if there were white men), since its subject is per accidens. The necessary contingent The necessary contingent is simply the necessary. This sense of “contingent” allows the inferences “If it’s necessary for something to be, it’s contingent for it to be” and “If it’s necessary for something not to be, it’s contingent for it not to be”. However, it’s easy to construct

62 63

Kilwardby ad A3 Part 2 dub.5 (8rb); Kilwardby ad A16 dub.5 (27ra). Kilwardby ad A3 Part 2 dub.12 (9ra).

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fallacious hypothetical syllogisms by mixing up the contingent as necessary (Q L) with the contingent as non-necessary (Q Q ), as shown in Fig. 1.1.64 If Lp then Q Lp If Q Q p then ~Lp If Lp then ~Lp Fig. 1.1. Fallacious argument with two senses of contingency

The non-necessary contingent Fallacies also arise in connection with the non-necessary contingent, for instance “If it’s impossible, it’s not necessary; if it’s not necessary it’s contingent; therefore if it’s impossible it’s contingent”. The nonnecessary contingent is not just anything that is not necessary; it’s that which is not impossible and not necessary.65 Kilwardby holds that “contingency is a certain potentiality for being”, and consequently “the necessary has less of the nature of the contingent than the non-necessary”. Thus: “… the contingent in general is said of the necessary and the non-necessary contingent, not just equivocally, nor just univocally, but analogically”.66 It is the primary kind of contingency (namely the non-necessary contingent) that Aristotle is talking about in his modal syllogistic.67 The definition Aristotle gives is “that which is not necessary, but which when supposed actual doesn’t imply any impossibility”.68 Kilwardby asks whether it’s appropriate to define one opposite in terms of another, as this definition appears to do. He points out that the question rests on a metaphysical understanding of definition as articulating the genus and differentia of the term to be defined; but the definition of contingency is not intended in that way. … a definition given by genus and differentiae indicating the thing’s essence doesn’t admit any opposition. It’s true however that some sort of a definition made through knowledge of the thing may well be given by the privation of opposites. For what is intermediate between two things is sometimes known by their privation. Now the contingent is between the Kilwardby ad A3 Part 2 dub.2 (8ra). Kilwardby ad A3 Part 2 dub.3 (8ra). 66 Kilwardby ad A13 dub.4 (20ra–b): “… contingens commune dicitur de contingenti necessario et non necessario non pure aequivoce nec univoce, sed analogice.” 67 Kilwardby ad A13 dub.2 (20ra). 68 A13, 32a18–20. 64 65

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necessary and the impossible in this way. The necessary stands only to being, and the impossible to non-being; but the contingent as determined here is in between the two. It can be (and in this way it differs from the impossible), and it can not-be (and in this way it differs from the necessary). And so, since it is intermediate by privation between the necessary and the impossible, it ought to be defined by their privation.69

Here is one way of interpreting Kilwardby’s two-way contingencypropositions. Universal two-way contingency-propositions have to be incompatible with the corresponding affirmative and negative necessity-propositions. These incompatibilities can be secured if we interpret a universal contingency-proposition as stating a pair of propositions, both of which contain an outer modality of possibility. These propositions state that it’s possible for the predicate to belong to all of the subject and it’s possible for it to belong to none of the subject. A particular contingency-proposition would then state that it’s possible for the predicate to belong to some of the subject, and it’s possible for the predicate not to belong to some of the subject, and it’s possible for the subject not to belong to some of the predicate. (The addition of this last clause gives the particular proposition the semantic symmetry that it needs in order to be convertible.) In these truth-conditions, there is no requirement that the subject or predicate be of a specific type—per se or per accidens.70 Amongst the propositions that Kilwardby takes to express true twoway contingencies, we find “Every white man is contingently an animal”.71 Even though this proposition might be thought to express a de dicto necessity, and to be analytically true, it does in fact satisfy the truthconditions we have just outlined. On the one hand, it’s possible that all white men are animals; and this possibility would be actualized if some 69 Kilwardby ad A13 dub.3 (20ra): “… definitio data per genus et differentias quae indicant essentiam rei non admittit quamcumque oppositionem. Verum tamen definitio qualiscumque facta propter rei cognitionem bene potest dari per privationem oppositorum. Illud enim quod medium est inter duo aliquando per privationem illorum cognoscitur. Sic autem est contingens inter necessarium et impossibile. Necessarium enim solum se habet ad esse, impossibile vero ad non esse. Contingens autem hic determinatum medium est inter haec quia potest esse et per hoc differt ab impossibili. Potest etiam non esse et per hoc differt a necessario. Sic ergo cum sit medium per privationem dictum inter necessarium et impossibile oportet ipsum definiri per privationem utriusque.” 70 In Thom, Medieval Modal Systems, no general interpretation is offered of Kilwardby’s non-necessary contingency-propositions, though there are interpretations of natural contingencies (Definitions 6.15 and 6.16, p. 104) and of indeterminate contingencies (Definitions 6.17 and 6.18, p. 110). 71 Kilwardby ad A14 dub.5 (22rb).

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actual man were white. On the other hand, it’s possible that no white men are animals; and this possibility would be actualized if there were no white men. Indeterminate vs. natural contingencies Kilwardby, following Aristotle, distinguishes two types of two-way contingency-proposition, the indeterminate and the natural: … contingency in the sense of the non-necessary divides into the natural and the indeterminate—though Aristotle doesn’t make this division here, but speaks of them in their common nature, comprehending them by the non-necessary. However, the non-necessary contingent which is genuinely indeterminate, or genuinely two-way, can always be converted to the same sense of contingency—as is clear from the immediately preceding example [sc. “It’s contingent for every, or some, man to be white, so it’s contingent for something white to be a man”]. And I call a “genuinely indeterminate contingency” that which is related utterly equally to being and non-being.72

What does it mean to be related equally to being and non-being? Some interpreters take this in a statistical sense.73 In that case, if whiteness for men were balanced equally between being and non-being, then exactly fifty percent of men would be white; but I don’t find that an obvious reading of Kilwardby’s words. Another reading is suggested by what he has to say about natural contingencies. These are typified by the natural tendency of old people to go grey: Next there is a further puzzle about the statement74 that when a man is, then either necessarily or for the most part he goes grey. For there is a doubt how this may be true, since men do not go grey for the most part, except in old age. And it is to be said that the act of going grey can be said of the movement towards going grey, or of the completed movement. If it is said of the movement towards greyness then a man goes grey always and of necessity when he is. For greyness comes from the incorporation of phlegm into the upper part of the head—which incorporation is caused 72 Kilwardby ad A3 Part 2 dub.5 (8rb): “… contingens non necessarium dividitur per natum et infinitum, licet Aristoteles non dividat ea hic sed loquitur de eis in ratione communi comprehendendo ea per non necessarium. Contingens autem non necessarium quod est infinitum vere vel ad utrumlibet vere converti potest semper in eadem acceptione contingentis, et patet per exemplum proximo dictum. Et dico contingens infinitum vere quod aequaliter penitus se habet ad esse et non esse.” See Lagerlund p. 43. 73 Lagerlund, Modal Syllogistics in the Middle Ages, 45. 74 A13, 32b4–10.

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by the diminution of natural heat, and this incorporation and diminution of heat exists always and uninterruptedly. But if it is said as the completed movement, then man goes grey frequently. For greyness comes to many if they last until old age.75

Evidently, a natural contingency such as the greying of the old is a twoway contingency, in that it’s possible that all the old will go grey and it’s possible that none will. But this is unlike other two-way contingencies, in that it is underpinned by a natural law, namely a natural movement towards greyness with age. So, it’s not merely possible that aged persons will go grey, it’s natural that they do. Here, we have an outer modality, not of necessity or possibility, but of naturalness. An affirmative natural contingency with predicate “a” and subject “b” would then be true iff it’s natural that whatever can be b can be a, and it’s possible that no b is a.76 A negative natural contingency would state that it’s natural that what is b need not be a and it’s possible that what can be b can be a. Given that naturalness entails possibility,77 a natural contingency-proposition implies the corresponding two-way contingency-proposition (as it should). All of this suggests another interpretation of Kilwardby’s “genuinely indeterminate” contingencies. These are just those two-way contingencies that are not natural contingencies; and they stand equally between being and non-being, simply in the sense that, not being natural necessities, they are not weighted one way or the other by an underpinning natural law. A genuinely indeterminate contingency with predicate “a” 75 Kilwardby ad A13 dub.8 (20va–b): “Consequenter dubitabitur ulterius de hoc quod dicit quod cum homo est aut de necessitate aut frequenter canescit. Dubium est enim quo modo hoc sit verum cum homo non {non ABrCm1E1E2F1F2KlO2O3P1V: om. Ed} canescat ut in pluribus nisi in senectute. Et dicendum quod ille actus canescere potest dicere motum in canitiem vel tantum motum completum. Si dicat motum in canitiem, sic semper et de necessitate canescit homo cum sit. Pervenit enim canities ex incorporatione flegmatis in superiore parte capitis, cuius incorporationis causa est diminutio caloris naturalis, et ista incorporatio et caloris diminutio semper sit et continuo. Si autem dicit tantum motum completum, sic ut frequenter canescit homo, in pluribus enim si maneant ad aetatem pervenit canities.” 76 Thom, Medieval Modal Systems (Definitions 6.15 and 6.16, p. 104) interprets a natural contingency as a single necessity to the effect that what falls under the subject may or may not fall under the predicate. In doing so, it fails to capture the bias of natural contingencies to one possibility rather than the other. 77 Naturalness doesn’t entail actuality, and so this is a deontic modality (like “It’s obligatory that”) rather than an alethic one (like “It’s necessary that”). Such deontic modalities do, however, entail possibility. See A.N. Prior, Formal Logic (Oxford: Clarendon Press 1962), 224, 226.

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and subject “b” is true iff it’s possible but not natural that all that can be b can be a, and it’s possible that no b is a.78 The core of contingency-propositions We have interpreted contingency-propositions as being composed of a pair of propositions having the same general structure as necessitypropositions—a core proposition governed by an outer modality. The outer modality is one of possibility or naturalness. But what of the core? Kilwardby notes an ambiguity here: “In contingency-propositions tobe-said-of-all has two senses—‘Everything that is contingently b etc.’, and ‘Everything that is b etc.’”.79 I take it that in the first of these senses, the predication is made about all that may or may not fall under the subject—that the contingency which is applied to the subject is two-way contingency. As we have seen, every contingency-proposition, in so far as it implies an affirmative possibility-proposition, implies something about what can fall under the subject; though, in so far as it implies a negative possibility-proposition, it concerns only what is actually under the subject. Kilwardby comments on the sense in which a contingency-proposition may be about what contingently falls under the subject: “And it is to be said that in contingency-propositions the mode of contingency falling on the extremes ampliates their supposition”.80 The doctrine of ampliation is thus explicitly invoked in his treatment of contingency-propositions, contrary to Lagerlund’s statement that “It is important to mention that there is no indication in the works of Kilwardby and Albert the Great that the theory of ampliation was applied to modal propositions used in the modal syllogistic”.81 Kilwardby takes “It’s contingent for everything white to be an animal” to be false if its subject is read as ampliated to the contingent; and he adds that it would be false “even if only animals were white”.82 78 Thom, Medieval Modal Systems (Definitions 6.17 and 6.18, p. 110) interprets an indeterminate contingency as two complementary possibilities. In doing so, it fails to capture the fact that indeterminate contingencies exclude natural contingencies and together with them exhaust the field of two-way contingencies. 79 Kilwardby ad A16 dub.1 (26vb): “Dici de omni in contingentibus duas habet intentiones, scilicet omne quod contingit esse b etc. et omne quod est b etc.” 80 Kilwardby ad A13 dub.11 (21ra–b): “Et dicendum quod in propositione de contingenti modus contingentiae cadens super extrema amplificat eorum suppositionem.” 81 Lagerlund, Modal Syllogistics in the Middle Ages, 47 n. 90. 82 Kilwardby ad A14 dub.2 (22ra).

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This proposition could, however, be true if it were read as an unampliated contingency. On that reading it would state that it’s possible that everything which can be white can be an animal, and it’s possible that nothing which is actually white has to be an animal. The first of these possibilities would be actualized if only animals had the potential to be white; and the second would be actualized if no animals were white. So contingency-propositions that ampliate their subjects to the contingent are semantically distinct from the contingency-propositions we have considered thus far. The meaning attributed to them by Kilwardby appears to be this: what may or may not fall under the subject may or may not fall under the predicate.83 Kilwardby asks whether these are the only senses that contingencypropositions have in the modal syllogistic: Lastly, someone will perhaps wonder why he doesn’t lay down a third sense of contingency-propositions, namely “Everything that is necessarily b is contingently a”. … And it is to be said that there is that third sense, just as has been argued from a comparison of the subject with what it necessarily inheres in; but he omits it because it is not separate from the second sense; on the contrary, it is understood in it. For, the other sense is sufficiently understood by this sense “Every that is b etc.”, because (as was said earlier) inherence is not differentiated between inhering without qualification and inhering of necessity.84

Negations of contingency-propositions Aristotle states at two places in the Prior Analytics that propositions such as “It’s contingent that no b is a” or “It’s contingent that some b is not a” should be regarded as affirmative.85 He compares these with propositions like “It is not-good” which, though they have a negated 83 This interpretation is put forward in Thom, Medieval Modal Systems, Definitions 6.13 and 6.14, 102. 84 Kilwardby ad A13 dub.12 (21rb): “Ultimo forte dubitabit aliquis quare non ponat tertiam intentionem propositionis de contingenti, scilicet hanc: Omne quod necesse est esse b contingit esse a. … Et dicendum quod ibi est illa tertia intentio sicut oppositum est ex comparatione subiecti ad ea quibus necessario inest, sed omittit eam quia non separatur a secunda intentione, immo per eam intelligitur. Quia enim {enim ACm1E1E2F1F2KlO2O3P1V: ibi Ed} inesse simpliciter et ex necessitate inesse non diversificat inhaerentiam, ut praedictum est, ideo per hanc intentionem: Omne quod est b etc. intelligitur sufficienter reliqua.” The reference is to Kilwardby ad A13, dub.10. 85 A3, 25b18; A13, 32a40.

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predicate, are actually affirmations. And he attributes the fact that contingency-propositions convert like affirmative assertorics to their being affirmations. Distinct from these are the negations of contingency-propositions; and Kilwardby says a little about them, stating truth-conditions for them: “The negative ‘It’s not contingent for no c to be b’ has two causes of truth, namely either because it’s necessary for some c to be b, or because it’s necessary for some c not to be b”.86 Given a nontrivial proviso, this result is consistent with the truth-conditions for twoway contingency-propositions stated earlier. A universal contingencyproposition with predicate “b” and subject “a” is true iff (1) it’s possible that whatever can be b can be a, and (2) it’s possible that nothing which is actually b must be a. The proposition is therefore false iff either (3) it’s necessary that something which can be b can’t be a, and (4) it’s necessary that something which is actually b must be a. Now, clause (3) states the truth-condition for a particular negative necessity-proposition; and (4) states the truth-condition for a particular affirmative necessity-proposition. The non-trivial proviso is that “a” and “b” are per se terms. If this proviso is not met, then the necessitypropositions in Kilwardby’s rule (stated above) may be per accidens necessities. He doesn’t give details of the logic of negated contingency-propositions. However, it’s clear that there are valid syllogisms with such negations as premises, and that such syllogisms can be proved valid in either of the ways displayed in Fig. 1.2. ⎫

~q p Qaba

C →

Qaba p q

Labi p ⎪ ⎪ ⎪ ⎪ ⎪ q ⎬ ⎪ Labo p ⎪ ⎪ ⎪ ⎪ ⎭ q

R1 →

Qaba p q

Fig. 1.2. Proof-patterns for syllogisms with negated universal contingency-premises

The first proof proceeds by the familiar process of Indirect Reduction. The second proceeds in accordance with a rule, R1, which depends on 86 Kilwardby ad A17 Note (29ra): “Haec enim negativa ‘Non contingit nullum c esse b’ duplici de causa vera est, scilicet vel quia necesse est aliquod c esse b vel quia necesse est aliquod c non esse b.”

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Kilwardby’s truth-conditions for negated universal contingency-propositions. As such, it might seem to be vulnerable to the proviso that those truth-conditions have to be stated in terms of per se necessities. However, this is not really a problem, since per accidens necessities are weaker than per se ones. Consequently, if “q” is implied by a pair of per accidens necessities (along with the second premise “p”), then a fortiori it will be implied by a pair of per se necessities. Similar remarks apply the proof of syllogisms with negated particular contingency-premises, as shown in Fig. 1.3. ⎫

Laba p ⎪ ⎪ ⎪ ⎪ ⎪ q ⎬ ⎪ Labe p ⎪ ⎪ ⎪ ⎪ ⎭ q

R2

→

Qabi p q

Fig. 1.3. Proof-pattern for syllogisms with negated particular contingency-premises

R2, which is used here, assumes that the negative “It’s not contingent for some b to be a” is true iff either that it’s necessary for all a to be b, or it’s necessary for no c to be b. And again, this rule is correct, whether the necessity-premises in question are per se or per accidens. Unrestricted assertorics An assertoric sentence is sometimes used to state that something is the case as-of-now; and sometimes to state something that holds without any restriction to present, past or future time. The latter usage was identified by Aristotle in the Prior Analytics via the Greek word hapl¯os which the Latins translated as simpliciter [without qualification, unrestricted, absolute]. As examples of assertorics that are not unrestricted because they are either accidental or as-of-now predications, Kilwardby cites the propositions “Socrates is a man”, “A white is a man”, and “This man is a man”. By contrast, he mentions “Some man is a man” as an unrestricted assertoric.87 Unrestricted assertorics play an important role in his account of the syllogism, and he uses them in more than one sense. In one sense, 87

Kilwardby ad A9, Note 1 (17ra).

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unrestricted assertorics are apodeictic propositions: “… the subject is essentially under the predicate. So it is a necessary proposition. Hence it is the same in reality as a necessity-proposition, even if it is not the same in mode”.88 He explains that by an essential predication, he understands the first and second senses of per se given in the Posterior Analytics, or propositions that reduce proximately to them.89 When the assertoric Minor premise in a necessity-syllogism is apodeictic, it is legitimate to substitute inferiors for the Major’s subject. If it’s necessary that all b are a, and (as a matter of necessity) all c are b, then it follows that it’s necessary that all c are a. Thus, by reading the Minor in such a syllogism as an apodeictic unrestricted assertoric, Kilwardby is able to substitute into a necessity-context, while still treating the necessityMajor as containing an outer mode of necessity. However, unrestricted assertoric propositions are not confined to apodeictic propositions, since: … those things are said to inhere unrestrictedly, that always or frequently inhere, having a cause of inherence in the subject, or of the subject’s inherence in the predicate. Hence natural contingencies are said to be unrestricted assertorics. And such contingency-propositions, when they are put as assertorics, are unrestricted assertorics.90

He goes on: … it is to be said that an unrestricted assertoric (as this is taken here) is not always necessary, nor does it have the force of a necessity-proposition. For an unrestricted assertoric requires merely that the nature of the terms cohere or are disjoint; but a necessary proposition requires necessity in addition to this coherence or disjunction. This is clear because a natural contingency, when supposed to be [the case], is not necessary, but it is an unrestricted assertoric. And so it doesn’t follow that if a necessityproposition can be concluded then so can an unrestricted assertoric— because it doesn’t have the force of a necessity-proposition.91 88 Kilwardby ad A9 dub.2 (16va): “… subiectum essentialiter est sub praedicato, et ita est propositio necessaria. Quare idem est secundum rem ei quae est de necessario, etsi non sit idem secundum modum.” The semantic equivalence between unrestricted assertorics and necessity-propositions is recognized in Thom, Medieval Modal Systems, 96, 99. 89 Kilwardby ad A15 dub.9 (25rb). Aristotle, Posterior Analytics translated with notes by Jonathan Barnes (Oxford: Clarendon Press, 1975) A4, 73a34–b3. 90 Kilwardby ad A15 dub.9 (25rb): “… illa quae semper insunt vel frequenter insunt, habentia tamen causam inhaerentiae in subiecto vel praedicati in subiecto, dicuntur inesse simpliciter. Unde contingentia nata simpliciter inesse dicuntur, et propositiones de tali contingenti si ponantur inesse, sunt de inesse simpliciter …” 91 Kilwardby ad A16, dub.5 (27rb): “… propositio de inesse simpliciter ut hic accipitur

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Kilwardby notes that, while the assertoric proposition corresponding to a natural contingency is unrestricted, that corresponding to an indeterminate contingency is as-of-now.92 He takes all affirmative unrestricted assertorics (the natural as well as the apodeictic) to imply that “the subject stands for everything that can be under it, for the predicate possibly inheres in all of these”.93 This appears to be the basic logical property of affirmative unrestricted assertorics.94 It belongs to both types of unrestricted assertoric, but with different outer modalities: if it’s per se necessary that what is b is a then it’s necessary that what can be b can be a, but if it’s naturally contingent that what is b is a then it’s natural that what can be b can be a. Negative unrestricted assertorics behave as follows. If they are apodeictic, they state that it’s necessary that no possible b is possibly a. Kilwardby says that such propositions state that “the predicate is removed from those things that are actually and [from those that] can be under the subject”.95 Thus, they have ampliated subjects. They are equivalent to necessity-propositions. There are also non-apodeictic unrestricted negative assertorics. “Nothing white is black” is an example.96 Kilwardby says that this proposition is unrestricted but does not have the force of a necessity-proposition.97 One point of interest is that the contradictory of an unrestricted assertoric is not itself an unrestricted assertoric. An unrestricted assertoric is governed by an outer mode of necessity or naturalness; the non semper est necessaria neque virtutem propositionis de necessario habens. Propositio enim de inesse simpliciter non plus exigit nisi quod rationes terminorum cohaereant vel discohaereant. Propositio autem necessaria exigit necessariam et talem terminorum cohaerentiam vel discohaerentiam. Et hoc patet cum contingens natum positum inesse non sit necesarium, est tamen {est tamen ABrCm1E1E2P1V: et tamen F 1F2O2O3: est Kl: cum tamen est Ed} de inesse simpliciter. Et ideo non sequitur quod licet {licet ABrCm1E1E2F1F2KlO2O3P1V: si Ed} propositio de necessario possit in conclusionem de inesse quod propositio de inesse simpliciter, quia non habet virtutem propositionis de necessario.” 92 Kilwardby ad A15 dub.6 (24va). Where Ed has “contingens autem natum erit contingens de inesse sequitur”, ABrE 1E2F1F2KlO2O3P1V have “contingens autem natum erit de inesse simpliciter”. 93 Kilwardby ad A15 dub.5 (24va): “… subiectum stat pro omnibus quae sub eo possunt esse, praedicatum enim omnibus illis possibiliter inest.” 94 See Thom, Medieval Modal Systems, 114. 95 Kilwardby ad A16 dub.5 (27rb): “… praedicatum removeri ab his quae actu et potentia sunt sub subiecto …” 96 Kilwardby ad A9 dub.7 note (17ra). 97 Kilwardby ad A16 dub.5 (27rb).

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contradictory of an unrestricted assertoric is therefore not governed by those modes but by a mode of possibility or of what is not natural.

Summary Kilwardby distinguishes assertorics from necessity- and contingencypropositions. Assertorics may be either as-of-now or unrestricted. Unrestricted assertorics may be either necessary or natural. A necessary unrestricted assertoric states that a categorical relationship (a relationship of inclusion, non-inclusion, exclusion or non-exclusion) holds necessarily between its terms; a natural unrestricted assertoric states that a categorical relationship holds naturally. “It is naturally the case that” is a deontic modality. The contradictory of an unrestricted assertoric is sui generis; it is not itself an unrestricted assertoric. He distinguishes per accidens from per se necessity-propositions. An affirmative per se necessity-proposition states that a categorical relationship holds necessarily between its terms, where these terms are taken as per se. A negative per se necessity-proposition states that a categorical relationship holds necessarily between its terms, where these terms are ampliated to the possible. Per se necessity-propositions may be considered indifferently as de dicto or as de re statements. The contradictory of a necessity-proposition is a (one-way) possibility-proposition. He distinguishes three senses of “contingency”: the generic contingent (i.e. one-way possibility), the necessary contingent and the nonnecessary contingent (i.e. two-way possibility). He distinguishes ampliated from unampliated contingency-propositions. A non-necessary contingency-proposition states a pair of one-way possibilities; one of these states an affirmative one-way possibility connecting the terms, while the other states a negative one-way possibility. The affirmative possibilitystatement is the contradictory of a negative necessity; and the negative possibility-statement is the contradictory of an affirmative necessity. The non-necessary contingency-proposition as a whole is always considered to be affirmative. Among non-necessary contingency-propositions, Kilwardby distinguishes indeterminate from natural contingencies; the difference is that in a natural contingency-proposition one of the paired possibility-statements states what is not just possible but natural.

chapter two SYLLOGISM

Kilwardby expounds three different readings of Aristotle’s definition of the syllogism. He doesn’t attribute these readings to anyone in particular; however the first corresponds closely to Boethius’s exposition, and the second is consistent with Aristotle’s own further remarks on the nature of the syllogism in the Prior Analytics. The third reading is in fact that advanced in the late twelfth-century Dialectica Monacensis. Kilwardby rejects this reading, but he doesn’t express a preference in relation to either of the first two, instead simply distinguishing two senses of “syllogism” corresponding to these two readings. However, his own discussions of Induction and Enthymeme and of the types of consequence demonstrate that his understanding of the nature of the syllogism is deeper and more subtle than what lies on the surface of either of the first two expositions of the Aristotelian definition. He goes far beyond Aristotle’s own discussion. For instance, he discusses the syllogism’s Final Cause, which he defines in terms of necessary consequence and the “showing” and “making-known” of the conclusion. He gives a subtle account of the perfection of perfect syllogisms, distinguishing perfection of well-being (which is possessed only by the direct moods of the first Figure) from perfection of being (which is possessed by all syllogisms). He allows that there are degrees of perfection, and he uses the Aristotelian definition of being-said-of-all or of-none to demarcate the most perfect syllogisms. The wider class of perfect syllogisms is identified by means of a rule of “assimilation”, that specifies which features of the premises are replicated in the conclusion of a perfect syllogism.

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chapter two Aristotle’s definition A syllogism is discourse in which, certain things being set out [positis], something else comes about [contingit] of necessity from their being so.1

Kilwardby asks why Aristotle gives Discourse [oratio] as the Genus of Syllogism. Surely this is not the proper Genus, but rather Argumentation or Reasoning.2 Kilwardby states that one’s response to the objection depends on one’s exposition of Aristotle’s definition, and he distinguishes three different readings of that definition as a definition by Genus and Differentia. First exposition The first reading assumes that Aristotle’s definition is meant to exclude “all opposite Species of Reasoning (namely Induction, Example and Enthymeme) and the Useless Conjugations and the Faults dealt with in Book 2 (including petitio principii, and non-causa ut causa)”: Thus the particular “certain things” excludes Enthymeme. By “set out” is understood arrangement in Mood and Figure, and this excludes the useless premise-pairs and Induction. “Of necessity” excludes Example, which possesses mere probability since it is a rhetorical argument (as Aristotle says in Book 1 of the Posterior Analytics).3 By “something other comes about” petitio principii is excluded—not as a sophistical ground but as a fault in Syllogism simpliciter that is dealt with in Book 2.4 By “from their being so” non causa ut causa is excluded.5 A1, 25b18–20. Kilwardby 4vb. Kilwardby ad A1 Part 4 dub.1 (4vb). Actually, Lambert of Auxerre, Logica (Summa Lamberti). Prima edizione a cura di Franco Alessio (Firenze: La nuova Italia editrice, 1971) Tract V, f. 91v, p. 102 gives Argumentation as the Genus, as does Roger Bacon, Summulae Dialectices Ed. R. Steele (Oxford University Press, 1940), 289.33–34. Roger (290.11–18), however, also quotes the Prior Analytics definition and explains Discourse as it occurs there as meaning “a sequence of expressions tending to a single End, namely to inducing belief or something else, according to how Cicero in the Rhetoric takes it, where Rhetoric is said precisely to be a series of Discourses tending to a single End, namely Persuasion. So Discourse is here the same as Argumentation, as in Oris Ratio [‘an oral reason’], and not as it taken in the Peri Hermeneias, nor according as it is a prior habit”. 3 Posterior Analytics A1, 71a9–11. 4 B16. 5 Kilwardby ad A1 Part 4 dub.1 (4vb): “Sic per particulam ‘quibusdam’ excluditur enthymema. Per hanc particulam ‘positis’ intelligendum est dispositionem in modo et figura, per illam excluduntur inutiles coniugationes et inductio. Per hoc quod dicitur ‘ex necessitate’ excludit exemplum quod solam habet probabilitatem cum sit argumentum rhetoricum, ut dicit Aristotoles in principio Posteriorum. Per hoc quod dicit ‘aliud 1 2

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This exposition is similar to that given by Boethius in his De syllogismo categorico. Of particular note are Boethius’s exclusion of single-premised arguments by the phrase “certain things”,6 and his exclusion of Induction by the phrases “set out” and “of necessity”.7 Of course, it is Aristotle’s (not Boethius’s) definition that is being discussed, and consequently the premises are not required to be conceded [concessis] (as in Boethius’s definition) but only to be set out [positis].8 But, even though the exposition is not identical with that of Boethius, I shall refer to it as Boethian, because of the similarities. According to this exposition, (1) Discourse is the Genus. (2) “Certain thing” indicates the Differentia that there is more than one premise (thus excluding Enthymeme as not a Syllogism). (3) “Set out” indicates a second Differentia—namely that the premises are arranged in a valid Mood and Figure (thus excluding the useless Conjugations and Induction). (4) A third Differentia is that the premises “necessitate” the conclusion (thus excluding Example). (5) A fourth Differentia is that the conclusion is “other than” the premises (thus excluding petitio principii). (6) A fifth is that the Conclusion results from the premises being so (thus excluding non-causa ut causa). Thus there are six types of Discourse: (1) complex discourse in Figure and Mood necessitating a conclusion other than the premises and dependent on the premises (i.e. syllogism), (2) complex discourse in Figure and Mood necessitating a conclusion other than the premises and not dependent on the premises (i.e. non-causa ut causa), (3) complex discourse in Figure and Mood necessitating a conclusion not other than the premises (i.e. petitio), (4) complex discourse in Figure and Mood not necessitating a conclusion (i.e. useless premise-pairs), (5) complex discourse not in Figure and Mood (i.e. informal argumentation), (6) simple discourse. This division of discourse into six species is shown in Fig. 2.1.

accidit’ excluditur petitio eius quod est in principio, non secundum quod est locus sophisticus sed secundum quod est peccatum circa syllogismum simpliciter, et determinatur in secundo sic. Per hoc quod dicit ‘eo quod haec sunt’ excluditur non causa ut causa {Per hoc quod dicit eo…causa Cm1E1E2F1F2KlO2P1: om. EdACrV }.” 6 Boethius, De syllogismo categorico in Opera Omnia II. Ed. J.-P. Migne. Patrologia Latina 64 (Paris: 1860), 821B–C. 7 Boethius, De syllogismo categorico, 822A–B. 8 Kilwardby ad B2 Part 2 dub.3 (57va) argues that the definition of syllogism should not include this Boethian addition.

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Fig. 2.1. First division of Discourse

In explaining how this scheme bears on the question whether syllogism ought to be defined as a kind of argumentation, Kilwardby appeals to Boethius’s definition of argumentation and to Priscian’s definition of discourse. Argumentation is the explication of an argument in discourse, and an argument is a belief-producing reason about a doubtful matter.9 “Therefore, argument is relative to the soul in which it produces belief; and so is syllogism, in so far as it is an argument.” On the other hand, discourse is a congruous ordering of expressions.10 “Therefore, syllogism, in so far as it is discourse, is relative to its integral parts.” This relativity he regards as “more essential” than the relativity to a believing soul.11 In using the expression “more essential” he is not ruling out an essential relationship between the syllogism and the pro9 The definitions of argument and argumentation come from Boethius, De topicis differentiis, 1174C. 10 Priscian, Prisciani Grammatici Caesariensis Institutionum grammaticarum libri XVIII ed. Martin Hertz Grammatici Latini ed. H. Keil II, III (Leipzig 1855). 11 Kilwardby ad A1 Part 4 dub.1 (4vb).

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duction of beliefs concerning doubtful matters. For, when he used this expression to describe the centrality of quality in determining a syllogistic proposition, this did not rule out an essential role for quantity; it merely drew attention to the fact that some propositions (not syllogistic ones) possess quality without quantity. Similarly here, in the case of belief-producing syllogisms, there is an essential relativity both to the conclusion and to the mind in which belief is produced. But there are some syllogisms (not belief-producing ones) in which the first relativity exists in the absence of the second. Second exposition According to a second exposition, Aristotle’s definition does not exclude Induction, Enthymeme and Example (these being reducible to syllogism), but only excludes the useless premise-pairs, petitio and non-causa.12 Kilwardby does not explain how the details in the definition support this interpretation; but we can see that on this reading there are four types of discourse, as shown in Fig. 2.2.

Fig. 2.2. Second division of Discourse

12

Kilwardby ad A1 Part 4 dub.1 (4vb–5ra).

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Kilwardby explains how the question is to be answered on this reading: In this sense, it is to be said to the foregoing question that if Argumentation or Reasoning had been put in the definition of Syllogism as its proper Genus, then the essential completing Differentiae, by which the other species of Argument are excluded from Syllogism, would have had to be added to that Genus, and by these the other species of argumentation would have been excluded from the syllogism.13

The point is that if Argumentation were the genus of Syllogism then “the other species of argumentation”—namely Enthymeme, Induction and Example—would have to be excluded from the definition of Syllogism; but they are not excluded from Syllogism on this reading, so Argumentation is not the genus of Syllogism. Third exposition Kilwardby mentions a third exposition, according to which the definition excludes all Sophistical Grounds, along with the other species of Argumentation.14 In fact this reading was adopted by the Dialectica Monacensis (a work attributed by De Rijk to an English author writing “not later than the last decades of the twelfth century”):15 The definition given can therefore make clear from what has been said that all other species of argumentation are excluded (namely Induction, Example, Enthymeme), and it also excludes the sophistical syllogism no matter what its cause (and this includes 13 fallacies).16

On this reading, “certain things” excludes Example and Enthymeme along with all fallacies in dictione; “set out” excludes all argumentation not expressed in Figure and Mood (including Induction); “necessar13 Kilwardby ad A1 Part 4 dub.1 (5ra): “Secundum istam intentionem dicendum est ad praedictam questionem quod si in definitione syllogismi poneretur argumentatio vel ratiocinatio, quod est proprium genus eius, oporteret tali generi addi differentias essentiales completivas syllogismi per quas excluderentur a syllogismo aliae species argumentationis.” 14 Kilwardby ad A1 Part 4 dub.1 (5ra). 15 L.M. de Rijk, Logica Modernorum Volume 2 Part 1 (Assen: Van Gorcum 1967), 414. 16 Dialectica Monacensis, in L.M. de Rijk, Logica Modernorum: a contribution to the history of early terminist logic Vol. 2 Part 2 (Assen: Van Gorcum, 1967) 453–638, 490:12–15: “Potest igitur manifestum esse ex predictis per diffinitionem datam, quod excluduntur omnes alie species argumentationis, scilicet inductio, exemplum, entimema, et preterea excluditur sophisticus sillogismus secundum omnes sui causas, et hoc quantum ad tredecim fallacias.”

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ily” excludes the fallacies of Accident, Consequent, secundum quid and ignorantiam elenchi; “other” excludes petitio; “because of those” excludes non-causa ut causa.17 Essentially the same reading is adopted by William Sherwood.18 Kilwardby rejects this reading: But this is not to be consented to. For Syllogism formally (as dealt with in this Book) is merely two propositions and three terms. Form and Figure and Mood can be saved in a sophistical Syllogism, as is clear from this: “Every dog runs, everything that barks is a dog, therefore etc.”, and in many others.19

This reading would appear to imply that a Syllogism faulty in matter is not a Syllogism—a proposition which Kilwardby condemned in 1277, and which he explicitly refutes in this Commentary when considering whether a syllogism from false premises is a syllogism. One of the arguments in favour of the to-be-condemned proposition runs as follows: A syllogism, since it is a certain composite ought to be composed of matter and form. Hence, if it is deficient in either, it will not be a syllogism. If, therefore, it is faulty in matter it will not be a syllogism. But it is said that a syllogism from false premises is faulty in matter, and this is true. Hence, it is not a syllogism.20

He qualifies the assumption that a syllogism from false premises is faulty in matter, distinguishing different senses in which something counts as a syllogism’s matter: … it is to be said that the material principles of the syllogism without qualification are two propositions (and if this is lacking there will be no syllogism); but of the ostensive syllogism [the material principles are] two true propositions. So, even though a syllogism with false premises is lacking in matter, it is not lacking in the matter of a syllogism without qualification but in the matter of an ostensive syllogism; and so, even Dialectica Monacensis, 489:13–490:11. Norman Kretzmann, William of Sherwood’s Introduction to Logic (Minneapolis: University of Minnesota Press 1966) III.1, 57–58. 19 Kilwardby ad A1 Part 4 dub.1 (5ra): “Sed ei non est consentiendum. Syllogismus enim formaliter, de quo determinatur in hoc libro, tantum modo sunt duae propositiones et tres termini. Forma autem et figura et modus potest salvari in syllogismo sophistico, sicut patet hic: Omnis canis currit, omne latrabile est canis, ergo etc., et in multis aliis.” 20 Kilwardby ad B2 Part 2 dub.2 (57rb): “Syllogismus cum sit quoddam compositum debetur compositio a materia et forma, quarum si deficiat altera syllogismus non erit. Si ergo peccat in materia syllogismi, non erit syllogismus. Sed dicitur quod syllogismus ex falsis peccat in materia, et hoc est verum. Quare non est syllogismus.” 17 18

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The Anonymus Aurelianensis III The exposition of the definition advanced in the manuscript Anonymus Aurelianensis III 22 doesn’t appear in Kilwardby’s discussion. This may be worth mentioning, because that manuscript is thought to contain the earliest Latin commentary on the Prior Analytics. In this exposition, “certain things” is said to exclude both Example and Enthymeme.23 “Set out” is said to pertain to the syllogism’s matter, and to exclude Hypothetical Syllogism as well as discourse in which nothing is either affirmed or denied.24 “Other” is said to exclude ridiculous syllogisms and cases where what is known is inferred from what is known, or what is known from what is unknown, or what is unknown from what is unknown.25 The word “something” in relation to the conclusion is said to exclude “syllogismi immodificati”, namely those having only indefinite or particular propositions.26 “Of necessity” is said to exclude Induction, which has only a necessity of things not of the combination.27 “From their being so” is said to exclude syllogisms that state too little or too much.28

21 Kilwardby ad B2 Part 2 dub.2 (57rb): “… dicendum quod syllogismo simpliciter sunt principia materialia, quae sunt duae propositiones, et si in hoc sit defectus non erit syllogismus, syllogismi autem ostensivi sunt duae propositiones verae. Quamvis ergo syllogismus ex falsis deficiat in materia, non deficit in materia syllogismi simpliciter sed in materia syllogismi ostensivi, et ideo, quamvis ex falsis sit, non tamen sequitur quod non sit syllogismus sed quod non ostensivus simpliciter.” 22 Sten Ebbesen, “Analyzing syllogisms, or Anonymus Aurelianensis III —the (presumably) earliest extant Latin commentary on the Prior Analytics, and its Greek model”, Cahiers de l’institut du moyen-âge grec et latin 37 (1981) 1–20. 23 Ebbesen, “Analyzing syllogisms”, 18:17–19. 24 Ebbesen, “Analyzing syllogisms”, 18:11–16. 25 Ebbesen, “Analyzing syllogisms”, 18:20–24. 26 Ebbesen, “Analyzing syllogisms”, 18:28–32. 27 Ebbesen, “Analyzing syllogisms”, 19:1–31. 28 Ebbesen, “Analyzing syllogisms”, 20:1–4.

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Kilwardby’s account Kilwardby doesn’t explicitly state what his own reading of Aristotle’s definition is. However, on the basis of scattered statements elsewhere in his commentary we can assemble a coherent account of the nature of the syllogism. First of all, he excludes hypothetical syllogisms as not being covered by Aristotle’s definition: And it is to be known that there are two types of syllogism, namely ostensive and categorical (whose necessity is from “to-be-said-of-all or none”) or hypothetical (whose necessity is rather from positing the antecedent or denying the consequent, or from the nature of opposition, or another topical relation). Here and throughout the first book Aristotle is speaking of the first type, not the second.29

In the case of Induction and Enthymeme, and in his remarks on different types of consequence, his account of the syllogism emerges as one that is highly nuanced. Induction Induction is treated differently by Aristotle in the Prior Analytics and in the Topics. In the Topics (and also in Boethius) it is treated as a progression from particulars to universals,30 and Boethius says that though it is readily believable “it is not as certain as syllogism”,31 thus emphasizing its opposition to the syllogism. Aristotle, on the other hand, while allowing that Induction proceeds from particular cases to the universal, emphasizes its reducibility to syllogism. He describes Induction as proving through c that a belongs to b, where b is the

29 Kilwardby ad A42–A44 dub.2 (49ra): “Et sciendum quod syllogismus duplex est, scilicet ostensivus et categoricus, cuius necessitas est per dici de omni et de nullo, vel hypotheticus, cuius necessitas magis est vel a positione antecedentis vel a destructione consequentis {vel a positione antecedentis vel a destructione consequentis Cm1E1E2F1F2 O2O3P1: vel a positione vel a destructione contingentis Ed} vel a natura oppositionis vel alia habitudine huius locali. De syllogismo primo loquitur hic Aristoteles et locutus est per totum primum huius, de secundo autem non.” 30 Aristotle, Topics in Sophistical Refutations and Topics translated by W.A. PickardCambridge, in The Works of Aristotle translated into English under the editorship of W.D. Ross vol. 1 (Oxford: Clarendon Press, 1928) I.12, 105a13–17; Boethius, De topicis differentiis 1183D–1184A. 31 Boethius, De topicis differentiis 1184A–B. Boethius (1184D) adds that Induction “takes its force” from the syllogism.

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Middle between a and c.32 In order to understand this, let us suppose a scientific syllogism with b as Middle between a and c, as in Fig. 2.3: aba bca aca Fig. 2.3. Scientific syllogism

Then the corresponding inductive syllogism might be one of those shown in Fig. 2.4: aca bca

aca cba

aba

aba

Fig. 2.4. Inductive syllogisms

In both cases it is proved through “c” that a belongs to b. But the first syllogism is invalid and the second valid. Accordingly, the inductive reasoning, if it is represented by the first argument, does not draw a necessary conclusion; but it does draw a necessary conclusion if it is represented by the second argument. The ambiguity arises because an inductive argument makes an assumption to the effect that the Minor premise is convertible, in other words that the enumeration of cases is exhaustive,33 and the Induction complete.34 Failing this completeness, all we have is an Example or a collection of Examples, and the inference is not necessary but merely probable.35 As Kilwardby says, “Many Examples make one Induction”.36 In discussing the question whether Induction is opposed to Syllogism or whether it reduces to Syllogism, Kilwardby says that the answer depends on how the definition of Syllogism is expounded. He distinguishes a general and a proper sense of “syllogism”: And it is to be said that Syllogism is said in two ways—in a general sense, and properly. In the general sense, Syllogism is not opposed to Induction; and taking “syllogism” this way, it’s true to say that Induction is a 32 33 34 35 36

B23, 68b17–18. B23, 68b27–29. See Thom, The Syllogism, 85–86. Kilwardby ad B23 dub.10 (79va). Kilwardby ad B24 dub.3 (80ra). Kilwardby ad B24 dub.4 (80ra): “… multa exempla unam faciunt inductionem.”

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syllogism. But speaking properly, Syllogism is opposed [sc. to Induction]. For, Syllogism properly speaking preserves the genuine nature of the syllogism, namely that it proves the first [term] of the third via the Middle; and Induction is opposed to this.37

When he speaks about a genuine syllogism proving the first of the third via the Middle, he must be thinking of the Middle term not according to the linguistic arrangement [secundum dispositionem sermonis], but according to reality and nature [secundum rem et naturam].38 Indeed Aristotle says that the syllogism through the Middle is prior and better known “in nature”.39 According to linguistic expression both the inductive and the non-inductive syllogisms prove a Major of a Minor via a Middle. But only the first fits Aristotle’s description, and proceeds according to reality and nature. In any event, it seems that Kilwardby doesn’t opt for a single exposition of the definition of Syllogism; rather, he applies different expositions depending on whether the definition is supposed to define Syllogism generally speaking or properly speaking. All this suggests that he endorses the first (Boethian) exposition for the proper sense, and the second (Aristotelian) exposition for the general sense. In his discussion of B22, he interprets Aristotle’s rule that “When a and b are in all of c, and b and c are convertible, then it’s necessary for a to be in all of b”40 as applying to Induction41—which would make the right-hand syllogism in Fig. 2.4 the inductive argument, in accord with the general sense of Syllogism. Again, in his discussion of B23, he states that the only necessity that Induction has, is what it draws from Syllogism;42 and it draws this necessity from syllogism because it reduces to a syllogism. This impression is strengthened by his comment about Induction that it is “virtually and fundamentally made through one of the three Figures, and has necessity only through them, and stands in potentiality to them, being rooted in them, even though it is 37 Kilwardby ad B23 dub.4 (79ra): “Et dicendum quod syllogismus dupliciter est {dupliciter est Cm1E1E2F1F2KlO2O3P1V: dicit dupliciter Ed}, scilicet communiter dictus et proprie. Syllogismo autem communiter dicto non opponitur inductio, immo sic sumendo syllogismum verum est dicere quod inductio est syllogismus. Syllogismo autem proprie dicto opponitur. Syllogismus autem proprie dicitur ubi salvatur vera natura syllogismi, scilicet quod probatur primum de tertio per medium, et huic opponitur inductio.” 38 Kilwardby ad B23 dub.9 (79rb). 39 B23, 68b35–37. 40 B22, 68a21–23. Kilwardby (77va). 41 Kilwardby ad B22 dub.1 (78ra). 42 Kilwardby ad B23 dub.1 (79ra).

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not primarily and per se in the Figures but is only reducible to them”.43 Looking at what Induction actually is, it is not a syllogism; but looking at what it virtually and fundamentally is, it falls into a syllogistic Figure. The actual perspective corresponds to the first exposition of the definition of Syllogism; the virtual perspective to the second. Enthymeme Aristotle and Boethius define the enthymeme in different ways—Aristotle as a syllogism from probabilities or signs,44 and Boethius as a syllogism that is lacking one premise.45 This difference is noted in the twelfth-century Dialectica Monacensis.46 Kilwardby doesn’t explicitly advert to the fact that these two authorities appear to define two different concepts of an enthymeme. In his questions on B27 he acknowledges the existence of both accounts but seems to find no conflict between them, affirming on one page that an enthymeme is a syllogism from probabilities or signs,47 and on the next that an enthymeme expresses only one premise.48 Nevertheless there is a certain duality in Kilwardby’s account of the enthymeme; and it parallels his double perspective on Induction. In A15 Aristotle states that nothing follows of necessity from a single thing, and that two are required.49 Commenting on this, Kilwardby distinguishes between premises in reason [apud rationem] and in language [apud sermonem]. Similarly, he holds that a syllogism must have three terms secundum rationem even if it has fewer secundum substantiam.50 Even if only one premise may be expressed in words, still in reason there must 43 Kilwardby ad B23 dub.7 (79rb): “… virtualiter et radicaliter fiunt omnes per praedictas figuras, quare non habent necessitatem nisi ab eis. Et in potentia {in potentia Cm1E1E2F1F2KlO2O3P1V: ita prima Ed} se habent ad illas ut radicantur in eas, non autem fiunt actualiter per figuras primo et per se, sed ex reductione.” 44 B27, 70a10–11. 45 Boethius, De topicis differentiis 1184B–C. 46 Dialectica Monacensis, 488:3–10. 47 Kilwardby ad B27, 70a10–11 (81vb). 48 Kilwardby ad B27 Part 1 dub.2 (82ra). Kilwardby adopts this reconciliation even though his text says merely “Entimema ergo est syllogismus ex ichotibus aut signis”, not interpolating “imperfectus” [ateles] as all the edited books available to Julius Pacius did (to his displeasure). See Julius Pacius, In Porphyrii Isagogen et Aristotelis Organum Commentarius Analyticus (Frankfurt: Apud Heredes Andreae Wecheli, Claudium Marnium & Iohan. Aubrium 1597) (264B). 49 A15, 34a17–18. 50 Kilwardby ad A25 Part 1 dub.1 (36rb).

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be two. The notion of an enthymeme as a form of arguing incorporates a contrast between what is expressed and what is in the understanding. An enthymeme may have only one express premise: “… but the Understanding does not admit a conclusion unless another proposition is at least understood along with it”.51 Taking the perspective of the Understanding, Kilwardby regards Enthymeme as “substantially the same as Syllogism”: Even if the sense of both propositions is not expressed in the enthymeme, it is understood in the enthymeme. Hence, in reality and following the train of reasoning [secundum rem et decursum ipsius rationis] there are two propositions here, just as in the syllogism. And in this way they are not opposed ways of arguing, but one is reducible to the other.52

From this perspective, an enthymeme is a syllogism; but from the linguistic perspective Enthymeme and Syllogism are opposed to one another. The linguistic perspective fits the first (Boethian) exposition of the definition of Syllogism; the perspective of the Understanding fits the (Aristotelian) second exposition. Ivo Thomas saw in these ideas “a fine example of logical psychologism” and finds Kilwardby’s position to be lacking a “purer and sounder logical standpoint”.53 I agree that Kilwardby’s theory of the syllogism cannot be contained within the bounds of pure logic. His is not simply a theory of syllogistic implication, but is also an attempt to account for the human activity of arguing syllogistically. Now, such an attempt is an interpretive activity; and in distinguishing between what people say explicitly when they present an argument and what must be imputed to them by way of extra assumptions, the interpreter is inevitably assuming some background logical theory. In Kilwardby’s case, one might think, the assumed background theory is the standard theory of the syllogism. Applying that theory to a case of reasoning from a single premise, the theorist will interpret the reasoning as resting on an unstated premise. So, it may be reasonable to point to the danger 51 Kilwardby ad A25 Part 2 dub.1 (37ra): “… tamen intellectus non admittit conclusionem nisi coaccipiat aliam propositionem adminus intellectam.” 52 Kilwardby ad B27 Part 1 dub.4 (82rb): “… substantialiter idem est syllogismus et enthymema. Quamvis non sit in enthymemate expressus sensus utriusque propositionis, intelligitur ex enthymemate una, unde secundum rem et decursum ipsius rationis duae sunt propositiones ibi, sicut in syllogismo. Et hoc modo non sunt oppositi modi arguendi, sed unum in alterum reducibile.” 53 Ivo Thomas O.P., “Maxims in Kilwardby”, Dominican Studies 5 (1954): 129–146, 132.

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of circularity in such an interpretive endeavour (and this danger exists with all interpretation); but it is not reasonable to accuse Kilwardby of psychologism. Be that as it may, Kilwardby offers several non-psychological arguments for the thesis that inference requires at least two premises: Every conclusion is inferred through some Middle. But a Middle exists only through a relation to two Extremes. Hence it’s necessary for the conclusion to be inferred through a Middle that is related to the conclusion’s two Extremes. But if the Middle is related to the conclusion’s two Extremes, two intervals are produced, and two propositions. So of necessity every conclusion is inferred from two propositions. Further, just as it is in genuine motion, so it is in rational motion. But genuine motion exists only from one Extreme to another through a Middle… Hence the motion of reason will be from Extreme to Extreme through a Middle. Hence when a conclusion is produced from Extremes, it is first necessary according to reason to compare a Middle to the two Extremes, and afterwards to conclude one Extreme of the other Extreme. Thus there will first necessarily be two propositions. Further, the conclusion is either affirmative or negative. If it is affirmative, one Extreme is concluded of the other through something agreeing with both, and that then will necessarily be compared to both—as is clear of itself—and then there will then be two propositions from this comparison. If it is negative, one Extreme will be denied of the other, and this will be through something agreeing with one but not with the other. So in order for one Extreme to be denied of the other, it’s necessary that something be compared with both—to one through agreement, to the other through difference. There will then necessarily be two propositions. Hence it’s clear that whether the conclusion is affirmative or negative, it’s necessary in reason for there to be two premises from which the conclusion is inferred.54 54 Kilwardby ad A15 dub.4 (24rb–va). “Omnis conclusio infertur per aliquod medium. Sed medium non est nisi per relationem ad duo extrema. Quare necesse est conclusionem inferri per medium relatum ad duo extrema conclusionis. Sed si medium sit relatum ad duo extrema conclusionis, facit duas dimensiones et duas propositiones. Quare omnis conclusio per duas propositiones infertur ex necessitate. Adhuc similiter est in motu vero et motu rationis. Sed motus verus non fit nisi ab extremo alio in extremum et per medium [Four or five words occur here, but there is little agreement among the Mss.] Quare motus rationis erit ab extremo in extremum per medium. Quare cum fiat conclusio ex extremis, necesse est secundum rationem comparare medium ad duo extrema et postea concludere unum extremum de extremo, et sic prius erunt necessario duae propositiones. Adhuc conclusio aut est affirmativa aut negativa; si affirmativa, concluditur unum extremum de altero per aliquod conveniens utrique, et illud tunc necessario comparabitur ad utrumque, sicut patet per se, et erunt tunc duae propositiones ex ista compa-

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The first and the third of these arguments seem to beg the question. How could we be sure that every inference requires a Middle term, or that it concludes one Extreme of the other “through” something else, unless we were already assuming that every inference has two premises? The second argument postulates an analogy between inference and physical motion; and indeed Kilwardby had proposed this analogy earlier in the commentary.55 The problem is that, even though the process of inference is like physical motion in certain respects, it appears (according to Aristotelian doctrine) to be unlike it in the very respect under consideration. While Aristotle holds that in the physical domain motion is continuous and thus cannot be composed of indivisible motions,56 he holds that in the realm of reason there are immediate propositions.57 Types of consequence In his questions on B4, Kilwardby draws a distinction between two types of inference—the essential (or natural) where the consequent is understood in the antecedent, and the accidental (or positive) according to which the necessary follows from everything.58 And he comments on ratione. Si sit negativa removebitur unum extremum ab altero, et hoc per aliquid uni conveniens et non alteri. Ad hoc ergo quod extremum de extremo removeatur necesse est aliquid ad utrumque comparare ita quod ad unum per convenientiam, ad alterum per differentiam, et erunt tunc necessario duae propositiones. Quare patet quod sive debuit concludi conclusio affirmativa sive negativa, necesse est apud rationem duas fieri praemissas conclusionem inferentes.” Thomas, “Maxims in Kilwardby” 131 gives a text of this passage based on O2 and O3. 55 Kilwardby ad A1 Part 1 dub.2 (3va). 56 Aristotle, Physics translated by R.P. Hardie and R.K. Gaye, in The Works of Aristotle translated into English under the editorship of W.D. Ross vol. 2 (Oxford: Clarendon Press, 1930) VI.1, 231b15 ff. 57 Posterior Analytics A2, 72a7. 58 Kilwardby ad B4 Part 2 dub.1 (59va). Christopher J. Martin, “Aristotle and the logic of consequences: the development of the theory of inference in the early thirteenth century”, in Ludwig Honnefelder, Rega Wood, Mechtild Dreyer, Marc-Aeilko Aris (eds.), Albertus Magnus und die Anfänge der Aristoteles-Rezeption im lateinischen Mittelalter (Münster: Aschendorff Verlag n.d.) 523–553 provides a detailed discussion of the distinction. Thomas, “Maxims in Kilwardby”, 137 presents a text based on O2 and O3. A possible source of Kilwardby’s terminology is Al-Ghazali’s Logic. This may have influenced some of the texts in the Logica Modernorum. Ghazali calls a concept essential when its subject cannot be understood in any way without it. (Charles Lohr, “Logica Algazelis: introduction and critical text”, Traditio: studies in ancient and medieval history, thought and religion 21 (1965): 223–290, 247). With such concepts he contrasts those for which it’s not possible to be posited [positivum] as essential. We can’t posit that Man

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accidental inferences such as “You are an ass, so you are not an ass”, where one opposite follows from another: Now, in such consequences, when one opposite follows from the other, it isn’t because one opposite posits the other; but the consequent posits itself on account of its own necessity and not on account of the antecedent. So, in natural consequences the antecedent posits its consequence; but in accidental consequences this is not necessary.59

Now, syllogistic consequences by definition are such that the conclusion follows on account of the premises; so they must be natural rather than accidental consequences. And there is evidence that Kilwardby took them to possess the defining characteristic of natural consequences, namely that the consequent is understood in the antecedent. For, in his commentary on Aristotle’s example of thinking that a mule is pregnant while knowing that all mules are infertile,60 he states that to know or opine the premises of a syllogism is to know or opine the conclusion, provided we think them through [pertractare].61 The Material, Formal and Final Cause of the syllogism Kilwardby applies the Aristotelian doctrine of causes to the syllogism— something which Aristotle himself does not do—with specific discussions about the syllogism’s material, formal and final causes. Among the propositions Kilwardby condemned in 1277 was the proposition that there is a single substantial form of Man. As G.J. McAleer observes:

is Animal, or Black is a Colour, or Four is a Number. The accidental can be posited [accidentale positivum est]. (Lohr, “Logica Algazelis: introduction and critical text”, 248). 59 Kilwardby ad B4 Part 2 dub.2 (59vb): “Quod autem in talibus consequentiis unum oppositorum sequitur ad alterum non est quia unum oppositorum ponit alterum, sed ipsum consequens propter necessitatem sui ponitur et non propter suum antecedens. In consequentiis ergo naturalibus antecedens ponit suum consequens, in consequentiis autem accidentalibus non est hoc necesse.” Thomas, “Maxims in Kilwardby”, 139 presents a text based on O2 and O3. 60 B21, 67a33–37. 61 Kilwardby ad B21 dub.5 (76va). At the same time, he comments ad B21, 67a32–37 (76ra) “So therefore it’s clear that it’s contingent to know both propositions through one Middle (and then, in a way, the conclusion is known); and yet it’s contingent to be ignorant of the conclusion in so far as it is actually considered. Thus it’s not unacceptable to know and to be ignorant of the same thing, because this knowledge is not contrary to that ignorance.” “Sic igitur patet quod contingit scire secundum unum medium ambas propositiones, et tunc scitur aliquo modo conclusio. Contingit tamen

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Robert was a defender of the common position in the Middle Ages that the human being is made up of a plurality of substantial forms. This position was a commonplace in medical literature until the seventeenth century.62

The plurality of forms is a doctrine that Kilwardby held in general terms, and he applied it to the syllogism: And it is to be said that there is an order in materials. For some are remote and unarranged, and some are proximate and arranged. And so it is in forms. Some are material forms, which are in potentiality to an ulterior form, and some are ultimate and completing forms. Thus we find an order in a syllogism’s materials and forms. For, in materials, the term is its remote and unarranged material, and the proposition is its proximate and arranged materials; and in forms, Figure is the incomplete form which is in potentiality to an ulterior form, and Mood is the ultimate form completing the syllogism.63

Taking the remote, unarranged and incomplete perspective, the terms are the syllogism’s material cause, and Figure is its formal cause; but taking the proximate, arranged and ultimate perspective, the propositions are the syllogism’s material cause, and Mood is its formal cause. As a consequence of this plurality of forms and matters (as we shall see later), an argument can exist in an incomplete state, not fully syllogized. The nature of the syllogism’s final cause can be broached via the question why Aristotle speaks of the syllogism as one discourse rather than many. To this question, Kilwardby offers a double answer: It is to be said that that with which something else coheres cannot be counted along with that thing; but the Minor proposition in a syllogism

ignorare conclusionem in eo quod actu considerat. Sic etiam scire et ignorare idem non est inconveniens quia haec scientia illi ignorantiae non contrariatur.” 62 G.J. McAleer, “Kilwardby, Robert”, in Thomas F. Glick, Steven J. Livesey and Faith Wallis (eds.), Medieval Science, technology and Medicine: An Encyclopedia (New York: Routledge 2005) http://www.routledge-ny.com/middleages/Science/kilwardby.pdf. 63 Kilwardby ad A4 Part 2 dub.4 (10va): “Et dicendum quod sicut ordo est in materiis, quaedam enim est remota et indisposita, quaedam autem propinqua et disposita, sic est in formis. Quaedam est forma materialis et in potentia ad formam ulteriorem, quaedam autem est ultima et completiva. Et sic invenimus in syllogismo ordinem in materia et in formis. In materiis quia terminus {terminus AE 1E2F1F2KlO3P1V: termi Ed} est materia eius remota et indisposita, propositio vero est materia propinqua et disposita; in formis etiam quia {quia ACm1E1E2F1F2KlO3P1V: qua Ed} figura est forma incompleta et in potentia ad ulteriorem formam. Modus autem est forma ultima syllogismi completiva.”

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chapter two coheres with the Major to the conclusion; and so it is not counted with it. And so it is to be said that Syllogism is a discourse and not many discourses. Further, it is said to be one discourse because of the unity of its End, namely the conclusion.64

Earlier, in expounding Aristotle’s definition, Kilwardby had noted the syllogism’s double directedness—towards its conclusion and towards the production of belief concerning doubtful matters. In the passage just quoted he takes up the theme of the syllogism’s directedness towards its conclusion; and earlier (in his Prologue) he had touched on the other theme, that of the directedness towards the production of belief. What, he asks there, is the function [opus] of the syllogism, and what makes a good syllogism good? It would not be right to say that the syllogism’s function is to produce knowledge or belief, since the syllogism in general (as opposed to concrete instances of the syllogism) does no such thing,65 though the function of the dialectical syllogism is to produce belief or opinion and that of the demonstrative syllogism is to produce knowledge. The syllogism in general is the instrument by which we acquire belief about a matter of opinion and knowledge about a knowable matter.66 The syllogism in general descends immediately into the demonstrative syllogism, in which is primarily found its goodness.67 However, the goodness of the syllogism in general abstracts from both dialectical and demonstrative cases. Its function is to prove [probare] something, and thus to infer it of necessity; and when it does this in one of its species, it produces belief or knowledge.68 64 Kilwardby ad A1 Part 4 dub.2 (5ra). The second part of the answer is notable for its invocation of the notion of a final cause. Albertus Magnus, Priorum Analyticorum, I.i.5, 294 expands on this reasoning: “Because both Premises are referred to one Discourse, namely the Conclusion, and are united in one End, they are not without qualification diverse, but are related to one End. And so the Syllogism is more acceptably said to be a Discourse rather than many Discourses. [“Quia ambae praemissae referuntur ad unam orationem quae est conclusio et quae in uno fine uniuntur, non simpliciter sunt diversa, sed relata ad unum finem. Et ideo syllogismus congruentius dicitur oratio quam orationes”.] 65 Kilwardby Prologue dub.8 (2vb). 66 Kilwardby Prologue dub. 8 (2vb). 67 Kilwardby ad A1 Part 1 dub.1 (3va). Where Ed has “benignitas”, AE E Cm CrF F 1 2 1 1 2 KlO2O3P1V have “bonitas”. 68 Kilwardby Prologue dub. 8 (3ra).

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Later in the commentary, he gives a fuller account. Commenting on A23, he states that the Final Cause of the syllogism is to show [ostendere] something about something.69 In his commentary on A24, he distinguishes two conditions on the syllogism—first that the inference is necessary, and second that it imparts knowledge [notificare].70 In several passages he pairs the terms “ostendere” and “notificare”; and in his discussion of B14, he argues that they go together: “… it can be said that there is an ostensive syllogism only when the premises make the conclusion known [notificent]; for, otherwise they do not show it [ostendunt]”.71 He says that there can be no syllogism without a necessary consequence, but that there can without showing and making-known.72 Thus it is that the syllogism’s directedness towards its conclusion is more essential than its directedness towards showing and making-known. When petitio occurs nothing is shown and no knowledge is imparted.73 By contrast, something can be shown in a pair of circular syllogisms, such as are displayed in Fig. 2.5.74 Kilwardby accepts the principle that what shows something is prior and better known than what is shown, and he accepts that this principle implies that in a pair of circular syllogisms the same thing is prior to itself; but he isn’t worried by this implication, so long as the same thing is prior to itself under different concepts [sub diversis rationibus]—as indeed is the case with circular syllogisms.75 p aba

r baa

r

p

Fig. 2.5. Circular syllogisms

Here, the ratio under which “p” is prior to “r” is different from that under which “r” is prior to “p”. In a petitio nothing is shown, but there may be a syllogism (in the sense that the inference is necessary): 69

Kilwardby ad A23 dub.1 (34vb). Kilwardby ad A24, dub.2 (35vb). 71 Kilwardby ad B14 dub.3 (69vb): “… dici potest quod syllogismus ostensivus non est nisi praemissae conversionem notificent. Aliter enim non ostendunt.” 72 Kilwardby ad A24, dub.2 (35vb). 73 Kilwardby ad A24, dub.2 (35vb). 74 Thom, The Syllogism, 203–206. 75 Kilwardby ad B5 dub.1 (60vb). 70

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chapter two When a conclusion follows of necessity, the proposition is not always demonstrated, but it has to be such that there is a showing [ostensio] and a making-known [notificatio]. And to this end the premises have to be suitable for showing the conclusion and making it known. This, however, is lacking when there is a petitio.76

Kilwardby agrees with Aristotle that the Disjunctive Syllogism “Man is either mortal or immortal, but he is not immortal, so he is mortal” is a petitio: “… such a division is not a dialectical syllogism, nor does it draw the desired conclusion without begging the question …”.77 What ought to be shown is simply assumed;78 and this is contrary to the definition of Syllogism. The same applies to non-causa ut causa: “These two [sc. petitio and noncausa ut causa] offend in that they do not show the proposed conclusion —as Aristotle says of non-causa ut causa in the Sophistical Refutations”.79 Similar considerations apply to syllogisms from opposed premises, such as the example in Fig. 2.6. (to be studied)(discipline)e (to be studied)(discipline)a (to be studied)(to be studied)o Fig. 2.6. Syllogism from opposed premises

Here there are only two terms, not the usual three. And Kilwardby asks therefore whether it is true, as Aristotle says, that every Syllogism must have three terms. His reply is that what Aristotle says is in fact true of all Syllogisms “not substantially [secundum substantiam] but conceptually [secundum rationem]”.80 Conceptually, the syllogism instantiates Cesare, and substitutes the same expression for the Major and Minor term. Now, if there can be a syllogism with opposed premises, and thus a syllogism with only two terms, why couldn’t there be a syllogism with just one term, such as is shown in Fig. 2.7? 76 Kilwardby ad B16 dub.3 (72vb): “… non semper demonstratur propositum quando ex necessitate sequitur conclusio, sed oportet quod fiat ostensio conclusionis et notificatio. Et adhuc exigitur ut praemissae sint natae ostendere conclusionem et notificare. Hoc autem deficit quando sit petitio.” 77 Kilwardby ad A31 dub.1, 43ra. 78 Kilwardby ad A31 dub.1, 43rb. 79 Kilwardby ad B17 dub.2 (74rb): “… peccant eo quod non ostendunt propositum, sicut dicit Aristoteles in Elenchis de non causa ut causa.” Aristotle, Sophistical Refutations 5, 167b35. 80 Kilwardby ad A25 Part 1 dub.1 (36rb); ad B15 dub.1 (71ra).

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(man)(man)e (man)(man)a (man)(man)e Fig. 2.7. Apparent syllogism with a single term

He replies: A syllogism is not made from a single term, even if it may be made from two, in that a syllogism’s conclusion has to be other than either of the premises, because “something other” has to “come about from their being so”, but this can’t be so if only one term is taken, as it can with two.81

The inference in Fig. 2.7 couldn’t fulfill the syllogism’s function of being suitable for showing the conclusion and making it known, because the conclusion is not other than the premises.82 Perfection In Aristotle’s theory of the syllogism the notion of perfection plays a key role because every syllogism of the system is reducible to a perfect syllogism. Every syllogism is perfectible, and here is Aristotle’s definition of a perfect syllogism: “I call a syllogism perfect if it requires nothing other besides the things assumed in order that its necessity is apparent”.83 Commenting on this definition, Kilwardby opines that the reason why Aristotle highlights the distinction between perfect and imperfect syllogisms, at the expense of other distinctions such as that between universal and particular syllogisms or that between affirmative and negative syllogisms, is that perfection concerns the syllogism per se, whereas the other distinctions are per accidens.84 If perfection concerns 81 Kilwardby ad B15 dub.1 (71rb): “… ex uno termino non fit syllogismus quamvis fiat ex duobus, eo quod in syllogismo oportet conclusionem esse alteram ab utraque premissarum quia oportet aliud accidere eo quod haec sunt, sed hoc non contingit si tantum unus sumatur terminus, sicut etiam in duobus.” 82 From the perspective of modern logic, it is interesting that Kilwardby allows the identification of two variables in a syllogistic form (as in Fig. 2.6) but not the identification of all three (as in Fig. 2.7). This means that his syllogistic falls in between the system 1A* (Thom, The Syllogism, 43–44), which doesn’t allow syllogisms with fewer than three distinct term-variables, and the system A (Thom, The Syllogism, 91), which allows syllogisms with a single term-variable. 83 A1, 24b22–24. Kilwardby (5ra). 84 Kilwardby ad A1 Part 5 dub.1 (5ra).

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the syllogism per se, how is it related to the syllogism’s Final Cause of showing a necessarily implied conclusion and making it known? All syllogisms—perfect and imperfect—have to satisfy the dual conditions of being necessary consequences and of being suitable for showing the conclusion and making it known. Inferences not satisfying these conditions—such as those displayed in Fig. 2.7—are not syllogisms at all. As Kilwardby might say, they lack the being of a syllogism. He draws a distinction between two sorts of perfection—perfection of being and perfecting of well-being. Imperfect syllogisms lack the well-being, not the being, of the syllogism. The syllogism in general needs nothing extrinsic in order to be necessary (or, we may add, to be suitable for showing the conclusion and making it known), but an imperfect syllogism needs something extrinsic to it if it is to be evidently necessary (and more generally, if it is to perform its function evidently).85 The distinction between perfection of being and perfection of wellbeing appears to come from Aristotle’s Metaphysics Δ16, which distinguishes two ways in which things can be called complete or perfect (teleion)—“some from their being without deficiency in respect of goodness (kata to eu) and not to be surpassed and having nothing to be found outside them, others in respect of being in general not to be surpassed in their various genera and having nothing outside them”.86 Albert’s Commentary on Metaphysics V. 16 reads Aristotle in this way, distinguishing perfection in regard to being from perfection according to virtue and standing well related in that regard.87 Kilwardby summarizes the doctrine as follows: 85 Kilwardby ad A1 Part 5 dub.3 (5rb). Kilwardby uses this distinction in his De Ortu Scientiarum, 527, where he says that whereas Aristotle was concerned with the esse of logic Boethius was more concerned with its esse bene and accordingly went into details about different types of definition and division, because he was addressing the young. In his discussion of rhetoric in the same work (604), Kilwardby applies this same distinction to discourse [oratio], whose esse includes the Major and Minor proposition, and whose esse bene includes any rhetorical elements that go beyond this bare minimum— things like explanatory or ornamental devices or techniques of concealment. In the same work (504) he states that demonstration is the strongest mode of syllogizing and its principal and final intent, which once possessed satisfies human curiosity. Albertus Magnus, Priorum Analyticorum, I.i.6 (294B) makes use of this same distinction, but he adds that the division of syllogisms into perfect and imperfect is not a division according to species but according to the way things appear to us. 86 Aristotle’s Metaphysics Books Γ, Δ and E translated with notes by Christopher Kirwan (Oxford: Clarendon Press, 1971), 1021b30–1022a3. 87 Albertus Magnus, Metaphysicorum Libri XIII, in Beati Alberti Magni Ratisbonensis Epis-

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Next, it may perhaps be enquired about that which says that the syllogisms in this Figure are imperfect. For, if the perfect is that which can produce another thing like itself (as Aristotle says),88 and the syllogisms in this Figure, and in the third, produce another thing like themselves— namely a first Figure syllogism by reduction—then it seems that the syllogisms in the second and third Figures are perfect. And if a syllogism is called perfect on account of the necessary inference of its conclusion, there will be perfect syllogisms in all Figures, since that is found in all Figures. And it is to be said that the perfect is that which attains its proper virtue. A virtue is that which perfects the possessor and renders its work good.89 Now, the work of a syllogism is to show and make known [ostendere et notificare]. And this work is done well through necessity and evident necessity. Hence the virtue perfecting a syllogism is necessity and evident necessity. This is found only in the first Figure. So the syllogism is said to be perfect only in the first Figure.90

copi Ordinis Praedicatorum Opera quae hactenus haberi potuerunt. Tomus tertius, ed. Petrum Jammy (Lugduni: Claudii Prost, Petri & Claudii Rigaud, Hieronymi de la Garde, Ioan. Ant. Huguetan filii, 1651) V.iv.1, 211A–B. 88 Aristotle, De Anima Books II and III, translated with introduction and notes by D.W. Hamlyn (Oxford: Clarendon Press, 1968) 2.4, 415a22 ff.: “it is the most natural function of living things, such as are perfect and not mutilated or do not have spontaneous generation, to produce another thing like themselves ….” 89 Compare Aristotle, Metaphysics Δ 16, 1021b23 ff.: “Again, things which have reached their fulfilment [telos], when it is worth while, are called complete [teleia] by virtue of having attained their fulfilment.” Kirwan translation. Kilwardby runs this passage together with the preceding one (1021b14ff.), where perfection is defined in terms of things, “own proper excellence [oikeias aretes]”. And W.D. Ross, Aristotle’s Metaphysics, a revised text with introduction and commentary 2 vols. Oxford: Clarendon Press, 1924. vol. 1, 332 also says “This sense of teleion is hardly to be distinguished from the second, and in the summary (l. 30–1022a1) no reference is made to it. It seems to be merely a restatement of the second sense from a slightly different point of view, viz. that of the connexion of teleion with telos”. 90 Kilwardby ad A5 dub.3 (12rb): “Consequenter forte dubitabitur de eo quod dicit syllogismum huius figurae esse imperfectum. Si enim perfectum est quod potest generare sibi simile, sicut dicit Aristoteles, syllogismus autem huius figurae, similiter et tertiae, generat ex se sibi simile quia syllogismum primae figurae per reductionem, videtur quod syllogismus secundae et tertiae figurae sit perfectus. Si autem dicatur syllogismus perfectus per necessariam illationem conclusionis, cum illa reperiatur in omni figura, erit in omni figura syllogismus perfectus. Et dicendum quod perfectum est quod attingit propriam virtutem. Virtus autem est quod perficit habens et opus eius bonum reddit. Opus autem syllogismi est ostendere et notificare. Hoc autem opus bene fit per necessitatem et evidentiam necessitatis. Quare virtus perficiens syllogismum est necessitas cum evidentia. Hoc tantum invenitur in syllogismis primae figurae, et ideo solus syllogismus primae figurae dicitur perfectus.”

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So, while the being of the syllogism is found even in the second and third Figures, its well-being is found only in the perfect moods of the first. One could put the point by saying that what the syllogism in general does, it does best in the first Figure. Why should the well-being of a syllogism depend on its evidentness, its obviousness to us? The idea might suggest an uncomfortably subjective element in the notion of syllogistic perfection. Alternatively, it might be related to Wittgenstein’s observation that a proof must be perspicuous.91 I believe that if we look closely at Kilwardby’s text we will find that he uses objective criteria to determine whether an inference is “evident”. One of these objective criteria lies in his reading of Aristotle’s definition of what it is to be said-of-all or of-none of a subject. To-be-said-of-all or of-none Now, for one thing to be in another as a whole is the same as for one thing to be predicated of all of the other. Now, we say “to be predicated of all” when there is nothing to be taken of the subject, of which the other is not said. And similarly “of none”.92

This celebrated statement of dici de omni is appealed to throughout Kilwardby’s exposition of the syllogistic. It can be elucidated by reference to the modern notion of monotonicity. In general, a subject-predicate form of proposition has upwards / downwards monotonicity in its subject / predicate if, together with a proposition stating that a third term includes (or is included in) the subject / predicate, it implies a conclusion of the same form as the original proposition, where the third term is substituted for the original subject / predicate. Formally, qual1quant1 a,b is monotonic:93 – downwards for the subject b iff qual1quant1 a,b ∧ bca → qual1quant1 a,c – downwards for the predicate a iff qual1quant1 a,b ∧ aca → qual1quant1 c,b 91 Ludwig Wittgenstein, Remarks on the Foundations of Mathematics edited by G.H. von Wright, R. Rhees, G.E.M. Anscombe, translated by G.E.M. Anscombe (Oxford: Basil Blackwell 1964), 65 f. 92 A1, 24b26–30. Kilwardby 5rb. 93 See Jan van Eijck, “Syllogistics = Montonocity +Symmetry +Existential Import” (draft paper), http://homepages.cwi.nl/~jve/papers/05/syllogic/Syllogic.pdf (Accessed 9.54 am, 18 July 2005), 2.

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– upwards for the subject b iff qual1quant1 a,b ∧ cba → qual1quant1 a,c – upwards for the predicate a iff qual1quant1 a,b ∧ caa → qual1quant1 c,b. Universal affirmative and universal negative assertoric propositions possess downwards monotonicity in the subject. This is a modern reformulation of the dici de omni et nullo, and it is one way of providing an objective criterion for the evident validity of the universal perfect moods in the first Figure. Interestingly, Kilwardby takes perfection to admit of degrees. He says that the universal moods of the first Figure are more perfect than the particular moods, not as to their necessity, but as to their evident necessity: “For, the particulars arise from the universals and draw their necessity from them originally”.94 He is here referring to the doctrine of A7 according to which the particular moods in the first Figure reduce to universal moods in the second, and thence to Celarent, as shown in Fig. 2.8. ace Celarent

cae

aba cbe

C →

aba bci aci

bce abe Celarent

bae bce

C aca →

abe bci aco

Fig. 2.8. Reduction of particular first Figure moods

However, he is careful to distinguish reducibility from perfectibility. He notes that in that two-stage reduction, the second Figure syllogisms 94 Kilwardby ad A7 dub.9 (14vb). The idea that perfection admits of degrees, and that the universal direct first Figure moods are more perfect than the particular ones can be found in Alexander of Aphrodisias, On Aristotle’s Prior Analytics 1.1–7, translated by Jonathan Barnes, Susanne Bobzien, Kevin Flannery, S.J., and Katerina Ierodiakonou (Ithaca NY: Cornell University, 1991), ad A7, 29b1–2, 113,7–15; and also in Philoponus ad A7, 29b1–2 (114,16–22). The latter is quoted in Günther Patzig, Aristotle’s Theory of the Syllogism: a logico-philological study of Book A of the Prior Analytics translated from the German by Jonathan Barnes (Dordrecht: Reidel 1968), 68.

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constitute merely an intermediate stage and not a stopping-point. The stopping-point is Celarent.95 In taking the universal first Figure moods to be more evident and more perfect than the corresponding particular moods, Kilwardby is voicing a perception that finds the universal first Figure moods to be more evident than the corresponding particular moods. The monotonicity property just described provides an objective criterion for this evidentness. Günther Patzig puts it like this: … “perfection” should be ascribed only to those syllogisms in which the relation between A and B can, by the mediation of that between B and C, be “transmitted” or “conveyed” to the pair A and C. And this stronger requirement is only satisfied by Barbara and Celarent: Darii has a in the first premiss and i in the conclusion, and Ferio has e in the first premiss and o in the conclusion.96

As Patzig saw, the same analysis can be applied to the universal first Figure LXL and QXQ moods: the universal necessity- and contingency-propositions of Aristotle’s modal syllogistic exhibit the same monotonicity as universal assertorics. Notice that the criterion for monotonicity is not purely syntactic. Monotonicity doesn’t exist unless certain inferences are valid. Thus it is partly a semantic property. This is important in relation to the universal LQL and XQX moods of the first Figure. The mere fact that the syntactic features of the Major premise are replicated in the conclusion doesn’t create monotonicity, because the conclusion doesn’t follow from the premises in these moods. Patzig’s approach also explains the perfection of the LLL, QQQ , or QLQ moods: in all these cases the modality of the Minor premise is such that the relation between Major and Middle terms is replicated in that between Major and Minor terms. But these cases do not fit the definition of downwards monotonicity under the subject, as set out above. In order to bring these cases under a notion of monotonicity, we have to broaden that definition, relativizing it to different inherencerelations as expressed in the Minor premise. Thus, we may define a subject-predicate proposition mode1qual1quant1a,b as possessing downwards monotonicity for the subject, relative to necessary inherence, iff mode1qual1quant1a,b ∧ Lbca → mode1qual1quant1a,c, 95 96

Kilwardby ad A7 dub.6 (14vb). Patzig, 68–69.

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and as possessing downwards monotonicity for the subject, relative to contingent inherence, iff mode1qual1quant1a,b ∧ Qbca → mode1qual1quant1a,c. Then universal necessity- or contingency-propositions exhibit such monotonicity relative to necessary inherence; and universal contingency-propositions exhibit it relative to contingent inherence. And these facts may be regarded as providing objective criteria for the perfection of the universal LLL, QLQ and QQQ moods in the first Figure. None of this explains the perfection of the particular XXX, LXL, QXQ , LLL, QLQ or QQQ moods. However, Kilwardby has a simple rule that provides an explanation: … the conclusion is part of the Major, and mostly in regard to the predicate, which they share. With regard to the subject, it is part of the Minor. And so it follows the Minor in states affecting the subject (such as universality and particularity), and the Major in states affecting the predicate (such as affirmative and negative, assertoric and modal).97

He uses the word “assimilation” to describe this sharing of states affecting the subject or predicate: … the conclusion is always assimilated to the Major in quality. It is, however, assimilated to the Minor in quantity, because quantity is known from the subject, and the Minor and the conclusion have the same subject.98

His rule of assimilation may be paraphrased by saying that a first Figure mood is perfect, relative to the type of inherence signalled by mode2, iff mode1qual1quant1a,b ∧ mode2qual2quant2b,c → mode1qual1quant2a,c. This rule identifies all the perfect syllogisms. If one wants to demarcate the “more perfect” universal moods of the first Figure, this can be achieved by using the criterion of downwards monotonicity under the subject (relative to a type of inherence).

97 Kilwardby ad A9 dub.5 (16vb): “… conclusio est pars maioris, et maxime secundum praedicatum in quo communicat cum ipsa, et quantum ad subiectum pars minoris, et ideo sequitur minorem in dispositionibus accidentibus subiecto eius, quae sunt universalitas et particularitas, maiorem autem in dispositionibus accidentibus praedicato eius, quae sunt affirmativum et negativum de inesse et de modo.” 98 Kilwardby ad A4 Part 2 dub.12 (11ra): “… conclusio semper assimilatur in qualitate maiori, minori autem assimilatur in quantitate, quia a subiecto cognoscitur quantitas, et idem est subiectum minoris et conclusionis.”

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The Figures Kilwardby subscribes to the Aristotelian doctrine that there are only three syllogistic Figures. At the same time he subjects this doctrine to critical scrutiny. He recognizes that the same material may be arranged in Figure and Mood in multiple ways. This becomes clear in his consideration of syllogisms from opposed premises, such as the one in Fig. 2.6. Next, it is enquired about the arrangement of the premises in the syllogism. For, with the same subject in both premises the syllogism will be in the third Figure. But with the same predicate in both the syllogism will be in the second. And so, every syllogism from opposites looks to the second and third Figure indifferently—which is absurd, as it seems. It is asked, then, how in general one knows when the syllogism is in the second and when in the third Figure, and in what way. And it is to be said that considering the premises before the conclusion is drawn [pertractationem], the syllogism’s matter is indifferently in the second Figure and the third. But through the act of drawing the conclusion [per voluntatem pertractantis] the conclusion is concluded in the second or in the third. And then it is clear where the syllogism is, because if subject is concluded of subject the syllogism is in the second Figure, and if predicate of predicate the syllogism will be in the third—one extreme always being concluded of the other and the Middle not entering into the conclusion.99

Why, he asks, do we need more than the first Figure, seeing that in the first Figure we can conclude all four forms of proposition? He reminds us of the distinction between perfect and “less than perfect” syllogisms. It is only in the perfect first Figure that all types of conclusion are deducible, but syllogisms in the other Figures are still valid, even if imperfect. 99 Kilwardby ad B15 dub.6 (71va): “Consequenter quaeritur de dispositione syllogismi ex praemissis, quia cum idem subiciatur utrobique erit syllogismus in tertia figura. Adhuc cum praedicetur utrobique erit syllogismus in secunda, et ita omnis syllogismus ex oppositis indifferenter spectat ad secundam et tertiam figuram, quod est inconveniens, ut videtur. Queritur ergo quo modo cognandum sit {cognandum sit Cm1E1F2KlO2O3P1: cognoscit sic Ed.) generaliter quando fit iste syllogismus in secunda et quando in tertia, et qualiter. Et dicendum quod considerando praemissas ante conclusionis pertractionem est materia syllogismi secundae figurae vel tertiae indifferenter. Sed per voluntatem pertractantis est ut concludatur in secunda {secunda Cm1E1E2F2KlO2O3P1V: prima Ed} vel in tertia. Conclusio autem iam patet ubi fit syllogismus. Quia si concludatur subiectum de subiecto, fit syllogismus in secunda figura, si praedicatum de praedicato, figurae tertiae erit syllogismus, eo quod semper {semper Cm1E1E2F2KlO2O3P1V: similiter Ed} concluditur extremitas de extremitate et medium non ingreditur conclusionem.”

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But, he goes on, shouldn’t there be a fourth Figure, in which the Middle is predicate in the first premise and subject in the second? He responds that what differentiates the Figures from one another is the “categorial order”: … it is to be said that a syllogistic Middle either has the first condition in the categorial order (and then it is the second Figure), or the last condition (and then it is the third Figure), or the middle condition in the categorial order (and then it is the first). And it’s necessary for the Middle under such a condition to be first a subject and lastly a predicate. For, if it were lastly a subject and first a predicate, either the propositions taken would have to be false (namely, if universals were taken), or particulars would be taken (and then there would be no syllogism). It’s clear then that since there are only the three stated conditions of the Middle, there can’t be another Figure where the Middle is first a predicate and lastly a subject.100

The categorial order is the order exhibited by entities belonging to one of the ten Aristotelian categories. As the Dialectica Monacensis puts it: It is to be known therefore that corresponding to the ten genera of things or natures, ten orders of predicable terms are to be distinguished. And each order of like-natured predicables is called a category.101

The ordering relation is that of superior to inferior. This relation is interpreted by Ivo Thomas as being that between the predicate of an affirmative proposition and its subject.102 But this reading is too broad, since it does not restrict superiors and inferiors to items falling into a category. The terminology of superiors and inferiors goes back 100 Kilwardby ad A4 Part 2 dub.2 (10rb–va): “… dicendum quod medium syllogisticum aut habet conditionem primi in ordine praedicabili {AF 1KlO2V ordine predicamentali; EdBrE 1E2F2O3P1 ordine predicabili}, et tunc est figura secunda, aut conditionem ultimi, et tunc est figura tertia, aut conditionem medii in ordine praedicabili, et sic est prima. Et necesse est medium sub tali conditione subici primo et praedicari de postremo. Si enim subiceretur ultimo et praedicaretur de primo, aut oporteret accipere propositiones falsas, scilicet si acciperentur universales, aut accipere propositiones particulares, et ex talibus non est syllogismus. Patet igitur cum non sint nisi tres dictae conditiones medii, non potest esse figura alia ubi praedicetur medium de primo et subiciatur ultimo.” The idea that the syllogistic Figures somehow reflect the categorial order can be found in twentieth-century scholars, such as F. Solmsen, Die Entwicklung der aristotelischen Logik und Rhetorik (Berlin, 1929), 54: “Thus the reduction of all the other figures to the first … in fact means precisely their reduction to the ontologically rational line of terms”. (Quoted in Patzig, 87 n. 36.) 101 Dialectica Monacensis, 514:6–9. 102 Thomas, “Maxims in Kilwardby”, 132.

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to Boethius, who distinguishes between most general genera, lowest species, and intermediate species, saying: “Of things, some are superior, some inferior, some in between”.103 This is to think of superior and inferior in absolute terms, and to describe intermediate species as neither superior nor inferior. In Peter of Spain the notions of superior and inferior have lost their absolute character, and intermediate species are described as both superior and inferior, relative to different entities: “For they are genera with respect to their inferiors, and they are species with respect to their superiors”.104 (Peter’s words here echo the Latin translation of Ghazali.)105 This relation holds among entities, not linguistic items; and when it holds, it holds essentially, as Peter says: “For every superior is of the essence of its inferior”.106 In saying that in the second Figure the Middle occupies first position in this order, and in the third Figure it occupies last position, whereas in the first it occupies the middle position, Kilwardby is obviously thinking of a progression downwards in the categorial order, from superior to inferiors. As we saw earlier, Kilwardby draws a distinction between the Middle according to the linguistic arrangement [secundum dispositionem sermonis] and the Middle according to reality and nature [secundum rem et naturam]; and it appears that he has this distinction in mind here, so that when he talks about the Middle in the categorial order, it is the Middle according to reality and nature and not the Middle in the linguistic order. We can construct a model corresponding to Kilwardby’s reasoning. Consider any three distinct entities a, b, c; and consider a relation R (with converse ) defined on them so that each of the entities is related by R or to at least one of the other entities. If an entity is related by R to something, let’s say that it is an origin. If something is related by R to it, say it is a destination. Then of our three entities, either (1) two have a common origin but no two have a common destination, or (2) two have a common destination but no two have a common origin, or (3) 103 Anicii Manlii Severini Boethii, De divisione liber: critical edition, translation, prolegomena, & commentary by John Magee (Leiden: Brill 1998), 32, 23–24: Rerum enim aliae sunt superiores, aliae inferiores, aliae mediae. 104 Peter of Spain, 19:24–25. 105 Lohr, “Logica Algazelis: introduction and critical text”, 98–99: “respectu superioris dicitur species, respectu vero inferoris dicitur genus.” 106 Peter of Spain, 22:20–21.

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no two have a common origin and no two have a common destination. These three situations are pictured in Fig. 2.9. a

bRc (1)

aRb (2)

c

aRbRc (3)

Fig. 2.9. Three-Figure ontological model

In situation (1), b is the common origin of both a and c. In situation (2), b is the common destination of both a and c. In situation (3), a and b are origins, and b and c are destinations, but no entity is either an origin or a destination for both the remaining entities. These situations are ontological correlates of the three Figures. They are the three conditions in the categorial order descried by Kilwardby, and they correspond respectively to the second, third and first Figure. In situation (1) two inferiors a and c are ordered below a common superior b; and no conclusion can be drawn about an ordering relation between a and c. In situation (2) a single inferior b is ordered below two superiors a and c; and a, as a consequence, overlaps with c. In situation (3) a lower inferior a is ordered below a higher inferior b which is ordered below a superior c; and a, as a consequence, is ordered below c. He says that in situation (3) it’s necessary that the Middle term is subject in the first premise and predicate in the second, otherwise the premises would be false if they were universal, and would be useless if they were not. This assumes a downward progression through the categorial order. And it is indeed true, as he says, that if we try to reverse this ordering, for example by stating that Animal is inferior to Man and Substance to Animal, we misrepresent the categorial order (if these statements are meant as universal) or else we have stated something that is syllogistically useless (if they are meant as particular). The argument would be that a fourth Figure is not possible, because even though we might mention the sequence Man-Animal-Substance in a different order, there is nothing in the nature of things to warrant any variation of the categorial order. Against this line of reasoning it might be urged that in focusing on ontological questions about the entities in Aristotelian categories, Kilwardby forgets that the syllogism is a linguistic entity—its genus being discourse. Being is just as it is; but discourse may be true or false, and his ontological construction of the three Figures is accordingly

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open to the type of objection that Łukasiewicz raised, namely that it assumes without warrant that we are dealing with syllogisms having true premises.107 The assumption is unwarranted because a syllogism may well infer a true conclusion from false premises, as Kilwardby himself recognizes when commenting on B2, adding that it’s not in virtue of their being false that such premises imply a truth.108 But is such a syllogism a genuine syllogism? He answers: And it is to be said that a syllogism from falsehoods [can be considered] in two ways—in one way as it relates to the conclusion, whether that is true or false (and in this way it concludes of necessity, and it is a genuine syllogism); in another way as it relates to a true conclusion inferred from a falsehood or falsehoods (and in this way there is no necessity in it, nor is it a genuine syllogism but only in a certain respect).109

Now, if the truth or falsity of the premises can’t be taken into account when determining whether something is a genuine syllogism, surely it shouldn’t be taken into account when determining the number of syllogistic Figures. Kilwardby himself recognizes as much when he writes: “Further, Figure is not the placing of the Middle in truth, but a certain arrangement for placing; hence the placing brings the Figure about [efficit]”.110 So it’s not clear how seriously Kilwardby means us to take his categorial construction of the three Figures. Elsewhere, indeed, he underplays the connection: … it doesn’t always happen that the Minor is taken under the Middle as it subsists and in the predicamental line, but it suffices that it be taken under it conceptually, in whatever way it’s truly said of it.111

The question can be resolved if we remember his general attitude to definition:

107 Jan Łukasiewicz, Aristotle’s Syllogistic from the standpoint of modern formal logic 2nd edition (Oxford: Clarendon Press 1957), 28. 108 Kilwardby ad B2 Part 2 dub.1 (57ra). 109 Kilwardby ad B2 Part 2 dub.2 (57rb): “Et ita dicendum quod syllogismus ex falsis dupliciter est: uno modo prout respicit quamcumque conclusionem sive veram sive falsam, et sic ex necessitate concludit et vere syllogismus est; alio modo prout respicit conclusionem veram tantum, illatam ex falso vel ex falsis, et sic non insunt ex necessitate neque vere syllogismus est, sed secundum quid tantum.” 110 Kilwardby ad A23 dub.1 (34vb): “Adhuc figura non est situs medii in veritate sed dispositio quaedam ad situm. Unde ipse situs efficit figuram.” 111 Kilwardby ad A36–A37 dub.3 (46va): “… non semper accipitur minor sub medio in prima figura sicut subsistens et in linea praedicabili, sed sufficit quod secundum rationem accipiatur sub eo, quoquo modo vere se habeat medium dictum de eo.”

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… it is to be said that if some property needs to be defined, it should be defined according to that manner by which it happens to those things in which it primarily inheres. For its primary and most excellent being is there.112

Assuming (as an Aristotelian realist well might) that the primary instances of the relationships exemplified by the syllogistic Figures are given in the categorial order, then it would be reasonable to define the Figures by reference to that order, even though there will of necessity be (non-primary) instances that depart from it. We should also bear in mind that Kilwardby in his later work, De Ortu Scientiarum, posits conceptual connections between the linguistic, the rational and the ontological orders. Logic, he says, is a rational science because it studies modes of reasoning; but it is also a linguistic [sermocinalis] science because it studies reasoning expressed in language. “Hence it is well said to be about rational discourse [sermone ratiocinativo], or about discursive reasoning [ratiocinatione sermocinata]”. In addition, he says, logic (like metaphysics) is about beings in general. Accordingly: “… the subject of logic is in one way said to be reasoning, in another way, reasoned discourse, and in a third way reasonable [ratiocinabile] being without qualification …”.113

Summary Aristotle’s definition sets the syllogism apart from other inferences— though there are disagreements about exactly how the definition should be interpreted. He also suggests that other types of inference (such as Induction and Enthymeme) are reducible to the syllogism. Kilwardby manages to combine both these perspectives. Interpretative disagreements aside, it’s clear that the definition of Syllogism states that the conclusion follows from the premises precisely by virtue of those premises. It follows from this that the syllogism is an essential rather than an accidental inference, in the sense that the consequent follows by virtue of the antecedent, rather than following by virtue of some feature intrinsic to itself. Kilwardby opines that the syl112 Kilwardby ad B8 dub.6 (64rb): “… dicendum quod si aliqua proprietas debeat definiri, debet definiri secundum illum modum quo contingit ei cui primo inest; ibi enim est esse eius excellentissimum et primum.” 113 Kilwardby, De Ortu Scientiarum, 578.

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logism also shares an epistemological feature with other essential inferences, namely that the consequent in understood in the antecedent—in the sense that to know or opine the premises is to know or opine the conclusion, provided we think them through. This opinion of Kilwardby’s demonstrates that his curiosity about the syllogism is not confined to its logic. And indeed, besides investigating its epistemology he also ventures into metaphysical analysis, distinguishing its proximate from its remote matter (propositions from terms), and its proximate from its remote form (Mood from Figure). These distinctions allow him to conclude that a syllogism may be incompletely formed—if it is shaped by a remote form but lacks a proximate form. He also takes a metaphysical stand on the issue of the number of syllogistic Figures. Kilwardby takes a teleological and a functional view of the syllogism, as directed towards showing its conclusion and making it known. Drawing on this view, he can explain why it is that question-begging arguments or arguments from opposed premises are not good syllogisms. The perfect syllogisms are preeminent both from an axiomatic point of view and in the degree to which they exemplify the syllogism’s Final Cause. Extending the earlier discussion by Patzig and drawing on Kilwardby’s notion of assimilation, I have offered an analysis of perfection in terms of monotonicity.

chapter three REDUCTION

In his discussion of A33, Kilwardby distinguishes four senses in which the word “reduction” is used with reference to syllogisms:1 – the process of turning a potential syllogism (a yet-to-be-syllogized sentence, arranged2 in a confused and disorderly way) into an actual syllogism (a syllogized sentence); – among already syllogized sentences, the reduction of a syllogism in one Figure to a syllogism, with the same conclusion, in any other Figure; – the reduction of imperfect Figures to perfect; – the reduction of imperfect Figures to primary perfect syllogisms. Kilwardby describes the chapters immediately prior to A45 as dealing with type 1 reduction, whereas A45 deals with the issue of reducing a “syllogized sentence” (syllogizata) that is in one Figure to another Figure.3 Robin Smith makes a similar observation, noting that “the subject of the Chapter is not ‘resolving’ an arbitrary given argument into a figured argument, but ‘leading back’ a given figured argument from one figure into another”.4 About type 2 reduction Kilwardby says that it “is not on account of the perfection of syllogisms, but so that it can be clear how many ways the same [conclusion] can be syllogized from the same Middle altered”.5 To a modern logician, A45 reads as if it is presenting ways of basing the syllogistic on axioms other than the perfect syllogisms of the first Figure.6 But this perspective shouldn’t lead us to confuse the ques1

Kilwardby ad A32–33 dub.1 (44ra–b). Ed has “disputata”, as do E 1O2O3P1V. However, Cm1E2F1F2Kl all have “disposita”. 3 Kilwardby ad A45, 50b5 ff., (49rb). 4 Aristotle, Prior Analytics translated, with introduction, notes, and commentary, by Robin Smith (Indianapolis: Hackett 1989), 177. 5 Kilwardby ad A32–A33 dub.1 (44rb): “… non est propter syllogismorum perfectionem, sed ut pateat quot modis idem syllogizari possit per idem medium alteratum.” 6 Thom, The Syllogism, 97–101. 2

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tion of axiomatization with that of perfectibility. Kilwardby notes that a syllogism can be perfected only by reduction to the first, not by reduction to any other, Figure. Accordingly, he notes that many of the reductions discussed in A45 do not count as ways of perfecting a syllogism.7 A45 doesn’t consider the possibility of reducing a syllogism to another syllogism in the same Figure, in the way that Camestres might be reduced to Cesare by converting its conclusion, or Celarent to Camestres, or Darii to Disamis: cae Cesare

abe aca

e --- conv

bce cbe

Camestres

aba

e --- conv

ace bce cbe

e --- conv

i --- conv Disamis

cai aci bca

i --- conv

abi bai

Fig. 3.1. Direct reductions of Camestres to Cesare, Celarent to Camestres, and Darii to Disamis

Kilwardby doesn’t mention this omission explicitly; but he does opine that Aristotle’s intention in A45 is to confine his discussion to cases where the starting syllogism and the end-syllogism share a conclusion.8 This restriction would explain why Aristotle doesn’t discuss indirect reductions in A45; and Kilwardby notes this.9 It would also explain (though Kilwardby doesn’t mention this) why Aristotle doesn’t mention subaltern or indirect moods. Type 3 and type 4 reductions are central to Aristotle’s project of constructing a theory of the syllogism, for it is through such reductions that imperfect syllogisms are perfected. Imperfect syllogisms may be perfected by two main methods—Direct and Indirect Reduction. Aristotle also makes sparing use of two other techniques—Exposition and Upgrading. Kilwardby has something interesting to say about all these techniques of reduction.

7 8 9

Kilwardby ad A45 dub.1 (49vb). Kilwardby ad A45 dub.2 (49vb). Kilwardby ad A45 dub.6 (50ra).

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Conversion He engages in a subtle and sustained discussion about the nature of conversion, as well as posing specific queries about particular Aristotelian laws of conversion. The nature of conversion He asks which branch of the art of Logic Conversion belongs to. Since it is neither a Definition, nor a Division, nor an Example, nor an Induction, nor a Syllogism, it seems that it must be an Enthymeme. Pursuing this hypothesis, he shows how Conversion can be represented in the form of an Enthymeme whose unexpressed premise states the Law of Identity (that all b is b). I interpolate my comments in his text: Which branch of the art does conversion belong to? For, it is not Definition or Division. Hence it seems that it will be an Argument. But it is not Example, since it does not infer like from like. Nor is it Induction, as is clear. Nor is it Syllogism, since it doesn’t have two propositions and three terms. So it is Enthymeme. But on the contrary, an Enthymeme has three terms, just like a syllogism, whereas conversion has only two. But then it is said that it is an Enthymeme and has three terms conceptually [secundum rationem], even though it has two in reality, just as it is with a syllogism from opposites which has two terms in reality, which two are three conceptually, e.g. “No learning is learned, all learning is learned, so no learning is learning”, and it is said to be similar therefore with conversion. But then it is asked, since every enthymeme is perfected by a syllogism, how it is that conversions as determined here reduce to a syllogism? And it may be said that the conversion of the universal negative is an enthymeme existing in potentiality with respect to a syllogism in the first of the second Figure [Cesare] by the addition of a Minor, thus: No b is a, every a is a, so no a is b. It is to be said also that the conversion of the universal affirmative is a possible enthymeme in respect of the first mood of the third Figure [Darapti] with the addition of a Major, thus: Every b is b, every b is a, so some a is b. And the conversion of the particular affirmative is a possible enthymeme in respect of the fourth of the third [Datisi] by the addition of a Major, thus: Every b is b, some b is a, so some a is b. Thus conversions are enthymemes, and are perfected by the said syllogisms.10 10

Kilwardby ad A2 dub.14 (6rb–7ra): “… dubitatur de conversione quae via artis est.

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The syllogistic proofs of the laws of assertoric conversion are shown in Fig. 3.2. Cesare

abe

aaa bae

Identity

Identity Darapti, Datisi

bba

aba,i bai

Fig. 3.2. Syllogistic proofs of assertoric conversions

He continues: But to the contrary, imperfect syllogisms, namely those in the second and third Figures, are perfected by conversion, and conversion if it is an enthymeme is perfected by an imperfect syllogism, as was just said; so if conversion is an enthymeme and is perfected by an imperfect syllogism, there will be a circular perfection, and the same thing will be perfecting itself, which is unacceptable, so it is unacceptable that conversion is an enthymeme. Further, the conversion of the universal negative if it is an enthymeme is perfected (as was said) by the first syllogism of the second Figure [Cesare], but that same syllogism is imperfect and is perfected by converNon enim est definitio nec divisio, quare videtur quod erit argumentatio. Sed non est exemplum cum non inferat simile ex simili, nec etiam inductio, ut patet, nec syllogismus quia non habet duas propositiones et tres terminos; ergo enthymema. Sed contra: enthymema habet tres terminos sicut syllogismus, conversio autem tantum duos. Sed dicendum quod est enthymema et habet tres terminos secundum rationem licet tamen duos secundum rem, sicut est de syllogismo ex oppositis qui tantum habet duos terminos secundum rem, qui duo sunt tres secundum rationem, {licet … rationem vel sim. ACm1CrE1E2F1F2KlO2O3P1V: sicut et aliquis Ed}. Verbi gratia, nulla disciplina est studiosa, omnis disciplina est studiosa, ergo nulla disciplina est disciplina. Dicet ergo quod similiter est de conversione. Sed tunc quaeritur cum omne enthymema perficiatur per syllogismum, qualiter conversiones hic determinatae reducuntur in syllogismum. Et dicit quod conversio universalis negativae est enthymema existens in potentia respectu syllogismi primi secundae figurae per additionem minoris sic: nullum b est a, omne a est a, ergo nullum a est b. Dicendum est quod conversio universalis afirmativae est enthymema possibile respectu primi modi tertiae figurae per additionem maioris sic: omne b est b, omne b est a, ergo aliquod a est b. Conversio autem particularis affirmativae est enthymema possibile respectu quarti tertiae per additionem maioris sic: omne b est b, aliquod b est a, ergo aliquod a est b. Sic igitur sunt conversiones enthymemata et per dictos syllogismos perficiuntur.” Ivo Thomas O.P., “Kilwardby on conversion” Dominican Studies 6 (1953): 56–76, 65–66 gives the text of this passage based on O2 and O3. The notion of the number of terms secundum rationem being greater than the number secundum rem can be explained by means of the notion of a substitution instance. See Thom, The Syllogism, 32–34.

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sion of the universal negative, so the conversion of the universal negative is perfected by itself, and this is unacceptable, so it is unacceptable to lay down that conversions are enthymemes.11

The proof of Cesare by Direct Reduction is shown in Fig. 3.3. e --- conv Celarent

abe bae

aca

bce

Fig. 3.3. Direct reduction of Cesare It might be said that conversion is an enthymeme and is perfected by a syllogism in the second or third Figure, and on the other hand a syllogism in the second or third Figure is perfected by conversion, but these perfections are different from one another, and so this is not unacceptable, because conversion if it is an enthymeme is perfected as to its necessity (for an enthymeme does not have necessity of itself but from the syllogism) whereas a syllogism is perfected by conversion, not as to its necessity but its evident necessity; and so it is in different ways that conversion perfects and is perfected by a syllogism, and there is nothing unacceptable.12

11 Kilwardby ad A2 dub.14 (6va): “Sed contra: syllogismi imperfecti, scilicet syllogismi secundae et tertiae figurae, perficiuntur per conversionem, et conversio, si sit enthymema, perficitur per syllogismum imperfectum, ut iam dictum est. Ergo si conversio sit enthymema et perficitur per syllogismum imperfectum, erit ibi circularis perfectio {perfectio ABrCm1CrE1F1F2KlO2O3P1V: imperfectio Ed} et idem seipsum perficiens. Sed hoc est inconveniens. Quare inconveniens est quod conversio sit enthymema. Adhuc conversio universalis negativae si sit enthymema perficitur, sicut dictum est, per primum syllogismum secundae figurae {per primum syllogismum secundae figurae: E 1F1O2O3P1: per syllogismum primum secundae figurae BrV : per syllogismum secundae figurae E 2F2K1: per syllogismum primae figurae Cr: per syllogismum primae et secundae figurae Cm1: regulam primum syllogismum Ed}. Sed ille idem syllogismus est imperfectus et perficitur per conversionem universalis negativae. Quare conversio universalis negative per seipsam perficetur. Sed hoc est inconveniens. Ergo inconveniens est ponere quod conversiones sint enthymemata.” 12 Kilwardby ad A2 dub.14 (6va): “Si dicetur quod conversio sit enthymema et illud perficitur per syllogismum secundae et tertiae figurae et e converso syllogismus secundae et tertiae figurae perficitur per conversionem, quia tamen diversa est haec et illa perfectio, ideo non sequitur inconveniens. Conversio enim cum sit enthymema perficitur per syllogismum quo ad necessitatem (enthymema enim de se non habet necessitatem, sed a syllogismo), syllogismus autem perficitur per conversionem non quo ad necessitatem, sed quo ad evidentiam necessitatis, et ideo diversimode perficit conversio syllogismum et perficitur a syllogismo, et ita non est inconveniens.”

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This reasoning appears to invoke the distinction between perfection of being and perfection of well-being outlined in Chapter 2. An enthymeme is perfected in the first sense when it is reduced to a syllogism; an imperfect syllogism is perfected in the second sense when it is reduced to the first Figure. But things are not quite as they seem, because the reasoning claims not just that the enthymeme is perfected in the sense of being reduced to syllogistic form but that it in addition it is made necessary by that reduction. The suggestion is that the syllogism causes necessity to inhere in the enthymeme: But on the other side, any necessity that a syllogism gives an enthymeme is a necessity that it already has, so the syllogism has necessity; and by the same token, any evident necessity which the enthymeme-conversion gives to a syllogism, it already has, so the conversion has in itself evident necessity and a fortiori has necessity, because just as the syllogism is with necessity so the conversion is with evident necessity. But the opposite was said in the response.13

Here Kilwardby builds on the previous suggestion that the syllogism causes necessity in the enthymeme, by invoking the Aristotelian doctrine that the cause of a Form’s inhering in a subject must already possess that Form. The argument on the other side continues: Some people agree with this, saying that conversion is not an enthymeme nor another argumentation nor does it prove anything but merely takes it for granted, for “No b is a” and “No a is b” are simply not different from one another but say the same thing with transposed terms. And they say that conversion is a property or relation of terms joined by “to be” or “not to be”. And this seems to be what Aristotle has in mind since he often says that a proposition converts in the terms, and thus he seems to be attributing conversion primarily to the terms; and this also appears in the definition of imperfect syllogisms when he says conversion is necessarily had through the terms taken (not the propositions taken); and Aristotle also appears to follow this usage at the end of the second Figure in the generation of uniform assertoric syllogisms,14 as can be seen by inspection. And this is said quite subtly.15 13 Kilwardby ad A2 dub.14 (6va): “Sed contra: si syllogismus det enthymemati necessitatem, non nisi necessitatem quam habet, quare syllogismus habet necessitatem; ergo eadem ratione si enthymema quod est conversio det syllogismo evidentiam necessitatis, hoc non potest esse nisi det evidentiam quam habet, quare conversio habet penes {penes ABrCm1CrE1E2F1F2KlO2O3P1: prius Ed} se evidentiam necessitatis, quare multo fortius necessitatem, quia sicut syllogismus necessitatem, sic et conversio, et adhuc necessitatis evidentiam; cuius oppositum dictum est in respondendo.” 14 A5, 28a4–7. Kilwardby 12ra. 15 Kilwardby ad A2 dub.14 (6va): “Huic consentiunt aliqui dicentes {Huic consentiunt

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He replies to the first of these objections by modifying the doctrine that a cause must already possess the Form caused, in such a way as to deflect the objection to conversion’s being an enthymeme: But if you wish to maintain the other path and hold that conversion is an enthymeme, you can reply to the reason on the other side thus. When it is said that if conversion produces evident necessity in a syllogism it has evident necessity in itself, such that evident necessity is in it, the argument is not valid. For it need not be the case that every cause of something has in itself the form caused as its own form; but it does need to be the case that it has something similar in a more excellent way. For example, the Sun is the cause of heat, but it doesn’t have heat as its own form,16 but rather has it as a cause (namely an efficient cause). Similarly, substance causes accidents but doesn’t need to have the accidents in itself as its form but has in itself something belonging to the accidents in a more excellent way. And similarly I say about conversion and syllogism: even though conversion makes necessity evident in a syllogism, it doesn’t follow that it has evident necessity in itself; but rather something belonging in a more excellent way, whatever that may be.17

aliqui dicentes BrCm1CrE1E2F1F2KlO2O3P1: Huic consueverunt aliqui dicere Ed} quod conversio non est enthymema nec alia argumentatio nec etiam probat aliquod, sed sumit tamen. Non enim diversum simpliciter dicunt ‘Nullum b est a’ et ‘Nullum a est b’, sed idem in situ transmutatum {transmutatum ABrCm1CrE1E2F1F2KlO3P1: transmutant O2: transmutetur Ed}. Et isti dicunt quod conversio est proprietas sive habilitas quaedam terminorum coniunctorum per esse et non esse, et istud videntur verba Aristotelis sapere cum dicit quod propositio convertitur in terminis. Videtur enim attribuere conversionem terminis primo. Hoc etiam videtur per definitionem syllogismi imperfecti cum dicit conversionem necessario haberi per sumptos terminos, non autem per sumptas propositiones. Hoc etiam videtur Aristotoles sumere versus finem secundae figurae in generatione syllogismorum uniformiter de inesse, sicut patet inspicienti. Et illud satis subtiliter dictum est.” 16 Aristotle, De Caelo translated by J.L. Stocks, in The Works of Aristotle translated into English under the editorship of W.D. Ross vol. 2 (Oxford: Clarendon Press, 1930) II.7, 289a29–34: “… the upper bodies are carried on a moving sphere, so that, though they are not themselves fired, yet the air underneath the sphere of the revolving body is necessarily heated by its motion, and particularly in that part where the Sun is attached to it. Hence warmth increases as the Sun gets nearer or higher or overhead.” 17 Kilwardby ad A2 dub.14 (6va): “Si tamen placet aliam viam sustinere, dicendum quod conversio est enthymema. Et potest responderi ad rationem in oppositum sic: cum dicit si conversio facit evidentiam necessitatis in syllogismo, ergo habet penes se evidentiam necessitatis, ita quod in ipsa sit evidens necessitas, argumentum non valet. Non enim oportet quod omnis causa alicuius habeat penes {penes ABrCm1CrE1E2F1F2 KlO2O3P1: prius Ed} se formam causati sicut formam sui, sed oportet quod habeat aliquid simile {simile Cm1CrE1E2F1F2KlO3P1V: substantiam Br: substantiae Ed} modo

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These last words are tantalizing; but I think it’s clear what Kilwardby means. The laws of conversion don’t possess evident necessity in the same sense as do the perfect syllogisms; but they do possess something that can endow a syllogism with that evident necessity. This is the power of re-ordering the terms in the syllogism, for it is the order of terms that the doctrine of perfection takes as determining the evidentness of a perfect syllogism. So the idea that conversion might be an enthymeme can be saved from the objection based on the supposed attributes of a cause. He raises a final difficulty: For, after the enthymeme is reduced to an imperfect syllogism, that syllogism has to be reduced further to a perfect syllogism by the conversion of some proposition which after conversion will be the same as the conclusion. And so in this syllogism thus perfected the conclusion will not be other than the premises—which is contrary to the definition of syllogism. This is clear of itself in any conversion if it is reduced to a syllogism. And so the ultimate perfection of conversion will be by conversion, and so the same thing will be prior to itself.18

The repetitive pseudo-arguments to which Kilwardby refers are shown in Fig. 3.4. e --- conv Celarent

abe bae

aaa bae

Identity

Identity Darii

aba,i bba

bai

a, i --- conv

bai

Fig. 3.4. Apparent syllogistic proofs of conversion tamen excellentiori. Verbi gratia, sol causa caloris est, non tamen habet calorem sicut formam sui, sed habet eum sicut causam, scilicet efficientem {efficientem ABrCm1CrE2 F1F2Kl: effective EdE 1O2O3}. Similiter substantia causat accidens, non tamen oportet quod habeat in se accidens sicut sui formam, sed habet in se aliquid accidenti conveniens modo tamen excellentiori. Sic similiter dico de conversione et syllogismo. Quamvis enim conversio faciat evidentiam necessitatis in syllogismo, non tamen sequitur quod in se habet necessitatem evidentem, sed quod aliquid conveniens modo tamen excellentiori, quicquid illud sit.” 18 Kilwardby ad A2 dub.14 (6va–vb): “Postquam enim enthymema reducitur in syllogismum imperfectum, adhuc habet ille syllogismus reduci ulterius in syllogismum perfectum {adhuc … perfectum ACm1CrF1F2KlO2O3P1V: om. BrEd}, et hoc per conversionem alicuius propositionis quae per conversionem erit eadem conclusioni. Et sic in illo syllogismo sic perfecto non erit conclusio aliud a praemissis, quod est contra definitionem syllogismi simpliciter. Istud autem per se patet in omni conversione si in syllogismum reducatur. Et sic adhuc ultima perfectio conversionis erit per conversionem, et erit idem prius se.”

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And so perhaps it is safer to say that conversion is not an argument. And then it should be said (as was said before) that conversion does not prove anything but takes it for granted (which, however, doesn’t seem hard to maintain). Or else it should be said that in conversion there is a consequence from one thing to another but not an argument. And then among consequences we would distinguish some whose terms are three in reality and conceptually, in such a way that they can constitute two propositions with a conclusion different from the premises (for of necessity this holds of every argument, in that every argument is reducible to a perfect syllogism). Conversion is not such a consequence, but such a consequence produces an argument. Other consequences are of one thing with another (or rather with its altered self), and these are sometimes made in two terms that do not produce a perfect syllogism having two propositions different from the conclusion, and such a consequence is not the consequence of a genuine argument but of that which is in a certain way related to an argument (se habet aliquo modo ut argumentum), and conversion is such a consequence.19

So he finishes by rejecting the assumption that conversion is any kind of argument (including Enthymeme), and yet he accepts that it can be represented in the form of an Enthymeme. He denies that conversion possesses the evident necessity of perfect syllogisms, and yet affirms that it has the power to perfect imperfect syllogisms. He denies that it draws its necessity from the necessity of the syllogism, and yet he affirms that it is a kind of necessary consequence.20 19 Kilwardby ad A2 dub.14 (6rb–7rb): “Et ideo forte securius est dicere quod conversio non sit argumentum. Et tunc oportet dicere, sicut prius dictum est, quod conversio nihil probat sed sumit, quod tamen videtur non {non BrCm1E1E2F2V: om. EdACrF 1KlO2 O3P1} difficile sustinere. Vel oportet dicere quod in conversione est consequentia unius ad alterum, non tamen argumentum. Et tunc distinguendum est de consequentia quod quaedam est in terminis quae sunt tres secundum rem et rationem ita quod possunt constituere duas propositiones et unam conclusionem diversam a praemissis—hoc enim est de necessitate cuiuslibet argumenti eo quod omne argumentum in syllogismum perfectum {in syllogismum perfectum ABrCm1CrE1F1F2O2O3P1V: om. Ed} reducibile est— sed talis consequentia non est in conversione {Sed … conversione E 1F1KlO2O3P1: om. EdBrCm1CrF2V}, sed talis consequentia est quae facit argumentum. Alio modo est consequentia quae est convertentia unius com altero, vel potius eiusdem cum seipso alterato, et ista aliquando sit in duobus terminis qui non faciunt syllogismum perfectum habentem duas propositiones diversas {diversas ABrCm1CrE1E2F1F2KlO2O3P1V: om. Ed} a conclusione, et ideo talis consequentia non est consequentia veri argumenti sed eius quod se habet aliquo modo ut argumentum. Talis autem est consequentia conversionis.” Kilwardby’s lengthy discussion of the nature of conversion is commented on by Thomas, “Kilwardby on conversion”, 72–76, nn. 15, 22 and 24. 20 Ebbesen, “Albert (the Great?)’s companion to the Organon”, 95 says that Albert

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Assertoric propositions Aristotle holds that universal and particular affirmatives convert to particular affirmatives, and universal negatives convert to universal negatives.21 Affirmative assertoric propositions As a counter-example to the convertibility of the universal affirmative, Kilwardby suggests that even if it’s true that something white is a man it isn’t true that every man is white. His solution is to distinguish conversion with a proposition from conversion to that proposition. The universal affirmative converts to a particular affirmative but doesn’t convert with it.22 He suggests a counter-example to the convertibility of particular affirmative propositions. Some fire is extinct, but it’s not true that something extinct is fire. He replies that per se the sentence “Some fire is extinct” is false; what is true is that something that was fire is extinct.23 Indefinite propositions obey the same conversion-laws as particulars.24 Aristotle’s discussion is focused on particulars rather than indefinites because they are more commonly used (especially in syllogisms). Also, indefinites can be seen as derivative upon particulars, by deletion of the explicit quantifier that is present in particulars.25 He also treats the conversion of singular propositions, in so far as it falls under the syllogistic art, as being comprehended under that of particulars.26

follows Kilwardby in all this, “except on one important point: he feels no qualms about accepting the claim that conversion is ‘taking the same, arranged in a different way’ and so he endorses that view without reservation, endorsing at the same time Kilwardby’s final solution, that there are two kinds of consequence”. 21 A2, 25a17–19, 20–22, 15–16. 22 Kilwardby ad A2 dub.9 (6rb). 23 Kilwardby ad A2 dub.10 (6rb). 24 Kilwardby ad A2 dub.4 (5vb–6ra). 25 Kilwardby ad A2 dub.5 (6ra). 26 Kilwardby ad A2 dub.6 (6ra).

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Negative assertoric propositions Kilwardby considers some standard objections against e-conversion. No old man will be a boy, but it’s not the case that no boy will be an old man. No boy was an old man, but it’s not the case that no old man was a boy. No citizen is female, but it’s not the case that no female is a citizen.27 Kilwardby suggests two solutions to these counter-examples. To start with, he appeals to a rule of appellation, according to which the propositions about old men and boys are ambiguous, depending on what the subject-term stands for. The rule in question is that a common term within the scope of a past-tense verb can stand for those things that are or for those things that were. He points out that if the subject of the converse has the same temporal reference as the predicate of the convertend, there is no counter-example. It’s only if the subject of the convertend stands for no-one who will be an old man but the predicate of the converse does, that the inference breaks down. He adds a second solution, according to which the intended temporal reference can be made explicit by reducing the proposition to one containing a present-tense verb, putting the words “present” or “future” into the subject or predicate. If no future old man will be a present boy then no present boy will be a future old man. Ebbesen points out that Albert jumbles these two solutions together in the formula “No presently existing or future old man is a boy”, which he claims is true and has a true converse.28 Regarding the sophism about female citizens, Kilwardby states simply that if “citizen” is used throughout unambiguously (for example for male citizens) then the objection collapses. He also raises a question about the conversion of reduplicative propositions such as “Nothing that is to be chosen is evil in so far as it is to be chosen”. He says that such a proposition doesn’t convert in such a way as to retain the reduplication, though it does convert if we just take the main terms. Thus “No evil is to be chosen” converts to “Nothing that is to be chosen is evil”. But “No evil is to be chosen in so far as it is evil” does not convert to “Nothing that is to be chosen 27 Kilwardby ad A2 dub.7 (6ra). The grammatical gender of “citizen” [civis] is masculine. For the rule of appellation, see Thomas, “Kilwardby on conversion”, 56–76. 28 Kilwardby ad A2. Albert. See Ebbesen, “Albert (the Great?)’s companion to the Organon”, 93–94.

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in so far as it is evil, is evil”. He points out that when we are making a conversion, we do not take everything that is in the subject and put it in the predicate; for instance we do not take the “Every” in “Every man is an animal”. The reason is that “every” modifies the subject in so far as it is a subject: “It is the same with a reduplication in the predicate. Reduplication is a condition of the predicate in relation to the subject, and in so far as it is a predicate”.29 The valid conversions are “No evil is to be chosen in so far as it is evil; so nothing that is to be chosen is evil” and “so nothing that is to be chosen is evil in so far as it is to be chosen”: “So that the reduplication is always made of the subject to the predicate; for otherwise there will be a false or nugatory proposition”.30 He raises the question whether Aristotle’s proof of e-conversion involves a circularity, since e-conversion is proved via i-conversion, and i-conversion is proved via e-conversion. He replies that there are three different ways in which e-conversion can be proved without assuming i-conversion. It can be proved by an expository syllogism as shown in Fig. 3.5: bc ac

Exposition

→

abi

bai abi

C →

abe bae

Fig. 3.5. Expository proof of conversion, with singular terms

(c is here assumed to be a sensible particular.) Second, it can be proved by a syllogism in Ferio, as shown in Fig. 3.6: Ferio

abe bai aao

→

abe bae

Fig. 3.6. Reduction of e-conversion to Ferio

Since these premises lead to “the greatest unacceptability, namely that something is denied of itself ”, we can infer that one premise implies the

29 Kilwardby ad A38–A40 dub.5 (47vb): “Eodem modo est de reduplicatione ex parte praedicati. Reduplicatio conditio praedicati est in comparatione ad subiectum et in quantum praedicatum est.” 30 Ibid. “… ita quod semper fiat reduplicatio subiecti ad praedicatum. Aliter enim erit propositio falsa et aliquando nugatoria.”

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opposite of the other. (This proof was known to Alexander.)31 Third, we can appeal to the maxim “A superior cannot be denied of that which belongs to its inferior”. Hence, when a is superior to c, and b is predicated of c, ex hypothesi it will be false to deny that a is said universally of b. Hence, when c is a certain a, and c is b, the first (namely “No b is a”) will be false.32

Regarding o-conversion, Kilwardby notes that, even though “No b is a” implies “Some b is not a”, and “No a is b” implies “Some a is not b”, it doesn’t follow that the second of these propositions implies the fourth. The fallacy is that of assuming that what follows from the antecedent follows from the consequent.33 Contraposition Boethius,34 unlike Aristotle, recognizes as valid the process of conversion by contraposition, as shown in Fig. 3.7. aba b a

a

Fig. 3.7. Conversion by contraposition

Kilwardby asks why Aristotle doesn’t assert this type of conversion in A3. He points out that strictly speaking contraposition isn’t a conversion-principle, since the original terms are not terms in the contraposed proposition, and that Boethius applies the name of conversion beyond its strict meaning.35 But, of course, this doesn’t address the question of its validity. For Kilwardby’s views on the validity of contraposition we have to turn to his commentary on A46, where he raises the following objection to it:

Alexander ad A2, 25a14–17 (34, 17–20). See Thom, The Syllogism, 94. Kilwardby ad A2 dub.8 (6rb): “Quare cum a sit superius ad c et de c praedicatur b, ex hypothesi falsum erit negare a dici b universaliter. Quare cum c sit aliquod a et c sit b, falsum erit primum, scilicet ‘Nullum b est a’.” 33 Kilwardby ad A2 dub.11 (6rb). 34 Boethius, Introductio ad syllogismos categoricos in Opera Omnia II. Ed. J.-P. Migne. Patrologia Latina 64 (Paris: 1860), 804A. 35 Kilwardby ad A2 dub.13 (6ra). 31 32

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chapter three Further, if every man is an animal, every non-animal is a non-man; but a stone is a non-animal, so a stone is a non-man—and if so then a stone exists; hence from the first, if every man is an animal, a stone exists. But this doesn’t follow.36

In reply, he acknowledges that “Non-man” means “A being that is not a man”; but, as we saw in Chapter One, he draws a distinction between beings according to nature (“such as wood or stone or the like”) and beings according to reason or understanding. He explains: “The case is similar with what we say when we say that, in seeing darkness I see nothing. For, in this way, in understanding ‘Non-man’ I understand nothing”.37 So, the proposition “Every non-animal is a non-man” doesn’t imply the existence of stones, and the objection to contraposition is answered. It seems that he accepts contraposition as valid. Necessity-propositions Affirmative necessity-propositions Kilwardby considers some apparent counter-examples to the convertibility of affirmative necessity-propositions. The propositions “All who are literate are of necessity men”, “All who are healthy are of necessity animals”, “All who are awake are of necessity animals” are true, but their converses are false. In reply, Kilwardby states that in the first of these propositions, the term “literate” stands for its supposita, not for the quality of literacy; however, in the converse proposition, “Some man is of necessity literate” it stands for that quality. The reason for this is that names38 such as “literate” stand for their supposita when they are 36 Kilwardby ad A46 Part 1 dub.10 (52rb–va): “Adhuc si omnis homo est animal, omne non animal est non homo. Sed lapis est non animal, ergo lapis est non homo. Et si hoc, ergo lapis est. Quare a primo, si omnis homo est animal, lapis est. Istud non sequitur …” 37 Kilwardby ad A46 Part 1 dub.10 (52va): “… est simile de eo quod dicimus quod videndo {videndo BrCm1E1E2F1F2KlO2O3P1V: vidente Ed} tenebras nihil video; sic enim intelligendo non hominem nihil intelligo.” 38 Ed (7ra) has “talis enim est natura abstractorum quod cum subiiciuntur stant pro suppositis cum autem predicantur stant pro qualitate et forma”. Lagerlund, Modal Syllogistics in the Middle Ages, 27 n. 37 refers to Cm1, which has “subiectorum et predicatorum terminorum” in the place of “abstractorum”. This (as he says) “makes much more sense, since Kilwardby is discussing concrete accidental terms”. The Cm1 reading is confirmed in ABrCrE 1E2F1F2KlO2O3P1V.

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in subject-position but stand for the quality or form when they are in predicate-position. And it is this different meaning of the term in its two occurrences that gets in the way of the conversion. This response assumes that the proposition “All who are literate are of necessity men” is true.39 He goes on to offer another reading—one that takes the proposition “All who are literate are of necessity men” to be false:40 Alternatively, it can also be said with fair credibility that those propositions with the name of an Accident as subject are necessity-propositions only per accidens, not per se. … Therefore, when Aristotle teaches that necessity-propositions convert, he is only teaching that per se necessitypropositions convert; however, the counter-examples that are produced are per accidens necessities, and so all the counter-examples fail.41

This is not really an alternative response. The two are complementary —the first ruling out per accidens necessities, the second ruling in per se ones. As we saw in Chapter One, per se necessity-propositions require that both their terms be per se. It is because of this symmetry that they obey the standard laws of conversion. Negative necessity-propositions In Chapter One we found Kilwardby requiring the same subject / predicate symmetry in affirmative and in negative necessity-propositions: both terms must be per se. Thus we would expect negative as well as affirmative necessities to obey the standard laws of conversion. And this indeed is Aristotle’s position, endorsed by Kilwardby: It happens in the same way for necessary propositions. For, the universal privative converts universally, while both affirmatives convert in part. For, if it’s necessary for a to inhere in no b, it’s necessary also for b to inhere in no a. For, if it’s contingent for some, it’s also contingent for some b to be a.42

See Lagerlund, Modal Syllogistics in the Middle Ages, 25–28. See Lagerlund, Modal Syllogistics in the Middle Ages, 28–39. 41 Kilwardby ad A3 Part 1 dub.4 (7rb): “Aliter etiam dici potest satis probabiliter, scilicet quod huius propositiones subicientes nomen accidentis non sunt per se de necessario sed per accidens tantum. … Quando ergo Aristoteles docet convertere propositiones de necessario, solum docet convertere propositiones de necessario per se; instantia autem facta est in propositionibus quae sunt per accidens de necessario, et sic pereunt omnes instantiae.” 42 A3, 25a27–32. Kilwardby 6vb. 39 40

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Aristotle’s proof of Le-conversion appears to assume the convertibility of particular affirmative contingency-propositions; but this conversion hasn’t been established at this point in the text. Kilwardby, however, doesn’t read Aristotle’s proof as resting on contingency-conversion: Since it is contingent that some a is b, nothing unacceptable follows from supposing it to be the case. And if it is supposed to be the case that some a is b, it converts; therefore some b is a. And this can’t stand with the first, namely “Necessarily no b is a”.43

In other words, he sees Aristotle’s proof as having the structure shown in Fig. 3.8. i --- conv

bai abi

R1

→

Mbai Mabi

C

→

Labe Lbae

Fig. 3.8. First proof of Le-conversion

This version of Aristotle’s proof presupposes rule R1 which licenses the inference from the validity of i-conversion to that of Mi-conversion. This rule might be seen as flowing from the principle that if “p” is possible then nothing impossible follows from “p”, since if “p” implies “q” then the possibility of “p” implies the possibility of “q”. We will return to this topic when we discuss the procedure of Upgrading later in this Chapter. Contingency-propositions As shown in Chapter One, we need to distinguish generic from specific contingency; within the latter we need to distinguish the necessary from the non-necessary contingent; and within the non-necessary contingent we need to distinguish indeterminate from natural contingency.

43 Kilwardby ad A3 Part 1 dub.2–3 (7ra): “Cum contingens sit aliquod a esse b, ponatur in esse et nihil sequitur inconveniens. Et si ponatur in esse sic ‘Aliquod a est b’, convertetur; ergo aliquod b est a, et hoc non potest stare cum prima, scilicet ‘Necessario nullum b est a’.”

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Generic contingency: one-way possibility-propositions The generic sense of contingency is one-way possibility; and in this sense of contingency, he takes both affirmative and negative contingency-propositions to convert, as shown in Fig. 3.9.44 Maba,i

Mabe

Mbai

Mbae

Fig. 3.9. Conversion of contingency-propositions, in the sense of possibility

This result is not quite what we would expect on the basis of the semantics for possibility-propositions outlined in Chapter One. We saw there that a possibility-proposition states a de dicto possibility. Now, such propositions are convertible when their dicta are convertible; so on this basis we may say that affirmative possibility-propositions, whose dicta state that some or all of what can fall under the subject can fall under the predicate, are convertible. But a universal negative possibility-proposition, whose dictum states that nothing actually falling under the subject has to fall under the predicate, does not exhibit such symmetry, and we cannot count it as convertible without further ado. If, however, we were to require that the subject-term is per se, so that it’s necessary that whatever is b has to be b, then the dictum (and consequently the whole possibility-proposition) becomes convertible. Kilwardby does not note the need for this further requirement. The propositions “Some musician is possibly a man” and “Something white is possibly black” are regarded by Kilwardby as being true. He reads Aristotle as illustrating the convertibility of negative oneway possibility-propositions through his example of the proposition “It’s contingent that no garment is white”, which he takes to convert to “It’s contingent that nothing white is a garment”.45 Since the adduced example is of the nature of two-way contingency, and also of the nature of contingency in general, he doesn’t teach that it converts according to its nature of two-way contingency, but according to its nature of contingency without qualification. And this is clear from the

44 A3, 25a39–40; 25b4–13. See Paul Thom, The Logic of Essentialism: an interpretation of Aristotle’s modal syllogistic (Dordrecht: Kluwer, 1996), 43–44. 45 A3, 25b10–11. See Thom, The Logic of Essentialism, 43–44.

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chapter three proof of conversion, when he takes “It’s necessary for something white to be a garment” as the opposite of the conclusion.46

Kilwardby points out that the convertibility of negative contingencypropositions in this generic sense doesn’t imply that negative contingency-propositions in one of the specific senses are also convertible.47 Specific contingency: necessity-propositions The first species of contingency-propositions is composed of necessitypropositions, the second of two-way possibility-propositions. Kilwardby holds that each of these classes can be considered either in so far as it exemplifies the genus (one-way possibility-propositions) or in so far as it exemplifies the specific features that define it as a sub-class: Contingency in the sense of the possible contains the necessary contingent and the non-necessary contingent under it. Specific contingencies can therefore be considered in two ways—either in so far as the nature of the contingent without qualification is found and preserved in both (and in this way both convert, and so the non-necessary contingent in so far as it is contingent, converts into the contingent in the sense of the possible, as will be clear later), or else in so far as the nature of the necessary or the non-necessary is superadded to the contingent without qualification (and in this way the necessary contingent converts just like a necessity-proposition, but the non-necessary contingent doesn’t convert, as is said in the text).48

He makes a similar observation about negative contingencies: “… negative contingencies taken in the general sense, or as necessities not-to46 Kilwardby ad A3 Part 2 dub.11 (8vb–9ra): “Cum ergo in inducto exemplo sit natura contingentis ad utrumlibet et natura contingentis communis, non docet convertere illud secundum naturam contingentis ad utrumlibet sed secundum naturam contingentis simpliciter. Et hoc patet per probationem conversionis eius cum dicit pro opposito conclusionis istam ‘Necesse est aliquod album esse tunicam’.” 47 Kilwardby ad A3 Part 2 dub.9 (8vb). 48 Kilwardby ad A3 Part 2 dub.10 (8vb): “Contingens possibile continet sub se contingens quod est necessarium et quod est non necessarium. Contingit igitur considerare utrumque contingens speciale dupliciter, scilicet aut quo ad naturam contingentis simpliciter reperiri et salvari in utroque—et sic utrumque convertitur, unde contingens non necessarium in quantum contingens est convertitur in contingens {in contingens AE 1F1F2KlO2O3P1V: in non contingens EdBrCm1CrE2} per possibile, sicut patebit consequentur—aut quantum ad naturam necessarii et non necessarii quam superaddunt contingenti simpliciter—et sic contingens quod est necessarium convertitur, sicut et propositio de necessario, sed contingens quod est non necessarium non convertitur, sicut dicitur in littera.”

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be, are said to be contingent not-to-be, and convert in the same ways as negative assertorics and necessities”.49 So, contingency-propositions, in the sense of the necessary, if they are considered simply as contingency-propositions, convert in the same ways as one-way possibility-propositions. This is to say that they convert into possibility-propositions as shown in Fig. 3.9. Equally, if they are considered in their specificity as necessity-propositions, they convert into necessity-propositions, as shown in Fig. 3.10. Laba,i

Labe

Lbai

Lbae

Fig. 3.10. Conversion of contingency-propositions, in the sense of necessity

Specific contingency: two-way possibility-propositions According to Aristotle,50 contingency-propositions in the sense of the non-necessary (i.e. as two-way possibilities) are subject to two different types of conversion—term-conversion and conversion by opposed quality.51 Conversion by opposed quality (complementary conversion) Kilwardby summarizes Aristotle’s doctrine succinctly: … affirmative contingency-propositions and assertorics have one [conversion]—term-conversion (which was spoken about above)—and another by opposed quality (which is spoken about here). Universal negatives have only that by opposed quality, and particular negatives have both …52

These results are what we would expect, given the affirmative / negative symmetries in the semantics for contingency-propositions as outlined in Chapter One. 49 Kilwardby ad A3 Part 2 Note (9ra–b): “Adhuc negative de contingenti sumpto secundum genus et secundum quod necessarium non esse dicitur contingere non esse, similiter convertuntur in negativis de inesse et de necessario.” 50 A13, 32a29–b1. 51 Kilwardby ad A3 Part 1 dub.1 (7ra). 52 Kilwardby ad A13 dub.6 (20rb): “… affirmativae de contingenti et de inesse habent conversionem unam in terminis, de qua superius locutus est, aliam per oppositas qualitates, de qua hic loquitur. Negativae vero universales habent tantum illam quae est per oppositas qualitates, particulares vero negativae habent utramque.”

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Term-conversion A two-way contingency-proposition may be considered as such (simply as belonging to that genus), or it may be considered as falling under the species to which it belongs (either as a natural contingency or as an indeterminate contingency). And it may be that it is convertible when considered as a two-way contingency proposition, but is not convertible when considered as a natural contingency, or when it is considered as an indeterminate contingency. Kilwardby gives a general account of this type of situation: And it is to be said that when some superior contains several inferiors under it, those inferiors can be considered in two ways—either in so far as the nature of the superior is preserved in the inferior, or in so far as the inferior adds something to the nature of the superior. If the inferior is considered in the first way, then the proper features of the superior are preserved in it; but not if it is considered in the second way. And this is clear by Induction in all things. The case is similar, therefore, with contingency.53

Contingency-propositions, in so far as they are considered simply as contingency-propositions, convert to one-way possibility-propositions, as in Fig. 3.9. If, however, they are considered in their specificity as two-way possibilities, affirmatives (but not negatives) convert, as shown in Fig. 3.11.54 Qaba,i Qbai Fig. 3.11. Qa-conversion

These conversions hold for contingency-propositions that ampliate their subjects to the contingent. We saw in Chapter One that these propositions state that what may or may not fall under the subject may or may not fall under the predicate. It’s clear then that such a proposition, whether it is universal or particular, implies that some53 Kilwardby ad A3 Part 2 dub.10 (8vb–9ra): “Et dicendum quod cum aliquod superius continet sub se plura inferiora, contingit illa inferiora dupliciter considerari, scilicet aut quo ad naturam sui superioris salvari in inferioribus aut quo ad naturam quam addit inferius supra superius. Si primo modo consideretur ipsum inferius, sic in ipso salvantur propriae passiones superioris, si secundo modo, nequaquam; et istud patet inducendo in omnibus. Similiter ergo est de contingenti.” 54 A3, 25a40–b3.

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thing which may or may not fall under the predicate may or may not fall under the subject. The conversions shown in Fig. 3.11 also hold for those contingencypropositions that do not ampliate their subjects. According to “some people”, says Kilwardby, affirmative two-way contingency-propositions do not convert to two-way contingency-propositions, but to one-way contingency-propositions. Contrasting himself with these “some people”, he says: “I think it is to be said that a contingency-proposition (in the sense of the non-necessary), speaking without any qualification, converts to a contingency in the same sense”.55 More generally he observes: “… affirmative contingencies in any sense of the word so far determined convert to the same sense of contingency, in the same way as affirmative assertorics and necessities”.56 Kilwardby gives a demonstration of the convertibility of universal affirmative two-way contingency-propositions, considered as such. If it’s contingent for every b to be a, it’s contingent for some a to be b. For, otherwise the opposite—“It’s not contingent for some a to be b”— will stand. Now, there are two ways in which this could be true—either because necessarily no a is b, or because it’s necessary for every a to be b. In the first case, it may be argued thus: it’s necessary that no a is b, therefore it’s necessary that no b is a—which is incompatible with the first. In the second case, it may be argued thus: it’s necessary that every a is b, therefore it’s necessary that some b is a—which is incompatible with the first (namely with “It’s contingent for every b to be a”).57

The reasoning is displayed in Fig. 3.12.

55 Kilwardby ad A3 Part 2 dub.5 (8rb): “… puto quod dicendum sit quod propositio de contingenti non necessario absolute loquendo convertitur in eadem acceptione contingentis.” 56 Kilwardby ad A3 Part 2 Note (9ra): “… affirmativae de contingenti in omni acceptione eius tam determinata convertunter in eadem acceptione contingentis eo modo quo affirmativae de inesse et de necessario.” 57 Kilwardby ad A3 Part 2 dub.4 (8ra): “… si contingat omne b esse a, contingit aliquod a esse b; quia si non, stabit oppositum, scilicet ‘Non contingit aliquod a esse b’. Haec autem potest esse vera dupliciter, scilicet vel quia nullum a necesse est esse b vel quia necesse est omne a esse b. Si primo modo, arguatur sic. Necesse est nullum a esse b, ergo necesse est nullum b esse a; sed hoc non potest stare cum primo. Si secundo modo, arguatur sic. Necesse est omne a esse b, ergo necesse est aliquod b esse a; et hoc non potest stare cum primo, scilicet cum hac ‘Contingit omne b esse a’”.

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Lbae (1)

Labe Qaba Lbaa

(2)

Labi

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

R2

→

Qbai Qaba

C →

Qaba Qbai

Qaba

Fig. 3.12. Purported proof of Qa-conversion

He relies here on R2—the rule we formulated in Chapter One (Fig. 1.3) to enable proofs of formulae with negated contingency-premises— which assumes that if Qbai is false, either Lbae or Lbaa is true. He also assumes the convertibility of necessity-propositions. Further, he assumes that (1) if Labe is true then Qaba is false and that (2) if Labi is true then Qaba is false. The proof assumes the validity of necessity-conversion; and Kilwardby elsewhere shows that such a conversion is valid only if its terms are per se.58 So, at most what the proof shows is that contingencyconversion is valid for per se terms. But the notion of a two-way contingency-proposition with per se terms has limited application. There are indeed true two-way contingency-propositions with per se terms, for instance “It’s contingent for all animals to be men”. (It’s possible that all animals are men; and this possibility would be actualized if there were no non-human animals. And it’s possible that no animals are men; and this possibility would be actualized if there were no men.) But true natural contingencies never have two per se terms: the subject may be per se (as in “It’s contingent that all men are musicians”), but never the predicate. What is worse, R2 is not correct for per se necessities, because the falsity of Qbai doesn’t guarantee the per se status of “b” and “a”, and therefore doesn’t imply the truth of either Lbaa or Lbae as per se necessitypropositions. The fact is that the proof of contingency-conversion does not work. However, contingency-propositions are convertible, as in Fig. 3.11. If it’s contingent for all (or some) b to be a, then it’s contingent for

58

Kilwardby ad A3 Part 1 dub.4 (7rb).

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some a to be b. These conversions are in accord with the semantics for contingency-propositions outlined in Chapter One. Universal two-way contingency-propositions, considered as such, also imply universal one-way possibilities, as shown in Fig. 3.13. Qaba,e Mbaa,e Fig. 3.13. Conversion of universal Q-propositions to universal M-propositions For, if it is said “It’s contingent for no man to be white”, this means two-way contingency. But if it converts thus, “It’s contingent for nothing white to be a man”, this means contingency in the sense of the possible, which indeed is contingency in general. Hence the sense here is “It’s not necessary that something white is a man”.59

The example is Aristotle’s,60 who takes it that “It’s contingent that nothing white is a man” is false because it’s necessary for something white not to be a man; and Kilwardby interprets this to refer to swans and snow, which are necessarily not human.61 Natural and indeterminate contingencies Aristotle doesn’t say much about the two species of two-way contingency—natural and indeterminate contingency—because “he is here determining term-conversions, and he doesn’t need to use term-conversion for natural or indeterminate contingencies separately for the perfection of syllogisms in the sequel”.62 Kilwardby fills out the details: … the non-necessary contingent which is genuinely indeterminate, or genuinely two-way, can always be converted to the same sense of contingency—as is clear from the immediately preceding example [sc. “It’s contingent for every, or some, man to be white, so it’s contingent for something white to be a man”]. And I call a “genuinely indeterminate contingency” that which is related utterly equally to being and nonbeing. 59 Kilwardby ad A3 Part 2 dub.12 (9ra): “Si enim dicatur ‘Contingit nullum hominem esse album’, hic dicitur contingens ad utrumlibet. Si autem convertatur sic: ‘Contingit nullum album esse hominem’, dicitur hic contingens pro possibili, quod quidem contingens commune est. Unde hic est sensus ‘Non necesse est aliquod album esse hominem’.” 60 A17, 37a4–8. 61 Kilwardby ad A17, 36b35–37a31 (28rb). 62 Kilwardby ad A3 Part 2 dub.7 (8va).

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chapter three A non-necessary contingency which is natural, however, converts only to a common contingency (which is possible)—as is clear from the counterexamples cited above as objections. And the reason for this is that a natural contingency has an appointed cause, which however can be impeded. Hence when something is predicated of a subject as a natural accident, it stands indifferently to being and not-being, since its cause can be impeded—as when we say it’s contingent for a man to be a musician. For, a man has a natural potentiality to become a musician. Hence his nature inclines him more to becoming a musician than to the opposite. Now, such a contingency, which has an appointed cause, determines its subject. Hence such a natural accident stands only to being when it is made subject to its subject—as when we say it’s contingent for a musician to be a man.63 It’s clear therefore that a natural contingency doesn’t ever convert to a contingency in the same sense. Another sign of this is that when a natural contingency converts by opposite qualities it only converts to a common contingency …64

If we symbolize natural contingency by “Q N ” and indeterminate contingency by “Q I ”, then we have the results shown in Fig. 3.14. QNaba,i

QIaba,i

Mbai

QIbai

Fig. 3.14. Conversion of natural and indeterminate contingencies

These results are consistent with the semantics for natural and indeterminate contingencies which we have attributed to Kilwardby. First of all, natural contingencies convert to one-way possibilities, and not to 63 Albertus Magnus, Priorum Analyticorum, I.i.13 (303A) amplifies: “For, there is no natural contingency whereby it’s contingent to be a man.” 64 Kilwardby ad A3 Part 2 dub.5 (8rb): “… contingens autem non necessarium quod est infinitum vere vel ad utrumlibet vere converti potest semper in eadem acceptione contingentis, et patet per exemplum proximo dictum—et dico contingens infinitum vere quod aequaliter penitus se habet ad esse et non esse. Contingens autem non necessarium quod est natum non convertitur {convertitur ABrCm1CrE1F1F2KlO3P1V: converti Ed} nisi in contingens commune, quod est possibile, et patet exemplum per instantiam superius positam in oppenendo. Et ratio huius est quod contingens natum habet causam ad ipsum ordinatam, quae tamen impediri potest. Unde cum praedicatur aliquod accidens sicut natum de subiecto, indifferenter se habet ad esse et non esse, quia causa eius potest impediri, ut cum dicitur ‘Contingit hominem musicum esse’. Homo enim habet potentiam naturalem ut musicus sit. Unde magis se habet ad hoc ut musicus sit quam {quam ABrCm1CrE1E2F1F2KlO3P1V: supra lineam add. O2: om. Ed} ad oppostium de sui natura. Tale autem contingens quod habet causam ordinatam determinat sibi subiectum. Unde tale accidens, scilicet natum, cum subicitur suo subiecto, tantum ad esse se habet, ut cum dico ‘Contingit musicum esse hominem’. Patet igitur quod contingens natum non convertitur usque in contingens eiusdem acceptionis.

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anything stronger. Supposing that it’s natural that everything which can be b can be a (and it’s possible that nothing which is actually b need be a), we may infer that it’s possible that something which can be a can be b. And this inference holds for all terms, per se, mixed or per accidens. But we can’t infer anything stronger than that; in particular, we can’t infer a two-way possibility-conclusion to the effect that it’s possible that something which can be a can be b, and it’s possible that something which is actually a need not be b. The second conjunct here doesn’t follow: it’s natural that something which can be a man can be literate, and it’s possible that something which is actually a man need not be literate; but it isn’t possible that something which is actually literate need not be a man. Similarly, supposing that it’s natural that nothing which can be b has to be a, and it’s possible that something which is actually b need not be a, we may infer that it’s possible that nothing which is actually a has to be b. In the second place, genuinely indeterminate contingencies convert to contingencies of the same type. Suppose that it’s two-way contingent that something which can be b can be a, and it’s possible that something which is actually b need not be a. We may infer that it’s contingent that something which can be a can be b, and it’s possible that something which is actually a need not be b. But, as with the conversion of twoway contingencies considered in general (i.e. abstracting from natural and genuinely indeterminate contingencies), this inference is subject to the condition that “b” be a per accidens term—though Kilwardby doesn’t point this out.

Indirect reduction Kilwardby notes that all syllogisms can be reduced indirectly, i.e. per impossibile.65 Indirect reduction of second and third Figure syllogisms to the first Figure proceeds along the lines shown in Fig. 3.15—though Kilwardby makes little use of these reductions, at least in the non-modal syllogistic. Signum etiam ad hoc est quod contingens natum, cum convertitur per oppositas qualitates, non convertitur nisi contingens commune.” See Lagerlund, Modal Syllogistics in the Middle Ages, 43. 65 Kilwardby ad A45 dub.4, (49vb).

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ab--- bc+ ac--ab--- bc+ ac---

C

→

C →

ab--- ac+

ab+ bc+

bc---

ac+

ac+ bc+

ab+ bc+

ab+

ac+

C

→

C →

ab+ ac--bc--ac--- bc+ ab---

Fig. 3.15. Indirect reduction of the second and third Figures to the first

He notes that when syllogisms are said to be proved per impossibile, this is using the expression in a broad sense, covering both per impossibile syllogisms proper and what Aristotle in B8–11 calls conversive syllogisms.66 He also distinguishes syllogisms per impossibile from syllogisms ad impossibile.67 From amongst syllogisms per impossibile, syllogisms ad impossibile, and conversive syllogisms, it is the latter that are most relevant to the perfection of syllogisms. First of all, a syllogism per impossibile is a type of hypothetical argument, whereas a syllogism ad impossibile is not. If it can be proved ostensively that no c is a, because every a is b and no c is b (Camestres), this proof can be re-cast as a proof per impossibile arguing that if some c is a and every a is b then some c is b (Darii), but it is false that some c is b (since no c is b), therefore the hypothesis (“Some c is a”) is false, i.e. no c is a. The per impossibile syllogism, he says, “concludes from the falsity of the conclusion to the falsity of the hypothesis”. But the ad impossibile syllogism is the inference to that false conclusion from the false hypothesis. Neither of these is the same as the conversive syllogism, which is what corresponds to the process of Indirect Reduction. He lists five ways in which conversive syllogisms differ from syllogisms per impossibile. Of these, the most important is the third, which is “that a conversive syllogism shows a consequence; but a syllogism per impossibile shows a point”.68 The purpose of a conversive syllogism is to demonstrate the validity of another syllogism, not to demonstrate its own conclusion. The principle of inference in Indirect Reduction

66

Kilwardby ad A5 dub.4 (12rb). Kilwardby ad A29–30 Part 1 dub.2 (41vb). 68 Kilwardby ad B11–B12 Part 1 dub.4 (67ra–b): “Tertia est quod syllogismus conversivus ostendit consequentiam, sed syllogismus per impossible ostendit propositum.” 67

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is “When one opposite is inconsistent with a premise or premises, the other opposite follows”,69 or “The consequent being destroyed, the antecedent is destroyed”.70 He expands on the point: And it is to be said that the conversive shows a consequence by running according to the maxim “The consequent being destroyed, the antecedent is destroyed, therefore the antecedent being posited so too is the consequent”. Now, the consequent is destroyed by the supposition of its contrary as well as its contradictory. It stands together with neither the contrary nor the contradictory in truth. So a conversive syllogism proceeds by supposing the contrary as well as the contradictory.71

The two types of conversive syllogism he here identifies correspond to the two types of Indirect Reduction, according to Rule K or Rule C, as shown in Fig. 3.16:72 p q r p q r

K →

C →

¬r q

~p ~r q ~p

Fig. 3.16. Rules of indirect reduction

Exposition We saw earlier that Kilwardby endorses a proof taking the form shown in Fig. 3.17, where “z” is a singular term.

69 Kilwardby ad A5 dub.5 (12rb): “… cum unum oppositorum repugnet praemissae vel praemissis, hoc non est nisi quia eius oppositum sequitur …” 70 Kilwardby ad B8 dub.1 (64ra): “… destructo consequente, destruitur antecedens …” 71 Kilwardby ad B11–B12 Part I dub.5 (67rb): “Et dicendum quod conversivus ostendit consequentiam decurrendo secundum hanc maximam sic: destructo consequente destruitur antecendens, ergo posito antecedente ponitur et consequens. Unde autem consequens destruitur tam per suppositionem sui contrarii quam per suppostionem sui contradictorii, neque simul stant contraria in veritate neque contradictoria. Ideo syllogismus conversivus procedit supponendo tam contrariam quam contradictoriam.” 72 See Thom, The Syllogism, 39–41.

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p bz az q

Exposition

→

p bai q

Fig. 3.17. Rule of Exposition, proving syllogisms with a particular affirmative premise

This is one form taken by expository proofs. In Chapter Five we shall find him endorsing expository proofs governed by the rules shown in Fig. 3.18: p Lane Lcna q p ~Laz Lcz

Exposition

→

Exposition

q

→

p Laco q p Laco q

Fig. 3.18. Rules of Exposition, proving syllogisms with a particular negative premise

We shall also find him declining to accord the status of a fully formed syllogism to arguments like that in Fig. 3.18, that have a singular middle term.

Upgrading Aristotle proves Barbara XQM by a procedure which has been dubbed Upgrading.73 Kilwardby glosses Aristotle’s proof as follows: So he proceeds thus. First, he states the assertoric Major and contingency-Minor conjugations with both affirmative—whose utility he shows by deduction to the impossible. For, the opposite of the Major may be concluded from the opposite of the conclusion and the Minor put as an assertoric, by an assertoric / necessity mixture in the fifth of the third [Bocardo], the Major being a particular necessity-[proposition] and the Minor a universal assertoric. And this is “These things having been determined”.

73 K.L. Flannery, “Alexander of Aphrodisias and others on a controversial demonstration in Aristotle’s modal syllogistic”, History and Philosophy of Logic 14 (1993) 201–214.

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Second, he further perfects the same conjugation by putting the Minor as an assertoric. For, it concludes in the first Figure from both assertorics to an assertoric conclusion, from which there further follows the contingency-conclusion that he intends. And this is “It is also possible by the first”.74

The outlines of these proofs are shown in Fig. 3.19. Bocardo LXX

Barbara

Laco bca abo

→

aba Qbca Maca

aba bca

X / M --- subord

aca Maca

→

aba Qbca Maca

Fig. 3.19. Aristotle’s Upgrading proofs of Barbara XQM

Strictly speaking, there are two distinct processes here. Rather than applying the same name to both, let’s apply the name Upgrading only to the second proof, reformulating the first as a two-stage proof, as shown in Fig. 3.20. Bocardo LXX

Laco bca abo

C →

aba bca Maca

Upgrading

→

aba Qbca Maca

Fig. 3.20. Analysis of Aristotle’s first proof of Barbara XQM

Kilwardby takes these proofs to be sound; and, as we have just seen, he sees them as forming part of a process of perfecting Barbara XQM. But the process whereby an inference is led back to a perfect syllogism is one which is governed by syntactic rules. We have a definition of the rule C which warrants the first step in the above proof of Barbara XQM. How do we define the rule of Upgrading which is supposed to warrant the second step? 74 Kilwardby ad A15, 34a34–b6 (23ra): “Procedit ergo sic: primo ponit coniugationes ex utrisque affirmativis, maiore de inesse et minore de contingenti, cuius utilitatem ostendit per deductionem ad impossibile. Ex opposito enim conclusionis et minore posita inesse concludit oppositum maioris per mixtionem necessarii et inesse in quinto tertiae maiore existente particulari de necessario et minore universali de inesse. Et hoc est Determinatus autem. Secundo perficit adhuc eandem coniugationem per positionem minoris inesse. Ex utrisque enim de inesse concluditur per primam figuram et

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Kilwardby gives a prescription for the construction of the whole proof of XQ syllogisms (including both steps): “the opposite of the Major may be concluded from the opposite of the conclusion and the Minor put as an assertoric”. This suggests the following rule: ~r X{b,c} ~p

→

p Q{b,c} r

Fig. 3.21. Purported rule for the proof of XQ syllogisms

But such a rule would license inferences such as Barbara XQX which Kilwardby shows to be invalid. Bocardo

abo bca abo

→

aba Qbca aca

Fig. 3.22. Apparent Upgrading proof of Barbara XQX

That this mood is invalid is shown by the following counter-example: (moving)(walking)a Q(walking)(man)a (moving)(man)a Fig. 3.23. Counter-example to Barbara XQX

The purported proof of Barbara XQX must be unsound, because (unlike the proof of Barbara XQM) it leads from an enabling syllogism the opposite of whose conclusion is not compossible with its premises, to an enabled syllogism the opposite of whose conclusion is compossible with its premises: … it is otherwise with the opposite of the contingency-conclusion and the opposite of the assertoric conclusion. For it was shown earlier that the opposite of the assertoric conclusion is compossible with the Minor, but is incompossible with the Minor put as an assertoric. However, the opposite of the contingency-conclusion is incompossible with both. For if every b is a then b is under a, and so “Of necessity some c is not a” and “For every c it’s contingent to be b” are not compossible—because if some c is of necessity not a then there is some c for which it’s impossible

modum conclusio de inesse, ex quo ulterius sequitur conclusio de contingenti quae intenditur. Et hoc est Est autem et per primam.”

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to be a, and if so then there is some c for which it’s impossible to be b, and so it’s not contingent for every c to be b if it’s necessary for some c not to be a.75

Therefore the rule in Fig. 3.21 doesn’t preserve validity. Kilwardby deploys this tactic to discredit purported proofs of a number of other syllogisms, including Celarent XQX,76 Cesare XQX,77 Cesare QXM and QXX,78 Darapti XQX79 and Bocardo XQX.80 The tactic is to be found as early as Alexander;81 and it is perfectly sound, but it doesn’t tell us what is wrong with the purported proofs; nor does it tell us exactly what is the transformation rule that warrants Aristotle’s proof by Upgrading. Nowhere does Kilwardby display the same lucidity regarding Upgrading as he does in relation to Indirect Reduction. In this respect, he is not alone: generations of logicians have struggled to say clearly and convincingly what is the syntactic rule underlying Aristotle’s procedure.82 In the absence of such a syntactic rule, Kilwardby is not justified in speaking of Aristotle’s procedure as “perfecting” the XQM syllogisms of the first Figure. It is possible, however, to construct a semantic account that explains why Upgrading is a sound procedure, by examining the context in which Aristotle uses the Upgrading procedure, and by reflecting on Kilwardby’s semantical analysis of the premises of XQM-1 syllogisms.

75 Kilwardby ad A15 dub.8 (25ra): “… aliter est de opposito conclusionis de contingenti quam de opposito conclusionis de inesse. Ostensum enim est prius quoniam oppositum conclusionis de inesse compossibile {compossibile AE 1E2F1O2O3P1: incompossibile EdCm1F2Kl: impossible Br} est minori, sed incompossibile cum minori posita inesse. Oppositum autem conclusionis de contingenti est incompossibile utrique. Si enim omne b sic est a, tunc est b sub a, et ita haec incompossibilia, scilicet ‘De necessitate aliquod c non est a’ et ‘Omne c contingit esse b’, quia si aliquod c non de necessitate est a, tunc aliquod est c quod impossibile est esse a, et si hoc, tunc aliquod est c quod impossibile est esse b. Et ita non contingit omne c esse b si necesse est aliquod c non esse b.” 76 Kilwardby ad A15 dub.8 (25ra). 77 Kilwardby ad A18 dub.3 (29va). 78 Kilwardby ad A18 dub.4 (29va). 79 Kilwardby ad A21 dub.2 (32va). 80 Kilwardby ad A21 dub.3 (32va). 81 Alexander of Aphrodisias, On Aristotle’s Prior Analytics 1.14–22, translated by Ian Mueller with Josiah Gould; introduction, notes and appendices by Ian Mueller (Ithaca NY: Cornell University Press, 1999), ad A16, 36b19ff. (217,8ff.). 82 See Thom, The Logic of Essentialism, 78–85 and Thom, Medieval Modal Systems, 111– 112.

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Aristotle’s Upgrading proof is presented in the context of a rule stating that when something follows from given premises, the possibility of the conclusion follows from the possibility of the premises.83 Here we have the modal principle that if “p” and “q” imply “r”, then the compossibility of “p” and “q” implies the possibility of “r” (a particular case of the principle that if the antecedent of an entailment is possible, so is the consequent);84 and Kilwardby observes that this principle is “used in the perfection” of Barbara XQM.85 He is aware that the principle concerns the joint possibility of a pair of premises, not just their individual possibility: Further, it seems that the false follows from truths, because the impossible (which will never be true) follows from two possibilities which are sometimes true. For, from “Every man is white” and “No man is white” it follows that no man is a man; and yet the premises are possible and the conclusion impossible. And it is to be said that Aristotle’s words are to be understood about two compossible truths. For, even if each in itself is possible, the impossible follows from their incompossibility; but the false never follows from compossible truths.86

In order to see the bearing that this principle has on the proof by Upgrading, we need to reflect on the semantical analysis of the premises in XQM-1 syllogisms. The Major premise there needs to be an unrestricted assertoric, according to both Aristotle and Kilwardby. Otherwise its subject “b” might stand only for those things that are actually b, whereas the same term as predicate of the Minor stands for what can be b—and in that case the Middle term would stand for more things in the Minor than in the Major. If the Major could be an as-of-now assertoric, we could construct a counter-example leading from truths to a falsehood, as in Fig. 3.24:

A15, 34a12–22. G.E. Hughes and M.J. Cresswell, An Introduction to Modal Logic (London: Methuen 1968), 37. 85 Kilwardby ad A15, 34a2–33 (22vb). 86 Kilwardby ad B18–20 dub.2 (75ra): “Adhuc videtur quod ex veris sequitur falsum, quia ex duobus possibilibus, quae aliquando sunt verae, sequitur impossibile, quod numquam erit verum. Ex istis enim ‘Omnis homo est albus’ ‘Nullus homo est albus’ sequitur quod nullus homo est homo, et tamen praemissae sunt possibiles et conclusio impossibilis. Et dicendum quod intelligendus est sermo Aristotelis de duobus compossibilibus veris. Quamvis enim utraque in se sit possibilis ex earum incompossibilitate sequitur impossibile, sed numquam sequitur falsus ex veris et compossibilibus.” 83 84

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(mare)(grey-haired)a Q(grey-haired)(old man)a M(mare)(old man)a Fig. 3.24. Counter-example to Barbara XQM, with as-of-now Major

But if the subject of “b is a” stands for what can be b, the proposition is an unrestricted assertoric, meaning something like “Everything that can be b can be a”.87 The Major must be unrestricted, but it may be an unrestricted assertoric of either of the types we distinguished in Chapter One: … it is to be known that the unrestricted assertoric proposition is taken differently in respect of a necessity and a contingency-conclusion in the generic sense. The unrestricted assertoric that, together with a necessityproposition, concludes to a necessity-proposition should be necessary in reality [secundum rem]. It doesn’t suffice that it is a natural contingency supposed to be actual. For example, everyone who is literate of necessity is a knower, every man is literate, but it’s not of necessity that every man is a knower. However, in respect of a contingency-conclusion the assertoric can be manifested by a natural contingency.88

Notice that he doesn’t require that the assertoric premise express the actualization of a natural contingency; he simply says that it can be of this type. His position is that the assertoric premise can be either an apodeictic unrestricted assertoric or a natural unrestricted assertoric. Further, he argues that the contingency-premise may be either a natural or an indeterminate contingency. He considers an argument that draws on the relationship between the Minor premises of the enabling and enabled syllogisms in Fig. 3.20. The argument assumes that the process of Upgrading shown in Fig. 3.20 works by putting the contingency-Minor of the enabled syllogism as an assertoric. Now, when a natural contingency is put as an assertoric, the assertoric is unrestricted; and when an indeterminate contingency is put as an asser87

Kilwardby ad A15 dub.5 (24va). Kilwardby ad A15, dub.9 (25rb): “… sciendum quod aliter accipienda est propositio de inesse simpliciter respectu conclusionis de necessario et aliter respectu conclusionis de contingenti communi. Illa enim de inesse simpliciter quae cum alia de necessario concludit conclusionem de necessario debet esse necessaria secundum rem, et non sufficit quod sit contingens natum positum inesse. Verbi gratia, omne grammaticum de necessitate est sciens, omnis homo est grammaticus—sit ita, sed non de necessitate omnis homo est sciens. Respectu autem conclusionis de contingenti potest illa de inesse manifestari a contingenti nato.” 88

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toric, the assertoric is as-of-now. But, the enabling syllogism (having a mixture of necessity- and assertoric premises) must have an unrestricted assertoric premise, and so the contingency-proposition in the enabled syllogism has to be a natural contingency. In reply he reminds us that the enabling syllogism takes an LXX form (a strengthened XXX syllogism), and as such it may have an as-of-now assertoric premise. That being so, the corresponding contingency-proposition (the Minor in the enabled syllogism) may be an indeterminate contingency: … the assertoric Major in the said conjugations does not appropriate the contingency-Minor so that it is a natural contingency. Hence, it could equally well be taken as an indeterminate or as a natural contingency.89

Now, there are in principle four possible combinations of premises— (1) an apodeictic unrestricted Major with an indeterminate contingency Minor, or (2) an apodeictic unrestricted Major with a natural contingency Minor, or (3) a natural unrestricted Major with an indeterminate contingency Minor, or (4) a natural unrestricted Major with a natural contingency Minor. Kilwardby doesn’t say that a possibility-conclusion follows in all four cases. In fact, such a conclusion follows in three of the cases—as is shown by the following considerations. (1) If “r” follows from “p” and “q”, then when “p” is necessary and “q” is possible, “r” is possible.90 We know from Aristotle that if the premises of a valid inference are compossible then the conclusion is possible;91 and we also know that where “p” is necessary, the possibility of “q” implies the compossibility of “p” and “q”.92 (2) If “r” follows from “p” and “q”, then when “p” is necessary and “q” is natural, “r” is possible. This case reduces to the first, since the natural must be possible. (4) If “r” follows from “p” and “q”, then when “p” is natural and “q” is natural, “r” is possible. This case depends on the fact that what is natural must 89 Kilwardby ad A15 dub.6 (24vb): “… maior de inesse in dictis coniugationibus non appropriat minorem de contingenti ut sit de contingenti nato. Quare poterit sumi adeo bene de contingenti infinito sicut de contingenti nato.” 90 This follows from two principles—one (noted earlier) stating that if the antecedent of an entailment is possible then so is the consequent, the other stating that when “p” is necessary and “q” is possible the conjunction of “p” and “q” is possible. For this second principle, see footnote 92. 91 A15, 34a16–22. 92 If “p” and “q” are not compossible, then it’s necessary that if p then not q. In that case, if it’s necessary that p it’s necessary that not q; and so, either it’s not necessary that p or it’s not possible that q. Transposing then, if it’s necessary that p and it’s possible that q, then “p” and “q” are compossible. This chain of reasoning is valid in system T. See Hughes and Cresswell, 30–40.

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be compossible with whatever else is natural. Any pair of natural laws must be able to actualized simultaneously. This follows from the fact that if a pair of natural laws are conjoined, the conjunction will also be a single natural law.93 Given that the conjunctive natural law must express a possibility, it follows that any two natural laws must be able to be actualized simultaneously. The fourth combination doesn’t yield a possibility-conclusion. (4) If r follows from “p” and “q”, then when “p” is natural and “q” is possible, “r” may not be possible. If all old people go grey and all who go grey are mares then all old people are mares. The Minor is natural, the Major is possible, but the conclusion is impossible. Kilwardby gives a description of one type of reasoning fitting pattern (2): whatever is the ontologically superior of a substance’s natural capacities may be inferred to be a possibility for that substance. He comments that the Major in an XQM-1 syllogism predicates a superior of its inferior; and when the inferior is contingent for some subject, sometimes the superior is necessary for that subject, and sometimes it is (two-way) contingent. Whiteness is contingent for man though its superior, colour, is necessary; walking is contingent for man and its superior, motion, is contingent; but in every case the superior is (one-way) possible for the subject.94 Here is a more general description of how cases (1), (2) and (3) apply to Barbara XQM. We have a valid core inference (shown in Fig. 3.25), whose Major may be embedded within a modality of necessity or naturalness, and whose Minor may be embedded within a modality of naturalness or possibility (provided that a natural Major is not combined with a merely possible Minor), while the conclusion is embedded within a modality of possibility. (what can be a)(what can be b)a (what can be c)(what can be c)a (what can be a)(what can be c)a Fig. 3.25. Core inference in Barbara XQM

This doesn’t give a syntactic rule justifying Aristotle’s Upgrading proofs. But, since those proofs are confined to the first Figure, such a rule 93 This is a general principle of deontic logic, being equivalent to the axiom “It’s permissible that either p or q iff either it’s permissible that p or it’s permissible that q”. See Prior, 221. 94 Kilwardby ad A15 dub.12 (25va).

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only needs to say that the XQM version of every first Figure thesis is a thesis. This formulation, however, doesn’t generalize to the other Figures, where not all XQM standard moods are theses.

Summary One syllogism may be reduced to another by a number of different methods. Direct Reduction is the method most widely used by Aristotle. This is a process in which a syllogistic premise is replaced by its converse or by another proposition that implies it, or else a syllogistic conclusion is replaced by its converse or by another proposition which it implies. Kilwardby devotes a great deal of effort to trying to understand the nature of the Conversions used in this process. His final conclusion—that Conversion is a consequence but not any sort of argument—recognizes that Conversion doesn’t neatly fall under any of the concepts provided by Aristotelian logic. Kilwardby faithfully summarizes the Aristotelian laws of conversion for assertorics, necessity-proposition, possibility-propositions and contingency-propositions. In the case of assertorics, necessity- and possibility-propositions, those laws state that universal and particular affirmatives convert to particular affirmatives, and universal negatives convert to universal negatives. The conversion of contingency-propositions, however, doesn’t work out on the same lines. Affirmatives convert in the same ways as assertorics, but negatives do not. The process of Indirect Reduction works by deducing the contradictory or contrary of one premise from the other premise together with the contradictory of the conclusion. The operation of this rule depends on the existence of a list of contraries. This is a matter that will become very important in Chapter Six when we come to consider mixed contingency / necessity syllogisms in the second Figure. The process of Exposition works by finding a shared term between the subject and predicate of a particular proposition, and syllogizing with reference to that term. The process takes two forms, depending on whether the shared term is a singular or a common term. While these processes can be described in syntactic terms, the process of Upgrading requires semantic understanding. The analysis suggested in this Chapter requires that we first identify a core embedded in a given inference, and then argue with regard to that core. The given inference is valid if it can be represented as arguing from the necessity

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of one of the core’s premises, and the possibility of the other, to the possibility of the conclusion. Equally it is valid if it can be represented as arguing from the naturalness of one premise and the necessity of the other to the possibility of the conclusion. Finally it is valid if it can be represented as arguing from the naturalness of both premises to the possibility of the conclusion.

chapter four THE ASSERTORIC SYLLOGISTIC Aristotle presents his doctrine of non-modal syllogisms in chapters 4–6 of the first Book of the Prior Analytics. Kilwardby faithfully expounds the doctrine, taking as given the Aristotelian list of perfect moods and the Aristotelian methods of perfecting the imperfect moods. This fidelity has led one modern scholar to claim that “Kilwardby added nothing of substance to the theory of the assertoric syllogism”.1 However, that judgment overlooks the fact that Kilwardby goes beyond the standard Aristotelian doctrine in several ways, supplying defences of numerous points—including the doctrine that there are just three Figures, and the order of the Figures and of the moods in each Figure. He also formulates general principles applying to all syllogisms, and special principles on the basis of which the fruitful premise-pairs in each Figure can be computed, and he defends the valid moods against pretended objections. Among the post-Aristotelian material brought to bear on the theory of assertoric syllogisms we find a distinction between expository and other syllogisms, a distinction between genuinely negative propositions and those that are not genuinely negative, a theory of the ontological order of entities that fall into an Aristotelian category, and sundry principles governing that order. Some of this material can be traced to writers in the Islamic world; and one Latin treatise of the twelfth century, the Dialectica Monacensis, was clearly influential on Kilwardby’s use of syllogistic principles to compute the number of useful premisepairs in each Figure. His integration of these various ideas into the assertoric syllogistic is in most cases convincing.

1 Henrik Lagerlund, “Medieval theories of the syllogism”, The Stanford Encyclopedia of Philosophy (Spring 2004 Edition), Edward N. Zalta (ed.), URL = http://plato.stanford.edu/ archives/spr2004/entries/medieval-syllogism/.

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Kilwardby lays down two principles that are common to all valid syllogisms: (P1) In a valid syllogism, one premise must be universal. (P2) In a valid syllogism, one premise must be affirmative.2

These rules come directly from Aristotle.3 At least one universal premise In his comments on A4 Kilwardby puts (P1) forward as a “common principle”; but it transpires in his comments on A24 that he doesn’t think it applies to absolutely all syllogisms: And it is to be said that Aristotle doesn’t mean that without a universal there is no sort of syllogism—for, a necessary conclusion is drawn from singulars in the third Figure—but that no syllogism competently related and arranged according to mood is produced without universals.4

He doesn’t count third Figure syllogisms from two singular premises as “competently related and arranged according to mood”. Such inferences may be valid, and may be in the third Figure; but he denies that these are Syllogisms properly arranged in a Mood. Elsewhere he states in general that “two propositions, made with three terms, necessarily determine a Figure through the placing of the terms, but don’t necessarily determine a Mood”.5 This agrees with his metaphysical analysis (noted in Chapter Two) of the materials and form that constitute a syllogism. A singular syllogism appears as one that is not fully formed, being endowed with a remote form (namely Figure) but lacking the completing form (Mood). He regards such arguments as “a knowable proof through something knowable falling under the senses”, such a proof being “obvious of Kilwardby ad A4 Part 2 dub.8 (10vb). A24, 41b6–7. 4 Kilwardby ad A24 dub.2 (35va–b): “Et dicendum est quod non est intentio Aristotelis quod sine universali nullo modo sit syllogismus—quia ex singularibus in tertia figura concluditur necessario—sed quod syllogismus competenter se habens et secundum modum dispositus non fiat sine universalibus.” 5 Kilwardby ad A28 Part 2 dub.2 (40vb): “… duae propositiones factae in tribus terminis per situm terminorum ex necessitate determinant figuram, sed non de necessitate determinant modum.” 2 3

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itself ” and not requiring a theory [doctrina] since we do not need a theory of what falls under the senses.6 In the light of this account, the “common” principle requiring a universal premise doesn’t apply to all syllogisms but only to fully formed ones, namely those in a Figure and Mood. At least one affirmative premise He notes that Aristotle doesn’t give a proof that an affirmative premise is required, but that this omission is not important because (P2) follows from the definition of syllogism. He offers two arguments for this view: For, when it is said in the definition of syllogism that something follows from their being so, it is signified that the premises are the cause of the conclusion. But no negation is a cause. So there has to be an affirmative. Or, it can be said that Aristotle, in showing that some universal is in a syllogism, shows that something in the syllogism stands to something else in it as whole to part. But taking a part under its whole cannot be with a negation, but with an affirmation. Hence, in showing that something in the syllogism is universal he has thereby shown that something is affirmative. And I am speaking of perfect syllogisms in the first Figure, from which all others come and to which they reduce.7

In the first of these arguments he is taking “cause” in a sense broader than what is required for a demonstrative syllogism—as he notes elsewhere: “… every proposition has a cause of its truth, but that doesn’t always suffice for demonstration, for the cause that demonstrates has to be prior and better known…”.8 In the second argument, he relies 6 Kilwardby ad A2 dub.12 (6ra): “… expositio est scibilis probatio per aliquod scibile subiacens sensui. Talis autem per se manifesta est et non oportet de ea habere doctrinam; de his enim quae sensui nota sunt non indigemus doctrina.” 7 Kilwardby ad A24 dub.1 (35va): “Quia cum dicitur in definitione syllogismi quod aliquid sequitur eo quod haec sunt, significatur quod praemissae sint causa conclusionis. Sed negatio nullius est causa. Quare oportet aliquam esse affirmativam. Vel dici potest quod per hoc quod Aristoteles hic ostendit aliquam esse universalem in syllogismo ostensum {ostensum ABrCm1E1E2F1F2KlO2O3V: affirmatio Ed} est quod aliquid in syllogismo se habet ad aliud in {in ACm1E1E2F1F2O2O3P1: cum Ed: om. BrV } ipso sicut totum se habet ad partem. Sed acceptio partis sub suo toto non potest esse cum negatione, sed cum affirmatione. Quare per hoc quod hic ostendit aliquam esse universalem in syllogismo relinquitur ostensum esse aliquam esse affirmativam. Et loquor de syllogismo perfecto, scilicet primae figurae, a quo omnes alii extrahuntur et in quem omnes reducuntur.” 8 Kilwardby ad A35 dub.2 (45vb): “… omnis propositio causam habet suae veritatis,

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on an assumption to which he frequently returns, namely that subject stands to predicate in a part-whole relation, at least conceptually (even though in reality subject and predicate may be inter-convertible).9 So, both arguments turn on a special sense of the key-term, “cause” or “part”. He offers another argument that is free of this semantic contrivance: It is to be said that every syllogism aims to deny something of something, or to affirm something of something. Now, one thing can be denied of another, only through some difference between them; and this difference is necessarily some of one of them that doesn’t agree with the other. So, since this difference has to be a Middle, it will necessarily be affirmed of one (or that one [affirmed] of it), though it is denied of the other (or the other of it). And so if the aim is to deny something of something, it’s necessary for one at least of the propositions to be affirmative. If, however, something has to be affirmed of something, this has to be through something agreeing with both. And so since this has to be a Middle, it’s necessary for both to be affirmative, since the Middle agrees with both Extremes. And so it’s clear how an affirmative, as well a negative syllogism necessarily has some affirmative proposition—and what is the reason for this.10

This is to say that, since negative and affirmative conclusions respectively are syllogized following the two patterns shown in Fig. 4.1, it is always necessary to have an affirmative premise. {a,b}+ {b,c}---

{a,b}+ {b,c}+

{a,c}---

{a,c}+

Fig. 4.1. The quality of premises and conclusion illa tamen non in omnibus sufficit ad demonstrandum; oportet enim causam quae demonstrare debet esse priorem et notiorem.” 9 Kilwardby ad B5 dub.2 (60vb). 10 Kilwardby ad A4 Part 2 dub.10 (10vb): “Dicendum quod omnis syllogismus intendit removere aliquid ab aliquo aut affirmare aliquid de aliquo. Unum autem non potest ab alio removeri nisi per aliam differentiam eorum. Haec autem differentia {differentia ABrCm1E1E2F1F2KlO2O3P1V: dicitur Ed} necessario est aliquid unius non conveniens tamen alii. Cum ergo haec differentia debet esse medium, necessario de altero affirmabitur vel alterum de ipso, quamvis negetur de reliquo vel reliquum de ipso. Et ita si intendatur aliquid removeri ab aliquo necessarium est adminus alteram propositionum esse affirmativa. Si autem debet aliquid de alio affirmari, oportet quod hoc fit per aliquod conveniens utrique. Et ideo cum illud debeat esse medium, necesse est utramque esse affirmativam cum medium sit utrique extremo conveniens. Et ita patet quomodo tam syllogismus affirmativus quam negativus necessario habet aliam propositionum affirmativam, et patet eius causa.”

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It might seem that the common principle (P2), like (P1), really applies only to a sub-class of syllogisms, namely the assertoric syllogisms, the pure necessity-syllogisms, and the necessity / assertoric mixtures, and that it doesn’t apply to the contingency-syllogisms (where, because affirmatives and negatives are equivalent, both premises may be negative). We should, however, remember Kilwardby’s view that negative contingency-propositions are not “genuinely negative”—a view that enables him to take (P2) as meaning that there are no valid syllogisms with two “genuinely” negative premises. This is the way Avicenna takes this principle, holding that a syllogism may be valid without having an explicitly affirmative premise provided that one of its negative premises implies an affirmative, and he explicitly mentions negative contingencypropositions as implying affirmatives.11 We shall see in Chapter Six that Kilwardby adopts a similar approach.

The first Figure Given that there are 16 possible premise-pairs in the first Figure, which ones yield a syllogistic conclusion? Kilwardby deals with the question by laying down principles from which the number of useful premisepairs can be deduced. It is important to understand that the term “principle” as used by him in this context is not simply a synonym of “rule” (as used, for example, by Peter of Spain in speaking of rules for each of the syllogistic Figures), or “property” (as used, for example, by Albert the Great in speaking of properties of the individual Figures). Kilwardby’s purpose in drawing up a set of principles is to allow a deduction to be made of the premise-pairs that comply with the stated principles; but this is not the purpose that either Peter or Albert has in mind when drawing up their lists of rules / properties. They simply want to identify rules or properties that truly apply to all syllogisms in the given Figure.

11 Ibn S¯ın¯ a, Remarks and Admonitions. Part One: Logic, translated from the original Arabic with an introduction and notes by Shams Constantine Inati (Toronto: Pontifical Institute of Mediaeval Studies, 1984), Seventh Method, 396.

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Principles Kilwardby lays down two principles peculiar to the first Figure: (P3) In first Figure syllogisms, the Major must be universal. (P4) In first Figure syllogisms, the Minor must be affirmative.12

He considers a putative counter-example to (P4): any animal that is not rational is irrational, an ass is not rational, so it is irrational. Here the Minor is negative, and yet it seems to be a valid syllogism in the first Figure. In reply he says that even if this reasoning holds for some terms, it doesn’t hold for all terms and thus is considered useless. His point is well-taken since the predicate of the negative Minor is “rational” and the subject of the Major is “not rational”, and these are different terms, so that the form of the argument can be represented as in Fig. 4.2. ad a bce aca Fig. 4.2. Invalid first Figure syllogistic form with negative Minor

This form holds for some terms but not for all. He also offers a second response: Or it can be said that the said syllogism has a compound [hypothetica] Major. Hence the whole antecedent of the Major is by way of Middle; and so the negation that is in the antecedent is a part of the Middle and doesn’t negate the Middle.13

This involves a different analysis of the counter-example, according to which the subject of the Major, and the predicate of the Minor are identical, so that the syllogism is formally valid. But it no longer constitutes a counter-example to (P4), since the Minor is now affirmative—as in Fig. 4.3.

12 Kilwardby ad A4 Part 2 dub.9 (10vb). Ghazali adds as a common principle that no syllogism can have a negative Minor and a particular Major. (Lohr, “Logica Algazelis: introduction and critical text”, 260.) 13 Kilwardby ad A4 Part 2 dub.8 (10vb): “Vel potest dici quod dicti syllogismi sunt ex maiore hypothetica, unde totum antecedens maioris est in ratione medii. Et ideo ipsa negatio quae est in antecedente est pars ipsius medii et non negat ipsum medium.”

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aba bca aca Fig. 4.3. Valid first Figure syllogistic form with negative terms

In his discussion of circular syllogisms he considers another seeming counter-example to the special rules of the first Figure. This is Aristotle’s inference “What a is in none of, b is in all of; c is in none of a; so b is in all of c”.14 He says that this seems not to be a syllogism since it violates both (P3) and (P4). His answer is that this inference is in the first Figure but is not in any Mood. (This recalls his judgment about singular syllogisms that they are in a Figure but not in any Mood.) His analysis is that the Major premise implies the proposition “If a is in none of c then b is in all of c”, so that we have a hypothetical inference “positing the antecedent”, as shown in Fig. 4.4: ace If ace then bca bca Fig. 4.4. First Figure hypothetical syllogism with negative Minor

(Positing the antecedent we would call Modus Ponens.) He continues: Nor is it unacceptable to infer in a hypothetical syllogism by positing the antecedent from a negative Minor, because the negation doesn’t fall on the whole Middle but is as a part of it. Nor is it unacceptable to infer an affirmative from a negative.15

As with the common principles (P1) and (P2), he offers a justification for both (P3) and (P4). One argument—namely that to-be-said-of-all implies these principles—was discussed in Chapter Two. He offers an additional argument for (P4): Now, of the remaining principle, namely that the Minor is affirmative, it is to be said that if the Minor were negative, either the Major would be negative (and then a syllogism would not be produced—for the stated 14 B5, 58a26–32. See Thom, The Syllogism, 205, where, however, the conclusion is mis-stated as “b belongs to all c”. 15 Kilwardby ad B5 dub.9 (61va): “Nec est etiam inconveniens inferre per syllogismum hypotheticum a positione antecedentis ex minore negativa, quia negatio non cadit super totum medium, immo quasi eius pars est. Nec etiam est inconveniens sic inferre affirmativam ex negativa.”

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chapter four reason), or the Major would be affirmative (and then there would be a fallacy of the consequent), because from the negation of a thing’s inferior there doesn’t follow the negation of the same thing’s superior.16

(P2) implies that if the Minor is negative the Major is affirmative. Thus he represents the Middle term as an inferior of the Major. An inferior is thought of as the antecedent to its superior. But that antecedent is denied in the negative Minor premise, where it is predicate. Thus the syllogism can be represented as committing the fallacy of denying the antecedent, as shown in Fig. 4.5. a←b b←c a ←c Fig. 4.5. Fallacy of denying the antecedent

He also advances other considerations in favour of (P3): Alternatively, it can also be said that if the Major were particular, the Middle could be more common than the Major Extreme. For, an inferior is predicated of a superior in part, affirmatively and negatively. And if this were so, it could happen similarly that the Major was negative and the Extremes convertible, or exceeding and exceeded. And a negative conclusion couldn’t follow—unless it was false—as it clear from the terms “man”, “animal”, “ass”.17

His point is that both “abi” and “abo” are true if a is inferior to b. Earlier we saw the terminology of inferiors and superiors as coming from the theory of categories. What Kilwardby is saying is that if in the first Figure the Major premise could be particular (whether affirmative or negative), then it could be a proposition like “Some animal is a man”, which predicates an ontological inferior of its superior. In that case two counter-examples can be constructed, as in Fig. 4.6. 16 Kilwardby ad A4 Part 2 dub.11 (11ra): “De reliquo autem principio, scilicet quod minor sit affirmativa, dicendum quod si minor esset negativa, aut maior esset negativa, et tunc non fieret syllogismus, cuius causa dicta est, aut maior esset affirmativa, et tunc fieret fallacia consequentis, quia ad negationem inferioris de aliquo non sequitur negatio superioris de eodem.” 17 Kilwardby ad A4 Part 2 dub.11 (11ra): “Aliter etiam dici potest quod si maior esset particularis, posset medium esse communius maiore extremitate. Praedicatur enim inferius de superiore particulariter affirmative et negative. Et si ita esset, posset similiter contingere quod maior fuerit negativa et extrema convertibilia vel {vel ABrCm1E1E2F1F2KlO2O3V: ut CrP 1: om. Ed} excedentia et excessa, et non posset sequi conclusio negativa nisi falsa, sicut patet in his terminis ‘homo’ ‘animal’ ‘asinus’.”

the assertoric syllogistic

(man)(animal)i (animal)(ass)a

(man)(animal)o (animal)(man)a

(man)(ass)i

(man)(man)o

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Fig. 4.6. First Figure syllogisms with particular Major premises

Given that there are independent reasons for holding that the Minor must be affirmative, these two examples complete the proof that no syllogistic conclusion follows if the Major is particular. In demonstrating the infertility of premise-pairs, Aristotle doesn’t appeal to principles. Instead he uses what has come to be known as the method of contrasted instances.18 He selects two trios of terms, both of which make the premises true, but one of which makes the Major true of all the Minor while the other makes it true of none. For example, the 8 premise-pairs that have a particular Major—ia-1, oa1, ie-1, oe-1, ii-1, oi-1, io-1, oo-1—are dealt with in three groups, using a mixture of categorial and denominative terms. First, he shows the premises ia-1 and oa-1 to be sterile, by means of the terms “good”, “state”, “prudence” and “good”, “state”, “ignorance”.19 Second, the does the same for the premises ie-1 and oe-1, using the terms “white”, “horse”, “swan” and “white”, “horse”, “raven”.20 Third, he despatches the premises ii-1, oi-1, oi-1, oo-1 with the terms “animal”, “white”, “horse” and “animal”, “white”, “stone”.21 In none of these cases does he use a trio of terms from a single category. And, while his objective is clear—namely, to show that the premises are compatible with either a universal affirmative or a universal negative proposition in which the Major Extreme is predicated of the Minor—he doesn’t give any clue about how to find the appropriate trios of terms. What Kilwardby has done is to draw all this material together as a single problem, and to indicate how to pick terms from a single category that will show that no syllogistic conclusion follows if the Major is particular.

18 19 20 21

Ross ad A4, 26a2–9. A4, 26a33–36. A4, 26a36–39. A4, 26b21–25.

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Useful premise-pairs Interestingly, each of the special principles for the first Figure implies one of the common principles, the satisfaction of (P3) implying that of (P1), and the satisfaction of (P4) implying that of (P2). So these special principles can stand as sufficient for the first Figure. Kilwardby, however, doesn’t recognise this fact, and proceeds to make use of all four principles to exclude the useless premise-pairs in the first Figure. The cases are divided into four, according as [1] both premises are universal, [2] both are particular, [3] the Major is universal and the Minor particular, or [4] the Minor is universal and the Major particular. Premise-pairs are then excluded if they conflict with (P1)(P4). The order in which the four cases are considered is determined by the principles that apply to them. Cases to which (P1) and (P3) apply are considered before those to which (P2) and (P4) apply: Now, if [2] both are particular, either both are affirmative, or both are negative, or the first is affirmative and the second negative, or vice versa; and all four of these combinations are useless, since from particulars nothing follows syllogistically (P1). But if [4] the Major is particular and the Minor universal, there are the same four combinations; and all are useless, because the Major in the first Figure has to be universal (P3). Now, if [1] both are universal and affirmative, there is no offence against any of the principles, and so the premise-pair is useful. But if both are negative, or the Major is affirmative and the Minor negative, there are two useless premise-pairs, since from negatives there is no syllogistic conclusion (P2). Also, in the first Figure there is no syllogistic conclusion from a negative Minor (P4). But if the Major is negative and the Minor is affirmative, as long as both are universal there will be a useful premisepair, because there is no offence against any principle. Now, if [3] the Major is universal and the Minor particular, either both are affirmative (and that is a useful premise-pair), or the Major is negative and the Minor is affirmative (and that is useful), or both are negative (P2) or only the Minor (P4)—and these two are useless. So, laying down the said principles, it’s evident that two syllogistic propositions which can be related to each one another in 16 moods make useful premise-pairs in only four ways, leaving 12 useless ones.22 22 Kilwardby ad A4 Part 2 dub.9 (10vb): “Si enim utraque sit particularis, aut utraque est affirmativa aut utraque negativa aut prima affirmativa et secunda negativa aut e converso, et omnes istae quattuor combinationes sunt inutiles quoniam ex particu-

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The four useful premise-pairs are shown in Table 4.1. aa

ai

ea

ei

Table 4.1. Useful first Figure assertoric premise-pairs

The idea of using syllogistic principles to exclude the useless premisepairs can be traced back to Alexander of Aphrodisias, who follows Aristotle’s division of premise-pairs into four cases, according as both are universals, the Major is particular and the Minor universal, the Major is universal and the Minor particular, or both are particular. Alexander doesn’t formulate a connected argument aimed at justifying the exclusion of all the useless premise-pairs in the first Figure; but in four disparate passages he does appeal to the special principles for the first Figure to exclude groups of useless pairs. He uses (P4) to exclude ae-1, ee-1,23 as well as ao-1, eo-1.24 He uses (P3) to exclude ia-1, oa-1,25 as well as ii-1, io-1, oi-1, oo-1.26 Alexander notes that either (P3) or (P4) can be used to exclude ie-1, oe-1.27 Alexander’s use of the device is notable for its reliance solely on the special principles for the first Figure, and also for its acknowledgement of the fact that some useless pairs can be excluded by more than one principle. laribus non syllogizatur. Si autem maior sit particularis et minor universalis, ibi sunt eaedem quattuor combinationes, sed omnes inutiles quia maior in prima figura debet esse universalis. Si autem utraque sit universalis et affirmativa, non est peccatum contra aliquod principium, et ideo coniugatio utilis. Si autem utraque sit negativa vel maior affirmativa et minor negativa, sunt duae coniugationes inutiles quia ex negativis non syllogizatur. In prima etiam figura non syllogizatur ex minore negativa. Si autem maior fuerit negativa et minor affirmativa, dummodo ambae fuerint universales, erit utilis coniugatio quae non peccat contra aliquod principium. Si autem propositio maior fuerit universalis et minor particularis, aut utraque est affirmativa, et est utilis coniugatio, aut maior negativa et {et ACm1CrE1E2F1F2O2O3P1V: aut Ed} minor affirmativa, et adhuc utilis, aut utraque est negativa aut tantum minor, et sunt duae inutiles. Suppositis ergo dictis principiis manifestum est quod duae propositiones syllogisticae, quae a sedecim modis se possunt habere, tantum quattuor modis coniugationes utiles facere possunt, duodecim autem inutiles faciunt.” 23 Alexander ad A4, 26a2–9 55,17 ff. 24 Alexander ad A4, 26a39–b10, 63,10ff. 25 Alexander ad A4, 26a30–33, 62,25 ff. 26 Alexander ad A4, 26b21–25, 68,15 ff. 27 Alexander ad A4, 26a30–33, 62,25 ff.

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The device occurs in a number of Arabic sources including the Logica Algazelis (Al-Ghazali’s Maq¯asid al-fal¯asifa). According to Charles Lohr, this work was known in the Latin world “from early in the second half of the twelfth century”.28 The cases are divided into four, according as the Minor is universal affirmative, particular affirmative, universal negative or particular negative. Ghazali argues that (P3) excludes ia-1, oa-1; (P1) excludes ii-1, oi-1; and (P4) excludes ae-1, ee-1, ie-1, oe-1, ao-1, eo-1, io-1, oo-1.29 This approach is quite distinctive. Ghazali’s and Alexander’s ways of dividing the cases are shown in Fig. 4.7 and Fig. 4.8.

Fig. 4.7. Ghazali’s division of first Figure premise-pairs

Fig. 4.8. Alexander’s division of first Figure premise-pairs

Alexander’s division is independent of the syllogistic principles, and so each case will include some fruitful and some sterile pairs. By contrast, Ghazali’s primary division draws on (P4), which determines that all pairs in cases 3 and 4 will be sterile. Ghazali also uses (P1) as well as (P3) and (P4), whereas Alexander relies solely on (P3) and (P4). 28 29

Lohr, “Logica Algazelis: introduction and critical text”, 228. Lohr, “Logica Algazelis: introduction and critical text”, 262–263.

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The idea is taken up in various Latin treatises in the twelfth century, including the Ars Burana. De Rijk dates this work to “the third quarter of the twelfth century”.30 The cases are divided into four, corresponding to the four principles (P1)–(P4). It is argued that (P2) excludes the pairs ee-1, eo-1, oe-1 oo-1; and (P1) additionally excludes ii-1, io-1, oi-1; and (P3) additionally excludes ia-1, ie-1, oa-1; and (P4) additionally excludes ae-1, ao-1; so that only the 4 useful premise-pairs remain.31 Here there is no division of the cases independent of the principles to be used in excluding useless pairs. With these comparators in view, we can see that Kilwardby’s approach is like Alexander’s in starting from Aristotle’s exhaustive division of the cases; however, unlike Alexander, and like Ghazali and the Ars Burana, Kilwardby invokes general as well as special syllogistic principles. However, the closest analogue to Kilwardby’s reasoning is found in another twelfth-century treatise, the Dialectica Monacensis. Here we find all the distinctive features of Kilwardby’s approach—the exhaustive division of cases, the deployment of general as well special principles: In the first Figure there are 16 premise-pairs, 4 useful and 12 useless. In order that we may ascertain which are useful, it is to be known as a rule that (P3) when the Major is particular, or (P4) the Minor negative, nothing follows in the first Figure. And this rule is to be understood solely in relation to direct conclusions. About this rule, we can have 16 premise-pairs, dividing them into 4 useful and 12 useless as follows. The two premised propositions are related either as [1] both universal, or [2] both particular, or the [3] Major universal and the Minor particular, or [4] vice versa. If [1] both universal, either both are affirmative, or both negative. If both affirmative, the premise-pair will be useful in the first mood of the first Figure [Barbara]. If both negative, the premise-pair is useless in relation to the same [rule]. If the Major is negative and the Minor affirmative, the premise-pair is useful in the second mood of the first Figure [Celarent]. If the other way around, the premise-pair is useless in relation to the same [rule]. Therefore if both are universal, in general we can have two useful and two useless. Again, if [2] both are particular, either both are affirmative or both negative, or the Major negative and the Minor affirmative, or vice versa. De Rijk, Logica Modernorum vol. 2 part 1, 398. Ars Burana in L.M. de Rijk, Logica Modernorum: a contribution to the history of early terminist logic Vol. 2 Part 2 (Assen: Van Gorcum, 1967) 175–213, 198:26–199:16. 30 31

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chapter four And either way, the premise-pair is useless. Hence if both are particular, we have 4 useless and none useful. Again, if [3] the Major is universal and the Minor particular, either both will be affirmative or both negative. If both affirmative, the premise-pair is useful in the third mood of the first Figure [Darii]. If both negative, it is useless in relation to the same [rule]. Again, if the Major is negative and the Minor affirmative, the premise-pair is useful in the fourth mood of the first Figure [Ferio]. If the other way around, it is useless in relation to the same [rule]. Again, if [4] the Major is particular and the Minor universal, either both will be affirmative or negative, or the Major negative and the Minor affirmative, or vice versa. And either way, the premise-pair is useless. So therefore we have 4 useless premise-pairs. So therefore according to the aforesaid rule we can have 16 premisepairs, 4 useful and 12 useless.32

The main difference between this treatment and Kilwardby’s is that whereas at each step of the argument Kilwardby explicitly mentions the principle that is being invoked to exclude a premise-pair, the Dialectica Monacensis never does so, merely mentioning at the beginning that the special principles for the first Figure are to be invoked. So, in one way 32 Dialectica Monacensis, 498:5–499:4: “In prima figura sunt sedecim coniugationes, quattuor utiles et duodecim inutiles. Ut autem habeatur quae sint utiles, sciendum pro regula quod maiore existente particulari vel minore negativa nihil sequitur in prima figura. Et regula ista solum intelligenda est de directe concludentibus. Iuxta hanc regulam dividendo possumus habere sedecim coniugationes, quattuor utiles et duodecim inutiles, hoc modo: duae propositiones praemissae aut sic se habent quod ambae sunt universales aut ambae particulares aut maior universalis et minor particularis aut e converso. Si ambae universales, aut ambae affirmativae aut ambae negativae. Si ambae affirmativae, utilis erit coniugatio in primo modo primae figurae. Si ambae negativae, inutilis est coniugatio circa eundem. Si maior negativa et minor affirmativa, utilis est coniugatio in secundo modo primae figurae. Si autem e converso, inutilis est coniugatio circa eundem. Igitur si ambae sunt universales, in universo possumus habere duas utiles et duas inutiles. Item. Si ambae sunt particulares, aut ambae affirmativae aut ambae negativae aut maior negativa et minor affirmativa aut e converso. Et sive sic sive sic, inutilis est coniugatio. Unde si ambae sint particulares, quattuor habemus inutiles et nullam utilem. Item. Si maior sit universalis et minor particularis, aut ambae erunt affirmativae aut ambae negativae. Si ambae affirmativae, utilis est coniugatio in tertio modo primae figurae. Si ambae negativae, inutilis est circa eundem. Item. Si maior negativa et minor affirmativa, utilis est coniugatio in quarto modo primae figurae. Si e converso, inutilis est circa eundem. Hic igitur habemus duas utiles et duas inutiles. Item. Si maior particularis et minor universalis, aut ambae erunt affirmativae aut negativae aut maior negativa et minor affirmativa aut e converso. Et sive sic sive sic, inutilis est coniugatio. Sic ergo habemus quattuor inutiles. Sic ergo secundum praedictam regulam possumus habere sedecim coniugationes, quattuor utiles et duodecim inutiles.”

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Kilwardby’s approach marks an advance, since it displays the logic of the reasoning more overtly. On the other hand, it lacks the elegance of the Dialectica Monacensis approach (and that of Alexander) in so far as it unnecessarily invokes the general principles. Syllogisms The valid syllogisms resulting from the useful premise-pairs are Barbara, Celarent, Darii and Ferio. These are perfect syllogisms. Applying the analysis of perfection proposed in Chapter Two, we see that the universal syllogisms exhibit downwards monotonicity for the subject. In all four syllogisms the conclusion is assimilated to the premises in the manner noted by which Kilwardby. In his discussion of A28, where Aristotle lists the methods whereby different forms of proposition may be concluded in a syllogism, Kilwardby summarizes the ways in which universal affirmatives and universal negatives may respectively be syllogized. His summary is a compendious wording of Barbara, Celarent, Cesare and Camestres: A universal affirmative is constructed through something agreeing universally with both extremes, which indeed must be placed as a Middle. But a universal negative is concluded through a Middle agreeing with one extreme and differing from the other universally—though this difference and agreement can alternately be in respect of the subject and the predicate or vice versa.33

About Barbara he elsewhere observes: … in every syllogism concluding a universal affirmative, whether or not it is a genuine demonstration, the Middle is under the Major extreme at least conceptually [secundum rationem ad minus], since it is as a part of it, even if it isn’t always a part in reality [secundum rem]. And I say “conceptually” because a subject as such always has the nature of an inferior in respect of a predicate.34 33 Kilwardby ad A28 Part 1 dub.2 (40ra): “… universalis affirmativa construitur per aliquid utrique extremorum conveniens universaliter, quod quidem necesse est esse medium positione. Universalis autem negativa concluditur per medium uni extremorum conveniens et ab alio differens, et hoc universaliter; sed haec differentia et convenientia potest alternatim esse respectu subiecti et praedicati aut e converso.” 34 Kilwardby ad A31 dub.2 (43rb): “… in omni syllogismo concludente universalem affirmativam, sive sit vera demonstratio sive non, ipsum medium secundum rationem adminus est sub maiori extremitate, cum ut pars eius sit, quamvis non semper sit pars secundum rem. Et dico secundum rationem quia subiectum in quantum huiusmodi semper habet rationem inferioris respectu praedicati.”

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In his discussion of A26, he gives an “ontological” statement of Barbara and Celarent, which assumes that the premises express relations within the categorial order: … if the inferior is in anything taken universally, the superior is in the same.35 … when the Middle and the Major extremes are disparate, whatever one of them is in universally the other is not in universally.36

Barbara As an apparent counter-example to Barbara, he considers the sophism: all bronze is natural, every statue is bronze, so every statue is natural. (This sophism is also found in the Dialectica Monacensis,37 in the section dealing with the Fallacy of Accident; and Kilwardby mentions it as committing this Fallacy in his De Ortu Scientiarum.)38 Unmoved, he maintains that Barbara is formally valid: “Hence it is to be said that the conclusion (namely, that every statue is natural) follows according to the art of this book. Or if not, then the first [premise] is to be denied”.39 He adds that the conclusion “Every statue is natural” is true, even if per accidens.40 Albert elaborates: “The conclusion is true only per accidens, if it is supposed that there is no statue that is not of bronze. For then a statue is natural, not through being a statue, but through being bronze”.41

35 Kilwardby ad A26 dub.2 (37ra): “… sequitur si inferius inest alicui universaliter sumpto quod superius insit eidem.” 36 Kilwardby ad A26 dub.2 (37rb): “… quando medium et maior extremitas sunt disparata, cuicumque unum inest universaliter non inest reliquum.” 37 Dialectica Monacensis, 587:28–30. 38 Kilwardby, De Ortu Scientiarum, 174. 39 Kilwardby ad A4 Part 2 dub.7 (10va): “Unde dicendum quod sequitur conclusio, scilicet ‘Omnis statua est naturalis’, secundum artificem huius libri, vel si non, neganda est prima.” 40 Kilwardby ad A4 Part 2 dub.7 (10vb): “… conclusio est {conclusio est ABrCm E F 1 2 1 F2O2O3P1: conclusio sit E 1: conclusio non est Ed} vera quamvis per accidens {quamvis per accidens AEdE 1E2F1O2O3P1: quamvis primo per accidens BrCm1F2}.”. 41 Albertus Magnus, Priorum Analyticorum, I.ii.3 (309B): “Tamen etiam conclusio per accidens vera est si ponatur quod nulla statua sit nisi ex aere. Tunc enim statua non per hoc quod est statua sed per hoc quod est aes est quid naturale.”

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Order of moods Kilwardby expresses a view (apparently with some reluctance [sed hoc non est multum curandum]) about the order in which the four direct moods in the first Figure should be listed. The universal moods should precede the particulars and the affirmatives should precede the negatives.42 We shall find him taking up this question in the other Figures too, with continuing expressions of reluctance.

The second Figure Definition of the second Figure When the same thing inheres in all of one and none of the other, or in all of both or none of both, I call such a Figure the second.43

Notice that this definition requires that one of the premises be negative. But, says Kilwardby, isn’t this too broad a definition? There are no valid moods in the second Figure with both premises negative, or both affirmative; but that shouldn’t enter into the Figure’s definition. He reminds us that Aristotle is here setting aside questions of Mood and looking only at the Figure.44 Since Mood is a function of quality, this remark implies that in Kilwardby’s view it can’t be part of the definition of a Figure that there be a negative premise. Albert expands a little on this: “In the useful conjugations, and in those that are useless, the position of the terms in place and in order is preserved. And this makes the second Figure”.45

42 Kilwardby ad A4 Part 2 dub.14 (11ra): “… sed hoc non est multum curandum, verum tamen, si placet, ordo per se patet, quia universale aut particulare et affirmativum aut negativum etc.” 43 A5, 26b34–36. Kilwardby 11ra. 44 Kilwardby ad A5 dub.1 (12ra): “Et dicendum quod Aristoteles ibi significat figuram tantum circumscribendo modum.” 45 Albertus Magnus, Priorum Analyticorum I.ii.6, 312A: “Tamen tam in utili coniugatione quam in ea quae inutilis est salvatur positio terminorum in situ et ordine. Et hoc facit secundam figuram.”

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Definition of Middle I call that the Middle which is predicated of both, and those of which this is predicated I call Extremes; the Major extreme I call the one placed next to the Middle, while I call the Minor the one situated furthest from the Middle. (The Middle is placed outside the extremes and is first in position.)46

Kilwardby distinguishes a literal sense of “middle” as that which is between extremes (and is thus middle in position), from a metaphorical sense as that which unites extremes in the understanding. The Middle in the first Figure is a middle in both these senses. But in the second Figure it is a middle only in the metaphorical sense. In position, the Middle is first in the second Figure, with the Major second and the Minor third. At least, this is how Aristotle sets the terms out in his counter-examples to invalid moods. Kilwardby says he doesn’t mind this, since such a positioning is accidental to the second Figure.47 Aristotle’s reductions Aristotle reduces second Figure syllogisms directly or indirectly to the first Figure in the ways illustrated in Fig. 4.9.48 e --- conv

ace

abe bae ac+ bc---

Celarent

aba

cae cbe bce

aba bc+ ac+

C

→

aba ac---

abe bc+

bc---

ac---

e --- conv

e --- conv

C

→

abe ac+ bc---

Fig. 4.9. Aristotle’s direct or indirect reduction of second Figure syllogisms A5, 26b36–39. Kilwardby 11ra. Kilwardby ad A5 dub.2 (12ra–b): “… sed de isto non est curandum; talis enim positio valde accidentalis est secundae figurae.” 48 The direct reductions are given at A5, 27a5–14, 32–36. The indirect reductions of Cesare, Camestres and Festino are signalled at A5, 27a14–15 and A7, 29b5–6. The indirect reduction of Baroco is given at A5, 27a36–b1. The remaining indirect reductions are given at B11, 61b19–23, 37–38; B14, 63a7–14, 16–18; A29, 45a28–31. See Thom, The Syllogism, 47–50. 46 47

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Kilwardby doesn’t comment on the Indirect Reductions (except for that of Baroco) in his commentary on A5; however, he does notice them in his discussion of second Figure conversive syllogisms.49 Principles Kilwardby appeals to the common principles, and in addition to two special principles governing the second Figure: Next, someone will enquire concerning the sufficiency of the moods in this Figure, why when there are 16 premise-pairs, only 4 are useful. This is to be solved by supposing the common principles we had before, and two that are proper to this Figure, of which one is that the Major is universal, and the second is that one of the propositions is negative.50

Thus we have the principles: (P5) In second Figure syllogisms, the Major has to be universal. (P6) In second Figure syllogisms, one of the premises has to be nega-

tive. The Dialectica Monacensis states two slightly different principles: “In the second Figure nothing follows when the Major is particular or when both are of the same quality”.51 This statement is redundant in so far as it includes the common principle proscribing any argument with two negative premises. Kilwardby derives these principles from the routes by which Aristotle perfects second Figure syllogisms. He argues: And it can be said that the syllogisms in this Figure descend from the negative moods of the first Figure by the simple conversion of the Major. Hence the Major in syllogisms of this Figure has to be universal and negative, since there the Major is universal and negative.52 49

Kilwardby ad B9 Note (64vb). Kilwardby ad A5 dub.8 (12va): “Adhuc consequenter dubitabit aliquis de sufficientia modorum huius figurae quare cum sint sedecim coniugationes tantum quattuor sunt utiles. Ad quod solvendum supponenda sunt communia principia prius habita et duo propria huius figurae, quorum unum est quod maior sit universalis, secundum quod altera propositionum sit negativa.” Equivalents of (P5) and (P6) are stated in Ghazali (Lohr, “Logica Algazelis: introduction and critical text”, 263). 51 Dialectica Monacensis, 499:7–8: “In secunda figura maiore existente particulari vel utrisque similis qualitatis nihil sequitur.” 52 Kilwardby ad A5 dub.9 (12va): “Et dici potest quod syllogismi huius figurae descendunt a modis negativis primae figurae per conversionem maioris simpliciter. Quare cum illic sit maior universalis et negativa, oportet in syllogismis huius figurae maiorem 50

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Evidently, the principles which Kilwardby’s reasoning establishes are those underlying Aristotle’s methods of perfection, not those underlying all possible methods of perfection. The importance of the distinction will become apparent in Chapter Six. This situation will be repeated throughout his commentary. Kilwardby will regularly argue for the principles underlying each class of syllogisms, along similar lines to those we have analyzed for second figure assertoric syllogisms. Kilwardby also supplies a second proof of (P5): Or better, it can be said that if the Major in this Figure were particular, the Major Extreme could be in more things than the Middle and the Minor Extreme. For an inferior can be predicated of its superior in part, affirmatively and negatively. And if this were so, a true negative conclusion could not be made, because a superior cannot be truly denied of its inferior. Neither could there be an affirmative conclusion, because that is not concluded in this Figure (as will already be clear). And so no conclusion could follow if the Major were particular. Now, the reason why a negative doesn’t follow is that the negation of a superior doesn’t follow from the negation of an inferior to the same. And this is clear from the terms “man”, “animal”, “ass”. For, even if some animal is a man, and no ass is a man, it doesn’t follow that no ass is an animal.53

Here Kilwardby supplies a recipe—using categorial terms—for constructing a counter-example to any purported second Figure syllogism with a particular Major. The instructions are as follows. Let the Major “b” be superior to both the Middle “a” and the Minor “c”. Then some b is a and some b is not a, some b is c and some b is not c. So, we have a counter-example, as shown in Fig. 4.10.

esse universalem et negativam {esse universalem et negativam BrCm1CrE2F2KlV: esse universalem AE 1F1O2O3P1: universalem negativam Ed}.” 53 Kilwardby ad A5 dub.9 (12va): “Vel melius dici potest quod si maior esset particularis in hac figura, posset maior extremitas esse in plus medio et minore extremitate {extremitate ACm1F1F2O2O3V: existente BrF 2P1Ed}. Potest enim inferius praedicari de suo superiore particulariter affirmative et negative. Et si ita esset, non posset concludi conclusio negativa vera, quia superius de inferiore non potest vere negari, nec etiam concludetur affirmativa, quia non concluditur in hac figura, sicut iam patebit. Et ita nulla conclusio posset sequi si maior esset particularis. Causa autem quare negativa non sequitur est quia non sequitur ad negationem inferioris de aliquo negatio superioris de eodem. Et hoc patet in his terminis ‘homo’ ‘animal’ ‘asinus’. Quamvis enim aliquod animal sit homo et nullus asinus est homo, non tamen sequitur quod nullus asinus sit animal.”

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abi ac bco Fig. 4.10. Second Figure syllogism with particular Major

Terms satisfying Kilwardby’s prescription are “man”, “animal”, and either “man” or “ass”, depending on whether we want an affirmative or negative Minor. These terms make the premises true and the conclusion false. He also gives a recipe for constructing a counter-example to any purported second Figure syllogism with two affirmative premises: … since the Middle is predicated of the Extremes, if both were affirmative a superior could be predicated of two disparate [terms] neither of which could be affirmed of the other. Also, a superior could be affirmed of two inferiors one of which was under the other, so that one could not be denied of the other. Hence it’s evident that neither an affirmation nor a negation can follow in this Figure, unless one of the propositions was negative.54

Here his prescription is to take two inferiors of a common superior. In order to show that no affirmative conclusion follows, let the inferiors be disparate. In order to show that that no negative conclusion follows, let one be inferior to the other. Fig. 4.11 illustrates the strategy. (animal)(man)a (animal)(ass)a (man)(ass)i (body)(man)a (body)(animal)a (man)(animal)a Fig. 4.11. Counter-examples to aa-2 moods

Aristotle states that in the second Figure the conclusion has to be negative.55 Kilwardby proposes a line of reasoning in support of this: 54 Kilwardby ad A5 dub.10 (12va): “… cum medium praedicetur de extremis, posset superius ad duo disparata praedicari de illis, quorum neutrum de altero posset affirmari. Posset etiam superius affirmari de duobus inferioribus quorum alterum esset sub altero, et ita non posset alterum de altero negari. Quare manifestum est quod nisi fuerit altera propositionum negativa neque sequitur affirmatio neque negatio {negatio ABrCm1CrE1E2F1F2KlO2O3P1V: negativa Ed} in hac figura.” 55 A5, 28a7–9.

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chapter four And it is to be said that this is because when one is negative a difference is signified between one Extreme and the other, through the Middle which belongs to one of them and not the other. Hence, since the conclusion is made from the Extremes, it’s necessary that it deny one Extreme of the other.56

He notes a further feature of the second Figure syllogisms, namely that the conclusion has the same quantity as the Minor. The reason for this, he says, is that the Minor and conclusion share a subject, and quantity is a determination of the subject.57 Useful premise-pairs Kilwardby uses his two principles to deduce the useful premise-pairs: When these are supposed, if subsequently the combinations are made of two syllogistic propositions by universal and particular and further by affirmative and negative, just as was done in the preceding chapter, it is clear that only four premise-pairs are useful and do not offend against any principle, but the other 12 do offend against some principle and accordingly are to be counted as useless.58

(P5) excludes 8 cases where the Major is particular (ia-2, ie-2, ii-2, io-2, oa-2, oe-2, oi-2, oo-2). (P6) additionally excludes aa-2, ai-2. The common principle (P1) requiring a universal premise is already implicit in (P5). The common principle requiring an affirmative premise (P2) additionally excludes ee-2, eo-2. 4 useful pairs remain. They are shown in Table 4.2.59

56 Kilwardby ad A5 dub.12 (12va): “Et dicendum est quod hoc est quia altera exsistente negativa significatur differentia unius extremitatis ad alteram per medium quod uni illorum contingit et alii non. Quare cum conclusio fiat ex extremis, necesse est ipsam removere unum extremum ab altero.” 57 Kilwardby ad A5 dub.13 (12va): “Amplius autem quaeret forte aliquis quare in hac figura conclusio semper assimiletur minori propositioni in quantitate. … Et dicendum quod idem subicit in minore et conclusione, et a subiecto distinguitur quantitas. Et ideo assimilatur conclusio minori in quantitati.” 58 Kilwardby ad A5 dub.8 (12va): “Quibus suppositis si postea fiant combinationes duarum propositionum syllogisticorum per universale et particulare et ulterius per affirmativum et negativum, sicut factum est in capitulo praecendenti, patet quod tantum quattuor coniugationes utiles sunt et non peccantes contra aliquod principium. Aliae autem duodecim peccantes sunt contra aliquod principium et ideo inutiles sunt reputandae.” 59 This reasoning is not presented in detail but summarized by Kilwardby ad A5 dub.8 (12va).

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ae

ea

ao

ei

135

Table 4.2. Useful second Figure assertoric premise-pairs

Syllogisms The valid syllogisms from these pairs of premises are Cesare, Camestres, Festino and Baroco. Order of moods The question of the order of the moods in the second Figure is one on which Kilwardby expresses an opinion, while at the same time saying that it is not of much importance. The universal moods descend from Celarent–Cesare by a process of conversion, Camestres by a more complex process of conversion and transposition. This added complexity places Camestres after Cesare in the order. The particular moods arise from the universals—Festino from Cesare by replacing the Minor by its subaltern, Baroco from Camestres by the same process.60

The third Figure Aristotle’s reductions Aristotle reduces third Figure syllogisms to the first Figure in the ways illustrated in Fig. 4.12.61

60

Kilwardby ad A5 dub.14 (12va–b). The direct reductions are given at A6, 28a17–22, 26–29; 28b7–14, 33–35. The indirect reductions are given at A6, 28b17–20; A7, 29a37–39, 29b19–21; A29, 45a31–33; B11, 61b11–15; B14, 63a19–23. See Thom, The Syllogism, 50–52. 61

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ac

bc+

ac+

cb+

ca+

bc+

ba+

ab

ab+ abuniv bc+ ac

C →

ac bc+ abpart

Fig. 4.12. Aristotle’s direct or indirect reduction of third Figure syllogisms

Kilwardby doesn’t comment on the Indirect Reductions (except for that of Bocardo) in his commentary on A6; however, he does notice them in his discussion of third Figure conversive syllogisms.62 Principles Kilwardby says that the two common principles (P1) and (P2) must be observed in this Figure63 along with one principle that is peculiar to the third Figure: “And it is to be said that the two common principles are to be supposed, and one principle proper to this Figure, namely that the Minor is affirmative”.64 Thus we have the principle: (P7) In third Figure syllogisms, the Minor must be affirmative.

The Dialectica Monacensis states a more complex principle: “In the third Figure nothing follows when the Minor is negative or when both are particular”.65 The second part of this is redundant, given that every syllogism must have a universal premise. Kilwardby argues for the correctness of (P7) on the basis of the ways in which Aristotle perfects third Figure syllogisms (see Fig. 4.12): “The third Figure descends from the first Figure by conversion of the Minor. 62

Kilwardby ad B10 Note (65ra). See his discussion of the first Figure. 64 Kilwardby ad A6 dub.2 (13vb): “Et dicendum quod supponenda sunt duo principia communia et unum principium proprium huius figurae, scilicet quod minor sit affirmativa.” 65 Dialectica Monacensis, 499:11–12: “In tertia figura minore existente negativa vel utraque particulari nihil sequitur.” 63

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Now, the Minor in the first Figure is always affirmative. So the Minor stays affirmative in the third Figure”.66 (P7) is of course a more specific version of one of the common principles, namely (P2) which states that at least one premise needs to be affirmative. So, the mention of that common principle is superfluous in this context; and Kilwardby’s prescriptions for the third Figure, while not incorrect, are not as elegant as they could have been. It is interesting to observe that Ghazali achieves that missing level of elegance by specifying only (P7) and (P1) for the third Figure.67 Aristotle states that in the third Figure the conclusion has to be particular.68 Kilwardby doesn’t prescribe this as a principle, but he does offer a line of reasoning in support of it: The Middle is subject to the Extremes. Hence it is like a part of the Extremes. And so one Extreme can’t be concluded through it, either affirmatively or negatively, of the other—except in part. For, from such a Middle [the Extremes] can’t be concluded either to agree or to differ— except in part. Further, since the Middle is subject to the Extremes, it could be inferior to them in such a way that the Extremes are subalterns. And then the Minor Extreme would be in more than the Major. Hence the Major will not be affirmed or denied of the Minor except in part. For, an inferior is not predicated of a superior except in part, as is clear for example with the terms “animal”, “body”, “man”.69

Here again he draws a counter-example from the categorial order, as shown in Fig. 4.13.

66 Kilwardby ad A6 dub.3 (13vb): “Tertia figura descendit a prima per conversionem minoris, minor autem primae figurae est tantum affirmativa, et ideo manet minor affirmativa in tertia figura.” 67 Lohr, “Logica Algazelis: introduction and critical text”, 265. 68 A6, 29a16–18. 69 Kilwardby ad A6 dub.5 (13vb): “Medium subicitur extremitatibus, unde est tamquam pars extremitatibus. Et ideo per ipsum non potest concludi unum extremum de altero nec affirmative neque negative nisi particulariter, quia non possunt concludi convenire per tale medium nec differre nisi secundum partem. Adhuc cum medium subiciatur extremitatibus, poterit esse inferius ad illas ita quod extrema sint subalterna, et tunc potest minor extremitas esse in plus maiore. Quare de minore non affirmabitur vel negabitur maior nisi particulariter, quia non praedicatur inferius de superiore nisi particulariter. Et patet exemplum in his terminis: ‘animal’ ‘corpus’ ‘homo’.”

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chapter four (animal)(man)a (body)(man)a (animal)(body)a Fig. 4.13. Categorial counter-example to aaa-3

The fact that the conclusion in the third Figure is always a particular proposition doesn’t prevent there being a universal premise, as Kilwardby notes: But this can equally be done whether [the Middle] is subject to the Major Extreme universally and the Minor particularly or vice versa, because in both ways the Middle is a part of both Extremes, and so can give rise to a conclusion in this Figure.70

Since we have a Figure that delivers only particular conclusions, can there be a Figure that delivers only universals? No, says Kilwardby: “The particular is in the universal as consequent in antecedent. And so it’s necessary that a Figure which concludes to universal also concludes to a particular”.71 We have a Figure that delivers only negatives, so why don’t we have a Figure that delivers only affirmatives? And it is to be said that the standing of the Middle to the Extremes is the cause of the conclusion. Now, the standing of the Middle in the second Figure, since it is above the Extremes, is not collected in an affirmative conclusion, as was said previously. But, the standing of the Middle in the first Figure, when it is between the Extremes in position, can conclude to an affirmation through agreement with both Extremes; or to a negative through difference with one of them. Similarly, the standing of the Middle under the Extremes in the third Figure can conclude to a particular affirmative through agreement with both Extremes; or to a particular negative through difference from one Extreme. Therefore, since there are only three syllogistic relations, the first and the third conclude to an affirmation as well as a negation. And so it’s clear that there isn’t any other Figure concluding only to affirmatives, as there is one concluding only to negatives.72 70 Kilwardby ad A6 dub.6 (14ra): “Sed hoc aequaliter potest facere sive subiciatur maiori extremitati particulariter et minori universaliter sive e converso, quia utroque modo est medium pars utriusque extremi. Et ideo potest in conclusionem huius figurae…” 71 Kilwardby ad A6 dub.7 (14ra): “… particularis est in universali sicut consequens in antecedente. Et ideo necesse est figuram quae concludit universalem concludere particularem.” 72 Kilwardby ad A6 dub.8 (14ra): “Et dicendum quod habitudo medii ad extrema

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Kilwardby illustrates (P7) by means of the categorial terms “animal”, “man”, “stone” (to show that no affirmative conclusion follows from a negative Minor), and “substance”, “stone”, “man” (to show that no negative conclusion follows): And it is to be said that the reason for this is that the Middle is under the Extremes. Hence if the Minor were negative, the Extremes could sometimes be separated; and then an affirmation would not follow, as for instance with the terms “animal”, “stone”, “man”. And sometimes the Extremes could be subalternated one to the other, the Minor being under the Major; and then a negation would not follow, as with the terms “substance”, “stone”, “man”—for “stone” will be Middle. And so, with a negative Minor, neither an affirmation nor a negation would follow. And so the Minor has to be affirmative.73

Concerning the second of these arguments, with its reliance on the reducibility of the third Figure to the first Figure by conversion of the Minor, Kilwardby is aware that this implies that in the perfecting syllogisms of the first Figure the Minor will always be a particular, but that this doesn’t mean that in the perfected syllogism of the third Figure the Minor has to be a particular. For, there are cases where the Minor in the third Figure is a universal affirmative, which converts to a particular affirmative in the process of Direct Reduction.74

causa est conclusionis. Sed habitudo medii secundae figurae cum sit supra extremitates, non confertur ad affirmationem concludendum, sicut prius dictum est. Habitudo vero medii {secundae … medii om. Ed} primae figurae cum sit inter extremitates secundum positionem potest per convenientiam ad utramque extremitatem concludere affirmativam sicut per differentiam ab altera illarum negationem {negationem AE 1E2F1F2O2P1V: negationum EdCm1O3}. Similiter habitudo medii in tertia figura sub extremitatibus potest per convenientiam ad utrumque extremum concludere particularem affirmativam sicut per differentiam ad alterum extremum particularem negativam. Cum igitur non sint nisi tres habitudines syllogisticae, prima et tertia concludunt affirmationem sicut negationem. Et sic patet quod non est aliqua figura concludens tantum affirmativam, sicut est aliqua concludens tantum negativam.” 73 Kilwardby ad A6 dub.3 (13vb): “Et dicendum quod ratio huius est quod medium est sub extremis. Unde si minor esset negativa, possent extrema aliquando esse separata, et tunc non sequeretur affirmatio, sicut in his terminis: ‘animal’ ‘lapis’ ‘homo’. Possent etiam extrema aliquando esse subalterna adinvicem minore tamen existente sub maiore, et tunc non sequeretur negatio, sicut in his terminis: ‘substantia’ ‘lapis’ ‘homo’, ‘lapis’ erit medium. Et ita minore existente negativa neque sequitur affirmatio neque negatio, et ita oportet quod sit affirmativa.” 74 Kilwardby ad A6 dub.4 (13vb).

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Useful premise-pairs From (P7) Kilwardby deduces the useful premise-pairs: When these things are supposed, if the combinations are made both by universal and particular and by affirmative and negative, just as was said in the other Figures, it will be clear that 10 premise-pairs offend against some principle, and 6 do not offend; and so only 6 are useful.75

On the basis of his prescriptions, we can exclude as useless the 8 premise-pairs that have a negative Minor (ae-3, ao-3, ee-3, eo-3, ie3, io-3, oe-3, oo-3), since they conflict with (P7). Given (P1), we can additionally exclude ii-3 and oi-3. There remain only 6 useful premisepairs. These are shown in Table 4.3. aa

ai

ea

ei

ia oa Table 4.3. Useful third Figure assertoric premise-pairs

Syllogisms The valid syllogisms are Darapti, Felapton, Disamis, Datisi, Bocardo and Ferison. Aristotle shows that those moods in this Figure that are subject to Direct Reduction reduce to the universal moods of the first Figure via their reduction to the particular moods in the first Figure.76 The reduction of Darapti to Celarent via Darii is shown in Fig. 4.14.

75 Kilwardby ad A6 dub.2 (13vb): “Quibus suppositis si fiant combinationes et per universale et particulare per affirmationem et negationem, sicut in aliis figuris dictum est, patebit quod decem coniugationes peccant contra aliquod principium, sex autem non peccant. Et ideo tantum sex sunt utiles.” 76 Kilwardby ad A7 dub.8 (15ra).

the assertoric syllogistic

abe Celarent

aca

bae

e --- conv K →

bce

bca aca

cbi abi

141

a --- conv Darii

Fig. 4.14. Reduction of Darapti to Celarent

Felapton, Disamis, Datisi and Ferison can similarly be reduced to Celarent via Darii or Ferio. Of course, Bocardo is not susceptible of this type of treatment. Strengthened moods Some of the moods in the third Figure follow from others by strengthening a premise: Darapti follows from Disamis or Datisi in this way, as does Felapton from Ferison, as shown in Fig. 4.15. sub Disamis

aca aci

bca

abi

Fig. 4.15. Reduction of Darapti to Disamis

Kilwardby acknowledges these facts, saying that they rest on the principle that what follows from the consequent follows from the antecedent.77 Order of moods The universal moods precede the particulars, and among them one reduces to the third mood in the first Figure (Darii) and the other to the fourth mood (Ferio). Thus Darapti precedes Felapton. Kilwardby sees the particular moods as arising from the universals through a process whereby a universal premise is replaced by the corresponding particular, as shown in Figure 4.16.78 The process, of course, is not 78 Kilwardby ad A6 dub.9 (14ra). Philoponus (In AnPr 105,28–31) reports that Theophrastus placed Datisi before Disamis on the ground that it required only one conversion, whereas Disamis requires two, in the process of Direct Reduction to the first Figure. The text is quoted in I.M. Bochenski, ´ La logique de Théophraste (Fribourg en Suisse: Librairie de l’université 1947), 65.

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one of logical consequence; rather the reverse (as Kilwardby noted earlier), since a universal premise is logically antecedent to a particular. A syllogism generated in this way he calls a syllogism by diminution.79

Fig. 4.16. Syllogisms by diminution

Kilwardby notes that the third Figure is unique is exhibiting this relation between universal and particular moods both under the Major Extreme and the Minor Extreme. In the other Figures it only happens under the Minor, as for instance Camestres generates Baroco.80

Indirect and subaltern conclusions In addition to the direct syllogisms recognized in A4–6, Aristotle in A7 recognizes the validity of two indirect moods in the first Figure, as shown in Fig. 4.17.81 Fapesmo

aba bce cao

Frisesomorum

abi bce cao

Fig. 4.17. Fapesmo and Frisesomorum

These premise-pairs were rejected as useless in Aristotle’s discussion of the first Figure in A4. Clearly, they do not obey the principles laid down for the direct first Figure: Frisesomorum has a particular Major, and both of them have negative Minors. While it is true, says Kilwardby, to say that a syllogism may have multiple conclusions, every syllogism has a primary conclusion, from which other conclusions may follow. He takes as an instance the mixture of 79 80 81

Kilwardby ad A6 dub.10 (14ra). Kilwardby ad A6 dub.10 (14ra). A7, 29a23–26.

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the necessary and the contingent, where two conclusions follow, one via the other.82 In the non-modal syllogistic, there are three subaltern first Figure moods, as shown in Fig. 4.18. Baralipton

aba bca cai

Celantes

abe bca cae

Dabitis

aba bci cai

Fig. 4.18. Subaltern first Figure moods And it is to be said that a diverse arrangement of the premises makes for a different conjugation. Now, the two moods which he gives here are arranged differently in the premises than the direct moods. So they are different conjugations from those … However, the other three are not different in their premises … and so he deals with them in Book 2 where he shows the power of the syllogism to have many conclusions.83

When the Major is negative and the Minor affirmative in the first Figure, there may be an indirect conclusion, but the situation is not one where the sole conclusion is the indirect one.84 Aristotle says that there are also indirect moods in the second and third Figures.85 Fig. 4.19 gives apparent counter-examples to these indirect moods. (ass)(man)e (ass)(animal)i

(ass)(man)e (animal)(man)i

(animal)(man)o

(animal)(ass)o

Fig. 4.19. Apparent counter-examples to indirect second and third Figure moods

Kilwardby acknowledges that these arguments are invalid, but he denies that these are the indirect second and third Figure moods that Aristotle means to be valid. Aristotle means the universal negative premise to be the Minor, not the Major. This implies that Aristotle meant the moods in Fig. 4.20 to be valid: Kilwardby ad A25 Part 2 dub.2 (37ra). Kilwardby ad A7 dub.1 (14va–vb): “Et dicendum quod dispositio diversa praemissarum facit diversam coniugationem. Duo vero modi quas hic dat aliter disponuntur in praemissis quam modi directi, et ideo sunt diversae coniugationes ab illis … et ideo in secundo determinat eas ubi ostendit potestatem syllogismi super conclusiones.” 84 Kilwardby ad A7 dub.5 (14vb). 85 A7, 29a26–17. 82 83

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Firesmo

abi ace

Faresmo

cbo

aba ace

Frisemo

cbo

abi bce

Fapemo

cbo

aca bce cbo

Fig. 4.20. Indirect second and third Figure moods86

Aristotle’s reductions Fapesmo and Frisesomorum are perfected “by conversion of both premises and transposition”, as shown in Fig. 4.21.87

a --- conv Ferio

aba

bce

bai

cbe

e --- conv

i --- conv Ferio

cao

abi

bce

bai

cbe

e --- conv

cao

Fig. 4.21. Direct reduction of Fapesmo and Frisesomorum

The indirect moods are perfected “in the second [Figure] by conversion of the universal negative and transposition, in the third by conversion of the affirmative and transposition—and in all cases the reduction is to the fourth of the first [Ferio]”,88 as shown in Fig. 4.22. ace Ferio

abi

cae cbo

e --- conv

i --- conv Ferio

aci cai bco

bce

a --- conv Ferio

aca cai

bce

bco

Fig. 4.22. Direct reduction of non-subaltern indirect second and third Figure moods

The second of the indirect moods in the second Figure differs from the other indirect moods in the second and third Figures in that its premises yield a direct conclusion in the first place, which is only then converted to an indirect one, as in Fig. 4.23.89

86 87 88 89

Kilwardby ad Kilwardby ad Kilwardby ad Kilwardby ad

A7 dub.2 (14vb). See Thom, The Syllogism, 52–55. A7 dub.2 (14vb). A7 dub.2 (14vb). A7 dub.3 (14vb).

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Camestres

145

aba ace

e --- conv

bce cbo

Fig. 4.23. Direct reduction of subaltern indirect second Figure moods

The indirect moods can only yield a particular negative conclusion. Kilwardby argues for this thesis by again appealing to the ontology of superior and inferior. In these moods the Minor term may be inferior to the Major. (In the first Figure, he instances the terms “animal”, “man”, “ass”; in the second Figure “man”, “animal”, “ass”; in the third Figure “animal”, “man” “ass”. These terms satisfy the premises yielding an indirect conclusion in each of the Figures, and in each case the Minor is inferior to the Major.) The indirect conclusion has to be negative (since there is a negative premise). The only way a superior can be denied of an inferior is in a particular proposition.90

Summary Kilwardby proposes the following principles for assertoric syllogisms, the first two of which apply to all syllogisms. (P1) (P2) (P3) (P4) (P5) (P6) (P7)

In every syllogism, one premise must be universal. In every syllogism, one premise must be affirmative. In first Figure syllogisms, the Major must be universal. In first Figure syllogisms, the Minor must be affirmative. In second Figure syllogisms, the Major must be universal. In second Figure syllogisms, one of the premises must be negative. In third Figure syllogisms, the Minor must be affirmative.

By using these he is able to generate exactly the sets of premise-pairs that Aristotle identifies. Most of the principles are to be found in the Dialectica Monacensis, and are also used there to determine the useful premise-pairs. But overall Kilwardby’s statement of them is superior in that it doesn’t contain redundancies.

90

Kilwardby ad A7 dub.4 (14vb).

chapter five NECESSITY-SYLLOGISMS

The chapters on modal syllogistic in the Prior Analytics are divided according to the premise-pairs of the potential syllogisms they discuss. The pure necessity-syllogisms are dealt with in A8, and the assertoric / necessity mixtures of the first, second and third Figures in A9– A11 respectively. Kilwardby’s interpretation of these moods continues the implementation of wider interpretive goals that had already been set in his approach to the assertoric syllogistic. Among those goals is the project of making explicit the logical links that he finds between different parts of Aristotle’s text. What he doing here is to make the presentation of Aristotle’s syllogistic more systematic. As Lewry observes in relation to Kilwardby’s commentaries, Kilwardby’s interest is evidently to discover the unity and coherence of Aristotle’s thought and to apply to logic and philosophical grammar principles, modes of analysis and analogies drawn from natural philosophy.1

Indeed, by using principles on which large sections of the syllogistic can be logically based, he is presenting it not only in a systematic fashion, but also somewhat in the manner of an Aristotelian science. Logic is indeed a science according to Kilwardby, and one which both studies the syllogism and (like all science) proceeds through syllogistic reasoning.2 The principles of syllogistic science, as it turns out, are already in the Prior Analytics in the form of Aristotle’s rules for each Figure and for each mixture of modalities; but they are there more in the manner of regularities observed in the data, than as principles from which the data are to be derived. In particular, Aristotle doesn’t make use of these rules to deduce which premise-pairs are useful and which useless. Kilwardby does so, and takes himself thereby to have established the completeness 1 2

Kilwardby, On Time and Imagination, xv. Kilwardby Prologue dub.8 (2ra).

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[sufficientia] of Aristotle’s results. The principles of this reasoning he infers from a consideration of the routes whereby Aristotle perfects imperfect syllogisms. Kilwardby’s interpretation of the modal syllogistic has another goal —that of providing semantical interpretations for the premises and conclusions of modal syllogisms. For clues to the meaning of these propositions he looks to Aristotle’s ontology. As we saw in Chapter One, he interprets necessity-premises as per se necessities, thus invoking the notion of a per se being. Assertoric premises (at least in the valid moods) he interprets as apodeictic unrestricted assertorics; and in the necessitysyllogistic he takes these to be the same in reality [secundum rem] as necessity-propositions, even though they differ in modality [secundum modum]. On one crucial point—concerning the validity of the first Figure XLL syllogisms that are rejected by Aristotle—Kilwardby’s position is especially interesting. Aristotle uses a counter-example to an XLL1 syllogism that requires something to be true merely as-of-now—the proposition that every animal is moving. However, his use of this type of counter-example gives rise to an interpretive problem, because asof-now counter-examples can equally be brought against the LXL1 moods which Aristotle accepts as valid. To escape this difficulty, Kilwardby postulates the existence of a rule of “appropriation” whereby the Minor in the LXL-1 moods has to be an unrestricted assertoric, whereas there is no such rule requiring the Major in the XLL-1 moods to be unrestricted. The notion of appropriation is metaphorical; but Kilwardby gives some indications as to the limits of its deployment: … the only proposition that can appropriate another so that it is taken according to its requirements is one which is Major in a perfect mood of the first Figure according to to-be-said-of-all or of-none, or for which it’s possible to become the Major in a perfect mood of the first Figure by reduction …3

By appealing to the notion of appropriation, he makes the meaning of assertoric propositions dependent on their inferential context, and in this way solves the interpretive problem at hand. The contextdependence of the assertoric propositions in mixed necessity-syllogisms 3 Kilwardby ad A15 dub.6 (24vb): “… solum illa propositio potest sibi aliam appropriare ut accipiatur secundum exigentiam eius quae est maior in modo primae figurae perfecto secundum dici de omni vel de nullo vel quae possibilis est ut per reductionem fiat maior in modo perfecto primae figurae …”

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may be compared with the context-dependence of terms, as understood in suppositio theory. Just as in suppositio theory a given predicate may impose certain requirements on what serve as its subject,4 so in Kilwardby’s inferential theory a given Major premise may impose certain requirements on what can act as its Minor: “Because the Major contains the whole syllogism, the Minor has to be according to its requirements, not the other way round”.5 However, this context-dependence has consequences for the character of the Kilwardby’s syllogistic system. While that system has a certain resemblance to an Aristotelian science, it cannot have the character of a formal system in the modern sense: the latter would require that all propositions have contextindependent meanings. As with his interpretation of the non-modal syllogistic, he augments the Aristotelian material in numerous small ways, all of which are designed to support and enhance that material. For example, he provides two alternative interpretations of the expository proofs of Baroco LLL and Bocardo LLL. Finally, the semantics that he has adopted leads him to question two of Aristotle’s counter-examples, and to replace them with ones of his own that are consonant with that semantics. This is an area of Aristotle’s text which Kilwardby has no scruples in amending, since he holds that the examples are not there because the discipline requires them but simply on account of the audience.6

4 For example, William of Sherwood, Introductiones in logicam (V.viii): “Subiectum autem quandoque supponit formam absolute, quandoque autem non, et hoc secundum exigentiam praedicati, secundum illud: ‘Talia sunt subiecta, qualia permiserint praedicata’”. Cited in Luisa Valente, “‘Talia sunt subiecta qualia predicata permititunt’: le principle de l’approche contextuelle”, in Joël Biard et Irène Rosier-Catach, La Traditiona Médiévale des Catégories (XIIe–XVe siècles), Actes du XIIIe symposium européen de logique et de sémantique médiévales (Avignon, 6–10 juin 2000) (Louvain–Paris, Éditions Peeters 2003) 289–311, 294 n. 14. Valente, 292 n. 6 also cites Roger Bacon, who dismisses the notion that the predicate imposes a “requirement” on the subject in his Summulae dialectices: “Et hoc dico quia credatur a multis quod talia praedicata vel subiecta faciant sua virtute huismodi ampliationem vel restrictionem et nituntur quaerere causam huismodi et infinitos errores propter hoc addunt putantes quod de necessitate ampliant talia verba et restringant, cum tamen sine illis in potestate nostra est et bene placito, ut transumamus nomina ampliando et restringendo sicut volumus”. 5 Kilwardby ad A9 dub.7 (16vb–17ra): “… quia maior continet totum syllogismum et ad eius exigentiam oportet minorem esse, sed non e converso.” 6 Kilwardby Prologue dub.7 (2vb).

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In A8 Aristotle states his doctrine of pure necessity-syllogisms. Each of the standard valid assertoric syllogisms remains valid when both premises and conclusion are changed to become necessity-propositions.7 Aristotle’s reductions The pure necessity-syllogisms in the first Figure are perfect. Aristotle says that being-said-of-all or of-none is the same for these syllogisms as for the perfect non-modal moods.8 Further, the standard moods are perfectible when they are expressed as pure necessity syllogisms. Aristotle deduces this conclusion from two premises. (1) Since the laws of conversion for necessity-propositions are the same as for assertoric propositions, all but two of the second and third Figure LL-moods reduce to the first Figure by modal conversion, along the lines shown in Fig. 5.1—in parallel with the direct reductions of assertoric syllogisms. Lace Le --- conv LLL --- 1

Labe Lbae

Lac

Celarent LLL

Lbc---

Lcae

Lcbe Lbce

Lbc+ LLL --- 1

Lab a

Lac

Lcbi

Le --- conv

Le --- conv

L --- conv

Labpart

Fig. 5.1. Aristotle’s reductions of LL-2 and LL-3 moods

(2) The two exceptions are Baroco LLL and Bocardo LLL, which are perfectible, though not in the same way as their assertoric counterparts; for these are perfected not by Direct Reduction but by Exposition, as shown in Fig. 5.2.

7 8

A8, 29b36–30a2. A8, 30a2–3.

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Camestres LLL Felapton LLL

Laba Lane Lbne

→

Lbco Lcna Lbca

Felapton LLL

Exposition

Lcna

Lane

Lbna

Barbara LLL Exposition

→

Labo

151

Laba Laco Lbco

Laco Lbca Labo

Fig. 5.2. Expository proofs of Baroco LLL and Bocardo LLL, with general terms

On this way of understanding modal Exposition, the process does not proceed by taking singular terms, but by taking “less universal” ones. On a second understanding, the proofs by modal Exposition can be represented as making use of syllogisms containing singular terms. These syllogisms are in the third Figure but not in any mood of that Figure, as shown in Fig. 5.3.9 Laba ~Laz ~Lbz

Lcz

Lbco

Exposition

→

Laba Laco Lbco

Lcz Lbca ~Laz

Lbz Labo

Exposition

→

Laco Lbca Labo

Fig. 5.3. Expository proofs of Baroco LLL and Bocardo LLL, with singular terms

Either way, the upshot is that the same syllogisms are perfectible whether they are assertoric or pure necessity-syllogisms.10

Kilwardby ad A8 dub.8 (16ra). See Thom, The Logic of Essentialism, 281. Kilwardby ad A8 dub.6–7 (15va–b) also considers indirect reductions of Baroco LLL and Bocardo LLL to Barbara LMM and Barbara MLM respectively. He struggles unsuccessfully to demonstrate the validity of these Barbaras. 9

10

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Principles It follows that the principles governing pure assertoric syllogisms are the same as those governing assertorics: And note that for the completeness [sufficientia] of the moods in the uniform generation of necessity-syllogisms the very same principles are to be supposed in every Figure as were supposed before in the generation of assertoric syllogisms; and in each Figure there will be just as many moods here as there, and no more.11

As the Dialectica Monacensis puts it: “It is to be noted therefore that a necessity-conclusion follows from two necessities in the first or the second or the third Figure”.12 This means that we have the following principles: (P1) In (pure necessity-)syllogisms, one premise must be universal. (P2) In (pure necessity-)syllogisms, one premise must be affirmative. (P3) In first Figure (pure necessity-)syllogisms, the Major must be uni-

versal. (P4) In first Figure (pure necessity-)syllogisms, the Minor must be affir-

mative. (P5) In second Figure (pure necessity-)syllogisms, the Major must be

universal. (P6) In second Figure (pure necessity-)syllogisms, one of the premises

must be negative. (P7) In third Figure (pure necessity-)syllogisms, the Minor must be affirmative. Syllogisms As we have seen, Aristotle accepts all the standard LLL-1 moods as perfect; and he takes all standard LLL moods in any Figure to be valid.

11 Kilwardby ad A8 Note (16ra): “Et nota quod ad sufficientiam modorum in uniformi generatione syllogismorum de necessario supponenda sunt principia eadem penitus in omni figura quae prius supposita sunt in generatione syllogismorum de inesse; et tot erunt modi hic per omnem figuram quot et ibi, et non amplius.” 12 Dialectica Monacensis, 500:15–16: “Notandum ergo quod ex duabus de necessario in prima vel in secunda vel in tertia figura sequitur conclusio de necessario.”

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Semantics Applying our analysis of perfection in terms of monotonicity, we see that the universal LLL-1 syllogisms exhibit downwards monotonicity for the subject, relative to necessary inherence. In all four LLL-1 syllogisms the conclusion is assimilated to the premises in the manner noted by Kilwardby. Given the semantics for necessity-propositions which we outlined in Chapter One, it’s clear that all standard LLL moods are valid, because each of them has a valid core inference whose premises and conclusion are embedded in an outer modality of necessity.

LX-1 moods Aristotle gives his doctrine of the first Figure assertoric / necessity-syllogisms in A9. Kilwardby summarizes the chapter: This part is divided into two. In the first, he sets out the universal premise-pairs, and in the second part the particulars (when he says “But in the particulars”).13 In the first, he proceeds thus. First, he lays down a certain rule according to which the premise-pairs proceed, saying that sometimes a necessity-conclusion comes about from one such premise; and because he says “sometimes”, he expounds this, saying that it is not with either proposition being a necessity, but the Major being a necessity and the Minor an assertoric—by which he means the rule that a necessity-conclusion follows when the Major is a necessity and the Minor an assertoric (and this is “But it comes about”).14 Second, he sets out the premise-pairs—first the useful, second the useless.15

Kilwardby sees the order of Aristotle’s presentation—the statement of a principle, followed by a determination of the useful and useless

A9, 30a33 ff. A9, 30a15 ff. 15 Kilwardby ad A9, 30a15–33 (16rb): “Et dividitur haec pars in duas. In prima ponit coniugationes universales, in secunda particulares cum dicit In particularibus autem. In prima sic procedit. Primo ponit quamdam regulam secundum quam procedunt coniugationes dicens quod aliquando accidit conclusio de necessario ex altera propositionum existente tali; et quia dixit quandoque, exponit hoc dicens quod non fit hoc utralibet propositionum existente de necessario sed maiore existente de necessario et minore de inesse tantum—in quo tamen intendit hanc regulam: maiore existente de necessario et minore de inesse sequitur conclusio de necessario; et hoc est Accidit autem. Secundo ponit coniugationes, et primo utiles, secundo inutiles.” 13 14

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premise-pairs—also as an order of logical sequence. In treating A9 this way, he makes a significant improvement on the original treatment. Aristotle had determined the useless premise-pairs by constructing a number of individual counter-examples; but in so doing he left open the question of completeness, because he nowhere argues that the useless premise-pairs which he identifies are the only useless ones. Kilwardby supplies an answer to this question by deriving the list of useless premise-pairs from the principles that Aristotle states for syllogisms in general and for this mixture in particular. Principles The special principle governing this mixture, then, is: (P8) In first Figure assertoric / necessity syllogisms, the necessity-propo-

sition must be Major.16 The Dialectica Monacensis states this principle as follows: “Note that in the first Figure a necessity-conclusion follows as often as the Major is a necessity[-proposition] and the Minor an assertoric”.17 Kilwardby also draws attention to the fact that the common principles (P1) and (P2) apply to the LX-1 syllogisms, as do the first-Figure principles (P3) and (P4).18 But what is the basis for (P8)? The syllogisms of this mixture are perfect; and Kilwardby bases (P8) on some general features of perfect syllogisms in the first Figure: … the conclusion is part of the Major, and mostly in regard to the predicate, which they share. With regard to the subject, it is part of the Minor. And so it follows the Minor in features affecting the subject (such as universality and particularity), and the Major in features affecting the predicate (such as affirmative and negative, assertoric and modal).19

16

Kilwardby ad A9 dub.5 (16vb). Dialectica Monacensis, 500:22–23: “Notandum in prima figura quod quotiens maior fuerit de necessario et minor de inesse, sequitur conclusio de necessario.” 18 For both these groups of principles, see Kilwardby’s treatment of the first Figure. 19 Kilwardby ad A9 dub.5 (16vb): “Et dicendum quod conclusio est pars maioris et maxime secundum praedicatum in quo communicat cum ipsa et quantum ad subiectum pars minoris. Et ideo sequitur minorem in dispositionibus accidentibus subiecto eius quae sunt universalitas et particularitas, maiorem autem in dispositionibus accidentibus praedicato eius quae sunt affirmativum et negativum de inesse et de modo.” 17

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Since modality is a feature affecting the predicate, the conclusion of a perfect first Figure syllogism must have the same modality as the Major premise, and so a necessity-conclusion follows only from a necessityMajor. Useful premise-pairs Kilwardby applies his principles to determine the useful premise-pairs: The combinations are made thus. Since there are two propositions in a syllogism, either both are universal, or both are particular, or the Major is universal and the Minor particular, or vice versa. Whence, in each case these four divide into another four by combinations of affirmation and negation. Thus, if both are universal, either both are affirmative, or both negative, or the Major is affirmative and the Minor negative, or vice versa; and similarly for the other members. And there will be 16 premise-pairs so far, of which each divides into two. Thus, if both are universal affirmative, either the Major is assertoric and the Minor a necessity[-proposition], or vice versa; and similarly if both are negative, and so on for the others. And so in total the premise-pairs are 32, with only 4 useful and 28 offending against some principle, as is easy to see. And if it is asked why he doesn’t set out all the useless premise-pairs, it is to be said that this is because almost all of them are had from things already predetermined. He touches on only four offending against the special principle of this mixture.20

It’s regrettable that, having gone into such detail setting out the preliminaries, he adopts such a casual attitude to the main business of ruling out the 28 useless cases. But we can supply the details. The application of (P1)-(P4) excludes all premise-pairs other than LaXa, XaLa, LeXa,

20 Kilwardby ad A9 dub.6 (16vb): “… fiant combinationes sic: cum duae propositiones sint in syllogismo, aut utraque est universalis aut utraque particularis aut maior universalis et {et ACm1CrE1F1F2KlO2O3P1V: aut EdE 2: eadem unde Br} minor particularis aut e converso. Deinde unumquodque istorum quattuor dividatur in alia quattuor per combinationes affirmationis et negationis sic: si utraque est universalis, aut utraque est affirmativa aut utraque negativa aut maior est affirmativa et minor negativa aut e converso; et similiter de aliis membris. Et erunt ita sedecim coniugationes quarum unaquaeque adhuc dividitur per duo sic: si utraque sit universalis et affirmativa, aut maior est de inesse et minor de necessitate aut e converso, et similiter si utraque sit negativa, et ita de aliis. Et ita in universo coniugationes sunt XXXII, tantum IV utiles, XXVIII {XXVIII ABrCm1E1E2F1F2KlO2O3: XXVI Ed} autem peccant contra aliquod principium, sicut patet intuenti. Et si quaeratur quare non ponit omnes coniugationes inutiles, dicendum quod hoc est quia fere omnes habentur per determinata; tantum quattuor tangit peccantes contra principium primum huius mixtionis.”

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XeLa, LaXi, XaLi, LeXi and XeLi. And, applying (P8) we can exclude XaLa, XeLa, XaLi and XeLi. (These are the four that Aristotle touches on.)21 This leaves the four cases shown in Table 5.1. LaXa

LeXa

LaXi

LeXi

Table 5.1. Useful LX-1 premise-pairs

Syllogisms Aristotle says that each of the standard first Figure syllogisms is perfect when stated with a necessity-Major, an assertoric Minor and a necessity-conclusion.22 Kilwardby comments that the whole power and inference of these syllogisms resides in the Major premise.23 As befits a perfect syllogism, these are justified by an appeal to being-said-of-all or of-none: It’s clear therefore from the nature of being-said-of-all and of-none, and from the rule “When one thing of another”, that when the Major is a necessity-[proposition] the Minor is appropriated so that it is an unrestricted assertoric; and so against such a premise-pair there are no counter-examples with an as-of-now assertoric Minor. But with a necessity-Minor, the Major can be either an as-of-now assertoric or an unrestricted assertoric, because the Minor doesn’t appropriate the Major to itself; and against such a premise-pair there may well be counterexamples with an as-of-now assertoric.24

Kilwardby identifies the rule “When one thing of another etc.” as coming from the introductory chapters of the Categories:25 21

6.

XaLa: A9, 30a23–25. XeLa: A9, 30a, 32–33. XaLi: A9, 30b2–3. XeLi: A9, 30b5–

A9, 30a17–23, 30b1–2. Kilwardby ad A9 dub.1 (16va). 24 Kilwardby ad A9 dub.7 (16vb): “Patet igitur quod per naturam eius quod est dici de omni et de nullo et per illam regulam: Quando alterum de altero, quod quando maior est de necessario, appropriatur minor ut sit de inesse simpliciter; et ideo contra talem coniugationem non est instare per minorem de inesse ut nunc. Quando autem minor est de necessario, quia minor non appropriat sibi maiorem, potest maior esse de inesse ut nunc vel de inesse simpliciter; et ideo contra talem coniugationem bene licet instare {instare ABrCm1CrE1E2F1F2KlO2O3P1V: stare Ed} per maiorem de inesse ut nunc.” 25 Kilwardby ad A9, 30a15–33 (16vb). 22 23

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Whenever one thing is predicated of another as of a subject, all things said of what is predicated will be said of the subject also. For example, man is predicated of the individual man, and animal of man; so animal will be predicated of the individual man also—for the individual man is both a man and an animal.26

The reference to the Categories is significant, since it places beyond doubt Kilwardby’s reliance on ontological doctrines expounded in Aristotelian works other than the Prior Analytics as a source for his interpretation of the syllogistic. The ontological slant is apparent throughout his commentary. Lagerlund is somewhat dismissive of Kilwardby’s appeal to beingsaid-of-all, commenting that he “seems to think” it “illuminates his point”.27 But by our analysis Kilwardby is right to claim perfection for the LXL-1 syllogisms, because Barbara and Celarent LXL exhibit downwards monotonicity for the subject, while the conclusions of Darii and Ferio LXL are assimilated to their premises in accordance with Kilwardby’s prescriptions. Semantics To every first Figure necessity / assertoric syllogism there corresponds a valid pure necessity-syllogism. So, the question arises “whether it is superfluous to add the mode of necessity to the Minor”,28 since without that addition the syllogism is valid (namely as an LXL-1 mood). If the addition is indeed superfluous, it seems that the first Figure uniform necessity-syllogisms commit a fallacy of non-causa ut causa. This is a good question, and one that poses itself for those modern analyses of modal syllogistic that take necessity-propositions to imply, but not to be implied by, the corresponding assertorics.29 A related question is whether the first Figure LLL moods are imperfect, seeing that they are Categories 3, 1b10–15. Ackrill translation. Lagerlund, Modal Syllogistics in the Middle Ages, 42. 28 Kilwardby ad A9 dub.2 (16va): “Adhuc videtur quod si ex una de inesse et altera de necessario sequitur conclusio de necessario, quod in generatione uniformi de necessario aliquid sit quod sit non causa ut causa conclusionis; apponitur enim modus necessitatis ad minorem {modus necessitatis ad minorem ACm1CrF1KlO3V: ad minorem modus necessitatis E 1: ad maiorem modus necessitatis BrE 2F2: ad minorem modus veritatis EdO2} sine quo adeo bene sequitur conclusio de necessario sicut cum illo.” 29 Many of the semantic proposals in Thom, Medieval Modal Systems, (Table 10.9 p. 199) are subject to this difficulty. Among the exceptions are the semantics proposed for the simpliciter systems of Campsall and Kilwardby. 26 27

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derivable from the LXL moods by virtue of the fact that their necessityMinors imply the assertoric Minors of the corresponding LXL moods.30 Either way, it seems that the premises of the first Figure LLL moods are stated in a form that is stronger than what is required in order to deliver the necessity-conclusion. Kilwardby denies that this is the case. He holds that the Minor in the LXL moods must be an unrestricted assertoric which is the same in reality as a necessity-proposition, even if it is not the same in mode; and he deals with the issue by stating that despite the syntactic differences there is no difference “in reality” between the Minor premises in the LLL and LXL moods: Whatever in reality [secundum rem] is the cause of the conclusion there [sc. in the uniform moods] is also the cause in the assertoric / necessitymixture. For, in the mixture the Minor is an unrestricted assertoric, as Aristotle says in the text. So the subject is essentially under the predicate. So it is a necessary proposition. Hence it is the same in reality as a necessity-proposition, even if it is not the same in mode [secundum modum].31

Kilwardby doesn’t explain what he means by “secundum rem”; but it seems he is talking about a proposition’s truth-makers in Aristotelian ontology. What makes the unrestricted proposition “Man is an animal” true is a relation of inferior to superior in the category of substance, whereby Man is essentially under Animal. This relation holds as a matter of necessity; and the same relation makes the necessityproposition “Man is necessarily an animal” true. So the two propositions are semantically identical even though syntactically they differ. Consequently, every standard LLL mood is valid because it contains a valid core inference each of whose premises contains an outer modality of necessity. The same analysis applies to the perfect LXL-1 moods and to the other mixed necessity / assertoric moods that reduce to them. Given that perfection implies validity, it’s clear that some interpretations of assertoric and necessity-propositions are not allowable in per30 This view is adopted in Patzig, 65. Patzig’s view is discussed in Richard Patterson, Aristotle’s Modal Logic: essence and entailment in the Organon (Cambridge University Press, 1995), 214–219. 31 Kilwardby ad A9 dub.2 (16va): “Quicquid enim secundum rem est causa ibi conclusionis est causa in mixtione de necessario et inesse. In mixtione enim illa minor est de inesse simpliciter, sicut dicit Aristoteles in textu. Et ita subiectum essentialiter est sub praedicato. Et ita est propositio necessaria. Quare idem est secundum rem ei quae est de necessario, et si non sit idem secundum modum.”

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fect LXL syllogisms. If in these syllogisms the assertorics could be as-ofnow then the necessity-premises would have to be per accidens, because a mixture of a per se necessity and an as-of-now assertoric, as in Fig. 5.4, doesn’t imply a per se necessity-conclusion.32 L(animal)(man)a (man)(white)a L(animal)(white)a Fig. 5.4. Apparent counter-example to Barbara LXL

So, the only semantic interpretations that preserve the monotonicity (and hence the validity) of the perfect LXL syllogisms are (1) as-of-now assertorics with per accidens necessities, or (2) unrestricted assertorics with per se necessities. We saw earlier that only per se necessities obey the laws of modal conversion. But those laws are needed in order for second and third Figure LX moods to reduce to the first Figure. So type (1) semantics is ruled out, and only type (2) semantics will be congruent with Aristotle’s results. It is not surprising, then, that in his exposition of A9 Kilwardby reads Aristotle’s expression for the Minor premise in LXL-1 moods (“for c is some of those things that are b” or “for c is under b”) as stating that the Minor extreme is essentially under the middle.33 Given this semantics, we can see how being-said-of-all applies to Barbara LXL: And it is to be said that in the first Figure the Major proposition contains the whole syllogism and its inference. For, both the conclusion and the Minor are a part of it. And so when the Major is a necessity[proposition], it appropriates the Minor to itself so that the latter has to be an unrestricted assertoric, and the Minor extreme is taken under the Middle essentially so that the Minor is in reality necessary. For, a part of the necessary as such cannot be other than necessary.34

32

Kilwardby ad A9 dub.6 (16vb). Kilwardby ad A9, 30a9–33 (16rb); ad A9, 30a33–b6 (16va). For the same interpretation of assertoric premises in the other Figures, see Kilwardby ad A10, 30b7–40 (17rb); ad A11, 31a18–b12 (18ra); ad A11, 31b12–33 (18rb); ad A11 dub.1 (18vb). 34 Kilwardby ad A9 dub.6 (16vb): “Et dicendum quod maior propositio in prima figura continet totum syllogismum et consequentiam eius, quia tam conclusio quam minor pars eius est. Et ideo maior cum sit de necessario appropriat sibi minorem ita quod oportet ipsum esse de inesse simpliciter et minorem extremitatem essentialiter accipi sub medio, ita quod minor sit secundum rem necessaria; pars enim necessarii in quantum huiusmodi non est nisi necessaria.” 33

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He says that since the conclusion is part of the Major it has to be necessary because a part of the necessary cannot be other than necessary. Now, it is worth pointing out that if the Minor premise in an LXL-1 syllogism were as-of-now, the conclusion might not be necessary in the same sense as the Major. If (as in Fig. 5.4) everything white is a man and of necessity every man is an animal, it follows indeed that of necessity everything white is an animal; but the necessity of the conclusion is, Kilwardby says, a per accidens necessity, whereas that of the Major was per se. The conclusion in this case is not part of the Major. In order for it to be so, the Minor has to be an unrestricted assertoric. The metaphor of the Minor’s appropriation by the Major is not fully clear; but Albert, instead of talking about appropriation, talks about “being within the ambit of its generality” [intra ambitum suae communitatis],35 and this gloss may be helpful. Let’s remember that Kilwardby is assuming the Major to be a per se necessity-proposition and the Minor an unrestricted assertoric. Such a Major states, of a class of beings identified (say) as the bs, that they, considered as such (i.e. from the point of view of a philosophical science of being as being), are as. Therefore it includes under its subject-term (within the ambit of its generality) any term “c” such that the cs, considered as such, are bs; and it implies that they will also be as. This is why a per se Major requires an unrestricted Minor. Only in such a Minor does the predicate apply to the subject considered as such. And in such a configuration, it is indeed true that the Minor forms part of the Major, and being such a part inherits the Major’s necessity. In an as-of-now Minor (such as “Everything white is a man”) the predicate doesn’t apply to the subject considered as such: it’s not true of white things considered as such that they are men. Kilwardby’s rules of appropriation amount to the adoption of the following rule: (F) In syllogisms complying with Aristotelian principles,36 an asser-

toric premise is to be read as unrestricted; in other contexts, an assertoric premise may be read either as an as-of-now assertoric or as unrestricted.

35 Albertus Magnus, Priorum Analyticorum, I.iii.4 (331B). Apparently Albert doesn’t notice that to follow Kilwardby on this point is incompatible with allowing as-ofnow counter-examples to Barbara LXL. Later, in discussing the second Figure at I.iii.6 (333B), he adopts Kilwardby’s expression “to appropriate”. 36 In the present mixture, the Aristotelian principle is (P8).

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We will see this rule being applied in all mixed syllogisms containing an assertoric premise. Throughout the assertoric / necessity-syllogistic he pursues this approach, taking an assertoric sentence’s signification to vary with its inferential context. Since this approach allows the one sentential form to have different meanings in different contexts, it implies that Aristotle’s assertoric / necessity-syllogistic is not to be understood as a fully formal system. Kilwardby refers to an unnamed interpreter who distinguishes between a valid and an invalid version of Barbara LXL: And someone may say that if the Minor proposition is an as-of-now assertoric so that something is taken under the Middle in which it inheres per accidens, a conclusion doesn’t follow; but if the Minor is taken as an unrestricted assertoric, it does follow.37

In fact, this distinction is made by Albert the Great, who says that Barbara LXL is valid if the Minor is unrestricted, but not if it is an as-of-now proposition.38 Kilwardby’s position on this is subtly different from Albert’s. He agrees with Albert’s judgment that an inference such as that in Fig. 5.4 is invalid. However, he doesn’t agree that such an inference instantiates Barbara LXL. According to Kilwardby, the rule of appropriation requires that all instances of Barbara LXL have an unrestricted assertoric premise, whereas instances of Barbara XLL may have either an unrestricted or an as-of-now assertoric premise. By contrast, Albert allows instances of either Barbara LXL or Barbara XLL to have either an unrestricted or an as-of-now assertoric premise. Invalid inferences As I read him, Kilwardby holds that since the assertoric Major in the first Figure XLL moods may be true merely as-of-now, those moods are invalid. Lagerlund, however, holds a different interpretation according to which Kilwardby “accepts—LL [sc. XLL] for the first figure”.39 Lagerlund’s view is not totally wrong. On the one hand, because the Major in the XLL-1 moods may be an as-of-now assertoric, those 37 Kilwardby ad A9 dub.6 (16vb): “Et dicet aliquis quod si minor propositio sit de inesse ut nunc ita quod accipiatur sub medio aliquid cui inest per accidens, non sequitur conclusio, sed si accipiatur minor de inesse simpliciter, sequitur.” 38 Albertus Magnus, Priorum Analyticorum, I.iii.3 (329B). 39 Lagerlund, Henrik, “Medieval Theories of the Syllogism”.

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moods are invalid; and both Aristotle and Kilwardby are aware of this fact. On the other hand, were we to specify that the Major in these moods has the sense of an unrestricted assertoric, then given Kilwardby’s doctrine of unrestricted assertorics, we would have to say the XLL-1 moods thus specified are valid, because according to that specification they would be semantically indistinguishable from the corresponding LXL moods. However, even in that case the XLL moods thus specified would not be perfectible, since there would be no way of reducing them to the perfect syllogisms of the first Figure. So, under the hypothesis that all the assertoric premises are unrestricted, it turns out that not all valid inferences are perfectible.

LX-2 moods Aristotle gives his doctrine of the second Figure assertoric / necessity syllogisms in A10. Kilwardby summarizes the chapter: Here he determines the mixture of necessities and assertorics in the second Figure, and it divides into two [parts]. In the first, he gives the universal premise-pairs; in the second, the particulars (when he says “Similarly will be related”).40 In the first, he proceeds thus. First, he gives a rule according to which the universal premise-pairs are made, saying that if in the second Figure the necessity[-proposition] is a universal negative the conclusion will be a necessity, but if the necessity was a universal affirmative, there won’t on this account be a necessity-conclusion in the second Figure. Second, he sets out the premise-pairs—first the useful ones, second the useless.41

As in his treatment of LX-1 moods, Kilwardby presents the principles governing this mixture as determining the useful premise-pairs. He also provides a justification for the principles, which is based on the ways in which LX-2 syllogisms are perfected.

A10, 31a1 ff. Kilwardby ad A10, 30b7–40 (17ra–b): “Hic determinat mixtiones necessarii et inesse in secunda figura. Et dividitur in duas. In prima dat coniugationes universales, in secunda particulares cum dicit Similiter autem se habebit. In prima sic procedit. Primo dat regulam secundum quam fiunt coniugationes universales dicens quod si in secunda figura universalis negativa sit de necessario, et conclusio exit de necessario; si autem universalis affirmativa fuerit de necessario, non propter hoc erit conclusio de necessario; et hoc est in secunda figura. Secundo proponit coniugationes, et primo utiles, secundo inutiles.” 40 41

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Aristotle’s reductions Kilwardby takes it that second Figure LX syllogisms can be reduced to the first Figure only if one premise is the result of converting a premise in a first Figure LX syllogism. This accords with Aristotle’s methods of perfecting these syllogisms, as shown in Fig. 5.5.42 Le --- conv LXL --- 1

Lace

Labe Lbae Lbc

ac

Celarent LXL

aba

Le --- conv

Lcae

Le --- conv

Lcbe Lbce

Fig. 5.5. Aristotle’s direct reduction of XL-2 moods

Principles In addition to the common principles,43 and the two special principles for the second Figure,44 there is one extra principle for this mixture. (P9) In second Figure assertoric / necessity syllogisms, one premise

must be a universal negative necessity-proposition.45 As the Dialectica Monacensis expresses this (a little more wordily): “In the second Figure a necessity-conclusion follows if a universal negative is taken as a necessity, whether it is Major or Minor, and the other is assertoric, and not otherwise”.46 Kilwardby derives (P9) from the fact that Aristotelian perfections proceed as in Fig. 5.5: And it is to be said that the syllogisms of this Figure descend from the second of the first [Celarent] and reduce to it …

Kilwardby ad A10 dub.1 (17va). A10, 30b9–18, 31a5–10. See Kilwardby on the first Figure. 44 See Kilwardby on the second Figure. 45 Kilwardby ad A10 dub.5 (17vb). 46 Dialectica Monacensis, 500:25–27: “In secunda figura si universalis negativa sumatur de necessario, sive sit maior sive minor, et altera de inesse, sequitur conclusio de necessario.” 42 43

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chapter five And note that from these things it’s obvious what the reason is for the principle that the necessity-proposition in this Figure is a universal negative.47

Useful premise-pairs Applying his principles, Kilwardby calculates the number of useful premise-pairs: These things being laid down, the combinations may be made according to what was said about the first Figure. And the premise-pairs number 32, but only 3 are useful and the other 29 offend against some principle and so are useless. And note that he doesn’t here enumerate all the premise-pairs but only certain ones offending against the special principle of this mixture in this Figure, because the others are clear from what was said before.48

We can reconstruct the reasoning as follows. Using (P1), (P2), (P5) and (P6) we eliminate all pairs other than LeXa, XeLa, LaXe, XaLe, LeXi, XeLi, LaXo, XaLo. (P9) can then be used to eliminate XeLa, LaXe, XeLi, LaXo, XaLo—these being the pairs that Aristotle shows to be useless by means of counter-examples.49 The remaining three pairs are useful and are shown in Table 5.2. LeXa

XaLe

LeXi Table 5.2. Useful LX-2 premise-pairs

47 Kilwardby ad A10 dub.1 (17va): “Et dicendum quod syllogismi huius figurae descendunt a secundo primae figurae et in illum reducuntur. … Et nota quod ex his manifesta est causa huius principii quod universalis negativa sit de necessario in ista figura.” 48 Kilwardby ad A10 dub.5 (17vb): “Quibus suppositis fiant combinationes secundum quod dictum est circa primam figuram. Et numerentur XXXII {XXXII ABrCm1E1E2 F1KlO2O3V: XXXI EdP 1: XXVII F 2} coniugationes, sed tantum III {III EdABrCm1E1E2 F1F2KlO2O3P1: II V } utiles; aliae autem, scilicet XXIX peccabunt contra aliquod principium et ideo inutiles sunt. Et nota quod non enumerat hic omnes coniugationes sed tantum quasdam peccantes contra proprium principium huius mixtionis in hac figura, quia alia patent ex praecedentibus.” 49 XeLa: A10, 30b7–9. LaXe: A10, 30b31–40. XeLi: A10, 30b7–9. LaXo: 31a10–15. XaLo: 31a15–17.

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Syllogisms From these premise-pairs, syllogisms can be constructed as shown in Table 5.3. LXL

XLL

Cesare50 Festino52

Camestres51

Table 5.3. LX-2 syllogisms

No form of Baroco is included in this list, and Kilwardby notes that even though the Figures and moods of uniform necessity-syllogisms exactly match those of assertoric syllogisms, there is no such matching when we compare mixed assertoric / necessity syllogisms with assertorics.53 Semantics Kilwardby sees the ways in which Aristotle perfects these syllogisms as implying a rule of appropriation that determines the sense of affirmative assertoric premises in this mixture: And it is to be said that the syllogisms of this Figure descend from the second of the first [Celarent] and reduce to it. Hence a universal negative in this Figure has the same power as in the first, because it descends from it and reduces to it. But the universal negative in the first Figure is the Major, which appropriates the affirmative to itself in such a way that it is an unrestricted assertoric and not as-of-now (as was said earlier). Hence too, the universal negative in this second Figure, which is in potentiality to becoming the Major in the first, has the same power of appropriating the affirmative to itself in such a way that it has to be taken as an unrestricted assertoric and not as-of-now. Hence every counter-example to the useful premise-pairs of this Figure, where the assertoric is an as-ofnow assertoric, is sophistical.54 50 51 52 53 54

A10, 30b9–13. A10, 30b13–18. A10, 31a5–10. Kilwardby ad A8 dub.4 (15va). Kilwardby ad A10 dub.1 (17va): “Et dicendum quod syllogismi huius figurae descen-

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Thus, he postulates the existence of a rule of appropriation by virtue of which the affirmative assertoric premises in Cesare LXL, Camestres XLL and Festino LXL have to be read as unrestricted assertorics; and by this means he dispatches ostensible counter-examples such as that shown in Fig. 5.6. L(stone)(animal)e (stone)(white)a L(animal)(white)e Fig. 5.6. Apparent counter-example to Cesare LXL

At the same time, he doesn’t postulate any rule of appropriation by virtue of which the assertoric premises in Camestres LXL, Baroco LXL and Baroco XLL have to be unrestricted; on the contrary, he says there is no such rule: “… in this Figure, a particular cannot appropriate to itself a universal, nor an affirmative a negative”.55 So, once again we see him operating with rule (F)—the rule that in syllogisms complying with Aristotelian principles, an assertoric premise is to be read as unrestricted, but in other contexts an assertoric premise may be read either as an as-of-now assertoric or as unrestricted. In the present context the Aristotelian principle is (P9). Invalid inferences Aristotle rejects Cesare XLL, Festino XLL, Camestres LXL, Baroco LXL and Baroco XLL as invalid; and Kilwardby agrees with these rejections, basing them on the non-existence of any rule of appropriation whereby the assertoric premise might be required to be unrestricted. Because the assertoric premises in the rejected moods are not covered by any rule of appropriation, those premises may be as-of-now assertorics, and so it is legitimate to pose as-of-now counter-examples to dunt a secundo primae et in illum reducuntur. Unde universalis negativa in ista figura habet eandem potentiam quam in prima, quia descendit ab ea et in eam reducitur. Sed universalis negativa in prima figura est maior quae appropriat sibi affirmativam ut sit de inesse simpliciter et non ut nunc, ut prius dictum est. Quare et universalis negativa in hac secunda figura, quae est in potentia ut fiat maior in prima, eandem habet potentiam ut appropriet sibi affirmativam ut oporteat illam sumi de inesse simpliciter et non ut nunc. Quare omnis instantia sophistica est quae fit contra coniugationes utiles huius figurae illa de inesse existente de inesse ut nunc.” 55 Kilwardby ad A10 dub.4 (17vb): “… in hac figura non potest particularis sibi appropriare universalem nec affirmativa negativam.”

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them (such as Aristotle’s counter-examples to Camestres / Baroco LXL and Cesare / Festino LXL56 shown in Fig. 5.7). At the same time, because the assertoric premises in the asserted moods are so covered, it is not legitimate to pose such examples for them (such as the one Kilwardby suggests against Cesare LXL,57 shown in Fig. 5.8) Camestres / Baroco LXL

Cesare / Festino XLL

L(animal)(man)a (animal)(white)e,o L(man)(white)e,o (animal)(white)e L(animal)(man)a,i L(white)(man)e,o

Fig. 5.7. Aristotle’s counter-example to Camestres / Baroco LXL and Cesare/ Festino XLL

L(stone)(man)e (stone)(white)a L(man)(white)e Fig. 5.8. Apparent counter-example to Cesare LXL

And yet, if we were to specify that the assertoric premise in Cesare XLL is to be unrestricted (as in “No raven is intelligent, every man is of necessity intelligent, so no man is possibly a raven”), it would be reducible to a Celarent XLL with an unrestricted Major; and while the latter is not perfect, it is, as we saw earlier, valid. Similar considerations apply to all the other rejected moods. Aristotle’s counter-examples Aristotle says that Baroco XLL can be shown invalid by the same terms that were used earlier to refute Camestres LXL (Fig. 5.7). Adapted to Baroco XLL, the intended counter-example would be as in Fig. 5.9.58 (animal)(man)a L(animal)(white)o L(man)(white)o Fig. 5.9. Aristotle’s counter-example to Baroco XLL 56 57 58

A10, 30b31–40; 31a10–15; 30b7–9. Kilwardby ad A10 dub.1 (17va). A10, 31a15–17.

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This counter-example is the only one in Aristotle’s discussion of LX2 moods where the assertoric premise is not as-of-now. Moreover, the example is not convincing, because if there is a white thing that cannot be an animal, then presumably it couldn’t be a man either, and so if the premises were true the conclusion would be true too. Whether this was Kilwardby’s reason for being dissatisfied with the example, we don’t know. We do know that, later when he is considering Aristotle’s counter-example to Ferison, he states that the proposition “It’s necessary for something white to be an animal” is only a per accidens necessity. Whatever his reason he proposes to vary the present example: … it is to be known that in order to show the inutility of this, the terms are to be ordered otherwise than before; for, “man” ought to be Middle, “white” the Major Extreme and “animal” the Minor, thus everything white is a man, of necessity some animal is not a man, but it doesn’t follow that of necessity some animal is not white.59

(man)(white)a L(man)(animal)o L(white)(animal)o Fig. 5.10. Kilwardby’s counter-example to Baroco XLL

So, he proposes a Major that is true merely as-of-now (as contrasted with Aristotle’s Major, which was necessarily true); and he proposes a Minor that states the per se necessity that some animal cannot be a man (whereas Aristotle’s Minor stated the per accidens necessity that something white can’t be an animal). So the premises of his example fit his semantics, whereas the premises of Aristotle’s example did not. But what of the conclusion? Recall that what was unconvincing about Aristotle’s example was that it seemed that if the premises were true the conclusion would be true too. Isn’t Kilwardby’s example open to the same objection? Certainly the conclusion, that some animal cannot be white, would be regarded as true by Aristotle, since he takes it that no raven is possibly white.60 Whether Kilwardby would agree isn’t clear. At 59 Kilwardby ad A10, 31a1–17 (17va): “… sciendum quod aliter ordinandi sunt termini quam prius ad ostensionem inutilitatis {inutilitatis AF 1KlO3P1V: utilitatis EdCm1E2F2O2: suppositis universalis Br} huius. Debet enim ‘homo’ esse medium et ‘album’ maior extremitas et ‘animal’ minor sic: omne album est homo, de necessitate quoddam animal non est homo; sed non sequitur ‘de necessitate quoddam animal non est album’.” 60 A16, 36b7–10.

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any rate, one could argue that even if the conclusion of Kilwardby’s example is true, it isn’t true because of the premises, and another example is easily constructed, where the conclusion is evidently false and the premises true, as in Fig. 5.11. All that is needed is a Major term that is accidental to whatever it inheres in—such as the term “approaching”. (man)(approaching)a L(man)(animal)o L(approaching)(animal)o Fig. 5.11. Third counter-example to Baroco XLL

LX-3 moods Aristotle gives his doctrine of the third Figure assertoric / necessity syllogisms in A11. Kilwardby summarizes: Here he determines this mixture in the third Figure; and it is divided into two [parts]. In the first, he determines it. In the second (when he says “It’s obvious therefore that an assertoric”),61 he concludes corollaries from what has been said. First, he determines the universal premisepairs, second the particulars (when he says “But if indeed one is universal”).62 In the first, he proceeds thus. First, he gives a rule according to which the premise-pairs proceed, saying that if either is a universal affirmative necessity in the third Figure—and if either is a negative necessity—a necessity-conclusion follows, and otherwise not (and this is “In the last”).63 Second, he sets out the premise-pairs, first the useful, second the useless.64

A12, 32a6ff. In Kilwardby’s text (18va) this is counted as part of A11. A11, 31b11ff. 63 A11, 31a18ff. 64 Kilwardby ad A11, 31a18–b12 (18ra): “Hic determinat hanc mixtionem in tertia figura. Et dividitur in duas. In prima determinat eam, in secunda cum dicit Manifestum igitur quoniam inesse concludit quoddam correlarium {quoddam correlarium Cm1E2F1F2KlO2O3V: quoddam correlatio Br: correlarium P 1: correlaria Ed} ex praedictis. Prima in duas. In prima {in duas. In prima om. Ed} determinat coniugationes universales, in secunda particulares cum dicit Si autem haec quidem universalis. In prima sic procedit. Primo dat regulam secundum quam procedunt coniugationes dicens quod si utraque fuerit universalis in tertia figura et affirmativa utralibet necessaria et si altera fuerit negativa et necessaria, sequitur conclusio de necessario, aliter non. Et hoc est In postrema. Secundo ponit coniugationes, et primo utiles, secundo inutiles.” 61 62

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Aristotle’s reductions Aristotle perfects the third Figure assertoric / necessity moods in the ways shown in Fig. 5.12.65 Kilwardby notes that the affirmative moods reduce to Darii LXL, and the negative ones to Ferio LXL.66 bc+ Darii LXL

Laca

cbi

conv

Labi

conv Darii LXL

ac+ cai

bc+ Lbca

Lbai Labi

Ferio LXL

Lace

Li --- conv

cbi

conv

Laba

Fig. 5.12. Direct reductions of LX-3 moods

Fallacious reductions Kilwardby poses puzzles about Felapton XLL and about Bocardo LXL, similar to ones he raised earlier about the second Figure syllogisms.67 Expository proofs are possible where the syllogism has a particular premise; but their existence doesn’t affect Kilwardby’s overall argument. Principles He states the two special principles for this mixture: (P10) In affirmative third Figure assertoric / necessity syllogisms, the ne-

cessity-premise must be a universal affirmative. (P11) In negative third Figure assertoric / necessity syllogisms, the neces-

sity-premise must be a universal negative.68 The Dialectica Monacensis doesn’t give a single rule, or a pair of rules, but simply summarizes each valid mood.69 Kilwardby bases his principles on the routes whereby Aristotle perfects syllogisms in this mixture:

65 66 67 68 69

A11, 31a24–37, 31b12–20, 35–37. Kilwardby ad A11 dub.1 (18vb). Kilwardby ad A11 dub.2–3 (18vb). Kilwardby ad A11 dub.5 (19ra). Dialectica Monacensis, 500:31–501:8.

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And it is to be said that the affirmative moods of this Figure descend from the third of the first [Darii] where the Major is a universal affirmative. And in the affirmative moods, they are reduced to the first Figure in such a way that whatever was a universal affirmative in the affirmative moods here [sc. in the third Figure] can become the Major in the third of the first … The case is similar with the negative moods, namely, which descend from the fourth of the first [Ferio] and reduce to that, where the Major is a universal negative … From these things, therefore, it’s obvious that two principles are to be laid down in this mixture in this Figure, one for the affirmative moods (namely, that the necessity is a universal affirmative), the other for the negatives (namely, that the necessity is a universal negative), and the necessity of these is clear from what has been said.70

Useful premise-pairs Kilwardby doesn’t show how to deduce the useful premise-pairs from these principles, saying only: And it is to be said that the common principles are to be laid down, and the special principle that everywhere accompanies the third Figure (namely, that the Minor is affirmative). In addition, it is to be laid down that in the affirmative moods the necessity is a universal affirmative, and in the negatives the necessity is a universal negative. These things being laid down, it’s clear from the combinations that there are 6 useful premise-pairs and 26 offending against some principle.71

However, it is easily shown that the common principles and (P7) rule out all but LaXa, XaLa, LeXa, XeLa, LiXa, XiLa, LaXi, XaLi, LeXi, 70 Kilwardby ad A11 dub.1 (18vb): “Et dicendum quod modi affirmativi huius figurae descendunt a tertio primae ubi maior est universalis affirmativa, reducuntur etiam in primam figuram ita quod quaecumque fuerit universalis affirmativa hic in modis affirmativis possibilis est ut fiat maior in tertio primae. … Similiter dico de modis negativis, scilicet quod descendunt a quarto primae et in ipsum reducuntur ubi maior est universalis negativa. … Ex his ergo manifestum est quod duo sunt principia supponenda in mixtione huius figurae, unum quantum ad modos affirmativos, scilicet quod universalis affirmativa sit de necessario, aliud quantum ad negativos, scilicet quod universalis negativa sit de necessario; et patet necessitas eorum ex iam dictis.” 71 Kilwardby ad A11 dub.5 (19ra): “Et dicendum quod supponenda sunt principia communia et istud proprium quod consequitur tertiam figuram ubicumque, scilicet quod minor sit affirmativa. Praeterea supponendum est quo ad affirmativos modos quod universalis affirmativa sit de necessario et quo ad negativos quod universalis negativa sit de necessario. Quibus suppositis patet per combinationes quod sex sunt utiles coniugationes et XXVI {XXVI ABrCm1E1E2F2KlO2O3P1: XXV F 1: XXXVI Ed: aliae sunt V } inutiles peccantes contra aliquod principium.”

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XeLi, LoXa, XoLa. (P10) then excludes LiXa, XaLi; and (P11) excludes XeLa, XeLi, LoXa, XoLa. The remaining 6 premise-pairs are useful and are shown in Table 5.4. LaXa

LeXa

LaXi

LeXi

XaLa

XiLa

Table 5.4. Useful LX-3 premise-pairs

Syllogisms The valid moods are shown in Table 5.5. LXL

XLL

Darapti72 Datisi74 Felapton76 Ferison77

Darapti73 Disamis75

Table 5.5. LX-3 syllogisms

Semantics Kilwardby follows the pattern we have seen in his treatment of assertoric / necessity-moods in the second Figure. He starts by laying down the means by which Aristotle perfects the moods of this Figure as shown in Fig. 5.12; and from this he infers rules of appropriation that determine the sense of the assertoric premise: And it is to be said that the affirmative moods of this Figure descend from the third of the first [Darii] where the Major is a universal affirma72 73 74 75 76 77

A11, 31a24–30. A11, 31a31–33. A11, 31b12–20. A11, 31b12–20. A11, 31a33–37. A11, 31b35–37.

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tive. And in the affirmative moods, they are reduced to the first Figure in such a way that whatever was a universal affirmative here [sc. in the third Figure] can be made the Major in the third of the first. Hence, just as the universal affirmative that is Major there [sc. in the first Figure] appropriates the other proposition to itself so that it has to be taken as an unrestricted assertoric when that universal affirmative is a necessity-proposition (as was said before), so too the universal affirmative in the affirmative moods of this Figure appropriates the other [proposition] and its converse to itself so that it’s necessary for it to be an unrestricted assertoric. I say “when the universal affirmative is an affirmative necessity-proposition”; and Aristotle frequently signifies this in the text, saying that it’s necessary that the Minor extreme is essentially under the Middle in the converse of the assertoric proposition. Similarly I say about the negative moods.78

(He mentions the converse because of the need to convert the assertoric premise as shown in Fig. 5.12.) Here he postulates the existence of rules of appropriation according to which the affirmative premises in the moods that Aristotle accepts have to be read as unrestricted. Rule (F)—that in syllogisms complying with Aristotelian principles, an assertoric premise is to be read as unrestricted; in other contexts, an assertoric premise may be read either as an as-of-now assertoric or as unrestricted—is again at work here, the relevant Aristotelian principles being (P10) and (P11). In his discussion of necessity/ assertoric mixtures in the third Figure79 he raises the question whether LX premises in Bocardo entail a syllogistic conclusion even though Aristotle denies this.80 He replies that the solution to this puzzle is the same as was given earlier to

78 Kilwardby ad A11 dub.1 (18vb): “Et dicendum quod modi affirmativi huius figurae descendunt a tertio primae ubi maior est universalis affirmativa, reducuntur etiam in primam figuram ita quod quaecumque fuerit universalis affirmativa hic in modis affirmativis possibilis est ut fiat maior in tertio primae. Unde sicut universalis affirmativa quae est ibi maior appropriat sibi reliquam propositionem ita quod oportet eam sumi de inesse simpliciter cum ipsa universalis sit de necessario, sicut in praecedentibus dictum est, sic et universalis affirmativa in modis affirmativis huius figurae appropriat sibi reliquam et conversam eius ita quod necesse est ipsam esse de inesse simpliciter. Dico cum {dico cum AE 1F1O2O3: dico autem si: BrCm1E2F2: dico quod KlV : Et hoc dico quod P 1: Hoc dico Ed} universalis affirmativa sit {sit ABrCm1E1E2F1F2KlO2O3P1V: est Ed} de necessario, et hoc significat Aristoteles frequenter in littera dicens quod necesse est minorem extremitatem esse sub medio essentialiter in conversa propositionis de inesse. Similiter dico de modis negativis …” 79 Kilwardby ad A11, 31b33–32a5 (17vb–19ra). 80 Kilwardby ad A11 dub.2–3 (18vb).

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the question about necessity/ assertoric premises in Baroco. His view is that Baroco XLL and Bocardo LXL would be valid if their assertoric premises were specified as unrestricted; but in the absence of any rule of appropriation requiring that the assertoric premise be unrestricted they are invalid. Invalid inferences In the absence of any such rules covering the assertoric premises in the rejected moods, as-of-now counter-examples (such as Aristotle’s counter-examples to Felapton XLL, Datisi XLL, Ferison / Bocardo XLL and Disamis / Bocardo LXL81 shown in Fig. 5.13) are effective against them; but similar counter-examples (such as the one Kilwardby suggests against Felapton LXL,82 shown in Fig. 5.14) are not effective against the valid moods. Felapton XLL

Datisi XLL

Ferison / Bocardo XLL

Disamis / Bocardo LXL

(good)(horse)e L(animal)(horse)a L(good)(animal)a

(awake)(animal)a L(biped)(animal)i L(awake)(biped)i (awake)(white)e,o L(animal)(white)i,a L(awake)(animal)o

L(biped)(animal)i,o (awake / moving)(animal)a L(biped)(awake / moving)i,o

Fig. 5.13. Aristotle’s counter-examples to Felapton XLL, Datisi XLL, Ferison/ Bocardo XLL and Disamis / Bocardo LXL

81 A11, 31b4–10, 27–33; 31b40–32a5. Aristotle comments that better terms should be taken to reject Felapton XLL; and Kilwardby ad A11, 31a18–b10 (18rb) explains that there is an equivocation on “good”, and that while it’s contingent for every animal to be naturally good, it’s not contingent for every animal to be morally good. 82 Kilwardby ad A11 dub.1 (18va).

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L(stone)(man)e (white)(man)a L(stone)(white)o Fig. 5.14. Apparent counter-example to Felapton LXL

And yet, if we were to specify that the Major in Felapton XLL is to be unrestricted (as in “No raven is intelligent, every raven is of necessity black, so something black is necessarily not intelligent”), it would be reducible to a Ferio XLL with an unrestricted Major; and while the latter is not perfect, it is, as we saw earlier, valid. Similar considerations apply to all the other invalid moods. In fact, all of the moods that Aristotle (and Kilwardby following him) takes to be invalid turn out to be valid if their assertoric premise is specified as being unrestricted;83 but, even if specified in this manner, none of them is perfectible. Aristotle’s counter-examples (awake)(white)e L(animal)(white)i L(awake)(animal)o Fig. 5.15. Aristotle’s counter-example to Ferison XLL

Aristotle’s example supposes that nothing white is awake, and it’s necessary that some white thing is an animal; consistent with these assumptions, he says it is possible for all animals to be awake.84 Kilwardby challenges the Minor premise that it’s necessary that some white thing be an animal. He points out that Aristotle himself elsewhere85 takes it to be contingent that nothing white is an animal. The solution he proposes hinges on distinguishing two senses of the offending proposition that it’s necessary for something white to be an animal, depending on what is principally signified by “white”. If the quality of whiteness is principally signified, then the earlier statement (that it’s contingent for 83 Thom, Medieval Modal Systems, 101 interprets Kilwardby’s necessity /assertoric syllogisms under the assumption that assertoric premises are uniformly read as unrestricted, and thus comes to the conclusion that all standard moods in each of the Figures is valid in the mixtures LXL and XLL. This is correct so far as it goes; but it doesn’t acknowledge that the assumption in question is rejected by Kilwardby. 84 A11, 31b40–32a4. 85 A10, 30b35.

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nothing white to be an animal) is true. If white substances (including swans) are primarily signified, then the current statement (that it’s necessary for something white to be an animal) is true. And on this latter reading, the proposition is “not really a necessity-proposition, but is more of a per accidens proposition”. Presumably, this means that if white things are considered purely in so far as they are white, it’s neither necessary nor impossible that they should be animals; but if they are considered as the substances that they are, it might well be necessary or impossible for particular ones to be animals. For swans it’s necessary, for snow impossible. Even with this ambiguity cleared up, Kilwardby remains dissatisfied with Aristotle’s example because it is an accidental necessity. That the white thing is necessarily an animal is true only because the white thing is a swan—and this latter circumstance is an accident, even if it is a necessary accident of swans that they are white. Accordingly, he proposes a better counter-example—one in which the Minor is a per se necessity:86 (white)(animal)e L(man)(animal)i L(white)(man)o Fig. 5.16. Kilwardby’s counter-example to Ferison XLL

Summary Kilwardby proposes the following principles for necessity-syllogisms: (P1) In (pure necessity-)syllogisms, one premise must be universal. (P2) In (pure necessity-)syllogisms, one premise must be affirmative. (P3) In first Figure (pure necessity-)syllogisms, the Major must be uni-

versal. (P4) In first Figure (pure necessity-)syllogisms, the Minor must be affir-

mative. (P5) In second Figure (pure necessity-)syllogisms, the Major must be

universal. (P6) In second Figure (pure necessity-)syllogisms, one of the premises

must be negative. 86

Kilwardby ad A11 dub.3 (18vb).

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(P7) In third Figure (pure necessity-)syllogisms, the Minor must be

affirmative. (P8) In first Figure assertoric / necessity syllogisms, the necessity-propo-

sition must be Major. (P9) In second Figure assertoric / necessity syllogisms, one premise

must be a universal negative necessity-proposition. (P10) In affirmative third Figure assertoric / necessity syllogisms, the ne-

cessity-premise must be a universal affirmative. (P11) In negative third Figure assertoric / necessity syllogisms, the neces-

sity-premise must be a universal negative. These principles yield exactly Aristotle’s results. Most of the principles can be found in the Dialectica Monacensis, but not all. The exception lies in the principles for LX-3 syllogisms. So Kilwardby’s statement of principles is more comprehensive than that in the Dialectica Monacensis. Moreover, the Dialectica Monacensis doesn’t use the syllogistic principles, as Kilwardby does, to deduce the useful premise-pairs. By enunciating the principles underlying Aristotle’s account of the necessity-moods, and by showing how Aristotle’s results follow from these principles, Kilwardby has disclosed both a certain sufficientia and a certain ordo in Aristotle’s text. To that extent, his interpretation must be counted a success. But he achieves this order by postulating rules of appropriation which have the effect of shielding another order from the reader’s view, namely the order that would result if we were to read assertoric propositions uniformly throughout as unrestricted. Within this other order there lies a system that Kilwardby does not want to attribute to Aristotle—a system in which there is no semantic difference between assertorics and necessity-propositions, and within which all mixed assertoric / necessity-inferences are equally valid.87 That system achieved explicit recognition only in the work of William Ockham.88

87 This is the “maximal” system that is expounded in Thom, Medieval Modal Systems, 98–101. 88 Thom, Medieval Modal Systems, 151–155.

chapter six CONTINGENCY-SYLLOGISMS

Throughout his treatment of the contingency-syllogistic, Kilwardby pursues his high-level program of supplementing Aristotle’s text in ways that will enhance its credibility and systematicity. On the syntactic plane, he continues to enunciate principles for each mixture of modalities in each Figure. He bases these principles on observable features of the perfectible inferences in each such mixture; and from them he deduces which premise-pairs in those mixtures are fruitful and which are not. Many of these principles are to be found in the Dialectica Monacensis, and some are in Aristotle; but Kilwardby’s statements are more comprehensive and more accurate. Moreover, Kilwardby finds fault with one of Aristotle’s principles, namely the principle governing mixed necessity/ contingency moods in the second Figure. He sees that Aristotle’s principle only covers inferences that reduce directly to the first Figure. He knows that some second Figure necessity/ contingency inferences reduce to the first Figure indirectly, and that therefore more premise-pairs in this mixture are fruitful than allowed for by Aristotle’s principle. Some of these premise-pairs yield conclusions consistent with the general principles governing the second Figure, and some do not. In the former case, Kilwardby’s observations result in a correction to Aristotle’s modal syllogistic. But this is not so in the latter case: Kilwardby doesn’t suggest that necessity / contingency premise-pairs which are not consistent with the general character of the second Figure should be added to the pairs Aristotle recognizes, thus expanding the range of second Figure necessity/ contingency syllogisms. On the contrary, he argues that these premise-pairs yield a consequence only by virtue of their terms and not because of their syllogistic form. On the semantic plane, he continues to state truth-conditions for different types of modal proposition, allowing their sense to fluctuate according to their inferential context. In particular, he follows Aristotle in requiring that the Major premise in the XQM-1 syllogisms has to be unrestricted, but allows that it may belong to either of the two types of unrestricted assertoric which he recognizes.

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In the pure contingency-syllogisms, Kilwardby states a Theorem specifying which metaphysical types of term can occur as subjects or predicates in contingency-propositions. He improves on Aristotle by constructing a sound proof that Cesare QQQ cannot be indirectly reduced to the first Figure. In the assertoric / contingency mixtures, he deploys the Substance / Accident distinction to solve logical puzzles, and he again uses categorial rules to exemplify patterns of logical inference. In the necessity/ contingency mixtures, he draws on popular beliefs about mythical creatures in order to illustrate a logico / ontological point.

QQ-1 moods The uniform contingency-syllogisms of the first Figure are discussed in A14, those in the second Figure in A17, those in the third Figure in A20. The contingency/ assertoric mixes in the three Figures are discussed respectively in A15, A18 and A21; the contingency/ necessity mixes in A16, A19 and A22. Where then, Kilwardby asks, are the discussions of the syllogisms with (one-way) possibility premises? There should be three chapters discussing pure possibility-syllogisms, three discussing possibility / assertoric mixes, three discussing possibility / necessity mixes, and three discussing possibility / contingency mixes—a total of 12 missing chapters!1 In response he argues that Aristotle has omitted nothing; and his argument is a metaphysical one. One-way possibility is a genus, and it is not actually anything other than its species, which are the necessary and the non-necessary contingent, “and so he neither could nor should determine the production of contingency-syllogisms otherwise than through their species”.2

1

Kilwardby ad A14 dub.1 (21vb). Kilwardby ad A14 dub.1 (22ra): “… quare nec potuit nec debuit aliter determinare generationem syllogismi de contingenti nisi secundum istas species”. 2

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Principles In order to deduce the useful premise-pairs, Kilwardby lays it down as a principle for this combination that the Major must be universal. This, of course, is just (P3), which applies to all first Figure syllogisms. The Dialectica Monacensis states an equivalent requirement for this mixture: It’s clear therefore that in the first Figure, if both [premises] are universal contingencies, there is always a syllogism, whether they are affirmative or negative. But with particulars, if the Major is universal there will always be a syllogism …3

The “common principles” enunciated earlier by Kilwardby continue to apply to contingency-syllogisms. A contingency-syllogism is required to have a universal premise, and to have an affirmative premise. However, the need for an affirmative premise has to be understood in the light of Kilwardby’s distinction between genuine and non-genuine negative propositions. Useful premise-pairs He states that there are 8 useful premise-pairs.4 This result is implied by the rule that the Major be universal. The pairs are shown in Table 6.1. QaQa

QeQa

QaQe

QeQe

QaQi

QeQi

QaQo

QeQo

Table 6.1. Useful QQ-1 premise-pairs

3 Dialectica Monacensis, 502:8–10: “Patet ergo quod in prima figura si utraque fuerit universalis et de contingenti, semper fit syllogismus, sive sint affirmativae sive negativae. In particularibus vero si maior sit universalis, semper fit syllogismus…” 4 Kilwardby ad A14 dub.6 (22rb).

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Syllogisms The valid syllogisms are listed in Table 6.2. QQQ Barbara5

QaQaQe-1

QaQeQa-16

QaQeQe-1

Celarent7

QeQaQa-1

QeQeQa8

QeQeQe-1

Darii9

QaQiQo1

QaQoQi-110

QaQoQo-1

Ferio11

QeQiQi-1

QeQoQi-1

QeQoQo-1

Table 6.2. QQQ-1 syllogisms

Semantics He reads the Major premise as ampliated to the contingent. As we saw in Chapter One, on such a reading the proposition states that what may or may not fall under the subject may or may not fall under the predicate. This reading enables him to conclude that Aristotle was right in accepting the pure contingency-syllogisms of the first and third Figures, and in rejecting those in the second. The interpretive strategy 5 6 7 8 9 10 11

A14, 32b38–33a1. A14, 33a5–12. A14, 33a1–5. A14, 33a12–17. A14, 33a21–25. A14, 33a27–34. A14, 33a25–27.

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is to formulate a reading that is based on sound logical theory and that verifies Aristotle’s results.12 The Minor premise, however, doesn’t need to be taken as ampliated, but could be true in the sense “Everything that is actually c may or may not be b”.13 Thus, the terms of a perfect syllogism like Barbara QQQ might be represented in either of the ways shown in Fig. 6.1.

Fig. 6.1. Two representations of the terms in Barbara QQQ

Evidently, in the first case it follows that what may or may not be c may or may not be a; and in the second it follows that what is actually c may or may not be a. In both cases there is a perfect syllogism, because there is downwards monotonicity for the subject, relative to contingent inherence. Propositions may be classified according to the types of terms they contain. Thus, Kilwardby distinguishes the following four classes of propositions: “Every term is either substantial or accidental. So [a proposition] has to be taken either with two accidental terms, or two substantial, or one substantial and one accidental”.14 Given that the Major premise is ampliated to the contingent, and given this classification of propositions, Kilwardby enunciates a theorem about the Major premise of a QQ-1 syllogism. Theorem. If the subject of the Major is an accidental term, and its predicate is a substantial term, then the Major is false.15 See Lagerlund, Modal Syllogistics in the Middle Ages, 47–48. Kilwardby ad A14 dub.3 (22rb). 14 Kilwardby ad A14 dub.2 (22ra): “Omnis enim terminus aut est substantialis aut accidentalis. Aut ergo habet maior propositio accipi in duobus terminis accidentalibus aut in duobus substantialibus aut in uno substantiali et altero accidentali”. 15 Kilwardby ad A20 dub.1 (31vb–32ra). Lagerlund, Modal Syllogistics in the Middle Ages, 48–50. 12 13

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He explains: For this is the sense: Everything for which being b is contingent is contingently etc. Hence, when a proposition like “Everything white is contingently an animal” is taken as Major, it is false, because it is taken in the sense “Everything for which being white is contingent is contingently an animal”, which is false. Even if only animals were white, still it’s contingent for many things to become white, for which being an animal is impossible, such as wood and stone and the like.16

Perfect syllogisms He proposes a counter-example to Barbara QQQ: Q(white)(literate)a Q(literate)(black)a Q(white)(black)a Fig. 6.2. Apparent counter-example to Barbara QQQ

His response draws on an ontological distinction: … it is to be said that the Major “For everything literate it’s contingent to be white” is true, because for everything for which it’s contingent to be literate, it’s contingent etc. But the Minor is ambiguous, because when it says “It’s contingent for everything black to be literate”, the term “black” is either taken primarily for a quality and secondarily for a substance (and then it is false), or the other way round (and then it is true because literacy could inhere in a substance which is now black). And when it concludes “So it’s contingent for everything black to be white”, the same distinction arises here. A true conclusion follows in the same sense in which the Minor is true, and a false conclusion in the same sense in which the Minor is false.17 16 Kilwardby ad A14 dub.2 (22ra): “… quia is est sensus: Omne quod contingit esse b contingit esse etc. Unde cum accipitur pro maiori talis propositio: ‘Omne album contingit esse animal’, haec est falsa, quia accipitur sub hac intentione: ‘Omne quod contingit esse album contingit esse animal’, quod falsum est. Quamvis nihil fuerit album nisi animal, contingit tamen multa fieri alba quae impossibile {impossibile ABrCm1E1E2F1F2KlO2O3P1V: possibile Ed} est esse animalia, sicut ligna lapis et huiusmodi”. Albertus Magnus, Priorum Analyticorum, IV.iv (344A) has “Even if it is supposed that everything that is white is an animal, still many things (such as wood and stone and a garment) can be white but none of them can be an animal”. [“Quia et si ponatur quod omne quod est album sit animal, tamen multa contingit esse alba, sicut lignum et lapidem et vestem, quorum nullum contingit esse animal”.] 17 Kilwardby ad A14 dub.2 (22ra): “… dicendum quod maior est vera, scilicet ‘Omne grammaticum contingit esse album’, quia omne quod contingit esse grammaticum contingit etc. Minor autem distinguenda est, quia cum dicit ‘Contingit omne nigrum esse

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Presumably, the Minor proposition “It’s contingent for everything black to be literate” is false when read as being primarily about the quality of blackness, because it’s not by virtue of being black that a black thing is contingently literate. Similarly, it’s not by virtue of being black that a black thing is contingently white. By contrast, if the Minor proposition is true when read as being primarily about black substances, because it is contingent that all such substances should be literate, then it’s equally contingent that all such substances should be white. So on Kilwardby’s analysis, if we maintain the same sense throughout, the conclusion is true whenever the premises are true, and the counter-example fails. Aristotle’s reductions of imperfect syllogisms When the Minor or both Minor and Major are negative, the syllogism is imperfect. Kilwardby regards these conjugations as being imperfect in mood (because in the first Figure a perfect mood has an affirmative Minor), and also imperfect as regards evident necessity.18 They are perfected in the ways shown in Fig. 6.3. Qbc--Qab+

Qbc+

Q --- conv

Qac

Q --- conv

Qab---

Qbc---

Qab+

Qbc+

Q --- conv

Qac

Fig. 6.3. Direct reduction of QQQ-1 syllogisms

Aristotle’s counter-examples Aristotle demonstrates the inutility of QiQa-1 premises by means of the counter-example shown in Fig. 6.4.19

grammaticum’, aut accipitur iste terminus ‘niger’ pro qualitate primo et pro subiecto secundario, et tunc falsa est, aut e converso, et tunc vera est quia substantiae quae iam nigra est contingit inesse grammaticum. Cum autem concludit ‘Ergo contingit omne nigrum esse album’, hic incidit eadem distinctio; et eo modo quo minor est vera sequitur conclusio vera, eo modo quo minor est falsa et conclusio falsa”. 18 Kilwardby ad A14 dub.4 (22rb). 19 A14, 33b3–17.

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Q(animal)(white)i Q(white)(man)a Q(animal)(man)i Fig. 6.4. Aristotle’s counter-example to QiQaQi-1

Now, Kilwardby asks, does “white” here stand for a white animal or for a white non-animal? If it stands for a white animal, the Major proposition will be necessary; if for a white non-animal, it will be impossible. His answer is subtle: And it is to be said that “white” is taken for a white substance which is an animal; and yet it is not a necessity-proposition. For, even if every man is an animal of necessity, still it’s contingent (not necessary) for every white man to be an animal. And that the particular “Some white is an animal” is contingent, is clear by conversion of the universal contingencyproposition into a particular contingency-proposition.

He notes that the prohibition against contingency-propositions with accidental subjects and substantive predicates applies only to universal propositions, not to particulars; so “It’s contingent that some white is an animal” can be true.20

QQ-2 moods According to Aristotle there are no valid uniform contingency-syllogisms in the second Figure. Kilwardby agrees, and he lays down a principle that implies this result. (P12) In second Figure pure contingency-syllogisms, one of the premises

must be negative without qualification [simpliciter negativa].21 He means negative propositions that are equivalent to affirmatives to be counted as not negative without qualification. He then argues that, since affirmative and negative contingency-propositions are inter20 Kilwardby ad A14 dub.5 (22rb): “Et dicendum quod accipitur album pro substantia alba quae est animal. Non {Non EdBrCm1E1E2KlV: haec AF 1: nec O2O3P1} tamen est propositio de necessario. Quamvis enim omnis homo sit animal de necessitate, tamen contingit omnem hominem album esse animal, et non est hoc necessarium. Et quod sic sit particularis ut ‘Contingit aliquod album esse animal de contingenti’ patet per conversionem universalis de contingenti, quae convertitur in particularem de contingenti”. 21 Kilwardby ad A17 dub.2 (28vb).

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convertible, (P12) implies that there cannot be a second Figure syllogism from a pair of contingency-premises. He points out that just as affirmative contingency-propositions are convertible into negatives, negatives are convertible into affirmatives. Given this two-way convertibility, he asks why Aristotle bases the lack of QQ-2 syllogisms on the convertibility of negatives to affirmatives, rather than on the convertibility of affirmatives to negatives—seeing that there are22 syllogisms with negative premises in the second Figure. He adds a counterfactual speculation: And I think that if there were a Figure that could syllogize only from genuine affirmatives (where affirmative contingency-propositions are not genuine affirmatives since they convert with negatives), there would not be syllogisms from affirmative contingency-premises in such a Figure.23

Arguments for the imperfectibility of Cesare QQQ That Cesare QQQ cannot be shown to be valid by Direct Reduction to the first Figure is evident, Aristotle argues, because such a reduction would proceed by conversion of the Major premise; however, that premise is a universal negative contingency-proposition, and such propositions are not convertible.24 Kilwardby endorses this argument.25 That Cesare QQQ cannot be shown to be valid by Indirect Reduction to the first Figure, is not so easy to demonstrate. Aristotle tries to show this by seeing what follows when we suppose the opposite of the conclusion, which he names as “It’s contingent that all c is b”. Combining this with the Major premise, we get syllogism x in Fig. 6.5: x

Qabe Qbca Qace

→ Qe||Qa

Qabe Qaca Qbce

y

Fig. 6.5. Erroneous Indirect Reduction of Cesare QQQ

22 Ed (28vb) has “ex negativis non fit syllogismus in secunda figura”. ABrCm E E F 1 1 2 1 F2KlO3P1V have “negativis fit syllogismus in secunda figura”. O2 has “negativis fit syllogismus in prima figura”. 23 Kilwardby ad A17 dub.2 (28vb): “Et puto quod si aliqua esset figura quae solum posset syllogizare ex vere affirmativis, quando affirmativae de contingenti non sunt vere affirmativae eo quod convertuntur cum negativis, non fiet syllogsimus in tali {tali ACm1E1E2F1F2KlO3P1V: extalibue Br: tertia Ed} figura ex affirmativis de contingenti”. 24 A17, 37a32–35. 25 Kilwardby ad A17 dub.1 (28vb).

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However, Aristotle says, x’s conclusion is compatible with y’s minor premise, “It’s contingent that every c is a”; in fact the two are convertible. So y (Cesare QQQ) can’t be reduced indirectly to the first Figure.26 Aristotle’s argument is fallacious, since it takes Qa- and Qe-propositions to be incompatible, whereas they are in fact equivalent. Kilwardby doesn’t say this; but he offers the following interpretation: In taking “It’s contingent for every c to be b” as the contrary, he takes the conclusion to be a contingency-[proposition] in the sense of the possible, because only such contingencies are implied in this Figure, and it is not supposed that it is a two-way contingency. And so it’s evident that he took the contrary correctly, because affirmative and negative contingencies in the sense of the possible do not convert.27

This interpretation effectively substitutes two other syllogisms for x and y. These are u and z in Fig. 6.6. Kilwardby’s argument supposes that Aristotle is trying to show that Cesare QQM (syllogism z) cannot be proved per impossibile, and that his procedure is to take the contrary of the Me-conclusion—namely the corresponding Ma-premise—along with the original major, thus reducing z to u (QeMaQe-1). u

Qabe Mbca Qace

K → Qe||Qa

Qabe Qaca Mbce

z

Fig. 6.6. Erroneous Indirect Reduction of Cesare QQM

This reduction is, however, also fallacious, because (pace Kilwardby) Me- and Ma-propositions are not contradictories, and Qe- and Qapropositions are not contraries. It is indeed true in a certain sense that if a is possible for all c then it’s not the case that a is possible for no c; however, the latter proposition is not an Me- but an Li-proposition. So, the reasoning which Kilwardby attributes to Aristotle doesn’t show that Cesare QQQ or QQM can’t be indirectly reduced to the first Figure; instead, it shows that Cesare QQL (syllogism v in Fig. 6.7) can’t be so reduced. A17, 37a35–37. Kilwardby ad A17 dub.6 (29ra): “… sumendo hanc ‘Contingit omne c esse b’ per contrariam conclusionis supponit conclusionem esse {supponit conclusionem esse BrCm1E1E2F1F2O2O3P1V: supponit esse conclusionem Kl: supponit conclusionem A: supposita conclusione Ed} de contingenti pro possibili quia tale contingens tantum in hac figura concluditur, et non supponit quod sit de contingenti ad utrumlibet. Et ita manifestum est quod bene sumit contrariam, quia non convertuntur affirmativa et negativa de contingenti pro possibili”. 26 27

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u

Qabe Mbca Qace

K → Qe||Qa

Qabe Qaca Lbce

189

v

Fig. 6.7. Erroneous indirect reduction of Cesare QQL

However, there is no good reason to think that Aristotle would want to show this. The syllogism which he wants to show can’t be indirectly reduced to the first Figure is Cesare QQQ. Leaving to one side the interpretation of Aristotle’s text, Kilwardby offers his own demonstration that Cesare QQQ can’t be indirectly reduced to the first Figure. The purported demonstrations canvassed thus far have not rested on an adequate account of the contradictory opposite of contingency-propositions. Kilwardby now states: The negative “It’s not contingent for no c to be b” has two causes of truth, namely either because it’s necessary for some c to be b, or because it’s necessary for some c not to be b. And if either of these is taken together with the Major it produces a mixture of contingency and necessity in the first Figure, which doesn’t destroy the Minor. And if either is taken together with the Minor, it produces a mixture of contingency and necessity in the third Figure, which doesn’t destroy the Major.28

Spelling this reasoning out, if Cesare QQQ were to be reduced indirectly to the first Figure, then because the contradictory of its conclusion is a disjunction of an Li- and Lo-proposition, and because the contradictory of its Minor premise is also a disjunction of Li- and Lo-propositions, it would reduce to the four first Figure syllogisms in Fig. 6.8. Qabe Lbci

Qabe Lbci

Qabe Lbco

Qabe Lbco

Laci

Laco

Laci

Laco

Fig. 6.8. Demonstration that Cesare QQQ cannot be indirectly reduced to the first Figure

28 Kilwardby ad A17 Note (29ra): “Haec enim negativa ‘Non contingit nullum c esse b’ duplici de causa vera est, scilicet vel quia necesse est aliquod c esse b vel quia necesse est aliquod c non esse b. Et si accipiatur altera illarum cum maiori fiet mixtio contingentis et necessarii in prima figura quae non interimit minorem. Si autem altera accipitur cum minori fiet mixtio contingentis et necessarii in tertia {tertia ABrCm1E1E2F1F2KlO2O3P1V: secunda Ed} figura quae non interimit maiorem”.

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None of these syllogisms is valid; and so indirect reduction to the first Figure is impossible. Through a correct analysis of the contradictory of a universal contingency-proposition, Kilwardby has here improved on Aristotle’s attempt to show that Cesare QQQ is not indirectly reducible to the first Figure.

QQ-3 moods Aristotle’s reductions Aristotle reduces the syllogisms of this mixture to the first Figure by the methods shown in Fig. 6.9.29 Qbc--Qac

Qbc QQQ --- 1

Qac Qcb Qab

QQQ --- 1

Qca Qbc Qba

QQQ --- 1

Qac---

Qbc+

Qac+

Qcb+ Qab

Qab Fig. 6.9. Aristotle’s reductions of QQQ-3 moods

These reductions make use of term-conversion for contingency-propositions; and as we saw in Chapter Three, ampliated contingencypropositions are indeed convertible in the standard ways. Principles Kilwardby lays down a single principle for useful premise-pairs of contingency-premises in the third Figure. It is simply one of the “common principles”, namely (P1), stating that there has to be a universal premise. The Dialectica Monacensis states that a contingency-conclusion follows from two contingency-premises in all six moods of the third Figure.30 This overlooks those moods where an affirmative contingencyproposition is replaced by a negative.

29 30

A20, 39a14–39b2. See Thom, The Logic of Essentialism, 65–67. Dialectica Monacensis, 504:4–6.

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Useful premise-pairs Kilwardby acknowledges that there are valid syllogisms with two negative premises, and with a negative Minor, because negative contingency-propositions are equivalent to affirmatives. He says there are 12 useful premise-pairs;31 and indeed there are only 12 ways in which the requirement that there be a universal premise can be satisfied. The 12 premise-pairs listed in Table 6.3.32 QaQa

QeQa

QiQa

QoQa

QaQe

QeQe

QiQe

QoQe

QaQi

QeQi

QaQo

QeQo

Table 6.3. Useful QQ-3 premise-pairs

Syllogisms The syllogisms are listed in Table 6.4. QQQ Darapti33

QaQaQo-3

QaQeQi-3

QaQeQo-3

31 Kilwardby ad A20 dub.2 (32ra). Ed is garbled, having “minor” several times instead of “quattuor”. The right numbers are found in ABrE 1E2F1KlO2O3P1V. 32 Apart from these 12, there are 4 premise-pairs that are infertile: QiQi, QiQo, QoQi and QoQo. Aristotle (A20, 39b2–6) shows that these pairs yield no syllogistic conclusion. 33 A20, 39a14–19.

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QeQaQi-3

QeQeQo-3

QeQeQi-335

Disamis36

QiQaQo-3

QiQeQi-3

QiQeQo-3

Datisi37

QaQiQo-3

QaQoQi-3

QaQoQo-3

Bocardo38

QoQaQi-3

QoQeQo-3

QoQeQi-339

Ferison40

QeQiQi-3

QeQoQi-341

QeQoQo-3

Table 6.4. QQQ-3 syllogisms

Darapti QQQ Kilwardby suggests a counter-example to Darapti QQQ. Q(man)(moving)a Q(horse)(moving)a Q(man)(horse)i Fig. 6.10. Apparent counter-example to Darapti QQQ

34 35 36 37 38 39 40 41

A20, 39a19–23. A20, 39a26–28. A20, 39a31–36. A20, 39a31–36. A20, 39a36–38. A20, 39a38–39b2. A20, 39a36–38. A20, 39a38–39b2.

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In response he points out that Darapti QQQ reduces directly to Darii QQQ (Fig. 6.9). But the Major in Darii QQQ , Qaca, must be understood as stating that everything that is contingently c is contingently a; therefore the same construal must be put on the Major in Darapti QQQ. However, in the proposed counter-example if the Major is read this way it is false, stating that everything that is contingently moving is contingently a man; and because it is false, the counterexample fails because it falls foul of the Theorem that true contingencyMajors cannot predicate a substantial term of an accidental term.

XQ-1 moods A15 deals with mixtures of assertoric and contingency-premises. Aristotle’s reductions The syllogisms fall into two classes. Barbara, Celarent, Darii and Ferio in QXQ are perfect.42 The XQM moods are imperfect but valid,43 and their proofs follow two patterns. The standard moods are proved by the Upgrading procedure that we discussed in Chapter Three, and their proofs can be represented as in Fig. 6.11. ab bc subord

ac

Upgrading

→

Mac

ab Qbc Mac

Fig. 6.11. Perfection of XQM-1 moods by Upgrading

The remaining moods are reduced to standard XQM moods by Qconversion, as in Fig. 6.12. Qbc--ab

Qbc+

Q --- conv

Mac Fig. 6.12. Direct reduction of non-standard XQ-1 moods 42 43

A15, 33b33–40; 35a30–35. See Thom, The Logic of Essentialism, 40. A15, 34a34–b1; 34b19–27, 30–31, 35–b2. See Thom, The Logic of Essentialism, 78–82.

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Principles In addition to (P3), which requires all first Figure syllogisms to have a universal Major, Kilwardby lays down one special principle for this mixture.44 (P13) In first Figure assertoric / contingency syllogisms, the Minor must

not be a negative assertoric. (P13) is stated in the Dialectica Monacensis: “If the Minor is taken as a negative assertoric, in no way will there be a syllogism”.45 That (P13) is correct can be seen by inspecting Fig. 6.11 and Fig. 6.12, and by noting that the Minor in the perfect moods is never negative. Useful premise-pairs The application of (P3) and (P13) to the 32 possible premise-pairs, excludes all but the following 12.46 QaXa

QeXa

XaQa

XeQa

QaXi

QeXi

XaQe

XeQa

XaQi

XeQi

XaQo

XeQo

Table 6.5. Useful XQ-1 premise-pairs

44 Aristotle doesn’t state these, though at A15, 35b20–21 he does state that when the Major is universal there is always a syllogism; but this is not the same as stating that when the Major is not universal there is no syllogism. 45 Dialectica Monacensis, 502:23–24: “Si minor sumatur de inesse et negativa, qualiscumquae fuerit maior, nullo modo erit syllogismus”. 46 Kilwardby ad A15 dub.14 (25vb).

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Syllogisms The valid syllogisms are shown in Table 6.6. QXQ Barbara47

XQM QaXaQe-1

Barbara48 XaQeMa-149

QeXaQa-1

Celarent50

Celarent51 XeQeMe-152

Darii53

QaXiQo-1

Darii54 XaQoMi-1

QeXiQi-1

Ferio55

Ferio56 XeQoMo-1

Table 6.6. QXQ-1 and XQM-1 syllogisms

A15, 33b33–36. A15, 34a34–b1. 49 A15, 35a6–11. 50 A15, 33b36–40. 51 A15, 34b19–27. 52 A15, 35a16–20. This mood is mis-stated as XeQeMa-1 in Thom, The Logic of Essentialism, 82. 53 A15, 35a30–35. 54 A15, 35a30–31. 55 A15, 35a30–35. 56 A15, 35a30–31. 47 48

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Semantics Perfect syllogisms Kilwardby interprets the contingency-propositions in the perfect QXQ moods—unlike those in the uniform contingency-moods—as unampliated: In contingency-propositions to-be-said-of-all has two senses—“Everything that is contingently b etc.”, and “Everything that is b etc.” It is taken in the first sense in the generation of uniform syllogisms, and in the second in the generation of syllogisms with a mixture of contingencyand assertoric propositions.57

As we saw in Chapter One, an unampliated universal contingencyproposition states two possibilities. One of these possibilities would be actualized if whatever can fall under the subject can fall under the predicate; the other would be actualized if nothing that actually falls under the subject must fall under the predicate. Kilwardby takes the Minor to be an unrestricted assertoric, and he states that Aristotle signals this by saying that the Minor is “under b”.58 This must mean that the Minor has to be an apodeictic unrestricted assertoric. That being so, Barbara QXQ-1 is valid. It contains two core inferences. One states that if whatever can be b can be a, and whatever can be c can be b, then whatever can be c can be a. The other states that if nothing that is b has to be a, and whatever is c is b, then nothing that is c has to be a. Both of these are valid; and validity is preserved if in each case the premises are embedded within outer modalities of possibility and necessity respectively, while the conclusion is embedded within an outer modality of possibility. (But notice that in analyzing the syllogism this way we have adopted two different readings for the apodeictic unrestricted Minor: in the first case we read “It’s necessary that whatever can be c can be b” and in the second we read “It’s necessary that whatever is c is b”. Thus, our analysis assumes that both these readings are available.)

57 Kilwardby ad A16, dub.1 (26vb): “… dici de omni in contingentibus duas habet intentiones, scilicet ‘Omne quod contingit esse b etc.’ et ‘Omne quod est b etc.’ Et secundum primam intentionem accipitur in generatione uniformi, secundum secundam in mixtione contingentis et inesse”. 58 Kilwardby ad A15 dub.3, 24rb. See A15, 33b34–35, and A. Rini, “Hupo in the Prior Analytics: a note on Disamis XLL” History and Philosophy of Logic 21 (2000): 259–264.

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The QXQ-1 syllogisms are perfect in the following sense: the conclusion results from the Major premise by substituting an inferior for the Major’s subject. The Minor premise (being apodeictic) places the Minor term as an inferior of the Middle; and the conclusion expresses a contingency, as does the Major premise. In other words, the universal QXQ-1 syllogisms (like the universal LXL-1 syllogisms) exhibit downwards monotonicty for the subject. The particular syllogisms, in both mixtures, follow Kilwardby’s prescriptions regarding the assimilation of the conclusion to the premises. Imperfect syllogisms As we saw in Chapter Three, the Major in these syllogisms has to be an apodeictic or a natural unrestricted assertoric. Consequently it will state, either as a matter of what is necessary or of what is natural, that its predicate possibly belongs or does not belong to what can fall under its subject. The Minor premise has to be a natural or an indeterminate contingency; but cannot be an indeterminate contingency if the Major premise is a natural unrestricted assertoric. In the affirmative moods, this means that its predicate possibly belongs to what can fall under its subject. The conclusion, then, will state that the Major term can (or cannot) possibly belong to what can fall under the Minor; and thus it will state an affirmative possibility—the denial of a negative necessity. Barbara XQM’s core inference states that if whatever can be b can be a, and whatever can be c can be b, then whatever can be c can be a. This is valid; and validity is preserved if the premises are embedded within outer modalities of necessity and possibility respectively, while the conclusion is embedded within an outer modality of possibility. Validity is also preserved if both premises are embedded within a modality of naturalness, or if the Major is embedded within a modality of necessity and the Minor within a modality of naturalness. The negative moods have to imply a conclusion which states a negative possibility—the denial of an affirmative necessity—and thus will have to imply that the Major term cannot apply to what actually falls under the Minor term. However, this fact doesn’t require us to read the contingency-Minor differently in the negative and affirmative moods. We can read the contingency-Minor in all first Figure moods as predicating something of what can fall under the Minor term, because a denial about what actually falls under that term will follow from a denial about what can fall under it.

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Neither affirmative nor negative XQM-1 syllogisms satisfy the criteria for perfection. However, on our analysis of Upgrading these syllogisms are provable by that method, because their core inferences are valid and the application of outer modalities to those core inferences preserves validity. In the moods of this mixture, Kilwardby again operates in accordance with rule (F)—that in syllogisms complying with Aristotelian principles—in this case (P13)—an assertoric premise is to be read as unrestricted; in other contexts, an assertoric premise may be read either as an as-of-now assertoric or as unrestricted. But this mixture differs from necessity / assertoric mixtures in that the unrestricted assertoric premise may be either natural or apodeictic. Invalid inferences Celarent XQX As mentioned in Chapter Three, Kilwardby cites a seeming Upgrading proof of Celarent XQX: Disamis

aci bca abi

C →

abe bca ace

Upgrading

→

abe Qbca ace

Fig. 6.13. Apparent Upgrading proof of Celarent XQX

(This argument was known to Alexander.) In reply, he refers us to a passage later in his comments on A16, where he refutes a similar purported proof of Barbara LQX by making his customary point that, while the opposite of the conclusion of the enabling syllogism is incompossible with the premises, this is not the case for the enabled syllogism.59 (This was also Alexander’s response to the purported proof.) He proposes a second argument for the validity of Celarent XQX. The Major is an unrestricted assertoric, and thus “in reality has the force of a necessary proposition, or a necessity-proposition” so that whatever is implied by LeQa-1 ought to be implied by XeQa-1; however, the former premises imply an assertoric conclusion, so the latter should too. In reply, he invokes his distinction between two types of unrestricted assertoric. The assertoric premises of imperfect XQsyllogisms do not have to be necessary, or to have the force of a 59

Kilwardby ad A16 dub.8 (27va).

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necessity-proposition; rather, they may express the actualization of a natural contingency. Thus, the argument from the validity of Celarent LQX fails.60 Kilwardby maintains that Celarent XQX is invalid on the ground that in a negative assertoric proposition the predicate is denied of what actually falls under the subject, not what contingently falls under the subject. He adds: When the Major is a negative assertoric and the Minor is a contingencyproposition, what follows is not the Major’s actual denial, but its potential denial, of those things that are contingently taken under the Middle via the Minor proposition. And so what follows by virtue of the premises is not an assertoric conclusion that in actuality denies the Major extreme of the Minor, but (I say) a contingency-conclusion.61

Here he reminds us that that XeQa-1 premises, while not entailing an assertoric, do entail a possibility-conclusion—one that expresses the Major’s potential denial of the Minor. This is what we would expect, given our analysis of Upgrading. Celarent XQQ Kilwardby observes that this inference in invalid. All that follows is a one-way contingency-conclusion: … when the Major is a negative assertoric the Middle and Major Extremes are related in the manner of opposites. And it may seem that there is a rule that if one opposite contingently inheres the other is contingently removed. And so it may seem that it ought to follow “So it is two-way contingent”, as the Minor is such a contingency. And so Aristotle shows that a contingency conclusion in the sense of the possible follows in the negative moods, and doesn’t show this in the affirmative moods but takes it as obvious.62 Kilwardby ad A16 dub. 5 (27ra). Kilwardby ad A16 dub.5 (27ra): “…cum maior est negativa de inesse et minor de contingenti, non sequitur actualis remotio maioris extremitatis ab his quae contingenter sumuntur sub medio per minorem propositionem, sed sequitur remotio potentialis. Et ideo non sequitur conclusio de inesse quae actu removet maiorem extremitatem a minori, sed conclusio de contingenti dico virtute praemissarum”. 62 Kilwardby ad A15 dub.12 (25va): “Quando enim maior est negativa de inesse, medium et maior extremitas se habent per dispositionem sicut opposita. Et apparet tamquam regula quod si unum oppositorum contingenter inest, et reliquum contingenter removetur. Et ita apparet quod debeat sequi ‘Ergo contingens ad utrumlibet’, sicut minor est de tali contingenti. Et ideo ostendit Aristoteles conclusionem de contingenti pro possibili sequi in modis negativis, et non ostendit hoc de affirmativis, sed supponit hoc tamquam manifestum”. 60 61

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Being grey-haired is naturally contingent for old people, but being grey-haired has many opposites; and some of these (such as being blond) are two-way contingent (though not naturally contingent) for old people, while some (such as being headless) are impossible. Accordingly, a two-way contingency does not follow.

XQ-2 moods In A18, Aristotle presents his views about second Figure syllogisms containing a mixture of an assertoric and a contingency-premise. Aristotle’s reductions XQ-2 syllogisms reduce directly to first Figure XQM syllogisms with a universal negative assertoric, in one of the ways exhibited in Fig. 6.14.63 ace e --- conv XQM --- 1

e --- conv XQM --- 1

abe bae

Qac

Mbc

abe

Qac---

bae

Qac+ Mbc

XQM --- 1

Qab

M --- conv

Q --- conv Q --- conv

XQM --- 1

cae

e --- conv

Mcb Mbc Qab---

ace

Qab+

cae

Mcb Mbc

e --- conv

M --- conv

Fig. 6.14. Direct reduction of XQ-2 moods

Notice that term-conversion for contingency-propositions is not used. Obviously, XQ-2 syllogisms cannot reduce directly to affirmative first Figure syllogisms, since they are all negative. Equally, they cannot derive from perfect XQ-1 syllogisms, because the latter have a contingency-Major, which would not convert to a universal major in the second Figure as required.64 63 Kilwardby ad A18 dub.1 (29va). A18, 37b24–35; 38a3–7. See Thom, The Logic of Essentialism, 82–83. 64 Kilwardby ad A18 dub.2 (29va).

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Principles In addition to the general requirement (P5) on all second Figure syllogisms that the Major must be universal, Kilwardby lays down one special principle for this mixture: (P14) In second Figure assertoric / contingency syllogisms, the assertoric

must be a universal negative.65 (P14) can be found in Aristotle,66 and also in the Dialectica Monacensis.67 Useful premise-pairs Applying (P5) and (P14), it’s clear that only the 6 premise-pairs listed in Table 6.7 are useful. XeQa

QaXe

QeXe

XeQe XeQi XeQo Table 6.7. Useful XQ-2 premise-pairs

65 66 67

Kilwardby ad A18 dub.6 (29vb). A17, 36b31. Dialectica Monacensis, 503:17–21.

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Syllogisms Aristotle recognizes the syllogisms in Table 6.8. XQM

QXM

Cesare68 XeQeMe-269 Camestres70 QeXeMe-271 Festino72 XeQoMo-273 Table 6.8. XQM-2 and QXM-2 syllogisms

Semantics Kilwardby uses the unrestricted nature of the assertoric premise to disallow a putative counter-example to Cesare XQM:74 (white)(animal)e Q(white)(horse)a M(animal)(horse)e Fig. 6.15. Apparent counter-example to Cesare XQM

He retorts that “every counter-example here where the assertoric is taken as an as-of-now assertoric is sophistical”. In fact, the patterns of reduction shown in Fig. 6.14 have implications for the semantic 68 69 70 71 72 73 74

A18, 37b24–28. A18, 37b29–35. A18, 37b29. A18, 37b29–35. A18, 38a3–4. A18, 38a4–7. Kilwardby ad A18 dub.3 (29va).

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interpretation of the contingency- and assertoric premises in XQ-2 syllogisms generally. Assertorics and contingency-propositions must be read as in the XQM-1 moods, since they reduce to those moods. Thus, in Cesare XQM an apodeictic or natural unrestricted assertoric must be paired with a natural or indeterminate contingency (but not so that a natural unrestricted assertoric is paired with an indeterminate contingency). One of its core inferences states that if nothing which can fall under the Major can fall under the Middle, and everything that can fall under the Minor can fall under the Middle, then nothing that can fall under the Minor has to fall under the Major. This is valid; and so is Cesare XQM. A similar analysis can be given for Festino XQM and for Camestres QXM. Invalid inferences Camestres XQM and Baroco XQM Camestres XQM and Baroco XQM are invalid. Each of these moods has two core inferences. The premises of one core inference affirm that the Middle can apply to whatever can fall under the Major and to what can fall under the Minor. The premises of the other core inference affirm that the Middle can apply to whatever can fall under the Major but doesn’t have to apply to what actually falls under the Minor. The first of these inferences is useless because it has two affirmative premises in the second Figure; the second is useless because it lacks a genuine Middle term. Thus the process of Upgrading gains no foothold here, since there is no valid core inference on which it can operate. Cesare QXM, Festino QXM and Baroco QXM Having earlier dismissed as-of-now counter-examples to Cesare XQM as sophistical, Kilwardby raises just such a counter-example to Cesare QXM:75 Q(white)(man)e (white)(animal)a M(man)(animal)e Fig. 6.16. Kilwardby’s counter-example to Cesare QXM 75

Kilwardby ad A18 dub.4 (29va).

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The explanation for this seeming inconsistency is that Kilwardby is as usual operating under rule (F). According that rule, in syllogisms complying with (P14), an assertoric premise is to be read as unrestricted; in other contexts, an assertoric premise may be read either as an as-of-now assertoric or as unrestricted. Cesare XQM complies with (P14); Cesare QXM does not. So, it is legitimate to pose an as-ofnow counter-example to the latter, but not to the former. Kilwardby’s counter-example occurs in a context where he is discussing what looks like an Upgrading proof of Cesare QXM along the lines of Fig. 6.17: Ferio XLX

abe Lbci aco

C →

abe aca Mbce

Upgrading

→

Qabe aca Mbce

Fig. 6.17. Apparent Upgrading proof of Cesare QXM

(He outlines a similar proof of Cesare QXX.) He dismisses the proof, stating that the Major “is not to be admitted as an assertoric”, because the opposite of the enabled conclusion is compatible with the Major but the opposite of the enabling conclusion is not. Cesare QXM is indeed invalid since it has two core inferences, neither of which is valid. One of them argues from two affirmative premises in the second Figure, and the other lacks a genuine middle. A similar treatment can be given for Festino QXM. Baroco QXM is also rejected by Kilwardby, and he offers an explanation of its invalidity: … the mixture of contingency- and assertoric [premises] never holds unless the assertoric [premise] is an unrestricted assertoric. Now, a particular negative assertoric, because it cannot be made either Major or Minor in the first Figure by reduction to the first Figure, is not appropriated so that it is an unrestricted assertoric; but it may well be as-of-now. And so there is not a useful conjugation in the mixture of contingencyand assertoric [premises] in the said moods.76

Its invalidity can also be demonstrated by showing the invalidity of its core inferences. 76 Kilwardby ad A22 dub.4 (33va): “… mixtio contingentis et inesse numquam tenet nisi illa de inesse sit de inesse simpliciter. Particularis autem negativa de inesse eo quod nec potest fieri maior nec minor in prima figura per reductionem in prima figura non appropriatur ut sit de inesse simpliciter, sed bene potest esse ut nunc. Et ideo in mixtione contingentis et inesse non sit utilis coniugatio in dictis modis.”

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XQ-3 moods Mixtures of assertoric and contingency-premises in the third Figure are discussed in A21. Aristotle’s reductions Aristotle’s direct reductions of the valid XQ-3 syllogisms proceed in one of the ways shown in Fig. 6.18. Qac+

Qbc [1]

XQM --- 1

ac Qcbi

[2]

Mabpart

Darii XQM

Qcai

bca

Mbai Mabi ac+

bc+ [3]

QXQ --- 1

Qacuniv

cbi

Qabpart

[4]

Darii QXQ

cai

Qbca

Qbai Qabi

Fig. 6.18. Aristotle’s direct reduction of XQ-3 moods

Moods with premise-pairs XaQa, XaQe, XaQi, XaQo, XeQa, XeQe, XeQi or XeQo reduce to the first Figure as in [1].77 Two further premise-pairs (QiXa, QoXa) reduce to the first Figure as in [2]; and the resulting syllogisms are Disamis QXM and QoXaMi-3. Four further premise-pairs (QaXa, QaXi, QeXa, QeXi, QiXa, and QoXa) reduce to the first Figure as in [3]; and the resulting syllogisms are Darapti QXQ , Datisi QXQ , Felapton QXQ , and Ferison QXQ.78 The two remaining premise-pairs (XiQa, XiQe) reduce to the first Figure as in

77 Aristotle (A21, 39b10–30) gives these reductions for Darapti, Felapton, Datisi and Ferison, and for XaQeMi-3 and XeQeMi-3, but he overlooks XaQoMi-3 and XeQoMi-3. See Thom, The Logic of Essentialism, 84–85. 78 These proofs are all to be found in Aristotle. (Darapti and Felapton QXQ: A21, 39b16–22. Datisi and Ferison QXQ: A21, 39b26–30.)

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[4]; and the resulting syllogisms are Disamis XQQ (reducing to Darii QXQ) and XiQeQi-3 (reducing to QeXiQi-1).79 There are also indirect reductions, and these proceed in the ways shown in Fig. 6.19.80 [5]

Lab Qbc ac

C →

ac Qbc Mab

[6]

Lab bc

K → L||Q

Lac

Qac bc Mab

Fig. 6.19. Indirect reduction of XQ-3 moods

Moods with premise-pairs XiQa, XaQi and XaQa reduce to the first Figure as in [5]; and the resulting syllogisms are Disamis XQM, Datisi XQM and Darapti XQM. Premise-pairs QoXa, QeXa, QiXa, QeXi, QaXi and QaXa reduce to the first Figure as in [6]; and the resulting syllogisms are Bocardo QXM, Felapton QXM, Disamis QXM, Ferison QXM, Datisi QXM and Darapti QXM. Of these, only Bocardo QXM is proved by Aristotle;81 and Kilwardby recognizes this reduction: … he lays it down in the text, showing its utility by deduction to the impossible thus: It’s contingent for some c not to be a, and every c is b, so it’s contingent for some b not to be a. If it doesn’t follow, the opposite will stand, namely “It’s not contingent for some b not to be a”, which is equipollent with “It’s necessary for every b to be a”. So an assertoric/necessity mixture is produced from this and the Minor, destroying the Major, thus: It’s necessary for every b to be a, every c is b, so it’s necessary for every c to be a. And it follows, because the Minor is an unrestricted assertoric.82

The first of these is to be found in Aristotle, but not the second (A21, 39b26–30). See Thom, The Logic of Essentialism, 75–77. 81 A21, 39b31–39. 82 Kilwardby ad A21, 39b31–39 (32rb): “… et hanc solam ponit in littera ostendens eius utilitatem per deductionem ad impossibile sic: Contingit aliquod c non esse a et omne c est b, ergo contingit aliquod b non esse a; si non sequitur, stabit oppositum, ‘Non contingit aliquod b non esse a’, quae aequipollet isti ‘Necesse est omne b esse a’. Fiat ergo mixtio necessarii et inesse ex hac et minore ad {ad F 1: om. Ed} destruendam maiorem sic: Necesse est omne b esse a, omne c est b, ergo necesse est omne c esse a. Et sequitur quia minor est de inesse simpliciter.” Thom, The Logic of Essentialism, 136 pro79 80

poses the following counter-example to Bocardo QXM:

Q(animal)(white)o (swan)(white)a M(animal)(swan)a

.

But Kilwardby would not accept this, because the Minor is an as-of-now assertoric. He would be equally unimpressed with the counter-example proposed by Buridan 4.3.21 (Thom, The Logic of Essentialism, 238):

Q(riding)(running)o (horse)(running)a M(riding)(horse)a

.

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Principles Aristotle states that when both premises are particular, there will be no syllogism83—which, of course, is equivalent to (P1). In addition to the general requirement (P1) on all syllogisms that there is a universal premise, Kilwardby lays down one special principle to delimit the class of XQ-3 premise-pairs that are useful. (P15) In a negative third Figure assertoric / contingency syllogism (i.e.

where the assertoric premise is negative), the assertoric must be universal and must be the Major premise.84 In relation to this mixture, the Dialectica Monacensis states simply that if either premise is a contingency, the other premise being assertoric, there is always a syllogism.85 This statement is too broad, not excluding any pair of premises in this mixture; but Kilwardby’s statement hits the mark. He states: It is to be said that all premise-pairs of this mixture in this Figure descend from the particular moods of the same mixture in the first Figure, with the Minor converted, as in the foregoing. Thus, those whose assertoric is negative descend from those having the same negative, and those whose assertoric is affirmative descend from those having it affirmative. But the moods of this mixture in the first Figure having a negative assertoric have it as a universal; hence it is the same here. Further, the negative moods here reduce to those that have a universal negative assertoric Major (as has been said). Hence, they have to observe that proposition which would become the Major. Such, however, can only be a universal negative assertoric. Hence, they have to observe a universal negative assertoric. But they can’t have that as Minor. Hence the Minor in the third Figure can’t be simply negative; nor can it in the first. Hence they have to have it as Major. And so it’s clear that in negative moods (namely where the assertoric is negative), it must be universal and Major.86 A21, 40a1–2. Kilwardby ad A21 dub.4 (32va). 85 Dialectica Monacensis, 504:7–9: “Utralibet enim existente de contingenti et reliqua de inesse, semper fit syllogismus.” 86 Kilwardby ad A21 dub.5 (32va–b): “Et dicendum quod omnes coniugationes huius mixtionis in hac figura descendunt a particularibus modis eiusdem mixtionis in prima figura minore conversa, sicut in praecedentibus habitum est, ita quod illae quae habent illam de inesse negativam descendunt ab his quae habent eandem negativam et illae quae habent illam de inesse affirmativam descendunt ab his quae habent illam affirmativam. Sed modi huius mixtionis in prima figura habentes illam de inesse negativam habent eam universalem. Quare et sic erit hic. Adhuc, modi negativi hic reducuntur in illos qui habent maiorem negativam de inesse et universalem, sicut iam dictum 83 84

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Even though Aristotle denies it,87 it seems that Darapti XQX is valid and can be proved by Upgrading. Celarent

abe bca ace

K →

aca bca

Upgrading

abi

→

aca Qbca abi

Fig. 6.20. Apparent Upgrading proof of Darapti XQX

Kilwardby replies in his usual fashion: And it is to be said that it doesn’t follow if the opposite is to be given and the Minor is not to be posited as an assertoric … And so, positing the Minor as an assertoric creates an unacceptable result that didn’t exist previously.

Useful premise-pairs (P1) excludes those premise-pairs that are made up of two particulars, namely XiQi, XiQo, XiQo, XoQo, QiXi, QiXo, QoXi, and QoXo. (P15) excludes those pairs that have a particular negative assertoric premise and those that have a negative assertoric Minor premise. The application of the two principles leaves 16 pairs, as shown in Table 6.9:88 XaQa

XeQa

XiQa

QaXa

QeXa

XaQe

XeQe

XiQe

QaXi

QeXi

XaQi

XeQi

XaQo

XeQo

QiXa

QoXa

Table 6.9. Useful XQ-3 premise-pairs est. Quare oportet quod observent eam propositionem quae possit fieri maior. Talis autem non est nisi universalis negativa de inesse. Quare oportet quod observent illam de inesse universalem negativam. Sed possunt habere eam minorem. Quare minor in tertia figura non potest esse simpliciter negativa, sicut nec in prima {sicut nec in prima AE 1F1KlO2P1: sed nec in prima O3: sicut nec prima Cm1E2: sicut neque prima Br: sicut in prima F 2V: sic nec ipsa Ed}. Quare oportet ut habeant eam maiorem. Et ita patet quod in modis negativis, ubi scilicet illa de inesse negativa est, oportet quod ipsa sit universalis et maior.” 87 A21, 39b7–9. 88 Kilwardby ad A21 dub.4 (32va).

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Syllogisms The resulting syllogisms are shown in Table 6.10. Some premise-pairs yield more than one conclusion. Whatever implies an affirmative contingency-conclusion also implies the corresponding negative. Thus while there are 16 fruitful premise-pairs there are more syllogisms. XQM

XQQ

QXQ

QXM

Darapti89

Darapti90

Darapti91

(Darapti)

XaQeMi92

QeXaQi

(QeXaMi)

Felapton93

Felapton94

(Felapton)

XeQeMo95

QaXaQo

(QaXaMo)

(Disamis)

Disamis96

Disamis

(XiQeMi)

XiQeQi

QoXaMi

Datisi97

Datisi98

(Datisi)

XaQoMi

QeXiQi

(QeXiMi)

A21, 39b16–22. Overlooked by Aristotle. See Thom, The Logic of Essentialism, 57. 91 A21, 39b16–22. 92 A21, 39b22–25. 93 A21, 39b16–22. 94 A21, 39b16–22. 95 A21, 39b22–25. Misprinted as ‘XeQeMi’ in Thom, The Logic of Essentialism, 84. 96 A21, 39b26–30. Lagerlund, “Medieval Theories of the Syllogism” mistakenly says that Kilwardby “does not manage to get -CC [sc. XQQ] and LCC [sc. LQQ] for Disamis in the third figure”. What is true is that the semantics proposed in Thom, Medieval Modal Systems fails to deliver Disamis XQQ as valid (pp. 104–105); but that is a fault in Thom, not Kilwardby. The fault results from the fact that in Thom, Medieval Modal Systems, particular contingency-propositions are not convertible. In Chapter One of the present work we defined particular contingency-propositions so that they would be convertible. 97 A21, 39b26–30. 98 A21, 39b26–30. 89 90

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XQQ

QXQ

QXM Bocardo99

Ferison100

Ferison101

(Ferison)

XeQoMo

QaXiQi

(QaXiMi)

Table 6.10. XQM-3, XQQ-3, QXQ-3 and QXM-3 syllogisms

Semantics The reductions shown in Fig. 6.18 have implications for the interpretation of the contingency and assertoric premises in this mixture. Rule (F) operates here again. In syllogisms complying with (P15), an assertoric premise is to be read as unrestricted. In reductions of types [1] and [2] the assertoric may be either a natural or an apodeictic unrestricted assertoric. In types [3] and [4] the assertoric must be an apodeictic unrestricted assertoric. In inferences not complying with (P15), an assertoric premise may be read either as an as-of-now assertoric or as unrestricted. Felapton XQM Kilwardby raises a question about the premise-pairs that Aristotle thinks are useful.102 He proposes an objection to Felapton XQM.103 (man)(moving)e Q(horse)(moving)a M(man)(horse)o Fig. 6.21. Apparent counter-example to Felapton XQM

99 A21, 39b31–39. Thom, Medieval Modal Systems, 104–105 has it (pace Kilwardby) that Bocardo QXQ and Disamis QXQ are valid. This lack of correspondence can be attributed to an erroneous analysis of Kilwardby’s contingency-propositions. 100 A21, 39b26–30. 101 A21, 39b26–30. 102 Kilwardby ad A21 dub.1 (32rb–32va). Ed has “Hic dubitatur primo de conjugationibus”, as do Cm1KlO3V. To this AE 1E2F2P1 add “utilibus”. 103 Kilwardby ad A21 dub.1 (32va). Ed has “Non tamen contingit omnem equum esse hominem”. ACm1E1E2F1F2KlO2O3P1 have “Non tamen contingit aliquem equum esse

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In accordance with rule (F) he replies that the assertoric Major should have been unrestricted, because in the first Figure syllogism to which Felapton XQM reduces the assertoric Major is unrestricted. Perfectible non-Aristotelian inferences With one exception, Aristotle accepts all the standard moods in the form XQM and QXM. The exception is Bocardo XQM, which falls foul of (P15). Nonetheless, Kilwardby suggests two proofs of this inference. One is by Indirect Reduction and Upgrading: Barbara LXX

Laba bca aca

C →

aco bca

Upgrading

→

Mabo

aco Qbca Mabo

Fig. 6.22. Apparent Upgrading proof of Bocardo XQM

His comment on this runs parallel to his comments on other spurious Upgrading proofs: … the opposite of the conclusion (whether it is an assertoric or contingency-conclusion) is consistent with a Minor whose Major is an as-ofnow assertoric, but when the Minor is put as an assertoric that is not compatible with either.104

Kilwardby also suggests an expository proof of the validity of Bocardo XQM: Qbca cna Felapton XQM

ane

Qbna Mabo

Barbara QXQ Exposition

→

aco Qbca Mabo

Fig. 6.23. Apparent expository proof of Bocardo XQM

hominem”. V has “Et tamen non contingit aliquem equum esse hominem”. Br has “Sed tamen non contingit aliquem equum esse hominem”. 104 Kilwardby ibid: “… oppositum conclusionis tam de inesse quam contingenti stare potest cum minori ex qua maior est de inesse ut nunc, sed cum minor ponatur inesse, ipsa est incompossibilis utrique.”

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He comments: It is to be said that it is useless, in that the universal negative taken under the particular can be an as-of-now assertoric, as can that particular; and so it creates a useless premise-pair.105

One of Bocardo XQM’s core inferences states that if something that can be c can’t be a, and whatever can be c can be b, then something that is b doesn’t have to be a. The other core inference states that if something that can be c can’t be a, and nothing which is c has to be b, then something that is b doesn’t have to be a. Neither core inference is valid; and so Bocardo XQM is not valid.

LQ-1 moods Mixtures of contingency- and necessity-premises in the first Figure are dealt with in A16. Perfectibility Like the XQ-1 moods, these fall into two classes. Barbara, Celarent, Darii and Ferio in QLQ are perfect.106 The LQM, QLM and LQX moods are imperfect but valid, and Aristotle reduces them directly to other Figures in the ways shown in Fig. 6.24.107 LLL --- 2

LLL --- 3

Lab Lac Lbc Lac Lbc+ Lab

K → L||Q

K → L||Q

Lab Qbc Mac Qab Lbc+ Mac

105 Kilwardby ad A21 dub.3 (32va): “… dicendum quod inutilis est eo quod universalis negativa accepta sub particulari potest esse de inesse ut nunc, sicut et ipsa particularis; et ideo facit inutilem.” 106 A16, 35b23–26. See Thom, The Logic of Essentialism, 40–41. 107 A16, 35b38–36a2, 36a8–17, 25–27, 34–b2. See Thom, The Logic of Essentialism, 58– 59, 69–71.

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Lab LXL --- 1

Lba Lbc

ac+

K → L||Q

Lab Qbc ac---

Fig. 6.24. Indirect reduction of imperfect LQ-1 moods

Principles In addition to (P3)’s requirement that all first Figure syllogisms have a universal Major, Kilwardby lays down one special principle for this mixture. (P16) In a first Figure necessity / contingency syllogism, the Minor must

not be a negative necessity-proposition.108 That this is correct can be seen from the fact (shown in Fig. 6.24) that the only cases where the Minor is a necessity-proposition reduce to third Figure LLL moods (where the Minor has to be affirmative), and by reflecting on the fact that the Minor in the perfect QLQ moods is never negative. There is no explicit statement of this principle in Aristotle. However, the Dialectica Monacensis does state an equivalent requirement: After this we must see about the mixture of the contingent and the necessary. It is to be noted therefore in this mixture that it is given as a general rule that when the Major is particular or the Minor is a negative necessity, there will never be a syllogism. If these two things are avoided, there will always be a syllogism.109

Useful premise-pairs Applying Kilwardby’s two principles, we can see that (P3) excludes all 16 pairs having a particular Major, namely Qi_/_/, Qo_/_/, Li_/_/ and Lo_/_/. Of the remaining 16, 4 are excluded by (P16), namely QaLe, QaLo, QeLe and QeLo. Thus there are 12 useful premise-pairs, and 20 useless, just as Kilwardby states.110 The useful premise-pairs are shown in Table 6.11. 108

Kilwardby ad A16 dub.13 (27vb). Dialectica Monacensis, 502:33–503:3: “Post haec videndum est de commixtione contingentis et necessarii. Notandum ergo in hac mixtione quod datur pro regula generali quod quando maior fuerit particularis vel minor fuerit negativa de necessario, numquam erit syllogismus.” 110 Kilwardby ad A16 dub.13 (27vb). 109

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QaLa

QeLa

LaQa

LeQa

QaLi

QeLi

LaQe

LeQe

LaQi

LeQi

LaQo

LeQo

Table 6.11. Useful LQ-1 premise-pairs

Aristotle mentions most of these—the exceptions being LaQo, LeQe and LeQo.111 Syllogisms From these 12 premise-pairs, we can construct the syllogisms shown in Table 6.12.

111 112 113 114 115 116

QLQ

QLM

LQX

LQM

Barbara112

(Barbara)

Barbara113

QeLaQa-1

(QeLaMa-1)

LaQeMa-1114

Celarent115

(Celarent)

Celarent116

(Celarent)

LeQeXe-1

(LeQeXe-1)

See Ross, Aristotle’s Prior and Posterior Analytics, 346. A16, 36a2–5. A16, 35b38–36a2. A16, 36a25–27. A16, 36a17–21. A16, 36a8–15.

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215

QLQ

QLM

LQM

Darii117

(Darii)

Darii118

QeLiQi-1

(QeLiMi-1)

LaQoMi-1

Ferio119

Ferio120

Ferio121

(Ferio)

LeQoXo-1

(LeQoXo-1)

Table 6.12. LQ-1 syllogisms

Modal subordination Kilwardby holds that whenever two conclusions follow from a pair of premises, one of the conclusions is primary and the other secondary.122 He applies this principle to LeQa-1: To this it is to be said that by the nature of the Major proposition what the Major actually denies of the Middle follows primarily in an assertoric conclusion, because the Major is actually denied of those things that are or can be under the Middle. Further, the nature of those conclusions is such that the assertoric precedes the contingencyproposition, as antecedent to consequent—as Aristotle signals in the text proving that the contingency-conclusion follows because it follows from the assertoric conclusion.123 And so the order of the conclusions is clear.124

A16, 35b23–26. A16, 36a39–b2. 119 A16, 35b23–26. 120 A16, 36a39–b2. 121 A16, 36a34–39. 122 Kilwardby ad A25 Part 2 dub.2 (37ra). 123 A16, 36a15–17. 124 Kilwardby ad A16 dub.5 (27rb): “Et dicendum quod ex natura maioris quae actualiter removet primam a medio sequitur primo conclusio de inesse, quia actu removet primam ab his quae sunt sub medio vel possunt esse. Adhuc, natura ipsarum conclusionum est quod illa de inesse praecedant illam de contingenti sicut antecedens suum consequens, sicut signat Aristoteles in littera probans conclusionem de contingenti sequi per hoc quod sequitur conclusio de inesse. Et sic patet ordo conclusionum.” 117 118

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Semantics Perfect syllogisms Kilwardby’s view is that the QLQ-1 moods, like the QXQ-1 moods, are governed by to-be-said-of-all and are perfect. This is consistent with our analysis of perfection. In the terminology of that analysis, these moods exhibit downwards monotonicity for the subject, relative to necessary inherence. Kilwardby holds that the Major premise in the QLQ-1 moods is an unampliated contingency-proposition in which “the Major extreme is contingently (not actually) said of all that is contained under the subject”.125 … the mixture of contingency and necessity, if the necessity is Minor, is regulated by to-be-said-of-all in the sense “Everything that is b etc.”, just as in the mixture of necessity and the assertoric, because in saying “What is b etc.”, the Minor is taken according to actual inherence …126

The contingency-Major is, then, unampliated. It might seem that these moods are not ruled by to-be-said-of-all and are not perfect: … to-be-said-of-all has two senses in contingency-propositions—either “everything for which being b is contingent etc.” or “everything that is b etc.” It is taken in the first sense in uniform syllogisms, and in the second sense in contingency/ assertoric mixtures. Hence it seems that a contingency/ necessity mixture is not ruled by to-be-said-of-all, and so will not be a perfect conjugation.127

Kilwardby solves the puzzle by referring to his earlier discussion of QXQ-1 moods:

125 Kilwardby ad A16 dub.4 (27ra): “… maiorem extremitatem contingenter dici et non actualiter de omnibus contentis sub subiecto.” 126 Kilwardby ad A16, dub.1 (26vb): “… mixtio contingentis et necessarii {contingentis et necessarii ABrCm1E1E2F1F2KlO2O3P1V: ex necessario Ed}, si minor sit de necessario, regulatur per dici de omni secundum istam intentionem: ‘Omne quod est b etc.’, sicut et mixtio contingentis et inesse, quia per hoc quod dico ‘Quod est b’ sumabitur minor accepta secundum actualem inhaerentiam…” 127 Kilwardby ad A16 dub.1 (26vb): “… dici de omni in contingentibus duas habet intentiones, scilicet ‘Omne quod contingit esse b etc.’ et ‘Omne quod est b etc.’ Et secundum primam intentionem accipitur in generatione uniformi, secundum secundam in mixtione contingentis et inesse. Quare videtur quod mixtio contingentis et necessarii {necessarii ABrCm1E1E2F1F2KlO2O3P1V: inesse Ed} non reguletur per dici de omni, et ita non erit ibi perfecta coniugatio.”

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To the first it is to be said that a necessity/contingency mixture, when the Minor is a necessity [-proposition], is regulated by being-said-of-all in the sense “Everything that is b etc.”—just like an assertoric / contingency mixture.128

In that earlier discussion he had considered the option of reading the contingency-Major in QLQ-1 moods as stating that whatever is necessarily b is contingently a: Lastly, someone will perhaps wonder why he doesn’t lay down a third sense of contingency-propositions, namely “Everything that is necessarily b is contingently a”. Further, if the sense “Everything that is contingently b etc.” makes for the uniform generation of contingency-syllogisms, and the sense “Everything that is b etc.” makes for the generation of assertoric/ contingency, by the same reason there will be a third sense making for a mixture with necessity-propositions. And this will only be “Everything that is necessarily b is contingently a”. So it seems that he has insufficiently given the senses of universal contingency-propositions. And it is to be said that there is, as objected, this third sense comparing the subject with those things in which it necessarily inheres. But he omits it, because it is not separate from the second sense but is understood in it. For, inhering without qualification and inhering of necessity are not different kinds of inhering (as was said before).129 So the other sense is sufficiently understood by this sense “Every that is b etc.”. And by the sense “Everything that is b is contingently a” we have sufficiently the sense of a mixed contingency-syllogism with an assertoric [premise], and of a mixed contingency-syllogism with a necessity-[premise], as is clear by reflection. For the assertoric that is taken in a mixture with a contingency-[premise] ought to be an unrestricted assertoric, not an as-of-now one, as will be clear in what follows.130 The response to the objections is then clear from this.131 128 Kilwardby ad A16 dub.1 (26vb): “Ad primum dicendum quod mixtio contingentis et necessarii {contingentis et necessarii ABrCm1E1E2F1F2KlO2O3P1V: ex necessario Ed}, si minor sit de necessario, regulatur per dici de omni secundum istam intentionem: ‘Omne quod est b etc.’, sicut et mixtio contingentis et inesse…” 129 The reference is to Kilwardby ad A13 dub.10 (21ra). 130 This is an important part of Kilwardby’s doctrine about mixed contingencysyllogisms, and also a clue to his understanding of unampliated contingency-propositions. 131 Kilwardby ad A13 dub.12 (21rb): “Ultimo forte dubitabit aliquis quare non ponat tertiam intentionem propositionis de contingenti, scilicet hanc: ‘Omne quod necesse est esse b contingit esse a’. … Adhuc, si haec intentio: ‘Omne quod contingit esse b etc.’ faciat generationem uniformem syllogismorum de contingenti et haec intentio: ‘Omne quod est b etc.’ faciat generationem mixtam contingentis cum inesse, eadem ratione erit intentio tertia faciens mixtionem cum necessario, et haec non est nisi ista, ut videtur,

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We saw in Chapter One that though Kilwardby allows a syntactic distinction between “What is b is necessarily a” and “What is necessarily b is necessarily a”, he doesn’t recognize a semantic distinction. Analogously, he sees no real difference between a contingency-proposition about what is without qualification b, and one about what is of necessity b. Now, because the Minor in a QXQ-1 syllogism has to be unrestricted, the Major in a QLQ-1 syllogism has the same sense as the Major in a QXQ-1 syllogism. In both cases, what is (without qualification, or necessarily) b is said to be contingently a. By this line of reasoning Kilwardby explains the perfection of Barbara QLQ and the other QLQ-1 syllogisms. The Major term “a” applies contingently to whatever is without qualification (i.e. necessarily) b; therefore it applies contingently to all c—given that all c is without qualification (i.e. necessarily) b. Imperfect syllogisms Negative LQX moods Kilwardby discusses two putative counter-examples to Celarent LQX:132 L(black)(white)e Q(white)(man)a

L(man)(stone)e Q(stone)(moving)a

(black)(man)e

(man)(moving)e

Fig. 6.25. Apparent counter-examples to Celarent LQX

Common to both counter-examples, he says is that they take contingency in the Minor as indeterminate contingency, where it should be taken as natural contingency in order to satisfy the requirements of the Major.133 ‘Omne quod necesse est esse b contingit esse a’. Quare, ut videtur, insufficienter dat intentiones universalis de contingenti. Et dicendum quod ibi est illa tertia intentio, sicut oppositum est, ex comparatione subiecti ad ea quibus necessario inest, sed omittit eam quia non separatur a secunda, immo per eam intelligitur quia ibi inesse simpliciter et ex necessitate inesse non diversificat inhaerentiam, ut praedictum est. Immo per hanc intentionem: ‘Omne quod est b etc.’ intelligitur sufficienter reliqua, et per hanc intentionem: ‘Omne quod est b contingit esse a’ habemus sufficientem intentionem syllogismi de contingenti mixto cum inesse et de eodem mixto cum necessario, sicut patet intuenti. Quia illa de inesse quae accipitur cum mixtione cum contingenti debet esse de inesse simpliciter et non ut nunc, sicut patebit in sequentibus. Patet igitur ex his responsio ad opposita.” 132 Kilwardby ad A16 dub. 5 (27ra). 133 Kilwardby ad A16 dub.5 (27rb).

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In Kilwardby’s proposed counter-examples the conclusions are asof-now; but he knows that there are other counter-examples where the conclusion is unrestricted (e.g. if of necessity nothing white is an Intelligence and it’s contingent that every man is white, then it doesn’t follow that no man is an Intelligence).134 In general, what follows from the premises of Celarent LQX is “the Major Extreme’s actual denial of those thing that are contingently taken under the Middle via the Minor proposition, and so a negative assertoric follows”. This assertoric conclusion is indeed unrestricted: … just as the necessity-Major in an assertoric / necessity mixture appropriates the Minor to itself so that it is an unrestricted assertoric, similarly the necessity-Major in a contingency/necessity mixture appropriates the Minor so that it is a natural contingency. This being so, it’s necessary in negative contingency moods with a necessity-Major that the conclusion be an unrestricted assertoric. So if we assume the opposite of such a conclusion, with the converse of the Major, we get a good mix with a necessity-Major and an unrestricted assertoric Minor. This is clear from the following example: Of necessity no musician is made of wood, it’s contingent that every man is a musician, so no man is made of wood. From this it’s clear that in the imperfect moods of this mixture the contingency-Minor ought to be taken as a natural contingency.135

This analysis of Celarent LQX, as leading from a natural contingencyMinor to an unrestricted assertoric conclusion, is plausible on the basis of our earlier analysis of natural contingencies. On that analysis, the Minor premise in Celarent LQX states inter alia that it’s not merely possible but natural that what can fall under the Minor term can fall under the Middle. Such a Minor premise does indeed link to the necessity-Major, which states that it’s necessary that nothing which can fall under the Middle can fall under the Major. Together, these 134 The Minor and conclusion are missing in Ed (27ra), but are found in ABrCm E E 1 1 2 F1F2KlO2O3P1V. 135 Kilwardby ad A16 dub.9 (27va): “… sicut in mixtione necessarii {necessarii AE F 1 1 O2O3: necessaria Ed} et inesse maior quae est de necessario appropriat sibi minorem ut sit de inesse simpliciter, sic in mixtione contingentis et necessarii maior quae est de necessario appropriat minorem ut sit de contingenti nato. Qua {qua EdAO2O3: quo E 1F1} sic existente necesse est {necesse est Ed: oportet O2: oportet necessario AE 1F1O3} conclusionem modi negativi contingentis {contingentis Ed: om. AE 1F1O2O3} ex {ex Ed: habentis AE 1F1} maior {maior EdE 1: maiorem AF 1O2O3} de necessario esse de inesse simpliciter. Unde si assumatur oppositum talis conclusionis cum conversa maioris, fiet bona mixtio ex maiore de necessario et altera de inesse simpliciter. Et hoc potest patere in exemplo sic: De necessitate nullum musicum est lignum, contingit omnem hominem esse musicum, ergo nullus homo est lignum. Ex quo patet quod in modis imperfectis huius mixtionis minor de contingenti tantum debet sumi de contingenti nato.”

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premises entail that it’s natural that nothing which can fall under the Minor can fall under the Major—a natural unrestricted assertoric conclusion—and Kilwardby rejects Celarent LQL on the ground that only a natural unrestricted conclusion follows.136 Against Kilwardby’s analysis it might be urged that it doesn’t fit the Aristotelian proof of Celarent LQX (given in Fig. 6.24). That proof rests on the fact that the conclusion of Celarent LQX is the contradictory of the unrestricted assertoric premise in Ferio LXL. Now, the contradictory of an unrestricted assertoric is not an unrestricted assertoric. An unrestricted assertoric states a necessity; and therefore its contradictory states a possibility, not a necessity. Yet Kilwardby takes the Aristotelian proof to be a good one, and comments that “if we assume the opposite of such a conclusion, together with the converse of the Major, we get a good syllogism from a necessity-Major and an unrestricted assertoric.”137 Thus, he takes the contradictory of an unrestricted assertoric to be another unrestricted assertoric. Now, there exists an alternative analysis which meets this objection. We might read the conclusion of Celarent LQX as stating the possibility (not the necessity) that what can fall under the Minor cannot fall under the Minor. This conclusion stands in the right relation to the assertoric Minor of Ferio LXL. But on this reading, there is no need (pace Kilwardby) to read the Minor of Celarent LQX as stating a natural contingency. If we read the Minor as simply stating an unampliated contingency, the inference will be valid. If it’s necessary that nothing which can be b can be a, and it’s possible that whatever can be c can be b, then it’s possible that whatever can be c can be a. This, however, is not Kilwardby’s analysis. According to him, the Minor cannot be other than a natural contingency, because of its appropriation by the Major. LQM moods If we read the contingency-premise in these moods as expressing a natural contingency, a possibility-conclusion follows in all moods. Given that we have a valid core inference, if we attach a modality of necessity to one premise and a modality of naturalness or possibility to the other, we may attach a modality of possibility to the conclusion. 136 Kilwardby ad A16 dub.6 (27rb). Thom, Medieval Modal Systems, 107 is at odds with Kilwardby here, taking the assertoric conclusion to be as-of-now. On Kilwardby’s interpretation, Aristotle never uses as-of-now assertorics in the modal syllogistic. 137 Kilwardby ad A16 dub.9 (27va).

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Invalid inferences Affirmative LQX moods Aristotle states that the affirmative LQX moods are invalid;138 and Kilwardby obligingly provides an argument for their invalidity.139 The reason why the negative first Figure LQX moods are valid and the affirmative ones are not, according to him, lies in the different truthconditions of negative and affirmative necessity-propositions. As we have seen, the negative necessity-proposition applies to all that falls under the subject, or that can do so. By contrast, the affirmative applies only to all that actually falls under the subject: The proposition “Every man is of necessity an animal” doesn’t say that whatever can be a man is an animal, but that whatever is a man is an animal. And so, when such a proposition is put as Major with a contingency-Minor, an actual affirmation about those things that are contingently taken under the Middle doesn’t follow. And so, neither an affirmative necessity-proposition nor an assertoric follows—because in both, the predicate is actually affirmed of the subject.140

Our analysis is in agreement with this result. If it’s necessary that whatever is b is a (and thus it’s necessary that whatever can be b can be a), and it’s natural or possible that whatever can be c can be b, then it’s natural or possible that whatever can be c can be a. Now, in the case where it’s natural that whatever can be c can be a, we have an unrestricted assertoric conclusion; but in the case where this is merely possible, we do not. So, an unrestricted conclusion doesn’t follow. (Notice that on the alternative analysis of natural contingencies suggested a moment ago, Barbara LQX is valid: if it’s necessary that whatever can be b can be a, and it’s possible that whatever can be c can be b, then it’s possible that whatever can be c can be a. So that alternative analysis, while it accommodates Aristotle’s proof of Celarent LQX and his rejection of Celarent LQL, does not accommodate his rejection of Barbara LQX.) A16, 35b26–28. Kilwardby ad A16 dub.10 (27vb). 140 Kilwardby ad A16 dub.7 (27va): “Haec enim propositio: ‘Omnis homo de necessitate est animal’ non dicit quod quicquid potest esse homo est animal, sed quicquid est homo est animal. Et ideo cum talis propositio posita est pro maiori et minori {minori AF 1O3P1: minor EdE 1O2} de contingenti, non sequitur actualis affirmatio de his quae contingenter sumuntur sub medio. Et ideo nec sequitur affirmatio de necessario nec de inesse, quia in utraque actualiter affirmativa praedicatur de subiecto.” 138 139

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Kilwardby asks whether Barbara LQX can be proved by Upgrading as in Fig. 6.26. Bocardo

aco bca abo

K → o||La

Laba bca

Upgrading

aca

→

Laba Qbca aca

Fig. 6.26. Apparent Upgrading proof of Barbara LQX

He rejects the process, in his usual way: The Minor is not to be put as an assertoric, because the opposite of the conclusion is not incompatible with the contingency-Minor but it is incompatible with it when it is put as an assertoric. And so if the Minor is put as an assertoric, things are put as inconsistent that were not previously inconsistent, and in this way an impossibility follows.141

On our analysis, Barbara LQX cannot be proved by Upgrading because such a proof would require that we use the principle “If the premises of a valid inference are respectively necessary and possible then the conclusion is either necessary or natural”—which is unsound.

LQ-2 moods A19 deals with premise-pairs combining a necessity- and a contingencyproposition in the second Figure. Aristotle’s reductions Kilwardby states that all the syllogisms in this mixture derive from imperfect LQ-1 syllogisms, and that this derivation proceeds by Leconversion.142 He has in mind those syllogisms that are perfected by Direct Reduction. The styles of Direct Reduction employed by Aristotle are shown in Fig. 6.27.143 141 Kilwardby ad A16 dub.8 (27va): “… non ponenda est minor de inesse, quia oppositum conclusionis non est incompossibile minori de contingenti, sed est incompossibile eidem positae inesse. Et ita si ponatur minor inesse, iam fiunt repugnantia quae prius non erat, et ideo sequitur impossibile.” 142 Kilwardby ad A19 dub.1 (30va). See Thom, The Logic of Essentialism, 71–73. 143 A19, 38a16–18, 21–26; 38b6–13, 25–27, 31–35. See Thom, The Logic of Essentialism, 59–61, 71–73.

contingency-syllogisms

Le --- conv

Lace

Labe Lbae

223

Qac

M(X )bc

LQM(X ) --- 1

LQM(X ) --- 1

Qab

(M ---) conv

Lcae

Le --- conv

M(X )cb M(X )bc

Fig. 6.27. Direct reduction of LQM-2 and LQX-2 moods

Kilwardby asks why no LQ-2 syllogisms derive from the perfect LQ-1 syllogisms, or from those that have an affirmative necessity-major. He replies: And it is to be said to the first that a syllogism in this Figure derives from a syllogism in the first Figure by simple conversion of the Major; but in the perfect moods of this mixture (namely of the necessary and the contingent), the Major isn’t convertible, since it is a contingencyproposition. To the second it is to be said that all syllogisms in this Figure are negative and derive from the first Figure by simple conversion of the Major. But in the imperfect moods of this mixture in the first Figure where the Major is a necessity-proposition, the Major doesn’t convert simply and the syllogism isn’t negative.144

He is right when he says that in the perfect LQ-1 syllogisms, for example Celarent QLQ , the Major is not convertible, because universal contingency-propositions are not convertible. In the first Figure moods that have an affirmative necessity-Major, such as Barbara QLQ or Barbara LQM, the Major premise is either not convertible at all, or is not convertible in such a way as to generate a syllogism in the second Figure. These facts adequately explain why no LQ-2 syllogisms reduce directly to perfect LQ-1 syllogisms. However, they do not touch in direct reductions; and Kilwardby (as we shall see) is aware that there are indirect reductions to the first Figure. 144 Kilwardby ad A19, dub.2 (30va): “Et dicendum ad primum quod syllogismi huius figurae descendunt a syllogismis primae figurae per conversionem maioris simpliciter, sed in modis perfectis {perfectis ABrCm1E1E2F1F2KlO3P1V: imperfectis EdO2} huius mixtionis, scilicet necesarii et contingentis, non est maior ita convertibilis, cum sit de contingenti. Ad secundum dicendum quod omnes syllogismi huius figurae negativi sunt et per conversionem maioris simpliciter a syllogismis primae descendunt, sed in modis imperfectis huius mixtionis in prima figura maiore existente de necessario neque convertitur maior simpliciter neque fit syllogismus negativus.”

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Principles Aristotle states that if the necessity-premise in an LQ-2 mix is a universal negative then the premise-pair will be useful.145 It is noteworthy that this is a statement of a sufficient condition for an LQ-2 premise-pair to be useful—not a necessary and sufficient condition. The Dialectica Monacensis also states the presence of a universal negative necessitypremise as a sufficient condition: “For, when the necessity [-premise] is a universal negative there is always a syllogism, no matter what the quantity and quality of the other [premise]”.146 Thus we have the principle: (P17) In a second Figure necessity / contingency syllogism, if the neces-

sity-premise is a universal negative, the premises are useful. Kilwardby takes the presence of a universal negative necessity-premise to be both sufficient and necessary where the inference is perfected by Direct Reduction.147 Useful premise-pairs Kilwardby knows that some LQ-2 inferences are perfected by Indirect Reduction, but he takes no cognizance of this fact in his calculation of the useful premise-pairs. The reason for this omission is that he is trapped by his own methodology. In every mixture of premises his approach is to take Aristotle’s principle for that mixture as the starting point from which he derives the list of useful premise-pairs. But in the case of LQ-2 premises, Aristotle’s principle relates only to inferences that are perfectible by Direct Reduction. So Kilwardby calculates the premise-pairs on that basis, even though he knows (and will later aver) that the list is incomplete because it doesn’t consider Indirect Reductions.

A19, 38b38–40. Dialectica Monacensis, 503:28–29: “Quando enim universalis negativa sumitur de necessario, semper fit syllogismus, qualiscumque vel quantumcumque fuerit reliqua.” 147 Kilwardby ad A22 Note (33va) states (P17) as necessary and sufficient for those moods that are perfected by conversion: “And I am talking here about the moods that are perfected by conversion” [Et dico hic de modis qui secundum conversionem perficiuntur]. 145 146

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In accordance then with (P5) and (P17)—treated as if it were a necessary and sufficient condition148—he says there are 6 useful premisepairs, just as in the preceding XQ mixture.149 These are shown in Table 6.13. LeQa LeQe

QaLe

QeLe

LeQi LeQo Table 6.13. Useful LQ-2 premise-pairs

Syllogisms The 6 fruitful premise-pairs generate the syllogisms in Table 6.14. LQX

LQM

Cesare150

(Cesare)151

LeQeXe-2152

(LeQeMe-2)153

148

QLX

QLM

Camestres154

(Camestres)155

QeLeXe-2156

(QeLeMe-2)157

Kilwardby ad A19 dub.1 (30va). Kilwardby ad A19 dub.14 (31va). Ed has 6 useful pairs (2 universal and 2 particular), and doesn’t give the number of useless ones. Albertus Magnus, Priorum Analyticorum, I.iv.24 (376A) has 6 useful pairs (4 universal and 2 particular) with 6 [sic] useless. BrCm1E1E2F1F2KlO2O3P1 have the right numbers, namely 6, 4 and 2. 150 A19, 38a16–18. 151 A19, 38a16–18. 152 A19, 38b6–13. 153 A19, 38b6–13. 154 A19, 38a21–26. 155 A19, 38a21–26. 156 A19, 38b6–13. 157 A19, 38b6–13. 149

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LQM

QLX

Festino158

(Festino)159

LeQoXo-2160

(LeQoMo-2)161

QLM

Table 6.14. LQX-2, LQM-2, QLX-2 and QLM-2 syllogisms

Semantics Because the LQX-2 and QLX-2 moods reduce to LQX-1 moods as shown in Fig. 6.27, Kilwardby takes the contingency-premises to state natural contingencies and the conclusions to be unrestricted assertorics.162 Camestres LQX, Baroco LQX, Cesare QLX, Festino QLX Aristotle rejects Camestres LQX, Baroco LQX, Cesare QLX and Festino QLX.163 Accordingly, LaQe-2, LaQo-2, QeLa-2, QeLi-2 are not included among the useful pairs in this mixture. It might seem that Camestres LQX reduces indirectly to Darii LXL, as shown in Fig. 6.28: Darii LXL

Laba bci Laci

K → Li||Qe

Laba Qace bce

Fig. 6.28. Purported indirect reduction of Camestres LQX

Similar reasoning (modelled on the Aristotelian proofs of the LQX-1 moods) could be applied to Baroco LQX, and to Cesare and Festino QLX.164 But Kilwardby states that these proofs don’t work, because in A19, 38b25–27. A19, 38b25–27. 160 A19, 38b31–35. 161 A19, 38b31–35. In Thom, Medieval Modal Systems, Table 6.4, Baroco QLX is attributed to Kilwardby. This is a mistake. Since Baroco QLX is valid on the semantics proposed in that work, that semantics should not be attributed to Kilwardby. 162 Kilwardby ad A19 dub.3 (30va–vb). 163 A19, 38b13–17, 27–29; 38a38–b5. 164 Kilwardby ad A19, dub.10 (31ra). For these reductions see Patterson, 196; Thom, 158 159

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the enabling syllogism the Minor needs to be an unrestricted assertoric.165 This is a good reason because if the Minor in the enabling syllogism is an unrestricted assertoric the conclusion of the enabled syllogism is the contradictory of an unrestricted assertoric, and therefore (as we saw in Chapter One) that conclusion is not itself an unrestricted assertoric—whereas Kilwardby wants the conclusion to be an unrestricted assertoric. He spends some time discussing Aristotle’s counter-example to Camestres LQX, shown in Fig. 6.29.166 L(moves)(waking)a Q(moves)(animal)e (waking)(animal)e Fig. 6.29. Aristotle’s counter-example to Camestres LQX

He starts by querying the underlying principle stated by Aristotle, namely that nothing prevents a belonging contingently to all b and necessarily to all c when c is under b. He distinguishes two ways in which one thing can be contained under another—either as an essential part which adds a Differentia to an essential whole, or as an accidental part which adds an Accident. He explains: If therefore c is contained under b as an essential part, as “man” under “animal”, then what Aristotle says is false—as the objection argues. But if it is contained under it as an accidental [part], this is not false and unacceptable—as is clear from the terms that Aristotle lays down. For, “waking” is under “animal” accidentally, and so nothing prevents Motion inhering contingently in every animal (and none), and yet inhering of necessity in everything waking.167

The Logic of Essentialism, 61, 76, 129, 63. Thom, Medieval Modal Systems, 14 n. 88, 89 says these moods are provable “by Aristotelian methods” even though Aristotle explicitly rejects them; and in Table 1.8 (p. 14) and Figure 6.4 (p. 109) counts these moods as Aristotelian syllogisms. But this is to confuse perfectible inferences with syllogisms. Besides, given the above-mentioned reductions, the assertoric conclusion in these moods has to contradict the assertoric premise in LXL-1 moods; but according to Kilwardby, the assertoric premise in LXL-1 moods is equivalent to a necessity-proposition, and thus the conclusion of the present moods is equivalent to a possibility-proposition. So (as we shall see later in this Chapter), there is reason to take Camestres LQM and related moods as perfectible but not syllogistic; but there is no reason (on Kilwardby’s reading of modal syllogisms) to take Camestres LQX and related moods in that way. 165 Kilwardby ad A19 dub.10 (31rb). 166 A19, 38a38–b5. 167 Kilwardby ad A19 dub.4 (30vb): “Si ergo c continetur sub b sicut pars eius essen-

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Thus Kilwardby accepts the principle underlying Aristotle’s counterexample if it is understood as applying to accidental containments, but not if it is understood as applying to essential containments. He argues that the counter-example’s premises are true: But further perhaps there will be a doubt about the truth of the proposition “It’s necessary for everything waking to be moving” which he assumes. For the same reason “It’s necessary that every animal moves” seems to be true … And it is to be said that waking is a certain sensing, and sensing is a certain movement, and so waking is a certain movement.168 And so it’s evident that the proposition is true since the subject is contained essentially under the predicate—but “animal” is not contained in this way under it. … It may be asked by what motion the waking are moved. It can be said that when the waking are said to be moving of necessity, “motion” is taken in a general sense. But “Every animal moves of necessity” is not true because “animal” is not contained essentially under the predicate as “waking” is. Or, it can be said that it is taken for local motion, since the waking are of necessity moved by some local motion. For, in sensing, something is emitted or is received in that which moves locally, as is clear in the case of sight, where something is emitted (whatever it might be). This is clear in the case of the basilisk (which kills just by looking),169 tialis, ut homo sub animali, falsum esset quod diceret Aristoteles, sicut oppositum est. Si autem continetur sub eo sicut pars accidentalis, non est hoc falsum et inconveniens, sicut patet in terminis quos ponit Aristoteles. ‘Vigilans’ enim sub ‘animali’ est, sed accidentaliter, et ideo nihil prohibet motum contingere omni animali et nulli, et tamen ex necessitate inesse omni vigilanti.” 168 Albertus Magnus, Priorum Analyticorum, I.iv.23 (372B) presents the argument a little more fully: “For it is clear that there is Motion of necessity in the waking, because Waking in actuality is none other than a motion of heat and the sense from inside to outside.” [“Nam patet quod vigilanti ex necessitate inest motus, quia vigilia secundum actum non est aliud nisi motus caloris et sensuum ad exterius ab intimis.”] Unfortunately, Albert goes on to say that it follows that everything waking is necessarily an animal—thereby missing the point that Aristotle is presenting a counter-example, not a proof of validity. 169 Albertus Magnus, De Animalibus, XXV, 666B–667A: “The basilisk … kills by looking, and everything on which its sight falls dies. But I don’t think that what Pliny and some others say is true, namely that the basilisk doesn’t kill a man by a look unless it sees the man first and is seen in return by the man, but that the man’s look kills the basilisk if the man sees it first. For this is unreasonable. Nor do Avicenna and the Philosopher Semerion tell this. Nor is the reason why the look kills, as some say, that rays coming from its eyes corrupt those things on which they fall, because it is not a judgment of natural things that rays come out from the eyes. Rather, the cause of the corruption is the visual spirit [spiritus visivus] which diffuses far and wide on account of its subtle substance, and this corrupts and kills all things.”

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and the wolf (which renders a man hoarse by looking),170 and of woman (who at the menstrual time stains a mirror with a bloody cloud just by looking).171

His view seems to be that if we adopt the perspective of metaphysical realism, then it is indeed necessary that the waking are moving, because the metaphysical realities involved are the state of waking and the state of movement, and one of these is essentially contained in the other. By contrast, it’s not metaphysically necessary that animals are in movement, because movement is an accident (rather than part of the essence) of an animal. So the premises of Aristotle’s example are indeed [“Basiliscus … interficit visu; moritur enim omne super quod cadit visus eius. Quod autem dicit Plinius et alii quidam, basiliscum non occidere hominem nisi prius viderit hominem quam e converso videatur ab homine et quod visus hominis, si prius videat basiliscum, interficiat basiliscum, ego non puto esse verum, quia rationem non habet, nec Avicenna et Semerion Philosophi qui experta loquuntur hoc narrant, nec est causa quod visu interficit quam quidam dicunt, quod videlicet radii egredientes ab oculis eius corrumpant ea super quae cadunt, quia non est sententia naturalium quod radii ab oculis egrediantur, sed potius causa corruptionis est spiritus visivus qui longe valde diffunditur propter substantiae subtilitatem et hic corrumpit et necat omnia.”] Kilwardby doesn’t enter the debate between the rays and the spirits. His point is that, whatever the precise details, sight involves local motion of something. 170 Albertus Magnus, De Animalibus XXII.ii.1, 601B: “The wolf is a known animal, ferocious and cunning, of which they say that if it sees a man come before it it takes away his voice; but if the man comes before it, it loses its boldness, and if the man loosens his clothes he recovers his voice.” [“Lupus est animal notum, ferox et dolosum, de quo dicunt quod si hominem videns praevenerit intuitu oculorum, vocem aufert, et si praevenitur ab homine, audaciam amittit, et si homo solverit amictum, vocem recuperat.”] 171 Kilwardby ad A19 dub.5 (30vb). “Sed ulterius forte dubitabitur de veritate huius propositionis: ‘Necesse est omne vigilans esse movens’ quam assumit; eadem ratione videtur esse vera ‘De necessitate omne animal movet’. … Et dicendum quod vigilare est quoddam sentire, sentire autem est quidam motus, et ita vigilans est quoddam movens. Et ita manifestum est quod propositio est vera. Continetur enim subiectum sub praedicato essentialiter, sed sic non continetur sub eo ‘animal’. … Et si quaeratur quod dandum est illud movens quod movetur motu qui est vigilia, dici potest quod cum dicitur vigilans de necessitate movetur, accipitur ‘motus’ communiter. Non tamen haec est vera: ‘De necessitate omne animal movetur’ eo quod ‘animal’ non continetur essentialiter sub praedicato, sicut continetur ‘vigilans’. Vel potest dici quod accipitur pro motu locali. Vigilans enim aliquo modo est movens localiter ex necessitate, quia in sentiendo aliquid extra mittit vel intra suscipit in quo fit motus localiter, sicut patet de visu, ubi aliquid extra mittitur, quicquid sit illud, sicut patet de basilisco, qui interficit solo visu, et de lupo, qui reddit hominem raucum visu, et muliere, quae tempore menstrui speculum nube sanguinea solu visu inficit.” Albertus Magnus, De somno et vigilia Tractatus I c. 6 (87A): “For if women look into very clear mirrors while undergoing menstruation, a certain surface is generated, like bloody clouds.” [“In speculis enim valde puris si contingat mulieres inspicere menstruis supervenientibus generatur superficies quaedam velut nubes sanguinea.”]

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true according to Kilwardby. Those premises are not true merely asof-now: the necessity that the waking are moving is grounded in the ontological reality that Waking is a form of Motion, and in the absence of any such ontological reality that would ground a necessary connection between being an animal and waking it is contingent (eternally contingent) that animals are waking. By contrast, if the conclusion is understood as stating an unrestricted assertoric to the effect that (necessarily or naturally) nothing that can be an animal can be awake, then it is false. So on Kilwardby’s reading of the assertoric conclusion, Aristotle’s counter-example shows that Cesare QLX and related moods are invalid. Baroco QLX He deals in the same way with what he calls the “difficult puzzle” [difficilis dubitatio] of Baroco QLX, as shown in Fig. 6.30:172 This mood, which must be rejected if (P17) is a necessary condition for the utility of LQ-2 premises (but which is not explicitly rejected by Aristotle), seems to be provable from Barbara QXQ: Barbara QXQ

Qaba bca Qaca

K → Qa||Lo

Qaba Laco bco

Fig. 6.30. Purported indirect reduction of Baroco QLX

Given his reading of the assertoric conclusions in all these inferences as unrestricted, he is bound to regard them as invalid (and thus not reducible to perfect syllogisms), because the unrestricted assertoric conclusion doesn’t contradict the unrestricted assertoric premise in the enabling syllogism.

172 Kilwardby ad A19 dub.11 (31rb). This proof is given in Thom, The Logic of Essentialism, 63.

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Perfectible non-Aristotelian inferences Camestres and Baroco LQM, Cesare and Festino QLM Let us now turn to Camestres LQM, Baroco LQM, Cesare QLM and Festino QLM. These are the only standard second Figure LQM or QLM moods that Aristotle rejects. Like their counterparts with assertoric conclusions, they superficially satisfy the requirement of (P6), that there be a negative premise. In all four cases there appears to be a negative contingency-premise. But, given Aristotle’s doctrine that contingency-propositions are never genuinely negative, these moods actually conflict with (P6); and since the satisfaction of (P6) is necessary in order for an inference to be syllogistic, these inferences are not syllogistic. They may not be syllogistic; but Kilwardby takes them to be perfectible. Fig. 6.31 shows how they can be reduced to first Figure LLL or QLQ moods:173 Darii, Barbara LLL

Ferio, Celarent QLQ

Laba Lbci,a Laci,a Qabe Lbci,a Qaco,e

K → Li,a||Qe,o

K → Qo,e||La,i

Laba Qace,o Mbce,o Qabe Laca,i Mbce,o

Fig. 6.31. Indirect reduction of Camestres and Baroco LQM, and of Cesare and Festino QLM

(Alexander had proposed an analogous proof for LaQaMe-2, a mood that differs from Camestres LQM only in expressing the contingencyMinor explicitly as an affirmative.)174 Kilwardby deals with the issue as follows: To all these it can be said that a negative contingency [-conclusion] (in the sense of the possible) follows as has been shown according to the relationship and consequence of the terms, just as a universal conclusion is said to follow in the third Figure when the terms are convertible, 173 Kilwardby ad A19 dub.10 (31rb). For the reduction of Cesare QLM, see Thom, The Logic of Essentialism, 73. Thom, Medieval Modal Systems, Table 6.7 (p. 111) is misleading in counting all the QLM-2 and LQM-2 moods as Aristotelian. The truth (as Kilwardby recognizes) is that while all are perfectible, not all are Aristotelian syllogisms. 174 Alexander ad A19 (240,4–11).

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chapter six or an affirmative in the second Figure. For, this is only thanks to the terms and not thanks to the form; and such a conclusion doesn’t follow by virtue of the premises arranged thus in such premise-pairs in such a Figure—because both are in reality affirmative, which is incompatible with the Property of the second Figure. Aristotle, therefore, respecting the arrangement of the premises and the form, together with the Property of the Figure, posits such premise-pairs as useless. For, a negative never follows immediately from affirmatives thanks to the form. So, attend to the fact that by the terms that Aristotle posits in the counterexample he doesn’t rule out that a negative contingency [-conclusion] (in the sense of the possible) may follow. For, it follows, respecting the consequence of the terms; but he posits the premise-pairs to be useless because they cannot yield such a conclusion from the form of syllogizing or the Property of the Figure. And this is Aristotle’s approach everywhere in this mixture.175

He believes that a possibility-conclusion follows, as is shown by the proof in Fig. 6.31. At the same time, he appeals to (P6), the principle that in the second Figure one premise must be negative, and to the Aristotelian doctrine that all contingency-propositions are affirmative, in order to argue that Camestres LQM and the associated moods are unsyllogistic. So his position is that these moods are not syllogistic, even though they are valid and indeed (as Fig. 6.31 shows) perfectible. To illustrate the possibility that an argument may be valid but unsyllogistic, he alludes to the fact that a second Figure argument consisting solely of affirmative propositions may be valid because of a relationship among the terms, but cannot be syllogistic. This, however, is not a good analogy, since Camestres LQM (unlike affirmative second Figure arguments) is formally valid. Now, we saw earlier that a formally valid argument (such as a Barbara XLL with an unrestricted Major) may be 175 Kilwardby ad A19 dub.10 (31rb): “Ad omnia haec dici potest quod negativa de contingenti pro possibili sequitur secundum habitudinem et consequentiam terminorum, sicut iam ostensum est, sicut dicitur in convertibilibus sequi conclusionem universalem in tertia figura et affirmativa in secunda. Hoc enim non est nisi gratia terminorum, et non gratia formae. Non tamen sequitur talis conclusio virtute praemissarum sic dispositarum in dictis coniugationibus et sub tali figura ex quo utraque secundum rem affirmativa est, quod repugnat proprietati secundae figurae. Aristoteles igitur respiciens dispositionem praemissarum et formam cum proprietate figurae ponit tales coniugationes inutiles esse. Numquam enim gratia formae ex affirmativis immediate sequitur negativa. Unde attendendum est quod per terminos quos ponit Aristoteles ad instantiam non excludit quin sequatur negativa de contingenti pro possibili, quia respiciendo ad consequentiam terminorim sequitur. Ponit tamen coniugationes inutiles esse quia ex forma syllogizandi et proprietate figurae non possunt in talem conclusionem. Et sic considerat Aristoteles ubique in his mixtionibus.”

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unsyllogistic because it is not perfectible. But this is not the case with Camestres LQM, which is perfectible. What Kilwardby needs to argue here is that Camestres LQM and the related moods are valid and perfectible but are unsyllogistic because they do not obey (P6), which is a requirement on all second Figure syllogisms. If that were so, then all standard second Figure LQM and QLM moods would be valid and perfectible, and we would have a maximal system of such moods.176 While Kilwardby clearly recognizes that such a system is possible, he does not attribute it to Aristotle; nor does he think that Aristotle’s system should be revised along these lines. We will return to this subject in the Appendix. Baroco QLM Turning now to what he calls the “more difficult puzzle” [difficilior dubitatio] of Baroco QLM—which conflicts with (P17) understood as stating a necessary condition, but does not conflict with (P6)—Kilwardby considers an indirect reduction to Barbara QLQ:177 Barbara QLQ

Qaba Lbca Qaca

K → Qa||Lo

Qaba Laco Mbco

Fig. 6.32. Indirect reduction of Baroco QLM

His reply is as follows: And it is to be said that the said conclusion follows as was shown. But he [Aristotle] doesn’t allow that these are useful unless they have a universal negative necessity [-premise], as is clear from what he lays down to be observed. And in truth, in deciding about the particular premise-pairs he doesn’t exclude the fourth mood.

In this laconic passage, Kilwardby makes three points. First, Baroco QLM is indeed reducible to Barbara QLQ. Second, Aristotle’s rule (P17) excludes the premises of Baroco QLM as useless. Third, Aristotle doesn’t explicitly rule Baroco QLM out as invalid. This is an accurate statement of the facts. But the implication is unmistakable: that there Such a maximal system is put forward in Thom, Medieval Modal Systems, 110–111. Kilwardby ad A19 dub.12 (31rb): “Et dicendum est quod dicta conclusio sequitur sicut ostensum est. Sed ipse non reputat hic utiles nisi quae habent universalem negativam de necessario, sicut patet ex illa quam ponit observandam; et in veritate in agendo de coniugationibus particularibus non excludit quartum modum.” 176 177

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are perfectible (and hence valid) inferences from premises which (P17) rules out as useless.178 Given Kilwardby’s recognition that (P17) is good only for Direct Reductions, and that Baroco QLM is provable by Indirect Reduction, it’s clear that he thinks Aristotle’s system of Q / L / M moods should be corrected so as to include this mood. In the Appendix we will delve further into this matter.

LQ-3 moods In Aristotle’s text, A22 deals with the combination of necessity- and contingency-premises in the third Figure. Aristotle’s reductions Kilwardby notes four different ways in which these syllogisms may be perfected: In the third Figure, [1] in the moods which after reduction have a contingency-Major, the conclusion is a contingency-[proposition] in the sense determined above. [2] But in those which after reduction have an affirmative necessity-Major, contingency is concluded in the sense of the possible. [3] And in those which after reduction have a negative necessity-Major, the conclusion may be a possibility- or an assertoric [proposition]. [4] Besides these, there are also useful moods in the fifth mood [Bocardo] that do not reduce by conversion, which yield a contingency-[conclusion] in the sense of the possible, namely those that have a particular negative necessity-[premise].179

These four ways are displayed in Fig. 6.33.180 Lbc+ QLQ --- 1

Qac

Lcb

L --- conv

L --- conv

i

QLQ --- 1

Qab

Lac Lca

Q --- conv

Qbc

Qba Qab

[1] 178 In his Note to A22 (33va), he remarks that there are second Figure necessity /contingency syllogisms which Aristotle omits. 179 Kilwardby ad A22, Note (33va). 180 A22, 40a13–23, 25–35, 39–b1; 40b2–6. See Thom, The Logic of Essentialism, 62–63, 73–77.

contingency-syllogisms Qbc+ LQM --- 1

Lac

Qcb

+

Q --- conv

Q --- conv

+

LQM --- 1

Mab+

235

Qac Qca Lbc+

M --- conv

Mba Mab

[2] Qbc [3]

LQX --- 1

Lac---

X / M subord

Qcb

Q --- conv

[4]

ab---

Barbara LQM

Laba Qbca Maca

C →

Laco Qbca Mabo

Mab---

Fig. 6.33. Direct or indirect reduction of LQ-3 moods

Additionally, there is Bocardo QLM, which reduces indirectly to the first Figure in a fifth way, as shown in Fig. 6.34.181 Barbara LLL

Laba Lbca Laca

K → La||Qo

Qaco Lbca Mabo

Fig. 6.34. Indirect reduction of Bocardo QLM

In fact, all QLM-3 moods are indirectly reducible in this way, though only Bocardo QLM is thus proved by Aristotle.182 Bocardo QLM is seemingly overlooked by Kilwardby in this Note (though it is recognized by him in his comment on 40b2–3 and in his computation of the useful premise-pairs).183 Principles The Dialectica Monacensis doesn’t state a principle from which the useful premise-pairs in this mixture can be deduced. So, while Kilwardby may have drawn on that text when composing the principles governing contingency/ necessity mixtures in the first and second Figures, he could not do so for the third Figure. In addition to (P1)’s requirement that

181 182 183

Thom, The Logic of Essentialism, 63 and 77. A22, 40b2–3. Kilwardby ad A22, 40a39–b16 (33ra).

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there must be a universal premise, he lays down one special principle for determining the useful premise-pairs. (P18) In a third Figure necessity / contingency syllogism, the Minor must

not be a negative necessity-proposition.184 That this is correct can be seen by inspecting Fig. 6.33 and Fig. 6.34. Useful premise-pairs Given his two principles, Kilwardby says that of the 8 pairs of universal premises, 2 are useless (QaLe and QeLe)—they conflict with (P18)— and the remaining 6 yield a syllogistic conclusion. If the Major is universal and the Minor particular, 6 combinations yield a conclusion and 2 don’t, namely QaLo and QeLo—they too conflict with (P18). Similarly if the Major is particular and the Minor universal:185 here again 6 yield a syllogistic conclusion and 2 don’t, namely QiLe and QoLe—they too conflict with (P18). The remaining 8 combinations are excluded by (P1). Thus we have the 18 pairs listed in Table 6.15.186 LaQa

LeQa

LiQa

LoQa QaLa QeLa QiLa

LaQe

LeQe

LiQe

LoQe

LaQi

LeQi

QaLi

QoLa

QeLi

LaQo LeQo Table 6.15. Useful LQ-3 premise-pairs

Note that some of the useful combinations lack an affirmative premise: this is true of LeQe, LeQo, and LoQe. Kilwardby has considered this, and he has an explanation of this apparent violation of (P2). As we saw 184

Kilwardby ad A22 dub.2 (33rb). This sentence is missing from Ed (33rb). ABrCm1E1E2F1F2KlO3P1V have “Similiter maiore existente particulari et minore universali sex erunt utiles et duae inutiles.” 186 Ed (33rb) says that this gives a total of 12 useless combinations and 12 that yield a syllogistic conclusion. But this is to ignore the cases where the Major is particular and the Minor universal. The total number of combinations should be 32, not 24. And if we take into account the syllogisms with particular Major, 18 yield a syllogistic conclusion, and the remaining 14 are useless. ACm1E1E2F1F2KlO3P1V have the correct tally. 185

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in discussing his treatment of pure contingency-syllogisms in the second Figure, he holds that some contingency-propositions are negative without qualification [simpliciter negativa], and some are not.187 The negative contingency-propositions in these all-negative moods are negative only in a qualified sense, because an affirmative premise has been replaced by a negative, through a process of conversion by opposite qualities. Syllogisms The syllogisms are shown in Table 6.16. QLQ

QLM

LQX

LQM

Darapti188 QaLaQo-3 (Darapti)

Darapti189

QeLaQo-3

LaQeMi-3190

LQQ

Felapton191 QeLaQi-3 (Felapton) Felapton192 Felapton193 QaLaQi-3

LeQeMo-3 Disamis194

(Disamis)

Disamis195 LiQaQo-3

(LiQeMi-3) LiQeQi-3 LiQeQo-3 Datisi196

QaLiQo-3 QeLiQo-3

(Datisi)

Datisi197 LaQoMi-3

187 Kilwardby ad A17 dub.2 (28vb). Ed is garbled at this point, repeating the phrase “aliam simpliciter negativam”. Instead of “non sunt simpliciter negative” it has “si sunt simpliciter negative”. ABrCm1E1E2F1F2KlO3P1V do not have these errors. 188 A22, 40a18–23. 189 A22, 40a13–16. 190 A22, 40a33–35. 191 A22, 40a18–23. 192 A22, 40a25–32. 193 A22, 40a25–31. 194 A22, 40a39–b1. 195 A22, 40a39–b1. Thom, Medieval Modal Systems, 106 rejects Disamis LQQ on the basis of a semantics that doesn’t provide for the convertibility of particular contingencypropositions. In Chapter One of the present work, we have sketched a semantics that makes such propositions convertible. 196 A22, 40a39–b1. 197 A22, 40a39–b1.

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QLM

LQX

Bocardo198

LQM

LQQ

Bocardo199 LoQeMo-3

Ferison200

QeLiQi-3

(Ferison)

Ferison201

QaLiQi-3

Ferison202 LeQoMo-3

Table 6.16. QLQ-3, QLM-3, LQX-3, LQM-3 and LQQ-3 syllogisms

Summary The special principles laid down by Kilwardby for contingency-syllogisms are: (P12) In second Figure pure contingency-syllogisms, one of the premises

must be negative without qualification [simpliciter negativa]. (P13) In first Figure assertoric / contingency syllogisms, the Minor must (P14) (P15)

(P16) (P17) (P18)

not be a negative assertoric. In second Figure assertoric / contingency syllogisms, the assertoric must be a universal negative. In a negative third Figure assertoric / contingency syllogism (i.e. where the assertoric premise is negative), the assertoric must be universal and must be the Major premise. In a first Figure necessity / contingency syllogism, the Minor must not be a negative necessity-proposition. In a second Figure necessity / contingency syllogism, if the necessity-premise is a universal negative, the premises are useful. In a third Figure necessity / contingency syllogism, the Minor must not be a negative necessity-proposition.

These principles deliver just the set of useful premise-pairs identified by Aristotle. In most of these cases, the principles that Kilwardby articu198 A22, 40b2–3. Thom, Medieval Modal Systems, 106 has Bocardo QLQ (Disamis QLQ) valid (pace Kilwardby); but this is due to a mistaken analysis of contingencypropositions. 199 A22, 40b2–3. 200 A22,40b2–3. 201 A22, 40b3–6. 202 Not in Aristotle. See Thom, The Logic of Essentialism, 77.

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lates can be found in the earlier Dialectica Monacensis. The exceptions are QQ-3 (where the Dialectica Monacensis principle is too weak), XQ-3 (where the Dialectica Monacensis principle is too broad) and LQ-3 (where the Dialectica Monacensis doesn’t state a principle). So Kilwardby’s statement of the syllogistic principles is superior in that it is more comprehensive and more accurate. Moreover, Kilwardby makes use of the principles to deduce the useful premise-pairs, which the Dialectica Monacensis doesn’t do. Kilwardby’s interpretation of the contingency-moods, like that of the necessity-moods, succeeds in displaying a certain ordo and sufficientia in Aristotle’s text. At the same time, he is a fine enough logician and an honest enough reporter to acknowledge that there are discrepancies between the set of syllogisms generated by Aristotelian principles and the system of perfectible inferences. There are two such discrepancies among the contingency moods. One concerns Camestres LQM and related moods. These conflict with the general character of the second Figure, and when added to the Aristotelian moods they generate a maximal system. Kilwardby is aware of the possibility of this system, but he is not recommending that Aristotle’s system be expanded to include it. What is needed is a method of blocking the Indirect Reductions of these moods within Aristotle’s system. He doesn’t supply that method; we shall do so in the Appendix. The second discrepancy concerns Baroco QLM. This mood does not conflict with the general character of the second Figure, but is inconsistent with (P17) when understood as a necessary and sufficient condition on fruitful second Figure contingency / necessity premise-pairs. In this instance Kilwardby is recommending a correction of Aristotle’s system. In order to make his recommendation good, what is needed is a revision of (P17). Again, he doesn’t supply this revision; and we shall do so in the Appendix.

appendix KILWARDBY AND MODERN LOGIC

It is a sign of Kilwardby’s genius that many of his ideas call for development within the framework of modern logic. These ideas include his notion of a “natural” proposition, his use of the notion of appropriation in systematizing the mixed assertoric / necessity-syllogisms, his doubts about mixed contingency/ necessity moods in the second Figure, and a comment he makes on a pons asinorum for modal propositions.

Propositions expressing natural laws In Chapters One and Three we saw that some propositional forms used by Kilwardby require a notion of what is the case by virtue of natural laws. The natural, in this sense, is logically similar to the obligatory in being a deontic rather than an alethic modality: it does not imply actuality. But it does imply possibility: “ought” implies “can”, and this remains true when the “ought” is one of natural rather than moral law. We also saw that the natural is a modality which is preserved by conjunction: when “p” is natural and “q” is natural the conjunction of “p” and “q” is natural. A semantics that reflects these facts can be developed along the following lines. We assume one proposition—“n”—which comprehensively expresses The Law. We define compliant worlds as worlds in which “n” is true. We do not assume that all worlds are compliant; but we do assume that there is at least one compliant world. We define “Φp” as meaning that “p” is true in all compliant worlds; and we stipulate that “It’s natural that p” is true iff Φp. What is natural is then what is necessitated by The Law “n”. But, if “p” is necessitated by “n” and “q” is necessitated by “n” then “p and q” is necessitated by “n”. Therefore, when “p” is natural and “q” is natural “p and q” is natural. This semantics delivers two key rules that we need for the purpose of validating Kilwardby’s results.

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Rule A1.

p q r

→

Φp Φp Φr

Proof. Suppose that Φp. Then “n” necessitates “p”. Suppose that Φq. Then “n” necessitates “q”. But “p” and “q” necessitate “r”. So, “n” necessitates “r”. So Φr. p q Lp Φq Rule A2. → r Φr Proof. Suppose that Lp. Then “n” necessitates “p”. Suppose that Φq. Then “n” necessitates “q”. But “p” and “q” necessitate “r”. So, “n” necessitates “r”. So Φr. Given our definition of “Φp” as “‘n’ necessitates ‘p’”, it follows that “Lp” implies “Φp”. The reason is that “Lp” implies that “n” necessitates “p”. So statements of necessity are a special case of statements of naturalness. Certain statements of naturalness (namely, those of the form “Whatever can be b can be a”) imply statements of indeterminate contingency. Suppose that it’s natural that whatever can be b can be a. Then “n” necessitates “Whatever can be b can be a”. Thus, since “n” is possible, it’s possible that whatever can be b can be a.

An Aristotelian system of L / X / M inferences Kilwardby is unique among the major medieval logicians in devising a system (KilwardbyA) that exactly captures the Aristotelian L / X / M syllogisms, whose base comprises the LXL-1 moods, Cesare LXL, Festino LXL, Camestres XLL, Datisi LXL, Ferison LXL and Disamis XLL.1 Of the seven major medieval L / X / M systems2 only two include or are included in Aristotle’s system. Buridan’s main system does not contain Aristotle’s, because it lacks Barbara LXL; but Aristotle’s system does contain Buridan’s, since it contains LLX-3, Darii and Ferio LXL, Festino LXL, Datisi and Ferison LXL, LXX-3, and Celarent XLX. The Kilwardby / Campsall simpliciter system (KilwardbyS) contains Aristotle’s system, since it is maximal. But Aristotle’s system does not con-

1 2

Thom, Medieval Modal Systems, 12. Thom, Medieval Modal Systems, 194–195.

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tain it, since Aristotle’s system lacks Barbara XLL. We can therefore expand Figure 10.2 of Medieval Modal Systems as follows:

Fig. A1. The medieval L / X/ M systems

Kilwardby achieves this perfect match by first interpreting necessitypropositions as per se and assertorics as simpliciter, and then applying his doctrine of appropriation. From a modern viewpoint, the doctrine of appropriation enters a system at the level of the formation-rules. We can safely represent Kilwardby as using standard formation-rules for individual proposition-forms, and as defining well-formed formulae in a standard way as a sequence of premises and a conclusion, separated by a sign of inference.3 What his doctrine of appropriation does is to place restrictions on what pairs of proposition-forms may be admitted as sequences of premises. Thus, Kilwardby’s language has an unusual feature: not every pair of propositions counts as a premise-pair in a well-formed formula. And, while this is an unorthodox way of building up the well-formed formulae of a system, it is one that admits of an exact formulation. This feature of Kilwardby’s language flows from the fact that he wishes to exclude certain premise-pairs (such first Figure pairs in which an unrestricted Major is coupled with a necessity-Minor) from consideration. As we saw in Chapter Five, if such pairs were included in his language then he would have to recognize XLL-1 syllogisms as valid; and accordingly they are excluded by his rules of appropriation. Thus, XL-1 premise-pairs are not admitted, nor are premise-pairs in the other Figures which would by conversion reduce to XL-1 pairs. Seen in this way, Kilwardby’s appropriation system can be given a formal characterization, albeit an ad hoc one.

3

Thom, The Syllogism, 21–26.

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appendix

An Aristotelian system of L / Q / M inferences Kilwardby appraises inferences containing categorical propositions in two different ways. Sometimes he judges an inference as perfectible or not perfectible, and sometimes as syllogistic or not syllogistic. These two ways of appraising inferences have conceptually distinct bases. Judgments of perfectibility are based on the existence of a syntactic method of reducing the inference to a set of perfect syllogisms. Judgments about whether an inference is syllogistic are based on a number of principles which Kilwardby (following Aristotle) formulates for each significant combination of modalized or unmodalized premises. Kilwardby also believes that the class of perfectible inferences properly includes the class of (Aristotelian) syllogisms. He points out this relationship of inclusion in connection with the mixed contingency / necessity inferences. Here, the syllogisms are the standard QLQ moods in the first Figure plus Darapti, Felapton, Datisi and Ferison QLQ; Celarent, Ferio, Cesare and Festino LQX; Barbara, Darii, Darapti, Felapton, Datisi, Bocardo, Ferison LQM; Disamis LQQ; Camestres QLX; Disamis and Bocardo QLM. (I have omitted syllogisms with weakened conclusions and syllogisms obtained by complementary conversion.) Besides these (as we saw in Chapter Six), Kilwardby takes Camestres and Baroco LQM, Cesare and Festino QLM, and Baroco QLM, to be perfectible but not syllogistic. These last inferences are provable by Indirect Reduction, as shown in Fig. A2.

Darii, Barbara LLL

Ferio, Celarent QLQ

Laba Lbci,a Laci,a Qabe Lbci,a

Barbara QLQ

Qaco,e Qaba Lbca Qaca

K → Li,a||Qe,o

K → Qo,e||La,i

K → Qa||Lo

Laba Qace,o Mbce,o Qabe Laca,i Mbce,o Qaba Laco Mbco

Fig. A2. Perfectibility of non-Aristotelian QL-2 inferences

Thus, while the standard axioms and rules may be adequate for generating the perfectible syllogisms, they are too strong to generate just

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the Aristotelian syllogisms. A problem exists, then, of modifying the standard axioms and rules so that they do not generate these renegade inferences. Kilwardby doesn’t offer a solution to this problem; but a solution may be sketched along the following lines. All the proofs in Fig. A2 (apart from the proof of Baroco QLM) depend on applications of rule K in which a superficially negative contingency-proposition is taken as contrary to an affirmative necessityproposition. However (as we saw in Chapter One), Kilwardby takes all contingency-propositions, other than those which deny the contingency of some state of affairs, to be affirmative. Consequently, if we compare the left-hand and right-hand inferences in Fig. A2, we find that the former possess a feature that is lacked by the latter. If you consider the sequence comprising the premises followed by the contradictory of the conclusion (the antilogism sequence),4 you find that in the left-hand inferences it contains exactly one negative member.5 Let’s call this feature The Distribution of Quality. The left-hand inferences possess it because they derive an affirmative conclusion from affirmative premises. The right-hand inferences lack it because they derive a negative conclusion from affirmative premises, and therefore their antilogism sequence contains no negatives. Now, the perfect syllogisms all possess The Distribution of Quality. So, if we can construct a system based on the perfect syllogisms, but having rules that preserve the Distribution of Quality, then we have a system which does not contain as theses any inferences lacking The

Thom, The Syllogism, 181. Thom, The Syllogism, 181–182. As we saw earlier, Kilwardby holds that every categorical proposition is either affirmative or negative. It is also orthodox Aristotelian theory that the contradictory of an affirmative is a negative and vice versa. In the case of assertoric propositions, it’s clear how to apply this division: a- and i-propositions are affirmative, e- and o-propositions negative. These results presumably hold for unrestricted as well as for as-of-now assertorics. Notice, however, that apodeictic unrestricted assertoric propositions do not come in contradictory pairs, since they are equivalent to necessity-propositions. Instead, the contradictory of an Xa-proposition is an Xo; that of an Xe is an Xi; that of an Xi is an Xe; and that of an Xo is an Xa. Of these contradictories, Xa and Xi are affirmative; Xe and Xo are negative. Similar comments apply to unrestricted assertorics which are not specified as apodeictic or as natural. Among necessity-propositions, I take La and Li to be affirmative; Le and Lo negative. Their contradictories are respectively Mo and Me (negative); and Mi and Ma (affirmative). Kilwardby takes all Q - and Q N -propositions to be affirmative. I therefore take all Qand QN -propositions to be negative. 4 5

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Distribution of Quality. What we need to do is to modify rule K so that r and ¬r have to be of opposite quality. The following modification is sufficient: p q ¬r q provided that r and ¬r are of opposite quality.6 KK: → r p ~ We can now distinguish two Kilwardby systems of L / Q / M inferences. There is a wider system (using rule K ) which includes the Aristotelian syllogisms plus Baroco QLM along with Camestres and Baroco LQM, and Cesare and Festino QLM. This system includes the inferences that Kilwardby describes as holding “thanks to the terms”. Then there is a narrower system (using rule K K ) which includes the Aristotelian syllogisms plus Baroco QLM. Principles for Kilwardby’s two L / Q / M systems Aristotle puts up the principles (P1)-(P18) as generating the premisepairs yielding a syllogistic conclusion. But these principles are not necessary and sufficient for either of Kilwardby’s L / Q / M systems. What modifications need to be made to these principles, in order to construct a set of principles generating the premise-pairs of the two Kilwardby systems? For the wider system, we need to abandon (P6). For the narrower system, we need to retain (P6), while adding the following modified versions of (P17): (P17a) In a second Figure necessity / contingency syllogism with a ne-

cessity-Major, the Major must be a universal negative 6 On the analyses of necessity- and contingency-propositions proposed by Marko Malink, “A reconstruction of Aristotle’s modal syllogistic” History and Philosophy of Logic 27 (2006): 95–141, an La-proposition is compatible with the corresponding Qa-proposition. For example, the proposition “Everything that is awake is necessarily moving” is true on Malink’s analysis, being a non-substantial essential predication (p. 109). But the proposition “Everything that is awake is contingently moving” is also true on his analysis, because it is true of every term “z” that if every z is awake then (1) either “z” or “moving” is not a substance-term, (2) “Every z is moving” is not a substantial essential predication, (3) “Everything moving is z” is not a substantial essential predication, (4) If either “z” or “moving” is a substance-term then some z is moving. Consequently, Lapropositions are compatible with Qi-propositions, and Li-propositions are compatible with Qa- or Qi-propositions. Given these results, Malink is able to block the proofs of Camestres and Baroco LQM, Cesare and Festino QLM, simply by requiring that in indirect reductions r and ¬r be mutually incompatible, without explicitly invoking the notion of quality.

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(P17b) In a second Figure necessity / contingency syllogism with a con-

tingency-Major, the Major is a universal contingency-proposition and the Minor must be negative. Table 6.13 of Chapter 6 then needs expanding, as in Tables A1 and A2. LaQa

LeQa

QaLa

QeLa

LaQe

LeQe

QaLe

QeLe

LaQi

LeQi

QaLi

QeLi

LaQo

LeQo

QaLo

QeLo

Table A1. LQ-2 premise-pairs yielding a conclusion in Kilwardby’s wider system

LeQa LeQe

QaLe

QeLe

QaLo

QeLo

LeQi LeQo

Table A2. LQ-2 premise-pairs yielding a conclusion in Kilwardby’s narrower system

Because the wider system sanctions the unrestricted use of rule K, whereas the narrower system using only rule K K, the difference between the two systems can be represented as a difference concerning what may be allowed as pairs of contraries. The wider system allows the pairs {La, Qi}, {Li, Qa}, {Le, Qi} and {Lo, Qa}—and their equivalents by complementary conversion. The narrower system allows only {Le, Qi} and {Lo, Qa}.

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A modal pons asinorum A27–28 summarizes Aristotle’s advice on how to construct a proof of a categorical proposition.7 Kilwardby asks how this art is acquired. He assumes that it cannot be innate, since that would be an unAristotelian position;8 and his answer is that the art is acquired, not through a preceding art, but by way of sense-perception and experience [experimentum]. The latter is defined by Kilwardby as “a collation made according to reason from many thoughts and retained in memory”; it is what immediately precedes a universal.9 By sense-perception, or by understanding mediated by sense and experience, we perceive that Animal is contained in Sensible, and Sensible in Body; and we know at once that Body inheres universally in Animal by way of Sensible. In this way we discover the rule that Aristotle gives for finding a middle term to demonstrate a universal affirmative proposition. Similarly with the other rules he gives. And thus we acquire the art of finding middle terms.10 In A28 Aristotle shows how to construct syllogistic arguments with true premises leading to true conclusions in each of the four forms of categorical proposition. In each case he shows how a proposition (be it universal or particular, affirmative or negative) can be deduced syllogistically from a pair of universal premises. These premises are formed from the subject and predicate of the proposition to be proved, along with terms that either contain one of those terms, or are contained in one of those terms, or exclude one of those terms. Thus the universal affirmative “a belongs to all e” can be proved syllogistically provided that there are terms “c” and “f ” such that a belongs to all c, f belongs to all e, and c = f. The universal negative “a belongs to no e” can be proved syllogistically provided that there are terms “b” and “h” such that b belongs to all a, h belongs to no e, and b = h, or else a belongs to no d, f belongs to all e, and d = f. The particular affirmative “a belongs to some e” can be proved syllogistically provided that there are terms “b” and “g” such that b belongs to all a, e belongs to all g, and b = g; or else a belongs to all c, e belongs to all g, and c = g. Finally, the parThom, The Syllogism, 73–75. Kilwardby ad A27 dub.3 (38va): “… ars inventiva medii est aquisita, nec tamen per artem praecedentem sed per viam sensus et experimenti”. 9 Kilwardby ad A29–30 Part 2 dub.1 (42va): “Sed dicitur experimentum hic collatio facta secundum rationem multorum pensatorum et memoriter retentorum”. 10 Kilwardby ad A27 dub.3 (38va). 7 8

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ticular negative “a doesn’t belong to all e” can be proved syllogistically provided that there exist terms “d” and “g” such that a belongs to no d, e belongs to all g, and d = g. Besides these six combinations, the remaining ones do not yield a syllogistic conclusion in which “a” is predicated of “e”. If b belongs to all a and f belongs to all e, nothing follows syllogistically, because if “b” were the same as “f ” this would be a pair of affirmative premises in the second Figure. If d belongs to no a and h belongs to no e, nothing follows syllogistically, because this is a pair of negative premises. And if a belongs to all c and e belongs to no d, where c = d, we have a useless third-figure combination. (Kilwardby questions this last statement,11 saying that what follows is that some a is not e. This of course is true, but it is not inconsistent with what Aristotle says, since Aristotle sets up the problem with “a” as predicate, not subject, of the conclusion; and Kilwardby acknowledges this.) Kilwardby asks why Aristotle says that in looking for universal premises to construct a syllogistic proof, we should prefer the most general and remote consequences of the subject e, given that earlier12 he had said that we should choose the nearest and most proper consequences of the subject.13 His answer is that earlier Aristotle had been taking the consequents of the subject in an absolute sense, not in comparison with the predicate;14 but in the present passage he understands the consequents of the subject not to be also consequents of the predicate but to be antecedents of the predicate. From a logical point of view, this is a sound observation from Kilwardby. If “f ”’ stands for the consequents of “e”, and “c” for those terms of which “a” is a consequent, then there is no chance that an f will be a c if the cs can be consequents of “e”. Kilwardby notes that in the exposition of the syllogistic logic as part of the logic of Judgment, Aristotle had devoted separate treatments to modal and to assertoric syllogisms; and he poses the question why we do not also have separate treatments for the Invention of modal and Kilwardby ad A28 Part 2 dub.4 (40vb). This reference is to A27, 43b22–29. Ed lacks Kilwardby’s dubium on this point. It is present in ACm1E1E2F1F2KlO2O3P1V. 13 Kilwardby ad A28 Part 1, dub.5 (40ra). Throughout this chapter Aristotle uses the expressions “consequent” and “antecedent” to mean terms that are universally true of a given term, or terms of which the given term is universally true. 14 Kilwardby ad A28 Part 1, dub. 6 (40ra): “… absolute et non comparando {Ed incomparando: ACm1E1E2F1F2KlO2O3P1V non comparando}illud consequens ad praedicatum”. 11 12

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assertoric syllogisms.15 His answer is that all syllogisms are originally generated from what it is to be said of all or of none; but, since there are different principles of conversion for modal and non-modal propositions, and the perfection of syllogisms depends on conversion, it was necessary to give a different treatment for the Judgment of modal and non-modal syllogisms. However, the art of Invention is the same for both modal and assertoric syllogisms, since in both cases it consists in finding antecedents, consequents and incompatibles of the Major and Minor terms. What Kilwardby says here is true; however what he does not say is that the relations of antecedence, consequence and incompatibility may be modalized or unmodalized. For instance, there is one class of terms that are predicated of the subject-term, and there is another narrower class of terms that are necessarily predicated of the subject-term. As a consequence, the problem of constructing a pons asinorum for modal propositions cannot be reduced to that of constructing the pons for assertorics. Kilwardby does not address this problem. The solution requires that in addition to the combinations of unmodalized premises considered in the standard pons, we must we look at combinations of modalized premises. We need to consider not only the actual consequents, antecedents and repugnants of “a” and “e”, but also their necessary and contingent consequents, antecedents and repugnants. There are 8 × 8 = 64 possible combinations, as against the 9 in the assertoric pons, and these are displayed in Fig. A3.

15

Kilwardby ad A28 Part 1 dub.1 (39vb).

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Fig. A3. Generalized pons asinorum

An unbroken line indicates a syllogistic premise-pair in Aristotle’s syllogistic (including Kilwardby’s correction of Aristotle’s L / Q / M system); a non-uniform broken line indicates extra premise-pairs in Kilwardby’s maximal L / X / M system; a uniform broken line indicates

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extra premise-pairs in Kilwardby’s maximal L / Q / M system. The following combinations are fruitful in Kilwardby’s system (including his correction of Aristotle’s L / Q / M system). To prove

Use

7–2 5–2 2–5 Particular affirmative assertoric 2–7 7–7 Particular negative assertoric 5–7 Universal affirmative necessity 6–1 6–2 Universal negative necessity 4–1 4–2 1–4 2–4 Particular affirmative necessity 6–6 6–7 7–6 1–6 2–6 Particular negative necessity 4–6 4–7 Universal contingency 8–3 8–2 8–1 Particular contingency 8–8 8–7 8–6 3–8 2–8 1–8 Universal affirmative possibility 7–3 6–3 Universal negative possibility 5–3 3–5 4–3 3–4 Particular affirmative possibility 7–8 6–8 5–8 4–8 3–6 3–7

Universal affirmative assertoric Universal negative assertoric

Mood Barbara Celarent, Cesare Camestres, Celantes Baralipton Darapti Felapton Barbara LLL Barbara LXL Celarent LLL, Cesare LLL Celarent LXL, Cesare LXL Camestres LLL, Celantes LLL Camestres XLL, Celantes LXL Darapti LLL Darapti LXL Darapti XLL Baralipton LLL Baralipton XLL Felapton LLL Felapton LXL Barbara QQQ , Celarent QQQ Barbara QXQ , Celarent QXQ Barbara QLQ , Celarent QLQ Darapti QQQ Darapti QXQ Darapti QLQ Baralipton QQQ Baralipton XQQ Baralipton LQQ Barbara XQM Barbara LQM Celarent XQM, Cesare XQM Camestres QXM, Celantes XQM Celarent LQM, Cesare LQM Camestres QLM, Celantes LQM Darapti XQM Darapti LQM Felapton XQM Felapton LQM Baralipton QLM Baralipton QXM

Table A3. Proof patterns for modal and non-modal propositions (including Kilwardby’s corrected L /Q/M system)

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The following combinations are fruitful in Kilwardby’s maximal L / X / M system but not in Aristotle’s syllogistic. To prove

Use

Universal affirmative necessity 7–1 Universal negative necessity 1–5 Universal negative necessity 5–1 Particular negative necessity

5–6

Mood Barbara XLL Camestres LXL Celarent XLL Cesare XLL Felapton XLL

Table A4. Extra proof patterns in Kilwardby’s maximal L /X/M system

The following combinations are fruitful in Kilwardby’s maximal L / Q / M system but not in Aristotle’s syllogistic. To prove

Use

Mood

Universal negative possibility Universal negative possibility

1–3 3–1

Camestres LQM Cesare QLM

Table A5. Extra proof patterns in Kilwardby’s maximal L /Q/M system

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INDEX OF DUBIA

Prologue Dub. 1

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

Sed tunc statim dubitabit aliquis cum in omni scientia fiat inventio et iudicium secundum quod unaqueque scientia differt propter quod iste due partes magis logice quam alii scientie attribuuntur Et tunc ulterius dubitatur propter quod in doctrinis specialibus non denominatur ars inveniendi et iudicandi sicut in logica Sed tunc ulterius dubitatur si inconveniens est simul determinare scientiam et modum sciendi Sed tunc dubitatur si enim logica simul tradat scientiam et modum sciendi Sed adhuc restat dubitatio quia cum scientia specialis modum et inveniendi et iudicandi non habeat determinare sed scientia communis Sed super hoc dubitatur non videtur quod diversimode tradatur ars inveniendi et iudicandi Quibus habitis adhuc dubitaret forte aliquis cum 5 sint partes rhetorice scilicet inventio dispositio etc, quare sola inventio de numero illarum ponitus pars logice Super hoc quod suppositum est syllogismum simpliciter esse genus subiecti in hac doctrina duplex occurit dubitatio Habito quod scientia debeat esse de syllogismo simpliciter et possit consequens est utrum presens scientia scilicet libri priorum sit de ipso

2ra

Dubitat hic quare promittit prohemium posteriorum cum prohemio istius libri cum scientie diverse sint Adhuc dubitatur quid intendat per demonstrationem et disciplinam demonstrativam et quare magis dicit circa demonstrationem et demonstrativa disciplina quam econverso Consequenter forte dubitatur de secunda parte prohemii et primo de hoc quod dicit Primum et deinde

3rb–3va

2ra 2ra 2ra–rb 2rb 2rb–2va 2va–2vb 2vb–3ra 3ra–b

A1 Part 1 Dub. 1 Dub. 2

Dub. 3

3va

3va

262 Dub. 4 Dub. 5 Dub. 6

index of dubia Adhuc dubitatur ad quod determinandum est hic quid propositio et quid terminus Consequenter queritur ad quid diffinitur hic syllogismus Ulterius forte dubitabit aliquis quare non determinat consequenter quid conversio sicut determinat quid sit dici de omni et dici de nullo

3va–3vb 3vb 3vb

A1 Part 2 Dub. 1

Dubitatur forte hic et primo de diffinitione propositionis et primo de genere eius Dub. 2 Deinde dubitaret forte aliquis de differentiis eius Dub. 3 Adhuc forte dubitaret aliquis quare magis ponuntur he differentie affirmativum et negativum ad circumloquendum differentiam talem quam alie differentie dividentes propositionem que sub disiunctione accepte cum propositione convertuntur Dub. 4 Sed tunc dubitatur quia maior diversitas est syllogismi penes has differentias propositionis universale particulare Dub. 5 Deinde queritur quare aliter diffiniatur hic propositio quam in topicis et quam enunciatio in libro periarmenio Dub. 6 Consequenter forte dubitatur de divisione propositionis Dub. 7 Adhuc forte dubitabit aliquis cum multiformis sit divisio propositionis Dub. 8 Adhuc dubitaret forte aliquis cum indefinita {indefinita CrE 2F1F2KlO2O3Ed: infinita ACm1E1P1V} et particularis propositio convertantur Dub. 9 Adhuc dubitatur cum indefinita {indefinita E 1E2F1F2KlO3P1VEd: infinita A} dicatur per privationem tam universalis quam particularis ut dicit in littera quare magis convertitur cum particulari quam cum universali Dub. 10 Adhuc forte dubitabit aliquis cum ars imitetur naturam ut dicit secundo physicorum in natura autem nihil est infinitum {infinitum ACm1E1E2F1KlO2P1VEd: indefinitum F 2O3} ut accipitur in tertio physicorum quomodo est in arte aliquid infinitum {infinitum ACm1E1E2F1F2O2O3P1VEd: indefinitum Kl} sicut propositio indefinita Dub. 11 Consequenter dubitatur de verificatione diffinitionis dicit enim quod demonstrativa propositio sumit et non interrogat dialectica autem econverso querit huius causam

4ra 4ra 4ra

4ra–4rb 4rb 4rb 4rb 4rb 4rb

4rb

4rb–4va

index of dubia Dub. 12 Adhuc dubitatur de minore propositione argumenti quod dicit quod omnis propositio syllogistica simpliciter est dialectica vel demonstrativa

263 4va

A1 Part 3 Dub. 1 Dub. 2

Dub. 3 Dub. 4 Dub. 5

Sed queritur quare magis dicit terminum voco quam terminus est Adhuc dubitatur cum similiter resolvatur syllogismus in propositionem sicut propositio in terminum quare magis diffiniatur terminus per hoc quod in ipsum resolvitur propositio quam propositio per hoc quod in ipsam resolvatur syllogismus Adhuc forte dubitabit aliquis quare diffinitur terminus per propositionem Deinde dubitatur de divisione termini scilicet quare potius dividit ipsum per subiectum et predicatum quam per nomen et verbum Adhuc dubitatur de divisione termini

4va

Sed dubitatur quare in diffinitione eius ponitur oratio que est genus non proprium et non ratiocinatio et argumentatio que est genus proprium Adhuc forte dubitatur quare non dicit quod syllogismus est orationes sed dicit oratio Adhuc dubitatur quod intendat per hanc particulam Positis etc Queritur quare non ponit hanc differentiam concessis sicut Boethius ponit Adhuc dubitabit aliquis qualiter conclusio est alius a premissis

4vb–5ra

Dubitatur hic forte cum sit syllogismus universalis et particularis affirmativus et negativus sicut perfectus et imperfectus quare tantum determinat quid est syllogismus perfectus et imperfectus Adhuc dubitatur cum perfectum et imperfectum sint passiones syllogismi quare determinat quid sint Adhuc dubitatur si syllogismus imperfectus aliquid indiget ut dicitur in sua diffinitione

5ra

4va

4va 4va 4va–4vb

A1 Part 4 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5

5ra 5ra 5ra 5ra

A1 Part 5 Dub. 1

Dub. 2 Dub. 3

5rb 5rb

264

index of dubia

A1 Part 6 Dub. 1 Dub. 2 Dub. 3

Potest ergo queri cuius conditio sit dici de omni scilicet utrum subiecti aut predicati Adhuc dubitatur quare magis dicitur in eius diffinitione quando nihil est sumere subiecti quam sub subiecto et que est diversitas inter hos sermones Adhuc dubitatur videtur enim quod ista diffinitio conveniat propositioni indefinite

5rb

Dubitaret forte aliquis de prima divisione propositionis quam ponit Adhuc dubitatur illa enim que ex necessitate insunt insunt, {insunt insunt CrE 1F1KlO3P1: insunt, ACm1E2F2O2VEd} et tam illa quam ex necessitate sunt quam illa que simpliciter insunt contingit inesse Adhuc dubitatur quare tantum istas tres divisiones ponit scilicet de inesse de modo affirmativum negativum universale particulare indefinitum Adhuc dubitatur cum doceat convertere propositiones iam divisas quare omittit indefinitam Sed tunc dubitatur si eadem est conversio particularis et indefinite quare magis docet illa in propositione particulari quam econtra Adhuc dubitari potest quare non ponit nec ostendit conversionem propositionis singularis Consequenter dubitatur de conversione universalis negative de inesse Ulterius forte dubitaret aliquis de modo probandi conversionem eius Consequenter forte dubitatur de conversione universalis affirmative de inesse Consequenter dubitatur de conversione particularis affirmative Consequenter dubitabit aliquis de particulari negativa Consequenter dubitari potest de conversione in communi Adhuc dubitatur quare non determinat conversionem per contrapositionem sicut facit Boetius Adhuc forte dubitatur de conversione qua via artis est

5vb

5rb 5rb

A2 Dub. 1 Dub. 2

Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9 Dub. 10 Dub. 11 Dub. 12 Dub. 13 Dub. 14

5vb

5vb 5vb–6ra 6ra 6ra 6ra 6ra–b 6rb 6rb 6rb 6rb 6rb 6rb–vb

index of dubia

265

A3 Part 1 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Hic forte dubitatur primo de hoc quod dicit propositiones de necessario similiter converti sicut propositiones de inesse Adhuc dubitatur de ostensione conversionis universalis negative de necessario Adhuc dubitatur aliquis de ostensione conversionis affirmativarum propositionum Ulterius forte dubitaret aliquis videtur enim esse instantia contra conversionem universalis affirmative de necessario

7ra 7ra 7ra 7ra–b

A3 Part 2 Dub. 1

Dubitaret forte aliquis primo circa distinctionem contingentis Dub. 2 Adhuc dubitatur quomodo necessarium dicitur contingere Dub. 3 Adhuc forte dubitaret aliquis nam cum non necessarium dicatur contingere impossibile autem est non necessarium ergo impossibile dicitur contingere Dub. 4 Consequenter forte dubitaret aliquis de conversione affirmativarum de contingenti Dub. 5 Queritur igitur quare Aristoteles separat conversionem negativarum de contingenti pro non necessario a conversione aliarum negativarum Dub. 6 Sed tunc forte dubitabit ulterius aliquis si contingens natum in affirmativis non convertitur in contingens natum sed in contingens commune Dub. 7 Sed tunc dubitatur quare Aristoteles non distinguit hic natum ab infinito sed confuse utitur eis Dub. 8 Consequenter forte dubitatur de conversione negativarum de contingenti et primo de hoc quod dicitur contingens non esse Dub. 9 Sed tunc forte dubitatur cum contingens non esse sit commune ad necesse secundum non esse ad possibile quod potest esse et non esse et contingens non esse in communi convertitur in terminis sicut dictum est Dub. 10 Sed tunc dubitabit forte aliquis cum superius inveniatur in quilibet inferiori et ubicumque invenitur ipsum superius etiam invenitur propria passio eius Dub. 11 Sed adhuc forte dubitaret aliquis videtur enim quod exemplum quod ponit de contingenti quod dicitur contingere eo quod ex necessitate inest pretendat quod contingens ad utrumlibet negativum convertatur

7vb–8ra 8ra 8ra 8ra–b 8rb–va 8va 8va 8va–b 8vb

8vb 8vb–9ra

266

index of dubia

Dub. 12 Consequenter dubitatur de eo quod dicit de contingenti nato et infinito Dub. 13 Adhuc dubitatur de hoc quod dicit particularem negativam de tali contingenti converti in terminis

9ra 9ra–b

A4 Part 1 Dub. 1

Sed hic posset forte aliquis dubitare de ordine

9va

A4 Part 2 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5

Dub. 6 Dub. 7 Dub. 8 Dub. 9 Dub. 10 Dub. 11 Dub. 12 Dub. 13 Dub. 14

Dubitaret forte aliquis primo de figura in communi Adhuc forte dubitabitur de multitudine figurarum quomodo tres sint Consequenter forte dubitatur de syllogismo ex quibus componatur Sed tunc dubitatur quomodo hoc sit cum unius rei una sit forma et non multe Ulterius forte dubitatur aliquis de quibusdam dictis in littera et primo de hoc quod dicit in syllogismis universalibus prime figure medium esse in toto primo et ultimum in toto medio Adhuc forte dubitatur quare dicit syllogismum talem esse perfectum ex exremitatibus Adhuc forte dubitatur de primo videtur enim quod inutilis coniugatio Adhuc forte dubitaret ulterius aliquis de hoc quod dicit inutilem esse coniugationem si minor vel utraque sit negativa Consequenter forte dubitatur de sufficientia coniugationum utilium prime figure Sed tunc ulterius forte dubitatur de principiis positis ad istam sufficientiam faciendam que sit causa duorum principiorum communium Consequenter queret aliquis que sit causa propriorum principiorum scilicet quare maior in prima figure sit universalis et minor affirmativa Adhuc dubitabit forte aliquis quare conclusio in hac figura semper assimilatur maiori in qualitate et minori in quantitate Adhuc solet queri quare Boetius ponit novem modos Adhuc solet queri ordo in modos huius figure

10rb–va 10va 10va 10va

10va 10va–b 10vb 10vb 10vb 11ra 11ra 11ra 11ra

index of dubia

267

A5 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9 Dub. 10 Dub. 11 Dub. 12 Dub. 13 Dub. 14

Hic forte dubitaret aliquis de hoc quod dicit secundam figuram esse quando medium de utroque extremo dicitur omni vel nullo Consequenter dubitatur de his que consequenter dicit in littera medium enim est id per quid unum extremum distat ab altero Consequenter forte dubitabitur de eo quod dicit syllogismum huius figure esse imperfectum Consequenter forte dubitabitur super hoc quod probat quartum modum per impossibile Sed tunc ulterius dubitabitur qualis sit necessitas huius ostensionis si unus oppositorum repugnat premissis ergo relique oppositorum sequitur Ulterius forte dubitaret aliquis qualiter sit verum quod dicit primum postremo omni et nulli inesse et tamen non invenire terminos omni inesse Sed tunc dubitabitur videtur enim quod contingit sumere terminos omni inesse quocumque se habeat particularis Adhuc consequenter dubitabit aliquis de sufficientia modorum huius figure Sed tunc queritur causa huius principii quod maior est hic universalis Consequenter queritur causa alterius principii scilicet quod altera dicitur esse negativa Sed tunc forte queritur si altera debet hic semper esse negativa quare differenter maior vel minor et non altera determinate Solet etiam hic queri quare hic tantum sequitur conclusio negativa Amplius autem queret forte aliquis quare in hac figura conclusio semper assimiletur minori propositioni in quantitate Adhuc multiplex solet esse oppositio de ordinatione modorum huius figure

12ra

Sed hic forte dubitaret aliquis de expositione {expositione ACm1CrE1E2F1F2KlO2O3P1V: propositione Ed} per quam perficitur quasdam syllogismos huius figure quid necessario sequitur ad propositiones universales quod sequitur ad singulares quando fit contractio subiecti in propositionibus universalibus ad hoc aliquid et singulare

13vb

12ra–b 12rb 12rb 12rb 12rb 12rb–va 12va 12va 12va 12va 12va 12va 12va–b

A6 Dub. 1

268

index of dubia

Dub. 2

Consequenter dubitaret de sufficientia utilium coniugationibus huius figure Dub. 3 Sed tunc dubitat ulterius aliquis quare oportet hic minorem esse affirmativam Dub. 4 Sed tunc ulterius dubitabit quomodo in tertia figura minor sit universalis Dub. 5 Consequenter queritur quare in hac figura concluditur tantum particularis Dub. 6 Consequenter forte dubitabitur de ista figura in comparatione ad alias et primo queritur quare in hac figura maior possit esse universalis et particularis Dub. 7 Adhuc forte aliquis dubitabit cum habeamus unam figuram quod concludit tantum particularem quare non habeamus aliam figuram concludentem tantum universalem Dub. 8 Adhuc dubitare posset aliquis iuxta predictam cum habeamus figuram concludentem solam negativam quare non habeamus aliam concludentem solam affirmativam Dub. 9 Adhuc dubitari solet de ordine modorum huius figure Dub. 10 Sed tunc forte dubitatur quomodo dicatur hic esse modi universales Dub. 11 Adhuc forte dubitatur aliquis quare in ista figura descendunt modi particulares ab universalibus ita quod tam a maiore quam a minore Dub. 12 Solet autem adhuc multipliciter opponi de ordine trium figurarum

13vb 13vb 13vb 13vb 14ra 14ra

14ra

14ra 14ra 14ra 14ra

A7 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Dub. 5 Dub. 6

Dubitaret forte aliquis cum det duas coniugationes indirectas quas ponit Boetius quare non ponit alias tres Adhuc forte dubitaret utrum isti duo modi indirecti indifferenter possunt esse in secunda et tertia figura sicut in prima Sed tunc queritur utrum eodem modo perficiantur in omni figura vel diversimode in diverse Deinde forte dubitabit utrum isti duo modi in omni figura maneant indirecti, illi duo scilicet quorum alter habet maiorem universalem affirmativam et alter minorem negativam Adhuc dubitabitur quare isti duo modi indirecti tantum possunt in conclusionem particularem Adhuc forte queret quare non sit modus indirectus quando maior est negativa et minor affirmativa sicut econtra

14va–b 14vb 14vb 14vb

14vb 14vb

index of dubia Dub. 7

Consequenter queret quare modi universales secunde figure non possunt reduci in universales prime per impossibile sicut possunt particulares secunde Dub. 8 Ulterius forte dubitabit aliquis qualiter particulares prime perficiuntur per universales secunde Dub. 9 Sed tunc etiam dubitabit aliquis quomodo particulares prime perficiuntur per universales eiusdem Dub. 10 Consequenter forte dubitabit de eo quod dicit particulares tertie reduci in universales prime

269 14vb 14vb 14vb 15ra

A8 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9

Hic forte dubitaret aliquis de pertinentia huius partis Consequenter contingit dubitare de sufficientia Adhuc dubitari potest de sufficientia queritur enim quare non considerat propositionem de impossibile Consequenter forte dubitaret aliquis de dictis in littera et primo de eo quod primo ostendit diversitas enim syllogismi non constitit nisi secundum figuram et modum Consequenter dubitabit de eo quod proximo ostenditur videtur enim repugnatur ei quod primo ostendit Consequenter queritur quare dicit quartum secunde et quintum tertie non reduci similiter sicut in illis de inesse Quare queritur utrum sic sit vel non Sed ulterius forte dubitaret aliquis de expositione quam ponit Aristoteles Adhuc cum fiat expositio subiecti particularis negative in dictis modis queretur quomodo fieri habeat expositio ad hoc quod sit in eadem figura

15rb 15rb–va 15va

Sed hic dubitatur primo de ipsa mixtione Adhuc dubitatur quia videtur hoc superfluere mixtio Consequenter dubitatur utrum maiore existente de inesse et minore de necessario sequatur conclusio saltem de inesse Sed tunc queritur quare Aristoteles talem mixtionem non determinavit Sed tunc quia ex iam dictis patet quod maiore existente de inesse et minore de necessario sequitur conclusio de inesse tantum econtra autem se habentibus sequitur conclusio de necessario et de inesse queritur causa huius

16va 16va 16va

15va

15va 15va 15va–16ra 16ra 16ra

A9 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5

16va–b 16vb

270 Dub. 6 Dub. 7

index of dubia Consequenter forte dubitaret aliquis quot sunt hic coniugationes Sed tunc dubitaret cum oporteat hic maiorem esse de necessario

16vb

Dubitabit hic forte aliquis quia videtur esse instantia contra coniugationes utiles ubicumque illa de inesse non est de inesse simpliciter Consequenter forte dubitabit de coniugationibus inutilibus videtur enim posse probari coniugatio quam reputatur inutilis iuxta primum modum per impossibile Consequenter forte dubitabitur quarto modo videtur enim esse utilis coniugatio in eo maiore existente de necessario sic Adhuc forte dubitaret aliquis circa quartum modum quod videtur esse coniugatio utilis cum particularis negativa est de necessario sic Consequenter forte queretur numerus et sufficientia utilium coniugationum huius figure

17va

Dubitatur hic primo de modis utilibus quia contra coniugationes utiles est hic instare sicut superius si illa de inesse accipiatur ut nunc Consequenter potest dubitari de coniugatione inutili iuxta secundum modum potest enim ostendi quod sit utilis per oppositum conclusionis Posset etiam hic dubitari de quinto huius easdem dubitationes et eodem modo quo dubitatum est circa quartum secunde Ulterius forte dubitaret aliquis de positione terminorum in fine capituli contra coniugationem inutilem iuxta sextum modum Consequenter queret aliquis de sufficientia utilium coniugationum in hac figura Ultimo vero dubitabit aliquis de eo quod dicit in correlario quod non sequitur conclusio de inesse

18vb

Hic solet dubitari de ordine Consequenter dubitari solet circa diffinitionem contingentis

19vb–20ra 20ra

16vb–17ra

A10 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5

17va 17va–b 17vb 17vb

A11 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

18vb 18vb 18vb 19ra 19ra

A12–13 Dub. 1 Dub. 2

index of dubia Dub. 3

Adhuc dubitaret forte aliquis cum contingens necessarium et impossibile opponantur et unum oppositorum non diffinitur per alterum Dub. 4 Consequenter forte dubitaret aliquis utrum contingens commune equivoce dicatur de contingenti necessario et non necessario Dub. 5 Consequenter forte dubitabitur de modo probandi quod necessarium sit contingens Dub. 6 Ulterius forte dubitabit aliquis de conversione quam docet Dub. 7 Consequenter forte dubitabit de hoc quod dicit hominem canescere non habere continuam necessitatem Dub. 8 Consequenter dubitabitur ulterius de hoc quod dicit quod cum homo est aut de necessitate aut frequenter canescit Dub. 9 Ulterius forte dubitabitur de eo quod dicit contingens natum non converti in idem genus contingentis Dub. 10 Ulterius autem dubitabitur forte de eo quod dicit circa dici de omni in contingentibus Dub. 11 Ulterius forte dubitabit quomodo illa de contigenti dat intelligere illam de inesse Dub. 12 Ultimo forte dubitabit aliquis quare non ponat tertiam intentionem propositionis de contingenti

271 20ra 20ra–b 20rb 20rb 20rb–va 20va–b 20vb–21ra 21ra 21ra–b 21rb

A14 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

Hic posset aliquis dubitare quidam de sufficientia in processu Aristoteli Consequenter forte dubitaret aliquis de utilibus coniugationibus Ulterius autem queret aliquis sub quo intentione universalis de contingenti accipi debet minor universalis in ista generatione Consequenter forte dubitabit de eo quod dicit hic esse syllogismum imperfectum Adhuc forte dubitabit aliquis de terminis positis ad coniugationem inutilem Ultimo forte queretur sufficientia modorum huius generationis

21vb–22ra 22ra 22ra 22ra 22ra 22ra

272

index of dubia

A15 Dub. 1

Dub. 2

Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9 Dub. 10

Dub. 11 Dub. 12

Dub. 13 Dub. 14

Sed hic forte dubitaret aliquis primo quia cum in mixtione necessarii et inesse non est utilis coniugatio nisi maiore existente de necessario quomodo et quare sit hic utilis coniugatio maiore existente de inesse et non de contingenti Sed tunc dubitatur quare maiore existente hic de contingenti sequitur contingens ad utrumlibet minore autem existente de contingenti non sed contingens pro possibili Consequenter dubitabit aliquis de coniugationibus perfectis Ulterius consequenter dubitabitur de eo quod dicit Aristoteles nihil ex necessitate sequitur ex uno Hic forte dubitabitur de coniugationibus imperfectis et videtur quod omnes inutiles sint Consequenter ulterius dubitabitur de modo perficiendi istas coniugationes Adhuc forte dubitabit aliquis de reliquo modo perficiendi istas coniugationes scilicet per primam figuram Consequenter dubitabitur de his coniugationibus utrum possint in conclusionem de inesse Adhuc forte dubitaret aliquis que dicantur inesse simpliciter cum dicat quod maior debet esse de inesse simpliciter et non ut nunc Consequenter forte queret utrum maior de inesse modi negativi {negativi ACm1E1E2F1F2KlO2O3P1V: necessarii Ed} imperfecti debeat esse de inesse simpliciter sicut maior de inesse modi affirmativi Sed tunc ulterius queret utrum conclusio coniugationis affirmative intelligenda sit de contingenti pro possibili sicut conclusio coniugationis negative vel non Sed tunc queritur quare Aristoteles mentionem facit de modis negativis contingentis dicens quod concludunt contingens pro possibili et non dicit de modis affirmativis Consequenter dubitatur ulterius de hoc quod dicendum est istos modos posse in conclusione de contingenti pro possibili Ultimo forte queretur sufficientia syllogismorum huius mixtionis

24ra

24ra–b

24rb 24rb–va 24va 24va–b 24vb 24vb–25rb 25rb 25rb

25rb–va 25va

25va–b 25vb

index of dubia

273

A16 Dub. 1

Sed forte dubitabitur hic primo circa predictas coniugationes videtur enim quod not sit perfecta coniugatio ex maiore de contingenti et minore de necessario Dub. 2 Consequens est questio quare magis perfecta est coniugatio ex suppositione necessarii sub contingenti quam econtra Dub. 3 Consequens est dubitatio quare modus perfectus concludit contingens secundum dictam determinationem {determinationem ACm1E1E2F1F2KlO2O3P1V: demonstrationem Ed} aliquando autem modus imperfectus contingens pro possibili et non concluditur idem contingens in omnibus Dub. 4 Sed adhuc dubitatur quare scilicet modi perfecti solum possunt in contingens secundum dictam determinationem et non possunt in conclusionem de necessario vel de in esse Dub. 5 Consequenter forte dubitabitur de modis imperfectis et primo de illis qui habent maiorem negativam de necessario Dub. 6 Sed tunc dubitabitur cum talis coniugatio possit in conclusionem de inesse et in conclusionem de contingenti utrum possit in conclusionem de necessario Dub. 7 Sed hic ulterius forte dubitaret aliquis de coniugationibus imperfectis ubi maior est affirmativa de necessario ibi enim dicitur conclusionem sequitur de contingenti solum Dub. 8 Dubitatur tamen utrum in tali coniugatione sequitur conclusio de inesse Dub. 9 Consequenter forte adhuc dubitabit de probatione quam ponit Aristoteles ad ostendendum conclusionem de inesse sequit Dub. 10 Adhuc autem forte dubitaret aliquis si tam in modis affirmativis quam negativis imperfectis maior de necessario appropriat minorem ut sit de contingenti nato Dub. 11 Sed tunc emergit dubitatio de eo quod dictum est hic in mixtione contingenti et inesse Dub. 12 Consequenter etiam forte dubitabitur de eo quod dicit quod in modo imperfecto habente maiorem de contingenti negativam non sequitur conclusio de inesse Dub. 13 Ulterius forte queretur de sufficientia modorum huius mixtionis

26vb

26vb 26vb–27ra

27ra

27ra–b 27rb 27rb–va

27va 27va–b 27vb

27vb 27vb 27vb

274

index of dubia

A17 Dub. 1 Dub. 2

Dub. 3

Dub. 4

Dub. 5 Dub. 6

Dubitaret forte aliquis quare ex utraque de contingenti non fiat syllogismus in hac figura Ulterius forte dubitaret quia cum affirmativa et negativa de contingenti convertantur dupliciter quare magis ostendit non fieri syllogismum ex contingentibus eo quod negative convertuntur affirmativis quam ostendat ex eis {eis ACm1F1F2O2KlP1V : eis non O3Ed} fieri syllogismum per hoc quod affirmative convertuntur negativis Sed tunc ultra dubitaret si enim non fiat syllogismus secunde figure nisi per aliam vere negativam cum negativa de necessario et negativa de contingenti pro possibili non sint vere negative non poterit fieri syllogismus in secunda figura aliquando tali negativa accepta quod enim falsum est Consequenter forte queret aliquis cum universalis affirmativa et negativa de contingenti convertuntur advicem et ars non admittit conversionem negative in terminis eo quod ipsa convertitur cum affirmativa quare admittit ars conversionem affirmative cum tamen convertatur cum negativa Consequenter forte queret quomodo se habeat hic consequentia quam facit Aristoteles necesse est aliquid a esse b ergo non contingit omne Adhuc forte dubitaret aliquis de eo quod probat syllogismum uniformem hic non posse ostendi per impossibile

28vb 28vb

28vb

28vb

28vb–29ra 29ra

A18 Dub. 1 Dub. 2

Dub. 3 Dub. 4 Dub. 5 Dub. 6

Primo de regula hic observanda queritur enim quare oportet hic illam de inesse esse universalem negativam Si queritur quare non descendit hic aliquis syllogismus a modo imperfecto habente maiorem affirmativam de inesse quare etiam a modis perfectis illius mixtionis non descendat hic aliquis syllogismus Consequenter forte dubitabit de coniugatione et primo de utilibus habent enim instantiam Ulterius forte dubitabit aliquis de coniugationibus inutilibus et primo de inutilibus ubi illa de inesse est affirmativa Adhuc forte aliquis dubitabit de quarto secunde ubi non vult fieri utilem coniugationem Consequenter forte queritur quot sunt hic syllogismi et qua sit eorum sufficientia

29va 29va

29va 29va 29va–b 29vb

index of dubia

275

A19 Dub. 1 Dub. 2 Dub. 3

Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9 Dub. 10

Dub. 11 Dub. 12

Dub. 13 Dub. 14

Sed hic forte dubitaret aliquis primo de principio hic observando scilicet quod propositio de necessario sit universalis negativa Sed tunc queritur quare a modis perfectis huius mixtionis in prima figura non descendit aliquis syllogismus secunde Consequenter dubitatur de modo probandi consequentiam conclusionis {consequentiam conclusionis ACm1E1E2F1F2KlO2P1V: concludendum conclusionis O3: coniugationis Ed} de inesse in modis utilibus huius mixtionis Ulterius forte dubitabit aliquis de quibusdam dictis in littera circa terminos quibusdam positos ad instantiam Sed ulterius forte dubitabitur de veritate huius propositionis necesse est omnem vigilans esse movens quam assumit Consequenter forte dubitabitur utrum in terminis prehabitis intendat Aristoteles quod stet affirmativa de necessario vel affirmativa de inesse Sed tunc ulterius dubitabitur utrum cum talibus premissis possit stare affirmativa de inesse Sed adhuc restat eadem dubitatio de terminis quas ponit Aristoteles Ulterius forte dubitabit aliquis de coniugationibus et primo de utilibus quia videntur habere instantiam Consequenter forte dubitabit de coniugationibus inutilibus videtur enim quod maiore existente affirmativa de necessario {de necessario ACm1E1E2F1F2KlO2O3P1V: om. Ed} sit utilis coniugatio Adhuc restat difficilis dubitatio de quarto secunde videtur enim utilis esse coniugatio si maior fuerit de contingenti et minor de necessario Sed difficilior est dubitatio de contingenti pro possibili quod videtur sequitur ex necessitate quia ex opposito conclusionis et maiore sequitur oppositum minoris in prima figura per mixtionem contingentis necessarii Et tunc queritur cum coniugationes utiles universales fiant iuxta primum modum secunde figure Consequenter forte queritur sufficientia syllogismorum huius mixtionis

30va 30va 30va–b

30vb 30vb 30vb–31ra 31ra 31ra 31ra 31ra–b

31rb 31rb

31rb 31rb–31va

A20 Dub. 1

Dubitatur hic quia videntur coniugationes utiles habere 31vb–32ra instantiam

276 Dub. 2

index of dubia Consequenter queritur sufficientia syllogismorum in ista generatione

32ra

Hic dubitatur primo de coniugationes utiles {utiles AE 1E2F2V: om. Cm1F1KlO3P1Ed} videntur enim habere instantiam Adhuc dubitatur utrum coniugationes utiles hic possint in conclusione de inesse Adhuc forte dubitabit aliquis de illis coniugationibus que habent maiorem particularem negativam de inesse et minorem universalem de contingenti, videntur enim utiles respectu conclusionis de contingenti Consequenter forte queritur sufficientia syllogismorum Sed queritur causa secunde suppositionis scilicet quod in modis negativis ubi illa de inesse negativa est oportet quod sit universalis et maior

32rb–va

Dubitatur hic et primo de primo modo ubi videtur sequitur tam conclusio de inesse quam de necessario Consequenter forte queretur sufficientia coniugationum hic habitarum Sed dubitabitur tunc quomodo in hac mixtione in tertia figura mixta sunt modi inutilis coniugationis illa de necessario existente particulari negativa Sed tunc forte dubitabitur quare magis in his mixtionibus sit utilis coniugatio in quarto secunde et quinto tertie particulari negativa existente de necessario quam ipsa existente de inesse Notandum quod tripliciter est mixtio

33ra–b

A21 Dub. 1 Dub. 2 Dub. 3

Dub. 4 Dub. 5

32va 32va

32va 32va–b

A22 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Note

33rb 33rb 33rb–va

33va

A23 Dub. 1 Dub. 2 Dub. 3

Dubitatur hic qualis sit demonstratio per quam ostendit quod tres sunt figure Consequenter forte queretur que sit differentia syllogismi ostensivi ad illum quod est ex hypothesi Consequenter forte dubitabit aliquis cum multiplex sit divisio syllogismi quia aut est affirmativus aut negativus aut universalis aut particularis aut dialecticus aut demonstrativus et sic de aliis quare magis ad ostendendum numerum figurarum assumit has differentias syllogismi ostensivi ex hypothesi quam aliquas alias

34vb 34vb 34vb

index of dubia Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8

277

Hic dubitari posset de eo quod dicit ex uno nihil sequit Amplius autem dubitabit forte aliquis de eo quod dicit syllogismum ad impossibile syllogizare conclusionem falsam Adhuc dubitabit quare ostendit de syllogismo ad impossibile ipsum esse ostensivum et fieri per predictas figuras Adhuc solet queri quare non contingit per positionem conclusionis arguere positionem alicuius premisse et per veritate conclusionis veritatem alicuius premisse Ultimo dubitabitur ad quod hic ostendit hoc correlarium scilicet quod omnes syllogismi perficiuntur per universales prime

34vb–35ra 35ra

Dubitabit hic forte aliquis quare Aristoteles non probat quod in omni syllogismo oportet esse aliquam affirmativam Consequenter forte dubitabit de sua probatione videtur enim opposita dicet Adhuc videtur nulla esse divisio sua cum dicit sine universali aut non erit syllogismus aut petitur quod est in principio Adhuc forte dubitabit super hoc quod dicit quod si idem ad sui probationem accipiatur erit petitio Adhuc dubitabit aliquis de intentione sua quod ad syllogizandum particulare oportet sumere universale Consequenter queritur quare universalis conclusio exigat utramque premissam universalem

35va

Dubitabitur forte hic de eo quod dicitur omnem syllogismum habere tres terminos et tantum tres Consequenter forte dubitatur de eo quod ponit duos terminos ad significandum medium unum Adhuc forte dubitaret aliquis de eo quod dicit ibi ostendendo minorem

36rb

Dubitabit hic de hoc quod dicit tres terminos facere duas propositiones

37ra

35ra 35ra 35ra

A24 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

35va–b 35vb 35vb 35vb 35vb

A25 Part 1 Dub. 1 Dub. 2 Dub. 3

36rb–va 36va

A25 Part 2 Dub. 1

278 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

index of dubia Consequenter forte dubitatur de eo quod dixit in syllogismo uno conclusionem esse medietatem propositionum Consequenter dubitatur de eo quod dicit quod semper numerus propositionum minor est numero terminorum in uno Adhuc solet queri quomodo uno termino addito addatur tantum una propositio Adhuc forte dubitatur de eo quod dicit quod addito termino extrinsecus ad solum ultimum non sit conclusio Ultimo dubitatur de eo quod dicit quod cum sumitur terminus intrinsecus ad unum solum terminum non sit conclusio

37ra

Dubitaret hic forte aliquis de eo quod vult universalem affirmativam esse difficiliorem aliis ad construendum Sed tunc ulterius forte dubitaret quare universalis affirmativa tantum in una figura syllogizetur Adhuc forte queretur quare pluribus modis et figuris syllogizatur particularis conclusio quam universalis Adhuc si queratur quare negativa particularis in pluribus syllogizetur quam affirmativa particularis

37va

Sed hic forte dubitaret aliquis qualiter Aristoteles determinet de inventione Adhuc forte dubitabit quare dat artem inveniendi medium at non dat artem inveniendi extrema Adhuc dubitabitur quomodo possit esse ars inventiva mediorum Consequenter forte queret de dictis in littera et primo de hoc quod dicit quod individuum non predicatur nisi secundum accidens Deinde forte queret de hoc quod dicit aliquando consequentia esse ut frequenter vel ut nunc Sed tunc queritur quare magis nominat ea antecedens et consequens quam subiectum et predicatum Adhuc forte queretur de eo quod dicit ad consequens non est addendum omne Adhuc forte dubitabit aliquis quare dicit ipsum omne additum ad consequens superfluere

38va

37ra 37ra–b 37rb 37rb

A26 Dub. 1 Dub. 2 Dub. 3 Dub. 4

37ra–b 37rb 37vb

A27 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8

38va 38va 38va–b 38vb 38vb 38vb 38vb

index of dubia Dub. 91

279

Adhuc queretur quare magis eligenda sunt consequentia propria et antecedentia propinqua quam remota

-

Hic forte dubitabitur cum in generatione syllogismorum diversum dederit artem circa illas de inesse et illas de modo quare non similiter facit hic in mediorum inventione Adhuc forte queretur quare plures sunt regule ad terminandum universalem negativam et unica ad terminandum universalem affirmativam Adhuc forte queretur quare non ponit regulam ad concludendum particularem affirmativam Eodem modo opponi potest de particulari negativa Adhuc forte dubitaret aliquis de terminis quos ponit Consequenter dubitabitur de idoneitate quam ponit in fine

39vb

Dubitatur forte de hoc quod dictum est ipsum ostendere sufficientiam sue artis seorsum in illis de modo ab illis que sunt ostensivi et ex hypothesi Consequenter queritur cum duplex sit dispositio formalis in syllogismo scilicet modus et figura et sunt inutiles inspectiones contra modum quare non sunt alie inutiles peccantes contra figuram Ulterius forte queretur cum ad modum exigantur duo scilicet qualitas et quantitas et sunt quidam inspectiones inutiles peccantes contra qualitatem quare non sunt similiter alique peccantes contra quantitatem Consequenter dubitabitur de una inutilium inspectionum Adhuc queri potest quare Aristoteles removet duas inspectiones que sunt ad contraria

40vb

A28 Part 1 Dub. 1

Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

39vb–40ra 40ra 40ra 40ra 40ra

A28 Part 2 Dub. 1 Dub. 2

Dub. 3

Dub. 4 Dub. 5

40vb

40vb

40vb 40vb–41ra

A29–A30 Part 1 Dub. 1

1

Dubitatur hic de eo quod dicit quod omne demonstratione ostensive potest demonstrari per impossibile et e converso

ACm1E1E2F1F2KlO2O3P1V: om. Ed.

41vb

280 Dub. 2 Dub. 3 Dub. 4

Dub. 5

index of dubia Consequenter queritur de eo quod dicit quod in syllogismo per impossibile oportet accipere medium terminum alium ab extremis Consequenter forte dubitatur de eo quod dixit syllogismum ex hypothesi qui fit secundum transumptionem termini fieri per principia dicta Adhuc forte dubitaret aliquis quare non ostendit syllogismum ex falsis et circularem et ostensivum et illum qui est ex oppositis fieri per dicta principia sicut illum qui est ad impossibile Ulterius forte dubitabitur de eo quod dicit hanc artem communem esse ad omnem scientiam et ad omnem propositum terminandum

41vb 41vb–42ra 42ra

42ra–42rb

A29–A30 Part 2 Dub. 1 Dub. 2

Dubitaret forte aliquis hic quomodo principia propria scientiarum per sensum vel experimentum possint accipi Adhuc dubitabitur quia dicit quod principiis propriis inventis per artem istam scimus inventa ordinare

42rb

Dubitabit forte aliquis circa primam rationem quam ponit in littera. Consequenter dubitabitur de secunda ratione quam ponit

43ra–b

Dubitabit forte aliquis primo de intentione huius capituli Sed tunc dubitatur potentia enim est ante actum quare oratio syllogizanda prior est quam oratio actu syllogizata Consequenter forte dubitabit aliquis de dictis in littera et primo de hoc quod dicit quod si habeamus syllogismorum generationem et medii inventionem et syllogismorum reductionem tunc habebimus sufficienter propositum

44ra–b

42va

A31 Dub. 1 Dub. 2

43rb

A32–A33 Dub. 1 Dub. 2 Dub. 3

44rb 44rb

index of dubia Dub. 4

Dub. 5 Dub. 6 Dub. 7 Dub. 8

281

Ulterius forte dubitatur de suo exemplo ubi dicit conclusionem sequi sed propositiones deficere {sed propositiones deficere Cm1E1E2F1KlO2O3P1V: sed propositiones differe F 1F2: si propositiones necessarie sint Ed} ut hic, non substantia interempta non interimitur substantia, sed partibus rei interemptis interimitur res, ergo partes substantie sunt substantie. Sed tunc forte queretur quare non declarat quomodo accidit decipi superfluitate sicut diminutione Consequenter queritur de secundo exemplo scilicet si est homo est animal et si est animal est substantia ergo si est homo est substantia. Consequenter dubitatur de hoc quod videtur Aristoteles velle nomen dividuum addi termino discreto sic, Omnis Aristomenes intelligibilis est etc. Sed tunc queritur cuiusmodi peccatum contra syllogismum est in dictis exemplis

44rb–va

Dubitaret forte aliquis utrum semper accidat peccatum contra syllogismum sumptis terminis non in concretione immo in abstractione Ulterius forte dubitatur de eo quod dicit hanc esse falsam in concretis de necessitate nullum sanum est egrum.

45ra–b

Dubitatur forte hic quia videtur velle opposita Adhuc dubitatur forte quia videtur quod omnis propositio mediata sit

45vb 45vb

Dubitatur hic primo an de obliquis syllogizari possit vel non Sed tunc dubitatur si ex obliquis syllogizetur quare modum syllogizandi in eis non tradit Aristoteles superius Adhuc dubitaret forte aliquis quia videtur talis syllogismus fallere per diversitatem medii Adhuc forte dubitatur de eo quod dicit quod utraque premissarum sumpta in obliquo {obliquo Cm1E1F1F2KlO3P1V: obiecto O2: aliquo Ed} conclusio potest esse de recto

46rb

44va 44va 44va–b 44vb

A34 Dub. 1 Dub. 2

45rb–va

A35 Dub. 1 Dub. 2

A36–A37 Dub. 1 Dub. 2 Dub. 3 Dub. 4

46rb 46rb–va 46va

282

index of dubia

Dub. 5 Dub. 6

Adhuc queritur quare non docet syllogizare ex omnibus obliquis Si autem queritur quare non docet syllogizare ex obliquis per tres figuras

46va

Dubitaret forte aliquis primo cuius proprietas sive conditio est reduplicatio Habito quod reduplicatio sit conditio ad predicatum addenda queritur utrum fiat syllogismus in aliquia figura cum reduplicationem maiori Consequenter forte queretur cum contingat negative syllogizare cum reduplicatione sicut et affirmative quare non docet hic Aristoteles Ulterius forte queretur utrum cum reduplicatione medii et minoris possit per omnes {possit per omnes Cm1E1E2F1F2KlO2O3P1V: quod oportet Ed} figuras syllogizari Sed tunc sequitur quod aliquis universalis negativa non convertitur simpliciter Sed tunc dubitatur de dictis in littera et primo de hoc quod dicit quod posita reduplicatione ad medium erit minor propositio falsa Ulterius forte dubitaret aliquis quare minor est non intelligibilis Ultimo forte dubitabit de exemplo quod ponit

47rb

Dubitaret forte aliquis cum hic {hic E 1E2F1F2KlO2O3 P1V: b Ed} propositio cui b inest illi omni inest a habeat intellectum particularis propositionis Adhuc dubitabitur quare non facit mentionem de illa que syllogizat negative in prima figura sicut de illa que affirmative

48rb

46va

A38–A40 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Dub. 5 Dub. 6 Dub. 7 Dub. 8

47rb–va 47va 47va

47va–b 47vb 47vb 47vb

A41 Dub. 12 Dub. 2

2

At this point F2 (33ra) repeats A38 dub.2 and dub.3.

48rb

index of dubia

283

A42–A44 Dub. 1

Dub. 2 Dub. 3 Dub. 4

Hic forte dubitabitur primo de preconcessione {preconcessione Cm1E2F2O2P1: preconfessione E 1F1KlO3Ed} queritur enim cum ad concludendum universale per particulare oporteat uti preconcessione quare non {non Cm1E1E2F1F2O2O3P1: om. KlEd} similiter ad interimendum universale per particulare Deinde dubitatur de hoc quod non vult syllogismos ex hypothesi posse reduci in syllogismum Consequenter dubitabit aliquis ubi Aristoteles determinabit modos syllogismorum ex hypothesi Sed tunc queritur quare non determinat in secundo huius de omni syllogismo ex hypothesi

48vb–49ra

Dubitatur hic cum in principio docuerit reductionem syllogismorum imperfectorum in perfectos hic autem docet reducere omnes figuras invicem Adhuc dubitatur quare non reducat singulos syllogismos omnium figurarum in alias figuras Sed queritur cum prima figura reducitur in secundam et econverso per conversionem maioris quare magis sic quam per conversionem minoris Adhuc queritur quare prima et tertia reducuntur invicem minore conversa Dubitatur adhuc de eo quod dicit solum quartum secunde et quintum tertie reduci per impossibile Adhuc queritur cum sic sit intentio de reductione et duplex sit reductio scilicet per conversione et per impossibile quare non docet hic reductionem ad impossibile pro reductione sicut conversionem

49vb

Sed forte dubitatur de eo quod dicit alterum istorum esse equale esse non equale non omni inest Consequenter dubitatur quare hic determinat consequentiam propositionum de terminis finitis et infinitis Adhuc forte aliquis dubitabit cum dicit Aristoteles sequitur est non bonum ergo non est bonum utrum intendat de propositionibus universalibus istam consequentiam vel non Sed contra Socrates fuit non iustus ergo non fuit iustus similiter erit non albus ergo non erit albus

51ra

49ra 49ra 49ra

A45 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

49vb 49vb 49vb 49vb–50ra 50ra

A46 Part 1 Dub. 1 Dub. 2 Dub. 3

Dub. 4

51ra 51ra–b

51rb

284 Dub. 5 Dub. 6 Dub. 7

Dub. 8

Dub. 9 Dub. 10 Dub. 11 Dub. 12 Dub. 13 Dub. 14

Dub. 15

Dub. 16 Dub. 17 Dub. 18 Dub. 19 Dub. 20

index of dubia Sed contra non sequitur est non duplum ergo non est duplum est non equale ergo non est equale Ulterius autem dubitaret forte aliquis de eo quod dicit istas duas negationes non esse bonum non esse non bonum alicui eidem inesse Consequenter dubitabit cum determinat consequentiam terminorum finitorum et infinitorum, finitorum etiam privatorum, et non determinat infinitorum privatorum que sit habitudo eorum De terminis autem infinitis {infinitis Cm1E1E2F1F2KlO2 O3P1V: finitis Ed} et privativis quia tangit hic Aristoteles immorandum est ut discutiatur eorum natura principaliter autem de infinitis disserendum est de quibus primo queritur an aliquis terminus possit infinitari vel non Adhuc est hic questio cum omne verbum significet agere vel pati quomodo verbum infinitum hoc {hoc Cm1E1F1F2O2O3P1V: non Ed} significet Hoc habito consequens est questio utrum unica negatio similiter et semel plures terminos infinitare possit His habitis consequens est questio que pars orationis infinitari potest His habitis ulterius forte dubitaret aliquis de infinitatione nominis et primo an omne nomen infinitari possit His habitis forte dubitaret ulterius aliquis de nomine infinito utrum aliquid ponat vel non Sed iuxta hoc incidunt due {due Cm1E1E2F1F2KlO3P2V: om. O2Ed} questicule una est in quo differunt nomen infinitum apud grammaticum et nomen infinitum apud logicum Alia questio est quare negatio aliquando in compositione cum nomine facit tantum pure negativum ut nemo aliquando autem infinitat ut non homo Solet autem queri per quam naturam nomini coniungitur negatio in compositione ad infinitandum Nunc autem restat querere de nomine privativo et primo quis terminus privari possit Consequens est questio que sit differentia inter nomen privativum et infinitum et pure negativum Et quia ulterius forte dubitaret aliquis de infinitatione non casualis sicut verbi queritur primo utrum omne verbum infinitari possit vel non Deinde queritur utrum verbum in omni differentia persone indifferenter infinitari possit ita quod differat a verbo pure negato

51rb 51rb 51rb

51rb–va

51va 51va–b 51vb–52ra 52ra 52rb–vb 52vb

52vb

52vb 52vb–53ra 53ra 53ra–b 53rb

index of dubia Dub. 21 Ulterius queritur de verbi tertie persone infinito utrum cum sit extra orationem aliquid differat a verbo pure negativo vel non Dub. 22 Hoc habito consequenter queritur utrum verbum infinitum in oratione differat a verbo pure negato

285 53rb–va 53va

A46 Part 2 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Dubitatur forte hic de pertinentia partis huius Ulterius forte dubitaret aliquis utrum affirmationis et negationis possit esse negatio Iuxta hic queritur utrum negationis sit negatio Sed tunc restat dubitatio de hoc quod dictum est negationem perimere aliam negationem que sit causa huius

54ra–b 54rb

Hic dubitaret forte aliquis de modis indirecte concludentibus Adhuc dubitaret quia hic dicit propositionem conversam diversam esse a convertente

54vb

Dubitaret forte aliquis quare dicit maiorem syllogizari de his que sunt sub medio et de his que sunt sub minori extremitate et non dicit ea que sunt supra maiorem extremitatem Adhuc videtur falsum quod dicit maiorem extremitatem de omnibus illis syllogizari eodem syllogismo Deinde forte dubitaret aliquis de his que dicit circa secundam figuram {figuram E 1E2F1F2KlO2O3P1V: om. Ed} Ulterius forte dubitabit de hoc quod dicit consequenter maiore per syllogismum non removeri a contentis sub medio Adhuc forte dubitabit quare non docet syllogizare maiorem extremitatem {maiorem extremitatem Cm1E1E2F1F2KlO2O3P1V: om. Ed} de contentis sub medio vel minore in secunda figura maiore existente affirmativa Ultimo queretur de hoc quod dicit in particularibus prime maiorem concludi de contentis sub medio non tamen per syllogismum

55rb

54rb 54rb

B1 Part 1 Dub. 1 Dub. 2

54vb

B1 Part 2 Dub. 1

Dub. 2 Dub. 3 Dub. 4 Dub. 5

Dub. 6

55rb 55rb–va 55va 55va

55va

286 Dub. 7 Dub. 8

index of dubia Et questio est de syllogismis tertie ubi maior aliquando particularis est Adhuc restat questio de duobis modis syllogismorum predeterminatis potentibus in plures conclusiones quid utilitatis habent

55va

Dubitatur hic cum quadruplex sit comparatio veri et falsi Adhuc dupliciter de eo quod ostendit falsum non sequitur ex veris

56ra

55vb

B2 Part 1 Dub. 1 Dub. 2

56ra

B2 Part 2 Dub. 1 Dub. 2

Dubitaret forte aliquis quomodo verum sequi ex falsis Ad dictorum evidentiam consequens est questio utrum syllogismus ex falsis sit syllogismus vel non Dub. 3 Ulterius forte dubitaret aliquis que sit utilitas syllogismi ex falsis Dub. 4 Sed tunc ulterius forte dubitabit aliquis quare hic determinatur de syllogismo ex falsis Dub. 5 Ulterius autem dubitaret aliquis de his qui sunt in littera sunt enim in universalibus octo coniugationes Dub. 6 Sed tunc dubitabitur utrum ille due combinationes quas omittit utiles sint respectu conclusionis vere Dub. 7 Et dubitatur tunc quare maiore in toto falsa non potest conclusio esse vera Dub. 8 Sed tunc queritur quare minore tota falsa sequitur verum Dub. 9 Adhuc dubitari posset de eo quod dicit Aristoteles in quibusdam exemplis Dub. 10 Dubitatur autem cum modi particulares ab universalibus veniant et de modis universalibus maiore tota falsa non sequitur verum quomodo in particularibus syllogismis maiore tota falsa sequatur verum

57ra–b 57rb–va 57va 57va 57va 57va 57va–b 57vb 57vb 57vb

B3 Dub. 1 Dub. 2

Dubitaret forte aliquis cum in universalibus sint octo combinationes Sed tunc queretur an utraque existente tota falsa possit in conclusionem veram

58va 58va

index of dubia

287

B4 Part 1 Dub. 1 Dub. 2 Dub. 3

Sed queritur quare omittit duas habentes alteram in 59ra–b toto falsam Sed tunc queritur utrum utraque possit in 58rb conclusionem veram Solet autem opponi in omnibus his figuris sic falsitas est 58rb privatio veritatis

B4 Part 2 Dub. 1 Dub. 2

Dubitaret forte aliquis de maiore propositione sue rationis videtur enim ad idem esse vel non esse sequi idem Deinde forte dubitabitur de hoc quod concludit illud pro inconvenienti si b non est magnum b est magnum

59va 59va–b

B5 Dub. 1

Dubitaret forte aliquis hic primo de circulari syllogismo in communi Dub. 2 Consequenter forte dubitabit aliquis utrum sit syllogismus nec ne Dub. 3 Ulterius forte dubitaret aliquis que sit utilitas talis syllogismi Dub. 4 Adhuc autem dubitaret forte aliquis que sit differentia syllogismi circularis hic determinati ad demonstrationem circularem quam docet Aristoteles in secundo posteriorum in capitulo de causis Dub. 5 Ulterius queritur de his que dicit in littera de diffinitione circularis syllogismi Dub. 6 Consequenter queritur de syllogismo circulari negativo Dub. 7 Sed queret aliquis quare non transponit illas negativas in affirmativis Dub. 8 Ulterius forte dubitaret aliquis de conversione quam facit negative in affirmativam Dub. 9 Consequenter forte dubitabitur de syllogismo quem facit scilicet cui nulli inest a illi omni inest b c nulli inest a ergo c omni inest b Dub. 10 Ulterius forte queritur de particularibus et primo queritur quare in tertio {tertio Cm1E1E2F1F2O2O3P1V: quarto Ed} modo non syllogizatur maior nec conversa maioris nec conversa minoris aut conclusionis Dub. 11 Consequenter dubitabit aliquis de modo particulari negativo

60va–b 60vb 60vb–61ra 61ra

61ra–b 61rb 61rb–va 61va 61va 61va–b

61vb

288

index of dubia

Dub. 12 Si autem queratur quare aliter hic transumpta est maior in affirmativam quam in syllogismis universalibus Dub. 13 Queritur ultimo quomodo iste syllogismus qui fit iuxta primum modum et ille qui fit iuxta quintum modum possit reduci in syllogismum simplicem et cathegoricum

61vb 61vb

B6 Dub. 1 Dub. 2

Dubitaret forte aliquis hic de hoc quod primo dicit hic affirmativam propositionem non posse syllogizari circulo Consequenter forte queretur quare negativa propositio secundi modi syllogizatur circulariter per figuram eandem et non sic negativa primi modi

62a–b

Dubitatur de hoc quod dicit syllogismum circularem in secunda primo ad concludendum universalem propositionem esse per tertiam figuram Sed dubitatur de eo quod ibi dicit omnes circulationes factas in tertia figura fieri per eandem figuram Deinde forte dubitabitur que sunt ostensiones in secunda et in tertia figura non concludentes in figura propria que circulares non sunt et que sunt ille que circulares sunt sed incomplete

63ra

Dubitaret hic forte aliquis de syllogismo conversivo utrum possit esse instrumentum ad artem pertinens Dubitabit forte aliquis consequenter de eius differentia ad syllogismum ex falsis Ulterius dubitabitur forte ad quem pertineat syllogismus conversivus et que sit eius utilitas Sed tunc queritur si ibi diffinitur et hic quomodo differenter Consequenter dubitabit aliquis de diffinitione hic posita et primo queritur quare diffinit convertere et non syllogismum conversivum Adhuc dubitabitur quia videtur diffinitio diminuta eo quod non extendit se nisi ad conversionem factam in prima figura Consequenter queri potest de oppositionibus quas ipse enumerat

64ra

62ra

B7 Dub. 1 Dub. 2 Dub. 3

63ra–b 63rb

B8 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7

64ra 64ra–b 64rb 64rb 64rb 64rb–va

index of dubia Dub. 8

289

Ultimo forte dubitabitur de eo quod docet syllogizare per contrariam conclusionis

64va

Sed hic forte queritur causa quare in prima destruitur maior per tertiam figuram Ulterius queritur quare in secunda non interimitur maior per tertiam figuram Queritur etiam quare maior in tertia interimitur per primam Queretur forte adhuc quare oppositum in prima figura sumitur in syllogismo conversivo loco propositionis interimende

65va

B9–B10 Dub. 1 Dub. 2 Dub. 3 Dub. 4

65va 65va 65va

B11–B12 Part 1 Dub. 1

Dubitaret forte aliquis primo ad quem pertineat syllogismus per impossibile Dub. 2 Ulterius autem dubitaret aliquis quare non diffinit hic actum syllogismi per impossibile Dub. 3 Sed tunc ulterius dubitabit de eius diffinitione Dub. 4 Adhuc forte dubitaret ulterius aliquis que sit differentia syllogismi per impossibile et conversivi Dub. 5 Sed tunc queritur cum iste syllogismus sit similis conversivo ut dicitur in littera quare non procedit supponendo contrariam Dub. 6 Consequenter forte dubitabit aliquis circa syllogismum conversivum cum processit determinando ipsum secundum modos diversos figurarum per ordinem quare non sic processit circa syllogismum ad impossibile Dub. 7 Ulterius dubitabit quare universalis affirmativa non ostendit per impossibile in prima figura sed in aliis Dub. 8 Consequenter forte dubitatur de intellectu quorundam dictorum in littera primo de eo quod dicit quod syllogismus ad impossibile procedit non presyllogizato opposito hypothesis Dub. 9 Deinde queretur forte de eo quod dicit ubi ostendit universalem negativam non demonstrari per impossibile supposita eius contraria Dub. 10 Item dubitatur de hoc quod dicit particularem negativam non posse per impossibile demonstrari sumpta eius subcontraria

66vb–67ra 67ra 67ra 67ra–b 67rb 67rb

67va 67va

67va 67va

290

index of dubia

B11–B12 Part 2 Dub. 1 Dub. 2 Note

Dubitaret forte aliquis hic quare demonstrando per impossibile in secunda figura semper accipitur extrinsecus unum sub subiecto hypothesis Ulterius dubitaret quare non docet contrarias negativarum conclusionum demonstrandarum sicut affirmativarum Notandum autem quod octo possunt hic fieri syllogismi per impossibile

68ra

Dubitaret forte aliquis hic quare semper {semper Cm1E1F2KlO2O3P1V: similiter Ed: om. E 2} assumitur in tertia figura verum extrinsecus et supra substantiam hypothesis

68va

Hic forte dubitaret aliquis quare non prius comparavit syllogismum circularem et conversivum ad ostensivum Sed nunc queritur quare comparat syllogismum per impossibile ad ostensivum secundum differentiam Demum dubitabitur de hoc quod dicit ostensivum syllogismum sumere propositiones veras et indubitabiles Consequenter forte queretur aliquis quare demonstrativum per impossibile in una figura non potest per eosdem terminos in eadem figura ostensive syllogizari Ulterius forte dubitaret aliquis utrum negativa demonstrata per impossibile in prima figura non possit aliter syllogizari ostensive quam in secunda {secunda Cm1E1E2F2KlO2O3P1V: prima Ed} Ulterius forte queretur etiam de particulari affirmativa demonstrata per impossibile in prima an aliter quam in tertia possit syllogizari ostensive in eisdem terminis Adhuc queret aliquis utrum per primam tantum ostensive syllogizantur conclusiones demonstrate per impossibile in secunda Adhuc forte queret idem de conclusionibus per impossibile demonstratis in tertia Adhuc queretur utrum idem sit demonstrare per impossibile prius ostensive syllogizatum et conversive syllogizare

69va–b

68ra 68ra

B13 Dub. 1

B14 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Dub. 5

Dub. 6 Dub. 7 Dub. 8 Dub. 9

69vb 69vb 69vb

69va

69va–70ra 70ra 70ra 70ra

index of dubia

291

B15 Dub. 1

Dubitatur primo de syllogismo ex oppositis an sit syllogismus Dub. 2 Ulterius forte queretur de cuius consideratione sit Dub. 3 Consequenter queritur de dictis in littera et primo quare cum aliis oppositionibus non {non Cm1E1E2F2KlO2O3P1V: om. Ed} connumeravit in littera oppositionem privativam et relativam Dub. 4 Deinde queritur quomodo verum sit quod dicit tres oppositiones esse secundum veritatem Dub. 5 Ulterius forte dubitabit aliquis de hoc quod dicit non fieri syllogismum negativum ex oppositis in prima figura Dub. 6 Consequenter queritur de dispositione syllogismi ex premissis Dub. 7 Consequenter forte queretur de tertia notabilitate quam ponit de multiplicatione syllogismi ex oppositis Dub. 8 Sed tunc queritur numerus et sufficientia utilium Dub. 9 Deinde queritur de comparatione syllogismi ex oppositis ad syllogismum ex falsis Dub. 10 Deinde queritur de hoc quod dicit opposita posse concludi per unum syllogismum

71ra–b 71rb 71rb

71rb 71rb–va 71va 71ra 71va 71va 71va–b

B16 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9

Hic forte dubitaret aliquis ad quid hic determinatur de petitione Sed tunc queritur que sit differentia petitionis que est secundum veritatem et que est secundum opinionem Ulterius forte queretur de his que dicit in littera et primo de hoc quod dicit quod in petitione non demonstratur propositum Consequenter dubitatur de eo quod dicit non demonstrari propositum eo quod id quod prius est per posteriora ostenditur Adhuc dubitatur consequenter quod non fit petitio quando per ignota arguitur ad eque ignota Consequenter forte dubitatur de hac principia esse nota per se Ulterius autem forte dubitaret aliquis de diffinitione petitionis Deinde forte queritur de secundo modo petitionis Deinde queritur de modo petendi quod est in principio

72va 72va–b 72vb 72vb 72vb 72vb–73ra 73ra 73ra 73ra

292

index of dubia

Dub. 10 Consequenter forte queretur cum sint tres figure et unaqueque syllogizatur et sint syllogismi universales et particulares adhuc affirmativi et negativi utrum in omnibus contingit indifferenter petere quod est in principio Dub. 11 Sed tunc queritur quot modis in omni figura petitur quod est in principio

73ra

73ra–b

B17 Dub. 1 Dub. 2

Dub. 3 Dub. 4 Dub. 5

Dub. 6 Dub. 7 Dub. 8

Dub. 9

Dubitabitur forte de pertinentia hoc partis et aliarum dictarum in hac Adhuc dubitatur de pertinentia huius partis cum plures sint loci sophistici quam petitio eius quod erit in principio et non causa ut causa quare isti duo determinantur hic et non alii Adhuc dubitatur cum hic determinet de petitione et de non causa ut causa et in elenchis similiter quare in topicis solum de petitione et de non causa Deinde forte queretur an sit differentia inter non causam ut causam et falsum et non propter hypothesim accidere Consequenter forte adhuc queritur quare magis probat falsum non propter hypothesim accidere in syllogismo per impossibile quam ostendit per petitionem accidere in syllogismis ostensivis Consequenter adhuc queritur utrum dictum peccatum contingit in syllogismis ostensivis Consequenter queritur de modis eius quod est non propter hoc accidere falsum Adhuc cum hic determinet modos eius quod est non causa et non causa dicatur quatuor modis sicut et causam videtur quod quatuor modi non cause debent esse Adhuc forte dubitabit aliquis quomodo accipiuntur modi {modi Cm1E1E2F2KlO2O3P1V: termini Ed} quos determinat

74ra 74ra–b

74rb 74rb 74rb

74rb 74rb 74rb

74rb–va

B18–B20 Dub. 1 Dub. 2 Dub. 3 Dub. 4

Hic queritur forte quomodo hic determinatur de oratione Adhuc videtur quod ex veris sequitur falsum Consequenter forte queretur quare docet hic instruere opponentem Adhuc forte queretur de elencho ad quem pertineat {pertineat E 1E2F2KlO2O3P1V: pertinet Ed}

74vb–75ra 75ra 75ra 75ra

index of dubia Dub. 5 Dub. 6

Deinde adhuc forte dubitabit aliquis quare non determinatur alicubi {alicubi Cm1E2F2KlO2O3P1V: alibi Ed} in libro de elencho Ulterius autem forte queret aliquis an sit eadem diffinitio syllogismi et elenchi

293 75ra 75ra

B21 Dub. 1

Dubitabit hic aliquis quomodo peccatum hic determinatum cadit iuxta syllogismum ex oppositis Dub. 2 Deinde queritur quare hoc peccatum nominatur fallacia opinionis Dub. 3 Deinde forte dubitabit aliquis de his que dicit in littera et primo de hoc quod dicit quod si aliquis opinetur omnes quatuor propositiones que sunt ex diversa coniugatione idem et secundum idem sciet et ignorabit Dub. 4 Consequenter dubitabit de his que sunt ex diversa coniugatione utrum scilicet contingit opinari simul ibi omnes quatuor propositiones Dub. 5 Sed tunc dubitatur utrum sit verum vel non Dub. 6 Deinde forte queretur utrum contingit simul opinari omnem d esse a et nullum d esse a Dub. 7 Ulterius forte dubitaret aliquis utrum contingat unam propositionem opinari secundum utrumque modum {modum P 1V: medium Cm1E1E2F2KlO1O2O3Ed} Dub. 8 Adhuc forte queret quomodo potest simul opinari duas propositiones secundum idem medium non tamen opinando conclusionem Dub. 9 Ulterius autem forte dubitabit aliquis de triplici modo sciendi Dub. 10 Deinde queritur quid scire cui ignorare opponitur Dub. 11 Consequenter forte queretur quomodo contingit in his opinari Dub. 12 Deinde dubitabit adhuc forte aliquis cum aliter se habeat oppositio in his que sunt ex eadem coniugationem quam in his que non sunt ex eadem coniugationem Dub. 13 Ultimo forte dubitabit aliquis de modo ostendendi quod contingit simul opinari bonum esse malum et bonum {bonum Cm1E2F2KlO1O2O3P1V: malum Ed} esse bonum

76rb 76rb–va 76va

76va 76va 76va–b 76vb 76vb 76vb–77ra 77ra 77ra 77ra

77ra–b

B22 Dub. 1

Dubitatur in principio ad quid introducuntur hic iste regule

78ra

294

index of dubia

Dub. 2

Sed dicet aliquis quod debet esse alia regula docens reductione exempli et deductionis et instantie Dub. 3 Adhuc forte dubitabitur de prima convenientia Dub. 4 Adhuc queritur quare magis ostendit in syllogismo negativo quod si medium convertatur cum extremis quod extrema convertantur quam in syllogismo affirmativo Dub. 5 Adhuc forte dubitatur de eo quod dicit in syllogismo negativo ex conversa minor et maior concludi conversam conclusionis Dub. 6 Adhuc forte dubitabit aliquis de eo quod vult ostendere consequentia maioris per consequentiam conclusionis Dub. 7 Adhuc forte dubitabit aliquis de ultima regula quam ostendit Dub. 8 Sed tunc queritur cum sit scientia moralis rationalis et naturalis et ipse determinat hic regulas pertinentes ad materiam rationalem et aliam particularem et ad materiam moralem quare non determinat aliquem particularem ad naturalem Dub. 9 Ulterius forte dubitabit aliquis de hoc quod supponit in littera scilicet quod magis eligendum magis fugiendum opponuntur Dub. 10 Ultimo forte dubitabit aliquis de exemplo eius

78ra 78ra–b 78rb

78rb 78rb–va 78va 78va

78va 78va

B23 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7 Dub. 8 Dub. 9

Primo dubitatur ad quid determinatur hic de inductione Adhuc cum prime species sint opposite sicut syllogismus et inductio quare non determinatur inductio in scientia separata a scientia syllogismi Sed tunc forte queretur quare non determinat sic artem faciendi inductionem Consequenter forte queret aliquis quomodo inductio in syllogismum reducitur Consequenter queritur quomodo sumenda sunt singula in ratione unius termini in hac reductione Deinde forte queret aliquis de inductione utrum inductio reducat in syllogismum cuiuscumque figure Consequenter dubitabit de his que dicit philosophus in littera et primo de hoc quod dicit aliquos modos arguendi a syllogismo fieri per predictas figuras Deinde queritur de inductionis diffinitione Adhuc illud de quo ostenditur quod est tantum in argumentatione et illud per quid medium quare si inductio primum ostendat de medio per terminum in inductione tantum erit medium et econverso

79ra 79ra 79ra 79ra 79ra–b 79rb 79rb 79rb 79rb

index of dubia Dub. 10 Deinde queritur de hoc quod dicit quod omnia singularia aggregant in tertia Dub. 11 Deinde queri posset de eo quod ipse dicit inductionem syllogismum esse propositionum immediatarum

295 79rb–va 79va

B24 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6

Dubitatur hic primo de diffinitione exempli Consequenter forte queritur de reductione eius in syllogismum Sed nunc dubitabitur de reductione illorum in duos syllogismos Consequenter queritur de reductione exempli in inductionem Sed tunc opponet aliquis si multa exempla reducuntur in unam inductionem et inductio una in syllogismum unum ergo multa exempla in syllogismum unum Consequenter queritur forte ad quem pertineat exemplum

79vb 79vb

Hic forte dubitabit aliquis de introductione deductionis Adhuc dubitabit quia videtur quod in inductione sit petitio eius quod est in principio Adhuc queretur ad quem pertineat deductio Adhuc forte queretur an fieri possit in aliis figuris deductio a prima Sed tunc dubitabitur circa modos eius et primo quare non est deductio vel argumentatio aliquem habens maiorem dubiam et minorem per se nota

80rb 80rb–va

Dubitatur hic a quibusdam propter hoc quod hoc iam determinatur de instantia Sed queri potest cum elenchus concludit contradictionem conclusionis et instantia similiter quomodo differunt Deinde dubitabit forte aliquis ad quem pertineat instantia Consequenter queritur quare magis diffinit instantiam quam instantivum syllogismum Deinde forte adhuc dubitabit aliquis de hoc quod dicit instantiam esse particularem

81rb

79vb–80ra 80ra 80ra 80ra

B25 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5

80va 80va 80va

B26 Dub. 1 Dub. 2 Dub. 3 Dub. 4 Dub. 5

81rb 81rb 81rb 81rb

296 Dub. 6

Dub. 7

index of dubia Deinde forte queretur de hoc quod dicit quod opposita concluduntur per solam primam et tertiam {concluduntur per solam primam et tertiam Cm1E1E2F1F2KlO1O2O3P1V: concluduntur per secundam figuram et tertiam Ed} Sed queritur quare non est instantia evidens in secunda figura

81rb–va

Dubitatur hic aliquis de pertinentia {pertinentia E 1F1KlO1O2O3P1V: particularia Ed} huius partis quia cum entimema sit ratio contracta ad materiam particularem {particularem Cm1E1F1F2O1O2O3: om. Kl: naturam Ed} pertinet enim ad rhetoricum secundum Aristoteles in posterioribus non debet hic determinari nisi determinetur de syllogismo simpliciter et in communi et cum queritur cui pertineat entimema Deinde forte queretur de reductione entimematis in syllogismum Deinde adhuc forte dubitabit aliquis que sit differentia ichotis et signi Deinde queretur forte de diffinitione entimematis Adhuc si entimema sit ex ichotibus et signis tunc in universalibus exiguntur plura signa et plures ichotes

82ra

Hic forte dubitaret aliquis cum Aristoteles doceat invenire medium syllogisticum et etiam medium similiter entimematis quare non docet invenire medium in inductione et exemplo Consequenter forte dubitabit ad quem pertineat huius signi inventio Sed tunc accidit dubitatio cum enim hec ars ad logicum pertineat videtur quod hec sufficientia debeat hic determinari Ulterius forte dubitabit aliquis de hoc quod supponit naturales passiones transmutare animam Adhuc dubitabit cum debeat ostendere habitum magnarum extremitatum signum esse fortitudinis hic supponit Sed tunc dubitatur quia videtur quod eadem ratione posset ostendi habitus magnarum extremitatum signum esse fortitudinis in leone Adhuc videtur quod sua ars nulla sit

83ra

81va

B27 Part 1 Dub. 1

Dub. 2 Dub. 3 Dub. 4 Dub. 5

82ra–b 82rb 82rb 82rb

B27 Part 2 Dub. 1

Dub. 2 Dub. 3 Dub. 4 Dub. 5 Dub. 6 Dub. 7

83ra 83ra–b 83rb 83rb 83rb 83rb

INDEX LOCORUM Albert the Great, De Animalibus XXII.ii.1 229 XXV 228 Albert the Great, De Somno et Vigilia I.6 229 Albert the Great, Metaphysicorum V.iv.1 63 Albert the Great, Priorum Analyticorum 5–6 I.i.5 58 I.i.6 62 I.i.13 98 I.ii.3 128 I.ii.6 129 I.iii.3 161 I.iii.4 160 I.iii.6 160 I.iv.23 228 I.iv.24 225 I.vii.10 17 IV.iv 184 Alexander of Aphrodisias, Prior Analytics commentary A2, 25a14–17 (34,17–20) 87 A4, 26a2–9 (55,17ff.) 123 A4, 26a30–33 (62,25ff.) 123 A4, 26a39–b10 (63,10ff.) 123 A4, 26b21–25 (68,15 ff.) 123 A7, 29b1–2 (113,7–15) 65 A16, 36b19 ff (217,8ff.) 105 A19 (240,4–11) 231

Algazel, Logica 228 247 248 260 262–263 263 265

124 55 56 118 124 131 137

Aristotle, Categories 3, 1b10–15

157

Aristotle, De Anima II.4, 415a22ff.

63

Aristotle, De Caelo II.7, 289a29–34

81

Aristotle, Metaphysics

Δ16, 1021b14 ff. 63 Δ16, 1021b23ff. 63 Δ16, 1021b30–1022a3 62

Aristotle, Physics VI.1, 231b15 ff.

55

Aristotle, Posterior Analytics A1, 71a9–11 42 A2, 72a7 55 Aristotle, Prior Analytics A1, 24a16–17 11 A1, 24b22–23 61 A1, 24b26–30 64 A1, 25b18–20 42 A2, 25a15–17 20, 84 A2, 25a17–19 84 A2, 25a20–22 84 A3, 25a27–32 89 91 A3, 25a39–40

298 A3, 25a40–b3 A3, 25b4–13 A3, 25b10–11 A3, 25b18 A4, 26a33–36 A4, 26a36–39 A4, 26b21–25 A5, 26b34–36 A5, 26b36–39 A5, 27a5–14 A5, 27a14–15 A5, 27a36–b1 A5, 28a4–7 A5, 28a7–9 A6, 28a17–22 A6, 28a26–29 A6, 28b7–14 A6, 28b17–20 A6, 28b33–35 A6, 29a16–18 A7, 29a23–26 A7, 29a26–17 A7, 29a37–39 A7, 29b5–6 A7, 29b19–21 A8, 29b36–30a2 A8, 30a2–3 A8, 30a12–13 A9, 30a15 ff. A9, 30a17–23 A9, 30a23–25 A9, 30a32–33 A9, 30a33ff. A9, 30b1–2 A9, 30b2–3 A9, 30b5–6 A10, 30b7–9 A10, 30b9–18 A10, 30b9–13 A10, 30b13–18 A10, 30b31–40 A10, 30b35 A10, 31a1ff. A10, 31a5–10 A10, 31a10–15 A10, 31a15–17 A11, 31a18ff.

index locorum 94 91 91 35 121 121 121 129 130 130 130 130 81 133 135 135 135 135 135 137 142 143 135 130 135 150 150 20 153 156 156 156 153 156 156 156 164, 167 163 165 165 164, 167 175 162 163, 165 164, 167 164, 167 169

A11, 31a24–37 A11, 31a24–30 A11, 31a31–33 A11, 31b4–10 A11, 31b11ff. A11, 31b12–20 A11, 31b27–33 A11, 31b33–37 A11, 31b35–37 A11, 31b40–32a5 A11, 31b40–32a4 A12, 32a6ff. A13, 32a18–20 A13, 32a29–b1 A13, 32a40 A13, 32b4–10 A14, 32b38–33a1 A14, 33a1–5 A14, 33a5–12 A14, 33a12–17 A14, 33a21–25 A14, 33a25–27 A14, 33b3–17 A14, 33a27–34 A15, 33b33–40 A15, 33b33–36 A15, 34a12–22 A15, 34a16–22 A15, 34a17–18 A15, 34a34–b1 A15, 34b19–27 A15, 34b30–31 A15, 34b35–b2 A15, 35a6–11 A15, 35a16–20 A15, 35a30–35 A15, 35a30–31 A15, 35a36–40 A16, 35b23–26 A16, 35b26–28 A16, 35b38–36a2 A16, 36a2–5 A16, 36a8–17 A16, 36a8–15 A16, 36a15–17 A16, 36a17–21 A16, 36a25–27

170 172 172 174 169 170, 172 174 172 170, 172 174 175 169 30 93 35 24, 32 182 182 182 182 182 182 185 182 193 195 106 108 52 193, 195 193, 195 193 193 195 195 193, 195 195 195 212, 215 221 212, 214 214 212 214 215 214 212, 214

index locorum A16, 36a34–b2 A16, 36a34–39 A16, 36a39–b2 A16, 36b7–10 A17, 36b31 A17, 37a4–8 A17, 37a32–35 A17, 37a35–37 A18, 37b24–35 A18, 37b24–28 A18, 37b29–35 A18, 37b29 A18, 38a3–7 A18, 38a3–4 A18, 38a4–7 A19, 38a16–18 A19, 38a21–26 A19, 38a38–b3 A19, 38b6–13 A19, 38b13–17 A19, 38b25–27 A19, 38b27–29 A19, 38b31–35 A19, 38b38–40 A20, 39a14–b2 A20, 39a14–19 A20, 39a19–23 A20, 39a26–28 A20, 39a31–36 A20, 39a36–38 A20, 39a38–b2 A21, 39b7–9 A21, 39b10–30 A21, 39b16–22 A21, 39b22–25 A21, 39b26–30 A21, 39b31–39 A21, 40a1–2 A22, 40a13–23 A22, 40a13–16 A22, 40a18–23 A22, 40a25–35 A22, 40a25–32 A22, 40a25–31 A22, 40a33–35 A22, 40a39–b1 A22, 40b2–6

212 215 215 168 201 97 187 188 200 202 202 202 200 202 202 222, 225 225 226, 227 222, 225 226 222, 226 226 222, 226 224 190 191 192 192 192 192 192 208 205 205, 209 209 205, 209, 210 206, 210 207 234 237 237 234 237 237 237 234, 237 234

A22, 40b2–3 A22, 40b3–6 A24, 41b6–7 A27, 43b22–29 A29, 45a28–31 A29, 45a31–33 A41, 49b14–16 A41, 49b17–20 B1, 53a11–12 B5, 58a26–32 B5, 58a27–28 B11, 61b11–15 B11, 61b19–23 B11, 61b37–38 B14, 63a7–14 B14, 63a16–18 B14, 63a19–23 B16 B21, 67a33–37 B22, 68a21–23 B23, 68b17–18 B23, 68b27–29 B23, 68b35–37 B27, 70a10–11

299 235, 238 238 114 249 130 135 17 17 14 119 14 135 130 130 130 130 135 42 56 51 50 50 51 52

Aristotle, Sophistical Refutations 5, 167a1–2 25 5, 167b35 60 Aristotle, Topics I.12, 105a13–17

49

Ars Burana 198:26–199:16

125

Augustine, De Genesi ad literam 6, 5, 8 4 Augustine, De vera religione 18, 35–36 4 Boethius, De syllogismo categorico 821B–C 43 822A–B 43 Boethius, De topicis differentiis 1174C 44

300 1184B–C 1183D–1184A 1184A–B 1184D

index locorum 52 49 49 49

Boethius, De Divisione 32, 23–24 70 Boethius, Introductio ad syllogismos categoricos 804A 87 Dialectica Monacensis 488:3–10 489:13–490:11 490:12–15 498:5–499:4 499:7–8 499:11–12 500:15–16 500:22–23 500:25–27 500:31–501:8 502:8–10 502:23–24 502:33–503:3 503:17–21 503:28–29 504:4–6 504:7–9 514:6–9 587:28–30

52 47 46 126 131 136 152 154 163 170 181 194 213 201 224 190 207 69 128

Ibn Sina, Remarks and Admonitions I.vii, 396 117 Kilwardby, De Ortu Scientiarum 174 128 504 62 527 62 578 73 Kilwardby, Prior Analytics commentary 1, 3–5 Prologue dub.7 6, 149 Prologue dub.8 58, 147 A1 Part 1 dub.1 58

A1 Part 1 dub.2 A1 Part 2 dub.1 A1 Part 2 dub.2 A1 Part 2 dub.3 A1 Part 2 dub.4 A1 Part 2 dub.5 A1 Part 4 dub.1 A1 Part 4 dub.2 A1 Part 5 dub.1 A1 Part 5 dub.3 A2 dub.4 A2 dub.5 A2 dub.6 A2 dub.7 A2 dub.8 A2 dub.9 A2 dub.10 A2 dub.11 A2 dub.12 A2 dub.13 A2 dub.14 A3 Part 1 dub.1 A3 Part 1 dub.2–3 A3 Part 1 dub.4 A3 Part 2 Note A3 Part 2 dub.1 A3 Part 2 dub.2 A3 Part 2 dub.3 A3 Part 2 dub.4 A3 Part 2 dub.5 A3 Part 2 dub.7 A3 Part 2 dub.9 A3 Part 2 dub.10 A3 Part 2 dub.11 A3 Part 2 dub.12 A4 Part 2 dub.2 A4 Part 2 dub.4 A4 Part 2 dub.7 A4 Part 2 dub.8 A4 Part 2 dub.9 A4 Part 2 dub.10 A4 Part 2 dub.11 A4 Part 2 dub.12 A4 Part 2 dub.14 A5 dub.1

55 11 12 12, 13 12 12 42, 44, 45, 46, 47 58 61 62 84 84 84 85 87 84 84 87 115 87 78, 79, 80, 81, 82, 83 93 90 20, 89, 96 93, 95 28 30 30 95 29, 32, 95, 98 97 92 92, 94 92 29, 97 69 57 128 114, 118 118, 122 116 120 67 129 129

index locorum A5 dub.2 A5 dub.3 A5 dub.4 A5 dub.5 A5 dub.8 A5 dub.9 A5 dub.10 A5 dub.12 A5 dub.13 A5 dub.14 A6 dub.2 A6 dub.3 A6 dub.4 A6 dub.5 A6 dub.6 A6 dub.7 A6 dub.8 A6 dub.9 A6 dub.10 A7 dub.1 A7 dub.2 A7 dub.3 A7 dub.4 A7 dub.5 A7 dub.6 A7 dub.8 A7 dub.9 A8 Note A8 dub.4 A8 dub.6–7 A8 dub.8 A9, 30a9–33 A9, 30a15–33 A9 Note 1 A9 dub.1 A9 dub.2 A9 dub.5 A9 dub.6 A9 dub.7 A10, 30b7–40 A10, 31a1–17 A10 dub.1 A10 dub.4 A10 dub.5 A11, 31a18–b12

130 63 100 101 131, 134 131, 132 133 134 134 135 136, 140 137, 139 139 137 138 138 138 141 142 143 144 144 145 143 66 140 65 152 165 151 151 159 153, 156 37 156 38, 157, 158 67, 154 19, 155, 159, 161 39, 149, 156 159, 162 168 163, 164, 165, 167, 171 166 163, 164, 171 159, 169

A11, 31b33–32a5 A11, 31b12–33 A11 dub.1 A11 dub.2–3 A11 dub.3 A11 dub.5 A13 dub.1 A13 dub.2 A13 dub.3 A13 dub.4 A13 dub.6 A13 dub.7 A13 dub.8 A13 dub.10 A13 dub.11 A13 dub.12 A14 dub.1 A14 dub.2 A14 dub.3 A14 dub.4 A14 dub.5 A14 dub.6 A15, 34a2–33 A15, 34a34–b6 A15 dub.1 A15 dub.3 A15 dub.4 A15 dub.5 A15 dub.6 A15 dub.8 A15 dub.9 A15 dub.12 A15 dub.14 A16 dub.1 A16 dub.4 A16 dub.5 A16 dub.6 A16 dub.7 A16 dub.8 A16 dub.9 A16 dub.10 A16 dub.13 A17, 36b35–37a31

301 173 159 159, 170, 173, 174 170, 173 20, 176 170 28 30 31 30 93 24, 25 33 22, 35, 217 34 35, 217 180 34, 183, 184 183 185 31, 186 181 106 103 22 196 54 39, 107 39, 108, 148 105 38, 107 109, 199 194 22, 34, 196, 216, 217 216 15, 26, 29, 38, 39, 199, 215, 218 220 23, 24, 221 27, 198, 222 219, 220 221 213 97

302

index locorum

36, 189 187 186, 187, 237 28 188 200 200 105, 202 105, 203 201 222, 225 223 226, 229 227 26 226, 227, 231, 232 A19 dub.11 230 233 A19 dub.12 A19 dub.14 225 A20 dub.1 183 A21 dub.1 210, 211 A21 dub.2 105, 191 A21 dub.3 105, 212 A21 dub.4 207, 208 A21 dub.5 207 A22, 40a39–b16 235 A22 Note 224, 234 A22 dub.2 236 A22 dub.4 204 A23 dub.1 59, 72 A24 dub.1 115 A24 dub.2 12, 59, 114 A25 Part 1 dub.1 52, 60 52 A25 Part 2 dub.1 A25 Part 2 dub.2 143, 215 A26 dub.2 128 A27 dub.3 248 A28 Part 1 dub.1 250 A28 Part 1 dub.2 127 A28 Part 1 dub.5 249 A28 Part 1 dub.6 249 A28 Part 2 dub.2 114 A28 Part 2 dub.4 249 A29–30 Part 1 dub.2 100 A29–30 Part 2 dub.1 248 A31 dub.1 60 A17 Note A17 dub.1 A17 dub.2 A17 dub.3 A17 dub.6 A18 dub.1 A18 dub.2 A18 dub.3 A18 dub.4 A18 dub.6 A19 dub.1 A19 dub.2 A19 dub.3 A19 dub.4 A19 dub.9 A19 dub.10

A31 dub.2 A32–33 dub.1 A32–33 dub.6 A33, 47b15–40 A34 dub.2 A35 dub.2 A36–37 dub.2 A36–37 dub.3 A36–37 dub.4 A38–40 dub.5 A41 dub.1 A41 dub.2 A42–44 dub.2 A45, 50b5 ff. A45 dub.1 A45 dub.2 A45 dub.4 A45 dub.6 A46 Part 1 dub.10 A46 Part 1 dub.13 A46 Part 1 dub.18 A46 Part 2 dub.3 B1 Part 1 dub.2 B1 Part 2 dub.1 B2 Part 2 dub.1 B2 Part 2 dub.2 B2 Part 2 dub.3 B4 Part 1 dub.3 B4 Part 2 dub.1 B4 Part 2 dub.2 B5 dub.1 B5 dub.2 B5 dub.9 B8 dub.1 B8 dub.6 B9 Note B10 Note B11–12 Part 1 dub.4 B12 dub.5 B14 dub.3 B15 dub.1 B15 dub.6 B16 dub.3 B17 dub.2 B18–20 dub.2 B21 dub.5 B22 dub.1

127 75 14 18 26, 27 115 13 13, 72 13 86 18 18 49 75 76 76 99 76 88 15 16 17 14 13 72 47, 48, 72 43 16 55 56 59 116 119 101 73 131 136 100 101 59 60, 61 68 60 60 106 56 51

index locorum B23 dub.1 B23 dub.4 B23 dub.7 B23 dub.9 B23 dub.10 B24 dub.3 B24 dub.4 B27, 70a10–11 B27 Part 1 dub.2 B27 Part 1 dub.4

51 51 52 51 50 50 50 52 52 53

Lambert of Auxerre, Logica Tract V f.91v 42 Pacius, In Porphyrii Isagigen et Aristotelis Organon Commentarius Analyticus 264B 52 Peter of Spain, Tractatus 19:24–25 70 22:20–21 70 81:12–18 23

303

Philoponus, Prior Analytics commentary A7, 29b1–2 (114,16–22) 65 Plato, Phaedo 65d 74a

20 20

Roger Bacon, Summulae Dialectices 289.33–34 42 290.11–18 42 Vincent of Beauvais, Speculum Quadruplex III.xl, 242 23 William of Sherwood, Introduction to Logic III.i 47 V.viii 149

GENERAL INDEX Accident, 20, 81, 89, 98, 180, 227, 230 actualization, 19, 107, 199 affirmation, 13, 16, 17, 36, 115, 133, 138–139, 140, 155, 221, 284 Al-Ghazali, 70, 118, 124, 125, 131, 137 Albertus Magnus, 1, 5, 6, 17, 34, 55, 58, 84–85, 98, 117, 128–129, 160– 161, 184, 225, 228–229 Alexander of Aphrodisias, 65, 87, 102, 105, 123–125, 127, 198, 231 ampliation, doctrine of, 34 antecedent, positing the ~, 49, 56, 119 appellation, rule of, 85 appropriation, 148, 160–161, 165– 166, 172–174, 177, 220, 241, 243 argumentation, 42–44, 80 Aristotle, 1, 3, 4, 6, 8, 11–14, 16, 17, 20–28, 30, 32, 35, 41–43, 45, 49, 51–52, 55–58, 60–64, 66, 73, 76, 80, 84, 86–87, 89–91, 93, 97, 100, 102–103, 105–106, 108–110, 113– 115, 119, 123, 127, 129–133, 135– 137, 140, 142–144, 147–150, 152– 154, 156, 159, 161–170, 172–177, 179–180, 182–183, 185–191, 193– 194, 196–197, 199–202, 205–215, 220–222, 224, 226–228, 230–235, 238–239, 242–244, 246, 248–249, 251–253 ars vetus, 3 assimilation, 41, 67, 74, 197 Augustine, 4 Augustinian theology, 2, 4 Averröes, 21 Avicenna, 117, 228 axiomatization, 76

Bacon, Roger, 5, 25, 42, 149 Barbara, 66, 103, 125, 127–128, 151, 252 LLL, 151, 231, 235, 244, 252 LMM, 151 LQM, 214, 223, 235, 244, 252 LXL, 157, 159, 160–161, 242, 270 LXX, 211 MLM, 151 QLM, 214 QLQ , 212, 214, 218, 223, 233, 244, 270 QQQ , 182–184, 252 QXQ , 193, 195–196, 211, 230, 252 XLL, 161, 232, 243, 253 XQM, 102–104, 106, 107, 109, 195, 197, 270 Baroco LLL, 20, 149, 150–151 LQM, 231, 244, 246 LQX, 226 LXL, 166–167 QLM, 233–234, 239, 244–246 QLX, 226, 230 QXM, 203–204 XQM, 203 being, perfection of ~, 41, 62, 80 being-said-of-all, 41, 150, 156, 157, 159, 217 being-said-of-none, 41, 150, 156 Blackfriars, 1 Bocardo, 102, 136, 140–141, 173, 192, 234 LLL, 149, 150–151 LXL, 170, 174 LXX, 103 LQM, 244 QLM, 235, 238, 244 QLQ , 238

306

general index

QXM, 206 QXQ , 210 XLL, 174 XQM, 211–212 Boethius, 2, 12, 41, 43–44, 49, 52, 62, 70, 87 Buridan, Jean, 206, 242 Cardinals, College of, 1 Camestres, xix, 76, 100, 127, 130, 142, 145, 252 LLL, 151, 252 LQM, 227, 231–233, 239, 244, 246, 253 LQX, 226–227 LXL, 166–167, 253 QLM, 225, 252 QLX, 225, 244 QXM, 202–203, 252 XLL, 165–166, 242, 252 XQM, 203 Campsall, Richard, 157 cause, 6, 19, 38, 46, 56, 80–82, 115– 116, 138, 158, 189, 228 accidental ~, 27 appointed ~, 98 efficient ~, 1 essential ~, 19, 26–27 final ~, 41, 56, 59, 62, 74 formal ~, 6, 57–58 inseparable ~, 19, 26, 27 material, 6, 57 separable, 27 Celarent, 65, 66, 76, 79, 82, 125, 127–128, 130, 135, 140–141, 163, 165, 208, 252 LLL, 150, 252 LQL, 220–221 LQM, 214, 252 LQX, 199, 214, 218, 220–221, 244 QLM, 214 QLQ , 212, 214, 223, 231, 244, 252 QQQ , 182, 252 QXQ , 193, 195, 252 XLL, 167, 253 XLX, 242 XQM, 195, 252

XQX, 105, 198–199 XQQ , 199 circularity, 54, 86 completeness (sufficientia), 3, 5, 50, 147, 152, 154 conclusion, 11–14, 17, 41, 43, 45, 48, 54, 56, 58–59, 60–64, 66–68, 71–76, 82–83, 92, 100, 102, 106, 108–111, 115, 117, 120–121, 125, 127–128, 132, 134, 137–138, 142– 144, 148, 150, 153, 154–161, 168, 175, 179, 184–185, 187–189, 196– 198, 204, 209, 215, 219, 220, 222, 230–233, 243, 245, 248–249 affirmative ~, 54, 116, 132–133, 139, 245 assertoric ~, 103–104, 198–199, 211–212, 215, 219–221, 226– 227, 230–231 contingency ~, 103–104, 107, 190, 199, 209, 211, 215, 231, 234 false ~, 100, 133, 169, 184 indirect ~, 142–143, 145 necessary ~, 50, 53, 114, 160 necessity ~, 152–156, 158–159, 162–163, 169 negative ~, 54, 116, 132–133, 139, 145 possibility~, 99, 108–109, 199, 220, 232, 234 subaltern ~, 142, 143 syllogistic ~, 110, 117, 121, 122, 173, 191, 236, 246, 249 conjunction, 108–109, 241 consequence, xv, 3, 41, 49, 55–56, 59, 62, 71, 83–84, 100–101, 110, 142, 179, 231–232 249–250 consequent denying the ~, 49, 101 fallacy of ~, 47, 87, 120 contingency, 28, 30, 32, 34, 94–95, 97–98, 189, 197, 199, 200, 207, 231, 234, 239, 245, 252 generic ~, 19, 28, 40, 90–91 indeterminate ~, 32–33, 39, 94, 97–98, 107–108, 197, 203, 218, 242

general index natural ~, 33, 38–39, 94, 98, 107– 108, 199, 218–220 necessary ~, 40 non-necessary ~, 40, 98 specific ~, 90, 92–93 contraposition, 87–88 contrasted instances, method of, 121 conversion, 77–87, 89, 93, 96–97, 110, 131, 135–136, 139, 141, 144, 186–187, 223–224, 234, 237, 243, 250 assertoric ~, 78 complementary ~, 93, 244, 247 contingency ~, 90–91, 93–94, 96 necessity ~, 96 term ~, 80, 93–94, 97, 190, 200 ~ to, 84 modal ~, 84, 150, 159 proofs of ~, 82, 86, 92 e-conversion, 20, 85, 86 i-conversion, 86 o-conversion, 87 Le-conversion, 90 Mi-conversion, 90 Qa-conversion, 94, 96 ~ with, 84 Darapti, 77–78, 140–141, 205, 252 LLL, 252 LXL, 252 LQM, 244, 252 XQM, 206, 252 XQX, 105, 208 QLM, 237 QLQ , 244, 252 QQQ , 192, 252 QXQ , 205, 252 QXM, 206, 209 XLL, 252 Darii, 66, 76, 82, 100, 126–127, 140– 141, 171–172 LLL, 150, 231, 244 LQM, 215, 244 LXL, 157, 170, 226, 242 QLM, 215 QLQ , 212, 215 QQQ , 193

307

QXQ , 193, 205 XQM, 205 Datisi, 77, 78, 140, 141, 205 LQM, 244 LXL, 242 QLM, 237 QLQ , 244 QXM, 206, 209 QXQ , 205 XLL, 174 XQM, 206 denial, 13, 16, 199 De Rijk, L.M., 6, 23, 46, 125 Disamis, 76, 140–141, 198, 209 LQM, 237 LQQ , 237, 244 LXL, 174 QLM, 244 QLQ , 238 QXM, 205, 206, 209 XLL, 196, 242 XQM, 206, 209 XQQ , 205, 209 discourse [oratio], 11, 42–45, 48, 57– 58, 62, 71, 73 Dominican Order (Order of Preachers), 1 Ebbesen, Sten, 5, 85 Edward I, King of England, 1 Eleanor (of Castile) (Queen of Edward 1), 1 enthymeme, 41–43, 45, 46, 48–49, 52–53, 73, 77–83 example, 42–44, 45–46, 50, 77 Felapton, 140–141, 205, 252 LLL, 151, 174, 252 LQM, 237, 244, 252 LXL, 174, 175, 211, 252 QLM, 237 QLQ , 244 QXM, 206, 209 QXQ , 205 XLL, 170, 174–175, 253 XQM, 210–211, 252 Ferio, 66, 86, 126–127, 141, 144, 171

308

general index

LQM, 215 LQX, 215, 244 LXL, 157, 170, 220, 242 QLM, 215 QLQ , 212, 215, 231, 244 QXQ , 193 XLL, 175 XLX, 204 Ferison, 140, 168, 205 LQM, 238, 244 LQX, 238 LXL, 242 QLM, 238 QLQ , 238,, 244 QXM, 206, 210 QXQ , 205, 210 XLL, 174–176 XQM, 210 Festino, 130, 135 LQM, 226 LQX, 226, 244 LXL, 165–167, 242 QLM, 231, 244, 246 QLX, 226 QXM, 204 XLL, 166–167 XQM, 202–203 figure, 4, 42, 46–47, 52, 57, 68–74, 114–115, 119, 129 form, 16, 57, 80–81 proximate ~, 74, 114 remote ~, 74, 114 syllogistic ~, 4, 47, 57 formation-rules, 243 forms, plurality of, 57 Gentile da Cingoli, 5 Giles of Rome, 2, 8 Gregory X, Pope, 1 Identity, law of, 77 imagination, 16, 24 induction, 41–43, 45–46, 48–52, 73, 77, 94 inference accidental ~, xv, 55–56, 73 essential ~, xv, 55–56, 73

inferior, 19, 23, 24, 69, 70–71, 87, 94, 109, 120, 127–128, 132–133, 137, 145, 158, 197 invention, 250 Judgment, 249, 250 Kilwardby, Robert, passim Lagerlund, Henrik, xv, 28, 32, 34, 88–89, 99, 113, 157, 161, 183, 209 law moral ~, 241 natural ~, 33, 109, 241 Lewry, Osmund, 3, 4, 147 Marenbon, John, xv matter, 12, 47–48, 68, 74 mixture, 102–103, 108, 113, 117, 121, 123–124, 126–128, 131, 135, 140, 142–143, 145, 147, 153–156, 162, 165, 169, 179, 185, 191, 194, 201, 205, 207–210, 214, 216, 222, 232– 233, 239, 243, 246–247, 251–252 assertoric/contingency ~, 193– 212 assertoric / necessity ~, 153–176 necessity/contingency, 212–238 useful ~, 113, 117, 122–123, 125, 134, 140, 147, 155, 162, 164– 165, 169, 171, 172, 177, 181, 190–191, 194, 201, 207, 213– 214, 224–225, 235–236, 238– 239 useless ~, 42–43, 45, 122–123, 125–126, 134, 147, 154–155, 169, 219, 230, 232, 236, 259 modality, 18–19, 66–67, 154–155 alethic ~, 241 deontic ~, 33, 40, 241 ~ of naturalness, 33–34, 39, 109, 197, 241 ~ of necessity, 21, 23, 26, 28, 33, 39, 109, 153, 157–158, 175, 197, 220 ~ of possibility, 28, 31, 33–34, 40, 109, 196, 197, 220 modus ponens, 119

general index monotonicity, 64–67, 74, 127, 153, 157, 159, 183, 216 mood, 4, 12, 42, 46–47, 57, 68, 74, 114–115, 119, 129 affirmative ~, 170–173, 197, 199, 221, 223 assertoric ~, 158, 161–162, 167– 168, 170 contingency ~, 179, 196–197, 239, 241 direct ~, 41, 65, 129, 141 imperfect ~, 113, 157, 193, 212, 216, 219, 223 indirect ~, 76, 141, 142, 144, 145 invalid ~, 104, 130, 161, 175, 221, 230 necessity ~, 170, 172, 179, 239, 241 negative ~, 131, 173, 197, 199, 207, 218, 219, 237 non-modal ~, 150 order of ~s, 129, 135, 141 particular ~, 65, 66, 135, 140–142, 207 perfect ~, 41, 65, 67, 113, 148, 152, 158, 172, 185, 194–195, 213, 216, 223–224 strengthened ~, 141 subaltern ~, 143, 145 universal ~, 65–67, 129, 135, 140– 142 valid ~, 43, 113, 129, 148, 158, 162, 170, 172, 174–175, 193, 212, 221, 233 Necessity, 26–28, 33, 39, 153, 197 accidental ~ 176 de dicto ~, xviii, 31, 40 de re ~, 40 evident ~, 63, 65, 79–83, 185 inner ~, 21 outer ~, 21 per accidens ~, 19–21, 27, 40, 89, 160, 168, 176 per se ~, xviii, 19–21, 23–24, 29, 40, 89 non-causa ut causa, 42, 43, 45, 47, 60 Nicholas III, Pope, 1

309

Ontology, 14, 19, 21, 26, 145, 148, 158 ordo, 177, 239 Oxford University of, 1, 2 Paris University of, 1, 2, 4 Patzig, Günther, 65, 66, 69, 74, 158 Peter of Spain, 70, 117 petitio principii, 43, 45, 47, 59, 60 Philoponus, 65, 141 Plato, 20 pons asinorum, 241, 248, 250 possibility, xviii, 19, 28–29, 32–34, 40, 90–91, 96, 105–106, 108–109, 111, 180, 196–197, 220, 239, 241, 252–253 Prantl, 5 premise, xix, 11, 13, 17, 36–37, 41, 43, 50–56, 58–61, 66, 68–69, 71, 73–74, 77, 82–83, 86, 101, 104, 106, 108, 110–111, 114, 116, 121–122, 141–145, 148, 150, 153, 155, 157–158, 163, 170, 180, 185, 187–188, 196–197, 199, 203, 207– 208, 215, 219–220, 222, 224, 228, 230, 232–234, 243–245, 248, 250 affirmative ~, 102, 114–117, 118– 122, 133–134, 136–137, 143, 145, 152, 155, 171–173, 175–176, 181, 185–186, 203–204, 213, 231, 236–237, 245, 249 assertoric ~, 38, 102, 104, 107– 108, 148, 153–156, 159–162, 165–168, 172–175, 198–200, 202–208, 210–211, 217, 219– 220, 227, 230, 238 contingency ~, 36–37, 96, 102, 107–108, 187, 190, 193, 197, 200, 203–205, 207, 210, 212, 216–222, 226, 231, 234, 247 false ~, 47, 71, 72 major ~, 13, 38, 62, 66, 102, 106, 108, 118–122, 126, 131, 143, 145, 149, 152–156, 159, 161, 171– 172, 176, 179, 181–183, 186–187,

310

general index

193–194, 197–201, 207, 211, 213, 215–220, 223, 234, 236, 238, 246–247 minor ~, 13, 24, 26, 38, 50, 57, 62, 66–67, 102, 104, 107–109, 118–123, 125, 133, 134–140, 142–143, 145, 148–149, 152– 156, 158–161, 163, 171, 173, 175–176, 183–185, 188–189, 191, 194, 196–197, 199, 204, 206–208, 211, 213, 216–222, 227, 231, 236, 238, 247 necessity ~, 37–38, 102, 148, 153, 155–156, 158–159, 170, 212, 217, 219–220, 223–224, 233–234, 246 negative ~, 102, 117, 118–120, 122, 125–126, 129, 131, 133, 136–137, 139–140, 142–143, 145, 152, 171, 176, 185–187, 191, 199, 207, 213, 224, 231–232, 238, 246– 247, 249 opposed ~s, 60, 68, 74 particular ~, 49, 118, 121, 132– 133, 142, 213, 236 possibility ~, 180 true ~, 72, 121, 133, 168–169, 230, 248 universal ~, 114–115, 118, 122, 126, 131, 136, 138, 141–142, 145, 152, 155, 171, 176–177, 181, 190–191, 194, 200–201, 207, 213, 224, 236, 246, 248 principles, 108, 113–114, 117–119, 122–127, 131–134, 137, 142, 145, 147–148, 152, 154–155, 160, 162– 164, 166, 170–171, 176–177, 173, 177, 179, 181, 190, 194, 198, 201, 207–208, 213, 224, 235–236, 238, 239, 244, 246, 250 Priscian, 44 proposition, xviii, 5, Chapter 1 passim, 47–48, 53–57, 60, 64, 69, 74, 77, 80, 82–83, 85, 87–89, 91, 107, 110, 114–115, 119–120, 125, 127, 138, 145, 148–150, 155, 158, 168, 173, 175–176, 182–183, 188–

189, 207, 221, 228, 241, 243, 246, 250 affirmative ~, 12, 13, 15, 17, 26, 27, 29, 36, 39, 54, 67, 69, 84, 91, 95, 110, 115–116, 119, 138, 140, 144–145, 152, 154, 165– 166, 170, 173, 177, 187, 207, 221, 232, 245, 248 apodeictic ~, 38 assertoric ~, 15, 19, 23, 27, 37– 40, 65, 67, 84–85, 93, 102–104, 107, 148, 150, 152–153, 157–158, 162–163, 165, 169, 173, 177, 180, 196, 199–200, 203–204, 208, 210–212, 216–217, 219– 222, 234, 245, 250 as-of-now ~, 19, 106, 126, 156, 159, 160, 165–166, 168, 198, 202, 204, 206, 210–212, 220, 245 categorical ~, xviii, 13, 244–245, 248 contingency~, 19, 22, 28, 31– 32, 33–38, 40, 66–67, 90–97, 108, 110, 117, 180, 186–191, 196, 199–200, 203, 209–210, 215–218, 222–223, 231–234, 237–238, 245–247 false (nugatory) ~, 27, 86 indefinite ~ 84 modal ~, 23, 25, 34, 179, 241, 250,, 252 necessity ~, 19–29, 31, 34, 36, 38–40, 66, 88–89, 92–93, 96, 102, 107, 110, 148, 150, 153–160, 162–163, 169, 171, 173, 176–177, 186, 197–199, 213, 217, 220–223, 227, 234, 236, 238, 242–243, 245, 252– 253 negative ~, 15, 19, 26–29, 31, 34, 36, 40, 65, 89, 91, 113, 121, 131, 133, 162–164, 169, 170–171, 177, 181, 186–187, 194, 197, 201, 207–208, 212–214, 219, 221, 233, 236, 238, 245–246, 248, 252–253

general index particular~, 77, 84, 90, 102, 110, 124, 139, 248 possibility ~, xviii, 19, 28–29, 34, 40, 91–94, 110, 227, 232, 234 “prosleptic” ~, 18 reduplicative ~, 85 syllogistic ~, 12, 45, 122, 134 true ~, 26–27, 47, 228 universal ~, 18, 36, 65–66, 77, 84, 97, 102, 121, 124–125, 127, 129, 139, 155, 162–164, 170–171, 173, 177, 186–187, 196, 201, 207, 212, 224, 233, 238, 246–248, 252–253 unrestricted ~, 19, 37–40, 106– 108, 148, 156, 158–162, 165– 166, 170, 173, 179, 196–198, 203–205, 210–211, 217, 219– 221, 226, 227, 230, 245 proposition-forms, 68, 127, 243 secundum rem, 16, 17 secundum vocem (secundum quid), 16, 17 psychologism, 53 Quality, xvii, xviii, 12–13, 15–16, 45, 67, 88–89, 93, 116, 129, 131, 175, 184–185, 224, 245–246 Reasoning, syllogistic, 23, 147 reduction, 63, 65, 69, Chapter 3 passim, 130, 135, 140–141, 144, 148, 150, 163, 170, 185, 187, 190, 200, 202, 204–206, 210, 222, 226–227, 231, 234 direct, 76, 79, 110, 130, 135, 139– 140, 144–145, 150, 163, 170, 185, 187, 193, 200, 205, 222, 224, 234 indirect, 36, 76, 99, 100–101, 105, 110, 130–131, 135–136, 151, 187–190, 206, 211, 213, 224, 226, 230–231, 233–235, 239, 244, 246 reduplication, 85, 86

311

Semerion, 228 Street, T., xv superior, 19, 69, 70–71, 87, 94, 109, 120, 128, 132–133, 137, 145, 158, 239 suppositio theory of, 149 syllogism ~ ad impossible, 100 affirmative ~, 61, 116, 200 assertoric, ~, 81, Chapter 4 passim, 147, 150–154, 157, 161–163, 165, 169–170, 175, 177, 194, 201, 238, 241, 249– 250 ~ by diminution, 142 circular ~, 14, 59, 78, 119 contingency ~, 110, 117, Chapter 6 passim, 247 conversive ~, 100–101, 131, 136 demonstrative ~, 58, 115 dialectical, ~, 58, 60 disjunctive ~, 60 expository~, 86 ~ from false premises, 47 hypothetical ~, 30, 48, 49, 119 inductive, ~, 50–51 imperfect, ~, 61–63, 68, 76, 78– 80, 83, 148, 185, 197–198, 218 necessity ~, 24, 38, 110, 117, Chapter 5 passim, 213, 224, 234, 236, 238, 241 non-modal ~, 113, 143, 149, 250 ostensive ~, 47–49 possibility ~, 180 potential ~, 75, 147 scientific ~, 50 singular ~, 114, 119 perfect ~, 13, 41, 61–63, 67– 68, 74–76, 78, 82, 83, 103, 115, 127, 148, 154, 156, 162, 183–185, 196, 216, 230, 244– 245 ~ per impossibile, 99, 100, 188 ~ with negative premises, 134, 187 syllogistic science, 147

312

general index

Teleion, 63 term accidental ~, 17, 88, 106, 183, 186, 193, 201, 211 adjectival ~, 15, 16 denominative ~, 19, 121 infinite ~, 16 major ~, 60, 66, 120–121, 127– 128, 130, 132–133, 138, 142, 168–169, 183, 197, 199, 216, 219, 250 minor, 51, 60, 66, 72, 121, 130, 132, 134, 137–138, 142, 145, 159, 168, 173, 196–197, 203, 220, 250 substantial ~, 183, 193 substantive ~, 15 ~ conversion, 93–94, 97, 190, 200 Thomas, Ivo, 53, 55–56, 69, 85 Thom, Paul, 5, 18, 21, 26, 31, 35, 38–39, 50, 59, 75, 87, 91, 101, 105,

119, 130, 135, 144, 151, 157, 175, 177, 190, 193, 195, 200, 205–206, 209–210, 212, 226, 230–231, 233– 234, 237–238, 242–243, 245, 248 transposition, 135, 144 upgrading, 76, 90, 102–107, 109–110, 193, 198–199, 203–204, 208, 211, 222 Vincent of Beauvais, 23 Viterbo, 1 Well-being, perfection of, 41, 62, 64 William Ockham, 177 William of Sherwood, 47, 149 Wittgenstein, Ludwig, 64 1277 propositions condemned in, 1, 15, 25, 47, 56

STUDIEN UND TEXTE ZUR GEISTESGESCHICHTE DES MITTELALTERS 3. Koch, J. (Hrsg.). Humanismus, Mystik und Kunst in der Welt des Mittelalters. 2nd. impr. 1959. reprint under consideration 4. Thomas Aquinas, Expositio super librum Boethii De Trinitate. Ad fidem codicis autographi nec non ceterorum codicum manuscriptorum recensuit B. Decker. Repr. 1965. ISBN 90 04 02173 6 5. Koch, J. (Hrsg.). Artes liberales. Von der antiken Bildung zur Wissenschaft des Mit-telalters. Repr. 1976. ISBN 90 04 04738 7 6. Meuthen, E. Kirche und Heilsgeschichte bei Gerhoh von Reichersberg.1959. ISBN 90 04 02174 4 7. Nothdurft, K.-D. Studien zum Einfluss Senecas auf die Philosophie und Theologie des 12. Jahrhunderts. 1963. ISBN 90 04 02175 2 9. Zimmermann, A. (Hrsg.). Verzeichnis ungedruckter Kommentare zur Metaphysik und Physik des Aristoteles aus der Zeit von etwa 1250-1350. Band I. 1971. ISBN 90 04 02177 9 10. McCarthy, J. M. Humanistic Emphases in the Educational Thought of Vincent of Beauvais. 1976. ISBN 90 04 04375 6 11. William of Doncaster. Explicatio Aphorismatum Philosophicorum. Edited with Annota-tions by O. Weijers. 1976. ISBN 90 04 04403 5 12. Pseudo-Boèce. De Disciplina Scolarium. Édition critique, introduction et notes par O. Weijers. 1976. ISBN 90 04 04768 9 13. Jacobi, K. Die Modalbegriffe in den logischen Schriften des Wilhelm von Shyreswood und in anderen Kompendien des 12. und 13. Jahrhunderts. Funktionsbestimmung und Gebrauch in der logischen Analyse. 1980. ISBN 90 04 06048 0 14. Weijers, O. (Éd.). Les questions de Craton et leurs commentaires. Édition critique. 1981. ISBN 90 04 06340 4 15. Hermann of Carinthia. De Essentiis. A Critical Edition with Translation and Commentary by Ch. Burnett. 1982. ISBN 90 04 06534 2 17. John of Salisbury. Entheticus Maior and Minor. Edited by J. van Laarhoven. 1987. 3 vols. 1. Introduction, Texts, Translations; 2. Commentaries and Notes; 3. Bibliography, Dutch Translations, Indexes. 1987. ISBN 90 04 07811 8 18. Richard Brinkley. Theory of Sentential Reference. Edited and Translated with Introduction and Notes by M. J. Fitzgerald. 1987. ISBN 90 04 08430 4 19. Alfred of Sareshel. Commentary on the Metheora of Aristotle. Critical Edition, Introduction and Notes by J. K. Otte. 1988. ISBN 90 04 08453 3 20. Roger Bacon. Compendium of the Study of Theology. Edition and Translation with Introduction and Notes by T. S. Maloney. 1988. ISBN 90 04 08510 6 21. Aertsen, J. A. Nature and Creature. Thomas Aquinas’s Way of Thought. 1988. ISBN 90 04 08451 7 22. Tachau, K. H. Vision and Certitude in the Age of Ockham. Optics, Epistemology and the Foundations of Semantics, 1250-1345. 1988. ISBN 90 04 08552 1 23. Frakes, J. C. The Fate of Fortune in the Early Middle Ages. The Boethian Tradition. 1988. ISBN 90 04 08544 0 24. Muralt, A. de. L’Enjeu de la Philosophie Médiévale. Études thomistes, scotistes, occamiennes et grégoriennes. Repr. 1993. ISBN 90 04 09254 4

25. Livesey, S. J. Theology and Science in the Fourteenth Century. Three Questions on the Unity and Subalternation of the Sciences from John of Reading’s Commentary on the Sentences. Introduction and Critical Edition. 1989. ISBN 90 04 09023 1 26. Elders, L. J. The Philosophical Theology of St Thomas Aquinas. 1990. ISBN 90 04 09156 4 27. Wissink, J. B. (Ed.). The Eternity of the World in the Thought of Thomas Aquinas and his Contemporaries. 1990. ISBN 90 04 09183 1 28. Schneider, N. Die Kosmologie des Franciscus de Marchia. Texte, Quellen und Unter-suchungen zur Naturphilosophie des 14. Jahrhunderts. 1991. ISBN 90 04 09280 3 29. Langholm, O. Economics in the Medieval Schools. Wealth, Exchange, Value, Money and Usury according to the Paris Theological Tradition, 1200-1350. 1992. ISBN 90 04 09422 9 30. Rijk, L. M. de. Peter of Spain (Petrus Hispanus Portugalensis): Syncategoreumata. First Critical Edition with an Introduction and Indexes. With an English Translation by Joke Spruyt. 1992. ISBN 90 04 09434 2 31. Resnick, I. M. Divine Power and Possibility in St. Peter Damian’s De Divina Omni-potentia. 1992. ISBN 90 04 09572 1 32. O’Rourke, F. Pseudo-Dionysius and the Metaphysics of Aquinas. 1992. ISBN 90 04 09466 0 33. Hall, D. C. The Trinity. An Analysis of St. Thomas Aquinas’ Expositio of the De Trinitate of Boethius. 1992. ISBN 90 04 09631 0 34. Elders, L. J. The Metaphysics of Being of St. Thomas Aquinas in a Historical Perspective. 1992. ISBN 90 04 09645 0 35. Westra, H. J. (Ed.). From Athens to Chartres. Neoplatonism and Medieval Thought. Studies in Honour of Edouard Jeauneau. 1992. ISBN 90 04 09649 3 36. Schulz, G. Veritas est adæquatio intellectus et rei. Untersuchungen zur Wahrheitslehre des Thomas von Aquin und zur Kritik Kants an einem überlieferten Wahrheitsbegriff. 1993. ISBN 90 04 09655 8 37. Kann, Ch. Die Eigenschaften der Termini. Eine Untersuchung zur Perutilis logica Alberts von Sachsen. 1994. ISBN 90 04 09619 1 38. Jacobi, K. (Hrsg.). Argumentationstheorie. Scholastische Forschungen zu den logischen und semantischen Regeln korrekten Folgerns. 1993. ISBN 90 04 09822 4 39. Butterworth, C. E., and B. A. Kessel (Eds.). The Introduction of Arabic Philosophy into Europe. 1994. ISBN 90 04 09842 9 40. Kaufmann, M. Begriffe, Sätze, Dinge. Referenz und Wahrheit bei Wilhelm von Ockham. 1994. ISBN 90 04 09889 5 41. Hülsen, C. R. Zur Semantik anaphorischer Pronomina. Untersuchungen scholastischer und moderner Theorien. 1994. ISBN 90 04 09832 1 42. Rijk, L. M. de (Ed. & Tr.). Nicholas of Autrecourt. His Correspondence with Master Giles and Bernard of Arezzo. A Critical Edition from the Two Parisian Manuscripts with an Introduction, English Translation, Explanatory Notes and Indexes. 1994. ISBN 90 04 09988 3 43. Schönberger, R. Relation als Vergleich. Die Relationstheorie des Johannes Buridan im Kontext seines Denkens und der Scholastik. 1994. ISBN 90 04 09854 2 44. Saarinen, R. Weakness of the Will in Medieval Thought. From Augustine to Buridan. 1994. ISBN 90 04 09994 8 45. Speer, A. Die entdeckte Natur. Untersuchungen zu Begründungsversuchen einer „scientia naturalis“ im 12. Jahrhundert. 1995. ISBN 90 04 10345 7 46. Te Velde, R. A. Participation and Substantiality in Thomas Aquinas. 1995. ISBN 90 04 10381 3 47. Tuninetti, L. F. „Per Se Notum“. Die logische Beschaffenheit des Selbstverständlichen im Denken des Thomas von Aquin. 1996. ISBN 90 04 10368 6 48. Hoenen, M.J.F.M. und De Libera, A. (Hrsg.). Albertus Magnus und der Albertismus. Deutsche philosophische Kultur des Mittelalters. 1995. ISBN 90 04 10439 9 49. Bäck, A. On Reduplication. Logical Theories of Qualification.1996. ISBN 90 04 10539 5

50. Etzkorn, G. J. Iter Vaticanum Franciscanum. A Description of Some One Hundred Manuscripts of the Vaticanus Latinus Collection. 1996. ISBN 90 04 10561 1 51. Sylwanowicz, M. Contingent Causality and the Foundations of Duns Scotus’ Metaphysics. 1996. ISBN 90 04 10535 2 52. Aertsen, J.A. Medieval Philosophy and the Transcendentals. The Case of Thomas Aquinas. 1996. ISBN 90 04 10585 9 53. Honnefelder, L., R. Wood, M. Dreyer (Eds.). John Duns Scotus. Metaphysics and Ethics. 1996. ISBN 90 04 10357 0 54. Holopainen, T. J. Dialectic and Theology in the Eleventh Century. 1996. ISBN 90 04 10577 8 55. Synan, E.A. (Ed.). Questions on the De Anima of Aristotle by Magister Adam Burley and Dominus Walter Burley 1997. ISBN 90 04 10655 3 56. Schupp, F. (Hrsg.). Abbo von Fleury: De syllogismis hypotheticis. Textkritisch herausgegeben, übersetzt, eingeleitet und kommentiert. 1997. ISBN 90 04 10748 7 57. Hackett, J. (Ed.). Roger Bacon and the Sciences. Commemorative Essays. 1997. ISBN 90 04 10015 6 58. Hoenen, M.J.F.M. and Nauta, L. (Eds.). Boethius in the Middle Ages. Latin and Vernacular Traditions of the Consolatio philosophiae. 1997. ISBN 90 04 10831 9 59. Goris, W. Einheit als Prinzip und Ziel. Versuch über die Einheitsmetaphysik des Opus tripartitum Meister Eckharts. 1997. ISBN 90 04 10905 6 60. Rijk, L.M. de (Ed.). Giraldus Odonis O.F.M.: Opera Philosophica. Vol. 1.: Logica. Critical Edition from the Manuscripts. 1997. ISBN 90 04 10950 1 61. Kapriev, G. …ipsa vita et veritas. Der “ontologische Gottesbeweis” und die Ideenwelt Anselms von Canterbury. 1998. ISBN 90 04 11097 6 62. Hentschel, F. (Hrsg.). Musik – und die Geschichte der Philosophie und Naturwissenschaften im Mittelalter. Fragen zur Wechselwirkung von ‘musica’ und ‘philosophia’ im Mittelalter. 1998. ISBN 90 04 11093 3 63. Evans, G.R. Getting it wrong. The Medieval Epistemology of Error. 1998. ISBN 90 04 11240 5 64. Enders, M. Wahrheit und Notwendigkeit. Die Theorie der Wahrheit bei Anselm von Canterbury im Gesamtzusammenhang seines Denkens und unter besonderer Berücksichtigung seiner Antiken Quellen (Aristoteles, Cicero, Augustinus, Boethius). 1999. ISBN 90 04 11264 2 65. Park, S.C. Die Rezeption der mittelalterlichen Sprachphilosophie in der Theologie des Thomas von Aquin. Mit besonderer Berücksichtigung der Analogie. 1999. ISBN 90 04 11272 3 66. Tellkamp, J.A. Sinne, Gegenstände und Sensibilia. Zur Wahrnehmungslehre des Thomas von Aquin. 1999. ISBN 90 04 11410 6 67. Davenport, A.A. Measure of a Different Greatness. The Intensive Infinite, 1250-1650. 1999. ISBN 90 04 11481 5 68. Kaldellis, A. The Argument of Psellos’ Chronographia. 1999. ISBN 90 04 11494 7 69. Reynolds, P.L. Food and the Body. Some Peculiar Questions in High Medieval Theology. 1999. ISBN 90 04 11532 3 70. Lagerlund, H. Modal Syllogistics in the Middle Ages. 2000. ISBN 90 04 11626 5 71. Köhler, T.W. Grundlagen des philosophisch-anthropologischen Diskurses im dreizehnten Jahrhundert. Die Erkenntnisbemühung um den Menschen im zeitgenössischen Verständnis. 2000. ISBN 90 04 11623 0 72. Trifogli, C. Oxford Physics in the Thirteenth Century (ca. 1250-1270). Motion, Infinity, Place and Time. 2000. ISBN 90 04 11657 5 73. Koyama, C. (Ed.) Nature in Medieval Thought. Some Approaches East and West. 2000. ISBN 90 04 11966 3 74. Spruyt, J. (Ed.) Matthew of Orléans: Sophistaria sive Summa communium distinctionum circa sophismata accidentium. Edited with an introduction, notes and indices. 2001. ISBN 90 04 11897 7

75. Porro, P. (Ed.) The Medieval Concept of Time. The Scholastic Debate and its Reception in Early Modern Philosophy. 2001. ISBN 90 04 12207 9 76. Perler, D. (Ed.) Ancient and Medieval Theories of Intentionality. 2001. ISBN 90 04 12295 8 77. Pini, G. Categories and Logic in Duns Scotus. An Interpretation of Aristotle’s Categories in the Late Thirteenth Century. 2002. ISBN 90 04 12329 6 78. Senger, H. Ludus Sapientiae. Studien zum Werk und zur Wirkungsgeschichte des Nikolaus von Kues. 2002. ISBN 90 04 12081 5 79. Fitzgerald, M.J. Albert of Saxony’s Twenty-five Disputed Questions on Logic. A Critical Edition of his Quaestiones circa Logicam. 2002. ISBN 90 04 125132 80. Darge, R. Suárez’ Transzendentale Seinsauslegung und die Metaphysiktradition. 2004. ISBN 90 04 13708 4 81. Gelber, H.G. It Could Have Been Otherwise. Contingency and Necessity in Dominican Theology at Oxford, 1300-1350. 2004. ISBN 90 04 13907 9 82. Bos, E.P. Logica modernorum in Prague about 1400. The Sophistria disputation ‘Quoniam quatuor’ (MS Cracow, Jagiellonian Library 686, ff. 1ra-79rb), With a Partial Reconstruction of Thomas of Cleve’s Logica. Edition with an Introduction and Appendices. 2004. ISBN 90 04 14009 3 83. Gottschall, D. Konrad von Megenbergs Buch von den natürlichen Dingen. Ein Dokument deutschsprachiger Albertus Magnus-Rezeption im 14. Jahrhundert. 2004. ISBN 90 04 14015 8 84. Perler, D. and Rudolph, U. (Eds.). Logik und Theologie. Das Organon im arabischen und im lateinischen Mittelalter. 2005. ISBN 90 04 11118 2 85. Bezner, F. Vela Veritatis. Hermeneutik, Wissen und Sprache in der Intellectual History des 12. Jahrhunderts. 2005. ISBN 90 04 14424 2 86. De Rijk, L.M. Giraldus Odonis O.F.M.: Opera Philosophica. Vol. II: De Intentionibus. Critical edition with a study on the medieval intentionality debate up to ca. 1350. 2005. ISBN 90 04 11117 4 87. Nissing, H.-G. Sprache als Akt bei Thomas von Aquin. 2006. ISBN 90 04 14645 8 88. Guerizoli, R. Die Verinnerlichung des Göttlichen. Eine Studie über den Gottesgeburtszyklus und die Armutspredigt Meister Eckharts. 2006. ISBN-13: 978-90-04-15000-3, ISBN-10: 90-04-15000-5 89. Germann, N. De temporum ratione. Quadrivium und Gotteserkenntnis am Beispiel Abbos von Fleury und Hermanns von Reichenau. 2006. ISBN-13: 978-90-04-15395-0, ISBN-10: 90-04-15395-0 90. Boschung, P. From a Topical Point of View. Dialectic in Anselm of Canterbury’s De Grammatico. 2006. ISBN-13: 978-90-04-15431-5, ISBN-10: 90-04-15431-0 91. Pickavé, M. Heinrich von Gent über Metaphysik als erste Wissenschaft. Studien zu einem Metaphysikentwurf aus dem letzten Viertel des 13. Jahrhunderts. 2006. ISBN-13: 978-90-04-15574-9, ISBN-10: 90-04-15574-0 92. Thom, P. Logic and Ontology in the Syllogistic of Robert Kilwardby. 2007. ISBN 978 90 04 15795 8.