Medieval Mereology

Medieval Mereology

B O C H U M E R STUDIEN ZUR PHILOSOPHIE Herausgegeben von Kurt Flasch - Ruedi Imbach Burkhard Mojsisch - Olaf Pluta

Band 16 DESMOND PAUL HENRY Medieval Mereology

B.R. GRÜNER AMSTERDAM · PHILADELPHIA 1991

DESMOND PAUL HENRY

Medieval Mereology

B.R. GRÜNER AMSTERDAM «PHILADELPHIA 1991

Library of Congress Cataloging-in-Publication Data Henry, Desmond Paul. Medieval mereology I Desmond Paul Henry. p. cm. — (Bochumer Studien zur Philosophie ; Bd. 16) Includes bibliographical references and index. 1. Whole and parts (Philosophy) — History. 2. Logic, Medieval. 3. Ontology — History. I. Title. I I . Series. BD396.H46 1991 111',82--dc20 91-26920 ISBN 90 6032 318 1 CIP

No part of this book may be translated or reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher. © by B.R. Grüner, 1991 ISBN 90 6032 318 1 Printed in The Netherlands B.R. Grüner is an imprint of John Benjamins Publishing Co. P.O. Box 75577 - 1070 A N Amsterdam ֊ The Netherlands

Contents Page-numbers are shown in brackets Preface (xi) Coded Booklist

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I N T R O D U C T I O N : MEREOLOGY, METAPHYSICS, A N D SPECULATIVE G R A M M A R (1) 0.1 Mereology, A n c i e n t and Contemporary (1) 0.2 Medieval Mereology and Metaphysics (2) 0.3 Intimations of Speculative Grammar (10) 0.4 The Grammar of Quiddity and Universals (14) 0.5 The Grammar of Parthood (20) 0.6 U n i t y of Medieval and Contemporary Approaches (21) 0.7 Summary (30)

1

THE E A R L Y MEDIEVAL INHERITANCE (32) 1.1 A i m s and Method (32) 1.2 Assets f o r E x p l o i t a t i o n (34) 1.3 Boethius on Division (37) .31 Outline (37) .32 The Thesis of Part-Whole D i s p a r i t y (38) 1.4 -parts and P a r t s - o f - X (46) 1.5 The Scandal of the N o n - D i s c r e t e Singular (56) 1.6 Temporal Parts (61) 1.7 Conclusion (62)

2

A B E L A R D A N D HIS CONTEMPORARIES 2.1 H i s t o r i c a l P r e l i m i n a r y (64) 2.2 Some C r u c i a l Distinctions (65) 2.3 P a r t s - o f - X and X - P a r t s (79) 2.4 I d e n t i t y and P r i n c i p a l Parts (92) 2.5 Increase and Decrease (116)

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Contents 2.6 The Temporal Dimension (139) 2.7 Master Peter's Mereology (151) 2.8 Porretan Mereological Scandals (180) .81 The O r i g i n a l i t y of G i l b e r t of P o i t i e r s (180) .82 Clarembald's Accusation (185) .83 The G i l b e r t i a n Innovations (190) .84 The Compendium of Porretan L o g i c (194) 2.9 Concluding Remarks on Section 2 (217) 3

A Q U I N A S » M E R E O L O G I C A L ASPECTS (218) 3.1 Metaphysical Background (218) .101 Introduction (218) .11 F r o m Elements t o F o r m (218) .12 Quiddity and the Individual (225) 3.2 Wholes and Parts (229) 3.3 Further Precisions (254) .31 M a t t e r and Part (255) .32 Essential or Integral? The Status of M a t t e r and F o r m (257) .33 F o r m - P a r t s and I n t e g r a l i t y (261) .34 'Part of ...' as a Functor (270) .35 Principal Parts (273) .36 A n a l o g i c a l l y I n t e g r a l Parts (282) 3.4 A n a t o m y of the Soul (279) .41 P a r t s - o f - X and X - P a r t s (279) .42 Organic Parts and the Mereological A x i o m a t i c (287) .43 The Soul in Quaestiones Disputatae (298) (308) .44 Eucharistic Mereology 3.5 A t t r i b u t i o n s and Actions (313) 3.6 Resurrection and I d e n t i t y (317) 3.7 N a t u r a l and A r t i f i c i a l (321)

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Contents 4

SOME B U R I D A N I A N THESES

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5

THE DE SOPHISTICIS ELENCHIS IN THE THIRTEENTH C E N T U R Y 5.1 Introduction (341) 5.2 Integral and Universal (343) 5.3 A t t r i b u t i o n s : f r o m secundum quid to simpliciter (344) 5.4 Mereology and Manifolds (356) 5.5 Around the L i a r Paradox (372)

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6

WYCLIF'S DEVIANT MEREOLOGY (383) 6.1 A W y c l i f f i a n Work on Universals: General Prospectus (383) 6.2 Parallels and Innovations (389)

7

CATEGOREMATIC A N D SYNCATEGOREMATIC 7.01 Transitional Introduction (406) 7.1 Syncategoremata as Functors (417) 7.2 Examples f r o m E a r l i e r SyncategoremataTreatises (420) 7.3 Nicholas of Paris on totus,''whole' (428) 7.4 Nicholas of Paris on Exceptives (443) 7.5 Ockham on Integral Wholes (453)

8

VENETIAN HARVEST (462) 8.01 Paul of Venice: L i f e and Style (462) 8.02 Scope of the Present T r e a t m e n t (462) 8.03 References and Cross-References (463) 8.04 E d i t i o n and Translation Policies (464) 8.05 The C a t e g o r e m a t i c / S y n c a t e g o r e m a t i c Distinction (464) 8.06 General Remarks on 'Whole' taken Syncategorematically (469) vii

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Contents

8.1 Truths D e r i v e d f r o m the Syncategorematic 'Whole' (472) 8.2 Falsehoods D e r i v e d f r o m the Syncategorematic 'Whole' (475) 8.31 A r g u m e n t Against the T r u t h of [8.12] (479) 8.32 A r g u m e n t Against the T r u t h of [8.11] (480) 8.33 A r g u m e n t Against the Falsehood of [8.23] (481) 8.41 Reply t o Objection [8.31] (485) 8.42 Reply t o Objection [8.32] (489) 8.431 F i r s t Reply t o [8.33] (491) 8.4311 C r i t i c i s m o f [8.431] (493) 8.4320 Second Reply t o [8.33] (496) 8.4321 F i r s t C o u n t e r - r e p l y t o the D e s t r u c t i v i s m of [8.4320] (497) 8.4322 Second C o u n t e r - r e p l y t o [8.4320] (499) 8.4323 T h i r d Counter-reply t o [8.4320] (500) 8.4324 Final Three Counter-replies to [8.4320] (500) 8.433 T h i r d Reply t o [8.33] (503) 8.5 Objections t o this Last Reply (505) 8.6 Replies t o these Last Objections (507) 8.7 Final Reply (512) 8.81 On 'Whole' taken C a t e g o r e m a t i c a l l y (518) 8.82 Arguments against the Proposed Conventions (526) 8.83 Replies to these Objections (530) 8.9 Postscript t o Paul of Venice (536) S I T U A T I O N A L REVIEW

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PRESUPPOSITIONAL E X P L I C I T A T I O N (541) 10.01 Prospectus (541) 10.1 Some P r o t o t h e t i c a l Functors (545)

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Contents 10.2 O n t o l o g i c a l A x i o m , D e f i n i t i o n s , and Theses (550) .21 Functors of Existence (551) .22 Functors of I d e n t i t y (557) .23 Functors of Inclusion (558) .24 Many-link Functors (559) .25 N a m e - f o r m i n g Functors and N o m i n a l Expressions (563) .26 A Q u i d d i t a t i v e '... is ...' (566) .27 A n A x i o m f o r Ontology (568) .28 Punctuation (571) .29 A b s t r a c t Nouns (575) 10.3 Mereology (578) .31 A x i o m System Based on 'pt( )' (579) .32 Theses Based on the 'pt( )' A x i o m a t i c (582) .33 Concerning 'outside of ...' (586) .34 On Overlapping (588) .35 A t o m i s t i c Mereology (589) 10.4 Conclusion (590) I N D E X O F NAMES A N D TOPICS

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Preface Mereology is the theory which deals with parts and wholes in the concrete sense, and this study follows its varied fortunes during the Middle Ages. Thus preliminary indications of how mereological parthood differs metaphys­ ically and grammatically from that encountered in various other contexts are followed by a brief sketch of the type of theory inherited by the early medievals from Boethius. The presentation of texts, translations, and analytic comments then ranges over Peter Abelard's extensive and hitherto almost unappreciated mereological theories, along with highly illum­ inating and sometimes controversial material from other twelfth-century sources, including Gilbert of Poitiers, Clarembald of Arras, Joscelin of Soissons, and 'Master Peter'. The effect of the subsequent recovery of Aristotle's Metaphysica on mereological theory is typified by sketches of the many and varied uses made of it by Aquinas, for example in conn­ ection with the Aristotelian soul, personal identity, and so forth. A brief sample of Buridanian treatment is followed by those applications made under the umbrella of thirteenthcentury comment on Aristotle's work on fallacious argument. The curiously odd and original theories of Wyclif when dealing with mereological 'universals' are brought to light. The final two historical sections are articulated in terms of what bec­ ame well-known as a central medieval distinction within the senses of 'whole', i.e. that between the categorematic (i.e. nominal) and syncategorematic (i.e. quantifier-incorporating) senses. After samples from Walter Burleigh, Nicholas of Paris, and William of Ockham, the final main exploitation of the distinction in Paul of Venice's Logica Magna is recounted. xi

Preface The e n t i r e p r o j e c t is rounded of by an account (especially designed f o r non-specialists) of the contemporary theories presupposed throughout the comment which has been made on the various t e x t s . Thus are sustained both methodological c l a r i t y and uninterrupted communication between medieval w r i t e r s and contemporary readers having an interest in such subjects as logic, metaphysics, theology, psychology, or linguistics, and t h e i r h i s t o r y . No knowledge of L a t i n or acquaintance w i t h logic is demanded of the reader. The L a t i n , however, is made available f o r those who may p r e f e r t o c o n ­ sult i t . No l o g i c a l symbolism is used in the m a i n body of t h e book, but some contemporary background is appended in the f i n a l section, so t h a t those interested may f o l l o w i t up. The t h e o r e t i c a l tools used f o r the a n a l y t i c phase of the investigation derive wholly f r o m the lectures and w r i t i n g s of Czeslaw L e j e w s k i , to whom I am here vastly indebted. In par­ t i c u l a r , his prolongations of the mereological theory of Lesniewski were the inspiration f o r the e x p l o r a t i o n of t h e many h i t h e r t o unknown or l i t t l e understood t e x t s of medieval logic and metaphysics wherein mereological discussion abounds. I.M. Bochenski also added his much appreciated viva voce en­ couragement. So abundant are t h e r e l e v a n t t e x t s t h a t select­ ion and exclusion has been an enormous p r o b l e m in a work where space p e r m i t s only the presentation of samples, and r e s t r i c t s the possibilities of discussion to the m i n i m u m . Those readers desiring larger backgrounds and wider-ranging discussion of present-day issues m i g h t consult, f o r example, the Parts and Moments volume (SPM in the coded booklist b e l ­ ow), edited by B a r r y S m i t h . On the h i s t o r i c a l side, the recent Cambridge History of Later Medieval Philosophy, edited by N. K r e t z m a n n , A . Kenny, and J . Pinborg (CLM in the booklist) cannot be bettered both as a source of m o r e general i n f o r m ­ ation and detailed bibliography. G. Kalinowski's i n t r o d u c t i o n t o his t r a n s l a t i o n o f relevant work o f Lesniewski (coded as

xii

Preface LF below) also has important details of the affiliations of the Lvov-Warsaw school, ultimate fount of so much of the letter and spirit of the theoretical background. Grateful acknowledgement has to be made to those edit­ ors and publishers of texts from which quotations have been made, of which the following are the principal (with further details in the booklist): L.M. de Rijk's Dialectica of Abelard, L. Minio-Paluello's Twelfth-Century Logic, the N.M. Häring texts of Gilbert of Poitiers and Clarembald of Arras, the Compendium of Porretan Logic (S. Ebbesen, K.M. Fredborg, and L.O. Neilsen), the Spiazzi editions of Aquinas' works, Sten Ebbesen's edition of Quaestiones Super Sophisticos Elenchos (author unknown), Ivan J. Mueller's Tracta tus de Universalibus of Wyclif, the texts of Nicholas of Paris and others from H.A.G. Braakhuis' volumes on thirteenth-century syncategoremata tracts, P. Boehner's editions of Ockham's and Burleigh's logic; other sources are made clear at appropriate points. I thank also the many friends who have provided mereological references, most of which have not yet been followed up in the present work, owing to their quantity. I partic­ ularly regret having been unable to follow through Constant­ ino Marmo's indications on Scotus, but Janet Schofield's ref­ erence from the Buddhist scriptures has been gratefully u t i l ­ ised. A large portion of this study owes its existence to the enterprise of Peter Geach and W. Kneale, as well as to the generosity of the British Academy. As editors of the series comprising Paul of Venice's Logica Magna they arranged for the Academy's kind provision of copies of those areas of the manuscript and early printed sources which were requisite for the work of editing, translating, and commentary which I und­ ertook over the space of many years. The mereological port­ ion of this extensive enterprise has been embodied as a sal­ ient section of the present book. Although the publication in full of the original results of my former task is now no

xiii

Preface longer in prospect, the Academy has r e c e n t l y again provided help in the f o r m of f u r t h e r financial support t o d e f r a y the production costs of the present p u b l i c a t i o n , and f o r this I wish to express m y g r a t i t u d e . Some earlier papers have evolved into sections of this book. Section  is founded on the corresponding work in The Vienna Circle and the Lvov-Warsaw School (ed . Szaniawski, K l u w e r , 1989), and Section 6 derives in p a r t f r o m m y W y c l i f paper in Die Philosophie im 14. und 15. Jahrhundert, In Memoriam Konstanty Michalski (1879 - 1947), ed. Olaf P l u t a , Bochumer Studien zur Philosophie, 10; Grüner, A m s t e r d a m , 1988; the section on 'Master Peter's Mereology' has evolved f r o m a s i m i l a r piece in De Ortu Grammaticae, a m e m o r i a l volume t o Jan Pinborg, edited by G.L. B u r s i l l - H a l l , Sten Ebbesen, and Konrad Koerner. F i n a l l y , there is some kinship between Sect­ ion 2.2 and the study of Abelard's m e r e o l o g i c a l t e r m i n o l o g y in the L . M . de R i j k Festschrift {Medieval Semantics and Metaph­ ysics, ed. E.P. Bos, Ingenium, Nijmegen 1985). I acknowledge w i t h thanks the editors' kindnesses in suggesting the pieces and in p e r m i t t i n g the use of these as f u r t h e r foundational material. Much gratitude is also due to the editors of the Bochum Studies in Philosophy ( K u r t Flasch, Rudi Imbach, Burkhard Mojsisch, and Olaf Pluta) f o r t h e i r kind acceptance of m y essay into t h e i r distinguished series. The exceedingly patient and fine e d i t o r i a l work of Olaf Pluta in connection w i t h t h a t series persuaded me t o entrust the work to t h e m , and has continued t o be a source of u n f a i l i n g support throughout a highly d i f f i c u l t period of gestation and production. I cannot thank h i m enough f o r his patience and forbearance over many arduous months of preparation. The physical preparation of t h e t y p e s c r i p t and its t r a n s ­ f o r m a t i o n into camera-ready copy has been most soundly based on the exceedingly skilled i n i t i a t i o n into the w o r l d of w o r d xiv

Preface processing which was a f f o r d e d to me by Gavin Mather, a d i s t ­ inguished expert in this f i e l d . Without his g i f t s of t i m e and advice the enterprise would have been impossible. In present­ ing the t e x t w i t h a ragged r i g h t (as opposed to the n o r m a l l y justified) f o r m a t , we are f o l l o w i n g the lead of a superb p r e d ­ ecessor in a kindred f i e l d . I t was in this manner t h a t in t h e i r great work on geometric s y m m e t r y , E.H. Lockwood and R.H. M a c m i l l a n (Cambridge U.P. 1978) broke down the boundary b e t ­ ween the page's t e x t and its m a r g i n in order to f a c i l i t a t e and encourage the provision of the doodles and diagrams which served as illustrations of the subject at points where d e f i n ­ i t e l y f i x e d and unequivocal diagrams, especially at the e l e m ­ entary l e v e l , are in no way mandatory: they m e r e l y serve t o make i n t u i t i v e l y concrete, in a quite contingent sort o f way, the general concepts under discussion at a given p o i n t . Much of mereology and its history is in e x a c t l y the same position. The concepts are general, and c a l l f o r no d e f i n i t i v e diagrams, but pedagogically helpful illustrations can intervene f o r readers whose intuitions require assistance in this manner. Scope and encouragement f o r the reader's own m a r g i n a l p r o v i s ­ ion of these sketches (as w e l l as f o r remarks, expletives, and so f o r t h ) according t o the taste of the reader, or in purs­ uance of the lead of the medieval w r i t e r whose w o r k is being studied, is afforded by this typographical device. (The o f f ­ beat word-splittings v e r y occasionally entailed by its use, in order t o avoid too extensive areas of w h i t e paper-space, are mereological novelties in t h e i r own r i g h t . ) Abelard's houses, rods, gnomons, miscellaneously-sorted heaps, and manifolds are examples only contingently connected w i t h the c e n t r a l theses in question. But they, as w e l l as A r i s t o t l e ' s syllables, coats, liquids, and elements, Aquinas' w h i t e surfaces and p o t e n t i a l l y resurrected m a t e r i a l bodily parts, Paul of Venice's bits of wood, disintegrating and continuing human beings, as w e l l as the m u l t i p l e choppings-up of the hapless Socrates into his xv

Preface principal and subordinate parts, along w i t h t h e perpetual example of t h e house, w i t h its w a l l s , r o o f , foundation, and Buridan's k i t c h e n , can a l l be diagrammed in the margins, i f desired, even though the range of application of the discourse at such points in no way necessarily ties one down t o any one such i l l u s t r a t i o n . Such doodles are m e r e l y optional c r u t ­ ches which the reader may supply at w i l l , though encouraged, one may hope, by the obviation of any typographical apartheid between t e x t and m a r g i n . I am glad to pay t r i b u t e t o the highly e x p e r t , and sometimes stern, help of my sister Pauline in the reading of the f i n a l proofs. This has been of great assistance in e x ­ punging m i s p r i n t s , along w i t h slacknesses and i n f e l i c i t i e s of expression and layout. I am acutely conscious t h a t such an close survey of the t e x t could w e l l portend a new level of a t t a i n m e n t in these departments, as f a r as m y own monographs are concerned. The u t t e r l y p a t i e n t and supremely p r a c t i c a l support of m y w i f e Louise has sustained m y w e l l - b e i n g throughout the years o f authorship in which not only was she herself subjected to illnesses each almost f a t a l , but also l a t t e r l y suffered periods of word-processor-widowhood w i t h great e q u a n i m i t y . To adequately express m y h e a r t f e l t g r a t i t ­ ude f o r t h a t support lies quite beyond the bounds of any capacity of mine. The recent and much to be lamented death of W i l l i a m Kneale has been a reminder of m y indebtedness t o h i m and t o Martha Kneale f o r t h e i r kind help and h o s p i t a l i t y . I would t h e r e f o r e l i k e t o dedicate this production t o the m e m o r y of this logician and historian of logic, and as an expression o f g r a t i t u d e f o r the encouragement which the t w o of t h e m provided so many decades ago. D.P.H.

XVI

Coded Booklist Works to which allusion is to be made w i l l be indicated by the i t a l i c i s e d code l e t t e r s which are coordinated w i t h the bibliographical details provided below. In the course of t h e i r use, the code letters w i l l generally be i m m e d i a t e l y f o l l o w e d by page, column, or paragraph numbers. References to lines or groups of lines in works or MSS which have lines already n u m ­ bered or which are usefully numerable w i l l be made by adding a f u l l point and t h e individual line-numeral, or the i n i t i a l and t e r m i n a l line-numerals when several lines are in question (e.g. AD 154.2.12). Where works r e f e r r e d to are in several volumes, or are divided into numbered books or parts, the Roman numeral of the volume, book or part w i l l be interposed between the code l e t t e r s and the modes of reference already described. The numerals at t i m e s d i r e c t l y annexed to the code ՝ACM՝ are those s e r i a l l y a l l o t t e d t o the paragraph numbers of the manual edition ( M a r i e t t i ) . The l i s t in no way purports t o be exhaustive of the f i e l d or t o invariably indicate the best editions of t e x t s . I t m e r e l y records those books or papers which happen to have been used in t h e composition of this w o r k . The t e x t s r e p r o d ­ uced are occasionally c o r r e c t e d in the l i g h t of b e t t e r editions. Non-standard L a t i n spellings have been r e t a i n e d , f o r the most p a r t , where these are used in the present-day editions quoted.

ACG

Aquinas, Thomas, Summa Contra Gentiles; (Editio Leonina Manualis, Casa E d i t r i c e M a r i e t t i ) T u r i n , 1934 ACM Aquinas, Thomas, In X I I Libros Metaphysicorum A r i s t o t e l i s C o m m e n t . , (ed. R. Spiazzi) M a r i e t t i , T u r i n , 1950 AD Abelard, Peter, Dialectica (ed. L . M . de R i j k ) Van G o r c u m , Assen, 1956 xvii

Booklist ADA ADG ADS AI ΑΡΗ

AQD AST AVA  BCL BHL BMS BP BPL BPP BS

BSC

Aquinas, Thomas, Quaestiones Disputatae; De Anima; V o l . II of AQD Anselm of Aosta and Canterbury, Dialogus de Grammatico {S I, pp. 140 - 68; reproduced in HCD, HDG) A r i s t o t l e , De Sophisticis Elenchis, Bekker, 1831 Minio-Paluello, L. Twelfth-century Logic II: Abaelardiana Ined., Ed. di Storia e L e t t e r a t u r a , Rome, 1958 Aquinas, Thomas, In Aristotelis Libros Peri Hermeneias et Posteriorum Analyticorum Expositio (ed. R. Spiazzi) M a r i e t t i , 1955 Aquinas, Thomas, Quaestiones Disputatae (ed. R. Spiazzi) M a r i e t t i , 1949 Aquinas, Thomas, Summa Theologiae, Biblioteca de Autores Cristianos, M a d r i d , 1951 Avicenna (Ibn Sina), Metaphysica sive Prima Philosophia, Venice 1495 (reprod. Bib. S.J., Louvain, 1961) Boethius, A.M.T.S., L o g i c a l Works as reproduced in PL 64 Buridan, John, Compendium Totius Logicae, Venice, 1499 (Minerva r e p r i n t , 1965) Bochenski, Ι.Μ., A History of Formal Logic (tr. and ed. by Ivo Thomas) N o t r e Dame U.P., 1961 Boethius of Dacia, Modi Significandi (ed. Pinborg and Roos) Copenhagen, Gad, 1969 Burleigh, Walter, De Puritate Artis Logicae (ed. P. Boehner) Franciscan Inst., St. Bonaventure, 1955 Bochenski, I.M., Précis de Logique Mathématique, F.G. Kroonder, Bussum, 1949 Boyle, Robert, Selected Philosophical Papers (ed. M.A. Stewart), Manchester U.P., 1979 Braakhuis, H.A.G., De 13de Eeuwse Tractaten over Syncategorematische Termen, Doctoral Dissertation in 2 vols. Leiden, 1979 Buddhist Scriptures, A new t r a n s l a t i o n by Edward Conze, Penguin Classics, 1959 xviii

Booklist BT

CC CIA CLM CLP COA DA DLM DP E ESE FSO

G GA

GTT

Boethius of Dacia, Quaestiones super Librum Topicorum (ed. N.G. Green-Pedersen and J . Pinborg) Opera V I , p a r t I, Gad, Copenhagen, 1976 Cahiers de l ' I n s t i t u t du Moyen-âge Grec et L a t i n , publiés par le directeur (University of Copenhagen) Cousin, V.,(ed.) Ouvrages Inédits d'Abélard, Imprimerie Royale, Paris, 1836 The Cambridge H i s t o r y of L a t e r Medieval Philosophy (ed. K r e t z m a n n , Kenny, Pinborg) Cambridge U.P., 1982 (ed. S. Ebbesen, K . M . Compendium Logicae Porretanum, Fredborg, L.O. Nielsen) CC 46, Copenhagen, 1983 Cousin, V., (ed.) Petri Abaelardi Opera, A u g . Durand, Pari: 1859 D a l Pra, M., (ed.) Pietro Abelardo, Scritti di Logica, La Nuova E d i t r i c e , Florence, 2nd ed. 1969 de R i j k , L . M . , Logica Modernorum, Van G o r c u m , Assen, 19 1967 Deely, J . N . , Tractatus de Signis: The Semiotic of John Poinsot, U n i v e r s i t y of C a l i f o r n i a , 1985 A b b r e v i a t i o n used f o r the p r i n t e d version of VLM in section 8 only Ebbesen, S. (ed.) Incertorum Auctorum, Quaestiones super Sophisticos Elenchos, Gad, Copenhagen 1977 Ockham's Theory of Propositions: P a r t II of the Summa Logicae, t r . A . J . Freddoso, H. Shuurman; i n t r . by A . J . Freddoso, N o t r e Dame U.P. 1980 Garlandus C o m p o t i s t a , Dialectica, (ed. L . M . de Rijk) Van Gorcum, Assen, 1959 Geyer, ., Peter Abaelards Philosophische Schriften, Beiträge zur Gesch. der Philosophie und Theol. des M i t t e l a l t e r s , v o l . 2 1 , 1919-33 Green-Pedersen, Niels J . , The Tradition of the Topics in the Middle Ages, Philosophia, Munich, 1984 xix

Booklist  Henry, D.P., 'Anselmian C a t e g o r i a l and Canonical Language', Les Mutations Socio-culturelles au tournant e des XI -XI Siècles: Etudes Anselmiennes, pp. 537 ֊ 548; Éditions du CNRS, Paris 1984 HCA H ä r i n g , N.M., Life and Work of Clarembald of Arras, Pont. I n s t i t u t e of Medieval Studies, T o r o n t o 1965 HCD Henry, D.P., Commentary on 'De Grammatico', Reidel 1974 HDG Henry, D.P., The 'De Grammatico' of St. Anselm: the Theory of Paronymy, N o t r e Dame U.P. 1964 HFS Hughes, Robert, The Fatal Shore, A history of the trans­ portation of convicts to Australia, 1787 1868, Collins H a r v i l l , London 1987 HGP H ä r i n g , N.M., The Commentaries on Boethius by Gilbert of Poitiers, P o n t i f i c a l I n s t i t u t e of Medieval Studies, Toronto 1966 HL Henry, D.P., The Logic of St. Anselm, Clarendon, O x f o r d 1967 HLM Henry, D.P., Medieval Logic and Metaphysics, Hutchinson U.L., London 1972 HQS Henry, D.P., That Most Subtle Question, Manchester U.P., 1984 HSG Henry, D.P., 'Singular Syllogisms of Garlandus C o m p u t i s t a ' Revue Internationale de Philosophie, 113, 1975 HSL Hispanus, Petrus, Summulae Logicales, (ed. I. M. Bochenski) M a r i e t t i , 1947 HT Hume, D., Treatise of Human Nature, (ed. Selby-Bigge) O x f o r d , 1896 HTT Henry, D.P., 'The Truncation of T r u t h - f u n c t i o n a l C a l c u l ­ ations', Notre Dame Journal of Formal Logic. Vol I I , no. 4, 1961 HUP Henry, D.P., 'Univerals and P a r t i c u l a r s ' , History and Philosophy of Logic 7, 1986 KAU K i n g , P.O., Peter Abailard and the Problem of Universals, Ph.D. Dissertation (two v o l . ) , Princeton 1982 xx

Booklist KB

K i n g , Peter, Jean Buridan' s Logic. The Treatise on Supp­ osition; The Treatise on Consequences; tr. with Introduction, R e i d e l , Dordrecht 1985 KLS K l i m a , Gyula, 'Libellus pro Sapiente - a c r i t i c i s m of A l l a n Bäck's A r g u m e n t against St. Thomas Aquinas' D o c t r i n e of the I n c a r n a t i o n ' , The New Scholasticism, V o l . L V I I I , 2, 1984 KSL K r e t z m a n n , N., William of Sherwood's Introduction to Logic, Minnesota U.P., 1966 KSS K r e t z m a n n , N., William of Sherwood's Treatise on Syncategorematic Words, Minnesota U.P., 1968 LAF Le Blanc, Audoënus V., A Study in the Axiomatic Found­ ations of Mereology, M.A. Thesis, U n i v e r s i t y of Manchester 1983 LAM Lejewski, C , 'A Note on a Problem concerning the A x i o m a t i c Foundations of Mereology', Notre Dame J. of Formal Logic, V o l . IV, no. 2, 1963 2nd ed., C l a r ­ LAS Lukasiewicz, Jan, Aristotle1 s Syllogistic, endon, O x f o r d , 1957 LAX L e Blanc, Audoënus V., 'Axioms f o r M e r e o l o g y ' , Notre Dame Journal of Formal Logic, V o l . 26, no. 4, 1985 LE Locke, John, Essay Concerning Human Understanding (1690), variously edited and r e p r i n t e d LEM L e j e w s k i , C , ' A C o n t r i b u t i o n t o the Study of Extended Mereologies', Notre Dame Journal of Formal Logic, V o l . X I V , no. 1, 1973 LF Lesniewski, Stanislaw, Sur les Fondements de la Mathém­ atique (fragments: t r a d u i t du polonais par G. Kalinowski) Hermes, Paris, 1989 LID LLE

L e j e w s k i ,  , 'On I m p l i c a t i o n a l D e f i n i t i o n s ' , Studia Logica, 1958 L e j e w s k i ,  , 'Logic and Existence', Brit. Journal for Philosophy of Science, V o l . 5, 1954

xxi

the

Booklist L·L·L Luschei, E.C., The Logical Systems of Lesniewski, N o r t h Holland, A m s t e r d a m , 1962 LLM Lejewski, C , ' A C o n t r i b u t i o n to Lesniewski's Mereology', The 5th Yearbook of the Polish Arts and Sciences Abroad, 1954 - 5. LMC L e j e w s k i , C , 'A Note concerning the N o t i o n of M e r e o l ogical Class', Notre Dame Journal of Formal Logic, V o l . X I X , no. 2, 1978 LNA Le Blanc, Audoënus V., 'New A x i o m s f o r Mereology', Notre Dame J. of Formal Logic, V o l . 26, no. 4, 1985 LNM L e j e w s k i , C , 'A Note concerning the N o t i o n of M e r e o l ogical Class. Postscript', Notre Dame Journal of Formal Logic, V o l . 2 1 , no. 4, 1980 LON Lejewski, C , ' O n t o l o g y : What N e x t ? ' , Language and Ontology, Hölder-Pichler-Tempsky, Vienna, 1982 LPP Leibniz, G.W., Philosophical Papers and Letters, ( t r . ed. L.E. Loemker) 2nd ed. R e i d e l , Dordrecht, 1969 LR L e j e w s k i ,  , 'On Lesniewski's Ontology', Ratio, V o l . I, no. 2, 1958 LSB L e j e w s k i , C , 'Studies in the A x i o m a t i c Foundations of Boolean A l g e b r a ' , Notre Dame Journal of Formal Logic, V o l . I, nos. 1 and 3; V o l . II no. 2 LSE L e j e w s k i , C , ' A r i s t o t l e ' s Syllogistic and its Extensions', Synthese, X V , 1963 LSS Lejewski, C , 'Syntax and Semantics of Ordinary L a n g ­ uage', A r i s t o t e l i a n Society Supplementary Volume X L I X , 1975 M A b b r e v i a t i o n used f o r MS Vat. L a t . 3132 version of VLM in section 8 only ML M i l l , J.S., A System of Logic (5th edn.) London, 1862 MM M c M u l l i n , E., (ed.) The Concept of Matter, N o t r e Dame U.P., 1963 MV M i l n e , A . A . , The Christopher Robin Verses, Methuen, London, 1932 xxii

Booklist NGL

NM

NPL NSZ

NTP 

OQ PBV PFL PL PLB QF RBL RFF ROM

Nuchelmans, G., 'Geulinx' Containment Theory of L o g i c ' , Mededelingen der Kon. Ned. Akad. van Wet., Deel 51, Nth. Holland, Amsterdam, 1988 Kenny, A . J . P . , Longuet-Higgins, H.C., Lucas, J . R · , Waddingt o n , C . H . , The Nature of Mind (The G i f f o r d Lectures) Edinburgh U.P., 1972 Nuchelmans, G., Late Scholastic and Humanist Theories of the Proposition, N t h . Holland, A m s t e r d a m , 1980 Nuchelmans, G., 'Stanislaus of Z n a i m (d. 1414) on T r u t h and F a l s i t y ' , Medieval Semantics and Metaphy­ sics, Ingenium, Nijmegen, 1985 Nuchelmans, G., Theories of the Proposition, North Holl­ and, A m s t e r d a m , 1973 Ockham, Wm. of, Summa Totius Logicae (ed. P. Boehner), The Franciscan I n s t i t u t e , St. Bonaventure, N.Y., 1957, 1962 Orenstein, A l e x . , Existence and the Particular Quantif­ ier, Temple U.P., Philadelphia, 1978 Perreiah, A . R . , 'A Bibliographical I n t r o d u c t i o n t o Paul of Venice', Augustiniana 17, 1967 P r i o r , A . N . , Formal Logic, (2nd edn.) Clarendon, O x f o r d , 1962 Patrologia L a t i n a , Series ed. J.P. Migne Pinborg, Jan, (ed.) The Logic of John Buridan, Museum Tusculanum, Copenhagen, 1976 Quine, W.V., From a Logical Point of View, H a r v a r d . 1953 L e w r y , P. Osmund, (ed.) The Rise of British Logic, Pont. Inst. of Medieval Studies, T o r o n t o , 1985 Russell, E.S., Form and Function: A Contrib. to the Hist­ ory of Animal Morphology, M u r r a y , London, 1916 Rosier, I., '"O Magister ...": G r a m m a t i c a l i t é e t i n t e l l i g i b ­ i l i t é selon un sophisme du X I I I s i è c l e ' , CC 56, 1988

xxiii

Booklist S SA

SAI SB SLM

SM SML SPM SR SS

SST

ST TAU VLM

S c h m i t t , F.S., (ed.) 5. Anselmi Opera Omnia, Nelson, Edinburgh, 1946 e.s. Sobocinski, ., "On Well-constructed A x i o m Systems', Yearbook of the Polish Society of Arts and Sciences Abroad, 1955 - 6 Spade, P.V., Peter of Ailly: Concepts and Insolubles, R e i d e l , 1980 Sinkler, Georgette, 'Roger Bacon on the Compounded and D i v i d e d Sense', RBL 145 - 171 Sobocinski, ., 'Studies in Lesniewski's Mereology', Yearbook of the Polish Society of Arts and Sciences Abroad, 1954-5 Salisbury, John o f , Metalogicon, (ed. C . C . J . Webb) O x f o r d 1929 Scarre, G., Logic and Reality in the Philosophy of John Stuart Mill, K l u w e r , D o r d r e c h t , 1989 S m i t h , Barry, (ed.) Parts and Moments: Studies in Logic and Formal Ontology, P h i l . V e r l a g , Munich, 1982 Sobocinski, ., 'L'analyse de l'antinomie Rusellienne par Lesniewski', Methodos, vols. I and II Sherwood, Wm. of., 'The Syncatgegoremata of W i l l i a m of Sherwood', (ed. J.R. O'Donnell) Medieval Studies, V o l . I I . 1941 Shapiro, H. and Scott, F., (eds.) 'Walter Burley's De Toto et Parte՝, Archives ďhistoire doctrinale et litt, du moyen âge 1966, V r i n , Paris, 1967 Sherwood, Wm. o f , Treatise on Syncategorematic Words (tr. N. Kretzmann) Minnesota U.P., 1968 Tweedale, M . M . , Abailard on Universals, N o r t h Holland, A m s t e r d a m , 1976 Venice, Paul of, Logica Magna, (Pars P r i m a ) , Venice, 1499

xxiv

Booklist VVS Venice, Paul of, Logica Magna, 2a pars, Tractatus de Ventate et Falsitate Propositionis, Tractatus de Significato Propositionis, (ed. F. del Punta; t r . w i t h notes by M . M . Adams) for the B r i t i s h Academy, O x f o r d , 1978 WCN Whitehead, A . N . , The Concept of Nature, Cambridge U.P., 1920 WIS Wiggins, D., Identity and Spatio-Temporal Continuity, B l a c k w e l l , O x f o r d 1967 WOT Whitehead, A . N . , 'The Organisation of Thought', Report of the 86th Meeting of the Brit. Assoc. for the Advancement of Science 1916, Presidential Address, pp. 355 - 63, London, 1917 WRP Whitehead, A . N . , Russell, ., Principia Mathematica to *56 (reprint) Cambridge U.P., 1962 WSP Wiggins, D., 'On being in the same place at the same t i m e ( w i t h one remark about categories and m a t e r ­ i a l i s m ) ' , Philosophical Review, 1968 WSW Woodger, J . H . , 'Science w i t h o u t Properties', Brit. J. for the Philosophy of Science, 7, 1951 WT W i t t g e n s t e i n , L., Tractatus Logico-Philosophicus, Routledge and Kegan Paul, London, 1922 WUE W y c l i f , John, On Universals (Tractatus de Universalibus) t e x t translated by A . Kenny, i n t r o . by P.V. Spade, Clarendon, O x f o r d , 1985 WUL W y c l i f , John, Tractatus de Universalibus, t e x t edited by Ivan J . Mueller, Clarendon, O x f o r d , 1985 WW Wood, Rega, Adam de Wodeham: Tractatus de Indivisibilibus, K l u w e r , Dordrecht, 1988

xxv

0. Introduction: Mereology, Metaphysics, and Speculative Grammar 0.1 Mereology, A n c i e n t and Contemporary 0.11 Mereology is, s t r i c t l y speaking, the theory of the p a r t in its concrete (as opposed t o abstract) sense; i t may hence, by i m p l i c a t i o n , deal w i t h wholes also, although this need not necessarily be so. Systematic work on these topics was p r o d ­ uced in 1916 by the Polish philosopher S. Lesniewski; his pap­ er on the general theory of sets was in t h a t year published in Moscow (cf. SLM, LF). My own acquaintance w i t h his mereol­ ogy is due to the teaching of Czeslaw L e j e w s k i , who studied under Lesniewski. 0.12 The technical expression of the t h e o r y , which c o n s t i t ­ utes an axiomatised set of co-ordinates f o r the precise l o c ­ ation of t h a t medieval m a t e r i a l w i t h which this book is t o be concerned, w i l l be made available in section 10. However, f i r s t approximations w i l l be exposed by means of i n f o r m a l explanat­ ions and suggestions f o r reading m e r e o l o g i c a l theorems in a sort of ad hoc English. In this way readers who are unused to expression in a r t i f i c i a l language, or who may even be averse to such expression, w i l l s t i l l be able to f o l l o w the unfolding of the h i s t o r y . On the other hand, those who p r e f e r a more f o r m a l exposition are catered f o r in section 10. So much, by way of a n t i c i p a t i o n , for the contemporary aspect of the res­ earch. 0.13 H i s t o r i c a l l y , however, mereological theorising dates 1

Section 0 back not only to medieval t i m e s , but also is of ancient date, occuring in Buddhist and A r i s t o t e l i a n works. Thus, according to the Buddhist scriptures, f r o m someone who says t h a t he has a r r i v e d on t h e scene in a c h a r i o t a c e r t a i n K i n g Milinda asks f o r an explanation of w h a t a chariot is. Can i t be the shaft to which the horses are harnessed? Is i t the axle? Is i t the wheels? Is i t the chassis? When all such questions concerning the concrete p a r t s , dealt w i t h individually, are answered neg­ a t i v e l y , a l i k e question concerning a combination of these components can s t i l l be answered negatively: the c h a r i o t is said t o be none o f these, presumably because neither the ind­ ividual components nor various combinations of components am­ ount t o a c h a r i o t . A l l these negative answers leave the king querying whether there is such a t h i n g as a c h a r i o t , beyond the mere sound of the w o r d : BSC 148 - 9. He presumably was w e l l aware of a fundamental confusion (see 1.4) as w e l l as a fundamental principle (cf. 1.32) both of which are in play here. Such a principle and other m e r e o l o g i c a l distinctions are encountered in A r i s t o t l e ' s Metaphysica (chapters 25 to 27 of Book ). 0.2 Medieval Mereology and Metaphysics 0.21 Nevertheless, the mereology covered by t h e present work remains w i t h i n the confines of the medieval L a t i n West, but w i t h o u t in any way c l a i m i n g t h a t i t was an exclusive property of t h a t West. The t e m p o r a l scope w i l l range f o r w a r d f r o m the t i m e of t h a t precursor Boethius whom we may sometimes c a l l 'the R o m a n ' (470 - 525 A . D . ) in order t o distinguish h i m f r o m another Boethius, namely Boethius of Dacia (flourished 1270 80), a Danish philosopher t o whom we are t o be indebted here­ under f o r his e x c e l l e n t l y clear elucidations. The m a i n f i n a l f i g u r e w i l l be Paul of Venice (c. 1369 - 1429) whose monumen­ t a l embodiment of what f o r h i m was all possible l o g i c , the Logica Magna, contains sections expressly devoted to m e r e o l ­ ogy (section 8 below). The l a t e r John of St. Thomas has a l s o .

2

Introduction been incidentally included, given his prologations of c e r t a i n of Aquinas' theses (section 3.75). 0.22 F o r a medieval v i e w o f the general s i t u a t i o n of m e r e o l ogy r e l a t i v e l y t o other disciplines we may already draw upon the e m i n e n t l y clear sketch of its character provided by B o e t h ­ ius o f Dacia. He regards i t as a t y p i c a l component of M e t a ­ physics, the science of being in general. His reasons f o r so doing are s t i l l quite remarkably cogent and i n t e l l i g i b l e , and we may hence allow h i m to speak f o r himself, as f o l l o w s : (A) Metaphysics teaches us about those things which are common to a l l beings, and which are not m e r e l y proper t o some p a r t i a l aspect of being. For even as, on the one hand, experts in a specific f i e l d o f study teach us about those things which are t h e i r own proper objects of knowledge, and do not ascend to those things which are common both t o t h e i r own objects of knowledge and t o other entities as w e l l , but presuppose t h e m as already covered by metaphysics, .... so also, on the other hand, t h e metaphysician teaches those things which apply in common to all beings, and does not descend t o the level of things exclusively appropriate to the p a r t i a l aspects of being. These pertain t o those specialised bodies of knowledge t o which the objects of knowledge are allocated. N e x t i t must be c a r e f u l l y noted t h a t the terms which c o n t r i b u t e to the expression of those most general t r u t h s (those which are said t o be universally applicable notions of the soul, or highest axioms) are not peculiar t o some specif­ ic body of knowledge, but are common t o all of t h e m . Thus these terms 'whole' and ' p a r t ' , whence is constructed the self-evident proposition, 'The whole is greater than its p a r t ' , are not peculiar to medical science alone, f o r in geometry is t o be found something which is r e l a t e d t o some other t h i n g in the way in which a whole is related t o a p a r t . Nor are those terms peculiar t o geometry, f o r in a r i ­ t h m e t i c , in music, in astronomy, and in all the other bodies

3

Section 0 of knowledge, there is t o be found something which p e r f o r m s the f u n c t i o n of the whole r e l a t i v e t o something f u l f i l l i n g the r ô l e of part. The same is t o be understood concerning the terms which enter into the composition of the other most general truths and highest axioms, whatsoever they may be. Hence these terms (whence are composed t h e f i r s t principles of demonstration which are utilised in a l l the m o r e special­ ised bodies of knowledge) are the concern of the metaphysic­ ian f o r he is concerned w i t h all those items w h i c h are are common to a l l and peculiar to none. This is also evident f r o m Metaphysica V, wherein A r i s ­ t o t l e deals w i t h these very general t e r m s . And because we become aware of complex principles t o the e x t e n t t h a t we become aware of t h e i r t e r m s (as the same philosopher r e m ­ inds us in Analytica Posterior a I, ch. 3) i t is the duty of the metaphysician to promulgate those most general t r u t h s or highest axioms to which appertain those t e r m s , and which i n ­ corporate them in t h e i r make-up. I t must also c a r e f u l l y be noted t h a t these most general t r u t h s , these universally app­ licable notions of the soul and highest axioms, are common to a l l more s p e c i f i c bodies of knowledge w i t h o u t being pec­ u l i a r to any one of t h e m . The ground of this f e a t u r e lies in the f a c t t h a t the terms whence the t r u t h s are composed are of the universally applicable sort. Thus this proposition, 'The whole is greater than the p a r t ' would be peculiar t o geometry were i t to be the case t h a t the t e r m s whence i t is composed were themselves peculiar to g e o m e t r y . (Metaphysica docet illa, quae sunt omnibus entibus communia, et quae non appropriantur alicui parti entis. Sicut enim artifices spec­ iales docent illa, quae sunt scibilibus propria, et non ascendunt ad illa, quae sunt suis scibilibus et aliis entibus communia, sed illa supponunt a metaphysico, .... sicut metaphysicus docet ea, quae sunt communia omnibus entibus. Ad illa autem quae sunt propria partibus entis non descendit. Illa enim pertinent ad scientias speciales, quibus approp-

4

Introduction riantur illa scibilia. Secundo considerandum est diligenter quod termini qui componunt illas dignitates quae dicuntur communes animi conceptiones vel maximae propositiones, non appropriantur alicui scientiae speciali, sed communes sunt omnibus. Isti enim termini 'totum' et 'pars' ex quibus componitur haec propositio per se nota, 'totum est maius sua parte' non appropriantur medicinae, quia in geometria invenitur aliquid se habens ad aliud sicut totum ad partem;  appropriantur geometriae, quia in arithmetica et musica et astronomia et in omnibus aliis scientiis invenitur aliquid ut totum, aliquid ut pars se habens ad aliud. Et eodem modo intelligendum de terminis componentibus alias dignitates et maximas propositiones, quaecumque fuerint. Ideo isti termini, ex quibus componuntur prima principia demonstrationum, qui­ bus utuntur omnes scientiae speciales, pertinent ad metaphysicum, quia ad ipsum pertinent omnia ea, quae sunt omnibus communia et nulli propria. Et hoc etiam appare է V. Metaphysicae, ubi Aristoteles tractat de istis terminis communibus. Et quia principia complexa cognoscimus, in quantum terminos cognoscimus, ut dicit Philosophus, I. Posteriorum, ideo metaphysicus declarare debet dignitates sive maximas proposit­ iones, ad quem pertinent termini, ex quibus componuntur. Considerandum etiam est diligenter, quod istae dignitates, communes animi conceptiones et maximae propositiones sunt communes omnibus scientiis specialibus, et nulli earum ap­ propriantur. Causa huius est, quia termini, ex quibus comp­ onuntur, sunt termini communes. Ista enim propositio, 'Totum ' est maius sua partes propria esset geometriae, si termini, ex quibus componitur, essent sibi proprii. BMS Q. 8, Dubium, lines 5 - 43, pp. 37 - 8) 0.23 Even now, centuries a f t e r Boethius of Dacia w r o t e these words, they can scarcely be b e t t e r e d as an i n f o r m a l sketch of the t h e o r e t i c a l s i t u a t i o n or l o c a t i o n , so t o speak, of the mereology which he here t y p i f i e s by 'The whole is greater than the p a r t ' . As is the case throughout the present 5

Section 0 essay, readers who come t o this m a t e r i a l w i t h an eye t o t a l l y innocent of any sort of philosophical sophistication are in a good position to grasp what is being propounded. They can agree t h a t p a r t i c u l a r sciences deal w i t h p a r t i c u l a r subjectm a t t e r s , each demanding its own special aptitude f o r the han­ dling of the m a t e r i a l in question. I t also seems p e r f e c t l y feasible to envisage a subject of study which makes e x p l i c i t the general presuppositions and notions common to a l l our bodies of knowledge and t o all our demonstrations (e.g. i n f e r ­ ences f r o m the given to the i n f e r r e d ) . This is the sort of thing outlined in the Presuppositional E x p l i c i t a t i o n contained in section 10. Those w i t h only a l i t t l e acquaintance w i t h e l ­ ementary geometry taught in the old-fashioned way w i l l have no d i f f i c u l t y in at least perceiving the possibility of g a t h ­ ering those general presuppositions into an axiomatised or at least s y s t e m a t i c a l l y ordered whole, rather in the way t h a t ge­ o m e t r y orders its own m a t e r i a l . The mereology at which Boeth­ ius of Dacia hints would then be p a r t of t h a t ordered whole, an axiomatised metaphysics, which at this innocent-eye stage seems a p e r f e c t l y feasible p r o j e c t . 0.24 Most u n f o r t u n a t e l y i t happens t h a t the c o m p a r a t i v e c l a r i t y and humane p u r i t y of this vision of what the Danish Boethius calls 'metaphysics' has been obscured by the sub­ sequent history of the case, as w e l l as by the lack, in suc­ ceeding periods, of an adequate n o t a t i o n f o r the accurate and unambiguous presentation of metaphysical theses. A g a i n , the moderns' obsessional over-concern w i t h theory of knowledge, i t s e l f possibly in p a r t begotten by lack of precisely t h a t speculative grammar w i t h which Boethius of Dacia was concer­ ned (HQS 230 - 1) leads to a quite unnecessary despair of the possibility of metaphysics as a body of knowledge which is not only most general (as in Boethius' description) but also expressly concerned w i t h the study of being-as-such. Or y e t again, in p a r t i c u l a r r e l a t i o n t o Boethius' own example of a metaphysical theorem which pertains t o mereology, namely 'The

6

Introduction whole is greater than its p a r t ' , only a l i t t l e sophistication based on elementary mathematics of the i n f i n i t e seems to f a l s i f y the theorem in at least one of its senses. Y e t m e d ­ ieval awareness of such possibility of f a l s i f i c a t i o n is, in f a c t , well-documented (CLM 564 - 591). 0.25 Indeed, i t transpires t h a t the Danish medieval philos­ opher was not being rashly o v e r - o p t i m i s t i c , a f t e r a l l . The metaphysics which he envisaged has actually been produced by the already-mentioned Polish philosophers in an u p - t o - d a t e f o r m , and is expressed in an accurate and unambiguous fashion by the use of contemporary a r t i f i c i a l language. Employed in the philosphical mode (as opposed to the f o r m a l i s t - m a t h e m a t ­ ical mode) such language exhibits precise details of its s t r ­ u c t u r e by means of a p p r o p r i a t e l y - d i f f e r e n t i a t e d n o t a t i o n . In other words, the language's grammar is o v e r t l y visible in the symbolism used. (An elementary exposition of how this a f f e c t s our account of the t h e o r e t i c a l s i t u a t i o n of mereology is t o f o l l o w in 0.4 below). The metaphysical theories expressed in this g r a m m a r - e v i n c i n g language, such as the already-mentioned mereology, and the general theory of being (ontology) do a c t ­ ually exist; they have been axiomatised in m u l t i p l e fashions, and hence are c e r t a i n l y possible. A l t h o u g h the account of medieval mereology is obviously f o r the most p a r t to be c o n ­ cerned w i t h mereology as such, some inkling of the general grammar and its ontological foundations are quite essential to a b e t t e r understanding of the scene. A f t e r a l l , the grammar is constantly required f o r the provision of reminders which focus a t t e n t i o n on the precise semantic status of the m e d ­ ieval discourse which is undergoing e l u c i d a t i o n . In any case, the medievals in general and Boethius of Dacia in p a r t i c u l a r , were acutely aware of the necessity f o r such reminders, and of t h e i r p r i m e philosophical importance. Hence some acquaint­ ance w i t h neighbouring and s y s t e m a t i c a l l y p r i o r f i e l d s , inc­ luding grammar and ontology, is quite essential i f the m e d i e v -

7

Section 0 al forays across such fields are to be appreciated and adjud­ ged in respect of t h e i r cogency. 0.26 This last point raises methodological issues which, as i t happens, were also covered by Boethius of Dacia. He was concerned w i t h the foundations o f grammar in its most general and universal f o r m . Such foundations, he believed, could be traced to metaphysics. The way in which such 'speculative grammar' may indeed be founded upon the present-day m e t a p h ­ ysics w i l l shortly be adumbrated. For the m o m e n t , however, any c l a i m to a capacity f o r objective adjudication as to sense and even as t o t r u t h when surveying medieval m a t e r i a l may sound somewhat overbold. Boethius o f Dacia made a corresponding c l a i m when dealing w i t h the status of t r a d i t i o n a l (as c o n t ­ rasted w i t h 'speculative') grammar. He based his c l a i m on the u l t i m a t e metaphysical foundation of his r a t i o n a l , l o g i c a l , speculative grammar. Grammar thus founded must, he believed, exhibit universally applicable and t o t a l l y cogent g r a m m a t i c a l structures, not peculiar t o any p a r t i c u l a r natural language. Here is e x e m p l i f i e d what he called the modus demonstratīvus, the demonstrative mode, f o r the explanation and exposition of grammar in general. W i t h this, he says, is to be contrasted the modus narra tivus, the m e r e l y h i s t o r i c a l or n a r r a t i v e mode of producing grammar. This is based not on the i m m u t a b i l i t i e s of metaphysics, but rather on ancient authorities who issued recommendations deriving f r o m the s h i f t i n g sands of the cus­ t o m a r y usage of the best authors; this is a foundation based on linguistic anthropology which investigates the past history of the p a r t i c u l a r language in question. I t was t o such found­ ations t h a t classical t r a d i t i o n a l grammarians such as Donatus and Priscian looked f o r the j u s t i f i c a t i o n of t h e i r rules (BMS Q.9, HQS §1.2). E x p l o i t a t i o n of the contrasting mode, the modus demonstratīvus, ensures t h a t the inadequacies of t r a d i t i o n a l grammar can be r e c t i f i e d or c o r r e c t e d . For example, p l u r a l i t ­ ies at the specific and individual levels respectively have diverse ontological foundations, and hence require r e c o g n i t i o n

8

Introduction as the diverse g r a m m a t i c a l forms which they accordingly e x e m ­ p l i f y . Ordinary grammar alerts us t o no such d i s t i n c t i o n , w i t h the result t h a t even philosophers can make howlers (BMS Q.68, HQS §1.2). 0.27 Now the application of this d i s t i n c t i o n between the demonstrative and the n a r r a t i v e modes t o the present enter­ prise may, in the f i r s t place, be precisely as envisaged by its author, i.e. in the f i e l d of r a t i o n a l or 'speculative ' grammar. The whole recent contemporary a n a l y t i c movement, f r o m Bertrand Russell onwards, stemmed f r o m the d i s t i n c t i o n between the real (or l o g i c a l , or deep) f o r m of an u t t e r a n c e and its apparent (or merely t r a d i t i o n a l g r a m m a t i c a l , or appar­ ent l o g i c a l , or surface) f o r m . 'Russell's m e r i t is to have shown t h a t the apparent logical f o r m of the proposition need not be its real f o r m ' was Wittgenstein's summary of the s i t ­ uation: WT 4.0031. The adjudicational a i m which raised the present methodological phase of our discussion is served in the f i r s t place by the i d e n t i f i c a t i o n of the logical (or 'spec­ u l a t i v e ' grammatical) f o r m of the passage of medieval m e t a p h ­ ysics or logic which is undergoing examination. Thus is made available an essential element in any check upon its sense. N e x t , given the a v a i l a b i l i t y of axiomatised versions of the appropriate present-day deductive metaphysics answering ex­ a c t l y t o Boethius of Dacia's specifications (BMS Q. 8, c f . HQS §1) i t becomes possible t o v e r i f y whether the thus g r a m m a t ­ i c a l l y c l a r i f i e d medieval theorem is or is not indeed a t r u t h of metaphysics. The way in which this enterprise can be c a r ­ r i e d through t o a c t u a l i t y has been inaugurated in HLM and HQS. There, both the g r a m m a t i c a l status and the t r u t h or o t h ­ erwise of such theorems were shown to be subject to s t r i c t and i n t e l l i g i b l e c o n t r o l . Once again, not m e r e l y metaphysics, but deductive or demonstrative metaphysics, is now available f o r the dual adjudications just outlined. These theorems are thus now adoptable not m e r e l y because some venerable author­ i t y happened to countenance t h e m , but simply because they are 9

Section 0 t r u e . A t the same t i m e independent medieval witnesses t o the t r u t h of present-day theses serve t o raise the status of the l a t t e r above the level of m e r e l y local considerations. I t is here, i f anywhere, t h a t a t r u l y perennial philosophy may be found. The next section (0.3) already has incidental examples of such witness. 0.28 In our survey of medieval mereology, definitions and theorems drawn f r o m present-day metaphysical theories w i l l be introduced in various forms and at various levels, as required by the c o n t e x t , f i n a l l y being s y s t e m a t i c a l l y made e x p l i c i t in section 10. However, in order t o give some immediate indic­ ation of the nature of the essential l o g i c a l , speculative grammar, lack of which may be responsible f o r some confus­ ions of modern philosophy {HQS §4.431) a b r i e f i n f o r m a l set of examples expressly designed to be relevant t o the s i t u a t i o n of medieval mereology w i l l next be provided. 0.3 I n t i m a t i o n s of Speculative G r a m m a r 0.31 These intimations are founded upon the highly lucid paper by C. Lejewski (LSS). The remarkable correlations b e t ­ ween t h a t paper and the proposals of Boethius of Dacia f o r his speculative grammar w i l l be noted as the exposition dev­ elops. 0.32 L o g i c a l grammar requires a logical language as its foundation or presupposition. The logical language m a y , in the f i r s t place, be a tidied-up version of some natural language in w h i c h , for example, standard meanings f o r c e r t a i n key e x ­ pressions have been laid down or understood. Or again, in addition t o such standard meanings, some of the a l t e r n a t i v e construals as t o parts of speech which ordinary grammar would p e r m i t are e l i m i n a t e d and the residual construal taken t o be the logical one. Thus the early medieval philosopher Garland the Computist reports t h a t logicians construe p a r t i c i p l e s exclusively as verbs, thereby e l i m i n a t i n g the nominal a l t e r ­ native which the ordinary grammar of L a t i n would p e r m i t : G 10

Introduction 48.13, 71.29.30. Anselm of Aosta and Canterbury used both of the t a c t i c s mentioned f o r the purposes of l i n g u i s t i c analy­ sis, i.e. f o r the expression of t h e t r u e , r e a l , or logical f o r m of an utterance which (he says) can be concealed by an integumentum, a c o v e r i n g , constituted by the m e r e l y apparent, o r d ­ i n a r y - g r a m m a t i c a l f o r m of t h a t u t t e r a n c e (De Casu Diaboli I). Non-logical ordinary usage (usus loquendi) is not only the foundation of ordinary grammar, but also of t h a t misleading integument. (For a general account of this Anselmian a n t i ­ cipation of what Wittgenstein described as 'Russell's m e r i t ' in the quotation provided in 0.27 above, see HL. A short summary of his a t t i t u d e and p r a c t i c e in this respect is provided in ). A generic t e r m which w i l l cover both this f i r s t (i.e. m o d i f i e d natural language) stage, and its second, or ' c a t e g o r i a l ' stage, later to be considered, is 'canonical language ' . 0.321 The canonical language of Western medieval philosoph­ ers whose w r i t i n g s are the subject of the present work was drawn f r o m L a t i n . A language such as L a t i n is said t o be heavily inflected because i t involves the profuse use of i n ­ complete forms, called 'stems' or ' r o o t s ' , which are duly sup­ plemented t o f o r m complete expressions. Thus the L a t i n v e r b stem ' a m - ' is completed by ' - o ' , ' - a s ' , ' - a t ' , ' - a m u s ' , ' - a t i s ' , and ' - a n t ' , in order t o express what in English comes out as 'I l o v e ' , 'thou lovest', 'he (or she) loves', 'we l o v e ' , 'you l o v e ' , and 'they l o v e ' , respectively. H e r e , in the English, i f we e l i m i n a t e the obsolete 'thou lovest', all the work of d i f f e r ­ e n t i a t i n g between the examples is p e r f o r m e d by separate ex­ pressions (the pronouns ' I ' , ' y o u ' , 'he' etc.) and the t e r m i n ational i n f l e c t i o n is m i n i m a l , in t h a t ' l o v e ' remains constant throughout save in the 'he loves' case, w i t h its simple a d d i t ­ ional ' s ' . Now a more heavily i n f l e c t e d language, such as the L a t i n here described, correspondingly much more readily gives rise to the notion of incomplete, or indeterminate linguistic expressions than does a r e l a t i v e l y l i g h t l y i n f l e c t e d one which tends t o produce most of its e f f e c t s by the juxtaposition of 11

Section 0 independent units such as t h e English pronouns, already recognisable as words complete in themselves. Hence i t comes about t h a t the m o r e heavily i n f l e c t e d medieval natural languages (including Latin) would tend to make t h e i r users take m o r e readily t o such a notion (HQS §3). I t is a notion which has now been once m o r e brought t o the fore in r e c e n t l y developed linguistic sciences. In a logical canonical language the incomplete expression may be called a functor. 0.33 Thus although what old-fashioned grammar t e r m s V e r b s ' (or, somewhat misguidedly taking the species f o r the genus, ՝doing-words') are in English w r i t t e n out (or otherwise i n ­ scribed) as units d i s t i n c t f r o m t h e i r appropriate accompan­ ying nouns or pronouns (and hence tend t o be viewed as dis­ t i n c t , independent units) they may nevertheless, in t h e i r t u r n , also be seen as what the late A . N . P r i o r genially described as 'sentences w i t h a hole in t h e m ' . This description is intended to restore somewhat a version of t h a t feeling of i n c o m p l e t e ­ ness displayed by the L a t i n verb-stems, and which may now be reproduced typographically, in the f i r s t place, by the use of dots. Thus '... runs', '...exists', '...does not e x i s t ' , are all gapped expressions which, when completed by the insertion of some name or other in the 'hole' indicated by the dots, f o r m assertive sentences, e.g. 'Marian runs', ' M a r i a n exists', 'Marian does not e x i s t ' . Those gapped expressions, of which t h e r e are i n d e f i n i t e l y more kinds, are e x e m p l i f i c a t i o n s of the functors mooted above, and much of the s t r u c t u r e of present-day l o g i c canonical language may be represented as f u n c t o r i a l in nature. 0.34 A f u r t h e r sample of the kinds of f u n c t o r possible arises f r o m the obvious f a c t t h a t one may also have a p o t ­ ential sentence w i t h t w o 'holes' in i t , rather than one, thus: .1

( )

is

( )

In this expression the use of brackets (rather than dots) is intended to indicate the appropriateness of c o m p l e t i o n by 12

Introduction names, as in the simple and obvious instance of '(Socrates) is (musical) ' . However, as this example i t s e l f hints, the canonical language now being presupposed is such t h a t 'name ' covers proper and common nouns as w e l l as adjectives, although abs­ t r a c t nouns are i m p l i c i t l y excluded. A p a r t f r o m this last ex­ clusion, this amounts to the broad, general, classical sense of 'name', as found in the antique grammarians Donatus and P r i s cian. Ordinary present-day grammar also p a r t i a l l y f o l l o w s this classical lead. Thus proper nouns such as 'Socrates', ' P l a t o ' , ' M a r i a n ' , ' G y u l a ' , and ' E v e l y n ' , common nouns such as ' d o g ' , 'horse' and ' g i r a f f e ' , as w e l l as adjectives such as ' b l u e ' , ' t a l l ' , and 'sour', are a l l possible nominal completions of the f u n c t o r in the envisaged a r t i f i c i a l l y m o d i f i e d canonical l a n ­ guage which is here being p r e f i g u r e d by the f o r m shown at .1 above. Hence f u r t h e r examples of a completed .1 would be ' H a m l e t is k i n g ' , ' K i n g is H a m l e t ' , 'Plato is c a p t a i n ' , 'Captain is P l a t o ' , ' K i n g is c a p t a i n ' , 'Fido is f a i t h f u l ' , and ' F a i t h f u l is F i d o ' . N o t a l l of these are p o t e n t i a l l y t r u e , however, and the t r u t h - c o n d i t i o n s have not y e t been spelled out: c f . section 10.2111. For the purposes of the canonical language, complex noun-expressions, among which are Russell's 'descriptions', are also appropriate completions. Such 'descriptions' are complex nominal expressions such as those e x e m p l i f i e d in 'The present king of France is the baldest man in B i r m i n g h a m ' . Names and other nominal expressions which are empty (i.e. r e f e r t o no thing) are also a d m i t t e d , as some of the examples just p r o v ­ ided i m p l y . 0.35 The functor shown at 0.34.1 is, of course, only one of i n d e f i n i t e l y many possible ' t w o - h o l e ' cases, and, f o l l o w i n g LSS, indices in the style of those used by K. Ajdukiewicz may now be adopted in order to indicate the g r a m m a t i c a l d i f f e r e n c e between these ' t w o - h o l e ' cases and the 'one-hole' cases. Thus the sign 's' is used f o r assertive sentence (or 'proposition') and V f o r name (in t h e broad sense mentioned). Then 's/n' is the categorial index of the functors which f o r m a proposition 13

Section 0 (hence the 's' t o the l e f t - h a n d side of the oblique stroke) f r o m a single name (hence the single V t o its right) as e x ­ e m p l i f i e d in 0.33. Using the same conventions, i.e. the product to the l e f t of t h e oblique stroke and the r e q u i r e m e n t f o r the a t t a i n m e n t of t h a t product t o its r i g h t , one has 's/(n n)' as the c a t e g o r i a l index of functors such as 0.34.1, which c a l l f o r t w o nominal completions in order t o f o r m a proposition. The c a t e g o r i a l indices delineate semantic categories. These c o r ­ respond t o , but go beyond, the 'parts of speech' of t r a d i t i o n a l significandi, grammar, and are to some extent akin t o the modi modes of s i g n i f i c a t i o n , of speculative grammarians such as Boethius of Dacia. A canonical language which allows one t o read o f f , w i t h o u t ambiguity, the semantic categories of its functors, may accordingly be called a categorial language, and is usually a r t i f i c i a l . Examples of some of the varieties of c a t e g o r i a l language are given in LSS 133 - 141. Some f u r t h e r elementary details of the one suitable f o r present purposes are given in section 10, as w e l l as in HLM II and HQS §0, §6. 0.36 F r o m all this i t is clear t h a t Boethius of Dacia's general p r o j e c t f o r speculative grammar, as outlined in BMS, is here being realised in a v e r y f u l l and e f f i c i e n t manner. One has the foundation of s c i e n t i f i c grammar upon a m e t a ­ physical basis, as opposed to grammar founded on a narrow l i n g u i s t i c anthropology. This realisation is made possible in the way described in LSS. D e f i n i t i o n s contained in the p r o g ­ ressively enlarging presupposed c a t e g o r i a l language make av­ ailable the f u r t h e r parts of speech needed f o r analysis. A f e w f u r t h e r details on these points w i l l be added below in section 0 . 6 , and the principles involved are to be invoked at various points throughout the present study, beginning w i t h the sec­ t i o n which now f o l l o w s . 0.4 The Grammar of Q u i d d i t y and Universals 0.41 The foregoing p r e l i m i n a r y allusions to semantic c a t e g ­ ories and t h e i r indices allow note to be taken of a t y p e of 14

Introduction semantic v a r i e t y which is of the greatest importance in m e d ­ ieval metaphysics in p a r t i c u l a r , and in philosophy in general. I t so happened, as a m a t t e r of h i s t o r y , t h a t during the middle ages general theories about the various kinds of things i n ­ volved in human speculation tended t o c l a i m t o concern the being of the type of t h i n g in question. Thus the d e f i n i t i o n of the 'quiddity' or ' n a t u r e ' o f a given sort of thing purported to express what is involved in being t h a t sort of t h i n g . Under such conditions the d e f i n i t i o n a l proposition ՝Man is rational animal՝, notwithstanding the apparently nominal completions of the gaps flanking its '... is ...', is not t o be construed as b u i l t around a f u n c t o r i a l '... is ...' of index s/(n n), but rather as having the sense, ՝To be man is to be rational an­ imal՝ in which the i t a l i c i s e d forms are no longer names, but i n f i n i t i v e s (i.e. 'to be ...' forms). Such i n f i n i t i v e s (as LSS 142 suggests and as the medievals would tend to agree) are more akin t o verbs (i.e. s/n forms) rather than t o names. Other quidditative statements about types of e n t i t y in general were expressed using p a r t i c i p i a l terms (e.g. f o r homo, man, one has humanitatem hab-ens, humanity-hav-ing) and we have already noted in passing t h a t medieval logicians took the p a r t i c i p l e to be v e r b - l i k e also. Given t h a t the c a t e g o r i a l index of the one-nominal-place verb has been established ab­ ove as ' s / n ' , then t h a t of an '... is ...' t a k i n g such verbs as its t w o completions w i l l obviously be ՝s/(s/n s/n)՝. Such an '... is ...' is nowadays s y s t e m a t i c a l l y and unambiguously d e f ­ inable in the categorial language here presupposed: c f . 10.26 and HQS §6.6. Looking at the m a t t e r in such a way solves the so-called 'problem of universals' f o r those who wish t o avoid nominalism (which insists on remaining at the s/(n n) level) and Platonism (which allegedly allows abstract nouns and other predicate-componential names to name l i t e r a l l y existing abstract e n t i t i e s ) . The non-nominalist, non-Platonic, but ' r e a l i s t ' medievals inclined t o the i n t e r p r e t a t i o n now being suggested, or at least require i t to obtain the results which 15

Section 0 they desire (HQS § 4 . 1 , HUP). 0.42 A t t e n t i o n has already been called elsewhere (HQS 234) to e a r l y medieval versions of this i n t e r p r e t a t i o n , as when the r a t i o n a l g r a m m a r i a n Peter Helyas elucidates abstract nouns such as albedo, 'whiteness', by means of the i n f i n i t i v e f o r m facere album, ' t o - w h i t e - m a k e ' , and says t h a t this novel voc­ abulary expresses form, i.e. quiddity. Such i n f i n i t i s a t i o n of a name corresponds e x a c t l y to t h e use of present-day functors having the same purpose (10.24), i.e. the c r e a t i o n of terms of semantic index s/n, which are appropriate t o discourse at the q u i d d i t a t i v e l e v e l . Other s i m i l a r t w e l f t h - c e n t u r y coinages are especially prominent in the work of G i l b e r t of P o i t i e r s and his school: CC 46, pp. 8, 9, 8 1 , HGP 90 - 91. Here the coinages are e x p l i c i t l y said to be expressive of the esse, the being, of the t h i n g in question. E a r l i e r , St. Anselm had worked t o make e x p l i c i t the possibilities of the i n f i n i t i v e facere ..., 'to do ...', as a very general verb: HQS 157 - 60. Aquinas ' notion of f o r m as actus essendi coheres w i t h all t h i s , and even the at f i r s t sight puzzling remarks of Stanislaus of Z n a i m on o n t o l o g i c a l t r u t h f a l l easily into place when the significance is realised of his ' i d e n t i f i c a t i o n of t h e state of a f f a i r s expressed by an a c c u s a t i v e - p l u s - i n f i n i t i v e phrase w i t h a universal f o r m expressed by an abstract noun' t o which Gabriel Nuchelmans calls a t t e n t i o n : NSZ 318. 0.43 A n u n f o r t u n a t e t r a d i t i o n a l d e f i n i t i o n of the universal as that which is predicated of many things or as that which is predicable of many things, has tended needlessly t o plunge the question o f the status of q u i d d i t a t i v e discourse (which is a precise e x e m p l i f i c a t i o n o f discourse concerned w i t h ' u n i ­ versals') into entanglement w i t h other questions expressed in terms of an a c t i v i t y called ' p r e d i c a t i o n ' . Now i t is p e r f e c t l y t r u e t h a t in respect of a l i m i t e d class of sentences f r o m a m ­ ong those based on t h e s/(n n) type of '... is ...' there m i g h t be some occasion t o split up such a sentence into t w o parts, so as t o see i t as its equivalent proposition which consists 16

Introduction of a s u b j e c t - t e r m (the i n i t i a l name) and a ' p r e d i c a t e ' (the rest of the proposition, index s/n). Thus 'Socrates is blue' splits up into 'Socrates' and '... is b l u e ' , the l a t t e r being the predicate. The exercise then continues (presumably on the supposition t h a t t h e proposition is true) t o ask 'Of what is the predicate "... is blue" predicated?'. To this the answer could in the present case be 'Socrates' (rather than '"Socrates'": cf. HQS §0.71). However, the c a t e g o r i a l language which we are presupposing in the present work has t h a t l a t ­ itude (concerning empty names and nominal expressions) des­ cribed in 0.34 above, and thus admits t h e use of empty names, i.e. names which r e f e r to nothing at a l l , an admission which was quite general in medieval t i m e s and gave rise t o much discussion: HQS § 0 . 1 , §2. R e l a t i v e l y to t h a t language i t is c e r t a i n l y possible t o countenance a t r u e subject-predicate proposition the s u b j e c t - t e r m of which is nevertheless e m p t y . (An e x t r e m e example would be Heidegger's 'The Nothing noths': HLM II §4, HQS §2.6911). In such a c o n t e x t the universal v a l i d i t y of the 'Of what is ... predicated?' sort of question i t s e l f becomes questionable. The f o u r t e e n t h - c e n t u r y John Wycl i f was alive to this s i t u a t i o n , as WUE 4 7 - 8 shows. This bringing into question is even more evidently j u s t i f i e d when the l i t t l e sophistication in speculative grammar acquired ab­ ove makes one alive to the possibility of propositions based on m a i n or c e n t r a l functors categorisable by s/(s/n s/n), as in 'Man is rational animal՝, regarded as a d e f i n i t i o n of the being of man. Pig-headed insistence on the v a l i d i t y of the question in such a type of case already obviously saddles one w i t h a m u l t i t u d e of previously unencountered e n t i t i e s such as the universal man. 0.44 A f u r t h e r example of a sentence at the q u i d d i t a t i v e level which this last-mentioned index also covers would be 'Man is a species'. Should someone accustomed to the l i m i t e d c o n t e x t exercise c r i t i c i s e d above unwarrantably feel impelled to c a r r y on w i t h t h e i r usual p r a c t i c e in respect of this 17

Section 0 sentence and ask, 'Of what is "... is a species" predicated in this proposition?', then i t should by now be clear t h a t this question is quite out of place in t h e enlarged contexts now shown t o be possible. Indeed, given the obviously p o v e r t y s t r i c k e n semantic presuppositions underpinning such a ques­ t i o n , i t is probably b e t t e r to abandon i t altogether, and t o concentrate instead on the c a t e g o r i a l s t r u c t u r e of propos­ itions. A f t e r a l l , predication is an action p e r f o r m e d by human beings, and to judge by the amazing antics described in the w r i t i n g s of present-day experts on this branch of anthrop­ ology, there seems t o be no l i m i t t o the range of oddities which i t may cover. Hence in order to avoid d i s t r a c t i n g excursions into such m a t t e r s , including wild-goose chases a f t e r t h e objects of which i t is assumed predicates must invariably be predicated, i t is more peaceful to m e r e l y investigate the l o g i c a l - g r a m m a t i c a l s t r u c t u r e of problem sentences such as 'Man is a species'. In the case of this example both terms are then, as we have seen above, and f o l l o w i n g the lead of LSS 142, analysable as v e r b - l i k e s/n f o r m s . Queries raised by philosophers about the status o f 'quiddities', 'universals ' , 'classes', 'species', and so f o r t h , and of discourse centred round t h e m (sometimes comprising abstract nouns), should in the f i r s t place be r e f e r r e d t o t h a t s t r u c t u r e . The one suggested above can, incidentally, be complicated in all manner of ways according t o the demands of individual interests underlying queries, as section 10 below and HLM, HQS, are all designed t o show. In any case, and in the last r e s o r t , i f one s t i l l insists on r e t a i n i n g the des­ c r i p t i o n of the universal as that which is predicable of many things, then those terms of index s/n which f i g u r e in q u i d d i t ­ a t i v e discourse as analysed above answer precisely to t h a t description. Concrete examples of medieval discussion on the details of this area are to be given in section 3.1. In the m e a n t i m e i t should be realised t h a t the above analysis of q u i d d i t a t i v e discourse also may be used t o cover medieval 18

Introduction t a l k about f o r m s , essences, and 'secondary substances ' . 0.45 The m e r e o l o g i c a l relevance of this excursion into the speculative grammar of q u i d d i t a t i v e discourse can be complex and m a n i f o l d . Thus the d e f i n i t i v e investigation of medieval mereology requires t h a t the unprecedented theorems propounded by G i l b e r t of Poitiers and his school concerning the r e l a t ­ ions between mereological and q u i d d i t a t i v e features of r e a l i t y should be analysed and assessed f o r t r u t h . This is an enter­ prise r e q u i r i n g all the powers of the proposed c a t e g o r i a l l a n ­ guage, which must plainly r i v a l or exceed those of technical language used by G i l b e r t i f the l a t t e r is ever t o be e l u c i d ­ ated f o r contemporary understanding. In contrast, there is a very simple and c o m p a r a t i v e l y w e l l - k n o w n point of relevance, the nature of which may be outlined as f o l l o w s : the general theory of parts and wholes happens, as a purely contingent h i s t o r i c a l f a c t , to have provided philosophers and logicians w i t h a construal of general discourse concerning groups, classes, or species and genera of things, which is a l t e r ­ native to the construals already mentioned above. Thus f a r , talk concerning such 'universals' has been analysed in terms of functors of index s/(s/n s/n) in order to provide i t w i t h a non-Platonic, non-nominalist location. But quite i r r e s p e c t i v e of such construals, there is at least agreement t h a t here we are dealing w i t h a ' d i s t r i b u t i v e ' type of whole, in the sense t h a t the ' u n i v e r s a l ' in question is equally and f u l l y d i s t ­ r i b u t e d among its elements, i.e. its e x e m p l i f i c a t i o n s . A n a l t e r n a t i v e way of looking at the e n t i r e universals question, however, was in terms of wholes which were called integral, the stock example of which was the individual house which consists of parts such as its walls, r o o f , and found­ ation. Here we are c l e a r l y on mereological ground, and i t was already standard p r a c t i c e to v i e w collective wholes such as a people, a f l o c k , or an army, in this same m e r e o l o g i c a l sort of way. F r o m this i t is but a short step to v i e w i n g humanity or mankind, and other such general wholes, as c o l l e c t i v e or i n t 19

Section 0 egral concrete objects of this sort. Such wholes, however, are no longer ' d i s t r i b u t i v e ' in the sense just noted, since t h e i r elements are by no means a l l of the sort betokened by the class's t i t l e ; hence the radical contrast between ' d i s t r i b u t i v e ' and ' c o l l e c t i v e ' wholes. The vocabulary of the theory of c o l ­ l e c t i v e wholes and of its logical foundations is subject t o diverse fluctuations during the middle ages. A t t e n t i o n w i l l t h e r e f o r e be given in f u t u r e sections t o recording those variations at the hands of authors such as Abelard (2.1) and W y c l i f (section 6). However, the present intent is m e r e l y to emphasise t h a t use of mereological notions f o r the provision of an a l t e r ­ native solution to a h i s t o r i c a l l y local p r o b l e m , usually described as the medieval problem of universals, is, f r o m the purely t h e o r e t i c a l perspective o f both the medievals and ourselves, a c o m p a r a t i v e l y minor m a t t e r , and v e r y f a r f r o m being the raison d'être of mereology i t s e l f . Indeed, i t is p e r f e c t l y possible t o have a mereological system which does not o v e r t l y incorporate the concept of collective class (in the general sense of complete collection) at a l l . I t should be emphasised t h a t this general sense of ' c o l l e c t i v e ' is to be adhered to below except in contexts where the m o r e narrow sense (e.g. f l o c k , army, etc.) is plainly t o be understood. Medieval w r i t e r s sometimes f o l l o w e d (7.21) or discussed (2.232) this more general sense of the mereological ' c o l ­ l e c t i v e ' . S t i l l , t h e i r p r e f e r r e d generic t e r m was usually 'integral'.

0.5

The Grammar of Parthood By way of a f i r s t appendix t o these i n t r o d u c t o r y r e m ­ arks, we may note a slight elaboration on the points of l o g ­ ical or speculative grammar which have been outlined above. This extension is one which i t is essential to bear in m i n d , and may be expressed as f o l l o w s . We have already noted t h a t

20

Introduction ' ...

is ...'-discourse concerning the being (esse) of things in general can be described as operating at a q u i d d i t a t i v e l e v e l , i.e. as having s/(s/n s/n) as its cardinal c a t e g o r i a l index. Hence, as already also noted, the d e f i n i t i o n a l proposition 'Man is rational animal' exemplifies this l e v e l . Here man is the species defined by the combination of animal (which is a gen­ us, or broader class) and rational (which is a differentia narrowing down t h a t broader class t o coincide w i t h man: c f . HQS §4). Now use of t h e w o r d ' c o m b i n a t i o n ' here betokens a r e l a t i o n between p a r t and whole, but not t h a t r e l a t i o n of p a r t and whole which is t y p i c a l l y m e r e o l o g i c a l . I t was, t o some e x t e n t , on the deliberate confusion of these t w o senses of part and whole t h a t K i n g Milinda based his sophistical queries mentioned at the outset above (0.13). Mereological theorems and other t r u t h s concerning nonq u i d d i t a t i v e parts and wholes are in f a c t most n a t u r a l l y and conveniently conducted at what may be called the ' n o m i n a l ' l e v e l , of which the n o m i n a l l y - f l a n k e d f u n c t o r '( ) is ( ) ' , of c a t e g o r i a l index s/(n n), is a sample. Integral wholes are objects susceptible of being r e f e r r e d t o by the c o m p l e t i v e names annexed t o such a f u n c t o r , and hence discourse con­ cerning such wholes is most economically pitched at this level. (One may conjecture t h a t g r a m m a t i c a l investigation of the convoluted theorems of G i l b e r t of Poitiers and his school m i g h t provide evidence t h a t this need not always be so.) P r o p ­ ositions which d e t a i l the integral make-up of m e r e o l o g i c a l wholes (as opposed to the q u i d d i t a t i v e make-up of d i s t r i b ­ u t i v e wholes) were said t o f u n c t i o n by enumeration of parts. K i n g Milinda was d i v e r t i n g 'What is i t ...?'-discourse in the d i r e c t i o n of such enumeration, w i t h the results observed above: c f . 3.23, 3.32, 7 . 0 1 , and HQS §4.231. 0.6 U n i t y of Medieval and Contemporary Approaches 0.61 A second appendix t o these i n t r o d u c t o r y remarks is simply a reminder t h a t the whole process of advance through 21

Section 0 contemporary issues which are here being raised, f o l l o w s the lines indicated in Boethius of Dacia's speculative grammar, so t h a t here and in the results which ensue f r o m this sort of i n v e s t i g a t i o n , medieval m a t e r i a l is not being f o r c e d into a r e m o t e and alien mould f o r i n t e r p r e t a t i v e purposes. For not only did this Boethius have the quite i n t e l l i g i b l e view of the nature o f metaphysics ( w i t h due regard t o its m e r e o l o g i c a l aspects) which has been observed in 0.2 above; he also show­ ed how the modi significandi, modes of s i g n i f i c a t i o n , as dis­ cerned by l o g i c a l grammar, could be founded upon the modi essendi, modes of being, which are e x p l i c i t l y said to be the concern of philosophers who produce metaphysics, a foundat­ ion which is t o be underpinned by our own section 10: (A) Someone is a grammarian by his acquisition of grammar. But no one can have the l a t t e r b e f o r e i t has been o r i g i n a t e d . Hence he who f i r s t made i t e x p l i c i t was no g r a m m a r i a n , but a philosopher c a r e f u l l y t a k i n g account of the intrinsic natures modes of s i g n i f i c a t i o n , of things. For the modi significandi, be they essential or incidental, generic or s p e c i f i c , derive f r o m the modes of understanding, and modi significandi are as diverse as are the modes of understanding whence they derive. But the modes of understanding themselves derive f r o m the i n t r i n s i c natures of things. Otherwise they would be mere i n t e l l e c t u a l f i g m e n t s , and this would appear t o be an impossible s i t u a t i o n . A g a i n , to the e x t e n t t h a t the modes of understanding are diverse, to the same e x t e n t must t h e r e be d i v e r s i t y between the modes of being whence they derive. Now the modes of being appropriate t o various items are known f r o m the i n t r i n s i c natures of things, and these the g r a m m a r i a n as such cannot scrutinise, but r a t h e r the m e t a ­ physician. In this connection i t should be understood t h a t as long as he abides in the contemplation of things themselves and the modes of being and understanding thereto approp­ r i a t e , then the grammarian is a philosopher. When, however, he joins thing w i t h w o r d , t r a n s f o r m i n g i t into the word's

22

Introduction s i g n i f i c a t e , and c o n t r i v i n g modes of understanding f r o m modes of being, and the modi significandi of the words f r o m the modes of understanding, he then begins to be a g r a m ­ m a r i a n . (Aliquis est grammaticus in habendo grammaticam. Ipsam autem nullus habere potuit ante suam generationem. Ideo qui invenit earn, non fuit grammaticus, sed erat philosophus proprias naturas rerum diligenter considerans. Modi enim significandi tam essentiales quam accidentales, tam generales quam speciales a modis intelligendi accepti sunt, et quanta est differentia inter modos significandi, tanta est differentia inter modos intelligendi, a quibus accepti sunt. Modi autem intelligendi accepti sunt a propriis modis essendi rerum. Aliter enim essent figmenta intellectus, quod non vid­ etur possibile. Et quanta est differentia inter istos modos intelligendi, tanta necessario debet esse inter modos essendi rerum, a quibus accepti sunt. Modi autem essendi appropriati rebus diversis cognoscuntur ex propriis rerum naturis, quas per se considerare non potest grammaticus, sed metaphysicus. Circa hoc intelligendum quod quamdiu ipse est in speculatione ipsarum rerum et modorum essendi, qui eis appropriantur, et modorum intelligendi, ipse est philosophus. Cum autem ipsam rem voci copulat faciendo ipsam vocis significatum, et modos essendi modos intelligendi faciendo, et modos intellig­ endi modos significandi vocis, iam incipit esse grammaticus. BMS Q . l , lines 38 - 58, pp. 6 » 7) 0.62 Now in distinguishing between the diverse types of '... is ...' which are g r a m m a t i c a l l y possible, we have not only been on the borders of discourse locating the various above-ment­ ioned modi essendi (cf. BMS Q. 40, HQS 131 - 2) but also have been presupposing the u l t i m a t e a v a i l a b i l i t y of a f u l l - b l o w ' theory of '... is ...', i.e. an '... is ...'-ology, so to speak, or in the m o r e usual nomenclature already encountered previously, an ontology. This c e r t a i n l y has, as part of its business, the metaphysical elucidation of modes of being, and i n t r o d u c t o r y explanations are available in our Presuppositional E x p l i c i t -

23

Section 0 ation (section 10), as w e l l as in LR, HLM, and HQS. I t is a source of definitions which s y s t e m a t i c a l l y introduce expres­ sions of new semantic categories, parts of speech, s t a r t i n g f r o m some p r i m i t i v e t e r m and developing the language accord­ ing t o o v e r t l y laid down rules of d e f i n i t i o n . I t is this which f a c i l i t a t e s the passage f r o m being a metaphysician t o being a grammarian so graphically here described by Boethius of Dacia. I t is because the systematic notation such as t h a t introduced in section 10 aims to display, w i t h o u t a m b i g u i t y , the semantic categories of its expressions, t h a t we are here dealing w i t h a canonical language which is at the same t i m e categorial. Such a language is also the u l t i m a t e basis f o r t h e s t r i c t l y c o r r e c t use of the c a t e g o r i a l indices which have been sketched above. These are the indices of the real or logical f o r m of u t t e r ­ ance, as made available and expressed in the c a t e g o r i a l lang­ uage of the ontology, the theory of being, t r a d i t i o n a l l y r e g ­ arded as c e n t r a l to metaphysics. Thus, y e t again, they are thereby t h e present-day counterparts of at least some of t h e modi significandi, modes of s i g n i f i c a t i o n , envisaged by Boet­ hius of Dacia: c f . BMS Q. 37, HQS §1.11, §0.54301. 0.63 Such parallels between the present-day l o g i c a l - m e t a ­ physical foundation f o r grammar and t h a t propounded by Boet­ hius of Dacia are f u r t h e r c o n f i r m e d by the way in which he saw his speculative grammar as having a type of universality which parallels t h a t of logic. This he makes quite o v e r t l y clear thus: (A) A body of knowledge expressed in various tongues is the same body of knowledge: i t enables the same t h i n g to be known, and in the same fashion. This is because i f the knowable and the mode of knowledge are i d e n t i c a l , then so also is the body of knowledge. But grammar yields i d e n t i c a l knowables in identical modes w i t h respect to diverse tongues. Hence a l l tongues involve one grammar. A g a i n , there is but one logic f o r all tongues, and hence also just one grammar. The antecedent proposition is obvious,

24

Introduction f o r i f there were not just one logic in many tongues, then we would have a logic s p e c i f i c a l l y diverse f r o m t h a t which the philosophers handed down to us when they handed down the logic in Greek which we possess translated into L a t i n , and so on w i t h other bodies of knowledge. But there is no such d i v e r s i t y . Hence [all tongues involve one g r a m m a r ] . A g a i n , when the essentials of a body of knowledge are identical f o r various people, then one has the same body of knowledge shared among those various people, even though the incidentals may vary. Now grammar is in this position w i t h respect t o the various tongues. When a speaker of Greek expresses some notion, then a L a t i n speaker can express a notion which is the same as f a r as e v e r y t h i n g essential to grammar is concerned, even though t h e i r expressions d i f f e r as t o the manner of combining utterances. But this is i n c i ­ dental, as is t o become subsequently apparent. The v e r d i c t on this question is t h a t all tongues are one in t h e i r grammar. The reason f o r this is t h a t the whole of grammar derives f r o m things. (It cannot be an i n t e l l e c t u a l f i g m e n t , f o r an i n t e l l e c t u a l f i g m e n t is something having no c o r r e l a t e in e x t r a - m e n t a l things). Now the nature of things is the same for a l l men, even though they have diverse t o n ­ gues; hence also the modes of being and of understanding are the same, hence also the modi significandi, modes of s i g n i f y ­ ing; hence also the modes of expression-forming or speaking display likenesses. And hence the e n t i r e grammar which r e l ­ ates t o one tongue resembles t h a t which relates t o another tongue. (Scientia una est in diversis idiomatibus, per quam idem scitur et eodem modo. Si enim idem est scibile et idem modus sciendi, una est scientia. Sed per grammaticam idem scitur et eodem modo in diversis idiomatibus. Ergo omnia Item: Una logica est in quoidiomata sunt una grammatica. cumque idiomate, ergo et una grammatica. Antecedens patet, quia si non esset una logica in omni idiomate, tunc nos aliam haberemus in specie quam philosophi nobis tradiderunt, cum

25

Section 0 in graeco logicam tradiderunt et in latino nos habemus ipsam translatam, et sic de aliis scientiis. Hoc autem est falsum. Ergo et cetera. Item: Cuiuscumque scientiae essentialia sunt eadem apud diversos, ipsa eadem est apud diversos, etiam si Grammatica est huiusdiversificetur in suis accidentibus. modi in diversis idiomatibus. Sicut enim graecus aliquem conceptum exprimit, sic et latinus eundem conceptum exprimit quantum ad omnia quae sunt essentialia grammaticae, licet orationes eorum differant in figuratione vocum. Ipsa enim acAd quaestionem dicencidentalis est, ut postea videbitur. dum, quod omnia idiomata sunt una grammatica. Et causa huius est, quia cum tota grammatica accepta sit a rebus (non enim potest esse figmentum intellectus; Ulud enim est figmentum intellectus, cui nihil respondet in re extra animam) et quia naturae rerum sunt similes apud omnes, ideo et modi essendi et modi intelligendi sunt similes apud omnes illos, apud quos sunt illa diversa idiomata, et per consequens similes modi significandi, et ergo per consequens similes modi construendi vel loquendi. Et sic tota grammatica, quae est in uno idiomate, est similis illi, quae est in alio idiomate. BMS Q.2, lines 22 - 50, pp. 11 - 12) For a f u r t h e r discussion of this thesis of Boethius, HQS §1.13 may be consulted. 0.64 Some readers may s t i l l be experiencing f u r t h e r qualms about the comparative s i m p l i c i t y , not to say n a i v e t y , of the prospectus unfolded above. A f t e r a l l , we are being promised axiomatised ontology and mereology, and many cross-references may be made to works of present-day 'symbolic l o g i c ' , which also figures in an elementary s o r t of way in the Presupposi t i o n a l E x p l i c i t a t i o n (section 10 below). F u r t h e r , i t transpires t h a t things indubitably l o g i c a l , such as c a t e g o r i c a l s y l l o g ­ i s t i c , are close neighbours of c e r t a i n axioms of ontology (LSE, HSG, HQS §2.5). F r o m this logical tenor of our groundwork a whole plethora of f u r t h e r objections can easily be generat­ ed. L e t us f i r s t of a l l take an obvious h i s t o r i c a l objection.

26

Introduction Can one r e a l l y accept t h a t this present-day, mechanical, quasi-mathematical body of what some describe as 'tautologies' (and hence as t e l l i n g us nothing about how things are) may be the counterpart of the venerable discipline of metaphysics as presupposed by someone such as Boethius of Dacia? Indeed, would any medieval thinker ever have considered f o r a single moment the rapprochement between logic and metaphysics which is here being envisaged? 0.65 Really t w o points are at issue here, w i t h questions about the nature of logic intermingled w i t h the o r i g i n a l l y h i s t o r i c a l query. L e t the h i s t o r i c a l point be considered f i r s t . I t was in f a c t common philosophical d o c t r i n e about the t i m e of Boethius of Dacia t h a t logic resembled metaphysics in being a most general body of knowledge (scientia communis; cf. HQS §1.134) and an independent case can s t i l l be made out for such a kinship (HQS §2.5, §2.7). The possibilities of such a rapprochement were discussed, f o r instance, by the f o u r t e e n t h century Bartholomew of Bruges when he was dealing w i t h the s u b j e c t - m a t t e r of logic (De Subiecto Logicae, ed. Sten Ebbesen and Jan Pinborg, CC 39). Although his lengthy discussion tends t o conclude against a t o t a l i d e n t i f i c a t i o n (and r i g h t l y , r e l a t i v e l y t o one then prevailing view of logic), nevertheless someone or other about t h a t t i m e was seeing t h a t abundant grounds could be invoked in its favour. Thus Bartholomew himself reproduces the argument t h a t the study of being is common to both metaphysics and logic. The argument continues: (A) That body of knowledge which analyses i d e n t i t y and d i v e r s i t y , likeness and unlikeness, c o n t r a r e i t y , d i f f e r ­ e n t i a t i o n , p a r t and whole, has being as its s u b j e c t - m a t t e r , as A r i s t o t l e maintains in the f o u r t h book of his Metaphysica; but logic is a body of knowledge of this s o r t , as is evident to the expert in logic, hence [being is the s u b j e c t - m a t t e r of logic, and hence logic and metaphysics would appear at least to overlap]. (Illia scientia quae considerat idem et diversum, simile, dissimile, contrarium, differens, partem, et totum, est

27

Section 0 de ente ut de s ubiec to, ut vult Philosophus 4 Metaphysicae; sed logica est huiusmodi, ut palam est scienti logicam; quare etc. CC 39, 37.69.71) 0.66 A present-day argument in favour of the same thesis makes the same point. Thus C. L e j e w s k i , speaking of the modern logicians Frege and Russell, says, ' T h e i r logical vocabulary accommodated, among other notions, the notion of i d e n t i t y , the notion of d i f f e r e n c e , the notion of existence, the notion of numerical i n f i n i t y . These notions are logical notions, and some of t h e m are already mentioned by A r i s t o t l e as character­ i s t i c notions of the science of b e i n g ' ; LON 173. 0.67 As regards the f u r t h e r p o i n t based on the discrepant natures of modern formal logic and metaphysics, i t is devoutly to be hoped t h a t philosophers at least w i l l be capable of distinguishing between a f o r m a l i s t approach t o axiomatised systems (which t r e a t s t h e m as uninterpreted arrays of rule-manipulable marks) and the metaphysical approach t o logical systems which sees t h e m as interconnected and interpreted bodies of t r u t h s , r i g h t f r o m the s t a r t . The f o r m a l i s t approach, e x e m p l i f i a b l e in the exposition of m a t h ­ e m a t i c a l set-theory, may be e x c e l l e n t l y suited to m a t h e m a t i c a l concerns, which may w e l l be quite d i s t i n c t f r o m those of p h i l ­ osophy, t r a d i t i o n a l l y engaged in the pursuit of t r u t h (and this requires i n t e r p r e t a t i o n of one's theories). As f o r the contention t h a t logic, even i f seen as i n t e r p r e t e d systems, nevertheless can only embody a host of empty tautologies, this tends to arise f r o m complete ignorance of any ongoing compendium of logic. The nature of logic is c e r t a i n l y not at all adequately e x e m p l i f i e d and exhausted by the f e w bleakly potted examples c u s t o m a r i l y produced by some philosophers who c l a i m to be investigating t h a t nature. People w i t h a v i e w of logic as t r i t e l y t a u t o l o g i c a l should have a go at proving (or disproving) some interestingly contentious m e r e o l o g i c a l thesis. Russell's notion t h a t definitions are m e r e l y typographical abbreviations may have contributed t o the t a u t o l o g y canard,

28

Introduction but a system using c r e a t i v e definitions of the sort presup­ posed in 10 below leaves i t f a r behind. I t also casts aside any notion t h a t axiomatised theories must invariably embody mechanically decidable theorems, pasture f o r the omniscient computer. 0.68 To t h e objection t h a t analysis in terms of logical systems and logical grammar is superfluous, since i t presupposes t h a t we understand the t e x t being analysed anyway, the r e p l y is t h a t such an objection trades on the a m b i g u i t y of ' u n d e r s t a n d ' . There are diverse grades of understanding. For example, most people can w e l l understand t h a t when I am here I am not there, but only very low-grade geography could emerge f r o m remaining at t h a t level of understanding. The more f i n e and accurate the coordinates in terms of which l o c ­ ation and r e l a t i v e directions are characterised, the b e t t e r the geographical understanding, and the more specialised the s k i l l involved in acquiring and e x p l o i t i n g such understanding. Correspondingly, one needs only a r e l a t i v e l y r u d i m e n t a r y und­ erstanding of n o n - i d e n t i t y t o realise t h a t there is some d i f ­ ference or other between saying t h a t one quiddity is not an­ other, and saying t h a t one individual is not another, or again, between b o t h of these and saying t h a t one given number is not some other. Such elementary understanding is c e r t a i n l y needed in order t o launch the metaphysical p r o j e c t of d e t e r ­ m i n i n g e x a c t l y the differences between the various cases, but i t by no means satisfies the need f o r the more r e f i n e d under­ standing necessary i f a l l the traps and p i t f a l l s surrounding non-identity are to be avoided. I t is in this way t h a t the m u l t i p l e obscurities which have been at the r o o t of most despair of metaphysics may in the end be obviated, a f t e r due investigation. 0.69 A order to position indices,

note of caution may, however, be inserted here, in avoid d i s t o r t i o n of perspective owing to the presup­ of Lesniewski systems, use of A j d u k i e w i c z c a t e g o r i a l and so f o r t h , as tools f o r the p r o j e c t of elucidating 29

Section 0 medieval mereology and other metaphysical doctrines. A r g u m ­ ents about the nature and status of those tools in themselves should not be allowed t o obscure the c e n t r a l i t y of t h a t p r o ­ j e c t . A f t e r a l l , only the very elementary reaches of these e l u c i d a t o r y systems are being appealed t o , and these are quite i n t e l l i g i b l e to non-specialist human beings of good w i l l who are interested in the main p r o j e c t . I t would be t o t a l l y beside the present point t o derogate t h a t elementary e x p l o i t ­ ation on the grounds of disputes concerning general foundat­ ional questions which are quite outside the ambit of the present p r o j e c t . 0.7 Summary In b r i e f , i t has been contended above t h a t the medieval section of the history of mereology may be recounted, under­ stood, and appreciated in the l i g h t of an i n t r o d u c t o r y e x p l ­ o i t a t i o n of the present-day f o r m of the discipline, as a x i o m atised by S. Lesniewski and his successors. The medieval p h i l ­ osopher, Boethius of D a c i a , has been shown to explain the t h e o r e t i c a l location of mereology in r e l a t i o n to metaphysics in a way which is compatible w i t h and corresponds to c o n t e m ­ porary axiomatised bodies o f knowledge, of t h e sort outlined in the Presuppositional E x p l i c i t a t i o n (section 10 below). Fur­ ther, a likewise elementary version of a contemporary f o r m of logical grammar, u l t i m a t e l y overlapping in its aims w i t h the speculative grammar established by Boethius o f Dacia (i.e. the resolution of g r a m m a t i c a l a m b i g u i t y by recourse t o quasiuniversal grammar) has already inaugurated the task of keep­ ing under c o n t r o l the l o g i c a l - g r a m m a t i c a l s i t u a t i o n of m a t e r ­ ial w i t h which a history of medieval mereology is going t o have to deal. A l t h o u g h by this present point applications of such grammar have been excessively e l e m e n t a r y , i t and the c a t e g o r i a l language c o n s t i t u t e d by Lesniewski's ontology on which i t is based are quite essential f o r any thorough understanding and appreciation of t e x t s in w h i c h q u i d d i t a t i v e -

30

Introduction level terms mingle with collective-class discourse (e.g. in the work of the ps-Joscelin of Soissons, or in the commentaries of Gilbert of Poitiers: cf. 2.7). It should be clear, of course, that the importation of Boethius of Dacia has not been for the purpose of using him as some sort of authority, but chiefly in order to bear witness to the continuity of the medieval and present-day disciplines, as well as to illustrate the sympathy between the two. In some respects such was also the intention of Lesniewski himself, as LSE reminds us.

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1. The Early Medieval Inheritance 1.1 A i m s and Method 1.11 A dual task is now in prospect. N o t only must some details of how medieval thinkers inherited mereology be p r o ­ vided, but i t is also necessary to begin t o introduce t e c h n ­ ical vocabulary and examples of the background presupposit­ ions required f o r the i d e n t i f i c a t i o n of those h i s t o r i c a l d e t ­ ails in t e r m s of exact t h e o r e t i c a l coordinates, so to speak. The f i r s t of these operations exemplifies the modus narrativus and the second the modus demonstratīvus as d i s c r i m i n a t e d by Boethius of Dacia (cf. 0.26 above). The present-day found­ ations of the l a t t e r modus are outlined in the Presuppositional E x p l i c i t a t i o n (sec. 10 below). For i n t r o d u c t o r y purp­ oses a m i x t u r e of the t w o is now proposed; the reasons f o r this m i x t u r e m i g h t be summed up by parodying a w e l l - k n o w n saying of Immanuel K a n t (1724 - 1804) and declaring t h a t (for persons who are not overskilled in grasping the i m p o r t of either t h e medieval t e x t s or the expressions of the contemp­ orary a r t i f i c i a l language used to express the mereology) nar­ ratives without demonstrations are (relatively) blind, where­ as demonstrations without narratives are (relatively) empty. The perhaps overstated t e r m s of this declaration of purely local i n t r o d u c t o r y p o l i c y may be expanded by considering t h a t a t r u l y s t r i c t and austere t r e a t m e n t would simply and i n i t i a l ­ ly lay down a present-day axiomatised version, couched in the c a t e g o r i a l language, of the ontological and mereological dep-

32

The Early Medieval Inheritance artments of the metaphysics envisaged by Boethius of Dacia and his fellow medievals. The historical material, replete in itself with surd elements could only then be subsequently in­ troduced and elevated to the level of analytic intelligibility by the identification and location of its contents in the light of the thus already stated and deductively organised systems. 1.12 However, readers not versed in contemporary logic could tend to find the atmosphere of the initial pure mereology so rarefied as to be unbreathable; examples, and especially r e l ­ evant historical examples, may hence here intervene as ball­ ast to bring the material down to the level of the earth of untutored understanding. Hence it is rather the way of nar­ rative tinctured with a modicum of demonstration (in the sen­ se of relativisation to the background systems of section 10's Presuppositional Explicitation) which resolves both horns of our dilemma. This mixture avoids leaving the reader somewhat in the dark as to the logical sense, import, and significance of the narrated text. Intimations of what exact analytic tech­ niques can make of the material are rendered available con­ currently with the presentation of the text. The narrative thus becomes discerning while the potential demonstrative framework begins at the same time to be filled out with appr­ opriate elucidatory and initiatory instances. 1.13 Rather more detailed grounding in the demonstrative task may then be postponed to the Presuppositional Explic­ itation (section 10) which contains the ultimate foundations, open for all to see, supporting any judgements which may be made as to the sense as well as to the truth of both the medieval theorems and our own provisional preliminary expres­ sions of them; these foundations w i l l all be available for optional perusal by those so inclined. Indeed, by the time section 10 is reached it is to be hoped that readers biassed from habits deriving from concentration on either side of the division between the narrative and demonstrative modes will 33

Section 1 have so e f f e c t u a t e d in t h e i r own persons a concrete fusion of the t w o aspects t h a t excursions across t h e i r boundaries w i l l no longer present any p r a c t i c a l d i f f i c u l t i e s . 1.2 Assets f o r E x p l o i t a t i o n 1.21 I t is now possible f o r t h w i t h t o build upon the scraps of logical grammar made available in the I n t r o d u c t i o n , so as to locate m o r e e x a c t l y the nature of our presuppositions on the demonstrative or t h e o r e t i c a l side. 1.22 F i r s t there is t o be recalled f r o m 0.41 the salient d i s t i n c t i o n between the n o m i n a l l y - t e r m e d and the f u n c t o r i a l l y t e r m e d senses of '... is ...', and of other two-gapped functors. Thus the proposition 'Socrates is humble' may be taken t o ex­ e m p l i f y an '... is ...' the gaps of which are f i l l e d by names (in the broad sense described in 0.34; c f . HLM §1.1, HQS §0.543). Its c a t e g o r i a l index is hence s/(n n), i.e. i t forms a proposition f r o m t w o names. Taking such an '... is ... ' as the p r i m i t i v e s t a r t i n g - p o i n t f o r our theory of '... is ...' (tech­ n i c a l l y called 'ontology') i t is possible t o define t h a t f u n c t ­ o r i a l l y - t e r m e d or ' q u i d d i t a t i v e ' sense of '... is ...'. Its index s/n). This d e f i n i t i o n a l exercise is p e r f o r m e d in is s/(s/n the presuppositional elucidation contained in section 10, w i t h details at 10.26. The resulting theory takes care of what in present-day t e r m i n o l o g y , which nevertheless follows c e r t a i n medieval patterns, may be called distributive wholes, and also serves t o elucidate medieval q u i d d i t a t i v e d e f i n i t i o n a l l y - f o u n ded discourse concerning genus and species. Distributive wholes are so called because the quiddity in question is f u l l y distributed t o each element of such a whole, as A b e l a r d is to r e m i n d us in due course (2.21). 1.23 Thus consider the quiddity or nature of man. This could be betokened by an abstract noun such as 'man-ness' in English. The 'class' of men, taken as a d i s t r i b u t i v e whole, is such t h a t 'man-ness' pertains to each o f its elements (or, in medieval t e r m i n o l o g y , 'subjective p a r t s ' ) . In contrast, i f the

34

The E a r l y Medieval Inheritance class of men is taken in what the medievals sometimes called an integral (or even ' c o l l e c t i v e ' ) sense, i.e. as the l i t e r a l l y complete c o l l e c t i o n of men, then t h a t c o l l e c t i o n , as w e l l as all m e n , are now among its elements (in the sense of D1 of 10.31) as also are the i n d e f i n i t e l y many variously-sorted objects (parts of men) which are likewise its elements. These l a t t e r objects qualify as elements of the i n t e g r a l or c o l l e c t ­ ive class of men because they have each a p a r t in common w i t h a man. Thus m y r i g h t hand is an element of the i n t e g r a l (or c o l l e c t i v e ) class of men because i t has a p a r t (e.g. the r i g h t thumb) which is at t h e same t i m e p a r t of a man. This contrasting l a t i t u d e as regards the natures of the elements of an integral whole is r e f l e c t e d in the present-day d e f i n i t ­ ion of complete collection or collective class (Kl( )) r e p r o d ­ uced as D2 in 10.3 below. Abelard (or someone else of his period) got p r e t t y near to making o v e r t this and other p r e ­ suppositions of the notion of i n t e g r a l whole (cf. 2.7). 1.231 Incidentally, i t is t o be r e g r e t t e d t h a t i t has proved impossible to avoid what look like allusions exclusively t o human persons of the male sex by the use of 'man' in the foregoing and subsequent examples. As 3.439 is to show, i t is f u t i l e t o move over to adjectives l i k e 'human', so as to produce neutral nominal expressions such as 'human being' (correlated w i t h the abstract noun 'humanity') which w i l l cover persons of both the male and the female sex. For example, m y finger is human; i t is also a being; hence i t is a human being. This illustrates the unavoidable breadth of 'human being', a point t o which 3.439 w i l l allude. For the moment i t can be said t h a t in the present context 'man' embraces woman, i.e. i t is m o r e l i k e the L a t i n 'һmo' in its broad sense, as when Pliny speaks of 'man's m i l k ' (lac hominis), and Juvenal allows a woman t o t r u l y say 'I am a m a n ' ('homo sum', inquit mulier). 1.24 The s l i g h t l y more sophisticated reader may have had m i l d qualms in l e t t i n g pass the notion of 'class' in its

35

Section 1 d i s t r i b u t i v e sense, when i t b r i e f l y f i g u r e d in 1.23. D i f f i c ­ ulties arise in connection w i t h l o c a t i n g what is being talked about when (for example) the class of men (in the n o n - i n t e g ­ r a l sense of 'class') is the topic of discourse. Indeed, the l i t e r a l i n t e r p r e t a t i o n of present-day class-calculus (or set theory) would take the class of men t o be a separate object over and above individual men. The same d i f f i c u l t y could be raised in connection w i t h the use of abstract 'universal' nouns such as 'man-ness', which occurred above. Does i t not have meaning by standing f o r an e x t r a 'universal' e n t i t y over and above individual men? I t has t h e r e f o r e to be emphasised, in the f i r s t place, t h a t as f a r as the present work is concer­ ned, the word 'class' has m e r e l y been imported as a handy, a l ­ ready e x i s t i n g , l o c u t i o n in the English language, and carries w i t h i t no i m p l i c a t i o n of the existence of objects such as classes, at least where the non-integral sense of 'class' is in question. Things go quite otherwise, of course, where those conc­ r e t e i n t e g r a l wholes, the c o l l e c t i v e classes defined in 10.31, are concerned. The elements of these are f a m i l i a r to a l l , and t h e r e f o r e we are acquainted w i t h the c o l l e c t i v e class of men (for instance) in the same sense as t h a t in which someone who perceives a p o r t i o n of a p a r t i c u l a r house at a p a r t i c u l a r t i m e is said t o perceive the house. (This point was made by some­ one o f the school of Joscelin of Soissons in the t w e l f t h c e n ­ t u r y : c f . 2.68(A)). The contrasting ' d i s t r i b u t i v e ' sense of class, as outlined above, need not invariably betoken classes of the m y t h i c a l s e t - t h e o r e t i c a l sort either, and does not do so here. Talk about classes in this d i s t r i b u t i v e sense, as w e l l as the use of abstract nouns such as 'man-ness' (employ­ ed above), may all be taken care of by w e l l - d e f i n e d discourse at the q u i d d i t a t i v e (i.e. f u n c t o r i a l l y - t e r m e d ) l e v e l , as r e ­ called in 1.22, and concerning which more technical d e t a i l is given in 10.26. Such t a l k , as thus construed, need bring no postulation of superfluous entities (sets, classes, P l a t o n i c -

36

The Early Medieval Inheritance a l l y - r e a l universals, abstract objects, and so f o r t h ) in its t r a i n . In point of f a c t , i t w i l l f o r the most p a r t be by way of contrast and occasional elucidation t h a t q u i d d i t a t i v e - l e v e l discourse w i l l be required in our analyses. Mereology proper can be conducted at the n o m i n a l l y - t e r m e d l e v e l , i.e. w i t h s/(n n) as the t y p i c a l c a t e g o r i a l index of the m a i n verb of propos­ itions involved in the discussion. 1.3 Boethius on D i v i s i o n 1.31 Outline 1.311 I t is t o the Roman Boethius (cf. 0.21) t h a t we now r e v e r t . In his great work on what he calls ' d i v i s i o n ' he attempts to bring together some of the senses in which he is to use this notion: (A) There are many ways in which division is said [to take place]. Thus there is the division of a genus into its species. A g a i n , there is the division which occurs when the [integral] whole is divided into its own parts, or again when a word having many meanings has its various meanings sorted out .... We undertake the division of a genus into its species when we assert t h a t one l o t of the animals are the r a t i o n a l ones, while the other l o t are the i r r a t i o n a l ones .... The [integral] whole is divided into its parts in so f a r as we resolve each thing into t h a t of which i t is composed, as when I assert '[In respect of] the house, the r o o f is one t h i n g , the w a l l another, and the foundation another'. (Divisio ... multis modis dicitur. Est enim divisio generis in species. Est rursus divisio cum totum in proprias dividitur partes. Est alia cum vox multa significans in significationes proprias recipit sectionem .... Genus dividimus in species cum dicimus animaliurn alia rational ia, alia irrationalia.... Totum in partes dividitur quoties in ea ex quibus compositum unumquodque resolvimus, ut cum dico, domus aliud esse tectum, aliud paries, aliud fundamentum.  877 B-D ) 1.312

This deliberately selective set of e x t r a c t s f r o m Boe-

37

Section 1 thius' t e x t c o n f i r m s the highly miscellaneous character of the processes subsumed by h i m under the t e r m ' d i v i s i o n ' . I t w i l l not do to simply gloss i t as ' c l a s s i f i c a t i o n ' , although in some places i t has t o do w i t h t h i s . We ourselves are scarcely to be concerned w i t h distinguishing between the meanings convey­ ed by ' e q u i v o c a l ' words (e.g. 'pen') and the e f f e c t of t h e i r use in speech on its logical and other features (e.g. d e t e r m i n a t ­ ion of t r u t h - v a l u e ) . Interesting though these and some of the other kinds o f division may be, they are f o r the most p a r t distractions f r o m the present mereological research. In f a c t the most notable d i s t i n c t i o n f o r our purposes is t h a t between t w o of the types of whole, i.e. the i n t e g r a l (or c o l l e c t i v e ) and the 'universal' (or q u i d d i t a t i v e ) . The l a t t e r is encount­ ered in the example of t h e genus animal being divided into the r a t i o n a l and i r r a t i o n a l animals. A m o n g the r a t i o n a l a n i m ­ als is, of course the species man. Now this entails t h a t man is somehow included w i t h i n animal rather in the way t h a t a p a r t is included in the whole. This is one sense in which a whole is divided into its parts. ( A t the same t i m e , paradox­ i c a l l y , i f one defines man as rational animal, this suggests, c o n t r a r y w i s e , t h a t animal is also somehow a p a r t of man). A l l this is q u i d d i t a t i v e m a t e r i a l , and t h e status of q u i d d i t a t i v e parts and wholes is bringable under c o n t r o l by construal in terms of v e r b - f l a n k e d f u n c t o r s , as recalled in 1.22 above. The other type of whole, the i n t e g r a l , e x e m p l i f i e d above by the case of the whole house, the parts of which are walls, r o o f , and foundation, is the m e r e o l o g i c a l whole, of the t y p e w i t h which we are c e n t r a l l y concerned 1.32 The Thesis of Part-Whole D i s p a r i t y 1.321 L e t us f o r the moment concentrate on Boethius' e x a m ­ ple o f the house, in order to b r i n g into r e l i e f its m e r e o l o g ­ ical significance. In accordance w i t h what has already been noted, the house's most s t r i k i n g f e a t u r e as contrasted w i t h a d i s t r i b u t i v e whole is the sortal v a r i e t y of its 'elements'. The

38

The Early Medieval Inheritance elements of the d i s t r i b u t i v e whole, elements which the m e d i e ­ vals called its 'subjective' parts, all have the name of t h e i r whole r i g h t l y applied t o t h e m , and are t o t h a t e x t e n t alike in kind. Thus the species man has as its elements, in the d i s t r i b u t i v e sense, the sundry persons E v e l y n , M a r y o n , Jo, Raymond, and so on, these being some of its ' s u b j e c t i v e ' parts, whereas in the integral sense the elements of the c o l l e c t i v e class of men are i n d e f i n i t e l y various in nature (e.g. Plato's l e f t f o o t during 400 B.C., m y r i g h t ear during 1960, t h a t object which is the combination of the t w o , m y finger nails today, your tongue yesterday, and all the bits of these and t h e i r combinations in turn). In this respect the i n t e g r a l elements of the c o l l e c t i v e class of men are akin in t h e i r sortal v a r i e t y t o the w a l l , r o o f and foundation of the house in which I am w r i t i n g these words. Most i m p o r t a n t l y , no one of these integral elements can be said to be the house. I n ­ deed, as a very loose p r e l i m i n a r y generalisation, w i t h 'x' and 'y' holding open spaces f o r names, one m i g h t say t h a t i f  is a p a r t of y (in the mereological sense of ' p a r t ' ) then  is not y. This is in strong contrast w i t h the d i s t r i b u t i v e e l e m ­ ents, each of which is its universal whole, in the sense t h a t the t e r m in question can be predicated of each: Jack is a man, Jo is a man, and so on. 1.322 However, i t f o r t h w i t h becomes appropriate to r e c a l l t h a t in the b e a u t i f u l l y incisive opening t o his pioneering work on Biology, the Historia Animalium, our predecessor A r i s ­ t o t l e enuntiated a c o u n t e r - d i s t i n c t i o n the echoes of which endure through the middle ages and are s t i l l v e r y much w i t h us. This is the d i s t i n c t i o n between like-natured or homogen­ eous p a r t s , on the one hand, and d i s s i m i l a r l y - n a t u r e d , or heterogeneous parts, on the other; all this is in a m e r e o l o g ­ i c a l c o n t e x t . Thus he says t h a t some parts of animals are simple, and these can be divided into s i m i l a r l y - n a t u r e d parts, as when flesh can be divided into pieces which are themselves flesh. Other parts are complex, and cannot be d i v 39

Section 1 ided into s i m i l a r l y - n a t u r e d parts; thus the hand cannot be ided into hands, nor the face into faces. The b i o l o g i c a l pur­ pose of these assertions may be t o enuntiate f o r the f i r s t t i m e the general d i s t i n c t i o n between tissues and organs (RFF 12 - 13). Independently of this biological i m p o r t , however, A r i s t o t l e ' s d i s t i n c t i o n continues t o inspire, d i r e c t l y or i n ­ d i r e c t l y , countless qualms over the ages about the thesis t e n t a t i v e l y mooted at the close of the last paragraph. Thus f r o m A r i s t o t l e we seem t o be obtaining a counter-example which runs: i f this flesh is p a r t of flesh, then this flesh is s t i l l flesh. 1.323 Thus again, Boethius, in the already-encountered De Divisione, repeats A r i s t o t l e ' s point by saying t h a t an animal is separable into those of its parts which have like-natured parts, as in the cases of the flesh and the bones, or a l t e r n ­ a t i v e l y into those parts which do not have like-natured parts, e.g. into hands and f e e t . The same goes, he continues, as far as a ship or a house are concerned (Animal separatur quidem in partes eas quae sibi similes habent partes, in carnes, et ossa; rursus in eas quae similes sibi non habent partes: in manus et pedes. Eodem quoque modo et navis et domus:  888). He has also in f a c t by this juncture already raised and a t t ­ empted t o s e t t l e the same point in connection w i t h another example which is supposed to serve as a f u r t h e r c o u n t e r - i n s t ­ ance to the p u t a t i v e m e r e o l o g i c a l thesis t h a t the integral whole is never predicable of the p a r t (as opposed to the d i s t r i b u t i v e whole such as a genus (e.g. animal) which is predicable of the ' p a r t ' (e.g. man) as when one asserts t h a t man is animal): (A) The species is always i d e n t i f i a b l e w i t h the genus, as man is i d e n t i f i a b l e w i t h animal .... In contrast the [integral] p a r t is not always i d e n t i f i a b l e w i t h the whole. For not only is the hand not i d e n t i f i a b l e w i t h the man, but also neither is the w a l l i d e n t i f i a b l e w i t h the house. A t any r a t e this is obvious enough in those things which have

40

The Early Medieval Inheritance dissimilarly-natured parts, but the same does not apply in those things having like-natured parts, as in the case of the bronze r o d , the parts of which are continuous and made up of the same bronze; here the parts seem to be the same as the whole. This, however, is a false impression, f o r while i t is t r u e t h a t parts of this sort are like [to the whole] in respect of t h e i r substance, they are nevertheless not q u a n t i t a t i v e l y the same.(Species idem semper quod genus est, ut homo idem est quod animal .... Pars vero non semper idem est quod totum. Neque enim idem est manus quod homo, nec idem paries quod domus. Et in his quidem quae dissim­ iles partes haben է, hoc darum est. Sed non eodem modo in his quae similes, ut in aeris virgula, cuius partes sunt continuae, et eiusdem sunt aeris, videntur idem esse partes quod totum est, sed falso; fortasse enim idem sunt partes  879D - 880A) huismodi substantia, non enim quantitate. 1.324 Here, having f i r s t e x e m p l i f i e d a version of the s t a n ­ dard d i s t i n c t i o n between d i s t r i b u t i v e and c o l l e c t i v e (or integral) wholes, Boethius has gone on to invoke q u a n t i t y in order to keep part and whole s t i l l d i s t i n c t when an i l l defined instance of these apparently exceptional homogen­ eously-parted integral cases is in question. C l e a r l y a oneounce slice of bronze rod is not to be i d e n t i f i e d w i t h the five-pound rod of which i t is a slice. I t is in his c e n t ­ u r i e s - l a t e r commentary on the foregoing t e x t t h a t Peter A b e l ard goes beyond Boethius' m e r e l y q u a n t i t a t i v e , contentual, d i s t i n c t i o n , as a basis f o r the non-assimilation of p a r t t o whole in such cases, and propounds a m o r e sophisticated l i n ­ guistic d i s t i n c t i o n , namely t h a t between the mere a t t r i b u t i o n in t e r m s of words having a s i m i l a r make up {in materia vocis) and the a t t r i b u t i o n of words having not only s i m i l a r make-up but also likeness of meaning (in significatione). Here is his commentary on the 'At any r a t e ....' ( ' E t in his ....') section of the above. I t w i l l be observed t h a t the c o m m e n t a r y at t i m e s takes the f o r m of actually repeating the words of the 41

Section 1 o r i g i n a l t e x t of Boethius, but w i t h explanatory interpolations: (A) A t any r a t e this [distinction between part and whole] is obvious enough in those wholes which have d i s s i m i l a r l y natured parts as f a r as both the make-up and the meaning of the words [used to express them] are concerned, but the same does not apply in respect of those wholes the parts of which are like-natured [only] as f a r as the make-up of the words is concerned. Hence he gives the instance of the r o d of bronze which has as its parts this bronze and this bron­ ze, w i t h this bronze as a p a r t only as f a r as the make-up of the words is concerned, r e l a t i v e l y t o the this bronze which is the whole taken as the bronze r o d . Hence i t would appear t h a t one can assert t h a t this bronze [part] is this bronze [whole]. But i t cannot be said t h a t this bronze, when i t is the whole taken as the bronze r o d , actually signifies many continuous bronzes a l l at once. But this bronze, when taken as a p a r t , signifies one single whole piece. Hence i t cannot ground a predication of its whole by means of the assertion t h a t i t is this [whole]. {Et in his totis quae habent partes dissimiles in materia vocis et in significatione hoc clarum est, quia partes non sunt idem quod totum; sed non est eodem modo in his totis quae habent partes similes in materia vocis. Et inde dat exempla in virgula aeris quae habet partes hoc aes et hoc aes et hoc aes in materia vocis quantum ad hoc aes quod est totum acceptum pro virgula aeris. Unde videretur quod posset dici hoc aes est hoc aes. Sed non potest dici quia hoc aes quando est totum acceptum pro virgula aeris significat plura aera simul continua. Sed hoc aes, quando accipitur pars significat totum unum frustum. Et ideo non potest recipere praedicationem sui totius ut dicatur hoc est. DA 169.18.29; c f . also section 2.69 below.) 1.325 I t is this in materia vocis ('as far as the make-up of the words is concerned ' ) q u a l i f i c a t i o n , which A b e l a r d l a t e r , at the appropriate point, brings to bear on the A r i s t o t e l i a n

42

The E a r l y Medieval Inheritance biologically-based thesis reproduced (as noted above) by Boethius. Abelard's description of the s i t u a t i o n incorporates t h a t q u a l i f i c a t i o n in such a way as to make i t clear t h a t the d i s t i n c t i o n between the t w o types of parts, the homogeneous and the heterogeneous, is m e r e l y apparent (generated by l i k e ­ ness of w o r d make-up) rather than r e a l : (A) [Some] parts into which an animal is separated have parts which are like t o t h e m as far as the make-υρ of the words is concerned. Thus an animal can be broken down into the flesh and the bones, and these parts in t h e i r t u r n have other parts which are like to t h e m in respect of the makeup of the words [used to describe t h e m ] . For one can assert: one p a r t of this flesh is this flesh, another p a r t is this flesh and another p a r t this flesh. Likewise in respect of this bone: one p a r t is this bone and y e t another part is this bone. In contrast, there is the whole which is divided into the parts which do not have parts s i m i l a r t o themselves, f o r one cannot assert: this hand has this hand as one p a r t and this other hand as another. Rather one asserts t h a t in respect of this hand, the thumb, index, middle, r i n g , and l i t t l e fingers are its various parts. In a s i m i l a r [heterogeneous] fashion the house and the ship are split up.(Partes in quas separatur animal habent partes similes sibi in materia vocis. Potest enim dici: carnis alia pars haec caro, alia pars haec caro, alia pars haec caro. Eodem modo: hoc os, aliud hoc os, aliud hoc os. Rursus div­ idi tur totum in eas partes quae non habent partes similes sibi. Non enim potest dici: huius manus, alia pars haec manus, alia pars haec manus. Sed potius dicitur: huius manus, alia pars pollex, alia pars index, alia pars medius, alia pars medicinalis, alia pars auricularis. Eodem quoque modo dividitur et domus et navis. DA 195.27.39; c f . AD 576.12.19.) 1.326 Lest this i n t i m a t i o n of the proposed thesis of p a r t whole disparity should s t i l l appear to be lacking in incis-

43

Section 1 iveness, i t may be added t h a t A b e l a r d repeatedly incorporates t h a t thesis in his definitions of possible sub-divisions of integral-whole types, such as the continuous and the noncontinuous: (A) The continuous whole is defined thus: a continuous whole is one which is made up of impredicable parts, in the sense t h a t they cannot have t h e i r whole predicated of t h e m , and such t h a t as long as the whole exists the parts are not susceptible of change of place, as is apparent in the case of a house. A non-continuous whole is one which is made up of parts which cannot have t h e i r whole predicated of t h e m , but w h i c h , w h i l e the whole exists, are s t i l l suscep­ t i b l e of change of place, as is apparent in the cases of a f l o c k and an army. The universal [i.e. d i s t r i b u t i v e ] whole is t h a t which is made up of parts which can have t h e i r whole predicated of each of t h e m . (Sic diffinitur continuum totum: continuum totum est illud quod constat ex partibus impraedicabilibus, id est non recipientibus praedicationem sui totius, non valentibus transmutari secundum localem manente ipso toto, ut potest videri in domo. positionem, Incontinuum totum est illud quod constat ex partibus non recipientibus praedicationem sui totius, valentibus transmutari secundum localem positionem, manente ipso toto, ut potest videri in grege et in exercitu. Universale totum est praedicationem illud quod constat ex partibus recipientibus sui totius singillatim. DA 193.36 - 194.6.) The f i n a l sentence is one of many Abelardian reminders of how integral and d i s t r i b u t i v e wholes (these l a t t e r being here called 'universal') are d i f f e r e n t i a t e d by the very c r i t e r i o n embodied in the thesis of p a r t - w h o l e disparity. Incidentally this passage shows t h a t one need not necessarily be mistaken in a t t r i b u t i n g the example of the house as a continuous whole to A b e l a r d , c o n t r a r y to what has been alleged (KAU I, 202). S t i l l , as we are to see in due course (2.231) Abelard does hold t h a t in a s t r i c t sense no a r t i f i c i a l constructs are c o n -

44

The E a r l y Medieval Inheritance tinuous wholes. 1.327 This, then, is the beginning of documentation t o the e f f e c t t h a t any biologically-based objection, or any other quibble against the characterisation of integral parts as heterogeneous, or at least disparate, in respect of t h e i r appropriate wholes, is u l t i m a t e l y sophistry. The same goes f o r objections using 'mass' terms (e.g. ' e a r t h ' , ' a i r ' , ' w a t e r ' ) as t h e i r basis. 1.328 Since, t h e r e f o r e , we are t o have centuries of testimony as t o the generalisable t e n a b i l i t y of p a r t - w h o l e disparity, testimony which abrogates any countenancing of apparent counter-examples, we can note the thesis of p a r t whole disparity as an essential feature of mereology, thus: .1

For a l l A and B, i f A is p a r t - o f - B , then i t ' s not t h a t A is 

The categorial-language version of this English approximation is deduced f r o m mereological axioms in 10.324 of the Presuppositional E x p l i c i t a t i o n . Very many technical remarks on the vocabulary and grammar of this thesis could be appropriate at this p o i n t , but since i t is f a i r l y i n t e l l i g i b l e f o r the pur­ poses of p r e l i m i n a r y exposition, such remarks are also r e l e g ­ ated t o 10.3. Suffice i t t o say t h a t t h e c a p i t a l or small l e t t e r s used in the manner of the 'A' and the ' B ' in .1 above are nominal variables, w i t h 'name ' being here understood in the broad classical sense recalled above in 0.34. Throughout our h i s t o r i c a l investigations, note w i l l o f t e n be made of the appreciation of this v i t a l thesis. Only exceptionally is i t apparently opposed (cf. 2.8 below). O f t e n the converse of the ' A is B' of the consequent is also assumed as being available; i f ' A ' and ' B ' are unshared names, then this is quite in order as f a r as the u l t i m a t e c a t e g o r i a l language is concerned.

45

Section 1 1.4 -parts and P a r t s - o f - X 1.41 There s t i l l remain three other features of m e r e o l o g i c a l wholes, the appreciation of which may be regarded as suggest­ ing a high level of t h e o r e t i c a l acumen, since even our c o n t e m ­ poraries have on occasion lacked the c a p a c i t y f o r such appr­ eciation (HQS §4.543). The f i r s t , which is to be studied in the present sub-section, may i n i t i a l l y be r e l a t e d t o a simple f e a t u r e of L a t i n grammar, namely the use of special t e r m i n a t ­ ions f o r names in order t o express t h e i r ' o f - a - . . . ' or ' o f - t h e ...' case (called the genitive case). Consider now the stock example of t h e house as an i n t e g r a l whole, and the d r a f t t h e ­ orem 1.328.1, which was roughly made e x p l i c i t in the previous section. F r o m t h a t theorem i t follows t h a t i f this w a l l is p a r t of this house, then i t ' s not t h a t this w a l l is this house. However, even though the w a l l is not this-house, i t is never­ theless (as long as the a p p r o p r i a t e l y - i d e n t i f i e d house of which i t is a part s t i l l exists) of-this-house, i.e. f o l l o w i n g what both Peter Abelard and the high medieval logician W a l t e r Burleigh are t o say, 'house' can s t i l l be predicated of its i n t e g r a l p a r t , but only in the genitive (i.e. 'of-a-house') mode. Under these circumstances the w a l l (like other integral parts actually c o n s t i t u t i n g an integral whole) may be said t o part. be a genitively-moded By way of a n t i c i p a t i o n of a more complete account of Abelard's (1.4) and Burleigh's (7.01) remarks, here is, f i r s t of a l l , the l a t t e r ' s appreciation of (i) the Thesis of Part-Whole D i s p a r i t y , made o v e r t above, and (ii) the possibility of genitively-moded p a r t p r e d i c a t i o n . His example is t h a t of a human person ('man'): (A) I t has to be maintained t h a t whole and part are taken in many senses. Of these one concerns the i n t e g r a l p a r t , of which the whole is not invariably predicated s t r a i g h t f o r w a r d ­ l y , but rather obliquely [i.e. in the g e n i t i v e ] . Thus the hand is p a r t - o f - t h e - m a n [in the integral sense of ' p a r t ' ] because 'The hand is the m a n ' is false, whereas 'The hand is [that]

46

The E a r l y Medieval Inheritance o f - t h e - m a n ' is t r u e . (Dicendum quod totum et pars multiplic iter capiuntur, quia quoddam est pars integralis de qua non predicatur omne totum in recto, sed in obliquo, et sicut manus est pars hominis, quia hec est falsa: 'Manus est homo'; sed hec est vera: ''Manus est hominis'. SST 302) In a way, one of Abelard's passages on g e n i t i v e l y - m o d e d parts is already more advanced than t h a t of the t e m p o r a l l y l a t e r Burleigh. This is because A b e l a r d r i g h t l y realises (AD 548) t h a t mere dependence on g r a m m a t i c a l v a r i e t y can y i e l d only a p r e t t y contingent c r i t e r i o n of d i s t i n c t i o n , and may even prove to be a d i s t r a c t i o n f r o m the c e n t r a l p o i n t , i.e. the d i s t i n c t ­ ion between integral and d i s t r i b u t i v e wholes (here called 'universal'). However, f o r the m o m e n t , here is a s u f f i c i e n t i l l u s t r a t i o n of the genitive accent which occurs when he is speaking of integral wholes: (B) However, the divison of such wholes is not expressed in the same case, i.e. the n o m i n a t i v e , as is the division of the universal whole. This blocks t h a t whole's being predicated of the individual parts in the n o m i n a t i v e case; this cannot be e f f e c t u a t e d f o r the integral whole. Rather is i t proper to express i t in the g e n i t i v e , thus: 'This small line is p a r t - o f t h i s - l i n e , t h a t other is another ...', or 'This man is p a r t - o f this-populace, t h a t other another ...' (Horum autem totorum profertur quo diviso non ita eodem casu, idest nominativo, divisio universalis. Quod quidem illud impedit quod ad singulares partes praedicationem, quae nominativo fit, non habet id quod est integrum, sed ita earn proferri genitivo conveni է: 'Huius lineae alia pars est haec lineola, alia illa', vel 'Huius populi alia pars hic homo, alia ille'. AD 547.29.34; c f . 2.65(A) below) 1.42 By this stage i t may w e l l be asked what can possibly be the promised weighty m e r e o l o g i c a l significance of this heavyhanded labouring of a t r i v i a l and obvious p o i n t . Strangely enough, this significance, although present and appreciated in medieval discussion, as we are seeing, nevertheless comes out

47

Section 1 most strongly in a society such as ours, based as i t is on the mass production of components f o r the construction of machines and other complex structures, given t h a t such comp­ onents sometimes become p a r t s - o f - t h e - s t r u c t u r e s . In such a s i t u a t i o n we are daily accustomed t o speak of machine-parts, and there are huge c o m m e r c i a l concerns specialising in the production, sale, and salvaging, of such things as c a r - p a r t s . The salient point of the adduction of these examples (mach­ i n e - p a r t , c a r - p a r t s , etc.) is t h a t such 'parts' do not, u n t i l incorporated appropriately in some relevant and complete ind­ ividual object, constitute parts-of whatever i t is t h a t they are components f o r . In other words, u n t i l t h a t i n c o r p o r a t i o n , Burleigh's and Abelard's genitive mode does not apply to t h e m . They are -parts as opposed t o p a r t s - o f - X ; the l a t t e r group, t h a t of the genuine, genitively-moded parts, may comprise, but is in no way exhausted by, actually-incorporated components in appropriate cases. Things f o r m e r l y p a r t s - o f - X can r e v e r t to being -parts (e.g. when a machine or building or organism is broken up). A l t e r n a t i v e l y , things f o r m e r l y p a r t s - o f - X may become -parts w i t h o u t ever having been components in the sense of ready-made incorporable parts. 1.43 I t is i m p o r t a n t t h a t the industrial examples used t o lead into the present d i s t i n c t i o n should be l e f t behind in t h e background, once they have served t h e i r purpose of presenting a reminder o f a common everyday sense in which something is an -part as opposed to a p a r t - o f - X . For general mereological purposes, p a r t s - o f - X which are actually-embodied compon­ ents in f a m i l i a r industrial or organic sorts of sense, stand on the same level as p a r t s - o f - X which may never have been described or delineated previously in any s o r t of way, howev­ er f a n c i f u l , e.g. today's top inch of m y r i g h t ear along w i t h a c y l i n d r i c a l section imagined as passing h o r i z o n t a l l y across the m i d d l e of my l e f t knee-cap of yesterday. Such i l l u s t r a t ­ ions, in order t o be i n t e l l i g i b l e , have t o be r e l a t i v i s e d , as here, t o ordinary vocabulary f o r things such as p a r t s - o f - m y -

48

The E a r l y Medieval Inheritance body, but the boundaries of t h a t vocabulary may c l e a r l y have to be trangressed or enlarged in order t o delineate some of the i n d e f i n i t e l y many such parts f o r which we h i t h e r t o have had no names or descriptions, simply because the occasion f o r t h e i r use has not previously arisen, although t h e r e is no reason why a new s c i e n t i f i c discovery of t o m o r r o w should not render some one or other of t h e m s u f f i c i e n t l y i m p o r t a n t t o be given its own d e f i n i t i o n and name. A b e l a r d is immensely con­ scious of this lack of vocabulary for theorising about i n t e g ­ r a l wholes and parts, as we are t o see (2.3). 1.44 The f u r t h e r importance of this accent on the d i s t i n c t ­ ion between -parts and p a r t s - o f - X (or, in other words, b e t ­ ween -parts and genitively-moded parts) is t h a t the presentday l i t e r a t u r e on parts and wholes is s t r e w n w i t h the wrecks of discussions which have f a i l e d to appreciate this c r u c i a l ambiguity of the word ' p a r t ' (HQS §4.543). M y own appreciation of the d i s t i n c t i o n stems f r o m the lectures of C. L e j e w s k i . I t is a d i s t i n c t i o n which enables one to evaluate y e t one more degree of sophistication displayed by w r i t e r s on mereology or on m e r e o l o g i c a l topics, and is hence w o r t h y of observation in its occurrences in the l i t e r a t u r e . In due course (2.3) f u l l e r details of Abelard's appreciation of the d i s t i n c t i o n are t o be placed on r e c o r d , but in the m e a n t i m e i t may be noted t h a t he r e i t e r a t e s i t most f o r c e f u l l y in y e t m o r e examples of comment Divisione. on Boethius ' De 1.441 Thus Abelard's f i r s t enuntiation of the d i s t i n c t i o n in his c o m m e n t a r y on Boethius' De Divisione occurs when a point which we need not pursue f o r its own sake is being raised about c e r t a i n likenesses between integral and d i s t r i b u t i v e wholes. Boethius had asserted t h a t while the i n t e g r a l whole can be split up into its parts, the converse does not hold (i.e. t h e parts cannot be s p l i t up in t e r m s of t h e i r respect­ ive wholes: the parts of a beetle are not beetles). L i k e w i s e , although a genus can be split up (notionally) into its spec­ ies, the species cannot be split up into genera: Nec partes 49

Section 1 in totum dividantur, quamvis totum separe tur in partes,  species secantur in genera, licet genus in species dividatur;  878C. To this rather fuzzy sort o f p o i n t , the exact i m p o r t of which need not detain us here, Abelard raises an objection which ingeniously incorporates what m i g h t count as instances of the prima facie u n i n t e l l i g i b l e converses r e j e c t e d by Boethius: (A) I t would appear t o be false t h a t parts cannot be s p l i t up in terms of t h e whole, and t h a t species cannot be split up in terms of a genus. For we can assert: one p a r t - o f - t h e house is the w a l l , another p a r t is the r o o f , and another p a r t the foundation. But we can also have a converse, namely: the w a l l is one thing when incorporated w i t h i n the [whole] house, and another t h i n g when outside the ambit of a [whole] house. The same s o r t of possibility of sub-classification applies to the foundation and the r o o f . A g a i n , i t can be asserted t h a t [the genus] animal has man as one o f its sub-species, and non-man as the other; this can then indeed have a converse, namely: non-man can have animal and non-animal as its sub­ divisions. The solution is t h a t although the parts may be s p l i t up in terms of t h e i r [integral] whole, the parts themselves are nevertheless not s p l i t up under the same name [ s t r i c t l y speaking]. For when we say t h a t the w a l l is one t h i n g when incorporated in the house and something else when outside the ambit of the house, we r e a l l y have here a division of some genus [e.g. w a l l - s t r u c t u r e ] into its sub­ species, and in consequence i t does not [ s t r i c t l y speaking] r e m a i n named in the same way. {Videtur esse falsum quod partes non possint dividi in totum et quod species non possint dividi in genus. Possumus enim dicere: domus alia pars paries, alia pars tectum, alia pars fundamentum. Poss­ umus convertere: paries alius in domo, alius extra domum; similiter potest dividi fundamentum et tectum. Ista potest dici: animal aliud homo, aliud et non-homo; et potest con­ verti: non-homo aliud animal, aliud non-animal. Solutio: licet

50

The Early Medieval Inheritance partes dividantur in totum, non tarnen dividuntur sub eodem nomine; quia quando dicimus: paries alius in domo, alius extra domum, est divisio generis in species et ita non rem­ anet sub eodem nomine. DA 163.20.31) 1.442 W i t h o u t pursuing the t e x t any f u r t h e r , and w i t h o u t too much speculation on the precise significance of 'does not remain named in the same w a y ' (՝non remanet sub eodem nom­ ine՝) i t is at least here clear t h a t Abelard is witnessing quite strongly t o the presence of a salient t w e l f t h - c e n t u r y d i s t i n c t i o n between the -part and the p a r t - o f - X . In this instance t h e paries in domo, the w a l l incorporated into the house, is a part-of-the-house, whereas the paries extra domo, the w a l l not incorporated into the house, is a house-part, i.e. an e n t i t y of a d i f f e r e n t species, which should hence s t r i c t l y speaking have a diverse name. Presumably our 'house-part' and 'part-of-the-house' could here count as f u l f i l l i n g such a requirement of d i v e r s i t y . 1.443 Lest i t should be thought t h a t this most i m p o r t a n t c r e d i t is being o v e r - l i g h t l y a t t r i b u t e d t o A b e l a r d , there is also his pertinent and lapidary comment on t h a t section of De Divisione wherein Boethius seems t o be in danger of not m e r e ­ ly losing sight of the specific d i s t i n c t i o n just-described, but also of giving a d e f i n i t e l y misleading expression of c o n t r a r y doctrine. Thus Boethius prolongs his d i s t i n c t i o n between dis­ t r i b u t i v e and integral wholes by adding the f o l l o w i n g con­ trast: (A) A g a i n , every genus is by nature p r i o r to its appropriate species, whereas the [integral] whole is posterior to its appropriate parts Hence also i t is t r u l y said: if the genus perishes, so also immediately do the species, whereas i f the species is abolished, the abiding genus s t i l l c o n t i n ­ ues in being. But the opposite applies to t h e [integral] whole, since i f the p a r t of the whole perishes, then the whole does not exist, even i f i t is only one p a r t which has been abolished. Conversely, i f the whole perishes, the parts 51

Section 1 may s t i l l r e m a i n about the place. Thus i f , f r o m a whole house, someone takes away the r o o f , then he does away w i t h t h a t whole which was there previously. S t i l l , w i t h the r o o f gone, the walls and foundations w i l l s t i l l abide. (Amplius genus omne naturaliter prius est propriis speciebus, totum autem propriis partibus posterius est Hinc quoque illud vere dicitur, si genus interimatur, statim species deperire. Si species interempta sit, non peremptum genus in natura consistere. Contra evinit in toto. Nam si pars totius perit, totum non erit, cuius pars una sit interempta. Sed si totum pereat, partes remanent distributae, ut si de integra domo quis abstulerit tectum, totum quod ante fuit interrupit. Sed perempto tecto, parietes et fundamenta constabunt.  879 ) 1.444 Here we have, surely enough, what may be described as an incautious p r e - t h e o r e t i c a l v i e w of the m a t t e r , and f o r a w h i l e A b e l a r d lets i t pass, since he wants to make a p o i n t about r e a l and imaginary part-hood (DA 167). But at the most c r i t i c a l point, where an aesthetically pleasing contrast has been drawn by Boethius as f r o m his Contra evenit in toto ('the opposite applies t o the [integral] whole') in the passage quoted above, Abelard undermines t h a t contrast at its c r u c i a l p o i n t , namely where the integral parts are said to remain about the place a f t e r the whole has perished. He repeats, but, most s i g n i f i c a n t l y , qualifies Boethius ' t e x t thus: (A) I f the whole perishes, the parts may w e l l r e m a i n about the place, but t h a t is t o say: those objects which had a previous career as parts may abide w i t h o u t being susceptible of t h e i r f o r m e r t i t l e , since they are not now asserted to be parts. (Si totum pereat, partes remanent distributae, scilicet res istae quae fuere partes permanent, sed non sub eo nomine, quia non dicuntur partes. DA 168.1.3) 1.445 Presumably, i f pressed, Abelard should also apply his s t r i c t sense of part (i.e. the g e n i t i v e , p a r t - o f - X sense) to the e l i m i n a t i o n of any Boethian thesis about parts in such a

52

The Early Medieval Inheritance s t r i c t sense being ' p r i o r ' to t h e i r wholes, as in the last but one quoted passage. In spite of this apparent c o m m i t m e n t , however, we are later t o see how he took on the r ô l e of a medieval ancester of modern a t o m i c theory when impelled in the d i r e c t i o n of such a thesis of p r i o r i t y (2.531 below). I t is a thesis which has a natural tendency t o align i t s e l f w i t h a t o m i c theory (in the classical sense, o f course). Perhaps the l a t e r general medieval adoption of w h a t was anciently a theory opposing i t s e l f to a t o m i s m , namely the d o c t r i n e of matter and form, the grounds of which are discussed in 3.11, explains the by then pervasive appreciation of the d i s t ­ i n c t i o n between -parts and p a r t s - o f - X . 1.45 The doctrine of m a t t e r and f o r m may for the moment be described as taking its rise f r o m everyday cases of the coming into being of objects constructed f r o m p r e - e x i s t i n g makings, or 'parts' (in the -part sense). The makings are the matter and the nature, essence, or quiddity of the con­ s t r u c t e d object is the form. This simple scheme is extended outwards f r o m its o r i g i n a l c o n t e x t in all sorts of ingenious fashions: cf. MM, HQS §5.21. §5.23, §5.32. F o r m a l discourse is q u i d d i t a t i v e discourse, o f t e n centred around the d e f i n i t i o n of what i t is to be such and such a s o r t of t h i n g , and all its exigencies may be covered by t a k i n g i t t o involve functors of s/n) index, as i n i t i a l l y described in 0.4 above, the s/(s/n and which are defined in 10.26. 1.46 What concerns us at this present point is the con­ c o m i t a n t appreciation t h a t the nature or f o r m of the con­ s t r u c t e d object only supervenes on its makings when they have acquired an appropriate composition ör s t r u c t u r e . What f o r an atomist is a mere rearrangement of changeless and p r e - e x i s t ­ ing elementary particles is f o r the adherent of the d o c t r i n e of m a t t e r and f o r m , or hylomorphism, a r a d i c a l change f r o m the non-being of a t h i n g of a given nature to there being a thing of t h a t given nature, hence t h e correspondingly r a d i c a l change in the status of its makings f r o m being -parts to be-

53

Section 1 ing p a r t s - o f - X . Surely enough, the a t o m i s t concedes t h a t the same change takes place, at least nominally, but the eternal changelessness and i n t r i n s i c i n d i f f e r e n c e to a l t e r a t i o n of the principal elementary atoms tends t o reduce its significance. But t h a t significance can be enhanced f o r an i n t e g r a l whole, whether its parts are seen as a t o m i c individuals or not, by comparing its composition to the d e f i n i t e l y f o r m a l (i.e. quidd­ i t a t i v e ) composition of genus and differentia so as to f o r m a species; the genus is now analogously viewed as ' m a t t e r ' or makings, and the differentia as s t r u c t u r e - i d e n t i f y i n g ' f o r m ' : c f . 0.5 above and HQS §4.3421, §5.24. Here is Boethius making this comparison: (A) A g a i n , the genus is the m a t t e r in respect of its species, f o r even as the bronze, having received its f o r m , becomes a statue, so also the genus, having received its dif­ ferentia becomes the species. Thus likewise the m u l t i p l e parts of the [integral] whole are the m a t t e r , and s i m i l a r l y the composition of those same parts is the f o r m . For even as the species is made up of genus and differentia, so also the [integral] whole is made up f r o m its parts. Hence i t comes about t h a t the whole d i f f e r s f r o m each of its parts because o f the very composition of those parts along w i t h the others, and likewise the species d i f f e r s f r o m the genus by the combination of the differentia. (Amplius quoque gen­ us speciebus materia est. Nam sicut aes accepta forma tran­ sit in statuam, ita genus accepta differentia transit in speciem. Totius vero partium multitudo materia est; forma vero earundem partium compositio. Nam sicut species ex genere constat et differentia, ita totum constat ex partibus; unde fit ut totum ab unaquaque parte sua, partium ipsa compositione differat, species vero a genere, differentiae coniunctione.  879 - D) 1.461 Leaving aside any f a c i l e expostulation which could be inspired by the highly impressionistic comparison of the t w o diverse cases (although i t may be about the only discussion-

54

The Early Medieval Inheritance orientation available at the present pre-systematic level) we may note that not only is the distinction between -parts and parts-of-X likely to be reinforced by that comparison, but also by the thesis of part-whole disparity ('the whole differs from each of its parts') which Boethius seems to be quite deliberately stressing here. Certainly both points emerge quite strongly from Abelard's attempt to construe topic-infer­ ential connections based upon the latter part of this Boethian passage: (A) The genus is the matter of its species, but the multiple parts, i.e. the many parts, are the matter of the whole; it is the combination of those same parts which is the form, if the whole is thus made up from the parts and from the com­ position Indeed, the multiple parts are the matter of the whole, and the combination of those parts is the form. If the [integral] whole thus is made up from parts and from combination, even as the species is made up from genus and differentia, then on the basis of likeness one can make an inferential connection. For one has the following reciprocal inference-justifying thesis: if the multiple parts are the matter, and the combination is the form, then the [integral] whole is made up of the parts and of the combination, even as the species is made up of the genus and the differentia; and if the [integral] whole is thus made up of the parts and of the combination in the same way as the species is made up from the genus and the differentia, then the multiple parts are the matter and the combination of the parts is the form . . . . Indeed, the form-composition brings it about that the whole differs from each and every one of its parts, giv­ en the combination of those parts, they being taken into ac­ count as involved in the whole because of the conjoined parts. (Genus est materia suis speciebus sed multi tudo partium, id est multae partes, sunt materia totius, sed compositio earundem partium est forma, si totum ita constat ex partibus et ex compositione Vere multitudo partium 55

Section 1 est materia totius et compositio earundem partium est forma. Si totum ita constat ex partibus et ex compositione, sicut species constat ex genere et ex differentia, tunc illud est a Convertitur enim sic: Si multitudo partium pari comitabili. est materia, et compositio est forma; tunc totum ita constat ex partibus et ex compositione, sicut species ex genere et differentia; et si totum ita constat ex partibus et ex comp­ ositione sicut species constat ex genere et ex differentia, earumdem tunc multitudo partium est materia et compositio partium forma est ... Quandoquidem compositio forma, unde fit ut totum differat ab unaquaque parte sua, compositione ipsarum partium, considerata in ipso toto propter partes coniunctas. DA 168.18.35) 1.462 Stress on the special status of g e n i t i v e l y - m o d e d p a r t s - o f - X , and on the thesis of p a r t - w h o l e d i s p a r i t y , could scarcely be more w e i g h t i l y expressed than i t is here, given the l i t e r a r y language then available. Allusions to the p r i m a c y of form in integral-whole contexts is destined t o endure w e l l beyond A b e l a r d . Thus we have Aquinas' t h i r t ­ eenth-century v e r d i c t : 'No separated p a r t shares in the f o r m of the whole, e.g. the hand separated f r o m the man does not share in the f o r m of humanity': ΑΡΗ I , i v , 43, p. 21. Further remarks on this topic w i l l be made at the appropriate j u n c t ­ ure in 3.41 below. 1.5 The Scandal of the Non-discrete Singular 1.51 As medieval thinkers were w e l l aware, the seeds of confusion are constantly latent in the n o n - t h e o r e t i c a l l y or­ iented nature of l i t e r a r y language and ordinary usage. St. Anselm of Aosta and Canterbury is an early witness t o such awareness: HL §2.13, §4. We have so far surveyed at least t w o areas in which the looseness of sense concerning part has to be overcome i f a viable mereology is to be produced. N o w , given the t r u i s m t h a t wholes tend t o consist of a p l u r a l i t y of parts, and the f u r t h e r t r u i s m t h a t t h a t parts must be s i n -

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The Early Medieval Inheritance gulars in some sense i f they are t o make up a p l u r a l i t y , we are now t o see how the mere elevation of an ordinary t e r m into a technical t e r m f o r singulars may embody an option which excludes essential t h e o r e t i c a l generality. 1.52 L e t us r e v e r t y e t again to the banal example of the house which has walls, r o o f , and foundations as its parts. C l e a r l y some parts may be made up of other parts. Thus the walls plus the r o o f make up a p a r t . But so also do t h e r o o f plus the foundation. We thus now have t w o parts which are c e r t a i n l y n o n - i d e n t i c a l , and hence deserve t o be counted among the m u l t i p l i c i t y of the parts of the house. Y e t they overlap in t h e sense t h a t they have a part in common, namely t h e r o o f . This notion is easily and obviously d i s t i l l a b l e into a mereological thesis defining such overlap thus: .1

For a l l A and B, A is an OVERLAPPER OF  i f and only i f , f o r some ,  is a p a r t of A and  is a p a r t of  (cf. 10.34 below, and SLM 38, D12)

(Here  is the common p a r t , the counterpart of the r o o f in the example). Conversely, the non-overlapping parts, such as the w a l l and foundations, have no p a r t in common, i.e. the one is outside the other, thus: .2

For a l l A and B, A is OUTSIDE  i f and only i f A exists, and  has a p a r t , and any p a r t of  is not a p a r t of A

(This English approximation is d e f e c t i v e in t h a t i t appears t o exclude simple or a t o m i c objects (i.e. objects which have no parts) f r o m the possibility of being outside one another: c f . 10.331, 10.35 below, and SLM 36, D5. This need not detain us here, as i t does not a f f e c t the point at issue). 1.53 Now things which satisfy .2 can be said t o be discrete (cf. 10.336, SLM 37, D10) in respect o f one another, and i t is apparent t h a t such discreteness has sometimes been taken to

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Section 1 be the hallmark of i n d i v i d u a l i t y or s i n g u l a r i t y (and hence, i n d i r e c t l y , of p l u r a l i t y ) . This assumption has even been e x ­ p l i c i t l y enshrined in the taking over of the existing L a t i n adjective 'discretus' as a technical indicator of individuals; c f . f o r example, note 82 of CLM 232, and the generally d i f f ­ used notion of 'discrete' suppostilo: CLM 194, KB 40-1. How then are we to name, or even t o b r i n g w i t h i n the ambit of such suppositio d o c t r i n e , the possibly non-discrete individuals c o n s t i t u t i n g the m u l t i p l i c i t y of an object's parts? Although the problem was r a r e l y confronted so d i r e c t l y , its essential f e a t u r e , namely the existence of overlapping parts, was c e r t ­ ainly recognised in the l i v e l y t w e l f t h and f o u r t e e n t h century discussions of a c e r t a i n 'Master Peter' and of W y c l i f respect­ i v e l y , as sections 2.78 and 6.232.233 are t o show. In the early period Abelard brings in the notion of overlap i n c i d e n t a l l y , when c r i t i c i s i n g those who characterise as ' p r i n c i p a l ' the parts which are not themselves parts of parts, e.g. walls, r o o f , and foundation in respect of the house. I t is easy enough, runs his c r i t i c i s m , to take any parts thus specified as ' p r i n c i p a l ' , and t o render t h e m secondary by appropriate dichotomy: (A) As f o r those who only c a l l ' p r i n c i p a l ' those parts which are [not] also parts of parts, I wonder why they c a l l the w a l l , r o o f , or foundation 'principal p a r t s ' . For i f we look at the house dichotomously, so t h a t on the one side we consider the foundation as one independent p a r t , and on the other we consider the w a l l - a l o n g - w i t h - t h e - r o o f , we then are l e f t w i t h either r o o f or w a l l as parts of the [ l a t t e r complex] p a r t . (Hi quidem qui principales partes eas solum vocant quae partium partes etiam <non> sint, miror quare parietem aut tectum aut fundamentum principales partes appellent. Nam si domus bifarium divisionem consideremus, ut illinc fundamentum per se partem unam constituamus, hinc autem alteran partem ponamus parietem simul et tectum .... , inveniemus aut AD 549.21.27) parietem aut tectum partes partis.

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The Early Medieval Inheritance 1.531 Here we have e x a c t l y the sort of example pressed into service above in order t o lead t o the notion of overlap. Y e t although here and elsewhere, as we are t o see in great d e t a i l later (2.4) A b e l a r d makes many allusions to cases which e m ­ body non-discrete parts, i t is only in the 'Master Peter' t r a c t , which has been a t t r i b u t e d t o h i m , t h a t the r e a l l y s a l ­ ient features of such a situation are brought to the f o r e . The outcome, perhaps because of untowardly persistent e n t a n ­ glement w i t h the t e r m ' d i s c r e t e ' in its medieval sense, is s t i l l going to prove c o m p a r a t i v e l y disappointing: cf. 2.78. For t h e m o m e n t , however, enough has already been said t o make i t plain t h a t the f o l l o w i n g position, a t t r i b u t e d to the later Henry of Harclay (CLM 429, 434), far f r o m being ' l o g i c a l l y necessary', is just plain false: (A) E v e r y t h i n g t h a t exists in r e a l i t y is essentially singular - i.e. logically incapable of existing i n , as a constituent of, numerically many simultaneously. 1.532 Further, in spite of the a t t r i b u t i o n of the 'Master Peter' t r a c t to A b e l a r d , there is s t i l l no question as to Abelard's own obsession w i t h the p l u r a l i t y of discrete e l e m ­ ents as the keynote f o r the c o n s t i t u t i o n of a m u l t i p l i c i t y : (A) A l l those things which are the same in t h e i r being are the same in their number, but the converse does not hold, since every p a r t is numerically the same as its whole, i.e. is not n u m e r i c a l l y diverse f r o m i t , but they are not the same in t h e i r being, since no being of a p a r t is the being of the whole. For we account no things numerically diverse unless they are discrete in respect of the whole quantity of their being, so t h a t neither is i t the case t h a t the one is the other, nor t h a t something of the one is something of the other. I t is in this fashion t h a t we account those things n u m e r i c a l l y diverse t o which d i v e r s i t y in numeration is properly allocated. This by no means applies t o the p a r t and to the whole, the number of which is in no way made m u l t i p l e . Thus no one counts some whole as one, and then by 59

Section 1 adding one of its parts counts i t as two; this is because the p a r t is embraced in the number of the whole. Those things are hence not numerically diverse (i.e. those things are the same) which are not discrete wholes in respect of the quantity of their being. (Et haec omnia quidem quae eadem sunt essenti aliter, eadem sunt numero, sed non convertitur, quia omnis pars cum suo toto est eadem numero, idest non diversa numero ab eo, sed non essentialiter eadem sunt, cum nulla partis essentia sit essentia totius. Nulla enim dicimus diversa numero, nisi quae tota quantitate suae essentiae discretae sunt, ut  unum sit aliud  aliquid de uno sit aliquid de alio. Sic igitur diversa numero dic­ imus, quorum diversitas numerando valet assignari. Quod nullo modo parti et toti convenit, cum nullo modo numerus valeat multiplicari. Nemo enim de toto dicit 'unum', et add­ endo partem illius dicit 'duo', cum in numero totius pars comprehensa sit. Sunt itaque non diversa numero, idest ea­ suae essentiae tota discreta dem, quaecumque non quantitate sunt. GA 558.21 - 559.4) Given this sort of statement, one may then ask how Abelard can deal w i t h the t w o - p a r t e d p a r t (wall-plus-roof) and its parts (wall or roof) each of which has parts in c o m ­ mon w i t h the f i r s t (two-parted) p a r t , and all of which f i g u r e in his passage previously quoted. Do they not all count as parts (in the p l u r a l ) , so t h a t we can thus use the p l u r a l noun 'parts' notwithstanding the overlap which makes t h e m non-dis­ crete? He is, a f t e r a l l , elsewhere quite prepared t o take cues f r o m the use of p l u r a l verbs in decisions as to oneness or manyness: 2.31(A). And i f one is prepared to count the w a l l - p l u s - r o o f part as one p a r t , the w a l l as another, and the r o o f as another (to which surely no objection can be made) then there would seem t o be likewise no objection to counting the whole house as one t h i n g , the walls, r o o f , and foundation as three m o r e , thus making at least four objects, t o which the many other discriminable parts involved may also be added.

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The E a r l y Medieval Inheritance The f a c t t h a t what look like cross-classifications may be intervening here is not relevant t o the present p o i n t ) . 1.533 A g a i n , i t is no use t r y i n g t o keep Abelard's accounts s t r a i g h t by the common present-day expedient of grasping at the ever-handy straw of ' t y p e ' or ' l o g i c a l levels ' (KAU I, 281) which has been so fashionable in the last h a l f - c e n t u r y . That mereological theses may be maintained consistently at the n o m i n a l l y - t e r m e d level has been argued in 0.5. I t may w e l l be t r u e , as a f a c t of local linguistic anthropology, or desir­ able, as a s t y l i s t i c recommendation, t h a t as A b e l a r d states, 'we [whoever t h a t may be] account no things n u m e r i c a l l y d i v ­ erse unless they are discrete in respect of the whole quant­ i t y of t h e i r b e i n g ' , as stated in the p e n u l t i m a t e l y quoted passage, but f r o m the point of view of a t r u l y general m e r eology, both this and the placing of parts and wholes at d i v ­ erse logical levels are quite unnecessary r e s t r i c t i o n s . I f one s t i l l wants t o m a i n t a i n t h e m f o r some reason or another, then the details can easily be spelled out by means of the section 10 suggestions, but w i t h f u l l consciousness of the r e s t r i c t i o n s thus being imposed on the generality of m e r e o l ­ ogical theory. Since the 'Master Peter' already-mentioned was w e l l acquainted w i t h c e r t a i n paradoxes which a d m i t t e d l y f o l l o w f r o m the m o r e general stance, f u r t h e r discussions may be postponed u n t i l the examination of his relevant t e x t in 2.7, and of the uncannily parallel theses of John W y c l i f (6.232, .233). 1.6 T e m p o r a l Parts 1.61 Accustomed as they were t o read and to comment on t h a t chapter 6 of A r i s t o t l e ' s Categoriae which deals w i t h q u a n t i t y , medieval philosophers were f a m i l i a r w i t h the notion of t e m p o r a l parts and wholes. However, t h e i r t r e a t m e n t of these tends t o range over topics such as measures of t i m e intervals (days, weeks, and so f o r t h ) , or, f o l l o w i n g A r i s t o t l e ' s h i n t , over t h e t e m p o r a l extent of actions. T r e a t m e n t of the 61

Section 1 t e m p o r a l dimensions of objects tended t o f a l l into the back­ ground compared w i t h t h e i r spatial aspects. Indeed, as we are to see, although Abelard had all the t h e o r e t i c a l apparatus needed f o r a sophisticated appreciation of the mereological aspect of t e m p o r a l dimensions, he just f a i l e d to bring i t e f f ­ e c t i v e l y t o bear: 2.6. The by no means t o t a l l y prevalent ass­ o c i a t i o n of verbs w i t h t i m e , deriving f r o m A r i s t o t l e ' s d e f i n ­ itions of the parts of speech, may be responsible f o r this sort of f a i l u r e : cf. HQS §3.241. 1.62 Of course, general mereology as such need have no t r u c k at all w i t h temporal and spatial parts and wholes; these are the province of subsequent theories, namely, Chronology and Stereology, w i t h which we are not p r i m a r i l y concerned. However, the level of i n f o r m a l p r e l i m i n a r y discussion (which may later govern notional choices of values f o r variables) is considerably enhanced by the presence of an appreciation of the t e m p o r a l , as w e l l as the s p a t i a l , aspects of things. This w i l l become amply evident when Abelard's peculiar contortions arising f r o m a lack of t h e o r e t i c a l c o n t r o l of such t e m p o r a l aspects are reviewed: e.g. 2.5, 2.6. 1.7

Conclusion Two sorts of a u t h o r i t y have coloured t h e foregoing acc­ ount. The one is h i s t o r i c a l , w i t h allusions to Boethius, A b e l ­ ard, and others. Notions have been traced back t o t h e i r prop­ onents by reference to original t e x t s . The other source of remarks has been the very nature of the t o p i c of parts and wholes. Here hints have been given of how t h e o r e t i c a l found­ ations, u l t i m a t e l y to be exposed in 10.3, m i g h t be considered, and coincidences between such hints and the h i s t o r i c a l t e x t s have been noted. Needless t o say, here and throughout this essay, i t is the theory, and not the h i s t o r i c a l a u t h o r i t y , which is the source of verdicts on the sense and t r u t h of mereological and other theses. In other words, as some m e d ­ ievals themselves would s t a t e , outside theology, the appeal t o

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The Early Medieval Inheritance the authority of an author is the lowest form of justification for a tenet. A point of doctrine is of interest not just bec­ ause Abelard (for example) stated that it was so (or not so), but rather because that point happens to find confirmation (or disconfirmation) in the general theory which may be ref­ lected in Abelard's arguments, or in the theoretical discussion of others dealing with that point. The arguments themselves are thus subject to rational tests, rather in the way that ar­ ithmetical truths are likewise subject. In the end, the theor­ etical truths ultimately to be defended are fully stated, us­ ing perspicuous categorial language, in the Presuppositional Explicitation contained in section 10. There a few primitive terms are the intelligible foundations which enable these truths to be held together in axiomatised systems. A t the same time they are foundations for the study of the subjectmatter of this entire book, open for all to see, and therefrom to advance. For the moment, however, while glancing (rather inad­ equately) at Boethius the Roman's great work on the several processes which he subsumes under the heading of 'division', and enjoying some assistance from the clarifications of the early medieval Abelard, it has proved possible to call atten­ tion to the thesis of the disparity of parts and wholes (1.32), to the distinction between -parts and parts-of-X (1.4), to what is, from the point of view of medieval vocab­ ulary, the scandal of those non-discrete singulars to which elementary mereology calls attention (1.5), and finally, to an anticipatory allusion concerning temporal parts (1.6). Such acquisitions, along with informal statements of mereological theses, have already constituted means of adjudicating upon the level of sophistication evinced in mereological texts. Many more such means are to be added as the exposition ad­ vances, thus yielding a heightened appreciation of the signif­ icance of the fields undergoing exploration.

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2. Abelard and His Contemporaries 2.1 H i s t o r i c a l P r e l i m i n a r y 2.11 I t is well-known t h a t in his Metalogicon (Bk. I I , ch. 17) John of Salisbury gives an account of t w e l f t h - c e n t u r y dis­ cussions of the 'universals' controversy and its h i s t o r i c a l s e t t i n g . I t suggests t h a t a mereological theory of universals was upheld by Joscelin, Bishop of Soissons. Surely enough, there does r e m a i n a substantial record (incorporating what was f o r m e r l y known as the Fragmentum Sangermanense, a s a l ­ ient piece of nomenclature t o be retained in 2.71 below) which alludes to Joscelin and does contain some interesting m e r e o l ­ ogical m a t e r i a l . I t also contains an even greater amount of very competent statements concerning q u i d d i t a t i v e discourse, as 2.7 is also to record, but these can only be considered i n ­ c i d e n t a l l y , and by way of contrast, as outlined in 0.5. Peter Abelard (1079 - 1142) does d i r e c t a set of arguments against a mereological theory of universals of the sort Joscelin m i g h t have supported. Those arguments have o f t e n been the subject of discussion, as in TAU 113 - 5, KAU P t . I I , ch. 8, HQS §4.5. 2.12 However, f r o m the point of v i e w of mereology as such, the f a c t t h a t i t may be used to provide one of many possible a l t e r n a t i v e elucidations of discourse at the specific and gen­ eric l e v e l , and hence of discourse involving universals (cf.

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Abelard and his Contemporaries 0.4) remains a c o m p a r a t i v e l y t r i v i a l t r u t h . The i n t e r e s t of the study of aggregate (or integral) wholes or c o l l e c t i v e classes would be unwarrantably diminished were i t m e r e l y t o have such elucidation as its sole point (0.45). I t is such a d i m i n u t i o n , based on the contingent accidental bias of the n a r r a t i v e mode (0.26), which has h i t h e r t o tended to conceal f r o m v i e w any realisation t h a t Abelard's mereology is by no means e n t i r e l y taken up w i t h applications to the universals question. He has a vast corpus of mereological discussions which r i v a l in volume and interest the pseudo-Joscelin's r e m ­ arks in the Fragmentum Sangermanense. Indeed, so vast is t h a t corpus t h a t i t must be emphasised f o r t h w i t h t h a t the description of Abelard's m e r e o l o g i c a l work which now follows constitutes only a v e r y p a r t i a l and low-grade clearing of some of the relevant t e r r a i n . I t has only been kept w i t h i n reasonable bounds by a ruthless but r e g r e t f u l refusal t o be entangled in those neighbouring fields which are, in the end, essential t o a f u l l appreciation of the s i t u a t i o n . 2.13 In any case, even the h i t h e r t o unnecessarily s t r i c t coupling of Abelard's mereology w i t h t h e universals question has already given rise t o the very ample and profound works of Tweedale (TAU) and K i n g (KAU). To these we shall be ind­ ebted in due course on all sorts of counts. Only a short note on Abelard's campaign against c o l l e c t i v e universals need hence be added to the account already provided in HQS §4.5. 2.2 Some C r u c i a l D i s t i n c t i o n s 2.21 The thoroughness of Abelard's appreciation of t h a t general d i s t i n c t i o n between integral (or c o l l e c t i v e or agg­ regate) wholes and d i s t r i b u t i v e (or universal) wholes (cf. 1.2) may f i r s t be c o n f i r m e d by f u r t h e r quotations. (We have a l ­ ready had some i n t i m a t i o n of his appreciation of aspects of this in 1.32). His trenchant and relevant account of the m a t ­ begins by an o v e r a l l dis­ t e r as i t appears in his Dialectica t i n c t i o n of t h e various ways in which wholes in general may

65

Section 2 be divided into parts. This, he says, m a y occur substantial­ l y (secundum substantiam), f o r m a l l y (secundum formam), or s u b s t a n t i a l l y - f o r m a l l y (secundum utrumque): AD 546.21.23. Substantial division is t r e a t e d f r o m AD 546.24 t o 555.19. Side-stepping any express elucidations of these t e r m s , i t is s t i l l more or less possible t o advance d i r e c t l y t o the inst­ ance o f substantial division which is now in question, i.e. the d i s t i n c t i o n between the integral whole and the d i s t r i b u t i v e whole (՝secundum diffusionem') (A) The whole w i t h respect t o substance is one t h i n g in so f a r as i t embraces quantity (being then said t o be an int՛ egral whole) and another in respect o f the distribution of a shared essence (being then a universal whole) as when a species is distributed among its individual members, thus: ' This is one m a n , t h a t is another, ....' This whole its universality by being predicated o f its ind­ evinces ividual elements, whereas such does not hold at a l l in res­ pect of an i n t e g r a l whole, since the p r e d i c a t i o n is then o f the aggregation of a l l the elements at once, e.g. 'this house ' is predicated of these walls, this r o o f , and this foundation taken all at once. A n integral whole cannot be a universal whole since u n i v e r s a l i t y does not involve elements in a q u a n t i t a t i v e sense, but rather consists in the d i s t r i b u t i o n of its common nature throughout t h a t m u l t i p l i c i t y of whose individual elements i t is predicated. The integral whole, in contrast, involves only a singular predication in respect of i t s e l f , as in the case of Socrates, who is made up of these members. (Totum autem secundum substantiam, aliud secundum comprehensionem quantitatis, quod dicitur integrum, aliud secundum diffusionem communis essentiae, quod universale est, ut cum species suis distribuitur individuis hoc modo: 'hmo alius hic, alius ille ...' hoc autem totum universale esse apparet quod de singulis praedicatur partibus, integrum autem minime, sed de omnibus simul col·lectis, ut haec domus de his parietibus, et hoc tecto et hoc fundam-

66

Abelard and his Contemporaries ento simul acceptis. Hoc autem quod integrum est, quippe partes in quantitate universalitas non habet, sed in suae communitatis diffusione permulta de quibus singulis praedicatur. Integrum autem singularem ad se praedicationem habet, ut Socrates qui ex his est compositus membris. AD 546.24 - 547.5, c f . 574 - 5 and DA 166, 193) 2.211 Here the p e r f e c t l y proper and usual Abelardian r e f ­ usal o f u n i v e r s a l i t y (in the ' d i s t r i b u t i v e ' sense) t o the i n t e g r a l whole is e x e m p l i f i e d (cf. GA 14, TAU 114), as also is his appreciation of the s i n g u l a r i t y of the i n t e g r a l whole (in so f a r as i t is a complete c o l l e c t i o n : c f . 10.321). The e x a m ­ ple o f the house, t h a t usual accompaniment of medieval dis­ cussions of the i n t e g r a l whole, recurs when the d i s t i n c t i o n between the t w o sorts of whole is repeated (but w i t h ՝totum secundum coniunctionem՝ used as an a l t e r n a t i v e t o ' i n t e g r a l ' ) and a f u r t h e r interesting and i m p o r t a n t elaboration, which is to have a most prominent f u t u r e , now appears: (A) But now we shall f i r s t deal w i t h commonplace rules der­ ived f r o m whole. Now we take ' w h o l e ' in t w o ways. We look at i t in one way in respect of its d i s t r i b u t i o n , and in a n ­ other in so far as i t involves c o l l e c t i v i t y . Thus t h a t w h i c h is a d i s t r i b u t i v e whole we c a l l 'generic', f o r one and the same whole generic substance exists in a l l of its species, as in t h e case of animal in man and in horse, or man in Socrat­ es and Plato Now, however, we can f o l l o w through t h e rules which apply to the i n t e g r a l whole. Of these t h e f o l l o w ­ ing is the f i r s t : I f the whole exists, any p a r t of i t must likewise exist. For example, i f the house is, its w a l l must also exist. On the other hand, i f the sign of q u a n t i t y is [taken t o be] p r e ­ f i x e d so as to gather together the individual parts, then not only is i t in respect of existence, but also as regards any other predicate t h a t 'the whole' covers any p a r t . Thus i f we assert:

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Section 2 I f the whole house is w h i t e (i.e. i f w h i t e applies to the house in respect of [all] its individual parts), then the w a l l is w h i t e , then the relevant rule is of the f o l l o w i n g sort: Whatsoever applies w h o l l y t o the whole (i.e. in respect of each of its individual parts) applies to each and every p a r t thereof. 'Totum' autem du­ (Nunc autem prius locum a toto tractemus. obus modis accipimus: aliud secundum diffusionem, aliud sec­ undum coniunctionem consideremus. Et quod quidem in diffus­ ione totum est, generale vocamus. Generalis enim substantia tota simul et eadem in omnibus suis speciebus existit, veluti animal in homine et in equo, aut homo in Socrate vel in Platone Nunc autem regulas eius quod integrum est totius exsequamur. Quarum prima haec est: toto existente, necesse est quamlibet eius partem existere, veluti si domus sit, et parietem necesse est esse. Si vero quantitatis signum apponitur quod singulas colligit partes, non solum circa esse, sed circa quodlibet praedicatum 'totum'' ponit quamlibet partem. Ut si ita dicamus: si tota domus est , idest si album convenit domui circa singulas eius partes, tunc paries est albus, regula autem huismodi erit: quidquid convenit toti totaliter, idest secundum singulas eius partes, convenit cuilibet partem eius: AD 339.30.35, 343.32 - 344.7) 2.212 In this passage, the apparently t r i t e d i s t i n c t i o n between t w o of the possible senses of the L a t i n word ՝totum' represents a f e a t u r e of medieval mereology which was to become immensely prevalent in l a t e r centuries. As A b e l a r d says, when the word ՝ totum՝, ' w h o l e ' , is used as a sign of the ' q u a n t i t y ' of the proposition (i.e. when occurring at the head of a c a t e g o r i c a l proposition in a manner s i m i l a r t o t h a t in

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Abelard and his Contemporaries which ՝omnis՝, ' a l l ' appears) then i t may cover any of the parts of the whole in question. For example, ՝Tota domus est alba', 'The whole house is w h i t e ' , must, i f the extension of the rule under discussion is to be m a i n t a i n e d , be understood as 'Each and every q u a n t i t a t i v e or i n t e g r a l p a r t of the house is w h i t e ' . On this analysis one may then infer t h a t the w a l l is w h i t e (it being granted t h a t the w a l l is one of the parts of the house). This quantificational construal of ՝totum՝ as 'Each and every ...' combined w i t h the m e r e o l o g i c a l t e r m 'pars', ' p a r t ' , so as to y i e l d a complex f u n c t o r having the e f f e c t of 'Each and every p a r t of ... is ...' l a t e r became w e l l - k n o w n as the 'syncategorematic' construal. This is because the L a t i n counterparts of the ' E v e r y ' , ' N o ' , and 'Some' of the proposi t i o n a l forms 'Every ... is ...', 'No ... is ...', 'Some ... is ...', and 'Some ... is not ...', each of c a t e g o r i a l index s/(n n), were among what were said t o be 'syncategorematic' t e r m s ; here we have Abelard suggesting t h a t 'whole' may on occasion join t h e i r company; later i t would also r e c e i v e t h e i r label, as 7.08 is t o r e c o r d . The functors just described were said t o f o r m categorical propositions, and were t h e centrepieces of categorical syllogistic (cf. LAS). The nominal or n a m e - l i k e expressions which suitably f i l l e d the gaps in these functors were said to be ' c a t e g o r e m a t i c ' t e r m s . Thus not only are ' t i g e r ' and ' f u r r y ' , in 'Every t i g e r is f u r r y ' , c a t e g o r e m a t i c t e r m s , but also the generic and s p e c i f i c names of i n t e g r a l wholes ('the whole ...', 'this whole house', and so on). 2.213 In short, the syncategorematic use of 'totum', 'the w h o l e ' , involves its incorporation w i t h i n a f u n c t o r , a gapped expression, which takes t w o names or name-like expressions as its completions, i.e. the categorial index of the resultant f u n c t o r is, like those others mentioned in the last paragraph, s/(n n). Subsequent treatises devoted to syncategoremata, or syncategorematic t e r m s , which we are later to consult in sec­ t i o n 7, are in e f f e c t analytic expositions of what are for the most p a r t p r o p o s i t i o n - f o r m i n g f u n c t o r s , usually t a k i n g t w o , 69

Section 2 but sometimes only one, of the nominal or quasi-nominal c o m ­ pletions. 2.214 However, in contrast, ՝totum . . / , 'the whole ...', may i t s e l f sometimes be seen as a nominal or n a m e - f o r m i n g ex­ pression, i.e. as i t s e l f akin in semantic category to t h e c o m p l e t i n g nominal expressions, of index n, which serve t o 'saturate' the syncategorematically-based functors mentioned above. Such completions were, as we have seen, said to be categorematic t e r m s , and ՝totum ...'. 'the whole ...' was also said to be used categorematically in a proposition such as, ՝Tota domus valet centum marchas՝, 'The whole house is w o r t h one hundred m a r k s ' . C l e a r l y this 'whole ...' must not be syncate g o r e m a t i c a l l y construed in the way outlined in t h e last par­ agraph, as this construal would y i e l d 'Each and every p a r t of the house is w o r t h one hundred m a r k s ' . This in t u r n would allow one t o i n f e r t h a t just an i n s i g n i f i c a n t f r a g m e n t of the house was also w o r t h one hundred marks, a result p l a i n l y not intended. On the c o n t r a r y , ' t o t a domus', 'the whole house', must in this context be seen as a nominal-type designation of a whole which constitutes the subject of the proposition. G r a m m a t i c a l l y speaking, ՝tota', 'the whole ...', is here, in its c a t e g o r e m a t i c use, a f u n c t o r which forms a name f r o m a name, i.e. its index is n/n: c f . 10.326. In short, as a p r e l i m i n a r y characterisation of what later became common medieval i n t e l l e c t u a l c o i n , one may say t h a t totus (or tota, or totum, according t o the g r a m m a t i c a l gender of the c o m p l e t i n g name or names) may sometimes be seen as founding a p r o p o s i t i o n - f o r m i n g f u n c t o r t a k i n g t w o names as completions, i.e. s/(n n) is its c a t e g o r i a l index. Sometimes, however, i t is to be analysed as a n a m e - f o r m i n g functor which when adjoined to a name forms a nominal expression, i.e. n/n is its index. In the l a t t e r case i t may approximate in sense to '... as a w h o l e ' , or t o 'the whole of ...'. In the f o r m e r , the s/(n n) sense, however, a distinguishing natural-language approximation is d i f f i c u l t to f i n d . This is because one would

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Abelard and his Contemporaries want to make i t plain by means of such an approximation t h a t by using i t t o express the sense of Totus Socrates est ... , 'The whole Socrates is ...', one would be l e t t i n g oneself in f o r 'Every integral p a r t of Socrates is ...' as its equivalent, even in cases wherein an insertion such as 'less than Socrates' would make the rendering in the l a t t e r sense obviously t r u e , but in the f o r m e r apparently false. There seems to be no such a p p r o x i m a t i o n , and Paul of Venice w i l l m a i n t a i n the c o n ­ sequently surprising e f f e c t of this syncategorematic sense, an e f f e c t which he is by no means the f i r s t t o n o t i c e (cf. 7.08), by discussing such cases in m u l t i p l e d e t a i l (8.1). 2.215 I t is because of the later t e r m i n o l o g i c a l prepon­ derance of the d i s t i n c t i o n between the c a t e g o r e m a t i c and the syncategorematic senses of totus, ' w h o l e ' , t h a t a t t e n t i o n is now being called t o Abelard's prescience in n o t i c i n g the dist­ i n c t i o n , which he goes on t o stress (although w i t h o u t the later terminology) in the next quotation, a continuation of his last passage. Thus he has suggested t h a t the r u l e which applies to the i n t e g r a l whole concerning the existence of its part ('If the whole exists, then any p a r t also exists') may even apply when predicates other than '... exists' are in question, e.g. one may have ' I f the whole house is w h i t e then any p a r t of i t is w h i t e ' . He is here extending a rule which is taken in the f i r s t place t o apply t o the integral whole as such, i.e. a rule which holds in respect of existence, and w h i c h , as described in the later t e r m i n o l o g y noted above, holds when totus is taken c a t e g o r e m a t i c a l l y . When, in this f u r t h e r example, the rule is now being extended to cover just any predicate other than existence (e.g. '... is w h i t e ' ) , the totus is now being taken, as he is t o go on to say, in the h i t h e r t o unencountered 'universal', i.e. syncategorematic, sense, w i t h 'each and every p a r t ' as its equivalent: (A) But probably this could be called the commonplace rule ' f r o m the universal w h o l e ' , rather than ' f r o m the i n t e g r a l whole', if, t h a t is to say, we take into account the sense 71

Section 2 r a t h e r than the words. For t h a t sense would appear to be as follows: I f each and every p a r t o f the house is w h i t e , then this [part] is w h i t e . For ' p a r t of this house ' is a wider t e r m than 'this [par­ t i c u l a r ] p a r t ' and is predicated of [each of] the individual parts of the house. (Sed hic fortasse locus magis ab univ­ ersali toto quam ab integro potest dici, si videlicet sensum potius quam verba pensemus. Talis enim sensus videtur ut: Si quaelibet pars domus sit alba, haec est alba. 'Pars' namque 'domus' universalius est 'hac parte',  de singulis partibus domus praedicatur: AD 344.8.12) The suggestion t h a t Abelard is here a n t i c i p a t i n g the later syncategorematic sense of 'whole' is f u r t h e r c o n f i r m e d by the parallelism between his 'secundum singulas eius partes',՝in respect of its individual p a r t s ' (2.211(A) above) and the 'quaelibet pars quantitativa vel integralis', 'every q u a n t i t a t i v e or i n t e g r a l p a r t ' used by Paul of Venice when defining t h a t sense (VLM 56ra, c f . 8.06). 2.22 More of Abelard's precisions are s t i l l available. Thus he is t o c o n f i r m yet again t h a t t h a t contrast between the ' d i v i s i o n ' of a genus into its species and of an i n t e g r a l whole into its parts, already encountered in Boethius' version (1.31), can be deepened by saying t h a t the f o r m e r depends on quality and the l a t t e r on quantity (cf. 2.21). Based on the ' a n a l y t i c ' nature of a d e f i n i t i o n is his c l a i m t h a t in respect of a given species a division cannot be e f f e c t u a t e d , even by reason, between the appropriate specific d i f f e r e n c e and its genus (cf. 0.5) whereas the parts of an i n t e g r a l whole may sometimes be discerptible by reason alone, or sometimes a c t ­ ually separable: (A) He who says t h a t this animal is a rational animal, where­ as t h a t animal is irrational illustrates how the nature of animal is sometimes possessed of r a t i o n a l i t y , and sometimes of i r r a t i o n a l i t y , so t h a t these t w o p e r t a i n inseparably to

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Abelard and his Contemporaries the species which they c o n s t i t u t e , and belong t o t h e i r v e r y nature w i t h o u t being separable t h e r e f r o m , even by reason. Hence i t r i g h t l y happens t h a t a division of the genus [e.g. animal] is brought about by q u a l i t y [e.g. r a t i o n a l i t y ] in those relevant species the nature of which is rounded o f f by a specific d i f f e r e n c e . On the other hand, the division of the whole in the integral sense does not involve the r e c e i v ­ ing of a q u a l i t y , but rather the b r a c k e t i n g together of the q u a n t i t y of a composite. For even i f the parts were t o t ­ ally q u a l i t y - f r e e , they would nevertheless c o n s t i t u t e a d i v ­ ision of the whole whose being they make up as long as they remained w i t h i n its compass. F u r t h e r m o r e , whether the parts are possessed of qualities or not, the properties of those qualities are not taken into account; r a t h e r , the notion of the m a t e r i a l parts alone is looked into when the whole is split into its [integral] parts. This, i t should be noted, can be investigated just as v a l i d l y when the parts can only be separated f r o m each other by a d i s t i n c t i o n of reason as when they can also be separated out in actual f a c t . (Qui dicit hoc profecto animal rationale animal esse, illud vero irrationale, animalis naturam modo a rationalitate, <modo ab irrationalitate> occupari demonstrat, quae ipsis inseparabiliter adhaerent speciebus quas creant, et in substantia insunt n ab eis vel ratione separari queunt. Unde recte generis div­ isio secundum qualitatem fieri contingit, in his quidem speciebus quarum substantia differentiis completur. Diviso vero totius integri scilicet, non ad qualitatum susceptionem pert­ inet, sed ad quantitatis compositi comprehensionem; etsi enim partes omni qualitate sint absolu tae, non minus substantiae totius divisionein facerent, cuius essentiam conficerent, dum in eius comprehensione remanerent, n, sive qualitatibus attendpartes informentur sive non, qualitatum proprietates untur, sed sola materialium partium comprehensio consider­ atur, cum in partes totum dividitur; quod quidem inspiciere licet tarn in his partibus quae sola ratione a se separari

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Section 2 possuni, quam in his quae etiam actu: AD 574.20.35) 2.221 This passage's d i s t i n c t i o n between actual and notional separability of an object's integral parts carries w i t h i t a reminder of the A r i s t o t e l i a n d i s t i n c t i o n between the c o n t i n ­ uous and the discrete, which Abelard now relates t o i n t e g r a l wholes thus: (A) Of those wholes which are said t o be i n t e g r a l or cons­ t i t u t i v e , some are continuous, as is this line, which has parts which are continuous, and others are not, as in the case of this people, the parts of which are apart one f r o m the other. (Horum autem totorum quae integra sive constit­ utiva dicuntur, alia sunt continua, ut haec linea quae continuas habet partes, alia non, ut hic populus, cuius partes disgregatae sunt: AD 547.27.29) 2.222 The case of the line, which may easily be supposed to have like-natured parts (i.e. smaller lines), d i f f e r s f r o m t h a t of, say, a heap of stones, in t h a t the f o r m e r ' s parts are not only continuous one w i t h the other, but also bear the same name, whereas a stone is not a heap. The parts of a line are thus taken t o be also lines. I t is assumed, likewise rather slackly, t h a t t h e parts of a bronze rod are each bronze rods, in t h e same s t y l e . Abelard is aware t h a t such slackly-expressed pseudo-homogenous cases breed s i m i l a r i t i e s of expression which could lead one to think t h a t d i s t r i b u t i v e , rather than c o l l e c t i v e , wholes were in question. As we have seen, the d i s t r i b u t i o n of a shared essence among its individual e l e m ­ ents in the case of a universal whole is expressed by 'This is one m a n , t h a t is another ....', f o r example: here the species name, in this case ' m a n ' , is predicable of each element of the species (2.21(A) above). But likewise, as we are now n o t i c i n g , since the parts of a line are allegedly each lines, and parts of a rod are each (supposedly) rods, ' l i n e ' and ' r o d ' can both be predicated of t h e i r respective parts and of t h e i r respect­ ive wholes as w e l l ; AD 576.19.22. So the usual thesis of disp­ a r i t y , according t o which the whole cannot be predicated of

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Abelard and his Contemporaries the p a r t where integral wholes are concerned (cf. 1.21), seems t o have collapsed at this point. However, the integral whole cases are shown by A b e l a r d to be s t i l l distinguishable, and this upon at least t w o grounds. F i r s t of a l l , the g e n i t i v e case (expressed by the appropriate t e r m i n a t i o n in L a t i n , and by the corresponding ' p a r t - o f - X ' locution in English) is appropriate in respect of each such p a r t , whereas only the n o m i n a t i v e (yielding t r a n s ­ lations such as 'this is one m a n ' , 'this is another man ' ) is appropriate in the d i s t r i b u t i v e - w h o l e cases (AD 547.27.34). Here the d i s t i n c t i v e g e n i t i v e is precisely the one w h i c h occurs in the ' p a r t - o f - X ' expression also stressed elsewhere by A b e l a r d (cf. 1.4, 2.65), and which is used as a constant contrast w i t h the '-part' expression (cf. 2.3). Secondly, A b e l a r d recommends more s t r i c t a t t e n t i o n to the exact ident­ i t y of w h a t is being predicated, since this shows t h a t the usual c r i t e r i o n of p a r t - w h o l e disparity c h a r a c t e r i s t i c of i n t e g r a l wholes is in f a c t s t i l l o p e r a t i v e . Thus i f we ident­ i f y the line in question as 'That line which is composed of these t w o lesser lines', or the r o d in question as 'That rod which is composed of these t w o lesser rods', then i t is c e r t ­ ainly not the case t h a t these expressions are predicable of the lesser lines or lesser rods respectively, at least w i t h the same sense of ' t h a t ' and 'these': AD 476.12.19; c f . 1.324 above. 2.23 Elsewhere, having denied t h a t the continuous is, in A r i s t o t l e ' s s t y l e , such t h a t its parts are joined to a common t e r m (on t h e ground t h a t this would lead to an i n f i n i t e r e g ­ ress and would e l i m i n a t e the possibility of a t w o - p a r t c o n t ­ inuum), Abelard continues: (A) However, i f we want t o define continuum p r o p e r l y , we assert t h a t the continuum is such t h a t its parts are placed together w i t h o u t any gap, i.e. i t has parts which are not rendered distant one f r o m the other either by the i n t e r p o l ­ ation of any other things, or by any separation of one from

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Section 2 the other. In this way we can take in a t w o - p a r t line or any other line you please. C o n t r a r i w i s e , the discrete is to be taken as being such t h a t its parts are apart f r o m one an­ other. (Si autem continuum proprie definire velimus, dīcamus id esse continuum, cuius partes sibi sine intervallo sunt insertae, hoc est, habet partes quarum nullam facit distantiam interpositio alterius rei vel ulla ab invicem divisio. Ut bipunctalem quoque et quamlibet lineam in continuo compreh­ end amus, discretum econtrario accipiendum est, illud cuius partes ab invicem distant: AD 73.24.30) 2.231 The odd consequences, mereologically speaking, of the connection of this sense of ' d i s c r e t e ' w i t h the possibility of numeration have already been adumbrated in 1.5, and w i l l be seen again in 2.7. In the meantime, the s p e c i f i c a t i o n of c o n t i n u i t y given above has consequences f o r human powers of e f f e c t u a t i o n . We cannot join together bodies in such a way t h a t there is no distance between t h e m . A l l we can do is t o disjoin or to aggregate: (A) The question whether every q u a n t i t a t i v e change is sub­ j e c t t o our operation is not t o be neglected. I t could be t h a t we are capable of e f f e c t u a t i n g the d i m i n u t i o n of any composite object, but w i t h o u t being correspondingly able to b r i n g together any object by way of increase. For I think t h a t there is no human operation capable of joining physical objects so t h a t no distance subsists between t h e m . Hence neither t h e continuous length of a line, nor the breadth of a surface, nor the bulk of a physical object, are w i t h i n the scope of our a c t i o n , although presumably we can increase the number of things which are combined together in the same place, as when we bring some f u r t h e r stones to add t o this pile of stones, or when we join together baulks and stones in the building of a house, or again, when we s t u f f the caulking between t h e planks in the construction of a ship. These are joined together by our operations, not by the creation of nature; indeed, these things we e f f e c t not by

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Abelard and his Contemporaries c r e a t i o n , but only by composition. This occurs when we c o m ­ bine into one s t r u c t u r e those things which as a result of natural c r e a t i o n are m u l t i p l e and s c a t t e r e d . (Non est praetermittendum utrum omnis secundum quantitatem motus nostrae sit operationi subiectus. Et fortasse cuiuslibet compositi diminutionem efficere valeamus, sed non ita secun­ dum quodlibet augmentum subiecta unire. Neque enim aestimo hominum operatione ulla ita coniungi corpora ut nulla sit inter ea distantia. Unde  lineae longi tudo continua vel superficiel latitudo vel corporis spissitudo nostrae sub­ jacent actioni, sed for tasse numeri multiplicatio secundum aggregationem ad eumdem locum; velut i cum huic acervo lapidum aliquem aliorum lapidem aggregamus vel ligna lapidibus coniungamus in compositione domus aut lignis stophas inserimus in constructione navis. Haec itaque nostra uniuntur operatione, non naturae creatione, et haec quidem secundum compositionem, non secundum creationem efficimus, dum ea quae secundum suae creationem plura sunt a divisa, nostra operatione in unam fabricam componimus: AD 431.10.23 ) 2.232 Thus (opines Abelard) the having of intervals between the parts is one of the marks of a man-made object. However, this is also a feature of what everyone agrees are examples of collective wholes (in the more common sense of the words) such as a f l o c k or a c r o w d . Hence one m i g h t argue t h a t m a n made objects such as houses and ships are likewise to be named ' c o l l e c t i v e ' wholes in t h a t m o r e f a m i l i a r sense. But c o n t r a r i w i s e , i t would s t i l l appear t h a t there are c e r t a i n i n t e g r a l wholes which in some sense are man-made, and y e t are continuous. To this second contention, which m i g h t be backed up by reference t o a c t i v i t i e s such as glass-making (also used by Boyle in a cognate c o n t e x t t o which allusion is made in 2.531(A)), a reply is provided which a t t r i b u t e s c r e d i t to God in such cases: (A) To c r e a t e is invariably foreign t o any action of ours, and is solely t o be a t t r i b u t e d t o God. N o r is i t the case,

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Section 2 when the ashes of hay have been put into the furnace so t h a t they may t u r n into glass, t h a t our action operates to e f f e c t u a t e any c r e a t i o n of the glass. Rather i t is God who, while we are lacking knowledge of the s t u f f involved in the nature of those things which we make ready, himself operates in a hidden manner, and brings into being a new substance. But when the glass has been created by God, i t is t r a n s f o r m ­ ed by us into many sorts of vessels, in the same way as we make up a house f r o m stones and logs already created by God. Nothing is here created, but rather are created things joined together. (Omnis creatio a nostris actibus est aliena et soli Deo ascribenda. Neque enim in fornace posito cinere feni ut in vitrum transeat, noster actus in creationem vitri quicquam operatur, sed ipse Deus nobis etiam physicam ignor­ antibus in natura eorum quae praeparavimus, occulte operatur a novam perficit substantiam. Ubi autem vitro a Deo creatum est, nostra operatione in vasa multifarie formatur, sicut ex lignis et lapidibus iam a Deo creatis domum componimus, nihil quidem creando, sed creata coniungendo: AD 419.36 420.6.) L i k e w i s e , only God, s t r i c t l y speaking, is the author of human generation: (B) Note also t h a t t o God alone pertains the act of gener­ a t i o n , i.e. t h a t bringing t o pass of substantial being which is c r e a t i o n . For although as a result of sexual a c t i v i t y on the p a r t of the father there is some d i s t i n c t p o r t i o n which becomes shaped into a human being, nevertheless, even when the f a t h e r has perished, the workmanship of nature s t i l l goes on operating w i t h i n the m o t h e r ' s womb, so t h a t the p o r ­ t i o n is enlivened and shaped into a human being. Hence the child i t s e l f is not the work of man, but of nature, i.e. of God, and the human operation of disposing the m a t e r i a l is purely accidental, as when a house or a sword is put t o g ­ ether w i t h o u t being at the same t i m e substantially gener­ ated. Nor is glass, which is said t o be a species of m e t a l ,

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Abelard and his Contemporaries the work of man, but r a t h e r of nature. For those things having been prepared by man which are essential f o r the c r e a t i o n of glass, i t is nature alone which operates upon the m a t e r i a l thus prepared so as t o t u r n t h a t m a t e r i a l into glass while we, f o r our p a r t , know nothing of the physics of the process. (Nota etiam quod solius Dei est generare, id est operatione sua in substantiam promovere, quod est creare. Nam etsi ex coitu patris quaedam portio separata sit quae formetur in hominem, patre tarnen defuncto non minus natura opifex operatur in visceribus matris de infuso semine, ipsum scilicet formando et vivificando in hominem. Unde puer ipse non hominis opus est, sed naturae, id est Dei, hominem autem operatio alterare tantum materiam videtur secundum accid­ entia, veluti dum domum componit vel gladium, non etiam in substantiam generare. Neque enim vitrum, quod species met­ alli dicitur, hominis est opus, sed naturae, quia praeparatis ab homine quae necessaria sunt ad creationem vitri in mat­ eria praeparata a nobis, sola natura operatur earn in vitrum convertens, nobis quoque physicam ignorantibus: GA 298.28.40.) Such Abelardian theses on the differences between human and divine (or natural) productions may w e l l be at the b o t t o m of the controversy between h i m and a c e r t a i n A l b e r i c , as dis­ cussed in TAU 148 - 152. I t is curious to note in passing t h a t the seventeenth-century philosopher Robert Boyle, w i t h whom comparisons are to be made in 2.531, spoke of the ashes of wood as the source of glass (BPP 74, quoted 2.531(A)) whereas the ashes of hay are given this c r e d i t in the abovepassage. quoted Dialectica

2.3 P a r t s - o f - X and -parts (cf. 1.4 above) 2.31 However, leaving aside any f u r t h e r details of gener­ ation and c r e a t i o n , one may r e v e r t t o the response which A b elard provides to the other, and h i t h e r t o neglected, argument mentioned above in 2.232, namely the one t o the e f f e c t t h a t 79

Section 2 artifical wholes such as houses and ships should count as collective wholes (in the more usual, old-fashioned, sense of 'collective') since, by his doctrine, they are non-continuous: (A) It is by the operation of man, and not by the creation of nature, that these [artificial] things are said to constit­ ute unities. Indeed, their names have been thought by some to be collective names, similar in sense to the names of a manifold, a flock, or a crowd, but this is not so. For the [artificial] things in question must undergo a certain com­ position and be brought all together in whatsoever way the unification has to be effected so that a house or a ship may be made. The collectivities, in contrast, can retain their characters, even if the parts are scattered. Thus this human unit living in Paris and that other one who abides in Rome constitute this pair. Hence mere multiplicity of units makes a manifold; in contrast, men assembled make a crowd, and an aggregation of irrational animals constitutes a flock. For we do not account distantly scattered animals to be a people, flock, or crowd; they must be gathered together. Further, a ship or a house is constituted not only by mere plurality [as was the manifold], nor even by gathering tog­ ether [as was the flock], but rather by the imposition of a fixed structure on things. For not just any old conjunction of materials constitutes a house or a ship; this constit­ ution occurs only if the parts are appropriately brought together. It is hence that i t comes about that the names of these latter kinds of object, whatever their mode of unity, happen not to be plural in grammatical number, as contrasted with those things which do not constitute such unity, e.g. the names of a plurality of units or animals. That this is so is obvious from our not following the singular-noun-withplural-verb pattern which holds in these [less structured] cases when we come to deal with those [more structured] cases. We do not follow our pattern of saying 'the people or the flock or the crowd come [in the plural]' by likewise 80

Abelard and his Contemporaries saying 'the house (or the ship) are made' in the case of such a r t i f i c a l wholes.(Haec itaque hominum operatione, non naturae creatione, una dicuntur; quorum quidem nomina, sicut numeri vel populi vel gregis vel turbae, quidam coUectiva esse aut­ umani, sed falso; haec enim quocumque modo uniri necesse sim­ est, ut domus vel navis fiant, quamdamque compositionem ul iuncta recipere; illa, etiam disgregata, suas retinent proprietates. Haec enim unitas hominis Parisius habitantis et illa hominis Romae manentis hunc faciunt binarium. Unde sola unitatum pluralitas numerum perficit, popul um vero hom­ inum conventus, vel turbam vel gregem irrationalium congregatio. Neque enim populum aut gregem vel turbam accipimus in animalibus per loca longinqua diffusis, sed simul congreg­ atis; navis autem aut domus non solum in pluralitate rerum aut congregatione cons istunt, sed in certa rerum compos itione. Neque enim, quoquo modo iungatur materia, domum aut navim efficit, nisi propria compositione uniantur membra. Unde non ita horum nomina, quoquo modo uniuntur, pluralia esse contingit, sicut eorum ex quibus unum non efficitur, ut diversarum unitatum aut diversorum animalium; quod quidem et ex eo manifestum est quod non ita istorum singularis numeri nominibus plurales verborum personas apponimus, sicut illorum. Non enim sicut dicimus: populus vel grex vel turba veniunt, ita dicimus domus vel navis fiunt, quae factitia sunt tota: AD 431.23 - 432.5.) As we are t o be reminded in 2.33, the exact point at which the f o r m e r l y scattered m a t e r i a l s actually become a house or a ship or whatever, is not a m a t t e r f o r d e t e r m i n a t i o n by m e r eologists, but by experts on houses and on ships, and so f o r t h . The definitions of the various sorts of things in question are t h e principles f o r a decision here. The manner in which a l i f e could depend upon such a decision is v i v i d l y i l l u s t r a t e d in Robert Hughes' superb account which covers the m a i n genocidai section of the B r i t i s h pioneering antipodean Gulag A r c h i p e l a g o , now known as Tasmania. In 1833 a prisoner, 81

Section 2 James P o r t e r , having just assisted in the building of the b r i g ' F r e d e r i c k ' , absconded in i t , and a f t e r various e x c r u c i a t i n g adventures, was f i n a l l y t r i e d in 1837 f o r p i r a c y , w h i c h could be a hanging m a t t e r . He ingeniously defended h i m s e l f by c l a i m i n g t h a t t h e object in which he had absconded was not r e a l l y a ship, since i t had not y e t been f o r m a l l y commiss­ ioned: ropes, boards, nails and t i m b e r s , even though assembled shipwise, s t i l l need not be a legal ship; t h e seizure m i g h t be t h e f t , but i t was not p i r a c y ; HFS 218. 2.32 F r o m his last-quoted t e x t i t is at least clear t h a t in the course of thus sorting out the i n t e g r a l wholes which he would s t i l l be prepared t o c a l l ' c o l l e c t i v e ' in the old, customary, sense, Abelard is now approaching a much m o r e n o t e w o r t h y t o p i c , his t r e a t m e n t of which has already been b r i e f l y encountered in 1.4 above. This is the d i s t i n c t i o n which we have rendered in English as being t h a t between X parts and p a r t s - o f - X . The house or the ship, he has been p o i n t i n g out, are not c o n s t i t u t e d by t h e m e r e aggregation of house-parts or ship-parts; as the passage stresses, due and appropriate composition is required so t h a t the f o r m e r l y p u t a t i v e -parts do in f a c t become p a r t s - o f - X (cf. HLM 124, HQS 253 - 4). In a l i k e s p i r i t , when glossing the Categoriae, he asserts: (A) There are, however, some names which signify many things at once, not on the basis of mere p l u r a l i t y , but because of a c e r t a i n aggregation; these are said to be collective or comprehensive names, such as 'populace', and 'bed o f roses'. A populace is not said to be in the same way as men [in the plural] are, i.e. on the basis of just any human p l u r a l i t y . Only those who are gathered together in one place are called a 'populace'. Hence the name 'populace' would appear t o be derived f r o m a c o l l e c t i o n of men. Now there are other names which s i g n i f y many things taken t o g ­ ether, but not m e r e l y on t h e basis of m u l t i p l i c i t y , nor y e t on the basis of aggregation, but r a t h e r because of the i m p -

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Abelard and his Contemporaries osition of a f i x e d u n i f i e d s t r u c t u r e on things, whether t h a t s t r u c t u r e be the product of nature, as is t h a t w h i c h shapes those human members which taken all together are said to be a man, or whether i t be a r t i f i c i a l , such as t h a t which per­ vades a house. Indeed, m e r e m u l t i p l i c i t y is not enough t o give us a genuine house. For i f walls and roof just lie about, gathered together in one place on the foundations, they are not hence said t o be a house unless such compon­ ents of the house acquire t h e i r s t r u c t u r e . The f i r s t t w o types of name mentioned above, i.e. those which are class­ i f i e d m e r e l y on a p l u r a l i t y basis and those which are c o l l e c t i v e , are construed w i t h plural verbs, f o r we say b o t h 'Men run' and 'Many r u n ' . In contrast, those which are p r o d ­ uced on the basis of a u n i t y involving a c e r t a i n s t r u c t u r a l togetherness are not construed w i t h p l u r a l verbs. A f t e r a l l , we do not say 'The house are' or 'The ship are՝. (Sunt autem quaedam nomina plura significantia simul non ex pluralitate sola, sed ex aggregatione quadam, quae col·lectiva vel comprehensiva dicuntur, ut populus, rosetum. Non enim quaelibet hominum pluralitas sicut homines dicitur, ita et populus, sed sunt. tantum populus vocantur qui in uno loco congregati Unde populi nomen a collectione hominum sumptum videtur. Sunt autem alia nomina res piures simul significantia non ex pluralitate sola vel ex aggregatione, sed ex certa rerum compositione, sive illa compositio sit naturae, ut illa quae est in humanis membris, quae homo omnia simul dicuntur, sive sit artificialis, veluti ea quae est in domo. Non enim ad proprietatem domus sufficit vel rerum pluralitas vel aggregatio. Si enim paries et tectum super fundamentum iacent aggregata in unum locum, non ideo domus dicerentur, nisi ea compositionem haberenť. Illa autem quae ad domum pertinent duo nomina, quae scilicet vel propter pluralitatem tantum reperta sunt vel col·lectiva sunt, cum verbis pluralibus construuntur. Nam et 'homines currunt' et 'piures currunť dicimus. Illa vero quae secundum quandam compositionis unitatem

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Section 2 inventa sunt, non construuntur cum pluralibus; non enim dicimus: 'domus vel navis sunt' : GA 171.4.22.) Given t h a t the imposed s t r u c t u r e in question follows t h e lines laid down in t h e appropriate d e f i n i t i o n , the consequ­ e n t l y strong connection between t h e X - p a r t s / p a r t s - o f - X d i s t ­ i n c t i o n and the d e f i n i t i o n o f the species in question is f u r ­ ther i l l u m i n a t e d when i n f e r e n t i a l relations between whole and p a r t are expressly discussed by A b e l a r d . He begins w i t h the r a t h e r broad question of a t t r i b u t i o n s in general (cf. 5.3), but soon narrows down t o t h e more manageable t o p i c of existence: (B) I t also o f t e n happens t h a t ' w h o l e ' allows consequences as t o t h e c o l l e c t i o n o f a l l the p a r t s , thus: I f something is predicated of t h e whole, then i t is also predicated of a l l the parts taken together. I f the whole is predicated of something, then a l l its parts taken together are likewise predicated. I f something is denied of t h e whole, then i t is denied of a l l t h e parts taken together. I f the whole is denied of something, then a l l its parts taken together likewise [are denied]. These principles are e x e m p l i f i e d in the f o l l o w i n g : I f t h e house exists, then t h e walls along w i t h the roof and along w i t h the foundations a l l exist. I f any object is the house, then i t is t h e w a l l , r o o f , and foundations a l l joined together. I f the house does not exist, then neither do those three joined together. I f something is not t h e house, then neither is i t those three joined together. In f a c t these consequences could be characterised as f o l l o w ­ ing upon the nature defined, rather than upon the whole as such. A f t e r a l l , house can only be understood as these three walls, roof, and foundation, endowed with the composite structure of a house. A n y other s t r u c t u r e operative on the joining of the parts is not relevant, but only t h a t which

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Abelard and his Contemporaries constitutes a house. Were this not so, the negative members of the aforementioned consequences would not hold t r u e , namely: I f the house does not e x i s t , then neither do those t h r e e [parts] joined together. I f something is not t h e house, then neither is i t those three [parts] joined together. This is because these things may a l l exist conjoined and y e t not have the s t r u c t u r e of a house. Thus likewise the d e f i n ­ i t i o n of a house is assembled in accordance w i t h this s t r u c ­ t u r e when these three things joined together are shown t o be collectionem a house. (Saepe etiam totum ad partium omnium antecedit, ut: si videlicet aliquid praedicatur de toto, et de omnibus partibus eius simul acceptis, vel: si totum de aliquo, et omnes simul partes, aut: partibus si aliquid a toto removetur, et ab omnibus simul, vel: si totum ab aliquo, et omnes simul partes, ut in subiectis exemplis continetur: si domus est, paries et tectum et fundamentum sunt, aut: si qualibet res est domus, et paries, tectum, fundam­ entum, simul iuncta, vel: si domus non est,  illa tria coniuncta sunt, vel: coniuncta. si aliquid non est domus,  est illa tria Quae quidem consequentiae non totius, sed definiti possunt dici; domus namque nihil aliud est intelligenda quam illa tria composιtionem domus habentia. Aliam enim in comunctione partium compositionem non recipimus, nisi illm quae domum fecit. Alioquin non essent verae praepositae con­ sequentiae inter negationes, quae videlicet dicunt: si domus non est, illa tria iuncta non sunt, vel si aliquid non est domus,  est illa tria coniuncta. Possen է enim haec tria et coniuncta existere et compos­ definitio itionem domus non habere. Sicque recte domus

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Section 2 secundum compositionem componitur, cum illa tria coniuncta domus esse monstrantur: AD 344.18 - 345.6.) 2.33 The strength of the c o n v i c t i o n which A b e l a r d here displays as t o the importance of this c r u c i a l d i s t i n c t i o n between -parts (which can subsist independently of t h e i r conjunction) and p a r t s - o f - X , the subsistence of which is bound up w i t h there being an X , and the conjoining of which must be in accordance w i t h the d e f i n i t i o n o f an X , is r e i n f o r c e d by alone the number of times he discusses i t in his Dialectica (e.g. at least AD 431-2, 547 (quoted in 1.41(B) above), 550 - 1 , 554.4.7, and 575, as w e l l as in the present context) and by the v e r y e x t r e m e lack of i n f e r e n t i a l connection which he sometimes imposes where -parts m i g h t be in question. For instance, notwithstanding concessions which he has made above, he refuses, when appropriately pressed, t o countenance the inference f r o m house t o wall, on the ground t h a t this last could be taken in its house-part sense. This refusal occurs in the s e t t i n g of hypotheticals having impossible antecedents: (A) I t is asserted, however, [by some opponent] t h a t a l ­ though what is propounded in the antecedent is impossible (i.e. t h a t only a house exists, in the sense t h a t there exists no other t h i n g , and t h a t the house is one thing) nevertheless a t r u e consequence can be based upon i t . For although the consequence: i f Socrates is a donkey, then he is an animal may have an impossible antecedent, i t is nevertheless t r u e as a whole. But i t m i g h t be f u r t h e r asserted t h a t the a n t e ­ cedent of this consequence, though impossible in i t s e l f , is nevertheless not impossible r e l a t i v e l y to t h e consequent, and indeed t h a t which i t necessarily entails is c e r t a i n l y c o m p a t ­ ible w i t h i t . I t is on this account t h a t the consequence: i f Socrates is a donkey then he is an animal is t r u e . But i t does not look as though the aforementioned consequence, namely: I f the house alone exists (i.e., and no other thing

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Abelard and his Contemporaries exists) the w a l l does not exist is t r u e in this sort of way. For in this case the antecedent is impossible, even r e l a t i v e l y to the consequent, since the existence of the house demands the being of the w a l l . On this account [the opponent may contend] t h e f o l l o w i n g seems t o be t r u e : i f the house alone exists, then the w a l l exists. But this is not so. For w h i l e there can be no house w i t h o u t a w a l l , there is no necessary entailment in the assertion: i f the house is, then the w a l l is. This is because the consequent is not i n t e l l i g i b l y contained in the antecedent. He who asserts 'The house i s ' attributes being t o the three things [walls, r o o f , and foundation] only in so f a r as they are taken all together [i.e. as p a r t s - o f the-house]; i t is not a t t r i b u t e d t o any o f t h e m on t h e i r own account [as house-parts]. That which is a t t r i b u t e d t o many things taken together need not be a t t r i b u t a b l e to each individual concerned, as when 'house' applies t o all of its parts taken together, but cannot properly be applied to any of them.(Dicitur autem quia impossible est illud quod in antecedenti proponitur, quod scilicet domus sit sola, idest nulla alia re existente, et quod unum est, sed non ideo minus facit veram consequentiam. Licet enim haec consequential si Socrates est asinus, est animal antecedens habeat impossibile, tarnen non ideo minus est vera. Sed fortasse dicitur quia huic consequentiae antecedens, licet in se impossibile sit, tarnen quantum ad consequens non quod est impossibile, quippe ipsum secum bene pateretur necessario exigit, ideoque vera est haec consequential si Socrates est asinus, est animai. Sed non ita illa supraposita: si domus est sola, idest et nulla alia res, paries non est vera videtur. Nam illud quod quidem antecedit impossibile domus est esse etiam quantum ad consequens, cum videlicet

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Section 2 existentia parietis exigat essentiam. Neque enim ut domus sit, paries potest non esse. Itaque etiam illa vera videtur: si domus est sola, paries est. Sed falso. Licet enim domus absque parie te non possi է exis­ tere, non tarnen ideo necessaria est consecutio quae ait: si domus est, paries est quippe in antecedenti consequens non intelligitur. Qui enim dicit 'domus est', tribus tantum simul acceptis esse attribuit,  nulli quidem eorum per se. Quod autem simul pluribus attribuitur, non necesse est singulis attribuì, ut cum rdomusr illis simul conveniat, nulli tarnen eorum aptari pot­ est: AD 346.3.28) C l e a r l y , f o r Abelard's point t o hold, the 'wall' ('paries՝) in the f i n a l hypothetical has t o be taken in the sense of ' w a l l qua house-part ' , i.e. per se, as A b e l a r d expresses i t . In t h a t sense the hypothetical need not be t r u e . (There are many other i m p o r t a n t points raised in this passage, of course, but to avoid g e t t i n g too much bogged down in t h e m here, t h e i r t r e a t m e n t has been concentrated elsewhere. Thus the last words embody the thesis of p a r t - w h o l e disparity (1.32). The 'house alone' paradox occurs in Master Peter's discussion (2.786), and mereological repercussions of 'solus', 'only ...', are t r e a t e d by Peter of Spain (7.316)). 2.34 The same awareness of the d i s t i n c t i o n between X - p a r t s and p a r t s - o f - X figures in Abelard's c r i t i c s m of Roscellin ! s 'madness': (A) I r e c a l l , however, t h a t m y teacher Roscellin propounded the mad opinion t h a t no thing consisted of parts. Thus he described parts in the same way as t h a t in which he describ­ ed species, i.e. as mere utterances. I f someone were to assert t h a t t h a t object which is a house is made up of other objects, namely the w a l l and the foundations [and t h e r o o f ] , then he used to a t t a c k their contention thus: i f t h a t object which is the w a l l is p a r t of t h a t object which is the house, then since the house is just the w a l l i t s e l f , along w i t h the

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Abelard and his Contemporaries r o o f and the foundation, the w a l l w i l l accordingly be p a r t o f - i t s e l f along w i t h the other parts. But how can the same thing be p a r t of i t s e l f ? A g a i n , every p a r t is n a t u r a l l y p r i o r t o its whole. But how can the w a l l be said t o be p r i o r t o i t s e l f and to the others when i t is in no way prior to i t s e l f ? (Fuit autem, memini, magistri nostri Roscellini tam insana sententia ut nullam rem partibus constare veliet, sed sicut solis vocibus species, ita et partes adscribebat. Si quis autem rem illam quae domus est, rebus aliis, parie te scilicet et fundamento, constare diceret, tali ipsum argumentatione impugnabat: si res illa quae est paries, rei illius quae domus est, pars sit, cum ipsa domus nihil aliud sit quam ipse paries et tectum et fundamentum, profecto paries sui ipsius et ceterorum pars erit. At vero idem quomodo sui ipsius pars fuerit? Amplius: omnis pars naturali ter prior est suo toto. Quomodo autem paries prior se et aliis dicetur, cum se nullo modo prior sit? AD 554.37 » 555.9.) Thus f a r we are confined t o Abelard's account of the m a t t e r , which leaves Roscellin in a rather paradoxical l i g h t . The point of the teacher's argument is said t o be supportive of the ' n o m i n a l i s m ' f o r which he is notorious, i.e. parts as w e l l as 'universals' are being reduced to mere utterances. This is y e t another aspect which must be l e f t aside at this point; i t w i l l be touched upon later in 2.7. C o n c e n t r a t i n g , t h e r e f o r e , on Roscellin's reductio ad absurdum argument d i r e c t e d against the partisan of real parts, we observe t h a t i t could be con­ strued in isolation as an example of what occurs when X - p a r t s and p a r t s - o f - X are confused, and i t is in this l i g h t t h a t Abelard views i t , thereby blunting his teacher's point: (B) However, this contention is undermined because when i t is said t h a t the w a l l is p a r t of i t s e l f and of t h e rest of the parts, this is granted, but w i t h the parts all taken together and joined, so t h a t when he says t h a t the house is those three things, he does not grant each one of t h e m [taken individually] to be the house, but only all of t h e m taken 89

Section 2 together and [duly] conjoined. Hence i t is not t r u e t h a t the w a l l , or one of the other things, is t h e house, but only the three taken all together. Further, and likewise, the w a l l is p a r t of i t s e l f - a n d - t h e - r e s t - o f - t h e - t h i n g s - w h i c h - a r e - a l l joined-together, t h a t is, of the whole house, but i t is not p a r t of i t s e l f on its own account [i.e. as a house-part]. And indeed, the w a l l is said t o be p r i o r t o i t s e l f - a n d - a l l - t h e others-joined-all-together, but i t is not t h e r e f o r e p r i o r to i t s e l f taken on its own account. For the w a l l existed [as a house-part] before being conjoined [to become p a r t - o f - t h e house], and every p a r t must exist by nature p r i o r t o f o r m i n g the c o l l e c t i v i t y w i t h i n which i t is embraced. (At vero haec argumentatio in eo debilitatur quod cum dicitur paries sui et ceterarum partium pars esse, conceditur, sed simul acceptorum et coniunctorum, veluti et cum domum dicit illa tria esse non singula concedit earn esse, sed simul accepta et coniuncta; unde neque parietem esse verum est, neque ali­ quid aliorum, sed illa tria simul. Sic quoque paries pars sui et ceterorum simul coniunctorum est, hocest totius domus, non sui per se, et prior quidem se et aliis simul coniunctis dicitur, non ideo per se; prius enim paries fuit quam illa coniuncta essent, et unamquamque priorem naturaliter esse oportet quam collectionem efficiat in qua comprehend atur: AD 555.10.19.) 2.35 C l e a r l y , Abelard's c r i t i c s m , like his c r i t i c i s m in the 'house alone ' case above, depends upon scrutinising (for ex­ ample) the w a l l qua house-part; in this sense i t can in no way be said t o be p a r t - o f - i t s e l f . However, qua p a r t - o f - t h e house i t may be said in some way to be p a r t - o f - i t s e l f (i.e. along w i t h the rest of the parts); this constitutes an i n t e r ­ esting f u r t h e r example of an Abelardian admission of nondiscrete objects (cf. 1.5, 2.78). His f i n a l p o i n t , concerning 'natural' p r i o r i t y (cf. TAU 115) requires various c l a r i f i c a t i o n s before i t can be accepted. I t may w e l l be simply a r e f l e c t i o n of 2.232(A), w i t h 'nature' c r e a t i n g and man combining. Or 90

Abelard and his Contemporaries again, i f 'naturally p r i o r ' means t h a t negation of t h e e x i s t ­ ence o f some p a r t - o f - X entails the negation of the existence of X ( r e l a t i v e l y t o a c e r t a i n appropriate s p e c i f i c a t i o n of X) then this too is acceptable enough. I f , however, he means t h a t all p a r t s - o f - X must somehow subsist independently as X parts p r i o r t o incorporation into X , then this is a highly contingent and unnecessary contention, based upon a l i m i t e d and f a m i l i a r set o f examples (houses, ships, and so f o r t h , coupled w i t h l i m i t e d conventional componential views as to t h e i r parts). Indeed, qua p a r t s - o f - X , parts only exist when the whole actually exists. As the medievals o f t e n remind us, not only is a dead man not a man, but also a dead hand is not a hand (in the p a r t - o f - m a n sense) and so on: c f . 3.41. H o w ­ ever, i t is at least evident t h a t throughout the foregoing passage there is stout defence of the thesis of p a r t - w h o l e disparity (1.32), and both Roscellin and Abelard c l e a r l y agree w i t h what is presented in modern dress at 10.323, i.e. nothing is a proper p a r t of i t s e l f . 2.36 A f u r t h e r index of Abelard's consciousness of the d i s t i n c t i o n between -parts and p a r t s - o f - X is perhaps t o be found in what appears t o be his refusal t o allow f l o u r t o be the m a t t e r or p a r t of bread (AD 415.6.7). Were one t o contend t h a t some types of makings abide as b e f o r e when compounded (e.g. a w a l l which was a house-part) but some others do not (e.g. f l o u r becomes crumbs when baked into bread) he would apparently reply t h a t t o say t h a t flour is a p a r t of bread involves absurdities as great as those involved in saying t h a t trees (rather than wood) are the makings or p a r t of a house (AD 575.11.36). He here insists t h a t the r ô l e of p a r t (proprietas partis) is strongly to be distinguished f r o m the same p a r t considered independently (in natura propri ae substantiae, i.e. as an -part rather than as p a r t - o f - a n - X ) . Hence, change f r o m a w a l l which is a house-part, t o a w a l l which is p a r t of-a-house, is just as m e r e o l o g i c a l l y radical as is the change f r o m f l o u r to bread. The d i s t i n c t i o n between the t w o 91

Section 2 sorts of change in the makings (i.e. between t h a t w h i c h , n o m ­ inally at least, abides as b e f o r e , and t h a t which does not) is hence i r r e l e v a n t as far as general mereology is concerned (cf. 3.31). Theories covering the various species of case may be l e f t t o appropriate specialists. 2.4 I d e n t i t y and P r i n c i p a l Parts 2.41 Y e t another obviously possible c l a s s i f i c a t i o n of mereological items would be of those which are s p a t i a l , those which are temporal, and those which are both. Most of Abel-ard's examples are clearly at least spatial, but he seems to have d i f f i c u l t y in endowing t h e m w i t h any u n i t y across t i m e , on account of their changefulness. This is largely because of his p r i n c i p l e t h a t no t h i n g has m o r e parts at one t i m e than at another: Nulla res uno tempore piures habet partes quam alio; GA 300.20.21, cf. AD 423.29.32. L i k e w i s e , when he comes to deal w i t h what he calls ' t e m p o r a l wholes' in a r a t h e r spec­ ialised sense, i.e. days, weeks, and so on, he finds t h a t they have features which are apparently incompatible w i t h estab­ lished mereological theses. His dealings w i t h t e m p o r a l wholes may be postponed t o 2.6. I t is when he is c o n f r o n t i n g the r i d d l e of what constitutes a ' p r i n c i p a l p a r t ' t h a t the m e r e o l ­ ogical aspects of change become somewhat tangled. 2.42 Is i t possible t o give a quite general s p e c i f i c a t i o n of what constitutes the p r i n c i p a l p a r t (or parts) of any object? I t is during Abelard's discussion of this question t h a t the i d e n t i t y across t i m e of s p a t i o - t e m p o r a l objects appears t o be threatened. C e r t a i n l y his adherence t o the p r i n c i p l e of the q u a n t i t a t i v e i d e n t i t y of parts mentioned in the last paragraph is already enough of a t h r e a t t o i d e n t i t y across t i m e , and his discussion of p r i n c i p a l parts is one of the m a i n places in which he seems t o c o n f i r m t h a t he is in the g r i p of the quite r a d i c a l consequences of t h a t p r i n c i p l e . The expository trouble here is t h a t the discussion becomes quite lengthy, and at its t e r m i n a t i o n Abelard appears to accept a v e r d i c t against 92

Abelard and his Contemporaries which he has given highly cogent arguments. However, the passages concerning these questions are of s u f f i c i e n t interest in themselves to authorise ample quotations, even though they are at times somewhat beside the main mereological point, which centres around Abelard's d i f f i c u l t i e s w i t h our p r e t h e o r e t i c a l impression that changing objects (ourselves included) persist through t i m e . 2.421 I t is taken f o r granted t h a t the present problem e m ­ braces change in a q u a n t i t a t i v e , usually s p a t i a l , sense, and t h a t the parts involved are understood t o be parts in the mereological sense. The medievals were constantly aware of the d i f f e r e n c e between this sense and t h a t encountered when speaking about the parts of a d e f i n i t i o n or of its c o r r ­ esponding nature or essence (cf. 0.5). 2.43 Abelard's own introduction to the question of principal parts, which now f o l l o w s , is i n t e l l i g i b l e enough, at least s u p e r f i c i a l l y . His position, to judge f r o m his abovementioned principle (2.41) and his various declarations throughout the succeeding discussions, would appear to be on the side of those extreme 'destructivists' who not only assert t h a t p r i n c i p a l parts are those the destruction of which ent­ ails the destruction of the whole, but also see just any q u a n t i t a t i v e change (here usually diminution) as d e s t r u c t i v e of the i d e n t i t y of the whole. What he later declares t o be his own u l t i m a t e v e r d i c t , however (AD 552.38 - 553.7, c f . 2.45 below) seems p r e t t y close t o the c o n t r a r y ' c o n s t i t u t i v e ' thesis, despite his having by then given c e r t a i n already a n t i ­ cipated excellent and interesting arguments t o r e f u t e i t , as we are s h o r t l y to see. Such a ' c o n s t i t u t i v e ' thesis is outlined in his introduction t o the topic. I t claims t h a t those parts are p r i n c i p a l which are not part of parts. N e x t is described a moderate version of the 'destructive' position; this accounts those parts principal the destruction of which destroys the substance, i.e. the sortal nature, of t h e i r whole. Such destruction is here sharply distinguished f r o m merely 93

Section 2 q u a n t i t a t i v e d i m i n u t i o n which does not destroy the object essentially or substantially: (A) The question arises as t o which parts should properly be called ' p r i n c i p a l ' and which should be called 'secondary ' . Some people look at this f r o m the constitutive point of v i e w , others f r o m the destructive. The f o r m e r c a l l those parts ' p r i n c i p a l ' which are not parts of parts, but only parts of the whole, as in the case of this man's soul and the body to which i t is joined, or in t h a t o f this house, when one has this w a l l and this r o o f and this foundation. On the other hand, those who look at the nature of t h a t w h i c h is p r i n c i p a l f r o m a destructive point of v i e w , assert t h a t p r i n c i p a l parts are just those which [on being destroyed] destroy the sub­ stance of the whole, e.g. the head, the c u t t i n g o f f of which destroys the m a n . They t h e r e f o r e put the stress on the des­ t r u c t i o n of the substance; [in contrast] the destruction of no m a t t e r what part is enough f o r the reduction of the quantity o f the whole, so t h a t i f this mere f i n g e r n a i l p e r ­ ishes, the q u a n t i t y of Socrates' body cannot r e m a i n the same, since he is now not so great as he was e a r l i e r . For even as a whole is increased by any addition, so also i t is d i m i n ­ ished when anything is taken away. But although on the r e m ­ oval of the f i n g e r n a i l the q u a n t i t y of the body does not r e m a i n constant, the essence of the Socratic substance does not, nevertheless, appear changed, since he is s t i l l said t o be Socrates and a man. However, he no longer comprises, q u a n t i t a t i v e l y speaking, all those things w h i c h were there b e f o r e , i.e. the nail along w i t h a l l the rest o f his parts. (Est autem quaestio quas principales, quas secundarias part­ es vocari conveniat; alii enim secundum Constitutionen!, alii Hi namque eas prin­ secundum destructionem has considerant. cipales vocant quae partium partes non sunt, sed tantum totius, ut in hoc homine animam et corpus, quibus coniungitur, vel in hac domo hunc parietem et hoc tectum et hoc fundam­ entum. Qui vero princip alitatem secundum destructionem con94

Abelard and his Contemporaries siderant, dicunt eas tantum principales esse quae substantiam totius destruunt, ut caput, quod abscisum hominem perimit. Substantiae vero destructionem ideo considerant: ad quant՛ itatis totius diminutionem cuiuslibet partis destructio suff­ icit, ut si hic etiam unguis pereat, quantitas Socratis corp­ oris eadem remanere non potest, cum tanta iam non sit quanta prius erat. Sicut enim per adiunctionem cuiuslibet totum crescit, sic per eius substractionem diminuitur. Cum autem ungue adempto quantitas corporis eadem non remanet, Socraticae tarnen essentia substantiae non videtur mutata; adhuc enim et homo et Socrates dicitur. Sed non omnia illa quae prius erant, secundum quantitatem esse contingit, idest unguem cum ceteris omnibus partibus: AD 549.4.20.) This last-drawn contrast between substantial destruction and m e r e l y q u a n t i t a t i v e reduction should be noted, since Abelard is l a t e r going t o be forced ( i t would appear) t o i d e n t i f y the t w o . In the m e a n t i m e there now f o l l o w s his m e r e o l o g i c a l l y s i g n i f i c a n t r e f u t a t i o n of the f i r s t , the c o n s t i t u t i v e p o s i t i o n . He points out t h a t i f the house has been divided into walls, r o o f , and foundation as its ' p r i n c i p a l ' parts, on the ground t h a t none is p a r t of a p a r t , but only p a r t of the whole, then i t simply needs a t w o - w a y s p l i t between, f o r example, t h e foundation on the one hand, and w a l l - p l u s - r o o f on the other, to make either the w a l l or the r o o f into a p a r t of a p a r t , and hence now no longer p r i n c i p a l . (It is this ingenious device of a dichotomy rendering secondary the allegedly p r i n ­ cipal p a r t s , which opens the way t o those non-discrete but nevertheless d i s t i n c t parts which were the t o p i c of ' . 5 above): (B) Now l e t us scrutinise both opinions. As f o r those who only c a l l 'principal parts' those which are not also parts of parts, I wonder why they c a l l w a l l , r o o f , or foundation 'principal p a r t s ' . For i f we look on the house dichotomously, so t h a t on t h e one side we consider the foundation as one independent p a r t , and on t h e other we consider the w a l l 95

Section 2 along w i t h the r o o f , we then are l e f t w i t h e i t h e r r o o f or w a l l as parts of parts. In this way we can, by an appropriate s p l i t t i n g - u p , render secondary any [allegedly] p r i n c i p a l p a r t of any composite. This can also be done in the case of this man, i f we take this f o o t of i t s e l f to be one p a r t and t h e r e s t , along w i t h the soul, as the other p a r t [so t h a t the head, f o r example, now becomes secondary]. A g a i n , any secondary p a r t can become p r i n c i p a l , as in the case of this f o o t , which turns out not t o be a p a r t of a p a r t according to the division mentioned. I t happens, t h e r e ­ f o r e , t h a t according t o various points of v i e w , the parts of the same composite are both p r i n c i p a l and secondary, and any such p a r t may have either sort as its p a r t . (Sed nunc quidem utramque sententiam perquiramus. Hi quidem princip­ ales partes eas solum vocant quae partium partes etiam <non> sint, miror quare parietem autem tectum aut fundam­ entum principales partes appellent. Nam si domus bifariam divisionem consideremeus, ut illinc fundamentum per se partem unam constituamus, hinc autem alteram partem ponamus parietem simul et tectum [et fundamentum], inveniemus aut parietem aut tectum partes partis. Sic quoque etiam quo­ libet composito quamlibet principalem partem secundum aliquam divisionem possumus habere secundariam, ut in hoc etiam homine, si hunc pedem per unam partem dicamus et residuum corpus cum anima alteram accipiamus, aut quamlibet secundarium principalem, ut hunc pedem qui nullius partis secundum hanc divisionem pars invenitur. Contingit itaque secundum diversos constitutionis respectus eiusdem compositi partes et et quaslibet utrumque habere: secundarias esse et principales A D 549.21.34.) 2.431 The point has already been made (in 1.5) t h a t once one starts operating w i t h dichotomies of the s o r t which A b e l ard is here considering, i t is v e r y easy to produce overlap­ ping parts, i.e. parts which are non-discrete and which t h e r e ­ f o r e are not numerable i f (like Abelard) one takes d i s c r e t e 96

Abelard and his Contemporaries ness as the criterion of numerable individuality or singular­ ity. Thus the two parts wall-and-roof and foundation-androof are clearly distinct and different parts of the house, but they are not discrete in the sense of having no overlap. Alternatively, wall and wall-and-roof are distinct but over­ lapping, and so on with the remainder of obvious combinations of situation (cf. 2.78, 6.2 below). However, given his position on numerability, the calling of attention to such overlaps can scarcely be Abelard's intent here. He wishes to overturn the constitutive position, and goes on to remind the reader that the contrast between that position and that of the destructivist is accentuated by the fact that what everyone would agree are in some sense parts of parts (e.g. the head as part of the body, itself part of the body-soul composite) are precisely the sort of part held to be principal by the destructivists, since no head means no man: AD 549.35.38. (On the theoretical position of overlapping, 10.34 may be consulted). 2.44 Then follows the exposition of what was evidently a central area of discussion at the time, as parallels from the Fragmentum Sangermanense are to confirm. Abelard, with his constant tenet that an identical object cannot have more or less parts at different times, as well as his corresponding resolutions of the paradoxes of increase later to be docum­ ented in 2.5, would certainly appear to be committed to the strong form of destructivism which he is now to describe. For the destructivists who are mentioned in 2.43(A) are mod­ erates, in that they stress the distinction between substant­ ial destruction and quantitative diminution, as was observed. For them it is the entailment of substantial destruction which is the mark of the principal part; no principal part means no substance of the original sort, e.g. no head means no man. Now, however, this rather misty criterion is superseded by positing that just any quantitative part is in fact a subs­ tantial and hence principal part, as far as the identity of 97

Section 2 the whole is concerned: (A) But f i r s t we must enquire as t o whether all the parts are such t h a t t h e i r destruction entails the destruction of the whole, even in its very substance. This would r e a l l y appear to be the case, since the destruction of any b i t of stone f r o m this w a l l also substantially destroys this house. For i f the stone-fragment is not, then this-house-comprisi n g - i t does not abide, indeed, does not even e x i s t ; but i f i t does not t h e r e f o r e e x i s t , neither does i t r e t a i n its e x i s t ­ ence. That on the destruction of this s t o n e - f r a g m e n t this house which comprises i t no longer is, becomes clear i f we f i r s t elucidate what is conveyed by 'this house'. For 'this house' conveys nothing more than a l l the parts of the house taken together. But as in its parts this stone-fragment is also contained, this house must be this s t o n e - f r a g m e n t plus all the other parts taken together. T h e r e f o r e , whosoever a t t r i b u t e s existence to this house must also be c o m m i t t e d t o a t t r i b u t i n g i t t o this stone and to a l l the other parts taken together; indeed, this house is just this stone and a l l the rest o f the parts taken together. A s , t h e r e f o r e , this house exists, so also must this stone-fragment and a l l the rest of the parts taken together. And i f this c o l l e c t i o n of t h e m a l l taken together exists, then i t is c e r t a i n l y necessary t h a t this stone-fragment should also be. For i f this s t o n e - f r a g ­ ment is not, then the c o l l e c t i o n of a l l the parts is n o t , t h a t is the c o l l e c t i o n of this stone-fragment together w i t h the rest. If, t h e r e f o r e , this house is, so also is this s t o n e - f r a g m e n t . Hence i f this s t o n e - f r a g m e n t is not, neither is this house. (Sed utrum omnes partes destructae svum etiam substantia destruant totum, prius est inspiciendum. Quod utique videtur, ut quislibet lapillus huius parietis destructus hanc domum in substantia quoque destruït. Si enim hic lapillus non est, haec domus quae ex ipso constat, in sua substantia non remanet, quippe n existit; si autem ideo non existit,  suam retinet existentiam. Quod autem 98

Abelard and his Contemporaries hoc lapillo destructo haec domus non sit quae ex ipso con­ stat, clarum est, si prius quid per 'haec domus' accipiatur, discernamus. 'Haec' itaque 'domus' nihil aliud accipitur quam omnes partes eius insimul acceptae. Cum autem et in part­ ibus eius hic lapillus [non] contineatur, oportet hanc domum esse hunc lapillum et omnes simul alias partes. Qui ergo profecto eamdem huic lapidi existentiam huic domui attribuit, et omnibus aliis simul partibus concedit; quippe nihil aliud est haec domus quam hic lapis et ceterae simul partes omnes. Cum itaque haec domus existat, oportet hunc lapillum et ceteras omnes simul partes. Quod si haec omnium collectio sim­ ul est, profecto et hune lapillum necesse est esse. Si enim hic lapillus non est, non est omnium collectio, idest huius lapilli simul et ceterorum. Si itaque haec domus est, et hic lapillus. Unde si hic lapillus non est, haec domus non est: A D 549.38 - 550.17.) 2.441 As an apposite quotation f r o m the contemporary pseudo-Joscelin is shortly t o remind us, this sort of c o n t ­ ention and many more c i t e d by Abelard obtain t h e i r e f f e c t by confining the r e f e r e n t of ' t h i s ' , as in 'this house', to an appropriate (and possibly momentary) t e m p o r a l p o r t i o n of the continuant in question. I t then follows a u t o m a t i c a l l y , w i t h 10.327 as the presupposed back-up thesis, t h a t even the slightest, minutest, ensuing q u a n t i t a t i v e change destroys the thus identified object. This in any case coheres with Abelard's principle of the identity of the number of parts: no thing can have more or less parts at one t i m e than at another (cf. 2.41 above). Another remark f r o m the same pseudo-Josc­ elin w i l l in due course also remind us t h a t in ordinary c i r c ­ umstances a m o m e n t a r y v i e w i n g or touching of a p a r t may quite n o r m a l l y be taken to j u s t i f y claims in respect of the spatio-temporal whole. This can apply even when the p a r t is t e m p o r a l . Estate agents or realtors would be vastly i r r i t a t e d were clients to renege on the basis t h a t they had only seen p a r t of the house, either because they only saw i t f o r ten 99

Section 2 minutes ( i f ' p a r t ' is taken in the t e m p o r a l sense) or because a segment of the e x t e r n a l rear was not v i e w e d , or again, bec­ ause a mouse has now nibbled a splinter away f r o m the door­ post, so t h a t what was at the t i m e of v i e w i n g a spatial part of the house has now become the whole house, spatially speak­ ing. Such, however, are consequences of the e x t r e m e destructi v i s t point of v i e w , and Abelard is conscious not only of t h e m , but also t h a t t h a t v i e w makes all parts, be they spatial or t e m p o r a l , equally p r i n c i p a l . A n y q u a n t i t a t i v e change, no m a t t e r how m i n u t e , brings into being a new house: (A) But we do not say t h a t i f this stone-fragment is r e m ­ oved then a house does not abide in the remaining parts, but only t h a t t h a t house which was made up f r o m t h a t pebble and the rest of the parts does not r e m a i n : t h a t p a r t i c u l a r house was made up of parts, and was r e f e r r e d t o as 'this house' in terms of the composition of all those parts. But when the substance of this house, which is based on a l l the s t u f f pertaining to i t , does not r e m a i n , then the house in question also necessarily perishes substantially. No longer w i l l one be able t o say, 'This house i s ' , of the house t h a t previously was, t h a t is, the c o l l e c t i o n of this s t o n e - f r a g ­ ment and the rest of the parts together; this can now only be said concerning the c o l l e c t i o n of the rest of the parts, a c o l l e c t i o n which is quite d i s t i n c t f r o m t h a t which previously existed, the d i s t i n c t i o n between these being akin t o t h a t of a p a r t to a whole whose quantity comprises t h a t p a r t : t h a t object which is now said to be 'this house' w i l l have been a part of the f i r s t house. (Non autem dicimus quod si hic lapillus auferretur, in residuis partibus domus non remaner­ et, sed non haec quae scilicet ex illo lapillo et ceteris partibus constaret; quae quidem singularis domus constituta fuerat ex partibus, et secundum omnium partium compositionem haec fuerat dicta. Ubi autem huius domus substantia, quae in tota ipsius materia est, non remanet, et ipsam domum in substantia necesse est perimi. Neque enim amplius dici pot100

Abelard and his Contemporaries erit h' domus est1 de ea quae prius erat, idest de collectione huius lapilli et ceterarum simul partium, sed de collectione tantum reliquarum partium; quae quidem collectio ab alia prorsus quae prius erat, diversa est, veluti pars a toto in cuius quantitate clauditur. IIIud enim quod modo haec domus dicitur, pars fuerit prioris domus: AD 550.18.29; the initial 'nunc' has here been replaced by the obviously appropriate 'non'.) 2.442 Various things could be noted here. We confine our­ selves in the first place to chiding Abelard for slipping in 'prorsus','wholly', when speaking of the distinction between the two states of the house: they are not wholly distinct, as he claims, but have very large overlaps. This is all of a piece with what has already been noted above about his coup­ ling of discreteness and countability. Next it may be obser­ ved that we are on the brink of a discussion about the way in which the major part (called the natura magna in the Frag­ mentum Sangermanense) of an object becomes the whole object when the remaining minor and possibly quite insignificant part of the object is eliminated. If this elimination does not change the object's nature, in a usual sort of sense, as when the mouse nibbles a mere splinter away from the house's door­ post, when Socrates trims his nails, when the cat loses a hair, or even its tail, then it looks as though a nascent house, Socrates, or cat, as the case may be, was lurking under the old whole, awaiting its bringing into being by the elim­ ination of the insignificant part. To the discussion of this natura magna topic time will be devoted in due course. For the present, it may be noted in passing that although, as already remarked, Abelard may be recounting rather than esp­ ousing this destructivist theory, his frequent difficulties with the mirror-image of the present question of diminution, i.e. the question of increase (HLM III §8, cf. 2.5 below) confirm that he does indeed find himself entangled with a destructivist position akin to that of our own contemporary 101

Section 2 Woodger, for whom proper names of objects extended in time are in fact common nouns covering many successive 'timeslices' of what would ordinarily be regarded as the objects in question: WSW 202 - 5. Paul of Venice describes and discuss­ es such a doctrine also: VLM 56a, on which see 8.33 and 8.432 below. 2.443 However, before going forward to a profounder insp­ ection of the many threads of discussion thus initiated, it is worthwhile to remain with that passage of Abelard's Dialectica on the consideration of which we are now embarked. Not only is there further evidence of the destructivism to which Abel­ ard himself is apparently committed, but also there occurs yet another fine argument sustaining the distinction between Xparts and parts-of-X (cf. 2.3, 1.4, and AD 344.34.35, 431.33.35). Thus he continues: (A) It is in this way, I believe, that one can show that if this stone-fragment is not, then this house is not, i.e. this stone-fragment together with the rest of the parts which are assumed to go to make up this house do not exist. And that this house does not exist when this stone-fragment is dest­ royed is quite evident from the fact that no object can be pointed to as 'this house', that is, as this stone-fragment and the other parts. But perhaps it might be objected: this stone-fragment having been removed, and now existing on its own outside the aggregated house (for after all the subst­ ance of a body cannot altogether be annihilated), this house which formerly was, still is; that is, this stone-fragment and the things which were parts along with it still exist. Now indeed this cannot be denied. But it does not thence follow that this house exists. Those things which lack house-like composition cannot be said to be 'this house', and for 'this house' to be, it is not sufficient that just its makings exist, otherwise even before the house was made the wood and the stones could be described as a house; for this the composite requires a certain structure. Hence it does 102

Abelard and his Contemporaries not follow that if this wood and these stones exist, then this house exists, unless they remain joined together with their parts duly disposed. It is this sort of composition, since it is of all the parts together, which must change when some part or other is removed. (Sic itaque ostensum esse arbitror ut si hic lapillus non est, haec domus non est, idest hic lapillus et simul ceterae partes quae in eius constitutione ponuntur non existunt; et bene hanc domum non esse apparet destructo hoc lapillo, cum nulla res haec domus esse monstrari possit, idest hic lapillus et ceterae partes. Sed fortasse dicitur remoto hoc lapillo et per se existente extra coniunctionem domus - neque enim substantia corporis omnino adnihilari potest - haec domus, quae prius erat, esse, idest hic lapillus et quae cum eo fuerant partes existere; quod quidem nec nos negamus. Sed non ideo hanc domum esse contingit. Neque enim haec domus dici poterunt quibus compositio domus defuerit,  ut haec domus sit, materiae suff­ icit existentia; iam enim ante fabricationem ipsam domum possen t vocari ligna ipsa et lapides; sed formae quoque nec­ essaria est compositio. Non itaque consequitur ut si haec ligna et lapides existant, haec domus existat, nisi scilicet secundum compositionem, quae in dispositione partium est, maneant. Quam quidem compositionem, cum sit omnium simul partium, necesse est mutari qualibet partium ablata: AD 550.29 - 551.6.) (In his disquisition on increase (2.531 below) Abelard himself appears at one point to come pretty near to the doctrine he rejects here, namely that the mere existence of the parts is the essential thing, with acquisition of their structure qua house being somehow accidental). 2.444 Then follows a contention leading to a verdict on the way in which destructivism affects the problem of 'principal parts' which launched us on the present discussion: (A) It also looks as though, from the nature of the princ­ ipal part, one can demonstrate as follows that when this 103

Section 2 stone-fragment is destroyed, then the house itself is quite destroyed. Thus were this stone-fragment not to exist, then neither will that of which it is a principal part exist; but this whole in its turn being thus destroyed, it is necessary that that of which it is a principal part be destroyed. This again having been destroyed, it is necessary that that of which this further part is a principal part should be des­ troyed, and there come to be destroyed as many of the ind­ ividual parts as there are wholes which are principal comp­ onents, until one arrives at the destruction of the wall itself; and when it is eliminated the house itself must per­ ish. In this way, therefore, the removal of no matter what part of any composite necessitates the elimination of the whole. Hence, from the destructive point of view all the parts are principal. (Videtur quoque et ex natura princip­ alis partis demonstrari ut hoc lapillo destruc to domus quoque ipsa omnino destruatur, hoc modo: si enim hic lapillus non fuerit, non erit illud cuius ipse est pars principalis, quo toto scilicet peremto necesse est et illud perimi cuius et ipsum erat pars principalis. Quo iterum destructo illud quoque destruí necesse est cuius haec etiam pars principalis fuera t to tiens que ex singulis partibus destrtis tota ipsa quae principaliter componunt, continget destruí, donec ad destructionem parietis veniatur; quo quidem interempto domum necesse <est> perimi. Sic itaque qualibet parte cuiuslibet compositi ablata totum necesse est intercipi. Unde omnes secundum destructionem principales esse convenit: AD 551.6.17.) 2.445 It is next claimed that in this last-quoted argument certain presupposed principles have not been explicitly stated. One of these is that not every whole happens also to be a part, as otherwise substance will expand to infinity: non omne totum partem contingat esse, quippe in infinitum substantia cresceret: AD 551.28.29. This is yet another illustration of Abelard's high degree of mereological soph104

Abelard and his Contemporaries istication, indicating as it does an awareness of the poss­ ibility of atomistic mereology, on which see 10.35 below. However, we can now pass on to the application of destructiv­ ism to the case of Socrates. It is concluded that, as fore­ seen above, the distinction between quantitative and subst­ antial destruction lapses; a quantitative destruction of no matter how small a part is equally a substantial destruction of the Socrates in question: (A) It has therefore yet again been established by this latest argument that the destruction of no matter what part necessarily entails the destruction of the whole also, in such a way that we admit that all parts are principal parts, and the same applies if we consider the results of destruct­ ion as applied to substances. For in the same way as on the removal of Socrates' nail a certain quantity remains in the rest of the body, but not that whole quantity which initially was in the whole object, so also a substance remains, but not that whole substance which subsisted in the first place, namely that to which the removed part of the nail pertained. Therefore the extent to which quantitative destruction oc­ curs when the nail is destroyed is no greater than that which occurs in the case of that substance of which the nail itself was a part. (Constat itaque et ex argumentatione novissima qualibet partium destructa totum quoque ipsum necessario des trui, ut scilicet omnes partes principales esse confiteamur, si vim quoque destructionis in substantia pensemus. Sicut enim ungue Socratis ablato, quantitatis quaedam in residuo corpore manet, sed non ea tota quae prius in toto fuerat, sic quoque substantia remanet, sed non ea tota quae prius exstiterat, cuius videlicet pars unguis ademptus exstiterat. Non itaque magis quantitatem ex ungue destruí contingit quam substantíam, cuius quidem substantiae unguis ipse pars fuerat: AD 552.6.14.) 2.446 This apparently extreme destructivism has now taken us well beyond the bounds of pre-theoretical commonsense, in 105

Section 2 that a succession of various objects each called 'Socrates' is produced by the various successive minimal changes in Socrat­ es from moment to moment. As a final attempt to escape this paradox there is propounded what looks like a recognitional criterion of identity: (A) But if, from the destruction of any part there ensues the abolition of the whole substance, what are we to say when we can see Socrates still there and still abiding in his substantial whole as a man, notwithstanding his loss of a nail, or even of a whole hand or foot? It therefore looks as though being this man does not involve the having of all his parts together, but only those without which he cannot be recognised; otherwise it looks as though there will be equivocation in the predication of his name from time to time, according to the increase or decrease of his body. (Sed quid dicemus, si ex cuiuslibet partis destructione totius substantiae sequitur interitus, cum ungue perdito aut tota etiam manu vel pede, Socratem remanere viderimus et in ipso adhuc hominis integram substantiam permanere? Non itaque hic homo in omnibus suis partibus simul consistere videtur, sed in his tantum praeter quas reperiri non potest; alioquin aequivoca eius praedicatio per diversa tempora videbitur secundum augmenta corporis sui vel detrimenta: AD 552.15.21.) 2.447 But (he continues) since there need be no definite point of recognition, e.g. we may recognise a person by the way they speak or gesticulate with their hand, we are there­ fore again pushed back in the direction of destructivism. We cannot, in general, designate some feature of a human being which could not happen to be the point of recognition. Hence it could well be the case that any removable piece, upon el­ imination, will destroy the man. Still, this illustrates that not every destruction of a man is a case of homicide: (A) But it is, on the grounds stated, necessary to admit that also on the removal of a hand, that man who first 106

Abelard and his Contemporaries existed no longer remains; hence whoever removes a hand or any part of a man would appear to be committing homicide. But of course not any destruction of a man is said to be homicide, but only that which expels the soul from its place; for homicide is the killing of a man, and the killing cannot take place without the expulsion of the soul. (Oportet i taque ut manu quoque ablata hic homo qui prius existebat, non maneat; unde quicumque manum abstulit aut quamlibet hominis partem, homicidium perpetrasse videtur. Sed fortasse non quaelibet hominis destructio homicidium dicitur, sed illa tantum quae animam sede sua expellit; est enim homicidium hominis interfectio, interfectio autem fieri non potest nisi expulsione animae: AD 552.32.37.) 2.45 Finally Abelard pronounces his verdict on the two positions, constitutive and destructive, in this matter of principal parts. It is difficult to see the difference between his preferred solution and that constitutive one against which he has so effectively argued in 2.43 above: (A) But now that both the opinions on principality of parts are tottering, we can conclude with what, according to our judgement, is now obvious. It appears to us that, as noted above, those parts should be called 'principal' which are such that on their being conjoined, the completion of the whole forthwith ensues, as when roof, foundation, and wall, are joined, then the house is immediately completed. The same does not apply to their compounded parts; for although all the parts of the roof are properly disposed, and likewise those of the wall and foundation, for the completion of the house there is still lacking the joining together of the components: wall, roof, and foundation. However, their due bringing together at once renders the house complete. (Nunc autem de principalitate partium sententia cassata quid nostro praeluceat arbitrio supponamus. Principales itaque partes, ut supra notavimus, nobis appellari viden tur quarum ad se coniunctionem totius perfectio statim subsequitur, ut 107

Section 2 tecto <et> fundamento aut pariete coniunctis, domus statim perficitur, sed non ita eorum partibus compositis; etsi enim in tecto omnes partes eius iam sint dispositae ac similiter in pariete aut fundamento, deest tarnen ad perfectîonem domus compositorum et parietis, tecti et fundamenti ad se invicem coniunctio, quorum quidem conventus domus perfectîonem stat­ im reddit: AD 552.38 - 553.) Whatever one may make of this declaration, there seems little doubt that Abelard's principles commit him to destruct­ ivism, as we have seen. Standing outside the controversy, however, we may still ask whether the already-mentioned nat­ ura magna argument holds good. According to this argument, which is adumbrated in Abelard's above-quoted discussion, and which comes explicitly to the surface in the Fragmentum Sangermanense, that object which is (say) the whole house minus the stone-fragment (and yet is still not itself the house) becomes the house upon the removal of the stone-fragment which was an integral part of the former house. It is the almost-total house (e.g. the whole house minus the still adhering stone-fragment) which is said to be the natura magna. The Fragmentum example outlines the case of Socrates who has as a part the-whole-of-himself-except-a-still-integrated-fingernail (all this being in this case the natura magna). It is then suggested that when that fingernail is removed by actual cutting off, the original Socrates stops existing, and the natura magna which earlier was not Socrates becomes Socrates. To lead up to this point the Fragmentum first imagines the construction (creatio) of new constructs (creaturae) out of elemental parts (points or atoms) until some part of Socrates is attained. The destruction of Socr­ ates as a whole, like the destruction of Abelard's house, recounted above in 2.44, turns on the destruction of this small constructed part, say a fingernail. Now, however, the pseudo-Joscelin makes explicit the point which was surfacing in Abelard's account, namely, that the 'natura magna', the 108

Abelard and his Contemporaries major identifiable object, which before the removal of the small part was constituted by everything in the whole except that small part, now itself becomes the whole. Thus in the case of the removal of the fingernail, the natura magna which earlier was not Socrates, becomes Socrates: (B) Thus therefore increasingly, with new constructs prog­ ressively being constructed, until some portion of Socrates is arrived at (e.g. the nails), you finally get some major identifiable object which will be part of Socrates, but which will not be Socrates, being identifiable as Socrates' make-up minus the nail; the nail is as much a part of Socrates as that major identifiable object also is. But now the des­ truction of that nail destroys a part of that object of which the nail is a part; that object is Socrates, and hence Soc­ rates is destroyed. But at the same time that major ident­ ifiable object which earlier was part of Socrates, but not Socrates, is, on the destruction of the nail, the Socrates which still abides, so that that which earlier was not Soc­ rates becomes Socrates. (Sic itaque crescendo novasque creaturas progressive creando, donec ad aliquam Socra tis perveniatur particulam, utpote ad ungues, et habebis unam magnam naturam quae erit pars Socratis, et non Socrates, quia in eius constitutione non est ungula, quae ungula pars est etiam Socratis cum alia magna parte.  autem ungula destructa destruitur pars illa naturae cuius ungula pars est, quae natura est Socrates, et ita destruitur Socrates. IIIa autem natura magna quae prius pars Socratis erat et non Socrates, destructa ungula, remanet Socratem, et ita quod prius non erat Socrates, fit Socrates: CIA 511, cf. KAU 149* - 150*.) We have already seen this kind of thing above in conn­ ection with the diminishing house, and Abelard advances an even more similar example when glossing Porphyry. Thus, hav­ ing discussed the way in which being a biped could still be an essential differentia of man, even though individual human 109

Section 2 beings may lack feet, he raises the mereological question: (C) There crops up another query as to whether that trunc­ ated man (which initially was part of the man when the man was still whole) is also a man, i.e. whether [i] within the make-up of the whole man there was that substance which is now the truncated man and which was then also a man who was the same living sensitive rational and mortal being that he now is, or whether [ii] the amputation brought about the existence of a man who previously did not exist. (Occurrit alia quaestio, cum post abscisionem pedum illa substantia curtata quae prius pars hominis, dum homo integer permanebat [homo sit], utrum in ipsa constitutione hominis integri illa substantia quae modo curtata est, homo etiam tunc esset et animata et sensibilis et rationalis et mortalis, sicut modo est, aut potius per abscisionem fiat homo quod prius non erat: GA 105.9.14.) 2.451 Here two alternatives are being suggested. Alter­ native (i) is the natura magna story in a version according to which there was, so to speak, a man waiting to be brought into the light, having been concealed within the former whole. Alternative (ii) is closer to the quotation in (B) above from the pseudo-Joscelin and also to the previous Abelardian story about the diminishing house. According to this alternative the natura magna was not a man, but becomes a man on the removal of the foot, or the nail, or whatever. One thus has a succession of men or houses, as the case may be, in the full destructivist fashion. The consequence of alternative (i) is that within every man there are indefinitely man men (and so on in other cases). This is regarded by Abelard as a reductio ad absurd­ um of alternative (i): (A) But if, within the make-up of the man there was also a man, then the same man is certainly made up of lots of men. For by multiple amputations there can be separated out the parts which will be distinct men, e.g. if we were also to cut 110

Abelard and his Contemporaries off the arms of someone who already has had his feet amp­ utated. But of course no one will grant that there are many men within one man, nor that there can be many men without their being many souls. (Sed si in constitutione quoque hominis homo erat, profecto idem homo ex pluribus constat hominibus, quia per plures abscisiones plures possun t separ­ an partes quae separatae homines erunt, ut si abscisis pedibus brachia quoque abscidamus. At vero multos homines in uno esse nemo concedit, nec multos esse sine multos animabus: GA 105.14.19.) Alternative (ii) is, as one might expect, accepted but then softened and modified on the lines of the 'not every destruction of a man is homicide' thesis encountered in 2.447(A) above: (B) But if by our amputatory action we bring it about that that which was not a man becomes a man, and by the same token a living being and an animal, then it also brings it about that the process of generation comes within the ambit of our operation. In fact there is no incompatibility here if even as our action in some way as it were destroys a man who previously existed (i.e. when, by means of the amput­ ation, we bring it about that those things which were the man in the first place are not at the same time also another man) so also our action brings it about that that which earlier was not a man becomes a man. For that destruction of a man which does not involve the removal of the soul, no more amounts to homicide than that making which does not involve the giving of a soul really amounts to the making of a man. Hence the body which is enlivened by the same soul is said to be always the same man, even though the bulk of its being at one time involves fewer parts (when the child is) and at another time involves more parts (at the time of youth). And the bodies which formerly were and now are not the same in respect of the bulk of their being, are adjudged to be the same in respect of the identical effects of the 111

Section 2 soul. Were this not so, then when the nails had been trim­ med we would say that there now was another human subst­ ance. For although this is true as far as the distinction of being is concerned, still, as far as the effects of the soul are concerned, another person is not said to be involved. (Quodsi per abscisionem nostram id facimus hominem quod non erat homo, similiter et animatum et animal, et ita nostrae quoque operationi generatio subiecta est. Sed non inconveniens est, si nostra operatio sicut hominem quodammodo destruit, qui prius erat, dum scilicet per abscisionem fac­ imus, ne sint homo illa simul, quae prius erat homo, ita etiam faciat, ut homo sit quod prius homo non erat,  tamen talis destructio hominis est homicidium facere, quae anim­ am non aufert,  talis effectio proprie dicitur hominem efficere, quae animam non confert. Unde idem semper homo esse dicitur corpus eadem anima vegetatum, licet essentiae quantitas modo minor sit partibus, dum puer est, modo major, dum iuvenis est, et licet eadem non sint corpora secundum quantitatem essentiae quod prius erat et quod modo est, pro eodem tarnen secundum eundem effctum animae ludicantur, alioquin abscisi unguibus aliam hominis diceremus substantiam, quod licet quantum ad discretionem essentiae verum sit, quantum tarnen ad effectum animae alia persona non dicitur: GA 105.19.34.) 2.452 In point of fact, Abelard's taking refuge in the unity of the soul throughout the successive beings brought about by truncation, so as to preserve the unity of the per­ son, obviously leaves him with the task of letting us know how one counts souls, and he does nothing about this. We thus still have the threat of an indefinitely large succession of Socrateses generated by any quantitative change, no matter how minute, within what would ordinarily be called 'Socrates'. We are hence still in the grip of a Woodger-style destruct­ ivism similar to that encountered in the pebble-losing house or the nail-losing Socrates. The whole story here comes near 112

Abelard and his Contemporaries to the account given by the eighteenth-century David Hume of the contrast between 'a succession of related objects' (in reality) and the 'notion of identity' which 'confusion and mistake' cause us to attribute to that succession: (A) Our propensity to this mistake is so great from the resemblance above-mention'd, that we fall into it before we are aware; and tho' we incessantly correct ourselves by reflexion, and return to a more accurate method of thinking, yet we cannot long sustain our philosophy, or take off this biass from the imagination. Our last resource is to yield to it, and boldly assert that these different related objects are in effect the same, however interrupted and variable. In order to justify to ourselves this absurdity, we often feign some new and unintelligible principle, that connects the objects together, and prevents their interruption or variation. Thus we feign the continu'd existence of the perceptions of our senses, to remove the interruption; and run into the notion of a soul, and self, and substance, to disguise the variation. But we may further observe, that where we do not give rise to such a fiction, our propension to confound identity with relation is so great, that we are apt to imagine something unknown and mysterious connecting the parts, beside their relation; and this I take to be the case with regard to the identity we ascribe to plants and vegetables. And even when this does not take place, we still feel a propensity to confound these ideas, tho' we are not able fully to satisfy ourselves in that particular, nor find any thing invariable and uninterrupted to justify our notion of identity: HT 254 - 5 (Bk I Sec. VI). We have in fact just been seeing Abelard act in the way here described by Hume, i.e. attributing human continuity in nonhomicidal types of human change (for instance) to the nonexpulsion of the soul. But now the Fragmentum intervenes at this point in such a discussion, and shows how to quite sim­ ply avoid such buck-passing tactics and Hume-style f lutter113

Section 2 ings by restoring coherence to the usual pretheoretical acc­ ount of changing objects. This simply presupposes that Soc­ rates (for example) is just the complete collection of both his spatial and temporal parts. He can then still remain Soc­ rates, and the same Socrates, although his being may now in­ volve fewer (or more) spatial parts than before. Thus, after a discussion which appears at one point to exploit the dist­ inction between a complete collection and a mere collection, i.e. one which need not be complete (10.329 below), the Fragmentum continues: (B) Upon the destruction of one part of Socrates, the rem­ ainders which survive do not become Socrates, but rather Socrates comes to be made out of them, and so this position does not force us to hold that a Socrates perishes or that a non-Socrates becomes a Socrates. All we have to hold is that while Socrates' being still abides, he is now made up of fewer parts. (Destructa una parte Socratis, ceterae quae rem­ anent non fiunt Socrates, sed ex ipsis fit Socrates, et ita secundum hoc neque cogimur dicere quod Socrates pereat, neque quod non-Socrates Socrates fiat; sed hoc tantum, salva essentia Socratis, consentimus ex istis, ut Socrates ex paucioribus existat: CIA 512, cf. KAU 150*.) There is hence no need to abandon our pre-theoretical picture of Socrates as an object or person who persists ac­ ross time. 'Socrates' names Socrates from the beginning to the end of his life, and is the proper name of the complete collection of the Socratic spatio-temporal parts. 2.453 This discussion of the effect of the truncation of the parts of a man, and even of his principal parts, has a rather sad human postscript. Abelard himself was the victim of a violent criminal amputation, directed at his genitals, instigated by the guardian of Heloïse, Abelard's pupil, mother of his child, and later his spouse. It seems evident that mereological work such as that briefly touched upon above had rendered the logician sufficiently well-known as an expert on 114

Abelard and his Contemporaries principal parts for it to be the basis of ribaldry from the side of his enemies, who were many. One of them, his notor­ ious nominalist master Roscellin, has been mentioned in pass­ ing above. At the close of what must be one of the most ab­ usive letters ever written, he draws on this branch of Abel­ ard's learning to insult him accordingly. Thus, having remind­ ed Abelard of his chequered career, he expresses doubts as to his identity. He appears, says Roscellin, to be neither a cleric nor a lay person, and would he not even be lying if he dared to call himself by the Latin name 'Petrus', which is of the masculine gender? For, concludes the letter: (A) I am quite sure ... that a noun of the masculine kind no longer signifies its usual object should that object be cut off from its kind. This is because proper names customarily relinquish their meaning should it happen that the things signified by them decline from their complete state. For neither is a house said to be a house when the roof or wall has been removed; rather would it be called an incomplete house. ... Having been deprived of that part which constitutes you a man, you are not to be called 'Peter', but rather 'incomplete Peter'. I had intended to sort out a host of true and obvious facts which would have been to your shame. However, since I am proceeding against an incomplete man, I am leaving incomplete the work which I had begun. (Certus sum ... quod masculini generis nomen, si a suo genere decid­ erit, rem solitam significare recusabit. Solent enim nomina propria significationem amittere, cum eorum significata cont­ ingent a sua perfectione recedere. Neque enim ablato tecto vel parietiete domus vocabitur domus, sed imperfecta domus vocabitur. Sublata igitur parte quae hominem facit, non Pet­ rus, sed imperfectus Petrus appellandus es. ... Plura quidem in tuam contumeliam vera ac manifesta decreveram; sed quia contra hominem imperfectum ago, opus quod coeperam imperf­ ectum relinquo: COA 803.)

115

Section 2 2.5 Increase and Decrease 2.51 Twelfth-century exemplifications of problems about integral parts and wholes covered a quite extensive range. In addition to the usual examples involving Socrates, the house, a people, a crowd, and a flock or army, there are to be enc­ ountered the quite general couples, trios, quartets, and so on, of the manifold objects considered in the Sententie secundum M. Petrum (cf. 2.7 below). These Sententie may not actually be Abelard's, but they are well within the spirit reported by Alberic of Paris, who said that when pressed Abelard would countenance aggregate wholes such as those composed from both himself and the then present queen of France, or from himself and the Apennine Mountains (TAU 151). For Abelard every multiplicity did indeed make one whole (omnis plura unum totum efficere). In his Dialectica no restrictions are imposed initially as regards proximity or remoteness of the parts of a given whole: omnia .... plura simul accepta, sive continuata sint sive disiuncta, totum sunt ad singula: AD 422.4.5. Indeed, we have already seen how man-made objects tend to be discontinuous, in any case (2.232). However, his generality lets in the possibility of the need for references to outlandish and hitherto unheard-of composites, and the vocabulary for dealing with them usually does not exist. This is going to be made overtly evident in the passage next to be quoted. Again, the often-repeated example of the pair of men who may be as far apart as are Paris and Rome (AD 64.20.21, 423.13.16, 431.28.29, 576.5.7) confirms that proximity of parts is not required for the countenancing of a given aggregate whole. However, this quite refreshing generality, in conjunct­ ion with Abelard's destructivist principle that no thing can have more or less parts at one time than at another (GA 299 - 300, AD 423.29.30), appears to lead him into strange places where questions of increase and decrease are concerned. 2.52 Still, on the way to these strange places it is at least fascinating to observe the manner in which Abelard att116

Abelard and his Contemporaries empts to evade the toils of natural language which must at some points inevitably lack nomenclature for some of the the­ oretically possible composites. Thus in the Glosses on the Categorise we are reminded that it is water which is said to increase by the addition of water, it is the heap of stones which is said to increase by the addition of stones, and so on. But what is it that increases when a piece of wood is added to the heap of three stones? The heap perhaps? What of the addition of a man to the stones? What of the addition of a horse to a group of men? Do we now say that physical objects or animals are increased? And although he shows how it is possible to devise ways of talking appropriate to at least some of the out-of-the-way cases, Abelard prefers to interpret increase with reference to the production of likemembered compounds such as the heap of stones. Still, in this type of case, as also in the more complex counterparts such as the house-decrease examples which came up in the discussion of 'principal parts' (2.44, 2.45 above) there is, according to Abelard, strictly speaking no one thing to which the increase (or decrease) may be attributed, but rather a succession of objects (stone-heaps, houses, and so forth, if we insist on such novel and misleading uses of these pre-theoretical names). This accords with the destructivist princip­ les which have been observed lurking in the background thro­ ughout the foregoing sections. Here now is the passage from the Glosses which usefully serves as an introductory quotat­ ion for such topics: (A) Now we deal with change of quantity, which involves increase and decrease. Such change is said to be quantit­ ative because for it to take place there is a change in the bulk or the amount of the substance of the thing, when some­ thing is taken away or added. Thus if I remove some part of water from water, then the water from which the other water is taken is said to diminish. Contrariwise, the same water is said to increase, should I add some other water. Thus, 117

Section 2 then, the amount of the bulk is either made greater by in­ crease or lessened by decrease. But I think that it is not unreasonable to raise the question as to what the thing is that is said to increase, and in what exactly the increase consists. For if someone asserts that to increase is to make something bigger than it was by the adjoining of something, then it looks as though nothing increases. There is just no thing which seems to become bigger by such addition than it previously was. For example, if a fourth stone is added to another three, and the heap which first of all consisted of three stones is said to be increased by the incorporation of the fourth stone, then this does not seem to be true. This is because the heap of three stones does not, on the adjoining of the fourth stone, undergo a quantitative increase over its previous state; it still is made up of three stones as before, nor has it more parts than before. At the same time and for the same reas­ on, neither does the fourth, the added, stone; nor yet again does the composite formed from the three stones and the ad­ ded one increase either. For now, as before, there remain four stones, and they are no more now than they were before. But perhaps the problem could be more easily solved if we spoke of increase only in respect of that which, when some­ thing is added, forms a composite of the same nature or pr­ operties as the original. For example, if water is added to water, then the water to which the addition is made is tran­ sformed into a composite which is still said to be water. Likewise, if to three stones a fourth is added, the threemembered heap of stones is said to increase when it is tran­ sformed into a bigger heap of stones by the addition of the fourth stone. Here the second and bigger heap is, like the first, still a heap of stones. However, if a log is added to the three stones, neither the heap of stones nor the stones are said to increase. since the lot tal in together is neither just stones, nor is it a heap of stones. Still, one might 118

Abelard and his Contemporaries say that the heap increased, since both what was there at first and what was made out of it are both called heaps. In contrast, if a stone is adjoined to a man or a man to a stone, then one cannot in like fashion say that the stone or the man increases; but one could speak of an increase if the resulting composite could be said to be of stones or could be said to be of men. Hence, if to two men a third is added, then men increase. But if to them a horse is added, then one cannot speak of an increase of men, but rather perhaps of animals, or of bodies. This is because the resulting whole can be said to be of animals (or bodies) as opposed to men, since the things which were there before were also called animals [or bodies]. But someone might object: if those animals which increase by the adjoining of the horse are men, then if the animals increase, the men must also increase. And we must indeed grant that those very beings which are men do increase owing to the adjoining of the horse, but not in respect of the number of men, but of animals. Hence, if we want to express ourselves properly, we should rather say that it is some­ thing other than men which increases; in a sense, however, and with the qualification mentioned, we can also say that the men increase. It is therefore obvious, from what has been said, that increase should rather be understood in the sense according to which, by means of an addition, something is transformed into a composite of the same sort as the original, as oppos­ ed to that sense which would stress the increase in the number of parts. No thing has more parts at one time than at another. the collection of three stones never has more parts owing to the addition of something; it has no more parts at the later stage than at the earlier. Similarly, although the earlier flesh of a child may be covered over [in the course of growth], that flesh does not increase in such a fashion that it now comprises more parts than it did bef119

Section 2 ore. For that which is added is not a part of that to which it is adjoined. (Nunc de motu quantitatis agamus, qui cremen­ to et diminutione comprehenditur. Qui ideo quantitatis dicitur, quod secundum hunc motum ipsa massa vel capacitas substantiae rei permuta tur, aliqua re subtracta vel addita. Ut si aquae partem aliquam subtraham aquae, decrescere dicitur, cui subtrahitur alia aqua. Crescere vero eadem aqua dicitur, si aquam aliam superaddam. Sic quippe ipsa massae capacitas vel extenditur crescendo vel minuitur decrescendo. Sed puto non irrationabiliter quaeri, quae res crescere dicatur, aut quid etiam sit crescere. Si quis autem crescere dicat adiunctione alicuius maius effici, quam prius esset, nulla crescere videntur. Nihil quippe est quod videa tur mai­ us fieri, quam prius erat ex additamento, ut si tribus lapidibus quartus addatur atque ideo acervus qui prius erat trium lapidum, crevisse dicatur adiunctione quarti lapidis, non vid­ etur verum, cum videlicet acervus trium lapidum adiuncto et­ iam quarto lapidum in quantitate sui non plus quam prius habeat, sed tribus lapidibus tunc quoque constat, sicut ante,  plures partes quam prius habet, sed  quartus lapis superadditus simili ratione crevit  ipsum quoque com­ positum ex tribus lapidibus et quarto adiuncto. Nam et ante sicuti nunc quattuor lapides manebant  plures modo sunt quam ante. Sic autem fortasse facilius solvetur, si vide­ licet crescere id dicamus quod per adiunctionem alterius transit in tale compositum quod a natura vel proprietate sua non recedit, veluti si aquae alia aqua superaddatur, aqua cui superadditum est, in quoddam transit compositum quod etiam aqua dicitur. Similiter, si tribus lapidibus quartus super­ addatur, acervus trium lapidum crevisse dicitur, dum in acervum maiorem lapidum transit per adiunctionem quarti lapidis, qui etiam secundus maior acervus sicut ille qui prius erat, acervus est lapidum. Si vero tribus lapidibus lignum addat­ ur, acervus lapidum non dicitur crevisse  etaim lapides, quippe totum simul coniunctum  lapides est  acervus 120

Abelard and his Contemporaries lapidum; acervus tunc fortasse creverit quia et quod prius erat et quod ex eo factum est, acervus dicitur. Quod si lapis homini vel homo lapidi coniungatur, nec homo neque lapis simili ratione crevisse dicitur, quod quidem esset, si compositum vel lapides vel homines dici posset. Si ergo duobus horninibus tertius addatur, homines crescunt. Si vero eisdem equus addatur, non crementum hominum, sed animalium et corporum fortasse dici potest; totum quippe ipsum animalia vel corpora dici potest, non homines, sicut et ea quae ante fuerant, animalia dicebantur. Sed oppone tur: si ea ananimaliaquae crescunt per adiunctionem equi, homines sunt, profecto ubi ea crescunt, oportet etiam homines crescere. Et nos quidem concedimus res ipsas quae homines sunt, crescere per adiunctionem quoque equi, non quantum ad numerum hominum, sed animalium. Unde si proprie loqui desideramus, alia potius quam homines crevisse dicamus, quodammodo tarnen, ut dictum est, homines quoque crevisse, ut determinatum est, possumus dicere. Manifestum est itaque ex his quae dicta sunt, magis in eo crementum accipi debere, quod aliquid videlicet per adiunct­ ionem alterius in consimile sibi compositum transit, quam in eo quod eius partes multiplicentur. Nulla enim res uno tem­ pore plures habet partes quam alio, ut collectio trium lapid­ um num quam abundat in partibus per coniunctionem alicuius, ut videlicet plures modo partes quam prius habeat, nec caro pueri quae prius erat occulta, recipit incrementum, ut pluribus modo constet partibus quam prius. Quippe illud quod super additum est, non pars eius cui adiunctum est: GA 299.11 - 300.25.) 2.521 The fanciful talk which occurs here, concerning the later flesh of a child which covers over the earlier layer, is quite uncannily echoed by a Nabokov character in his Lolita (Pt. II, ch. 3) who describes how the later state of the flesh of young women ('heavy, low-slung pelvis, thick calves ...') may be seen as f the coffin of coarse, female, flesh within which 121

Section 2 my nymphets are buried alive'. This is, in its way, a literary reminder that the modes of expression chosen to express inc­ rease may be indefinitely many. Even when the choice is nar­ rowed down to the production of like-sorted-compound descr­ iptions in the style mooted by Abelard in this last passage, the track to be taken still remains indefinite. Context and purposes, real or imaginary, may govern suggestions. One can easily invent settings in which most of the items of his inc­ reasingly miscellaneous list of additions to the heap could be covered. Thus headings such as 'debris from the disastrous flood', 'washed-up flotsam and jetsam', or 'surviving items from the wreck', could suffice. At the other extreme, the limit of logical austerity could ensure coverage of the case by means of disjunctively-articulated nomenclature, e.g. 'a heap of either stones or wood or men or horses'. There is a point in the Latin of this last text which might even tolerate a translation of this sort; thus si compositum vel lapides vel homines dici posset has been translated above as, 'if the resulting composite could be said to be of stones or could be said to be of men', and the context makes it fairly certain that this is what Abelard intends to say. Had he for a mom­ ent glimpsed the possibility of disjunctive nomenclature, the sense would then indeed be, 'if it were possible to call the composite "either stones or men"'. This is, however, highly unlikely. 2.53 We have, of course, already seen a great deal of Abelard's struggle with changes in the number of parts of an object across time. This was under the heading of 'Principal Parts' in 2.4 above, and the principle which is repeated in the last-quoted passage, to the effect that no object ever has more parts at one time than at another, impeded his efforts to make sense of the process of such changes, apart from the destructivist way out. He recurs to the mirror-image of the problem of decrease (prominent in 2.4) in a further lengthy passage on increase in the Dialectica, next to be quoted. This 122

Abelard and his Contemporaries probably represents his final effort to come to terms with the problem of identity across quantitative change-processes. First in this passage comes a trenchant statement of the pro­ blem, a statement which repeats the paradox underlined in the last-quoted passage, namely that there would appear to be no one thing to which we can point in the increase situation and say that it is that thing which increases. After all, the number of parts involved in the situation is the same. His tone of voice may suggest that he is now discontented with his previous verdict, and is going to move on to something more sophisticated: (A) It looks as though no increase can take place by the conjoining of something else; indeed, it looks as though no­ thing increases. For when one thing is conjoined with an­ other, neither that which is conjoined nor that to which it is conjoined increase, since neither has more parts than it previously had. Yet neither does the whole which is compos­ ed of these appear to increase either. It still keeps only those parts which it previously had. For in the same way as the two combined form together a whole in relation to each part, so also when disjoined, they were the whole in relation to the same parts. For all pluralities when taken together, whether adjacent or disjoined, are the whole in relation to each of their parts. The same contention may be advanced in respect of decrease. What is it, then, that we assert to in­ crease or to diminish? (Non enim aliquod augmentum fieri in aliquod videtur per adiunctionem alicuius, quippe nihil augeri videtur. Neque enim cum aliquid alicui apponitur, illud quod appositum est crevit, neque illud cui appositum est, cum plures partes quam prius non habeat. Sed  totum quod ex eis coniunctum est, crevisse videtur. Eas enim tantum partes adhuc retinet quas prius habebat, illud scilicet quod est adiunctum et cui est adiunctum. Sicut enim illa duo coniuncta simul totum sunt ad singula, sic et disiunctum totum erant ad eadem. Omnia enim plura simul accepta sive continuata 123

Section 2 sint sive disiuncta, totum sunt ad singula. Similiter et de detrimento potest opponi: quid igitur augeri vel diminuì dicemus? AD 421.34 - 422.6.) 2.531 Clearly visible here, yet again, is Abelard's usual principle that just any sort of plurality can be taken to form some whole, even if the parts are disjoined, as noted in 2.51. Coupled with the point from the previous quotation to the ef­ fect that there is no item the parts of which are increased, that principle easily confirms the conclusion that nothing in­ creases or diminishes. The same effect, as has been observed in 2.41, may be obtained by accentuating the principle that no object, if it is to be the same object, has any variation in the number of its parts. But something hitherto not brought into prominence is making its appearance. As well as the in­ sistence on the sameness of the number of parts, there is al­ so a tendency towards denying that real novelty ensues upon the combination of parts. This is the sort of conclusion rea­ ched by the seventeenth-century philosopher and scientist Robert Boyle (1627 - 1691), founder of the Royal Society in the kingdom of England. He too would insist that the bring­ ing together of formerly scattered parts does not really br­ ing into being anything which, substantially speaking, was not already there previously. True, human beings may say that something is generated or produced de novo, as Boyle puts it, but this is a mere matter of nomenclature or 'denomination', and not a real change in the things themselves. A further interesting, but much more minor, coincidence in Boyle's words to this effect, which now follow, is the presence of the example of glass-making, already noted in a similar context in Abelard's writing (2.232(A)): (A) ... we say that a body belonging to that species, as suppose a stone or a metal, is generated or produced de novo - not that there is really anything of substantial produced, but that those parts of matter that did indeed before pre­ exist, but were either scattered or shared among other bod124

Abelard and his Contemporaries ies, or at least otherwise disposed of, are now brought tog­ ether and disposed after the manner requisite to entitle the body that results from them to a new denomination, and make it appertain to such a determinate species of natural bodies, so that no new substance is in generation produced, but only that which was pre-existent obtains a new modification or manner of existence. Thus when the spring and wheels and string and balance and index & c , necessary to a watch, which lay before scattered, some in one part, some in another, of the artificer's shop, are first set together in the order requisite to make such an engine to show how the time pass­ es, a watch is said to be made - not that any of the ment­ ioned material parts is produced de novo, but that till then the divided matter was not so contrived and put together as was requisite to constitute such a thing as we call a watch. And so, when sand and ashes are well melted together and suffered to cool, and there is generated by the colliquation that sort of concretion we call glass, though it be evident that its ingredients were both pre-existent and do but by their association obtain a new manner of existing together: BPP 45. (The italics here shown are based on Boyle's own script.) It is precisely this sort of contention, i.e. that the parts are already there and are just rearranged, so that no­ thing new is really made, or (in Abelard's case) nothing really increases, which if endowed with a privileged theoretical st­ atus relatively to other modes of description, can swamp the theoretical priority of that transition from X-parts to partsof-X which the history of mereological sophistication shows to be absolutely crucial (cf. 1.4, 2.3, 2.443, 3.41). In con­ sequence, nothing really happens when making or increase would ordinarily be said to have taken place in respect of some object, except that previously non-assembled quasi-componential parts are put together or appropriately relocated, as in the cases of Boyle's watch and Abelard's example of the 125

Section 2 men who are first outside and then inside the house: 2.542(A) below. The geometry of configuration thus provides the lang­ uage for the ultimate and unrevisable description of the uni­ verse. Destruction is then, as Boyle goes on to remind us, the dissociation of parts, and nothing more: (B) Thus, if a stone falling upon the watch break it to pieces, as, when the watch was made, there was no new subst­ ance produced - all the material parts (as the steel, brass, strong, &c.) being pre-existent somewhere or other (as in iron and copper mines, in the bellies of those animals of whose guts men use to make strings) - so not the least part of the substance of the watch is lost, but only displaced and scattered; and yet that portion of matter ceases to be a watch as it was before: BPP 46. 2.5311 Ultimately, therefore, both for Boyle and for Abelard at the present state of his intellectual musings, there is presupposed a constant stock, as it were, of object-parts wh­ ich are rearranged by generation (or increase) or by corrup­ tion (or decrease). Exactly what it is which comes to be (or to increase) or to perish (or to decrease) is a matter of what is said, and this need not, except where the ultimate and unrevisable descriptions are concerned, impinge on 'the least part of the substance' of things (as Boyle puts it). The incr­ easing and decreasing objects countenanced by pretheoretical human considerations already have had difficulty in surviving Abelard's theoretical scrutiny, given his principles that every manyness makes a whole, and that no thing ever has more or less parts. The present reductions of increase to conjunction and of decrease to disjunction leave room only for diversity of place, so that increase and decrease come to resemble the processes of condensation and rarefaction respectively. Rar­ ity, density, smoothness, and roughness, are all in fact ment­ ioned by Abelard in the controversy as to whether these are qualities or positionally-connected attributes: AD 1 0 0 - 1 . 2.532 Yet whatever the experts may say in their theories, 126

Abelard and his Contemporaries there still lingers the pretheoretical impression that somehow or other increase involves an increase in the number of parts (in some as yet unclear sense) and decrease conversely invol­ ves a decrease in the number of parts. The central passages which now follow may well be in the nature of a final revisionary interjection, designed to do more justice to that impr­ ession, since the 'therefore 1 of the closing part of the whole passage (2.55(A)) would intermesh much better and more relev­ antly with the last-quoted section of Abelard's text (2.53(A)) than with the material now to follow. Certainly Abelard's way of working, as described by the learned editor of the Dialect­ ica (AD XXII - XXIII), as well as the change of accent in the direction of the discussion, would cohere with the supposition of such an interjection. 2.54 However, leaving aside such remoter considerations and conjectures, we may revert to Abelard's text. It certainly does at first sight appear to be moving towards a reconcil­ iation with the pre-theoretical view. Increase, it is said, does involve the having of more parts, so that the whole has more parts than the parts each have. This having of more parts must, says Abelard, apply to objects which have parts in common, otherwise we are not dealing with increase. Thus of two men, one may have more parts than the other, but it would be improper to say that we are hence dealing with a case of increase. In contrast, a whole has more parts than its part, but the whole has parts in common with its parts. (Verbally, this appears to be an anticipation of a central clause in the contemporary definition of collective class: D2 of 10.31.) Indeed, he continues, the whole has all the parts of an indiv­ idual part and some extra ones as well. This excess is said to constitute the essence of increase: (A) This question should be taken quite straightforwardly, and it should be asserted that the increase takes place in the composite with respect to the simple parts, since the former has more parts. It contains all the parts of the 127

Section 2 simple part plus some more, in the way that a sextet cont­ ains all the parts of a quartet and the parts of a couple in addition. Hence it is rightly said to be greater in respect of an increase whether in relation to the quartet or the couple which it contains. Now 'greater' or 'less' are not always properly understood in respect of an increase. The only proper meaning of 'incr­ ease' involves one thing's having more quantitative parts than another. 'Having more parts' here means 'Embracing all the parts of the other, and having some more in addition', and this can only be encountered in a part-whole situation. Herein necessarily all parts of the parts are parts of the whole, and every whole is also the whole of the sum of its parts. For as all the elements of each part also go to make up the whole itself, the whole must have quantitatively more parts than each of its several individual parts, that is to say, it embraces all those parts and has certain others as well - and this amounts to having more parts in addition. One thing cannot have more parts in addition than another when none of those parts is had in common. Thus when one man is said to be bigger than another, this cannot properly be said in respect of additional parts, since the two have no common parts, but rather in respect of the excess of dimens­ ion, because this man stretches further in height or breadth than the other. Hence also, he who interprets increase as implying a greater degree of extension, as when some taller man or other is credited with an increase realative to some shorter man, interprets 'increase' in a grossly improper sen­ se. For being increased, properly speaking, involves a grow­ ing of the thing itself by the adjoining of some quantity. (Sed simpliciter quidem accipiendum est et dicendum augment­ um esse in composito respectu simplicium, cum plures habeat partes. Easdem enim omnes et plures continet, sicut senarius omnes partes quaternarii et insuper binarii. Unde maior sec­ undum augmentum, sive hoc quaternario sive illo binario quem 128

Abelard and his Contemporaries continet, recte dicitur. Non autem semper 'maius' et 'minus' proprie secundum augmentum accipimus. Proprie namque tantum augmentum accipimus in eo quod in partium quantitate suarum ab alio abund­ at. In partibus autem 'abundare' dicimus 'omnes alterius partes comprehendere et insuper aliquas habere'; quod quidem in toto ad partem tantum poteris inspicere. Omnes enim par­ tes partium totius esse necesse est et omne totum totius partium quoque totum est. Cum autem omnia cuiuslibet partis membra ipsum quoque constituant totum, necesse est totum a singulis partibus suis in quantitate partium abundare, idest easdem omnes et insuper quasdam habere, quod quidem est plures in augmento partes habere. Non enim plures partes sec­ undum augmentum hoc habent quam illud, cum nullae fuerint communes. Unde cum quislibet homo alio maior dicitur, non secundum augmentum partium dici potest proprie, quia nullas dimensionem, habent communes partes, sed secundum excedentem quia scilicet extensa est magis longitudo huius aut latitudo quam illius. Quodsi quis etiam secundum dimensionis excellentiam augmentum accipiat, ut quislibet longior homo respectu brevioris augeri dicitur, multum improprie augmentum accipit. Augeri namque proprie dicitur per adiunctionem alicuius quantiatem massamque ipsam substantiae crescere: AD 422.6.30.) 2.541 At first sight, provided a gloss of the sort furn­ ished in the introductory summary (2.54) is accepted, there is little here to which even pretheoretical commonsense might take exception. Again, it is interesting to be able to call attention to Abelard's mention of 'common parts', not only for the already-noted anticipation of the collective class defin­ ition (D2, 10.31), but also because this is yet another Abelardian passage (cf. 2.431) which ensures that we are dealing with non-discrete objects which are nevertheless taken as multiple when we speak of a whole and any one of its parts: cf. also 10.336. This, as earlier pointed out in 1.5, is at 129

Section 2 loggerheads with a common medieval practice of denominating the separately numerable as 'discrete'. Of course, it would be very easily possible to raise distracting questions here about what is to count as a part, how such parts are to be counted, possible confusion between part and component, and the admis­ sion of infinitely many parts making nonsense of talk about 'more' or 'less' in this context. The ps-Joscelin's dealings with 'atoms' and 'points' in Fragmentum Sangermanense show that he might be ready to take some such matters into consid­ eration in contexts such as the present one, and that an at­ omic mereology, nowadays available (LEM), would thereby have an interesting field for analytic exploitation. But all this would be an unnecessary distraction in the elucidation of this present passage, which already is to present its own obscur­ ities. For although Abelard declaims strongly at its close against those who wrongly talk about increase in respect of two distinct objects such as the two men, one of whom is bulkier than the other, his own theoretical position could amount to a precise analogue of that two-man situation, were it not for the common-part qualification. What we ordinarily call an increasing object must, on Abelard's principles, be a succession of distinct but generally bulkier objects, every part of that succession having a part in common with the increasing object. 2.542 Again, in this last passage Abelard has already been using 'proprie', 'properly', to accentuate two diverse aspects of increase. First he called attention to the having of more parts, but then, towards its close, he stresses that purely adjunctive aspect which was so prominent in his preliminary query as far as this Dialectica passage is concerned (2.53(A)), and which is to return to prominence yet again further on, with the revived exclusion of the having of more parts (2.55(A)), once this present central section is done with. In the meantime, the purpose of the example of the expansion of a rectangle by the addition of its gnomon seems to be an 130

Abelard and his Contemporaries attempt to unite the two 'proper' aspects mentioned, i.e. both the more-partedness and the adjunctive. (The gnomon is the L-shaped triplication of an original rectangle whence is constructed, by due adjunction, a larger rectangle proport­ ionally similar to the original.) The process may be imagined as being prolonged so that the original rectangle becomes more and more enlarged, thus providing sortai continuity to the successive outcomes. Abelard, however, seems ultimately to prefer not to say that it is the original rectangle which is being enlarged, but merely that increased rectangles are being brought into being. This is doubtless because of his constant principle, shortly to be repeated yet again, that no thing has more or less parts at one time than it has at the other: Nulla .... res plures habet in aliquo tempore partes aut pauciories quam in alio (2.551(A) below). He also repeats his previous assertion (2.54(A) above) that each successive whole in an increase-situation contains all the parts of each of its parts taken severally, and also all the parts of all the rem­ aining parts: (A) For example, a rectangle is said by Aristotle [Categ. 15a 30-1] to grow by composition, that is to say the adj­ oining, of its gnomon, and when the latter is disjoined one has a diminution of that rectangle. For there are more parts in the rectangle when the gnomon has been adjoined than in the rectangle which remains when the gnomon is disjoined. The nature of a gnomon is such that it has three [identical] rectangular parts [arranged L-wise] so that when a fourth such rectangle is juxtaposed [within the angle of the Lshape] a complete rectangle ensues Now it is this [larger] rectangle which, either in respect of that rectangle which is added (i.e. its quarter) or in respect of the gnomon adjoined to the latter, is said to be increased. For it is not the case, as some people think, that the increase is to be credited to that [original] quadrilateral to which a four­ th part is [three times] adjoined, but to the composite form131

Section 2 ed from the former and the gnomon. For, as already asserted above, no part is increased in the constitution of anything, but rather the composite grows in respect of its components, and this is because the [composite] whole contains that which each of its [several] parts contains, and also, in addition, whatsoever any of the other parts contain.(Ut quadrangulum crescere Aristoteles dixit composito, idest adiuncto, gnomone, quo etiam subtracto diminuitur. Plures enim sunt partes in quadrangulo adiuncto gnomone quam in eo quod relinquitur de quadrangulo separato gnomone. Consistit autem gnomonis perfectio in tribus quadranguli partibus, quibus si quarta adiungatur, perficitur quadrangulus Qui quidem quadrangulus respectu illius quadranguli qui est adiunctus, quartam partem vel gnomonis cui adiunctus est, augerі dicitur. Neque enim, ut quidam putant, augmentum referendum est ad illum quadrangulum cui quarta pars adiungitur, sed ad comp­ ositum ex ipso et gnomone. Non enim, sicut diximus, aliqua pars in constitutione cuiuslibet augetur, sed ipsum composit­ um respectu componentium crevit, cum videlicet totum contineat quod unaquaeque pars comprehendit atque insuper quicquid quaelibet aliarum partium continet: AD 422.30 - 423.7.) Here, at the close, the otherness of parts to which reference is made is probably an allusion to the newly adjoined parts, although it could also be taken to allude to the new combinations of possibly non-discrete parts created by the adjunction. 2.55 We next revert, after this perhaps revisionary inter­ jection (cf. 2.532), to the mainstream of the contentions of the present passage, i.e. the Boyle-type accent on disjunction and conjunction of already-extant parts. The having of more parts has, in the last two passages, been interpreted as the production of successively bulkier objects, each of which is a whole having more parts than each of its constituent parts. (HLM III §8 deals in more detail with this phase.) Now it is to be the conjunction of previously-scattered parts which are 132

Abelard and his Contemporaries seen as the ultimate basis of increase. The example of the change of place of the two men, in the passage next to be quoted, is exactly what is required to underpin Boyle's point that nothing substantially changes, as far as reality is conc­ erned, and no matter what familiar usage may say. Descript­ ions of transpositions of already-existing parts are now the privileged and unrevisable descriptions, so that the world is at last made safe for the favourite intellectual activities of the 'speculative geometers' of whom Descartes speaks so happ­ ily in Meditation VI. Here is Abelard's version of the Boylean position: (A) For any composite to be increased, it must have more parts in some place than it had before in that particular place, and to be diminished involves having less than before with the result that increase involves capacity and bringingtogether in the same place, and not merely a [greater] plur­ ality of parts. For its parts are not more, but the compos­ ite has more parts in this place than it had before. Thus, should there be two men, one inside a house and one outside, those two are a composite in relation to each, and if the one who is outside is brought into the house, then that com­ posite has more parts within the house than it had before, but it has not on that account more parts than it had bef­ ore. (Est igitur augeri quodlibet compositum plures partes ipsum habere in aliquo loco quam prius habere t in eodem, et diminui pauciories habere quam prius, ut secundum capacitatem et aggregationem in eumdem locum augmentum consistat, non tantum secundum pluralitatem partium. Ipsae namque par­ tes plures non sunt, sed ipsum compositum plures partes hab­ et in hoc loco quam prius. Veluti si duo sint homines, unus intra domum et unus extra, illi duo unum sunt compositum ad singulos et si exterior in domum abstrahatur, idem compos­ itum plures partes in domum habet quam prius; sed non ideo plures partes quam prius: AD 423.8.16.) 2.551 Although one must be grateful for this superficial 133

Section 2 clarity of exposition, Abelard's own distinctions can be brought to bear against it. This accent on change of place of parts as the ultimate feature of increase veils the vital distinction, common to all mereological insight across the centuries, between X-parts and parts-of-X (cf. 1.4, 2.3). In the foregoing passage the talk is of parts in a vague, gener­ al, quasi-componential sense, without any underlining of the change of status from X-parts to parts-of-X when the parts become incorporated into X. What is presupposed here (and in the next extract from Abelard's text) is a disjunctive sense of 'part' which goes well with the notion of a previously ex­ isting component, i.e. as either-X-part-or-part-of-X. It is this which facilitates the accent on the 'sameness of parts' both before and after their incorporation into the whole, while suitably combining that accent with another reminder of the non-existence of the composite (and hence also of partsof-the-composite) before the whole-generating incorporation: (A) But the thesis that increase is to be taken with ref­ erence to the capacity of a place and conjunction [therein] not only goes against usage, but also against authority. Hence when it is said that, 'If something is added to something or other...', i.e. 'adjoined to something', as when we say: If something is added to something or other, then a greater whole is made, then this must not be taken as meaning that the composite is made bigger than it first was, but rather that it has been made bigger than the individual parts by the adjoining of any one of them; for previously the composite just did not exist. And if other parts which previously existed are inv­ olved, the number of these will not be diminished; they were the same parts without the conjunction as they were when conjoined. Hence the comparison embodied in increase is not that of the composite with its [earlier] self, but rather of the whole with its individual parts. Hence Boethius [in the 134

Abelard and his Contemporaries above-quoted maxim] rightly incorporates the word 'whole' [as opposed to saying 'it is made bigger']. For no thing has at some time more or less parts than at some other time. When more parts come about through a change of substance by inc­ rease, this substance cannot be said to be or to have been that which had fewer parts. (Quod autem secundum loci capacitatem atque coniunctionem augmentum accipiatur, non solum usus, sed auctoritas contra dicit, cum dicitur, 'si quid cuilibet rei sit additum ...', hoc est, adiunctum in aliquo. Unde cum dicimus: Si quid cuilibet rei sit additum totum maius efficitur, non ita est accipiendum ut maius fiat compositum quam prius esset, sed maius quam singulae partes effectum est per adiunctionem cuiuslibet ipsarum; prius namque compositum non erat. Et si aliae partes quae prius erant accipіantur, non erit numerus earum diminutus; eaedem enim extra coniunct­ ionem erant partes quae intra. Fit itaque augmenti compar­ ado non de composito ad se ipsum, sed de toto ad singulas partes. Unde recte 'totum' Boethius apponit. Nulla enim res plures habet in aliquo tempore partes aut pauciores quam in alio. Ubi enim plures fiunt partes mutata secundum augmen­ tum substantia, non potest haec substantia dici esse vel fuisse illa quae pauciores habuit: AD 423.18.32.) 2.552 The penultimate sentence of this extract, expressing as it does Abelard's constant principle of the unchanging num­ ber of parts in an identical object is practically a wordfor-word repitition of that statement of his common principle which occurs in the gloss on the Categoriae: Nulla enim res uno tempore plures habet parts quam alio, GA 300.20.21; cf. 2.52(A) above. The passage as a whole comes pretty near to being in some ways a stronger version of Boyle's understand­ ing of the axiom, Corruptio unius est generatio alterius, et e contra, 'The perishing of one thing is the coming into being of another, and vice versa', as it occurs in the following 135

Section 2 contention of the seventeenth-century philosopher: (A) These and the like examples may teach us rightly to understand that common axiom of the naturalists, 'Corruptio unius est generatio alterius, et e contra'; for since it is acknowledged on all hands that matter cannot be annihilated ... and since also the coalition of any competent number of these parts is sufficient to constitute a natural body.... it can scarce be otherwise but that the same agents that shat­ ter the frame or destroy the texture, of one body will, by shuffling them together and disposing them after a new man­ ner, bring them to constitute some new sort of bodies ... : BPP 46. 2.553 Here Boyle seems to be insisting that a diversity of sort must ensue in order to fulfil the axiom quoted. This scarcely seems a necessary consequence of his position, nor is it a necessary interpretation of the otherness mentioned in the axiom. The object which perishes may well be replaced by another object of the same sort. Indeed, Abelard's examples of the increasing rectangle and the diminishing house concern what are, on the destructivist theory which seems to be Abelard's, cases of the perishing of the one individual of a given species and its replacement by another of the same species. Further, as the ps-Joscelin of Soissons has pointed out (2.452 above) such a counter-commonsensical and destructivist view of change is by no means necessary, and this scarcely con­ stitutes a reinforcement of Boyle's insistence on successive sortai diversity. 2.56 The close of the lengthy extract from the Dialectica which is now undergoing scrutiny contains an argument which prolongs the point made in 2.55(A) above. This was to the effect that no increase in the number of parts is required for the increase of an object. Provided that the sense of 'part' is disjunctive, i.e. it is taken as 'either an X-part or a part-of-X', then the increase can validly be described as taking place merely by the bringing together of such parts 136

Abelard and his Contemporaries into their appropriate places. What this next quotation in effect shows is that if one opts to take a 'thing' to be a thus disjunctively-specified part, then one can also claim that the composition which contributes to an increase does not bring more things into being. But if one still recalls that increase means more parts (as in the example of the quadrilateral) then it looks as though something odd ensues, i.e. more parts are required, but more things cannot be in­ volved. In fact, of course, it is the tomfoolery with the presupposed sense of 'part' which is at the bottom of the oddity: (A) But it would seem that the composition of things can accomplish nothing towards an increase, since more things are not brought into being by the conjoining. For if you adjoin one thing to three things, their number does not seem to increase, except perhaps with reference to place, in the sense that there are more substances in this place than there were before, when one was not aggregated with the other at this place. Or perhaps, where those things which do not require conjunction, but which are naturally discrete, are concerned, as in the case of manifolds, composition effect­ uates nothing in the nature of an increase, as opposed to what holds for those which come into being by being juxtap­ osed, as in the case of the quadrilateral which is the due juxtaposition of gnomon and quadrilateral. For the gnomon and the quadrilateral would not together yield a quadril­ ateral, were the two former to remain discrete. Hence juxta­ position seems to be involved in the increase of the quadr­ ilateral, without at the same time involving an increase in the number of things. (Videtur autem compositio rerum nihil ad augmentum perficere, cum non sint plures res per coniunctionem effectae. Si enim unum tribus adiungas, non videtur eorum numerus crevisse, nisi forte secundum locum, quod sc­ ilicet plures in hic loco sunt substantiae quam prius, cum nondum unus duobus huc esset aggregatus. Ac fortasse in 137

Section 2 his rebus quae coniunctionem non exigunt, sed naturaliter discretae sunt, sicut numeri, nihil operatur ad augmentum compositio, sed in his quae per continuationem fiunt, ut in quadrangulo quod ex gnomone et quadrangulo coniungitur. Non enim quadrangulum unum redderet gnomo et quadrangulus, si discreta manerant. Unde ad quadranguli augmentum coniunctic quoque videtur operari, non ad numerum rerum: AD 424.32 424.5.) 2.561 Having indeed got himself into the position described at the beginning of this extract, Abelard would seem to be faced with the possibility of change of bulk across time which is not mere condensation or rarefaction, and yet which does not incorporate change in the number of things involved. This is so because of the above-mentioned presupposed inter­ pretation of 'thing' as 'either an X-part or a part-of-X'. Whence would ensue identity across time, whereas his theory would at other points deny it. For example, in the course of his exposition of the argument for a succession of houses as bits drop off the original (2.443(A)) he explicitly and rightly has opposed any insistence on such continued identity precis­ ely when things get to the stage of a lack of house-like com­ position of the dissociated parts (e.g. a former part-of-thehouse has now definitely become a house-part when it drops off the main body of the house). Now here at the close of the present discussion a like stress on conjunction or comp­ osition has come into its own once again, at least as far as essentially-structured aggregates such as a house or the increasing quadrilateral are concerned (as contrasted with loosely-structured pairs and trios of men, for example: 2.51, 2.7). But at the same time his final apparent elimination of increase in the number of 'things' in such conjunctive cases is quite sophistically based on the continued use of the dis­ junctive sense of 'thing'. When, as here, the 'things' are parts, and their comparative numeration is in question, then an at least triple ambiguity is hidden in the undertaking, 138

Abelard and his Contemporaries since 'parts' now takes in either parts-of-X or X-parts or parts-of-X-or-X-parts, with correspondingly diverse results. In the latter, the disjunctive sense, the number of thingparts abides (and Boyle seems inclined in this direction). Where 'part' means X-part> these decrease in number as they become incorporated (e.g. in the increasing rectangle of Abelard's example). Where 'part' means 'part-of-X', these increase in number as the ex-X-parts, duly incorporated, are transmuted into parts-of-X. Such, then, are just a few of the ambiguities at the bottom of Abelard's inconsistencies. These few, howevever, are those of most mereological interest. 2.6 The Temporal Dimension 2.61 A further prolongation of the difficulty experienced by Abelard in coming to terms with literally changing objects (increasing or decreasing) which endure through time, is to be found in his puzzlement when dealing with temporal wholes in a rather specialised sense, i.e. such 'things' as days, weeks, years, and so on. Fortunately, in his Glosses on the Categoriae he makes it clear that he is not dealing with time in the most abstract of senses, but rather with the durations of concrete objects. True, he finds fault with what he calls the common verdict (communis sententia) according to which each object has its own time, but this is only in order to sugg­ est that there is one time which is still correlated with the world of temporal objects as a whole: GA 184.38 - 185.10. This at least suggests that he would still connect temporal nomenclature with the before and after, the earlier and later time-segments, of concrete objects. Hence when, as in the passage next to be quoted, his discourse at first sight app­ ears to be about disembodied days, hours, and so on, this could be a false impression. At any rate, as we are to see in due course, there is little difficulty in transposing his prob­ lems about such temporal topics into the field constituted by the temporal parts and wholes of concrete objects. His leng139

Section 2 thy passage on temporal wholes in the Dialectica follows imm­ ediately upon the verdict on 'principal parts' (2.4 above). Here are the first few lines: (A) I shall now take a look at the nature of certain parts and wholes which appear to have features which are incom­ patible with those described above, even though such wholes are admitted to be of the same kind in so far as they are said to be integral. These are temporal wholes, such as this day, which is said to be made up of these twelve hours, and to be their collective whole. But this whole is alleged to exist in a fashion opposed to that applicable to other int­ egral wholes, in that [i] any of its parts suffices to establish the whole, and [ii] the destruction of the whole destroys all of its parts. For if the first hour is, then the day is said to exist; wh­ ence also, if the day were not, its first hour would not be said to exist. (Libet quorumdam totorum et partium naturam inspicere quae superioribus adversa videtur, cum quibus tamen eiusdem proprietatis in eo conceduntur quod integra dicuntur, ut sunt tota temporalia, velut haec dies quae ex his duodecim horis componi dicitur et ad eas totum constitutivum esse. In hoc autem toto econtra dicitur esse quantum ad al­ ia integra, in eo scilicet quod quaelibet pars eius ipsum ponit et ipsum destructum quamlibet partium suarum destruit. Si enim prima est, et dies esse dicitur, unde et si dies non fuerit, prima esse negabitur: AD 553.8.15.) 2.62 Leaving aside the remainder of this highly significant and important passage until 2.67 below (cf. HQS §4.53 for a full serial comment) it suffices for the moment to note that Abelard himself has provided material for the solution the difficulties concerning the theses numbered (i) and (ii) ther­ ein. Let us first remind ourselves of the basis of his qualms. In the case of (i) he is worried because he believes 140

Abelard and his Contemporaries t h a t this is, if taken generally, the false converse of the following thesis which he has already accepted and which was discussed above in 2.21(A), i.e., (iii) The existence of the whole entails the existence of any part thereof: AD 343.34.35. Now he certainly wants to retain (iii) since its equivalent, i.e. t h e non-existence of the part entails the non-existence of the whole (AD 346.31.34, 348.26.27) forms the basis of the paradoxes of destructivism which have been noted above. At the same time we are already in possession (cf. 2.32(B)) of Abelardian material which revolves around the already-noted thesis: (iv) If something is denied of the whole, then it is denied of all the parts taken together: AD 344.22.23. It was in connection with this thesis t h a t he noted how parts may still in some way subsist, even after the existence of the whole is denied, e.g. walls, roof, and foundations may somehow (e.g. qua house-parts) still abide, although not app­ ropriately joined together to make a house: 2.32(B) above, AD 344.33 - 345.8. In (ii), however, we have the expression of a case in which denial of the existence of the whole is incomp­ atible with the continued existence of the parts which pertain to that whole; in no sense whatsoever, not even as, e.g. dayparts, can they abide. 2.63 In point of fact the pervasive distinction, twice enl­ arged upon above (1.4, 2.3), between X-parts and parts-of-X, of which Abelard is well aware, can be brought t o bear to render all of (i), (ii), (iii), and (iv) true, and to remove his qualms in connection with any of these. We have seen how, if t h e r e be a part-of-X, then there must be an X for it to be a part of; whence (i) is generally true; no longer need it be seen as the possibly false converse of (iii). It is of course now understood t h a t 'part thereof' in (iii) must exclusively take on the sense of 'part-of-X', since otherwise it or its 141

Section 2 equivalent which states that the non-existence of the part entails the non-existence of the whole, can easily be given false instantiation. The non-existence of a certain car-part (e.g. an appropriate bumper) by no means entails the non­ existence of my car. Likewise (ii) is perfectly in order provided that parts-of-X are in question. It does so happen that in certain cases, X-parts remain about the place after the demise of X, as in the instance of Abelard's disjoined house-parts, noted above (2.32(B)). This need not affect the truth of (iv) at all, as Abelard is well aware, when parts-ofX are in question. 2.64 The equally interesting passage on time in Abelard's Glosses on the Categoriae covers somewhat the same ground, initially, as that which figures in the Dialectica passage broached above. However, the Glosses can here be interpreted as alluding adversely to his own position in the Dialectica: cf. note 2 of GA 186. Certainly, as the passage already quo­ ted from the Dialectica (2.61(A)) shows, he there puts the ac­ cent on the difference between temporal and other integral wholes. Now, however, he tends to wish to avoid such a thes­ is of differentiation: (A) But even as time, as a subject of discourse, is meas­ ured in a way diverse from that appropriate to the other qu­ antities, as we have shown above, so also it is a continuum in another fashion, i.e. as involving a succession of trans­ itory parts. And on this basis, time is said by some people to differ in nature from other integral wholes. For [they, like him in 2.61(A) above, say that] the persistence of other wholes requires that their individual parts should exist, and that it is not the case that when just one part exists the whole also necessarily abides. It is also accordingly agreed that upon the destruction of any one part the whole must al­ so be destroyed, but that there is no need for the part to be destroyed when the whole has been eliminated [cf. (i) and (ii) of 2.61(A)]. Yet they insist that the opposite of all 142

Abelard and his Contemporaries this applies to that composite which is time. For [they say] there is no necessity that the first hour should exist, given the existence of a day, but rather the opposite; neither is there any necessity that the day should not exist when the first hour does not exist, but rather the opposite. This gives them further ground for separating out the nature of that composite whole which is time from the natures of all other composites, and they declare that all those features which authorities in logic attribute to wholes in general apply only to those wholes made up of abiding parts. (Sed sicut diverso modo a ceteris quantitatibus tempus mensura tur subiectum, sicut monstravimus, ita et aliter continuum est, per successionem scilicet transi toriarum partium. Et insuper a quibusdam dicitur aliam naturam habere ab aliis integris. Cum enim ceteris totis permanentibus necesse sit singulas eorum partes existere, sed non una existente parte necesse est totum manere. Unde etiam illud constat quod una qualibet parte destructa necesse est totum destruí, sed non destructo toto partem perimi. E contrario dicunt contingere in hoc composito quod est tempus. Non enim, cum dies est, nec­ esse primam esse, sed e converso, neque cum prima non est, non necesse est diem non esse, sed e converso. Unde etiam naturam huius compositi quod est tempus, a ceteris dividunt et ista quae auctoritas in totis generaliter assignat, ad ea tantum tota reducunt quae permanentes habent partes: GA 186.36 - 187.8.) 2.65 Thus far, points maintained in the Dialectica text are here being recounted as something on which Abelard would cast doubt. Now follows his own verdict as far as his present Glosses go. It has features not at all present in the Dial­ ectica argument, especially the alternative construal of exis­ tence-claims in respect of the day as being mere figurative locutions having parts of the day as their quasi-referents. Thus a distributive rather than an integral whole (cf. 2.21(A)) is then in question, with parts-of-the-day as the distributed 143

Section 2 or diffused elements. In the course of the discussion a very firm grasp of the distinction between X-parts and parts-of-X is yet once more being displayed. He here certainly, yet ag­ ain (cf. 1.42(A)), anticipates by well over a century, Walter Burleigh's stress on how the use of the genitive case in the Latin version of 'parts-of-X' can be a pointer to discourse concerning integral rather than distributive wholes (cf. 1.4). The text continues accordingly: (A) We, for our part, do not agree in allowing some comp­ osite to exist when even just one of its parts does not ab­ ide in existence. For given that 'day' is just the name of the twelve hours it must be the case that when they make up a day, all the twelve hours are shown together; for indeed a day is asserted to be precisely those twelve hours. But how can such a multiplicity taken together be one thing wi­ thout each of its elements existing? Or how, when they do not make up a unity, can they remain all together? Thus he who takes the day to be the subject of his discourse and as­ serts 'The day exists', must [absurdly] include the existence of the first hour [for example] in his understanding [of what he is saying]. If, however, he is taken to be speaking only figuratively, so that he means to say, 'A part of the day exists', there would seem to be no necessity that that part should be the first hour; indeed, part of the day embraces also each of the individual parts after the manner of a un­ iversal whole [as opposed to an integral whole; cf. 2.21(A)], in the way that man comprises Socrates. Hence in the same way as it is not necessary, given that a man exists, for that man to be Socrates, so also given that a part of the day ex­ ists, it need not be that first hour. Exception might be taken to this contention, however, were we to put the str­ ess on our actually uttering '...of-the-day'. Under such conditions how can something be said to be part of-the day or part of-the house unless the house or the day does act­ ually have the part? But how are they each to have the part 144

Abelard and his Contemporaries in question unless they actually exist? Yet how are they to exist unless, as already noted, the existence of each and every one of their parts is granted? Hence also it appears to be the case that some part or other of-the-day cannot exist unless the first hour also is, and so on in respect of any of the others, if we persist in stressing that it is part of-a-day that is in question, i.e. that we are here dealing with a relation [between actual parts and their whole] rather than with the quiddity of the thing. (Nos in hoc non cons­ entimi, ut velimus aliquid compositum existere umquam una eius parte non permanente. Cum enim 'dies' nomen sit duo­ decim horarum, oportet cum dies fiunt, duodecim horas simul ostendere, quippe nil aliiud dies dicitur quam illae duodecim. Quomodo autem plura simul aliqua possint esse, nisi unumquodque eorum sit, aut quomodo, cum unum non fiunt, omnia simul poterunt permanere? Si quis ergo diem subiciat, dicens: 'dies est', oportet et primam esse intelligi. Si vero figuram in locutione faciat, ac si diceret 'pars diei existit', non videtur necessarium, ut prima sit, quippe pars diei ceteras quoque singulas partes continet quasi totum universale, sicut homo Socra tem. Unde non necesse est homine existente Socra tem esse, nec similiter parte diei existente hanc esse, nisi forte vim faciamus in eo quod dicimus 'diei'. Utquid enim dici diei vel domus pars poterit, nisi domus vel dies partem habeat? Quomodo autem partem habebunt, nisi sint? Quomodo autem erunt, ut dictum est, nisi unaquaque partium existente? Itaque id quoque verum videtur, ut pars aliqua diei esse non possit, nisi et prima sit et quaelibet, si vim faciamus in eo quod partem diei dicimus, in relatione scilicet, magis quam in essentia rei: GA 187.8.26.) 2.66 This is an excellent explicitation of two of the poss­ ibilities here. One may treat discourse about a day as integ­ ral-whole discourse, with stress on the genitive case of the locution 'part-of-the-day' (cf. 1.41 above). This makes the discourse mereological, covering part-whole relations. On the 145

Section 2 other hand allusions to parts of the day can be taken distr­ ibutively, i.e. allusion is being made to either this, that, or the other part of the day, without any necessity that the whole in question should actually be there. We here have discourse based on the 'distribution' of the essence or quid­ dity, i.e. of what is involved in being a day or part of a day, rather than on a concrete integral whole called 'a day' or on its concrete parts. It is for the quidditatively based inter­ pretation that Abelard now opts in this context. However, this option, he goes on to claim, then makes the integralwhole interpretation merely figurative. He fails to remind us, however, that this verdict will make a great deal of other discourse about temporally-extended physical objects also fig­ urative, and that this is an undesirable consequence. If so much becomes figurative, what then is to be literal? Here is his verdict leading to this consequence, and on the whole matter, insofar as the Glosses on the Categorise are conc­ erned: (A) Our decision at this point is that we are just dealing with [the distribution of] the quiddity of the part, and that when we say that a part of the day exists this amounts to alluding to either this stretch of time which we call the first hour, or that other stretch of time .... and so on; hence it is <not> necessary that the whole [day] exists. We do not in that case put the accent on the relation between the part and the [integral] whole, but rather on the quiddity of the thing taken purely intrinsically, as we would be doing were we to assert that if walls exist it is not necessary that a house should exist. Hence as far as the theory of time [as an integral whole] is concerned we deny that if some part-of-the-day exists (in the sense of either the first or the third or some other of the hours) then [the integral whole exists and hence, e.g.] the first exists [as the genitive 'part-of-the-day' might suggest] . And this confirms that it is just a customary figure of speech which 146

Abelard and his Contemporaries is being used when it is asserted that the day exists, for it amounts to saying that some part of those parts which per­ tain to a day exists; neither is it day which is the sent­ ence-subject, but rather a-part-of-the-day. Indeed, were one to take it that day is the subject, then [contrary to what is the case] never would theorems concerning parts and wholes of composite objects, of the sort based on the nature of other [integral] types of whole fail to apply to temporal wholes [as in fact they do fail, as 2.61(A) claims] . But my verdict is that men make up a figurative locution when they say that a day exists, for there is nobody who would give the name 'day' either to part of the day or to various times of day taken together, nor would anyone be so stupid as to judge that the many times of day all exist together simult­ aneously. (At vero nos hoc loco non nisi essentiam partium attendimus, cum dicimus partem diei existere, ac si diceremus: vel hoc tempus quod primam vocamus, vel illud, et cet­ era, unde <non> necesse est totum esse. Nullam vim facimus in relatione partis vel totius, sed in essentia rerum tantum,  si diceremus: si paries est, non necesse est domum esse. Itaque et istud negamus quantum ad essentiam temporum, quod si pars aliqua diei sit, id est vel prima vel tertia vel aliqua ali arum, prima est. Et haec quidem figura consuetudo sermonis, cum dicitur 'dies est', ac si diceretur: pars aliqua earum quae attenduntur in die est; nec dies subditur, sed pars diei. Si vero dies subicetur, numquam ceterorum totorum natura secundum argumenta totius et partis in compositis fallent temporibus. At figuram homines dicimus fig­ urare, cum dicunt 'dies est', quia nullus est qui 'diem' part­ em aliquam diei vocet sive plures simul horas  aliquis adeo desipit quod plures simul horas arbitretur existere: GA 187.25.40.) 2.67 The interpretation provided above of allusions to dies (the day, a day, day) as being really allusions to parts of the day, with these allusions being understood in a distrib147

Section 2 utive sense, is noteworthily absent from the main Dialectica passage, which concludes by extending the final part of the last extract's insistence on the figurative nature of disc­ ourse concerning such temporal wholes into an insistence on their being fictitious, and not real wholes, owing to the nonsimultaneity of their parts: (A) There never are many parts simultaneously in time, since there never exist many of them at the same time. Whence time is never made up of many parts, since neither the first nor the middle nor the last parts linger on; always only one of them is actually there. Nor can it ever properly be said that a day exists, although figuratively we might say that it exists through a part; that is, a part of the day exists. Yet even this is not properly a part, since it exists alone and is never compounded [into an actual and simultaneous whole]. For if 'day' is just the name of sev­ eral diverse hours, how can we properly say that a day ex­ ists unless the many hours continue in being? Every comp­ osite is the same as all its parts taken together; hence whatsoever can be attributed to the whole is also attrib­ utable to all its parts taken together, and conversely; again, to whatsoever the whole is attributable, all its parts taken together are attributable, and conversely. If, there­ fore, we are to stress the truth in this matter, such wholes should not be admitted. Philosophers should rather deal with them as if they were dealing with wholes, assembling into a pseudo-whole things which were past and things which are to be, so that they may show their nature, as if something might be made up from that which is and that which is not. Hence things which in fact are not wholes at all are for theoretical purposes viewed as if they [really] were wholes. {Plures autem simul partes in tempore numquam sunt, quia plures simul numquam existunt. Unde  tempus ex pluribus partibus umquam consisit,  prima scilicet manente  media  ultima, cum semper una tantum exstiterit. Nec 148

Abelard and his Contemporaries umquam dies esse proprie dici potest, sed figurative,  si per partem ipsum existere dicamus, idest partem ipsius existere, nec etiam proprie partem, quippe non componit, cum sola sit. Si enim 'dies' tantum diversarum horarum nomen sit, quomodo ipsum existere proprie dicemus, nisi plures horae permaneant? Omne enim compositum idem est cum omnibus suis partibus simul collectis; quare et quicquid toti attribuitur, et omnibus simul partibus, et econverso; cuicumque totum, et partes omnes simul convenient, et e converso. Si itaque rei veritatem insistamus, oportet ista tota non esse confiteri, sed tarnen quasi de totis philosophos de eis egis­ se, secundum hoc scilicet quod ea quae praeterita erant vel futura, cum eo quod praesentialiter est consideratione sua quasi unum colligebant, ut eorum naturam ostenderent, ac si ex eo quod non est aliquid esset. Quae itaque in re tota non sunt, secundum tarnen eorum considerationem quasi tota accipiuntur: AD 554.17.35.) 2.671 Once again Abelard here fails to remind us that exactly the same considerations apply to the the temporal aspect of spatial objects. If the non-simultaneity of the parts of a day makes the day into a figurative or fictitious object, then the same applies to quite banal temporally-ext­ ended objects such as this table. And yet again, if these considerations are thus extrapolated to common-or-garden everyday objects, so that they become figurative or fict­ itious, what, in the end, is non-figurative or non-fictitious? It is only in the prolongation of the last-cited text from the Glosses on the Categoriae that Abelard appears to go some way in the direction of an admission that parity of treatment may be extended to the objects of everyday life: (A) Hence as far as things in reality are concerned, it can never be truly and properly be said that a day exists, or that a whole [object] exists, or a quantity exists, or even that a piece of bronze exists, or that something or other exists in its entirety. However, we regard such a thing as 149

Section 2 if it were a true whole, and hence according to the mind's way of conceiving as though something permanent were in question, even though from the point of view of how things truly are, the passing by of parts is discriminable. For everything that the mind conceives is looked upon as if it were present. He who knows what is to come looks upon it as now thought to be present, and as if it were a thing already in being, thanks to the organisation effectuated by the mind. (Unde in rei ventate numquam vere et proprie dici potest dies esse vel totum esse vel quantitas vel etiam aes vel aliquid omnino esse. Sic tarnen ipsum attendimus, quasi vere totum esset atque ideo secundum animi conceptionem ut de permanente agitur, licet et secundum rei veritatem transitus partium distinguatur. Omne enim quod animus concipit, quasi praesens attenditur. Qui enim id quod futurum scit esse, considerat, iam ut praesens cogitat, et quasi ita iam sit res, animo constituit: GA 187.40 - 188.6.) This severely 'as if' passage, with its Kantian-style anticip­ ations, also comes pretty close to the type of discourse used by the eighteenth-century Scottish philosopher, David Hume, who speaks of 'fictions', 'mistakes', and 'errors' attributable to the generality of non-philosophical mankind in ascribing identity to objects (animal, vegetable, mineral) which really consist of a 'succession of related objects':   , VI (p.255). 2.68 More full-blown than Abelard's is the ps-Joscelin's appreciation of the fact that a spatio-temporal object such as a house, which endures through time, is likewise said to exist without any restriction of that existence to what is perceived during one short time-interval. Analogies such as that of of an army's being said to be up against a wall when only one section is thus engaged are used as further reminders of the legitimacy of such modes of speech: (A) So likewise I am said to touch a wall, not because every bit of me is applied to the wall; rather it is because of my touching it with perhaps only a fingertip that I am 150

Abelard and his Contemporaries said to be performing the touching. In like manner some army may be said to apply itself to a wall or to some area or other, not because each and every person in the army is up against the wall, but only some one or other of those belonging to the army. (Sic enim dicor tangere parietem, non quod singulae partes mei parieti haereant, sed forsitan sola summitas digiti, qua haerente dicor tangere. Eodem quoque modo exercitus aliquis dicitur haerere muro vel aliquo loco, non quod singulae personae excercitus illi haereant, sed aliquis de exercitu: /CALMI, 164*, lines 540-3.) 2.681 It is in this sort of way that a great many of the features of what Abelard calls 'temporal wholes', as described by him, could not only be seen in a less paradoxical light, but could also with justice be applied to temporal wholes in the larger and more general sense, such as the house. His unease on the score of his own 'temporal wholes' and their relegation to the realm of philosophers' fictions or merely figurative utterances, are appropriate companions to the destructivist type of theory from which he seems unable to extract himself, to judge by the passages observed above. 2.7 Master Peter's Mereology 2.71 Allusions have already been made to the mereological insights of the ps-Joscelin of Soissons (2.11). These were written down in a manuscript referred to by Victor Cousin in the nineteenth century as the Fragmentum Sangermanense but which is now Codex Paris, Bibliothèque Nationale lat. 13.368. A further copy of this material occurs in the manuscript cod­ ex Orléans, Bibliothèque Municipale 266, along with other log­ ical passages among which is the immensely rich Sententie Se­ cundum Magistrum Petrum, edited by L. Minio-Paluello in A/,

151

Section 2 whence extracts are to be drawn below. The fact that the psJoscelin and the Sententie, both of which contain highly sign­ ificant and important mereological discussions, thus occur to­ gether in the same codex is in itself already an indication of the high level to which studies of part and whole had already been carried in the twelfth century. 2.711 It is, of course, tempting to assume that the Master Peter who is said to be the author of the Sententie is indeed Master Peter Abelard. The learned editor in AI rightly prov­ ides reminders as to similarities of content and style which could tend to justify this assimilation: AI XLI. The general setting involves a defence of logical discussion of res, 'th­ ings' against those who claim that one must restrict oneself to the discussion of voces, 'utterances', if logical paradox is to be avoided. Such a defence does indeed reflect an Abelardian position of which echoes have already been encountered above in his refutation of one aspect of Roscellin's 'insane' championing of voces (2.34). However, to avoid a lengthy epi­ cycle lacking in mereological cost-effectiveness, this broader issue of the contrast between res and vox will not be pursued here. Chapter IV of Martin Tweedale's study (TAU) may profit­ ably be consulted on that issue. 2.72 We have already acquired a sufficient realisation of the highly general nature of Abelard's approach to appreciate that the apparent paradoxes which figure in the Sententie co­ uld well be grounded in that approach. At least one of the alleged paradoxes has its origin in an expression of what has been called Abelard's 'very extravagant claim' (TAU 151) that any pair of objects, no matter how separate or disparate, are together a third object. In 2.31(A) above we have seen how for him two human units who are scattered in Paris and Rome, and who make a pair, may constitute a case of this claim: AD 431.28.29; cf. AD 64.20.21, 576.5.7 and his preliminaries to his treatment of the problem of increase as described in 2.51. The testimony as to the debate between Abelard and Alberic of 152

Abelard and his Contemporaries Paris has been used above to underline the breadth of this claim (cf. TAU 150 - 1) which is also overtly stated as a general principle in the course of one of the already-enc­ ountered most weighty passages of the former on the inc­ rease-problem: Omnia .... plura simul accepta, sive continuata sive disiuncta, totum sunt ad singula, 'Every multiplicity, taken together, be it continuous or discrete, forms a whole in respect of its individual parts', AD 422.4.5. It is this very principle which is now to be attributed hypothetically to the adherents of res (amongst whom would presumably be Abelard) in the present Sententie: si rem ex re constare dicimus, et omnia plura unum totum efficere ... , 'if we assert thing to be made up from thing, and every many to constitute one whole ...': AI 114.13.14. This attribution forms the basis of an objection against the res proponents, and the main part of the section of the Sententie which deals with wholes is engaged in replies to this and other objections. The first of the objections posed on behalf of 'those who take "whole" to be a mere utterance' (qui 'totum' solummodo vocem esse confitentun AI 114.7.8), as against those who would deal with res, runs as follows: (A) [Objection 1] It is obvious that a six is made up of a four and its half (i.e. of the four). Yet this turns out to be false if we look at what holds at the level of things (res). For no self-subsisting four can make up a six by means of its own half. Another couple needs to be contr­ ibuted from the outside. (Constat senarium ex quaternario et dimidietate eius (scilicet quaternarii) perfici: quod falsum est si hoc in rebus conspiciamus; nullus quippe qu­ aternarius, qui per se est, per dimidiam partem sui senarium perficere possit nisi alter binarius extra addatur: AI 114.8.12.) The point of the objection is that a generally admitted num­ erical thesis goes wrong if we insist on interpreting it as though it were about things (res). Thus we take 'A six is a 153

Section 2 four plus half of the latter 1 to mean that a six-(thing) is made up of a four-(thing) and of its half (i.e. of two(things)). However, one sense of 'res' is concrete object, and it is obvious that combining four such with their half will not normally be taken to yield a set of six concrete objects. Hence the numerical thesis cannot be maintained if understood at the level of res, 'things', at least if these are understood as concrete objects. 2.73 Now there still subsists a certain vagueness in this attribution of the fault to the combining of res with res to make a whole, since there are many senses of 'res'. The Abelardian principle which is bound up with such senses, and wh­ ich states that every multiplicity makes a whole, might be in­ voked here, but in fact is only overtly stated and incontrovertibly used in the next objection. However, it will be adv­ antageous from an expository point of view to advance direct­ ly to the reply to the objection which has just been quoted, a reply which defends a res interpretation by rightly bringing forward highly significant distinctions in the senses of 'res'. In so doing it serves to locate the general general semantic possibilities of the situation in a most admirable and accur­ ate way. It was in order to circumvent any pre-emptive lim­ itation of those possibilities that 'senarius' and 'quater­ narius' have been non-commitally translated above as 'a six' and 'a four' rather than by definitively individual-level loc­ utions drawn from alternative nominal forms such as 'a sextet' or 'a six-fold manifold' or 'a six-thing', and so on. Indeed, the dual sense which the reply (on behalf of various senses of 're5') allocates to the pronoun 'its' (of the expression 'its half') in the following first section of the reply, confirms this requirement of translational non-commitment: (A) [First reply to Objection 1]. When the six is said to be made up of the four and its half, then the force of these words allows 'its' to be related to the name 'four' in two ways, i.e. as alluding either to individual objects or to the 154

Abelard and his Contemporaries species (or essence) of the four. This is because a relative expression can allude to its antecedent expression in either of two ways: (i) with reference to individual and as it were discrete objects, i.e. in such a way that individual sameness is in question, or (ii) without any individual import and just with reference to sameness of essence, i.e. with reference to the accord or resemblance of things at the level of the reason basic to the original introduction of the antecedent nominal expr­ ession in question. (Cum dicitur senarius perfici ex quater­ nario et dimidietate eius, istud 'eius' ad hoc nomen 'quaternarium' duobus modis referri potest secundum vim verborum, hoc est vel secundum personam vel secundum speciem sive na­ turam quaternarii. Duobus quippe dictio relativa ad premiss a refertur: modo quidem personaliter et quasi discrete (secun­ dum, scilicet identitatem persone), modo indifferenter  simpliciter secundum identitatem nature, id est convenientie vel similitudinis rerum iuxta causam inpositionis premissi nominis: AI 115.19.28.) The relevance of this distinction to the senarius-rblm is readily apparent in a preliminary sort of way. That problem need by no means scare us into adopting some sort of rival 'vox','utterance', solution. Remaining all the time on the side of 'res', one may realise that a six can be said to be made up of a four and its half not only when we are dealing with concrete individual objects (sense (0), but also in the truth-yielding manner of sense (ii), i.e. at a general level, revolving around what is involved in being six-fold, beingfourfold, being-half-of-fourfold, and so on. This level of discourse deals, as the passage says, with the essence or qu­ iddity of things, and has hence, in accordance with a mediev­ al usage, already been dubbed 'quidditative' in 0.4 above. 2.74 However, the deeper significance of such distinctions of level may be brought out in the manner already outlined in 155

Section 2 our 0.3 - 0.5 above. It is then possible to give a logicalgrammatical characterisation of the contrast being made in this reply to the first objection, using the Ajdukiewicz-style 'categorial indices' introduced in 0.3. It was there suggested that investigation of the status of quidditative discourse in non-nominalist (i.e. non-vox) style shows it to involve the use of verb-like functors as terms, with infinitives (e.g. 'to be ...') or participles (e.g. 'being ...') as their naturallanguage counterparts. Thus the terms of 'To live is to be' or 'Living is being' ultimately have the s/n index, though their own internal structures may be varied and complex, as HQS has shown. The '... is ...' of such sentences is accord­ ingly a functor which forms a proposition from two verbs (or verb-like expressions) and its index is thus s/(s/n s/n). With this may be contrasted the s/(n n) which pertains to the more familiar nominally-termed '... is ...' of propositions such as 'Socrates is literate', 'Marcus is Tully', and so forth. All the functors in question figure in or are definable in terms of the categorial language adopted in HQS and sketched in section 10 below. This provides an assurance that the comparatively loose and untechnical language of the present Sententie, as well as of the present comment, may in the end be rigorously analysed and backed up by co-ordinates display­ ing the highest degree of logical intelligibility. 2.75 Thus we now have points of reference for coming to terms with the multiple medieval fashions of expressing resdiscourse, be it at the concrete-thing level (as in (i) of the last-quoted passage), or be it at the level of quidditative discourse, as in (ii) of that passage. Indeed, given the lim­ itations of natural-language expression, the contrast between discourse at the s/(n n) level (wherein the names may name concrete objects) and that at the s/(s/n s/n) or quiddit­ ative level (wherein the terms are not really names at all) for the interpretation of '... is ...' could hardly be better stated than in the quoted passage. The expression 'personal­ 156

Abelard and his Contemporaries iter et quasi discrete*, ('individually and as it were discr­ etely') of the (i) of that passage not only anticipates the later medieval 'suppositio personalis' terminology for charact­ erising discourse at the nominally-termed level of '... is ...' (which comprises names alluding to individuals: CLM IV, 9) but also coheres with Abelardian usage such as that found in his Glosses on the Topica. There 'personaliter ab omnibus al­ ­­s rebus discreta' ('individually discrete from all other th­ ings') is used to point somewhat the same contrast as that which is now in question: DA 235.9.10. The twelfth-century compendium of Porretan logic (cf. 2.84 below) maintains the 'discrete' terminology in the same way on pp. 41, 48, 54, 84, and 90 of CC 46. In the present passage, therefore, a trans­ lation which has a concrete ring about it (e.g. 'A sextet is made up of a quartet and its half') would help to bring out the s/(n n) complexion of the false sense of the problemsentence. In contrast, 'Being sixfold is being a combination of the fourfold and its half', might help to point in the direction of the quidditative s/(s/n s/n) sense, which expr­ esses a truth. That the possibilities on the side of natural language still remain various, however, is confirmed by the present text's continuing to stress the distinction in further terms. 2.751 However, before going on to inspect such further expressions, it is worthy of note that our Master Peter, like the Gilbert of 2.8 below, has correctly sensed that the theory of integral wholes, a theory which deals with concrete collec­ tions of objects, (e.g. the various manifolds mentioned on one interpretation of the problem sentence) will most conveniently be couched at the nominally-termed level (as indeed is our contemporary theory of parts and wholes: cf. 0.5, 10.3, and HQS §4.5, §6.9). Further, as the corresponding contemporary sys­ tems amply demonstrate (10.26) the other, and more familiar, theory of wholes, namely the theory of distributive wholes, suitably incorporates quidditative-level discourse with its 157

Section 2 verb-like terms. Abelard is well aware of this contrast bet­ ween integral and distributive wholes: 2.21 above; cf. HQS 239, AD 546.24 - 547.5, 339.30.35, 574 - 5, and DA 166, 193. That this contrast is also the concern of the present text, and that mereological issues are centrally relevant thereto, is of course confirmed by the already-quoted 'totum' in the opening words of our section of the Sententie as well as by the ind­ ubitably mereological texture of the next two objections. There still remain, however, yet more replies to the first ob­ jection, i.e. we are still concerned with the two levels of discourse brought into play in the last-quoted extract from the Sententie: (A) [2nd Reply to 1st Objection]. Indeed, Boethius asserts that the nature is 'a likeness of the birth of things', as if he were openly proclaiming that these same things are of one nature, and resemble each other in the working of their nature. Whence it is that we say that this name 'man' is just the name of a nature, in that simply as a result of its introduction into the language it is naturally common to many things in the sense that they resemble each other by their nature (since each of them is a rational mortal anim­ al). In contrast, we say that the name 'Socrates' is that of an individual object rather than of a nature, since by it the discreteness of an object is indicated, as opposed to the li­ keness with which many things are endowed, thanks to the li­ keness of some same nature. (Naturam quippe Boethius dicit 'similitudinem rerum nascentium'  si aperte dicat easdem res esse unius nature que operatione nature similes sunt ad invicem. Unde hoc nomen quod est 'homo' nature dicimus, quod ex una ipsius inpositione commune est naturaliter multis re­ bus secundum hoc quod invicem sibi naturaliter sunt similes (in eo scilicet quod unaqueque eorum sit animal rationale mortale); hoc vero nomen 'Socrates' persone potius dicimus quam nature, quia per ipsum discret io persone monstratur, non convenientia multarum rerum quibus datum sit secundum 158

Abelard and his Contemporaries similitudinem alicuius nature: AI 115.31 - 116.11.) We plainly have here a further attempt to distinguish between discourse concerning a nature (or essence or quiddity, here glossed in the Boethian allusion as Wittgensteinian 'family resemblance') on the one hand, and discourse concerning named individuals such as Socrates on the other. Incidentally, the present recurring mention of similitude in accounts of the quidditative is also typically Porretan (cf. 2.75 above and CC 46, p. 50). This point will be taken up in 2.792 below. 2.752 It is at this sort of juncture that a precise charact­ erisation of the logical-grammatical difference between 'There is exactly one man' (where the 'man' is that of 'Man is a sp­ ecies') and 'There is exactly one Socrates' would be of help. Contemporary categorial language has no difficulty in this respect: 10.21, 10.26; cf. HQS 180 - 1. All the present text can do, however, is to continue with ad hoc hints, the next of which turns on appropriate replies to correspondingly diverse questions: (A) [3rd Reply to 1st Objection]. Again, we assert that the question 'What is man?' concerns the nature, whereas we ass­ ert that 'Who is this?' regards the individual. Whence it comes about that we reply to the first with the words relev­ ant to the nature or likeness (namely that which is common to many on account of their mutual likeness one to the oth­ er) e.g. 'man', 'animal'. In the second case, however, we respond with the word designating the personal individual or discrete object (i.e. that which signifies one thing determin­ ately and discretely) e.g. 'Peter', 'Marcia'. (Et hanc interrogationem 'Quid est homo?' nature dicimus, istam autem 'Quis est iste?' vel 'Que est ista' dicimus esse persone. Unde, ad illud, vocabulum nature seu similitudinis respondemus (quod videlicet multorum commune est secundum hoc quod sibi invicem similia sunt), ut 'homo', 'animal'; ad istud vero, vocabulum persone seu discretionis (quod videlicet unam rem determinate ac discrete significat), ut 'Petrus', 'Marcha': 159

Section 2 AI 116.11.18.) Here not only is the theme of contrast reinforced, but two further assets are acquired. First there is the reminder of the relation between the question 'What is it?' ('Quid sit ...?', literally, 'What may ... be?' in the old-fashioned subjunctive style) and the nature or essence. The Latin version of this question is, of course, the origin of the term 'quidditative', which not only has a well-known medieval counterpart, but was also adopted in 0.4 as a label for the functorially-termed discourse (e.g. '... is ...' of index s/(s/n s/n)) which corresponds thereto. Secondly, this passage's use of the qua­ lification 'determinate' for the signification of individual objects yields, by way of contrast, the indeterminate as the status of terms of the quidditative '... is ...', and this is exactly right. For the s/n of the last-shown index is the in­ dex of a verb, indeed the index of an incompleteness, an in­ determinacy, which fulfills the requirement implied by our text. In any case, this implication of functorial indeterm­ inacy for terms at the quidditative level is yet another of the items in the present text which was to have a well-marked future medieval career: HQS §5.1. 2.753 It follows (continues the reply) that as a particular case of the two-level distinction, statements of identity can be pitched at either of those two levels. Thus 'Marcus is the same object as Tully' is at the individual-object level ( 'se­ cundum personam': AI 116.19.21). In terms of our logical gra­ mmar, its main functor, i.. e. ... lo the same object as ...' is of index s/(n n), with its nominal terms indicating the individ­ uals in question. Here the singular identity as defined in 10.221 identifies the functor intended. The contrasting case is exemplified when natures or species are being identified as the same (e.g. those of Peter and Mark, insofar as they are both men or animals: AI 116.21.23). Here the analogous funct­ or of identity would have s/(s/n s/n) as its index. The th­ irteenth-century logical grammarian Boethius of Dacia has a 160

Abelard and his Contemporaries corresponding contrast when distinguishing the grammatical variety of plurals from an analytic point of view. For him the '... are ...' of 'Socrates and Plato are men' is a part of speech diverse from that of the '... are ...' of 'Man and donkey are animals': cf. 2.763(A). We are now seeing how the first is nominally-termed, but the second, the quidditative statement, is not: cf. HQS §4.35. 2.754 The same duality of identity statements is next ill­ ustrated from Trinity-lore. 'God-the-Father is-not God-theSon' negates identity at the individual-personal level, whereas the two persons in question have their common quiddity, i.e. their Godhead, so that we have once again a contrast between the two levels, i.e. 'secundum eandem personam' and 'secundum naturam' respectively: AI 116.26 - 117.7. In each of these cases there is no choice as to the correct (i.e. truth-pres­ erving) analysis. 2.755 However, duality of truth-preserving possibilities persists with reference to the interpretations of the Latin sentences, 'Homo sculpit qui pingit' and 'Homo sculpit et idem pingit': AI 117.8.16. Relatively to these Latin sentences, the categorial indices may usefully be brought to bear so as to bring out the difference between the nominally termed (i.e. s/(n n)) and the non-nominally-termed (i.e. s/(s/n s/n)) in­ terpretations. English versions of the same sentences do not stand in so great a need of such clarifications, though the same possibilities of variety subsist. Thus 'The man who sculpts also paints' or 'The man sculpts and the same man paints' are perhaps pushed by their definite articles (non­ existent in the Latin) in the direction of nominally-termed main functors, with 'the man' as a nominal term. Use of ital­ ics without the articles might provide the contrasting sense of some functor of identity at the quidditative level, e.g. 'Man sculpts and also paints'. (Two subsidiary points are being illustrated here. First, for certain ad hoc, local purposes, English may serve as a canonical language. Secon161

Section 2 dly, in general, there may be myriads of alternative ways of resolving the difficulties raised in the text undergoing anal­ ysis. However, a more consistent and sympathetic solution is that which is relativised to a unitary set of logical systems which in their turn are close in spirit to the medieval solut­ ions. These desiderata are fulfilled in the systems and meth­ ods which are being exemplified in the present remarks, and made explicit in 10.2.) 2.756 Most elegant and interesting is the presentation which then follows in the Sententie of a contrasting instance in which the duality possible in the previous case is not ap­ propriate: (A) But when we assert 'No resembler is that which it res­ embles', then if this proposition is to be true, the relation in question must be taken solely in the sense of singular identity [cf. 10.221], i.e. no thing is individually identical with that which it resembles. This hence obviates our basing the truth that Socrates resembles [some] man, e.g. Plato, on a relation of identity pitched at the man-as-such level. It merely requires that Socrates should not be that [individual] man whom he resembles. (Nam cum dicimus 'Nullum simile est illud cui est simile', si vera sit enuntiatio, accipienda est relat io tantummodo secundum identitatem persone, ut videlicet nulla res sit ea personaliter cui similis sit. Ut, cum Socrates sit similis homini (verbi gratia Platoni), non tamen ideo verum est simpliciter quod non sit homo, sed tantum quod non sit ille homo cui est similis: AI 117.16.22.) Here any singular identity between an object and that which it resembles is being excluded, and stress is rightly being laid on the truth that the resultant negation of identity does not operate at the quidditative level. From an English point of view, some of the trouble in interpreting this case could be said to stem from the lack of articles, once again, in the original Latin. Here the difference between resembling man and resembling a [certain] man is not directly visible, with 162

Abelard and his Contemporaries the result that the passage's confinement to the consequences of resemblance at the level of a singular identity (10.221) which is at the individual 'personal' level, is an efficient enough way of making the point. The specification of such identity (even when the latter is negated) ensures that the non-quidditative s/(n n) is the relevant index, exactly as the argument requires. If this logical grammar of the pres­ ent case is accepted, it yields interesting concomitant and independent confirmation of a claim made elsewhere (HQS 230) that Descartes' discussion as to whether his 'idea of the sun' (a quidditative matter) more resembles the actual sun (an individual object) than does the sun as sensed, is a piece of nonsense. (For an Abelardian discussion on resemblance AD 360 may be consulted). 2.76 Thus far the main point of the various arguments has been to illustrate how it is possible to interpret in terms of res (things) discourse somehow involving part-whole probleminferences, and how this can be done without the inconsistency with which supporters of the vox (utterance) interpretation of such part-whole talk threaten when they produce their examp­ les. The res, in their turn, have been appropriately allocated either to the individual ('personal') level, or to the quiddit­ ative level, i.e. that of the nature or species, as the case requires or allows. The latter level need not be taken in a Platonic realist sense, notwithstanding its anti-vox status, as 0.4 has illustrated. Among the res considered at the individ­ ual ('personal') level have been the concrete versions of a six (i.e. the sextet), a four (i.e. the quartet), and the latter's half, all of which are susceptible of consideration as var­ iously-related concrete collective classes in the mereological sense. 2.761 Now, however, we come to a case which will be un­ equivocally mereological, namely that of the house (made up of walls, roof, and foundations); this, as already observed, is the usual medieval illustration of an integral whole (or con163

Section 2 crete collection, complete collection, or collective class in the contemporary sense). Our text will here overtly and rightly state that consideration at the concrete level is ap­ propriate, as it was in the resemblance-case. We thus are to have a further precise identification of the level at which mereological questions proper are suitably pursued (i.e. in terms of functors of index s/(n n), typically), and also an explicit contrast of this with the quidditative-level discus­ sion of wholes, parts, identity, non-identity, and so on, as contained in some of the foregoing examples. All this is an­ other facet of the already-encountered usual Abelardian con­ trast between the integral whole (or 'collective class') and the distributive whole (i.e. the non-mereological sense of 'class' represented by the quidditative level of discourse; cf. 2.21 and HQS 239). At any rate, the opening sentences of the next continuing extract make clear that the prime point is the individual ('personal') level of discourse (so that s/(n n) would be the typical main-functor index): (A) This last case [i.e. that of the non-identity of resemblers] is like that of the assertion, 'If some whole is, every part of that whole is' (or '... of the same whole ...'). Here the relationship must be at the level of the individual object. For although every house is (a) whole, it is never­ theless not true that if (a) house is, then every part of (a) house is; otherwise it would follow that if this house is, every part of that other house is. What the rule is capable of showing is that if (a) house is, then it is every part of that house which is, and not of house in general. Likewise, neither is it true that if this house is, then this wall is (notwithstanding the fact that this wall is part of (a) house); rather, if a house exists, it is the wall of that same house which is. This is because although every house is made up of some wall, nevertheless not every house is made up of this wall. (Similiter et, cum dicimus 'si aliquod totum est, quelibet pars illius totius est' vel 'eiusdem 164

Abelard and his Contemporaries totius', ad personam facienda est relatio. Quippe, cum omnis domus totum sit, et paries sit pars domus, non tarnen verum est quod, si domus est, quelibet pars domus est; alioquin sequeretur quod, si hec domus est, quelibet pars illius alterius domus est (hoc scilicet per regulam ostendi potest quod si domus est, quelibet pars illius domus est que est, non domus simpliciter). Nec hoc similiter verum est quod, si hec domus est, hic paries est (licet hic paries pars sit dom­ us); sed, si domus est, paries eiusdem domus est que est. Omnis quippe domus ex aliquo pariete constat, sed non omnis ex hoc: AI 117.22 - 118.3.) Here the struggle is against the ambiguities of an articlefree Latin in a context where it is open for 'house' to be taken quidditatively rather than 'personally'. It is the 'house' of 'this house' which must be understood, and not the 'house' of 'House is a species of building', for example. Thus is elegantly confirmed the nominally-termed nature of the mereological discourse indicated by the use of the example of the house. 2.762 Other examples of the two levels of res-centred discourse continue to be given. 'Woman, who damned, also saved' has to be taken generically: circa naturam sexus muliebris: AI 118.4.9. In his Theologia Christiana Abelard deals with the same example, and agrees that a numerical identity of reference of 'woman' is to be eliminated (since Eve damned, but Mary saved) but is not so ready to speak about a mere sameness of nature on the side of the truthyielding interpretation. He rightly, and cautiously, restrains himself to an identity of definition of what it is to which the name 'woman' alludes in the two cases: COA 486. 2.763 A further potentially significant significant con­ cession on the part of the Sententie in connection with the first argument is to the effect that there are in fact three ways in which 'Socrates is what he is' may be taken: (і) 'in­ differenter' in the sense that it may be taken either way, i.e. 165

Section 2 at the quidditative or individual level; (ii) ''personaliter' i.e. at the nominally-termed level, or (iii) at the quidditative level: AI 118.18.23. The inclusion of the 'indifferenter' case completes the present text's anticipation of Boethius of Dacia's three-fold classification of levels of discourse which he distinguished in connection with the types of plurality. Two such levels had already been noted above, and the indetermin­ ate third possibility is mentioned in Boethius's apt summary of the situation which may now be quoted: (A) Three types of plural number are possible, namely the one which shows the thing as purely many, without making de­ terminate whether a quiddditative or individual plurality is in question; then there is the one which indicates the thing as multiplied in terms of a plurality of quiddities, and the third possible plural number is that which indicates the thing as multiplied in terms of a plurality of individuals. (Tres numeri plurales sunt possibiles: unus scilicet desig­ nans rem ut multíplicatam simplicíter non determinando utrum per plures species vel per plures individua, et alius design­ ans rem ut multíplicatam per plures species, et tertius num­ erus pluralis possibil is design ans rem ut multíplicatam per plura individua: BMS 167.54.59.) 2.77 However, having thus simply noted that by now the Sententie have anticipated and exploited all three of Boethius of Dacia's possible states of discourse in the discussion arising from the initial problem concerning 'totum', we may now concentrate on the two other mereological problems which were also raised. The semantic status of such problems has, as anticipated above, by now been thoroughly established, and the theorising overtly involves manyness taken personaliter, i.e. at the level of concrete individuals. (Subsequent dis­ cussion is to show that in a sense the undividability etymologically contained in the word 'in-divid-ual' could turn out to be inappropriate). 2.771 Here, then, is the second objection, allegedly origin166

Abelard and his Contemporaries ating, as before, from the side of those who restrict discus­ sion in this region of wholes and parts to the level of voces, utterances, rather than of res (things). Its aim is still to show that the res people commit themselves to untenable pos­ itions. (A) [Objection 2]. Not only this, but a further objection [to analysis at the level of things] appears to be possible. For suppose we assert: (i) Thing is made up from thing, and (ii) Every many makes a whole [cf. AD 422.4.5], then it certainly follows that: (iii) Whenever two objects are established, then those two make up one whole which is diverse from its individual components. Under these circumstances, whenever there are two there are three. This three is made up from the two in question and the whole composed from the two. This whole is diverse from each individual component, and hence, whenever there are two there are three, i.e. the two in question and the whole of them, which is not only diverse from each individual compon­ ent, but also susceptible of diverse predicates, since neither one nor the other of the parts is identical with that whole. But yet again, as every multiplicity makes a whole diverse from its individual parts, the whole just discussed will make one whole along with its two parts, and this further whole yet again with its parts, and so on to infinity. (Sed sic et istud obiciendum videtur: quod si rem ex re constare dicamus, et omnia plura unum totum efficere, utique, ubicumque erunt duo instituta, ipsa quippe duo unum totum facient quod est diversum a singulis illis; et ita ubicumque sunt duo, sunt tria (ipsa videlicet duo et totum ipsorum, quod est diversum a singulis Ulis et predicatione remotum, cum neque hec pars sit ipsum totum neque illa); sed rursus, cum omnia plura un­ um totum efficiant diversum a singulis Ulis, ipsum totum iterum cum duabus partibus suis unum totum efficiet, et ill167

Section 2 ud rursus aliud cum partibus suis; et sic in infinitum: AI 114.13.23.) This allegedly absurd argument in fact embodies an outstand­ ing piece of mereological insight. I have heard a thesis somewhat akin to part of this maintained in our own day by Czeslaw Lejewski, and John Wyclif may have glimpsed an aspect of it (6.2 below). It was recalled in 2.72 above that the principle numbered (ii) in the translation is definitely Abelardian. As we are to see in 2.781, however, the ressupporting reply supplied by Master Peter scarcely appreciates the passage's mereological insights, as neither would those who used to trendily claim in Oxford discussion during Gilbert Ryle's heyday that 'I have a right-hand glove and a left-hand glove and a pair of gloves' was akin to 'She came home in a sedan chair and a flood of tears', in that both exemplified type-fall nonsense. 2.78 The third and (for us) final supposedly reductio ad absurdum argument against res-type discourse contains further mereological sophistication: (A) [Objection 3]. Besides, we assert that this trio is made of these three [discrete] elements, and any two of those elements yield a pair. Who then can deny that this trio is made up of [at least] two pairs? For indeed the common element along with this element makes up one pair , so that the trio contains [at least] two non-identical pairs, as this pair is not that pair, since the latter has a proper part which the former lacks. (For if this pair's make-up involves this el­ ement, and that pair's make-up does not involve this same element, then by a necessary logical consequence it is ob­ vious that that pair is not this pair). (Preterea hunc ternarium dicimus constare ex his tribus unitatibus, et due unitates unum binarium quelibet reddunt. Quis neget hunc ter­ narium ex duobus binari is constare? Eadem quippe unitas cum hac unitate unum binarium <et cum illa alium binarium> fac168

Abelard and his Contemporaries it; et ita ternarius duos in se continet binarios diversos ab invicem, cum hic binarius nullo modo sit ille, cum talem habeat partem quam ille non habet. (Si enim bic binarius hac unitate constat, ille vero hac eadem non constat, profecto ille hoc non esse convincitur per necessarium sillogismi conplexionem): AI 114.23 - 115.5; cf. COA II, 487, quoted in 2.786 below.) The mereological notions of part and whole presupposed by such anti-res objections are in fact highly sophisticated, notwithstanding their status as supposed absurdities put up by the vox-people to demolish the claims of res-discourse here. This becomes obvious when it is noticed that the sense of 'part' here presupposed is such that when two or more th­ ings are said to be parts of some whole, they need not be to­ tally outside one another, i.e. they need not be discrete (a term familiar to present-day mereologists: 10.336). Thus the various successive wholes of objection 2 certainly overlap; so do the pairs selectable from the trio of this third objection (as Leibniz realised in his observations on just this case: LPP 77). Wyclif too will reproduce analogues of both the the last two objections, although with suspect additions: 6.2 below. The comparative rarity of such realisations of the overlapping (or 'non-discrete') possibility is what makes them remarkable: cf. 1.5. Relatively to an aloofly general present-day theory of part and whole, there is little at which to baulk in objec­ tions 2 and 3. Some of it may sound a bit odd to the unprac­ tised ear, but it certainly should not frighten one into ab­ andoning res-talk in this area, as the vox-supporters would suggest. 2.781 Unfortunately the otherwise highly competent author of the Sententie fails to achieve the required level of gen­ erality, since he goes on to resolve those objections in terms which fail to take full account of their mereological import. Clearly, in a fully general mereology, as 1.5 has suggested, it is desirable to have a sense of 'part' which need not entail 169

Section 2 discreteness. Only subsequently need discrete objects be con­ sidered as a more specialised topic. Instead, the argument now puts all its stress on a discreteness-requirement for parts of a manifold. Both individual things and individual parts are all seen as discrete in the sense of non-overlap­ ping: cf. 10.34. The actual Latin word 'discrete' or its derivative occurs at least four times on the couple of pages with which we are now concerned (AI 1 1 5 - 6 ) and the cognate 'personaliter' is even more numerous. This association is to be found also in Abelard, e.g. DA 235. The consequent defects of the quite unnecessary replies to objections 2 and 3 are thus patently obvious: (A) [Reply to Objection 2: 2.771(A) above]. That which we mentioned above as an objection against us, and which invol­ ved infinity, amounts to nothing. Indeed, when it is said that 'every many integrally makes a whole' we take the many to have elements which are mutally diverse not only in resp­ ect of their predicates, but also in respect of all the elem­ ents of their make-up (i.e. they are such that the one does not enter into the make-up of the other, nor does it embrace anything of the other). Thus the whole which is made up of the two parts is not diverse from those parts which it cont­ ains in respect of the being of its bulk. Hence that whole does not involve a part which is in some way diverse from those two parts. (Quod autem de infinitate supra quoque opposuimus nichil est. Quippe, cum dicitur quod 'quelibet plura unum totum integrali ter conficiunt' talia plura accipimus que diversa ad invі sunt, non solum predicatione verum etiam tota continentia constitutionis (ut videlicet alterum in constitutione  alterum  aliquid alterius comprehendat); totum autem quod ex duabus partibus constat non est ab Ulis quas continet diversum secundum essentie sue capac­ itatern. Ideo illud aliquid totum non confici tur quasi diversa pars ab Ulis: AI 119.25 - 120.3.) Here, as subsequent text is to confirm, the key word is 'div170

Abelard and his Contemporaries erse', which is to be contrasted with 'different'. (Aquinas has a version of this contrast: HQS 194, 278.) Certainly the div­ erse is here most competently described as being the non-ov­ erlapping. Hence continuity with the previous discussion would have been better preserved had our author persisted in his use of 'discrete' (as described above) instead of switch­ ing to the present 'diverse'. At any rate, this terminology still encapsulates the central point at issue. As forecast, 'part' or 'thing', where there are many such, here quite unnecessarily conveys the implication of mereological discr­ eteness, and relatively to such stipulations, it is true, as the passage implies, that we cannot (for instance) count the whole as one thing and one of the parts as another, so that objection 2 fails. Of course, as a record of linguistic an­ thropology, as opposed to the requirements of aloof theor­ etical generality, the 'we' of the passage may well truly record how non-theoretically inclined folk would take, 'Every many makes a whole'. They may well assume discreteness in the elements of the many: but presumably such anthropology is far from being the point at issue. Such a local linguistic point certainly should not be allowed to impinge on general discussions such as the present one purports to be. 2.782 It is in the reply to objection 3 that the actual contrast between diversity and difference is made explicit: (A) [Reply to Objection 3]. In this way is refuted that which was said above [sic] about 'being made up of two div­ erse couples', since 'diverse' does not really apply to them in this case, for they incorporate a common part in their bulk. Thus 'diverse' is only properly used at the level of individual objects when the discrepancy between them is not merely in respect of predicates, but involves also an absol­ ute discreteness in the being of their bulk, so that there is no discord between the latter and the former of these, in accordance with what was granted above [sic] ...., namely that it is possible for someone to have a house alone, i.e. they 171

Section 2 would not have anything other than a house, t h a t is, anything so diverse from a house t h a t it would be totally distinct from it, not just on the predicate level, but also in respect of its whole make-up. There are plenty of things which dif­ fer specifically or generically , but which are not diverse in the way described, as in the instance of a man and his hand . . . . Thus, when we say t h a t house is adequately split into wall, roof, and foundations, thereby giving the impres­ sion t h a t it can have no parts other than these, then 'other than' must here be taken to allude to those [other] p a r t s ' being discrepant from those specified, not only in respect of their predicates, but also as being diverse from them in th­ eir contribution to t h e make-up of the house, i.e. in such a way t h a t they are not contained in those three, nor do they contain those three, nor do they have any same part in com­ mon among any of the t h r e e of them. If these conditions are not fulfilled, then it will be false that walls, roof, and foundation are the sole three parts of house. (Hoc etiam modo refellitur quod dictum est 'confici ex duobus diversis binariis', cum 'diversi' recte non dicantur cum eandem contin'Diversum' itaque eant in quantitate sue essentie partem. personaliter dicimus proprie quod, non solum predicatione, verum etiam tota essentie sue quantitate discretum est (ut videlicet nichil aliud sit in isto quam in illo secundum quod superius ... concessimus aliquem posse habere solam domum, ut videlicet non haberet aliud a domo - hoc est ita diversum a domo ut, non solum predicatione, verum etiam constitutione ab ea penitus disiunctum sit). Multa itaque specie seu gen­ ere differunt que hoc modo diversa non sunt, sicut homo et manus ipsius ... . Sicut, cum dicimus 'domum sufficienter dividi per parietem et tectum et fundamentum',  si nullas alias partes preter istas tres habeat, 'alias' oportet intel­ lig i non solum predicatione remotas ab invicem, verum etiam diversas continentia constitutionis, ita scilicet ut  in partem istis contineantur  istas contineat,  eandem 172

Abelard and his Contemporaries cum aliquo horum trium communicent (alioquin falsum esset istas solas tres esse partes domus): AI 120.4.25.) 2.783 In spite of the already-indicated general misdirect­ ion of its thrust, this passage nevertheless still contains some prodigiously good mereological intuitions. But first a few remarks to assist in understanding it as a piece of prose may be offered, before any detailed comment on its content is made. Thus inspection of the text of objection 3, as we now have it, and as reproduced above (2.78) will show that it does not contain the word 'diverse' as alleged by this present passage, but this scarcely affects the point of the argument. Next, the mention of its having been granted that someone can have a house alone does not allude to the present text. Abelard was seen to deal with the 'house alone' case in 2.33 above, and it recurs in 2.786(C) below. Its present import is in any case clear enough: it is rather low-grade wit for someone to deny that it is possible for someone to have just a house, and to base this denial on the house's also involving parts. The witticism's shaft is avoided by ensuring that 'just ...' or '... alone' here merely excludes extrinsic discrete div­ ersity. The example of the man and his hand which are diff­ erent in kind but not discrete occurs also in Abelard's Gloss­ es on Porphyry (GA 535 - 6) and in the Theologia Christiana, of which more in a moment. The gloss is, however, quite del­ iberately indecisive on whether this non-discrete pair should be said to be numerically diverse. The categorical decis­ iveness of the present Sententie on this account is hence most significant. Questions such as these are to be coupled with discussions of the syncategorematic 'solus ...', 'only ...' in the next century, as 7.316 is to show. 2.784 From a mereological point of view, however, the int­ erest of the last passage does indeed lie in its appreciat­ ions of diversity and difference. Use of the former to char­ acterise mereological discreteness is excellently done. Still, one cannot but reiterate that the notion itself is misapplied 173

Section 2 in responding to the objections. Neither in those responses nor in the case of the three-parted house is it necessary that part should be specified as discrete, thus producing a more restrictive notion than that of part in the unrestricted sense initially used in mereology. However, it is an inter­ esting mereological enterprise to specify part in this dis­ crete sense, and to compare it with part in the more general sense which will of course allow that it is impossible for there to be just two objects (as in objection 2) and that there are at least three pairs as parts of the trio (as in objection 3). 2.785 It is in a work which is definitely Abelard's, namely his Theologia Christiana, that the same sort of question is ventilated, but without the distinction between diversity and difference, at least as far as terminology goes. What Abelard does is to speak of diversum sive differens (the diverse or the diffent) as if those these terms could be used inter­ changeably. Then ensue distinctions which lack the incisiveness of that between the diverse and the different which has been observed in the present Sententie. Again, although the Theologia has the distinction between numerical and ess­ ential diversity, it is suggested that the second of these only may be the basis of numerical diversity, and that there is no necessity here: (A) But we say that all those things are essentially div­ erse from each other which are mutually discrepant to the point of this one not being that one, as in the cases of Socrates' not being Plato, nor Socrates 1 hand Socrates. Thus likewise we say that all discrepants are essentially diverse when the being of the one is not the being of the other, even though it may be involved with the being of the other, as the hand with the man, or the wall within the house. Therefore even as Socrates is other in his being than is Plato, so also the hand of Socrates is other than Socrates, and every part is other than its whole But of those 174

Abelard and his Contemporaries things which are essentially diverse, some are also numer­ ically diverse, whereas some are not so at all. (Dicimus autem ea ab invicem essentialiter diversa quaecumque ita ab invicem dissident, ut hoc non sit illud, ut Socrates non est Plato,  manus Socratis Socrates. Sic etiam quaelibet dis­ parata diversa essentialiter dicimus, cum videlicet essentia unius non sit essentia alterius, quamvis tarnen sit de essen­ tia alterius, ut manus de homine, vel paries de domo. Sicut ergo Socrates aliud est essentialiter quam Plato, ita manus Socrates quam Socrates, et quaelibet pars quam suum totum Horum autem quae essentialiter diversa sunt, quaedam numero quoque diversa sunt, quaedam minime: COA 486 - 7; Theol. Christiana, Lib. III.) 2.786 However, the treatment of numerical diversity immed­ iately invokes the notion of discreteness, in a manner similar to that observed in the Sententie, and using similar examples: (A) For indeed we say that those things are numerically diverse which are so mutually discrete in the whole being of their quantity that not only is it the case that this one is not that one, but also neither is the one a part of the oth­ er, nor does it have a part in common with the other. Hence all and only those things are numerically different which are totally discrete in their being, whether they are only numer­ ically discrepant, as in the case of Socrates and Plato, or whether they differ specifically, as this man and that horse ... . (Diversa quippe numero dicimus, quae adeo tota essentiae suae quanti tate ab invicem discreta sunt, ut non solum hoc non sit illud, verumne unum alterius pars sit, vel eamdem cum eo communicet partem. Haec itaque sola et omnia numero sunt differentia, quae tota quantitate suae essentiae dis­ creta sunt, sive solo numero ab invicem distent, ut Socrates et Plato; sive etiam specie, ut hic homo et ille equus ...: COA 487.) We have here cut off the rest of Abelard's list, as questions could be raised as to whether his examples really come under 175

Section 2 his general description. Thus far, at any rate, discreteness is being recognised. Unfortunately it is now to be made the basis of all numerical distinction, thus overtly excluding manyness of overlapping objects in precisely the same way, and with reference to the same examples, as do the Sententie: (B) There are hence some cases of essential diversity which do not attain numerical difference, as in the case of the house and the wall, and, in general, any integral whole and its own part . . . . We also take diversity in this [dis­ crete] sense when we deny that there are various couples in a trio. For although the third element makes one couple with each of the other two elements, and this couple is not that other couple, we nevertheless deny them to be diverse couples. This is because they are not mutually discrete in respect of their bulk or volume, given that they have one element in common. (Sunt itaque quaedam essentialiter div­ ersa, quae non sunt numero differentia, utpote domus et par­ ies, et quodlibet integrum totum cum propria parte sua ... . Hoc etiam modo diversa sumimus quando negamus in uno tern­ ario diversos esse binarios. Nam licet tertia unitas cum unaquaque aliarum duarum unitatum unum binarium efficiat, et hic binarius non sit ille, diversos tarnen binarios dici neg­ amus, cum non sit tota capacitate vel continentia suae esscommunicent: entiae discreti, cum eamdem scilicet unitatem COA 487.) Here we are clearly very close to the reply given in 2.781(A) above, which couples discreteness and diversity in a way which has hitherto been notably absent from the Theologia Christiana, though prominent in the Sententie. The kinship of these two works is further evidenced by the former's going on to supply the 'house alone' case which was missing from the latter (cf. 2.782(A)): (C) But also, when we say that someone possess a house alone, i.e. a house and no other thing diverse from the house, it is numerical rather than essential diversity which 176

Abelard and his Contemporaries we understand here. For he who possesses a house also has in his possession the wall, which is not the house. But although the wall is essentially diverse from the house, nevertheless, since it is at the same time embraced within that house's bulk, one does not have numerical diversity in this case. (Sed et cum dicimus aliquem solam domum possid­ ere, hoc est, domum et nihil aliud diversum a domo, diversum numero potius quam essentialiter intelligimus. Qui enim domum possidet, etiam parietatem, qui non est domus, in pos­ sessione habet. Sed cum sit paries essentialiter diversus a domo, in qua non est diversus numero, cum videlicet in ipsa quanti tate domus comprehendatur: COA 487.) 2.787 Although we have here the same point being made as in the last-quoted Sententie passage (2.782(A)), the vocab­ ulary is notably different. There also recurs the alreadynoted contrasting Abelardian indecisiveness on part-whole numeration, an indecisiveness which is, of course, a pointer in the direction of a more aloofly general mereological apprec­ iation of these matters: (A) Every part is indeed essentially diverse from its whole, but not numerically diverse therefrom. Nor yet, per­ haps, is it hence numerically the same, unless one might per­ chance take sameness of number in a negative sense, i.e. as not involving numerical diversity. (Pars quippe quaelibet diversa est essentialiter a suo toto, sed non diversa numero, nec fortassis eadem numero, nisi quis forte numero idem neg­ ative dicat, hoc est non diversum numero: COA 487 - 8.) 2.788 Then comes a hint of that persuasion in the direct­ ion of Socratic manyness (for example) which arises from consideration of the natura magna concept found in the iden­ tity discussions of both Abelard and the ps-Joscelin (2.442, 2.45 above): (A) For if the part is said to be numerically diverse from the whole, then it may be granted that Socrates is predicat­ ed of many numerically different objects. Thus given that 177

Section 2 this whole, namely the composite of the hand and the rest of the body, is Socrates, it is also to be granted that the rem­ ainder of the body apart from the hand is Socrates, i.e. is vivified by this soul. (Si enim pars numero diversa diceretur a suo toto, profecto Socrates de differentibus numero praedicari concederetur, cum videlicet hoc totum ex manu scilicet et reliquo corpore compositum sit Socrates, et ipsum quoque reliquum corpus a manu deinceps Socrates sit dicendum, hoc est hac anima vivificatum: COA 488.) 2.79 In the background of this whole discussion there still lurks the possibility to which Abelard makes occasional allusions in passing, namely, that if the numbered units in these objections are not themselves in-dividuals in the style of those mereological atoms, the definition of which was in­ deed glimpsed by Abelard himself (2.445), then in each case there will be indefinitely more parts involved, thus giving rise to more countables, even discrete ones. Here 'atom' is being used in the general sense outlined by the definition 10.352 below, of which the following is a vernacular render­ ing: For all a, a is a mereological atom if and only if a exists and any proper or improper part of a is the same object as a. The final vital point of this definition, i.e. the identity of part and whole where the atom is concerned, is made very plain in the final clause of Abelard's remark, 'No composite is comprised in one of its parts, nor would that be a part which was not exceeded by the quantity of the whole; if, however, there were only one part [i.e. in the atomic case] then that part would be wholly identical with the whole', Nullum enim compositum una contentum est parte; neque enim pars esset quam totius quantitas non excederet; sed si una tantum esset, idem profecto cum toto fieret: AD 554.15.17; cf. 3.2941 and 7.52 below, also HQS 242 - 3. For a full-blown later medieval treatment of 'indivisibles', WW may be most usefully con178

Abelard and his Contemporaries sulted. 2.791 The paradox of the Sententie, at least in relation to the question of their Abelardian authorship, is that the ob­ jections to which the piece is mainly engaged in replying evince a higher degree of mereological generality than do the replies. Again, the objections are for the most part coherent with well-known, if not notorious, Abelardian theses. Yet those sometimes misdirected replies are not lacking in their own sort of mereological acumen. The fact that the object­ ions are attributed to the partisans of voces, utterances, could suggest that the piece echoes the time when Abelard was hearing the 'insane' arguments of his master Roscellin (cf. 2.34). 2.792 There is, however, another most lively twelfthcentury school, namely that of Gilbert of Poitiers, of which we are to hear something in the next section. This school, as already noted above, uses vocabulary close to that of the Sententie at crucial points. To follow through a conjecture that someone of Gilbert's school produced the Sententie would be too distracting an enterprise. However, as we are to see in 2.81, one prime characteristic of that school is the 'simil­ itude' theory of universals. This involves not merely the usual distinction between nominally-termed and quidditative discourse, but also the confinement of the latter to individ­ uated cases. This means that for Gilbert, strictly speaking, there is no general abstract whiteness (for example); there only exist a lot of individual whitenesses (e.g. of this pen, of this swan, and so on) which resemble each other. If we loosely talk about a general nature (e.g. of whiteness) then it is really to the group-likeness of the individuated abs­ tracts that allusion is being made. 2.793 Now in the 3rd reply to the 1st objection quoted above (2.752) precisely these features are all present, and in a Gilbertian vocabulary. Thus the expressions vocabulum nat­ ure and vocabulum persone parallel the pro natura and pro 179

Section 2 persona of Gilbert's commentary on Boethius' Contra Euticem et Nestorum, ch. 4, 105, for example: HGP 309. The similia of that reply is common Gilbertian coin. The possibility of tak­ ing 'Socrates is what he is' either personaliter or at the quidditative level has been noted as mentioned by the Sent­ entie (2.763 above) and presupposes, yet again, a character­ istically individuated 'Socraticity' at that quidditative level. Again, mereology abounds not only in the work of Gilbert him­ self, but also in the comparatively recently published prof­ ound work on logic from someone of his school, whence we are shortly to draw material: CC 46; cf. 2.84 below. 2.8

Porretan Mereological Scandals

2.81 The Originality of Gilbert of Poitiers 2.811 Gilbert, Bishop of Poitiers (who is also called 'de la Porree' or 'Porretanus') lived from 1076 to 1154, and was hence a contemporary of Abelard (1079 - 1142). However, his output tends to take a form different from that of Abelard, in the sense that his now recognised works are commentaries on the theological opuscula of the Roman Boethius. Still, this by no means diminishes their philosophical importance, and we are in any case most fortunate in now having avail­ able a purely logical-philosophical treatise from the hands of some disciple of his. Among its other unprecedented acc­ omplishments is a distillation and reduction to order of many theses which are otherwise heavily immersed in Gilb­ ert's commentaries. This treatise is the Compendium Logicae Porretanum, recently edited by S. Ebbesen, K.M. Fredborg, and L.O. Nielsen, in CC 46. It is to be referred to hereunder as CLP or as the Compendium.

180

Abelard and his Contemporaries 2.812 Before proceeding to further detail on the mereological front, it will be helpful to make use of our speculat­ ive grammar outlined in 0.3 and 0.4 in order to intelligibly situate Gilbert's prime metaphysical innovations. We saw in 0.4 how the main functor of stock propositions at the quidd­ itative level was parsable as a functor which forms propos­ itions from proposition-forming functors; the categorial in­ dex of such a main functor is hence usually s/(s/n s/n). Natural-language examples of such quidditative propositions (which were said to be concerned with 'universals' ; cf. 0.4) would be 'Man is rational animal', 'Humanity comprises rat­ ionality and animality','Being a man is being a rational an­ imal', 'To-be-man is to-be-rational-animal', or 'Forming the class of men is forming the class of rational animals', as well as 'Man is a species', 'Animal is a genus'. 'Rationality is a differentia', and so on, with a corresponding multiplic­ ity for each of the various vernacular correlates in other 'natural'languages. HQS §4.2 provides full textual details of Boethius' seminal accounts of these notions. From the point of view of the Latin, with which we are, in this West­ ern medieval area, chiefly concerned, the 'to be an χ is to be a y' correlate (i.e. 'esse χ est esse y') with its highly significant infinitive terms, usefully reflects the doctrine, transmitted by Boethius, that a definition states the being, the esse, of a sort of thing (HQS §4.3). This infinitivelytermed correlate usefully shows forth the linguistic level of the quidditative definitional frame (HQS §4.331). Owing to the already-mentioned then current description of the 'univ­ ersal' as 'that which is predicated of many things', it was assumed, in effect, that the gaps shown by means of the 'x' and the 'y' in the correlated form would be completed by sh­ ared nominal expressions (i.e. by our common nouns and cert­ ain adjectives). 2.813 Gilbert's crucial innovation was to go to the oppos­ ite extreme with regard to those nominal expressions. For 181

Section 2 him any such sharedness is accidental to the status of quid­ ditative discourse. Indeed, if one keeps to a strict lingu­ istic depiction of what actually is, then in effect the gapfillers in the quidditative frame have ultimately (i.e. in his presupposed canonical language) to be unshared nominal expr­ essions. As the Porretan Compendium succinctly expresses this most important point, 'universals are nothing other than singular forms', universalia non sunt alia quam singulares forme: CLP 41.50.51. There is no such thing as whiteness in general, but only particular individual whitenesses. 2.814 In this Porretan logic the special terminology for the topic of the terms of quidditative-level discourse cent­ res on the Latin 'subsistentia', which would appear to be a neat and typically medieval composite from the Latin stem 'subsist-' and one of the words covering quiddity, i.e. 'ess­ entia'; that composite has been truncated to 'subsistentia'. This last is in turn to be contrasted with the key-express­ ion associated with nominally-termed discourse such as Socr­ ates is a man'. This key expression is 'subsistens', which even more obviously looks like another neat composite of the stem 'subsist-' with the name-forming termination '...ens' ('..ent', '...ing', 'being', 'existent', 'entity'). Now the usual rather lack-lustre translations of Gilbert's 'subsist-entia' and 'subsistens' are 'subsistence' and 'subsistent' respect­ ively. These latter scarcely reproduce either the philos­ ophical associations or the declensionally-grounded disinctiveness of their Latin originals. However, not only can the associations and the distinctiveness be restored, but the diversity of semantic levels can also vividly be displayed if the translations become 'subsistessence' for the quidditatively-signifying 'subsistentia', and 'subsistentity' for the concrete individual-object term 'subsistens'. The Compendium trenchantly presents the contrast between subsistessences and subsistentities, with accent on the singularity of both, thus: 182

Abelard and his Contemporaries (A) On the reason why nothing may be said to pertain to one thing that pertains to another. Given that there are two categories of things, one of which contains the subsis­ tentities and the other the subsistessences, and since every one of the subsistentities is singular, hence so likewise (but not, however, on the same grammatical ground) every one of the subsistessences is singular. For even as indiv­ idual objects are mutually discrete, so also are individual subsistessences. But as every subsistentity is something or is in some fashion by its subsistessence, it follows that Socrates is other than Plato because of his subsistessence. This is more evidently so in the case of incidental attrib­ utes. Thus given that it is by his whiteness that Socrates is white, who could maintain that Plato is white by the same whiteness [as that of Socrates] when the two are div­ erse white objects? Hence it comes about that there is a one-to-one correlation between white objects and whiteness­ es. Likewise it is by his humanity that Socrates is a man, is also some particular man, and is a man other than Plato; after all, no one asserts that Plato is a man by the human­ ity of Socrates. Hence there is also a one-to-one correl­ ation between men and humanities. You can go on and infer in the same terms concerning everything substantial and in­ cidental. Therefore as a property [in the strict sense] is that which pertains [uniquely], so no form which pertains to one thing is also in another, and so also, nothing which pertains to one thing also pertains to another. (Ratio quare dicitur nil esse in uno quod sit in alio. Cum duo sint gen­ era rerum et unum sit subsistentium, aliud subsistentiarum, sicut quodlibet subsistentium est singulare, eodem modo non tarnen eadem dicendi causa - quelibet subsistentia est singulare. Nam sicut singula subiecta discreta sunt inter se, sic et subsistentie singule. Sed cum quodlibet subsist­ ens sua subsistentia sit aliquid vel alicuiusmodi, Socrates sua subsistentia est aliud quam Plato. Quod in accidental183

Section 2 ibus apertius vides. Cum enim sua albedine Socrates sit albus, quis dicet albedine eiusdem Platonem esse album, cum sint diversa alba? Inde est quod tot albedines sunt quot alba. Eodem modo humanitate sua Socrates est homo et est aliquis homo et est alius homo quam Plato; nemo enim dicit humanitate Socratis Platonem esse hominem. Quot ergo hom­ ines, tot humanitates. At sic et eadem ratione de quolibet substanţiali et accidentali collige. Cum ergo sola propr­ ietas sit id quod inest et nulla forma que est in uno est in alio, ita nichil quod sit in uno est in alio: CLP 41.57.70.) 2.815 Sufficient reminders of the logical grammar have now been given to underpin ventures into the characteristic Porretan mixture of quiddity and mereology which is soon to be made explicit. The Compendium amply supports the sugg­ ested s/n, i.e. verb-like, interpretation used above for the terms of the quidditative: CLP 4 - 10. 2.816 Enough has also been said to eliminate the usual present-day contention that mere denial of sharedness of attributes is sufficient to make a thinker into a 'nominalist' as far as universals are concerned. A major aspect of medi­ eval nominalism is an insistence on remaining at the nametermed level for the parsing of quidditative discourse, so that s/n and s/(n n) will be typical indices of all nametermed main sentential functors, notwithstanding the quidd­ itative context of the utterance of some of them. It is this insistence which goes counter to the Porretan admission of, and distinction between, both the quidditative subsistessences and the nominally-referred-to subsistentities. And quite independently of anything noted above concerning the grammar of his discourse, no one would contend that Gilbert of Poitiers was a nominalist, even though he and his school would certainly hold a position which by contemporary crit­ eria would be nominalistic. For the Compendium extract pro­ vided has already illustrated that Porretans would agree 184

Abelard and his Contemporaries that when I see two black writing-pens, 'I do not see just one colour in these two pens. I see two colours, two black­ nesses, one here and the other there, one the blackness of this pen, this blackness, and the other the blackness of that pen, that blackness. The two blacknesses are exactly alike, to be sure, but I only have to look at them to realize they are two nonetheless*: WUE xii. So unless some new thesis concerning early medieval nominalism is being mooted, it will scarcely do to go on to assert, on the grounds just cited, that 'If this is what I am inclined to say, and if I would be inclined to say similar things in other such cases, then I do not believe in universals and am said to be a "nominalist". There is no single blackness shared by or common to the two pens. Indeed, there are no shared or common entitites at all in the way realists suppose*: WUE xiii. For we have just seen how the Compendium's denial of shared attributes need not at all ential denial of a non-nominally-termed level of discourse, i.e. need not entail nominalism. The individual blacknesses are Porretan subsistessences which, though evid­ enced by perception, are not subsistentities. Quine's twent­ ieth-century criterion of nominalism, based as it is on den­ ial of the sharedness of attributes, and which is here being imported for use on medieval material, is quite irrelevant in this context: cf. 0.4, 6.14. 2.82 Clarembald's Accusation 2.821 Gilbert of Poitier's later contemporary, Clarembald of Arras, feels that well-known mereological truisms enable him to score an easy point when he accuses the bishop of going absurdly wrong in his theory of part and whole. Thus when speaking of objects which are material, and hence comp­ osite, in his commentary on the Roman Boethius' De Trinitate, Clarembald says: (A) Moreover, the remainder which are in matter are not what they are, i.e. they are composite or can be compounded. 185

Section 2 This is because of no composite can we truly say, 'It is this', its parts standing in the way of this being true, be­ cause each of the composites is its parts [each of which is a 'this']. I am not saying that each such is this part or that part taken individually, but this and that taken tog­ ether. For it is certainly not the case that some integral part or another can be predicated of its whole, although the bishop of Poitiers would have been prepared to grant that integral parts could be predicated individually or sep­ arately of their wholes; this involved his making up for himself a false logical principle, or at least certainly putting a false construction on it, namely: Every generic ' attribute of the integral part is predicated of its whole. He thereby rightly understood the integral part's generic attribute to imply the quality of the part, but he wrongly applied the principle. For he was applying the aforemen­ tioned rule so as to cover even the predication of integral parts of the whole by the use of substantival words, as in 'Socrates is his head'. But had he been willing to have the predication effected by means of adjectival words, exception need not have been taken to his aforementioned rule, since the integral part's quality can be predicated by adjectival words, as in 'Socrates is headed'. Substantival names, how­ ever, can in no wise be used for the predication of integral parts of their whole. {Porro reliqua quae in materia sunt non sunt id quod sunt, i.e. composita sunt vel componi possunt. Etenim de nullo compositorum vere dicere possumus 'Hoc est', prohibentibus eius dictionis veritatem partibus suis cum singulum compositorum sit suae partes. Non dico haec pars singular iter vel illa, sed haec et illa simul acceptae. Neque enim pars aliqua vere integralis de toto suo praedícari potest, quamvis Episcopus Pictaviensis partes integrales singularіter  separatim de toto suo praedícari concederei, falsum sibi in logica fingens aut certe male in­ telligens princípium quod est hoc: Omne genus partis integ186

Abelard and his Contemporaries ralis praedicatur de suo toto, genus partis qualitatem par­ tís bene intelligens, sed prave assignans. Quippe sic praedictam regulam assignabat in partium íntegralium praedicatione ut secundum ipsam etiam vocabulis substantivis partem de toto integro praedicaret, ut 'Socrates est suum caput'. Verumtamen si per vocabula adiectiva earn praedicationem fi­ eri voluisset, praedicta eius regula refutari non debere t, quoniam quali tas partium íntegralium vocabulis adiectivis de toto praedicari potest, ut 'Socrates est capitatus'. Subst­ antivis vero nominibus partes integrales de toto nequaquam praedicari possunt: HCA 126, §§48 - 9.) 2.822 Here two points are initially obvious. In the first place, if Clarembald's accusation is correct, then Gilbert is scandalously trespassing against the traditional mereological doctrine (1.328 above) which vetoed the interpredication of part and whole. Both 'The house is its wall', and 'Socrates is his head' would be adjudged false by all mereologists, me­ dieval, modern, and contemporary, who are not of Gilbert's school. The other point is not prominent to the same extent, but we have apparently encountered an aspect of it in Abelard's work (2.211, 2.32) when he discusses the conditions under which the property of the whole is also applicable to the part. Gilbert seems to be here depicted as having pron­ ounced a thesis concerning the converse case, i.e. concerning the conditions under which the property of the part is also truly applicable to the whole. He has, says Clarembald in the just-quoted passage, allowed that the generic attribute (in Latin, 'genus' in some non-strict sense) of the part appl­ ies also to the whole. Now this is a thesis which, under at least one interpretation, has hitherto been commonly disall­ owed on the basis of the example transmitted by Boethius; this concerns the Ethiopian who has white teeth but is not hence said to be white. Aristotle had already made this point in his Topica, which resurfaced in the West about 1120 (CLM 46) and the Compendium's discussion will be quoted in 187

Section 2 2.845(B). At first sight that Compendium would appear to do as Boethius of Dacia was to do in this connection, i.e. to simply say that for a quality which pertains to the part to be predicated also of the whole, it is sufficient that it should pertain to the whole's more principal part: non oportet quod insit illi subiecto secundum quamlibet partem eius, sed sufficit inesse secundum partem principaliorem: BT 136.24.26. The Compendium has 'gratia sue principalis par­ tis', on the basis of its principal part': CLP 36.4. However rational a solution this may superficially sound, the al­ ready-encountered difficulties in deciding on a specific­ ation of 'principal part' (2.4) show that this merely trans­ fers the problem into another equally intractable area, and in any case this discussion may turn out not to be exhaust­ ive of the central point at issue. 2.823 Exactly how such adjectival applications are to be regulated will later also be discussed in commentaries on one of the fallacies listed in De Sophisticis Elenchisi 5.3 below. In Gilbert's time, this Aristotelian work was only just making its appearance. Already the problem was greatly agitating Gilbert and his contemporaries, as Clarembald's present remarks and the allusions made in the Compendium (2.85 below) make clear. 2.824 Now although the several natures of the two bones of contention outlined in 2.822 above are thus each tolerably clear taken in themselves, the precise structure of the in­ terconnections which Clarembald seems to be attributing to them is not so clear. Some of the trouble could be Gilbert's own apparently obscure modes of expression, for which Clar­ embald attacks him in another salient paragraph, claiming that the word-juggling is quite deliberate obfuscation. At the same time Clarembald confirms that the central scandal is the claim that any integral part can be predicated on its own and independently of its whole: HCA 127, §50. We have seen above how Clarembald also suggests an excellent way of 188

Abelard and his Contemporaries avoiding this scandal, namely the replacing of the subst­ antival names of the parts (e.g. 'head', 'foot', 'leg') by their corresponding adjectives (e.g. 'headed', 'footed', 'legged') for the purpose of such predication. Thus while 'Socrates is his head' is indeed mereologically inept, 'Soc-rates is headed' (or 'footed', or 'legged') would seem to be generally in order, provided we are not in a footballing con-text. For the moment, however, several points are by no means clear. Why, for instance, did not Gilbert, an immeasurably more original thinker than Clarembald, adopt this simple way out? Yet again, what is the connection between Gilbert's mereol­ ogically deviant thesis and the other problem (i.e. attrib­ ution of the quality of the part to the whole)? It looks very much as though the answer to these queries, and one of the impulses behind Gilbert's notorious obscurity, may be found in his other great development, namely the production of theses concerning the interrelation between the quiddity of objects and their mereological aspects. Gilbert's text contains a hint that the Roman Boethius has already used the vocabulary of this mix: HGP 91, §61. Investigation of Gilb­ ert's work in this field hence requires not only, as is ob­ vious, a systematic mereological background, but also a so­ urce of expressions and theses concerning the quiddity, the being, the esse, of objects, and this, as we have already recalled in 2.815, takes us to propositions founded on funct­ ors having s/(s/n s/n) as their categorial index. We are, for the first time in the history of the history of thought, in principle equipped with the means of clarifying and prec­ isely adjudicating on such mixed mereological and esse thes­ es, and 10.2 and 10.3 provide full information on the tech­ nical details. Hence it is now possible to go forward with a full awareness of the nature of the general area, and to see what Gilbert and his disciples have to say for themselves on these topics.

189

Section 2 2.83 The Gilbertian Innovations 2.831 There is, in the first place, no doubt that Gilbert did enuntiate principles which would ground the deviations detailed in 2.82 above, and as alleged by Clarembald. Thus, resembling almost word-for-word the latter's version of Gil­ bert we do in fact have Gilbert's saying, 'Every generic att­ ribute of the part must be the generic attribute of that wh­ ich is made up from the part', omne genus partis eius quod ex ea constat, genus esse necesse est: HGP 92, §64. Further, the mix between the quidditative terms formed from 'esse ../, 'being ...', and mereological vocabulary clearly figures in the argument which is going from this point to support the even more scandalous predication of the part in respect of the whole. Thus we have, 'Whatsoever is the esse of any part is also the esse of that which is made up from that part, and it is indeed asserted of that whole as well as of the part', Quicquid ... est esse cuiuslibet partis, eius etiam quod ex illa parte constat est esse. Et de eo sicut de parte uere dicitur. HGP 94, §72. Now it is possible, given the centur­ ies-old distinction between mereological and quiddative sen­ ses of 'part' (0.5 and HQS 190) to provide a totally unob­ jectionable interpretation of this last principle. Thus ani­ mal is a quidditative 'part' of man, and there is hence no difficulty in saying that the esse, the being, of man invol­ ves the being of animal, its quidditative part. Indeed, one can go on to predicate this sort of part of its whole, as in 'Man is animal', the sense of which is adequately provided for in our categorial language: 10.26. Gilbert himself takes up the distinction between the two types of part (HGP 92 3) and no one, including Clarembald, would be in the least disturbed by this sort of quidditative inference. Hence the mereological interpretation must be the source of all the scandal. 2.832 In fact Gilbert is overtly engaged in carrying for­ ward his mix of the mereological and the quidditative in 190

Abelard and his Contemporaries such a way that, as his last-quoted statement hints, it is to found the notoriously deviant predication of mereological part of its mereological whole. The way in which that mix operates in this rôle emerges in the argument next to be qu­ oted. In it Gilbert fishes up an allegedly logical principle which he interprets in such a way that it extends the scope of that already-quoted rule of his, which runs, 'Every gener­ ic attribute of the part must be the generic attribute of that which is made up from the part': cf. 2.831, HGP 92, §64. The extension now in question makes the generic attribute mentioned in the last-quoted translation into the proprium, the property (in the strict medieval sense (HQS 197)) which this principle permits to be predicated of the whole, even though it is still strictly the exclusive proprium of the part. For example, although being coloured is an exclusive proprium of body, it is nevertheless correct also to predic­ ate colour of man (of which body is in some sense a part). It is here that the convenient new extended principle interv­ enes. That principle runs as follows: of that of which the exclusive proprium of something is predicated, that to which the proprium pertains is also predicated. On such a basis not only the attribute, but also that of which it is the proprium is predicated of the whole. In this case, therefore, body (the ground of the attribution of colour to man) is also predicated of man. This chain of reasoning is correl­ ated with the esse in question, and hence presumably invol­ ves the mixed principle propounded in HGP 94 and also quoted in 2.831 above: (A) There is that self-evident axiomatic proposition of the logicians in which they assert: Of whatsoever is predic­ ated that which pertains to something as an exclusive prop­ rium, of that also is predicated that to which the proprium pertains. Since therefore colour (which pertains exclusive­ ly to that whence whatsoever is truly said to be body is body) is predicated not just of that part of man which is 191

Section 2 body, but also of man, it is also necessary that the esse of body, to which colour exclusively pertains, i.e. corporeity, should be predicated not merely of that part of man, but also of man himself . . . . A man is therefore a body, not because of the body from which he is made up, but on acc­ ount of the esse of that body. (IIIa autem dialecticorum maxima propositio per se nota est qua dicunt: De quocumque predicatur id, quod soli alicui adest, proprium, de eodem predicatur id cuius est proprium. Quoniam igitur color qui soli illi adest unde est corpus, quicquid uere dicitur 'corpus' - non modo de hominis illa parte que corpus est, sed etiam de homine predicatur, necesse est ut etiam illud esse corporis cui soli adest color, i.e. corporalitas non modo de hominis illa parte uerum etiam de homine predicetur ... . Est igitur homo corpus non ab eo, ex quo ipse constat, corpore, sed ab illius corporis esse: HGP 9 4 - 5 , §§76 - 8.) 2.833 Similar treatment is meted out to 'spirit', which in spite of its non-quantitiative nature, we have to assume to be another integral part of man for the purpose of following the argument. The use of this as an example of course mak­ es no difference to the mereological aspects of the disc­ ussion. The general impression is that Gilbert makes feat­ ures of the esse to be responsible for his deviant mereology (for it must be realised that the head adduced in Clarembald's example is as much an integral part as are the body and the spirit (or soul) now used as examples). 2.834 Gilbert also points out how his use of the alleged logical principle involving propria results in the alignment of already-admitted rules for the 'division' of combined adjectives (cf. HCD 248 - 250) with parallel (but hitherto uncountenanced) rules for the 'division' of combined substantive parts. (The Clarembald of 2.821(A) is plainly expressing shock at Gilbert's treatment of substantives in predication). Thus even as 'The man is coloured and wise' separates into 'The man is coloured' and 'The man is wise', 192

Abelard and his Contemporaries so also, for Gilbert, 'The man is body and soul' separates into 'The man is body' and 'The man is soul': HGP 96, §80. This is a very interesting assimilation of the two cases, suggesting as it does that adjectives and substantives belong to the same semantic category in Gilbert's presup­ posed canonical language. However, if we really have now moved into the mereological area, his thesis is still (as we are constantly noting) yielding deviant results. The corres­ ponding exercise which would yield the deviation recorded by Clarembald would simply involve saying, 'Socrates is his head, trunk, and limbs'. (Here head, trunk and limbs are taken to be the integral parts constituting the collective class or complete collection of Socratic proper parts). One certainly goes against the common mereological veto on assimilating parts and wholes (1.32) when one admits the ensuing infer­ ence of 'Socrates is his head'. 2.835 However, Gilbert has a way out of such difficulties. It involves an appeal to a distinction the terminology of which dates back at least to the Roman Boethius, as Krempel's work on its history reminds us (cf. DP 478). This is the dis­ tinction between esse (being) and dici (being asserted): ali­ ud ... est solitarium esse, aliud solitarium dici, 'It is one thing to be in isolation, but another to be asserted to be in isolation', says Gilbert: HGP 96, §81. So although one may presumably assert 'Socrates is his head', this is still comp­ atible with his being his non-isolated head, trunk, and limbs. Various other ways of glossing Gilbert's use of this distin­ ction are also possible. It would logically be most gratif­ ying if it could be shown to amount to the distinction bet­ ween the real (i.e. esse) and the apparent or merely gram­ matical (i.e. dici) form of an utterance. Then the 'real' form of 'Socrates is his head', would be, 'For some x, Soc­ rates is his head and x', wherein this χ would cover the re­ maining parts of the collective class of Socrates' integral parts; (compare HQS §5.2). Various construals of the copula 193

Section 2 '... is ...' (or its Latin counterparts) might also be brought to bear. However, such analyses are beyond the scope of the present superficial and preliminary narrative. 2.836 Hence in the absence of any immediate ingenious justificatory analyses, it looks as though Gilbert's content­ ions can still be put out of court by usual examples. Thus his initial principle that every attribute of the part must be an attribute of that which is made up from the part (2. 831) is falsifiable by the familiar case of the white-tooth­ ed Ethiopian who is not hence said to be white: 2.822, cf. 2.845. Neither is it clear that Gilbert's attempt to add a restriction which will rescue him from the consequent infer­ ence of human incorporeality from the fact of the incorporeality of the spiritual part of man (HGP 86 - 6) really suc­ ceeds. The full analytic examination of Gilbert's thesis is a vast work which would only be the initial step in following up the mereological aspects of the numerous other later com­ mentaries on Boethius' De Trinitate. None of this can be un­ dertaken here. However, a more expressly logical treatise by some disciple of Gilbert's calls for at least a preliminary survey of certain of its mereological features, in the hope that it may throw more immediate light on his theses, and it is to this treatise that we may now turn. 2.84 The Compendium of Porretan Logic. 2.841 As Clarembald's criticism has shown, and as the Compendium is now to confirm, quite a stir was caused by Gilbert's deliberately unorthodox mereology. Other persons and schools are said to be involved, thereby adding to the evidence for a twelfth-century mereological ferment which, as already seen, also influenced Abelard, the ps-Abelard, and Joscelin of Soissons. The Compendium itself is a most rem­ arkable work, well worthy of study and analysis for its own sake. Within the mereological and the mixed quidditativemereological zone, Gilbert's treatment of what Clarembald saw 194

Abelard and his Contemporaries as scandalous is naturally followed up avidly in this work of a disciple of the former. However, it turns out that an in­ teresting preliminary takes us to a view of creation and be­ coming which not only contrasts with the already-observed Abelardian stance (2.232) but also, yet again, puts us in the presence of material displaying affinities with certain of the views of Robert Boyle, physicist of the modern era. 2.842 Indeed, the heading of the first extract now to be provided is obviously tending to support a theory of the conservation of matter across time, and it is this, rather than any of Gilbert's tinkering with the esse-mereological mix, which is used here to found once more the scandalous 'Socrates is the body' It is notable that this disciple of Gilbert is clearly self-conscious about the necessity to def­ end his master's innovatory modes of expression: (A) The reason why every body there ever was is also to be in the future. In this connection the reason should be noted why creation, generation, and making, are to be dist­ inguished one from the other. Creation is the formation of the thing, but not from pre-existing matter, as with the soul. Here creation is, as it were, concretion, for both the matter and the form are made from nothing and their unific­ ation is effected as they are created. Generation is an ingression into substance, i.e. into substantial being, as the water which was becoming wine was ingressing into the sub­ stantial being of wine [in the course of the miraculous tr­ ansformation by Christ at the marriage-feast]. Making, in contrast, is the artificial changing of the state or shape of something, as when a statue is made from bronze or a cleric from a layman. So since at the creation of the wor­ ld the four elements were created from nothing, they are, as it were, the leaven of all things, because they were ad­ equate material for all bodies, and thus all bodies were cr­ eated in them. For the elements are the seedbeds of all bodies, and after their creation we do not read of a creat195

Section 2 ion of any body having taken place. But it is said that every day God creates new souls. Hence all bodies date back to the beginning of the world. This is what Ovid [sic] was talking about when, retailing the doctrine of the philosophers, he says that nothing is born of nothing, and nothing can revert to nothing. For they said that everything came from pre­ existent matter, and contrarywise, nothing ever finished being. Since, therefore, that which is this statue is body, and as it is a creature of the creator of all things, and the time when it was created by God cannot be assigned to any time after the creation of the world, it follows that creation conferred corporeal being on that which is this statue. Now even as we observe the changing of the elements (for fire is condensed into air, and conversely air is rar­ efied into fire; again, air is condensed into water, and conversely water is attenuated into air, whereas water is condensed into earth, while earth itself is dissolved into water) so also we observe the changing of that which is this statue. For having first been earth, it was changed into the hard state of stone (hence there ingressed upon it the substantial being of stone) and in this way gener-ation conferred the being of stone upon it ... . By human ministration (and hence by an author or maker) heat is ap­ plied to the stone, whereby the stone is melted and thus bronze is made. Hence the same process of making confers being on the bronze, and after the bronze, on the statue. Thus it comes about that our chief philosopher ass­ erts, 'Creation conferred bodily being on Socrates, gener­ ation, the being of a man, and making, that of a cleric'. For he granted that Socrates was first a body before being a man. Again, Socrates will be a body again after he has stopped being a man; for generation confers nothing more on him than what corruption takes away. Hence it is that we 196

Abelard and his Contemporaries say that [saint] P e t e r is in Rome, and t h a t the same P e t e r is in heaven. While our chief's following agrees with this truth, undiscriminating prating militates against its being pro­ claimed. For the unlearned assert, nay they swear, t h a t our opinion is a source of falsehood. Still, having seen t h a t every body had prior being, you can understand by the same mental intuition that no body altogether gives up its being. (Ratio quare dicatur omne corpus quod fuit esse et fore. Ad hoc notandum quoniam aliud est creatio, aliud generatio, aliud factura. Creatio est rei plasmatio ex non preiacente materia, ut anime: ibi enim est creatio quasi concretio; fit enim tam materia quam forma ex nichilo et in creando eorum adunatio. At generatio est ingressus in substantiam, id est in substantiale esse, ut aqua que fiebat vinum ingrediebatur substantiale esse vini. Factura vero est alicuius rei artificiosa immutatio status aut figure, ut si de ere fiat statua vel de laico littera tus.  і taque in mundi creatione creata ex nichilo fuerint quattuor elementa, quasi omnium elevamenta, quia erant omnium corporum sufficiens materia, in eis omnia corpora fuerunt creata. Nam elementa omnium corporum sunt seminaria, post quorum creationem non legitur alicuius corporis facta esse creatio. Sed dicitur quia Deus cotidie creat novas animas. Ergo omnia corpora sunt ab initio mundi. Quod Ovidius refert retractando philosophorum: ait 'gigni de nichilo nichil, in nichilum nichil posse revertí'. Dicebant enim omnia esse ex preiacente materia, nichil vero interire. Cum i taque id quod est hec statua corpus sit, et cum crea toris omnium creatura sit, et non sit assignare post mundi creationem quando creatum sit a Deo, ei quod est hec statua creatio contul it esse corpus. Sed cum videas elementorum mutationem, ignis enim spissatur in aera et converso aer rarescit in ignem, aer spissatur in aquam et converso aqua tenuatur in aera, aqua 197

Section 2 vero condensatur in terrain, terra vero dissolvitur in aquam; eodem modo id quod est hec statua, cum prius fuerit terra, versum fuit in lapidis duritiem; іtaque ingressum fuit substantiale esse lapidis, et sic generatio contulit illi esse lapidem ministro et per hoc quodammodo auctore vel factore homine accessit lapidi calor quo sol­ utus est lapis et ita es factus est. Itaque factura eadem contulit esse es et post eri statuam. Inde est quod ille summus noster ait philosophus: creatio contulit Socra ti esse corpus, generatio hominem, factura littera tum. Concessit enim Socrates prius fuisse corpus quam hominem; item Socratem fore corpus postquam desieri t esse homo: nichil enim ei aliud generatio contulit quam quod corruptio abstulit et converso. Inde est quod Petrum dicimus esse Rome et eundem in celo. Cui veritati cum consentía t posteritas, confiteri non permittit indiscreta garruli tas; asserì t indoctus immo iurat nostram opinionem seminare abusionem. At viso quod est omne corpus quod prius fuit eodem mentis instinctu comprehendas quod nullum corpus omnino suum esse amittit: CLP 33.21 - 34.58.) 2.8421 The contrasts, echos, and anticipations displayed in this passage are so manifold that comment could be extended almost indefinitely. Leaving aside the attribution to Ovid of a Persius quotation, along with the presence of some unattributed Ovid, the chief point remains this alternative route towards the mereologically objectionable predication of the part in respect of the whole, as in 'Socrates is his body'. This will be given further mainstream comment in due course. In the meantime, as CC 47 points out in an appendix (pp. 142 - 3) there are resemblances between the distinct­ ions here propounded and those of Simon of Tournai (who died about 1203). Even the vocabulary of 'ingression' which occ­ urs in this last passage is also there. In the translation provided that word has been transliterated from the original 198

Abelard and his Contemporaries Latin because of its uncanny resemblance to A.N. Whitehead's widespread use of the expression in our own day: e.g. WCN 144, 148. We also have what appears to be a remarkable doc­ trinal contrast with Abelard. Thus while the Porretan bel­ ieved that physical creation took place originally and once and for all, Abelard (as we have seen in 2..232) claimed that it still went on, by means of divine action, in processes such as glass-making (and hence presumably in the extraction of metal of the sort described in the passage above). 'Auth­ or', 'maker', and 'minister', are the Porretan's words for the metal extractor, whereas Abelard credits man only with the disposing of materials to make way for God's creative action in such cases: 2.232, cf. AD 419.36 - 420.6 and GA 298.36.40. The seventeenth-century Robert Boyle was skirting around the same points, with 'nature' (as with Abelard on occasion: 2.231(A)) now taking God's place, when he said: (A) As for what may be objected, that we must distinguish between factitious bodies and natural, I will not now stay to examine how far that distinction may be allowed. For it may suffice for our present purpose to represent that, wh­ atever may be said of factitious bodies where man does, by instruments of his own providing, only give figure, or also contexture, to the sensible (not insensible) parts of the matter he works upon - as when a joiner makes a stool, or a statuary makes an image, or a turner a bowl - yet the case may be very differing in those other factitious prod­ uctions wherein the insensible parts of the matter are alt­ ered by natural agents, who perform the greatest part of the work among themselves, though the artificer be an ass­ istant by putting them together after a due manner. And therefore I know not why all the productions of the fire made by chemists should be looked upon as not natural but artificial bodies, since the fire, which is the grand agent in these changes, doth not, by being employed by the chem­ ist, cease to be and to work as a natural agent; and since 199

Section 2 nature herself doth, by the help of the fire, sometimes aff­ ord us like productions that the alchemist's art presents us: BPP 74-5. John Wyclif also discussed such topics in his De Universal­ ibus, especially in its twelfth chapter, which deals with creation, and concludes that creation is the origin of all natural change: creatio est princípium omnis motus naturalis: WUE 1 3 3 - 9 , WUL 289 - 300. 2.8422 Although all such points have obvious mereological repercussions, it is the already-noted Porretan claim, made in the last-quoted Compendium passage, which is more relev­ ant to our central mereological concern. This is obviously propounded with an uneasiness which echoes objections, and asserts that Socrates is his body (or a body). Even more scandalous are the past and future tensed, prenatal and post mortem versions of this claim. While it is true that at one point Gilbert himself does say that a man may stop being a man or an animal, but go on being a body or spirit (HGP 320 - 1), nevertheless his vocabulary is not quite as tidily org­ anised as is that of the foregoing passage from the Compen­ dium. Indeed, Gilbert himself was prepared to say that the 'figure' of the bronze is created by the smelting of the rock-ore: HGP 88. 2.843 The Compendium's version of the already-observed Gilbertian arguments on the way to the scandalous attribut­ ions of parts to wholes proceeds by way of a distinction between principal and adjectival parts. At first the dist­ inction appears to be pitched at the quidditative level, say­ ing that genera such as body and animal are principal, and their adjectival propria secondary. However, use of examples such as colour's being secondary in respect of body, and knowing's in respect of spirit (CLP III.5) soon carries the discussion into the mereological field, with Gilbert's prin­ ciple that every attribute of the part must be the attrib­ ute of that which is made up from the part (HGP 92, §64 and 200

Abelard and his Contemporaries 2.831 above) and its extension (2.832 above) plainly in evidence: (A) But even as Socrates' being said to be coloured is grounded in a part, so also his being said to be rational is grounded in a principal part. Hence, since the soul is a more worthy part than the body, and Socrates is susceptible of predication of things secondary, so that he is said to be both coloured and rational, our philosopher was in harmony with usage when he asserted that Socrates is a body, but he was more in agreement with reason when he said that Socr­ ates is a spirit. In any case, usage grants that a man en­ tirely given over to reason and contemplation may be said to be altogether a spirit. Some of his disciples, repeating his reasoning, still follow this usage, but at the same time usage and detraction push certain of them, as it were fear­ ful, into silence concerning this truth. (Sed sicut gratia partis Socrates dicitur coloratus, ita gratia sue principalis partis dicitur rationalis. Unde, quoniam dignior pars est anima quam corpus, et Socrates suscepit predicationem secundariorum ut dicatur et coloratus et rationalis, noster philosophus consentiendo usui dixit Socratem esse corpus; magis vero acquiescens rationi dixit Socratem esse spiritum. Habet autem usus ut concedatur de hornine omni rationi et contemplationi dedito ipsum totum esse spiritum. Quem qui­ dam suorum adhuc secuntur imitatores rationis; sed quosdam usus upote tímidos et detractio hanc veritatern tacere cogit: CLP 36.3.11.) 2.8431 Although this argument is so unsure of itself that it has to search for support from questionable linguistic anthropology certified as 'usage', it has elements which, rel­ atively to its principles, are not totally unfeasible, esp­ ecially if expressed in article-free Latin, so that we can gloss it as follows. Nobody objects to Socrates' being said to be coloured, but this attribution is 'grounded in a part', namely the body, of which colour is a proprium. Hence that 201

Section 2 ground should also be predicable of Socrates, yielding 'Socr­ ates is body'. Likewise, 'Socrates is rational' is grounded on yet another part of Socrates, i.e. his spirit, of which rationality is a proprium. Hence, once again, Socrates is spirit. 2.8432 However, the full force of Gilbert's already-rec­ ounted characteristic mixture of the quidditative and the mereological is reflected in the immediately succeeding passage of the Compendium: (A) The reason why the following are asserted: (i) it is not that because a man ceases to be, hence something ceas­ es to be, (ii) nor is it the case that because a statue be­ gins to be, hence something begins to be. For, in his com­ ment on the De Trinitate of Boethius, the master asserts, 'Creation brings it about that the subsistessence pertains in such a way that that to which it pertains is thereby rendered something, whereas concretion adapts to the same subsistessence a nature of a further type in such wise that that to which it applies is not simple'. Thus ... creation first of all conferred upon Socrates that subsistessence whereby he is signified by this term 'substance', whereas concretion, i.e. substantiation into a further substantial thing, i.e. corporeity, adapts a nature of a further type to a subsistessence of the original sort. Thus came about the generation of Socrates into man­ kind. For one part of the man having first been prepared, i.e. the body, then the soul having been created and forth­ with infused (indeed, infused at creation), the adjoining of those two parts completed [the man] Socrates, and in like fashion by their disjunction they destroyed [the man] Socr­ ates. Now should anyone, were it possible, connumerate Soc­ rates' body and soul along with the rest of things, they would not come across anything which was not by nature [as opposed to generation] already there before. Likewise, when the parts joined together by generation have been dissoc202

Abelard and his Contemporaries iated by corruption, nothing is to be found to be non-ex­ istent which existed earlier. Hence also our philosopher says that Socrates is still soul and body, but not man, even as in the first place he was body before being man. (Ratio quare dicantur hec: non quia homo desinit esse aliquid des­ init esse,  quia statua incipit esse, aliquid incipit es­ se. Ait enim magister in librum Boetii de Trinitate, 'Cre­ ado subsistentiam inesse facit, ut cui inest ab ea aliquid sit; concret io vero eidem subsistentie naturas posterions rationis accomodat, ut cui cum illa insint simplex non siť. Ut... Socrati creatio primo contulit hanc subsistentiam qua significatur hoc termino 'substantia'; ac concretio, idest corporeitatem, substantiatio in aliud substantie, scilicet subsistentie priori naturas posterions rationis accomodat. Et sic Socrates generatus fuit in hominem: nam una parte hominis prius preparata, id est corpore, anima modo infundendo creata - immo in creando infusa - due ille part­ es adiuncte Socratem perfecere; eodem modo disiuncte Socratem destruxerunt. At si quis, si fieri posset, animam Socratis et corpus eiusdem ceteris connumerasset rebus, non inveniret quid esset quod naturaliter prius non fuisset; eodem modo corruptione disiunctis partibus generatione coniunctis non est invenire quid non sit quod prius fuerit. Inde etiam noster philosophus ait Socratem adhuc esse anim­ am et corpus, sed non hominem, sicut prius fuit corpus quam homo: CLP 36.13 - 37.30.) 2.8433 As far as thesis (i) (i.e. it's not that because a man ceases to be, hence something ceases to be) at the open­ ing of this passage is concerned, any admission of Socrates' destruction seems to be irrelevant unless it applies to Soc­ rates qua man, or qua generated (as opposed to created or natural object). Insertions to this effect have been added to the translation. Under these circumstances, and relat­ ively to this sort of terminology, an intelligible theory of man (or of Socrates) is perhaps emerging, but it parts comp203

Section 2 any with our usual ways of talking, according to which Socr­ ates would be the Socrates-man: cf. HQS 5.144. Indeed he (or it) is now being described as the being which begins to be a something as a product of substantiality to which is added corporeity, and which is then generated into mankind by the creation and infusion of his soul, finally becoming both a bodily non-man and a spiritual non-man after death, but thr­ oughout all these mutations, still remaining a something: hence the truth of thesis (i). This elaborate theory coheres with what was said above in the first quotation from the Compendium (2.842) and with Gilbert's own words (2. 832(A)). The discussion is yet another illustration of the extent to which the question De Destructione Socratis ('On the Dest­ ruction of Socrates') agitated thinkers of the period. Voiced by Abelard, it is also prominent in the Fragmentum Sangermanense (2.45) and the reference in the following continuance of the last passage to a 'Master Ivo' (perhaps Ivo of Chartres: CC 46, XIX - XX) further betokens the controversial heat which it engendered. In that continuance we move beyond th­ esis (i) of the last-quoted passage, and proceed to the proof of thesis (ii) therein: It's not that if a statue begins to be, then something begins to be. Thesis (i) is still report­ ed as under attack on the score of outlandishness, and it is here that Master Ivo is mentioned: (A) But Master Ivo went against this obvious truth, and while granting that [the man] Socrates ceased to be, never­ theless denied that he is something, a verdict which pert­ ains to logic rather than to theology. On the same grounds as before, when a statue is made by the contriving of a sh­ ape in its material, our reason cannot find anything now created or generated but which was not there before. Rath­ er, everything in question was created beforehand, and hence was something, it being definitely not now something for the first time. For our reason here discovers nought save that which is made, which cannot be asserted to be that wh204

Abelard and his Contemporaries ich pertains to nature. Hence it is not because a statue begins to be that something begins to be. (Sed hanc nudam magister Ivo obduxit veritatem, concedendo Socratem desinere esse,  tamen aliquid, quod magis est logicum quam teologicum. Eapropter, cum ex ere fiat statua materiei forma impressa, ratio non invenit aliquid nunc esse creatum vel generatum quod prius non fuerit. Sed cum quidlibet sit creatum ante, et ideo aliquid, istud vere non nunc prima est aliquid (ratio enim nichil repperіt nisi facturam, quam non potest dicere naturam); i taque non quia statua incipit esse aliquid incipit esse: CLP 37.31.38.) 2.8434 In the passage quoted in 2.842(A) above, the Comp­ endium has stressed that all bodies date back to the creat­ ion, but a statue is the product of comparatively recent making. Now, however, the quite unnecessary restriction, already implied in the previous quotation, is being overtly placed on the term 'aliquid','something', so that it is now to apply only to natural, created, things, and not to made things. True, this coheres with the Compendium's appendix (also in CC 46) which gives a special place to 'natural' pre­ dicates. But such restrictions are unlikely to be cost-eff­ ective, given the notorious variety of opinions concerning creation, and especially concerning exactly when it takes place. Abelard, Wyclif, and Boyle have already been cited above on this controverted point. The boundary between the natural and the made may likewise fluctuate according to op­ inion, as Boyle's passage in 2.8421(A) illustrates. Under such circumstances, to keep one's logic aloof from such var­ iety of opinion, and untainted by local controversy, 'some­ thing', whether in its quantificational, nominal, or quidd­ itative rôle,as distinguished in HQS §2.6, should not have this restriction imposed upon it. The verdicts on the statue and the man may then be aligned together without paradoxical diversification. Both are instances in which the makings (perhaps pre-discernible as man-parts and statue-parts) bec205

Section 2 ome parts-of-a-man and parts-of-a-statue by generation and making respectively: cf. 1.4, 2.3. In both cases something comes into being, begins to be, and given that the doctrine is here being propounded in terms of a natural language, it is a pointless conceit to tinker thus with so useful and co­ mmon a word as 'something', and the same goes for its var­ ious counterparts in diverse languages. An artificial categ­ orial language can bring out the medievally-recognised syst­ ematic ambiguity of 'something' (along with that of 'one', 'thing', and 'being': HQS §4.0). This will be a much more trenchant categorial matter than the merely lexicographical proposal here shakily erected on the shifting sands of opin­ ions about exactly what is created, and when. Needless to say, these last passages display throughout a total lack of appreciation of the distinction between -parts, which may be about the place when X does not exist, and parts-of-X, which are only there when X is there: cf. 1.4, 2.3. 2.844 In the next section of the Compendium is encoun­ tered a further possible argument in favour of the sort of Gilbertian thesis to which Clarembald objected. However, certain preliminary reminders are necessary if the passage is to be intelligible on a first reading. We have already witnessed in 2.843 above the tendency to drift to and fro between quidditative and mereological senses of 'principal' and 'adjectival' or secondary parts. Now an analogy is to be drawn between the two which requires acquaintance with a mereological position described, and perhaps espoused, by the Abelard of 2.4. It is a position also described and critic­ ised by Paul of Venice later in the Middle Ages, and taken up yet again by Woodger in our own day: WSW, cf. HQS 246, and 8.432 below. It was termed 'Destructivism' above, and entails that, for instance, the destruction of the slightest part of an object results in the destruction of that object (and possibly the starting of another object). In Abelard's version, this really amounts to the abolition of any distin206

Abelard and his Contemporaries ction between principal and secondary parts (mereologically speaking). The loss of the slightest pebble makes the earl­ ier house into another house. So although one may initially talk of a principal part (the absence of which destroys the whole, e.g. Socrates' head) and a secondary part (e.g. Socr­ ates 1 hand, the destruction of which need not destroy Socr­ ates), in point of fact both types of part turn out in the end to be equally principal as far as destructivism is con­ cerned. 2.845 Also in this next-quoted section of the Compendium an analogy between this mereological destructivism (which operates, of course, at the 'subsistentity' level) and what holds at the quidditative ('subsistessence') level, is insin­ uated. Given this and the analogous fact that all attrib­ utes, whether essential or incidental are, in Porretan doct­ rine, individuated (2.816), then the following discussion of the quasi-Gilbertian thesis, 'Whatsoever is essential to the part is also essential to the whole' (cf. 2.831) becomes more readily intelligible: (A) In the same way as the very least particles contr­ ibute to the make-up of the [mereological] whole, so like­ wise all the forms of the parts, just as much the substant­ ial ones as the incidental ones, contribute to the indiv­ idual-constituting property. Taking one case will show what is intended in the rest. For even as the hand-of-Soc­ rates is a secondary [mereological] part-of-Socrates, so also the whiteness-whereby-the-hand-of-Socrates-is-white, since it is attributable only to Socrates, is an individ­ uating attribute-of-Socrates. It is in like manner that whatsoever accident pertains to the part pertains also to its whole. (Sicut minime particule sunt in constitutione totius, sic in constitutione personalis proprietatis sunt omnes forme partium, tam substantiales quam accidentales. Quod in uno consideratum in ceteris sit manifestum. Cum enim secundaria pars Socra tis sit manus Socratis, albedo 207

Section 2 qua Socra tis manus alba est, cum non sit in alio quam in Socrate, in Socrate est [secundario] ; et similiter quodlibet accidentale quod parti inest, et toti illius: CPL 37.43 - 38.49.) The import of this (on the basis of which the textual amen­ dment has been suggested) would appear to be as follows: individuated properties, even if in the first place they ap­ ply to mereologically secondary parts, also pertain to the whole. Thus far this is familiar Gilbertian ground, as 2.832 has shown. This conclusion would appear to lead, continues the Compendium, to an ensuing difficulty, since contrary to this conclusion and as pointed out by Aristotle, we still do not call someone 'white', for example, merely because some part such as their set of teeth, happens to be white. (Oth­ erwise, the semantics of a large number of public notices in racially discriminatory countries would have to be radically revised in order to obtain the desired effect. 'Whites only' would let in too many!) (B) In this connection Aristotle asserted [Topica Bk II, ch. 1], 'It is difficult to find a case in which a thing is named from its accident, or convertibly so named', meaning thereby, as it were, 'It is difficult to prove so and so to be the white merely on account of whiteness occurring in it'. This he said because although the accident of the part pertains to the whole, nevertheless the name it thus makes applic­ able to the part does not always apply also to the whole to which it is related as a part. (Unde Aristoteles dicit: 'Difficile est fieri denominationem ab accidente vel conv­ erti', quasi diceret, 'Difficile est aliquid probari album quoniam ei insit albedo'; quod ideo dixit quoniam accidens partis inest toti, non tarnen semper nomina tio quam confert parti, et toti cui inest secundum partem: CLP 38.49.53.) The accent on naming (nominatio) here might suggest that the distinction between dici and esse described in 2.835 may be operating. Thus although the whiteness may pertain to (in208

Abelard and his Contemporaries sit) the whole, the corresponding adjective ('white' ), taken nominally, need not be predicated of it. (Here is an inter­ esting extension of the problem of paronymy (i.e. roughly, of adjectival terms) so engagingly dealt with by Anselm of Aos­ ta and Canterbury in his De Grammatico: cf. HL §3, HDG, HCD, HQS §3.3). 2.846 The learned allusion to Aristotle's then lately rev­ ived work having thus been incorporated, the discussion rev­ erts to the ontologicai question on the side of esse (being ...) as opposed to dici (being said to be ...), thus: (A) Hence as the whiteness-of-Socrates'-hand pertains to Socrates, and as that whiteness goes along with the corpor­ eity which is part of the individual quiddity of this hand. it must be that the same [individuated] whiteness is joined to the quiddity of this hand. Hence it comes about that as this whiteness pertains to this quiddity whereby the hand of Socrates is his hand, it is necessary that even as that whiteness pertains to Socrates, so also the quiddity where­ by his hand is his hand pertains to him. (Itaque, cum alb­ edo manus Socratis insit Socrati, et illa adsit corporeitati que est pars individualis essentie manus huius, oportet ut eadem albedo huic essentie manus adiaceat; unde fit ut cum hec albedo adsit huic essentie, qua Socratis manus est man­ us, oportet ut cum illa albedo insit Socrati, etiam essentia qua manus eius est manus insit eidem: CLP 38.53.38.) Here we not only have (from the previous extract) whiteness pertaining to Socrates on account of his white hand, but al­ so, thanks to the importation of the hand's quiddity as the 'genus partis', we are now on the brink of Socrates' being his very hand ('the quiddity whereby his hand is his hand pertains to him'). Whatever one may think of the argument, it is clearly attempting to underpin Gilbert's principles as viewed in 2.83 above, and also runs parallel to the one env­ isaged by Clarembald when he accuses Gilbert of misinter­ preting, 'Every generic attribute of the integral part is 209

Section 2 predicated of its whole', Omne genus partis integralis praedicatur de suo toto: 2.821. Indeed, Clarembald would appear to have in mind the two-step process (from predication of attribute to predication of the part which has the attribute) just noted when he uses 'etiam', 'even' (or 'also') in his description of Gilbert's misuse of this rule to cover the predication of integral parts, so that under its authority 'he would predicate the part of its integral whole, even using substantive terms' (cf. 2. 821(A)). It is now also becoming even more apparent that genus partis, which I have translated as 'generic attribute of the part' in the quoted maxim, incorporates that latent ambiguity which allows it to mean either 'attribute applicable to the part' (e.g. 'white­ ness'), or 'attribute which is the abstract correlate of the part-name' (e.g. 'hand-ness'). 2.847 The point of Clarembald's shocked 'etiam may thus be seen as part of the broader history of paronymy, to which allusion was made before the last-quoted passage. Where adjectival terms are concerned, no one has any qualms in saying that, for example, whiteness (albedo) pertains to Socrates, so that Socrates is said to be white (albus); this use of the concrete adjective ensues upon the assertion of the attribution of the corresponding quality betokened by the abstract noun, 'whiteness'. The effect of the Compen­ dium's last-quoted argument is to incorporate the partattribute in an enlarged abstract attribute, so that whitecorporeal-Socratic-hand-ness (which incorporates the attrib­ ute which is the abstract correlate of the part-name, and for which, as a whole, we lack a more compact usage-approved abstract noun) is available should one wish to actually use it to found a predication of the corresponding concrete nom­ inal expression, 'white-corporeal-Socratic-hand' directly of Socrates. Though that 'quiddity whereby his hand is his hand' is thus said to pertain to him, we do not yet follow the paronymous parallel and concretely assert that Socrates 210

Abelard and his Contemporaries is such a hand. For the moment the Compendium coyly draws back from inferring such a scandalous consequence. As we have just noted, the move from abstract to concrete can sometimes be tolerated where the usual adjectival paronyms such as 'white' are in question, but (as Clarembald's shocked 'etiam' hints) a corresponding move on the non-adjectival, i.e. substantival, side (using, for example, 'hand' or 'head' as predicate terms) sounds either like a sin against logical gr­ ammar, or like a move productive of mereological falsehoods. 2.8471 This may explain why the Compendium, having shown everything on the side of esse (being) which would authorise that sin, nevertheless modestly holds back from it by temp­ orarily refusing to let it be said (dici), as suggested by Gilbert's own distinction between esse and dici recorded above in 2.835: (A) Yet I may not say that Socrates is the hand. This is because the quiddities are not predicated of all those th­ ings which share in them (and neither are accidents always thus predicated) but only of those things which principally have a share in them. Hence, when Aristotle said that name-taking issued from genus and species, he understood that this applied to those things by which they were princ­ ipally shared, e.g. this genus animal is predicated of that by which it is principally shared. (Non tamen dicam Socratem esse m anum, quia essentie non de omnibus predicantur a quibus participantur (sicut nec accidentia), sed de illis a quibus principaliter participantur. Unde cum dicit Aristoteles denominationem fieri a genere et specie intellexit a quibus principaliter participantur, ut hoc genus Animal de eo predicatur a quo principaliter participatur: CLP 38.58.63.) It is after this appeal to the non-mereological sense of 'principal', encountered above (2.843, CLP II, 5), and perhaps by a back-handed extension of Aristotle's Topica doctrine to the mereological parts, and not merely to the quidditative parts, that the author tentatively lets in the novel predic211

Section 2 ation of integral parts, with a suggestion that our custom­ ary language is inadequate to express such predications: (B) Alternatively it can be said that even as the genera and species of the parts may be shared in by their wholes [given the basis described above: 2.844, .845] one does indeed have convertible naming along with that sharing. However, there is a lack of suitable and customary modes of expression whereby this may be effectuated. Hence we have now suitably elucidated, 'Whatsoever is substantial to the part is also substantial to the whole'. (Vel dici potest quod cum genera partium et species participentur a totis, revera convertibilis denominatio cum participatione. Sed commoditas et idonei tas vocabulorum quibus hoc fiat deficit. Dictum est ergo quomodo intelligi conveniat 'quicquid est substantiale parti et toti': CLP 38.63.57.) 2.8472 In point of fact, what Clarembald has displayed in his criticism quoted in 2.821(A), is his realisation, to which both Gilbert and the author of the present Compendium pass­ ages seem obtusely blind, that it is at least possible to extend language in an intelligible and often usage-approved fashion in order to express the predication of the 'being' of the parts. Thus in order to express the Compendium's con­ clusion at this point on the side of esse (but using the ex­ ample of the head, rather than that of the hand) Clarembald has rightly said that the 'attribute of the integral part', even in the sense of 'attribute which is the abstract correl­ ate of the concrete part-name', can be said to pertain to the whole, but only with the ensuing predication of adjectivallyformed names, as in 'Socrates is headed': 2.821(A). Here the parallel with the above-described adjectival paronymous sit­ uation is maintained. The move from the abstract essence (the 'quality of the individual part') to the corresponding concrete name (in the broad classical sense of 'name') need not result in the predication of a substantival concrete ('head', 'hand'), but (as a parallel to the move from 'white212

Abelard and his Contemporaries ness' to 'white', recalled in 2.847) of an adjectival concrete ('headed', 'handed', 'legged', and so on). This takes one out of the mereological, and back into the area of the more us­ ually-encountered distributive whole: 2.21(A). 2.8473 The fact that the Compendium uses the hand as an example where Clarembald mentions the head, makes no mat­ erial difference between the two cases, especially from Gil­ bert's point of view. For him and the Compendium the hand, head, or any individual insignificant fragment are all equal­ ly principal, given the destructivist analogy which is being drawn here: 2.844, .845. Clarembald, not wishing to be drawn into such details, uses the head as a commonly recognised principal part. In any case, whether or not one adopts the destructivist principle cited at the opening of the argument quoted in 2.845(A) concerning the very least particles (min­ ime particule), the head is intuititively principal in a str­ onger mereological sense of 'principal' than is the hand, sin­ ce there are lots of people without hands. Hence the ident­ ification of Socrates and his head is more striking, whatever one's presuppositions as to principality. It might also be that the escape into 'capitatus' ('headed') could be consid­ ered a slight degree more acceptable from the point of view of usage than would be the Latin correlates of 'legged', or 'handed'. 2.8474 The following through of Clarembald's original obj­ ection (2.821(A)) has already served to indicate briefly the mereological threads of the Compendium. There is, however, much more which still remains to be made explicit. For the moment, nevertheless, a further concluding sample must suff­ ice. Abelard had mentioned the possible thesis that artif­ icial objects are just collectives which happen to have had their parts juxtaposed (2.310), and this thesis is now not only discussed, but linked with yet another attempt to und­ erpin the characteristic predication of the part, as champ­ ioned by Gilbert. Once again, the names of groups of scholars 213

Section 2 bear out the formidable dimensions of the discussion, alth­ ough who the 'Coppausists' were remains mysterious: (A) The reason why every contiguous whole may be said to be many (i.e. its parts) but this does not apply to any con­ tinuous whole. Of wholes, some are universal, some integr­ al, some virtual. Animal exemplifies the universal whole, house the integral, and the soul is a case of the virtual whole. For although the soul lacks a binding-together of parts, it does not lack an admixture of natural capacities, such as rationality, the capacity for desire, and the power of growing. Integral wholes, however, can be divided into those which have their parts gathered together, as tree, and those which have their parts separated, as flock. Now of the wholes which have their parts gathered together, some are continuous, such as stone, others are contiguous, such as cuirass, and others successive, such as time. Now the continuous whole is that the parts of which are so natur­ ally admixed that they are believed to hold together, or at least to grip one another, so that they are said to be con­ tinuous, as with a stone, which is truly said to be one thing. In contrast, the contiguous whole is that the parts of which are by nature disjoined, but which are put into con­ junction by a craft, as in the case of a cuirass or a piece of cloth. In these instances part is put close to part and gathered together so that a whole is thence effectuated. However, there is in fact no difference at all between a whole of this contiguous sort and one which has its parts separated, or which is collective [in the old-fashioned 'fl­ ock' sense], except that the contiguous one has its parts more close together, whereas the parts of the other are more disjoined, whether because of the nature of the craft, or by mere chance. Why, then, is the scattered-part whole many, as contrasted with the unity of the contiguous whole? Let us take it that we have eight stones here. If these 214

Abelard and his Contemporaries are not yet closely joined together, then we still have a scattered-part whole which is made up from those stones. If, however, they are brought closer together, they constit­ ute a whole which will not be identical with those parts, and this does not seem right . . . . In fact, as nearly all schools of thought treat the scatt­ ered-part wholes as many, so the contiguous whole may be said to be many, although this is denied by the Montanists, or at least only granted by a few, such as the Nominalists and Coppausists, who grant that just about any whole is id­ entical with its parts. But all those upholders of our sch­ ool of thought who have not looked into the reason for the truth of the thesis proposed above do not admit it. Never­ theless, if that reason is more closely inspected by them, then on the basis of that proposed thesis which they are compelled to grant, both 'Socrates is the body', and 'Soc­ rates is the spirit', ensue. (Ratio quare dicatur omne totum continguum esse plura (scilicet suas partes), sed nullum co­ ntinuum. Totorum aliud universale, aliud integrale, aliud virtuale. Universale est ut animai, integrale, ut domus, virtuale ut anima. Licet enim anima careat partium comp­ age, non tarnen caret naturalíum potentìarum concretíone, ut rationabilitatis et concupiscibilitatis et vegetationis. Sed totorum integralìum aliud congregativum suarum partium ut arbor, aliud disgregativum ut grex. Sed totorum congregativorum aliud continuum ut lapis, aliud contiguum ut lor­ ica, aliud successivum ut tempus. Totum itaque continuum est cuius partes sic sibi invicem sunt concrete naturaliter ut sese invicem continere vel saltem tenere credantur, unde continuari dicuntur, ut in lapide, de quo vere dicitur quod sit unum. At totum contiguum est cuius partes naturaliter disiuncte, arte vero sunt coniuncte, ut videtur in lorica vel in tela; ibi enim pars parti contiguatur et aggregatur ut totum in­ de efficiatur. At cum inter huiusmodi totum quod dicitur 215

Section 2 contiguum et totum disgregativum sive coUectivum adeo nul­ la sit differentia nisi quod partes huius ad se invicem sunt propiores, illius vero vel arte vel casu disiunctiores, que est ratio istud plura esse quod est disgregativum, illud vero unum quod est contiguum? Sint enim hic octo lapides; si nondum stricte sunt coniuncti, adhuc est totum disgreg­ ativum, quod est illi lapides; si vero strictius iungantur, inde fiet totum quod non erit Ule partes; quod non videtur. At cum fere ab omnibus totum disgregativum dicatur plura, et totum contiguum dicatur esse plura; quod tarnen a Mont­ anis negatur  a raris conceditur, nisi a Nominalibus et Coppausis, qui de omni fere toto concedunt quod sit sue partes. Sed hec positio non ab omnibus nostre doctrine professoribus recipitur, a quibus ratio veritatis non inv­ ­stigatur. Sed ex hac positione, 'Socrates est corpus', 'Socrates est spiritus' idem concedere coguntur si ab eius ratio subtilius disquiratur: CLP 38.71 - 39.100.) The point here seems to be that body and spirit are two parts of a contiguous whole, i.e Socrates, who, by the Porretan doctrine, may be said to be severally each of these. This is backed up by a claim that really there is no differ­ ence between a contiguous and a scattered-part whole, as illustrated by the case of the eight stones: both types of whole equally deserve a plural characterisation. Hence each part of that plurality can be predicated of that plurality. Thus Socrates is the contiguous body and spirit; hence he is body; hence also he is spirit. The statement that nearly all schools of thought treat the scattered-part wholes each as many is at least verified by Abelard's position in 2.31(A). The great volume of grammatical discussion on just this point may be gauged from I. Rosier's recent conspectus: ROM 25.

216

Abelard and his Contemporaries 2.9

Concluding Remarks on Section 2 It is deeply to be regretted that the exigencies of this hasty preliminary survey must lead to the close of these un­ satisfactorily brief remarks on some published twelfth-cent­ ury mereological texts. Each of the phases covered or ment­ ioned, particularly the mereological aspects of the works of Abelard, Joscelin of Soissons, Gilbert of Poitiers, and the Porretan Compendium, urgently demands the production of at least one separate full-length book. The addition of further hitherto unpublished texts continues to make the situation even more urgent. Yet all this richness is itself only the beginning of an intellectual stream which in the coming med­ ieval centuries is to outstrip still further the narrative and analytic capacities of the present initial survey.

217

3. Aquinas — Mereological Aspects 3.1

Metaphysical Background

3.101 Introduction Mereology is itself, by common medieval consent, part of Metaphysics, as 0.2 has intimated. It so happens that very ample background metaphysical material is easily accessible where Thomas Aquinas, a canonised member of the Order of Pr­ eachers (Dominicans), who lived from about 1225 to 1274, is concerned. The ecclesiastical-political reasons for such acc­ essibility are related in CLM 845. Advantage may now be tak­ en of the availability of Thomas' treatment of some more gen­ eral relevant presuppositions when he comments on Aristotle's Metaphysica. In the selection of texts from that commentary (ACM) which follows, allusions to 'the Philosopher' are in fact to Aristotle (about 384 -322 B.C.). Almost the full corpus of this thinker's works was, by Aquinas' time, becoming available to the Latin West. Translations into Latin made it possible for the Westerners to go beyond the more purely logical Aris­ totelian heritage bequeathed by the Boethius: CLM 4 5 - 7 9 .

3.11 From Elements to Form 3.111 Since, in what follows, attention will at times be focussed on Aristotelian 'form' as a unity-endowing coeffic­ ient for wholes in general, and in particular where Aquinas' equation of form and soul in organisms is concerned, it will 218

Aquinas - Mereological Aspects be of interest to draw upon Aquinas' Aristotle-based pleas in favour of such a notion of form, especially at points where mereological theorems also intervene, as in the first extract below (3.113(A)). This contains one of the theorems which follows from the already-noted theorem to the effect that if A is a proper part of B, then it's not that A is  (1.328.1, 10.324). From this it also obviously follows that if A is a proper part of B, then it's not that  is A. It is either this, or its immediate consequence that nothing is a proper part of itself (10.323) which is a present-day correlate of Aquinas', 'In all such instances it must be the case that the composite itself is not those things whence it is compounded', as it occurs in that extract. First, however, some further preliminary remarks need to be made. 3.112 Given that the extract's examples of the syllable and of composites involving the Aristotelian elements (earth, air, fire, and water) are mere illustrations of quasi-mereological generalities, it is clear that their particular associations (e.g. the sometimes chemical style of talk about the compos­ ition of elements) should not be allowed to obfuscate those generalities. When it is said below that 'upon the separation of the elements flesh does not remain, and upon the separat­ ion of the letters the syllable does not remain', the defic­ iencies of both these examples enable the one to serve as a corrective for the other. Thus for the purpose of the disc­ ussion, the names of the Aristotelian elements figure as mere labels for parts which can be clawed away from one another, rather in the way in which the letters of the syllable may be disassembled. Such element-labels are handy because the hom­ ogeneity of flesh (cf. 1.4, 1.32) would otherwise involve one in the distinction between the whole parcel of flesh and its individuated potential gobbets, which are also flesh. Such a distinction might nowadays be shown by typographical aids such as numerical or alphabetical subscripts of a sort not readily available to Aristotle and Aquinas. At the same time 219

Section 3 the labels 'fire' and 'earth' serve to distinguish between the flesh-parts which they may be taken to designate, as opposed to parts-of-flesh which are all flesh and coeval with the ab­ iding fleshy whole: cf. 1.4, 2.3. Such a service is absent where the letters are con­ cerned, since here one finds oneself saying, as does the text, that 'the letters do remain, after the dissolution of the syl­ lable'. This conveys an impression of the continued identity of the letters, so blinding one to their transition from parts of-a-syllable to syllable-parts, upon the dissolution of the syllable. The rôle of stressing such transition, of which Aquinas and all the Aristotelians were in some areas immense­ ly conscious, as the organic examples below are to show in 3.41, was borne by the notion of form as the ensurer of a hook on which could be hung the 'part-of-a- ...' genitivelyformed functor: 1.41. The form, essence, or quiddity is the truly structural counterpart of the specification covering the sort of thing the name of which completes that functor. As such, form-discourse can, as it were, take on a life of its own, in the complex of definitions, genera, species, and diff­ erentiae, which is provided by the predicables: 0.4, HQS §4.3. Such discourse can also enmesh the student of these matters in the centuries-old questions about how abstract quiddity and concrete individuals are related. A brief interjection on such points will hence shortly be added in 3.12. Incidentally, it should scarcely be necessary to point out that the Arist­ otelian elements should not be confused with the mereological 'element of ...' as defined in Dl of 10.31 below. 3.113 In the meantime, here, as promised, is Aquinas stres­ sing the non-identity of the whole and its elements. He then is to be saddled with the task of explaining how the elements of a whole change their identity upon incorporation in, or ex­ pulsion from, the whole which they compose. This he explains by the intervention of 'something' over and above those elem­ ents. (It is because this text is a commentary upon some 220

Aquinas - Mereological Aspects words of Aristotle that it incorporates mentioned expressions, enclosed in quotation marks.) (A) Hence the Philosopher says that because something is composed from something 'as an all', i.e. is wholly one, and not in the manner of a heap of stones, but rather in the style of a syllable, it is a unity in the absolute sense. In all such instances it must be that the composite itself is not those things whence it is compounded, even as a syllable is not its elements. Thus the syllable 'ba' is not identical with the two letters V and 'a'; likewise neither is the flesh identical with [its elements] fire and earth. This he proves in the following manner: because 'when dissolved' (i.e. when those things whence is made the composite are separat­ ed one from the other) 'this' (i.e. the whole), does not rem­ ain after the dissolution. Thus upon the separation of the elements flesh does not remain, and upon the separation of the letters the syllable does not remain. 'In contrast, the elements ...', i.e. the letters, do remain after the dissolution of the syllable, as also the fire and earth abide after the dissolution of the flesh. Hence a syllable is something over and above its elements, and that whereby the syllable is a syllable consists not merely in the vowel and consonant which are its elements, but in something else.. Thus also likewise flesh is not just fire and earth, or the hot and the cold under the influence of which the elements are mingled, but is also something else, whereby flesh is flesh. (Dicit ergo Philosophus, quod quia aliquid est sic ex aliquo compositum 'ut omne', idest totum sit unum, et non hoc modo sicut cumulus lapidum, sed sicut syllaba, quae est unum simpliciter; in omnibus talibus oportet quod ipsum compositum non sit ea ex quibus componitur, sicut syllaba non est elementa. Sicut haec syllaba, quae est A, non est idem quod hae duae literae  et ;  caro est idem quod ignis et terra. Et hoc sic probat: quia 'dissolutis', idest divisis ab invicem, his ex quibus fit compositio, 'haec', scilicet totum, non 221

Section 3 adhuc remanet post dissolutionem. Sicut iam divis is elementis non remanet caro, et divisis literis non remanet syllaba. 'Elementa vero', idest literae, remanent post dissolutionem syllabae, et ignis et terra remanent post dissolutionem carnis. Igitur syllaba est aliquid praeter elementa; et non sol­ um est elementa, quae sunt vocalis et consonans, sed alterum aliquid, per quod syllaba est syllaba. Et sic similiter et caro non solum est ignis et terra, aut calidum et frigidum, per quorum virtutem elementa commiscentur, sed etiam est aliquid alterum per quod caro est caro: ACM lib. VII, lec. XVII, 1674.) 3.114 The difficulty of thus talking about 'something over and above' without lapsing into the linguistically-generated illusion that here we have a something which is of the same type as the various somethings which make up the whole (i.e. the elements, the letters, and so on), is portended by the atomist-type argument, based on the principle that 'everything which is, is either an element or is compounded from elem­ ents'. This argument threatens an infinite regress: (A) Next, when he says, 'If therefore it is necessary ...', he raises a doubt about the main part of the discussion. For it has been shown that in flesh and syllable there is something over and above the elements in question. But it would appear that everything which is, is either an element, or is compounded from elements. If, therefore, there must be something that is in the flesh and the syllable which is other than their elements, and which is either an element or is compounded from elements, then the following awkwardness­ es ensue. If that something is indeed an element, then the same reasoning as before can be applied both to it and to the other elements. For it will have to be counted along with the others. Flesh will then be compounded from this something which we said was over and above the elements, but which we now suppose to be an element, and also from fire and earth [the supposed elements of flesh]. And because 222

Aquinas - Mereological Aspects it has already been proved that in every unitary composite there must be something over and above its elements, the same question will arise again about what is now over and above. If it is an element, then yet again flesh will con­ sist of the first element which was over and above, as well as of the original elements, and now again of some further one. Hence in this way the process will extend to infinity, and this is incoherent. (Deinde, cum dicit 'Si igitur necesse intendit. ...', movet dubitationem circa id quod principaliter Ostensum est enim quod in carne et syllaba est aliquid pr­ aeter elementa. Videtur autem, quod omne quod est, aut sit elementum, aut ex elementis. Si igitur necesse est id aliquod, quod est in carne et syllaba praeter elementa, aut elem­ entum esse, aut ex elementis, sequuntur haec inconvenienti a. Si quidem enim sit elementum, iterum erit eadem ratio et de hoc et de aliis elementis. Connumerandum enim aliis erit. Erit enim caro composita ex hoc quod diximus praeter elem­ enta et nunc ponimus elementum esse, et ex igne et terra. Et quia iam probatum est quod in omni compositum quod est unum, oportet esse aliquid aliud praeter elementa, erit eadem quaestio adhuc de illo alio; quod si sit elementum, iterum caro erit et primo alio elemento et ex elementis, et adhuc ex aliquo alio. Quare sic ibit in infinitum; quod est inconveniens: ACM ibid. 1675,1676.) 3.115 Any attempt to dodge the regress by making the 'something over and above' into an element-dependent some­ thing still fails: (A) If, therefore, we take it that this thing over and above which has been found is not itself an element, but comes from the existing elements, it is obvious that it is not made up from one element only, but from several elem­ ents. For were it not from many elements, but from one only, it would follow that the whole would be that very element. For that which is made up from water only is real­ ly just water. But if the thing is made from several elem223

Section 3 ents, then yet again the same reasoning will apply as in the cases of the flesh and the syllable, i.e. there will be within it something over and above the elements whence it is com­ posed. And thus yet again there will be an infinite regress. (Si ergo istud aliud inventum, non sit elementum, sed sit ex elementis; palam est quia non est ex elemento uno tantum, sed ex pluribus elementis. Quia si non esset ex pluribus, sed ex uno tantum, sequeretur quod esset illud idem elem­ entum totum. Quod enim est ex aqua tantum, est vere aqua. Quare si sit ex pluribus elementis, iterum eadem ratio erit de hoc quae et de carne et syllaba, quia scilicet erit in eo aliquid praeter elementa ex quibus est. Et de hoc iterum redibit quaestio eadem. Et sic iterum procedit in infinit­ um: ACM ibid. 1677.) 3.116 The dialectic continues, with the upshot that the 'something over and above' is indeed the quiddity (ACM 1678). This is not an element but a principle (ACM 1679). Further, we can call the quiddity a substance (in the 'secondary' sen­ se,cf. 0.44) and come to realise its type-distinction from the elements. It is their 'formal' principle, as distinct from their 'matter', i.e. their material make-up: (A) This sort of thing is not an element but rather a for­ mal principle, for an element is said to be that into which something is split 'inexistently', i.e. intrinsically, as in the case of the material make-up, e.g. the elements of the syll­ able BA and A and B. Hence, as the aforementioned principle will not be material, but formal, it will not be an element. Again, it is at one stroke obvious both (i) what kind of principle is the [secondary] substance, and (ii) that that principle is neither an element nor composed from elements. Thus is resolved the foregoing problem* (Quae etiam natura non est elementum sed principium formale. Elementum vero dicitur id in quo aliquid dividitur 'inexistens', idest est intrinsecum, sicut in materiam, puta elementa syllabae BA sunt , . Unde cum praedictum principium non sit materiale, 224

Aquinas - Mereological Aspects sed formale, non erit elementum. Et sic simul patet, et qua­ le principium est substantia; et quod neque est elementum, neque ex elementis; in quo solvitur dubitatio praemissa: ACM ibid. 1680.) 3.117 In our (and the Aristotelian) natural language we are now launched upon the countenancing of a 'formal princ­ iple', i.e. the quiddity, and the time is going to come when not only linguistic pressure ('Surely the quiddity is some­ things) but also doctrinal desiderata (e.g. with reference to the survival of the formal principle when identified in app­ ropriate cases as the human soul) will subject thinkers to speculative stress from which facile relief may be gained by succumbing in the graceful manner which the flexibility of unguarded quasi-ordinary language, when thus press-ganged into crewing a distant theoretical voyage, is only too ready to encourage. The quiddity, (or form, or essence) is then left as a vaguely odd sort of 'thing', the exact logical-gram­ matical status of which is only rarely made determinate. How­ ever, by means of reference to its coordinates relative to more unproblematic discourse about stocks, stones, animals, and other concrete objects, it is in fact possible to begin to make that status somewhat more determinately intelligible. Aquinas' own approach to such clarifying coordinates, of the sort ultimately exposed in 10.2, may accordingly now be sam­ pled, still drawing on the commentary on Aristotle's Metaphysica. 3.12 Quiddity and the Individual 3.121 Sentences about (for example) man in the quidditat­ ive sense (e.g. 'Man is animal', 'Man is a species') as opposed to sentences about particular human persons, plainly somehow involve complex grammatical-logical levels, given that there is something strangely mixed-up about 'Man is a species, and Socrates is a man; hence Socrates is a species', as Boethius long ago pointed out: 0.4, HQS 1 5 - 1 6 . It is hence instruct225

Section 3 ive to look at one of the myriad characterisations of this contrast of levels. It should be remembered that, as intimat­ ed above, animal (and hence human) forms (in the quidditative sense) are now being identified as the 'souls' of such beings. This must be accepted for the purpose of seeing the point of some of the following further comment on the Aristotelian Metaphysical (A) It should, however, be realised that this composite which is an animal or a man can be taken at two levels, i.e. either as a universal [quiddity] or as an individual object. It is taken as a universal when dealing with man or animal in general, but as an individual where Socrates and Callias are concerned. Hence he [i.e. Aristotle] asserts that the man and the horse which are thus found in individuals, but which are referred to generally as man and horse, are not [second­ ary] substances, i.e. are not just the forms in question, but are each a whole composed from both determinate material make-up and determinate form, though this latter itself is certainly not taken individually, but on the universal level. For man [at the quidditative level] is some composite from soul and body, but not from this-soul and this-body, whereas the individual entails some composite 'involving basic mater­ ial make-up', i.e. individuating material make-up. For ex­ ample, Socrates is something composed from this-soul and this-body, and likewise in the cases of other individuals. (Sciendum tarnen, quod hoc compositum, quod est animal vel homo, potest dupliciter sumi; vel sicut universale, vel sicut singulare. Sicut universale quidem sicut homo et animal. Sicut singulare, ut Socrates et Callias. Et ideo dicit quod homo et equus et quae ita sunt in singularibus, sed univer­ saliter dicta, sicut homo et equus 'non sunt substantia', idest non sunt solum forma, sed sunt simul totum quoddam com­ positum ex determinata materia et determinata forma; non quidem ut singulariter, sed universaliter. Homo enim dicit aliquid compositum ex anima et corpore, non autem ex hoc an226

Aquinas - Mereological Aspects ima et hoc corpore. Sed singulare dicit aliquid compositum 'ex ultima materia', idest materia individuali. Est enim Soc­ rates aliquid compositum ex hac anima et hoc corpore. Et si­ militer est in aliis singularibus: ACM 1490.) 3.122 The various points here being raised have been res­ olved in detail by reference to the most obvious of primitive notions, as adumbrated in 0.4, and as enlarged on in 10 below. Suffice it to say summarily that the talk about form, quidd­ ities, essences, universals, and secondary substances, all of which may involve abstract nouns, is analysable in terms of clearly-defined quidditative functors, such as those of 10.26. In contrast, discourse about concrete individuals is usually found at the nominally-termed level employed at the primitive stage of the axiomatic system of 10.2111. To say that the solution to the 'problem of universals' turns on the relation, as outlined by definitional clarifications in 10.26, between discourse at the quidditative level and discourse at the nom­ inally-termed level does not entail that that problem is thus being 'reduced' to a purely notational matter. Provided that the propositions thus related are true (i.e. may be approp­ riately tested by reference to how things are) then the exer­ cise concerns both reality and discourse; cf. HUP. 3.123 The 'determinate' material make-up which is the indiv­ iduating make-up as described in 3.121(A), is called elsewhere 'designated' material make-up (materia signata) and is embr­ oiled in an unprofitable and irritating homonymy whereby the same word 'materia' (translated herein usually as 'material make-up' in connection with concrete allusions) is used to designate the generic part of a definiens constructed from a genus and differentia (so to speak). Thus when man is def­ ined as rational animal, then the genus animal is the mater­ ial, and rational is the formal, part of the definiens: HQS 216, 220. It is to such quidditative 'matter' that reference is being initially made in the next paragraph of Aquinas' com­ mentary, with the determinate and individuating matter also 227

Section 3 receiving mention. Mereologically speaking, however, the most important thread here being initiated is the suggested align­ ment of matter and party precisely at that point where the passage stresses that the concrete individual is in question: (A) Thus it is evident that matter is part of the species. By 'species' we here understand not just the form [i.e. the differentia], but the entire quiddity. It is also equally evident that matter [in the sense of material make-up] is a part of that whole which 'derives from the species and the matter', i.e. is part of the individual, and this portends the essence of the species embodied in this determinate material make-up. The result is that matter is part of that which is composite, and that which is composite may itself be either universal or it may be individual. (Sic igitur patet quod materia est pars speciei. Speciem autem hic intelligimus non formam tantum, sed quod quid erat esse. Et patet etiam quod materia est pars eius totius quod 'est ex specie et materia', idest singularis, quod significat naturam speciei in hac mat­ eria determinata. Est enim materia pars compositi. Compos­ itum autem est tam universale quam singulare: ACM 1491.) 3.124 A later passage in the same commentary not only rec­ ognises and confirms the treacherous equivocation of 'materia' in such contexts, but also confirms its status at the concrete level as representing the integral (i.e. mereological) parts of the object: (A) It should be known that although the genus and the matter may be nominally identical, the latter is not always taken in the same sense. For matter [in the material make­ up sense] is an integral part of the object, and hence cannot be predicated of the object. Thus one cannot assert that a man is his flesh and bones. In contrast [at the quidditative level], the genus is predicated of the species [e.g. Man is animal]. So there must be some way in which the genus can signify the whole. (Sciendum est autem quod, licet idem sec­ undum nomen possit esse genus et materia, non tarnen idem 228

Aquinas - Mereological Aspects eodem modo acceptum. Materia enim est pars integralis rei, et ideo de re praedicari non potest. Non enim potest dici quod homo sit caro et os. Genus autem praedicatur de spec­ ie. Unde oportet quod significet aliquo modo totum: ACM 1546.) Here the last sentence would, if pursued to its conclusions, take us into a discussion remarkable because it yet again confirms overt medieval recognition of compound indeterminate functorial expressions which are needed as terms at the quid­ ditative level. Patrick of Ireland's version of such recognit­ ion has already been extensively analysed in HQS 273 -282. 3.125 The present passage goes on to exploit converse an­ alogies from the use of bronze as material make-up and as constituting the integral parts of a concrete object. It does so in a way which will become relevant when the topic of this Aquinate alignment between matter (as material make-up) and integral parts, is examined in more detail in 3.31.

3.2 Wholes and Parts 3.21 It would appear that for Aquinas the central and basic classification of types of whole is still that which distinguishes between the distributive (or 'universal') and the integral, and which has already been prominent in Abelard's Dialectica (2.2 above). Indeed, the distinction is still cen­ tral to present-day theory. The 'distributive' whole, one may recall, is so called because of its application to each (i.e. its 'distribution') of its elements (in a non-Aristotelian sen­ se of the latter). Thus each member of the universal (or distributive) whole, or class, of man is a man. In contrast, it is by no means the case that each element of the integral whole of men is a man. Aquinas forbears from making the then rather trendy and later pervasive distinction between categorematic and syncategorematic senses of 'whole', as dealt with in section 7 below. The distinction between universal and 229

Section 3 integral wholes was trenchantly enumerated first by Aristotle, as Aquinas' comment on the Metaphysica makes clear: (A) He lays down two types of whole when he says that the whole is expressed in two ways. The first alternative cov­ ers the fashion in which each and every one of the contents of the containing whole are 'that unitary thing itself', i.e. that containing whole itself, as occurs in the case of the universal whole which is predicated of every one of its parts. The other alternative is when a unity is constituted from the parts in such a way that it is not the case that every one of the parts is that unitary thing. This is the explanation of the integral whole, which is not predicated of any of its integral parts. (Ponit duos modos totius, dicens quod totum dicitur dupliciter: aut ita quod unumquodque contentorum a toto continente sit 'ipsum unum', scilicet ipsum totum continens, quod est in toto universali de qualibet suarum partium praedicato. Aut ex partibus constituatur unum, ita quod non quaelibet partium sit unum illud. Et haec est ratio totius integralis, quod de nulla suarum partium integ­ ralium praedicatur: ACM 1099.) Also here noteworthy in the last sentence is the continued adherence to that characteristically mereological denial that the whole may be predicated of a part, a denial which has persisted among the medievals (1.32) and reappears in Lesniewski's present-day system (10.324). 3.211 The sense in which the universal (or distributive) whole may thus be said to be a whole at all is, after all, questionable enough, and this vital fact is reflected in the repeated 'quasi','as it were', of Aquinas' remarks in the next paragraph: (A) He analyses the [two] aforementioned types of whole, beginning with the first, when he says that the universal is that which is predicated 'also wholly', i.e. in common over all. Here it is being said to be as it were some single whole on account of its being predicated of every one, in 230

Aquinas - Mereological Aspects the universal manner, and thus as it were containing many things as parts, in so far as it is predicated of each and every one of them. And all of these constitute a unity within that universal whole, in the sense that each of them is that one whole. Thus animal 'contains' man and horse and god because they 'are all animals', i.e. animal is predicated of each. (Exponit praedictos modos totius, et primo primum, dicens quod universale 'et quod total iter', idest communi ter praedicatur, dicitur quasi sit aliquod unum totum ex hoc quod praedicatur de unoquoque, sicut universale, quasi multa continens ut partes, in eo quod praedicatur de unoquoque. Et omnia illa sunt unum in toto universali, ita quod unumquodque illorum est illud unum totum. Sicut animai continet hominem et equum et deum, quia 'omnia sunt ammalia', idest quia animal praedicatur de unoquoque: ACM 1100.) Here, as is an alternative possible process, the 'distribution' of the universal whole is being exemplified by reference to the various species which are the 'subjective' parts of the genus. (The term 'god' is here used in one of the old Greek senses.) It may be recalled that more usually it is the ind­ ividuals of the species which are said to be its 'subjective' parts, with the species itself considered in its turn as a un­ iversal whole. Clearly there are the seeds of a still-perv­ asive confusion here: HQS §4.35. 3.212 Again, when certain further elaborations on parts are being made, the main division is still depicted as being that between the 'subjective parts of the universal whole', on the one hand, and the varieties of integral part on the other. For in ACM 1097, quoted in 3.23(B) below, all the other types of part mentioned are explicitly contrasted with such 'sub­ jective' parts. Then all those other types are said to be in­ tegral parts. The result is that for Aquinas integral wholes and parts embrace a wide variety of cases which he does not hesitate to extend even further by way of analogy. Approxim­ ately speaking, he appears to be willing to allow the notion 231

Section 3 of integral whole to embrace all those cases in which there is no predication possible either way between the part and the whole. The house is not the wall, neither is the wall the house, for example. This does indeed reflect well-known mereological theses of which a reminder was again provided in the previous paragraph. However, the breadth and variety of instances which it accordingly captures within its ambit go well beyond the mereological, as 3.36 is to indicate in more detail. 3.213 The first way in which, according to the commentary, there may be parts, is that which will later (ACM 1101) be said to be the most obvious, namely the quantitative: (A) In the first way that is said to be a part when it is that into which something is quantitatively divided. Of this, one sense is exemplified when that is said to be a part of something no matter what the lesser quantity into which the greater quantity is divided may be. For that which.repres­ ents the reduction of a quantitity is said to be a part of it, as when two things are in some sense a part of a trio. A further sense is exemplified when only that lesser quant­ ity which is proportional to the greater quantity is said to be a part. In this sense the two things are not part of the trio, but the two are in this sense part of a quartet, since two couples make a quartet. (Primo modo pars dicitur in quam dividitur aliquid secundum quantitatem: et hoc dupliciter. Uno enim modo quantumcumque fuerit quantitas minor, in quam quantitas maior dividitur, dicitur eius pars. Semper enim id quod aufertur a quantitate dicitur pars eius; sicut duo aliquo modo sunt partes trium. Alio modo dicitur solum pars quantitas minor, quae mensurat maiorem. Et sic duo non sunt pars trium; sed sic duo sunt pars quatuor, quia bis duo sunt quantuor: ACM 1093.) Here we are dealing with a sub-class of the integrals. The distinction thus made between proportional and non-proport­ ional parts of the quantitative whole is repeated in ACM 232

Aquinas - Mereological Aspects 1371. There follows the customary contrast between quantit­ ative and quidditative parts: (B) In the second way those things into which something is divided in a non-quantitative way are said to be parts. It is in this way that the species is said to be part of the genus. For the latter is divided into species, but not into parts of a quantity, as occurs with a quantity. For the whole quantity is not in each one of its parts, whereas the genus is in each of its species. (Secundo modo ea dicuntur partes in quae dividitur aliquid sine quantitate; et per hunc modum species dicuntur esse partes generis. Dividitur enim in species, non sicut quantitas, in partes quantitatis. Nam tota quantitas non est in una suarum partium. Genus autem est in qualibet specierum: ACM 1094.) The familiar distributive example is here being given at the genus-level. Thus the genus, such as animal, is equally 'dis­ tributed' throughout its sub-species. It is this feature which separates the distributives out from the integrals, all of which, including the already-mentioned quantitative whole, involve parts of primary substances, i.e. concrete individuals, as ACM 1097 is to confirm: 3.23(B). In the continuation of the present text, which now follows, Aquinas is alluding to the two levels of sense assignable to the notions of matter and form, i.e. at both the individual and quidditative levels, as already noted in 3.123: (C) In the third way those things into which some whole is divided or out of which it is composed are said to be parts. This whole may either be the species or something sharing in the species, i.e. the individual. For there are, as has been said, some parts which are parts of the species, and some parts which are material parts, and these are parts of the individual. Thus the bronze is part of the bronze sphere or of the bronze cube in the sense that it is the material into which the species is received. Hence the bronze is not part of the species, but part of that which participates in the 233

Section 3 species. (Tertio modo dicuntur partes, in quas dividitur, aut ex quibus componitur aliquod totum; sive sit species, sive aliquid habens speciem, scilicet individuum. Sunt enim, sicut dictum est, quaedam partes speciei, et quaedam partes materiae, quae sunt partes individui. Aes enim est pars sphaerae aereae, aut cubi aerei, sicut materia, in qua species est recepta. Unde aes non est pars speciei, sed pars habentis speciem: ACM 1095.) Here, as ACM 1097 is to confirm in 3.23(B), a most proper distinction is being drawn between being part of the species (as when the quidditative 'matter' constituted by the genus animal is the material 'part' of the species man) and being part of that which participates in the species, i.e. being part of some individual member of the species. In this last case the matter is the material make-up of the object in a quite familiar sort of way, and in view of its consequent mereological bearing a separate investigation of Aquinas' 'matter' as the integral part or parts will be undertaken below in 3.31. 3.22 However, Aristotle's own development of the notion of 'form' (now an 'integral' part to be contrasted with 'matter') as recorded in 3.11 above, already makes it very plain that the distinction between element and element (in a mereological sense), including the case wherein material make-up is in question, is typologically diverse from the distinction bet­ ween the element and the form. Correspondingly, although matter and form can in a very loose sense by said to be 'parts' of a primary (i.e. concrete) substance, they are so in such very specialised ways that it could be misleading to deal with them as though they were simple conceptual neigh­ bours of the parts of the house, notwithstanding the fact that they do superficially satisfy the mereological theses about the whole not being predicated of its parts, and vice versa: 3.21, 10.31 (Al), 10.324. The man is not his material make-up, nor is he his man-ising (i.e. the act of being a man, 234

Aquinas - Mereological Aspects which is precisely the 'form' in question), and these negations hold also in the reverse directions. But while there are quite straightforward instances of the first-mentioned neg­ ation (the man is not his leg, nor is his leg the man) the statement of the non-identity of the man and his form is not at all on the same logical level. Indeed, its import is to the effect that it just does not make any sense at all (i.e. the logical grammar is incoherent) to bring the concrete man and his 'form' together into the same elementary proposition, whe­ ther by way of affirmation ('Socrates is his form') or of neg­ ation ('Socrates is not his form'). This discrepancy of sense must be borne in mind when the consequences of Aquinas' iden­ tification of soul and form (where animals are concerned) are surveyed in 3.4. In the meantime, the following argument ab­ out the sense in which the form-soul can be said to be an in­ dividual object (hoc aliquid, this-something) may, on one pos­ sible analysis, serve to further deepen the suspicions about the treatment of form as an integral part: (A) Objection 1. That which is subsistent is said to be a this-something. But it is not the soul which is a thissomething, but rather the composite formed from the soul plus the body. Hence the soul is not subsistent ... . To the first objection the reply is that this-something can be taken in two ways: in one of these ways it covers any subsistent; in the other way it covers a complete subsistent fulfilling the nature of some species or another. In the first way it excludes the inherence of the incidental and of the form of the material make-up; in the second way, it also excludes the incompleteness exemplified by a part. This is why hand can be said to be this-something in the first way, but not in the second way. Hence as human soul is part of human spec­ ies, it can be said to be this-something in the first of these two ways, as a quasi-subsistent, but it cannot be said to be a this-something in the second of these two ways; in the latter way it is the composite of soul and body which is 235

Section 3 said to be this-something. (1. Quod enim est subsistens, dicitur hoc aliquid. Anima autem non est hoc aliquid, sed compositum ex anima et corpore. Ergo anima non est aliquid subsistens... . Ad primum ergo dicendum quod hoc aliquid potest accipi dupliciter: uno modo, pro quocumque subsist­ ente: alio modo, pro subsistente completo in natura alicuius speciei. Primo modo, excludit inhaerentiam accidentis et formae materialis: secundo modo, excludit etiam imperfectionem partis. Unde manus potest dici hoc aliquid primo modo. sed non secundo modo. Sic igitur, cum anima humana sit pars speciei humanae, potest dici hoc aliquid primo modo, quasi subsistens, sed non secundo modo; sic enim compositum ex anima et corpore dicitur hoc aliquid; AST I, q. 75, art. 2, arg. 1.) A perusal of this passage's two characterisations of thissomething would make it appear that although the first sense of this-something is said to cover any 'subsistent', it is the 'subsistessence' promoted by Gilbert of Poitiers, rather than his concrete 'subsistentity' which is here in question: cf. 2.813. In other words, given the exclusion of incidentals and material forms as here propounded, the quidditative level must be the one applicable; such an exclusion corresponds to the 'insusceptibility of contraries' used by Boethius to indicate that level: HQS 205 - 6. It is hand in the quidditative sense which is the this-something in the first way, and which is, like the soul, a quasi-subsistent. In the second sense the this-something is the concrete complete object, the Porretan 'subsistentity', and an integral whole in a quite familiar way, with a corresponding concreteness of parts. Aquinas' De Anima art. 1, . entirely confirms that incompleteness of the functorial quidditative ('non habens in se completam speriem') which the 'hand' example might incidentally tend to obfuscate. Given Boethius' incontrovertible version of the exclusion of incidentals just-mentioned, the categorial discomforts which would arise from making the soul-form an integral part on a 236

Aquinas - Mereological Aspects par with other integral parts begin to become apparent, and 3.32 will further scrutinise them. 3.23 Finally in this list of types of part and whole comes the odd sense in which the parts mentioned in a definition are parts of the thing (or sort of thing) undergoing defin­ ition: (A) In the fourth way, those things which are laid down in the definition of any thing, and which are parts of its exp­ lanation, are said to be 'parts'. In this sense animal and biped are parts of man. (Quarto modo dicuntur partes quae ponuntur in definitione cuiuslibet rei, quae sunt partes rationis sicut animal et bipes sunt partes hominis: ACM 1096.) Whatever one may make of this fourth and rather unusual sen­ se of 'part' to which attention had long ago been called by Boethius ( 1097A - ; cf. 0.5 above and HQS 4.321) sufficient is evident to make the contrast enuntiated in Aquinas' summ­ ary of cases in 3.213 reasonably intelligible. In this present and fourth sense the genus (e.g. animal) is indeed part of the definition of the species (e.g. rational animal, i.e. man). In the second sense (3.213(B)), however, given the 'distribution' of the genus among its 'parts', i.e. its subspecies, it is now the species which can be said to be 'part' of the genus. For example, man is 'part' of animal (quidditatively speaking), given that man is included in animal. (The sense of this inc­ lusion is analysed in 10.262 below.) Here is Aquinas' summary of such a situation: (B) From this it is evident that according to the fourth way the genus is part of the species, whereas in another way, namely the second, the species is part of the genus. For in the second way the part is taken as the subjective part of the universal whole, but in the other three ways it is taken as the integral part. However, in the first way it is taken as the part of a quantity, whereas in the other two of the three ways it is taken as the part of a substance. 237

Section 3 However, this is in such a fashion that the part in the third way is part of the thing, be it part of the species or part of the individual. In the fourth way it is part of the expl­ anation. (Ex quo patet quod genus quarto modo est pars spec­ iei; aliter vero, scilicet secundo modo, species est pars gen­ eris. In secundo enim modo sumebatur pars pro parte sub­ iectiva totius universalis; in aliis autem tribus pro parte integrali. Sed in primo pro parte quantitatis, in aliis autem duobus pro parte substantiae; ita tarnen, quod pars secundum tertium modum est pars rei; sive sit pars speciei, sive pars individui. Quarto autem modo est pars rationis: ACM 1097.) 3.24 Lest there should still subsist any doubt that form and matter are taken by Aquinas to be integral parts of a substance, ACM 1472 may be consulted. It is there asserted that, 'the part of a substance is not only matter, but also the form, and that whence something is composed'. Pains are there taken, as also in ACM 1473, to suggest that the 'mater­ ial' part of the definition is not a part in the integral way. This accords with ACM 1094 and 1097, as quoted in 3.213(B) and 3.23 (B) above. But even with this qualification, the standing difficulty of seeing the form as a part like other parts, as in 3.22 above, and the special position of Aquinas, given his identification of soul and form in animal cases, both generate especially interesting possibilities of dis­ cussion, as 3.4 is to illustrate. 3.25 The next task to be undertaken in this introductory section to the Aquinate treatment of mereology will involve a few reinforcing reminders of his adoption, elsewhere than when commenting on the Metaphysica, of the above-described classification of parts and wholes on the integral side. Thus in the Summa Theologica he speaks of 'part' being used as in the Metaphysica, with 'part of the essence' (pars essentiae) and 'quantitative part' (pars quantitatis) as the two main subdivisions: (A) 'Part' is taken in a twofold sense, as is stated in 238

Aquinas - Mereological Aspects Metaphysica V, i.e. as part of the essence and as quantit­ ative part. Now the parts of the essence are (as far as the nature of things is concerned) form and matter, their logical counterparts being the differentia and genus ...; because quantity belongs over on the side of the matter, the parts of the quantity are the parts of the matter. (Duplex est pars, ut dicitur in V Metaphys., scilicet pars essentiae, et pars quantitatis. Partes quidem essentiae sunt, naturaliter quidem, forma et materia; logice autem, genus et differentia ... quia quantitas se tenet ex parte materiae, partes quantitatis sunt partes materiae: AST III, q. 90, art. 2, c.) The interconnection here stated between quantity and matter is especially significant, given the Aristotelian underlining, expressed in ACM 1101, of the quantitative as the most obvi­ ous area where integral parts and wholes are encountered: cf. 3. 213. In AST I, q. 8, art. 2, ad 3, one also finds form and matter described as the parts of the essence, but genus and differentia, qualified as the 'logical' parts of the essence in the foregoing extract, are now simply 'parts of the species'. Further, the connection between matter and quantity which we have just seen stressed, would appear to be comparatively un­ derplayed in that same q. 8 of AST I, on which see 3.31 below. 3.26 Exemplifying yet again the central mereological thesis of part-whole disparity (1.32, 3.21), and mentioning the poten­ tial whole as a further type of whole, is this next passage which deals with Augustine of Hippo's characterisation of the essence of the soul: mens, notitia et amor sunt substantialiter in anima, 'mind, understanding, and love pertain to the essence of the soul': (A) Or, as some claim, this [just quoted] way of talking is verified in respect of the way in which the potential whole is predicated of its parts, such a whole being a half-way house between the universal whole and the integral whole. For the universal whole imbues every one of its parts to the full extent of that whole's essence and power, as animal im239

Section 3 bues man and horse, and this is why it is correctly asserted of its several parts. The integral whole, however, does not imbue every one of its parts, either in respect of its whole essence or of its whole power. This why it is in no way as­ serted of its individual parts. It can, however, be predic­ ated of them all taken together, even though this is improp­ er, e.g. we say that the wall, roof, and foundation are the house. In contrast, the potential whole imbues its several parts to the full extent of its essence, but not to the full extent of its power. Hence it comes about that in a certain way such a whole can be asserted of each part, but not with as full a propriety as the universal whole can be when it is asserted of its parts. It is in this fashion that Augustine holds that memory, understanding, and will are the one ess­ ence of the soul. (Vel, sicut quidam dicunt, haec locutio verificatur secundum modum quo totum potestativum praedicatur de suis partibus, quod medium est inter totum univers­ ale et totum integrale. Totum autem universale adest cuilibet parti secundum totam suam essentiam et virtutem, ut animal homini et equo: et ideo proprie de singulis partibus praedicatur. Totum vero integrale non est in qualibet parte, neque secundum totam essentiam, neque secundum totam virt­ utem. Et ideo nullo modo de singulis partibus praedicatur; sed aliquo modo, licet improprie, praedicatur de omnibus simul, ut si dicamus quod paries, tectum, et fundamentum sunt domus. Totum vero potentiale adest singulis partibus secundum totam suam essentiam, sed non secundum totam virt­ utem. Et ideo quodammodo potest praedicari de qualibet parte; sed non ita proprie sicut totum universale. Et per hunc modum Augustinus dicit quod memoria, intelligentia et voluntas sunt una animae essentia: AST I, q. 77, art. 1, ad 1.) 3.261 As well as its positive features, this passage disp­ lays a highly significant negation, i.e. 'The integral whole ... does not imbue every one of its parts, either in respect of its whole essence or of its whole power'. Here, from a verbal 240

Aquinas - Mereological Aspects point of view at least, the almost precise negation of Aug­ ustine's oft-quoted account of the soul is being consciously attached to the integral whole. For Augustine says that the soul is 'wholly in the whole, and wholly in every one of its parts', in toto est tota, et in qualibet eius parte tota est: VI De Trinitate, . 6. Aquinas quotes this description on innumerable occasions, even extrapolating it to describe God's imbuement of the whole and of every part of the universe. The relevant Latin of the present passage's description of the integral whole could well be a negative recollection of Aug­ ustine's text. From a theoretical point of view such material will clearly have to be recalled when, in 3.4 below, the topics of the soul and mereology are brought into detailed relation. For the moment, we may simply note the remarkable coincidence between this distributive Augustinian account of the soul within the parts of an integral whole ('.... in every one of its parts') and the likewise distributive qualification of every part of the integral whole in general which is cont­ ained in Lesniewski's original definition of the 'collective class of X-es' ('every part of which has a part in common with an X': D2 of 10.31). This will also be recurred to in 3.42. Incidentally, the 'impropriety' of saying that house is walls, roof, and foundation, which is alleged in the last ext­ ract, probably relates to what the ancients called the 'descriptional' nature of this characterisation; it does not follow their canonical pattern of a correctly organised definition: HQS 211 - 2. 3.27 There could still subsist some uneasiness about the present proposed alignment of Aquinas's doctrines with those of contemporary thought. However, it is safe to assume that where at least some of his integral wholes are concerned, we are definitely on mereological ground. Some other allusions of his to integral wholes are certainly venturesome exten­ sions, as when virtues are treated mereologically: 3.36. An easily accessible place (outside ACM) where discussion of 241

Section 3 integrals abounds is in AST III, q. 90, although, as 3.61 is to remind us, questions about Aquinas' authorship at this point could be raised. At any rate, in article 1, a subsequently unopposed 'sed contra1 gives a useful general description of parts: they are that whence the completeness of something is integrated; partes sunt ex quibus perfectio alicuius integ­ rator. The corpus continues: (A) The parts of a thing are what the whole is divided into in respect of its material make-up, for the parts are related to the whole even as matter is to form. This why (in Physica II) the parts are allocated to the genus of mat­ erial explanation, whereas the whole is allocated to the gen­ us of the formal cause. Wherever, therefore, there occurs some plurality where the matter is concerned, there also is to be found some sort of parthood. (Partes rei sunt in quas materialiter totum dividitur: habent enim se partes ad totum sicut materia ad formam; unde in II Physic. partes ponuntur in genere causae materialis, totum autem in genere causae formalis. Ubicumque igitur ex parte materiae invenitur aliqua pluralitas, ibi est invenire partium rationem: AST III, q. 90, art. 1, .) It is in this way that the already-observed connection bet­ ween material make-up and mereological parts is counterpoised by relating the whole to the Aristotelian form. This does not mean that the mereological whole is identified with that form. Rather, in terms of present day theory of the complete coll­ ection (or collective class or integral whole), the latter is said to be of so-and-so's (with a nominal variable as the notational counterpart of the 'so-and-so', so that in the concrete we would have, for example, the complete collection of men). Now the 'form of man' is discussed in quidditative discourse. Terms such as 'being a man', 'humanity', 'manising', 'to man-ise' are there typical, with predicate-variables for expressions of index s/n being the generic notational counterparts of such terms. Such terms are in their turn 242

Aquinas - Mereological Aspects 'brought down to earth', so to speak, in the systematic lang­ uage by means of a functor such as 'trm< >', defined in 10.251, and read in English as 'term satisfying ...'. This gives a nominal expression corresponding to the 'form' intim­ ated by the s/n type of completion appropriate to that funct­ or. The termination '...-er' in English may be seen as one correlate of this name-forming functor, since from, e.g. the s/n indexed term 'man-ise', we thereby have 'man-ise-er', i.e. the equivalent of 'man'. So 'complete collection of man-iseers', or 'complete collection of humanity-havers' (and so forth) would be expressions for integral wholes which would make explicit the position of the 'forms' in question (i.e. of 'to man-ise', or of 'humanity-having') relatively to the relev­ ant complete collection, i.e. throughout exemplifications of D2 of 10.31. 3.28 Still remaining with the same question of AST, we find that the corpus of article 3, while echoing some of 3.26(A), contrasts the positively 'distributive' relation of the 'subjective' parts to their whole (exemplified in the following text at the quidditative level: cf. 3.211) with the contrast­ ingly negative possibilities of the integral whole: (A) In each of the subjective parts there is present the whole power of the whole object, all at once, and in equal measure. Thus the whole import of animal in so far as it is animal, is verified in every species of animal, which simult­ aneously and in equal measure constitute the divisions of animal . . . . Integral parts ... are such that their nature demands that the whole does not imbue each individual part, either in respect of its whole power, or in respect of its nature, but rather the whole lot taken together. (Partibus subiectivis singulis adest tota virtus totius, et simul, et aequaliter: sicut tota virtus animalis, in quantum est animal, salvatur in qualibet specie animalis, quae simul et aequal­ iter dividunt animal ... . Partes integrales ...ad quarum rationem exigitur ut totum non adsit singulis partibus neque 243

Section 3 secundum totam virtutem eius, neque secundum essentiam, sed omnibus simul: AST III, q. 90, art. 3, c.) The point here being made is that the complete collection of the parts-of-the-man-Socrates (for example) does not have the-man-Socrates as each part (thus contrasting with the animal case, taken distributively, as described). This is obvious enough, as is also the insistence on the whole lot's being taken together. The definition of complete collection (i.e. integral whole) as provided in 10.31 (D2) not only looks after this last point in the first clause of its definiens ('Every a is an element of the complete collection of a's) but also makes, in its second clause, that positive and distrib­ utive characterisation of the parts so as to replace (so to speak) the Aquinate negations inspired by the distributiveintegral contrast. As noted in 3.261, that clause is to the effect that every element of the complete collection of a's has an element in common with an a. Aquinas' own approaches to the import of this clause will be exemplified in due cour­ se when the mereological aspects of his theory of the soul and the Eucharist are investigated in 3.4. Being 'ensouled' will turn out to be related in all sorts of ways to features covered by the contemporary axiomatised mereology made fully overt in the Presuppositional Explicitation (10.3). 3.281 A final extract from q. 90 reveals a remarkable re­ alisation of some further possibilities of extension as far as mereology is concerned. Thus Stereology (theory of space), Chronology (theory of time), along with Kinematics and Biology figure in the reply to the third objection, which contained a reminder that the whole is made up equally and simultaneously of its integral parts. (On the traps of simul, 'simultaneously', in Abelard's remarks on integral wholes, see HQS §4.535.) The thesis expressed by the objection in question is not so much rebuffed as most properly qualified and extended by reminders of the further structures (here called 'orders') within which mereological parts may figure: 244

Aquinas - Mereological Aspects (A) All integral parts have a certain mutual order. On the one hand some have an order which is situational, whether it be that they follow one upon the other (like the parts of an army [on the march]), whether they are in contact one with the other (like the parts of a heap), whether they are att­ ached to one another, as are the parts of a house, or wheth­ er again, they form a continuum as do the parts of a line. On the other hand some in addition display an order of caus­ al importance, e.g. the parts of animals of which the first in order of causal importance is the heart, with the other parts also mutually dependent on one another in a certain order of causal priority. In a third fashion parts are or­ dered according to a temporal order, as in the cases of the parts of time and of motion. (Omnes partes integrales hab­ ent ordinem quendam ad invicem. Sed quaedam habent ordinem tantum in situ, sive consequenter se habeant, sicut partes exercitus, sive se tangant, sicut partes acervi, sive etiam colligentur, sicut partes domus, sive etiam continuentur, si­ cut partes lineae. Quaedam vero habent insuper ordinem vir­ tutis, sicut partes animalis, quarum prima virtute est cor, et aliae quodam ordine virtutis dependent ab invicem. Tertio modo ordinantur ordine temporis, sicut partes temporis et motus: AST III, q. 90, art.3, ad 3.) Here we have as fine a preliminary characterisation of the further ramifications of metaphysical mereology as one could wish for at this stage, and some of its features have not only already been encountered (e.g. Abelard and his contem­ poraries on temporal wholes in 2.6) but will also often be relevant below if only by way of illuminating concrete ex­ ample. 3.29 Also relevant in a general sort of way are the all­ usions sometimes made to the distinction between the contin­ uous and the non-continuous (i.e. discrete, cf. 1.5, 2.78) types of integral wholes. It is nowadays plain that over a whole range of cases a decision as to which object is continuous 245

Section 3 and which non-continuous can turn upon variations in the ext­ ent of human powers of observation (by means of improved microscopes, for example). This fact alone should remove any lingering objections to the countenancing of collective obj­ ects having their elements spatially or temporally separated one from the other. At the same time the definition of dis­ creteness provided in 10.336 need not depend for the effect­ iveness of its application upon technical progress in the dis­ crimination of parts. For the discreteness here in question covers that of non-overlapping objects such as the various 'subjective' members of the distributive class of human be­ ings, and discernment of this is dependent upon no degree of technical advance, except perhaps at the foetal level. A short extract now confirms that Aquinas considered the Arist­ otelian position to countenance as parts of the integral whole those sections of a continuum which are merely potential, i.e. not actually physically discriminated, as are some of the parts of a non-continous whole: (A) Now the parts whence a whole is constituted can pert­ ain to the whole in two ways, i.e. in one way potentially and in another way actually. Thus parts are in a continuous whole potentially, whereas they are in actuality where a non-continuous whole is concerned, as when stones actually form a heap. The continuous exhibits a greater degree of unity, and hence is a whole to a greater degree, than is the non-continuous .... However, if we stress the parts, they exhibit parthood to a greater degree when they are actually [discrete] parts than when they are merely potentially [dis­ crete] parts. (Partes autem ex quibus constituitur totum dupliciter possunt esse in toto. Uno modo in potentia, alio modo in actu. Partes quidem sunt in potentia in toto cont­ inuo; actu vero in toto non continuo, sicut lapides actu sunt in acervo. Magis autem est unum, et per consequens magis totum, continuum, quam non continuum... . Tarnen si respiciamus ad partes, magis sunt ipsae partes, quando sunt actu, 246

Aquinas - Mereological Aspects quam quando sunt in potential ACM 1102 - 1103.) 3.291 A few paragraphs later there occurs the distinction already mentioned in 1.32 between homogeneous integrals (lat­ er called alľs by Aristotle) and the heterogeneous ones (which he is to say are more properly called 'wholes'): (A) He lays down the second set of distinctions. For when the quantity in question is such that there is an ordering of parts, so that there is a beginning, middle, and end (these constituting the essence of position) it must be the case that all such wholes evince positional continuity in their parts. But as far as position of parts is concerned, the positional whole is found to have a threefold set of possibilites. For [in the first place] some wholes exist in which diverse position of parts does not bring about any diversity, as is obvious in water. For no matter what the extent to which the parts of the water are transposed, they remain totally unchanged. The same goes for other liquid things like oil, wine, and suchlike. (Secundam diversitatem ponit. Cum enim ita sit quod in quantitate sit ordo partium, quia est ibi principium, medium et ultimum, in quo ratio pos­ itionis consist it, oportet quod omnia tota ista continuam habent positionem in suis partibus. Sed ad positionem part­ ium totum continuum tripliciter se invenitur habere. Quaedam enim tota sunt in quibus diversa positio partium non facit diversitatem, sicut patet in aqua. Qualitercumque enim transponantur partes aquae, nihil diffferent; et similiter est de aliis humidis, sicut de oleo, vino, et huiusmodi: ACM 1105.) It is at this point that Aristotle goes on to make a point which, as Aquinas rightly says, may arise from Greek idiom, i.e. such a homogeneous whole is said to be all its parts, as opposed to the whole of its parts. In contrast, 'the whole' is alleged to apply in the non-homogeneous cases. No such dist­ inction appears to hold in English, although a final decision on this point is the business of the usage expert, and not of 247

Section 3 the mereologist. The discussion then continues: (B) On the other hand there are some wholes in which the position of the parts does make some difference, as in man, in every animal, in house, and things of this sort. One does not get a house from just no matter what arrangement of the parts, but to the extent that the parts are appropriately ordered, and the same goes for man and animal.... Yet again, there are some wholes in which both states of affairs obt­ ain, because position in some respect brings about a differ­ ence intrinsic to them.... These are those things in which transposition of the parts leaves the material unaffected, while the form or shape does not remain the same. This is obvious in the case of wax which remains wax (although not of the same shape) no matter how much the parts may be transposed. The same goes for a piece of clothing and all things which are alike in their parts while taking on diverse shapes. (Quaedam vero sunt in quibus positio diťferentiam facit, sicut in hornine, et in quolibet animali, et in domo et huiusmodi. Non enim est domus qualitercumque partes ordinentur, sed secundum determinatum ordinem partium; et simil­ iter  homo  animal Quaedam vero sunt in quibus contingunt ambo, quia positio quodammodo facit differentiam in eis ...et ista sunt in quibus facta transpositione partium manet eadem materia, sed non eadem forma sive figura; ut patet in cera, cuius qualitercumque transponantur partes, nihilominus est cera, licet non eiusdem figurae; et similiter de vestimento, et de omnibus que sunt similium partium, retinentium diversam figuram: ACM 1106 - 1107.) Aquinas then goes on to provide a logical reason which may account for the Greek idiom which he had observed in operat­ ion: (C) The reason behind this diversity [in modes of express­ ion] is that 'all' has a distributive function, and on this account needs an actual manyness, or an imminent potential­ ity thereto; further, because a likeness of parts is involved 248

Aquinas - Mereological Aspects [in such wholes as are said to be alľs], they are split up into parts having a likeness to the whole, and there comes about a multiplication of the whole in such cases. For if every part of water is water, in every water are many wat­ ers, at least potentially. Likewise in one manifold there are many actual units. In contrast, 'the whole' signifies a coll­ ection of parts within some single object, and hence of such cases 'the whole' is properly asserted, since in them, from all the parts taken together, there comes about one complete object, to the completeness of which none of its parts att­ ains, as in the cases of a house and an animal. (Ratio aut­ em huius diversitatis est quia omne distributivum est, et ideo requirit multitudinem in actu, vel in potentia propinqua; et quia ea sunt similium partium, dividuntur in partes consimiles toti, fitque ibi multiplicatio totius. Nam si quaelibet pars aquae est aqua, in unaquaque aqua sunt multae aquae, licet in potentia; sicut in uno numero sunt multae un­ itates in actu. Toturn vero significat collectionem partium in aliquo uno; et ideo in illis proprie dicitur totum in quibus, ex omnibus partibus acceptis simul, fit unum per­ fectum, cuius perfectio nulli partium competit, sicut domus et animai: ACM 1108.) 3.292 Abelard's discussion of the bronze-rod case, as well as the flesh and bone examples (1.324, .325) have suggested that although some types of whole do have parts which are of the same sort as the whole (i.e. are homogeneous in this res­ pect) there is a certain superficiality, mereologically speak­ ing, about initially making too much of the contrast between these and the heterogeneous whole. True, leaving atomic the­ ory on one side, every part of this water is a water, whereas not every part of this man is a man. There should, at the appropriate point, be no problem in expressing this distinc­ tion mereologically. But as emerges from Abelard's discuss­ ions, not every part of this-water is this-water in the same sense of 'this' throughout. Only our contingent lack of cap249

Section 3 acity for readily discriminating between diverse parcels of water glosses over their individuated diversities. 3.293 In contrast, 3.291(B) contains, because of the easilydiscriminable diverse natures of the examples and of their parts, a transparently intelligible basis for the distinction between X-parts and parts-of-X (1.4, 2.3), the discussion of which, in Aquinas' terms, will be taken further in 3.41. At the same time 3.291(A) and (B) above should not mislead one into thinking that the fundamental distinction here being propounded is between the componential (e.g. a house) and the non-componential (e.g. a parcel of water) sorts of whole. Such a distinction may indeed be made in due course, but it scarcely need detain us at the aloofly general early stages of mereological theory. Indeed, at this stage the various sorts of whole can be shown to have a great deal in common. For it is characteristic of all integral wholes (or complete collections), comprising also both the wax and the clothes types of case (3.291(B)) that each of their elements has an element in common with an object of the sort in question, i.e. every element of a part-of-X has an element in common with an X. The second clause of the definiens of complete coll­ ection in the Presuppositional Explicitation (D2 of 10.31) gives an exact elucidation of the intention of the ordinarylanguage approximation contained in the previous sentence. The upshot is that at this very general level, the 'likeness of parts' which 3.291(C) ascribes exclusively to the alľs (i.e. the homogeneous wholes) may also thus be extended to the wholes (i.e. the heterogeneously-parted cases). Hence, para­ doxical as it may initially sound, there is a sense in which heterogeneously-parted wholes are homogeneous (i.e. as just described), as well as a sense in which admittedly homogen­ eous wholes are still heterogeneous (as recalled in 3.292). Both these facets of integral wholes are provided for in the mereological axiomatic of 10.31. 3.294 The same distinction between homogeneously-parted 250

Aquinas - Mereological Aspects and heterogeneously-parted sorts of whole recurs in AST I, q. 11, art. 2, which is a discussion of the senses in which the one and the many are to be contrasted. It would appear, says objection 2, that they should not be contrasted, since unities make up a multitude, and no contrasting item is made up of that with which it is contrasted: nullum oppositum constituitur ex suo opposito. To this objection is given the reply which is quoted below. In it, the distinction between the two sorts of whole and the non-identity of part and whole in the heterogeneous cases, both quite standard points (1.32), lead to the association of multitude with the heterogeneous parts only. This is the basis for extending the usual contrast between the parts and the whole to that between the unities which make up the multitude (i.e. the whole) and the multit­ ude-whole itself. However, even though it is true that each of the component unities may be a non-multitude (and to that extent the 'opposite' of multitude), it is not this oppositeness which qualifies them for parthood of the multitude, but their being entities of the appropriate and positive sort. This parallels the case of the house (a multitude by the des­ cription suggested) which is composed of parts (unities) whose qualification for parthood lies in their being appropriately composed, shaped, and adjoined physical objects, and not just in their being non-houses (as indeed they are, given the com­ mon mereological non-identities cited). There are, after all, one might add, lots of non-houses which in no way could ever be qualified to be parts-of-houses: (A) There are two sorts of whole: on the one hand there is the homogeneous whole, which is made up of like-natured parts; on the other hand there is the heterogeneous whole, which is made up of parts having unlike natures. Further, in every homogeneous whole the whole is made up of parts hav­ ing the same form as that of the whole, e.g. every part of a water is a water, and the make-up of a continuum out of its parts is of this sort. In contrast, in respect of every het251

Section 3 erogeneous whole, every part lacks the essence of the whole, e.g. no part of the house is the house, and no part of the man is the man. It is this sort of whole which is a multit­ ude. In so far, therefore, as the multitude's part lacks the essence of the multitude, the multitude is made up of unit­ ies in the same way as the house is made up of non-houses. However, it is not the case that the unities make up the multitude in respect of their being of an undivided nature, in so far as they are contrasted with the multitude, but rather because of the style of their being, in the way that the parts of the house make up the house because each is a physical object of a certain sort and not just because they are each a non-house. (Duplex est totum: quoddam homogeneum, quod componitur ex similibus pártibus; quoddam vero heterogeneum, quod componitur ex dissimilibus partibus. In quolibet autem toto homogeneo, totum constituitur ex partib­ us habentibus formam totius, sicut quaelibet pars aquae est aqua; et talis est constitutio continui ex suis partibus. In quolibet autem toto heterogeneo, quaelibet pars caret forma totius: nulla enim pars domus est domus,  aliqua pars hominis est homo. Et tale totum est multitudo. Inquantum ergo pars eius non habet formam multitudinis, componitur multitudo ex unitatibus, sicut domus ex non domibus: non quod unitates constituant multitudinem secundum id quod hab­ ent de ratione indivisionis, prout opponuntur multitudini, sed secundum hoc quod habent de entitate; sicut et partes domus constituunt domum per hoc quod sunt quaedam corpora, non per hoc quod sunt non domus: AST I, q. 11, art. 2, ad 2.) This decision allows that the heterogeneous whole is made up from its many diversely-sorted unities, but stresses that that diversity constitutes the basis of multitude, and not merely the multiplicity of the parts. So one can conserve that contrast between unity and multitude which is being mooted here, while rejecting it as the sole essential for the con­ stitution of the heterogeneous whole. (Immaterial to the 252

Aquinas - Mereological Aspects argument, but somewhat odd in appearance, is the restriction of the title 'multitude' to heterogeneous wholes, since there surely might be occasion for speaking of a multitude of likenatured (i.e. homogeneous) simple objects. In the end, of course, both the homogeneous and heterogeneous wholes stand on the same theoretical basis, relatively to general mereology: 3.292.) 3.2941 Much of the material in the last-quoted passage is susceptible of more thorough mereological analysis. The contrast between units and multitude perhaps points towards the notion of mereological atom, as discussed in LEM. This would give the compietest contrast between the unities (con­ sidered de ratione indivisionis, as the text has it) and the multitude made up from them, i.e. the mereological class of mereological atoms. Aquinas certainly considers the notion of mereological atom (3.3352) as also did Abelard (AD 554.15.17, cf. HQS §4.52). Some relevant points from LEM are reproduced in 10.35. 3.295 A further consequence of the distinction between the homogeneous and the heterogeneous arises in connection with the soul's being the form of the whole organism and of its every part. This, suggests objection 7 in article 10 of the Quaestiones Dísputatae de Anima, seems to make an animal into a homogeneous whole, as is fire, wherein the form imbues the whole and every part, each of which is also fire. Aquinas restores the distinction by replying that each homogeneous bit of the fire performs all the operations of fire, whereas not every operation performed by the whole animal is to be found in each part of an animal. However, the sense in which the parts of all wholes, even the heterogeneous ones, all have a homogeneous aspect, has just been recalled in 3.293, and is displayed in the axiom system sketched in 10.3. It could hence be (cf. 3.42) that the objection is here not totally beside the mark, and that each animated part is, adjectivally (or 'denominatively') speaking, animal. This is what tends to 253

Section 3 emerge from Aquinas' interpretation of the Augustinian thesis that the soul (for Aquinas the form of the whole) is wholly in the whole and in every part. This contrasts with the acc­ ent on the way in which the part lacks the form of the whole ('caret forma totius') which is so prominent in his De Anima objection. Further, the mereological homogeneity mentioned may turn out to be exactly what is required to ground the required adjectival sense, as 3.293 and 3.42 suggest. 3.296 Finally, the distinction between the homogeneous and heterogeneous is to re-surface prominently in the discussion about the resurrection of the body at the end of the world. As we are to see in 3.642, questions of personal identity connected with this eventuality lead to the decision that reconstitution of the disintegrated parts of the body can be slack enough where body-parts which are homogeneous are con­ cerned (e.g. flesh can have its bits juggled about without loss of identity) but the bits must not migrate from one species of part (e.g. flesh) to another (e.g. bone), i.e. across the boundaries of non-homogeneous parts. 3.3

Further Precisions

3.301 Although by this point Aquinas' views on parts and wholes in general, and relatively to some present-day mereol­ ogical theses in particular, have already been roughly sketch­ ed, there still remains a need for a more thorough investig­ ation of certain topics. What, for example, of the question already broached in 2.36 as well as in .123, .124, .213, and .27 of the present section 3 as to the sense in which Arist­ otelian 'matter' (in the 'material make-up' sense) may be aligned with mereological parts? The next section (3.31) covers this. Again, while there is no doubt that for Aquinas

254

Aquinas - Mereological Aspects the integral part embraces both form and matter (in the Aris­ totelian sense), nevertheless the diversity of sense encoun­ tered in 3.22 when the standard mereological negation of id­ entity between part and whole is extended to that between the form and the whole requires a check on Aquinas' classificat­ ions here: 3.32 deals with this. In 3.35 the perennial problem of the 'principal part' will be faced yet again. Abelard's lively account (2.4) is there prolonged in this Aristotelian context. In 3.36 some of Aquinas' extremely venturesome ext­ rapolations of mereological modes of expression into fields hitherto rarely associated with integral wholes will be touch­ ed upon. 3.302 Another highly important distinction, as we have con­ stantly been reminded, is that between X-parts and parts-ofX: 1.4, 2.3. In view of the prominent and characteristically Aristotelian treatment of this topic in a biological context, the central account of Aquinas' stance here will be postponed to 3.41, wherein that context is in question. 3.31 Matter and Part 3.311 From Boethius onwards the Latin West had become accustomed to talking about matter both in connection with the generic part of properly formulated definitions (0.5, HQS §4.3421) as well as in its more down-to-earth sense of mater­ ial make-up. We have seen in 3.213 and 3.27 how Aquinas is well enmeshed in this tradition, and initially the identific­ ation of an object's mereologically proper parts with its mat­ erial make-up in the basic Aristotelian sense would appear to be a promising approach. The correlation between quantity and the matter of physical objects points in the same direc­ tion, given the Aristotelian recognition of the quantitative as the typically integral: 3.213. Add to this the determinacy said to follow upon the quantitative matter's being imbued by its appropriate form, and the root of the genitive ('part-of...') sense of 'part' in this context is quite clear, as when 255

Section 3 Aquinas says that the matter of each natural object takes on a determinate quantity in accordance with its correlation with a determinate form (cuiuslibet rei naturalis materia accipit determínatam quantitatem secundum comparationem ad formam determínatam: AST III, q. 74, art. 2, ad 1; cf, 1.4). 3.312 This last quotation is probably an echo of a remark made in Aquinas' commentary on the Metaphysica about our say ing that bronze is the matter out of which a statue is made. We thus express ourselves in terms of the plain and unadorned word 'bronze', it is suggested, instead of the strictly neg­ ative nominal mode which is really intended ('non-statueesque-bronze') because we happen to have no name for the lack-of-determination-as-a-statue which is here in question: propter hoc quod est innominata priva tio, aliquando simplici nomine materia significatur materia cum privatione, ut ... aes accipitur pro aera in figurato, cum dicimus quod ex aere fit statua: ACM 1546. Here we are also in the neighbourhood of an acute awareness, extending beyond the more obvious biolog­ ical cases, of the way in which the transition from X-part (e.g. the unshaped bronze) to part-of-X (as in the determinately figured bronze) is veiled by the familiarly unguarded use of the same term ('bronze') to cover both phases, i.e. during the statue's time the bronze is the material make-up of the statue, whereas outside the statue's time it may be described as the material make-up for the statue. Boyle's modern philosophical atomism likewise tended to veil such transitions: 2.531, cf. 2.56, 3.294, 8.42. 3.313 The quantitative style of indeterminateness of Xparts just outlined has a more general counterpart when those X-parts have becomes parts-of-X. One then has the individual object (e.g. Socrates) which (or who) is not just a man (for example) but also all sorts of other 'things' which can be connected with the material make-up, the parts-of-Socrates (tall, pink, curly-haired, and so forth). In short, Socrates is also a-man-and-x, where x is likewise indeterminate, covered 256

Aquinas - Mereological Aspects only by the particular quantifier (HQS §5.2) prior to the det­ erminate unpacking of at least some of the cases of χ approp­ riate to Socrates' individuation. This counterpart represents yet another immensely important strand in the analysis of Aquinas' exploitation of Aristotelian 'matter', but plainly does not, of itself, yield 'matter' in the 'material make-up' sense. The latter, as we have seen in 3.22 above, satisfied the mereological requirement that one cannot predicate the whole of the (material) part (1.32, 3.21, 10.324) even in the homogeneous cases: 3.292, 3.293. As Aquinas puts it, 'Nothing is its own matter', Nihil est sua materia: AST I, q. 39, art. 2, ad 5. However, the former, the '... and x', sense of 'matter', does not, in the concrete, satisfy this negative thesis. Soc­ rates is a man-and-curly-haired, hence Socrates is curlyhaired. True, when we go abstract and say that the curlyhairedness, pinkness, tallness, and so forth serve as parts of the individuating features, the materia signata, the desig­ nated matter, whence ensues Socrates' being the particular human being that he is, as contrasted with any other, then one does, as it happens, obtain an apparent analogy with the mereological situation: Socrates is not his pinkness. In point of fact, however, this latter analogous-looking negation is founded on logical grammar, and is hence altogether distinct from the negations founded on the mereological thesis of part-whole disparity: 1.32. 3.3131 These few remarks hence turn out to merely deepen the sense of the complexity of the place occupied by 'matter' in Aquinas' usage. Certainly the 'material make-up' sense does not exhaust it, as neither does the '... and-x' construal. Some progress may perhaps be made by looking at the matterform pair, as is now to be undertaken in the next section. 3.32

Essential or Integral? The status of Matter and Form. 3.321 It has already been noted in 3.2 and 3.24 that Aqu257

Section 3 inas' commentary on the Metaphysica, starting from the commor dichotomy between 'distributive' and 'integral', goes on to subsume under the latter heading both the quantitative and the essential (or 'substantial') parts as sub-species. As a result both form and matter, in the Aristotelian senses, are essential parts which, though integral, are to be contrasted with the quantitative. This tends to militate against two other factors, namely the admission that integral wholes are most evident in the quantitative field (3.213), and the quant­ itative nature of matter where physical objects are concerned. For form is not a quantitative element, as the original arg­ ument in the Metaphysica has made clear (3.11), whereas mat­ ter at the non-quidditative level and in physical objects 'takes on its determinate quantity to the extent that it is imbued with a determinate form': AST III, q. 74, art. 2, ad 1; cf. 3.311 above. It is in sense-perceived objects that the notion of matter qua mereological part has its most proper application: 'The parts are related to the whole in the style of matter ... and matter is most properly talked about in connection with sense-perceived objects': partes se habent ut totum secundum rationem materiae ... Et ... materia magis proprie dicitur in sensibilibus; In Post. Anal. Bk. II, lec. ix [5], 494, p. 360a. However, any inclination, consequent upon this sense of matter, to see it and its correlated form as both disparate parts on a par with indisputably integral parts such as individual arms and legs has now to be resisted in a way which would not have arisen had matter and form not been classified as integral parts in the first place, as 3.22 has shown. 3.322 Walter Burleigh's later separation of integral and essential parts into coordinate species (7.07) is doubtless designed to avoid the above-noted inconveniences. William of Heytesbury and William of Ockham appear to follow Burleigh in this {CLM 238, n. 101) with Albert of Saxony (like Buridan in section 4 below) trenchantly identifying the integral and the 258

Aquinas - Mereological Aspects quantitative. This Albert also accordingly denied that the soul and body (cases of form and matter for Aquinas) of a human being are integral parts: CLM 239, n. 105. A like den­ ial issues from Peter of Spain's doctrine, as recorded in 7.3161 below. 3.323 Reverting, however, to the sorting-out of Aquinas' classifications, there occurs a distinction between parts of the species and parts of the matter when the Metaphysica's listing of senses in which something may be said to be made out of something is being examined. Here the notion of matter intervenes at the concrete level as a pointer towards essent­ ially unspecified concrete parts of the material make-up, as contrasted with the parts designated as essential by the definition (cf. 0.4): (A) Where parts are concerned, some are parts of the spec­ ies, and some are parts of the matter. Now the parts of a species are said so to be which are such that the complete­ ness of [instances of] the species depends upon them, and which are such that without them [instances of] the species cannot exist. This is why such parts are comprised within the definition of the whole object, as with soul and body within the definition of animal, and as with the inclusion of angle in the definition of triangle, and as with letters in the definition of syllable. In contrast, the parts of the matter are said to be those which are such that the exist­ ence of [instances of] the species does not turn thereon; rather such parts are inessential to the [existence of memb­ ers of the] species, e.g. it is inessential to a statue that it should be made out of bronze or of any other [particular sort of] material. Likewise it is inessential to the circle that it happens to be broken down into two semi-circles, and likewise inessential to the right-angle that an acute angle should happen to be a part of it. This is why parts such as the ones mentioned are not comprised within the definition of the relevant specific whole, but rather the opposite . . . . 259

Section 3 Whence, therefore, it is obvious that in this way something is said to be made from something in a primary and proper sense. (Sunt enim partium, quaedam partes speciei, et quaedam partes materiae. Partes quidem speciei dicuntur a quibus dependet perfectio speciei, et sine quibus esse non potest species. Unde et tales partes in definitione totius ponuntur, sicut anima et corpus in definitione animalis, et angulus in definitione trianguli, et litera in definitione syllabae. Partes vero materiae dicuntur ex quibus species non depend­ et, sed quodammodo accidunt speciei; sicut accidit statuae quod fiat ex aere, vel ex quacumque materia. Accidit enim circulo quod dividatur in duos semicírculos; et angulo recto, quod angulus acutus sit eius pars. Unde huiusmodi partes non ponuntur in definitione totius speciei, sed potius e converso ... . Sic ergo patet quod sic quaedam dicuntur ex aliquo fieri primo et proprie: ACM 1089.) 3.324 As it stands, this passage might be construed as un­ necessarily restricting the description 'parts of the material make-up' to those which have no necessary theoretical conn­ ection at all with the sort of thing which is in question. In fact, it is probably no more than a hint as to the sort of 'part' which should be included in the quidditative definition. From a general mereological point of view, the main relevance is to be found in the final part of the passage, which stres­ ses that being a part of-... is always being a part-of-X, with 'X' thus reminding us that the 'definition' of the non-included parts would comprise an allusion to the species in question, whereas the definition of the species does not overtly emb­ race such parts. Thus the bronze is part-of-the-statue, with statue as the species, understanding of which is essential to the understanding of the whole description, 'part-of-thestatue'; nevertheless, the species statue as such does not overtly comprise any bronze make-up. D2 of 10.31 covers this situation. 3.3241 Thus whether X be the Statue of Liberty or the corn260

Aquinas - Mereological Aspects plete collection of men, our knowledge of New York's famous artifact, or of what is involved in being a human being, acts as a hook on which to hang the appropriate sense of 'element of X', and hence enables one to identify such parts in the concrete. We may here, yet again, find ourselves in the reg­ ion of the 'innominata' encountered in 3.312 above. But where 3.312 was concerned with what may be a privation normally un­ named, but expressible by means of the name-forming functor 'non-...' (semantic category /, cf. 10.252) as in 'nonstatuesque bronze', we are now encountering what may be the normally unnamed but still generically expressible by the likewise n/n functor 'part of ...'. True, some parts do have their own species-names which are individualised, as in 'this arm', 'this toe', and so forth, but indefinitely many normally do not. Thus 'this-left-arm-and-this-fourth-toe-on-the-rightfoot-in-1989' is one such which, with appropriate individual­ isations of the 'this' could call attention to previously unnamed parts of a contemporary human being, and (as it hap­ pens, analogously) of the Statue of Liberty. This exemplifies the myriads of unheard-of (hence innominata, 'unnamed') combinations which constitute elements in the mereological sense and which are covered by the appropriate clause ('every element has an element in common with an X') of the D2 in 10.31. This clause shows how previously unheard-of combinations of possible bits of X, as well as unenvisaged chunks of statue-composing stuff, like the bronze, count as elements of X. 3.33 Form-parts and Integrality 3.331 Having thus been just reminded of the axiomatic's gather-all distributive characterisation of the indefinitely many possible sorts of mereological parts within a given int­ egral whole, a momentary reversion may now be made to the qualms revived in 3.301 concerning Aquinas' having made the Aristotelian 'form' into an integral part. Now there is no 261

Section 3 doubt at all that such a form does, in a superficial sort of way, satisfy some of the recognised theses of mereology. Thus it appears to satisfy 10.324, which may be roughly rend­ ered, 'If A is a proper part of B, then it's not that A is B', i.e. the 'Thesis of Part-Whole Disparity', broached in 1.32. For example if the form dogness is a proper part of Fido, then it's not that dogness is Fido. But here, as has been urged above, the superficiality of the apparent satisfaction is very evident from the abstract style of the noun 'dogness', thus used to designate a 'form'. This is in fact not a noun at all (10.29) except for adherents of the simpliste 'stand­ ing-for' school of meaning (3.117), and the non-identity of Fido and his dogness (and its parts) is just a consequence of the logical grammar of the situation: 3.22. These consid­ erations demonstrate how a merely apparent satisfaction of the thesis of part-whole disparity will not suffice to char­ acterise mereological parts. 3.332 The topic thus raised, i.e. that which concerns the 'parts' of a form (e.g. of dogness) becomes even more promin­ ent in connection with that distributive characterisation of parts recalled in 3.3241: every element of the integral whole of the X's has an element in common with an X. How, or in what sense, could an Aristotelian form (e.g. dogness) have a proper-or-improper-part (i.e. an element) in common with a dog? Such an element in common must be found if the form is to be an integral part of the dog (as Aquinas says it is) in the mereological sense. In what sense, therefore, does a form have parts? 3.333 Aquinas deals most thoroughly with this question when the status of the soul (i.e. the form) of an animal is being discussed. This will be covered at length, in its mer­ eological aspects, in 3.4. For the moment, a few relevant extracts suffice to advance towards an express treatment of the senses in which a form has parts. Of this form which is the animal soul Aquinas, overtly claiming to follow Aristotle, 262

Aquinas - Mereological Aspects asserts 'if it is the form of every part, it must be in every part, as opposed to being in the whole alone or being in one part alone': ADA art. 10, c., fully quoted in 3.435 below. Here one may surely add, 'or being itself a part on a par with those it imbues' as a further disjunct to be negated. But such a facile spinning out of negations need not by any means be the end of the matter. The relation of the form to every part is said to consist in its being 'in' each part. Our armoury of many-link functors suggests that where the sign  represents some linguistic correlate of the form in question, then saying that each part is a 'π-er' (or, technic­ ally, is a 'trm', cf. 10.251) makes that form to be 'in', or to imbue, every part. However, the internal structure of the 'π' needs further precision in this context. Suppose, for sim­ plicity's sake, that Fido is the only dog, and hence identical with the complete collection of dogs. Plainly, 'π' cannot here be '... -is-a-dog', since this would make each proper part of the dog into a dog, whereas only that improper part, Le. the whole, is a dog. Something like '... is endowed with doggy nature' is needed as the substituend for . This allows each element of Fido to be doggy, but not necessarily a dog. It is in this indirect, or 'denominative', adjectival, or paronymous sort of way that the 'form of dog' may in a very remote sense, be said to be 'part of' the dog Fido. Each part is a haver of dogginess, but without any necessary commitment to being a dog, which Fido as a whole is. (On the topic of den­ ominatives, or paronyms, see HQS 160 - 73, HL §3, HDG, and HCD). In fact Fido as a whole has that sort of dogginess which can cover both being a dog and (as in the case of the parts) merely having dogginess. 3.334 What then is this dogginess, which may thus comp­ rise, or need not invariably exclude, full-blown dogness (i.e. being a dog)? As so often happens, and as Plato's Socrates long ago remarked, the answer has been in front of our noses almost from the beginning: having an element in common with a 263

Section 3 dog, an instance of the by now oft-reiterated distributive clause of the definition of the collective class (or integral whole) (3.324 and D2 of 10.31) represents the mereological an­ atomy of the dog's soul, (and so on for other animals, includ­ ing man). Since, as Dl of 10.31 lays down, element is here proper or improper part, the full-blown '... is a dog' holds when that element ÍS the improper part (i.e. the whole) of the integral whole of the dog (i.e. Fido) which is in question. Fido is an improper part of Fido since Fido is identical with Fido. The more general '... is imbued with dogginess' holds for each proper part of Fido (i.e. 'part' in the usual nonwhole sense) as well as for Fido as a whole. 10.31 of the Presuppositional Explicitation relates all these considerations to totally unmysterious primitive terms, axioms, and definit­ ions. They will recur when the anatomy of the soul is being more fully dealt with in 3.4. 3.335 Everything which has just been said is not incompat­ ible with (although it supplements in detail) Aquinas' express general declaration on the typology of parts which is also to be found in the corpus of article 10 of his Q.D. de Anima (ADA 319). It is towards the elucidation of that general declar­ ation that the remarks hitherto made have been directed as preliminaries. That his declaration confirms almost every detail of these preliminaries may now be shown. 3.3351 First of all, let us recapitulate. Qualms were rais­ ed as to the sense in which an Aristotelian 'form' can be said to be an integral part of an object. These took us to the axiom and definitions for mereology, and these in turn to the fact that the elements of an integral whole each have an el­ ement in common with a being or beings of the species the integral whole of which is in question. Hence if the form is to be an element or part in the mereological sense, that form must itself have elements or parts in order to satisfy the relevant definition of integral whole. But in what sense may the form thus be said to have elements or parts? At the 264

Aquinas - Mereological Aspects outset of his passage containing a triply-distinguished ans­ wer to this question, Aquinas states an inessential ('den­ ominative', 'paronymous', 'adjectival') sense in which a form such as whiteness may be said to have parts. This type of having is grounded simply on the fact that an originally unified haver-of-whiteness (i.e. white-ise-er, or 'trm', structured as defined in 10.251, cf. 3.333) such as a surface is quantitatively disaggregated into lesser surfaces which are each themselves havers-of-whiteness (or whites): (A) There turn out to be three ways in which wholeness can be attributed to some form, these being correlated with the three ways in which it may come about that something has parts. In one way a thing has parts in respect of the partitioning of quantity, i.e. insofar as it is partitioned in respect of number [of parts] or size [of parts]. Now whole­ ness in respect of number or size is not to be attributed to a single form, except perhaps in an inessential sort of way, as exemplified in forms which just happen to be partitioned incidentally by a partition of the continuum [which they imbue], as occurs with a whiteness, for example, upon the partitioning of the continuous surface [which it imbues]. (Potest autem attribuì totalitas alicui formae tripliciter, secundum quod tribus modis convenit aliquid habere partes. Uno enim modo aliquid habet partes secundum divisionem quantitatis, prout scilicet dividitur numerus aut magnitudo. Uni autem formae non competit totalitas numeri  magnitudinis, nisi forte per accidens, puta in formis quae per accidens dividuntur divisione continui, sicut albedo per divisionem superficiei: ADA art. 10, c. p. 319a.) Such parts, i.e. the various whites resulting from quantitative partitioning, do contribute to the satisfaction of the mereol­ ogical definition of the integral whole in that they each have an element in common with a white surface. This corresponds to the distribution of that open sort of dogginess which may embrace even full-blown dogness, and which was encountered in 265

Section 3 the previous discussion; it is possessed by both parts and wholes in the dog world. 3.3352 Our previous findings suggest that it is inevitable that when Aquinas is making distinctions as to sorts of parts, there should occur a mention of quidditative parts. Surely enough, there now follows the familiar identification of mat­ ter and form (at the quidditative level: 3.12) with genus and differentia respectively, both then being parts of the whole quiddity. (He in fact uses the 'essential parts' vocabulary encountered in 3.321, in order to refer to the quidditative parts.) What is new and mereologically interesting in the present context, however, is the mention of mereological at­ oms, i.e. objects having no parts. Abelard had already inc­ identally envisaged such objects (AD 554.15.17) and LEM shows how present-day mereology takes cognisance of both the atomic and atomless possibilities of mereology: 10.35. Aquinas bel­ ieved that such objects, which were called by the medievals 'simple substances', existed in great profusion (HQS §5.24). He makes the point that both such simples and the composites can be alike in possessing natures which are not simple: (A) In another way something may said to be a whole in respect of the essential parts of the species, as when the matter and form are said to be parts of the composite [quid­ dity]. Genus [i.e. the matter] and differentia [i.e. the form] are in a certain sense parts of the species. It is in this sense that wholeness is attributed even to the essences of objects having no parts, i.e. on account of the completeness of the essence. Here the common feature is that even as composite objects share in the completeness of their species on the basis of their essential principles, so also the sim­ ple substances and forms exhibit their completed species on their own similar grounds. (Alio modo dicitur aliquid totum per comparationem ad partes essentiales speciei; sicut mater­ ia et forma dicuntur partes compositi: genus et differentia attribuitpartes quodammodo speciei. Et hic modus totalitas 266

Aquinas - Mereological Aspects ur etiam essentiis simplicibus ratione suae perfectionis; eo quod sicut composita habent perfectam speciem ex coniunctione principiorum essentialium, ita substantiae et formae simplices habent perfectam speciem per seipsas: ADA art. 10, c., p. 319a.) 3.3353 The third section of Aquinas' distinctions goes on to enlarge on the first case which, as we saw, distributed the having of an element in common with an X throughout the parts which comprised the integral whole of the X's. In the example of the white surface made up of white surfaces, every element thereof has an element in common with the whole white surface. Now the question is raised as to whether, and in what sense, if any, the whiteness as a whole may be att­ ributed to a part of that surface. (The reason for this question's being raised is because a sense in which the soul as a whole is in every part of the body is being sought.) That question, replies Aquinas, may be understood in the first place quantitatively, in which case it is obvious that no part mereologically comprises all the white parts. It is the int­ egral whole (i.e. the complete collection) which contains them all, and each part has part only of the whole stretch of whiteness-imbued surface. 3.3354 There is, however, continues Aquinas, another sense in which the whole whiteness is in every part, i.e. on the assumption that we are dealing with a single hue of white­ ness, then each part of the surface exemplifies the whole of the definition of that hue, i.e. as the first clause of the presupposed definition of integral whole (D2 of §10.31) rem­ inds us, when we are dealing with the complete collection (call it 'A') of relevant white surface-parts (let their shared name be V ) then every a is an element of A. Each of those white surface-parts has just as much a share in the full def­ inition of that hue of whiteness as has the others. However, things go differently where it is a question of the capacity of the whole; this is plainly not wholly in every part; e.g. 267

Section 3 each has not the same power to dazzle the sight as does the whole (where we are still dealing with the white-surface). Like remarks apply to the heating powers of the parts and whole of a fire: (A) In the third sense, something is said to be a whole relatively to the parts of its capacity or power, and such parts are allocated in correlation with the splitting up of its workings. Thus suppose some form to be given which is split up by the division of its continuum, and the question is raised about whether it is as a whole in every part of its physical object, e.g. whether the whiteness as a whole is in a part of its surface, then [i] if the whiteness is correlated with the quantitative parts (so that the resulting totality only pertains to the whiteness accidentally) then that whiteness is not wholly in every part, but its whole is in the whole, and its part in the part. [ii] If, in contrast, the question is interpreted as relating to the wholeness which is connected with the species, then the whole is in each part, assuming that the whiteness is just as intense in a given part as it is throughout the whole. But now it is the case that in respect of its capacity one does not have the whole in every part. For the whiteness which is in a part of the surface does not have the same dazzling-power as does the whiteness that is in the whole of the surface; the same sort of thing applies when the heat which is in a little bit of a fire cannot do as much heating as does the large fire. (Tertio modo dicitur totum per comparationem ad partes vir tutis, seu potes tatis; quae quidem partes accip­ iuntur secundum divisionem operationum. Si qua igitur forma accipiatur quae dividitur per continui divisionem, et quaeratur de ea utrum sit in qualibet parte corporis tota, utpote utrum albedo sit in qualibet parte superficiel tota: si ac­ cipiatur per comparationem ad partes quantitativas (quae quidem totalitas pertinet ad albedinem per accidens), non est tota in qualibet parte, sed tota in toto, et pars in parte. 268

Aquinas - Mereological Aspects Si autem quaeratur de totalitate quae pertinet ad speciem, sic tota est in qualibet parte; nam aeque intensa est albedo in aliqua parte sicut in toto. Sed verum est quod adhuc secundum virtutem non est tota in qualibet parte. Non enim potest tantum in disgregando albedo quae est in parte superficiei, sicut albedo quae est in tota superficie; sicut neque tantum potest calor qui est in parvo igne ad calefaciendum, sicut calor qui est in magno igne: ADA art. 10, c. p. 319a b.) 3.3355 These last 'capacity' instances raise questions with which we are not yet entirely equipped to deal, but enough has been said to make it plain that the mereological axioms and definitions of 10.31 reproduce Aquinas' intuitions in a systematic fashion, consonantly with the most stringent con­ temporary demands concerning axiomatised theory-construction. On Aquinas' own showing, although it is true that the 'form' is involved in being-in or 'imbueing' the mereological parts, it is scarcely helpful to make that form itself into a part on a par with those mereological parts. 3.336 However, the verdict on the original question which led to the examination of the present distinctions becomes a preliminary to an important clarification of Aquinas' doctrine of the animal anima, 'soul'. For he goes on so say, in this same corpus, that as in the case of the 'form of white' exam­ ined above, the soul, as the form of the animal body, 'imbues every part of the body', and 'as a whole is in every part of the body in respect of that wholeness associated with the fulness of the species' (cf. 3.43). The former of these last two excerpts is at least saying that every element of a man has an element in common with a form-of-man-ness-have-er (i.e. a trm) , and this is well in accordance with the definition of the integral whole: D2 of 10.31. As 3.334 has already suggested, the suitability of the Lesniewskian element (i.e. proper-or-improper-part, defined Dl, 10.31) for Aquinas' purposes at this point could not be more amply obv269

Section 3 ious. 'Element'qua 'improper part' covers the whole man's body being imbued with the form of man-ness (so that we have a whole man) whereas qua 'proper part' (i.e. in the usual sense of part) it covers the parts' being imbued with that form but without each of those parts' being a man. Thus each part is man-ish, but not a man. Only the whole (the 'part' in an improper sense which contributes to that of 'element') is a man. Use of 'element' covers both contingencies, as 10.31 confirms. 3.337 Thus is clarified the talk about the form of the body, i.e. the soul in animals, 'imbueing every part of the body'. At the same time Aquinas is protected from what could other­ wise be the damaging consequences of the second of the two last excerpts, which was to the effect that the soul as a whole is in every part of the body. This would at first sight appear to commit him to each part of a man being a man, for example. In fact that second excerpt does display an appr­ opriate note of caution: it is in respect of that wholeness associated with the fulness of the species (secundum totalitatem perfectionis speciei) that the form as a whole is said to be in every part of the body. He certainly does not want the fulness of the species (perfectio speciei) to imbue every part (or every part would be a man). Mereological imbuement, as implied by 'totalitas', avoids just this threat, while still allowing the whole to be a man. The way in which further details of his various avoidances in this respect are related to the mereological substructure will be examined in 3.4, which is more expressly devoted to the doctrine of the animal form or soul. 3.34 'Part of ...' as a Functor 3.341 Throughout the above rapprochement between Aquinas' various applications of Aristotelian doctrines on 'form' and 'matter' and present-day mereology, it has been taken for granted that 'part of ...' is a name-forming functor (categ270

Aquinas - Mereological Aspects orial index /3//2) which calls for completion by the name of the whole to which the part is subordinate. For the record, but without detailed analysis, which would take us too far from the central theme, here is a passage which stresses the dep­ endent nature of parts in a way which confirms Aquinas' agr­ eement with this point, and throws light on a great deal more also, especially as regards the contrast between the integral whole (amongst which are his 'substantial' wholes) and its proper parts in respect of the individuation-terminology which is here his primary theme: (A) Hence in order that what may be granted and what den­ ied in respect of such topics may be known, the following points are to be noted concerning names having to do with individuation . These may be names of 'first imposition' [i.e. terms used at the non-quidditative, nominally-termed level], such as 'person', and 'hypostasis', i.e. they signify the objects themselves; other names having to do with individ­ uation may be names of second imposition [i.e. at the quidd­ itative level] such as 'individual', 'supposit', and so on, used to signify the notion of individuality. Of such names [having to do with individuation], some have only to do with the category of substance (e.g. 'supposit' and 'hypostasis'), these not being applied to incidentals, and there is also 'person' applied within the scope of the nature of rational beings, and even 'natural object', according to Hilary's theory. Some others, in contrast, have to do with indiv­ iduation no matter what the category in question; for exam­ ple, 'individual', 'particular', and 'singular', which are also applied within the scope of the incidental [i.e. the nonsubstantial]. Now it is a characteristic feature of subst­ ance that it subsists of itself and in itself, whereas it is characteristic of an incidental to be in another. And hence those names which have to do with the individuation of sub­ stance only have their place where things which subsist of themselves and in themselves are concerned. This is why 271

Section 3 they are not even used of the parts of substances; it is because such parts do not exist in themselves but in the whole in question, although they are not mere incidentals. However, those names which have to do with individuation, irrespectively of whether it be in the substantial or the incidental area [as mentioned above] can appropriately be used of parts. Thus one cannot say that this [living] hand is a person, a hypostasis, or a supposit, whereas one can say that it is something particular, singular, or individual. For although the hand belongs to the category of substance, it is nevertheless not said to be a hypostasis, or a supposit, or a person, because it is not a complete substance subsist­ ing in itself. {Ut igitur sciri possit quid in talibus concedendum sit et quid negandum, considerandum est quod nominum ad individuationem pertinentium, sive sint nomina primae impositionis, sicut 'persona' et 'hypostasis', quae significant res ipsas, sive sint nomina secundae impositionis (sicut 'in­ dividuum', 'suppositum', et huiusmodi), quaedam eorum pertin­ ent ad solum genus substantiae, sicut 'suppositum' et 'hypo­ stasis', quae de accidentibus non dicuntur, et 'persona' in rationabili natura, et etiam 'res naturae', secundum acceptationem Hilarii. Quaedam vero pertinent ad individuationem in quocumque genere, sicut 'individuum'. 'particulare', et 'singulare', quae etiam in accidentibus dicuntur. Est autem substantiae proprium ut per se et in se subsistat; accident is autem est in alio esse. Et ideo illa nomina que pertinent ad individuationem substantiae, in illis solum locum haben t que per se et in se subsistunt. Et proper hoc etiam de partibus substantiarum non dicuntur, quia non sunt in seipsis sed in toto, quamvis non sint in subiecto. De quibus tarnen dici possunt nomina ad individuationem pertinentia convenienter tam in substantiis quam in accidentibus. Non enim potest dici quod haec manus sit persona, vel hypostasis aut suppos­ itum; quamvis dici possit quod sit aliquid particulare, sing­ ulare, vel individuum. Manus enim etsi pertineat ad genus 272

Aquinas - Mereological Aspects substantiae, quia tarnen non est substantia completa in se subsistens, non dicitur hypostasis aut suppositum vel per­ sona: AQD II, p. 427a - b; De Unione Verbi Incarnati, art. 2, .) One consequence of this which is made explicit in the disputed question De Spiritualibus Creaturis, article 2, ad 16, (AQD II p. 378) is that strictly speaking the human soul, being in some sense a part, is neither a hypostasis, nor a person, nor even a hoc aliquid (i.e. a 'this-something', an individual concrete object): cf. 3.22.

3.35 Principal Parts 3.351 Without in any way claiming to exhaust Aquinas' treatment of the topic of what constitutes the 'principal part' of an integral whole, the following samples of his remarks may serve to give some notion of his position as compared with that of Abelard (2.4). 3.352 There are certain obvious candidates for priority and principality qua parts, such as the parts necessary for the existence of an object of the species in question, given its definition: ACM 1482. This makes the parts covered by the definition relevant to a given form into 'prior' parts: oport­ et quod partes formales sint priores quolibet composito: ACM 1486. However, materially speaking, where the human body is concerned, there is a forecast of what is to be observed bel­ ow in 3.41. There it becomes evident that consciousness of the distinction between X-parts and parts-of-X (as covered in 1.4, 2.3, as well as in 3.41) ensures that the quasi-logical sense of 'prior' used just now for the form-relevant features is no longer dominant, and that the literal sense which now comes into play can be used to eliminate any pre-X temporal priority of the parts-of-X. No parts-of a given human or animal are temporally prior to it, although some parts which are still subject to this non-prior status, like all the rest, 273

Section 3 are nevertheless necessary components if there is to be an animal of the sort in question at all. The thesis that a dead finger is no more a finger than a pictured one (and so on for other parts) brings out very strongly the distinction between a part-of-X (e.g. the living finger) and an X-part (e.g. the amputated finger): (A) It is hence evident that parts of the body are in a certain sense prior 'to the entire whole', i.e. to the comp­ osite, and yet in a certain sense are not. There are some which are just as prior as the entire composite, in the sense that the animal made up from them is assembled from them. But they are by no means prior in the sense in which that is asserted to be prior to something which can exist without the existence of that other thing. For the parts of the body cannot exist separated from the animal, because a fing­ er does not remain a finger under no matter what circum­ stances. A finger which is cut off or dead can only be said to be a finger in an equivocal manner, as is a sculptured or pictured finger. But there is a sense in which parts of the sort mentioned are posterior to the composite animal, since the animal can exist without the finger. But there are some parts which, although they are not prior to the whole animal in the sense that they can exist without it, are nevertheless fully cotemporal with it. For even as these parts themsel­ ves cannot exist without the integral animal, so also the animal cannot exist without these. It is parts of this sort which are the principal parts of the body, and in which the form (i.e. the soul) is primarily in evidence, i.e. the heart or the brain. (Patet igitur quod partes corporis sunt prior­ es 'simul toto', idest composito quodammodo, et quodammodo non. Sunt quidem priores sicut simplex composito, in quantum animal compositum ex eis constituitur. Sunt autem non priores secundum modum quo dicitur esse prius id quod poť est esse sine alio; non enim partes corporis possunt esse separa tae ab animali; non enim digitus quocumquemodo se 274

Aquinas - Mereological Aspects habens est digitus. Iile enim qui est decisus, vel mortuus, non dicitur digitus nisi aequivoce, sicut digitus sculptus vel depictus. Sed secundum hanc consider ationem buiusmodi partes sunt posteriores composito animali, quia animal sine digito esse potest. Sed quaedam partes sunt, quae licet non sint priores toto animali hoc modo prioritatis, quia non pos­ sunt esse sine eo, sunt tamen secundum banc considerationem, simul, quia sicut ipsae partes non possun t esse sine integro animali, ita  integrum animal sine eis. Huiusmodi autem sunt partes principales corporis, in quibus primo consistit 'forma', scilicet anima; scilicet cor vel cerebrum: ACM 1487 9.) 3.353 Quite apart from the discussion of 'principal parts', much of what is said here could be taken as a salutary rem­ inder of how things are, and is to be extended in all sorts of ways in 3.41 and 3.5. Thus not only is a dead eye no longer an eye, but its behaviour in laboratory experiments is no longer seeing, but only simulated seeing; the latter is just not seeing, even as a simulated currency note is just not a currency note. Even less is a photosensitive cell or set of cells hitched to an ambling computer capable of seeing. The conceptual isolation of 'sense'-operations so that they can bombinate in the disembodied near-vacuum of the mythical 'mind' countenanced by the moderns' 'theory of knowledge' is all of a piece with, and has only tended to encourage, the linguistic aberrations of the present-day purveyors of 'art­ ificial intelligence'. Again, one only needs 'thinking' to be done by 'the brain' (as opposed to the thoughts of human per­ sons) and from this abstract launching pad computatorial ana­ logues of brain mechanisms go into orbit disguised as intell­ igent thought without any difficulty whatsoever. Give the imposing machine a human name (e.g. 'Eliza Weizenbaum', NM 92, 97) and engage in what can be described as speech-trans­ actions with it, and the illusion is complete. 3.354 Of course, from a practical point of view it can be 275

Section 3 linguistically convenient to use the former owner's name in connection with a disconnected limb or organ, or even for the whole corpse. Pathology laboratory talk would become intoler­ ably complex without such linguistic contrivances, and perhaps the same may apply in computer laboratories. But to treat such talk as non-poetic would be like taking seriously and literally the child's claim that her marvellous new doll really does weep tears and micturate. 3.355 Another set of Aristotelian dicta on principal parts may be derived obliquely from the philosopher's discussion of what is involved in being maimed, mutilated, lacking, or even truncated. (These terms are translations of Aquinas' own suggestions for possible translations of his transliterated 'colobon': ACM 1109.) Seven things are required of a whole if it is to be colobon. It must be quantitative (ACM 1110), het­ erogeneously-parted (ACM 1111), and the removed parts must be less than half the whole (ACM 1112). From this it looks as though the greater part is (or tends to be) the principal part. Again, the part which remains should not have an iden­ tity other than that which it had prior to truncation. This allegedly eliminates numerically-characterised objects from truncation: ACM 1113. Thus presumably a trio lacking a mem­ ber would in fact be a duo, as opposed to a truncated trio, whereas a cup without its handle is still a cup, although a mutilated one. Here the principal part is the sortai identity warrant for the type of object in question. Abelard has al­ ready shown (2.4) how contingent this criterion may be. Now although dissimilarity of parts (as observed in ACM 1111, noted above) is a requisite for a possible mutilation, there are still some dissimilarities of parts which are such that removal of one does not yield a mutilated object. Thus a dozen-parted manifold has duos and trios as its dissimilar parts, but removal of some one or two of these does not mutilate but changes the identity of the manifold qua mani­ fold. This is because a manifold, qua manifold, is like276

Aquinas - Mereological Aspects parted, and hence not a candidate for mutilation (ACM 1114), as was observed before (in ACM 1113). 3.356 Next comes a spelling out of what was involved in the already-mentioned requirement of heterogeneity (noted above as originating from ACM 1111). The position of the various parts should make a difference to the whole (e.g. as in a house) as opposed to what obtains in mass-wholes such as water, wherein no particular structure of the parts is demanded for the identity of the whole: ACM 1115. Finally, only the continuous can be mutilated. This contention is supported by the example of the discontinuous notes of a scale: ACM 1116. Abelard's suggestion that only the natural, or the divinely-effectuated, as opposed to the artificial in its humanly-contrived aspects, is really continuous (2.231, 2.232) would, by this criterion, mean that most artifacts could not be mutilated, and this would scarcely be helpful. From the point of view of the quest for principal parts, Aquinas' criterion of continuity is here suggesting that the principal parts must be continuous with the non-principal parts. Once again, this is exceedingly dubious, especially given the varieties of proximity which can count as constituting continuity. 3.357 On the side of the parts, so to speak, there are also certain requirements for their removal to amount to a maim­ ing. Thus the removal of the part must not be such as to destroy the whole: a decapitated man is not just maimed: ACM 1117. At the same time the removed parts must be in the ex­ tremities, e.g. a hand or a foot: ACM 1117. However, removal or lack of self-regenerating parts is not mutilation: neither the skinhead nor the bald person is mutilated: ACM 1118(2). Whence we are in effect being told that vital, central, and non-self-renewing parts tend to be principal. 3.358 As the final item in this hasty sampling, the succint capacity-related criterion for principal parthood from AST may be added: 277

Section 3 (A) To the fifth objection the reply is that one part of the body is said to be more principal than the other because of the diverse powers whereby the organs are parts of the body. That which is the organ of the more principal capacity is a more principal part of the body, as also is that which subserves the more principal capacity. {Ad quintum dicendum quod una pars corporis dicitur esse principalior quam alia, propter potentias diversas quarum sunt organa partes corpor­ is. Quae enim est principalioris potentiae organum, est principalior pars corporis: vel quae etiam eidem potentiae principalius deservit: AST I, q. 76, art. 8, ad 5.) 3.36 Analogically Integral Parts 3.361 It has already been observed in 3.212 that Aquinas applied the notion of the integral somewhat beyond the quant­ itative sphere to which, by his own admission (3.213) it most readily applies. To pursue in detail his similarly rather stretched subsumption of virtues and their parts under that same notion would be a lengthy matter, but must at least be noted in passing. Thus in AST I-II, q. 57, art. 1, ad 5, Cicero's claim that memory of things past, understanding of the present, and provision for the future, are parts of the virtue of prudence {Rhetorica II, ch. 53) is accounted for by making these three into quasi-integral parts of prudence. Here the presupposed definition of integral whole which requires that all the parts should make up that whole is the fount of the analogy: quodammodo comparantur ad ipsam [sc. prudentiam] sicut partes integrales, inquantum omnia ista requiruntur ad perfectionem prudentiae: cf. D2 of 10.31. 3.362 In AST II-II, q. 48, art. 1, c, having reiterated the three types of part (integral, subjective, and potential, as in 3.26(A) above) Aquinas exploits all three types for the purp­ ose of assigning parts to virtues. By now, owing to the use of the integral analogy ('ad similitudinem partium integral278

Aquinas - Mereological Aspects ium ') prudence has acquired no less than eight quasi-integral parts. The same similitude is exploited throughout the rem­ ainder of AST II—II· Indeed, the whole of q. 79 is devoted to the quasi-integral parts of the virtue of justice, as is evid­ ent from the corpus of its first article. In q. 80, ad 2, the same notions recur. The integral whole is mentioned when the relation of 'epieikeia' to iustitia is dealt with: q. 120, art. 2, c. . Again, q. 128, art. 1, c, and q. 143, art. 1, c, both exploit the notion of integral part for the mapping of various virtues. Justifications of such extensions could depend,somewhat on the senses in which virtues may be said to be composed from integral parts. Since this material is largely a lost world for the general contemporary reader, it would be point­ less to speculate on its cogency at this point. Clearly, how­ ever, the definitional nature of the discourse in question requires that (for Aquinas) it should be pitched at the quidd­ itative level, as opposed to the nominally-termed level norm­ ally appropriate to mereological theory: 0.4, 10.36.

3.4

Anatomy of the Soul

3.41 Parts-of-X and X-Parts 3.411 As was suggested in 3.302, a sketch of some aspects of Aquinas' vital distinction between parts-of-X and X-parts, already touched upon in 1.4 and 2.3, has been postponed until the present juncture, where the examples afforded by living integral wholes serve to stress that distinction in the most radically possible way. 3.412 It is the Aristotelian identification of the anima ('soul') of the animal (i.e. the 'ensouled') with its form or actuality, which tends to ground the immense force that that distinction displays in Aquinas' mereology at this point. It is most important for the contemporary reader to realise that 279

Section 3 the form-soul is not an object in the way in which the Cart­ esian mind-soul is (3.117). Indeed, it is nothing so myster­ ious, nor need it have any supernatural or theological assoc­ iations. The mystery, if any, is rather in the logical gram­ mar, at least for those lacking in the elements of our present-day counterpart of speculative grammar. Systematic­ ally speaking, the non-Platonic and non-nominalist Aristotelian medievals may be depicted as seeing quidditative discourse, with its verb-like terms, the categorial index of which is minimally s/n, as somehow prior: cf. 0.4. The definitions thus showing what is involved in the act of being-a-so-and-so would obviously and appropriately be expressed at this level, and using such verb-like terms: HQS 4.3. Concrete objects in general (typically alluded to by means of names in name-term­ ed sentences) are simply the embodiments (individualised into their various appropriate material make-ups) of such acts. This is why 'trm' or 'π-er', defined at 10.251, wherein V expresses the act-form, serves as a reminder, categorised as n/(s/n), of such embodiment. For living beings in particular, the act-form is said to be the anima, the soul, of the ensoul­ ed, the animal. The soul, strictly speaking, is thus the act (or actuality, primary actuality) of a living body: anima est proprie actus corporis viventis: AQD De Potentia, q. 3, art. 9, ad 8. 3.4121 From the point of view of general ontology and mereology, there is no difference between form in general and the particular cases of form here in question, i.e. those of living beings. However, practically speaking, the particular case of the animate tends to involve objects which have a peculiarly direct way of signalling their being, i.e. their aliveness. Vivere viventibus est esse, i.e.' To live is to be (as far as living beings are concerned)' is Aquinas' oft-used verb-termed prime link in this chain of reminders. Hence when X no longer signals its aliveness, one is immediately aware of the pres­ ence of mere X-parts, as opposed to parts-of-X. The inanim280

Aquinas - Mereological Aspects ate (e.g. a house) may be much more vague in such matters. Hence ensues the incisiveness of the organic examples which now follow: cf. also 3.432. 3.413 The first passage broaches part of the sense which Aquinas gives to his often-repeated quotation from Augustine of Hippo's De Trinitate VI: 'The soul is in each [living] body both wholly in the whole and wholly in each of its parts'. More detailed attention will be focussed on this borrowing below. For the moment it is used to illustrate the powerful impetus it supplies to the distinction between the part-of-X (e.g. the eye which is part of the animal), and the X-part (e.g. the former eye which is now no longer an eye upon the death of the animal): (A) But since the soul is at one with the body in being its form, it must imbue the whole of the body and every part thereof. For the soul is not an incidental form of the body, but rather the essential form. Now the essential form not only ensures the completeness of the whole, but also of each part. For since a whole is made up from parts, a form of the whole which does not give being to each of the parts of a physical object is a form which is just componential and arrangemental, as in the case of the form of a house, and such a form is incidental. In contrast, the soul is an ess­ ential form, and hence it can only be the form and actuality not only of the whole, but of every part as well. This is why, the soul having departed, in the same way as an animal and a man are not then asserted to be there save in an equ­ ivocal sense (in the style of a pictured or stone-fashioned animal), so also it is with the hand and eye, or flesh and blood, as the Philosopher says [De Anima II, ch. 1, 419b, 9 17]. The symptom of all this is the fact that when the soul has gone, no part of the body pursues its peculiar functions, as contrasted with every thing which abides within its part­ icular species, and thus abides in possession of the operat­ ive power of that species. For the actuality is to be found 281

Section 3 in that of which it is the actuality. Hence it must be that the soul is in the whole body and in each part thereof. (Sed quia anima unitur corpori ut forma, necesse est quod sit in toto, et in qualibet parte corporis. Non enim est forma corporis accidentalis, sed substantialis. Substantialis autem forma non solum est perfectio totius, sed cuiuslibet partis. Cum enim totum constat ex partibus, forma totius quae non dat esse singulis partibus corporis, est forma quae est compositio et ordo, sicut forma domus: et talis forma est accidentalis. Anima vero est forma substantialis: unde oportet quod sit forma et actus non solum totius, sed cuius­ libet partis. Et ideo, recedente anima, sicut non dicitur animai et homo nisi aequivoce, quemadmodum et animai pictum vel lapideum; ita est de manu et oculo, aut carne et osse, ut Philosophus dicit. Cuius signum est, quod nulla pars corpor­ is habet proprium opus, anima recedente: cum tamen omne quoc retinet speciem, retineat operationem speciei. Actus autem est in eo cuius est actus. Unde oportet animam esse in toto corpore, et in qualibet eius parte: AST I, q. 76, art. 8, .) 3.414 Aquinas' further arguments in favour of the thesis of the last-quoted sentence will be viewed in much more detail in 3.42, when its connection with the mereological axiomatic is to be investigated. While there is, of course, a difference between the animal and the house cases herein differentiated by Aquinas, it has to be repeated that from the general mer­ eological perspective that difference is at this point irrel­ evant. Abelard has already made clear his consciousness of the difference between house-parts and parts-of-the house (2.3), and Aquinas would surely agree to this non-organic par­ allel to his animate instances, as 3.312 has suggested. Such ultimate agreement is exemplified by a passage which begins by apparently leaving out of account the way in which the form of the whole informs also the parts in the organic cas­ es, but quickly rectifies the situation by adverting to prec­ isely those cases somewhat in the manner of the last-quoted 282

Aquinas - Mereological Aspects passage. Then comes the extension to the non-organics, with the line-parts and the parts-of-the-line appropriately dist­ inguished, rather in the style of Abelard's rod (1.232(A)): (A) The being of the whole is not that of any one of its parts. Hence when the whole is destroyed either the parts altogether cease to be, as do the parts of animals when their whole is destroyed, or, if they do remain, [the nature of] their actual being is other than it was, as when the being of the part of a line is other than that of the whole line. (Esse totius non est alicuius suarum partium: unde vel pars omnino desinit esse, destructo toto, sicut partes anim­ alis des truct animali; vel, si remanent, habent aliud esse in actu, sicut pars lineae habet aliud esse quam tota linea: AST II-I, q.4, art 5, ad 2.) 3.4141 Again, in the commentary on De Interpretatione, the general principle that no separated part has the form of the whole (nulla ... pars separata habet formam totius) sounds general enough to cover all cases, both organic and inorganic, even though the annexed illustration happens to be that of the non-man-ness of the separated hand: ΑΡΗ 43. Summa Contra Gentiles, however, seems to reduce that generality. Indeed, the strength of the contrast here drawn between the organic and the inorganic (house) cases seems to imply that when something ceases to be a house, then its parts 'may abide in the same species', whereas in fact they certainly cease to be parts-of-the-house and become house-parts, and this is surely a quite radical and relevant change of species: (A) For the soul is the form of the whole body in the same way as it is the form of the several parts. For were it the form of the whole but not of the parts, it would not be the essential form of such a body, as occurs with the form of a house; in this case the form of the whole is not the form of the individual parts, and is hence an incidental form. The proof that the soul is the essential form of the whole and of the parts also is furnished by the fact that both the 283

Section 3 whole and the parts owe their sortai natures to it. Hence on its departure neither the whole nor the parts abide in the same species. For the eye of a dead being, or the flesh thereof, only go by these names in an equivocal fashion. If, therefore, the soul is the actuality of the several parts (for the actuality is attributed to that of which it is the actuality), it only remains to be concluded that it is essen­ tially attributed to every part of the body. (Sic autem anima est forma totius corporis quod etiam est forma singularium partium. Si enim esset forma totius et non partium, non esset forma substantialis talis corporis: sicut forma domus, quae est forma totius et non singularium partium, est forma accidentalis. Quod autem sit forma substantialis tot­ ius et partium, patet per hoc quod ab ea sortitur speciem et totum et partes. Unde, ea abscedente, neque totum neque partes remanent eiusdem speciei: nam oculus mortui et caro eius non dicuntur nisi aequivoce. Si igitur anima est actus singularium partium (actus autem est in eo cuius est actus) relinquitur quod sit secundum suam essentiam in qualibet parte corporis: ACG II, ch. 72.) 3.415 It may here be noted incidentally that this same chapter of ACG restricts itself to a dual typology of whole, i.e. the quantitative and the essential, instead of the three which are usually adduced in this context, i.e. bringing in also the potential or capacitative, as in 3.26(A). The overlap between quantitative concrete parts and the physical make-up aspect of Aristotelian matter has already been noted in 3.31, and the complications which the unguarded use of this overlap may engender are to be glimpsed in the discussion of bodily resurrection. Thus (to anticipate the full discussion in 3.6) suppose such matter to be cannibalised, so that all the physical parts of (say) Mr. A. first become Mr.-A-parts (assuming slaughter and dismemberment prior to the ingestion of Mr. A by Mr. B), but finally finish up as parts-of-Mr.-B. Thus the 'same' matter has been both the parts-of-Mr.A and 284

Aquinas - Mereological Aspects the parts-of-Mr.-B. What then (says the following objection) can resurrection-theory make of this? (A) Whatsoever is genuinely of the human essence in one man can wholly become part of the material constitution of another man who eats the former for food. If, therefore, all the materially-constituted parts are to undergo resurrection in someone, it would follow that there is to rise in one person that which is genuinely of the human essence in an­ other, and this does not cohere with the situation. (Quidquid est de veritate humanae naturae in uno hornine, totum potest esse pars materiae in alio hornine qui eius carnibus vescitur. Si ergo omnes partes secundum materiam resurgant in aliquo, sequitur quod resurget in uno id quod est de ver­ itate humanae naturae in alio. Quod est inconveniens: AST Suppl. q.80, art. 5, s.c. 2.) 3.416 Still, a useful suggestion for making one type of differentiation between the X-part (here in the rôle of mat­ erial make-up before subsumption under its relevant essential form) and the part-of-X (qua actual material make-up) is also incidentally thrown up in the discussion of resurrection. Material make-up qua X-part ('before the reception of the essential form') has indeterminate dimensions, but having bec­ ome part-of-X, it acquires both determinate dimensions and determinate situation: (A) Where the material make-up of things generable and perishable is concerned, one should understand indeterminate dimensions before the reception of the essential form. Hence it is that division in respect of dimensions of this [ind­ eterminate] sort which properly pertains to the material make-up [before incorporation]. But complete and determinate quantity supervenes upon the material make-up in the wake of the essential form. Hence the division which is made in respect of the determinate dimensions is with reference to the [nature of the] species, especially when the having of a determinate lay-out of the parts is of the essence of the 285

Section 3 species; for example, the sort of thing that obtains in the human body. (In materia generabilium et corruptibilium dimen­ siones interminatas oportet intelligere ante receptionem formae substantialis. Et ideo divisio quae est secundum huiusmodi dimensiones proprie pertinet ad materiam. Sed quantitas completa et terminata advenit materiae post formam subst­ anti alem. Unde divisio quae fit secundum dimensiones term­ inatas, respicit speciem: praecipue quando ad rationem spec­ iei pertinet determinatus situs partium, sicut est in corpore humano: AST Suppl. q. 80, art. 5, ad 3.) 3.417 This same accent on determinateness of situation as a possible mark of part-of-X-hood recurs in the commentary on the Metaphysica. Of parts Aquinas there says: (A) Of parts there are some which are such that their sit­ uation does make a difference, as in the cases of a man and of every animal, as well as in a house, and things of this sort. For a house is not just parts arranged anyhow, but rather a determinate ordering of the parts, and the same goes for a man and an animal. (Quaedam vero sunt in quibus positio differentiam facit, sicut in hornine, et in quolibet animali, et in domo et huiusmodi. Non enim est domus qualitercumque partes ordinentur, sed secundum determinatum ordinem partium; et similiter  homo  animal: ACM 1106.) 3.4171 This passage is useful in its alignment of both the house and the organic cases, where cessation of part-of-Xhood upon destruction of X is concerned. It shows that Aqu­ inas' tendency to separate out the organic from the inorganic when making points about the soul, as in the next quotation, need not in the end, according to him, eliminate the inorganic cases from the scope of the distinction now in question: (A) For as the human body, or the body of any other anim­ al, is a certain natural whole, it may be said to be one bec­ ause it has one form whereby it is rendered complete, and this not merely in the style of the aggregated or composite 286

Aquinas - Mereological Aspects whole of the sort that occurs with a house and other such objects. Hence it must be that every part-of-a-man (and part-of-an-animal) receives its being and specification from the soul in its rôle as the proprietary form. This is why Aristotle says [in De Anima II, text 9] that when the soul has departed, there remains neither eye nor flesh nor any other part, except in an equivocal sense. (Cum enim corpus hominis, aut cuiuslibet alterius animalis, sit quoddam totum naturale, dicetur unum ex eo quod unam formam habeat qua perficitur non solum secundum aggregationem aut compositionem, ut accidit in domo, et in aliis huiusmodi. Unde oportet quod quaelibet pars hominis et animalis recipiat esse et speciem ab anima sicut a propria forma. Unde Philosophus dicit quod recedente anima, neque oculus neque caro neque aliqua pars remanet nisi aequivoce: ADA art. 10, c.) 3.4172 This separation out of the inorganic house from the organic cases may rely in part on the contingent linguistic fact that the wall-of-a-house, upon becoming a house-wall (when the house has perished) is still thus said to be a wall. This contrasts, evidently, with his tightened-up language cov­ ering organic perishings ('a dead hand is not a hand', and so forth) but need not by any means abolish the general appl­ ication of the distinction between X-part and part-of-X, as his own elucidation of the continued use of the same word 'bronze' both within and outside of the statue's time made saliently clear in 3.312. The organic cases should simply be seen as peculiarly evident exemplifications of that distin­ ction: cf. 3.4121. We have already seen how the commentary on the Metaphysica insists on this tightened-up language when discussing 'principal parts': ACM 1488, 3.352(A). 3.42 Organic Parts and the Mereological Axiomatic 3.421 The central point of this present section was adumb­ rated in 3.333 - 3.337 above, when the soul qua form was or­ iginally introduced, and more recently in 3.41. The general 287

Section 3 situation is quite simply that Aquinas' delineations of the structure of organisms stresses the distinction between Xparts and parts-of-X (as we have just observed in detail) by reference to the way in which the organism's form (or actual­ ity; cf. 3.412 above) imbues both the whole and the part of the organism. Now in one way this has obviously reinforced some of the observations made in the light of present-day mereology (in 3.334), but in another way, such imbuement may have appeared to weaken somewhat the universally-held mereological thesis of part-whole disparity (1.32, 10.324) which is a reminder about the non-identity of part and whole (where nondistributive wholes are in question: 2.21). After all, as the proponent of the third objection in the Summa Theologica passage to be quoted in 3.423(A) below is to claim, does not this imbueing make every part of the animal's body into an animal? 3.422 Here we are already up against the original Latin's article-free ambiguity. The 'animal' of the objector's Latin 'sequitur quod quaelibet pars corporis sit animal' might (slackly) be read as plain 'animal' in English, without any article, so that there would be nothing startling in every part-of-the-animal's being animal, i.e. vaguely animal in nature. (Strictly 'animalis', an adjective, would then be required as the Latin counterpart, and this we do not have here.) But plainly the objector's intended sense of the Latin 'animal' is here 'an animal', in which case ensues that inord­ inate multiplication of organisms within a single organism which is the basis of the objection. But although the latter reading is thus the objector's presupposition and point, the former looks very much like what Aquinas really wants, i.e. the animal nature's somehow permeating every bit of the org­ anism, 'denominating' each such bit, as indeed it 'denominates' each bit of the complete collection of animals. The suggest­ ion has already been made in 3.334 above, that mereology's definition of the integral whole (D2 of 10.31) according to 288

Aquinas - Mereological Aspects which every element (as defined in Dl of 10.31) of the coll­ ective class constituted by this-animal has an element in common with this-animal (but without necessarily being the animal) suffices to express precisely, and in a minimal sort of way, the manner of this permeation. Thus are facilitated both the required non-identity of proper parts and the whole, and the ensoulment of each proper part of the animal by its form (or 'soul'), but without each such part's being an animal. Only the 'improper part', i.e. the whole animal, is an animal. The extent to which Aquinas' frequently used, already-ment­ ioned, comparisons with the way in which a given colour imb­ ues a surface impinge upon his agreement with this contemp­ orary mereological feature may now be examined in slightly more detail by recourse to appropriate quotations. 3.423 Thus three of the objections to the thesis that the soul is in every proper part of the body run as follows: (A) [Objection] 2 ... . The soul is in the body of which it is the actuality. But it is the actuality of an organic body. Hence it can only be in an organic body. But it is not the case that every part of the body of a man is an organic body. Hence the soul is not wholly in every part of the body. [Objection] 3. Besides, in De Anima II it is asserted that even as a part of the soul is related to a part of the body (e.g. as the power of sight is related to the pupil of the eye) so also is the whole soul related to the whole body of the animal. Hence if the whole soul is in every part of the body, it follows that every part of the body is an anim­ al. [Objection] 5. Besides, if the whole soul were to be in every part of the body, then every part of the body would be immediately dependent upon the soul. Hence one part would not be dependent upon the other, nor would one part be more principal than the other, and this is obviously false. Hence it is not the case that the soul is wholly in each 289

Section 3 part of the body. (2 Anima est in corpore cuius est actus. Sed est actus corporis organici. Ergo non est nisi in corpore organ­ ico. Sed non quaelibet pars corporis hominis est corpus organicum. Ergo anima non est in qualibet parte corporis tota. 3. Praeterea, in II De Anima dicitur quod sicut se habet pars animae ad partem corporis, ut visus ad pupillam, ita anima tota ad totum corpus animalis. Si igitur tota an­ ima est in qualibet parte corporis, sequitur quod quaelibet pars corporis sit animal. 5. Praeterea, si in qualibet parte corporis esset tota anima, quaelibet pars corporis immediate dependeret ab anima. Non ergo una pars dependeret ab alia,  una pars esset principalior quam alia: quod est manifeste falsum. Non ergo anima est in qualibet parte corporis tota: AST, q. 76, art. 8.) 3.4231 Here objections 2 and 3 convey well the feeling that ensoulment of each part makes the part of the same sort as the whole. Objection 5 shows the distributive ('Every part ...') entailments of ensoulment, and claims their incompat­ ibility with some familiar assumptions on principal parts. Next the sed contra is to reiterate Augustine's words from De Trinitate about the soul's being wholly in the whole and in every part, and the corpus of the article will then begin by putting aside any notion of the soul's being a mover in what was later to be the Cartesian style. Then follows the pass­ age already quoted above in 3.413(A) as a strong statement of the distinction between part-of-X and X-part. Here that dis­ tinction, undoubtedly sound, and often made explicit by other thinkers (e.g. 1.4, 2.3) is used as the basis of, and is at this point the substantive content of, the thesis that the soul is in the whole (since a dead animal is not an animal) and in every part (since, for example,, a dead hand is not a hand). This section of the corpus, having already been quoted in 3.413(A), is here omitted, and the excerpt below will accord290

Aquinas - Mereological Aspects ingly pass on to the beginning of the subsequent elucidation, based on the three types of whole: quantitative, essential, and potential (or capacitative), already encountered in 3.3354: (A) But in a contrary sense [to the above-quoted object­ ions] there is Augustine's assertion in De Trinitate VI that 'the soul is in each [human] body, both wholly in the whole and wholly in each of its parts'. [Corpus of the article] I reply by asserting that (as has been said elsewhere already) were the soul to be at one with the body only as a mover, it could be said that it was not in each part of the body but only in the one whereby it would move the others .... In support of the thesis that the soul as a whole is in every part of the body the following considerations may be brought to bear, namely that because a whole is that which gets divided into parts, there are three sorts of wholeness corresponding to three sorts of division [into parts]. There is first the sort of whole which is div­ ided into quantitative parts, e.g. the whole line or the whole physical object. Next there is the sort of whole which is divided into parts intrinsic to the explanation and essence, e.g. that which is defined is divided in respect of the parts of the definition, and a composite being is divided into matter and form. The third sort of whole is the potential one, which is split up into the parts of its capacities. (Sed contra est quod Augustinus dicit in VI De Trinitate, quod anima in quocumque corpore et in toto est tota, et in qualibet eius parte tota est. Respondeo dicendum quod, sicut in aliis iam dictum est, si anima uniretur corpori solum ut motor, posset dici quod non esset in qualibet parte corporis, sed in una tantum, per quam alias moveret Et quod tota sit in qualibet parte eius, hinc consideran potest, quia, cum totum sit quod div­ idi tur in partes, secundum tripl icem divisionem est triplex totalitas. Est enim quoddam totum quod dividitur in partes quantitativas, sicut tota linea vel totum corpus. Est etiam 291

Section 3 quoddam totum quod dividitur in partes rationis et essentiae: sicut definitum in partes definitionis, et compositum resolvitur in materiam et formam. Tertium autem totum est pot­ entiale, quod dividitur in partes virtutis: AST I, q. 76, art. 8.) 3.424 The immediately-following text of the corpus (not to be quoted here) makes the point that forms, strictly speaking, are not quantitative, except incidentally in respect of the exemplifications of certain of them (e.g., as already seen, the whiteness of a given surface can incidentally be divided by the division of that surface). More relevant is the ensuing comparison, which will be quoted next, between the cases of whiteness and of the soul in the light of the triple distinct­ ion drawn at the close of the last quotation, calling attent­ ion to quantitative, essential, and potential types of whole­ ness and parthood. It turns out, upon inspection, that in the cases of both the whiteness and of the soul, only the essent­ ial type of wholeness applies wholly to both whole and part. Quantitative wholeness of the whiteness is present only in the whole white surface, and not in each of its parts, and the soul does not even have the incidental connection with quant­ ity that the whiteness has. Again, the dazzling capacity of the wholeness is greater in the whole white surface than in its parts, and in a vaguely analogous fashion the powers of the soul display a variability with respect to the various parts of the man. It is only when we come to consider essen­ tial completeness that we can attribute it to both whole and part in both the cases of the whiteness and of the soul, al­ though a certain qualification has here to be added on acc­ ount of the intrinsic diversity of parts in the human example: (A) So now if it were to be asked whether whiteness was wholly in the whole and in every part thereof, it would be necessary to make distinctions. For if the allusion here is to that quantitative wholeness which whiteness happens to have incidentally, this would not be present in every part of 292

Aquinas - Mereological Aspects the surface. The same would apply to the wholeness of its capacity, since the whiteness which is in the whole surface has a greater effect on the sight than the whiteness which is in some small part thereof. But if what is meant is the sortai and essential wholeness, then the whole whiteness is in every part of the surface. But since the soul is not subject to to quantitative wholeness, either intrinsically or incidentally .... it only remains to be said that the whole soul is in every part of the body in respect of that wholeness which involves comp­ leteness on the essential side, but not, however, in respect of wholeness of capacity. For it is not the case that in respect of each of its powers it is in each part of the body, but rather in respect of things seen it is in the eye, in respect of things heard, it is in the ear, and so on. Still, it should be noted that since the soul [unlike the whiteness] requires a diversity of [kinds of] parts, it is not to be correlated in the same way with the whole and with the parts. Rather it is to be correlated with the whole primarily and absolutely, in so far as that whole is the proper and proportionate completable thereof, and with the parts secondarily, in so far as they are ordered with refer­ ence to the whole. {Si ergo quaereretur de albedine, utrum esset tota in tota superficie et in qualibet eius parte, dis­ tinguere oporteret. Quia si fiat mentio de totalitate quant­ itativa, quam habet albedo per accidens, non tota esset in qualibet parte superficiel. Et similiter dicendum est de totalitate virtutis: magis enim potest movere visum albedo quae est in tota superficie, quam albedo quae est in aliqua eius particula. Sed si fiat mentio de totalitate speciei et essentiae, tota albedo est in qualibet superficie i parte. Sed quia anima totalitatem quantitativ am non habet,  per se  per accidens, ut dictum est; sufficit dicere quod anima tota est in qualibet parte corporis secundum totalit­ atem perfectionis et essentíae; non autem secundum totalit293

Section 3 atem virtutis. Quia non secundum quamlibet suam potentiam est in qualibet parte corporis; sed secundum visum in oculo, secundum auditum in aure, et sic de aliis. Tarnen attendendum est quod, quia anima requirit diversitatem in partibus, non eodem modo comparatur ad totum et aa partes: sed ad totum quidem primo et per se, sicut ad prop­ rium et proportionatum perfectibile; ad partes autem per posterius, secundum quod habent ordinem ad totum: AST I. q. 76, art. 8.) 3.4241 Here, as elsewhere, what would otherwise be just another boringly banal piece of narrative reproduction of Aquinas' oft-recounted doctrine may be restored to demon­ strative life by the means already invoked. The 'complete­ nessf of essential nature here in question is under theoret­ ical control thanks to the definitions of quidditative funct­ ors and their functorial terms (0.4, 3.12, 10.26). The conn­ ection between such terms (which can entirely account for the status of an abstract such as whiteness and the similarly placed 'form of animal' (i.e. soul)) and their concrete exem­ plifications is shown (for instance) by the definition of 'trm< >' (or '...-er' in the local vernacular) in which place is left for the completion of the gaps by means of the functor­ ial or quidditative terms in question, as in 'whiteness-haveer', 'man-ness-have-er', and so forth: 3.333, 10.251. What, however, is to be made of the bold and overt alignment Aquin­ as here makes between the way in which 'the whole whiteness is in every part of the surface', and the analogous way in which 'the soul is in every part of the body'? For while it happens to be the case that every part of the white surface is itself a white surface, it is not by any means the case that every part of the man is a man. What is needed is some fashion of permeation of the form in question such that this difference between the two cases vanishes under the aegis of mereological generality. 3.425 All that is needed to make explicit this requisite 294

Aquinas - Mereological Aspects manner of permeation is to follow Aquinas' lead in respect of the definitely mereological instance of the whole surface, and to apply it to the animal or the human case, but in the same mereological vein. The whole essence of whiteness is to be somehow involved in each part of the surface even as the whole essence of man is to be somehow involved in each part of (say) the man. Yet it will not do just to concretise these essences merely by saying that even as each part of the surf­ ace is a whiteness-have-er, so also each part of the man is a man-ness-have-er, since it is this which unwarrantably makes each part of the man into a man. Rather, consider the coll­ ective class (or complete collection, or integral whole) of men. Each of those men, as the definition of collective class shows (D2 of 10.31) is of course an element (i.e. a proper or improper part) of the collective class of men. However, as that definition also shows, being an element of the collective class of X's need not exclusively involve being precisely an X, but calls minimally only for the having of an element in common with an X. This yields exactly that loose sense of imbuement of the part which Aquinas' example suggests and his involvement of the 'wholeness of essence' requires. Such of the elements as are not themselves full-blown men neverthe­ less each have a part in common with a man, e.g. the elbow is both part of the arm and part of Socrates, i.e. is a part which both the arm and the man have in common. It is in this sense that the whole essence imbues every part without nec­ essarily making each such part into a man. 3.426 Aquinas has not yet carried through in all its details this sort of appreciation of the mereological situation apart from his attempt to escape from the threat of each part of an animal's being an animal by reference to the various 'powers' of the soul (on which see also 3.53 and 3.55 below) or to the organic dominance of the animal whole, as at the conclusion of the last-quoted passage. He still has the corresponding ob­ jections 2, 3, and 5, as quoted in 3.423(A), to deal with, but 295

Section 3 his replies scarcely advance mereological understanding, ex­ cept perhaps in the reply to objection 3, which deals with the situation of the improper part, i.e. the whole man, seen as a mereological whole, as opposed to the m e r e having of a part in common with the man, which applies to the proper part. Indeed, it could be t h a t a new t h r e a t now also looms: that 'first and proportionate completable' which tends to be used as a principal keynote in the now following replies to those objections, could just as well be taken to be that collective class of man-stuff animated by an Arabic-style world-soul (3.445, CLM 595-6) which it is precisely his aim to avoid: (A) To the second objection t h e reply is that the soul is the actuality of an organic body insofar as the l a t t e r is its first and proportionate completable. To the third objection the reply is that an animal is that which is composed from the soul and the body as a whole, the l a t t e r being the soul's first and proportionate completable. It is in this sense t h a t the soul is not in the p a r t . Hence there is no need for the part of an animal to be an animal. To the fifth objection the reply is that one p a r t of the body is said to be more principal than the other on the basis of the diverse powers displayed by the organs in so far as they are parts of the body. That which is the organ evincing the m o r e principal capacity is a more principal part of the body, as also is that which subserves the same more principal capac­ ity. (Ad secundum dicendum quod anima est actus corporis organici, sicut primi et proportionati perfectibilis. Ad secundum dicendum quod animal est quod componi tur ex anima et corpore toto, quod est primum et proportionatum eius perfectibile. Sic autem anima non est in parte. Unde non oportet quod pars animalis sit animal. ... Ad quintum dicen­ quam dum quod una pars corporis dicitur esse principalior alia, propter potentias diversas quarum sunt organa partes corporis. Quae enim est principalioris potentiae organum, est principalior pars corporis: vel quae etiam eidem potent296

Aquinas - Mereological Aspects iae deservit: AST I, q. 76, art. 8.) 3.427 This last reply is a useful clarification of the true import of the talk about 'powers of the soul' in the corpus of the article, as quoted in 3.424(A) above, wherein it was said that the soul is not in each part of the body in respect of all its various powers. Strictly speaking, given that the soul is an Aristotelian 'form' (as are also, e.g. whiteness and heat) it is improper to speak of such forms 'doing', or being able to do (i.e. having a capacity or power for) anything. Aquinas has a vivid remark on this (notwithstanding his view of the offices apparently assigned to 'pure forms' in the sup­ erlunary area of the universe) when he speaks of heat (i.e. for him. the 'form of heat', as opposed to various 'hots') not being that which heats, properly speaking: calor nullo modo caiefacit, proprie loquendo: AST I, q. 75, art. 2, ad 2. It is, of course, the hot object which does the heating, according to Aquinas. We here have the polar opposite of the later Cart­ esian 'mind'-soul, which turns out to be the only real cust­ omary immediate mover {pace Malebranche) within our world of physical objects. For Aquinas it is, at the concrete level, the man who sees by means of his eye, who feels by means of his hand, and so on. These are the operations of the parts which are attributed to a whole which is concrete, and not formal or abstract: operationes partium attribuuntur toti per partes: cf. 3.53(A). 3.428 Further confirmation that the soul-form does not lit­ erally do anything, even vivifying, is contained in the foll­ owing extract from De Potentia, along with what may be a rem­ inder that this is a sub-lunary ('corporeal') matter, as al­ ready noted: (A) However, in corporeal creatures one scarcely ever or never comes across an action performed by some essential nature, except by the mediation of something incidental thereto (e.g. it is by the light that is in it that the sun illuminates). On the other hand, it is indeed the case that 297

Section 3 when the soul vivifies the body, this is done by the essence of the soul. However, although vivification is thus express­ ed after the fashion of an action, it does not really come under the class of actions, since it is a primary rather than a secondary [i.e. the more usual sense] act. (In creaturis autem corporalibus vel vix vel numquam invenitur aliqua act­ io alicuius naturae substantialis nisi mediante aliquo accid­ ente: sol enim mediante luce quae in ipso est, illuminat. Quia vero anima vivificat corpus est per essentiam animae. Sed vivificare, licet per modum actionis dicatur, non tarnen est in genere actionis, cum sit actus primus magis quam sec­ undus: AQD De Potentia, q. 2, art. 1, ad 6, p. 26b.) 3.429 In continuing this survey of Aquinas' expressions of the way in which the soul-form permeates the body, our scrut­ iny's task will now comprise the search for any trace of even more detailed mereological ways of characterising that form's relation to the body-parts than has been encountered above. After all, the Summa Tbeologica, whence has been drawn the main text so far consulted on this matter, was described by him in its prologue as being expressly designed for beginners. It will therefore be interesting to see if his more special­ ised works are of use in assisting this search. 3.43 The Soul in Quaestiones Disputatae 3.431 For a first further sample of Aquinas' permeationtheory, one may draw upon the corpus of article 4 in his 'Dis­ puted Question' entitled 'On Spiritual Creatures' (De Spiritualibus Creaturis). Here, once again, the problem is whether the whole soul is in every part of the body. First comes a strong reiteration of the distinction between the X-part and the part-of-X, this being, as we have seen, the fundamental import for Aquinas of his pleas in favour of the affirmative answer: (A) I reply by asserting that the truth on this matter

298

Aquinas - Mereological Aspects depends upon the foregoing ones. For it was first shown that the soul is at one with the body, not as a mere mover thereof, but as its form. Next it was shown that the soul does not presuppose other essential forms which are in the matter, and which are responsible for the essential being of the body or of its parts. On the contrary, both the whole body and all its parts have their essential and specific being by the soul, in such a way that upon its departure, even as there no longer remains a man, or an animal, or a living being, so also there no longer remains a hand or an eye or flesh or bone, except in the equivocal way encount­ ered in their pictured or stone versions. Upon this basis, therefore, given that every actuality pertains to that of which it is the actuality, it has to be the case that the soul (which is the actuality of the whole body and of all the parts) should imbue the whole body and every part there­ of. (Respondeo dicendum quod Veritas huius quaestionis ex praecedentibus dependet. Ostensum est enim prius quod anima unitur corpori non solum ut motor, sed ut forma. Posterius vero ostensum est quod anima non praesupponit alias formas substantiales in materia, quae dent esse substantiale corpori aut partibus eius; sed et totum corpus et omnes eius partes habent esse substantiale et specificum per animam, qua rec­ edente, sicut non manet homo aut animal aut vivum, ita non manet manus aut oculus aut caro aut os nisi aequivoce, sicut depicta aut lapidea. Sic igitur, cum omnes actus sit in eo cuius est actus, oportet animam, quae est actus totius corp­ oris et omnium partium, esse in toto corpore et in qualibet eius parte: AQD II 385b.) Those versed in the general history of medieval soul-theory will here recognise the Aquinate 'unity of the form' content­ ion, for which he is well known. Not so well known, however, is the mereologically vital distinction which that contention is here supporting, as we have seen. 3.432 There then follows what is, in effect, part of Aquin299

Section 3 as' explanation for the already-noted greater vividness, as it were, of the distinction between X-parts and parts-of-X in the case of animals, as compared with the less vivid exem­ plifications found in inanimate physical objects (cf. 3.4121). The latter, he claims, have a greater uniformity of types of operation, on account of their correspondingly greater unif­ ormity of types of part: (A) However, there is a difference between the relation of whole to soul [in this animate sort of case], and the relat­ ion of whole to part [in general]. For soul is the actuality of the body, primarily and irrespectively, whereas it is the actuality of the parts only in so far as they are subordin­ ated to the whole. In support of this contention it has to be appreciated that because the material make-up exists on the basis of the form [which imbues it], that material make­ up must be such as is appropriate to the form in question. At the level of the perishable things which surround us, the forms of lower beings, which have correspondingly more feeble capacities, evince only a few operations, for the ex­ ecution of which no variety of parts is required; this is the sort of thing evident in all inanimate physical objects. In contrast, the soul, being a form of higher and greater capacity, is capable of being the basis of various operat­ ions, the execution of which calls for the non-uniformity of the parts of the body. Hence every soul calls for a diver­ sity of organs in the parts of that body of which it is the actuality; in so far as the diversity is greater, to that extent is the soul more of a higher level. Thus, therefore, it comes about that the forms of lower beings only have their material make-up completed in one sort of way, whereas the soul has correspondingly multiple ways, so that from unlike parts is made up the wholeness of the body, of which the soul is the first and absolute actuality. (Sed tarnen aliter se habet totum ad animam, et aliter ad partes eius. Anima enim totius quidem corporis actus est primo et per se, 300

Aquinas - Mereological Aspects partium vero in ordine ad totum. Ad cuius evidentiam considerandum est, quod, cum materia sit propter formam, talem oportet esse materiam ut competit formae. In istis rebus corruptibilibus formae imperfectiores, quae sunt debilioris virtutis, habent paucas operationes, ad quas non requiritur partium dissimilitudo; sicut patet in omnibus inanimatis corporibus. Anima vero, cum sit forma altioris et maioris virtutis, potest esse principium diversarum operationum, ad quarum executionem requiruntur dissimilies partes corporis. Et ideo omnis anima requirit diversitatem organorum in partibus corporis cuius est actus; et tanto maiorem diversitatem, quanto anima fuerit perfection Sic igitur formae infimae uniformiter perficiunt suam materiam; sed anima difformiter, ut ex dissimilbus partibus constituatur integritas corporis, cuius primo et per se anima est actus: De Spiritualibus Creaturis, art. 4, c; AQD II, 385b.) 3.433 That the lower animals resemble inanimate objects, in that they too display a tendency towards homogeneity of parts, is pointed out in the reply to the 5th objection in q. 3, art. 12 of the Q.D. de Potentia. There such animals are likened to wood, stones, water, and air, in this respect. 3.434 Reverting, however, to the De Spiritualibus Creaturis, we find the same triple distinction of wholes into quantitative, essential and potential, as was propounded in the AST text quoted in 3.4231 above. This is likewise used to attribute the second, the essential, type of wholeness to the whole and to every part. A threat corresponding to the one also encountered in AST hence also emerges: does not the thus pervasive imbuement by (say) dogness, make each part into a dog? Exclusion of the third, the capacitative, type of wholeness from every part, evades the threat and is also here linked with diversity of types of part and their respectively correlated operations. However, a new point emerges here: certain human operations cannot be correlated with any part of the human body: 301

Section 3 (A) Now the soul (and especially and human soul) is not spread out into matter, so that the first-mentioned [i.e. the quantitative] sort of wholeness does not apply to it. It hence remains that it is in respect of the wholeness of the essence that it can be said to be just wholly in every part of the body, although wholeness of capacity must here be excluded in this respect. This is because the parts are diversely completed by the soul, so as to bring about various operations, and there is even one of its operations, namely understanding, which is not correlated with any part of the body. Hence, looking at wholeness of soul in this way (i.e. as far as capacity is involved) not only is the soul not wholly in every part, but also it is not wholly in the whole. For the capacity of the soul goes beyond the measure of the body. (Anima autem, et praecipue humana, non habet extensionem in materia; unde in ea prima totalitas locum non habet. Relinquitur ergo quod secundum totalitatem essentiae simpliciter enuntiari possit esse tota in qualibet corporis parte, non autem secundum totalitatem virtutis; quia partes difformiter perficiuntur ab ipsa ad divers as operationes; et aliqua operatio est eius, scilicet intelligere, quam per nulla partem corporis exequitur. Unde sic accepta totalitate anima secundum virtutem, non solum non est tota in qualibet parte, sed nec tota in toto: quia virtus animae capacitatem corporis excedit: Q.D. de Spiritualibus Creaturis, art. 4, c. p. 386a.) At this point, either the non-literal style of attribution of actions to the soul observed in 3.428 is now being abandoned, or the soul is itself to be seen as an agent, with understanding as its action. In either event, this cannot but cause ripples in the originally calm Aristotelian doctrine of form, as observed in the Metaphysica (3.11). 3.435 The Q.D. de Anima follows in its corpus a line similar to that exemplified in the two works quoted above, but with a continued tendency to separate out artificial wholes from the natural ones as far as the application of the distinction bet302

Aquinas - Mereological Aspects ween X-part and-part-of-X is concerned, and as noted above in 3.4121 and 3.432. Also worthy of note are various of the accompanying arguments. Thus on the persistent topic of how it comes about that the soul-form (e.g. of man) can wholly imbue every part without making every part into a man (cf. 3.423 - 3.436), one has objection 7: (A) Objection 7. Every form which imbues some whole and also every part of it, grounds the naming of the whole and the naming of its every part, as is evidenced by the form of fire; for every part of fire is itself fire, whereas it is not the case that every part of an animal is an animal. Hence the soul does not imbue every part of the body. (Omnis forma quae est in aliquod tota et qualibet parte eius, denominat totum et quamlibet partem, sicut patet de forma ignis; nam ignis quaeiibet pars, ignis est. Non autem quaelibet pars animalis animal est. Non ergo anima est in qualibet parte corporis: AQD De Anima, art. 10, ob. 7, p. 318a.) To this the reply is in terms of the variety of operations which differentiate the human case from that of the objection's fire: (B) To objection 7: the reason why it's not that every part-of-an-animal is an animal in the way that every partof-a-fire is a fire, is that all the operations performed by fire are found to occur in every part of fire, whereas not every operation performed by an animal is to be found in every one of its parts, especially where the higher animals are concerned. (Ad septimum dicendum quod ideo non quaelibet pars animalis est animal, sicut quaelibet pars ignis est ignis, quia omnes operationes ignis salvantur in qualibet parte ignis; non autem omnes operationes animalis salvantur in qualibet parte eius, maxime in animalibus perfectis: AQD De Anima, art. 10, ad 7, p. 320a.) This remark leads to the topic of the relation between parts and operations, on which see 3.5 below. 3.436 Another argument, of a type already recorded in 3.413 303

Section 3 - 3.414, is that which prefaces the driving of a wedge between the natural animal and the artifical house (for example); this is broached in objection 16: (A) Besides, even as a man and an animal are wholes of a certain sort made up of diverse parts, so also is a house. But the form of house does not imbue every part of the house, but only the whole. Hence also the soul, which is the form of the animal, is not wholly in every part of the body, but only in the whole. (Sicut homo et animal est quoddam totum ex diversis partibus consistens, ita et domus. Sed forma domus non est in qualibet parte domus, sed in tota. Ergo et anima, quae est forma animalis, non est tota in qualibet parte corporis, sed in toto: AQD De Anima, art. 10, ob.16, p. 318b.) In his reply to this Aquinas allows himself to be swayed in the direction indicated by the objection, as far as the house is concerned: (B) The form of the house, like other artificial forms, is an incidental form, and hence does not endow the whole and every part with being and specification. Nor is this whole a unity in an irrespective sense, but a unity by aggregation. In contrast the soul is the essential form of the body, endowing both part and whole with being and specification, and the whole made up from the parts is a unity in the irrespective sense: hence the cases are not the same. (Forma domus, sicut et aliae formae artificiales, est forma accidentalis: unde non dat esse et speciem toti et cuilibet parti; neque totum est unum simpliciter, sed unum aggregatione. Anima autem est forma substantialis corporis, dans esse et speciem toti et partibus; et totum ex partibus constitutum est unum simpliciter; unde non est simile: AQD De Anima, art. 10, ad 16, p. 320b.) It is worth noting that Aquinas' description, at this juncture, of the status of the artificial house, is precisely the opposite of that which occurs in his commentary on the 304

Aquinas - Mereological Aspects Metaphysics (ACM 1673, quoted 3.74(A) above). Whereas in the present context being-one-irrespectively is denied of the house and of other artificials (neque totum est unum simpliciter), the Metaphysica-commentary attributes such irrespective unity to unities deriving from form, composition, and order (Et tunc oportet quod totum compositum sit unum simpliciter). Among these is the house. It may well be said that in the commentary Aquinas was merely elucidating Aristotle, and not presenting his own doctrine. Be that as it may, it still remains that Aquinas' authoritative later disciple, John of St. Thomas, will hang all the weight of his own extension of irrespective unity to artificials upon the just-mentioned Metaphysica text: cf. 3.7 below. 3.437 Another objection which yet again claims that Aquinas' position must attribute uniformity (e.g. in being-a-dog) to both whole and part, runs as follows: (A) Objection 17. The soul gives being to the body in so far as it is the form of the latter. Now it is the form of the latter in accordance with its essence, which is simple. Hence it is by its simple essence that the form gives being to the body. But from one naught but one derives naturally. If, therefore, the soul is in every part of the body insofar as it is the form thereof, it would follow that it endows each part of the body with being in a uniform manner. (17. Praeterea, anima dat esse corpori in quantum est forma eius. Est autem forma eius secundum suam essentiam, quae simplex est. Ergo per suam essentiam simplicem dat esse corpori. Sed ab uno non est naturaliter nisi unum. Si igitur sit in qualibet parte corporis sicut forma, sequetur quod cuilibet parti corporis det esse uniforme: QD De Anima, art. 10, ob. 17, p. 318b.) The reply which Aquinas produces to this objection is interesting on all sorts of counts. Thus it takes advantage of what originally appeared to be a non-strict way of talking (cf. 3.427, 3.434) whereby the soul is said to do things to 305

Section 3 the parts of the ensouled body. Again, the involvement of the notion of final 'cause' (i.e. the end or purpose) in organic contexts is appropriate and significant: (B) The reply to the seventeenth objection is that the soul, although it is essentially one and simple, nevertheless has an aptitude for diverse workings. And because it naturally endows those things which it completes with being and specification in so far as it is in fact essentially the form of the body (for those things which exist according to the nature of things exist for some purpose) it cannot but be the case that the soul should establish a diversity of parts to the degree demanded by the various workings in question. And it is indeed true that because of this sort of variety (the explanation of which lies in a goal, and not merely in the form) it is in the make-up of living things that goaldirected natural working is more apparent than it is in the case of other natural objects which are such that one form completes in a single-track manner that which has to be completed. (Ad decimumseptum dicendum quod anima, quamvis sit una et simplex in essentia, habet tarnen virtutem ad diversas operationes. Et quia natur aliter dat esse et speciem suo perfectibili in quantum est forma corporis secundum essentiam, ea autem quae sunt naturaliter sunt proper finem, oportet quod anima constituat in corpore diversitatem partium, prout congruit diversis operationibus. Et verum est quod propter huiusmodi diversitatem, cuius ratio est ex fine, et non ex forma tantum in constitutione viventium magis apparet quod natura operetur propter finem quam in aliis rebus naturalibus, in quibus una forma uniformiter perficit suum perfectibile: AQD De Anima, art. 10, ad 17, p. 321a.) 3.438 Tentatively, therefore, it would appear that these cursory inspections of more specialised texts on the soul scarcely show any mereological advance on the position recorded in section 3.42. The various manoeuvres observed, 306

Aquinas - Mereological Aspects which have the aim of avoiding the apparent consequences of the soul's being wholly in every part of the body, while calling attention to relevant associated points, do not appear to converge on the simplest solution, which appeals to no further theories beyond the mereological. Obviously, further research is needed to establish whether this is really so. Suffice it to repeat what was suggested in 3.336 and 3.425 above, namely that by the adoption of the definition of the integral whole provided in 10.31, it is possible to allow that the form-soul (e.g. of dog) imbues every element (of the collective class of dogs, and indeed of the individual dog Fido) in the simple sense that every element has an element in common with a dog (as opposed to necessarily being a dog). Something of this sort seems to be needed in order to avoid, at the earliest possible theoretical stage, the odd consequences of some of the extremely strong statements of imbuement by organic essential form, such as the one in Summa contra Gentiles: 'in an unqualified way, such a form has to be asserted to be wholly in every part of the body', sed absolute dicendum est earn totam esse in qualibet parte corporis: ACG II, ch. 72. 3.439 Considerations of the sort raised above, such as that every part of an animal has to be (adjectivally) animal, every part of a dog has to be doggish (or doggy), and every part of a man has to be human, unfortunately militate against a wellintentioned linguistic contrivance, mentioned in 1.231, which is designed to evade the male-oriented appropriation of the word 'man' to cover all members of the human race, be they male or female. (Latin was in the more fortunate position of having the word 'vir' to stress that a male human person is in question.) The contrivance involves using 'human being' as a general term which is neutral in respect of both sexes, thus evading the male bias of 'man'. However, we may now appreciate more strongly that not only is every part of a human being a being, but it is also human, so that every such part is also a human being, but not therefore necessarily a 307

Section 3 member of the species man. There are thus lots of human beings which are not human beings in the required sense of whole person. It may be added that use of the adjective 'human' (typically, in English, in the plural) will not remedy the situation in a medieval context, owing to the period's distinction between substantives and paronyms: cf. HQS §3.32. For such reasons, the present essay has regretfully had to stick to 'man' as a translation of 'homo' used in the general sense to refer to a member of the species man. Alternatively, 'human person' has sometimes been used, although this could cause trouble when personal post mortem survival is being discussed by the moderns; it would not trouble Aquinas: cf. 3.62. 3.44 Eucharistic Mereology 3.441 The distributive imbuement of every part of the organic whole by the whole essence, as exemplified in the quotation from ACG at the close of 3.438, recurs in a highly interesting fashion when Aquinas is discussing the manner of Christ's presence in the Eucharist, under the appearances of the consecreated bread and wine. He defends the doctrine that the whole Christ is beneath every part of those appearances, and here there is clearly an at least verbal parallel with his theories of formal imbuement described in the foregoing paragraphs. Such a doctrine is clearly liable to call forth an objection such as the following: (A) The body of Christ, being organic, has parts which are determinately distant one from the other. For the having of a determinate distance between the individual parts is of the very essence of an organic body, as eye is distanced from eye, and eye from ear. But this cannot be so if the whole Christ is under every part of the appearances;, for it would then have to be the case that under every part there would be every part, so that where one part was there would also be another. Hence it is not possible that the whole 308

Aquinas - Mereological Aspects Christ is under every part of the host [i.e. the consecrated bread] or under every part of the wine contained in the chalice. (Corpus Christi, cum sit organicum, habet partes determinate distantes: est enim de ratione organici corporis determinata distantia singularium partium ad invicem, sicut oculi ab oculo, et oculi ab aure. Sed hoc non posset esse si sub qualibet parte specierum esset totus Christus: oporteret enim quod sub qualibet parte esset quaelibet pars; et ita, ubi esset una pars, esset et alia. Non ergo potest esse quod totus Christus sit sub qualibet parte hostiae vel vini contenti in calice: AST III, q. 76, art. 3, ob. 2.) 3.442 No a t t e m p t can here be made to follow through the whole doctrinal network to which this is an objection. Hence no elucidation of the central t e r m s which figure in the following corpus of Aquinas' discussion of this point is here offered. Hence also the first sentence of the following quotation need only be understood in so far as it is a preface to the second sentence's application of his usual thesis (i.e. t h a t the essential type of wholeness imbues every part) to the case of homogeneous wholes ('under every p a r t of air is the whole nature of air ...'). Assimilation of the sacramental case to this case is used to resolve the original problem: (A) I reply by saying that, as is evident from the assertions made above, because the essence of the body of Christ is in this sacrament in virtue of the sacrament, whereas dimensive quantity is present in virtue of t h e concomitantly real, it hence follows that the body of Christ is in the sacrament in the essential fashion, i.e. in the fashion in which the essence is under the dimensions, i.e. not in t h a t fashion whereby the dimensional quantity of some body is under the dimensional quantity of its place. For it is obvious that the whole nature of the essence is beneath every part of the dimensions under which it is contained, even as under every part of air is the whole nature of air, 309

Section 3 and under every part of bread is the whole nature of bread. And this holds irrespectively of whether the dimensions are in fact split up, as when air is split up or bread cut up, or even whether they are in fact not split up, remaining only potentially divisible. And hence it is obvious that the whole Christ is under every part of the appearance of bread, even when the sacramental bread remains whole. {Respondeo dicendum quod, sicut ex supra dictis patet, quia in hoc Sacramento substantia corporis Christi est ex vi sacramenti, quantitas autem dimensiva ex vi realis concomitantiae, corpus Christi est in hoc Sacramento per modum substantiae, idest, per modum quo substantia est sub dimensionibus; non autem per modum dimensionum, idest, non per ilium modum quo quantitas dimensiva alicuius corporis est sub quantitate dimensiva loci. Manifestum est autem quod natura substantiae tota est sub qualibet parte dimensionum sub quibus continetur; sicut sub qualibet parte aeris est tota natura aeris, et sub qualibet parte panis est tota natura panis. Et hoc indifferenter sive dimensiones sint actu divisae, sicut cum aer dividitur vel panis secatur; vel etiam sint actu indivisae, divisibiles vero potentia. Et ideo manifestum est quod Christus totus est sub qualibet parte specierum panis, etiam hostia integra manente: AST III, q. 76, art. 4, c.) Thence is derived a reply to the original objection: (B) To the second objection the reply is that determinate distance of the parts within an organic body is based on the dimensional quantity thereof, for the nature itself, which is appropriate to the substance, is prior even to the dimensional quantity. (Ad secundum dicendum quod illa determinata distantia partium in corpore organico fundatur super quantitatem dimensivam ipsius: ipsa autem natura substantiae praecedit etiam quantitatem dimensivam: AST III, q. 76, ad 2.) 3.443 In the first objection to the next article (numbered 4) the thesis thus defended is used as the basis for a reductio ad absurdum, and this, together with its reply is here 310

Aquinas - Mereological Aspects reproduced chiefly on account of its exploiting, yet again, the usual Aquinate contention, observed at length above, that the nature of a substance is wholly in the whole and wholly in every part. Here, however, a useful overt statement of the status of the quantitative (and hence of the primarily mereological) in the case is also provided: (A) Objection 1. It has been said above (in article 3) that the whole body of Christ is intrinsically contained in every part of the consecrated sacramental bread. But there is no dimensional quantity which is wholly contained both within some whole and also within each one of its parts. It is hence impossible for the whole dimensional quantity of Christ's body to be contained in this sacrament . . . . The reply to the first objection is that the fashion in which every thing exists is made determinate in respect of that which pertains to it intrinsically, and not in respect of that which pertains to it only incidentally, e.g. it is in respect of its being white that some physical object exists for the capacity for sight, and not in respect of its being sweet-tasting, even though it is the same physical object which is both white and sweet. Hence the sweetness exists for the capacity of sight in respect of the whiteness, and not in respect of the sweetness. Hence because it is in virtue of this sacrament that the substance [presumably in the sense of essence] of the body of Christ is on the altar, its dimensional quantity is nevertheless there only concomitantly, and as it were incidentally. Hence the dimensional quantity of the body of Christ is not in this sacrament in respect of its own fashion, i.e. such that the quantity is wholly in the whole and its individual parts in the individual parts, but rather in the fashion of a [secondary] substance, the nature of which is wholly in the whole and wholly in every part. (Dictum est enim (a. 3) quod totum corpus Christi continetur sub qualibet parte hostiae consecratae. Sed nulla quantitas dimensiva tota continetur in ali311

Section 3 quo toto et in qualibet parte eius. Est ergo impossibile quod tota quanti tas dimensiva corporis Christa contineatur in hoc Sacramento... . Ad primum ergo dicendum quod modus existendi cuiuslibet rei determinatur secundum illud quod est ei per se, non autem secundum illud quod est ei per accidens: sicut corpus est in visu secundum quod est album, non autem secundum quod est dulce, licet idem corpus sit albus et dulce. Unde et dulcedo est in visu secundum modum albedinis, et non secundum modum dulcedinis. Quia igitur ex vi sacramenti huius est in altari substantia corporis Chrisi, quantitas autem dimensiva eius est ibi concomitanter et quasi per accidens, ideo quantitas dimensiva corporis Christi est in hoc scaramento, non secundum proprium modum, ut scilicet sit totum in toto et singulae partes in singulis partibus; sed per modum substantiae, cuius natura est tota in toto et tota in qualibet parte: AST III, q. 76, art. 4, arg. 1.) 3.444 Though much of these quotations concerning the Eucharist must here remain unanalysed, it is at least clear that Aquinas' usual theory of formal imbuement of the parts by the whole nature is being invoked. What is not clear is whether the application of the definition of the integral whole (D2 of 10.31), which was suggested as a handy clarification of that Aquinate theory, would be of help in this Eucharistic context. This would have to be a matter for further investigation. However, the application, not only of that definition, but of the whole mereological axiomatic, turns out to be immensely relevant to one aspect of Wyclif's theory of the Eucharist, to which allusion is to be made in 6.262 below. 3.445 Another cognate point requiring clarification is the analogy Aquinas draws between the way in which the whole soul (i.e. the form) wholly imbues every part of the body, as described above, and the way in which God imbues everything and every individual object: sicut anima est tota in qualibet parte corporis, ita Deus totus est in omnibus et singulis: AST I, q. 8, art. 2, ad 3. This Thales-like assertion has doubtless 312

Aquinas - Mereological Aspects to be taken in the capacitative sense which was also distinguished above, although, even so, the pantheistic impression remains quite striking. Alternatively, this could appear to place God in the position of the medieval Arabic world-soul, already mentioned in 3.426.

3.5 Attributions and Actions 3.51 Although broached by Abelard (2.515) it is in the field of fallacy-theory that the criticisms of attributions to the whole, of the properties of the part (or vice versa) are most abundantly exemplified. There the sometimes deceptive inference from an attribution made secundum quid (in a certain respect) to the same attribution made simpliciter (irrespectively) turns out to have a facet which concerns inferences from part to whole, as 5.31 is to illustrate. Attention has recently been called to theological aspects of such inferences by G. Klima (KLS) and Aquinas' obvious familiarity with the mereological aspects of the fallacy literature may be noted here by means of a sample passage which has the banal example of the Ethiopian whose part (i.e. set of teeth) is white, but who may not hence be said to be white irrespectively (i.e. as a whole). However (and here we come to the crucial and questionable type of case) Aquinas is among those who hold that when the attribution is to a part which is the only one susceptible of attributions of the sort in question, then the attribution may also be made to the whole, as in the case of the adjective 'crispus' ('curly'): Socrates has curly hair, hence Socrates is curly: (A) It should hence be appreciated that something is said to be so-and-so in the irrespective and proper sense when it is so-and-so in respect of itself. In turn, something is said to be so-and-so in respect of itself which is thus in respect of its whole, rather than what it is in respect of its part, for the part is not absolutely the same as the 313

Section 3 whole. Since the same holds in the other direction, any sameness of the two is merely relative. This is why that which suits something in respect of the whole suits it more in an irrespective sense than that which suits it in respect of the part. This is why, if something is by nature suited to something both in respect of the whole and in respect of the part, then if it [actually] pertains to the thing only in respect of the part, it is said to pertain to it only in a certain respect and not irrespectively, as in the case of the Ethiopian who has white teeth and is thus [i.e. in a certain respect] said to be white. The matter is quite otherwise, however, when the attribute in question is of such nature that it cannot but pertain to the part; under these circumstances someone is said to be curly irrespectively, given that he has curly hair. (Sciendum est ergo, quod simpliciter et proprie dicitur aliquid esse tale, quod est secundum seipsum tale. Dicitur autem aliquid esse secundum seipsum tale, quod est secundum totum, magis quam quod est secundum part em; quia pars non est simpliciter idem toti. Ipsum autem cum sit reciprocum, est relativum identitatis. Et ideo quod convenit alicui secundum totum, magis convenit ei simpliciter quam quod convenit ei secundum partem. Unde si aliquid natum sit convenire alicui secundum totum et partem, si conveniat ei solum secundum partem, dicitur convenire ei secundum quid, et non simpliciter. Sicut si dicatur Aethiops albus qui habet albos dentes. Secus autem est de eo quod non est natum inesse nisi secundum partem, sicut aliquis dicitur simpliciter crispus, si habeat capillos crispos: QD de Unione Verbi Incarnati, art. 3, c; AQD II, p. 430b.) 3.52 While there is no doubt that the linguistic convention here described as applying in the case of crispus may happen to hold in some natural language or another, there would appear to be no necessity that this should do so univerally. Scotus, however, is one of those who gives a logical justification to the convention, holding that 'Socrates is curly' can 314

Aquinas - Mereological Aspects only amount to 'Socrates is curly in respect of his head': KLS 214, n.24. 3.53 Thus, for Aquinas, where attributions are concerned, those which apply to the part do not generaly justify the same attribution to the whole, save in exceptional cases like that of 'crispus'. Still, there is, according to the same author, a sense in which precisely the opposite holds in respect of actions. The operation carried out by means of the part is to be attributed to the whole. This applies both as far as those organs which may be said to be parts-of-theanimal in a quite unexceptional way are concerned, as well as when the form-soul (whose status as a part is on a somewhat different level, cf. 3.117, 3.22) is in question: (A) Or it may be asserted that to act independently is appropriate to that which exists independently. But sometimes something can be said to be an independent existent (provided it does not inhere in an incidental manner or in the manner of a form of the material make-up) while at the same time being a part. Nevertheless, something is described properly as independently subsisting when it neither inheres in the way just mentioned, nor exists as a part. It is in this latter way that neither the eye not the hand can be asserted to be independently subsistent, and hence neither are they independent operators. Hence also it is that the operations of the part are attributed to the whole through those parts. For we assert that the man sees by means of the eye and feels by means of the hand . . . . It hence may be said that the soul understands, even as the eye sees, but it is most strictly properly asserted that it is the man who can understand by means of the soul. ( Vel dicendum quod per se agere convenit per se existenti. Sed per se existens quandoque potest did aliquid si non sit inhaerens ut accidens vel ut forma materialis, etiam si sit pars. Sed proprie et per se subsistens dicitur quod neque est praedicto modo inhaerens, neque est pars. Secundum quem modum oculus aut 315

Section 3 manus non posset dici per se subsistens; et per consequens nec per se operans. Unde et operationes partium attribuuntur toti per partes. Dicimus enim quod homo videt per oculum et palpat per manum ... . Potest igitur dici quod anima intelligit sicut oculus videt, sed magis proprie dicitur quod homo intelligat per animam: AST I, q. 75 art. 2, ad 2.) 3.54 This doctrine is, of course, a mere sample of the constant medieval Aristotelian thesis that actions are attributable to the whole singular object (actiones sunt suppositorum). It is Socrates who sees, understands, hears, smells, or feels by means of the various parts involved. Thus in AST I alone one may easily pick out statements and exemplifications of this principle, e.g. q. 39, art. 5, ad 1, and q.40, art. 1, ad 3. Also incidentally noteworthy as intrinsic to, and as a concomitant of, this thesis, is the above-quoted passage's refusal to posit an omnicompetent 'mind' (which not only understands, but also sees, hears, feels, and smells) in the style of the Cartesian mind-ego. Such a mind makes licit, by its relative vacuity of specification ('res cogitans', 'a thinking thing'), those previously unheard-of eye-free seeings, ear-free hearings, and so on, which found modern 'theory of knowledge'. Friar Thomas, and the myriads like him, accustomed to chant the familiar psalm In exitu Israel, which gains its dramatic effect by telling of the idols' having eyes, but not seeing, and having ears, but not hearing (oculos habent, et non videbunt; aures habent, et non audient) would take the ear-hearing and eye-seeing connections as second nature (and the corresponding quidditative and linguistic networks accordingly). The 'minds' of modern philosophy are the idols' odd obverses: eyes they have not, but they see; ears they have not, but they hear: oculos non habent, sed videbunt; aures non habent, sed audient - philosophical monsters even stranger than those idols of the gentiles. 3.55 According to Aquinas' commentary on the Metaphysica it is by their proper operations that the various organic 316

Aquinas - Mereological Aspects parts are definitionally distinguished. The organic whole is in its turn defined in terms of its formal ensoulment, as has frequently been observed above: (A) For the organic body can only be defined in terms of the soul. And it is in this respect that the soul is said to be the quiddity of physical objects of this sort. And that this is indeed the case is evidenced by the fact that if someone would properly define a part of any animal whatsoever, they cannot properly define it save in terms of its exclusively appropriate operation, as when it is asserted that the eye is that part-of-the-animal whereby it sees. The very operations of the part would just not be in evidence in the absence of sense or motion or other workings of parts of the soul. Hence it must be that the soul is used when some part of the body is being defined. (Corpus enim organicum non potest definiri nisi per animam. Et secundum hoc anima dicitur quod quid erat esse tali corpori. Et quod hoc sit verum patet per hoc quod si aliquis bene definiat cuiuscumque animalis partem, non potest earn bene definire nisi per propriam operationem. Sicut si dicatur quod oculus est pars animalis per quam videt. Ipsa autem operatio partium non existit sine sensu vel motu vel aliis operationibus partium animae. Et sic oportet quod definiens aliquam partem corporis, utatur anima: ACM 1484 - 5.)

3.6

Resurrection and Identity

3.61 After his mystical experience during December in the year 1273, the still comparatively young Thomas Aquinas declared that his own written works seemed to him to be a lot of rubbish. The immense Summa Theologica had not yet been completed, although what was already written was unprecedentedly imposing, so this event and Aquinas' death in 1274 occasioned the task of a disciple (perhaps Reginald of Piperno) in comp317

Section 3 leting the plan of the book. This completion, consisting of at least the 'Supplementum', and based upon Aquinas' early Commentary on the Sentences, has at times been criticised as an inferior production, but it at least displays a full awareness of still fashionable problems such as whether instantaneous motion across space is possible (Suppl. q. 84, art. 3) and whether two bodies can be in the same place at the same time (q. 83, arts. 2, 3, and 4; cf. WSP). Appetising though the prospect of full analyses of these and of more run-of-themill questions such as that concerning personal identity, which here also figures, would be, present limitations permit no more than a sampling of their flavour either within or on the borders of mereology. 3.62 The topics mentioned figure in the Supplementum's discussion of bodily resurrection, and the identity-problem is here, of course, particularly acute. We have just seen enough of Aquinas' doctrines to realise that he would tend not to use the facile Cartesian equation of 'the mind' and the self in order to ensure personal continuity and survival. Without the body, the soul-form, Aquinas' nearest approach to 'the mind', is no person; neither the definition nor the name apply to it: non competit ei neque definitio personae, neque nomen; AST I, q. 29, art. 2, ad 5. Bodily resurrection is hence quite essential to personal survival, and we are hence also faced with the problem of exactly which bodily parts must be reconstituted, and whence reconstituted, in order to ensure that the same person does indeed survive. 3.63 For instance, on the assumption that we are dust and will return to dust (pending the resurrection) the question is raised as to whether it has to be that the dust-particles of the human body should by the resurrection become again that part of the body which had originally disintegrated into those particles: AST Suppl. q. 79, art. 3. Plainly we are here in an area which will embrace also those contemorary conundrums of ours concerning reconstituted coffee-pots, and so forth: HQS 318

Aquinas - Mereological Aspects §4.543, WIS 13. An obvious difficulty ensuing upon a negative decision on this question is raised in objection 2: (A) Besides, a diversity of matter brings about a numerical diversity. But if the particles of dust do not return to the same parts, then the individual parts are not reconstituted from the same matter as that whence they were previously constituted. Hence they will not be numerically the same. But if the parts are diverse, so also will the whole be diverse, for the parts are compared to the whole even as matter is compared with form, as is obvious from Physica Bk. II. Hence there will not be numerically the same man, and this goes against the truth of the resurrection. (Praeterea, diversitas materiae facit diversitatem in numero. Sed, si pulveres non redeat ad easdem partes, singulae partes non reficientur ex eadem materia ex qua prius constabant. Sed si partes sunt diversae, et totum erit diversum: quia partes comparantur ad totum sicut materia ad formam ut patet in II Physic. . Ergo non erit idem numero homo. Quod est contra veritatem resurrectionis: AST Suppl. q. 79, art. 3, ob. 2.) Here familiar Aquinate points about individuation by matter (3.123, HQS §5.2) and the approximation of one aspect of Aristotelian matter to mereological parts (3.31) are being brought to bear. 3.64 The corpus of the article distinguishes and discusses various possible considerations, the simplest invoking the distinction between homogeneously-parted and heterogeneouslyparted wholes: cf. 1.32. A switch in the particles of parts of the former sort would not affect identity, but where the latter are concerned this would do so: (A) As far as a man is concerned, diversity of parts can be taken in two ways. In one way one has the diverse parts of the homogeneous whole, as with the diverse parts of flesh or the diverse parts of bones; in another way one has the diverse parts of the diverse species involved in the heterogeneous whole, such as are the bone and the flesh [the one 319

Section 3 in respect of the other]. Thus were it to be said that a part of the material make-up will be re-incorporated into some other part pertaining to the same species, this would only bring about some change in the situation of that [material] part. But a change in the situation of [such] a part does not result in a change of species where homogeneous wholes are concerned. Hence under these circumstances, if the material make-up of one part is transferred to another part, no obstacle to the identity of the whole is generated ... . If, however, it is said that the material make-up of one part returns to another part which is of another species, then under such circumstances not only is the situation of the part necessarily altered, but its identity also. (In homine possunt accipi diversae partes dupliciter: uno modo, diversae partes totius homogenei, sicut diversae partes camis, vel diversae partes ossis; alio modo, diversae partes diversarum specierum totius heterogenei, sicut os et caro. Si ergo dicatur quod pars materiae redibit ad aliam partem speciei eiusdem, hoc non faciet nisi varietatem in situ partium. Situs autem partium varia tus non variat speciem in totis homogeneis. Et sic, si materia unius partis redeat ad aliam, nullum praeiudicium generabitur identitati totius. ... . Si autem dicatur quod materia unius partis redit ad aliam partem alterius speciei, sic de necessitate variatur non solum situs partium, sed etiam identitas earum: AST Suppl. q. 79, c. ) Articles 1 and 2 of q. 80 go on to make comparisons between the nature of the soul and the nature of Art (the latter in the quite special medieval sense of an appropriate personal capacity). This constitutes a topic eminently worthy of expansion, were the operation cost-effective for the mereological theme, which unfortunately it is not. Most immediately relevant, however, is the comparison of the principal instruments ('organs') of some particular art to the principal parts of the body. The arrangements for protecting these instrum320

Aquinas - Mereological Aspects ents, such as the sheath for a knife, then correspond to the secondary bodily parts, such as hairs or fingernails. 3.65 Articles 4 and 5 of the same question turn upon what genuinely constitutes human nature (de veritate humanae naturae), this being an important adjunct to the highly pervasive medieval discussion of Veritas rei, the truth of things: cf. HL 234. However, this sort of Veritas is here said to embrace all the matter of the body, so that if everything which thus genuinely constitutes human nature is to be resurrected, then a gigantic bulk of matter will be resurrected for each adult person. Appropriate restrictions are therefore imposed in the corpus of article 5. 3.66 In the same vein, article 4 deals suitably with Adam's rib (which became Eve: arg. 2), beef-eaters (arg. 1), cannibals (arg. 4), and someone rather gruesomely nourished exclusively on human embryos (arg. 5). Basically, the various relations which X-parts having a past career as parts-of-Y may have to the parts-of-X which they become, leave plenty of scope for detailed present-day analyses in the light of the theory of collections (complete or non-complete). The same sort of scope is even more applicable where articles 2 and 3 of q. 83 are concerned. Here the overlap of physical objects, as covered by 10.34, at points and boundaries (especially in the reply to the 5th objection of article 2) is overtly countenanced. However, the level of discussion of this important possibility is scarcely as lively, and not at all as purely mereological, as was that of twelfth-century thinkers (1.5, 2.431) or as that of the later Wyclif will be seen to be in 6.2 below.

3.7

Natural and Artificial

3.71 It has already become evident that in general Aquinas tends to accentuate the distinction between parts-of-X (e.g. a 321

Section 3 living hand) and X-parts (e.g. the former hand, now strictly no longer a hand, having been amputated) where animal wholes are concered, but gives the appearance of refusing to apply that same distinction when the non-animal cases are used by way of contrast with the organic case: 3.413(A), 3.4141(A), 3.436. Now while there are obviously all sorts of contrasts between the definitely animal and the definitely inorganic, they both still come under the same general principles, mereologically speaking, including the vital distinction just recalled. Aquinas' tendency to exempt the non-animal from the scope of that distinction is doubtless connected with his theory of the substantial, as well as with the more dramatic way in which parts-of-animals become mere animal-parts, as compared with the perhaps hazy passage from parts-of-a-house to house-parts. The former wall of a house may still remain a wall, or even a house-wall, after the house, technicallyspeaking, has stopped being there. Again, the organic whole is the natural whole, whereas the house represents the artificial, thus bringing us up against a contrast already encountered in 2.8421, 2.8474, and 3.436. There are moments when the basic community of the various types of case, mereologically speaking, is recognised, as observed in 3.414(A), 3.416, 3.417. This last citation occurs in the commentary on the Metaphysica. Significantly enough, as we are shortly to see, it is upon the material from the Metaphysica-commentary that John of St. Thomas, a later noted disciple of Aquinas, bases his smoothing over of this otherwise apparently dissociative tendency in his master. 3.72 Thus when quiddity is being discussed, the Metaphysica-commentary does indeed present the quiddities of both a house and a man on the same level in their bearing on the transition from X-parts to part-of-X: (A) When we ask, 'What is the house?', this is the same as if we were to ask, 'Why are these, i.e. the stones and wood, the house?'. It is because of this, namely, [in Aristotle's 322

Aquinas - Mereological Aspects words], 'The parts of the house show forth what the being of a house is', i.e. because of the fact that the quiddity of house imbues the parts of the house ... . And likewise, when we ask, 'What is the man?', this is the same as if we were to ask, 'Why is this object, i.e. Socrates, a man?'. To this a reply is, 'Because he is imbued by the quiddity of man'. This amounts also to asking why a physical object thus arranged, i.e. in the organic manner, is a man. For that bodily arrangement is the matter of the man, even as the stones and the walls are the matter of the house. (Cum quaerimus, Quid est domus? idem est ac si quaereremus, 'Propter quid haec, scilicet lapides et ligna, sunt domus?'. Propter haec, scilicet, 'quia partes domus existunt id quod erat domus esse', idest propter hoc quod quidditas domus inest partibus domus Et similiter, cum quaerimus, 'Quid est homo?', idem est ac si quaereretur, 'Propter quid hoc, scilicet Socrates, est homo?'. Aut etiam idem est ac si quaereretur, 'Propter quid corpus sic se habens, ut puta organicum, est homo?'. Haec enim est materia hominis, sicut lapides et la teres domus: ACM 1666 - 7.) 3.73 True, in 1680 of ACM there is a separation out of naturals as true substances, as contrasted with artificials, but that this scarcely abrogates the foregoing is suggested by ACM 1672 - 3, where the contrast between the irrespective unity (unum simpliciter) and the unity in a certain respect (unum secundum quid) allocates the house to the former, as contrasted with the latter's exemplification in a heap of stones: (A) As regards the first point he [i.e. Aristotle] hints at a certain distinction which applies where the composition of something from many things is concerned. For sometimes the composition from many is effectuated in such a way that the whole made up from the many is a certain unity, as with a house made up from its parts, and a compound body from its elements. On the other hand the composite is sometimes eff323

Section 3 ectuated from the many in such a way that the whole thus compounded is not a unity absolutely speaking, but only in a certain respect, as in the obvious example of a heap or pile of stones. For here the parts are parts in actuality, because they are non-continuous. Hence absolutely speaking this is a multiplicity, but it is a unity only in a certain respect, i.e. in so far as the many parts are mutually gathered together in their place. (Circa primum innuit quamdam distinctionem compositionis alicuius ex multis. Quandoque enim ex multis fit compositio, ita quod totum compositum ex multis est unum quoddam, sicut domus composita ex suius partibus, et mixtum corpus ex elementis. Quandoque vero ex multis fit compositum, ita quod totum compositum non est unum simpliciter, sed solum secundum quid; sicut patet in cumulo vel acervo lapidum, cum partes sunt in actu, cum non sint continuae. Unde simpliciter quidem est multa, sed solum secundum quid unum, prout ista multa associantur sibi in loco: ACM 1672.) 3.74 If the house is, as here represented, an irrespective or absolute unity, then so also, a fortiori, is the human being, and no mereological cleavage is here recognised by Aquinas. In fact, even the heap of stones need not, at the most general level, be treated as a mereological exile. The important point is that the man and the house may now be assumed to be assimilated to the extent of ensuring that the latter artifact is subject to the distinction between X-part and part-of-X. The next paragraph allows the irrespective or absolute unities to include those based on form (i.e. substances in Aquinas' sense), on composition (comprising artificials, such as the house), and on order (e.g. syllables and ordered manifolds), with only the heap and the populace relegated to the contrasting status of secundum quid (in a certain respect) unities. Yet again, it is to Aquinas' relative toleration at this point that attention is now being called: (A) Now the explanation behind this distinction is as foll324

Aquinas - Mereological Aspects ows. Sometimes the composite is placed in its species on the basis of its unity; this unity may be either the form (as in the compound body), or the composition (as is shown in the case of the house), or the order (as shown in syllables and manifolds). In these cases the composite whole turns out to be a unity in the absolute sense. On the other hand, the composite is sometimes placed in its species merely on the basis of the collocation of the many parts, as is shown in the heap and the populace, and other things of this sort. In such cases the composite whole is not a unity in the absolute sense, but only in a certain respect. (Huius autem diversitas ratio est, quia compositum quandoque sortitur speciem ab aliquo uno, quod est vel forma, ut patet in corpore mixto, vel compositio, ut patet in domo, vel ordo, ut patet in syllaba et numero. Et tunc oportet quod totum compositum sit unum simpliciter. Quandoque vero compositum sortitur speciem ab ipsa multitudine partium collectarum, ut patet in acervo et populo, et aliis huiusmodi; et in talibus totum compositum non est unum simpliciter, sed solum secundum quid: ACM 1673.) 3.75 This tolerance in the matter of absolute (or irrespective) unities (which in fact need not, of course, exclude the application of general mereology to those unities 'in a certain respect' (secundum quid)) is avidly seized upon by John of St. Thomas, as is rightly observed by John Deely, in his recent comment on, and felicitous reproduction of, the following relevant text in his DP. Thus to the just-quoted Aquinate paragraph John of St. Thomas adds: (A) This teaching is most to be adhered to when sorting out the various types of absolute and incidental unity. Nor is it the case that all these are said to be one in the same way; rather there are various grades of unity. Thus an angel [i.e. a simple substance] or a heavenly sphere, which undergoes no change, is a more complete unity than are animals or living beings, which continually undergo increase or 325

Section 3 decrease, and also more so than is a river, which exhibits neither material nor formal identity between that which now flows within it and that which was flowing there twenty years ago, but only has an identity based on the location through which it flows. Likewise artifacts are only one from the point of view of an artificial form. Nevertheless all these are embraced within the specification of irrespective unity in terms of St. Thomas' rule according to which those things are one, irrespectively speaking, when they are placed in their species on the basis of some unity, whether that unity be of form, composition, or order. Only those things which are not placed in their species on the basis of one thing, but of a multitude, are merely unities in a certain respect. (Quae doctrina valde observanda est ad discernendum varios modos unitatis simpliciter et per accidens; nec enim omnia ista dicuntur uno modo unum, sed sunt diversi gradus. Perfection enim modo est unum angelus et coelum, qui nullam varietatem recipiunt, quam animalia seu viventia, quae continuo incremento et decremento transeunt, et quam fluvius, qui nec in materia nec in forma est idem, qui modo fluit, ab illo, qui fluebat a viginti annis, sed solum ratione loci, per quem fluit, et similiter artefacta sunt unum solum ex parte forma artificialis. Et tarnen haec omnia includuntur sub ratione unitatis simpliciter ex illa regula S. Thomae, quod ea sunt unum simpliciter, quae sumunt speciem ab uno, sive illud unum sit forma, sive compositio, sive ordo. Solum autem sunt unum secundum quid, quae non ab uno sumunt speciem, sed a multitudine: DP 496, n. 134.) Although John will still contrive to follow Aquinas to the extent of making an army or a city (like his master's heap or populace) into secundum quid unities, his espousal of Aquinas' notion of indeterminate matter (cf. ACG IV, 31, and HQS §5.2) for real unities opens the way to a great deal of latitude in this respect: (B) However, if there is determinate specification on the 326

Aquinas - Mereological Aspects formal side, while the specification on the side of the material make-up remains indeterminate, there results, absolutely speaking, some definite species as far as the object is concerned, even as there results the real unity of the river even though its water is not the same, but sometimes this lot and sometimes that lot. Nevertheless, because it flows in the same manner and sequence relatively to its banks, it remains the same river. The same goes for the unity of place when various successive surfaces come within the same boundaries. It is in this way that the various unities can, thanks to the discrimination exercised by the intellect, follow one upon the other, while remaining within the bounds of the same ultimate explanation. It is in this way that they have a real unity, formally speaking, although not materially. (Si autem formalitas determinata sit, sed materialis designatio sit indeterminata, simpliciter datur species determinata in re, sicut datur unitas fluvii in re, licet materialiter aqua non sit eadem, sed modo ista, modo illa; quia tarnen fluit sub eadem habitudine et successione alveum, manet idem fluvius. Et similiter est de unitate loci succedentibus diversis superficiebus in eadem distantia. Sed diversae unitates possunt succedere per designa tionem intellectus in eadem ratione ultimae, et sic habent unitatem in re formaliter, licet non materialiter: DP 497 n. 134.) Finally, and in the same vein, the countenancing of per se unity based on actually-parted wholes, and including artifacts, is confirmed thus: (C) From many actual entities there does not come about one independent unity by means of the unity of a form, but this can still occur by means of a unity of order and by means of a unity of measure; thus from several actual entities there comes about one artifact, and likewise one sacrament, and one banquet. These latter do have an independent unity, as opposed to a merely incidental unity of the sort that is based on a multiplicity and not on some one factor, 327

Section 3 be it form, or composition, or order. (Ex pluribus entibus in actu non fit unum per se unitate formae, bene tarnen unitate ordinis et mensurae, sicut fit ex pluribus entibus in actu unum artefactum et unum sacramentum et unum convivium Habent enim ista unitatem per se, quatenus opponitur unitati per accidens, quae sumitur a multitudine et non ab aliquo uno, sive forma, sive compositione, sive ordine: DP 496, n. 134.) 3.76 Other discussions of the distinction (or lack thereof) between natural and artificial have already been incorporated in 2.8421 and 2.8474 above. Much still remains to be recorded and discussed on this matter in so far as it figures in Aquinas' writings, but as the interest is here for the most part purely narrative, no more will now be added. Even more remains to be said on nearly every topic raised in this exceedingly cursory and selective survey of some mereological strands in the work of the Angelic Doctor, so called not on account of his genuinely excellent nature, but rather because of his advances which embrace a certain section of atomic mereology, almost entirely neglected above.

328

4. Some Buridanian Theses 4.1 The Compendium Totius Logicae (coded as BCL) by John Buridan (first half of fourteenth century), as printed in Venice in 1499, contains also a commentary by the later disciple of his, Jan Dorp. The text on which Dorp comments is a remaking of the Summulae Logicales by the thirteenth-century Peter of Spain, of which HSL is a printed version. Jan Pinborg's paper in PLB recounts the nature of this remaking. In fact the Buridanian text in Treatise VI of BCL and 5.18 of HSL agree quite closely in essentials at the relevant mereological point. Hence fourteenth-century extensions and developments of thirteenth-century material are here being witnessed. 4.2 Unfortunately, at the present juncture, it is only possible to give a highly superficial impression of this material. A not inconsiderable obstacle is the questionable status of the Latin text of BCL in the Venice edition. Again, ideally necessary would be further explanations of two features of the theoretical background used in Dorp's discussions. The first of these concerns allusions to the medieval doctrine of the Topica originally produced by Aristotle. The still controversial nature of the medieval versions of these 'topics' is discussed in Eleonore Stump's contribution to CLM (273 - 99). Niels J. Green-Pedersen's extensive and author329

Section 4 itative work (GTT) further confirms the varieties of possible status here. For present purposes a 'topic' can be taken to be closely associated with a 'locus', i.e. a commonplace rule. The other feature is the Buridanian appeal, allegedly in the spirit of 'nominalism', to a theory of deduction which mentions, rather than uses, the terms which are its subject-matter. Now such a theory is perfectly feasible and often useful (e.g. LAS §32). However, its categorial and indiscriminate use already tends to render needlessly long-winded and obfuscatory even theses which correspond to simple propositional logic. Since this latter is more basic than the ontology and mereology of 10.2 and 10.3 below, the still greater opacity of its application to these two fields may well be appreciated. Hence in the brief comment below reference will be made directly to the mereology of 10.31 as theoretical background, notwithstanding Buridan's or Dorp's corresponding nominalistic allusions to the terms of such theory. Such allusions in any case often result in theses which sound excessively odd, as we are to note in passing in 4.6. 4.3 As if all this were not enough, it has also to be realised that in Buridanian theory the linguistic 'terms' in question are 'parts' of a 'mental language' with which the author of these present remarks is not acquainted. Its nature has to be worked out by appropriately skilled metaphysicians or psychologists {PLB 79) although it apparently can be seen as having something in common with a categorial language having the same background function as the one presupposed for the expression of the theoretical spine of the present work: PLB 80, cf. 0.3, 10.015. Clearly, we have by now accumulated a mass of presuppositional riddles the toilsome clarification of which would prove to be totally lacking in cost-effectiveness relatively to the mereological project at this juncture. 4.4 All that is proposed here, therefore, is a preliminary and superficial impression, centred on a prima facie translat330

Buridanian Theses ion of Tr. VI of the Venice BCL. The only Latin to be quoted will be that of the section-beginnings, for service as catchwords. Supplemental numerals have been added for ease in cross-reference. The allusions to the 'constructive' and ' d e s tructive' natures of theses [3] and [5] plainly call attention to the affirmative or negative forms of the antecedents and consequents of the inferential forms in question at those points: (A) (Totum integrale ...) According to our author the integral whole is the one composed from parts having quantity, e.g. the house from wall, roof, and foundation. [1] The topic based on the integral whole concerns its relation to its part, and [2] The topic based on the integral part concerns its relation to its integral whole. [3] The topic based on the integral whole (cf. [1]) holds constructively only; e.g. the [whole] house exists; hence the wall [which is a part thereof] exists. [4] Theorem: The positing of the integral whole entails the positing of every part thereof [cf. D2 of 10.31]. [5] The topic based on the integral part (cf. [2]) holds destructively only; e.g. the wall [as part of the house] does not exist; hence the [whole] house does not exist. [6] Theorem: The remotion of the part of an integral whole entails the remotion of the whole thereof [cf. D2 of 10.31]. 4.401 Abelard, it may be noted, had considered [3] (cf. 2. 211(A)) and [4] (cf. 2.21(A)). 4.41 The discussion attributed to Dorp begins by casting doubt on the 'constructive' claims of [3] and [4]. The first argument, i.e. [7.1], turns on the nature of 'topics' and their nominalist interpretation; this is hence omitted here, although its final negative statement about 'terms' has been reproduced because it is an example of how such term-oriented discussion is irrelevant to mereology, notwithstanding the eminently true linguistic fact which is stated in this particular instance. 331

Section 4 The second argument, i.e. [7.2], is plainly correct in its example. Next come doubts about the 'destructive' claims made in [5] and [6] above. The first doubt [8.1] adduces a case of part to whole inference (i.e. from foot to animal) which seems valid qua inference. The second, i.e. [8.2], appears to rely on the situation of the homogeneous whole (cf. 1.32). (A) (In ista parte ...) In this section our author makes decisions about the topic based on the integral whole in relation to its part, and so on. For the showing forth of this: [7] (Dubitatur primo ...) First it is doubted whether this topic does hold constructively ([3], [4]). [7.1] It is argued that it does not .... for ... between this term 'house' and this term 'wall' the relation of integral whole to integral part does not hold. [7.2] (Secundo sic ...) In the second place the following is not valid: the man exists; hence the foot exists. For it is established that there are men without feet. And yet that topic [3] supports it ... and the opposite [of this criticism] is argued by our author in his text [3], [4]. [8] (Dubitatur secundo ...) The second doubt is whether the topic based on the integral part and arguing to its whole does hold only destructively (cf. [5], [6]). [8.1] Against its thus holding it is argued as follows: it follows validly that the foot exists, hence the animal exists. Yet here is a constructive argument from the part of the integral whole to its whole so that [the doubt is thereby justified; cf. [14.3] below]. [8.2] (Secundo sic ...) In the second place, as follows: let there be supposed some water, of which one part is called 'A'. Under these conditions it does indeed follow: A exists, hence water is. Nevertheless we have here an argument from the part of the integral whole to the whole ... for every part of water is water. But the opposite would appear to hold according to our author in his text [5]. 332

Buridanian Theses 4.51 In a beautifully systematic way the prerequisites for solutions to these difficulties are laid down. Thus in the [9.1]-series below, the 'possessed of quantity' requirement for an integral whole, which was posited at the outset in 4.4(A), is now surmised to be too restrictive. Perhaps, it is suggested, it is better to revert to Boethius' original definition of the integral whole as that which is made up of many, and leave it at that. This reversion, as [9.13] is to point out, will enable the atomic mereology (cf. 3.3352, 10.35) of simple substances such as angels to come within the ambit of the integral whole. Unfortunately it is here taken that such atoms must be non-quantitative, and even if this is accepted as regards space, there is surely no reason to exclude temporal quantity therefrom. But even this fragment of technical medieval angelology is comparatively lucid when contrasted with the mysteries of the mental language urged as a further supporting point in [9.131]. 4.52 A second precision, i.e. [9.2], concerns the sense of 'part' in this context. The details of the third precision, i.e. [9.3], are here omitted, as they turn on the nature of 'topics' and details of the theory of deduction to which our Presuppositional Explicitation in section 10 below has not advanced. A glimpse of the way in which that theory would express its mereological theses has been provided in the closing words of this [9.3] below. Finally, the [9.4]-series makes an attempt to be precise about the perennial problem of 'principal parts' (cf. 2.4, 3.35). (A) [9] (Pro solutione ...) For the solution of these doubts it is to be noted: [9.1] First, that the integral whole is described [in 4.4(A)] as that which is composed of parts which are possessed of quantity. [9.12] But Boethius, when describing the integral whole, left out the 'possessed of quantity' detail, and said [simply] that the integral whole is that which is made up of many. 333

Section 4 [9.13] And this definition presented by Boethius is more general, and perhaps better, for given the double manifold consisting of two angels, every unit is part of that manifold, and is no other sort of part than an integral one, but nevertheless neither of those units is possessed of quantity, since each of them is an indivisible angel. [9.131] Again, if one posits a mental proposition composed of a simple subject and predicate and a simple copula, this proposition would seem to be an integral whole relative to these three parts, and yet none of them is possessed of quantity. [9.2] (Secundo notandum ...) Secondly it is to be noted that the integral part is that whence, along with some other, there results one whole. [9.3] (Tertio notandum ...) Thirdly it is to be noted that ... the topic from the name of the integral whole is the topic relying upon the status of the inferred term standing in the relation of the name of the part of the integral whole. It is in this way that the definition of the topic of the integral whole should be expressed. [9.4] (Ulterius notandum ...) Further it is to be noted that the part of the integral whole is twofold: [9.41] One sort is principal, and it is that without which the whole does not subsist (e.g. the foundation of a tower, or the heart as related to the animal). [9.42] The other is less principal, and is that without which the being of the whole persists (e.g. the kitchen relatively to a house). 4.6 The conclusions based on such precisions begin by drawing on these decisions concerning the principal parts, and as such (in the [10]-series below) are reasonably intelligible. It is in the second conclusion (i.e. the [ll]-series below) that we once again join company with distinguished forerunners going back at least to Abelard. Thus [11.1] allows inferences from whole to principal part, as did Abelard in 334

Buridanian Theses 2.211. At this point, however, probably owing to his preoccupation with proper nominalist expression of the principles involved, as in [11.11], Buridan fails to attain to Abelard's acute realisation, recorded in 2.21 above, that we have here probably strayed back into the realm of quite ordinary standard inference from the general ('Every part-of-X exists') to the individual ('Hence this part-of-X exists'). Indeed, Abelard has also stated grounds for overriding the distinction between principal and other parts here (2.4 above). Needless to say, the analysis of the allegedly proper nominalist expression which is provided in [11.11] yields so obvious a falsehood if taken literally ('de virtute sermonis', as the mentalist nominalists themselves would say) that it could ground doubts as to the ultimate profitability of such persistent efforts to remain term-centred. As it stands, [11.11] makes it look as though the existence of the name of the integral whole allows one to infer the existence of the name of the principal part, and this is obviously ludicrous. Of course, there is a place for theory of deduction which concerns itself with expressions, where needed, but unless this is properly carried through in due order it is worse than useless. Its extension, as here, to mereology, borders on the perverse, and is certainly generative of unnecessarily penitential exercises, as compared with the simplicity of straight mereology at this point (10.3). (A) [10] (Istis notatis ...) These points having been taken into account, [10.1] The first conclusion is that the topic based on the name (cf. [9.3]) of the integral whole, and proceeding to the name of the integral part which is less principal (cf. [9.42]) does not hold constructively. For this does not follow: 'The man exists; hence the foot exists' (cf. [7.2]). [10.2] Likewise the topic based on the name of the less principal part of the integral whole does not hold destructively (cf. [8]). For this does not follow: 'The kitchen 335

Section 4 does not exist; hence the house does not exist'. [11] (Secundo conclusio ...) Conclusion the second: [11.1] The topic based on the name (cf. [9.3]) of the integral whole, and which proceeds to the more principal part, does indeed hold constructively in respect of the verb 'to be', used as the predicate. For this does indeed follow: 'The animal is; hence the body is', and likewise this follows: 'The house is; hence the foundation is': cf. [7.1]. [11.11] The theorem whereby these consequences hold is: Given the integral whole there is also given its principal part. This in turn is analysed thus: If the verb 'to be' used as a predicate is verified of the name of the integral whole, then that same verb is verifiable of the name of the more principal part. 4.7 Thus far the structure, and indeed the content, of this present text has been comparatively clear and intelligible. The same goes for the [12]-series which initiates the next piece of text. Remarks similar to those made on the [11]series are obviously still appropriate. Thus [12.1] is simply the obverse of the [11.1], and the Abelardian query as to its direct mereological relevance (2.215) can accordingly be extended to this point. The commonsense query about the 'nominalist' expressions in [12.1] can also be extended here on the same lines as the remarks made in 4.6 concerning [11.11]. However, while [13] makes the sort of elimination which predecessors (7.315) had agreed must be made, as in [13.2], nevertheless it perhaps does so by a form of overkill based on yet another quite legitimate point in [13.1]. Thus the latter rightly says that not merely the existential '... is', but all its consequences (e.g. '... is a being') may figure in the constructive arguments from whole to part. But apparently all other predicates are to be eliminated from validity of sequence in such arguments, and for all we know at this point, there could be too strong an elimination here, even though it certainly does have the already-noted virtue of 336

Buridanian Theses eliminating the by then notorious case which infers from the worth of the whole house to a like worth of its part [13.2]; cf. 10.326. To the first two original arguments the replies given in the succeeding [14]-series are superficially straightforward enough: (A) [12] (Tertio conclusio...) Conclusion the third: [12.1] The topic which is based on the name of the more principal part, and which proceeds to the name of the integral whole does indeed hold destructively in respect of this verb '... is', used as a predicate. For the following does follow: 'The heart is not; hence the animal is not'. Likewise the following does follow: 'The foundation is not; hence the house is not'. [12.11] The theorem whereby the [applicable] topic-description here holds good is as follows: On the removal of the integral part which is that whereby the integral whole subsists, that integral whole is itself removed. This in turn is analysed thus: If the verb '...is' used as a predicate is truly negated of the name of the integral part, it will also truly be negated of the name of the integral whole. [13] (Dubitatur tertio ...) In the third place there is this doubt: might it not be the case that these topical descriptions ([11], [12]) hold good in respect of predicates other than the verb '... is' in its use as a predicate? [13.1] To this the reply is that they do hold in respect of those other predicates which follow upon this verb '... is', used as a predicate, but not as far as other verbs which do not follow upon it are concerned {non respectu aliorum consequentium). Hence even as this follows: 'The house is; hence the wall is', so also this follows: 'The house is a being (or beings); hence the wall is a being (or beings)'. [13.2] The second point [concerning the other verbs] is obvious, because this does not follow: 'The house is worth one hundred francs; hence the wall is worth one hundred francs'. 337

Section 4 [14] (Tunc respondetur ad rationes ...) And now replies to the arguments can be made: [14.1] The first argument [7] indeed proves that no topic description ought to be said to be the one from the integral whole. [14.2] To the second [7.2] it is replied that it does prove that from the name of the integral whole to the name of the less principal part a logically valid consequence cannot be drawn, and this has been granted in the first conclusion [10.1]. 4.8 Finally, there are replies to the purported proofs ([8.1], [8.2]) that one can argue from the part to the whole affirmatively as well as negatively (cf. [5]). Worthy of note in this connection is the comment in [14.31] below to the effect that the organic example ('The foot exists; hence the animal exists'), originally occurring in [8.1], though valid, does not owe its validity to the mereological theorem mentioned. Still, the inference does ultimately rely on the important basic mereological distinction between X-part and partof-X (1.4, 2.3, 3.41). The organic 'foot' shows that a part-ofX is strictly in question, and hence entails the whole X in a way that the inorganic 'foundation', for example, could fail to do, since its latitude of sense extends both to house-part (with no house existing) and to part-of-the-house (which entails the whole existing house). We have seen how 'a dead hand is no hand', and so forth (3.413) ensure the existence of the whole X in the obviously organic part-of-X cases. But in fact any part-of-X calls for the existence of X (10.321) whether the X in question is organic or not. To that extent we do here have a general mereological thesis in the background, but the bridging supplement that a foot, if it is to be a foot in the proper sense, must be a proper part of an existing animal, is also presupposed. 4.9 It is hence with somewhat more justice that the example of the inference from the individual parcel of water, 338

Buridanian Theses called 'A', to plain water, is removed from the mereological sphere in the [14.32]-series below. The inference from 'A is this water' to 'A is water' may indeed be justified by reference to the plain relation between universal and 'subjective' part, as described in the immediately preceding section of the VIth Treatise of BCL. One suspects, however, that it is the converse of the topic 'from the universal whole to its subjective part', mentioned in [14.3211] below, which is really in question here. The passage finishes off with a useful contrast between the integral whole and the universal whole. This stresses the interdependence of part-of-X and X, as opposed to the independence of that universal which in our categorial language (but not Buridan's) functions at the quidditative level (0.4). Boethius of Dacia had by this time made the same point (5.46). (A) [14.3] (Ad alias rationes ...) As far as the other arguments are concerned, which aim to prove that the topical description which is based upon the part of the integral whole holds constructively [8] as well as destructively [5], the reply is: [14.31] (Ad primam ...) To the first [8.1], when it is asserted that the sequence, 'The foot exists; hence the animal exists' is a valid inference, this is granted, but it is asserted that that consequence does not owe its validity to a topical description involving the inference from the integral part to the integral whole. For although the term from which the inference takes its rise is indeed the name of a part of the integral whole, and the term thence inferred is indeed the name of the whole, nevertheless that inference is not valid in virtue of that relation. [14.32] (Ad secundum ...) To the second [8.2] when it is said that it does follow [in virtue of the topic based on the integral part], it is [in reply to this contention] denied that that inference is validated by the topical description which goes from the name of the integral part to the name 339

Section 4 of the whole. [14.321] For although the term on which the inference is based is indeed the name of the integral part, and the inferred term is the name of its whole, nevertheless the inference is not validated on the basis of that relation. [14.3211] But then someone may ask which topical description does validate that inference. The reply is that it is validated by the topical description from the universal whole to its subjective part, for in the case described above [8.2], water stands in the position of the more general class relative to the A. [14.4] (Dubitatur quarto ...) There is, in the fourth place, a doubt raised about the difference between the integral whole and the univeral whole. To this the reply is that the difference is as follows. The integral whole is made up from its parts, and is those parts all taken together. Neither does it remain in being without those parts. But the universal whole is one simple term, which is not made up from its subjective parts. Indeed, it can subsist independently of those subjective parts. (The quidditative counterpart of this final assertion is obvious enough when duly analysed, as in HQS §4.0. The more general topic of the distinction between the universal and the integral whole was initially dealt with in 1.3, and corresponds to the distinction between distributive and collective wholes respectively.) 4.91 There is, of course, much more of mereological interest in BCL. Indeed the next section of Treatise VI (corresponding to HSL 5.19, 5.20) makes plain that distinction between the quantitative whole (in the 'logical' sense of quantity) and the still predominantly quantitative (in a more usual sense) integral whole. This goes back ultimately to Boethius, but William of Sherwood and Peter of Spain (HSL loc. cit.) have in the meantime taken it in their stride: 5.22, 7.05, 10.23.

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5. The De Sophisticis Elenchis in the Thirteenth Century 5.1 Introduction 5.11 An extremely lively part of the new logic produced by the moderni of the thirteenth century and later, is to be found in their discussions of that apparent appendix to Aristotle's Topica which is known as De Sophisticis Elenchis, 'On Sophistical Refutations'. As the name implies, the work has something to do with logical fallacies. L.M. de Rijk's Logica Modernorum (DLM) recounts in detail the incursion of such fallacy-doctrine into the medieval Latin West. The sample which is here to be sifted for mereological material appears to originate from the last third of the thirteenth century, and hence is, from the narrative point of view, justifiably placed after the material from Aquinas in section 3 above. Indeed, from the theoretical point of view it will also be useful to be able to refer back to our already-acquired acquaintance with Aquinas' commentary on the Metaphysica of Aristotle, because the author of the text now to be explored makes some corresponding Aristotelian mereological allusions which will accordingly not be in need of a new introduction. 5.12 Exactly who was that author of the Quaestiones on the De Sophisticis Elenchis (coded as ESE) which is now in question, is somewhat of a mystery. The initial supposition that he may have been Boethius of Dacia (i.e. a Dane) has happily 341

Section 5 ensured the text's superb edition by Sten Ebbesen, with costs borne by the generosity of the Institutum Carlsbergicum, in the fine series Corpus Philosophorum Danicorum Medii Aevi. The lengthy introduction discusses the question of authorship without any definitive conclusion, although a certain Peter of Ireland is mooted as a possible candidate (ESE XI, LV-LIX). He might turn out to be the same person as the 'Patrick of Ireland' to whom attention was called in HQS 273 - 80. Were either of these attributions to prove correct, this would be some small compensation for the lack of mention of philosophy in a recent history of the Celts and their culture, even though John the Scot Eriugena and Duns Scotus are well-known Celtic medieval philosophers. 5.13 As the whole of ESE is replete with tantalisingly attractive topics it becomes immensely difficult to insulate one's mereological preoccupations from the other themes. This, however, must be attempted, even if it means, for example, resisting the temptation to linger too lengthily around the Paradox of the Liar (cf. 5.5 below). A specimen of the work's distinction between universal and integral wholes is provided in 5.2. By now, attempts by Abelard (2.211, 2.32) and the Porretans (2.82, 2.85) to settle questions about attributions to parts and wholes have become codified under discussions of fallacious inferences from attributions of properties to a part (an instance of 'secundum quid', 'in a certain respect') to the same attributions to the whole (i.e. an instance of ' simpliciter', 'irrespectively', 'without qualification'). The white-toothed Ethiopian and the curly man are by now usual cases, encountered in Aquinas (3.5). These are discussed here, with an extension to the case of the parti-coloured shield (5.3). Aspects of Master Peter's discussion of numerical manifold problems (2.7) are extended to 'Five are two and three' (5.4). Hitherto unencountered, however, is the discussion covering the relation between mereology and those

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paradoxes which are sometimes said to be grounded in selfreference (5.5). 5.2 Integral and Universal 5.21 The incisive clarity of the ESE commentary is already evident from the following distinction which covers already-familiar ground, in that it separates out the integral from the distributive whole (cf. 2.21), but noteworthily associates the distributive whole with the 'all ... ', 'omnis ...', functor, and hence here gives preference to the label 'universal' for such a whole. The contrasting attachment of 'the whole ...', 'totum' to the integral case, yields what was at one time a common distinction between ' totum' and 'omnis'. Needless to say, the theoretical affinity with present-day mereology is maintained in respect of the accent on the non-identity of part and whole (cf. 1.32, 10.324) and in the presence of clauses similar to those found in the present-day definition of the integral whole (D2 of 10.31): (A) Whole can be taken in two ways. There is one sort of whole which is the universal whole, every part of which is that very whole of which it is a part, and the indication of this sort of whole is the sign 'All ...'. The other is the whole in respect of which it is not the case that every part is that whole of which it is a part, taken on its own; rather it is constituted by every part with the cooperation of the others, i.e. all the parts taken together, and this is the integral whole, the indication of which is this sign 'the whole ...'. (Duplex est totum: est enim quoddam totum universale, cuius quaelibet pars ipsius est ipsum suum totum, et huius totius est hoc signum 'omnis' dispositio; aliud est totum cuius non quaelibet pars est suum totum per se, sed quaelibet pars cum adiutorio alterius, id est omnes partes simul sumptae, et hoc est totum integrale cuius dispositio est hoc signum 'totus': ESE Q.46, lines 237 - 243, p. 97.) 5.22 At various places earlier in the work, the syllogism 343

Section 5 (cf. LAS) is said to depend on rules involving quantity: ESE Q.16.22.23, Q.16. 127.131, Q.20.26.32. One might take this as simply being a reference to what is known as 'logical' quantity, this being the all-hood and some-hood represented in the four standard forms of the traditional categorical propositon, e.g 'all ... is ...', 'some ... is ...', Thus taken the remarks now in question are no news. It would be unusual for the allusion to quantity to be a reference to that quantitative area (in the ordinary sense of the word) which, as all tend to agree, provides the most characteristic application for mereological theorems; Aquinas (3.213) and Buridan (4.51, 4.91) have been quoted on this. Yet intriguingly enough, the part-whole relation is not only connected with the did de omni syllogistic principle in the place referred to by the last-cited ESE reference, but is also unequivocally elsewhere related to the integral whole: 'Every syllogism holds by virtue of the did de omni rule; but where the did de omni applies, there also applies the relation between the integral whole and its part'; ESE Q.19.12.15. Could this be a sign that a mereological interpretation of syllogistic is being presupposed? William of Sherwood, in contrast, had been quite clear about the distinction between logical and integral quantity (KSL 81) and Boethius of Dacia presupposes it in 5.46 below; cf. 7.05. 5.23 A further sample of the distinction between universal and integral wholes is to be made available in connection with the discussion of 'Five are three and two' in 5.3 below: ESE Q.831, arg. 2. There will also be the opportunity to renew acquaintance with several other themes (attribution, Xparts and matter, and so forth). It is to the ESE verdict on attribution that we now turn. 5.3 Attributions: from secundum quid to simplidter 5.31 It is on the basis of the white teeth (but now without any overt mention of Aristotle's Ethiopian from ch. V of De Sophisticis Elenchis) that whiteness is attributed to the 344

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person as a whole in the opening of Q.90 of ESE. Here one s t arts off from whiteness 'in a certain respect', but concludes simpliciter, 'irrespectively'. Much ingenuity is expended in cooking up support (by way of initial objection to the right solution) for this familiar fallacy's mode of operation. The second argument in its favour, for example, tries to t r a d e on an inference from principal part t o whole, using 'If the chest is healed, then the man is healed' as its analogue: cf. 3.35, 5.54(A). The fourth argument displays somewhat of an affinity with the Porretan claim recorded in 2.845. Since it is only in the replies (5.33) to these preliminaries that their substantial point becomes really evident, the best t a c t i c for the moment is simply to record these fallacious supporting arguments without more ado: (A) Next the question is raised whether this follows, i.e. 'He is white as far as his t e e t h are concerned; hence he is white'. In favour of an affirmative verdict: 1st. According to Aristotle's Topica [II, ch. 11, 115b 12 13] t h a t which happens to be the case in a certain respect happens irrespectively. But being white as far as t e e t h are concerned happens to be the case in a certain respect, hence it happens irrespectively. Hence [the inference is valid]. 2nd. Besides, according to the 5th book of Aristotle's Physica [ch. 1, 224a, 21 - 33] the man is said to be healed because his chest is healed; for, as it is there expressed, something is expressively transposed in t h r e e fashions, i.e. [i] irrespectively and [ii] in a certain respect, and [iii] with the irrespectives being either primary or not primary, and [in this last case] on the basis of a part, as occurs when something's holding in respect of a part is used as a ground for inferring its holding in respect of t h e whole; e.g. if the chest is healed, then the man is healed. Hence likewise, if he is white in respect of the part he will be white irrespectively. 345

Section 5 3rd. Besides, there is the inferential proof based on the principle that oppositional negation of the consequent justifies the inference of the oppositional negation of the antecedent. Thus if [as far as the contentious inference is concerned, one denies the consequent, asserting] he is not white irrespectively, then it ensues that there is no whiteness whereby he is white, and hence [by the principle now being adduced] he is not even white in respect of the teeth. 4th. Besides, there is a greater concurrence between the part and the whole than there is between an object and its non-essential attribute. But that which pertains to the nonessential attribute pertains also to object in which it inheres, and hence that which pertains to the part [such as 'it is white'] must also pertain to its whole. (Consequenter quaeritur utrum sequitur 'est albus secundum dentes, ergo est albus'. Videtur quod sic: 1. Quia per Aristotelem in Topicis quod contingit secundum quid contingit simpliciter. Sed esse album secundum dentes contingit secundum quid, igitur contingit simpliciter. Quare et cetera. 2. Praeterea per Aristotelem quinto Physicorum homo dicitur sanari quia thorax sanatur nam, ut ibi dicitur, aliquid transumitur tripliciter, scilicet per se et per accidens, et per se dupliciter, aut primo aut non primo sed per partem, ita quod cum aliquid insit parti, inerit toti, ut si thorax sanatur, homo sanatur. Ergo similiter si est albus secundum partem, erit albus simpliciter. 3. Praeterea probatio consequentiae quia oppositum consequentis infert oppositum antecedentis, ut si non est albus simpliciter, nulla albedine est albus, quare nec secundum dentes. 4. Praeterea maior est convenientia inter partem et totum quam inter subiectum et accidens, sed quod inest accidenti 346

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inest subiecto, quare quod inest parti inerit toti: ESE Q.90.1.18, p. 205.) 5.32 The body of the question gives a business-like decision on the matter, citing Aristotle's characterisation of the fallacy along with a very present-day style of falsification of the hypothetical ground on which depends the inference which is now in question. This hypothetical ground is, of course, 'If he has white teeth, then he is white', and, as 10.13 reminds us, the production of a case in which the antecedent ('He has white teeth') is true while the consequent ('He is white') is false will suffice to falsify the whole of this 'if ... then ...' proposition. This is here effectuated by pointing out that in general, 'the opposite of the consequent', in this case 'It's not that he is white', 'is compatible with the antecedent' (i.e. with 'He has white teeth'), it being then understood that there should be no difficulty in producing instances in which the antecedent is true and the consequent false: any black ghetto would suffice to produce such instances. Then, in the section to which '[i]' has been added in the translation below, the familiar crispus, 'curly', case, already encountered in Aquinas' version (3.5) is adduced as one in which the part to whole attribution may be made. Another example is also thrown in, namely that of the external appearance's being white, licencing the whole person's being said to be white. The section numbered '[ii]' correctly exposes the nature of the fallacious inference: (A) Aristotle maintains the opposed position, intimating that here there is a fallacy involving the interrelation between in a certain respect and irrespectively. The verdict is that the inference in question does not follow, because the opposite of the consequent is compatible with the antecedent, and the fallacy involving the interrelation between in a certain respect and irrespectively is perpetrated. However, since that which pertains to the part does sometimes give its name to the whole, but sometimes 347

Section 5 does not, the following points should therefore be understood. [i] There are some inessential attributes which, relatively to the object to which they apply, are uniquely correlated with the part to which they apply, as with curliness, which has the head as its unique correlate, or at least principally pertains to this [particular] part. A similar instance is that of white, which although it occurs elsewhere than in the surface appearance, is nevertheless principally to be found in that surface appearance. In such cases one can infer from the inessential attribute uniquely correlated with the given part to the application of that attribute to the whole. For example: 'He is curly in respect of his head; therefore he is curly', and 'He is white in his external appearance; therefore he is white'. [ii] There are, however, some other inessential attributes which lack the unique correlation with a part of the body, as in the case of the whiteness which [unlike the case of the white surface appearance] fails to uniquely fix a correlation with some definite part, so that it does not follow that if he is white just in respect of any old part, then he is therefore white without qualification. (Opposition vult Aristoteles innuens quod hic est fallacia secundum quid et simpliciter. Dicendum ad hoc quod non sequitur, quia oppositum consequentis potest stare cum antecedente, et est fallacia secundum quid et simpliciter. Quia tarnen aliquando illud quod inest parti totum denominat, et aliquando non, ideo intelligendum quod aliqua sunt accidentia quae in subiecto cui insunt partem sibi determinant, ut crispitudo determinat sibi caput vel saltern principaliter est in ista parte, ut licet album reperiatur alibi quam in facie, tarnen principaliter reperitur in facie, et in talibus sequitur ab accidentibus determinantibus sibi partem usque ad totum, ut 'est crispus in capite, ergo est crispus' - 'albus in facie, ergo est albus1. Accidentia autem alia sunt quae sibi partem non determinant 348

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in corpore, ut albedo non determinat sibi aliquam partem determinate, et ideo non sequitur quod si est albus secundum aliquam partem, ergo est albus simpliciter: ESE Q. 90.19.34, pp 205-6.) 5.33 Whatever one may think of this interesting attempt to have it both ways in the 'white'-case, it at least looks as though the ensuing replies to the original objections are sound enough. The second reply incorporates an appeal to the notion of principal party and the fourth appears to be appealing to the crispus, 'curly', kind of case of which a reminder was provided in [i] of the last passage. In general, for kindred discussions, KLS 2 1 3 - 5 , especially notes 19 - 26, may be consulted. Here are the replies: (A) Whence may be resolved the objections: The 1st. When Aristotle asserts in his Topica that that which happens to be the case in a certain respect (and so on, [as in 5.31(A)]) it has to be riposted that here inessential attributes as described under [i] are in question. The 2nd. As far as the next objection is concerned, I point out that the chest is here the hearty this being the dwelling-place of life in those things which possess a heart, and the place such that, if health is there to be found, then the whole body will be healthy. Hence it comes about that health tends principally to be correlated with the chest, and it is in such cases that the inference of the sort in question is sound; in other types of case, however, it is not sound. The 3rd. As far as the next objection is concerned: when it is claimed that from the oppositional negation of the consequent may be inferred the oppositional negation of the antecedent, it must be counterclaimed that this does not hold true, given that that which is asserted irrespectively is that which is asserted without any [qualifying] addition. Thus when someone is said to be whitey this is propounded irrespectively. In contrast, when someone is said to be white-toothed, white is then attributed only in a certain 349

Section 5 respect. Therefore, when the denial 'He is not white' is effectuated, we do not here have a denial of white which is other than altogether irrespective. But there is no necessity that if he is not white irrespectively, then he must not be white in a certain respect, and so the inference is not valid. The 4th. To the last objection I reply that although there is a greater concurrence of nature between part and whole [than there is between an object and its inessential attribute], nevertheless it is not in so far as it is the whole that the part concurs with the whole. In contrast, the inessential attribute concurs with the object to which it applies as a whole, and it is more on this account that the whole takes its name from what concurs with the inessential attribute as opposed to what concurs with the part; this applies most especially to the inessential attribute which is uniquely correlated with some particular part [such as curly]. (Per hoc ad rationes: 1. Cum dicitur per Aristotelem in Topicis 'quod contingit secundum quid' et cetera, dicendum quod loquitur hic de accidentia us primo modo dictis. 2. Ad aliud dico quod thorax est cor quod est domicilium vitae in habentibus cor, ubi si fuerit sanitas erit totum corpus sanum, et ideo principaliter determinat se sanitas ad thoracem, et in talibus concessum est quod valet ista consequentia, in aliis vero non. 3. Ad aliud, cum dicitur quod oppositum consequentis infert oppositum antecedentis, dicendum quod non est verum, quia simpliciter dico quod nullo addito dico; unde cum dicitur 'albus', simpliciter dicitur, sed cum dicitur 'albus dentem', secundum quid dicitur, et ideo cum negatur 'non est albus', non est negatio albi nisi solum simpliciter. Sed non oportet si non sit albus simpliciter quod non sit albus secundum quid, et ideo consequentia non valet. 4. Ad ultimum dico quod licet totum magis conveniat cum 350

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parte essentialiter, tarnen non secundum totum convenit cum toto; sed accidens secundum totum convenit cum subiecto, et magis propter hoc denominat totum quod convenit accidenti quam quod parti, et maxime illud accidens quod partem aliquam sibi determinat. ESE Q.90.35.55, pp 206 - 7) 5.34 In the same vein, Q.91 goes on to discuss the example of the shield which is half white and half black (presumably in respect of two intrinsically unbroken areas of the outward-facing surface). This is later to be discussed also by Scotus, and has the theological implications brought out by KLS 213 (text and n. 23). The text of this Q.91 presents scarcely any difficulties, and the whole is hence now presented without any commentative break. Suffice it to say that the second objection is illegitimately trading on what has already been settled as regards the crispus, 'curly', case, in the previous question (5.23(A)[i]). Also the reply part of the third argument uses the vocabulary incorporating simpliciter ('irrespectively') and secundum quid ('in a certain respect') to good effect when relativising the scope of descriptive assertions: (A) Next the question is raised as to whether an object which is white in respect of one half and black in respect of the other should be said to be white irrespectively or black irrespectively. That the answer is in the affirmative would appear to follow thus: 1. From what Aristotle says in his Topica [II, ch. 4, 111a, 33 - 6] [we learn that] that which comes within a genus comes also within the ambit of some species or other thereof [cf. 0.4 above]. Hence as a shield is coloured [i.e. comes under a genus] in a manner such as the object described [i.e. as parti-coloured], it is obvious that it will be either white irrespectively or black irrespectively. 2. Besides, it would appear to be the case that some part which is less than the half can govern the application of a 351

Section 5 name to the whole, as when the curliness of the hair of the head [which is less than half] governs the name given to the whole body. Hence, a fortiori, the inessential a t t r i b u t e of a half can thus apply. Hence the shield must be said t o be white irrespectively on the basis of its being white in one half. 3. Besides, the folowing is a true sequence: 'Air is above relatively to the earth, and below relatively t o the heavens, and hence is both above and below' [these being two opposites]. Hence likewise this follows: 'This is white in respect of one half and black in respect of t h e other half, and hence is both white and black' [these being two opposites]. The contrary is claimed by Aristotle in the context adduced [in the first objection]. The verdict is that the object in question is neither white irrespectively nor black irrespectively, nor is it coloured by a single colour, for on t h e same grounds as those whereby it could be asserted to be white it could also be asserted t o be black. Hence the alternatives are either [i] it will be simultaneously both white irrespectively and black irrespectively, or [ii] white in a certain respect and black irrespectively, or [iii] conversely [i.e. white irrespectively and black in a certain respect], or [iv] it is white in a certain respect and black in a certain respect. Now [i] is out of court, for then contraries would pertain actually to the same thing at the same time, and this is incoherent. Nor should the shield be said t o be as in [ii], i.e. white irrespectively and black in a certain respect, because its converse (i.e. [iii]) may be just as cogently asserted. Hence it is necessary t o assert [iv], i.e. t h a t it is white in a certain respect and black in a certain respect, and hence it does not follow t h a t it is either white or black irrespectively. [Replies to objections] 1. When it is argued t h a t 'that which takes its name from

352

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the genus also takes the name of some species or other of the genus', then this is true and can apply to several species. And hence when the shield is said to be coloured, it can be said to be either [i] white irrespectively or black irrespectively, or [ii] white in a certain respect or black in a certain respect. 2. In reply to the next objection I assert that as the shield is both black and white in respect of its halves, of which neither is dominant over the other, there is hence no compelling reason why it should be said to be white irrespectively rather than black irrespectively, and hence it is not said to be either the one or the other of these. After all, whiteness is no more dominant in one half than blackness is in the other half. But this is not the case where the [adduced example of] curliness is concerned, since this is fixedly correlated with the head, and principally pertains thereto. It is on this account that it governs the application of the name to the whole. 3. In respect of the next objection I reply in one alternative way thus: what follows is not 'hence it [i.e. the air] is both above irrespectively and below irrespectively', but rather, 'It is above in a certain respect and below in a certain respect'. This is because that which is above in respect of the earth is not above irrespectively, neither is that which is below in respect of the heavens below irrespectively. Alternatively one may reply that the relative status of these terms [i.e. 'above' and 'below'] does not narrow down the natures of those things to which they are relativised [i.e. they are already implied in the terms to which they are here applied, as in 'Heavens above'], and hence they may be applied irrespectively without any fallacy being perpetrated. In contrast, the partialising effect of an inessential attribution does so narrow down the scope of the inessential attribute's application that the irrespective inessential attribution does not follow in this fashion. 353

Section 5 (Consequenter quaeritur utrum aliquid quod est secundum unam medietatem album et secundum aliam nigrum debeat dici album simpliciter vel simpliciter nigrum. Quod sic videtur: 1. Per Aristotelem in Topicis, nam quod est in genere est in aliqua eius specie. Cum igitur scutum tale sit coloratum, manifestum est quod erit simpliciter album vel simpliciter nigrum. 2. Praeterea videtur quod pars minor medietate possit totum denominare, ut crispitudo capitis denominat totum corpus, quare fortiori ratione accidens medietatis. Igitur scutum debet dici simpliciter album, cum scilicet in medietate sit album. 3. Praetera sequitur, 'Aer est sursum in comparatione ad terram et deorsum in comparatione ad caelum; ergo est sursum et deorsum'. Ergo similiter sequitur 'Hoc est albus secundum unam medietatem et nigrum secundum aliam medietatem, ergo est album et nigrum'. Oppositum vult Aristoteles in littera. Dicendum quod nec simpliciter est album nec simpliciter nigrum nec aliquo uno colore coloratum, quia qua ratione diceretur album, eadem ratione diceretur nigrum. Igitur aut simul erit album simpliciter et simplicter nigrum, aut album secundum quid et nigrum simpliciter aut econverso, aut quod est album secundum quid et nigrum secundum quid. Non primo modo, quia tunc contraria inessent simul eidem in actu, quod est inconveniens; nec debet dici album simpliciter et nigrum secundum quid, qua eadem ratione diceretur econverso; et ideo necesse est dicere quod sit album secundum quid et nigrum secundum quid, et ideo non sequitur quod sit album aut nigrum simpliciter. 1. Et quod arguitur 'quod denominatur genere denominatur aliqua eius specie' verum est, vel speciebus multis. Et ideo, cum scutum dicitur coloratum, dicetur vel album simpliciter vel nigrum simpliciter, vel secundum quid album et secundum 354

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quid nigrum. 2. Ad aliud dico quod cum scutum sit album et nigrum secundum medietates quarum neutra est principalior altera, ideo non est maior ratio quare magis diceretur album simpliciter quam nigrum simpliciter, et ideo non dicetur nec sic nec sic; nam albedo non est magis principalis uni medietati quam nigredo alii. Sed non est sic in crispitudine, quae determinat sibi caput et principaliter sibi inest, et ideo totum denominat. 3. Ad aliud dico uno modo quod non sequitur '..ergo est simpliciter sursum et simpliciter deorsum ', sed 'secundum quid sursum et deorsum secundum quid'; quod enim est sursum ad terram non est sursum simpliciter, nec quod est deorsum ad caelum est deorsum simpliciter. Alio modo dicendum quod isti termini respectus non deminuunt de ratione illius cuius sunt termini, et ideo inferunt simpliciter sine fallacia, sed partialitas accidentis deminuit de ratione accidentis et ideo isto modo non infert accidens simpliciter: ESE Q. 91, pp. 207 -9.) 5.35 Finally it may be noted that we appear to have, in the course of Q.101 of ESE, some repetition of Abelard's appreciation, recorded in 2.32, that existential attributions differ from others in that they do run symmetrically from part to whole, and vice versa. We do not have, 'If the whole is worth $100, then so is the part', nor yet, 'If the part is white, then so is its whole', but we do have, 'If the whole exists, then so does its part', and its converse. This last point makes explicit what is bound up with the stress on the genitive 'part-of-X' construction: cf. 1.4, 2.3, 3.41, 10.321, and D2 of 10.31. Here is the brief sentence on all this: (A) Although a part is not existentially separable out from its whole, nevertheless it can be that some attribute is appropriate to the part in so far as it is a part, and thus not appropriate to the whole; hence that part may in such a sense [i.e. other than existentially] be distinguished from 355

Section 5 its whole. (Licet pars in esse non distinguatur a toto, tamen quia aliquid potest competere parti secundum quod est pars, quod non competit toti, ideo potest sic pars distingui a toto: ESE Q.101, ad 1, p. 236.28.30.) 5.4 Mereology and Manifolds 5.41 Thus far, these mereological discussions from ESE have been pleasantly incisive, but relatively pedestrian. Now, however, two changes of scene emerge, of which the first (Q.831) has to do with the truth of Quinque sunt duo et tria, 'Five are two and three'. This numerical example is of great interest in itself, given the present-day philosophical work on the analysis of such propositions, which owes so much to the inspiration of the Whitehead and Russell Principia Mathematica. Further, the author of ESE brings a new variety of points to bear in a most lively fashion. It hence behoves us to focus on the mereological material without giving way to the temptation to stray too far into neighbouring fields, both contemporary and medieval. Some such straying is, however, inevitable if the full import of the material is to be appreciated. Peter M. Simon's 'Number and Manifolds' in 5PM usefully conveys further background discussion. 5.42 First it may be noted that the proposition in question can be construed as being concretely concerned with a five-fold object (i.e. a quintet) as opposed to the number five (i.e. what is involved in being five). The former, the five-fold object, is a manifold to which reference may be made by names in name-termed propositions. In this sense it is subsumable under quantitative mereology. The other construal, which makes the proposition to be more abstract in its import, in that it concerns the number five, could be said to be about the Aristotelian 'form' of quintets - what it is that they have in common - and is 'designated' by the non-nominal terms of relevant quidditative propositions. This form is called the 'forma quinariV ('the form of five-foldness') in the 356

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corpus of the question (5.436(A) below), and this shows that we here have an appreciation of the dual possibilities now being outlined. As it stands, however, the proposition's plural verb (Quinque sunt ..., 'Five are ...') makes it apparent that the manifold, as opposed to the number in the formal sense, is its topic. 5.43 Next, and still by way of preliminary, note may be taken of those grammatical possibilities of the proposition now in question which have in fact already been the subject of investigation in ESE's previous question. Its sense, it has been said, can be compounded ('Five are two-and-three') or divided ('Five are two and five are three') with diverse corresponding parts of speech for the '... and ...' in each case: (A) On this subject it is asserted that this [proposition] is ambiguous in respect of composition and division, because of the fact that the '... and ...' can be either a functor joining terms or a functor joining propositions; if it joins terms, then the proposition is taken in the compounded sense, and is, according to some people, true (although this will be investigated later); if, however, it joins propositions, then it is taken in the divided sense and is false. That the conjunction [... and ...] can join its completions in [just] these two ways would appear to be ensured by the fact that it is of the very nature of a conjunction that it should always join expressions of like category. (Ad hoc dicitur quod haec est multiplex secundum compositionem et divisionem, ex eo quod li et potest copulare inter terminos vel inter propositiones: si inter terminos, sic est composita et vera secundum quosdam, de hoc tarnen videbitur iam; si autem inter propositiones, sic est divisa et falsa. Quod autem coniunctio hoc modo et illo possit ea copulare apparet sic, quia de ratione copulationis semper est unire similia: ESE Q.830.34.40, p. 343.) 5.431 This piece of analysis is neat enough for an immediate prima facie identification of the two senses mentioned. 357

Section 5 In the compounded sense, the '... and ...' is a functor which forms a name from two names; i.e. n/(n n) is its index. In the divided sense it forms a proposition from two propositions, so that s/(s s) is its index. That the latter sense gives falsehood is obvious from the falsehood of the two then remaining conjuncts: i.e. 'Five are two' and 'Five are three'. However, the apparent truth of the compounded senses is exactly what is to be looked at when Q.831 is presented in the next set of extracts. The final remark in this lastquoted passage, concerning the likeness of semantic category of the conjunction's two completions, has presumably been added to eliminate the possibility of further variety here; this would expand the range of cases which might be considered. This insistence on likeness of 'logical type' of completions around an '...and...' used to be supported, when Gilbert Ryle's talk of 'category mistakes' was fashionable, by calling attention to a case in which that insistence was neglected, giving the absurdity of 'She returned home in a sedan chair and a flood of tears'. It should be understood that the simple examples in the text quoted above by no means exhaust the huge range of types of case which the distinctions between compounded and divided sense may cover. That, however, is another matter, and Georgette Sinkler's useful study (SB) illustrate its scope. 5.432 After these preliminaries, attention may now be directed to Q.831, wherein the alleged truth of the proposition in its compounded sense (i.e. 'Five are two-and-three') is questioned. Support for its truth purports first to come from contrasts between the compounded and divided senses. The relevance of this will be dismissed easily in the reply. However, the real core of the mereologically relevant support comes from the second argument, wherein not only is the customary distinction between universal and integral wholes yet again expressed quite well, along with appropriate theorems, but the problem proposition itself is definitely allocated to 358

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the mereological field, and is thus said to be true. And indeed, if the pair and the trio are elements of the corresponding quintet, then that quintet is a collection or an integral whole of those elements: 10.329. (The contrast between the collection which is thus identifiable with its elements, and the mythical class (or set) of set-theory, which is typologically divorced from its elements, corresponds to the medieval contrast between integral and universal wholes when the universal whole is taken in the Platonic sense of an abstract object.) However, the misguided riposte to such proper mereological support for the proposition is later going to founder on what the author of ESE thinks is a smart refutational application of the distinction between X-parts and parts-of-X, as we are to see in 5.45. For a start, however, the rather unevenly meritorious text of these initial supporting arguments in favour of the proposition's truth in the compounded sense may now be excerpted from Q.831: (A) Consequently the question is raised as to whether the following, namely, 'Five are two and three', is true in the compounded sense, i.e. in so far as it involves a conjoined predicate. That the answer is affirmative would appear from the following: 1. Because, according to Aristotle in the second part of this work [ADS ch. 23, 1769a, 12 - 14], where composition deceives, division resolves, and vice versa. 'Five are two' [and 'Five are three'] are false under the divided sense, hence the proposition should be true in the composite sense [i.e. 'Five are two-and-three']. 2. Again, according to Aristotle in the fifth book of the Metaphysica, in the chapter [26] on the whole, there occur two sorts of whole, i.e. the universal and the integral. These are alike in that both involve a certain union of parts which as it were subsist as a whole in virtue of a certain unification; however, they differ in that the universal whole 359

Section 5 is exemplified in any one of its parts taken separately, and is predicated of any one of its parts taken separately, whereas the integral whole is exemplified in all of its parts taken together. In the present case, however, a five is a sort of thing integrally composed of a two and a three, and hence it can be predicated of two-and-three conjunctively. Hence the following is true: 'A two-and-three are five'; hence also is its converse [i.e. 'Five is two-and-three']. (Consequenter quaeritur utrum ista 'quinque sunt duo et tria1 vera sit in sensu composito, scilicet prout est de copulato praedicato. Et quod sic videtur: 1. Quia per Aristotelem in secundo huius ubi fallit compositio ibi solvit divisio et econverso, et hoc non esset nisi oratio falsa in sensu composito vera esset in sensu diviso et econverso. Nunc autem illa 'quinque sunt duo1 et cetera falsa est in sensu diviso, quare vera erit in sensu composito. 2. Item per Aristotelem quinto Metaphysicae capitulo de toto duplex invenitur totum, scilicet universale et integrale, quae conveniunt in hoc quod in utroque est quaedam unio partium tamquam in totalitate existente unione quadam, different autem in hoc quod totum universale salvatur in qualibet sui parte divisim et praedicatur de qualibet parte sui divisim, totum autem integrale salvatur in omnibus sui partibus coniunctim. Nunc autem quinque est quid integratum ex duobus et tribus, quare de duobus et tribus potest coniunctim praedicari. Haec ergo est vera 'duo et tria sunt quinque1, quare et sua conversa: ESE Q. 831.2.19, pp. 344-5) 5.433 Thus far the material and its mereological or other intent is tolerably clear. The allusion made to the Metaphysica was covered in our 3.21 above. Now, however, some others of the multiple possible interpretations of a proposition having the form 'A is B and C' begin to intervene, both in the counter-objections and in the body of the question. First 360

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of all, the counter-objection alludes to the Metaphysica (AVM) of the Arabic philosopher Ibn Sana, known to the medieval Latin West as Avicenna. Probably the fifth chapter of the third book (De certificanda quidditate numeri et de definitione suarum specierum) of AVM inspires the rather showy but arguably irrelevant initial set of distinctions. They are prefaced by the rash claim that they exhaust the possibilities of analysis in this situation. The first analysis (numbered [i] in the translation below), according to which the two and the three are attributed to the five aequaliter, in the same measure, results in a reversion to the already-eliminated false propositionally-termed interpretation of the '... and ...', (giving 'Two are five and three are five'). Here the '... are ...' is apparently taken as an identity, given that the passage mentions these converse forms of the two component propositions which thus have 'five' as their predicate terms. Perhaps something like 'Cicero is Tully and Marcus' would reproduce the intended type of construal-possibility here, with the '...is . . . ' a singular identity in each of the divided components ('Cicero is Tully' and 'Cicero is Marcus') as defined in 10.221. The second proposed construal (numbered [ii] below) might be something akin to 'Socrates is coloured and white' wherein the two predicate terms are compatible with each other, and are separately assertible of the subject; in the numerical case, however, this interpretation again yields falsehood, as in [i]. The third analysis, numbered [iii], just swivels round the two predicate terms, giving a further similar falsehood in the numerical case. Hence it looks as though the proposition cannot be true under any of the interpretations, which are supposed to be exhaustive: (A) The opposite is contended by the line of argument adopted by Avicenna in his Metaphysica, and I take up the argument on his sort of ground, as follows. There cannot but be three alternative interpretations of the proposition 'Five are two and three'. Either [i] it is interpreted as 361

Section 5 attributing a two and a three on an equal basis to the five, and in this sense it is false because it means that two are five and three are five. Or alternatively [ii] it is interpreted as principally attributing the three to the five themselves, not in an absolute sense, but in so far as they are affected by the two. Here again it is false, because the affectation of the three by the two does not abrogate the nature of those three, and from some referent affected in a non-abrogatory fashion there follows that referent itself; hence this follows: 'Five are two and three; hence five are three' [and this is plainly wrong]. Or alternatively [iii] it can be interpreted as principally attributing two to those five in so far as they are affected by the three, and this interpretation is just as false as are the foregoing. Hence falsehood ensues upon any mode of interpretation. (Oppositum arguitur per intentionem Avicennae in Metaphysica sua et arguo ratione eius: huius enim propositionis 'quinque sunt duo et tria' non potest haberi nisi triplex intellectus: aut intellegetur duo et tria aequaliter esse attributa ipsis quinque, et sic est falsa quia significat quod duo sunt quinque et tria sunt quinque; aut intellegitur quod tria attribuuntur principaliter ipsis quinque, non tarnen absolute, sed ut disposita sunt per duo, et sic adhuc est false quia dispositio trium per duo nihil deminuit de ratione ipsorum trium, et ab aliquo supposito cum dispositione non deminuente sequitur ipsum, quare sequetur, 'quinque sunt duo et tria, ergo quinque sunt tria'; aut potest intellegi quod duo principaliter attribuuntur ipsis quinque ut sunt disposita per tria, et iste intellectus falsus est sicut et praecedentes; quolibet igitur modo est falsa: ESE Q. 831.20.32, p. 345.) 5.434 The question's corpus, which next follows, still opposes the truth of the contested proposition, but without overtly drawing on the allegedly Avicennan distinctions just quoted. Rather, there is a sometimes misguided exploitation 362

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of the already-encountered alignment of Aristotelian 'matter' (in the 'makings' sense) with mereological parts, as mentioned in 3.31 and 2.36. This then casts the duo and the trio in the rôles of proper parts, and it would have been quite easy for the author to adduce forthwith the common thesis of the nonidentity of wholes and parts (1.32, 10.324) or of matter and the whole, as having some bearing on his negative aim. This, however, is postponed until later (5.436 below) and there follows instead a slanted interpretation of a section of Aristotle's Metaphysica which, as 3.11 has shown, is moving towards the explicitation of the notion of 'form', but which is here taken to ground not just the common thesis of non-identity of whole and part, but also the non-identity of the whole and all the parts. 5.435 What the author evidently has forgotten is that the last-mentioned non-identity is afflicted by the fatal ambiguity of 'part' when expressed in ordinary parlance: it may be an allusion to part-of-X (in which case the non-identity with the whole fails to hold) or it may be X-part (in which case the non-identity does indeed hold). The way in which Aristotle's examples of the elements (in the Aristotelian sense) and the letters have the disadvantage of masking this ambiguity was described at length in 3.11 above, and it is precisely into this latent trap that the argument falls: for example, the wrong conclusion is drawn from 'when the syllable perishes the letters still abide', with the abiding letters still being seen, in their state as X-parts, as being relevant to the non-identity of whole and part (in the part-of-X sense). So having thus obtained the non-identity of X and all the X-parts, the argument slides into the non-identity of X and all the parts-of-X, a falsehood which allows the negation of the contended proposition. This is in turn backed up by another apparent misinterpretation of Aristotle; this, however, could issue from a mistranslation, and hence will not be pursued here: 363

Section 5 (A) In reply to all this it is asserted that this, namely 'Five are two and three', when taken literally and in the composite sense, is false, for the five is diverse from a couple and from a trio, and hence is not two and three. For in a predicative proposition asserting that this is [this], the predicate should not be other than the subject, but should be numerically identical with it. A five, however, is other than a couple and a trio because a five is made up of a couple and a trio in the rôles of material and integral parts, and towards the end [ch.17, 1014b, 11 - 28] of the VIIth book of the Metaphysica Aristotle proves that in general a composite is other than its component parts, rather in the way that a mixture [of Aristotelian elements] is other than its admixed elements, and a syllable other than the letters which make up the syllable, because when the mixture perishes the admixed elements still abide, and when the syllable perishes the letters still abide. Hence five is other than two and three, for taken literally, five are not a two and a three taken conjunctively, and Aristotle expressly hints at this in the Vth book of the Metaphysica [ch. 14, 1020b, 7 - 8 ] when he says that six are not two threes from the point of view of formal predication, but just plain six. Two and three are related to five as are twice three and six, hence five are not two and three from the point of view of formal predication. (Ad hoc dicitur quod haec falsa in sensu composito 'quinque sunt duo et tria' de virtute sermonis: quod enim aliud est a duobus et tribus, illud non est duo et tria; in praedicatione enim dicente hoc esse , praedicatum debet esse non aliud a subiecto sed idem numero cum eo. Quinque autem aliud est a duobus et tribus tamquam ex partibus materialibus et integralibus, <et> probat Aristoteles in fini semptimi Metaphysicae quod univeraliter compositum est aliud a partibus componentibus, ut mixtum aliud est a mixtibilibus et syllaba aliud est quam litterae componentes syllabam, quia corrupto mixto adhuc manent mixtibilia et cor364

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rupta syllaba adhuc manent litterae. Igitur quinque aliud est a duobus et tribus, quare quinque de virtute sermonis non sunt duo et tria coniunctim accepta, et hoc expresse innuit Aristoteles quinto Metaphysicae, dicens quod sex non sunt bis tria formali praedicatione, sed semel sex. Ita se habent duo et tria ad quinque sicut bis tria ad sex, ergo quinque non sunt duo et tria formali praedicatione: ESE Q.831.33.49, p. 346.) 5.436 In further defence of the rejection of 'Five are two and three', the continued identification of parts with matter leads to the perfectly valid observation that some sort of linguistic device which stresses that the whole is made-outof, or made-from, the parts, should preferably be used to predicate the incorporated matter of the whole (e.g. the box is made-of-wood, or wooden). We have seen Clarembald's recommendations on just this point in 2.821, and were there able to assume that appropriate vocabulary of this sort was available. Since then, analysis of Aquinas' insistence on the way in which organic forms imbue organic parts has suggested a mereological way in which, with suitable constant terms inserted into the background axiomatic, that 'denominative', adjectival style of vocabulary might be further analysed (3.42). It looks as though some such analysis is now being attempted in the present case. Allusion is made to the same text of the VIIth book of Aristotle's Metaphysica as was drawn on in 5.435(A) above to ground the distinction between the composite and its makings, but it now rightly adverts to the central point of that Aristotelian text, i.e. the formal or quidditative aspect of the composite (cf. 3.11). This is, however, said to be still of no help in rendering acceptable the predication of the matter-parts of the whole, and so the contested proposition, alleged to involve such predication, is again out of court: (A) Again, in the IXth book of the Metaphysica [ch. 7, 1049a, 18 - 20] Aristotle proves that a predication of the 365

Section 5 form, 'This is that' does not yield a truth when used to express the matter's being predicated of its composite, but 'This is of that' does so. Hence 'The box is wood' is not a true predication, whereas the following is true, namely, 'The box is wooden'. Now, however, a fivefold sort of object is made up of a two and a three after the fashion of material parts, whereas its form is in the nature of an addition to its parts, even as, in general, in every composite the form or quiddity is something added to the component parts, as is proved in the VIIth book [ch. 17, 1041b, 8 - 28] of the Metaphysica. Hence as that which is fivefold would signify the quiddity as involving the aggregation made up of a two and of a three in the rôle of material parts, and of the form of fivefoldness thereto superadded, it would stand for that composite whole in a discursive context; hence in the present case it is of such a composite that the matter is being predicated; yet such predication [of matter, with the whole as the subject] is impossible, hence [it's not that five are two and three]. (Iterum in nono Metaphysicae probat Aristoteles quod materia vere non praedicatur de composito praedicatione dicente hoc esse illud sed hoc esse illius, unde non est praedicatio vera 'arca est lignum', haec tarnen est vera 'arca est lignea', Nunc autem quinque quid compositum <est> ex duobus et tribus tamquam ex partibus materialibus, forma autem eius quid additum partibus eius, sicut univeraliter in omni composito forma et quod quid est <est> additum aliquid partibus componentibus, ut probatum est septimo Metaphysicae. Cum igitur hoc quod est quinque significet suum quod quid est aggregatum ex duobus et tribus tamquam ex partibus materialibus et ex forma quinarii superaddita, illud totum compositum in oratione supponet; hic igitur materia praedicatur de composito, et talis praedicatio est impossibilis, quare et cetera: ESE Q.831.49.62, p. 346.) 5.44 The impression that our author is bluffing his way forward against the contended proposition for the sake of the 366

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startling effect of his negative position is deepened by the last section of the corpus, now to follow (as well as by the succeeding reply to the second original supporting argument). The suspicion raised above, namely that 'part' is being sophistically used exclusively for, or at least to comprise, Xparts, is now confirmed by the allusion to 'stones and wood' as being potentially the house, i.e. qua house-parts and not parts-of-the-(actual)-house. So granted that the predicate '... are two and three' is to be taken in the 'quintet-part' sense (as opposed to the 'part-of-the-quintet' sense), then of course such parts only potentially amount to an actual quintet. It is immensely strange that here and in reply to the second objection, the writer does not jerk himself out of this insistence on X-parts (and hence on potentiality, relatively to the actual quintet, or house, or whatever) in order to concentrate on the parts-of-X (presupposing the actuality of X). In this latter sense, a quintet is of course a trio-anda-duo, but the correctly sensed presence of indefinitely many other parts of this integral whole may be the basis of the continued (but necdless) accent on potentiality: (A) And this is what Aristotle expressly asserts in Book VIII [of Metaphysica, ch. 2, 1043a, 14 - 16], i.e. that he who says 'A house is stones and wood' is talking only about a potential house, since he is talking only about the matter of a house, and the matter is only potentially the composite formed from it. Hence, when we assert five to be two and three, restricting ourselves [as in the example of the house] to predicating the material parts of those five, then that predication only holds true when the five are viewed potentially. Now the five as propounded in an absolute sense [in the proposition under discussion] do not involve allusion to a five in in a potential sense, but only in an actual sense, hence [it's not that five are two and three]. (Et hoc expresse dicit Aristoteles octavo Metaphysicae quod qui dicit <domum esse lapides et ligna> dicit domum in potentia quia 367

Section 5 solum materiam domus dicit; materia autem solum est in potentia ipsum compositum, ergo cum dicimus quinque esse duo et tria solum praedicando partes materiales de ipsis quinque, solum vera est praedicatio illa de quinque in potentia. Nunc autem quinque absolute dicta non supponunt quinque in potentia dicta, sed solum quinque in actu, quare et cetera: ESE Q.831.62.70.) 5.45 There still remain the two replies to the initial supporting arguments (5.432), and of these the second is the most relevant to seeing what is going on here. The first reply confines itself to rightly pointing out that the contrast between compounded and divided senses necd not be designed just to exhaust truth-possibilities. However, the second reply obtusely argues against the integral whole's being the collective class of its parts on the quite overt basis that 'part' can here designate X-part (e.g. the scattered house-parts are mentioned as cotemporal with the non-existence of the house). This is an extension of the same confusion as that already observed in the corpus of the question (5.435 above). Indeed, there are perhaps signs of further confusion in the interpretation of 'coniunctim', 'conjointly', which then allows pre-constructional roof-and-foundation contact to count as conjunction which justifies the integral whole's name being predicated thereof. This is apparently taken as a reductio ad absurdum proof that the name of the integral whole is not predicated of all the conjoint parts, and the reply continues with a quite proper reminder that due arrangement and composition of all parts is necessary for such predication to be justified. However, if 'part' had been taken in the sense of 'part of ...' from the outset, then that reminder would have been otiose. Adoption (or presupposition) of the appropriate definition of integral whole (D2 of 10.31) ensures that both the 'part of ...' sense, and the relevant 'order and arrangement' can be operative. Here, then, are the replies (of which the first is excellent and the second con368

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fused in the manner described) to the objections reproduced in 5.432 above: (A) 1. The reply to the first argument in favour of the thesis is that when Aristotle says, 'where composition deceives [division resolves]', he does not convey thereby that discourse which is false in the composite sense is true in the divided sense, and vice versa, but rather that when the opponent [in an argument] wishes to impute an impropriety on the part of the respondent's contention [if it is taken] in the composite sense, then that respondent should reply that his contention was made in the divided sense (and vice versa), and so the impropriety was not to be attributed to him. 2. The reply to the other argument is that Aristotle was denying the thesis that the [name of] the integral whole can be predicated of all its parts conjointly, for the roof-andfoundation may be a [conjoint] part [and still not have 'house' predicated of them]; alternatively these parts may even be extant, but in a scattered manner while the house does not exist, so that for the house to be, not only are these parts required, but also their arrangement and composition. It still remains true that the universal and integral wholes are [as the objection stated] differentiated in so far as the universal whole is predicated of each of its parts taken severally, whereas the integral whole is not. It does not follow from this, however, that the integral whole is predicated of all its material parts taken together. (1. Ad primam rationem dicitur quod non intelligit Aristoteles per hoc 'ubi fallit compositio' et cetera quod oratio falsa in sensu compos ito vera sit in sensu diviso et econverso, sed quod cum opponens vult concludere inconveniens respondenti in sensu composito, debet respondens dicere quod earn dedit in sensu diviso et econverso, et tunc <non> accidet ei inconveniens. 2. Ad aliud dicitur quod non vult Aristoteles quod totum 369

Section 5 integrale praedicatur de omnibus suis partibus coniunctim; possible enim est esse partem tectum et fundamentum, vel ipsis existentibus divisis non existente domo, unde ad esse domus non solum exiguntur partes istae, sed etiam ordo et compositio ipsarum. Sed verum est quod in hoc differunt totum universale et integrale, quod totum universale de quacumque suarum partium praedicatur divisim, totum autem integrale non; et non sequitur ex hoc quod totum integrale de omnibus suis partibus materialibus praedicetur coniunctim: ESE Q.83l.7l.86, p. 347.) 5.46 The final sentence of this text, with its identification of parts as 'material parts', suggests that this terminology, with its implied allusion to Aristotle's 'matter' (in the sense of 'makings'), may be responsible for the confusing assumption that 'part' may cover both X-part and part-of-X. Parts in both senses may count as such 'matter'. It is being assumed that until the trio and the duo are duly certified as actually constituting the quintet, then they may be in the position of quintet-parts, and hence indeed may not make up the quintet, at least as yet. This is the relevance to the central question here. However, the slackness evinced in allowing X-parts to possibly count as parts-of-X tends to confirm Sten Ebbesen's refusal to allow that ESE is a work by Boethius of Dacia. This Boethius stresses most strongly the necessity for the whole's existence as a precondition for the existence of its parts, as well as the rôle of the parts in identifying the whole; this is just what ESE at this point lacks. That Boethius is also thus distinguishing the integral whole from the 'quantitative' whole in the sense of logical quantity may also incidentally be gathered from the following extract from his discussion of Aristotle's Topica. Herein the logically quantitative whole is typified by 'Every man ...' (cf. 10.23), and the immediate problem has been whether from a proposition thus formed one can infer to the corresponding proposition with a non-empty proper-name subject: 370

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(A) The reply to the second objection is that 'Every man runs, hence Socrates runs' does not reflect a dialectical commonplace rule, the reason being that the only dialectical relation [which some have attempted to bring to bear] here is that of the whole to the part. Now [thus using the integral as the paradigm] there is no relation between whole and part which holds without the whole's existing. For those entities are parts-of-the-whole which follow upon the existence of the whole, and which are such that without them the whole cannot be. If, however, there are parts of some whole which fail to come up to this specification [as in the case of X-parts], then this is something incidental in respect of that whole. Now that [logically] quantitative whole represented by 'Every man ...' can carry on being without that part which is Socrates. Neither can Socrates be inferred from it, nor does that whole's signification actually contain that of Socrates. Hence there is no dialectical relation [of the integral part-whole sort] between that [pseudo-] whole represented by 'Every man ...' and that part which is Socrates, although some who have merely glanced at the case do not see this. However, the following involves the [logically] quantitative commonplace based on the square of opposition: 'Every man runs; hence some man runs'. (Ad secundum dicendum quod hic non est locus dialecticus: 'omnis homo currit, ergo Socrates currit'; et ratio huius est quia hic nulla est alia habitudo dialectica, nisi totius ad partem. Nunc autem non est habitudo totius ad partem, sine qua totum esse potest; illae enim sunt partes totius, quae ad totum sequuntur et sine quibus totum esse non potest. Si autem alicuius totius aliae sunt partes, hoc accidit toti. Nunc autem illud totum in quantitate, quod est 'omnis homo', esse potest sine hac parte, quae est Socrates, nec Socrates ad ipsum sequitur, nec in significatione eius actu continetur. Ideo non est habitudo dialectica huius totius quod est 'omnis homo1 ad hanc partem quae est 'Socrates', licet quidam parum consider371

Section 5 antes hoc non vident. Sed hic est locus a toto in quantitate propter oppositas causas: 'omnis homo currit, ergo aliquis homo currit': BT II, q. 6, ad 2, p. 124.144 - 125.157.) 5.47 This is only one part of Boethius' discussion which is extremely acute, and could have an important bearing on John Stuart Mill's contention that the categorical syllogism committed the fallacy of petitito principii ('begging the question': ML II, 3). To pursue the conversation in that direction would be too much of a distraction. Fortunately chapters 2 and 3 of Geoffrey Scarre's recent book (SML) take up the controversy in a way which will facilitate any such pursuit. 5.48 The fact that what on the face of it is an arithmetical truism has proved capable of generating so much debate, must surely give pause to those who would want to lightly classify propositions as disputable and indisputable (these being trendy substitutes for the old and equally dubious 'synthetic' and 'analytic' respectively). 5.5 Around the Liar Paradox 5.51 Another highly tempting distraction, namely that constituted by the possibility of following through ESE's treatment of the Paradox of the Liar, must also be resisted. P.V. Spade's outline of medieval riches here is to be found in CLM 246 - 253, and more recently the same author and Sten Ebbesen have produced more comments, bibliography, and texts in CC 56 ('More Liars'). However, certain striking connections between mereology and the self-reference which is allegedly involved in the paradox, as will transpire below, are set up in these discussions, and it is the approach to those connections that must now first be outlined. 5.52 The text to which attention is chiefly to be devoted is in Q.95 of ESE. Here is raised the still very live problem about whether a term occurring in some stretch of speech can stand for the whole of which it is a part (and hence whether

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it can in that sense be self-referent). It is, of course, such an allusion to part and whole which brings the discussion within the ambit of mereology, although this theory is clearly not the only background body of presuppositions to which reference should ideally be made in this connection. Also necded is theory of deduction, which is about propositions and their interrelations. In its most down-to-earth form such a theory concerns propositions (or other parts of speech) as exemplified in actual tokens, i.e. individual inscriptions or other concrete significant utterances. Such are the objects counted when an editor counts the words in a document. Contrasted with such tokens are 'types', and these are what are being counted when it is said that a dictionary covers so many thousand words. A medieval correlate of this token-type distinction is going to be found in the present ESE Q.95. 5.53 A concrete token-based view of propositions was common, if not altogether general, in medieval times, and much of the corresponding theory of deduction was a Latin medieval production (with the fourteenth-century Robert Holkot as a prominent proponent thereof). A theory embracing the possibility of this sort of view has been the topic of copious lectures by C. Lejewski; these make it clear that a further book would be required to expound it properly. Hence, for the present analyses, merely ad hoc remarks, without any further theoretical backing from the Presuppositional Explicitation (Section 10) will have to suffice. (On the amazingly varied and lively medieval theories of the proposition, the two volumes (NTP and NPL) from the pen of G. Nuchelmans may most profitably be consulted. Particularly relevant to Holkot's tokenism is NTP 206.) 5.54 First of all the flavour of the present topic, as expressed in Q.95 of ESE, may be gathered from the dialectical opening of that question. Here both denial of, and support for, the thesis of the possibility of self-reference are based on Aristotle's Metaphysical 373

Section 5 (A) And so next it is asked whether a term located in a piece of discourse can stand for that whole of which it is a part. Against this thesis it is argued: 1. That the Philosopher in Metaphysica IV says that he who denies speech posits speech. But in the utterance of the denial of there being any speech, speech is used to deny the existence of speech. If, however, one who thus denies does affirm speech, then it would appear that the denial does not apply to that [particular piece of] speech, and that consequently the term ['speech'] does not stand for its whole. Hence [a term located in a piece of discourse does not stand for that whole of which it is a part]. 2. Again, a term does not stand for something which does not share in the appropriate form. However, a whole does not share in the form of the part, hence the part does not stand for the whole, because the integral part and its whole do not have the same form. The contrary would appear to hold from what Aristotle says at the end of Metaphysica IV [ch. 8, 1012b, 13 - 18] namely that he who says that everything is true, or alternatively, that everything is false, destroys his own contention, for he who says that everything is false says that he himself utters a falsehood, and at the same time posits its contrary [i.e. that he is telling the truth], hence [a term located in a piece of discourse can stand for that whole of which it is a part]. Again, the whole can stand for the part, as is obvious from what Aristotle says in the Physica [V, ch. 1, 224a, 25 - 6]. For he there says that a man is made healthy if his chest is made healthy; hence a part can stand for the whole. (Consequenter quaeritur utrum terminus in oratione positus possit supponere pro illo toto cuius est pars. Et arguitur quod non: 1. Quia dicit Philosophus in quarto Metaphysicae quod qui 374

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negat loquelam ponit loquelam. Sed dicendo nullam loquelam esse hic negatur esse a loquela. Si tarnen sic negans affirmet loquelam, tunc videtur quod pro ipsa non negatur, et per consequens quod terminus non supponat pro toto, quare et cetera. 2. Item terminus non supponit pro aliquo quod non participât formam eius, sed totum non participat formam partis, ergo pars non supponit pro toto, quia pars integralis et suum totum non habent eandem formam. Oppositum videtur per Aristotelem in fine quarti Metaphysicae: qui dicit omnia vera esse vel omnia falsa esse, destruit se ipsum, quia qui dicit omnia esse falsa dicit se dicere falsum, et ponit contrarium, quare et cetera. Item totum potest supponere pro parte sicut patet per Aristotelem in libro Physicorum, quia dicitur quod homo sanatur si thorax sanatur, quare pars poterit supponere et pro toto: ESE Q.95.1.19, p. 224.) 5.55 This second contrary argument plainly relies on both a stretched sense of 'stand for', as well as some thesis about principal parts and attributes (2.4, 3.35 above). No further direct allusion is made to this comparatively remote point. 5.56 For the better understanding of the question's corpus, a few reminders are required. It opens with what would appear to be a version of the type/token ambiguity, as outlined above. An expression taken in the integral sense is a token (and hence like any other mereological part or whole may be the object of discourse at the nominally-termed level). Alternatively, a word like 'false', though itself a token, can have the import of a universal, of which individual (token) falsehoods are the 'subjective' parts. In other words, the traditional distinction between integral and universal wholes (2.21 above) is here being brought to be bear in the linguistic field. The 'formal', universal, import of a theoretical word like 'false' may in turn itself be the topic of quidditative discourse. 375

Section 5 5.561 A key-word which appears in the first few lines of the corpus is 'dictum', literally, 'a thing said'. As G. Nuchelmans makes clear (e.g. NTP 169 - 176) this term has a highly convoluted medieval history. Fortunately, the previous Quaestio in ESE clearly uses the word to refer to what is said by token propositions: ESE Q.94.24.37. Hence if the token is not there, neither is the dictum.; again, if the dictum is not there, then neither is truth nor falsehood. First, then, is a record of one core-point which figures in the resolution of the puzzles constituted by the example of 'I am uttering a falsehood', this being one starting point for the Paradox of the Liar. (A) But when it is thus asserted, 'I am uttering a falsehood', there is no attribution of falsehood to some dictum, because there just does not exist some dictum to which the falsehood can be attributed. Neither, therefore, is falsehood attributed [by the proposition] and hence neither does it utter a truth determinately, nor does it utter a falsehood determinately. (Sed dicendo 'dico falsam' non est attributio falsi ad aliquod dictum, quia non est aliquod dictum cui falsum attribuatur, nec igitur falsum attribuitur nec igitur dicit verum determinate nec falsum determinate: ESE Q.94.32.35.) (And now the nature of the paradox is becoming clearer: both the truth and falsehood of 'I utter a falsehood' may apparently be inferred from that utterance, for if it is true, it is false, and if it is false, then it is true. This unpalatable consequence is here being evaded by what some might construe as an apparent rejection of the principle of bivalence for propositions. Contrary to such a principle, in this case the negation of the truth of the proposition does not entail its falsehood, and neither does its falsehood entail its truth.) 5.562 It could well be that the prominence of the problem of self-reference, which is our main present concern, is inspired by some attempt to supply the missing dictum in this solution 376

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to the Liar Paradox. The basis of that attempt would then be what Peter of Ailly is to characterise (one hundred or so years later) as 'a famous opinion ... every sentence signifies itself to be true' (SAI 65, cf. note 649 and, in general P.V. Spade's excellent editorial introduction to that volume). Certainly a variant expression of the 'famous opinion' is given in the corpus of the Q.94 whence was drawn the last quotation: omnis propositio asserit se esse veram, 'every proposition asserts its own truth'; Q.94.15.16. This would explain the interest in self-reference contained in the Q.95 with which we are now engaged. Otherwise the latter question remains unlinked, save by what might be a misprint, as we are to see in 5.564, with its predecessor's paradox-centred concern. Be that as it may, it is to the already-broached Q.95 that we now revert. 5.563 It has already been gathered from the preliminary arguments quoted in 5.54 above, that this Q.95 is tending to converge on 'All things are false' as yet another paradoxical proposition which is self-destructive (in the sense that if true it implies the opposite). A connection between 'All things are false', and 'I assert the false' has also been there set up in the 'contrary' section, on the alleged authority of Aristotle. The corpus now goes on to propound its version of the type/token distinction described above, and it is to the two last-quoted paradox-generating propositions that that distinction is now apparently first applied thus: (A) I assert that in the process of asserting 'All things are false', the word 'false' already takes on the rôle of an integral part of the following stretch of speech, i.e. 'I assert the false'; yet at the same time the word 'false' [in the latter] conveys the import of the universal whole in respect of all false things; hence if this is a false assertion, it is a subjective part of that which is the false, and it is in this fashion that the word 'false' can stand for the whole stretch of speech of which it is an integral part, i.e. 377

Section 5 in so far as it conveys the import of the universal in respect of that whole. This is because the unrestricted universal term stands for no matter which sharer in its conveyed form. (Dico quod sic dicendo 'omnia sunt falsa' iam li falsum habet rationem partis integralis huius orationis 'dico falsam', et tarnen li falsum habet rationem totius universalis ad omnia falsa; quare si hoc est dictum falsum est pars subiectiva eius quod est falsum, et ita li falsum potest supponere pro tota oratione cuius est pars integralis secundum quod habet rationem universalis respectu illius, quia terminus universalis non contractus supponit pro quolibet participante forma eius: ESE Q.95.20.27, pp. 224 - 5.) 5.564 As already mentioned, the previously alleged connection between 'All things are false' and 'I assert the false' here appears to be assumed as actualised, so that the token 'false' is available in both propositions. Exactly why the second proposition ('I assert the false') is thus imported is still nevertheless unclear, unless the 'I' is supposed to ensure integral concreteness. The argument could apparently proceed without it, i.e. on the assumption that the words 'dico falsum' are just not there, or that they are a scribal error for a repetition of 'omnia sunt falsa'. Still, the central gist is clear enough. 'False', taken as a token, is an integral part of whichever of these concrete propositions is here supposed to be actually extant. At the same time its meaning (relatively to 'false' qua type, or 'universal') conveys the import exemplified in all instances of falsehood, and hence the concrete propositon in question exemplifies (is a pars subiectiva, subjective part, of) that universal. As the passage goes on to conclude, this presupposes that there is no restriction imposed on the sense of 'false' which would prevent its standing for the very proposition in which it occurs, and of which it is hence an integral part. 5.57 The passage continues by extending such a presupposition of non-restriction to the case of 'There exists a prop378

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osition', so that it can stand for itself, and then goes on to examine the dire consequences of any bar on the self-reference thus authorised. Hence if, presumably on a quite presentday kind of suspicion that self-reference generates unacceptable paradox, self-reference were to be barred so that the 'proposition' of 'There exists a proposition' would not be self-referent, but would still refer to every other proposition, then it would stand for the proposition 'No proposition exists'. But now we have two contradictories, namely, 'There exists a proposition' and 'No proposition exists', and the change of accompanying structural logical signs around the word 'proposition' which these two exhibit does not entail a change in what this word stands for in the two propositions. Hence the 'proposition' of 'No proposition exists' must be self-referent in the sense that it refers to 'No proposition exists' (as did the 'proposition' in 'There exists a proposition'). Presumably one could carry out the converse process starting from 'No proposition exists'. The upshot is that any such bar on self-reference is self-refuting in that it leads to self-reference. (Of course there is in this argument a suspicion of confusion between signifying and standing for (cf. HQS 91 - 6) but as this is not germane to our central theme here, further comment at this point is eschewed). Here is the passage's continuation containing this material: (A) Thus, as in the assertion of 'There exists a proposition' this stretch of speech contains no restriction on the word 'proposition' which would make it stand for some determinate proposition, it will stand for every proposition, and hence for this very one. Should it be asserted that it stands for every proposition other than this one, then it would stand for the following, namely, 'There exists no proposition'. But the term ['proposition'] stands for the same things in both of these two contradictory propositions. Hence in this, namely 'There exists no proposition', the word 'proposition' does stand for the proposition 'There exists no prop379

Section 5 osition', for the [syncategorematic; cf. 7 below] sign does not make that which is not stood for to be stood for, nor does it make that which is stood for not to be stood for. Besides, unless a part could stand for that whole of which it is a part, then Aristotle would be propounding a falsehood in the text 'he who asserts that he is asserting the false (and so on) asserts the false irrespectively, even though he thereby tells the truth' [ADS ch. 25, 180b 2 - 7]. For unless the word 'false' could stand for this expression 'I assert the false', he would in no way be asserting the truth. (Cum igitur sic dicendo 'propositio est' in hac oratione nihil est contrahens li propositio ut stet pro aliqua determinate, stabit pro omni propositione, igitur et pro ista. Si dicatur quod stat pro omni propositione alia ab hac, ergo stabit pro ista 'nulla propositio est', et terminus pro eisdem supponit in duabus contradictoriis, quare in ista ' propositio est' li propositio supponit pro hac propositione ' propositio est', quia signum non facit de non supposito suppositum nec de supposito non suppositum. Praeterea nisi pars possit supponere pro toto cuius est pars, falsum diceret Aristoteles in littera 'qui dicit se dicere falsum et cetera, falsus est simpliciter, quo autem verus'. Nisi autem li falsum supponeret pro hac oratione 'dico falsum' nullam veritatem diceret: ESE Q.95.27.40, p. 225.) 5.58 The final contention here, namely that Aristotle himself presupposes the licitness of self-reference, tends to confirm the interpretation of the initial part of this passage which is embodied in the translation here presented. As the Latin edition in ESE shows, alternative construals could well be possible. 5.59 The consequent replies to the originally quoted objections to the thesis of self-reference (5.54 above) are still of interest. The first deals with the paradox of speech-based denial of the existence of speech: does it except itself from that denial? This is resolved by a distinction 380

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between the content of the denial and the speech act involved in the denial. As far as content goes, the denial can deny speech without positing speech (and hence the question of exception does not arise). In respect of the speech act itself, speech must be posited (and hence presumably must be excepted, or understood as excepted). The vague (though mereological) form of the second objection now inspires a reply which seems to draw upon the beginning of the abovequoted corpus, as quoted in 5.563(A), with its type/token distinction . In so far as it is an integral part of 'There exists a proposition', there is no reference by the word 'proposition' to the whole proposition, but the same word has a meaning which brings in all propositions. Hence qua type ('natura') there is a sort of self-reference. (Once again, the suspicion of confusion between significatio and suppositio, meaning and reference, will not be pursued here.) (A) The reply to the first contention is that when the following, namely, 'No speech exists', is asserted, two aspects arise for observation. There is, on the one hand, that which is negated (and this is relevant to this very stretch of speech), and on the other hand attention has to be paid to the very act of negating (and this relates to the person who is making the utterance). It is in this second respect that speech is posited, for one cannot deny that there is speech save by the use of speech, and it is in this sense that one posits speech. The reply to the other contention is that in the course of the assertion of 'A proposition exists', the word 'proposition', in so far as it is an integral part [of that proposition] does not stand for this stretch of speech, i.e. 'A proposition exists'. Nevertheless, that word can stand for the latter in so far as that word presents the universal whole in respect of that proposition, and in this sense it can do the same for any other proposition. In this latter sense it is obvious that there is no incoherence in the part 381

Section 5 standing for the whole of which it is a part, since the ['subjective'] part [of 'proposition' qua distributive universal] itself embodies the nature of a propositional complex. {Ad primum argumentum dicendum quod sic dicendo 'nulla loquela est' contingit duo considerare, scilicet ipsum negatum quod est ex parte ipsius orationis, et sic non ponitur aliqua loquela, et est considerare actum negandi qui est ex parte proferentis et sic ponitur loquela, nam non potest negare loquelam nisi per loquelam, et ita ponit loquelam. Ad aliud dicendum quod dicendo sic 'propositio est' li propositio secundum quod est pars integralis non supponit pro hac oratione 'propositio est', potest tarnen supponere pro ista secundum quod est totum universale ad ipsam, sicut et pro quacumque alia, et ita patet quod non est inconveniens partem supponere pro toto cuius est pars, nam pars habet in se naturam complexionis: ESE Q.95.41.53, pp. 225 - 6.) 5.6 There, most regretfully, the superb questions of ESE must be left without that further analysis which they so eminently merit. For more texts on the same question, the already-mentioned CC 56, pp. 205 - 227 may be consulted. The 'famous opinion' described in 5.562 as in effect claiming that every statement states that the statement itself is true, is prolonged into the seventeenth century by Arnold Geulincx (1624 - 1669, cf. NGL 20), and has, of course, reared its head yet again in the present century.

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6. Wyclif s Deviant Mereology 6.1 A Wycliffian Work on Universals: General Prospectus 6.11 At this juncture the recent publication of an edition and translation of a remarkable work by the Yorkshire logician and theologian John Wyclif (d. 1384) serves to direct a t t e n t ion to some of the most lively mereological ventures of the whole medieval era. The Tractatus de Universalibus has been edited by Ivan J. Mueller (WUL), and a companion volume (WUE) contains its English translation by Anthony Kenny, with an introduction by Paul Vincent Spade. 6.12 A preliminary clarification may first be offered to assist those already familiar with the Wyclif Society's editions of this m a s t e r ' s works. They may be already baulking at the claim, implied above, that his De Universalibus is edited in WUL for the first time. Is it not the case (one may ask) that the Society's 1905 volume of Miscellanea Philosophica II has his De Universalibus on pp. 1 - 1 5 ? It turns out, however, that this 1905 edition is in fact a work by Stanislav of Znojmo (or Znaim) who, with Hus, later carried forward the r e former's doctrines in Prague. An illuminating paper by G. Nuchelmans in the recent de Rijk Festschrift (NSZ) reveals the immense philosophical interest of this Stanislav's output, e s pecially on the perennial medieval topic of ontological truth. 6.13 More sophisticated misapprehensions may arise from Wyclif's having apparently referred not only to the WUL t e x t , 383

Section 6 but also to one of the sections of his De Ente ('Purgans errores circa universalia in commun?, Wyclif's Latin Works, vol 33) as his De Universalibus. The sorting out of such bibliographical matters is carried out superbly in the eighty or so pages of the editor's introduction to the volume of the Latin text, for all of which is due the gratitude of the scholarly world. 6.14 There can be no doubt about the genuineness of the claim, made in an identical note prefixed to each of the new volumes mentioned, that the present treatise 'is the climax of Wyclif's work as a philosopher, and contains the seeds of his distinctive theological ideas'. One should not be misled by the title De Universalibus, nor by elementary present-day ideas about 'universals', into thinking that Wyclif here simply puts forward a naïve Platonic-type realism. Were this to be the case, then the interest of the work would be considerably reduced. In fact, as Peter O. King has pointed out in his monumental thesis on Abelard's theory of universals, such a theory in the medieval setting necd have little in common with its present-day counterpart: KAU Vol I, pp 2 - 8; cf. 2.816 above. It is hence unfortunate that the logical preface to the translation simply takes over (WUE xii - xiii) a variant of W.V.O. Quine's contemporary criterion for 'realism' from his essay 'On What There Is' (QF 1 - 19). This involves the presupposition of a totally unsuitable view of quantification (in the contemporary logical sense) among other things: cf. HQS §2.3. The inepititude of the consequent characterisation of nominalism has already been noted in 2.816. 6.15 Nevertheless, one cannot but sympathise totally with the tentative nature of P.V. Spade's introductory approach (p. xi) and one must be grateful for his thorough exploration of 'predication' and his presentation of a fourteen-item list of other salient topics (usefully underpinned by textual references). The scope, boldness, and novelty of this work of Wyclif's, from any point of view, be it medieval or contem384

Wyclif's Deviant Mereology porary, is quite breath-taking. For example, the essence of Russell's thesis that a number is a class of relevantly similar classes is encapsulated in the remark that the 'species of tenfoldness is common to every number by which substances are formally characterised as ten' (WUL ch. 9, 11. 53 - 4). Again, this same chapter 9 continues to be aware of the category mistakes (in the contemporary sense) which can arise in the counting of universals and individuals. Thus 'No species and its individual can be counted formally together as things of the same species' (WUL ch. 9, 11. 224 - 6) is a salutary repetition of John of Salisbury's 'non connumerantur' verdict on the same point in his Metalogicon, or of Boethius of Dacia's insistence on the categorial distinctions between the parts of speech involved in 'Socrates and Plato are men' and 'Man and donkey are animal': cf. 2.762 above and HQS §4.35. I; chapter 11 the discussion of doing as opposed to acting (already adumbrated in the Anselmian Lambeth fragments) which was extended at length by Aquinas in the course of his contrast between Art (in the broad medieval sense) and Morality ('prudence'), is beautifully connected with the universals question and with considerations about the nature of creation as such. Then suddenly, in chapter 13, we find ourselves faced with the discussion of a precise anticipation of Descartes' thesis of constant re-creation at every new instant. 6.16 Such gesticulations in the directions exemplified by the astonishing variety of competently treated topics could be continued indefinitely, and many others can still easily be tied up with other medieval or modern material. However, there is at least one topic which Professor Spade, in his introduction to WUE, rightly finds immensely puzzling, and which marks a point at which (perhaps for theological reasons) Wyclif's discourse becomes venturesome in a way which stretches intelligibility to the limit. This is the discourse concerning the 'aggregate man' (Spade's topic (7) to which allusion is made on p. xlvi of his Introduction, and to which 385

Section 6 may be added the case of the 'aggregate chalice' shortly to be mentioned). This Introduction suggests (in its note 102) Goodman's 'calculus of individuals' as a present-day theory covering such an area. For our part, it is still to be the elementary end of Lesniewski's mereology which will serve both to situate many of Wyclif's discussions within the ambit of the medieval distinction between distributive and collective wholes, as broached in 1.22 above, and at the same time to highlight the paradoxical nature of some others. Thus from present-day mereology one cannot derive the thesis that the integral whole of (Eucharistic) chalices is a chalice, or that the integral whole of the men is a man. It would generally be grossly pointless and misleading to adopt such theorems. Yet Wyclif is to countenance both, as we are to see in detail below. 6.17 Whatever one may make of such theses from a theological point of view, and although one may deprecate the misleading new uses to which old nouns are here being put, their extensions in chapter 11 show that Wyclif is here venturing upon pionecr investigations into the actions and passions of objects such as the collective class of men. There is obviously such a class; it may well be misleading to call it a man, but there is no doubt that we and our members are parts of it. Attempts have even been made to depict it, or its parts, as in the misleading title-page to the first edition of Hobbes' Leviathan, but how to bring literal discourse concerning this object's doings and undergoings under systematic control is, to say the least, a novel exercise. We are to witness Wyclif undertaking exactly that exercise, nevertheless, in the manner detailed in 6.24 below. Even the enlargements on these mereological points which are to be provided below are but the merest preliminary approximations to any tenative appreciation of his work in this field. 6.18 Prior to those appreciations, there is a translational point affecting WUE which necds to be clarified, as it conn386

Wyclif's Deviant Mereology ects with that fundamental distinction between quidditative and nominally-termed discourse central to our logical grammar (0.4 and HQS §4.0). That grammar and its ensuing analyses assume that intellectus, understanding, and significatio, are quidditative matters (0.4, 10.26, and HQS §4.4) so that the Latin linguistic cognates of intellectus must all be seen as hanging together, so to speak. However, in WUE, the network of linguistic interconnection which holds constantly between intellectus, intelligere, and their related parts of speech throughout the Latin of WUL is deliberately broken to an extent which goes beyond mere stylistic variation where the translation in WUE is concerned. 'Intellectus' is rendered as 'intellect' or 'understanding', whereas 'intelligere' becomes 'to think of', or 'to have in mind' (and so on for other expressions derived from the verb) but never 'to understand' (and so on). Thus WUL 6 4 - 5 expresses some comparatively prosaic and evidently acceptable theses on the relation between intellectus, intelligibilitas, and intellectio, whereas the English (WUE 16) appears to propound some suspect connections between the intellect, the thinkable, thinkability, and actual thinking, the three last of which are not at all identical with the intelligible, intelligibility, and actual understanding respectively. Similar opacities attributed to Wyclif by the translation have this same divide as their basis. For example, could he really want to say (WUE p. 169, 1.129) that 'neither the Creator nor the creature can think of something which cannot be'? This creature who is writing these words thinks of such things quite often, and so God can certainly do so. Surely enough, consultation of the Latin (WUL 358) shows that all that is in question here is a trite alignment between the unintelligible and the impossible. Needlessly odd-sounding locutions also depend on the same policy, e.g. 'what Aristotle has in mind are extramental realities' (WUE p. 11). Any future researcher wishing to enlarge on the contrast between integral and universal wholes as they figure in this 387

Section 6 work of Wyclif's should bear these translation-conventions in mind when using the English of WUE at the points in question.

6.2 Parallels and Innovations 6.21 Except in those cases involving so-called 'mass' terms such as 'water', and which betoken homogeneous wholes (1.32), we have not yet encountered logicians who have been tempted to hold that the integral whole of the so-and-so's is itself a like so-and-so. However, in WUL, as already hinted, Wyclif does employ provocative instances of this unusual thesis. What we have encountered, however, in 2.7 above, are versions of certain of the paradoxes which Wyclif associates with that thesis; these medieval antecedents occurred in the pseudo-Abelardian Sententie Magistri Petri (AT), as we are shortly to recall. 6.22 Wyclif's status as a distinguished scholar and academic make it impossible for him to have been ignorant of the generally quite acceptable medieval tradition on integral wholes which we have hitherto recorded. Indeed, in WUL the customary distinction between the integral and the universal (or distributive) whole is quite properly traced back to Boethius (B 887D; cf. 1.22, 1.311(A)) and a thesis which tends to run counter to the provocative one mentioned just now is adduced in support: (A) And this is what Boethius asserts in his De Divisione when he separates out division of the meanings of a term from division of the integral whole into its integral parts, and these in turn from division of the universal whole into its individual exemplifiers. He also asserts that the second of these types of division differs from the third in that the integral whole is not any of the integral parts of itself, whereas the universal whole is any one of its parts. (Et hoc est quod dicit Boethius in lihrum Divisionum ubi ponit quod alia est divisio qua terminus dividitur in sua signif388

Wyclif's Deviant Mereology icata, alia divisio qua totum integrale dividitur in suas partes integrales, et alia divisio qua totum universale dividitur in suas partes subiectivas. Et in hoc, inquit, differt divisio media a tertia quod totum integrale non est aliqua sui pars integralis, sed totum universale est quaelibet sui pars: WUL ch. 4, lines 70 - 7.) Thus, according to this common doctrine, as we have it in 1.32, 2.21, and elsewhere (cf. 10.324) no proper part of a house is to be identified as a house, whereas every 'subjective1 part of the universal or distributive class of men may be identified as a man. Here the house is the common medieval example of an integral whole, the proper parts of which are walls, roof, and foundation. 6.221 Now Wyclif is not only to deviate from this common doctrine; he is also, and simultaneously, to embroil himself in an argument running parallel to the following one, already narrated and analysed in 2.7 above, which bases itself on the Abelardian principle that every many makes a whole: (A) Not only this, but another objection appears to be possible. For suppose we assert: Thing is made up from thing, and Every many makes a whole, then it certainly follows from these that: Whenever two objects are established, then those two make up one whole which is diverse from its individual components. Under these circumstances, wherever there are two, there are three. This three is made up from the two in question and from the whole which is composed from those two. This whole is diverse from each individual component, and hence, whenever there are two, there are three, i.e. the two in question and the whole of them, which is not only diverse from each individual component, but also susceptible of diverse predicates, since neither the one nor the other of the parts is identical with that whole. But yet again, as every multiplicity makes a whole diverse from its individual parts, the whole just discussed will make one whole along with its two parts, and this fur389

Section 6 ther whole yet again with its parts, and so on to infinity. (Sed sic et istud obiciendum videtur: quod si rem ex re constare dicamus et omnia plura unum toturn efficere, u tique, ubicumque erunt duo instituta, ipsa quippe duo unum totum facient quod est diversum a singulis Ulis; et ita ubicumque sunt duo, sunt tria (ipsa videlicet duo et totum ipsorum, quod est diversum a singulis Ulis et predicatione remotum, cum neque hec pars sit ipsum totum neque illa); sed rursus, cum omnia plura unum totum efficiant diversum a singulis illis, ipsum totum iterum cum duabus partibus suis unum totum efficiet, et illud rursus aliud cum partibus suis; et sic in infinitum: AI 114.13-23; cf. 2.7 above for further discussion.) 6.23 It is in his analogue of this already venturesome enough argument that Wyclif does not confine himself to the consideration of such very general objects as the duos and trios of this passage, but extends the same point to those aggregate individuals of his, such as the aggregate men or aggregate persons which exemplify his deviance. For him any collection of men, including the complete collection, is a man. Such oddly-named aggregate individuals are often met in De Univeralibus, as in the example of the complete collection mooted in chapter 7: 'All the individuals of the species man make up one aggregate man, and it is thus that the species which is communicated to the individual exemplifiers is distinguished from that lot', Omnia individua hominis sunt unus homo aggregatus, et ita species communicata suppositis distinguitur ab eisdem: WUL ch. 7, lines 353 - 5. As this sentence shows, the aggregate man is categorially distinct from the distributed or 'communicated' species, and hence cannot but constitute the collective whole. It is the following objection, dialectically raised by Wyclif against the use of such aggregates, which contains the analogue of the argument from the Sententie quoted in 6.221: (A) First, as regards the singulars, which are what come 390

Wyclif's Deviant Mereology first in our awareness: it would seem, from what has been said about the aggregate man, that given the existence of two or three men, infinitely many are thereby posited. For given Peter and Paul, one is given these two and a third man of the aggregate sort. Then in the same way as these three are given, there is also to be granted a fourth aggregate from these, and so on to infinity. Nor can any reason be adduced as to why, having obtained the aggregated third from the other two, one should not by the same process obtain an aggregated fourth from any three. Nor can any limit be assigned to the process, for let it be the case that the series of men has attained the number A, there still results from all these men a man additional to all the men making up that number, and so on without limit. (Et primo de singularibus quae praecedunt in nostra notitia. Videtur enim, iuxta dicta de homine aggregate, quod positis duobus vel tribus hominis, eo ipso ponuntur infiniti, nam positis Petro et Paulo ponuntur illi duo et homo tertius aggregates, et eo ipso quo ponuntur illi tres, ponitur quartes aggregates ex Ulis, et sic in infinitum. Nec dicetur ratio quare ex duobus resultat tertius aggregates, quin per idem ex quibuscumque tribus resultat quartes aggregates, nec est status signandus, quia esto quod ad A numerum hominum consequentia ex omnibus illis resultat homo praeter omnem hominem de Ulo numero. Et sic sine fine: WUL ch. 9, lines 6 - 1 8 . ) 6.231 Now while contemporary mereology might, with suitable corrections, and notwithstanding specious worries about typefallacies, be capable of making something of the progress towards the indefinitely many objects generated in the pseudo-Abelardian passage requoted above, the corresponding infinity of men generated by this last objection is plainly suspect in most of its details, especially as regards the notion of aggregate men. (These, incidentally, are not to be confused with the extra 'men' generated by a 'third man' type of contention which can be directed against that version of 391

Section 6 the Platonic theory of universals according to which the latter are paradigms of their exemplars.) Yet Wyclif only slightly resists the terms of this objection in the subsequent discussion, and by a further singular coincidence produces the analogue of yet another argument from the same earlier pseudo-Abelard, as we are to see in due course. In the meantime, the following is the quite moderate limitation of the threatened regress, as propounded by Wyclif: (A) I assert ... that given Peter and Paul as two elementary men, there thence ensue three men, but no more. Thus it is that the disreputable number two, the first to lapse from onehood, is in this respect the most unproductive of numbers. For the positing of three elementary persons pertaining to the human species does amount to the positing of seven men, and so on increasingly ... In response, therefore, to the argument based around the three men, namely, Peter, Paul, and the aggregate man, I assert that there is no succession of men beyond that which is thus achieved ... And should the reason for the difference between the cases be demanded by asking why (given that from Peter and Paul one obtains also a third but nevertheless aggregate man) one does not by the same sort of process, on the basis of these three, obtain a fourth aggregate man, then the reply is that every multiplicity of individual human beings presupposes elementary men, who are the bases of such multiplicity. Hence, as the fourth made-up man would not be an elementary man, since it is given that only one of its two components would involve elementary men, so also neither would it be an aggregate man, because an aggregate man only emerges from the original couple. Hence one cannot make out how a fourth human person would be constituted. (Dico ... quod positis Petro et Paulo, duobus simplicibus hominibus, ponuntur eo ipso tres homines, sed non plures. Ideo binarius infamis, primo recedens ab unitate, est maxime sterilis inter numeros quoad illud, cum positis tribus personis simplicibus speciei 392

Wyclif's Deviant Mereology humanae ponuntur septem homines, et ita crescendo ... Respondendo itaque ad argumentum factum de tribus hominibus, scilicet Petro et Paul et homine aggregate, dico quod non est processus hominum ulterior quoad illud ... Et si quaeratur causa diversitatis: quare, positis Petro et Paul, ponitur tertius homo aggregatus, quin per idem, positis aliis tribus, ponitur quartus aggregatus? dicitur quod hoc ideo quia omnis singularis multitudo hominum praesupponit homines simplices, radices talis multitudinis. Ideo, cum quartus homo fictus non foret homo simplex quia tantum alter istorum ponitur homo simplex, nec foret homo aggregatus quia tantum ille binarius est homo aggregatus, ideo non est fingendum quomodo foret quarta persona hominis: WUL ch, 9, lines 59 - 64, 66 68, 76 - 85; cf. CC 58, p. xi, note 2, on 'the disreputable number two' herein mentioned.) 6.232 Up to this juncture the point appears to be that aggregate men must have an immediate basis in the counting of individual human beings, as opposed to counting some mixture of individuals and aggregates. Indeed, providing that a slight ambiguity of sense, shortly to be mentioned, is duly settled, it looks very much as though Wyclif's aggregate man could be depicted simply as combinatorial man: quot combinationes, tot homines seems to be the rule in their case. This seems to be confirmed by the following continuation of the last passage: (A) In contrast, it is agreed that given Peter, Paul, and Linus, one couple is composed from a count of Peter and Paul, another from Peter and Linus, and a third from Paul and Linus, although they have in common the fact that each of these pairs can subsist without any of the others, and it is also agreed that the trio composed from taking these three [elementary men] together is in turn distinct from any such pair. This gives us three elementary men and four aggregate men, because there are four distinct numerical combinations of the [elementary] men. However, in the first example one 393

Section 6 obtains no numerical combination of men other than the aforesaid couple, as Peter along with Paul and Peter are just Peter and Paul, since no elementary man is part of one numerical group who is not part of the other. Hence even as Peter and Paul are the same as Paul and Peter, so also are Peter and Paul the same as Peter along with Paul and Peter, no matter how one may run through the numeration of the same men, and from the same [two] men one does not obtain a multiplicity of aggregate persons. (Sed constat quod positis Petro, Paulo, et Lino, alius binarius est numerus ex Petro et Paulo, alius autem ex Petro et Lino, et tertius ex Paulo et Lino, licet communicent intantum quod singulus istorum binariorum posset esse sine reliquo, et constat quod ternarius ex istis tribus simul distinguitur a quolibet tali binario. Et sic habemus tres homines simplices et quattuor aggregatos, quia quattuor distinctos numeros hominum. Sed in priori casu non est dari numerum hominum nisi dictum binarium, cum Petrus, Paulus, et Petrus sunt praecise Petrus et Paulus eo quod nullus simplex homo est pars unius numeri qui non est pars alterius. Ideo, sicut idem est Petrus et Paulus et Paulus ac Petrus, sic idem est Petrus et Paulus et Petrus, Paulus ac Petrus quomodocumque quis calculaverit eosdem homines. Et de eisdem hominibus non est multitudo personarum aggregatarum: WUL ch. 9, lines 86 - 102.) 6.2321 (A short note on the translation provided is relevant to the point now at issue. From the doctrine thus far expounded, it seems evident that 'ex istis tribus', 'from taking these three', in the first sentence cannot possibly be an allusion to the three couples, since this would not only open the way to a huge regress generating many more men than the seven Wyclif admits, but would also violate the principle just laid down to the effect that aggregate men are constructed directly from individuals. Hence the words 'elementary men' have been interposed to make this clear. One may also suspect that 'de eiusdem hominibus', 'from the same men', towards 394

Wyclif's Deviant Mereology the end of the passage must refer to the same two men, and an interpolation has accordingly been suggested to avoid the impression that from any group of elementary men one gets only one aggregate man. The whole example of the three elementary men militates against any such impression.) 6.233 By now Wyclif's notion of an aggregate man is thus somewhat clarified in so far as it is confirmed that only elementary men can constitute an aggregate man. Composites overtly including aggregates as their parts do not count as aggregate men, or so it would seem. The whole exercise is odd in its employment of old words such as 'man' in a new way, the point of which is not yet evident. Yet the mereological profit is not totally zero, as may be confirmed by a brief discussion of the generalised form of a kindred situation which figures in yet another passage from the pseudoAbelardian Sententie whence a quotation has been drawn in 6.221 above. Here, instead of Wyclif's Peter, Paul, and Linus, we have the trio of objects from which two at a time are selected to form non-identical pairs. The fact that only two such pairs are mentioned in the pseudo-Abelard, rather than the three which are possible and which are detailed by Wyclif in the last-given citation from De Universalibus, scarcely affects the mereological significance of both the kindred cases, which is as follows. An outstanding feature of mereological objects, already exemplified in the above-quoted Sententie extract and its Wycliffian parallel, is that they may have parts in common, i.e. they may be non-discrete or overlapping, and to that extent may be in the same place at the same time. Both the men aggregated from Wyclif's pairs of men who are members of a human trio, and the pairs selectable from the general trio of objects now to be recalled, have precisely this property of being each distinct from the others, as is stressed, and yet being non-discrete. It is worth repeating here that this notion of non-discrete singulars, or individuals, very occasionally considered by our own 395

Section 6 contemporaries (cf. HQS 257 - 8) was even more rarely made explicit by medieval philosophers because of the simple linguistic fact that one of their words for individual was the Latin counterpart of the word 'discrete'. Hence talk of a non-discrete individual would sound quite incoherent. Yet here is the Sententie's correspondingly highly significant and wholly overt description of just that possibility: (A) Besides, we assert that this trio is made up of these three [discrete] elements, and any two of those elements yield a pair. Who then can deny that this trio is made up of [at least] two pairs? For indeed the common element along with this element makes up one pair [and with that other element another pair], so that the trio contains within it [at least] two non-identical pairs, as this pair is not that pair, since the latter has a proper part which the former lacks. (For if this pair's make-up involves this element, and that pair's make-up does not involve this same element, then by a necessary logical consequence it is obvious that that pair is not this pair). (Preterea hunc ternariurn dicimus constare ex his tribus unitatibus, et due unitates unum binarium quelibet reddun t. Quis neget hunc ternarium ex duobus binariis constare? Eadem quippe unitas cum hac unitate unum binarium <et cum illa alium binarium> facit; et ita ternarius duos in se continet binarios diversos ab invicem, cum hic binarius nullo modo sit ille, cum talem habet partem quam ille non habet. (Si enim hic binarius hac unitate constat, ille vero hac eadem non constat. profecto ille hic non esse convincitur per necessarium sillogismi conplexionem): AI 114.23 - 115.5.) 6.234 The exceedingly advanced notion of part and whole here presupposed, and which must also be credited to Wyclif's parallel contention, has been the subject of comment in 1.5 and HQS 240 - 2. Leibniz later systematised the theory of such combinatorial objects: LPP 77 - 8. The reply provided by the pseudo-Abelardian Sententie (quoted in 2.781 above) makes 396

Wyclif's Deviant Mereology observations which tend precisely to stress the non-discreteness of the pairs. This it does by reminders of the nondiversity of the pairs. Such reminders are most useful in themselves, but unfortunately are there used in a misguided fashion so as to convey the implication that one cannot enumerate non-discrete or non-diverse objects, which is, of course, quite false: cf. 1.5. 6.235 To return, however, to Wyclif and his strangelynamed aggregate 'men': he persists in using them as topics of discourse throughout the De Universalibus. Thus we have, 'all men collectively are one person, as also are any two, however selected' (WUL ch. 8, lines 202 - 3), 'the aggregate person is not any one of its integral parts ... but is an aggregate formed from all of them, and this is truly an individual substance of a rational nature' (ibid. lines 344 - 7). Wyclif appropriately extends the distinction between the application of predicates to the collective whole, on the one hand, and to the shared species, on the other: 'It is truly concluded that apart from the collective capacity for laughter, one must also grant the human species which is the subject of a shared capacity for laughter appropriate to that species' (WUL ch. 11, lines 540 - 3). The admission of this collective capacity naturally leads on to the questions of collective actions in general, and of actions performed by collective objects such as his 'men' in particular. It is here that Wyclif once again tends to take a rather unusual tack, for although, for example, there is no difficulty in regarding the ordinary human being as the collective class of his spatial and temporal parts, and although the doings and undergoings of those parts, or of the whole may (subject to appropriate rules) be at times attributed to the whole (cf. 3.5), the case is more complex where his aggregate men are concerned. 6.24 The difficulty of attributing actions and undergoings to such a 'collective whole person of men' is strongly suggested as an objection in chapter 11, lines 24 - 29, p. 239; 397

Section 6 cf. WUE p. 107. Approaches to a reply extend from lines 203 onwards throughout the same chapter, and are sufficiently enmeshed with non-mereological material to demand separate treatment of a sort which cannot be pursued here. However, the following extracts may convey something of the flavour of Wyclif's endeavours in this direction: (A) Likewise it is asserted that the species is formally described in whatsoever way some individual within it is positively described, not in so far as it is the species [which is being described] but rather in so far as the individuals which are its [integral] parts are concerned. Thus the human species is begotten in this place, and is subject to perishing in this other place; it laughs here, and laments there, not in so far as it is a species, but rather in its elements. (Correspondenter dicitur species denominari formaliter quomodocumque denominatur positive aliquod eius individuum, non secundum quod species, sed secundum eius individua quae sunt eius partes. Ut species humana hic generatur, ibi corrumpitur, hic ridet, ibi dolet, non secundum quod species, sed in suis suppositis: WUL ch. 11, lines 349 55.) Here the extensionally-slanted view which leads to the aggregate man is at least in evidence. Already it could be that the aliquod eius individuum, 'some individual within it' permits interpretation as an allusion to an aggregate individual. For the next sentence, shortly to be quoted, claims to be connected with the foregoing remarks, and describes them as alluding to idem individuum, which may incorporate a reference not only to 'the same individual' (i.e. the self-identical individual) but alternatively to the same (i.e. aggregate) individual as was mentioned in the just-quoted passage; this is the aliquod eius individuum, which is now to be definitely identified with the individual susceptible of all the apparently troubling contrary characterisations. This cannot but be an aggregate (i.e. mereological) whole. Whatever the 398

Wyclif's Deviant Mereology soundness of this conjecture, it is at least wholly certain that the next few sentences confirm that we are here concerned with an individual object having quantitative parts (i.e. a form of mereological whole) which is contrasted with the species and its distributive (or 'subjective') parts. This is the contrast described in Wyclif's own allusions to De Divisione quoted above (6.22). Subjective parts are such that each of them bears the species-name (e.g. every subjective part of the species donkey is a donkey). The implied contrast is with the aggregate singular object or mereological whole, the quantitative parts of which (as has been noted above) necd not bear the species-name: (B) Nor is there any weight in the accordingly connected counter-objection: if on a given temporal occasion the individual now in question is susceptible of contrary characterisations in respect of its various quantitative parts, how does it come about that the species should not take on contrary characterisations in respect of its various subjective parts? Thus would emerge the absurdity that an identical thing on a given temporal occasion, and in an identical respect, would be characterised by opposite forms. (Nec obicitur efficaciter in consequentia instantia contra illud, quia: si idem individuum potest simul et semel denominari denominationibus contrariis secundum partes quantitativas diversas, quomodo non sic species reciperet denominationes contrarias secundum partes subiectivas diversas? Inconveniens itaque foret quod idem simul et semel, secundum idem, denominaretur formis contrariis: WUL ch. 11, lines 356 - 63; compare Aquinas in ADA art. 10, arg. 12.) 6.25 It is in the immediately following statements that Wyclif's full defence of the attribution of actions to an aggregate man is exemplified, both as regards its pionecring merits and its evidently accompanying linguistic strains. In the former respect there is no question that there is such a thing as the collective class, or aggregate or integral whole 399

Section 6 of mankind; each of us is a part of it, and so are our parts either singly or in any sort of combination (e.g. Socrates'left-eye-in-450-B.C.-and-my-trachea-in-June-1944). However, given such an unlimited and vast miscellany of the sorts (i.e. species) of objects then in question qua parts, occasions for the production of sortally-related accounts of the doings and undergoings of some of those objects will tend to be rather rare, unique, or non-existent, if only owing to such objects' having no species-name. Wyclif evidently thinks that he has some such occasion, and defends the ensuing sort of discourse: (A) We are in fact not committed to any unnatural prodigy by these assertions, for a collective man, such as a populace, does have so many heads and other members as would indeed be unnatural in a normal person. Such a collective man acts and undergoes actions quite properly relatively to what is suitably attributable to a person of this aggregate sort. Nor is it the case that the head of a populace is a normal head; rather it is an aggregate head, but still a head of the people. And so it goes correspondingly in respect of all the other organs; they are attributed to the populace by means of appropriate sortally-related characteristics. As regards those things which do not carry with them such a relation (as when the goings-on of just any part of the populace are attributed to the populace) they are still rightly attributed to the populace, but only in respect of the part in question. These senses are not incompatible, although the analogies they involve are pretty heavy going. (Nec sequitur monstruositas ex his dictis, nam homo collectus ut populus habet quotlibet capita et alia membra quae forent nimis monstruosa in persona simplici, et agit et patit decenter secundum quod convenit personae huiusmodi aggregatae. Nec est caput simplex caput populi, licet illud caput sit populi, sed caput aggregatum. Et correspondenter de aliis organis tri buendum est populo in denominatione speciali. IIIa autem, quae non connotant relationem talem, sunt attribuenda populo, ut oper400

Wyclif's Deviant Mereology atio cuiuslibet partis populi attribuitur populo, sed secundum partem. Nec contradicunt istae sententiae, sed laborant in aequivocis: WUL ch. 11, lines 364 - 75.) From this it would appear that there are, as it were, two levels of attribution in respect of aggregates of men, at least. There is that level of discourse which tends to follow the rules appropriate for dealing with specifically denominated parts (e.g. arm, head). There is also that level which relates to the actions of individual quantitative parts, and here, presumably, almost anything can happen. When I stumble over Socrates' foot I can, in respect of that part of the populace, be said to have stumbled over the populace; presumably the populace may also be said to have stumbled over part of itself, since I too am part of the populace. It is here that the linguistic going is becoming really heavy, as Wyclif remarks: laborant in aequivocis! 6.26 Thus without in any way condoning Wyclif's new uses for old words (e.g. 'man', 'head', and so on) we may at least begin to find them to some degree intelligible. Various further questions still subsist or newly intrude. They fall roughly into three sorts. First, do these new uses apply only in organic or personal cases? Given that we now are to hold that an aggregate man is a man, can this rule be extended without limit? Is an aggregate of beds a bed? Secondly, may some indication be given of simple cases in which desirable theological repercussions are the grounds for these linguistic manoevres? Thirdly, as an Oxford don Wyclif must surely have been aware of the many previous attempts by well-known philosophers such as Peter of Spain and Duns Scotus to make more precise the conditions under which the description of the part may be extended to the whole. Indeed, as we have seen in 5.3, this sort of discussion had been common medieval coin at least since the time of the recovery of Aristotle's De Sophisticis Elenchis, and this under the heading of the fallacy of argumentation from an attribution made in a certain respect 401

Section 6 (secundum quid) to one made simpliciter, i.e. without qualification, irrespectively. How does Wyclif's present position relate to this earlier work? 6.261 While this is scarcely the place in which to pursue the third of these queries, the first two may each receive an initial simple exemplary answer by adducing suitably elementary cases. Thus as far as theological relevance is concerned, there is that gospel story of the good Samaritan who (as in the stained glass of Chartres and elsewhere) typifies the Saviour, with the result that the traveller to whom the Samaritan ministers must be seen as the aggregate of the saved. This traveller (who in the story is going down from Jerusalem to Jericho) is used by Wyclif to illustrate one of the senses, the mereological sense in fact, which (he claims) may result from the application of restrictive functorial signs (e.g. of 'a certain ...') to common nouns: (A) This type of restriction is used by the uncreated Wisdom in chapter 10 of St. Luke's gospel, where it is related, 'A certain man was going down from Jerusalem to Jericho'. This man is that single man aggregated from the whole number of those who are to be saved and who, even as he went in a downwards direction because of our first and carnal parents, so also alone went by a higher way owing to that unique spiritual parent and head of the Church, who is Christ. (Et tali modo limitandi utitur Sapientia increata dicens Lucae 10: 'Homo quidam descendebat ab Jerusalem in Iericho'. Qui est ille unus homo aggregatus ex toto numero salvandorum, qui, sicut descendit in primis parentibus carnalibus, sic ille solus ascend it in unico parente spirituali et capite Ecclesiae qui est Christus: WUL ch. 8, line 5 0 - 6 . ) 6.262 The next example again illustrates a definite theological point, and at the same time (in response to another of our queries) suggests that where theological considerations make it desirable, it is indeed the case that inorganic objects may follow the deviant rule already observed in the 402

Wyclif's Deviant Mereology case of man, in that the aggregate of eucharistic chalices is itself a chalice. Here, once again, one may speculate as to the detail of the theological point; is its aim to preserve Christ's rôle as the unique high priest? The passage occurs in a discussion of the possible referents of a demonstrative pronoun such as 'this': (A) Hence there is no doubt that Holy Writ affords grounds for holding that demonstratives may perform their function in respect of either individual objects or collective objects. The church would not assert, on the authority of Christ, 'taking this excellent chalice into his holy and venerable hands, he blessed it' unless the pronoun ['this'] alluded to the whole collection of chalices within which the blood of the Lord is consecrated. It was that aggregate chalice which Christ handled by means of that part used in the service of those who were present at the Last Supper, and all those chalices constitute one chalice in respect of that office. (Ideo indubie tenendum est ex fide scripturae sacrae quod potest esse demonstratio singularis de demonstratio generalis. Aliter anim non diceret Ecclesia auctoritate Christi quod 'accipiens hunc praeclarum calicem in sanctas et venerahiles manus suas henedixit' nisi nomen demonstret totum genus calicum, in quibus sanguis Domini consecratur. Illum autem calicem aggregatum sumpsit Christus in manus suas secundum partem illam ministrantem eius in cena domini, et omnes illi calices sunt unus calix quoad illud officium: WUL ch. 7, line 256 - 65.) One may easily see how a whole mereological theory of the Eucharist could be constructed along these lines, even without a Wycliffian type of expansion of the sense of 'chalice' (or of 'bread'). 6.27 For the record, and to terminate this brief preliminary survey of Wyclif's mentions of aggregate objects in De Universalibus, a few other citations may be adduced. Thus we have aggregate persons: 403

Section 6 (A) It is however granted that two, three, four, and, in brief, all men, are each in respect of these various numerical groups, one person. This is because the authority of holy men and the assertion made elsewhere, but here presupposed, support the thesis that by sharing in the species all men collectively constitute one aggregate human person. The same applies also to every pair of men, every trio of men, and so on for no matter which numerical group of men. (Conceditur tarnen quod duo homines, tres, quattuor, et breviter omnes homines, sunt singulus is torum numerorum una persona, nam iuxta sanctos et declarationem factam alibi quae hic supponitur, participatione speciei omnes homines collectim sunt una persona hominis aggregata, et, per idem, omnes duo homines et sic de tribus et quolibet numero hominum: WUL ch. 8, lines 174 - 80.) In this same chapter 8 of WUL the appropriate collective bulk of the aggregate human race is mentioned: homo magnus, qui est totum genus humanum (lines 379 - 80). More discussion of the logic of the collective man then follows (lines 380 96). The next chapter, from which extracts have been given above, pursues the question of regresses generated by the thesis of aggregation. It seems to be hinted that mass terms (e.g. the four element-names) and continuants, may give rise to infinite collections (ch. 9, lines 216 - 24). Then the very large mereological object, the collective class of every created being, becomes the topic of a discussion starting out from generally admitted collectives, such as a people (lines 255 66). Here also is a hint that the fall of mankind in Adam may be viewed mereologically (lines 233 - 54). We have already seen something of the collective actions broached in chapter 11 (lines 348 - 426, 454 - 60, 540 - 52). Collective classes and the Trinity figure in lines 611 - 21 of the same chapter. The comparative 'perfections' of the individual man and the collective 'man' surface in chapter 12 (lines 115 32). The composition of objects from object-parts as their 404

Wyclif's Deviant Mereology 'matter' is contrasted with creation, using the familiar mereological house (lines 415 - 35, cf. 569 - 79 and section 3.31 above). This by no means necessarily exhaustive compilation of references at least underlines Wyclif's continued attention to this aspect of metaphysics. 6.28 Clearly, a more complete estimation of Wyclif's reasons for this concern with the collective still remains to be worked out. Previous discussions of the already-mentioned secundum quid et simpliciter inferences also necd to be brought to bear (cf. 5.3). As Gyula Klima has recently reminded us, the duality of Christ's nature evoked much subtle discussion concerning such inferences, which may have mereological overtones (KLS). However, these are all questions which offer a hope of maintaining intelligible contact with Wyclif's modes of discourse. His endowing of a collective object with the same name as some one or other of its parts (where mass-terms are not in question) appears to be rather more dubious. His Purgans errores circa univers alia in communi, which appears in volume 33 of his Latin works, and to which allusion is suggested in note 65 of chapter 9 in WUL, scarcely appears to help in this connection. Nevertheless, his collective men are certainly human beings; thus the present-day use of the latter description is once again put out of court as a neutral term to cover just male and female homines: cf. 1.231, 3.439.

405

7. Categorematic and Syncategorematic 7.01 Transitional Introduction 7.011 Thus far the historical location of the medieval m e r e ology has followed a more or less serial temporal order from Boethius through to John Wyclif's lively contribution. Throughout, it has been possible to progressively explain the theoretical relevance of the material in a reasonably continuous sort of way. Now, however, there is to be undertaken, after this present introductory piece, a move which, from a temporal point of view, is a sort of epicycle - a looping backwards in time so as to r e v e r t to the contemplation of a line of thought the initiation of which was observed in the writings of Abelard (2.212 above). This thread had continued to become quite prominent early in the thirteenth century (i.e. even before the Aquinate works touched on in section 3 above) and thus ran concurrently with those works and all that succeeded them, finally to become mereologically dominant up to and beyond the treatises of Paul of Venice, which are to be scrutinised in section 8. 7.02 Initially it is possible to view both types of t r e a t m e n t in the work of the same author, namely Walter Burleigh (1275 - 1344, or thereabouts), Material of a sort which enmeshes with what has hitherto been observed as the dominant strain is to be found in Shapiro and Scott's edition 406

Categorematic and Syncategorematic of his remarkable De Toto et Parte. True, interesting and novel features show up in this text, but none of them brings it into the further tradition with which we are, after this introduction, to be predominantly concerned. The work begins with what, given the usual correlation of essence with form, looks initially like a pretty sweeping alignment of the integral whole with matter, so that the contrasting essential or formal aspects which the text mentions should then involve, as previously observed in 0.5 and 3.31, segments of definitional statement, i.e. this would then be a purely quidditative business. We have already noted the trouble with such an alignment of integral parts and wholes with matter in the sense of material make-up: it tends to blur the distinction between X-parts and parts-of-X (3.31). However, the position first to be described could still be possible and not hitherto altogether unheard-of. Here is the text's opening which apparently outlines it: (A) It should be understood that whole and part are taken in many senses. Part is sometimes taken as the essential part which Aristotle and Averroes call the 'qualitative part' in many contexts, and they call the integral part the 'quantitative part'. In a corresponding manner, whole is taken as whole in respect of form and whole in respect of matter, and contrastingly for part there is part in respect of form and part in respect of matter. (Intelligendum quod totum et partes multipliciter capiuntur. Pars enim aliquando capitur pro parte essentiali quam Philosophus et Commentator multis locis vocant partem qualitativam et partem integralem vocant partem quantitativ am. Et isto modo accipitur totum pro toto secundum formam et toto secundum materiam; et per oppositum, pars, pars secundum formam et pars secundum materiam: SST 300.) 7.03 However, instead of the expected relegation of the formal to the quidditative, it now becomes, quite unprecedentedly as far as we are concerned, identified with what has 407

Section 7 hitherto been familiar as the principal part, i.e. a definitely mereological feature of reality. Hence it appears that the 'isto modo' ('in a corresponding manner') of the last-quoted text is designed to steer the discourse in the direction of allusions to the integral and quantitative part (as contrasted with the essential (or quidditative) and qualitative part). The above neutral translation should therefore become rather the fully committed 'in this manner', with the 'this' pointing to the integral and quantitative whole which has been the last-mentioned item at that point. Accordingly, the examples now given to illustrate parts in respect of form and material parts, show that these are both being taken as subdivisions of the integral whole, with the former being the principal, and hence invariable, parts, and the latter the changeable parts: (A) Parts in respect of form are those which always remain the same, as long as the whole remains the same complete object. Material parts are those which ebb and flow. In illustration of the former, the hand, head, and suchlike always remain the same as long as the whole remains the same. In illustration of the latter, the flesh and marrow ebb and flow while the whole remains the same. And even as part is talked about in two ways, i.e. in respect of matter and in respect of form, so also whole is talked about in two ways, namely in respect of form and in respect of matter. (Partes secundum formam sunt quae semper manent eedem, dum totum manet idem et perfectum. Partes materiales sunt que fluunt et refluunt. Exemplum primi: manus, caput, et huiusmodi semper manent eedem, dum totum manet idem. Exemplum secundi: ut caro, medulla fluunt et refluunt toto manente eodem. Et sicut pars dicitur dupliciter, scilicet secundum materiam et secundum formam, sic totum dicitur dupliciter, scilicet, secundum formam et secundum materiam: SST 301.) A weakness in the mode of expression here employed is evident in the description of the parts in respect of form as those 408

C a t e g o r e m a t i c and Syncategorematic which remain the same as long as the whole remains the same, with the extension of this description into the examples. Obviously even the material parts also remain the same as long as the whole remains the same (in a familiar sense of 'same'), so this description tends t o be rather inept as it stands. Indeed, it now becomes possible to suspect t h a t the author of this t e x t is becoming the victim of a most serious and unfortunate confusion. He has just talked of formal and material parts as changeable and unchangeable concrete (and hence mereological) parts respectively. Although this vocabulary is in itself unhelpful, since really principal and nonprincipal parts are in question, and can easily be referred to as such, there still necd be no objection to it in principle, provided the conventions are made clear by means of examples such as the ones provided up to the present point. Yet in its continuation, next to be quoted, the t e x t takes up afresh the normal medieval use of form as t h e principle of sortal identity, and this as a quidditative and not a concrete, integral, factor. The confusion becomes worse confounded by the connection of this standard application of the notion of form with the author's foregoing deviant non-quidditative application, a connection effectuated by means of the connective 'unde ('whence'). This gives a most misleading impression of a continuity which is just not there. In fact, far from the formal and material continuing to be subdivisions of the integral, as they initially were, they now become embedded in what would otherwise be a most helpful exposé of the rôle of Aristotelian form as the index of identity of kind across temporally located material increases and decreases. (Thus Abelard might have found this particular exposé a useful suggestion where his increasing and decreasing houses, heaps, and persons were concerned: cf. 2.5 above). (B) Whence a man in youth and old age is the same as far as his form is concerned, and has altogether the same [form-] soul, but is not the same in respect of matter, since 409

Section 7 he has one matter at one age and another at another, for the matter of the food converted into the nature of the thing increases the matter. Relatively to that which increases and decreases, new matter is continually being acquired and the old is being lost, so that there is not always the same man there as far as matter is concerned, even though he remains always the same in respect of the form. (Unde homo in iuventute et senectute est idem totum secundum formam, et habet eamdem animam omnino, sed non est idem secundum materiam, quia unam materiam habet in una etate et aliam in alia, quia materia alimenti conversi in naturam rei auctat materiam. In respectu illius quod augmentatur et diminuitur, continue adquiritur nova materia et deperditur antiqua, et ita non idem homo semper secundum materiam, quamvis semper remaneat idem secundum formam: SST 301.) These last words illustrate how a quidditative truth (sameness of form across time) can unwarrantably be concretised so as to suggest that the form is just another component, a part-of-..., which happens to be an unchanging component. This could underlie its assimilation to the principal part in the previous section. 7.04 The effect of such unwarranted concretisation continues by what may be its intrusion into the Aquinate identification of pure forms with superlunary objects (a hyper-Platonic hypostatisation: cf. HQS §5.24). At any rate, instead of trading on the quasi-logical basis of that hypostatisation in order to enginecr the changelessness of such objects, as does Aquinas, it is to the sameness of principal (here called 'formal') parts, as contrasted with secondary, material make-up, parts, that appeal is now made: (A) Whence follows one notable point, i.e. that the superlunary bodies tend more to abide the same than do the living things here below, for the superlunary bodies always remain entirely the same, whereas the living things here below only remain the same in respect of the form, and not in respect 410

Categorematic and Syncategorematic of their m a t t e r . (Ex isto sequitur unum notabile, scilicet, quod corpora supercelestia magis marient eadem quam animata hic inferius, quia corpora supercelestia semper m anen t eadem omnino, sed animata hic inferius solum manent eadem secundum formam et non secundum materiam: SST 301.) 7.05 Next follows the distinction between quantity in the integral sense and quantity in the logical sense (10.23) and other connected m a t t e r s which have already been noted in connection with William of Sherwood and Boethius of Dacia: 5.22, 5.46. It is on this kind of note t h a t the first p a r t of the present De Toto peters out. 7.06 Then the entire discussion of parts and wholes begins anew, as though what is now to follow were a second and quite distinct attempt to bring the topic under intellectual control. Indeed, it might as well be the work of some other author which now commences. At any r a t e , the level of the discussion now becomes altogether superior to the confusion hitherto observed in the first section of the text, and much more worthy of the Burleigh to whom it is attributed. Thus it initially comprises a beautifully incisive characterisation of the integral whole as involving parts which are 'parts of ...', i.e. the genitively-expressed sense of p a r t ' already exploited in 1.41. At the same time a concrete version of the thesis of part-whole disparity (10.324) is given: (A) It has to be asserted t h a t whole and part are taken in many senses, for. t h e r e is a certain type of part which is the integral, of which every whole is not predicated nominatively, but only in an oblique case. In this style the hand is part-of-the man, for this is false: 'The hand is the man'; however, this is true: 'The hand is of-the-man', (Dicendum quod totum et pars multipliciter capiuntur, quia quoddam est pars integral is de qua non predicatur omne totum in recto sed in obliquo, et sic manus est pars hominis, quia hec est falsa, 'manus est homo', sed hec est vera, 'manus est hominis': SST 302.) 411

Section 7 7.07 Next comes what has hitherto been noted as a common and most proper contrast, namely that between the integral and the 'subjective' part (cf. 1.321). Here the subjective part is taken in its enlarged sense, it being exemplified by both the individual, in so far as it is a member of its species, and by the species, in so far as it is a 'part' of the genus: (A) A part is so called in yet another sense when it is said t o be 'subjective', and it is called this relatively to the universal whole; thus also that which is logically contained is a part of its logical container. For example, man is a subjective part of animal, and animal is the universal whole relatively to man. Likewise, individuals are subjective parts of their species, and the species is the universal whole relatively to its individuals. Now it is of the parts in this subjective sense that every whole is predicated nominatively [as opposed to genitively], as in the case of 'Man is animal', and it is thus that the subjective p a r t is distinguished from the integral part. (Alio modo dicitur pars subiectiva, et ista dicitur respectu totius universalis et sic per se inferius est pars sui superioris. Verbi gracia: homo est pars subiectiva animalis et animal est totum universale respectu hominis. Et similiter individua sunt partes subiective speciei, et species est totum universale respectu individuorum. Et est pars subiectiva ista de qua predicatur omne totum in recto ut, scilicet, est hoc: 'homo est animal1, et per hoc distinguitur pars subiectiva a parte integrali: SST 302.) Aquinas' infelicitous classification of form and m a t t e r as parts coming under the integral heading (3.24 above) is next avoided by using a separate heading - the essential - for such parts: 'In yet another way a part is said to be an essential one, and it is thus that the m a t t e r and the form are parts of the composite subject', Alio modo dicitur pars essentialis, et sic materia et forma sunt partes subiecti compositi; SST 302. Exactly how this declaration could ever 412

Categorematic and Syncategorematic cohere with the initial extracts from this text, which were quoted above, is unclear. There, on one reading, and certainly in the light of the annexed examples, the formal and material parts were both taken quantitatively, whereas the next paragraph now to be quoted makes the formal parts into quidditative parts, which are said to be qualitative. This latter is the rather more useful and usual alignment, and once again its apparent lack of coherence with the first sections of the present text could give grounds for doubt as to sameness of authorship: (B) In a fourth way there is said to be the quidditative part, and in this sense the parts of the definition are quidditative parts of the species, and the whole which incorporates those parts is said to be the quidditative whole or the definable whole, as is obvious from the first book of the Physica. Parts in this sense are the forms showing forth the quiddity of that which is defined (according to Averroes on Metaphysica VII). There he asserts that every part of the definition is a form in respect of what is defined, even though one part, of the generic sort, is the material part relatively to the other part, i.e. relatively to the specific differentia among the essential parts. And in many contexts Averroes calls the quidditative parts the qualitative parts, and the integral parts he calls quantitative parts. {Quarto modo dicitur pars quidditativa, et isto modo partes diffinitionis sunt partes quidditative speciei et totum habens illas partes dicitur totum quidditativum vel totum diffinibile, ut patet ex primo Phisicorum. Partes sic dicte sunt forme declarantes quidditatem diffiniti per Commentatorem septem Metaphisice, ubi dicit quod quelibet pars diffinitionis est forma respectu diffiniti, licet una pars similis gener i sit pars materialis respectu alterius partis, scilicet, respectu differentie, partis essentialis. Et partes quidditativas vocat Commentator in multis locis partes qualitativas et partes integrales vocat partes quantitativ as: 413

Section 7 SST 302.) 7.08 Here the reminder that the matter and form of the definiendum may be correlated with the genus and differentia respectively, as delineated in the definition, is common doctrine already noted in 0.5; cf. HQS §4.3421. Although the De Toto et Parte contains much more logically interesting material, we may at this point leave it. This work has been used for the provision of a final illustration of a continuance of the line of approach which we have hitherto been witnessing. If it is the work of Burleigh, then it is indeed by the same author as the one who is to provide the initial illustration of that other tendency which is to be the central thread of the reseaches contained in the remainder of the present book. Thus here is what is definitely Burleigh's exemplification of the distinction between wholes in the categorematic and syncategorematic senses: (A) Now follows talk about this sign 'whole', which effectuates a distinction of integral parts. In this connection it should be realised that this word has a twofold sense in its discursive use. For in one sense it is taken categorematically, and in another syncategorematicaily. In the first sense it coincides with the complete, whereas in the second sense it is used in such a way that it amounts to every integral part. In this vein the question is raised about the following problem-sentence: The whole Sortes is less than Sortes. This may be proved to be the case as follows: Sortes' foot is less than Sortes; Sortes' hand is less than Sortes, and so on for all the other parts. Hence the whole Sortes is less than Sortes. It may be disproved thus: the whole Sortes is less than Sortes; but the whole Sortes is Sortes; hence Sortes is less than Sortes. But this conclusion is false, while the minor premiss is not false, so the major must be false. Solution: the first-stated proposition has several 414

Categorematic and Syncategorematic equivocal senses, on account of the possibility that 'whole' may be taken in the categorematic sense, in which case that proposition is false, for then the import which is conveyed is that the complete Sortes made up from all his parts is less than Sortes. However, if 'whole' is taken in the syncategorematic sense, then it is true, for then the import conveyed is that any part of Sortes is less than Sortes. (Sequitur de hoc signo 'totus', quod distribuit pro partibus integralibus. Et est sciendum, quod in oratione dupliciter accipitur. Nam uno modo tenetur categorematice, alio modo syncategorematice. Primo modo idem est quod 'perfectum', secundo modo sic tenetur pro qualibet parte integrali. Et secundum hoc quaeritur de hoc sophism ate: Totus Sortes est minor Sorte. Probatur sic. Pes Sortis est minor Sorte, et manus Sortis est minor Sorte, et sic de aliis omnibus partibus, ergo totus Sortes est minor Sorte. Improbatur sic. Totus Sortes est minor Sorte; totus Sortes est Sortes; ergo Sortes est minor Sorte. Conclusio falsa, et non minor, ergo major. Solutio. Prima est multiplex secundum aequivocationem, eo quod li 'totus' potest teneri categorematice, et sic est falsa, quia denota tur, quod perfectus Sortes ex suis partibus syncategorematice, est minor Sorte. Si autem intelligatur sic est verum, quia denotatur, quod quaelibet pars Sortis est minor Sorte: BP 256.) 7.081 This brief text is a gateway to our resumption of what was seen in 2.212 above as an Abelardian thread in the history of mereology. From the present point onwards, Norman Kretzmann's polished jewel of an essay comprising medieval dealings with Burleigh's problem-sentence (or sophisma) is a noteworthy pionecr back-up concerning the history of the case (CLM 230 - 245). Burleigh himself is far from being an innovator in the use of the key distinction: we have already seen, and are to see in more detail, that it was well in 415

Section 7 evidence long before he was even born. However, he has provided a clear and incisive typical exemplification of its development, and it is to the details of that exemplification that attention may now be directed. 7.082 First, as regards the wording of Burleigh's text, the central point should readily be apparent. 'The whole ...' in what he here calls the categorematic sense conveyes 'the complete ...' . Hence in this sense the propositiion, 'The whole Sortes is less than Sortes' is obviously false, since it amounts to, 'The complete Sortes is less than Sortes 1 , with its terms appropriately nominal in nature. Burleigh has gone to the trouble of underlining that obvious falsehood by the use of its even more obviously false nominally-termed consequence ('Sortes is less than Sortes') so as to found a reductio ad absurdum in the disproof section of his discussion. The contested proposition's proof in the contrasting syncategorematic sense of 'whole', with its built-in non-nominal quantifier 'Every ...', is apparently let pass by him. This sense is alleged to amount to, 'Every integral part of Sortes is less than Sortes', and this in itself does indeed seem unobjectionable. For the moment, therefore, we too might let it pass, although the earlier William of Sherwood apparently had not done so: KSS 40 - 1. The suspicion in fact subsists that this allegedly alternative sense of 'the whole ...', the earlier history of which was observed in 2.21, has only managed to continue its intrusion onto the scene by the initial taking of 'every integral part' as its quasi-nominal (i.e. categorematic) equivalent. Certainly the sophisma's shockvalue relies on its being at least read categorematically at the initial stage, even if its logical form is subsequently said to be syncategorematic (i.e. quantifier-incorporating). There also subsists the flavour of a puerile example of the old-fashioned 'fallacy of composition', wherein that shockvalue is somehow maintained by listing the bits which are less than the whole so as to attribute to the whole the pred416

Categorematic and Syncategorematic icate which thus applies to the bits taken severally. (A usual illustration of the fallacy is to point out that we would scarcely want to argue that because this, that, and the other, small tree will not shade me from the sun (when they are taken individually), the whole clump will therefore not do so; we seem to be doing the same sort of thing here.) The medievals themselves would certainly not let the matter rest in this state, and have plenty more to say on the subject of this sophisma, as N. Kretzmann shows (CLM 230 - 245) and as we are to see hereunder. 7.1 Syncategoremata as Functors 7.11 A description has already been provided in 2.21 of the way in which Abelard adumbrated the distinction between the categorematic and syncategorematic senses of 'totum' in the mereological area, but without his using this nomenclature. However, the concept of the syncategorema in general is of so vast an importance that it merits a pause for a reiteration of its import. The term itself figures in the early sixthcentury grammarian of Latin, the well-known Priscian, who tells us that logicians admit syncategoremata or consignificantia as well as their two basic parts of speech (name and verb): HQS §0.54, BS I, 2. The medieval logicians in due course embarked upon huge surveys of the syncategoremata. What exactly is their nature? 7.12 In fact, for the most part, they are precisely those incomplete expressions, or functors, which are still of the greatest importance in contemporary logic and logical grammar: 0.3 above, 10.012, and HQS §0.543 enlarge on this point. Indeed, so important are they that there may be some contemporaries of ours who are in necd of a reminder that functors are by no means uniquely present-day innovations. There are myriads of medieval characterisations which serve to eliminate any such impression of exclusive modernity. An ample collection of such characterisations is to be found in the most int417

Section 7 eresting volumes, coded as BS, which constitute H.A.G. Braakhuis' study of thirteenth-century syncategoremata. Thus, according to Nicholas of Paris, syncategorematic expressions are so called because they have meaning along with other expressions, and what they signify is completed by something else, and does not stand alone: cum alio significat, et suum significatum cum alio complet, et non per se: BS II, 6. Hence, he continues, their meaning, generally speaking, involves a certain intrinsically incomplete fashion of signifying something, that incompleteness being completed by the bringing to bear of its completion: Unde significatio generalis eorum est quidam modus significandi aliquid incomplete per se, quid modus terminatur ad adiunctum: BSII, 6. Or again, as N. Kretzmann quotes the same Nicholas in the Cambridge History's section on this topic (CLM 212 - 16), syncategorematic expressions 'signify in the manner of a disposition of things, and of terms signifying things. Every disposition, however, is indefinite in itself, and is made definite by that which it disposes'; CLM 214. It is the Latin 'infinita which has here been translated as 'indefinite'; it could also be taken as 'incomplete', with the correlative 'finita' as 'complete'; cf. CLM 214, note 12. 7.13 However, in medieval times, neither grammarians nor logicians were invariably prepared to follow through such useful generic classifications into all their consequences, nor have present-day historians drawn upon them. The result is confusion in all areas, for the discussion of which it is unfortunately not here possible to tarry. Still, even the most untutored glance at the contents-list of the Kretzmann translation and commentary on William of Sherwood's 'Treatise on Syncategorematic Words' (KSS) suffices to establish the functorial complexion of many of the expressions chosen for attention. Thus chapter xiv deals with 'non- ...' in so far as it may be either a functor forming a nominal expression (so that its index is n/n) or a functor forming a proposition 418

Categorematic and Syncategorematic (with index then s/5). Chapter xiii discusses the various functorial possibilities of 'is', e.g. as '... is ...' (index s/(n n)), or as '... is' (index 5/n). Again, 'All ...', 'Every ...', 'Both ... and ...', 'No ...', 'Neither ... nor ...', 'Only ...', 'If ... then ...', and '... and ...', are among the many other components of variously-termed proposition-forming functors also treated in KSS. Here, therefore s/(n n) and s/(s s) are among the indices in question. 7.14 So much, then, for the more noteworthy general medieval situation of syncategorematic expressions. The specifically mereological case occurs when totum, 'whole' is described (e.g. in KSS ch. ii) as susceptible of either categorematic or syncategorematic interpretation, in a fashion similar to that encountered in the Burleigh example (7.07 above). However, a little thought will make it apparent that both interpretations make 'the whole ...' into a functor, the one name-forming, the other proposition-forming, so that here we have an instance wherein something obviously functorial, in the name-forming case, is nevertheless exiled by the medievals from the syncategorematics. This could be because they baulked at making every name-forming (as opposed to proposition-forming) functor into a syncategormetic case, simply because the main contrast before their minds was that between certain proposition-forming functors, which are seen by them as syncategorematic, and the names or nominal terms (some comprising name-forming functors) which completed them; these latter hence tended to be lumped together as contrastingly categorematic, notwithstanding their functorial nature. The general medieval preoccupation with the contrast between those syncategoremata of traditional syllogistic, such as 'Every ... is ..', 'No ... is ...', and so on, and their nominal (and hence for them obviously categorematic) completions would tend to cause the assimilation of a name-forming functor having the sense of 'the whole ...', to the categorematics also, in spite of its ultimately functorial nature. 419

Section 7 7.2 Examples from Earlier Syncategoremata-Treatises 7.21 It has to be emphasised that the scope and variety of the treatises on syncategorematic terms are so vast as to allow only the most superficial impression, even of their mereological aspects, t o be conveyed here. Indeed, precisely the same might be said of H.A.G. Braakhuis' volume (BS) alone. From these is now gratefully drawn the following passage on 'whole' ascribed to a certain Robert Bacon; it is said to probably date from the first decade of the thirteenth century (BS I, 513). Still, it is clearly a highly sophisticated piece of work, with whole in the c a t e g o r e m a t i c sense being stressed as the complete collection of integral parts, as in D2 of 10.32, thereby correcting some possible confusion due t o Abelard (HQS §4.524). Indeed, the definition here provided of the integral whole in terms of 'outside of ...' (a very useful present-day primitive functor) has a contemporary counterpart in the definition provided in 10.332. Again, the passage's terminological assimilation of 'collective' to 'integral' coheres with the present work's use of 'collective' as in 'collective class' (0.45, 10.31) so that it has a sense broader than t h a t restricted to wholes having parts specified as discrete, as in the cases of a flock, an army, or a people. This r e s tricted sense sometimes persisted in traditional logic and grammar. The final section of the quotation is also most important, with its confirmation that the c a t e g o r e m a t i c sense of 'totus Sortes' amounts to 'the whole Sortes', i.e. is a singular compound nominal t e r m . This corresponds to thesis 10.31, which would confirm that t h e r e is exactly one collective or integral class of the parts of Sortes; this is, of course, Sortes. This leads us to the other point to be made, i.e. the equivalence of 'Sortes' and 'the whole Sortes'; this was r e produced in Burleigh's discussion (7.08 above). Not only does this have a counterpart in thesis 10.325 (the identity of Sortes and the collective class of Sortes, for example) but may also be said to ground a medieval type of thesis about 420

C a t e g o r e m a t i c and Syncategorematic the 'superfluity' of 'the whole ...' in this context; this would parallel the alleged superfluity of 'every . . . ' i n 'every Sortes', according to Garland the Computist (10.27, cf. HQS §2.551). We shall have occasion to r e v e r t to such allegations of 'superfluity' when some of Ockham's theories are being touched upon below in 7.51, 7.52. Here now is the beginning of the passage from Robert Bacon: (A) The sign [i.e. 'every'] which effectuates a distribution of the subjective parts of a [secondary] substance having been dealt with, we now have to proceed to discuss t h a t sign which effectuates a distribution of the integral parts of a [primary] substance, i.e. 'whole'. Hence first its meaning is to be discussed, and afterwards its inferential force in both propositional and argumentative contexts. Its meaning is evidenced thus: 'whole' is either c a t e g orematic or syncategorematic. If it is c a t e g o r e m a t i c , it then signifies the wholeness of t h e object, and in this sense it is usually said to be the formal whole, or the collective whole, or the integral whole, and in this sense whole is the same as complete, and is characterised as follows: the whole is that outside of which there is nothing. If it is a sync a t e g o r e m a t i c expression then it signifies the wholeness of the subject in so far as it is the subject; in this sense it is distributive, and thus is usually said to be the material whole. From what has been said above it is evident that the following, i.e. 'The whole Sortes is white', has two senses, in that the 'whole' may either be a c a t e g o r e m a t i c or a syncategorematic t e r m . If it is c a t e g o r e m a t i c it is a singular t e r m , since 'the whole Sortes' and 'Sortes' are mutually interchangeable. If it is syncategorematic, it is a universal t e r m . (Dictum est de illo signo quod distribuit inter partes Dicendum est de illo signo quod dissubiectivas substantie. tribuit inter partes integrales substantie, scilicet 'totus'. Dicendum est ergo primo de eius significatione, et postea de 421

Section 7 eius potestate, tarn in propositione, quam in argumentatione. Significatio eius patet sic: 'totus' aut est categoreuma aut syncategoreuma. Si categoreuma, tunc significat totalitatem rei, et sic solet dici totum formale, vel totum collectivum sive integrum, et hoc modo idem est 'totum' et 'perfectum', et sic deseribitur: totum est cui nihil est extra. Si sincategoreuma: tunc significat totalitatem suhiecti inquantum subiectum; et sic est distributivum et sic solet dici totum materiale. Per iam dicta patet quod hec est duplex: 'totus Sortes est albus', ex eo quod li 'totus' potest esse categoreuma vel sincategoreuma. Si categoreuma: singularis est, quia 'totus Sortes' et 'Sortes' convertuntur. Si syncategoreuma, univeralis est: BS I 121 - 2.) 7.211 The reason behind the final sentence's implying that 'totus Sortes', 'whole Sortes' forms a universal proposition when taken in the syncategorematic sense is, of course, that sense's interpretation as already shown above in Burleigh, i.e. as 'Every integral part of Sortes is ...' (7.08 above). This latter has the form of the 'universal affirmative' of traditional logic. In turn, this connects with Abelard's 'universal' and 'distributive' senses of totus, as recorded in 2.21; cf. 1.22. 7.22 Thus far, provided that any final judgement as to the cogency of the syncategorematic type of interpretation for 'whole' is for the moment suspended, one can at least follow with comparative ease the points which are being made. The same could apply as far as the point to be made early in the next passage is concerned, i.e. that the syncategorematic interpretation cannot be applied to 'whole ...' when it occurs as the second term ('predicate term') of some '... is ...'- proposition. However, consequences following upon the broad generalities of a theory couched in an appropriate categorial language are in fact not going to be captured by the still lively discussion. Nevertheless, the push towards such a 422

Categorematic and Syncategorematic theory is evidenced in the initial statement of two principles. The second of these, concerning the inapplicability of 'restriction', or 'amplification', to a 'discrete' term is simply a slight elaboration of the Garlandian 'superfluity' thesis mentioned before the last-quoted passage. Not only is 'every' superfluous in 'Every Sortes ...' (an example of amplification, with 'Sortes' as the discrete term) but so also is 'some' in 'Some Sortes ...' (an example of restriction). However, troubles follow thick and fast upon the conclusion that the subject-term of a proposition having the form 'totus ... est ...' ('(the) whole ... is ...') taken in its syncategorematic sense, must be a 'discrete' term or an equivalent thereto (cf. 1.5, 2.78, and 10.336). Let us first peruse the passage up to this point: (A) Now we turn to its inferential force. In order to have the skill to deal with the sign 'whole' we presuppose two principles: (i) the sign 'whole' exercises its distributive effect upon integral parts as such, and not merely incidentally so; (ii) a discrete term can neither be restricted nor amplified. However, it should be known that 'whole' may figure either in the subject-position or in the predicate-position. The predicate-position is eliminated, since 'whole' is a distributive sign (our intention here being to deal with it in its syncategorematic use) and it must hence figure in the subject-position. Now the subject-term is either common or discrete. It cannot be common for the following reason: [cf. (i) above] 'whole' is such that it exercises its distributive effect upon integral parts as such, and not merely incidentally so; hence it cannot exercise its distributive effect upon [the import of] a common term. Hence the subject-term must be either discrete or equivalent to a discrete term, as in the case of 'the whole Sortes'. (Sequitur de potestate eius. Ad habendum igitur cuiusmodi sophisticatio fit in hoc signo 'totus', supponimus duo principia, scilicet quod hoc 423

Section 7 signum 'totus' distribuit inter partes integrales per se et non per accidens; item: terminus discre tus neque potest restringi nec ampliari. Sed sciendum quod 'totus' aut ponitur ex parte subiecti aut ex parte predicati. Non ex parte predicati, cum sit signum distributivum -- intendimus enim de eo secundum quod est sincategoreuma --; ergo ex parte subiecti. Subiectus ergo terminus aut est communis aut discretus. Non communis; probatio: distribuit enim inter partes integrales per se et non per accidens; ergo non potest distribuere in termino communi. Ergo terminus subiectus aut erit discretus aut equipollens discreto, ut 'totus Sortes': BS I, 122.) 7.221 This accent on the 'distribution' of integral parts is quite in order relatively to the allegedly syncategorematic sense of 'whole' which is in question here. Again, the corresponding integral whole (collective class) embodying those integral parts must certainly be a single object (i.e. be covered by a so-called 'discrete' or unshared term). But it becomes apparent from the examples used by the author (e.g. 'Sortes') that only single individual objects as delineated in usual pre-theoretical talk are to be admitted. A non-continuous single object such as the integral whole or collective class of men, for example, is thereby disqualified. 7.23 Again, from the rest of the passage, which now follows, it is evident that the integral parts which (by the last extract) are 'distributed', do not count as 'individuals of a kind' (since these latter are now to be excluded from the distribution). This is a pity, since it implicitly excludes that clause of the definition of the collective class according to which, as 3.261 and 3.42 above reminded us, and as D2 of 10.32 confirms, every part of the collective class of A's has a part in common with an A. To this extent such 'distributed' parts are 'individuals of a kind'. However, if it is the distinction between the like-sorted 'parts' of 'classes' in a distributive sense, and the possibly variously-sorted parts 424

Categorematic and Syncategorematic of classes in a collective sense, which is being accentuated here, then this is a perfectly valid point, in spite of its apparent exclusions which have just been indicated. Indeed, as a summary s t a t e m e n t of the distinction between the rnereological inclusion of parts within a 'discrete' object on the one hand, and the so-called class-inclusion and class-membership, along with their quidditative analogues on the other, this single sentence has no rival in the history of logic: (A) Granted all this it is evident t h a t there cannot in this situation be a multiplicity which would permit a distribution exercised upon either the individuals of a kind or upon the kinds of individuals; neither can the distribution be exercised upon specific parts [of a genus] or upon numerical [i.e. individual] parts [of a species] because an unshared t e r m does not relate to parts of any such sort. (Hoc secsupposito patet quod non potest esse hie multiplicitas undum quod possit fieri dis tributio pro singulis generum vel pro generibus singulorum; nec quod possit fieri distributio secundum pro partibus secundum species vel pro partibus numerum, quia terminus discretus non habet huiusmodi partes: BS I, 122.) 7.231 The same mixture of points which are undoubtedly valid, relatively speaking, but unnecessarily limited in the light of ampler theory, will be seen to continue when the already-encountered thesis of the impossibility of amplifying or restricting a singular t e r m (7.22(A) above) is now brought to bear for the purpose of proving that the composite t e r m (e.g. 'totus Sortes', '(the) whole Sortes') does not involve any multiplicity, as contrasted with the sort encountered in 'Every ... is ...'. This appears at first sight to be incompatible with the s t a t e m e n t made above in 7.21(A) t h a t the syncategorematic use of 'totus' makes such a composite into a universal t e r m , so that 'the whole Sortes ...' then has the sense, 'Every integral part of Sortes ...'. One can, however, rescue the author from this lapse by assuming his failure to 425

Section 7 realise that his excellently salient distinctions made in the previously-quoted paragraph do not mean that terms indicating mereological 'parts' are excluded from completing 'Every ... is ...' functors; they may do so in exactly the same way as do the more usual specific, generic, and individual terms. 7.24 There are two other remaining points in the passage next to be quoted, namely inferences from broad to narrow classes (or vice versa), and the 'improper' use of shared names as opposed to unshared names with the 'totus ...', 'the whole ...' functor. Of these the first is simply a reflection of the fact that while all men are animals, not all animals are men. The second alludes to Boethius' having sometimes given the impression that individual human beings were the integral parts of the species man: B 877D, 887D. From a concrete point of view there would be nothing wrong with this provided that 'species' were taken as a mereological class, and it were to be realised that individuals of that sort by no means exhaust the types of parts discernible within the mereologically collective class of men. However, the present author's already-noted misguided inference that since collective classes are individual objects, they can only be individuals such as Sortes, leads him to exclude shared names such as 'man' from the 'totus' context. He will allow 'totus Sortes', with 'Sortes' as a proper name, but not 'totus homo', 'the whole (of) man', in spite of its possible use to refer to the whole of the collective class of men, and its hence being also an unshared-nominal expression (cf. 10.321). It is Boethius' apparent countenancing of 'totus homo' which is being relegated to the realm of the 'improper'. Abelard at least left open the possibility of taking 'man' in a collective sense, even though his preferred mereological example is that of the usual house (cf. 2.4 and HQS §4.51). Here now is the passage which embraces all these further points concerning '(the) whole ...': (A) Again, the second principle [(ii) of 7.22(A) above] 426

Categorematic and Syncategorematic makes it evident that there cannot be any multiplicity deriving from the composite term [formed from 'the-whole-..'] of the sort that was associated with the sign 'every ...'. The proof of this is that every discrete term is incapable of being amplified or restricted; hence the incorporation of the simple term [e.g. 'Sortes'] or of the composite term [e.g. 'the-whole-Sortes'] makes no difference on this score. Again, from the foregoing its inferential force in a piece of reasoning is evident. It permits the inference from the more narrow to the more broad, since whichever of these is in question is individuated and amounts to an unshared term in so far as the sign 'the whole..' is adjoined thereto. In the opposite direction it does not permit the inference from the more broad to the narrower, so that 'This whole animal ...' does not ground 'This whole man ...'. All the foregoing points are to be understood as referring to 'whole' when it is used in its proper sense. It is possible for it to be improperly adjoined to a common term, with the stress on that commonness, in the style which Boethius exploits when, in his De Divisione, he says that particular men are integral parts of man as such, and that man is divided into such 'parts', as it were in the way in which a whole is divided into its parts. (Item. Per secundum principium patet quod non possit esse aliqua multiplicitas ex termino composito, sicut fuit in hic signo 'omnis'. Probatio: omnis terminus discretus neque potest ampliari neque restringi; ergo nichil differt adiungi termino simplici et composito. Item. Ex predictis patet de potestate sua in argumentatione. Tenet ab inferiori ad superius, cum utrumque sit individuatum et habens vim discreti, secundum quod ei adiungi tur hoc signum 'totus'. Econverso non tenet a superiori ad inferius, ut 'totum hoc animal; ergo totus iste homo '. Omnia autem que predicta sunt intelligenda sunt de 'toto' 427

Section 7 quando tenetur proprie. Improprie enim potest adiungi termino communi, secundum quod est commune eo modo quo dicit Boethius in libro Divisionum, quod particulares homines sunt partes integrales hominis simplicter, et homo dividitur in has tamquam totum in partes: BS I, 122 - 3. 7.241 Conclusion. This sample of early thirteenth-century syncategoremata-tracts' attention to totus, 'whole', usefully illustrates that no matter how high the standard of mereological discussion may be, the lack of that precision provided by present-day axiomatised theories, such as those sketched in section 10, leads to a hit-and-miss situation where determinacy and consistency of results is concerned. 7.3 Nicholas of Paris on totus, 'whole'. 7.301 Still remaining within the ambit of syncategorematatracts, we now view yet another sample of mereological acumen, this time as displayed by the mid-thirteenth-century author, Nicholas of Paris, in his lengthy work as edited by H.A.G. Braakhuis (BS II). 7.31 It has to be stressed that this tract as a whole is replete with superb philosophical and logical material, and that only the mere surface of its mereological aspects is being here lightly touched upon. Thus although, as subsequent samples are to show, the discussion of totus intervenes in several connected contexts, it will perhaps be better to start at the point towards the end of the work where totus is treated more or less for its own sake. The text and translation have been divided into sections for easy correlation with the preliminary comment which now follows. Thus in [i] of (A) below is to be found an echo of a point made above in the second Burleigh text (7.08(A)) namely that 'Sortes' and 'the whole Sortes' have the same import (cf. 10.325 for theoretical underpinning); but this is wrongly said to make 'the whole' into a useless (and hence non-significant) expression in this context. In [ii] the same identity of imp428

Categorematic and Syncategorematic ort is exploited to stress what has been identified above as the categorematic sense of totus Sortes, 'the whole Sortes', to the exclusion of the syncategorematic or 'distributive' sense, which alludes to the parts. Thus, it is argued, is excluded the analysis of 'The whole Sortes is white' as 'Every part of Sortes is white'. (It is noteworthy that at this comparatively developed state of the history of the syncategoremata-tracts, the original vocabulary stressed by Abelard (2.21 above) is still retained. Thus in [ix] of 7.315(A) below, totus ... /whole ...', is seen as a functor which, when completed by a name, may be construed either 'distributively' (i.e. as a syncategorematic proposition-forming functor such as 'Every integral part of .... is ...') or 'collectively' (i.e. as a categorematic nominal expression formed from a name, as in 'The whole Sortes'). These distributive and collective characterisations of totus are verbally similar to those used by Abelard in 2.2 above. It is curious that this present text, devoted as it is to what by then were commonly called syncategoremata, and prefaced as it is by an introduction displaying a full consciousness of such terms' logical status, should not here apply to totus the then currently modish terminology of the categorematic and the syncategorematic. After all, such terminology had already been here applied by the earlier Robert Bacon (7.2 above). In [iii], as contrasted with [ii], an attempt is made to turn totus, 'whole' in a parts-relevant direction. This is done by an appeal to the logic of mereological exceptions, of which there is a great deal in the present tracts, as a later sample is to illustrate (7.4). Analysis of 'The whole Sortes apart from the foot', it is contended, demands that 'the whole ...' should refer to Sortes' parts in a way that 'Sortes' does not. In yet another contrary direction, [iv] expoits an interesting analogy between the functorial nature of 'whole ...', and that of other adjectival names. For example, 'white ...' requires ultimate completion by a primary substance-name, as 429

Section 7 does 'the whole ...'. In both cases the reference of the outcome is to the primary substance in question, and this, it is presumed, excludes part-reference. An interesting facet of this phase of the discussion is its reminder of the fundamentally functorial status of adjectives, avidly discussed as from Anselm of Canterbury and Aosta onwards (HQS §3.3). Here now are the first four sections of the text in question. The remainder will be continued after further comment: (A) Now follows a treatment of the expression 'whole', wherein the following are to be investigated: what it signifies, what its inferential force is, and certain expressions which give rise to difficulties on account of its incorporation therein. [i] The first point to be investigated is whether the expression 'whole' signifies something or nothing. If it signifies nothing, then it is placed in speech uselessly. Besides, adoption of this position would make it out not to be a part of speech. [ii] If it does signify something then this can only be the totality. Whence it would appear that this does not follow, i.e. 'The whole Sortes is white; hence every part of Sortes is white', for [the whole] Sortes is not some part of him; hence the term 'The whole Sortes' does not stand for some part of Sortes; hence the expression, 'The whole Sortes is white', should not be analysed as, 'Every part of Sortes is white'. [iii] Besides, we can very well assert and agree on, 'The whole Sortes is white apart from the foot'. Now when an exception is [thus] made in respect of something, it must be the case that the excepted object is embraced within the bounds of what is stood-for by the term whence is effectuated the exception. Hence a part of Sortes [i.e. the foot] is comprised within what this term 'The whole ...' here stands for, yet this part is not comprised within what this term 'Sortes' stands for, for 'Sortes' stands just for Sortes. 430

Categorematic and Syncategorematic Hence the part is what is stood-for within the range of this term 'the whole'; hence 'the whole' stands for the parts of Sortes. [iv] On the contrary: 'whole' is an adjectival expression; but every adjectival thing participates in the same substance as does its substantival; hence the term 'Sortes' stands for Sortes, and in like fashion the expression 'the whole' can stand for Sortes; hence it necd not stand for Sortes' parts. (Sequitur de hac dictione 'totus'. Unde videndum est quid significet; et quam potestatem habeat; et de quibusdam locutionibus que difficultatem habent propter appositionem eius. [i] Primo videndum est utrum hec dictio 'totus' aliquid significet aut nichil. Si nichil: ergo inutile ponitur hec dictio 'totus' in locutione. Preterea: secundum hoc non esset pars orationis. [ii] Si aliquid: sed nonnisi totalitatem. Ex hoc videtur quod non sequatur: 'totus Sortes est albus; ergo quaelibet pars Sortis est alba', quia [totus] Sortes non est aliqua pars sui; unde ille terminus 'totus Sortes' non supponit pro aliqua parte Sortis; ergo illa 'totus Sortes est albus' non debet sic exponi: quelibet pars Sortis est alba. [iii] Preterea, bene possumus dicere et competenter: 'totus Sortes est albus preter pedem'; sed quando fit exceptio pro aliquo, debet quod res excepta comprehendatur in suppositione termini a quo fit exceptio; ergo pars Sortis comprehendatur in suppositione huius termini 'totus'; sed non comprehenditur in suppositione huius termini 'Sortes', quia 'Sortes' non supponit nisi pro Sorte; ergo comprehenditur in suppositione huius termini 'totus'; ergo 'totus' supponit pro partibus Sortis. [iv] Contra: 'totus' est dictio adiectiva; sed omne adiectivum eandem substantiam habet quam et suum substantivum; ergo ille terminus 'Sortes' supponit pro Sorte, similiter hec dictio 'totus' supponet pro Sorte; ergo non supponet pro partibus Sortis: BS IL 432.2 - 433.12.) 431

Section 7 7.311 The next section of the continued exposition of the text is numbered [v] below, and is full of highly important mereological and ontologicai material. The functorial nature of the field is very much appreciated by Nicholas, or whoever the ultimate author of this objection may have been. He draws an analogy between resemblance, on the one hand, and totality and partiality on the other. In all these cases, he says, there is completion (of a functor) by reference to some further feature of things extrinsic to the functor. Thus (as we may now gloss his words) where A resembles B, we may say that A is a resembler, where 'resembler' must in the nature of the case implicitly also mean in full 'resembler of ....', (This is partly akin to what goes on in the case of the functor '...er', which forms a name from a verb, with 'trm< >' as the gapped nominal counterpart in the categorial language: cf. 10.251.) Likewise, the point continues, the definition of 'whole' (in the sense of integral whole or collective class) must indeed be defined (relatively to 'part' as a primitive term) in terms of 'part of ...', as Nicholas' words state, and as the definition of the functor 'Kl( )' in D2 of 10.31 confirms. One could also agree with the converse statement concerning parts in relation to the whole which the text enuntiates; such agreement could be on the basis of an appropriate alternative set of definitions, or perhaps merely because of the way in which 'element of ...' figures in the definition of 'Kl( )'. 7.312 Be this as it may, our objector in [v] is still being unnecessarily pernickety. While it is eminently true, as he says, that the functor 'this resembles ...' is incomplete, and while it may be admitted that somewhat the same does go for 'whole...', as he is insisting, in the sense that relatively to its full definition as 'integral whole of ...' one might indeed, in view of the definition's content (D2 of 10.31) say that 'whole of the elements of ...' would be a fuller ordinarylanguage expression of the definiendum at this point, it is 432

Categorematic and Syncategorematic still by no means necessary to retain this expression. Alternatively, but connectedly, relatively to the available theoretical background constituted by 10.31, one could say that the objector is here rather obtusely insisting that 'the whole Sortes' should invariably be expressed so as to incorporate overt reference to the integral whole of the elements of Sortes, as opposed to the integral whole of Sortes. In fact, as theorems 10.325 and 10.327 show, Sortes is identical with the collective class of Sortes, which in turn is identical with the collective class of the elements of Sortes: cf. also section [vii] of 7.313(A) below. So while due acknowledgement may be made in gratitude for the text's lively reminder about functors and about such theorems, the trivial fact is that an expression defined in terms of a given functor necd not continue unfailingly to make that functor overtly explicit for evermore. (A) [v] Again, even as a resemblance is that whereby something is related to some other thing, so likewise totality and partialness are each that whereby something is related to something else; hence even as that which resembles is described in relation to something else, so also the whole is described in relation to the parts, and the parts in relation to the whole. Hence even as the following is incomplete, i.e. 'this resembles ...', since that in relation to which it is said to resemble is not stated, so also the following is incomplete, i.e. 'The whole Sortes is white', since it does not make explicit the part-oriented aspect. ([v] Item, sicut similitudo est illud per quod aliquid refertur ad aliud, ita totalitas et partialitas sunt illud per quod aliquid refertur ad aliud; ergo sicut simile ad aliud dicitur, ita totum dicitur respectu partis et pars respectu totius; ergo sicut hec est imperfecta, 'Hic est similis', cum non dicatur respectu cuius dicatur simile, et ita hec est imperfecta 'totus Sortes est albus', cum non dicatur ibi respectu partis: BS II, 433.13.19.) 433

Section 7 7.3121 Before continuing the textual comment a slight postscript may be adjoined concerning this foregoing section [v] and the remarks already made thereon. The entire discourse here presupposes that we are ultimately dealing with proper parts (to which the thesis of part-whole disparity (10.324) applies in a non-vacuous way, i.e. they do not merely maintain the truth of the thesis by making both its antecedent and consequent false). If the possibility of atomic objects having no proper parts is mooted (10.35), then the relations of whole and proper part no longer here apply except vacuously. This is precisely the possibility which is to be raised by William of Ockham where God is concerned, since God, by the usual description, has no parts in the mereological sense (cf. 7.51, 7.52 below). Under these circumstances, even as 'Every Socrates is ...' has been said to have a superfluous 'every' on account of the unshared 'Sortes' (cf. 7.52, 10.27), so also 'The whole God is ...' is to be said to have 'the whole ...' superfluously prefixed, because of the divine 'simplicity', i.e. nonpartedness. Whatever one may think of Ockham's contention, it at least serves as an oblique reminder that the present insistence on the part-oriented aspect of 'the whole ...' necd not have invariable logical force. 7.313 However, to revert to the remainder of the present text: sections [vi] to [viii] are next to be reproduced. At one point in [vii] the objector is to continue to insist on the non-equivalence of Sortes and the integral whole of the parts of Sortes. This insistence (which is opposed to [v]) is used to rebut the claim made in [vi] that 'whole' and 'all' do not differ save in the nature of the things distributed by these signs. (The details are, in fact, here somewhat obscure.) The resolution of the case in [viii] at least goes some way in the direction of a discussion of the ways in which integral wholes and parts are interrelated, with perhaps some appreciation of the difference between collective class, or complete collection (D2 of 10.31) and mere collection 434

Categorematic and Syncategorematic (10.329). (A) [vi] Again, some people say that the expression 'whole' sometimes signifies universally, as does 'all'; hence there is no difference between 'whole' and 'all' apart from the fact that 'whole' extends its effect to integral parts, whereas the sign 'all' extends to the things in question which are stood-for [by the term which follows upon it]. [vii] But on the contrary: given that [from [vi]] the term 'whole' extends its application to integral parts, then when 'the whole Sortes' is propounded, it must be the case that those parts are intellectually grasped in the import of some explicit utterance; but they cannot be thus grasped unless they are signified; hence they are signified either by the term 'whole' or by the term 'Sortes'. But this cannot be by the term 'Sortes', since 'Sortes' does not signify the parts of Sortes, but Sortes himself. Hence they are signified by this term 'whole', so that 'whole' is the same as 'every part'. It follows from this that in the same way as 'every part of Sortes' and 'not every part of Sortes' are congruous expressions, so the same ought to go for 'whole Sortes' and 'not whole Sortes'; but this is not so. [Hence, contrary to [vi], 'whole' does not extend to integral parts.] [viii] The solution: we lay it down that even as the expression 'partly' signifies the part in the condition imparted by the expression 'in respect of ...', so likewise, therefore, even as partly has to be analysed as in respect of the part (and likewise as the adverb 'here' signifies a place in the condition signified by the expression 'in ...', and so has to be analysed as 'in this place') so also 'wholly' signifies the part in the condition imparted by the expression 'in respect of ...', but now with universality annexed; 'partly' and 'wholly' only differ in respect of breadth of application. Hence even as 'partly' has to be clarified thus: partly is the same as in respect of a part, so also we have: wholly is the same as in respect of every part. Our verdict 435

Section 7 is the same as far as the expression 'whole' is concerned, since [as in the case of 'part' and 'partly'] the only difference between 'whole' and 'wholly' lies in their case-endings. ([vi] Item, dicunt quidam quod hec dictio 'totus' quandoque significat universaliter, sicut 'omnis'; unde 'omnis' et 'totus' non differunt, nisi in hoc quod hec dictio 'totus' distribuit in partes integrales, hoc autem signum 'omnis' inter supposita inventa. lvii] Sed contra: cum fiat distributio per hunc terminum 'totus' inter partes integrales, cum dicitur 'totus Sortes', debet quod partes ille intelligantur per aliquam vocem repertam; sed non possunt intelligi nisi significentur; ergo significantur per hunc terminum 'totus', vel per hunc terminum 'Sortes'; sed non per hunc terminum 'Sortes', quia 'Sortes' non significat partes Sortis, sed ipsum Sortem; ergo significantur per hunc terminum 'totus'; ergo idem est 'totus' quod 'quelibet pars'; sicut ergo congrue dicitur 'quelibet pars Sortis' et 'non quelibet pars Sortis', ita debet dici 'totus Sortes' et 'non totus Sortes'; quod falsum est. [viii] Solutio. Dicimus quod sicut hec dictio 'partim' significat partem in habitudine designata per hanc dictionem 'secundum'; - unde sicut habet resolvi , idest: secundum partem; et similiter hoc adverbium 'hic' significat locum in habitudine significata per hanc dictionem 'in', unde sicut habet resolvi 'hic', idest: in hoc loco; - sic 'totaliter' significat partem in habitudine significata per hanc dictionem 'secundum', et cum universalitate; 'partim' et 'totaliter' non different nisi in raritate; unde sicut 'partim' sic se habet resolvi: 'partim' idest: secundum partem, eodem modo 'totaliter', idest: secundum quamlibet partem. Eodem modo dicimus de hac dictione 'totus', quia 'totus' et 'totaliter' non differunt nisi in casu: BS II, 433.20 - 434.18.) 7.314 Finally, in the section numbered [ix] (given in 7.315(A) below) the example about the house's being worth one hundred marks is wheeled in to give an immediate point to the dist436

Categorematic and Syncategorematic inction between the categorematic and syncategorematic senses of totus, 'whole', although, as already noted, the old terminology of collective and distributive, used as far back as Abelard, is here retained. Thus the problem concerns the difference between 'The whole Sortes is white' (which is supposed to allow a 'descensus' to 'Every part of Sortes is white', or to 'Sortes in respect of every part of him is white') and 'The whole house is worth one hundred marks' which does not allow a corresponding inference to 'Every part of the house is worth one hundred marks'. Our author resolves the problem by making the collective/distributive (or categorematic/syncategorematic) distinctions within the sense of 'whole'. The collective (or categorematic) sense does not allow a descensus to the parts (and hence is applicable to the case of the hundred-mark house). The distributive (or syncategorematic) sense allows the descensus to the parts and hence applies where the wholly white Socrates is concerned. In point of theoretical fact, of course, it is possible to transcend all these cases in the light of the contemporary mereology exposed in 10.3. Thus 10.326 allows the wall (for instance) to be an element of the house, and the house to be white, with the conclusion that the wall is an element of the collective class of white objects. This necd by no means entail that the wall itself is white, thus avoiding the rather questionable descensus which [ix] below is to approve and support. Likewise, the same 10.326 allows the wall to be an element of the house and the house to be worth one hundred marks, with the conclusion that the wall is an element of the collective class of objects which are worth one hundred marks, but without any implication that the wall is itself worth one hundred marks. What one can further prove, given the definition of collective class (D2, 10.31) is that every element of the white one-hundred-mark house has a part in common with a white object and with an object worth one-hundred marks. No difficulties follow in either case. Indeed, relatively to the 437

Section 7 general theory of part and whole both stand on exactly the same level. It is only when too elementary or restricted a background theory of inferential forms is brought to bear that the one (e.g. the white case) seems to work, and the other (the hundred-marks case) does not. 7.315 Peter of Spain seems to have gone some way in the more enlightened direction just suggested: CLM 236, note 96. His doctrine may have been at the root of the functorial question inherited from [v] in 7.312(A) above, and which section [x] of the next extract now takes up. Is 'the whole ...' of 'the whole Sortes' incorrectly completed by just 'Sortes'? Should it not be completed so as to read 'the whole of the integral parts of Sortes'? Various suggestions were made (in the comment provided above) as to how this question could be resolved, the principal one being that the objector's qualms were misplaced, given the identity of Sortes with the integral whole of Sortes, as well as with the integral whole of the parts of Sortes, these being identities underpinned by theorems 10.325 and 10.327. Indeed, the first sense of [x] below could initially be construed as pointing in this direction, but the examples which follow do not altogether bear this out. At the same time (and without regard to the first, the genus/species example) it might also be taken to be merely expanding on the distinction between the categorematic (collective) and syncategorematic (distributive) senses of whole. The genus/species example has an unfortunately non-mereological and purely quidditative tinge about it, although it might at a pinch be considered to make the author's point at the contingent level of some alleged usage. The upshot of the last sentence of [x] is that 'whole' in the 'universal' sense (which [ix] has claimed to be applicabled to both the collective and the distributive interpretations of 'whole') does not remain incomplete even if it makes no overt reference to parts. Thus 'the whole Sortes' is in order as it stands, and necd not be expanded to 'the whole of the parts 438

Categorematic and Syncategorematic of Sortes ...'. (A) [ix] A further point: from this, namely 'The whole Sortes is white', I can infer the following: 'Sortes is white in respect of every part'; from 'The whole house is worth one hundred marks' I cannot likewise infer 'Hence each part of the house is worth one hundred marks'. The reply to this is that the expression 'whole' is sometimes taken distributively and sometimes collectively; thus sometimes it signifies on a universal and divisible basis, and sometimes universally but conjointly. This latter is the interpretation under which one cannot infer to the distributively individual items, [x] As to the query raised [in [v] of 7.312(A)] above about 'The whole Sortes is white', we lay it down that the expression 'whole' sometimes signifies wholly, and is applied in respect of wholeness, and sometimes signifies the part universally, and is applied in respect of the part. Now when it signifies wholly, then it has to be qualified by means of an expression indicating the part, and it is taken in this sense in the assertion, 'The genus is the whole relatively to the species'. In contrast, when 'whole' signifies the part universally, then it does not have to be qualified by means of an expression indicating the part. On this basis I assert that the following, i.e. 'The whole Sortes is white', is not incomplete, because here the expression 'whole' is taken with reference to the part, ([ix] Item. Ex hac 'totus Sortes est albus' possum inferre hanc 'Sortes secundum quamlibet partem est albus', similiter ex hac 'tota domus valet centum marchas' non possum inferre: 'ergo quelibet pars domus valet centum marchas'. Ad hoc dicendum quod hec dictio 'totus' quandoque sumitur distributive, quandoque collective; unde quandoque significat quoniam universaliter et divisim, quandoque universaliter et coniunctim. Et propter hoc non potest fieri descensus in distributione. [x] Ad hoc quod queritur 'totus Sortes est albus', dicimus quod hec dictio 'totus' quandoque significat totaliter et 439

Section 7 imponitur a totalitate, quandoque significat partem universaliter, et imponitur a parte. Quando autem significat totaliter, tunc exigit determinari per dictionem designantem partem, et <sic> sumitur cum dicitur: 'genus est toturn ad speciem'. Cum vero significat partem universaliter, tunc non exigit determinari per dictionem designantem partem. Dico ergo quod hec non est imperfecta 'totus Sortes est albus', quia hec dictio 'totus' sumitur ibi secundum partem: BS II 434.20 - 435.14.) 7.316 As a sort of appendix to Nicholas' dealings with t h e hundred-mark house, noted above, here are P e t e r of Spain's remarks on a problem already broached by Abelard (2.33) and 'Master P e t e r ' (2.786(C)) concerning the use of exclusive terms such as 'only' in an integral-whole context. Can one own a house alone (i.e. only a house); does this exclude ownership of its wall? This sort of question is generalised by P e t e r of Spain. He asks, 'whether an exclusive expression used of the whole excludes the part, or vice versa, ... e.g. 'only the wall ..., hence it's not t h a t the house ...'; utrum dictio exclusiva addita toti excludat partem, vel econverso ..., ut 'tantum paries, non ergo domus'; BS I, 272. In the course of dealing with this problem, itself here p a r t of a magnificently extensive general disquisition on exclusives, P e t e r also brings in a one-hundred-pound-house, but in a r a t h e r questionable sort of way. 7.3161 Thus in spite of his saying t h a t he is dealing with incidental features of integral wholes and parts, most of his first and second groups of such features in fact represent their theoretical, and hence not merely incidental, aspects. As such, section [i] and [ii] of the next passage to be quoted are excellent. The elimination of the soul as an integral part contrasts with both Gilbert of Poitier's doctrine (2.833) and with the consequences of Aquinas' having made the form (and hence the soul, in animate cases) an integral part (3.24). P e t e r here agrees with the doctrine of the t r a c t attributed 440

Categorematic and Syncategorematic to Walter Burleigh (7.07 above), namely that we are at this point dealing with an essential part, which is altogether another matter. Unfortunately, knowing that he is saddled with the house-price case, and noticing that a result there desired (i.e. no inference from whole to part, or vice versa) prevails in the cases under section [i], he adopts the expedient of throwing the house-price case into that section's list, with the required result then, as it happens, emerging at the end of [iii]. True, all this is part of the general project of dealing with 'only ...', so that the ultimately required rejection is of 'Only the house is worth one hundred pounds; hence the wall is worth one hundred pounds'. In fact his section [i] contains material which incorporates a rejection of the same consequence, but in its more usual form, i.e. without the preliminary 'only'. However, his actual example in [iii] ('Only the house is worth a hundred pounds; hence the wall is not [worth this amount]'), although construable as a consequence of the required rejection, is in the end here expressed in such a form that quite ordinary and non-mereological considerations about 'only ...' as such would give this same result: (A) It is to be held that there are three kinds of incidental features to be encountered in the integral whole and in its part. Thus: [i] some features are such that they are only suited to the integral whole, e.g. being made up of parts, and having all its parts of a quantitative sort. On these grounds the soul is not an integral part, but rather an essential part; or again, being worth one hundred pounds is suited to the whole [house] and not to the part. [ii] Other features are suited only to the part and not to the whole, e.g. being less than the integral whole, or following from the integral whole, or the property of being a part, or of being conjoined with other quantitative objects to form a quantitatively larger object. 441

Section 7 [iii] In the third style are certain such [incidental] features which pertain indifferently to whole and to part, as e.g. white, black, warm, cold, wet, dry, and suchlike. It is in respect of the first of these sorts of features that the application of an exclusive expression to the integral whole excludes its application to the part, e.g. 'Only the house is worth a hundred pounds; hence the wall is not [worth a hundred pounds]'. However, as far as the other two kinds of incidentals are concerned, the application of the exclusive expression to the part always excludes the whole, but not vice versa. (Et dicendum quod tria genera accidentium reperiuntur in toto integrale et sua parte, quia [i] quedam accidentia sunt que tantum conveniunt toto integrali, ut componi ex partibus integralibus et habere partes omnes secundum quantitatatem, et propter hoc non est anima pars integralis, sed pars essentialis; et valere centum libras convenit toti et non parti. [ii] Alia sunt accidentia que conveniunt parti tantum et non toti, ut esse minus vel sequi ex toto integrali vel partialitas vel coniungi alii quanto ad faciendum malus secundum quantitatem. [iii] Tertio autem modo sunt quedam accidentia que indifferenter conveniunt toti et parti, ut sunt album, nigrum, calidum, frigidum, humidum, siccum, et consimilia. Et in primo genere accidentium dictio exclusiva adiuncta toti integrali excludit partem, ut 'tantum domus valet centum libras, non ergo p aries '. Et in aliis duobus generibus accidentium dictio exclusiva adiuncta parti semper excludit totum, et non econverso: BS I, 275.) 7.3162 There is plainly no reason at all (except perhaps for totally irrelevant economic ones) which compels acceptance of the statement (under [i]) about the whole house's price of one hundred pounds not being applicable to the wall as well. (It might be the wall having da Vinci's 'Last Supper' painted upon it.) Presumably Peter of Spain's verdict on Peter Abelard's 442

Categorematic and Syncategorematic only applied to the house (2.33, 2.786(C)) would be covered by the passage's closing words: the existence of only a house is not incompatible with the existence of its parts. 7.4 Nicholas of Paris on Exceptives 7.41 After this glimpse of a way in which mereology entered into the syncategoremata-tracts' vast discussions of exclusives, a similarly relevant sample of their treatment of 'exceptives' may now be provided. The text is gratefully drawn, once again, from H.A.C. Braakhuis' edition (BS II). 7.42 Verbally, the medieval treatment of exceptives revolves for the most part around the Latin 'praeter' chiefly in its 'except for ...' sense. The literature is very extensive, and as usual it is immensely difficult not to get entangled in its more general (as opposed to its purely mereological) aspects. The equipment for a thorough and axiomaticallybased analysis of the whole vast field, mereological and nonmereological, is outlined in section 10, and was fully popularised many years ago (e.g. in HLM ; cf. also HQS §6). Such analysis would be a lengthy but fascinating enterprise, and since it has not yet been carried out, it necds a great effort not to allow oneself to be over-much involved in its highly interesting non-mereological stretches. Typical examples of exceptive sophismata are Omnis homo praeter Sortem currit, 'Every man except for Sortes runs', and Tota domus praeter parietem est alba, 'The whole house except for the wall is white'. Of these the second is usually taken to require an interpretation in terms of the mereological whole. The sequence of certain medieval analyses proposed for some extremely complex exceptive cases, including that of Omnis homo praeter Socratem excipitur, 'Every man except Socrates is excepted', is ingeniously narrated by N. Kretzmann: CLM 218 - 30. 7.43 A few somewhat inadequate preliminary notes on peculiarities thrown up by the medieval doctrine on exceptives in general must first be supplied. There is agreement between 443

Section 7 Nicholas of Paris and William of Ockham (for example) that an exceptive without an initial quantifier such as 'Every ...' is 'incongruous' or 'improper', e.g. Homo-praeter-Sortem currit, '[A] man-other-than-Sortes runs'; BS II 139 - 41, O II 279 (ch. 18). The ultimate origin of this allegation of impropriety may lie in the oft-repeated claim that an exception made by means of an exceptive renders true what would otherwise be a false universal proposition (O II, 278 (ch. 18)). The sentence just given as an example of impropriety can, in a contrasting sort of way, be true both with and without the exceptive clause, and this may be why they prefer to exclude it (by means of the allegation of impropriety) from the ambit of the set of cases which they prefer to see as relevant. The deductively catastrophic results of this unnecessarily convoluted delineation of boundaries are excellently brought out by Ockham (O II, 279). The acceptance by both the authors mentioned of homo-alius-a-Sorte, 'man-other-than-Sortes', as the equivalent of homo-praeter-Sortem, 'man-except-for-Sortes', ought to have allowed them to use this compound term like any other, without incongruity, at least for elementary inferential purposes. Its general structure is that of 'manand-non-Sortes', and this is easily analysable in terms of definitions 10.252 and 10.253 below. Repercussions of the alleged incongruity will be observed in Nicholas' next-quoted remarks on exceptive integrals (7.45(A)). His claim that it is an incongruity resembling that of omnis Sortes ...., 'Every Sortes ...' (BS II, 141.17) echoes things said by the eleventhcentury Garland the Computist (cf. 10.27 and HQS 2.551) and may be related to some of Ockham's remarks in due course (in 7.52 below). Abelard discussed similar qualms concerning Omnis Aiax ..., 'Every Ajax ...': DA 236 - 7. 7.44 Nicholas, Ockham, and William of Sherwood all agree that 'except for' may be taken either exceptively, as just described, or diminutively, as in 'Ten except for five are five'; BS II, 173 - 5, O II 278.17 (ch. 18), KSS ch. x. We 444

Categorematic and Syncategorematic have already come across a way in which this sort of numerical case may be treated mereologically (5.4), and Nicholas too may invoke mereological doctrine. However, this diminutive sense is yet another thread of the rich tissue of medieval mereological exceptive doctrine which must, regretfully, be left on one side for the purposes of the present brief preliminary foray. Without further ado just a few paragraphs from Nicholas of Paris' work on exceptions from integral wholes may now be subjected to a somewhat superficial scrutiny. 7.45 Already enough has been said to make intelligible the qualms which Nicholas is now to express about the sensemaking status of the integral exceptive propositions of the type with which we are concerned; a proposition of this sort appears to be akin to the type which, as we have seen in 7.43, Nicholas and Ockham both denounce as improper or incongruous. Thus 'Man except Sortes is white', lacking as it does the allegedly requisite initial 'distribution' effectuated by a universal quantifier (e.g. 'All...'), is the common example of such incongruity. And now the present integral example, i.e. '(the) whole house except the wall is white' may be seen as apparently identical in form with the incongruous case, not only because it could, in principle, take such a quantifier at its outset, but also on the basis of the generally admitted equation of '(the) whole house' and '(the) house' (cf. 7,08, 7.21, 10.325 and BS II, 139). For it then becomes the glaring parallel of the 'Man except Sortes is white', as the sed contra, 'on the negative side', part of the following discussion is only too eager to interject in response to the initial defence of the congruousness of the integral house example: (A) Having viewed exception as effectuated in connection with the universal whole, now exception as effectuated with the integral whole is to be viewed; e.g. someone might assert, 'The whole house except for the wall is white'. 445

Section 7 Thus it can first be asked whether an exception of this sort makes sense. On the affirmative side: it would appear to be so because an exception is the taking-away of one part from some whole with reference to a third element in the situation. Hence as that thing which is the whole house is a whole taken as a whole, and that thing which is a wall is a part taken as a part, it can indeed be put aside by an exceptive expression. Again, exception is the removal of an occasion of falsehood involving the part as contrasted with the whole; thus should the wall not be white, [but the rest white] and were 'The whole house is white' to be asserted, then this would be false, and this would only be because of the part which is the wall; hence the putting aside of that part [by means of the exceptive clause] will result in a true proposition. But such putting aside can be effectuated by means of the exception. Hence exception can be effectuated within an integral whole. On the negative side: whole house and house are identical; but the following does not make sense, i.e. 'House except the wall is white'; hence also the other [does not make sense, i.e. 'Whole house except the wall is white']. Proof: even as 'Every man...' is related to 'man', so also is 'whole house ....' related to 'house'; but this does not make sense, i.e. 'Man except Sortes ...'; hence the same goes for the following: 'House except the wall ...'. (Viso de exceptione facta in toto universali, videndum est de excepto facto in toto integral!, sicut cum diceretur, 'tota domus preter parietem est alba', Potest igitur queri utrum talis exceptio sit congrua. Et quod sic videtur, quia exceptio est abstractio alicuius partis ab aliquo toto respectu tertii; ergo cum hoc quod est tota domus sit totum ut totum, et hoc quod est paries sit pars ut pars, per dictionem exceptivam potest ex trahi. Item: exceptio est remotio instantie falsitatis, que est partis contra totum; sed pariete non albo, si dicatur 'tota 446

Categorematic and Syncategorematic domus est alba', hec est falsa, et non est nisi pro hac parte que est paries; ergo remota parte illa erit vera; sed removen potest per exceptionem; ergo potest fieri exceptio in toto integrali. Sed contra: idem est 'tota domus' et 'domus'; sed hec est incomgrua 'domus preter parietem est alba'; ergo et alia. Probatio: sicut se habet 'omnis homo' ad 'homo', ita se habet 'tota domus' ad 'domus'; sed hec est incongrua 'homo preter Sortem'; ergo et hec 'domus preter parietem': BS II 170.11 171.6.) 7.451 Mereologically speaking, the identity put forward by this initial sed contra, 'on the negative side ...', may indeed be underpinned by 10.325. The house and the collective class of the house are the same. But the objection necds the dubious support of the incongruity thesis concerning the unquantified exceptive, as noted above. However, the analogy with the non-integral 'incongruous' case brings in factors more complex than Nicholas realises. For although a nonmereological homo-praeter-Sortem, 'man-except-Sortes', may indeed be simply rendered in terms of the categorial language as being of the form 'man-and-non-Sortes' (with the symbolism introduced in 10.252, .253), and this accords with the medieval homo-alius-a-Sorte, 'man-other-than-Sortes' analysis (cf. 7.43 above), nevertheless the corresponding mereological domus praeter-parietem, 'house-except-for-the-wall', must be seen as 'house-outside-of-the-wall', which raises a whole host of more complexly interesting connections; cf. 10.33. Yet although Nicholas' definition of the integral whole in terms of outside-of (noted in 7.21, cf. 10.332) would already arm the investigator with the relevant foundation for a systematic and definitive understanding of integral exceptions, the task cannot be undertaken in these preliminary notes, and hence still awaits execution. Worthy of note in this last passage also is the insistence on the exception's having to be made from the whole as such (totum ut totum, 'the whole taken as a 447

Section 7 whole'). This will be taken up in a later portion of the passage, having figured in an earlier section of Nicholas' tract. Finally, it is evident that the already-noted general doctrine that the exceptive must make what would otherwise be a false proposition into a true one has also been brought to bear: cf. 7.43. 7.46 In the continuation of the text, which is to be quoted in 7.462, the objections to the congruity of the integral exceptive turn on the distinction already encountered (in [ix] of 7.315(A)) between the distributive (i.e. syncategorematic) and collective (i.e. categorematic) interpretation of 'whole ...'. In the light of this distinction, the totum ut totum, 'whole as such', doctrine, to which allusion was latterly made, is applied to a query about a further extension of the analogy between the integral and the non-integral whole. 7.461 Thus the aim of the argument is to show that there is something dubious about exception when applied in respect of the non-distributively interpreted integral whole. The base used to ground the doubt is that of, 'The whole Sortes is white apart from his hand', with the added supposition that Sortes has no feet. The analogous non-integral case suggested is that of 'Every man apart from Sortes is white'. Into the discussion is then injected the highly unfeasible principle that individual exemplifications ('subjective parts') and their universal (i.e. non-integral) wholes are related in the same way as are integral parts and their integral wholes - a principle which does away with the thesis of part-whole disparity for integrals. The analogy between the two cases cited is said to be rendered void by the discrepancy between the possibility of inferring that Plato (another 'subjective part' of man) is white from the second exemplary sentence, and the impossibility of inferring that Sortes' foot is white from the first example and its context. This impossibility arises, it is alleged, because in this case 'the whole Sortes' is not being used to allude to the whole as such. Presumably the 448

Categorematic and Syncategorematic whole as such should, according to the objector, comprise the missing feet, and it forthwith becomes uneasily evident that a trivial equivocation is the basis of the present discussion, i.e. 'whole' may sometimes be used in such a sense that it does not apply to a maimed person (as the footless Sortes is assumed to be in this case). This is clearly so much beside the point that no further time necd be wasted on either the argument's details or its general configuration. That an overt previous inclusion of the foot as a part of Sortes is the only way of inferring to that foot is stressed in the final reply to the objections (i.e. at the close of the nextquoted passage). 7.462 The ensuing reply to the objections against the congruity of exceptive integrals will stoutly and rightly reject the analogies alleged to hold between integral and non-integral ('universal') wholes. The common distinction based on the predicability of the universal whole of the part (according to which every part of man is a man) and the non-predicability of the integral whole of its part (so that it's not the case that the wall is the house) is correctly adduced to drive a wedge between the integral and the universal wholes. This wedge is correspondingly backed up by 1.328 (part-whole disparity) and ultimately by 10.324. Still in the business of driving the same wedge is the reminder provided (in the penultimate argument) of that universality of actually extant parts encapsulated in the notion of the integral whole (and borne out by the definition of collective class provided in D2 of 10.31). Any corresponding universality can be absent where a shared named such as 'man' is used to allude to its exemplifying 'parts', unless, of course, a universal quantifier (e.g. 'all ...') is overtly prefixed. No such prefixing to 'the whole ...' is essential for the integral whole to comprise all its parts. Hence the analogy between house and whole house, on the one hand, and between man and every man on the other, just does not hold. 449

Section 7 Here now is the text embodying all these points: (A) Or the question can be raised as to how it comes about that it makes sense to except something from a non-distributed integral whole, whereas it cannot be excepted from a non-distributed universal whole. In support of this point a question is raised about the following skill-testing puzzle: suppose that Sortes has no feet and that he is wholly white except for his hand; then, 'The whole Sortes is white except for his hand' is propounded. Proof in favour of this proposition runs as follows. The proposition 'The whole Sortes is white' is false, yet the only thing telling against its truth is the hand; hence except for the hand it would be true; hence the original proposition would appear to be true. Contrary to that original proposition, [however, is the following proof; given that] the whole Sortes is white except for his hand; therefore Sortes' foot is white. That this follows is obvious from [i] the fact that one has a valid sequence in: 'Every man except Sortes is white; therefore Plato is white', and from [ii] even as subjective parts are to the universal whole, so also are integral parts to the integral whole, with the result that [iii] even as it follows thus: 'Every ... except Sortes ..; hence Plato ...', so also it ought to follow thus: 'The whole Sortes ... except the hand ...; hence Sortes' foot ...'. Now if it is asserted that [iii] does not follow, it is thence obvious that [here] that which is the whole Sortes is not the whole taken as whole, and that hence [by the rule proposed earlier: 7.45(A)] an exception cannot thence be effectuated, and thus we have the same result as at the opening of the discussion [i.e. it does not make sense to except something from an integral whole]. The verdict on all this is that an exception from an integral whole can be effectuated both when it is taken universally [i.e. syncategorematically] and when it is not thus taken, for the being of an integral whole requires the positing of all of its parts. Hence the whole Sortes is 450

Categorematic and Syncategorematic identical with every part of Sortes, and this identity abides when he is alluded to universally, and when he is not thus alluded to [i.e. when the expression is interpreted categorematically]; for when the completion [e.g. Sortes] is used to complete the sign ['the whole ...'] he is then signified as being the whole as a whole [or the whole as such]. Hence it is not the case that the same goes for the universal whole as goes for the integral whole, because the universal whole is fully exemplified in each of its ['subjective'] parts, nor is there any requirement that there should be many such [exemplifiers]. Hence when it is objected [as above, on the negative side] that even as man is related to every man, so also is house related to whole house, the reply is that this is not a true contention, for the term 'man' portends only a potential multitude, the potentiality of which is brought into actuality by the addition of the sign of universality ['every . . . ' ] . In contrast, the term 'house' [already] portends an actual multitude, and hence a universal[-implying] sign [such as 'whole'] added to this term signifies only the mode of standing-for, and does not bring it about [as actual]. As for the other objection about its not following thus: 'The whole Sortes ... hence the foot of Sortes ...', the reply is that in one sense it does follow, but in another sense it does not. For if it [i.e. 'whole'] is taken so as to involve the expression 'part', then one does have a valid sequence when it is asserted thus: 'The whole Sortes is white; the foot of Sortes is a part of Sortes; hence the foot of Sortes is white'. But since 'whole' is taken on its own [without explicitation about parts in the problem-sequence], in this sense [e.g. as 'unmaimed'] it does not follow. (Vel queritur quare fiat exceptio congrue a toto integrali non distributo, cum non possit fieri a toto universali non distributo. Ad cuius evidentiam queratur de hoc sophismate: posito quod Sortes careat pedibus, et sit totus albus preter manum, tunc 451

Section 7 proponitur hec, 'Totus Sortes est albus preter manum'. Probatio: 'totus Sortes est albus1, hec est falsa, et non est instantia nisi in manu; ergo excepto pro illa erit vera; ergo prima erit vera. Contra: totus Sortes est albus preter manum; ergo pes Sortis est albus; et quod sequatur patet; quia sequitur: 'Omnis homo preter Sortem est albus; ergo Plato est albus', quia sic se habent partes subiective ad totum universale, ita partes integrales ad totum integrale; sicut ergo sequitur: 'omnis preter Sortem; ergo Plato', ita debet sequi: 'totus Sortes preter manum; ergo pes Sortis'. Si dicatur quod non sequitur, ergo manifestum est quod hoc quod est totus Sortes non est totum ut totum; ergo ab eo non potest fieri exceptio; ergo idem quod prius. Dicendum ad hoc quod a toto integrali potest fieri exceptio et sumpto universaliter et non sumpto, quia ad esse totius integralis exigitur positio omnium partium eius. Unde idem est 'totus Sortes' quam 'quelibet pars Sortis', et idem sumptum universaliter et non sumptum, cum tarnen sumitur cum signo, significatur esse totum ut totum. Non ergo simile de toto universali et de toto integrali, quia totum universale servatur in qualibet sui partes, nec exiguntur plures. Quod ergo obicit quod sicut se habet 'homo' ad 'omnis homo', ita 'domus' ad 'tota domus', dicendum quod non est verum, quia in hoc termino 'homo' nulla est mul titudo nisi potentia, que potentia reducitur in actum per additionem signi universalis; sed in hoc termino 'domus' est multitudo partium actu, unde signum universale additum termino solum significat modum supponendi et non facit. Quod obicitur quod non sequitur: 'totus Sortes; ergo pes Sortis', dicendum quod uno modo sequitur, alio modo non, quia, si sumitur cum hac dictione 'pars', sequitur dicendo: 'totus Sortes est albus; pes Sortis est pars Sortis; ergo pes Sortis est albus'; sed quia per se sumitur, sic non sequitur: BS II, 171.8 - 172.16.) 452

Categorematic and Syncategorematic 7.5 Ockham on Integral Wholes 7.51 An example of the way in which the distinction between the c a t e g o r e m a t i c and syncategorematic senses of 'whole' comes to figure in a logical summa, as contrasted with a syncategoremata-tract, may now be drawn from chapter 6 of the s e c ond part of William of Ockham's Summa Logicae. First, and on the c a t e g o r e m a t i c side, comes the already-encountered equation of Sortes and the whole Sortes; cf. 7.08, 7.21 above, and 10.325. This would by now be t r i t e enough were it not that a fascinating note on atomic mereology is added: a like equation of God and the whole God could be improper because of the l a t t e r ' s implication of divine many-partedness: (A) As far as the sign which is distributive in respect of integral parts is concerned (of the sort to which this sign 'whole' is supposed to pertain) it should be realised that 'whole' can on occasion be taken categorematically and on occasion syncategorematically. If it is taken categorematically, then it means the same as 'completed', or 'made up from all its p a r t s ' , and hence in respect of the actual meaning of the word its addition or non-addition [to a nominal term] makes no difference. Hence whatever is asserted of the t e r m when it is combined with 'whole' taken categorematically is also asserted of it when it is not combined with 'whole'. For example, if the whole Sortes runs, then Sortes runs, and if the whole man is an animal then the man is an animal. However, it could be said t h a t 'whole' is only correctly adjoined to a t e r m which in some way implies something which is composite. On this basis the following, namely, 'The whole God is seen' may perhaps not be well said, since it would appear to imply t h a t God is made up of parts. Under such circumstances the inferential sequence which goes from the t e r m taken without 'whole' to the t e r m taken in combination with 'whole' will no longer follow. Thus 'God is seen; hence the whole God is seen' will not follow, because the consequent would imply t h a t God is made up of various 453

Section 7 distinct objects. (De signo distributivo pro partibus integralibus, cuiusmodi ponitur hoc signum 'totus', est sciendum quod 'totus' potest aliquando sumi categorematice, aliquando syncategorematice. Si sumatur categorematice, tunc significat idem quod 'perfectum' vel 'compositum ex omnibus suis partibus' et sic quantum ad veritatem sermonis tantum valet non addere quantum addere. Unde quidquid dicitur de termino sumpto cum 'totus' categorematice sumpto, dicitur de illo sumpto sine 'totus'. Sicut, si totus Sortes currit, Sortes currit, et si totus homo est animal, homo est animal. Tarnen posset dici quod non convenienter additur nisi termino import anti aliquo modo aliquid compositum; et ideo forte non est dictum proprie: 'Totus Deus videtur', quia videtur importare Deum componi ex partibus; et tunc non sequitur consequentia a termino sumpto sine 'totus' ad terminum sumptum cum 'totus', sicut non sequitur: Deus videtur, igitur totus Deus videtur, quia tunc in consequente implicaretur Deum componi ex aliquibus distinctis: O II ch. 6, p.240.1 - 241.16.) 7.52 These reservations on the part of Ockham about using 'whole' of God might hold good if one clung to some of the ordinary-language associations of the Latin word 'totum', or the English 'whole', but he has himself given excellent theoretical reasons for not playing this sort of game in a logical context, wherein no such restriction necd be placed on the completions of the functor corresponding to 'the whole ...'. Indeed, from a general point of view such a restriction would inconveniently render mereological theory inapplicable where atomic (i.e. no-parted) objects are concerned (cf. 3.2941 above, and 10.35). More importantly (and puzzlingly) his own forceful strictures on those who (on analogous ordinary-language grounds) insist that tria appellata, three referents, are required where 'Every ...' propositions are concerned, have just been delivered in his previous two chapters at this point. His arguments against those who maintain the incongruity of ' Every created understander is an angel', when the under454

Categorematic and Syncategorematic standers are but one angel and no man, can clearly be adapted against his own present case for the incongruity of 'whole God ...'. The eleventh-century Garland the Computist had been at pains to argue similarly against 'Every Sortes .•.', with incidental results of special logical interest (cf. 7.43, 10.27, and HQS §2.551). Doubtless the later widespread tria appellata people were of his persuasion. Ockham, in contrast, has properly pointed out that all the truth-conditions for his 'Every ...' proposition about the angel are fulfilled, even if it turns out that only one referent is in question, and similarly we may remind ourselves, in opposition to Ockham's own present tentative qualms, that the same applies to 'the whole God'. For whether Gabriel is an atomic object or not, it follows from theses 10.325 and .327 that Gabriel, the collective class of Gabriels, and the collective class of Gabriel's elements, are all identical, notwithstanding the scandalous sound that this may have to the ears of dedicated set-theorists. There is therefore no necessary implication of Gabriel's non-atomicity when the equations about the whole Gabriel are enuntiated, nor, in general, necd any bar be placed on talking about the collective class of (or constituted by) any particular atomic object. 7.53 Next, and conceivably with tongue in cheek, Ockham trots out the usual syncategorematic interpretation of 'whole' which the medievals have let themselves in for, and which we have been following in detail since 7.082 above. As he blandly reminds the reader, the whole Sortes' running then yields each bit of Sortes also running, and this calls for the further reminder, contained in his succeeding text, that the truth about each bit requires predicates suitable to each. We may thus see for ourselves that Sortes' elbow doesn't run, so the syncategorematic interpretation of 'The whole Sortes runs' gives a falsehood. This, it may be noted, is an example which contrasts interestingly with the case of the usual 'The whole Sortes is less than Sortes', interpreted syncategorematically, 455

Section 7 and also mentioned here, since each proper bit of Sortes is certainly less than Sortes. This occasional yielding of true consequents in the course of some descensus from the whole ... to each part of ..., is about the only apparent virtue of the unfortunate syncategorematic interpretation, and in the present example even that descensus necds the shocking antecedent about the whole Sortes' being less than Sortes. 7.54 There is, incidentally, no question here of the syncategorematic interpretation's encapsulating any original pretheoretical usage of the Latin word 'totus', usually translated as 'whole', which is somehow lost in that translation into English, but which would somehow make the Latin 'Totus Sortes est minor Sorte* unshocking in a way in which 'The whole Sortes is less than Sortes' is not. The tactic of leaving such problem-sentences in the original Latin when translating the present chapter into English, adopted in FSO 104, could necdlessly convey a misleading impression to the contrary. The peculiarity is not in the Latin, but in the underlying logic. As we are now seeing, and as we are to continue to see, medieval logicians have simply thrown in their lot with that faulty logic and allowed the usually contingent vagaries of pre-logical usage (without any organised preliminary research on the matter) to take the blame (or credit, depending on one's polemical aims) for any shock delivered to the ordinary reader. The nearest contemporary parallel is Quine's allegation that 'everyone' will agree with the (obviously false) answer 'Everything' to the question 'What is there?', when in fact he is just hoping that some usage or other will bear out his own restricted doctrine of logical quantification which underlies that false answer: cf. HQS §2.3. 7.541 Here, then, is the first section of Ockham's words on the syncategorematic interpretation of 'the whole ...': (A) However, if 'whole' is taken syncategorematically, it is thuswise a single sign, distributive in respect of the integral parts, indeed, in respect of the parts, properly so call456

Categorematic and Syncategorematic ed, of that which is conveyed by the term to which it is adjoined. Thus the following proposition, 'The whole Sortes is less than Sortes' is equivalent to this: 'Every part of Sortes is less than Sortes', and the following: 'The whole Sortes runs' is equivalent to this: 'Every part of Sortes runs'. Under such circumstances, every proposition of this sort is incapable of being true unless the predicate happens to be suitable to each part of that whole which is designated by the term to which 'whole' is adjoined. (Si autem Ii 'totus' teneatur syncategorematice, sic est unum signum distributivum pro partibus integralibus, immo pro partibus proprie dictis ipsius importati per terminum, cui additur, ut ista propositio: 'Totus Sortes est minor Sorte', aequivaleat isti: 'Quaelibet pars Sortis est minor Sorte', et ista: 'Totus Sortes currit', aequivaleat isti: 'Quaelibet pars Sortis currit'; et tunc quaelibet propositio talis non posset esse vera, nisi praedicatum conveniret cuilibet parti illius totius importati per terminum, cui additur: O II, ch. 6, p. 241.17.24.) 7.55 Finally there come some important questions of principle, as well as part of the process of throwing 'totus' (in Latin) to the wolves of usage when the logical going gets tough. Without an explicitly enuntiated categorial language, some segments of the natural language in terms of which the logic is being expressed become consecrated as proper-speak (proprie loquendo), with consequent relegation of others to the contingency of logically deviant usage. All this is done in an ad hoc way to suit local argumentative convenience. No assurance is possible that what then counts as the presently deviant usage may not, in the light of more exact logic, or for differing argumentative purposes, turn out at some other juncture to be the most profitable usage for local ends. This, as it happens, is the sort of outcome which emerges in respect of usages discussed in the coming continuation of Ockham's chapter, and it reflects the central weakness of chat-analysis in general. 457

Section 7 7.551 Thus first come the points of principle which affect the nature of the 'distributed' parts of the syncategorematic whole. Both medievals and contemporaries agree in distinguishing between the senses of 'part' where integral wholes (or collective classes) and where 'distributive' (in the contemporary sense) wholes are concerned. The latter embraces what is nowadays called 'class membership', and all the 'parts' of such a class-whole are homogeneous and bear the class name: 1.32. Thus, using the neutral term 'element' in the sense intimated in 1.23 and 1.32, one may say that every element of the class of men is a man. This is why one may describe this sort of class as 'distributive': the nature in question is homogeneously distributed throughout its elements. Parts thus distributed, as Ockham is now to remind us, were called 'subjective' parts, although he seems to think that the sense of 'part' in his presupposed proper-speak at this point has in fact to be stretched in order to countenance as parts such so-called subjective parts, and there may well be something in what he says, since that is why the more loose term 'elements' was inserted into the last sentence. He rightly stresses that when 'part' is used of the parts of an integral whole these (unlike the 'subjective' parts) may well be quite miscellaneous in their various natures; it is here that the word is 'properly' used, he claims, then going on to stretch that miscellany much too far, as we are to see in a moment. For the time being, however, we may interject a realisation that here there is in fact a certain agreement between medievals and contemporaries: it is true that when speaking of the whole a certain distributive aspect ranging over objects which may be very miscellaneous in their sorts, does intervene. It may be towards this that the medieval syncategorematic interpretation of 'the whole ...' is tending in its blind sort of way. Unfortunately, except perhaps in the organic cases envisaged by Aquinas (3.4 above), no generally definitive and agreed specification of the common nature appropriate to the integral 458

Categorematic and Syncategorematic part (other than its being a 'part of ...', cf. 1.4, 2.3, 3.41) was attempted, whereas the present-day definition of integral whole (or collective class) of X reminds us that every element of such a whole has an element in common with an X. (Here 'element' can comprise the 'proper part'.) It is to an element thus specified that a descensus can indeed be made. Each part of the collective class of beetles has a part in common with a beetle, and hence (unlike the elements of the 'distributive' class) necd not itself be a beetle; cf. 10.326. 7.552 Following his version of another salutary reminder of the distinction between collective (or integral) and 'distributive' (in the medieval and present-day sense) wholes, Ockham creates a stretched local proper-speak sense of 'part' to mischievously show his contempt for people like Burleigh who are keen on type distinctions of the sort propounded in 7.07 above. He suggests that the 'proper' sense of part would cover not just the generally-recognised miscellany of sorts of integral parts, but also what Burleigh had separated out as 'essential' parts, so that the 'proper' syncategorematic intepretation of 'The whole Socrates is a part of Sortes' would not only range across parts of the same logical type (e.g. hands and feet) but also across those not even of the same logical or metaphysical type (e.g. matter and form, as well as hands and feet). This is exactly the point raised in 3.32, when an alarm was sounded about Aquinas' being ready to slip into this sort of confusion of types: (A) Thus taken [i.e. syncategorematically] 'whole' is also properly a sign and is asserted to be distributive of integral parts, while other [syncategorematic quantificational] signs are distributive of subjective parts. Such an assertion has to be taken in the sense that the other signs are distributive of the things that are subsumable under the term, and these are not parts, strictly speaking, but only are so in a broad and extended sense. But the sign 'whole' is distributive of parts in the strict sense of parts-disc459

Section 7 ourse whether or not those parts are all of the same type or of diverse types; this distribution is effectuated in respe c t of those parts which are brought into play by t h a t sign to which the sign 'whole' is added. Thus in 'The whole Sortes is part of s o r t e s ' , the distribution is effectuated in respect of no m a t t e r what sort of Socratic part, e.g. in respect of m a t t e r and form, as well as of hands, feet, and so on for all the others. (Tunc proprie est signum et dicitur distribuere pro partibus integralibus, et alia signa pro partibus subiectivis. Quod sic est intelligendum, quod alia signa distribuunt pro contentis sub termino, quae non sunt proprie partes, sed tantum large et extensive accipiendo partes. Sed hoc signum 'totus' distribuit pro partibus proprie loquendo de partibus, sive sint partes eiusdem rationis sive alterius, et hoc pro partibus illius', quod importatur per ilium terminum, cui additur hoc signum 'totus'. Sicut in ista: 'Totus Sortes est pars Sortis', fit distributio pro qualibet parte Sortis, scilicet pro materia et forma et pro manibus et pedibus et sic de omnibus aliis: O II, 241.25.35.) 7.553 Having thus threatened us with an enumeration of parts which, as 3.22 reminded us, would enmesh us in the grossest sort of type-fallacy, Ockham relents to the extent of making the distinction between this omnium gatherum sense and the integral sense of 'part'. However, he still recklessly gives the impression that he is merely narrating this distinction as being just based on local usage or contingent stipulation: (A) Notwithstanding, it should be realised t h a t at times (whether it be on the basis of the import of the words, or of the usage or convention of some user or another, this is not my concern) 'whole' effectuates its distribution only in respect of integral parts, and not in respect of those essential parts, in the style of which m a t t e r and form are called ' p a r t s ' . Sometimes, in contrast, 'whole' distributes 460

Categorematic and Syncategorematic for all parts whatsoever, be they integral, essential, or of any sort. (Verumtamen sciendum, quod aliquando - sive de virtute sermonis, sive ex usu, vel placito alicuius utentis, non curo - 'totus' tantum distribuit pro partibus integralibus, non pro partibus essentialibus, cuiusmodi vocantur materia et forma; quandoque autem distribuit pro omnibus partibus, sive sint integrales sive essentiales sive qualescumque: O II, p. 241.36.41.) And there, most regretfully, we must part from Ockham. The by now customary warning may be added, i.e. that this preliminary sample gives only a faint idea of his use of mereological notions in philosophy, logic, theology, and politics.

461

8. Venetian Harvest 8.01 Paul of Venice: Life and Style. The life of Paul of Venice extended from 1369 to 1429. He was an Italian Augustinian friar, prolific in logical works, with the truly mammoth Logica Magna, whence some crumbs are now to be drawn, as his most impressive production. Other details may be found in PBV and VVS. Some idiosyncrasies of his latemedieval disputational style will soon become evident from the text reproduced and translated hereunder. It has to be admitted that that style is at times irritating, and the text's voluminousness is not at all directly proportional to its theoretical profitability. It is on account of this difference of style that it has been thought useful to slightly adapt the layout of the present account in order to fit in with Paul's modes of expression. The nature of the adaptations will shortly be explained. 8.02 Scope of the Present Treatment. Hereunder Treatise 14 of Part I of the Logica Magna is edited, translated, and somewhat elucidated by means of notes and comments. The context is such that Treatise 14 deals with 'whole', Treatise 15 with 'always' and 'eternal', and Treatise 16 with 'infinite'. These three treatises were endowed with a certain unity by 462

Venetian Harvest the subsumption of their subject-matter under the common heading of 'Terms which are sometimes taken syncategorematically, and sometimes taken categorematically'. Given the points made in the last section concerning such a division of types of term, this heading is scarcely a good portent, and certainly causes Paul to waste a great deal of time. Fortunately, he often surmounts its attendant disabilities, as we are to see. (Incidentally, the version of the contents of Logica Magna I given in VVS ix, and in BHL, both derived without checking from the tabulation of the 1499 edition, should hence be revised to show that Treatise 14 covers 'whole' in both its syncategorematic and categorematic senses. As it stands, that version only mentions what in the Logica Magna is in fact the less amply-treated categorematic use). Although the comment will continue to be of a comparatively elementary sort, with presupposed theoretical backing from section 10, nevertheless Paul's extensive and convoluted text of itself is often insufficient to carry the present-day reader along the thread of the argument. The elucidation hence takes the form of a single connecting commentary within which stretches of the text are serially interspersed. 8.03 References and Cross-references. References to works external to the whole of this present book will continue to be made by the letter-codes annexed to the titles in the book list provided at the outset. Internal cross-references will be made by means of the usual section numbers, as heretofore. Within this Paul of Venice section, however, such section and sub-section numeration will be prefixed not only to the comment, but also, with the addition of square braces ('[' and ']') to the correlated English and Latin texts with which it is interspersed, and in particular to the key propositions within those texts. Additional parenthetical allusions will also be provided to the pages and lines of the manuscript of the work preserved in the Vatican Library (coded as M) as well as to 463

Section 8 the 1499 printed edition (coded as £). Such indications take the form of the code-letter followed by the page-number, its side (i.e. recto (r) for the front side, or verso (v) for the rear side). Then follow the column-indications, with a or b for first or second, with finally the line-number, in the case of the manuscript source, i.e. Vat. Lat. 2132. Since the Mreferences are presented at the heads of sections of the text presented below, they may be of service for occasional precise allusions, as well as for allusions to Logica Magna text outside the mereological treatise. The E-references have been added at various points so as to be be of help to readers wishing to consult this early printed version. Readers hitherto unacquainted with the common conventions now being brought to bear may care to be reminded that they result in the association of the following ordered sequence of signs with each page-number: ra, rb, va, and vb. Finally, an example would be that of 'M 71 vb 49', which alludes to page 71 of M, obverse, column b, line 49. 8.04 Edition and Translation Policy. In view of the density of the subject-matter, every variation between the manuscript and the printed edition has been taken into account in order to provide, in the end, the prima facie probable Latin text which is presented hereunder. The critical apparatus showing, among other items, such textual variants, though produced when the editing was first undertaken, has been omitted here as its complexity is scarcely a cost-effective contribution to that elucidation of mereological doctrine which is the primary end of the present work. 8.05 The Categorematic/Syncategorematic Distinction. A certain unity is first assigned to Treatises 14, 15, and 16 of Part I by their already-mentioned subsumption under the heading of terms which are sometimes taken categorematically and sometimes syncategorematicaily. Although from the point of 464

Venetian Harvest view of a theoretically satisfactory exposition of the material, this unity may be specious or restrictive, it is nevertheless necessary to recall its basis in order to guage the effect which it is to have on the treatment of the topics thus subsumed. It has already been recorded in 7.11 that the notion of the syncategorematic was attributed by the grammarian Priscian to the dialectici (K II, 54.5) and that it usually covers the functorial form-words of elementary propositions, as opposed to the usually nominal completions of the formwords, such completions being the categorematic terms of the propositions. Hence M 74vb 27, to be quoted in [8.0501] below, shows our author giving 'man', 'animal', and 'ass' as examples of categorematic terms. That 'totus' ('whole'), 'semper' ('always'), 'ab aeterno' ('from eternity') and 'infinitus' may each, in their syncategorematic use, serve as a constant term contributing to the make-up of a functor which, when completed by one or more names forms a proposition, is more evident from some of Paul's illustrations than from others. (In respect of 'totus' see the present [8.05] and [8.81] below. In respect of 'semper' and 'aeternum', see M 73 vb 51, and for 'infinitus' see M 75ra 58. The texts of these three last-mentioned allusions are not comprised within the present compendium.) Common to the various descriptions is the feature attributed to the syncategorematic occurence of the terms, namely, their standing at the head of a proposition in such a way as to affect the main copula thereof, i.e. parts of the verb 'to be', whether explicit or implicit. This distinguishing feature can be made to work well enough for 'totus', which Paul examines in its syncategorematic occurrence in Tota haec propositio est vera The whole of this proposition is true. However, the translation is already here having to be manipulated in order to keep Paul's criterion working literally in the English, and even as it now stands, that translation is 465

Section 8 unsatisfactory, unless one is going in for the witty shock tactics to be described in the next phase of the comment. For the moment, however, it suffices to note that this Latin proposition is doomed to forever lack any English translation which will show forth what Paul takes to be its syncategorematic sense, namely, 'Every quantitative part of this proposition is true', as required by his stipulations ([8.06], cf. 7.54 above). Manipulation is also necded to ensure that the 'in infinitum' of Aliquo homine in infinitum maior est Plato fulfills the syncategorematic specification when translated. In order that this sentence should contain a syncategorematic 'in infinitum' it must, according to Paul, be understood as, 'Infinitely greater than some man is Plato'. This does indeed ensure that the 'infinitely' occurs before the copula 'is', but from the point of view of the translation alone, there seems to be no reason why the expression on each side of that 'is' may not be transposed without changing the sense of the whole proposition. That Paul realises that this sort of thing is happening is evidenced by his admission that in all of the 'semper', 'aeternum', and 'infinitum' cases, the word-transpositions involved necd not necessarily induce variations from one sense to another (cf. M 73 vb 51, M 75ra 58) in contrast to what he claims is the case when 'totum' is in question (cf. 8.06, 8.81). Thus although the general range of possibilities attributed to the syncategorematic/categorematic distinction is historically of the highest importance, it already looks as though Paul's exploitation of it as a means of unifying the various cases, of which 'totus', 'whole' is one, is not very successful in accomplishing its aims. Paul's own dissatisfaction may perhaps be more definitely evidenced in his recurrence to the whole question when composing the 'from eternity ...' treatise. There 'whole', along with other terms to be mentioned in our text [8.05] below will 466

Venetian Harvest be said to be a term which is neither categorematic nor syncategorematic, but ' praegnans' (clearly better rendered as 'expansive' rather than as the 'pregnant' adopted in VVS 21, line 15, cf. note e, p. 255). Here, at any rate, is a translation of the text which covers this reclassification, the sense of which necd not now be pursued further: [8.0501] [M 74vb 25] There arises a query as to the quantity of the following propositions: Always there was a man From eternity God was Eternally the world will be and likewise for similar propositions. For the purpose of a reply to this query let it be presupposed that certain terms are categorematic, certain are syncategorematic, and certain are neither categorematic nor syncategorematic, but rather expansive terms. The categorematic terms are simple mental intentions or the spoken or written terms thereto subordinated, e.g. 'man', 'animal', 'ass', and so on. Syncategorematic terms are simple acts of the soul, or the spoken or written terms thereto subordinated, e.g. 'all ...', 'none ...', 'any ...', 'both ...', 'other ...', 'some ...', and so on. Expansive terms are of two sorts. Certain are composite, and others are simple. The composite ones are those which are made up from a categorematic and a syncategorematic term, e.g. 'some man', 'all men', 'one of them', 'both of them', and so on. However, the simple expansive terms are those which are not made up in this way, but in fact come under a single mental term made up of a categorematic and a syncategorematic one, e.g. 'always', 'from eternity', 'sometime', 'everywhere', 'somewhere', 'the whole', 'infinite', 'something', and 'nothing'. It is obvious that each of these is a simple 467

Section 8 term, as are 'man' or 'animal', and yet each comes under one composite mental term. Thus: 'always' comes under 'at all times' 'from eternity' comes under 'before' (or 'after') 'all finite times' 'sometime' comes under 'at some time' 'everywhere' comes under 'in all places' 'somewhere' comes under 'in some place' 'whole' comes under 'every quantitative part' 'infinite' comes under 'greater than no matter how much' 'something' comes under 'some being' 'nothing' comes under 'no being'. [M 74vb 52] Now none of the composite terms cited can be said to be categorematic or syncategorematic, since it is not clear why they should be categorematic rather than syncategorematic; hence the simple terms which come under the composite ones are not said to be categorematic or syncategorematic, but rather, in exact terminology, to be expansive terms. From all this anybody can easily resolve the query by saying that none of the propositions which are in question has a quantity in the logical sense of 'quantity'. Clearly we are here becoming enmeshed with the mysteries of 'mental' once again (cf. Buridan in Section 4 above), and this is another reason for not carrying the comment any further. Enough has been displayed to confirm the unsatisfactoriness, even to Paul's mind, of the distinction between categorematic and syncategorematic terms. For the moment, however, reverting to the treatise which is to be analysed, we find him introducing and using the distinction in the following manner: [8.05] In order to link more intelligibly the earlier and later stages, one must go on to recollect some facts about certain terms, namely that they are sometimes taken categorematically, and sometimes taken syncategorematically, as is the case with: 468

Venetian Harvest 'whole' (in all its genders) 'always' 'from eternity', to which may be added 'infinite' (in all its genders). These are now to be discussed in turn, beginning with the first. [8.05] (M 71 vb 49, E 56ra) Ut autem priora cum posterioribus clarius intelligantur, memoriae restat reducere quorundam terminorum notitiam: quod quandoque categorematice quandoque syncategorematice tenentur, ut sunt hi: totus, tota, totum semper, et ab aeterno, quibus annectitur: infinitus, infinita, infinitum, de quibus omnibus a primo incipiendo est successive dicendum. 8.06 General Remarks on 'whole' taken Syncategorematically. Paul's t e x t , having thus allegedly grouped together the t e r m s in question under the two headings, is now t o begin t h e t r e a t m e n t of 'whole' in its syncategorematic use. It may be of assistance to recall and amplify on points already made previously. Simply at the level of translation it has long been evident t h a t the m e r e absence of articles such as 'a' and ' t h e ' , is one of the many features of the original Latin which c r e ates ambiguities. Paul resolves these initially, as did his predecessors, by m e r e stipulation. He will thereby be enabled to propose the truth of what, even in the original Latin, are those highly startling propositions of a by now familiar type, e.g. those numbered [8.11] to [8.17] under [8.1] below. Thus 'Totus Sortes est minor Sorte' [8.11], which looks like the Latin version of 'The whole Sortes is less than Sortes', has 469

Section 8 in fact to have the less startling sense of 'Any part of Sortes is less than Sortes'. Indeed, a sophisma contained in William of Sherwood's treatise on syncategoremata (ST 54, SS 40) shows how the former version results in the conclusion that Sortes must hence be less than Sortes, as also does Burleigh's text quoted in 7.08 above. And although Paul is now to follow the usual stipulation that 'whole' as a syncategorema here has the full sense of 'every (or any) quantitative part ....', and is thus able to preserve the truth of the above-quoted [8.11], William had long ago shown his discontent with this interpretation, and would prefer there to construe 'totus Sortes' as 'Sortes considered in respect of each and every part' (Sortes ita quod quaelibet pars: ST 54, SS 40). Ockham, however, if only for example's sake, seemed content enough with the equivalence now proposed by Paul (O I, 31.114.118; but cf. 7.53 - 7.541 above). There is at least a general agreement that when taken syncategorematically 'totus' is some sort of quantifier-embodying functor. Peter of Spain dealt with the same sophisma under the heading of the De Distributionibus treatise (numbered xii) in his Summulae Logicales, but did not there bring to bear the terms 'categorematic' and 'syncategorematic', although an earlier thirteenth-century treatise, attributed to Robert Bacon, had already used this terminology in Paul's fashion when dealing with 'whole': BS I, 122; cf. 7.2(A) above. Finally, Paul's definitive association of the integral whole with quantitative parts is worthy of note, in view of the history of vaccillations on this point (3.212, 3.24, 7.07 above). Here now is the continuation of Paul's text: [8.06] Concerning 'whole' taken syncategorematically. 'Whole' is taken syncategorematically when it is placed at the head of the whole proposition, so as to have a bearing on the principal copula of the sentence in which it occurs. Under these circumstances it is substitutionally equivalent to 'Every amount', and 'whole' (in 470

Venetian Harvest the various Latin genders) may be exchanged for 'Every quantitative (or integral) part'. For example: The whole Sortes is something human becomes: Every quantitative part of Sortes is something human. [Again] This whole proposition is true is equivalent to: Every quantitative part of this proposition is true. This whole piece of wood is being burned is equivalent to this: Every amount of this piece of wood is being burned, or, as in the earlier examples, to Every integral part of this piece of wood is being burned. I do not, however, wish to assert that when 'whole' is prefixed to the whole proposition it may indifferently be taken categorematically or syncategorematically, any more than may 'every' or 'no', when they are prefixed to the whole proposition. For no one would assert that the proposition: Every man is an animal was subject to doubt or necded to have its senses distinguished, and that it could be true or false according as to whether it signified an indefinite [proposition] or a universal [proposition], and so on. Quite the reverse: when asserted, it is granted without any quibbling, and the following, namely: It is not the case that every man is animal is forthwith denied. In the same way I declare that an unqualified response is to be given when propositions of the sort which are governed by the word 'the whole ...' (in all its Latin genders) are asserted. [8.06] [Ml1vb 55] De .toto. syncategorematice tento. 471

Section 8 Sumitur enim ly .totum. syncategorematice quando toti propositioni praeponatur, principalem copulam suae enuntiationis determinando et tunc huic oratione complexae .omne quantum. convertibiliter aequipollet, et ly .totus, vel .tota. cum ly .quaelibet pars quantitativa vel integralis. convertitur. Verbi gratia: Totus Sortes est aliquid hominis convertitur in istam: Quaelibet pars quantitativa Sortis est aliquid hominis. Tota haec propositio est vera aequipollet huic: Quaelibet pars quantitativa huic propositionis est vera. Totum istud lignum comburitur aequivalet isti: Omne quantum istius ligni comburitur, vel, ut prius: Quaelibet pars integralis istius ligni comburitur. Non enim volo dicere quod ly .totum. toti propositioni praepositum possit indifferenter sumi categorematice vel syncategorematice, sicut nec ly .omnis. vel .omne., .nullus. vel .nullum., cum toti propositioni praeponitur. Nullus enim diceret quod haec propositio: Omnis homo est animal esset dubitabilis vel distinguenda, et quod posset esse vera vel falsa sic significando indefinita vel universalis et cetera. Immo, cum proponitur sine quacumque distinctione conceditur, et hoc: Non omnis homo est animal continue negatur. Eodem modo dico ad huiusmodi propositiones determinatas per ly .totus., .tota., .totum., quod cum proponuntur sine distinetione aliqua responde tur. 8.1 Truths derived from the syncategorematic 'whole'. These have been numbered [8.11] to [8.16] in the text and 472

Venetian Harvest translation below, and the first three of them follow obviously enough from the usual equivalence now adopted by Paul. Thus the prima facie shocking 'The whole Sortes is less than Sortes' [8.11] when interpreted as 'Every quantitative part of Sortes is less than Sortes', is thereby rendered true. A query might be raised concerning [8.14], especially if considered in the alternative mooted in the text, according to which it may be understood as 'The whole proposition a is a syllable'. If this is to be true in its syncategorematic interpretation, then one must assume some sort of division into principal parts only, of which the syllable is one. But plainly certain parts of a proposition necd be neither syllables nor even meaningful utterances (dictiones); blobs of ink or noise-segments, for example, necd be neither syllables nor as such meaningful, but they are still quantitative parts of the proposition. The looseness of the sense of 'part' presupposed at this point, in spite of Paul's previous stipulations, becomes even more evident in the last three of the alleged truths. Thus 'The whole Sortes is less than some part of him' [8.15], can only become true in the sense of 'Every part of Sortes is less than some part of him' if 'part' may mean something like 'component' or 'combination of components other than the whole', and a more or less fixed range of such components is assumed. Then the analysis, namely, 'Every quantitative part of Sortes is less than some part of him' can be seen as true provided that each component is a lesser 'part' than the part which consists of that component plus some other. This at least evades the obvious counter-example which would be provided by the maximal natura magna sort of part, which would consist of Sortes minus one single atom. Of this part it is untrue to say that it is less than some proper part of Sortes. Likewise, in respect of 'The whole Sortes is bigger than some part of him' [8.16] when it is endowed with its 'Every quantitative part of Sortes is bigger than some part of 473

Section 8 him' interpretation: unless there is to be an infinite regress of smaller and smaller parts, this necds to be kept true by some supposition such as that the 'some part' is an atom, and that 'every quantitative part' refers to parts qua components other than atoms. This would suffice to make true that analysis: cf. 10.35. As will become evident, however, these interpretational hypotheses by no means exhaust the possible senses of 'part' which may be brought into play in the present context, and hence are by no means definitive. The final alleged truth, 'The whole Sortes is equal to some part of him' [8.17] is presumably grounded in the fact that every quantitative part of Sortes is equal to itself. Throughout the translations presented below 'the whole' has been used for 'totus' in order to conserve the deliberately startling effect of the original. If there existed alternative English versions which would make these sentences obviously true they would be beside the point, as 8.05 and 7.54 have indicated. At any rate, use of 'the whole of ....' is just what is required to give what looks like the categorematic sense, the misplaced assumption of which is at the root of the present startling effects (except perhaps in the case of [8.6]). As the ps-Robert Bacon had rightly said, if the 'Totus Sortes' of ' Totus Sortes est albus' is taken categorematically, then a singular proposition (cf. 10.2111) results: BS I, 122. Use of 'the whole of ...' ensures just such appearance of singularity which is strictly out of place where the present syncategorematically-interpreted propositions are concerned. [8.1] From all this I deduce: (1) The truth of propositions such as the following: [8.1] The whole Sortes is less than Sortes [8.2] The whole Sortes is part of Sortes [8.3] The whole Sortes is something pertaining to a man [8.4] The whole proposition a is a meaningful utterance (or a syllable) - letting 'a' be the proposition 474

Venetian Harvest 'A man exists' [8.5] The whole Sortes is less than some part of him [8.6] The whole Sortes is bigger than some part of him [8.7] The whole Sortes is equal to some part of him. The same applies to an infinite number of such propositions, every one of which is obviously true when construed in accordance with the equivalents proposed above. [8.1] Ex his infero lo: veritatem talium propositionum: [8.11] Totus Sortes est minor Sorte [8.12] Totus Sortes est pars Sortis [8.13] Totus Sortes est aliquid hominis [8.14] Tota a propositio est dictio vel syliaha (sit a illa: homo est) [8.15] Totus Sortes aliqua sui parte est minor [8.16] Totus Sortes aliqua sui parte est maius [8.17] Totus Sortes alicui suae parti est aequalis et sic de infinitati quarum quaelibet est vera, ut patet per sua convertihilia prius nominata. 8.2 Falsehoods derived from the syncategorematic 'whole'. These have been numbered [8.21] to [8.26] below. First comes a proposition which the article-free Latin's looseness, in the absence of the stipulated syncategorematic sense, could even allow to be true. Thus 'totum' in [8.21] could allow either a whole of what is in the world or the whole of what is in the world to be in your eye (or in your wallet). The first alternative, which the unwary reader assumes could hold, is now put out of court by means of the syncategorematic sense, as Paul goes on to explain (M 72ra 28). Clearly it is then not the case that every quantitative part of what is in the world is in your eye (or in your wallet). The remaining propositions, i.e. [8.22] to [8.26], are what would be obvious truths, were it not for the syncategorematic 475

Section 8 sense in which 'totum' is now being taken, as Paul once again goes on to explain. That all are thus prima facie truisms makes their falsehood all the more startling. Thus [8.22] is the unguarded statement of the usual truth-condition for an 'Either ... or ...' proposition, i.e. at least one of its component propositional parts must be true for the whole to be true. Then 'Either triangles are trilateral or I am a donkey' is true as a whole. However, the present syncategorematic interpretation makes every part true, which allows the dragging in as true the alternative which is the false one in our example. Worse still, any non-propositional bit of the disjunctive may also incongruously be said to be true. [8.23] would likewise be unobjectionable, given the commonly admitted identity of whole Sortes and Sortes (cf. 10.325). Again, [8.24] is a common adage of traditional syllogistic, but given that 'distribution' there has to do with 'logical quantity' (cf. 5.22), it is unclear whether the syncategorematic interpretation is to involve that sense, or the usual sense covered by the category of quantity. Finally, without going into the doctrine of the suppositio of terms, although Paul's lack of comment on [8.25] and [8.26] leaves an undesirable gap, it is at least generally evident that application of the syncategorematic sense of 'totum' to subject and predicate terms of certain propositions, may render false generalisations which would otherwise be true; cf. HLM III, §1. [8.2] Then secondly I infer the falsehood of propositions such as the following: [8.21] The whole that is in the world is in your eye (or - in your wallet) [8.22] The whole disjunctive proposition is true when one or other of its parts is true [8.23] The whole Sortes is a man (or - an animal) [8.24] The whole subject of a universal proposition is distributed [8.25] The whole subject of an exclusive proposition has 476

Venetian Harvest merely confused suppositio [8.26] The whole predicate of a universal affirmative or an indefinite proposition has merely confused suppositio (or - determinate suppositio) (M 12ra 28) That all these are false is obvious since: The first [8.21] means that every amount that is in the world is in your eye (or - in your wallet); but this is false, since neither the heavens, nor England, nor some man, is in your eye (or in your wallet). The second [8.22] signifies that every part of a disjunctive proposition is true when one or other of its parts is true, and this is false. For the following, namely 'You are a donkey', is part of a disjunctive proposition, and yet is not true when one or other of that proposition's parts is true. One may also deny that the whole of a disjunctive proposition, one or other of whose parts is true, is itself true [E 56rb] and this on the grounds that not just any part of a disjunctive proposition of which one or other of the parts is true, is true. This is because the following, namely 'You are a donkey', though part of a disjunctive one or other of the parts of which is true, is nevertheless false. The third [8.23] signifies that every quantitative part of Sortes is a man (or - an animal), and this again is false. The foot or the hand are each a quantitative part of Sortes, and yet neither is a man (or - an animal). The fourth [8.24] is likewise false, since it is not the case that every quantitative part of the subject of a universal proposition is distributed, since such parts are letters or syllables, and these are not distributed [in the logical sense]. Likewise it is obvious what has to be said about the other two propositions [8.25], [8.26]. [8.2] [M 12ra 22] 2o: infero falsitatem talium All

Section 8 propositionum : [8.21] Totum quod est in mundo est in oculo tuo (vel in bursa tua) [8.22] Tota disiunctiva est vera cuius altera pars est vera [8.23] Totus Sortes est homo (vel animal) [8.24] Totum subiectum propositionis universalis distribuitur [8.25] Toturn subiectum propositionis exclusivae stat confuse tantum [8.26] Totum praedicatum universalis affirmativae vel indefinitae stat confuse tantum vel determinate. [M 72ra 28] Haec omnia sunt falsa, ut patet. la [8.21] significat quod omne quantum quod est in mundo est in oculo tuo (vel in bursa tua) quod est falsum, quia nec coelum nec Anglia nec aliquis homo est in oculo tuo (vel in bursa tua). 2a [8.22] significat quod quaelibet pars disiunctivae est vera cuius altera pars est vera, quod iterum falsum est quia hoc .tu es asinus. est pars disiunctivae et tarnen ipsa non est vera cuius altera pars est vera. Negatur etiam quod tota disiunctiva cuius altera pars est vera est vera, quia non quaelibet pars disiunctivae cuius altera pars est vera est vera [E 56rb] quia hoc .tu es asinus. est pars disiunctivae cuius altera pars est vera, et tarnen ipsa est falsa. 3a [8.23] propositio significat quod quaelibet pars quantitativa Sortis est homo vel animal, quod iterum est falsum, quia tarn pes quam manus est pars quantitativa Sortis et tarnen non est homo nec animal. 4a [8.24] consimiliter est falsa quia non quaelibet pars quantitativa subiecti propositionis universalis distribuitur, ut sunt litterae et syllabae quae non distribuuntur.

478

Venetian Harvest Et sic patet quid de alias duas [8.25] [8.26] est dicendum. 8.31

Argument against the Truth of [8.12] The general form of the argument which now follows is that of a reductio ad absurdum. It operates by joining the [8.12] now in question to an obvious truth so as to derive a conclusion which, when subjected to the syncategorematic analysis is allegedly seen to be false. Whence is inferred the falsehood of the [8.12] on which that conclusion depends. The trouble with this process in the present case is the way in which it exploits the ambiguity in the Latin words 'pars quantitativa hominis' as they occur in the expanded conclusion 'quaelibet pars quantitativa hominis est pars Sortis'. Translation of the Latin of this expanded conclusion as 'Every quantitative part of some man [i.e. Sortes] is a part of Sortes' yields a truth. However, read as 'Every quantitative part of any man is a part of Sortes', it is obviously false. As this objection, which is based on the latter of these two readings, points out, there are plenty of parts of Plato which are not parts of Sortes. [8.31] Arguments against some of the theses advanced above. First one may prove that The whole Sortes is part of Sortes [8.12] is false, since the following is valid: The whole Sortes is part of Sortes But every Sortes is a man Therefore the whole man is part of Sortes. The inferential form is valid, since it involves arguing affirmatively from something subordinate to that which subordinates it, yet the conclusion is false. Hence the premisses taken as a whole must be false. Now the minor is not false; hence the major is. The falsehood of the conclusion is obvious, since it is not the case that 479

Section 8 every quantitative part of a man is a part of Sortes; there are many, indeed infinitely many, parts of Plato which are not parts of Sortes. [8.31] [M 12ra Al] Contra quaedam dicta arguitur, et lo probando hanc esse falsam: Totus Sortes est pars Sortis [8.12] quia sequitur: Totus Sortes est pars Sortis Sed omnis Sortes est homo Igitur totus homo est pars Sortis. Consequentia bona quia arguitur ab inferiori ad suum superiorem affirmative, et consequens est falsum, igitur et antecedens; non minor, igitur maior. Quod antecedens sit falsum patet quia non quaelibet pars quantitativa hominis est pars Sortis; multae, immo infinitae, sunt partes quantitativae Platonis quae non sunt partes Sortis. 8.32 Argument against the Truth of [8.11] Here yet another ambiguity is in question, namely that of 'pars quantitativa Sortis', 'quantitative part of Sortes'. In the present case, this is being used as a term to cover the matter of Sortes, which is in some sense his part. Thus taking the matter as a quantitative part yields a part of Sortes which is not less in quantity than Sortes, thus falsifying the equivalent of [8.11], namely, 'Every quantitative part of Sortes is less than Sortes'. We have by this time already seen several phases of the history of such a generally disadvantageous equation of parts and matter (3.31, 2.36 above). [8.32] Secondly one may object by arguing that the following is false: The whole Sortes is less than Sortes [8.11]. For: It is not the case that every quantitative part of 480

Venetian Harvest Sortes is less than Sortes. Hence: It is not the case that the whole Sortes is less than Sortes. The inferential form is here obvious, and the premiss is provable on the grounds t h a t the m a t t e r of Sortes, which is one of the parts of his composite whole, is a quantitative part, and involves length, breadth, and height, all jointly contributing to the quantity of his body, which nevertheless is not less than Sortes; hence [it is not the case t h a t just any quantitative part of Sortes is less than Sortes]. [8.32] [M 12ra 55] 2o arguitur probando illam esse falsam: Totus Sortes est minor Sorte [8.11] quia Non quaelibet pars quantitativa Sortis est minor Sorte igitur: Non totus Sortes est minor Sorte. [M 12ra 58] Patet consequentia et antecedens probatur: nam materia Sortis quae est altera pars compositi est pars quantitativa, quia longa, lata, et profunda concurrens ad compositionem corporis quanti et tarnen non est minor Sorte; igitur et cetera. 8.33 Argument against the Falsehood of [8.23] It is at this point t h a t the discussion moves into an area wherein questions are raised which definitely turn upon the theory of part and whole as such, i.e. upon mereology. The objection is to propound a deliberately garbled and convoluted aspect of a by then long-extant discussion on the question of identity across t i m e . We have already seen how, as early as the period of Abelard, in the twelfth century, at least two possible positions had emerged concerning what became known as the question De Destructione Socratis ('On 481

Section 8 the Destruction of Socrates'). We saw in 2.4 how Abelard described in great detail (AD 550 - 1) a position akin to that of our contemporary Woodger (WSW) according to which any addition (or subtraction) of any part of a quantitative whole entails the destruction of that whole, and may entail its replacement by another. One thus has a succession of successively perishing entitites covered by what now have to be outrageously classified as common names, e.g. 'this-house', 'Socrates', or 'Winston Churchill', as Paul of Venice is also later to recount in detail below ([8.4320]). In contrast, the Fragmentum Sangermanense (CIA 505 - 50) surmounts the paradoxes of such 'destructivism', as it was called above, by means of the reminder that even with fewer parts, Socrates is still Socrates. In other words, there is no reason why an abiding entity should not be seen as being the collective class of its temporal as well as of its spatial parts (cf. 2.45 - 2.451 above, and HQS §4.532). We have also seen how it was in the discussion of such a situation that there had arisen what the present text describes as the 'common practice' (which we now realise goes at least as far back as Abelard's time) of taking the case involving the spatial whole of the present-Socrates (or of Abelard's present-house), and designating some small part thereof as about to be lopped off, e.g. his finger, or a stone-fragment from the house. Prior to the removal of the finger or the fragment, the spatial whole apart from the finger or the fragment is discriminated as the appropriate natura magna, major identifiable object, in the terminology of the Fragmentum Sangermanense. It is such a natura magna which in the present instance is denoted by means of the letter 'a'. One thus has the situation outlined in the syllogism given below, i.e. Immediately after this instant a will be Sortes But a is a part of Sortes Hence a part of Sortes will be Sortes. 482

Venetian Harvest On the assumption that one wishes to avoid the paradoxes of destructivism, and hence to bring temporal as well as spatial parts into the discussion, then the major premiss must here be adjudged as clearly misleading: that a will be cospatial with a later temporal part of Sortes is all that may be strictly inferred. The histories of a and of Sortes are diverse. In particular, a was no man, whereas Sortes cannot but have been a man throughout his career (cf. HQS §4.55). In the proof which is to be offered of the last-quoted syllogism's major premiss, a is identified with a later Sortes, an identification which can only hold under the destructivist fashion of describing the situation. This same identification also allows the inference to 'A part of Sortes is Sortes', which in turn founds both 'Some part of Sortes is Sortes', and 'Some part of Sortes can be Sortes'. Whence by the sleight of hand displayed in the first syllogism of the present section of text shown below, one has 'Every part of Sortes is Sortes', and from this in turn is derived the required conclusion, 'The whole Sortes is the man Sortes'. But this last affords no resting-place, since it had been shown to be false in [8.23] by recourse to the syncategorematic interpretation of 'whole'. [8.33] Thirdly one may prove the following to be true: The whole Sortes is the man Sortes (or - the animal Sortes). This is because: Some part of Sortes is Sortes There is no more reason why this should hold of one part rather than of them all Therefore every part of Sortes is Sortes. The inferential form [here used] is obvious, and the major of the two premisses may be proved thus: Part of Sortes can be Sortes But no part of Sortes can be other than it already is Hence it is already part of Sortes. The major premiss of this is obvious if, in accordance 483

Section 8 with the common practice, I take the whole of the remainder of Sortes apart from his finger (which is supposed to cease at the present moment to be part of Sortes) and denote that remainder by V , so as to argue thus: Immediately after this instant a will be Sortes, a is a part of Sortes; therefore a part of Sortes will be Sortes. The major premiss of this is proved thus: a is composed of the matter remaining when the finger is removed, and of the intellective soul, and is Sortes. Immediately after this [instant] Sortes will be composed of that matter and that form. Therefore [immediately after this instant a will be Sortes]. It can also be proved that: A part of Sortes is Sortes, provided one adopts the same supposition [as above], since: a is Sortes a is a part of Sortes Hence a part of Sortes is Sortes. The inferential form [here used] is obvious, and the major is proved thus: a is a man who is a composite formed from a body and an intellective soul This man is no other than Sortes Therefore [a is Sortes]. [8.33] [M 12rb 1] 3o probatur hanc esse veram: Totus Sortes est Sortes homo vel animal. Nam: Aliqua pars Sortis est Sortes, et: Non est maior ratio de una quam de qualibet. 484

Venetian Harvest Igitur quaelibet pars Sortis est Sortes. Patet consequentia et antecedens pro maiori parte probatur, nam: Pars Sortis potest esse Sortes, sed Nulla pars Sortis potest esse aliud quam iam est, Igitur et iam est pars Sortis. Antecedens patet in casu commune signandum totum residuum Sortis praeter digitum qui desinat esse pars Sortis per positionem de presenti. Tunc noto illud residuum a digito per a, et arguo sic: Immediate post hoc a erit Sortes, et a est pars Sortis Igitur pars Sortis erit Sortes. Antecedens sic probatur: a est compositum ex materia residua a digito et anima intellectiva, et est Sortes; Immediate post hoc erit compositum ex illa materia et illa forma; Igitur et cetera. In eodem casu potest probari quod: Pars Sortis est Sortes, quia: a est Sortes, et a est pars Sortis; Igitur pars Sortis est Sortes. Consequentia patet et minor probatur. Nam: a est homo quia est compositum ex corpore et anima intellectiva, et: Non alius homo quam Sortes. Igitur et cetera. 8.41

Reply to Objection [8.31] The reply to the preceeding [8.31] produces a set of rules intended to filter out certain inferences which are 485

Section 8 invalid when expressions such as 'totum animal', 'whole animal', are in question. The rules look impressive enough, but their ad hoc nature, and the possibility of future unintended applications thereof leading to further dispute, suggest that it would have been simpler to concentrate on some direct distinction, e.g. that between 'the whole of some man' (or 'the whole of some animal') and 'the whole of any man' (or 'the whole of any animal'). Thus, or in some other way, the senses of 'totus homo' or of 'totum animal' could have been clarified. It is certainly not cost-effective, relatively to the central mereological project, to occupy several chapters for the analysis of the rules as they stand. They are hence left without detailed comment, although those already immersed in the history of medieval logic may find them interesting. From the examples provided it is evident that the 'subordination' on which the rules turn is that of an individual (the subordinated) to its specific and generic distributive classes (the subordinators). [8.41] Replies to these Objections. In reply to the first [8.31] I deny the validity of the inferential form, because although the argument takes place affirmatively through a middle term, it nevertheless relies on the previous distribution of the subordinating term. This happens because the term 'man' has confused and distributive suppositio in the conclusion, and occurs not in recto, but in obliquo [using the equivalences suggested under [8.06]]. Hence the following rules which hold in respect of the other distributive signs should be followed: The first rule is that from the subordinated to that which subordinates it and is distributed by the word 'whole', the inference is not valid. Hence this does not follow: The whole Sortes is something of Sortes Hence the whole man (or - the whole animal) is 486

Venetian Harvest something of Sortes. This is because it would then be necessary [under the syncategorematic interpretation] to grant that just any part of a donkey or of a horse was something of Sortes, and this is false. The second rule: an argument which proceeds by means of an appropriate middle term from that which subordinates and is distributed by the word 'whole' to that which is subordinated, is valid. For this follows: The whole man is not-white, and is Sortes Hence the whole Sortes is not white. Likewise, this follows: The whole man is black, and is Plato Hence Plato is black. The third rule: an argument which proceeds from that which is subordinate to its subordinator undistributed by the word 'whole', and by means of an appropriate middle term, is valid. For one may grant: It is not the case that the whole Sortes is known by me, and Sortes is [a man] Hence it is not the case that the whole man is known by me. Again: It is not the case that the whole of wood a is seen by you, and the wood a is just one piece of wood. Hence it is not the case that the whole of wood is seen by you. From this follows: The fourth rule: from a subordinator undistributed by the word 'whole' the argument to that which is subordinated is not valid. Hence this does not follow: It is not the case that the whole man is less than 487

Section 8 Sortes, and Sortes is a man Hence it is not the case that the whole of Sortes is less than Sortes. (It is obvious in this case that the premiss is true and the conclusion false) [cf. [8.32] and [8.42]]. [8.41] [M 12rb 20] Ad haec omnia respondetur: Ad primum: nego consequentiam, quia licet arguitur cum medio affirmative, arguitur tarnen cum distributione praecedente superius, eo quod ille terminus .homo, stat in consequente confuse et distributive, non in recto, sed in obliquo. Unde observandae sunt haec regulae quae observantur in aliis signis distributivis: Prima est quod ab inferiori ad suum superius distributum per li .totum. non valet argumentum; unde non sequitur: Totus Sortes est aliquid Sortis Igitur totus homo vel totum animal est aliquid Sortis, quia tunc oporteret concedere quod quaelibet pars asini vel equi esset aliquid Sortis, quod est falsum. Secunda regula est: a superiori distributo per li .totum. ad suum inferius cum debito medio est bonum argumentum. Sequitur enim: Totus homo non est albus, et Sortes est Igitur totus Sortes non est albus. Similiter sequitur: Totus homo est niger et Plato est Igitur totus Plato est niger. Tertia regula est: ab inferiori ad suum superius non distributum per li .totum. cum debito medio est bonum argumentum. Sequitur enim: Non totus Sortes cognoscitur a me et Sortes est Igitur non totus homo cognoscitur a me. Non totum a lignum videtur a te, et lignum a est 488

Venetian Harvest unum lignum Igitur non totum lignum videtur a te. Ex isto sequitur: Quarta regula: quod a superiori non distributo per li .totum. ad suum inferius non valet argumentum. Unde non sequitur: Non totus homo est minor Sorte, et Sortes est homo Igitur non totus Sortes est minor Sorte. Patet quod in casu antecedens est verum et consequens falsum. 8.42 Reply to Objection [8.32] This reply usefully, if not always intelligibly, removes the Aristotelian 'matter' of Sortes from the status of quantitative part, when used in association with the likewise Aristotelian notion of 'form' (on which see 3.24, 3.32). It has already become evident from 3.312, for instance, that such a use of this notion of matter may tend to tone down the crucial distinction between X-parts and parts-of-X. The relegation, now to take place, of matter to the qualitative or essential type of part, thereby taking it outside the scope of mereology, was adumbrated by the Burleigh quoted in 7.07 above. [8.42] In reply to the second objection [8.32] I deny that It is not the case that every quantitative part of Sortes is less than Sortes. I also deny that The matter which is one or other of the parts of the composite is a quantitative part. In order that a reply framed in this fashion may be understood, it should be noted that Sortes has parts of two types, namely: [i] Qualitative: this is one or other part of the compo489

Section 8 site, i.e. the whole consisting of that matter, with which is combined the soul, and [ii] Quantitative: this is less than the whole, containing less matter than the whole. It is this which with another quantity constitutes a whole of a certain amount, and of which none is potentially something else. From this it is clear that the matter of Sortes, or his body, is not said to be a quantitative or integral part, since it is perfectible by the soul, and is in potency with respect to the soul; this holds in spite of the fact that the two together make up a body which has quantity. Likewise it can be said of the matter of a horse or an ass, that while it is also a certain amount, it is nevertheless not a quantitative or integral part; this is because it is not less than the whole, being in fact made actual by the form, and in potency with respect to the latter. Strictly speaking it is called a qualitative or essential part. Hence [it is not the case that the matter which is the other part of the composite is a quantitative part]. [8.42] [M 12rb 47] Ad secundum argumentum [3.2]: Nego quod: Non quaelibet pars quantitativa Sortis est minor Sorte. [E 56 va] Et etiam nego quod: Materia quae est altera pars compositi est pars quantitativa. Et ut huiusmodi responsio intelligatur est notandum quod in Sorte est duplex pars, scilicet: Qualitativa, quae est altera pars compositi, videlicet tota illa materia cum qua componitur anima, et Quantitativa, quae est minor toto, minorem continens materiam quam totum quae cum alia quantitate constituit 490

Venetian Harvest aliquod totum quantum quarum nulla est in potentia ad aliam. Ex hoc patet quod materia Sortis vel eius corpus non dicitur pars quantitativa vel integralis, quia perfectibilis est per animam, et in potentia respectu eiusdem, non obstante quod ambo constituantur corpus habens quantitatem. [M 72rb 6l] Consimiliter potest dici de materia equi vel asini quod est quanta, non tarnen pars quantitativa vel integralis, cum non sit minor toto, immo actuatur per formam et est in potentia respectu eiusdem, sed praecise vocatur pars qualitativ a vel essential is; quare et cetera. 8.431 First reply to [8.33] In effect one now has the theoretical elaboration of the Sangermanense thesis [8.33], which allows identity across time in spite of changes in bulk. Thus the whole of the remainder apart from the finger is in no sense a candidate for being Sortes, since the latter involves all his parts (and not just some particular spatio-temporal segment or other): as the second alternative of the passage below has it, the soul (i.e. the form) joins with all the matter and not just with any part thereof. This agrees with Aquinas' view (3.4). Again, if one confines oneself to purely material considerations (as opposed to formal, quidditative, ones) then plainly Sortes can in no way be identified with the remainder-matter. Thus two incorrect construals of the remainder in question are accounted for. It is then confirmed (M 12va 11) that changes in matter necd not affect the identity of the 'composite' in question, e.g. tree-matter might become Socratic matter. (The text regards this as a fanciful supposition: a Sortes using a tree-derived cellulose slimming-diet might nowadays exemplify this fancy). [8.431] To the third objection [8.33] various people give subtle replies. 491

Section 8 First there is, to the case supposed, one reply which involves a clarification of what you understand by the whole of the remainder apart from the finger. Either you understand a single composite formed from the r e m a i n d e r - o f - t h e - m a t t e r - a p a r t - f r o m - t h e finger and from the intellectual soul, and under this interpretation we have an impossible case, since the soul joins with the [whole of the] m a t t e r , and not [as in this objection] with any part thereof; or you understand only the m a t t e r remaining apart from the finger, and do not count the intellectual soul along with it, and then this present reply asserts that Sortes will never be that matter, and consequently neither will he be that which remains apart from the finger. Now it can indeed properly be granted that immediately after this instant Sortes will be composed from that residual m a t t e r and that soul, and t h a t for the moment he is not thus composed. And although immediately after this instant the soul may be compounded with m a t t e r other than t h a t with which it is now compounded, nevertheless the composite will remain identical, since a change of m a t t e r does not change the composite. (Indeed, were the soul (to speak fancifully) to be compounded with a t r e e , that composite would still be identical with the composite which at present is Sortes.) [8.431] [M 72va 2] Ad tertium argumentum [8.33] diversi subtiliter respondent. distinguendo Dicit enim primo una responsio ad casum quid intelligis per totum residuum praeter digitum: vel unum compositum est materia residua a digito et anima intellectiva, et sic casus est impossibile, quia 492

Venetian Harvest anima componitur cum materia et cum nulla eius parte; vel intelligis solum materiam residuum a digito non connumerando animam intellectivam, et tunc dicitur quod Sortes numquam erit illa materia et per consequens nec illud residuum a digito. [M 72va 11] Verumtamen bene conceditur quod: Sortes immediate post hoc erit compositus ex illa materia residua et illa anima, et tarnen iam non est sic compositus, et licet immediate post hoc componet cum alia materia quam iam componitur, tarnen manebit idem compositum, quia varietate materiae non varietur compositum. Immo, si per imaginationem illa anima componeretur cum arbore, adhuc illud compositum esset idem homo qui nunc est Sortes. 8.4311 Criticism of [8.431] Two of the alleged presuppositions of the Sangermanense-type view [8.431] are next made explicit and subjected to attack. First is attributed some sort of opposition between the form's information of a matter as a whole, and its information of the part of the matter. Secondly, this view is also said to call for no limit on the change of matter, as long as the form (the intellectual soul in the case of man) remains the same; hence ensue two absurd consequences. First the begetter (Sortes père, with his intellectual form informing one lot of matter) and the begotten (Sortes fils, the matter of whom is informed by the form of his appropriately perishing father) may be identical. Second, given a single form informing two matters an identical man can meet, fight, kill, and bury himself. (There are indefinitely many possible troubles, of course, with these contentions. They could be taken as yet further examples of what may go wrong when notions having a perfectly intelligible starting point 493

Section 8 (as in e.g. 3.1 above) are allowed to take on a life of their own, so to speak, without the background guardianship of systems such as those in 10. Or again, the absurdities could be the result of an unwarranted preparedness, akin to what has been observed in Wyclif (6.2), to stretch what is involved in being a man, beyond the limits normally envisaged in the theory of man. Certainly the Sangermanense theory as such necd have nothing to do with them.) [8.4311] Now although this reply is a subtle one, it nevertheless supposes two things which are not taken in a physical sense. In the first place, it supposes that the intellectual soul is combined with the whole of the matter and not with any part of it. But against this: every part of the aforementioned matter is living; hence every such part is actuated by the form. Hence the form informs and actuates every part, and hence is combined with every such part. In the second place, this reply involves saying that whatever the extent to which the matter of a man may be changed, as long as the intellectual soul remains the same, he always is the same man. But I prove that this is not the case, for, if one grants this reply [and its second accompaniment, as just stated], then there follow two impossible conclusions. The first is that an identical man can generate himself. This is proved thus. The matter from which he comes does not pertain to the being of a man. Now suppose that Sortes generates and produces a foetus which remains in the womb until the instant when the soul is to be infused into it; then I require that God brings it about that Sortes perishes and infuses Sortes' soul into that body. Under these circumstances it is clear that that which is begotten is Sortes, and that which begets is Sortes. This is the result of supposing 494

Venetian Harvest that for the numerical identity of Sortes, nothing is required save identity of soul, and not of matter, and so on. The second conclusion which follows from the reply now in question is that the same man will go from Oxford to Rome, waylay himself, have a fight with himself, and end up by killing himself, finally coming back to England and telling his friends how he buried himself in Rome. This conclusion follows immediately from the fictitious supposition that the same soul can actuate two matters, one in Rome and the other in Oxford. Hence [the reply [8.431] is invalid]. [8.4311] [M 12va 19] Licet haec responsio sit subtilis, ipsa tarnen ponit duo quae non sunt physice dicta: Primo enim, ponit quod anima intellectiva componitur cum tota materia et cum nulla eius parte. Contra: quaelibet pars dictae materiae vivit, ergo quaelibet talis actuatur per formam. Igitur forma informat et actuat quaelibet partem et per consequens cum quaelibet tali componitur. Secundo, dicit haec responsio quod quantumcumque varietur materia hominis, dummodo maneat eadem anima intellectiva, semper est idem homo. Sed probo quod non, quia data ista responsione sequuntur duae conclusiones impossibiles: Prima est quod idem homo potest seipsum generare. Probatur: nam ex quo materia non est de esse hominis. Ponatur quod Sortes generet et producat foetum qui remanet in utero usque ad instans infusionis animae. Tunc volo quod deus corrumpat Sortem et infundat animam eius illi corpori. Isto posito, patet quod ilium generatum est Sortes et illud quod generat est Sortes, ex quo ad identitatem numeralem Sortis non requiritur nisi identitas animae et non materiae, et ceterae. Secunda conclusio est ista sequens ex hac responsione: 495

Section 8 quod idem homo transibit ab Oxoniense usque Romam et ibidem obviabit sibi ipsi et cum se ipso pugnabit et demum interficiret seipsum et demum reverteretur ad Angliam et dicet amicis suis quomodo se sepelivit seipsum Romae. Haec conclusio statim sequitur supposito per imaginationem quod eadem anima actuaret duas materias, unam Romae et aliam Oxoniense, quare et cetera. 8.4320 Second Reply to [8.33] Now comes what is, in fact, the Woodgerian destructivist thesis (cf. 8.33). According to this thesis, as here outlined, the Sortes which is the later a, the later natura magna, is in fact one of many parts of a single Sortes. But each such part is allegedly a Sortes, and would presumably correspond to a spatial and temporal part of the single whole Sortes. To evade the obvious conclusion that 'Sortes' is now a common noun which names many objects, each of which is a Sortes, the supporters of this view claim that such a noun cannot but signify many things, no one of which is part of another: but 'Sortes' now names many things, one of which (the collective class of all Sortes' parts) has many Sorteses (i.e. the many temporal parts) as its parts. So the things named by 'Sortes' are not all mutually exclusive. Hence 'Sortes' is not a common noun if the latter is as was specified (i.e. with all its referents non-overlapping). [8.4320] The second reply [to objection 8.33] asserts that Sortes is a part of Sortes, but that the Sortes in question is not part of himself, but rather of another Sortes, the situation being that Sortes is one per se being composed of many things, every one of which is a Sortes, and every one of which is part of a single Sortes who is Sortes in such a fashion as not to be part of any other Sortes. Consequently, under the circumstances outlined, a is both Sortes and is a part of 496

Venetian Harvest Sortes. But then there are infinitely many things in the composite consisting of a and the finger, and every one of them is both Sortes and a part of Sortes. And if on the basis of this it is asserted that the word 'Sortes' is now a common term, since one application thereof signifies many men and is predicated of these same men, this inference is denied. For such people [as support the present reply] assert that for a term to be a common term it must signify many things, no one of which is a part of another, and this does not obtain in the situation described. [8.4320] [M 72va 42] Secunda responsio dicit quod Sortes est pars Sortis, sed non idem Sortes est pars sui ipsius, sed unius alterius Sortis, ita quod Sortes est unum ens per se compositum ex multis, quorum quodlibet est Sortes, et quodlibet illorum est pars unius Sortes qui sic est Sortes quod non est pars alterius Sortis, ita quod in casu isto a est Sortes et pars Sortis, immo infinita sunt in composito ex a et digito, quorum quodlibet est Sortes et pars Sortis. Et si ex hac arguitur quod li .Sortes. sit terminus communis quia unica impositione significat plures homines et de eisdem praedicatur, negatur consequentiam. Dicunt enim quod ad hoc quod aliquis terminus sit communis, requiritur quod significat plura quorum unum non est pars alterius, qualiter non est in propos ito. 8.4321 First Counter-reply to the Destructivism of [8.4320] The last-described situation may be put out of court (it is now claimed) by considering the classification, relatively to pre-theoretical modes, of the various stages of the process being considered. First we have Sortes still in possession of his soon-to-be-amputated-finger, a stage now to be denoted by 'b' and then we have Sortes devoid of his finger, and already, as earlier, denoted by 'a'. One can assume that b is a 497

Section 8 man; if a is not of the same species (i.e. man) then neither is it the Sortes-man. If a is of the same species (i.e. man), then both it and b are primary substances, totally discrete, neither of which is p a r t of the other. (One point of this last alternative is to re-establish 'Sortes' as a common noun, contrary to the a t t e m p t , in [8.4320], to evade this allegedly absurd consequence of the destructivist position). In general, the unease here displayed about the classificatory status of the various man-stages, is most significant and proper (cf. HQS §4.551). [8.4321] Against this reply [8.4320] one may counterobject as follows: Let b be the composite formed by a and the finger. I ask whether a and b belong to the same subordinate species or not. If they do not, and b is a man, then a is not a man, nor is it Sortes. If they do belong to the same subordinate species, it follows t h a t they are therefore individuals and come under the human species and are primary substances; hence a as well as b is a demonstratively-indicated object (for it is proper to a primary substance to be a demonstratively-indicated object). Under these circumstances it follows t h a t neither of them is part of the other [and hence 'Sortes' may be a common noun]. [8.4321] [M 72va 55] Contra istam responsionem arguitur sic: et sit b compositum ex a et digito; quaero ergo si a et b sunt eiusdem speciei specialissimae vel non. Si non, et b sit homo, igitur a non est homo [E 56vb] nec Sortes consequenter. Si sunt eiusdem speciei specialissimae, igitur sunt individua et supposita speciei humanae et primae substantiae et per consequens tam a quam b est hoc aliquid quia proprium est primae substantiae esse hoc aliquid, et si sic, sequitur consequenter quod neutrum illorum est pars alterius.

498

Venetian Harvest 8.4322 Second Counter-reply to [8.4320] Another feature of the destructivist position, as outlined in [8.4320], is that it makes an object such as Sortes into a composite of infinitely many parts, as was there made explicit. Hence the term 'man' covers an infinity, as also does 'animal' (since Sortes has both of these predicable of him). But now one cannot say that 'animal' is a term more extensive in its application than is 'man' (and this goes contrary to the pretheoretical assumption on such extensions). Again, such infinities cannot be said to perform any given action, either as a whole (e.g. 'Every man runs') or as a part (e.g. the fingerless Sortes, the a) since actions pertain to the supposition, the whole of some individual, and not to its parts (cf. 3.5). Except in the case of synecdoche, the taking of the part for the whole as a literary device, it is Sortes who is said to chop the wood, not his hand. [8.4322] Secondly, it would follow that the term 'animal' is no more extensive than the term 'man', since both would have infinitely many things coming under them. Indeed, it would follow that all of the following are impossible: Every man runs Every many argues Every man speaks Every man eats Every man drinks This is because it is impossible for infinitely many men to perform the same action: indeed, under the circumstances a does just nothing, since the performance of actions is to be attributed to the objects coming under a term, and not to their parts. [8.4322] [M 12va 62] Secundo sequitur quod non communior est ille terminus .animal, quam ille terminus .homo., quia uterque illorum habet infinita supposita. Immo sequitur quod quaelibet istarum est impossibilis: Omnis homo currit Omnis homo disputat 499

Section 8 Omnis homo loquitur (vel comedit, vel bibit) quia non est possibile quod infinita homines faciant eundem actum. Immo a in casu isto nihil agit, quia actiones sunt suppositorum et non partium. 8.4323 Third Counter-reply to [8.4310] The theory of suppositio allows of inference from, e.g., 'All men are animal' to 'This man is animal and that man is animal and ....' (and so on throughout all the men). The term 'animal' is here said to have confused and distributive suppositio (cf. HLM III, §1). But where there are infinitely many men (as [8.4320] allows) the required determinacy of the conjunction resulting from such a descensus to the singulars ('this man ...' and 'that man ...', and so forth) fails, because an infinite number is a number greater than any assigned finite number (as [M 75rb 4] is to reiterate). This is taken to be a reductio ad absurdum of the proposed position. It is certainly an interesting juxtaposition of the elucidation of quantified discourse with the possibilities of infinity. [8.4323] In the third place it would follow that from the terms 'man' or 'animal', when they have confused and distributive supposi tio, one would not be able to have a descensus to all their singulars; this is obvious, since their corresponding singulars are then infinite in number, and one is not able to have a descensus to infinitely many things; therefore [the reply is absurd]. [8.4323] [M 72vb 7] 3o sequitur quod sub ille termino .homo, vel .animal, stante confuse et distributive, non contingit descendere ad omnia eius singularia. Patet quia infinita sunt singular ia illorum sed non contingit descendere ad infinita, ergo [et cetera]. 8.4324 Final three Counter-replies to [8.4320] In the fourth reply the full force of destructivism in 500

Venetian Harvest the matter of successive temporal parts is exposed. It follows from that theory that any addition brings about a further entity, e.g. a growing Sortes is in fact a succession of Sorteses, and presently existing men only now begin to exist (given that each changes in some way all the time). These and other obvious consequences are here used as reductiones ad absurdum of destructivism. In the fifth reply the consequences are continued in respect of action-descriptions, commitments from past lifehistory, and so on. In the sixth and final reply there is a reversion to the question concerning the threatened common-noun status of a proper noun such as 'Sortes'. It is here claimed that des­ tructivism cannot make its claim that 'Sortes' is not a common term cohere with medieval logical suppositio theory. In the absence of a detailed analysis of that theory (for which see ULM III, §1) the rather interesting, though non-mereological, details of this claim cannot be followed up here. [8.4324] In the fourth place, according to the supposition adopted, it would follow that from any addition whatsoever made to Sortes there would emerge another Sortes and another man. The argument is obv­ iously valid upon inspection of the supposition, but the conclusion is false, since it would follow that you were never baptised, that you never saw me before, and that you never ate or drank, or did anything at all; this is because you never existed at all, it being granted that you now begin to be a by means of an addition effect­ uated for the first time. In the fifth place it would follow that on any day infinitely many men would be generated, and indeed that the same man could not object or reply more than once, likewise as regards being insulted or walking. Hence in the course of a disputational exercise it is open to you to grant, to deny, or to put in doubt anything you 501

Section 8 please, and yet also to deny that you had granted, or denied, or doubted those very same things. Indeed, you could deny [in the course of a disputation] that you are committed to anything, that any position is to be attr­ ibuted to you and taken to be relevant to the question in hand, since all such claims would be false and irrat­ ional, as is obvious. Yet the inference is proved on the grounds that at no matter what instant some new frag­ ment is added to [or subtracted from] every man, and this for the first time. Finally one may object on the basis of the opinion put forward by the person who asserts that there are infin­ itely many men in Sortes, every one of which is a Sor­ tes, and yet who also holds that the word 'Sortes' is not a common term. Thus I select the proposition, 'Every one of these is Sortes', uttered while pointing to those in­ finitely many men in Sortes. I then ask whether the 'Sortes' which occurs in the predicate has merely con­ fused suppositio or discrete suppositio [cf. HLM III, §1]. Assume it to be merely confused; 'Sortes' is then a com­ mon term, and this negates the position opined. Suppose it is discrete: 'Sortes' then stands for a or for b or some other part. If it stands for b, then any of the things pointed to is b; however, a is one of those things; hence a is b, and consequently is not distinct from b. But this is contrary to the opinion described. Should it be asserted that the word 'Sortes' stands for a or for some other part, one may argue as before, and show how this opinion leads to a contradiction. [8.4324] [M 72vb 10] 4o: Ex ista positione sequitur quod quacumque additione facta ad Sortem resultat alius Sortes et alius homo; consequens patet intuenti positionem, et consequens est falsum quia tunc sequeretur quod numquam fuisse baptizatus nec prius vidisses me nec umquam comedisti nec bibisti nec aliquam actionem fecisti, quia 502

Venetian Harvest numquam fuisti dato quod iam incipias esse a per additionem factam nunc primo. 5o: Sequitur quod in qualibet die generantur infiniti homines, immo quod non contingeret eiusdem hominem bis disput are vel respondere, iniuriari, vel ambulare, et sic in arte obligatoria habes concedere et negare vel dubitare quidquid tibi placet, et negare te concessisse, negasse, vel dubitasse idem; immo quod tu non es obliga tus  aliquid est tibi positum et ad rem admissum, quae omnia sunt falsa et contra rationem, ut patet. Et con­ sequentia probatur quia in quaelibet instanti cuilibet homini est aliqua ρarticula primo addita. Ultimo arguitur contra fundamentum opinionis dicentis quod infiniti sunt homines in Sorte quorum quilibet est Sortes et tarnen li .Sortes, non est terminus communis. Et apio istam propositionem .Quilibet istorum est Sor­ tes. demonstratis illis infinitis homini bus in Sorte; tunc quaero si li .Sorte, a parte praedicati supponit confuse tantum vel discrete; si confuse tantum igitur est term­ inus communis, quod negat positio. Si discrete, igitur stat pro b vel a vel aliqua alia parte. Si pro b igitur quaelibet illarum est b. Sed a est aliquis illarum, ig­ itur a est b et per consequens non distinguitur a a b , quod est oppositum opinionis. Si autem dicatur quod li .Sortes. stat ibi pro a vel pro aliqua alia parte, arguatur ut prius deducendo opinionem hac ad contradictionem. 8.433 Third Reply to [8.33] After the dialectical discussion, Paul brings forward what he considers to be the proper reply to destructivism. He upholds what is, in effect, the Sangermanense thesis (cf. 8.33), according to which Sortes is a being which endures as the same Sortes, notwithstanding part-removals, and as 10.323 confirms when it says that nothing is a proper part of itself. 503

Section 8 Thus Sortes is in no circumstances a part of Sortes. In par­ ticular, the natura magna constituted by Sortes-apart-fromhis-finger, which appears to become Sortes on the cutting off of the finger, is not the Sortes-man, but may be seen as fir­ st a spatial and then as a temporal part of Sortes. Whether parts such as the finger are added or subtracted makes no difference to the theoretical situation. [8.433] The third reply, a reply which is the true reply, asserts that Sortes is not a part of Sortes, and denies that a is Sortes or a man, given the conditions outlined. This is because for something to be said to be a man it is not enough that it should just be a composite of body and intellectual soul. It is further required that the thing should be an object which is a per se unity, one which does not subsist by being a part of something or other. This description does not apply to a, since it is not a per se unity. In accordance with this reply, the following conclusion may be granted: a and b will exist tomorrow, but tomorr­ ow they will not be a and b, but rather, to speak more exactly, they will be a. This is based on the assumption that the whole of what remains apart from Sortes' finger is a, and that the finger is b which will be cut off tomorrow. Again, it does not follow that because a and b will be Sortes, but a and b are not Sortes, therefore Sortes is not Sortes, it now being granted that Sortes is a to whom the part b is to be joined immediately after this. This is no more valid than the inference from 'Something white will be Sortes' and 'No white thing is Sortes' to the conclusion that Sortes is not Sortes. The same applies to many other similarly invalid arguments. [8.433] [M 72vb 43] Tertia responsio, quae vera est, dicit quod Sortes non est pars Sortis,  a in casu isto est Sortes vel homo, quia ad hoc quod aliquid 504

Venetian Harvest dicatur homo non sufficit quod sit compositum ex corpore et anima intellectiva. Sed cum hoc requiritur quod sit aliquid per se unum non existens pars alicuius; modo sic non est de a, cum ipsum non sit per se unum. Iuxta hoc conceditur ista conclusio quod a et b erunt cras, sed non erunt a et b, sed praecise a, supposito quod totum residuum praeter digitum Sortis sit a, et digitus , qui cras erit abscissus. Item non sequitur: a et b erunt Sortes, sed a et b non sunt Sortes, igitur Sortes non est Sortes, supposito quod Sortes sit a cui immediate post hoc adveniet sibi pars h, sicut non sequitur: aliquid album erit Sortes, et nullum album est Sortes, igitur Sortes non est Sortes, et sic de aliis multis, et cetera. 8.5 Objections to this Last Reply We have now arrived at the lead-in to the conclusion of Paul's treatment of the syncategorematic sense of 'whole'. This takes the form of three objections to [8.433], these being of various levels of mereological relevance. In the first of the three, i.e. [8.51], there occurs a peculiarly extraneous complexity, in that for some reason Paul conjures up an imaginary objector who would reformulate the usual definition of animal as animated sensitive substance, so that it would become anima ted sensible substance instead. This revised definition is then used to found what could be a mereological point, relatively to the above discussion, namely that the per se unity of Sortes, which ensures that he is not a proper part of himself, applies to animals in general, and hence a clause to this effect should be added to the definit­ ion of animal. But this will then lead to what is taken to be an absurdity, namely that 'animal', contrary to common doct­ rine, will become a 'connotative' term (roughly speaking, an adjectival rather than a substantival term), since part of its definition takes us into another (non-substance) category, and 505

Section 8 it is no longer an absolute term. (On connotative terms, see HQS 5 96 - 123, 168.) In [8.52], the second objection, one foundation of the illusion that the natura magna is a substance in the full sense is reiterated: how can the remainder-apart-from-thefinger be classified, save as man or animal? After all, it is a compound of body and an intellective soul. In like fashion, [8.53] will insist that classifications of the entity with which one is faced both before and after the amputation of the finger in no wise differ. Hence the natura magna is a man or an animal. (Unless there is some mix-up in the letters used for the examples, it now appears that the hapless a (i.e. Sortes-less-a-finger) is to have yet another finger removed.) [8.51] Against this reply [8.433] the following objections are raised: [8.511] 1) It thence follows that this definition is not sound, namely: animal is animated sensible substance, and that we should rather say that animal is animated sensible substance which does not subsist as part of some other thing which is a per se unity. It thence follows that the term 'animal' is a connotative term, since there occurs in its definition a term of some other category, namely the term 'per se'. [8.512] 2) [One may also object] as follows: a is composed of a body and an intellective soul, hence a is an animated sensitive substance; hence a is in the category of substance, and hence under some species. But it would not appear to come under any species other than man or animal. Hence a is man or animal, and this conclusion is contrary to the supposition originally adopted. [8.513] 3) Were a to have a finger cut off, then a would be Sortes; but the latter would still rightly now be described as a composite of intellective soul with 506

Venetian Harvest matter, as he formerly was. Hence Sortes now is Sortes (a man or an animal) and then was Sortes on grounds identical in both cases. Hence [a is man or animal]. [8.51] [M 72vb 59] Contra istam responsionem arguitur sic: [8.511] Nam data illa sequitur quod haec definitio non esset bona: animal est substantia animata sensibilis, sed oporteret dicere: animal est substantia animata sensib­ ilis non existens pars alicuius alterius quod sit per se unum, et sic sequitur ilium terminum .animal, est connotativum, postquam in definitionem eius ponitur term­ inum alterius praedicamenti, scilicet ille terminus .per se. [8.512] 2o sic: a est compositum ex corpore et anima intellectiva, igitur a est in praedicamento substantiae et per consequens sub aliqua specie. Sed non videtur sub qua specie nisi sub specie animalis vel hominis. Igitur a est homo vel animal, quod negatur positio. [8.513] 3o, si digitus esset abscissus ab a, a esset Sortes. Sed aeque bene compontir anima intellectiva cum materia nunc sicut tunc faceret. Ergo [E 57ra] qua ratione a esset tunc Sortes et iam est Sortes homo vel animal; quare et cetera. 8.6 Replies to these Last Objections 8.61 In reply to [8.51] one now has [8.61], which disposes of the cavil concerning the definition of animal, and Paul then suggests totally sensitive animated substance as a definition which covers the requirements of the objection, but without making 'animal' into a connotative term. At any rate, this is one way of obtaining the desired effect, i.e. of ensuring that the natura magna is not Sortes, but only part of Sortes, whatever its temporal situation (i.e. whether before or after the finger-amputation). If it is still insisted that this proposal continues to make 'animal' connotative, then there 507

Section 8 are other possibilities (e.g. animated substance which is a sensitive individual object) which evade this consequence. The most important section of this reply, however, is the discrimination effected between salient features of the theory of animal and those of theories centred around mass-terms such as 'air' and 'water': any part of air is air; any part of water is water. However, it is not the case that any part of an animal is an animal. Hence although a temporal segment of a man may be said to be made up from body and intellectual soul, this does not make that segment into a man. We are thus rightly reminded that the central background to the whole discussion may be located in the thesis of part-whole disparity (1.32, cf. 10.324). [8.6] Replies to these objections [to the third reply , i.e. to [8.33]]. [8.61] To (1) [i.e. to [8.51]] I reply that animated sensible substance is not the definition of animal. This is because it is certain that a plant is not an animal, and yet it is an animated sensible substance which can be smelled, touched, and felt. Again, neither will the following serve, namely, animated sensitive substance, since a is an animated sensitive substance, and yet is not an animal since it is not some suppositum or other. At the same time I do not on this account want to main­ tain that the definition of animal should be sensible or sensitive animated substance not subsisting as a part of some other thing, and so on, since there then would fol­ low the unpalatable conclusion deduced above. In fact the definition of animal is totally sensitive animated substance; herein the reference to the total substance is intended to convey a single substance which is an actual individual this. Hence it is obvious that a is not an animal, since it is not a total substance, but only a partial one in respect of Sortes himself; and so on. 508

Venetian Harvest Should it again come about that [someone were to assert that] this whole [composite, and supposedly substantival] term was a term belonging to some other category, namely Relation, Quality, or some other such, then it has to be laid down that the definition of animal is animated substance which is a sensitive individual object, and then both the objections and the difficulties will be obliterated. For the case of animal is not like that of air or of water; any part of air is air, and any part of water is water, but it is not the case that any part of a man or an animal is a man or an animal. Hence it is granted that there is something compounded from body and intellectual soul which is neither a man nor an animal. [8.6] [M 13ra 11] Ad haec argumenta respondetur: [8.61] Ad lum [8.51] dico quod haec non est definitio animalis: substantia animata sensibilis: quia certum est quod planta non est animal, et tarnen est substantia animata sensibilis, quia potest sentiri tangi et palpari; immo  ista: substantia animata sensitiva, quia a est substantia animata sensitiva, et tarnen non est animal, cum non sit suppositum aliquod. Nec propter hoc volo quod haec sit definitio animalis: substantia animata sensibilis vel sensitiva non existens pars alicuius alterius, et cetera, quia tunc sequeretur inconveniens deductum. Sed definitio animalis est: substantia animata totalis sensitiva, intelligendo per substantiam totalem unam substantiam hoc aliquid in actu. Et per hoc patet quod a non est animal, cum non sit substantia totalis sed partialis in ipso Sortes, et cetera. [M 73ra 24] Et si iterum fiet quod de illo termino totalis quod esset terminus alterius praedicamenti, puta relationis, qualitatis, vel alicuius, dicatur tunc quod haec est definitio animalis: substantia animata hoc aliquid sensitiva - et tunc cessabunt argumenta cum 509

Section 8 quaestionibus. Non enim est sic de animali sicut de aere vel aqua: quaelibet enim pars aeris est aer et qualibet pars aquae est aqua, sed non quaelibet pars hominis vel animalis est homo vel animal. Conceditur igitur quod aliquid est compositum ex anima intellectiva quod non est homo nec animal. 8.62 In response to [8.52] the reply now given, i.e. [8.62], likewise claims that although the natura magna may be said to be an animated sensitive substance, this does not make it into a substance in the primary sense, i.e. an individual object (in some pre-theoretical sense of 'object'). But there could perhaps be some derivative sense in which it may be said to be of the species of its whole. Still, one cannot properly say that Sortes and some natura magna which is his part (spatial or temporal) are both of the same species. [8.62] To (2) [i.e. [8.52]] I reply by denying the validity of the inference whereby it is deduced that a is in the category of substance, for not every animated sensitive substance is in the category of substance; rather, to speak more exactly, this applies only to the animated sensitive substance which is an individual object. But on the contrary [to this specification] a is not an individual object, except perhaps were it to be maintained that a could be in the category of substance and a member of the human or animal species in some reductive sense, similar to that in which matter and form have to be assigned to the same species as the composite which they constitute. Still, it would not follow from this that man should then be the true description of a as well as of b; for a species is not truly used to cover any save its own supposita, and is therefore truly asserted of b and not of a. This is because b is a suppositum, being a primary substance, as opposed to a; from which the solution [to 8.52] is 510

Venetian Harvest obvious, [8.62] [M 73ra 34] Ad 2um [8.52] nego consequentiam in qua infertur quod a est in praedicamento substantias; non enim omnis substantia sensitiva animata est in praedicamento substantiae; sed praecise substantia hoc aliquid animata sensitiva: modo a non est hoc aliquid nisi forte diceretur quod a esset in praedicamento substantiae et in specie humana vel animalis per reductionem, sicut materia et forma dicuntur esse in eadem specie cum composito. Nec ex hoc sequitur quod ly .homo, debeat verificari de ly .a. sicut de , quia species non verificatur nisi de suis suppositis, ideo verificatur de b et non de a, quia b est suppositum cum sit substantia prima, et non a; et sic patet solutio. 8.63 To [8.53] it is replied that it is true that when the finger has been cut off, then the result may be indicated as being Sortes. However, the earlier temporal stage of the natura magna was certainly not an individual substance with per se unity, so that there is no parity between the two objects. [8.63] To (3) [i.e. [8.53]] a reply can be made by ad­ mitting that when the finger has been cut off a would then be Sortes; but when it is said that the grounds [for such assertion] are identical in both cases, and so on, I deny the likeness of the two cases, since when the finger is cut off, a would be an object having per se unity, but when the finger was not cut off, a would not be an object having per se unity, since on the contrary, for something to be Sortes it has to be an individual this in the same way as it has to be a man or an animal; hence [[8.53] fails]. [8.63] [M 73ra 46] Ad Sum respondetur concedendo quod digito abscisso a esset Sortes, sed quando dicitur eadem ratione et cetera, nego similitudinem, quia digito 511

Section 8 abscisso a esset aliquid per se unum, sed digito non abscisso a non esset aliquid per se unum, immo ad hoc quod aliquid sit Sortes requiritur quod sit hoc aliquid, sicut requiritur ad hoc quod sit homo vel animal; quare et cetera. 8.7 Final Reply This is the final reply coming within the ambit of the section which started out as an attempt to cover the syncategorematic sense of 'whole'. The next main item will acc­ ordingly be the section dealing with the categorematic sense of whole, i.e. the [8.8] texts. In the meantime, this final reply is described by Paul as being subtle, but later [8.713] he is to say that it is not as true as the third reply, i.e. as [8.433]. Paul himself scarcely helps by his use of abbreviatory letters. Thus how one reads the part [i] of this final reply depends on how one reads those letters. The use of 'iam ... in casu' seems to point to the previous readings, i.e. 'a' abb­ reviates 'Sortes apart from the finger', and 'b' is for 'Sortes with the finger*. Hence when it is said that 'a is already as much Sortes as b is, and yet is not a part of b any more than it is a part of Sortes, notwithstanding the name's signifying both', this may be taking a in its later temporal part (after the cutting off of the finger). Plainly it is not then part of an earlier temporal part which included the finger, alth­ ough both the temporal part b and the post-amputational a could be successively indicated demonstratively as being Sor­ tes. But this demonstrative indication is taken to indicate a as the whole Sortes, with the result that a cannot be said to be a part of Sortes, since this would make a distinct from Sortes, and the same thing cannot be distinct from itself. This, at any rate, is just one of the various hypothetical interpretations of the first part of this final reply. On the other hand Paul now goes on to give new senses to 'a' and 'b': thus 'a' is explicitly allocated the task of 512

Venetian Harvest abbreviating 'the hand of Sortes', ' b ' is to be used for 'the rest of Sortes apart from his hand', and 'c' is to be for 'the compound of the hand and the rest'. If these newly announced senses are to be retroactive, then the point of part [i] of this final reply has now to be revised as follows. One can demonstratively indicate both the hand prior to the absciss­ ion, and the rest both before and after the abscission, and in each case truly say, 'This is Sortes'. Yet obviously it's not that a is a part of b, and one must, says Paul, also deny that a is a part of Sortes. While this last point may clearly be granted after the abscission, this banal inference may not be the aim of part [i] of the argument. Ultimately, its aim app­ ears to be the avoidance of the inference to 'It's not that Sortes is Sortes', which would apparently result from the dem­ onstratively-based assertions outlined when combined with a's being a part of b and also a part of Sortes. Given the further obscurities of the suppositions invol­ ved, there are clearly many more possible interpretations, several of them presupposing quite interesting theses, which could be imputed to the reply as thus far examined, as well as to Paul's statement in [ii] about the impropriety of saying that  is compounded from a and b. Leaving such interpret­ ations aside, however, we may now pass on, as does Paul, to his granting in [iii] that b is  and b is Sortes, while it's not that b is a part of c. This combination could once again arise from the fact that b and  may be demonstratively ind­ icated as being Sortes. At any rate, this would explain the subsequent apparently paradoxical inference that Sortes with­ out his hand has two disparate quantitative measurements, and indeed cannot be restricted to a single such measurement. This is also said to be a way out of the destructivist thesis, which results in an infinity of men or horses. [8.7] A final reply is the one which is distinguished from all the others given above by its superior elegance and greater elaboration. 513

Section 8 [i] This reply supposes that under the circumstances outlined a is already as much Sortes as b is, and yet is not a part of b any more than it is a part of Sortes, this notwithstanding the name's signifying both. For there is just no sort of signification in this world which could bring it about that the same thing is dis­ tinct from itself. [ii] Now when it is supposed that a is Sortes' hand, b the rest of him, and  the whole compound formed from a and b, this can be granted, although the supposition is unsuitable and involves an incorrect assumption. For it is incorrect to say that  is compounded from a and b in the same way as it is improper to say that an identical object is compounded from itself, and that Sortes is Sortes but his hand is not. [iii] However, I let pass the assumption and I grant that b is  and is also Sortes, and I deny that b is a part of  If now it is further assumed that Sortes measures exactly eight feet, as far as his quantity goes, and that the hand, for the sake of the argument, measures only one foot, it can now be granted that b quantitatively measures eight feet, and also measures seven feet; I deny, however, that b alone is seven feet in respect of quantity; this is because I assert that b is of the same amount as c, notwithstanding your assum­ ption as to measurement. This is at any rate more eleg­ ant than having to grant that in one man are an infinity of men and only one intellective soul. It is also better than having to grant that he who buys a single horse for ten shillings buys infinitely many for the same mon­ ey, all such admissions being absurd. [8.7] [M 73ra 53] Ultima responsio quae inter ceteras superius nominatas pulchrior et magis sophistica esse dignoscitur ponit [i] quod iam a in casu illo est tam Sortes quam b et 514

Venetian Harvest non pars b sicut  pars Sortis, non obstante quaecumque significatione. Nulla enim significatio mundi potest facere quod idem a seipso distinguatur. [ii] Cum ponitur enim quod a sit manus Sortis et b residuum et  totum compositum ex a et , admittitur licet casus sit incongruus et improprie ponatur. Improprie enim dicitur quod  componitur ex a et b, sicut improprie dicitur quod idem componitur ex se, et quod Sortes est Sortes et non manus. [iii] Verum tarnen admitto casum et concedo quod b est  et Sortes, et nego quod b est pars c. Et si ponatur quod Sortes sit octopedalis quantitate praecise, et manus, gratia argument!, pedalis solum, conceditur quod b est octopedalis quantitatis, et etiam septempedalis quantitatis; nego tarnen quod b solum est septempedalis quant­ itatis; quia dico quod b est tantum sicut c, non obstante assignatione tua. Et hoc est pulchrius dicere quam concedere quod in uno homine sunt infiniti homines et solum una anima intellectiva, et qui emeret unum equum pro decem solidis emeret infinitas pro eadem pecunia, quae omnia sunt absurda et cetera. 8.711 The next stretch of text in this final reply now first supposes  to be the whole spatio-temporal Sortes, whence is cut off the hand a; this leaves  still existing after the abscission. However, the concomitant denial that  is  im­ mediately after this event now seems to presuppose that  is merely a pre-abscissional temporal segment of Sortes. It hence looks as though it is by thus varying the sense of 'c' that Paul's version of this allegedly subtle reply gains its force. [8.711] Again, if a is cut off from  in such a way that immediately after this moment a will not be a part of Sortes, one may grant that  will exist immediately af­ ter this moment, but  is Sortes; hence [c will exist 515

Section 8 immediately after this moment]. However, it has to be denied that  will be  immediately after this moment, in the same way as it has to be denied that immediately after this moment Sortes will have a finger [of the hand]. For immediately after this moment  will not be the remainder of Sortes along with the hand. [8.711] [M 73rb 10] Item, si abscideretur a a c ita quod immediate post hoc non erit a pars Sortis, concederetur quod  immediate post hoc erit, quia Sortes immediate post hoc erit, sed c est Sortes, igitur et cetera. Neg­ atur tarnen quod  immediate post hoc erit c, sicut negatur quod Sortes immediate post hoc habebit digitum. c enim immediate post hoc non erit residuum cum manu. 8.712 The same kind of vaccillation continues in the penul­ timate section of this last reply. Thus a now becomes Sortes-without-the-finger, and b becomes Sortes-with-the finger, the latter being the posterior of the two states of affairs. Identification of a with Sortes yields the conclusion that a will be b, but a will not be a after the addition. Clearly this last paradox arises from a switch to a as a mere temporal part of Sortes. [8.712] Likewise, should Sortes not have a finger, this state of affairs being a, and there were to be added to him a finger which along with a would constitute b, then it may be granted that a will be b, since Sortes will be b But a is Sortes. Therefore [a will be b]. Nevertheless I deny that a will be a when it will be b, since when a will be b, a will have a finger; but at that time it will not be a being lacking a finger, and hence at that time it will not be a for the reason that a is identical with Sortes lacking a finger. [8.712] [M 73rb 17] Similiter, si Sortes non haberet 516

Venetian Harvest digitum qui esset a et sibi adveniret digitus qui simul cum a esset b, concederetur quod a erit b, quia: Sortes erit b Sed a est Sortes Igitur et cetera. Nego tamen quod a erit a quando erit b, quia quando a erit b, a habebit digitum, ergo pro tunc non erit a, ex quo a nihil aliud est quam Sortes non habens digitum. 8.713 In this ultimate section of the last reply, the letters continue to be used in the same way (i.e. with a as the nonfingered Sortes, and b as Sortes with a later-adjoined fin­ ger). These are taken to be temporal parts, and it is rightly denied that (under those circumstances) a is b. Next, using 'b' and 'b', in effect, to indicate Sortes-as-a-whole over his diverse stages of development, it can be asserted that b was a. Then implicitly reverting to the temporal-part mode, one has the result that neither a nor b was was both a and b, since Sortes never both had and had not the finger in quest­ ion. Nevertheless, using b as the occasion for designating the whole Sortes, one can say that b was both a and b, i.e. an earlier temporal stage of that whole was without a finger, and a later one had a finger. Paul's final reiteration of the superior subtlety of the present reply, and his preference for the third one, i.e. [8.433] above, suggests his justified suspicion of this sort of switching procedure. [8.713] Again, after the addition of the finger I deny that a is b, since Sortes without the finger is not Sortes with the finger simply because Sortes-withoutthe-finger does not [then] exist. Nevertheless I grant that b was a, since Sortes-with-the-finger was Sorteswithout-the-finger, and I deny that either a or b was b and a, since there was no instant at which Sortes was both with the finger and without the finger. I grant, however, that b was a and b, since Sortes has been both 517

Section 8 without the finger and with the finger. The same goes for many other points which follow upon the holding of this opinion which appears to be more subtle than the rest. Nevertheless the third [8.433] is the truer and the more easily maintainable. [8.713] [M 73rb 24] Item, post adventum digiti nego quod a sit , quia Sortes sine digito non est Sortes cum dig­ ito, quia Sortes sine digito non est. Concedo tarnen quod b fuit a, quia Sortes cum digito fuit Sortes sine digito; et nego quod a vel b fuit b et a, quia in nullo instanti fuit Sortes cum digito et sine digito. Concedo tarnen quod b a et b fuit, quia Sortes [E 57rb] sine digito et cum digito fuit. Et sic de multis aliis quae sequuntur sustinendo banc opinionem quae prae ceteris subtilior videtur. Verumtamen tertia est verior et levius sust­ en tabilis. 8.81 On 'whole' taken Categorematically Although the official start is here made on 'whole' taken as a term in its own right (i.e. categorematically) as opposed to being taken as a functor incorporating a universal quantifier (i.e. syncategorematically; cf. 8.06), the categorematic sense has, in effect, been the object of study in the later part of the material which was brought under the syncategorematic heading, the scope of which has just concluded. The present categorematic section, in which that study might have been usefully developed, turns out to be comparatively disappointing. This is because, as often occurs, Paul is throughout engaged in a struggle with the ambiguities of the Latin expressions now in play. At the present point he is initially attempting to overcome these ambiguities by comparing the inferential possibilities of certain examples of the two sorts of case, categorematic and

518

Venetian Harvest syncategorematic. He is compelled to provide clumsy criteria for categorematicity which turn upon the way in which the term 'whole' does not affect the main copula of the prop­ osition when it appears after that copula. A language such as the Latin which he is using, in which word-order is in principle irrelevant to meaning, scarcely sustains such criteria. [8.81] The term 'whole' taken syncategorematically having been dealt with, we may now deal with the same term when it is taken categorematically. For it is taken categorematically when it cannot affect the principal copula of the proposition [in which it occurs]. This happens when it appears after the copula or when it precedes the copula in a limited and rest­ ricted rôle, and is thus equivalent to 'some' (in all its genders) or to 'being', in so far as its attachment to the term to which it is adjoined is concerned. Hence the proposition: Sortes is a whole man signifies that: Sortes is some man. The two following are also convertible: Some form of words is a whole proposition Some form of words is some proposition. Again, Some term is a whole substantive noun is equivalent to: Some term is some substantive noun. Again, the following proposition: In your eye is a whole that is in the world means that In your eye is some thing or being that is in the world. [8.81] [M 73rb 34] Dicto de illo termino .totum. syncategorematice tento, iam dicatur de eodem dum 519

Section 8 categorematice sumitur. Tenetur enim categorematice dum copulam principalem suae propositionis determinare non potest, et hoc contingit quando copulae post ponatur, vel eidem introcluse et restricte praeponitur; et tunc alicui istorum: .aliquis. .aliqua. . aliquod. vel .aliquid. aut .ens. aequipollet iuxta inhaerentiam termini cui adiungitur. Unde haec propositio: Sortes est totus homo significat quod: Sortes est aliquis homo. Aliqua oratio est tota propositio convertitur cum ista: Aliqua oratio est aliqua propositio. Item: Aliquis terminus est totum nomen substantivum aequipollet huic: Aliquis terminus est aliquod nomen substantivum. Item haec propositio: In oculo tuo est totum quod est in mundo significat quod: In oculo tuo est aliquid vel ens quod est in mundo. 8.811 Correspondingly, when the categorematic 'whole' appears before the main copula, it must be somehow limited and rest­ ricted to avoid its affecting that copula. Comparisons with syncategorematic cases are provided. These suggest that at least some of the distinctions which Paul is here laboriously spinning out could perhaps be covered in English by the dif­ ference between 'a whole' and 'the whole (of)'. In some examples the latter is required, and tends to be assigned to the syncategorematic reading; the former may be aligned with some of the categorematic examples, hence its equation with 'some' and 'being' in the text. [8.811] It is taken in like [i.e. categorematic] manner 520

Venetian Harvest when it precedes the copula in a limited and restricted manner because of a proceding term. Hence to assert: Some whole Sortes is a man signifies that: Some Sortes-being is a man, or alternatively that: Someone who is a Sortes-someone is a man. The same applies in other cases such as: Some whole copulative proposition is true Some whole substantive is a name Some whole which is in the world is in your hand. If, however, it is not limited in this fashion when it precedes the copula, but rather determines the copula, it is then taken syncategorematically rather than categorematically, and amounts to [what follows from] the equivalents previously expounded above (8.06). Under these circumstances, were one to assert: Sortes is bigger than a whole part of himself or: In your eye the whole of what is in the world may be seen or: The whole Sortes is less than Sortes, then in all of these the word 'whole' (in all its genders) is taken syncategorematically, and all of them apart from the second are true. For the first means that: Sortes is bigger than any quantitative part of himself which is true. The second means that: In your eye all the quantity that is in the world may be seen and this is false. The third means that: Any quantitative part [of Sortes] is less than Sortes which again is true. However, were the assertions to be 521

Section 8 formulated as follows: Sortes is bigger than some whole part of himself or In your eye some whole which is in the world is to be seen or Some whole Sortes is less than Sortes, then in each of these the word 'whole' is taken categorematically on account of its restriction and the limitation of its force and power, because of the addition of another preceding term, and each of them apart from the third is true, as is obvious given that one Sortes is all Sortes [and hence a fortiori is some Sortes]. [8.811] [M 73rb 48] Consimiliter sumitur quando copulae introcluse et restricte per terminum praecedentem praeponitur, ut sic dicendo: Aliquid totus Sortes est homo Significat quod: Aliquod ens Sortes est homo, vel: Aliquis qui est aliquis Sortes est homo, et sic de aliis ut: Aliqua tota copulativa est vera Aliquod totum substantivum est nomen, et Aliquod totum quod est in mundo est in manu tua. Si tarnen non sic introcluderetur dum copulae praeponitur sed ipsum determinaret non categorematice sed syncategorematice habendo aequipollentiam superius traditam, ut sic dicendo: Sortes tota sui parte est maior, vel In oculo tuo totum quod est in mundo videtur, vel sic Sorte totus Sortes est minor, In qualibet istarum .totus. .tota. .totum. syncategorematice tenetur, et quaelibet illarum praeter 522

Venetian Harvest secundam est vera. Prima enim significa t quod: Sortes quaelibet sui parte quantitativa est maior quod verum est. Secunda significat quod: In oculo tuo omne quantum quod est in mundo videtur, quod est falsum. Et tertia significat quod: Sorte quaelibet pars quantitativa est minor quod iterum verum est. [M 73 va 1] Verumtamen si diceretur sic: Sortes aliqua tota sui parte est minor, vel: In oculo tuo aliquod toturn quod est in mundo videtur, aut sic: Sorte aliquis totus Sortis est minor, in qualibet istarum li .totum. categorematice tenetur propter sui reclusionem et suae virtutis et potentiae limitationem ex additione alterius temini praecedentis: et quaelibet illarum praeter tertiam est vera, ut patet dato quod unus Sortes sit omnis Sortes. 8.812 The considerations noted above are confirmed by the way in which a proposition formerly using 'whole' syncategorematically [8.21], and which was false under that construal, now becomes true when converted so as to place 'whole' after the copula. Thus 'Totum quod est in mundo est in oculo tuo', i.e. 'The whole of what is in the world is in your eye' was seen as false, whereas 'In oculo tuo est totum quod est in mundo' is now said to be true, meaning that it has the Eng­ lish sense of 'In your eye is a whole which is in the world'. Without Paul's rather ad hoc conventions (or stipulations), the Latin itself would permit the two to be equivalent. The Eng­ lish, the articles of which could commit one either way, acc­ ording to the choice of V or 'the', has no necd of his conventions. So that although our author has said that the preposition of 'totum' to a whole proposition does not leave place for indifference as to syncategorematic or categorematic interpretation [8.06], the prima facie possibility of the equ523

Section 8 ivalence of the two Latin sentences just quoted would, with­ out his conventions, place him in just such a situation of indifference. Indeed, the Latin construals he has now given in [8.81] for the categorematic sense of 'whole', namely its replacement by 'aliquis' ('someone'), and so on, or by 'ens', could equally well be used to resolve the indifference which is in fact attributable to cases which he apparently has claimed to be unambiguously syncategorematic. [8.812] From these considerations I infer the truth of cases such as: In your eye is a whole that is in the world In my hand is a whole of the world's money In my mind is a whole of the wisdom, knowledge, and prudence which pertains to the world (or to God) For the first signifies that: In your eye is something that is in the world; this is true because therein is the pupil, or alter­ natively, some awareness or impression caused in the eye. The second signifies that: In my hand is some of the world's money, which again is true, given that I have a shilling or a penny in my hand. The third signifies that: In my mind is some wisdom, knowledge, and prudence pertaining to the world (or to God); this again is true. Whence [the suggested conventions are valid]. However, if from these it is concluded that: The whole of what is in the world is in your eye The whole of the world's money is in my hand The whole of the wisdom, knowledge, or prudence of the world is in my mind, then the inference has to be denied, since the word 'whole' is taken categorematically in each premiss, but syncategorematically in each conclusion. Likewise, suppose someone were to argue as follows: 524

Venetian Harvest Nothing which is in your eye is outside your eye But in your eye is the whole [of] that [which is] in the world Hence nothing is outside your eye, and consequently: Your eye is the whole world, then this is not a syllogism correctly set out as regards its mood and figure. The minor must be taken as, The whole [of] that [which] is in the world is in your eye. Under these circumstances the conclusion could follow, but this minor premiss would have to be denied. [8.812] [M 73 va 7] Ex his infero tales esse vera: In oculo tuo est totum quod est in mundo In manu mea est tota pecunia mundi In mente mea est tota sapientia, scientia, et prudentia huius mundi vel divina. Prima enim significat quod: In oculo tuo est aliquid quod est in mundo quod verum est, quia ibi est pupilla, vel notitia aliqua, aut species in oculo causata. Secundo significat quod: In manu mea est aliqua pecunia mundi, quod iterum verum est, dato quod in manu habeam solidum vel denarium. Tertio significat quod: In mente mea est aliqua sapientia, scientia, et prudentia huius mundi vel divina quod iterum verum est, ut appareat; quare et cetera. Et si ex his concluditur quod: In oculo tuo totum est quod est in mundo, et quod In manu mea tota pecunia mundi est, vel quod In anima mea tota sapientia, scientia, vel prudentia mundi est, negatur consequentia, quia in antecedente ly .totum. tenetur ca tegorematice et in consequente 525

Section 8 syncategorematice. Similiter si sic arguitur: Nihil quod est in oculo tu est extra oculum tuum Sed in oculo tuo est totum quod est in mundo Igitur nihil est extra oculum tuum et per consequens: Oculus tuus est totus mundus; cum non sit Syllogismus in modo nec in figura, debet enim sumi pro minori: Totum quod est in mundo est in oculo tuo et tunc consequeretur consequentia et negaretur minor. 8.82 Arguments against the Proposed Conventions 8.821 The first of the counter-arguments now takes up the points which were in fact urged in the comment provided ab­ ove, such as those concerning the arbitrariness of Paul's proposed conventions. 'Whole', it is now counter-claimed, can function in either of the two ways (categorematic or syncategorematic) whether or not it actually precedes the copula. Thus the 'whole' of 'Some whole Sortes is Sortes' should, by Paul's criteria [8.81], undergo construal as categorematic in order to be true. But plainly this particular affirmative proposition, like any other of the form 'Some a is b' has its corresponding indefinite (of the form 'a is b') which ensues from the removal of the 'some', and which can be inferred from it. By the original conventions [8.06] this ensuing indefinite must now have its 'whole' construed syncategorematically, and this gives a result which is obviously false, since it amounts to 'Every quantitative part of Sortes is Sortes'. To avoid this inference of falsehood from truth, categorematic construal of the version which lacks the 'some', i.e. '.. whole Sortes is Sortes', is called for; but this goes against those original conventions. [8.821] Arguments against some of these assertions, [i] [One may argue] by proving that the word 'whole' (in all its genders) can stand syncategorematically just as 526

Venetian Harvest well on the side of the predicate as it can on that of the subject, and that it can stand categorematically just as well on the side of the subject as it can on that of the predicate. Thus I take the following proposition: Some whole Sortes is Sortes. This is one particular proposition, and hence has corresponding to it a single indefinite proposition, which is no other than: [a] whole Sortes is Sortes. This, then, is its corresponding indefinite. The particular proposition is true by what was said above [8.811], hence also this indefinite one. However, the latter cannot be true if the word 'whole' is taken syncategorematically; hence it must here be taken cat­ egorematically. [8.821] [M 72va 30] Contra quaedam dicta arguitur, [primo]probando quod ly .totus. .tota. .totum. ita syncategorematice potest stare a parte praedicati sicut a parte subiecti, et ita categorematice a parte subiecti sicut a parte praedicati. Et capio istam propositionem: Aliquis totus Sortes est Sortes. Haec est una particularis, ergo sibi [E 57 va 30] corresponde t una indefinita, sed non alia quam ista: Totus Sortes est Sortes. Igitur haec est sua indefinita. Sed illa particularis est vera per prius dicta; igitur et haec indefinita. Sed ipsa non potest esse vera sumendo li.totus. syncateg­ orematice: igitur sumendo categorematice. 8.822 The remaining arguments against the proposed convent­ ions may now be presented in a single block of text. Thus argument (ii) in Paul's numeration is an obvious variant on his first effort, in that one passes from a universal propos­ ition assumed to be true to its corresponding indefinite, the initial component of which in the Latin is 'totus'. However, 527

Section 8 in the case in question, i.e. '.. whole copulative (proposition) is true*, from the syncategorematic construal demanded by Paul's original conventions in [8.06] results an absurdity, namely 'Every quantitative part of the copulative proposition is true'. Paul's argument (iii) goes on to outline the manner in which 'whole' can function in the same way as does 'all' or 'no' in the predicate part (i.e. in the post-copular region) of a proposition, thus leading to the claim that 'whole', like the plainly and universally admittedly syncategorematic 'all' and 'no', may also be taken syncategorematically in that region, contrary to the now proposed conventions. Again, argument (iv) suggests that the 'mental' construal (cf. section 4 above) could incorporate stipulations which would ensure that the categorematic or syncategorematic alt­ ernatives could obtain both in the subject and the predicate parts of a proposition. [8.822] (ii) One may also argue thus: Every whole copulative proposition is true Hence a whole copulative [proposition] is true. The inferential consequence is obvious,since it goes from the universal [proposition] to its corresponding indefinite [proposition]; but assuming the truth of the premiss for the sake of the argument, the conclusion is hence also true, but only if the word 'whole' is taken categorematically, since in a syncategorematic sense that conclusion would be impossible. Hence [the proposition must be taken in a categorematic way]. (iii) In the same way as some distributing terms are related to the predicate, so also is the word 'whole' related when no reason for differentiating between the two cases can be given. But some distributing signs are taken syncategorematically on the side of the predicate, hence [the same should apply to 'whole', and this goes against the now proposed conventions]. The minor of 528

Venetian Harvest this argument is obvious because such signs perform their distributing function just as well on the side of the predicate as they do on that of the subject, as in 'You are all man', and 'You are no donkey', since both the 'man' and the 'donkey' here have the same confused and distributive status [relatively to suppositio doctrine; cf. HLM III, §1]. (iv) It is argued as follows: 'whole' (in all its gen­ ders) is just an act of the mind which the mind executes as it pleases. But the mind is just as unfettered in relation to the subject [of a proposition] as it is to the predicate, and vice versa. Hence the mind can just as well bring it about that the word 'whole' stands categorematically in relation to the subject as it does to the predicate; also it can just as well bring it about that this word is syncategorematic in relation to the predicate as it is to the subject. But this is opposed to what was laid down and its consequences, at least if some reason for the diversity [between the two sets of signs] is not provided. [8.822] [M 73va 39] 2o: arguitur sic: sequitur: Quaelibet tota copulativa est vera Ergo tota copulativa est vera. Patet consequentia ab universali ad suam indefinitam; sed antecedens est verum in casu, ergo et consequens, sed non nisi teneatur ly .totum. categorematice, quia syncategorematice esset impossibile; ergo et cetera. 3o: sicut se habent aliqua distributiva a parte praedicati sic et ly .totus, cum non possit dari causa diversitatis. Sed aliqua signa distributiva a parte praedicati sumuntur syncategorematice; igitur et cetera. Minor patet quia aeque bene distribuunt a parte praedic­ ati sicut a parte subiecti; dicendo .Tu es omnis homo. , .Tu es non asinus., tam ly .homo, quam ly .asinus. stant confuse et distributive. 529

Section 8 4o arguitur sic: ly .totus. .tota. .totum. non est nisi actus animae quae anima facit ad placitum. Sed aeque libera est anima a parte subiecti sicut a parte praedicati et econtra. Ergo ita bene potest facere anima ly .totum. stare categorematice a parte subiecti sicut a parte praedicati, et ita syncategorematice a parte praedicati sicut a parte subiecti, quod est oppositum positionis et cetera, vel si non detur causa diversitatis. 8.83 Replies to these Objections These replies are now presented in a single stretch of text, still retaining Paul's numeration. Thus in reply to (i) Paul counters the move from 'Some whole Sortes is Sortes' to the allegedly corresponding indefinite '... whole Sortes is Sortes' by a reductio ad absurdum claim that this is like the move from 'Some all man is man' to the classification of 'All man is man' as an indefinite. This is because the 'whole' of the original proposition has, he says, the same syncategorematic and distributive rôle as has the 'all' of the lastquoted proposition. Hence he introduces a rule for the prod­ uction of the corresponding indefinites in such cases, and takes it that this suffices to block the offending inferences otherwise suggested as being possible. In the reply to (ii) it is stated that the same reply can be made as that just proposed in respect of (i) Reply (iii) counters the syncategorematic construal of 'whole' when it occurs in the predicate part of the propos­ ition by redefining the syncategorematic in such a way that it comprises within its very nature an influence upon the main copula of the proposition. This influence is then absent when 'whole' occurs in the predicate. Finally, reply (iv) makes overt the exigencies which are embodied in the forms of words employed in propositions when quantifiers, including the negative ones, are involved. This makes it plain that the arbitrary mind-controlled construals 530

Venetian Harvest are out of place. [8.83] Replies to these arguments. To (i) I reply that the indefinite alleged to correspond to the particular in question does not in fact so corr­ espond, but rather this one: The whole being Sortes is Sortes. For no-one would assert that the following: All man is man was an indefinite proposition. Yet this could in fact be proved to be the case if the argument put forth in the first place was valid. One could then take the following particular: Something all man is man (or Some all man is man) and then assert that it had no corresponding indefinite other than this one: All man is man. [and this is absurd because here we have a universal, and not an indefinite proposition]. I hence lay it down that whenever a distributing sign intervenes between the sign denoting particularity [i.e. 'some'] and the substantive [in this case 'man'], the corresponding indefinite proposition is arrived at by means of a prefixed transcendental term, as in A being all man is animal A being every donkey runs. The same follows in respect of other like cases. To argument (ii) the same kind of reply can be made, i.e. that here the argument is not in fact going from a universal to its corresponding indefinite. Should a query be raised as to which is the indefinite here, then it can be constructed as in the foregoing reply. In response to the third argument I deny that some distributing terms are taken syncategorematically in 531

Section 8 relation to the predicate. Then should it be argued that such terms distribute in a mobile fashion in rel­ ation to the predicate, and hence are taken syncategorematically, I would deny that this consequence holds. Hence as far as the whole case is concerned I assert that in the proposition You are all man the 'all' has just as much distributive force as it would have in relation to a subject-term, and yet is not taken syncategorematically in the way that it is taken in rel­ ation to a subject-term. For when it occurs before the subject-term it affects the main copula, but it does not do so when it occurs in the predicate. Now I lay it down that a distributing sign is taken syncategorematic­ ally when it determines the main copula, and categorematically when it does not determine that main copula. Hence the following is not a proper inference: The 'all' is here a syncategorematic term Hence the 'all' is here taken syncategorematically. Rather one necds to add 'determining the main verb' to the premiss [in order that this inference should follow], and so on. (iv) I reply by denying that the inference is valid, for by using the same form of argument I could prove that in the following proposition: All men are animal the 'all' can be taken categorematically as well as syn­ categorematically, and this because a sign of this sort is an act of the mind [and under the mind's control as far as its status is concerned]. But under these circ­ umstances the proposition would be false, since the word 'animal' would be taken determinately, and it would sig­ nify that some animal is all-men, which is false. Fur­ ther, everyone would have to raise doubts about that proposition when it was asserted; they would have to 532

Venetian Harvest ask whether the 'all' should be taken categorematically or syncategorematically, and this goes contrary to all manners of expression, be they ancient or modern. Again, it would follow that two contradictories could be simultaneously false, namely: Every-man is an animal, and Non-every-man is an animal. Indeed, they could also be simultaneously true, as in You are a man, and You are not a man. [For] on the assumption that the 'not' does not have the predicate as its scope, the latter would mean: You are-not [some] man which is true. Again, it would follow that in the following proposition: Every man is animal the 'man' could have merely confused suppositio and the animal' confused and distributive suppositio [cf. HLM III, §1]. This would be based on the contention that 'all' is an act of the mind, and that the mind is just as effective in relation to [controlling what goes on in] the subject as it is in relation to the predicate. Here I declare that when the mind lays down a distributing term on the side of a predicate it cannot but bring it about that such a term is taken categorematically, since it cannot make that term affect the preceding copula. But when the mind prefixes the term to the whole prop­ osition it cannot but bring it about that it affects the principal verb, and hence cannot but bring it about that it is taken syncategorematically. For the mind is free in this respect, namely, it can make a distributing sign occur categorematically or syncategorematically because it can place such a sign in an earlier or later position [in the proposition] as it pleases; but once it has put 533

Section 8 the sign in position, the mind is not free to bring it about that it should be taken in such or such a way. This is obvious from the example, for the mind is cap­ able of arranging for the 'man' to be predicate as well as subject (and vice versa), before the conjunction of 'man' with the other signs. But once the arrangement has taken place the mind is not free in this respect. For the mind cannot bring it about that in the prop­ osition: Man is animal the 'man' should be the predicate and 'animal' the subject. In this way the reply to the argument is obvious. [8.83] [M 73va 56] Ad haec argumenta respondetur: Ad lum dico quod illi particulari assignate non correspondet talis indefini ta, sed hocEns totus Sortes est Sortes. Nullus enim diceret quod hoc: Omnis homo est homo esset indefinita, quod tarnen probaretur si argumentatio prior teneret sumendo hanc particularem: Aliquid omnis homo est homo (vel Aliquis omnis homo est homo) dicendo quod non posset assignari aliqua indefinita nisi haec: Omnis homo est homo. Dico ergo quod quandocumque inter signum ρarticul are et substantivum mediat signum distributivum assignanda est indefinita per terminum transcendentem praepositum, ut: Ens omnis homo est animal Ens quilibet asinus currit Et sic de aliis consequenter. Ad 2um argumentum respondetur eodem modo: quod non arguitur ab universali ad suam indefinitam. Et si quaeritur de indefinita assign atur ut prius. 534

Venetian Harvest Ad 3um nego quod aliqua distributiv a teneatur syncategorematice a parte praedicati. Et tunc ad argum­ entum: distribuunt mobili ter a parte praedicati: ergo tenentur syncategorematice: nego consequentiam: unde pro tota dico quod in illa propositione: Tu est omnis homo ly .omnis. distribuit aeque bene sicut a parte subiecti, non tarnen tenetur syncategorematice sicut a parte sub­ iecti. Quia dum praeponitur subiecto determinat copulam principalem, et non sic facit a parte praedicati. Voco enim signum distributivum syncategorematice teneri quando determinat copulam principalem, et categorematice quando non determinat eandem. Ideo non sequitur: Ly .omnis. est hic syncategorema Igitur ly .omnis. tenetur hic syncategorematice. Sed oportet addere in antecedente, .determinans verbum principale., et cetera. Ad 4um nego consequentiam: per idem enim probarem quod in ista propositione: Omnis homo est animal ly .omnis. posset stare ita categorematice sicut syncat­ egorematice ex quo tale signum est actus animae. Et sic propositio esset falsa, quia ly .animal, staret determ­ inate et significaret quod aliquid animal est omnis homo, quod est falsum. Immo quilibet haberet dubitare illam cum proponeretur, quaerendo an ly .omnis. teneretur cat­ egorematice vel syncategorematice, quod est contra  modum loquendi, tam modernorum quam antiquorum. Item, sequeretur quod duo contradictoria essen t simul falsa, videlicet: Omnis-homo est animal, et Non-omnis-homo est animal. Immo simul vera, ut: Tu es homo, et Tu -es homo, 535

Section 8 posit quod ly .non. non caderet supra praedicatum, ipsa enim tunc significaret quod: Tu homo -es quod est verum. Item sequeretur quod in ista propositione: Omnis homo est animal ly .homo. posset stare confuse tanum, ly 'animaV confuse et distributive, ex quo ly .omnis. est actus animae et ita potens est ipsa anima a parte subiecti sicut a parte praedicati. Dico ergo quod quando anima ponit terminum distributivum a parte praedicati non potest facere quin categorematice teneatur: quia non potest per ipsum copulam praecedentem determinare. Sed quando ipsum praeponit toti propositioni non potest facere quin cadat super verbum principale. Ideo non potest facere quin syncategorematice teneatur. Anima enim in hoc est lib­ era facere signum distributivum stare categorematice aut syncategorematice, quia potest ipsum praeponere vel [E 57vb] post ponere sicut placet sibi. Et post positionem eiusdem non est libera facere ipsum taliter vel taliter teneri, t patet in exemplo. Nam anima potest facere quod ly .homo. sit ita praedicatum sicut subiectum, et econtra, antequam ipsum cum altero componat, sed post compositionem eiusdem non est sic libera. Anima enim non posset facere quod in hac propositione: Homo est animal ly .homo. esset praedicatum et ly .animal. subiectum. Et sic patet solutio ad illud.

8.9 Postscript to Paul of Venice At this point, as at so many others through the present book, it is necessary to stress that the commentary which has been provided could be indefinitely enlarged. However, noth­ ing mereological which Paul has had to say may not in prin536

Venetian Harvest ciple be covered by the axioms and theorems of the approp­ riate part of the Presuppositional Explicitation (10.3). Paul, on the other hand, is now concluding by yet again busily passing responsibility to theories such as that of suppositio or of the capacities of the mind, or the mysteries of 'mental', and so forth. In fact one does rather get the impression that the net­ work of responsibilities set up throughout his vast Logica Magna, of which only a minuscule sample has been edited ab­ ove, is similar to that so glaringly produced by the likewise vast networks of present-day political and administrative organisations. In all of these areas, the cunning disclaimer of the A.A. Milne character who consults his teddy-bear about the answer to the arithmetic problems, is the indispensable responsibility-evading groundwork of the whole process: 'And then it doesn't matter what the answer ought to be, 'Cos if he's right, I'm right, and if he's wrong it isn't meV: MV 173. True, the disclaimer cannot be quite so total in Paul's case, since he is elsewhere to be the purveyor of the material to which cross-reference is made. Yet the impression of 'buckpassing' in the vain hope that everything will somehow come out well in the end does tend to obtrude at many points. The suspicion that such a diffusion of responsibility may be taking place has for the most part been mooted in order to point the contrast with the policy embodied in the present work, wherein section 10 systematically bears the weight of all relevantly ultimate cross-referencing. As has been claimed in another context, 'The buck stops here'. In that explicitation a few elementary aspects of scrupulouslyconstructed systems are all that remain to be understood in order that the rest of the elucidatory comment presented may not only be intelligibly grounded, but also theoretically acceptable.

537

9. Situational Review 9.i From the point of view of what the Boethius of Dacia exposed in 1.11 would have called the narrative mode, a limited and partly chronologically-ordered account of some features of medieval mereology has now been sketched in a preliminary sort of manner. No ingenuity at all would be required to interject a further cloud of considerations and material which might conceivably be vitally relevant to each of the texts presented above. In point of fact, at this pre­ liminary stage the very existence of the narrative has been dependent upon resistance to any such interjections; without their reduction to a minimum the enterprise would smother al­ most from the outset. The counterpart of this self-denying ordinance in what that same Boethius would term the demon­ strative mode has been the relativisation of technical comm­ ent to the fields covered by the presuppositions made explicit in section 10, and these are pretty austere; but at least they are explicit. 9.2 From this explicitness and from a determination to avoid that buck-passing diffusion of responsibility which was desc­ ribed just now in 8.9, ensues the pedagogical consequence that in principle it is possible for anyone, even though initially devoid of mereological skill, to attain to a systematic grasp of the medieval material, a grasp which will be based on firm theoretical foundations. Such an attainment will not be susc­ eptible of abrogation by any element of the cloud of possible 538

Situational Review interjections mentioned above, especially if they are issued without the support of similarly coherent foundations. 9.3 Nevertheless, a superficial anticipatory glance, on the part of both the learned philosopher and of the unlearned be­ ginner, at the general complexion of the presuppositional explicitation soon to be offered in section 10, could only too readily suscitate a dreadful foreboding that they are about to be subjected to a battering about the head with a burst of that sort of unanalysed (and hence unintelligible) pseudomathematical gobbledygook which is sometimes inflicted upon hapless students of philosophy in general, and upon students of the medieval linguistic disciplines in particular, suppos­ edly as a deus ex machina which will resolve all problems of interpretation. From this sometimes well-justified fear could ensue a determination to stick to the familiar reassurance of chat-analysis, the insufficiency and contingency of which has been one of the main general impressions emerging from the material surveyed in the foregoing work. We have seen, for example, how the medievals relied on such analysis, surrep­ titiously or overtly canonising as 'proper-speak' ('proprie loquendo', e.g. 7.55) a given sense of a key term such as 'part' or 'whole', in accordance with local argumentative convenience. There is then no guarantee at all that similar convenience, or mere inertia under the treacherous pressure of unorganised usage, will not dictate or permit some other sen­ se at some other juncture, and so on throughout all the chan­ ges and chances of succeeding centuries. Such is a partial explanation of why it is only in a halting sort of way that medieval mereology progresses, if indeed it progresses at all. This also contributes to an explanation of why metaphysics takes on the appearance of that series of mock-contests for the abolition of which Kant supplied his own rather radical recipe in the Critique of Pure Reason. 9.4 The relativisation of the demonstrative aspect of the investigation to those principles sketched roughly in section 539

Section 9 10 is not by any means performed in the formalist manner, i.e. by reference to what are in principle unanalysed and uninter­ preted systems which just happen to have a configuration or interpretation somehow usable in the metaphysical context. On the contrary, the systems used are dedicatedly interpreted from the outset, with the move from the pre-theoretical to the theoretical stage being carried through by means of exp­ lanations which ground each of those primitive terms whereby other terms in the system are defined. Such explanations may rely for their success on a whole range of commonplaces, no one of which is in any way sacrosanct. Practical success in conveying the sense of the primitive term is the main aim at this level, even though unexamined positions which may later prove unacceptable, or which have no direct bearing (or no bearing at all) on the main topic, happen to have been used to accomplish this aim: cf. HQS §2.70. Neither is it necessary for present purposes to become over-entangled in controversy between rival schools of philosophical foundations for math­ ematics, or concerning the merits or otherwise of the various interpretations of quantificational notation (10.2113, .2115; cf. HQS §2.3). Even if there subsists an unwillingness to accept as metaphysical truths those theses couched in the canonical, categorial language shortly to be introduced in section 10, they may still be exploited as co-ordinates more exact than those of pre-theoretical language for the analysis of medieval themes. Throughout, only the very elementary and light-weight ends of the ontological and mereological spectra are to be exploited for such analyses, as will become clear as the exposition unfolds.

540

10. Presuppositional Explicitation 10.01 Prospectus 10.011 In accordance with the strategy outlined in 1.1, the presuppositions to which demonstrative comment has above been informally relativised are now to be made explicit. Those who prefer to put their trust in pre-theoretically disorganised ways of talking may well have been satisfied by the rough sketches of the appropriate background theories which have been provided, and hence either necd go no further, or alter­ natively necd do no more than consult the English approximat­ ions provided to all the logical linguistic expressions and theorems which are introduced below. 10.012 What, then, are the advantages of going any further into a somewhat more well-organised version of those theor­ ies? A sketch of the initial lines of the proposed advance may give some notion of the greater exactitude which may result therefrom, and may take its cue from the simple logical grammar broached in 0.3. There the informal notion of incom­ plete linguistic expression was introduced and exploited. The present-day terminology for such an expression is 'functor'. The medievals themselves included such expressions system­ atically among their syncategoremata or 'syncategorematic 541

Section 10 terms', as was observed in 7.1. The key contrast here is between what is necded to complete such a functorial expres­ sion (i.e. its completion, sometimes unhelpfully called its 'argument' nowadays) and that which is the product of the adjunction of such a completion. Thus the functor '...sits' lacks a name which, if supplied, produces a proposition (i.e. assertive sentence), such as 'Socrates sits'. One is hence here in the presence of a functor which forms a proposition from a single name. The categorial indices of functors may be expressed by means of the oblique stroke '... / ...', this being flanked (on its right) by the sign for what is required for the form­ ation of the appropriate outcome, and (on its left) by the sign for that outcome. The signs now in question are 's' (for 'proposition') and V (for 'name'), the latter being understood in its more usual classical ordinary grammatical sense, and thus covering both common and proper nouns, as well as ad­ jectives, but here to the exclusion of abstract nouns: HQS §2.33, LSS. Thus the categorial index of the functor '...sits' would be 5/n, i.e. it is here shown as a functor which, upon completion by one name, forms a proposition. Similarly, the index of a familiar sort of '... is ...' would be s/(n n), since 'Socrates is captain', is one of its possible completions. 10.0121 Strictly speaking, and in the end, the primary foun­ dation of the 'semantic categories' or 'parts of speech' delin­ eated by such indices lies in the canonical or categorial lan­ guage, which has yet to be exposed. Until due exposition therein has taken place, the category in question is not yet systematically available for the making explicit of the real or logical form of any utterance susceptible of analysis in terms of that language. The status of the present and other examples is hence of a preliminary educational sort, and not strictly official. In general, in the present section, the mush of verbiage surrounding the introduction of the primit­ ive terms is of this primarily pragmatic sort, and necd not 542

Presuppositional Explicitation itself be defensible as ultimately presuppositional. That privilege pertains to the systems which emerge from the mush by way of those primitives. 10.013 An initial characterisation of the path to be taken in the proposed explicitatory process may now be informally sketched in terms of the just-described categorial indices. Thus first of all, in 10.1, is to be presented a brief and fragmentary introduction to some of the language of Protothetic, the typical functors of which form propositions from propositions. Hence the counterpart of '... or ...', with propositional completions intended, will be of index s/(s s), as also will be those of 'if ... then ...', '.. if and only if ...', and '... and ...', all of which are to be introduced below. Here also are to be found functors with single propositional com­ pletions, as in 'It is not the case that ...', the index of which is s/s, since it forms a proposition from a proposition (as contrasted with another type of negation, namely the nominal, which forms a name from a name: see the material around 10.252 below for a slight amplification on this part­ icular contrast). 10.014 In 10.2 a theory which presupposes the Protothetic mentioned in 10.1 advances to functors which take as their completions not propositions (index V) but rather names (index V ) , and to a theory in which such nominally-completed functors are typical. Here a functor like '.. is ..', which forms a proposition from two names, and hence has the index s/(n n), is used as a primitive term for the theory, which is called 'Ontology', i.e. theory of '.. is ... -ness', or 'be-ing'. From that primitive will be systematically obtained, by means of suitable definitions, functors among which are those requ­ ired for the exact expression of that quidditative discourse touched upon in 0.4. For example, 10.261 defines the '... is* ...' which takes as its terms not names (as does the justmentioned primitive) but rather verbs, or verb-like express­ ions. For instance, given that, as noted above, s/n is the 543

Section 10 index of a verb which forms a proposition from a single name, one has s/(s/n s/n) as the index of that '... is* ...' defined in 10.261, and which is so vital for the expression of the form of quidditative discourse. Also derivable within this Ontology are theorems relevant to traditional syllogistic and Boolean Algebra (on which see LSE and LSB respectively). 10.015 Next, in 10.3, is to be presented the axiomatised theory of part, or 'Mereology', which is to provide the final back-up to the main research of the present volume. Indeed, readers who are familiar with the notation of the 'Propositional Calculus', which shares functors with P r o t o t h e t i c , and who are prepared to grasp a few of the definitions from Onto­ logy, should be able to pass forthwith to the Mereology of 10.3 without further ado. In this area the primitive t e r m will be 'pt( )', readable in English as 'part of ...', or as 'proper part of ...'. Evidently this functor forms a name or nominal expression from a name (as in 'part-of-Socrates') so t h a t its categorial index is n/n. Evidently also, therefore, we have here a move beyond Ontology by the addition of this further term. Definitions then go on to provide all other requisite mereological functors, and built into the whole arr­ angement now being described is that radical distinction bet­ ween the quidditative sense of part, which is provided for within Ontology, and the mereological part; this distinction was reiterated throughout the middle ages. So also was the pervasive distinction between part-of-X and X-part, as noted in 1.4, 2.3, 3.4. 10.016 In short, for those who wish to pursue this philos­ ophically fundamental exercise, t h r e e primitive t e r m s (one each for Protothetic, Ontology, and Mereology respectively) will help to found the categorial language (cf. 0.3) for the expression of those theories whereby may be made intelligible all the contentions of the present work which have been used as background guidance through the mereological aspects of medieval metaphysics. The mode of becoming acquainted with 544

Presuppositional Explicitation the various phases of the categorial language will be akin to that encountered in the learning of a foreign language. There will be no question of preliminary formalist rules of formation, and definitions will be creative of new forms which make possible inferences otherwise unachievable. Scrupulous examination of the resultant language may be subsequently made, in the form of 'terminological explanations': LLL ch. 7. In the course of the present exercise the '10'part of the numerous numbered cross-references will often be omitted below for brevity's sake. 10.1 Some Protothetical Functors 10.11 Since the initial aim is here not to expound a theory, but merely to somehow impart the sense of certain proposition-forming functors (some of which could serve as primitive terms) in an ad hoc intuitive sort of way, the device of 'truth-tables' may be slackly adapted for this pur­ pose. In particular, one may use the binary digits 'l' for 'true' and '0' for 'false', somewhat after the style of Woodger in the first volume of Analysis, or of Bochenski in BPL. It is then forthwith possible to list exhaustively the various functors which form propositions from two propositions (i.e. of index s/(s s)) by recourse to the 'truth-matrix', which shows all the ways in which the two propositions indicated by 'p' and 'q' may be true or false, i.e. either both false, or the first one true and the second false, or the first false and the second true, or both true, as shown by the successive two-member binary digital columns above the horizontal line in the array which next follows. Below that horizontal line are exhibited all the possible ways in which one may make selections from among those ways of being true and false. In other words, each row of digits below the horizontal line picks out (by means of its 1-digits) the columns of truthpossibilities of which it indicates the acceptance, and (by

545

Section 10 means of its 0-digits) the columns of truth-possibilities of which it indicates the rejection: p: 0 1 0 1 q: 0 0 1 1

[0]

000 0

[1]

0 0 0 1 characterises p.q (... and ..., conjunction)

[2]

00 10

[3]

00 11

[4]

0 100

[5]

0 10 1

[6]

0 110

[7]

0 1 1 1 characterises p V q (... or ..., disjunction)

[8]

1 0 0 0 Sheffer function; possible single primitive

[9]

1 0 0 1 characterises p≡q (.. if and only if ..)

[10]

10 10

[11]

1 0 1 1 characterises p  q (if ... then ...)

[12]

1100

[13]

1101 546

Presuppositional Explicitation [14]

1 1 1 0 Sheffer function; possible single primitive

[15]

1 1 11

10.12 Part of the point of such an array in the present context may be further explained as follows. Below the horizontal line the rows which (as already noted) exhaustively list the sixteen ways in which the four alternative columns of truth-possibilities displayed above the line may be select­ ed (by means of a 'l'-digit) or rejected (by means of a '0'digit), are numbered from [0] to [15]. Here only a few such selections from among those which characterise commonly-used or otherwise noteworthy functors of index s/(s s) necd be considered. Thus, with reference to the columns above the horizontal line, the row [1] selects (by means of its single 'l') the combination of the truth of the two propositions here indicated by 'p' and 'q', and rejects (by means of its three '0' digits) all the other three possible combinations of truth and falsehood for 'p' and 'q'. (Here 'p' and 'q' are being used to typify propositional 'variables', i.e. they are used in expres­ sions such as the 'p.q' now being characterised, in order to hold open places for propositions on each side of the functor ' . '). Plainly, with its exclusive adherence to the final column of truth-possibilities, this selection [1] characterises propositional conjunction, requiring for its truth the truth of both its conjuncts. That dot which is the symbol of this functor may in English be read as '...and..' 10.13 In contrast, selection [7] picks out the case in which all the alternative truth-possibilities except those in the first column above the line are accepted as true. This lets in ρ or q or both, symbolised by ' V ', and read as '..or...' in English. 10.14 Selection [11] is also hospitable to all possibil­ ities except one, in this case the third column, in which ρ is true and q is false, this latter combination being rejected. 547

Section 10 Discussions of great length have become traditional at this point in order to decide whether the functor here being char­ acterised really does correspond to some English use of the 'if ... then ...' often used as its reading version. This point will not be pursued here, except to say that the rejection of the second column, which has ρ t r u e and q false, is the int­ uitively prime feature of this functor. The old song's 'If you love me then everything's O.K.' can remain ' t r u e ' throughout all possible combinations of the first and second completions of its embedded 'if ... then ...' functor except in the event of your loving m e and its not being the case t h a t everything's O.K. 10.15 Selection [9] characterises what is, in effect, t h e conjunction of the conditional as characterised by [11] with its reciprocal 'q  p' covered in fact by [13], hence the t i t l e 'biconditional' which is sometimes given to this form of logical equivalence. 10.16 A further functor now to be characterised differs from those shown above in that it is of index 5/5, i.e. it forms a proposition from a single proposition, as opposed to the two hitherto exemplified. For this we require only the two alternative truth-values for ρ above the horizontal line, and the effect of the functor on those below the line is then shown thus: p: 0 1

~(p): 1 0

(it's not that ...; propositional negation)

This shows how, when it's not that p , one has ~(p) as true, and conversely, when ρ holds, then ~(p) comes out as false. 10.17 It is well-known that the so-called Sheffer fun­ ctions, which are covered by [8] ('not-(... or ...)') and [14] ('not-(... and ...)') above may each be used as a single 548

Presuppositional Explicitation primitive to provide the definitions of all the remaining functors and also for axiom-foundation. But although we thus now have our single primitive t e r m , and although corres­ ponding axioms are easily available (as PFL Appendix I shows), it must be realised t h a t rules of inference, definition, and of extensionality are also required for appropriate theoryconstruction here. Each of these calls for lengthy intro­ ductions which would be rather misplaced in the present context. (Rules of extensionality allow t h a t when two items are the same, they are such t h a t whatever is predicated of the one is also predicated of the other: quaecumque sunt idem, ita se habent, quod quidquid praedicatur de uno praedicatur et de alio, as Aquinas puts it in AST I, q. 40, art. 1, ob. 3.) Some idea of the importance of attention being given to rules of definition may be gained from the fact that with diverse rules of definition it is possible to have single primitives other than the Sheffer ones {LID). Enough has been said, however, to make plain the nature of the area within which well-known and commonly-admitted theorems featuring typically propositional variables may be obtained. That is all t h a t is required for the present account. And although Lesniewski's nomenclature ('Protothetic') has been borrowed as a heading for this material, it should be realised t h a t these notes scarcely reflect the nature of the actual systems propounded by him and by his followers. 10.1701 Note on the binary array in 10.11. Although it has no bearing on the presuppositional aim of the present section, it may still be of interest, for those who have the leisure so to do, to note that the succcessive columns above the horizontal line in t h e array are, if taken to indicate binary numbers with the least significant digit topmost, successively equivalent to the decimal numbers 0 to 3, and in general, where η is the number of proposit-ions in question, their 2n truth-possibilities may then be arrayed by thus columnising the successive binary equivalents of the numbers 0 to 2 n -1 549

Section 10 inclusively. Indeed, a further illustration is immediately available within the array above. Were we dealing with the truth-possibilities of four propositions, then it therefore happens t h a t the anticlockwise rotation of the quadruple columns of digits below the horizontal line (so that the columns themselves now become horizontal) would yield the truth-matrix for four propositions. Indeed, the squarebracketed numbers shown at the left-hand side of each row in the original 10.11, are each precisely the decimal equivalents of what have now become their corresponding columns, viewed as bi-nary numbers with the least significant digit uppermost, and ranging from 0 to 2 - 1 (i.e. 15), exactly in accordance with the just-mentioned recipe for making explicit the 2 n truth possibilities, in this case where η = 4. The arithmetical convention thus embodied in the construction of arrays which can serve as truth matrices for 'two-value' logic, to which I was first introduced by W. Mays, may be usefully exploited when the selections (each of 2 n digits) are presented independently of the then implicitly understood m a t r i c e s . Further, when n is so great as to impose the consideration of an inordinately lengthy set of digits in each such selection, then one can express the 2 n digits of each such row in terms of binary maps of maps of maps (and so on) of the appearance (for example) of each of the 16 patterns noted by [0] to [15] above, and the theorems required for operating on the resultant multi-level arrays are available in HTT.

10.2 Ontological Axiom, Definitions, and Theses 10.201 Elementary discussions of the material within this section have already been provided in many earlier monographs (HDG, HLM, HCD, HQS), and derive entirely from the work of  Lejewski, such as his LR. A rudimentary introduction, sufficient only to give the senses of the main t e r m s of 550

Presuppositional Explicitation Ontology, and to lead into the Mereology of the next section, is now to be presented. Throughout, the accent will be on moves from the already-known to the unknown, relatively to what has already been encountered. Thus the system of categorial indices reminds us that names may be used either to complete functors which form propositions (as in the case of '...exists', which has s/n as its index) or to form names or nominal expressions (as with 'non- ...', which in this context forms names from names, and hence has n/n as its index). These two possibilities lead to two corresponding schemas or frames for the expression of the definitions which are clearly going to be required. Since, as has been seen, Protothetic typically has propositions as the completions of its funct­ ors, the definitional frame dealing with the first type of functors just-mentioned, i.e. those which form proposit-ions (from names) may be called the protothetical frame, and simply involves an equivalence of the sort characterised by line [9] of the 10.11 array, as in .211, .21161, .21162, and so forth, below. The other schema will be explained and brought into action in due course.

10.21 Functors of Existence. However, still in conn­ ection with definition, certain further features of the canonical or categorial language which is being construct-ed must be touched upon. From a pedagogical point of view it will be better to actually display a concrete specimen of a relevant definition, so that the point of those further features may become more evident. Thus there are three functors now to be defined which express aspects of existence. Of these, the definition of the first may be exhibited (along with a fashion of reading it in English) thus:

551

Section 10 .211

[a]: ex(a) .≡. [3 b] . b e a (For all a: there exists at least one a if and only if, for some b, b is a; cf. LR Tl)

Here the English corresponding to the definiendum (the term introduced by the definition) has been rendered in bold type so as to emphasise the point of the exercise, i.e. the definition of the functor 'ex( )' in the categ-orial language which here corresponds to that English. The latter is just a teaching aid, and implies no claim that the categorial language here gives the only possible correct analysis of the English, a claim which would obviously be a piece of linguistic lore which is both false and in any case not in question here. The same goes for all the other English correlates provided as introductory aids in the pages which now follow. 10.2111 Three more basic features of .211 and its fellow expressions remain to be explained. The first is the pre­ sence of the functor ' . . . E . . . ' , of index s/(n n), and entitled 'singular inclusion', which may in English have '.. is ...', or '.. is a ...' as approximate correlates. Here we have the primitive term now being used to found Ontology, it being assumed that Protothetic is already given (as may be gathered from the presence of the proto-thetical functor '≡' in .211). The sense of this primitive and of other functors which form propositions from two names may be informally shown by the use of the Ontological Table (LR and HLM II). The nearest natural-language correlate of the '... E ...' now in question may be found in languages which have no articles, such as classical Latin. Thus the Latin 'est' of 'Socrates est philosophus', 'Socrates is [a] philosopher', is a partial representation of the sense intended. Alternatively, drawn from LTD, one may have the following specification: a proposition of the type 'Eb' will be regarded as true if and only if the expression which stands in the place of 'a' is 552

Presuppositional Explicitation an unshared noun-expression which designates, and if the only object thus designated happens also to be designated by the noun-expression which stands in place of 'b' The '.. E ...' here specified is the primitive t e r m of the axiom first used for Ontology by Lesniewski (LR T34); this axiom will be exposed and explained below. It is by no means t h e shortest possible axiom, nor does it involve the only possible primitive t e r m , as LR shows. Specifications such as the one provided should not be allowed to convey the impression that a present-tensed '... is ...' is here being described. The primitive '.. 6 ...' is tenseless, a possibility envisaged in detail by Boethius of Dacia: HQS §3.241. Questions about t h a t which is earlier, present, and later, may accordingly most suitably be left to be dealt with after Chronology (Theory of Time) has been established by the addition of just one more primitive t e r m to the three already envisaged above: cf. HQS §2.563, §5.131. 10.2112 The second basic feature which expression .211 exemplifies above is its use of lower-case italics (e.g. ' a ' , 'b') as variables. Hitherto (apart from the use of 'p' and 'q' as propositional variables) small strings of dots have been used to show the uncompleted nature of functors undergoing discussion, as when '... sits' is said to be a functor of index s/n. Now, instead of dots, the italics initially perform the same office of holding open the places in or around the adjacent functorial sign, but with the added advantage of both specifying and correlating the possible completions of those places. Thus the use of the lower case italics indicates that the appropriate completions (in the case of the functor '... E . . ' , for example) are names or name-like expressions in the broad classical sense of name, as already indicated. Such expressions may also be simple (as 'Socrates') or complex (as 'the baldest man in Birmingham'). The correlative function of variable-letters lies in their indicating t h a t were such nominal expressions (in this case) ever to be substituted for 553

Section 10 the variables ('a', 'b', and so on), the same name would occupy the places of the same variable throughout each such substitution. This does not, however, apply to the sections of .211 and its like which are enclosed in the small square brackets. 10.2113 Indeed, those sections comprising the small square brackets are the third basic feature which .211 exemplifies, and are technically called 'quantifiers'. As the suggested English renderings indicate, such quantifiers are intended to show the all-hood (first squarely-bracketted expression) or somehood (second squarely-bracketted expression) of the possible substituends of the corresponding variable-signs controlled by those expressions in the subsequent part or parts of the whole assertion. Thus the square-bracketted 'a' of expression .211 intimates that that expression holds in respect of no matter which nominal substituend for 'a' throughout that expression. Contrastingly, the 'b' of '[3b]' intimates that .211 holds from that point onwards in respect of at least one nominal substituend. The effect of all this is, for instance, to elucidate 'there exists at least one Socrates' in terms of the '.. is ...' in question, as 'Something or other is Socrates'. It is in this sense that the '... is ...', or, more strictly, the '...6 ...', is a primitive term whereby the functor 'ex( )' is being introduced into the categorial language, as well as being defined. 10.2114 Those who have no previous acquaintance with the sort of quantificational notation described will, it is hoped, be able to follow the import of the two quantifiers thus introduced, and to see the point of calling that which holds for all the 'universal' quantifier, and that which holds only for some the 'particular' quantifier. A reminder that the substitutable names or nominal expressions may be nonabstract names in the classical sense, simple or complex, and thus either referential or non-referential (i.e. 'empty', as is 'Pegasus') completes for such readers the outline of the three 554

Presuppositional Explicitation features exemplified. However, those already acquainted with the use of quantifiers may be conscious that the usage here presented differs in a certain respect from that of B. Russell and W.V.O. Quine, in that these latter presuppose non-empty (i.e. referential) unshared (i.e. non-common, or proper) and simple names as the only possible substituends for their corresponding nominal variables, with quite unnecessarily awkward consequences which will not be expatiated upon here: cf. HLM II, § 2.25, HQS §2.31 - § 2.33, §2.6; HQS §6.2 suggests further reading, with C. Lejewski's 'Logic and Existence' (LLE) as the seminal paper in this area. 10.2115 Three further quantificational points still re­ main to be noted. First, on account of its confining it­ self to non-empty (i.e. referring) proper-nominal substit­ uends for its nominal variables, the Russell-Quine type of quantification is called 'restricted', whereas the contras­ ting Lesniewski-style is called 'unrestricted', since its possible nominal substituends are not thus confined, as noted. This leads to the second of these final points on this sub­ ject, namely that quantification using the '  ' in the restricted sense is called 'existential', and may be read off in English as 'There exists a ... such that ...'. This is simply because if an expression thus quantified holds for some substituend, the latter must be one which refers to an existing object, whereas this necd not at all be the case where the ' 3 ' is used in the unrestricted sense, since an empty substituend may be in question, hence the 'for some ...' reading accordingly here employed. Thirdly, although this discussion has been conducted in terms of the substituends for the variables involved within the expressions of the categorial language, this necd not be taken to entail that the 'range' of the quantifiers is in fact constituted by such substituends, notwithstanding the willingness of some to defend looking at the matter in such a light, as described in Alex Orenstein's lively work (OQ). 555

Section 10 10.2116 All the same basic features (use of '... E ·..' as primitive, quantification over nominal variables) may be observed in the definitions of two further functors of existence, as follows: .21161 [a]:. sol(a) .≡:[bc]: be a .  e a .  . b E c (For all a: there exists at most one a if and only if for all b and c, if b is a and  is a then b is c; cf. LR T5) .21162 [a]: ob(a) .≡.[ b], a E b (For all a: there exists exactly one a if and only if for some b, a is b; cf. LR T16) The 'sol( )' of .21161 is inspired by the Latin 'solus', 'alone', and the 'ob( )' of .21162 derives from the alt­ ernative '... is an object' reading of this functor. This reminds us that being an object is being that of which there is exactly one. A further and hitherto quite unmentioned feature of the categorial language is becoming increasingly prominent, especially in .21161; this feature is the dotting used as punctuation so as to make determinate those propositional articulations which could otherwise be susceptible of alternative interpretations. However, as the English approximations provisionally make clear those articulations, explanation of the dotting may be postponed until the moment when a sufficient store of examples for consideration has been accumulated. A further point to note is the use of the functor '... E ...' in order to announce the existential import of an expression, as opposed to the Russell-Quine use of '3' for that purpose, with its then accompanying a 'There exists a ... such that ...' rendering; this was mentioned in 10.2115. The Russell-Quine presupposition of non-empty (i.e. referential) names as substituends for the nominal variables means that in their systems there would be a superfluity in 556

Presuppositional Explicitation the use of existence-announcing functors of the sort above defined. As the first essay in QF points out, denial of the existence of anything becomes a problem when restricted quantification is presupposed; the language itself commits one to the existence of 'everything'. In contrast, the unrestricted quantification used in the present work not only permits an escape from this undesirable situation, but also encourages the refined and notationally independent expres­ sion of existence by means of functors such as those defined above. There also accrue several other benefits: the 'theory of descriptions' no longer necd be a necessary prerequisite for the denial of existence, and that difficulty known as 'Russell's Paradox' is easily avoided, at least in one of its many versions (HQS §2.64, HLM 44 - 6). 10.22 Functors of Identity. Still remaining with the protothetical definitional frame, one may now deal with some expressions of sameness, both individual (as in 'Cicero is Tully' or 'Socrates is the same object as the white woollyhaired son of Sophroniscus') or general (as in '(A) mermaid is identical with (a) being (upper-) half woman, (lower-) half fish', thus expansively expressed to avoid the Magritte inversion). Of these the first is the Singular Identity, defined thus: .221

[ab]: a = b . = . a e b . b e a (For all a and b: a is the same object as b if and only if a is b and b is a; cf. LR T25)

As the presence of '... e ...' in the definiens, or defining part, makes clear, this sort of identity only holds true when the substituends for 'a' and 'b' are non-empty proper names which refer to the same object. However, while such names are not barred for the truth of the next functor to be defined, it is designed to embrace also the cases where 557

Section 10 shared names (empty or non-empty) contribute to its truth. Such Weak Identity is defined thus: .222

[ab]:. a o b . = :[c]: ce a .=. c eb (For all a and b: a is weakly identical with b if and only if, for all ,  is a if and only if  is b: cf. LR T27)

An alternative reading in English of the functor here defined would be 'Only all ... is ...'. A third form of identity of similar s/(n n) index is Strong Identity: this form requires non-empty terms for its truth: HQS §6.5441, HLM II §4.3.11, LB T26. 10.23 Functors of Inclusion. These follow the lines and nomenclature of the identities which have just been covered. Corresponding to the singular identity of .221 above is that Singular Inclusion, symbolised by the '...  ...', which has been adopted as the primitive term for the present system. Corresponding to the weak identity of .222 above is the Weak Inclusion, defined thus: .231

[ab]:. a cb . = :[c]:  ea . D. ce b (For all a and b: all a's are b's if and only if, for all c, if  is a then  is b: cf. LR T19)

Likewise there is Strong Inclusion, with an existence clause added to the definiens of .231, as in LR T18. We are now on the borders of Categorical Syllogistic, be it singular syllogistic (HSG) or the more familiar version, founded upon inclusions such as the ones mentioned, and their negations (HQS §6.542, LSE, LAS). Within syllogistic the 'All ...' or 'Every ...' and 'No ...' propositions were said to be 'universal' in their quantity (as opposed to the 'particular' forms 'Some ....' and 'Some ... is not ...'). Such is the location of the 558

Presuppositional Explicitation 'logical 1 quantity which the medievals were at pains to distinguish from mereological quantity: 4.91, 5.22, 5.46 and 7.05 all mention this point. 10.24 Many-link Functors. Still remaining within the ambit of definitions based on the protothetical frame (10.21) we now pass to a type of functor which is of prime philosoph­ ical importance, in that it enables analyses of statements at the level of quidditative discourse to be systematically exp­ ressed. In section 0.4 an outline of the situation of such discourse in the history of medieval thought was given. In turn, the situation of the functors now in question relatively to the usual expressions of contemporary logical teaching may be sketched as follows. It is customary to introduce Protothetic in the first place, since, as we have seen, its functors are required for subsequent theorising (as in the case of the equivalence ' . . ≡ . . . used in our definitions). The same applies to its theses. (Strictly speaking, the precise and scrupulous Protothetic of Lesniewski, of which we have really seen nothing, is not what is usually brought in initially, but

559

Section 10 rather some slacker, inexplicitly quantified 'propositional calculus', or something such.) Next it is customary to go on to 'predicate calculus', in which the typical terms inter­ related by protothetical functors are propositions broken down into the predicate ('... sits') and the name (e.g. 'Socrates') whence they are constituted, with 'Ø...' (read as 'phi ...'), and '...' (read as 'psi ...') as predicate variables, and V , 'y', ... (or 'a', 'b' ...) as nominal variables, so that ' a' (or ' (a)') shows the typical propositional structure here considered. Under these circumstances, 'Ø...' is of index s/n. This procedure, however, masks precisely that phil­ osophically vital area centred around the possible inner structures of the predicates thus blankly notationed as 'Ø.··', '...', and so forth. One could regard much of the definit­ ional material thus far presented in these initial steps into Ontology as breaking into that blankness, in so far as the means for overtly showing forth possible predicative inner structures are thereby constituted. Tools for exploiting these facilities at critical points are now at hand. (Of course, if one is not concerned with philosophy in general, or with metaphysics in particular, but rather with the task of getting on to the exposition of a subset of mathematical formalisms, then the intellectual close-focus now being prop­ osed may be quite insignificant.) The instances of 'many link' functors now to be listed are all capable of forming, as it were, individual types of structure within the general 'Ø...' type of form. The instances now in question have each an intrinsic gap which is completable by a name or name-like expression so as to yield a verb of index s/n. We are thus dealing with functor-form­ ing functors of index (s/n)/n. So much for the essential, intrinsic, aspect of the matter. However, from an extrinsic point of view, one ends up with proposition-forming functors of index s/n, as shown by the way in which this last index figures to the left-hand side of the previous categorial ind560

Presuppositional Explicitation ex. In short, variously structurable verbs, logically specif­ ied cases of 'φ...', are the ultimate outcomes of the exercise. It is when such functors are themselves, in their turn, used as terms for a suitably defined '... E ...' (10.26 below) that we have the basis for the analysis of medieval quidditative discourse. Firstly, and based on the isolation, as it were, of the '... o a' part of  'b  a' form of proposition (cf. .222 above), one has the definition of the functor 'C1{ }', thus: .241

[ab]: Cl{a}(b) . = . b o a (For all a and b: b's form the 'class' of a's if and only if b is weakly identical with a)

The mention of 'class' in the proposed reading (which also accounts for the form of the notation) is, of course a mere manner of speaking, and should by no means be taken to count­ enance the existence of any mythical class-entities. As poin­ ted out in LSS, this functor is often the correlate of names used at the quidditative level (e.g. the 'man' of 'Man is a species') or of natural-language expressions such as 'being a ...', 'to be a ...', 'esse ...', '...ens' (taken non-nominally, i.e. 'ut verbum', as the medievals said at this point), and so on, acc­ ording to language and context. Given the usually definitionally-associated nature of quidditative propositions, and the basic medieval thesis that esse cuiuslibet rei in definitione consistat, 'the being of a so-and-so is expressed by the def­ inition', such a 'to be ...' functor, along with the appropriate '... is ...' of .26 below, are exactly what are required in this context. (For fuller discussions of this further vital nexus between contemporary theory and medieval patterns of disc­ ourse see HCD n3.800b and HQS §4.3. As a solution to the 'universals' problem, or to the question as to the status of 'possible world' discourse, this 'esse ...', 'to be a ...' level of discourse, hitherto unknown or neglected by this century's 561

Section 10 contemporaries, opens up unlimited present-day philosophical prospects: HQS §2.34.) Even as the last definition yielded a verb-forming functor based on the identity '... o ...', so now the next yields one which is based on the inclusion '... c...' (.231 above). Thus in an analogous fashion, the functor ' c{ }' is now defined in terms of the truncation of 'b c a', i.e. as a codification of '... c a ' , thus: .242

[ab]: c {a}{b) . = . b c a (For all a and b : b's are being included among the a's if and only if all b's are a's)

It is clearly possible to continue this process of endowing the final part (index s/n) of a proposition formed from a main functor of category s/(n n) with the facility for taking off on a life of its own, so to speak, as an independent term. Thus the '... Ea' part of 'bEa' gives us the 'E{a}' of the following: .243 [ab]:E {a}(b) . = . b Ea (For all a and b : b a-ises if and only if b is a) The enigmatic quasi-English version here ventured may be ill­ uminated by an example instantiating this definition, i.e. 'Socrates deput-ises if and only if Socrates is a deputy'. This shows how, in the first place, there is here continued a further variant of the general guarantee that the categorial language has various verbs corresponding to every name or nominal expression. In the case of this functor, not only are examples countenanced by usual language admitted, as when from 'deputy' one here has 'deputises', but also quite unheardof examples (e.g. from 'man' one has 'manises', and so on). This is why the initially usual termination '-ises' has been suggested as the closest one can get to producing a non-tech562

Presuppositional Explicitation nical version of the definition's intention; placing a name in the gap immediately provides a corresponding verb (and so on, although in a more specialised fashion, for the previous two definitions also). The universally-available verbs of the last-given definition at least ensure that appropriate term­ inal completions are available for the next functor to be defined.

10.25 Name-forming Functors and Nominal Expressions. Whereas the last set of definitions has guaranteed the avail­ ability of various verbs corresponding to each noun, there now follows a converse counterpart, whereby every verb has its corresponding nominal form. Since the characteristic var­ iables of the theses of Ontology are ones which take names as their substituends, the ontologicai definitional frame is ap­ propriate for the introduction of new names or name-forming functors into the system. Like the protothetical frame hith­ erto employed, it still has the equivalence ' . . . ≡ . . . ' (index s/(s s)) as its main functor, but now the presence of '...G ...' on one side of that main functor is such as to require that the existential import thus inserted should be balanced by an existential clause (usually 'a E a') on the other side: cf. HQS §6.532 and references there given. The stress on this clause may seem to be a trivial matter. William of Ockham, however, was profoundly aware of its vital contribution to definitions (HQS §2.6) and B. Russell paid part of the price for the lack of it with his well-known paradox: HQS 99 - 101. Here follows the introduction of the name-forming functor 'trm< >', which yields a nominal form (index n/(s/n)) correlatively derived from the verb'Ø( )' (index s/n) when the latter completes the gap in the thus newly-introduced functor:

563

Section 10

It has to be admitted forthwith that the use of the notation 'trm ...', and its suggested English reading as 'term satisfying ...' are extremely odd in the absence of some acquaintance with the archaeology of early twentieth-century logical term­ inology. It just so happens that the 'term satisfying ...' exp­ ression is enshrined as a name-forming functor in the monum­ entally foundational Principia Mathematica of Russell and Whitehead. Were it not for this precedent, the simplest Engl­ ish expression of the functor now being defined would be just '... -er' (or ' -er'). Thus would be more clearly expressed its intent, i.e. the provision of the nominal expression '.. -er' corresponding to the verb ' φ( )', as when one has 'runn-er' corresponding to the verb '...run', and so on. Perhaps because '... -er' would suggest an Anglo-Saxon proprietorship, the thus classically-embedded 'trm< >' has persisted, with filial piety and linguistic neutrality as possible grounds for its retent­ ion. For 'term ...' unfortunately suggests that the name of a name is being introduced, and this is not, of course, in any way intended here. Propositional negation (index 5/5) has been introd­ uced in 10.16, and is the counterpart of English functors such as 'It is not the case that ...' or 'It's not that ...' However, usage suggests that non-propositional negations, such as 'ill­ iterate', 'ungrammatical', 'indefatiguable', and 'inept' also need their counterpart in the categorial language. Here now is that counterpart in its most elementary n/n form:

564

Presuppositional Explicitation As in .251, the existential clause 'aEa' here balances the equation in an intuitively obvious way. Lack of this and the other equally simple equipment now available has led in our own day to gross confusion about the 'Principle of the Excl­ uded Middle', with the consequent claim as to the necessity for a 'Theory of Descriptions' to rescue contemporary logic therefrom. Ockham and his fellow medievals had all this area well under control exactly in accordance with the contempor­ ary lines which have just been indicated {HQS 1 0 3 - 7 , cf. HLM II §4) and KB 27 gives ample independent confirmation of Buridan's familiarity with the two types of negation. (It is therefore a great pity that the eminent narrator of this confirmation there claims that the distinction 'cannot clearly be drawn in modern logic'. It is restricted quantification, summarised in 10.2115, and not 'modern logic', which is ultimately to blame for this present-day falling-away on the part of some philosophers from theoretical control such as that exemplified by Ockham and Buridan in these vital fields.) Although much remains to be said concerning these and other definitionally-introduced nominal expressions, as the cross-references provided in HQS §6.55 amply indicate, two more items suffice for present purposes. First comes a def­ inition óf ' . . .  . . . ' , i.e. a name-forming '... and ...' (index n/(n n)), as contrasted with the proposition-forming '... and ...' (index s/(s s)) of 10.12, thus:

Here the existential clause 'a E a' no longer need be overtly stated on the right-hand side of the definitional proposition, simply because it would be redundant, given the ' . . . E . . . ' of the last two conjuncts. Next, a nominal expression of the highest theoret565

Section 10 ical significance (HQS §2), in that it is 'empty', i.e. refers to no thing, is introduced:

For present purposes this will merely be exploited in deciphering Nicholas of Paris' definition of whole in .337 below.

10.26 A Quidditative '... is ...' A preliminary view of the status of 'quidditative' discourse (e.g. 'Man is animal', 'Man is a species') was provided in 0.4. Functors which serve to show the status of terms for such discourse were introduced in 10.24. Now an '... is ...' suitable for completion by such terms, as suggested by C. Lejewski, may be introduced. Since those terms are in fact verb-like expressions (as are the infinitives in 'To-be-a-man is to-be-included-among-theanimals') the categorial language takes account of this feat­ ure by its use of verb-variables ( and ­ index s/n) to flank the ' ' in the definition below. It would, of course, have been possible to asterisk this ' ', as is done in the English correlate, in order to notationally dist­ inguish it from the primitive name-flanked ' ' (index s/(n n)) whence have flowed our definitions. However, the context of its appearance (e.g. flanked by or their substituends from 10.24) is taken to be sufficient to show the distinction. Here now is the definition of a quidditative-level '... ..', as required:

566

Presuppositional Explicitation a, φ holds of a, and ψ holds of a, and also for all b and c, if φ holds of b and φ holds of c, then b is weakly identical with c) Since the terms of the '... Ε ...' here defined are each of index s/n (i.e. verbs, predicates), the '...G ...' itself here has the index s/(s/n s/n). The definition may be seen as cons­ tructed by analogy with the primitive ' . . . ε ...' the initial t e r m of which has to be an unshared name referring to an ob­ ject, in order for a true proposition to ensue. Hence like­ wise the initial t e r m of this quidditative '...ε ...' should pick out just one 'class' so to speak, e.g. when something of the form of the functor defined in 10.241 is t h a t t e r m . The weak identity defined at 10.222 is here used to ensure this effect. Still, the present '...G ...' has, of itself, no existential import (unlike our primitive). Medieval defin­ itions, s t a t e m e n t s in terms of the predicables (HQS §4.2), or which have as t e r m s infinitives or participles (understood functorially and non-nominally) are all parsable in terms of this ' . . . E ...'. Thus 'Man is (included in) animal', when .241 and .242 supply the structure for the t e r m s , has the form: .262

Cl{man}ε c (Being-a-man

{animal} is* being-included-in-the-animals)

Abstract nouns, suitably translated (10.29), may also be t e r m s of the quidditative ' . . E ...' All theorems which hold at the nominal level (i.e. t h a t of the primitive '..ε . . ' ) hold analog­ ously at the quidditative level, but not conversely. All def­ initions at the nominal level may likewise have quidditative analogues, and hence, in particular, one may have analogous but non-existential senses of 'being', and so forth. Medieval expressions of the contrast between discourse at the quidditative and nominally-termed levels do, 567

Section 10 of course, abound: HQS §3, §4. Thus the functorial {s/n) and hence incomplete nature of the quidditative (or essential) terms of the intemporal non-existential '... is* ...' defined in .261 is contrasted with the completeness of the concrete beings to which reference may be made by the terms which complete our primitive '... is ...': essentiae dicuntur res incompletae respectu entium, et solum entia dicuntur completa, 'essences are said to be things incomplete relatively to con­ crete objects, and only concrete objects are asserted to be complete*; CC 55, p. 149. Or again: duplex est ens vel esse, sc. quidditativum et esse actualis existentiae. Esse quidditativum ... est esse indifferens ad praesens, praeteritum et futurum, sicut vult Avicenna quod domus heptangula est univ­ ersale etsi non sit; 'Entity or being is twofold, i.e. quiddit­ ative and the being of the actual existent. Quidditative being ... is being indifferent with respect to past, present, and future, as Avicenna holds in respect of the heptagonal house which is* at the universal level, even though it does not exist'; CC 55, p. 157. Here not only is the contrast facilitated by .261 being stressed, but also the non-exist­ ential nature of the quidditative '... is* ...', thus coinciding with the sense of the definiens in .261.

10.27 An Axiom for Ontology. Strictly speaking, of course, the axiom now to be advanced should have been enuntiated at the outset of the present phase, since it effectuates the off­ icial introduction of the primitive '...G ...' which has been exploited in the foregoing definitions. Such initial enuntiation could, however, have been somewhat baffling in the course of an exceedingly elementary introduction such as the present one, and hence has been postponed until now. Although there is no necessity at all that any ax­ iom for Ontology should happen to be intuitively accessible to elementary understanding, the one which now follows may in 568

Presuppositional Explicitation fact be simply explained, even in medieval terms. Thus it is, as a whole, a sort of self-elucidation of the primitive '...G ...', and this in the following sense. Consider first of all the propositional form 'All a is b'. This (or its definitely existential correlate 'Every a is b', which adds an existence clause to the 10.231 case) was said by several early medieval philosophers to amount to a singular inclusion (i.e. just plain 'a is b') when the 'a' is a non-empty proper noun. A concrete example may be taken from the work of the Liège logician Gar­ land the Computist, whose life straddles the time of the Nor­ man Conquest. In his Dialectica he says that 'Every Cicero is a man' amounts simply to 'Cicero is a man'. This he summ­ arises as the 'superfluit "omnis"' ('superfluity of "Every"') situation, on the simple ground that removal of the 'Every' from 'Every Cicero is a man' leaves us with the equally true 'Cicero is a man', i.e. a singular inclusion. Details are given in HQS §2.55. For our part, we may use the examples just mentioned in order to form the implied equation, i.e. .271

Cicero is a man if and only if Every Cicero is a man

This is, in fact, an instance of the general form of the required axiom, in the sense that precise analysis of the right-hand member of this equation reveals the axiom's det­ ailed form. Given that the proper-nominal and referential status of 'Cicero' has to be made explicit, that right-hand member may be unpacked thus: .272

Every Cicero is a man if and only if (i) There exists at least one Cicero, (ii) All Ciceros are men, and (iii) There exists at most one Cicero.

Here (i) ensures that 'Cicero' is not an empty name, but this could leave open the possibility that it was shared, so that (iii) is required to eliminate this possibility. Now we can 569

Section 10 reap the profit of having available definitions of the funct­ ors figuring in the three numbered sub-propositions. Thus the 'There exists at least one ...' of (i) has its counterpart in 10.211; the 'A11 ... are ...' of (ii) is covered by 10.231, and the 'There exists at most one ...' of (iii) by 10.21161. One can hence first amplify .271 in the light of the analysis (authorised by .21161) of its last sub-proposition, as made visible in .272, thus: .273

Cicero is a man if and only if (i) There exists at least one Cicero, (ii) All Ciceros are men, and (iii) For all  and d, if  is Cicero, and d is Cicero, then  is d.

Next one can follow the lead here given,in (iii), and use the primitive '.. is ...' throughout as the sole functor of index s/(n n), thanks to .211 and .231, which are here applicable: .274

Cicero is a man if and only if (i) For some ,  is Cicero, and (ii) For all c, if  is Cicero then  is a man, and (iii) For all  and d, if  is Cicero and d is Cicero, then  is d.

Thus, by an informal intuitive use of the definitions ment­ ioned and prematurely propounded, it is now possible to grasp the sense and point of what in fact comes systematically bef­ ore the official use of those definitions, i.e. the required axiom for Ontology. For that axiom, which now follows, is just the generalisation of what was pretheoretically and intuitively grasped in .274: [ab]:: a ε b . = :.[3 ],  ε  :.[]:  e  . .  e b :. [cd]:  e  . d e a . . c ε d (For all a and b: a is b if and only if for some ,  is a,

.275

570

Presuppositional Explicitation and for all c, if c is a then c is , and for all  and d, if  is a and d is a, then  is d) A more compendious quasi-English reading may be based on the stages reached in .272 and .273 above, thus: .2751 For all a and b: a is b if and only if there exists at least one a, and all a's are b's, and there exists at most one a. The deductive system of Ontology may now advance from axiom and definitions to further truths either intrinsic to Ontology itself, as illustrated in LR, or as confirmatory (or otherwise) elucidations of medieval theses, as in HQS. Or again, and in furtherance of the latter type of undertaking, Ontology is to be extended by the addition of another single primitive term, so as to yield the Mereology which is to elucidate yet more relevant medieval theses of the sort covered by the present work. Finally, this Ontologicai section of the categorial language is already bringing into systematic existence even more of those diverse parts of speech to which the relevant categorial indices quoted above have borne witness. Illustrations of the success of this axiom in grounding deductions in an intuitive fashion are contained in HLM II §5, and in the many works of C. Lejewski.

10.28 Punctation. Readers who are spectators rather than actors in this logical theatre could well find any account of the conventions for the dotting here used as punctuation somewhat of a bore, and in that case may indeed pass on, as the sense of our theses in this particular work is usually evident enough without such lore. However, those who antic­ ipate a future study of works to which reference has been made will find mastery of those conventions of great service, 571

Section 10 as well as a provider of a certain intellectual exhilaration; my former colleague and computer pionecr Alan Turing wrote a large piece of work on their formalisation. Hence for the use of those so inclined, and in view of the proximate availab­ ility of an exemplary expression of some complexity, namely that axiom above-stated, a brief excursion into the use of dots as logical puncation may now be undertaken. It is already evident t h a t those dots show forth the scope or extent either of the completions of a given protothetical functor (cf. 10.1) or of the quantifiers (10.2114) here shown by enclosure in the lesser of the two sorts of square brackets or braces. The main difference between these two types of scope is that that of the quantifiers always extends to the right only, whereas the completions of the proposition-forming functors (of index s/(s s)) extend to either side thereof, and punctuation is necded in both cases to show how far, or within what limits, that scope extends. For example, the final ' . .  . . . ' of .275 is clearly not the functor on which t h e whole expression is hinged, but only relates to its immediately adjoining propositional forms. Or again, the quantifier '[3 c]' therein only relates to its immediately following propositional form, and not to the whole of the remaining right-hand side of the ax­ iom. The observant will also have noted t h a t although the proposition-forming '... and ...' (index s/(s s)) as defined in 10.12, was originally shown by means of a single dot, it has evidently proved convenient to use groups of dots at '... and . . ' - p o i n t s in complex expressions to represent simultaneously both the conjunction and t h e punctuation of a force approp­ r i a t e at that point. Initially it might appear t h a t the easy way out of the punctuation difficulties which obviously must be faced would be to adopt various orders of brackets and braces of the sort familiar in common algebra (HLM 33). However, these not only exhaust the typographically available stock of brack572

Presuppositional Explicitation ets, which are necded elsewhere, especially in many-link fun­ ctors (10.24), but also yield only about three levels of str­ ength, whereas dotting, when mastered, can cater for indef­ initely many. Such mastery is therefore a worthwhile enter­ prise. Previous discussions have been presented in HLM II §3 and HQS §6.3, §6.54332. For present purposes the following may suffice. First, and neglecting the use of dots for the combined '.. and ...' and punctuation purposes, and given that the lesser never stops the greater, it is laid down that the scope of dots adjoining an s/(s s) functor is never stopped by an equal number of dots, but only by a greater. Thus the three dots immediately following the '≡' of .275 are not out­ numbered by any group of dots throughout the remainder of the expression on the right-hand side of that '≡', so that the latter is thereby shown to be the cardinal functor for the whole expression, i.e. the whole proposition as from the end of the initial quantifier turns upon it. For the single dot to the left of the '≡' has its scope quickly arrested by the four quantifier-adjoining dots to the left of ' a e b' making the latter into the brief completion of the left-hand side of the cardinal equivalence-functor. The scope of the quantifier-following dots just en­ countered is invariably to their right, and unlike that of the functor-adjoining dots is stopped by an equal number of dots, but only when the latter adjoin s/(s s) functors. Thus the initial four dots of .275 ensure that their foregoing univ­ ersal quantifier covers the whole expression. As it happens, there is no equal number of functor-adjoining dots in .275 to illustrate the stoppage-rule just noted. Thus far, then, one may say that in respect of dotnumbers the s/(s s) functor-adjoining dots are 'strongest', in that they are never stopped by equals, whereas the quantif­ ier-adjoining dots are Mess strong', in that they are stopped by equals, but only when those equals adjoin s/(s s) fun­ ctors. They are thus not stopped by equals which are 573

Section 10 '·... and . . ' - f u n c t o r s , or which follow immediately upon another quantifier. The '... and . . ' - f u n c t o r dot or groups of dots may now be perceived as the 'weakest' of the t h r e e sorts of cases, in t h a t its scope is always brought to an end by an equal number of dots. Thus the three dots before the '[c]' of .275 are stopped in their scope by the t h r e e equivalence-adjoining dots to their left, and t o their right by the t h r e e '... and . . ' - d o t s preceding the last main section of .275. In short, the general rule, and the intuitively economical one, is that the scope of a dot or group of dots (in the case of functors of index s/(s s) other than the '... and . . / functor) extends outwards from the functor which it adjoins or (in the case of a dot or group of dots repres­ enting the '.. and ..'-functor) in both directions, or (in the case of a dot or dots immediately following a quantifier) t o its right. Such scope extends to a point where a greater number of dots occurs (or, should no such encounter occur, to the end or ends of the expression). As a further economy measure, one has the rule concerning scope-stoppage by an equal number of dots, as outlined above, and which may be briefly summarised thus. (For this purpose 'functor' alludes to cases of index s/(s s) only.) Thus to the question, 'To what extent and in what cases do equal numbers of dots t e r m i n a t e scope?', there are t h r e e answers: (i) Where functors other than conjunction are in question: never. (ii) Where quantifier-following dots are concerned: sometimes, i.e. only when that equal number of dots adjoins a functor. (iii) Where dots representing the propositional conjunction are concerned: always, whatever the status of those equals. We may hence now repeat the .275 axiom, but with the relevant Roman numerals aligned with each dot or group of dots so as 574

Presuppositional Explicitation to indicate which of the three just-stated cases are applicable:

This arrangement brings out well the way in which the stren­ gth (cf. (i)) of the hook-flanking single dot of the final section surmounts the weaker dot-as-conjunction (cf. (iii)) to its left. Naturally those who remain at the elementary reach­ es of our theories will rarely have the occasion to get bey­ ond expression the articulations of which may be coped with by means of common algebraic brackets. However, such things as single axioms for Mereolology, or the arrangement of proofs (as in HLM II §5, for example), call for uses of dots wherein the above-outlined conventions are indispensable.

10.29 Abstract Nouns. In a sense, of course, the question of the status of abstract nouns (terminating often in '-ness' or '-ty') has been with us ever since our first venture into the quidditative in 0.4. Exactly what may be decided concerning that status depends a great deal on the various intra-linguistic contexts in which they may occur, and the structure of such contexts may vary from one natural lan­ guage to another. Hence the present note relates primarily to such nouns as they occur in English usage, comprising some technical cases also. There is, in general, a noticeable contrast between 575

Section 10 the usage-grounded availability of abstract correlates of nouns or names where these are adjectival, and the situation where substantives are concerned. Thus the adjectives 'blue', 'tall', 'sweet', 'heavy', and the like all have familiar abstract correlates framed by the use of '-ness' as a termination, as in 'blueness', and so forth. The -'ity' termination appears in 'enmity', 'corporeity', and '-hood' gives 'bachelorhood', for example. However, if one goes beyond usage, then the ruthless abstractor can still create correlates of the substantives by adapting '-ness' as a means thereto, as in 'treeness', 'tableness', and so on, on which see HDG §4.1, LE III, ch. 8, §2. The philosophical bearing of these trite bits of grammatical lore is rich, various, and much too extensive for survey at this point. Only a few details of the above-ment­ ioned quidditative connection will be touched upon here. Thus in an English-speaking philosophical context (e.g. QF, Essay I) one may find assertions such as 'Pinkness is predicated of Socrates', these being allegedly derived from 'Socrates is pink', and so on. Now there is no particular difficulty about the grammatical status of the latter's parts. Its '... is ...' approximates to our primitive '...E ...'. The former sentence, however, leaves us with the abstract 'pinkness' as something somehow related to Socrates, and perhaps having a status (for all we know) analogous to that of the mud described in 'Mud is thrown at Socrates'. Since the latter leaves us with an object, namely the mud, related to Socrates, so, it might appear, does the former, with pinkness as the related object in its case. Hence comes about the postulation of abstract entities or 'universals', in accordance with what is sometimes said to be a 'Platonic' or 'Realist' position. However, facilities are now available for the use of such abstract nouns as terms in true propositions, but without any such Platonic commitment. Such propositions are framed at the level of the verb-termed quidditative '... is* ...' of 10.26. The terms themselves will be logically verb-like, 576

Presuppositional Explicitation rather than name-like, with details according to context. Thus the 'pinkness' will be related to '... is pink' in the following way. The latter is a predicate (i.e. a case of 'Ø ...') nominalisable, by means of 10.251, as 'trm', or 'Ø-r''. This may then be taken up as the completion of quidditative term-functors such as those made available in .241 and .242, so that its sense becomes, e.g. 'Being  -', 'To be  Ø-', or, in the concrete case, 'Being a term satisfying ... is pink' (or 'To be a term satisfying ... is pink'). This enables true statements to be made about pinkness (e.g. that it is* a colour) without any necessity of postulating some object named by this abstract noun. The original rather slack and inconsiderate talk about pinkness being predicated of Socrates, from which we started, can likewise be relativised to the quidditative-level facility just outlined. By this stage, however, it is local low-grade philosophical usage which is becoming the topic of discussion, and there is no necd to get bogged down in that. Another use of abstract nouns is that according to which some object is said to 'share in', 'participate in', or to 'have' the ...ness which is in question. This use was exp­ lained by St. Anselm in its Latin version ('... habens albedinem', and so on). The special logical interest of this part­ icular Latin version is that it permits a dual reading, either as a name ('whiteness-haver') or as a verb ('having whiteness') owing to what became codified as the distinction between '...ens ut nomen' ('...-er as a name') and '..ens ut verbum', ('...-ing as a verb'). (Notice how the structure of the two versions, Latin and English, is already parting company; hence the frequent necd to relativise problem cases to their orig­ inal natural language (from the narrative point of view) as well as to the unified background categorial language (from the demonstrative point of view) if proper historical and theoretical appreciations are to become clearer.) In the nom­ inal reading of the situation the categorial language makes 577

Section 10 available, for example, 'trm<e{white}>' which, like 'white-iseer', is the nominalisation of the appropriate verb in accord­ ance with 10.251, and shows the structure of the nominallyunderstood habens albedinem of the Latin. In contrast, the verb-like ('ut verbum') interpretation of the Latin participial expression 'habens albedinem', as 'being an ...is-white-ise-er' comes out as 'Cl{trm}', in accordance with 10.241. Thus interpreted, it is suitable to become a term completing the quidditative '... is* ...'. The Anselmian and other medieval exploitations of this contrast are sketched, with further references, in HQS §3.4. For other suggestions about the interpretation of abstract expressions, HQS §6.7 and HLM HI §6 may be consult­ ed. 10.3 Mereology 10.301 Building upon the foundations of the two previous sub-sections, the present one is to yield Mereology, the theory of parthood. This will be effectuated thanks to the addition of one further primitive term, namely 'pt( )', a functor which forms a nominal expression from a nominal expr­ ession, so that its categorial index is n/n. Its sense in English will be rendered below as 'part of ...'. (Note that this 'pt( )' corresponds to the 'ppt( )', i.e. 'proper part of ...' used in HQS §4.5, §6.9.) 10.302 The line of exposition to be adopted here is based most immediately on notes taken during lectures delivered in Manchester by Czeslaw Lejewski, supplemented by published papers of his (LLM, LAM, LMC, LNM) and of S. Lesniewski (LF). On the axiomatic foundations of mereology both the thesis (LAF) and papers {LNA, LAX) of Audoënus Le Blanc embody fur­ ther superb researches. Throughout all these works ample bibliographical information (e.g. concerning the writings of V.F. Rickey and B. Sobocinski) is made available. I have continued, as in HLM, HQS, the use of those lower-case var578

Presuppositional Explicitation iables, introduced in the sections above, for nominal terms, notwithstanding Lesniewski's original preference for a mixture of upper and lower. Adherence to the lower case is here int­ uitively and pedagogically the simplest option, and constit­ utes a notational variant of no theoretical importance. In­ deed, use of the mixture could convey the quite false impres­ sion that some sub-species within the semantic category of names is being built into the system, and this is certainly not the case; cf. 10.2112.

10.31 Axiom System based on 'pt( )'. Although the follow­ ing system fails to come up to the aesthetic desiderata prop­ ounded in Sobocinski's work on well-constructed axiom syst­ ems (SA), it nevertheless has the twin virtues of intuitive simplicity (thereby interlocking closely at several points with medieval doctrines) and historical interest, in that it is an,expression of Lesniewski's original mereological principles. First comes a principle universally underwritten by common sense and by medieval tradition:

Next the commonsense intuition is further underlined by means of a statement of the result of successive containments of parts:

In present-day terminology, this axiom expresses the transitivity of 'part of ...'. 579

Section 10

Presuppositional Explicitation their integral whole which is a, even if there happens to be just one b, such as Socrates, in which case the 'improper part' encapsulated in 'el( )' comes into operation, since Socrates is identical with his integral whole; cf. .325 below. The final (iv) becomes more intuitively acceptable when we have witnessed the medieval struggles to express the minimum which may be inferred about the parts relatively to a whole con­ cretely composed of objects of a specified sort or kind. We know that every integral part of the integral whole of the dogs need not be a dog, since, e.g. it may be a doggy paw. Still, the parts must all be somehow 'doggy' in the case of dogs, and human in the case of man, and so forth. This four­ th phase of the definition expresses just what is required here, e.g. every element of the collective class of the dogs has an element in common with a dog; every element of the collective class of men has an element in common with a man (i.e. is human), and so on. (This reminds us, yet again, of how 'human being' will not serve as a sexually neutral version of man in its Latin (homo) sense. Each of us consists of countless beings, every one of them human; cf. 1.231.) Note that the 'Kl( )' notation introduced above corresponds to the 'ccl( )' which was used in HQS. The functor defined in D2 may now be used to express the uniqueness of the collective class (or entire integral whole) of the so-and-so's. Any two candidates for the office of being a given collective class turn out in fact to be one identical object:

581

Section 10 Finally, axiom A4 ensures the existence of the collective class of the b's provided that there exists at least one b :

So concludes this rendering of the original Lesniewskian foundations. Their aesthetic inelegance (clearly obviated by later, especially single, axioms) lies in the use of the defined term 'K1( )' in A3 and A4, but in no way aff­ ects their deductive cogency. On single axioms for mereology, LLM, LAX, LNA and LAF are among the sources which may be consulted.

10.32 Theses based on the 'pt( )' Axiomatic. Given the presence of nominal negation in Al above, definition 10.252 allows the inference from Al of:

Since, by 10.21162, one may plainly infer 'ob(b)', 'there exists exactly one b') from the concluding clause of this present thesis, the latter is a reminder that 'pt( )', 'part of ...', calls for unshared noun expressions as its completions. In English (as opposed to Latin) this unsharedness is often made explicit by the use of 'the ...', e.g. 'the collective class of men'. This, in the spirit of .321, eliminates an ambiguity which in medieval article-free Latin could lead to misunder­ standings, as 10.327 is to remind us. See also 4.8 for a dis­ cussion around .321. Next, from Al and 10.252 (the definition of nominal 582

Presuppositional Explicitation negation) one has a thesis often upheld by the medievals, namely:

For example. if the roof is part of the house, then it's not that the house is part of the roof. From this in turn it follows, by reductio ad ab­ surdum (WRP *2.01), that no thing is part of itself:

Abelard overtly mentions this (2.34(A)). Aquinas seems to have it in mind at the point mentioned in 3.111, and Paul of Venice relies on it in 8.433. Again, and in yet another famil­ iar medieval style, from this and .321 it follows thus:

Thus if the wall is a part of the house, it's not that the wall is the house. 1.32 deals with some early medieval disc­ ussions of this thesis, which is there described as that of part-whole disparity. Also covered by this thesis is material in the following sub-sections of section 3, i.e. .113, .21, .22, .26, .291, .292, .313, and .331. Other examples are in 4.41, 5.21, 7.06, and 7.3121. From this point onwards little attempt will be made to provide even remote indications of the established proofconnections between the axiom system and the theses enuntiated below, since in many cases intermediate theses are 583

Section 10 involved which are not made explicit here. This applies in the first place to:

In other words, every existing object is the collective class of itself. Next comes a thesis which, among other things, contributes towards an answer to the medievally oft-raised question of how the application of an adjective (e.g. 'white', 'worth a thousand marks') to the whole affects what can be predicated of the parts. If the house as a whole is white, one might be tempted to infer that its parts were also white, whereas there is no such temptation to apply the price of the whole house to the individual parts also. What generalisation may be made which will show the most that can be inferred concerning the parts in both these types of case? The foll­ owing may help here:

Thus if b is the whole house and if that house is white ('bE c') then its element a is an element of the collective class of white objects (but this by no means makes it in consequ­ ence white); likewise, if the house is worth a thousand marks, then its element is an element of the collective class of things worth a thousand marks (but this by no means makes it in consequence worth a thousand marks). In short, in neither case need there be authorised the direct predication, in res­ pect of the part, of the adjective applicable to the whole: cf. 584

Presuppositional Explicitation [13.2] of 4.7. Further to the line initiated in .325, one also has:

Contrastingly:

Finally in these notes on material based on 'pt( )' as primitive term, one has the definition of the functor sometimes perhaps confused in Latin with collective class of ..., owing to the absence of articles in that language:

The definitional expression to the right of the equivalence may be expressed more briefly in terms of its two main clauses thus: (i) exactly one a exists, (ii) every element of a has an element in common (i.e. e) with a b (i.e. the d which is a b). Comparison with the definition of collective class (i.e. D2 of 10.31) will show that the mere collection now being covered lacks the property of containing all the b's (i.e. objects of the sort on question). Abelard appears to have had trouble in sorting out the difference between the 585

Section 10 collective class (i.e. the complete collection, the entire integral whole) and some collection or other, i.e. a mere collection as here characterised: HQS 4.524. The collective class of men is unique, whereas lots of things can count each as a collection of men. Use of the Latin 'totum integrum', 'integral whole', lacking any of the distinguishing articles available in English, may cover both these cases.

10.33 Concerning 'outside of ...' As the literature to which reference was made in 10.302 makes clear, the functor 'pt( )' (i.e. 'part of ...' or 'proper part of ...') is far from being the sole possible primitive term for Mereology. Another alternative is 'extr( )', (i.e. 'external to ...', 'outside of ...'). This is of particular interest since Nicholas of Paris invoked the same notion when characterising the integral whole, as may now be seen. Thus in the first place 'extr( )' may, for our own local familiarisation purposes, be defined in terms of the 'el( )' of Dl (10.31 above):

Next, whether in continuation from this last definition, or as an item in an axiomatic system based on 'extr( )' as primit­ ive term, the following is a definition of collective class ('K1( )') in terms thereof:

586

Presuppositional Explicitation and only if a is a, and for all c, a is outside  if and only if, for all d if d is  then d is outside c) Now Nicholas of Paris, when defining what he himself calls the 'collective or integral whole' (totum collectivium sive integ­ rum) says that such a whole is that outside of which there is nothing: totum est cui nihil est extra: 7.21(A). This is a negative correlate of .332, but can also be taken in various other connected senses, as .333 and .337 are to illustrate. Thus there now follow some other theses involving 'extr( )' First:

In brief, nothing is outside of itself. This applies also, of course, when a is some collective class, as indeed is each in­ dividual (cf. .325 above). This last thesis could hence be a further counterpart of Nicholas of Paris' remark that the col­ lective class is that outside of which there is nothing, i.e. nothing of that whole collective class. Or again, we have:

The functor 'extr( )' may be used also to charact­ erise discreteness, a notion which has been in play ever since our first encounter with it in the discussion of Master Pet­ er's Mereology (2.7): 587

Section 10

Finally, a rather more literal interpretation of Nicholas' words would go almost exactly parallel to Sobocin­ ski's definition of the Universe ('U'):

The V\' is here the nominal 'non-being' defined in .254. If Nicholas' words 'cui nihil est extra', 'outside of which there is nothing', are taken in their fullest possible import, then they indeed come close to a Latin version of the definiens within .337. What we are also incidentally glimpsing at this point is how the recently documented and hitherto unsuspected vast medieval and contemporary metaphysical investigations concerning the notion of nothingness {HQS §2) may be prof­ itably extended also into the mereological field.

10.34 On Overlapping. The notion of mereological overlapping is one which sometimes occurs in medieval discussions (e.g. 1.52, 3.65) and may easily be defined in terms of 'el( )' thus:

588

Presuppositional Explicitation Several decades ago it was shown by C. Lejewski (in LLM) that the functor here defined could in fact be used as a primitive term in a single axiom for mereology. In the thirteenth century Peter of Auvergne entered into the present conver­ sation by characterising the discrete as the non-overlapping: illa quantitas est discreta cuius partes non copulantur ad aliquem terminum communem: CC 55, pp. 45 - 6. This exem­ plifies yet another of the rare medieval contexts in which the possibility of overlapping objects is considered, although on this occasion only negatively; cf. 1.5.

10.35 Atomistic Mereology. A survey of the ways in which general mereology may be extended by the addition of the con­ cept of mereological atom is provided in LEM. For example, one may continue on from the definition of overlapper (.341) to the definition of atm in the style of H. Hiz, thus:

589

Section 10 theme investigated at enormous lengths by medieval philosoph­ ers. For some inkling of their vast explorations, Rega Wood's work on Adam de Wodeham's Tracta tus de Indivisibilibus (coded as WW) may be most usefully consulted.

10.4 Conclusion. This excursion through the book's basic presuppositions has been somewhat long drawn out, owing to the project of making those presuppositions intelligible to readers justifiably suspicious of the inscrutabilities of any categorial language blankly presented as a quite unexplained deus ex machina which is supposed to solve all difficulties except the one constituted by its very presence on the page. Other versions of such bridge-building have been made exten­ sively available in previous works (HLM, HQS, HCD, HDG). The principles are the same throughout, i.e. the distance between familiar pretheoretical discourse and the technical, canonical, discourse is rendered easily surmountable by anyone who wish­ es to check on the senses of the primitive terms of the cat­ egorial language. On these are based the theories which ser­ ve to delineate exact coordinates within whose bounds may be pitched the interpretation of hitherto inexactly situated philosophical discourse, past or present. But is the investigation of such rather dull preliminaries really worth while? An analogy based on a Bergsonian thesis may be coopted in order to suggest a reply to this question. Thus when it is said that every new word which a child utters is a hecatomb of living beings, this saying may be adapted to the unguarded proliferation of tech­ nical terms (some of which may be vituperated as ' jargon' by their deprecators) within philosophical discourse. Unless such terms are explicitly introduced in the light of an exact (though flexible) canonical language, then it is a hecatomb of precise discourse-possibilities which ensues. Kant's view that traditional metaphysics is a field for mock battles may 590

Presuppositional Explicitation then indeed not only be maintained, but extended to the other branches of philosophy as well.

591

Index Hereunder the bold-printed numerals each allude to all the sections and subsections whose numbers commence with t h a t numeral. Other numerals refer to pages. Personal names such as 'John of Salisbury' are listed in the order dictated by the given name (in this case 'John', for example), and not under the place-name. Abelard, P e t e r , 2.1 - 2.4, i, iii, iv, ν , xii, xiii, xiv, xix, 35, 41-5, 46-7, 48-52, 55-6, 58, 59, 62, 63, 152, 163, 173, 174-8, 180, 194, 199, 213, 216, 217, 282-3, 313, 335, 342, 384, 389, 396, 422, 426, 440, 442, 444, 482, 589 abstract entities, 37 abstract nouns, 10.29, 15, 16, 34, 36, 37, 227 Adam, 321, 404 Adam de Wodeham, xx Adams, M.M., χ adjectives, 430 aggregates, 385, 390-405 Ajdukiewicz, K., 13, 29, 156 Alberic of Paris, 116, 152-3 Albert of Saxony, 258 'alone', 86-8, 173, 176-7, 440 analysis, linguistic, 29, 457, 539 angels, 334 Anselm of Aosta and Canterbury (St.), ii, xv, 10-11, 16, 56, 385, 430, 577, 578 anthropology, linguistic, 14, 171, 201

592

Index Aristotle, 3.1, i, v, xiii, 1, 28, 39, 61, 62, 131, 208, 209, 211, 341, 344, 345, 349, 352, 354, 356, 363, 364, 369, 373, 401 art, 320, 385 artificial, the, 44-5, 76-7, 80-2, 205, 213, 277, 286-7, 302, 304-5 and the natural, 3.7 atomism, 53-4, 57, 130, 222, mereological, 10.35, 104-5, 178, 253, 434, 453, 455 Augustine of Hippo (St.), 241, 281, 290-2 authority, 62-3 Averroes, 407, 413 Avicenna, xiii, 361, 362 axioms, of Protothetic, 549 of Ontology, 10.27, 26 of Mereology, 3.42, 10.31, 26, 578 of Metaphysics, 2-4

Bäck, Allan, xvi Bacon, Robert, 420-8, 470, 474 Bartholomew of Bruges, 27 being, 15, 28, and asserting, 193-4, 209, 211, and definition, 561 and grammar, 22 and living, 280 and quiddity, 190 Bergson, E., 590 biology, 244 Bivalence, Principle of, 376 Bochenski, I.M., ii, xiii, 545 Boehner, P., iii, xiii 593

Index Boethius, A.M.T.S., 1.3, i, xiii, 2, 49, 54, 62, 63, 72, 134-5, 158, 159, 180, 181, 185, 194, 218, 225, 237, 333, 388, 406 Boethius of Dacia, xiii, xiv, 2-7, 9, 10, 14, 22, 24-6, 27, 30, 31, 32, 33, 160-1, 166, 339, 340, 341, 370-2, 385, 411, 426-7, 429, 538, 553 Bochum Studies in Philosophy, iv Booklist, xii-xx Bos. E.P., iv Boyle, Robert, xiii, 77, 124-6, 132-3, 135-6, 196, 199-200, 205 Braakhuis H.A.G., iii, xiii, 420, 428 British Academy, iii Buddhist Scriptures, xiii, 1 Burleigh, Walter, i, iii, xiii, 46-7, 48, 258, 406-17, 420, 422, 428, 441, 459, 470 Bursill-Hall, G.L., iv

Categoriae (of Aristotle), 82, 131, 135 categorial indices, 13-14, 15, 24, 34, 156, 542-3 categories, Aristotelian, 82, 131, 135 category mistake, 358 category, semantic, 14, 24, Christ, 402, 405 Chronology, 244, 553 Cicero, M.T., 278 Clarembald of Arras, 2.82, i, iii, 192-3, 206, 211, 212-3, 365 class,

collective, 20, 31, 32, 36, 386, 425, 580 distributive, 35-6, 424 mythology of, 18, 36, 359, 561 collection, 19, 41, 65-7, 214, 229-30, 244, 295, 420, 429-30, 585, complete, 20 594

Index Compendium Logicae Porretanum, 2.84, 180 composition and division, 358-60, 368-9 continuity, Abelard on, 2.23 Cousin, V., v, xiv, 151

Dal Pra, M., xiv Deely, J.N., xiv, 325 definition, 15, 28, 84, 563 creative, 29, 545, and esse, 561, matter and form of, 54, 227-8, 413-4, ontologicai, 554-71, parts of, 237, protothetical, 549, 559, definitional frames, 559, 563 del Punta, F., xx de Rijk, L.M., iii, iv, xii, xiv, 341, 383 Descartes, R., 133, 163, 280, 290, 297, 316 descriptions, 241 descriptions, theory of, 557 De Sophisticis Elenchis, 5, xiii, 188, 401 destructivism, 2.43, 2.44, 110-13, 116, 117, 136, 151, 206, 213, 482-5, 496-7, 501-3 diagrams, v dictum 376-7 differentia (predicable), 21, 413 diminutives, 444-5 discreteness, 57-8, 76, 129, 157, 174, 175, 245, 395, 423-4, 427-8, 588, and plurality, 59-61, 76, 96-7, 168-70, 177 diversity and difference, 170-8 division, 1.3, Abelard on types of, 2.22 of adjectives, 192-3, 595

Index Aquinas on types of, 291-2, mereological, 172 Dorp, Jan, 329, 330, 331

Ebbesen, Sten, iii, iv, xiv, 27, 180, 342, 370, 372 element (Aristotelian), 195-7 element (mereological), 580 ens: ut nomen, ut verbum, 577 essence, 19 (cf.quiddity) Eucharist, 3.44, 386, 403 exceptives, 430-1, 7.4 exclusives, 440 existential import, 568 expansive terms, 467-8 extensionality, 549

facere (to do), 16 fallacies, 5.3, 341 fictions (figmenta), 25 Flasch, K., xiv form, and action, 297-8 Aristotelian, 3.11, 3.22, 494-6, and identity, 409-10, and matter, 53, 412, and number, 356, and organism, 3.33, 3.4, 253, pure, 410, unity of, 299 Fragmentum Sangermanense, 64-5, 97, 101, 108-9, 113-4, 130, 151, 204, 482, 491, 493, 494, 503 Fredborg, K.M., iii, xiv, 180 Freddoso, A.J., xiv

596

Index functors, 11-13, 160, 432-3, of existence, 10.21, of identity, 10.22, of inclusion, 10.23, name-forming, 10.25, 243, 419, 432, name-termed, 227, 280, 294, primitive, for mereology, 579, for ontology, 552, 568, for protothetic, 549, proposition-forming, 12-13, 419, and syncategoremata, 7.1, 429, verb-termed, 0.4, 280, 294

Garland the Computist, xiv, xv, 10, 421, 444, 455, 569 Geach, P.T., iii generation, 195, 197, 202-3 genocide, 81 genus (predicable), 2 1 , 54, 439 Geulincx, Arnold, 382 Geyer, ., xiv Gilbert of Poitiers, 2.8, i, iii, xiv, 19, 21, 31, 157, 179-80, 217, 440 gnomon, and increase, 130-2, 137-8 God, 78-9, 196, 197, 199, 231, 312-3, 434, 453-4 Goodman, N., 386 grammar, deep, 9, 11, and logic, 9, 24-6, logical, 34, and metaphysics, 7-14, 24, and part, 0.5, and philosophy, 22-3, and quiddity, 0.4, 597

Index grammar (continued) speculative, 0.3, 6, 8, 14, 22, 30, traditional, 8-9, universal, 24-6, 30 Häring, N.M., iii, xv Heidegger, M. 17 Heloïse, 114 Helyas, P., 16 Henry, D.P., xv Henry of Harclay, 59 Historia Animalium, 39 Hobbes, Thomas, 386 Holkot, Robert, 373 Hughes, Robert, xv, 81 Hume, David, xv, 113 hylomorphism, 53

identity, and change, 2.4, 2.5, personal, i, 254, 493-521, quidditative, 160, and resurrection, 3.6, singular, 160, 162, 558, and soul, 111-13, 318, strong, 558, weak, 558 Imbach, R., iv inclusion, 10.23, 38 increase and decrease, 2.5, 76, 103 individuation, 3.12, 58, 271-3, 319 infinitives, as terms, 16, 181 innominate (unnamed privations), 256, 261 Institutum Carlsbergicum, 342 598

index intellectus, 6.18 '... is ...', 23, 336, theory of, 23-4, with nominal completions, 34, with verb-completions, 34, 566-7 Ivo of Chartres, 204

John of St. Thomas, 2-3, 305, 325-8 John of Salisbury, xix, 64, 385 John the Scot Eriugena, 342 Joscelin of Soissons, i, 31, 36, 64-5, 99, 108, 110, 130, 136, 150-2, 177, 194, 217 Juvenal, 35

Kalinowski, G., ii Kant, I, 32, 150, 539. 590 Kenny, Α., ii, xiv, xviii, xx, 383 King, P.O., xv, xvi, 65, 384 Klima, Gyula, xvi, 313, 405 Kneale, Martha, vi Kneale, W., iii, vi Koerner, ., iv Kretzmann, N., ii, xiv, xvi, 415, 418, 443

language, anthropology of, 14, 171, artificial, 1, 7, canonical or categorial, 11, 14, 19, 24, 30, 32, 45, 161, 206, 540, 544-5, 590, inflected, 11, mental, 330, 467-8, 533, 536, 537, and usage, 11, 460-1, 539 599

Index Latin, ii, 11 Le Blanc, A.V., xvi, xvii, 578 Leibniz, G.W., xvii, 396 Lejewski, C , ii, xvi, xvii, 1, 10, 28, 49, 168, 373, 551, 571, 578 Lesniewski, S., ii, xvi, 1, 29, 30, 31, 386, 553, 555, 559, 578, 579 Lewry, Osmund P., xviii Liar, Paradox of the, 5.5, 342 Locke, John, xvi Lockwood, E.H., v logic, ii, 24-6, and grammar, 24-6, mathematical, 28, 540, and metaphysics, 26-8, symbolism of, ii, 26-7, 539 Lolita, 121 Longuet-Higgins, H.C., xviii Lucas, J.R., xviii Lukasiewicz, Jan, xvi Luschei, E.C., xvii Lvov-Warsaw School, i, iv

Macmillan, R.H., ν Malebranche, N. de, 297 'man', sense of, 1.31, 3.439 manifolds, 5.4, 116, 152-79, 342 Marmo, Constantino, iii Mastger Peter, 2.7 Mather, Gavin, ν

600

Index matter, conservation of, 195-8, and dimensions, 285-6, and form, 53, 224-5, 284-5, and genus, 54-6, 227-8, and individuation, 257, and parts, 3.31, 228-9, 365-6 Mays, W., 550 McMullin, E., xvii Mereology, 0.1, 0.2, 10.3, 554, atomistic, 10.35, 104-5, 178, 253, axiomatic of, 250, axiomatic and organism, 3.42, and Eucharist, 3.44, and form, 3.41, and manifolds, 5.4, and metaphysics,3-6, 244-5, and quiddity, 190-2, 194, 202, 206 Metaphysics (of Aristotle), 285-7, 3.1, 3.2, i, 2, 3-4, 373-5, 413 metaphysics, 3.1, 3.2, i, ii, 22, 23, axioms of, 2-4, and logic, 26-8, 540 methodology, 1.1, 8 Michalski, ., iv Milinda, King, 2, 21 Mill, J.S., xvii, 372 Minio-Paluello, L., iii, xiii, 151 modi significandi, 22 modus narrativus/demonstrativus, 8-9. 32-3. 538 Moisisch, ., iv Mueller, I.J., iii, xx, 383

601

Index names, 13, 553-4, abstract, 10.29, 15, 16, 34, 36, 37, 227, and adjectives, 13, and descriptions, 13, empty (fictitious), 13, 17, of first imposition, 271 of names, 564, privative, 256, 261, of second imposition, 271, natura magna, 101, 108-12, 177-8, 482, 504, 506, 510-12 natural and artificial, 3.7, 285-7 negation, nominal, 564-5, propositional, 548 Nicholas of Paris, 7.3, 7.4, iii, 418 Nielsen, L.O., iii, xiv, 180 nominalism, 15, 89, 185, 330, 335 nothing, 17, 588 Nuchelmans, G., xviii, 373, 376, 383

O'Donnell, J.R., xix Ontology, 10.2, 30, 330, 543 axiom for, 10.27, definitions, 10.21 - 10.26, definitional frames, 551, 563, primitive t e r m for, 543, and quantification, 554-6, and punctuation, 10.28 Orenstein, Alex, xviii outsidehood, 10.33, 57, 420, 421-2, 447 overlapping, 10.34, 57, 59, 96-7, 169, 171, 246, 321 Ovid, 196-8

602

Index paronymy, 210-11, and organic form, 263, 265, 288-9 part, 0.5, analogically integral, 3.36, common,129, 250, 262-4, 269-70, 295, 307, 424, 437, 581, common, and increase, 127-30, component, 48, 473, of definition, 237, 413, essential, 3.32, 407-8, 412, of form, 3.33, homogeneous/heterogeneous, 39-45, 116-7, 247, 249-54, 300-1, 388, 319-20, improper, 264, 296, 580, material, 3.31, 233-4, 242, 258, 259-60, 365, 370, 407, 480-1, 489-91, and mutilation, 276-7, 449, operation of, 3.5, ordering of, 244-5, 'part of ...' as functor, 3.34 potential, 246, principal, 2.4, 3.35, 58, 188, 206-7, 211, 290, 335, 375, 409, 410, priority of, 90-1, 124-5, proper,434, 580, quantitative, 232, 237, 238-9, 255-7. 407-8, quidditative, 413, of species, 259, specific, 38, 58, subjective, 34, 39, 231, 237, 243, 339, 375, 378, 412, 421, 448, of surface, 265-9, temporal, 1.6, 2.6, of virtues, 278-9,

603

Index part (continued) X-part and part-of-X, 1.4, 2.3, 3.41, 125, 134, 136, 139, 141-2, 145, 206, 256, 260-1, 273-4, 288, 290, 298-9, 322-3, 338, 363, 368, 407, 410, 432, part-whole disparity, 1.32, 3.21, 44, 88, 177, 187, 239-40, 262, 583, part-whole predications, 3.5, 5.3, 186-7, 200, 208, 212, 337-40, 342, 401, 405, 434, 437-8, 584 Patrick of Ireland, 229, 342 Paul of Venice, 8, i, iii, ν , xviii, xix, xx, 2, 102, 206 Perreiah, A.R., xviii Persius, 198 person, 273, collective, 397, 404 P e t e r of Ailly, 377 P e t e r of Auvergne, 589 P e t e r of Spain, xv, 88, 259, 340, 401, 438, 440, 442, 470, Pinborg, Jan, ii, iv, xiii, xiv, xviii, 27, 329 Plato, 384, and universals, 15, 36, 576 Pliny, 35 plural, grammatical, 80-4, logical, 161 plurality, and contiguity, 214-6, and multitude, 251, 253 types of, 165-6 Pluta, Olaf, iv Poinsot, John, xiv possible worlds, 561 predicables, 37-8 predication, 16-18, of adjectives in respect of whole, 188-9, 190-4 presuppositions, 10, ii, 23, 30 604

Index Prior, A.N., xviii, 12 Priscian, 417, 464 pronoun, multiple senses of, 154-5 property (proprium), 183, 184, 191, 201-2 proposition, name-termed and verb-termed, 280, existence of, 377-82 Protothetic, 10.1, 543, 559 prudence, 278 punctuation, 10.28

quantification, 416, 539, 554-6, and existence, 555, restricted and unrestricted, 555 quantity, and identity, 2.4, logical, 340, 344, 370-2, 411, 476, 559, and part, 232, 238-9. 255, and universality, 66-7 question, quidditative, 18, 159-60 quiddity, 0.4, 3.12, 10.26, 21, 30-1, 34, 36, 38, 53, 146-7, 155, 158-62, 163, 165, 180, 181, 189, 194, 209, 236, 322, 438, and definition, 260, and form, 242, 294, 410, and functorial incompleteness, 236, 568, and intellectus, 387 Quine, W.V., xviii, 384, 456, 555, 557

Reginald of Piperno, 317 references, styles of, xii, 8.03 605

1.5

Index resemblance, 162-3, 432-3 resurrection, bodily, 3.6, 254, 284-5 Rickey, V.F., 578 Rosier, I,, xviii, 216 Russell, ., xx, 9, 11, 28, 385, 555, 557, 563, 564, his paradox, 557 Russell, E.S., xviii Ryle, G., 168, 358

Samaritan, good, 402 Scarre, G., xix, 372 Schmitt, F.S., xix Scott, F., xix, 406 Scotus, Duns, 314, 342, 351, 401 Stanislav of Znaim, xviii, 16, 383 Stereology, 244 Stewart, M.A., xiii Stump, Eleonore, 329 subsistentity and subsistessence, 182-4, 207, 236 substance, 238, 271, m a t t e r and form as integral parts of, 237-8, secondary, 19, 224-5, simple, 266 superfluit 'omnis', principle, 421, 434, 444, 455 suppositio, 58, 157, 476-8, 501, 385, 561, 576, syllogistic, 343-4, 419, 559 symmetry, geometric, ν syncategoremata, iii, 7.1 Szaniawski, ., iv

Tasmania, 81 t e r m s , primitive, 542 Thaïes, 312 606

Index theology, ii, 62, 386 Thomas of Aquino ('Aquinas'), St., 3, i, vi, xii, xiii, xvi, 3, 16, 56, 171, 344, 365, 367, 385, 406, 410, 412, 440, 459 Thomas, Ivo, xiii Topics, xiv translation, 469-70, 474, 479, 402-3 Trinity, 161, 404 truth, ontological, 16, -tables, 545-50 Turing, Alan, 572 Tweedale, M.M., xix, 65, 152 type-token distinction, 373, 375-6, 378

understanding, degrees of, 29 unity and plurality, 323-8 universals, 0.4, 34, 38, 64, 89, 179. 181, 184-5, 227, 230, 231, 237, 384, 385, 561, 576 as singular forms, 182, 185 usage (usus loquendi), 11, 460-1, 539

variables, nominal, 45, 553, predicate, 242, propositional, 553 verbs, Latin, 11, English, 12, as t e r m s , 184 Vienna Circle, iv Waddington, J.R., xviii 607

Index Webb, C.C.J., xix Whitehead, A.N., xx, 199, 564 wholes, and aggregates, 385, 390-405, in Aristotle's Metaphysica, 3.2, integral, 19, 37, 39, categorematic/syncategorematic, 2.21, 7, 8, 414-5, 420-2, 448, 455-7, 464-9, 518-26, collective, 19, 41, 65-7, 214, 229-30, 244, 295, 420, 429-30, collective objects, 400-1, and artificials, 80-4, (cf. 'artificial'), contiguous, 214-5, continuous, 44, 214-5, 245-7. 2511-2, 277, definition of, 250, 307, discontonuous, 116, 243-4, distributive, 34, 35, 39, 41, 44, 65-7, 143, 146, 157, 229-31, 243-4, 248, 290, 5.2, 388, 421, 429-30, 458-461, 470, essential, 284, fictitious, 148-51, homogeneous/heterogeneous, 39-45, 116-7, 247, 249-54, 300-1, 319-20, 388, inferential connection with part, 67-72, 84-90, 140-1, 164-5, 167-73, 243-4, 267-9, 4.4, 423-4, 426-8, 432-3, 441-2, integral, 19, 37, 39, 158, 232, 238, 241, 331, 337 340, 5.2, 359, 388, 420, 435-6, mass-, 277, 388, 404, 508-10, material, 421-2, organic, 3.42, 3.43, proportional, 232, quantitative, 284, 292-4, 333-4, 344, 470-2, temporal, 2.6, 245, universal, 5.2, 388-9, 448, 450-2 608

Index Wiggins, D., xx William of Heytesbury, 258 William of Ockham, i, iii, xviii, 258, 421, 434, 444, 445, 7.5, 563 William of Sherwood, xvi, xix, 340, 411, 418, 444, 470 Wittgenstein, L., xx, 11, 159 Wood, Rega, xx Woodger, J.H., xx, 102, 112, 206, 482, 545 Wyclif, John, xx, 1, 17, 58, 61, 111,6, 168, 169, 200, 205, 406, 494

609