Paul of Venice: Logica Magna: The Treatise on Insolubles (Dallas Medieval Texts and Translations, 27) [Translation ed.] 9789042949409, 9042949406

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d a l l a s m e d i e va l t e x t s a n d t r a n s l at i o n s

27

Paul of Venice Logica Magna: The Treatise on Insolubles

Edited with an Introduction, English Translation, and Commentary by Stephen Read and Barbara Bartocci

PEETERS

Paul of Venice Logica Magna: The Treatise on Insolubles

DALLAS MEDIEVAL TEXTS AND TRANSLATIONS

EDITORS

Kelly Gibson (University of Dallas) Philipp W. Rosemann (National University of Ireland, Maynooth) EDITORIAL BOARD

Charles S. F. Burnett (Warburg Institute); Marcia L. Colish (Yale University); Kent Emery, Jr. (University of Notre Dame); Hugh Bernard Feiss, O.S.B. (Monastery of the Ascension); Donald J. Kagay (University of Dallas); Theresa Kenney (University of Dallas); James J. Lehrberger, O.Cist. (University of Dallas); James McEvoy (†); Bernard McGinn (University of Chicago); James J. Murphy (University of California, Davis); Jonathan J. Sanford (University of Dallas); Baudouin van den Abeele (Université catholique de Louvain); Nancy van Deusen (Claremont Graduate University); Bonnie Wheeler (Southern Methodist University)

SPONSORED BY

DALLAS MEDIEVAL TEXTS AND TRANSLATIONS

27

Paul of Venice Logica Magna: The Treatise on Insolubles

EDITED WITH AN INTRODUCTION, ENGLISH TRANSLATION, AND COMMENTARY BY

Stephen Read (University of St Andrews) and Barbara Bartocci (University of St Andrews)

PEETERS LEUVEN - PARIS - BRISTOL, CT 2022

Cover illustration: The beginning of Paul of Venice’s Quadratura in MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2133, fol. 1r. By kind permission of the Vatican Library.

A catalogue record for this book is available from the Library of Congress. © 2022 – Peeters – Bondgenotenlaan 153 – B-3000 Leuven – Belgium. ISBN 978-90-429-4940-9 eISBN 978-90-429-4941-6 D/2022/0602/102 All rights reserved. No part of this publiction may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

To the memory of Walter Senner OP, my beloved mentor in medieval studies and in my life (B.B.) To the memory of Desmond Paul Henry, who first encouraged my interest in medieval logic and Paul of Venice (S.R.)

Frontispiece: The beginning of the sections on insolubles in Paul of Venice’s Logica magna (MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2132, fol. 236r). By kind permission of the Vatican Library.

Table of Contents Forewordix Acknowledgmentsxi Introduction1 1. Life and Works 1 2. The Structure of the Logica Magna3 3. Obligations 11 4. Insolubles 15 5. Paul’s Theories of Insolubles 36 6. The Authorship of the Logica Magna37 6.1 External Evidence 37 6.1.1 Perreiah’s External Evidence Reconsidered 37 6.1.2 Theories of Insolubles in Italy (from the End of the Fourteenth to the Mid-Fifteenth Century) 40 6.2 A Reassessment of Surviving Manuscript Sources 48 6.2.1 Parvus error in principio maximus est in fine48 6.2.2 Which Scribe Might Have Written Vat. Lat. 2132? A Working Hypothesis 53 6.3 Stylistic and Doctrinal Discrepancies between the Logica Magna and Paul’s Authentic Works 56 6.4 An Evaluation of the Hypotheses on the Authorship of the Logica Magna59 7. Editorial Principles 61 8. Conspectus Siglorum, Signorum, et Abbreviationum 61 8.1 Conspectus Siglorum 61 8.2 Conspectus Signorum 62 8.3 Conspectus Abbreviationum in Apparatu Critico 62 Bibliography63 Insolubilia: Latin Text and English Translation Chapter 1: Various Opinions on the Insolubles Chapter 2: The Author’s Opinion Chapter 3: Objections and Replies

76 79 135 175

viii

Table of contents

Chapter Chapter Chapter Chapter Chapter

4: 5: 6: 7: 8:

The Familiar Insoluble On Propositions which Appear not to be Insolubles On Quantified Insolubles On Non-Quantified Insolubles On Merely Apparent Insolubles

191 225 237 251 257

Commentary277 Appendices367 A: Quadratura (excerpts) 369 B: Sophismata aurea, Sophism 50 401 Notes on the Appendices439 Indices443

Foreword Self-reflexivity is one of the distinguishing marks of the human mind, which possesses the ability not only to direct its attention to objects, as non-rational animals do, but to reflect upon that attention itself. Moreover, Aristotle already argued that the human mind is able to become aware of itself only in such an intentional act as it is bent back upon another, not self-directed one. Only God, he famously suggested in the Metaphysics (XII.9), is capable of self-thinking thinking which is not thinking anything other than itself. In the Summa contra gentiles (IV.11), Thomas Aquinas was later going to build an entire cosmic hierarchy upon degrees of selfreflexivity. Yet even though (a particular kind of) self-reflexivity constitutes nothing less than the central and characteristic dignity of the human mind, it can be fraught with problems. Again, these were not just discovered yesterday. Augustine spent many pages of his dialogue De magistro analyzing examples of self-reflexive speech in which words themselves form the object of attention. How would one be able to refute someone—he asked his dialogue partner, his son Adeodatus—who jokingly suggested that a lion came from the mouth of someone who said the word “lion”? Now this works better in Latin, which has no article, than it does in English, but the point is nonetheless clear: self-reflexive language (in which elements of language are mentioned) has its own rules, operating at a different level from language that is used in the ordinary way. We have come to distinguish these levels conventionally by indicating mentioned terms through quotation marks or italics. That way, “He said lion” is clearly marked as different from, “He said (a) lion.” And so we do not have to imagine lions leaping from people’s mouths. Insolubles—the subject matter of this book—represent another case of self-reflexive language. In a proposition like “Socrates is saying a falsehood,” if the proposition is taken to refer not to some other proposition, but to itself, we will run into a paradox: for if the statement is true, it is false; but if it is false, then it is true! How to deal with this kind of issue formed the subject matter of medieval discussions and investigations which continue to be of interest to logicians in our own day. In the present volume, Stephen Read and Barbara Bartocci have painstakingly edited, translated, and analyzed one of the most important medieval treatises on insolubles, to be found in the Logica magna by the Augustinian friar Paul of Venice. Read and Bartocci’s work fascinatingly combines medieval scholarship with discussions of the logical issues themselves; thus in their introduction and notes they move from

x Foreword

c­ onsideration of manuscripts, sources, and influences to matters of translation and, indeed, the not so unimportant question as to whether Paul of Venice was actually right in his analysis of insolubles. So, the Dallas Medieval Texts and Translations series is able to present to the reader of this volume a distinguished contribution both to medieval scholarship and to the literature on an important logical problem. Philipp W. Rosemann November 2021

Acknowledgments We are particularly indebted to the Leverhulme Trust for awarding a Research Project Grant “Theories of Paradox in Fourteenth-Century Logic: Edition and Translation of Key Texts” (RPG-2016-333) to enable the work to be carried out that has led to the publication of this volume. We are particularly grateful to members of the Medieval Logic Reading Group for giving so freely of their time and expertise in weekly meetings during sessions 2017–18 and 2018–19 discussing the text and translation in detail. We are also grateful to the Steering Committee for the Leverhulme project (Professor E. Jennifer Ashworth, Professor Alessandro Conti, and Dr. Mark Thakkar) for their advice and encouragement. We must also thank the Arché Research Centre, the Department of Philosophy in the School of Philosophical, Anthropological, and Film Studies, and the University of St. Andrews for their support while the research for this volume was carried out. We were also helped by Dr. Miroslav Hanke of the Czech Academy, who allowed us to see his preliminary transcription of the 50th sophism from Paul’s Sophismata aurea. We are grateful to the staff of the Biblioteca Apostolica Vaticana for their assistance and particularly to Dr. Paolo Vian, who provided us with valuable palaeographical and codicological advice. We have also benefitted in later stages of our project from the endeavors of those who are involved in working at the DigiVatLib service and made digital images of medieval manuscripts available online. Their labors have the potential to improve knowledge of medieval thought radically.

Introduction 1. Life and Works1 Paul of Venice (Paulus Nicolettus Venetus) was born in Udine in 1369. We know little about his childhood and early youth, but it is likely that he received his early education in his home town. He joined the Order of the Hermits of St. Augustine around the age of fourteen, entering the Venetian convent of St. Stephen as a novice—which is the reason why he is called “of Venice.” Here he studied philosophy following classes, among others, on the logica antiqua, that is, the texts of the Old and New Logic (logica vetus and logica nova, for which see infra, § 3). From the end of 1387 until August 1390, Paul continued his philosophical studies at the studium generale of the order in Padua. Subsequently, from August 1390 to December 1393, Paul studied philosophy and theology at the studium generale of the order in Oxford. It was during his Oxonian stay that he became committed to the philosophical realism of Wyclif and the socalled “Oxford Realists” like Robert Alyngton, Johannes Sharpe, Roger Whelpdale, and John Tarteys.2 During this period, moreover, he may have started working on, and perhaps completed, one of his most influential works, the Logica parva or Summulae. This short and unsophisticated introductory textbook of logic focuses on the theories of terms, supposition, consequences, proofs of propositions, obligations, and insolubles.3 The compilatory character of the book led the fifteenth-century commentator Antonio Cittadini da Faenza to doubt its originality, and even the authorship of the Logica parva. In the prologue to his rather polemical commentary on the Logica parva, written in the 1470s, Antonio reports and seems to give credit to

1  For a more comprehensive and detailed account of Paul’s biography and related secondary literature and for a full list of his numerous writings, see Bottin, “Logica e filosofia naturale”; Perreiah, Paul of Venice, and Conti, Esistenza e verità, 9–19. 2  See Conti, Esistenza e verità, 9. 3  The date of composition of the Logica parva is not uncontroversial. Conti hypothesizes that Paul wrote it while in Oxford, and Bottin that he wrote it in Oxford or immediately after his return to Padua (see Bottin, “Logica e filosofia naturale,” 90; Conti, Esistenza e verità, 9 n. 33). Perreiah, Paul of Venice, 6 and 13, connects the Logica parva to Paul’s teaching in Padua as lector formatus and thus postpones its composition to the years 1396–99.

2 Introduction

stories according to which Paul’s booklet was a mere compilation of earlier British texts: Some authoritative people say that this booklet was found in Britain, where the study of logic and philosophy once flourished, among very old writings, so that it is certain that this opusculum was made up of these ⟨texts⟩ before Paul was born.4

At the beginning of 1394 we find Paul again present at the conventual studium in Padua, first as advanced student (or cursor) and subsequently, from 1396 to 1400, as beginning teacher (lector formatus). It is likely that over this four-year period as lector, Paul composed the Logica magna.5 The Quadratura seu quattuor dubia records part of Paul’s teaching activity when he was bachelor (1402–05);6 it solves 200 logical problems, to each of which corresponds one chapter, with the 200 chapters arranged under four doubtful questions (dubia), each with 50 chapters. In the following years Paul was bachelor in theology, taught at the conventual studium, and wrote on theological themes (for example, the Super I Sententiarum Johannis de Ripa lecturae abbreviatio, written in 1400–01) as well as on Aristotelian works (for example, the Conclusiones Ethicorum and the Conclusiones Politicorum in 1402–05). He became Doctor of Arts and Theology around 1405/06. By that time, he had completed his commentary on Aristotle’s Posterior Analytics and started his teaching career at the University of Padua, which was the main university in the Venetian Republic at the time. He was active there for fifteen years, during which time he composed, among other works, the Summa philosophiae naturalis (1408), the massive commentary on Aristotle’s Physics (1409), and plausibly the Sophismata aurea, which is a collection of 50 sophisms written for the training of students. It seems likely that  “Ferunt autem quidam, non auctoritate indigni, hunc libellum in Britannia, ubi olim et dialecticae et philosophiae studia floruerunt, in antiquissimis litteris compertus esse, ut ex illis constaret prius opusculum hoc extructum fuisse quam Paulus venetus natus esset” (Cittadini, In Dialectica minori Pauli Veneti, MS. Vatican City, Biblioteca Apostolica Vaticana, Urb. lat. 1381, fol. 2r). Cittadini’s doubts about Paul’s authorship of the Logica parva, which might be connected to the polemical nature of his commentary, emerge in several places of his commentary. So, for example, in explaining the intentio auctoris of the Logica parva, Cittadini adds, “whoever the author really was” (quicumque tandem is fuerit, fol. 2v). 5  For an overview of the content of the Logica magna, see § 2 below, and for the authorship of the Logica parva and the Logica magna and Perreiah’s dating of this work, see § 6 below. 6  Conti, Esistenza e verità, 12, and Bottin, “Logica e filosofia naturale,” 91, provisionally propose the years 1396–99. According to Perreiah, the “Quadratura is the record of his performance as an opponent (opponens) during his year of opposition which took place at Padua probably from October 1399 to July 1400” (Paul of Venice, 13). Our dating is based on the colophons of MSS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2133 and Vat. lat. 2134 (labelled B and V respectively: see Sigla Codicum in § 7 below): “Expliciunt determinaciones sophistice cum tabulis earundem acte per me fratrem Paulum de Veneciis ordinis fratrum heremitarum sancti Augustini dum essem lector in conventu Paduano ac bacellarius eiusdem sacratissimi ordinis” (B fol. 141rb, V fol. 161rb). 4

Introduction3

the Sophismata are connected with Paul’s teaching activity at the University of Padua, which began around 1405.7 In the introduction to this work (discussed below, § 6), Paul refers to “the crowd of students” by which he was greatly overwhelmed: this would fit better an academic environment with many students, rather than a mere conventual studium. Moreover, in Padua students certainly had to solve sophisms in the classroom, and these were more or less dependent on Heytesbury’s Sophisms.8 While he was still a professor at the University of Padua, Paul was assigned important religious and diplomatic roles. In May 1409, Pope Gregory XII named him prior general of the Hermits, a post from which Paul resigned shortly afterwards in February 1410. Furthermore, between 1409 and 1412 Paul served as ambassador to Hungary, Ulm, and Cracow for the Venetian Republic. But things changed greatly. Although he had been sent on diplomatic missions to represent the Serenissima (that is, the Venetian Republic) just a few years before, in 1415 Paul ran into trouble with the Venetian government, due to his failure to meet his academic duties fully, and eventually in 1420 the Venetian Council of Ten decided, for reasons that remain unclear, to ban Paul from the Venetian lands (including Padua) and exile him to Ravenna for five years. Nonetheless, in the following years Paul had a rewarding and productive career as preacher, teacher, logician, and philosopher. In 1420–24 he was at the Augustinian convent of Siena, of which he eventually became regent, taught at the city’s university, and seems at this time to have completed his extensive commentary on Aristotle’s Metaphysics. In 1424 Paul spent a short period of time in Bologna, before being assigned to the convent of the Hermits in Perugia. There he remained until 1428, teaching at the university and finishing his commentaries on Porphyry’s Praedicabilia and the Old Logic. In June 1428 the Venetian Republic allowed the elderly and sick Paul to return to Padua, where he died on 15 June 1429. 2. The Structure of the Logica magna The Logica magna is preserved in one complete manuscript, MS. Vatican City, ­Biblioteca Apostolica Vaticana, Vat. lat. 2132 (hereafter, M),9 in two manuscript 7  Perreiah, Paul of Venice, 6, dates the Sophismata to the years 1410–14, whilst Bottin, “Logica e filosofia naturale,” 95 and Conti, Esistenza e verità, 12, hypothesize they were written between 1396 and 1399. 8  See, for example, the sophisms presented and solved at the beginning of the questions on Peter of Spain’s Summaries of Logic written by Blaise of Parma, Paul’s predecessor as professor in Padua (Blaise of Parma, Questiones super Tractatus Logice Magistri Petri Hispani). 9  Available online at https://digi.vatlib.it/view/MSS_Vat.lat.2132.

4 Introduction

f­ ragments, and in an incunabulum published in Venice in 1499 (hereafter, E).10 The two fragments are, first, a substantial one: MS. Venice, Biblioteca Nazionale Marciana, Lat. 6.30 (2547), fols. 44ra–61rb, containing the whole of treatise 23 from Part I (on future contingents); second, MS. Parma, Biblioteca Palatina, Parmense 1023, which contains a very short passage from treatise 1 of Part I. In the appendix to his translation of Paul’s Logica parva, Alan Perreiah rightly criticizes the table of contents given in E (fol. 200r) as inaccurate and misleading, in presenting the work as consisting of 23 distinct treatises in Part I and 15 in Part II.11 That table summarizes the headings given recto on each folio and in-line at the start of each section. There are no in-line section headings in M, but many folios have a similar heading in the upper right corner recto of the folio.12 An attempt at a better structure was partially, but not fully, successful. It proposed a division of the planned twenty-volume British Academy edition into seventeen fascicules, eight for Part I and nine for Part II.13 Alan Perreiah presents his own analysis of the structure, identifying twelve chapters (five in the first part, seven in the second), the longest of these being chapter 3, titled “De probatione terminorum,” covering treatises 3–20 and corresponding closely to the chapter with that title in the Logica parva and to the treatises in De Rijk, “Semantics in Richard Billingham and Johannes Venator.”14 What follows is a fresh attempt to portray the actual structure of the text. Note in particular the different position of the treatise on syllogisms in M and in E: in M it is placed right after the discussion of subject-predicate (often called “categorical”) propositions; in E it comes much later, after the discussion of molecular propositions and the account of truth, falsehood, signification, and modality, even though in his discussion of syllogisms Paul focusses on assertoric, that is, non-modal, categorical syllogisms:

 Available online at https://gallica.bnf.fr/ark:/12148/bpt6k603439.image.  Paul of Venice, Logica parva, trans. Perreiah, 329. 12  They occur on fols. 5–8, 13, 16, 20, 23–127, 129–215, 217, 219, 223, 225–7, 229–45. Headings appear to have been clipped on fols. 216, 218, 220, 222, 224, and 228. 13  This unpublished list was circulated to some of the contributors in the late 1970s. 14  See Paul of Venice, Logica parva, trans. Perreiah, 330, and Paul of Venice, Logica magna (1499, fol. 101ra): “Assuming the proofs of terms and what follows from them, which I thought should have been said in the first part of the current work …” (Suppositis probationibus terminorum cum apendentiis eorumdem, que in prima parte mee orationis sophistice censui fore dicenda …). 10 11

25vb

28va

34va

20ra

22ra

26vb

19

2: On Supposition

1: On Terms

fasc.

2: de suppositionibus terminorum

I:2

i: Material Supposition

v: Supposition of Relatives

iv: Supposition in respect of ampliative words

iii: Personal Supposition

ii: Simple Supposition

c. iv: quarta divisio terminorum

c. iii: tertia divisio terminorum

c. ii: secunda divisio terminorum

1: de terminis16

treatises and chapters in E

iv: Immediate and Mediate Terms

iii: Common and Discrete Terms18

ii: Naturally Significant and Arbitrarily Significant Terms

i: Categorematic and Syncategorematic I:115 Terms

chapters

15

 Edited and translated by Kretzmann in Logica magna: Tractatus de Terminis. Kretzmann does not appear to have known about the second witness in MS. Parmense 1023. 16  M has de significatione termini in the upper right-hand corner on fols. 5r–6r and de significato termini in the upper right-hand corner on fols. 7r–8r. 17  The text in M jumps from fol. 4vb to fol. 9va, working backwards from fol. 9vb to fol. 9ra and jumping from fol. 9rb back to fol. 5ra; then from fol. 8vb to fol. 10va and from fol. 10vb back to fol. 10ra and from fol. 10rb (given as fol. 10vb in Kretzmann’s edition, 126) to fol. 11ra. 18  M has de communitate termini in the upper right-hand corner on fol. 13r. 19  M has de suppositionibus in the upper right-hand corner on fol. 20r, suppositiones on the recto sides of fols. 23–34 and suppositiones relativorum on the recto sides of fols. 35–40. Perreiah edited and translated the first three chapters in Logica magna: Tractatus de Suppositionibus, omitting the final two chapters without explanation.

20rb

16rb

12va

23ra

10vb

8ra

16ra

9vb17

3vb

18ra

3ra

2ra

Part I

folios in E folios in M sections

Introduction5

43vb

46ra

48va

50ra

52ra

53rb

55vb

60va

62vb

68ra

34ra

35vb

38ra

39ra

41ra

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47va

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viii: On “Maximum” and “Minimum”

vii: On Superlatives

vi: On Comparatives

v: On the “just as” Construction

iv: Reduplicatives

iii: Exceptive Expressions

3: On Terms that Ren- i: On the Term “differt” der Supposition Conii: Exclusive Expressions fused20

I:4

I:3

13: de obiectionibus et solutionibus argumentorum

12: de maximo et minimo

11: de superlativis

10: de comparativis

9: de dictione sicut

8: de reduplicativis

7: de regulis exceptivarum

6: de dictionibus exceptivis

5: de regulis exclusivarum

4: de dictionibus exclusivis: “in hoc capitulo”

3: de terminis confundentibus

20

 At the start of the seventh chapter of this section, on sicut (E fol. 42ra, M fol. 53rb), we read: “Having shown ‘differs,’ ‘other ⟨than⟩,’ ‘not the same,’ exclusives, exceptives, and reduplicatives in the ⟨previous⟩ part, it consequently remains to treat of other expressions having the power to render supposition confused, and first of the expression ‘just as’” (Ostenso in parte de differt, aliud, non idem, de exclusivis, exceptivis, et reduplicativis, restat consequenter de aliis dictionibus vim confundendi habentibus pertractare, et primo de hac dictione sicut). M has termini habentes vim confundendi in the upper right-hand corner on the recto side of fols. 41–45, exclusivarum regule on fol. 46r, and regule exclusivarum on the recto side of fols. 47–48 and 50–51, termini confundibiles on the recto side of fols. 54–58, and termini confundentes on the recto side of fols. 59–77, adding de maximo et minimo on the recto side of fols. 63–70, semper on fol. 73r, and ab eterno on fol. 74r.

40va

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6 Introduction

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89vb 90va21 93ra

97va

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70va 71ra 73ra

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82ra 91ra

v: On “Begins” and “Ceases”

i: On Hyper-Intensional Terms I:6 ii: On Modal Propositions iii: On Hyper-Intensional Propositions22 iv: On Compounded and Divided Senses 6: On Knowledge, Doubt, i: On Knowledge and Doubt I:724 Necessity, and Contin- ii: On Future Contingents I:825 gency

5: On Intensionality

iv: On “Immediate” and “Mediate”

iii: On “Infinite”

4: On Terms that can be i: On “Whole” taken Categoremati- I:5 taken categore-matically cally or syncategore-matically ii: On “Always” and “Eternal”

22: de scire et dubitare 23: de necessitate et contingentia futurorum

21: de sensu composito et diviso23

de terminis officiabilibus 19: de propositione exponibili 20: de propositione officiabili

18: de incipit et desinit

17: de isto termino immediate

16: de isto termino infinitum

15: de semper et aeternum: “Secundus terminus qui …”

14: de toto cathegorematice tento

22

21

 M in marg.: propositio de modo.  M has conclusiones de scire in the upper right-hand corner of fol. 96r, conclusiones de significare on fol. 97r. 23  M has de sensu composito in the upper right-hand corner on the recto side of fols. 98–99, 101–15, and 117–27. On fol. 100r it reads dubium an propositio modalis(?) sit alicuius quantitatis, and on fol. 116r dubium an omnia futura de necessitate eveniant. 24  See Logica magna: Tractatus de scire et dubitare, ed. and trans. Clarke. 25  See Logica magna: Tractatus de necessitate et contingentia futurorum, ed. and trans. Williams. Strangely, and with no explanation, Williams did not use the second witness from MS. Venice VI.30 (2547), despite mentioning it at xiii n. 2.

71rb

56ra

Introduction7

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176rb29

2: On Syllogisms

iv: On Irregular Syllogisms

iii: On the Third Figure28

ii: On the Second Figure

i: On the First Figure

vii: On the Conversion of Propositions

vi: More on the Square of Opposition

v: On the Equivalence of Propositions

iv: On the Square of Opposition

iii: On the Quantity of Propositions

1: On Subject-Predicate i: On the Definition of “Proposition” Propositions26 ii: On Subject-Predicate Propositions

II:7

II:2

II:1

13: de syllogismis capitulum27

8: de conversione propositionum

7: capitulum de natura situatorum in figura (de lege positorum in figura)

6: capitulum de equipollentiis

5: de figuris propositionum (capitulum de figura)

4: de quantitate propositionum

3: de propositione in genere

2: de propositione cathegorica

1: de propositione

26

 M has de enuntiatione vel propositione in the upper right-hand corner on fol. 129r and de propositione on fols. 131r–133r, argumenta contra primum principium on fol. 134r, regule contra⟨dicto?⟩riarum on fols. 135r–137r, de propositione universali on fol. 138r, de propositione particulari on fol. 139r, de propositione singulari vel indefinita on fol. 140r, de propositione singulari on fol. 141r, regule alie contra⟨dicto?⟩riarum on fol. 142r, regule contra⟨dicto?⟩riarum on fols. 143r–144r, de modo propositionum de inesse habencium on fol. 145r, regule propositionum se habencium equivalenter on fol. 146r, de triplici materia propositionum on fols. 147r–148r, and de materia contradictoriarum on the recto side of fols. 149–155. 27  In E, the chapter on syllogisms is placed after the sections on molecular propositions and on truth, signification, and modality. 28  De tertia figura capitulum E. 29  Labelled “164.”

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Part II

8 Introduction

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180va32

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iii: On the Modality of Propositions

ii: On the Significate of the Proposition

4: On Truth, Significa- i: On Truth and Falsehood tion, and Modality34

x: Rules of Inference33

ix: On Entailment Propositions

viii: On Conditional Propositions

vii: On Disjunctive Propositions

vi: On Conjunctive Propositions

v: On Properly Molecular Propositions

iv: On “Although”-Propositions

iii: On “Why”-Propositions

3: On Molecular Propo- i: On “When”-Propositions sitions ii: On “Where”-Propositions

Sequitur de rationalibus tractare

II:7

12: de necessitate contingentia possibilitate et impossibilitate propositionum

11: de significato propositionis capitulum

II:635 10: de veritate et falsitate propositionum

II:5

II:431 De condictionali capitulum

De disiunctiva

Capitulum de copulativa

Capitulum de explectiva

De causali

De localibus

II:330 9: de hypotheticis propositionibus

31

30

 See Logica magna: Tractatus de hypotheticis, ed. and trans. Broadie.  See Logica magna: Capitula de conditionali et de rationali, ed. and trans. Hughes. 32  M has regule consequentiarum de forma sive formalium in the upper right-hand corner on fols. 184r–185r and 188r, and regule consequentiarum tam de forma quam formalium on fols. 186r–187r. 33  M has de terminis tam pertinentibus quam impertinentibus in the upper right-hand corner on fols. 189r–191r, de terminis se habentibus se (se delevit?) tamquam superius et inferius on fol. 192r, de termino inferiori et superiori on fols. 193r and 196r–201r, de termino superiori et inferiori on fols. 194r–195r. 34  M has de veritate et falsitate in the upper right-hand corner on the recto side of fols. 202–210, de adequato significato on fol. 211r, de possibilitate et impossibilitate on fol. 212r, de necessitate et impossibilitate on fols. 213r–214r, and dubium an eadem propositio sit vera et falsa on fols. 215r–216r. 35  See Logica magna: Tractatus de veritate et falsitate propositionis et tractatus de significato propositionis, ed. and trans. del Punta and McCord Adams.

164rb

124va

Introduction9

221va

234vb

236ra

238va

240rb

241rb

243ra

243va

244rb

244va

181ra

191va

192rb

194rb

195vb

197rb

197vb

198rb

198vb

199ra

6. On Insolubles37

5. On Obligations

viii: On Merely Apparent Insolubles

vii: On Non-Quantified Insolubles

vi: On Quantified Insolubles

v: On Covert Insolubles

iv: On the Famous Insoluble “Socrates says a falsehood”

iii: Objections and Replies

ii: The Author”s Opinion

i: On Previous Opinions

iii: On Depositio

ii: On Positio

i: On Obligations in General

II:9

15: de insolubilibus

II:836 14: de obligationibus

37

36

 See Logica magna: Tractatus de obligationibus, ed. and trans. Ashworth.  M has insolubilia in the upper right-hand corner on the recto side of fols. 236–242 and 244, de non apparentibus insolubilia [sic] on fol. 243r, and apparentia insolubilia on fol. 245r.

216va

177ra

10 Introduction

Introduction11

3. Obligations The final two treatises of the Logica magna are respectively devoted to obligations and insolubles. These are two distinctive genres of treatise in medieval logic in the Latin West. The study of logic began in earnest among European thinkers in Latin Christendom only in the twelfth century, and although Peter Abelard made immense strides in logical understanding, the real spur to this new work came only towards the end of Abelard’s life, around 1130, through contact with the Muslim world, largely through Spain. The flourishing of learning which the Muslim world saw in the eighth and ninth centuries through the translation of Aristotle’s works (among others) into Arabic was repeated when Aristotle’s philosophy, not least his logic, became available in Latin, at first in translations from Arabic versions, later, following renewed contact between the western (Roman) and eastern (Orthodox) churches, directly from the Greek originals. Until this time, only Aristotle’s Categories and De interpretatione, together with Porphyry’s and Boethius’s introductions to his logic (known collectively as the logica vetus—the Old Logic) had been known in the West. Over the century from around 1130, the rest of Aristotle’s Organon (the Prior and Posterior Analytics, the Topics, and the Sophistical Refutations—known as the logica nova, the New Logic) were rendered into Latin and began to circulate in centers of learning in Europe. This stimulated original contributions from western thinkers, particularly the development of theories about logical aspects of language, to fill out and underpin Aristotle’s own analyses. These contributions were termed logica modernorum (“logic of the moderns”) or logica moderna (“modern logic”). It was developed under four main headings: (i) properties of terms (proprietates terminorum): theories about meaning (signification and what is usually rendered as “supposition”—suppositio), truth, and other semantic notions;38 (ii) theories of consequence (consequentia), that is, inference and validity of arguments; 39 (iii) (logical) obligations: disputations of a very specific sort, described below; (iv) insolubles: semantic and epistemic paradoxes (see § 4 below). This work reached its greatest heights in the fourteenth century in the years leading up to the compilation of Paul’s Logica magna. The unique genre of obligations has strong affinities with, and may well have been modelled on, Aristotle’s discussion of disputations in his Topics. Nonetheless,  See, for example, Read, “Medieval Theories: Properties of Terms.”  See, for example, Read, “The Medieval Theory of Consequence.”

38 39

12 Introduction

­ ristotle’s treatment in Book 8 concerns much more general disputations than the A heavily circumscribed and formal ones that we find described in medieval obligations literature. To anticipate the issue of the authenticity of the Logica magna (to be discussed in § 6 below): Ashworth argues that the strong similarities between (and peculiarities of) the discussions of obligations in the Logica magna and the Logica parva are strong evidence that these works have the same author.40 In both cases, Paul—assuming the author is Paul of Venice—is strongly influenced by a series of treatises on obligations by Albert of Saxony, William Buser, Ralph Strode, Peter of Candia, and Peter of Mantua. Saxony (writing in the 1350s) and Buser (1360s) were both in the so-called English nation at the University of Paris; at the same time, Strode was writing in Oxford; Candia, originally from Crete, studied there and in Padua and Paris; and Mantua was among those exposed to English logic in Padua, possibly as a result of the papal schisms in the latter decades of the fourteenth century. It is common practice to identify two main doctrines of obligations, the responsio antiqua (or “old response”) and the responsio nova (“new response”), although there are also individual differences in each author’s account. The responsio antiqua was codified from earlier accounts by Walter Burley in his treatise on the subject, written in Oxford in 1302.41 Paul and those influencing him all follow Burley’s line. Although there is still disagreement about the exact purpose of obligational disputations, they had an immense influence on the language and structures of other treatises—in particular, treatises on insolubles; so it is useful to understand Burley’s theory of obligations. If, as many argue, their purpose was the training and testing of students’ logical abilities, this would explain the prevalence of the language of obligations throughout philosophical treatises.42 There are two parties to an obligational disputation, an opponent and a respondent. The opponent presents the respondent with a background scenario (casus),43 briefly sketched, and an obligatum, a proposition (as we will translate the Latin term propositio) that is usually false in the scenario. Burley distinguishes six types of obligational disputation. The principal one is called positio while the proposition at its heart is the positum. The respondent may accept or reject the obligation, but Paul’s first rule, following standard practice, lays down that the obligation should be  See Ashworth, “Paul of Venice on Obligations.”  See Green, An Introduction to the Logical Treatise De obligationibus; partially translated into English in Burley, “Obligations.” 42  See, for example, Sinkler, “Paul of Venice on Obligations,” and Pironet, “William Heytesbury and the Treatment of Insolubilia,” 259. 43  See Pironet, “William Heytesbury,” Insolubilia,” 257.

40 41

Introduction13

accepted provided it is not intrinsically impossible.44 Paul focuses here on positio, since he simplifies Burley’s six types of obligation into two, positio and depositio.45 The logical obligation placed on the respondent by accepting the positum (in the responsio antiqua) is that he must respond to a subsequent series of propositions (proposita) presented by the opponent in only one of three ways according to strict rules:46 (a) any proposition proposed by the opponent which follows from the positum, from any previously granted proposition, and from the opposite of any previously denied proposition is deemed relevant and should be granted;47 (b) any proposition proposed by the opponent which is inconsistent with the positum, with any previously granted proposition, and with the opposite of any previously denied proposition is deemed relevant and should be denied;48 (c) any other proposition proposed by the opponent is deemed irrelevant (impertinens) and should be granted or denied according as it is known to be respectively true or false in the scenario, or doubted (or uncertain, that is, doubt should be expressed about it) if its truth is unknown or not specified in the scenario. Paul gives an example.49 Suppose that you are in Oxford, and I posit to you: “Every man is in Rome.” This is not intrinsically impossible, so you should accept the positio. Then the obligational disputation performed correctly by the respondent runs as follows: Opponent You are in Rome. You are not a man.

Respondent denied granted

Reasoning irrelevant and false relevant and following

Note, however, that if these two propositions had been proposed in the opposite order, they would have correctly elicited the opposite response: Opponent You are not a man. You are in Rome.

Respondent denied granted

Reasoning irrelevant and false relevant and following50

 See Logica magna: Tractatus de obligationibus, 51.  See ibid., 39. 46  See rules 5-10 ibid., 57–65. 47  Such propositions are called pertinens sequens (“relevant and following”). Ibid., 24. 48  Such propositions are called pertinens repugnans (“relevant and inconsistent”). Ibid., 24. 49  See ibid., 61. 50  “You are in Rome” must be granted because it follows from the positum—”Every man is in Rome”— and the opposite of “You are not a man,” which you denied. 44 45

14 Introduction

Much of the subtlety and difficulty in obligations lies in the interplay between propositions that are relevant and those that are irrelevant, in particular, spotting those that are relevant despite appearances. This seems to have been one reason why Roger Swyneshed, writing in Oxford in the 1330s, proposed a simplified set of rules forming what came to be known as the “new response.” Another reason may have been a peculiarity of the “old response” already pointed out by Burley,51 namely that, having accepted a false positum, the respondent can quickly be forced into granting any other falsehood consistent with it. Take this example: given that you are in Oxford and not a bishop, I posit “You are in Rome” to you. Then we can proceed like this: Opponent You are not in Rome or you are a bishop. You are in Rome. You are a bishop.

Respondent granted

Reasoning irrelevant and true

granted granted

the positum relevant and following

Roger Swyneshed is less well known than his namesake Richard, one of the leading Calculators, whose work anticipated some of the revolutionary mathematical physics of the seventeenth century. Roger was one of the Calculators too, although he was more influential in logic, especially through his work on obligations and insolubles. As we will see in § 5, in the Logica magna Paul followed and developed Swyneshed’s ideas on insolubles, but he did not follow the new response regarding obligations. The simplification Swyneshed introduced was to restrict relevance to following from or being inconsistent with the positum alone. As a consequence, responses do not depend on the order in which propositions are proposed. Thus, in the first obligation above, “You are not a man” will be correctly denied under the new response as irrelevant and false whenever proposed, and so too will “You are in Rome.” In the second obligation, “You are a bishop” will be correctly denied as irrelevant and false. Indeed, all responses will be independent and, as Ashworth observes, only onepremise inferences will feature, from the positum or to its opposite.52 This certainly simplifies obligational disputations and makes them easier for students to pass, if that in fact was their purpose. But it has one strange consequence, of which the proponents of the new response tried to make a virtue, if not indeed a badge of honor: “Having granted the parts of a conjunction, the conjunction need not be granted,

51 52

 See Burley, “Obligations,” 391, § 3.61.  See Ashworth, “Autour des Obligationes de Roger Swyneshed,” 357.

Introduction15

nor having granted a disjunction, need either of its parts be granted.”53 This suffices for our present purposes to explain the frequent recourse to the language of granting, denying, and doubting scenarios and propositions in Paul’s Insolubles.54 Ashworth notes Paul’s dependence on Strode’s Obligations.55 Strode presented a robust criticism of Swyneshed’s proposed revisions to the theory. Strode’s treatises were also familiar to Peter of Mantua, who also influenced Paul strongly.56 4. Insolubles Although the Liar Paradox and similar puzzles were well known and much discussed in antiquity, the medieval interest in them seems to be quite independent of those discussions, and largely ignorant of them.57 On the one hand, their paradoxical nature seems not even to have been properly recognized until the end of the twelfth century, and the only reference from antiquity that is regularly cited is Aristotle’s discussion of the oath-breaker in his Sophistical Refutations, chapter 25. The oathbreaker first says that he will break his oath and then proceeds to fulfill that oath by breaking a subsequent one. To be truly paradoxical, it would need to be one and the same oath that he both fulfills and breaks. This is what we find in the classic case of the Liar Paradox when someone says, “I am lying” (where this is all he says, or at least he means to refer to that utterance) or, “This utterance is false” (referring to that very utterance). If the proposition is true, then it must be false (for that is what was said), so it is not true (since it cannot be both), and consequently by reductio ad absurdum it really is not true, and so is false (assuming it is either true or false, and so if not true, then false). But given, as we have just proven, that it is false, it is surely true (since that is what was said). Thus we have proven both that it is true and that it is false (indeed, that it is both true and not true), which is paradoxical (literally, beyond belief). Something has surely gone wrong. But what is the mistake? We can divide medieval discussions of the insolubles—logical paradoxes such as the Liar—into two main periods, before Bradwardine and after Bradwardine. Thomas Bradwardine wrote his treatise on insolubles in Oxford in the early 1320s,  Swyneshed, Obligationes, 257: “Propter concessionem partium copulativae non est copulativa concedenda nec propter concessionem disjunctivae est aliqua pars ejus concedenda.” 54  For more information on obligations, see, for instance, Spade and Yrjönsuuri, “Medieval Theories of Obligationes,” and Dutilh Novaes and Uckelman, “Obligations.” 55  See Paul of Venice, Logica magna: Tractatus de obligationibus, xi–xii. 56  Strode’s treatise on obligations was examined in detail in Ashworth, “Ralph Strode on Inconsistency in Obligational Disputations” and Dutilh Novaes, “Ralph Strode’s Obligationes.” 57  See Spade and Read, “Insolubles,” §§ 1.1–1.2. 53

16 Introduction

and it seems to mark a sea change in the solutions that were mainly favored. Up until the 1320s two types of solution were the focus of attention, restrictio and cassatio (though only two treatises are known which favored the latter). Supporters of restrictio (restrictivism or restrictionism) proposed an outright ban on self-reference— in some authors, a blanket ban, in most a limited ban under which self-reference may not be combined with a negative (or privative) expression. Walter Burley, a prime advocate of the latter form of restrictivism, wrote: According to earlier writers, there are three sources of insolubles. The first source comes from the combination of an intentional verb with the expression “false” or anything convertible with it or with the denial of “true”… Concerning the first source it should be recognized that whenever the same act has reflection on itself with a privative determination, namely, with the determination “false” or anything convertible with it, then the act is restricted … For a term is restricted when it does not imply its superior [that is, any term it falls under] … It should be recognized that a part can never supposit for a whole of which it is part when, putting the whole in place of the part, there results reflection of the same on itself with a privative determination. So if one says “I say a falsehood,” the term “falsehood” does not supposit for “I say a falsehood” because, if the whole is put in place of the term “falsehood,” there results reflection on itself with a privative determination.58

Bradwardine attacked several versions of restrictivism mercilessly in chapters 3–4 of his Insolubles. While this was the pre-eminent proposal for solving the insolubles before his attack, it had few proponents thereafter.59 An anonymous later author wrote: But because there are both many authorities and strong arguments against this opinion, we can probably say that whereas in propositions where an act has reflection on itself or where a part supposits for the whole of which it is part there is a tortuous and unusual way of speaking or thinking, nonetheless, it is possible where what supposits for the whole is a general or universal term, e.g., “Every proposition is false,” but never when the term  Burley, Insolubilia, 271–2, §§ 3.01–3.03, correcting the text against De Rijk, “Some Notes on the Medieval Tract De Insolubilibus,” 90: “Et secundum antiquos, tres sunt radices insolubilium. Prima radix provenit ex conjunctione verbi pertinentis ad motum animi cum hac dictione falsum vel cum suo convertibili vel cum negatione veri. … Circa primam radicem est sciendum quod quandocumque idem actus reflectitur supra se ipsum cum determinatione privativa, scilicet cum hac determinatione falsum vel cum suo convertibili, tunc actus ponitur diminutive … Terminus enim ponitur diminutive quando non infert suum superius … sciendum quod nunquam supponit pars pro toto cuius est pars, quando, posito toto loco partis, accidit reflexio eiusdem supra se ipsum cum determinatione privativa. Ideo sic dicto dico falsum, iste terminus falsum non supponit pro hac: dico falsum, quia, posita hac tota loco huius termini falsum, accidit reflexio supra se cum determinatione privativa.” 59  A rare exception is found in Walter Segrave’s Insolubles; see Spade, The Medieval Liar, 113. 58

Introduction17

is singular, e.g., “This is false,” referring to itself, or “You are an ass or this disjunction is false,” or “This inference is valid, so you are an ass,” referring to that very inference itself, and so on.60

Bradwardine also dismissed cassationism (cassatio) as nonsense, just as its supporters dismissed insolubles as nonsense. The verb “to cass” (archaic, and derived from the Latin cassare) means to render null and void. Cassationism is regularly rejected in thirteenth-century treatises (indeed, as mentioned, it is defended in only two that survive). Bradwardine concluded after only a brief discussion: “the view of the nullifiers is sufficiently nullified.”61 Cassationism received its most extensive and wellargued defense in John Dumbleton’s Summa Logicae, written in the late 1330s and 1340s.62 Other theories were proposed and defended before Bradwardine’s revolution. He considers another couple in the fifth chapter of his treatise. But it was Bradwardine’s own proposal, along with his detailed and extensive criticism of restrictivism, which initiated a whole new era in discussions of the insolubles. His aim was to develop a solution to the insolubles which placed no restriction on self-reference or the theory of truth. His first and third postulates were: (1) “Every proposition is true or false” (and, implicitly, not both). (3) “The part can supposit for the whole of which it is part and its opposite and for what is convertible with them.”63

Along with his definitions of a true proposition as “an utterance signifying only as it is” and of a false one as “signifying other than it is,” along with some basic logical principles, he was able to establish his second conclusion, that “any proposition that signifies itself not to be true or to be false, also signifies itself to be true

60  Pironet, “William Heytesbury and the Treatment of Insolubilia,” 324: “Sed quia tam multae auctoritates quam fortes responsiones sunt contra istam opinionem, ideo probabiliter dici potest quod quamvis sit extortus et inconsuetus modus loquendi sive intelligendi in propositionibus ubi actus reflectitur super seipsum vel ubi pars supponit pro toto cujus est pars, tamen hoc est possibile ubi illud quod supponit pro suo toto est terminus communis sive universalis, ut hic ‘omnis propositio est falsa.’ sed numquam ubi talis terminus est singularis, sicut ‘hoc est falsum,’ eodem demonstrato, ut hic ‘tu es asinus vel haec disjunctiva est falsa,’ vel ‘haec consequentia est bona, ergo tu es asinus,’ eadem consequentia demonstrata, et sic de aliis.” 61  Bradwardine, Insolubilia, § 5.6: “Opinio ergo cassantium satis est cassata.” 62  See Spade, The Medieval Liar, 63–5. 63  Bradwardine, Insolubilia, § 6.3: “Suppositiones autem sunt sex. Quarum prima est ista: quelibet propositio est vera vel falsa … Tertia est ista: pars potest supponere pro suo toto et eius opposito et convertibilibus earundem.”

18 Introduction

and is false.” The basis of this claim was his novel proposal, encapsulated in his second postulate: (2) “Every proposition signifies or means as a matter of fact or absolutely everything which follows from it as a matter of fact or absolutely.” 64

The ingenious use of this postulate in proving his second conclusion is well worth studying, as is his application of it in analyzing a succession of insolubles. However, although most (if not all) subsequent writers on insolubles owe a debt to Bradwardine, few followed him completely, and in particular, it seems to have been his powerful second postulate that was not popular. Two alternative proposals that were presented in Oxford in the 1330s—both responding to Bradwardine’s idea but in different ways—dominated subsequent discussion of the insolubles. They were due to William Heytesbury and Roger Swyneshed. Discussion of the insolubles was shot through, as noted in § 3, by the language of obligations. But in Heytesbury’s case, it was not just the language (of granting, denying, etc.) but the whole structure of obligational disputations which framed the discussion. Most insolubles need a background scenario—minimal as in the Liar, which assumes that the Liar utterance is self-referential (and not just apologizing for a lie just told, for example); or more extensive in such cases as the common medieval scenario where a stock character called “Socrates” says one and only one thing, namely, “Socrates says a falsehood”; or again elaborate examples where, say, a landowner, troubled by vagabonds, has set up a gallows by a bridge over a river dividing his lands, decreeing that everyone who wishes to cross the bridge must declare their business and where they are going, to be let across if they speak truly but hung on the gallows if they lie: only to be confronted by Socrates (yet again) saying that his sole business is to be hung on the gallows.65 Heytesbury takes each insoluble to be the positum in an obligation. He first distinguishes an insoluble scenario from an insoluble proposition: A scenario of an insoluble is one in which mention is made of some proposition such that if in the same scenario it signifies only as its words commonly suggest, it follows from its being true that it is false and vice versa … [A]n insoluble proposition is one of which mention is made in some insoluble scenario such that if in the same scenario it signifies

64  “Secunda est ista: quelibet propositio significat sive denotat ut nunc vel simpliciter omne quod sequitur ad istam ut nunc vel simpliciter” (ibid.). 65  The example about crossing the bridge was a real stock in trade: see, for example, Bradwardine, Insolubilia, § 8.8.1; Paul’s Insolubles edited here, § 5.4; and Cervantes, Don Quixote, Part II, chap. 45.

Introduction19

only as its words commonly suggest, it follows from its being true that it is false and vice versa.66

Heytesbury’s solution turns on the specification of the precise signification of the insoluble. The scenario may specify that the insoluble signifies only as its terms commonly suggest (precise sicut termini communiter pretendunt) or it may leave this question open, having the insoluble signify as its terms commonly suggest but not necessarily only in that way (non sic precise).67 If it were left completely open, Hunter and others realize that the respondent could then do no more than doubt the positum, since he would not know how it signified.68 If the opponent adds precise, then the scenario should not be accepted, for we noted in § 3 that the first rule of obligations is that no intrinsically impossible obligation should be accepted. However, with this restriction the usual contradiction—that the positum is both true and false, or should be both granted and denied—is immediately forthcoming. Heytesbury’s third rule applies when the opponent does not add the precise restriction. In that case, he says, the obligation should be accepted, the insoluble should be granted as following, but that it is true should be denied. Recall the earlier proof that “I am lying” is both true and false. What is common to Bradwardine’s and Heytesbury’s solutions (and most others) is that they accept the first leg of the argument, using reductio ad absurdum to infer that the insoluble is false, but they find some way to block the second leg, arguing from its falsity, already granted, to its truth. Bradwardine, for example, blocks this move by reminding his reader that the truth of a proposition requires that the proposition signify only as it is, and since it signifies both that it is false (by the meaning of “lie” or “false”) and that it is true (by his second conclusion), it cannot signify only as it is (since it cannot be both true and false); so it is false. Heytesbury is more discreet. He writes: But if someone asks what the proposition uttered by Socrates signifies in this scenario other than that Socrates says a falsehood, I say that the respondent will not have to respond to that question, because from the scenario it follows that the proposition will signify other  “[C]asus de insolubili est ille in quo fit mentio de aliqua propositione quae, si cum eodem casu significet praecise sicut verba illius communiter praetendunt, ad eam esse veram sequitur eam esse falsam et e converso … [P]ropositio insolubilis est de qua fit mentio in casu insolubili quae, si cum eodem casu significet praecise sicut verba illius communiter praetendunt, ad eam esse veram sequitur eam esse falsam et e converso” (Heytesbury, Insolubilia, ed. Pozzi, 236; Pironet, “William Heytesbury and the Treatment of Insolubilia,” 284). For an alternative translation, see Heytesbury, Insolubles, trans. Spade, 46. 67  See Heytesbury, Insolubilia, ed. Pozzi, 238; Pironet, “William Heytesbury,” 284–5. 68  See, for example, Hunter in Pironet, “William Heytesbury,” 303. 66

20 Introduction

than that Socrates says a falsehood, but the scenario does not specify what that is and so the respondent does not have to respond any further to what was asked.69

The respondent is required only to say “I grant it,” “I deny it,” “I doubt it” and so is not obliged to enter into discussion of what the positum may or may not signify— that is for the opponent to do. Later authors came to distinguish the exact or primary signification of a proposition (what the words commonly suggest, or “according to the common institution of the idiom,” or “the common institution of grammar”)70 from its secondary or consequential signification, the two combining to make up its principal or total signification. But caution is needed, for these terms are often used slightly differently by different authors (and are often translated differently by different translators). A later author, identified by Spade as “Robert Fland,” but arguably properly called “Robert Eland,”71 presented Bradwardine’s and Heytesbury’s solution and invited the reader to choose between them: So ⟨these⟩ two responses [sc. Bradwardine’s and Heytesbury’s] are better than the others for solving insolubles. Therefore the respondent should choose one of them for his solution to the insolubles.72

What seems to have happened, however, is that the popular solution was to combine them. We find such a solution in a number of treatises, several anonymous, including that of pseudo-Heytesbury, so called by Spade because his treatise is so closely modelled on that of Heytesbury,73 and in treatises ascribed to John of Holland and to John Hunter (Johannes Venator).74 It is the solution commonly found in the

 Heytesbury, Insolubilia, ed. Pozzi, 240: “Si autem quaeratur in casu illo quid significabit illa propositio dicta a Sorte aliter quam quod Sortes dicit falsum, huic dicitur quod respondens non habebit illud seu illam quaestionem determinare, quia ex casu isto sequitur quod ista propositio aliter significet quam quod Sortes dicit falsum, sed casus ille non certificat quid illud sit et ideo non habet respondens quaesitum illud ulterius determinare.” See also Pironet, “William Heytesbury,” 286, and for an alternative translation, see Heytesbury, Insolubles, trans. Spade, 49–50. 70  See Pironet, “William Heytesbury,” 324–5: “dum solum significat juxta communem institutionem idiomatis in quo est disputatio”; “significet secundum communem institutionem grammaticae.” 71  See Read and Thakkar, “Robert Fland, or Elandus Dialecticus?” 72  Fland, “Robert Fland’s Insolubilia,” 65: “Ideo duae responsiones sunt meliores aliis ad insolubilia solvenda. Eligat ergo respondens unam istarum pro sua solutione ad insolubilia.” 73  See Spade, The Medieval Liar, 35–6. 74  The treatises by pseudo-Heytesbury and John Hunter as well as another, anonymous treatise are edited in Pironet, “William Heytesbury”; the treatise by John of Holland in John of Holland: Four Tracts on Logic, 123–46; and another anonymous treatise in Spade, “An Anonymous Tract on Insolubilia from Ms Vat. lat. 674.” 69

Introduction21

teaching manuals at Oxford now known as the Logica Oxoniensis,75 and is the basis of Paul of Venice’s solution in his Logica parva (see § 5 below). When it comes to the third rule, instead of refusing to specify what the additional signification is, pseudo-Heytesbury writes: “It must be said that … ‘A falsehood exists’ signifies conjunctively, namely, that a falsehood exists and that that very proposition is true.”76 Hunter spells this out more clearly. Suppose Socrates says, “Socrates says a falsehood” and there is no other proposition, call it A. The claim is that A is false. Then we might be tempted to argue as follows: A is false, Socrates says A, so Socrates says a falsehood. The inference is valid, the premises are true, so the conclusion is true too. But Socrates says the conclusion. So Socrates says the truth.

Not so, Hunter replies: It should be denied that Socrates says the conclusion. Rather, he says another proposition similar in sound but not in signification, because what was said by Socrates signifies that Socrates says a falsehood (and thus that it itself is false, since Socrates said nothing else) and that it itself is true. But any other similar proposition which is not said by Socrates signifies only that Socrates says a falsehood.77

But none of them offer any argument, as Bradwardine had done, to show that this additional signification is that the proposition itself is true. We might call this the modified Heytesbury solution. Roger Swyneshed’s solution was at root very different. His aim was to provide a solution without the suggestion of hidden meanings, but taking the expressions at face value, so that the principal signification is just what it is commonly taken to be, what the words commonly suggest (though this phrase, which seems to originate with Heytesbury, is not used by Roger). Rather, he focused on the fact that all the insolubles entail their own falsehood (although, of course, not only insolubles do  See De Rijk, “Logica Oxoniensis.”  Pironet, “William Heytesbury,” 292: “Et eodem modo dicendum est … ⟨quod⟩ ‘falsum est’ significat similiter copulative, videlicet quod falsum est et quod eadem propositio est vera.” 77  Ibid., 305: “Aliter arguitur sic, et probo illam propositionem esse veram quam Socrates dicit, et pono quod A sit illa. Tunc arguitur sic: A est falsum, Socrates dicit A, ergo Socrates dicit falsum. Illa consequentia est bona, et antecedens est verum, ergo et consequens; et Socrates dicit consequens, ergo Socrates dicit verum. Solutio: negandum est quod Socrates dicit consequens, sed dicit unam aliam sibi consimilem in voce et non in significatione, quia illa quae dicta est a Socrate significat quod Socrates dicit falsum, et sic, quod ipsamet est falsa, cum Socrates nihil aliud dicat, et quod ipsa eadem est vera. Sed quaecumque alia consimilis quae non est dicta a Socrate significat praecise quod Socrates dicit falsum.” 75

76

22 Introduction

that). Where Bradwardine and Heytesbury demanded for truth that everything a proposition signified, including any hidden secondary or additional signification, should obtain, Swyneshed proposed that truth should require a proposition not to entail its own falsehood: There are four definitions … The second is this: a true proposition is a proposition not falsifying itself, signifying principally as it is either naturally or by an imposition by which it was last imposed to signify. Third definition: a false proposition is an utterance falsifying itself or an utterance not falsifying itself signifying principally other than it is either naturally or by an imposition by which it was last imposed to signify.

Then Swyneshed defines an insoluble: [A]n insoluble as put forward is a proposition signifying principally as it is or other than it is ⟨which is⟩ relevant to inferring itself to be false or unknown or not believed,78 and so on.79

Swyneshed derives three famous iconoclastic conclusions from his theory: (1) There is a false proposition that signifies principally as it is. (2) There is a formally valid inference where the false follows from the true. (3) There are two mutual contradictories that are both false.80

The Liar serves to illustrate and support all three claims. For “This very proposition is false” does signify as it is, for it signifies that it is false (and, for Swyneshed, that is all it signifies) and by Swyneshed’s account of truth and falsehood, it is false, for it falsifies itself, that is, it entails its own falsehood. It also serves to establish the second claim; for consider the inference: The conclusion of this inference is false. So the conclusion of this inference is false.  Pozzi, Il Mentitore e il Medioevo, 182, adds non before creditam.  Swyneshed, Insolubilia (ed. Spade, 185–6; ed. Pozzi, 182): “Post illa sequuntur quattuor definitiones seu descriptiones … Secunda est haec: Propositio vera est propositio non falsificans se ipsam, principaliter sicut est significans naturaliter aut ex impositione vel impositionibus, qua vel quibus ultimo fuit imposita ad signifìcandum. Tertia definitio: propositio falsa est oratio falsificans se vel oratio non falsificans se principaliter aliter quam est significans naturaliter aut ex impositione vel impositionibus, qua vel quibus ultimo fuit imposita ad significandum. Quarta est haec: Insolubile ad propositum est propositio significans principaliter sicut est vel aliter quam est pertinens ad inferendum se ipsam fore falsam vel nescitam vel non creditam, et sic de singulis.” 80  See ibid. (ed. Spade, 188–9; ed. Pozzi, 186–8): “Aliqua propositio falsa significat principaliter sicut est … In aliqua consequentia bona formali ex vero sequitur falsum … [D]uo contradictoria sibi mutuo contradicentia sunt simul falsa.” 78 79

Introduction23

The conclusion is a version of the Liar, asserting of itself that it is false. Swyneshed claims that the inference is valid, for the premise signifies exactly the same as the conclusion, namely, it predicates the same property (being false) of the same thing (the conclusion). (In § 5 below we will consider what Swyneshed’s account of validity might have been.) Nonetheless, the premise is true, since it correctly (by his lights) says that the conclusion is false, and the conclusion is false, because it falsifies itself. The third claim is in some ways the most puzzling and surprising. How can contradictories both be false? Did Aristotle not introduce the notion of contradictories as pairs of propositions that cannot both be true and cannot both be false? Not so, according to Whitaker, who reminds us that what Aristotle actually wrote was: As men can affirm and deny the presence of that which is present and the presence of that which is absent and this they can do with reference to times that lie outside the present: whatever a man may affirm, it is possible as well to deny, and whatever a man may deny, it is possible as well to affirm. Thus, it follows, each affirmative statement will have its own opposite negative, just as each negative statement will have its affirmative opposite. Every such pair of propositions we, therefore, shall call contradictories, always assuming the predicates and subjects are really the same and the terms used without ambiguity.81

Whitaker claims that, in the subsequent chapters of De interpretatione, Aristotle proceeds to argue against what Whitaker calls the “rule of contradictory pairs” (according to which, of any pair of contradictories, one is true and the other false). For in chapter 7 Aristotle gives examples of pairs both of which are true, and in chapter 8 there are examples of pairs each of which is false. In chapter 9 (regarding the future sea-battle) the rule of contradictory pairs fails for future contingents.82 Swyneshed argues for his third claim by again taking “This proposition is false” and its pair “This proposition is not false,” each referring to the former. The latter, he says, denies of the former what the former affirms of itself. So by Aristotle’s account, they are a contradictory pair. But the former is false because it falsifies itself while the latter is false because it says, falsely, that the former is not false. Thus, we have a pair of contradictories both of which are false. Swyneshed adds, “or unknown or not believed and so on” at the end of his definition of insolubles in order to include what are now called epistemic paradoxes,  Aristotle, De interpretatione, chap. 6, 17a27–33 (trans. Cooke, 123–5).  See Whitaker, Aristotle’s De Interpretatione, 79.

81 82

24 Introduction

which the medievals included under the title “insoluble.” The most famous example is perhaps the Knower paradox, in the forms, “This proposition is not known” or, “You do not know this proposition.” Suppose it were known. Then it would be true, and so not known. Hence by reductio ad absurdum, it is unknown. That is, we have just proven that it is unknown, which is what it signifies. So it is true, and moreover, since we have proven it, we know that it is true and so it is known. Swyneshed’s response is to question the second leg of the argument. Let A be the proposition “A is unknown”: “A is unknown” should be granted, and it should be granted that I know A to signify principally in this way. And the inference, “therefore, I know A” should be denied. But it is necessary to add that A is not relevant to inferring itself not to be known. And if that is added, it should be denied. For it follows directly, “A is unknown, therefore, A is unknown.”83

So far we have looked only at developments at Oxford, following Bradwardine’s proposal. There were remarkably similar developments at Paris, remarkable not least for the fact that their differences suggest that there may not have been any direct influence. We can perhaps divide them again into two branches, the first stemming from John Buridan’s ideas, the second from Gregory of Rimini’s. Buridan’s solution to the insolubles is nowadays perhaps the most famous of all medieval solutions, having been discussed extensively over the past fifty years.84 In fact, Pironet showed that Buridan’s ideas developed over the course of three or four decades in some five works.85 His early suggestion was that every proposition signifies its own truth. This is strongly in contrast with Bradwardine’s solution, which claimed only that insolubles signify their own truth. Like Bradwardine, and unlike the modified Heytesbury solution, Buridan offered a proof of his broader claim that every proposition signifies its own truth: For every proposition is affirmative or negative. But each of them signifies itself to be true or at least from each it follows that it is true. This is clear first concerning affirmatives, for every affirmative proposition signifies that its subject and predicate supposit for the same, and this is for it to be true … Secondly, it is clear concerning negatives, for a

 Swyneshed, Insolubilia (ed. Spade, 209): “Admisso casu, concedenda est illa a nescitur. Et concedendum est quod ego scio a sic principaliter significare. Et neganda est consequentia ‘igitur, scio a.’ Sed oportet addere quod a non est pertinens ad inferendum se ipsum fore nescitam. Et si illud addatur, illud est negandum. Nam sequitur immediate ‘a nescitur; igitur, a nescitur.’” 84  See, for example, Zupko, “John Buridan,” § 4. 85  See Pironet, “John Buridan on the Liar Paradox.”

83

Introduction25

negative does not signify that the subject and predicate supposit for the same, and this is for the negative proposition to be true.86

Then propositions such as the Liar are self-contradictory, signifying both that they are true and that they are false, and so are simply false: Regarding this proposition, “I say a falsehood,” I grant that it is false … because it not only signifies that it itself is false, but also, from the general condition of a proposition, it signifies that it itself is true and is not true.87

However, in later works, Buridan had doubts about this claim. In his Sophismata, composed some twenty or more years later, in the 1350s, he rehearsed his earlier view before criticizing and revising it: For some people have said (and so it seemed to me elsewhere) that although this proposition [“Every proposition is false”] does not signify or assert anything according to the signification of its terms other than that every proposition is false, nevertheless, every proposition by its form signifies or asserts itself to be true … But this response does not seem to me to be valid, in the strict sense … rather, I [am going to] show that it is not true that every proposition signifies or asserts itself to be true.88

His objection was that this claim either implies that every proposition is metalinguistic, in always talking about the truth of some proposition (which he believes is 86  Buridan, Quaestiones Elencorum, 92: “[N]am omnis propositio est affirmativa vel negativa. Modo quelibet illarum significat se esse veram vel saltem ad quamlibet illarum sequitur eam esse veram. Patet hoc primo de propositione affirmativa, nam propositio affirmativa significat subiectum et predicatum supponere pro eodem. Et hoc est ipsam esse veram … Secundo patet hoc de negativa, nam negativa non significat esse idem pro quo supponunt subiectum et predicatum. Et hoc est propositionem negativam esse veram.” Although Buridan claimed that a proposition is true if it signifies as it is, he considered this phrase seriously misleading and cashed it out in terms of supposition: an affirmative proposition is true if subject and predicate supposit for the same, and a negative proposition is true if subject and predicate do not supposit for the same. See, for example, Buridan, Sophismata, chap. 2: “On the Causes of the Truth and Falsity of Propositions” (Summulae de dialectica, 845–62, especially the Fourteenth Conclusion, at 858–9). 87  Buridan, Quaestiones in primum librum Analyticorum Posteriorum, qu. 10, cited in Pironet, “John Buridan on the Liar Paradox,” 295 n. 4: “De ista propositione, ‘ego dico falsum,’ concedo quod est falsa … quia ipsa non solum significat se esse falsam. Sed etiam ex commune condicione propositionis, significat se esse veram, et non est vera.” 88  Buridan, Summulae de dialectica, 967–8; Buridan, Summulae de practica sophismatum, 154–5: “Aliqui enim dixerunt, et ita visum fuit [mihi alias corr.] quod licet ista propositio secundum significationem suorum terminorum non significet vel asserat nisi quod omnis propositio est falsa, tamen omnis propositio de forma sua significat vel asserit se esse veram … Ista tamen responsio non videtur mihi valere de proprietate sermonis … Sed ostendo illud non esse verum, scilicet quod omnis propositio significat vel asserit se esse veram.”

26 Introduction

not so), or commits one to the postulation of significates, some real correlate of the proposition (which to Buridan was anathema).89 Buridan’s revised view was that every proposition implies its own truth, or at least would do so if it existed: Therefore, we put this otherwise, in a manner closer to the truth, namely, that every proposition virtually implies another proposition in which the predicate “true” [would be] affirmed of the subject that supposits for [the original proposition]; and I say “virtually implies” in the sense in which the antecedent implies that which follows from it.90

Buridan’s account is one of the well-known solutions to the insolubles which Paul of Venice does not include in his survey of fifteen alternative solutions in the treatise on “Insolubles” in his Logica magna. But he does include Albert of Saxony’s account, which is very similar to Buridan’s early view.91 Paul also omits Gregory of Rimini’s solution from his survey of previous opinions, which is again surprising, since Gregory is one of the few authors to whom Paul refers by name in the Logica magna, and indeed, in the 1350s Gregory had been prior general of the Augustinians, Paul’s own order. But Gregory’s solution was taken over and adapted by Peter of Ailly, whose view Paul discusses, comments on, and criticizes at length. To understand Gregory’s approach, we need to recall that the medievals, following Aristotle’s lead, divided language into three levels: written, spoken, and mental. 92 Boethius cited Aristotle as saying: “[S]poken [words] are signs (notae) of impressions (passionum) in the soul, and those which are written of those which are spoken.”93 By the fourteenth century, it was common to speak of a complete parallelism of mental, spoken, and written propositions. Spoken and written propositions signify by human imposition (ad placitum), whereas mental propositions signify naturally. Gregory seems to have inferred that only spoken and written propositions can be insolubles as the paradoxical situation cannot infect mental language. Consequently, (spoken and written) insolubles correspond to non-insoluble mental propositions— in fact, to a conjunction of two such mental propositions, the first of which captures  These correlates are the notorious complexe significabilia, whose existence Buridan strongly contested. See, for example, Klima, John Buridan, § 10.2. 90  Buridan, Summulae de dialectica, 969; Buridan, Summulae de practica sophismatum, 155: “Ideo dicitur aliter, propinquius veritati, scilicet quod quaelibet propositio implicat virtualiter aliam propositionem qua de subiecto pro ea supponente affirmaretur hoc praedicatum ‘verum’: dico ‘implicat virtualiter’ sicut antecedens implicat illud quod ad ipsum sequitur.” 91  See, for example, Spade and Read, “Insolubles,” § 3.6. 92  See Aristotle, De interpretatione, chap. 1, and Read, “Concepts and Meaning in Medieval Philosophy.” 93  Boethius, Commentarii in librum Aristotelis Peri Hermeneias, 36: “sunt ergo ea quae sunt in voce earum quae sunt in anima passionum notae et ea quae scribuntur eorum quae sunt in voce.” 89

Introduction27

the primary or customary signification of the insoluble while the second says that the first conjunct is false. For example, taking the Liar again, the spoken proposition, “This proposition is false,” referring to itself, call it A, corresponds to the conjunctive mental proposition whose first conjunct says that (the spoken proposition) A is false, and whose second conjunct says that the first conjunct is false. Neither conjunct of the mental proposition is self-referential, nor is either insoluble or contradictory. In fact, the first conjunct is true (A is false) and the second conjunct is false (since it says falsely that the first conjunct is false); thus, the whole mental conjunction is false, and so the corresponding spoken proposition A is false too.94 Spade suggests that Gregory’s solution was a development of Bradwardine’s, and that Marsilius of Inghen’s was as well.95 However, Marsilius’s solution (probably developed in the 1360s) has strong similarities to the modified Heytesbury solution discussed above. Spade quotes Marsilius (in translation), discussing the common example where there is only one Socrates, who says only, “Socrates says a falsehood”: The reply is that the sophism is false. For it amounts to the conjunction “Socrates says a falsehood and it is false that Socrates says a falsehood.” But that is false on account of its second part. Therefore, although it is always as is signified by its first conjunct, nonetheless it is not as is signified by its second conjunct.96

So in general, an insoluble is expounded as a conjunction whose first conjunct expresses what the terms commonly suggest and whose second conjunct contradicts this and says that is false.97 But since insolubles falsify themselves, that second conjunct says that it is false that it is false, that is, that it is true, as Bradwardine, Heytesbury, and their successors proposed. Gregory’s solution, probably dating from the 1340s, was taken up and adapted by Peter of Ailly in his treatise on Concepts and Insolubles, written in 1372. Whereas Gregory claimed that insolubles were false, corresponding to false mental conjunctions, Peter argued that the phenomena are better explained by realizing that insolubles are equivocal, both true and false, corresponding to two different mental  No text on insolubles by Gregory survives, and so this is a reconstruction of Gregory’s view by Spade and others. See Peter of Ailly, Concepts and Insolubles, trans. Spade, 6–7. 95  See ibid., 6. 96  Ibid., 98 n. 56. See MS. Vatican City, Biblioteca Apostolica Vaticana, Pal. Lat. 995, fol. 73v: “Responditur quod sophisma est falsum, valet enim tantum sicut hec copulativa: sortes dicit falsum et falsum est sortem dicere falsum, modo ista est falsa pro secunda parte. Et ideo licet semper ita sit sicut ipsa significat prima significatione, tamen non est ita sicut ipsa significat secunda significatione, scilicet reflexione falsitatis, scilicet quod falsum sit sortem dicere falsum.” 97  Ibid., fol. 71r: “when it is said of some utterance that it is false, the sense is that is not as the utterance suggests” (quando de aliquo dicto dicitur esse falsum sensus est quod non est ita sicut istud dictum pretendit). For more information on Marsilius’s solution, see Spade and Read, “Insolubles,” § 4.1. 94

28 Introduction

­ ropositions—the two conjuncts from Gregory’s theory, now to be seen not as conp joined but as each separately corresponding to the insoluble. For example, A, in saying of itself that it is false, is true (in corresponding to the true mental proposition which says that A is false), but in saying that what it says (i.e., the mental proposition which says that A is false) is false, it is false. Now you see it, now you don’t; Peter tries to capture the flip-flop behavior that insolubles exhibit.98 5. Paul’s Theories of Insolubles We will consider the authenticity of the Logica magna in the next section (§ 6). Assuming for the moment that Paul was the author, we can find discussions by him of the insolubles in at least four works, his Logica parva,99 his Logica magna (in the treatise presented in the current volume), in the Quadratura (in at least four of its 200 chapters, two of which are edited and translated in Appendix A below), and in the final sophism, no. 50, in his Sophismata aurea (which can be found in Appendix B). We saw in § 4 that two of the leading solutions to the insolubles in the fourteenth century were the modified Heytesbury solution (adapting Heytesbury’s distinctive solution) and Roger Swyneshed’s. Paul follows both of these solutions in different works: the modified Heytesbury solution in the Logica parva and in the Quadratura, and Swyneshed’s in the Logica magna and the Sophismata.100 Recall that Heytesbury, and his followers such as Hunter and Holland, distinguish an insoluble scenario in which it is specified that the insoluble signifies only as the terms suggest (in which case the scenario, or obligatio, should not be accepted), from a scenario where that exclusion clause is omitted (in which case the insoluble and its scenario can be accepted and the insoluble denied). Paul follows this division at the start of the chapter on insolubles in his Logica parva, marking the distinction as that  See Spade’s comment in Peter of Ailly, Concepts and Insolubles, 12–13.  Translated in Logica parva, trans. Perreiah, and edited in Logica parva, ed. Perreiah. 100  There are few summaries and presentations of Paul’s solution(s). A very brief account of that in the Logica magna is given in Spade, The Medieval Liar, 83–4, and a slightly fuller, but confused and misleading one in Bottin, Le antinomie semantiche nella logica medievale, 148–51, who conflates the solutions in the Logica magna and the Logica parva. The account of Paul’s view in Bocheński, History of Formal Logic, 247–51, is also muddled and misleading: after correctly reproducing a selection of Paul’s divisions and assumptions, Bocheński writes: “Paul’s own solution is very like that of the eleventh [i.e., Albert of Saxony’s] and twelfth [i.e., Heytesbury’s] opinions, and so we do not reproduce his long and difficult text,” giving instead a one-page summary. This summary bears no relation whatever to what Paul writes in the Logica magna, not even to the passages Bocheński has cited from it, nor to Heytesbury, but is similar in many ways to Albert’s solution (see Albert of Saxony, “Insolubles,” 346–7). 98

99

Introduction29

between what is simply, or unrestrictedly, an insoluble (insolubile simpliciter) and what is an insoluble restrictedly (insolubile secundum quid—note that Perreiah translates this as “according to a condition”). Then he presents two conclusions. The first is that no simple or unrestricted insoluble should be accepted: No scenario from which what is unrestrictedly an insoluble arises should be accepted. For example, if anyone proposes that “Every proposition is false,” signifying only in that way, is the only proposition, the scenario should not be accepted because a contradiction follows.101

On the other hand, any scenario from which there arises what is an insoluble restrictedly, that is, without the exclusion clause, should be admitted: Every scenario from which what is restrictedly an insoluble arises should be accepted; and one grants the proposed insoluble by saying that it is false. For example, suppose that “This is false” is a proposition referring to itself which signifies as the terms suggest—call it A. Then the scenario is accepted, and A is granted, and it is said that it is false. If one argues like this: “A is false, therefore it signifies other than is the case,” I grant it. “But A only signifies that it is false, therefore that it is false is not so.” I deny the minor premise, and if it is asked what else it signifies, I say that it signifies that A is true, and that is the reason why A is false. So it should be said that every proposition which is an insoluble restrictedly signifies conjunctively, namely, as its terms suggest and that it is true.102

This is the modified Heytesbury solution, following Heytesbury in accepting an insoluble scenario only if it is left open that the insoluble proposition has a secondary or additional signification, and (unlike Heytesbury) specifying that additional signification as asserting its own truth. Paul spells this out towards the end of chapter 6: It should be noted that an insoluble has two significates, one exact (adaequatum) and one principal. The exact significate is a subject-predicate significate similar to the insoluble utterance. For example, the exact significate of “Socrates says a falsehood” is Socrates 101  Logica parva, ed. Perreiah, 132: “Prima conclusio est ista. Numquam admittendus est casus quo trahit originem insolubile simpliciter. Ut si poneretur quod illa ‘Omnis propositio est falsa’ esset omnis propositio sic praecise significans, non est admittendus casus quia sequitur contradictio.” An alternative translation can be found in Logica parva, trans. Perreiah, 240. 102  Logica parva, ed. Perreiah, 132–3: “Secunda conclusio. Omnis casus quo originatur insolubile secundum quid est admittendus. Et conceditur insolubile propositum dicendo ipsum esse falsum ut ponendo quod ista propositio ‘Hoc est falsum’ demonstrato se ipso significet sicut termini praetendunt quae sit A. Tunc admittitur casus, et conceditur A, et dicitur quod est falsum. Et si arguitur sic ‘A est falsum, igitur significat aliter quam est,’ concedo. Sed A non significat nisi quod hoc est falsum; igitur non est ita quod hoc est falsum. Nego minorem. Et si dicitur quod ergo aliud significat, dicitur quod significat A esse verum, et ratione cuius A est falsum. Unde dictum est quod quodlibet insolubile secundum quid significat copulative, videlicet sicut termini praetendunt, et quod ipsum est verum.” An alternative translation can be found in Logica parva, trans. Perreiah, 241.

30 Introduction

s­aying a falsehood or that Socrates says a falsehood. But the principal significate is a ­compound significate, for example, that Socrates says a falsehood and that the proposition is true.103

We find the same approach to insolubles in Paul’s Quadratura. This work is not about squaring the circle or quadrature, but is a highly formal and artificial series of two hundred sophisms arranged in four parts of fifty chapters each.104 The reason for the strange title Quadratura is that each chapter “is fortified with four conclusions and as many corollaries or more” (munitur quatuor conclusionibus et totidem aut pluribus correlariis). Each of the four main parts focuses on a particular question: First, whether the same inference can be both valid and invalid; secondly, whether the same proposition can be both true and false; thirdly, whether disparate things are verifiable of the same thing; fourthly, whether two incompatibles can be both true or both false.

The fifteenth chapter of Part I is explicitly concerned with insolubles, specifically with the following argument: This inference ⟨call it B⟩ is valid: A will signify only that everything ⟨that is or will be⟩ true will be false, so A will be false; and this inference ⟨B⟩ is invalid. So the question is true.105

Paul shows that the exclusion clause in the scenario (“A will signify only that …”) must be amended. This follows from his second conclusion: There is some proposition [namely, “Every proposition is false”] signifying principally purely predicatively which at some time will signify principally in a compound way. Nonetheless, there will be no change in it, nor will any new imposition be added to it.106

For “when it will be the only proposition it will signify principally that every proposition is false and that it is true, just like other insolubles, whose significations reflect wholly on themselves.” He continues, in discussing the third conclusion, to 103  Logica parva, ed. Perreiah, 149: “Notandum quod insolubile habet duo significata: unum adaequatum et unum principale. Adaequatum significatum est significatum categoricum simile dicto insolubilis. Unde adaequatum significatum illius ‘Sortes dicit falsum’ est istud Sortem dicere falsum vel quod Sortes dicit falsum. Principale autem significatum est significatum hypotheticum ut Sortem dicere falsum et illam propositionem esse veram.” An alternative translation can be found in Logica parva, trans. Perreiah, 255. Note that the principal significate is not just the second conjunct, as Perreiah says on 256, but the whole conjunction (as Perreiah recognized on 106). 104  As explained in § 1 above. 105  See Appendix A, § 1.15.1, pp. 368–9 below in this volume. This insoluble can also be found in Richard Ferrybridge’s Consequentiae: see the excerpt in Pozzi, Le ‘Consequentiae’ nella Logica Medievale, 262-71. 106  Appendix A, § 1.15.2.2, p. 376–7 below.

Introduction31

remark that “this conjunctive significate is called the principal significate of A, although it is not the exact ⟨significate⟩ but only the first part ⟨is⟩.”107 Without the exclusive phrase (that is, that A signifies only that everything true will be false), argument B is invalid, but if it is retained, Paul does not accept the scenario, “because it implies a contradiction.”108 In response to a later sophism, the second in Part II, Paul spells out his use of “exact significate” and “principal significate” in greater detail, and links them to the notions of truth and falsity: Finally, then, it should be said that it is because it immediately signifies a truth that any proposition is true, and it is because it immediately signifies a falsehood that any proposition is false, where outside the case of insolubles “immediately” means the same as “exactly.” But in the case of insolubles it means the same as “principally.” Hence “A man is an animal” is true because it immediately signifies a truth, that is, it exactly ⟨signifies⟩ the truth that a man is an animal; but “This is false,” referring to itself, is false because it immediately signifies a falsehood, that is, it principally ⟨signifies⟩ a falsehood, namely, that this is false and that this is not false.109

However, whereas in the Logica parva and the Quadratura Paul subscribes to the modified Heytesbury solution to the insolubles, in the Logica magna he defends a version of Swyneshed’s solution. Surprisingly, the modified Heytesbury solution does not appear among the fifteen solutions that Paul considers in his first chapter (in varying detail) and rejects, whereas Heytesbury’s own solution is considered, being the first solution to which Paul devotes more than a few lines.110 One of Paul’s objections to the solution turns, in fact, on Heytesbury’s reluctance to specify what the additional signification is which renders an insoluble false. This, and many of the other objections which Paul levels against Heytesbury’s view, are drawn from Peter of Mantua’s Insolubles (or possibly from a third text on which they both draw). Suppose “A falsehood is said” signifies principally (that is, wholly and exactly) that God exists. Heytesbury had claimed that a proposition could have a further signification in addition to what it standardly signifies, so presumably it could signify something completely different, such as, that God exists. If so, it would be necessarily true. But Heytesbury’s view is that “A falsehood is said” is false as uttered in the proposed scenario. Yet the conclusion of an inference which is clearly valid and whose premise is in doubt should not be denied—for if one denies the conclusion  Ibid., § 1.15.2.3, pp. 380–1.  Ibid., §ad 1.15.1, pp. 386–7. For further discussion of this example, see Read, “‘Everything true will be false:’ Paul of Venice and a medieval Yablo paradox.” 109  Ibid., § 2.2.3, pp. 396–7. 110  See Logica magna: Treatise on Insolubles, § 1.12, pp. 92–3 below in this volume. 107 108

32 Introduction

of an inference one recognizes to be valid, one is committed to denying the ­premise.111 So Heytesbury’s solution proposing such hidden and unspecified significations is unacceptable. Paul devotes the whole of the first chapter of his treatise on insolubles in the Logica magna to the rejection of these other proposed solutions. The treatment can be seen as falling into four groups, the first three groups corresponding to three sources on which Paul draws. First, he runs rapidly through seven of the eight alternative solutions that Bradwardine considers in his treatise,112 for the most part summarizing almost verbatim Bradwardine’s own criticism. He then turns to Heytesbury’s criticism of alternative solutions, starting with the second of the four solutions considered by Heytesbury (the first one Heytesbury rejects is Swyneshed’s, which Paul will himself accept), that of John Dumbleton. The next (third on Heytesbury’s list) is Kilvington’s, and then Paul comes to Bradwardine’s own solution (Heytesbury’s fourth). Thus, the first ten solutions considered are all from Oxford or, at least, were discussed at Oxford in the two decades from the early 1320s to the early 1340s. With the eleventh solution, Paul turns to his third source, namely, Peter of Mantua. The eleventh solution is Albert of Saxony’s, who presented it at Paris in the early 1350s, and possibly the same as John Buridan’s own early solution,113 the first view discussed by Mantua; and next to Heytesbury’s (Mantua’s second), as noted above.114 Before proceeding to the third view discussed by Mantua, Paul considers Peter of Ailly’s solution at some length, apparently drawing directly on Ailly’s own treatise, to the discussion of which Paul appends (without distinguishing it by number) a criticism of Mantua’s solution. Finally, Paul turns to a rejection of restrictivism, the first solution rejected by Bradwardine and the third by Mantua. However, Paul deals with the specific form given to it by Walter Segrave in Oxford in the 1320s or early 1330s, who in fact defended restrictivism in the face of Bradwardine’s objections,115 attributing the solution to the fallacy of accident, a suggestion not dealt with by either Bradwardine or Mantua. After this extended discussion of alternatives (occupying a quarter of his treatise), Paul sets out to develop his own solution, based firmly on Roger Swyneshed’s proposals from the 1330s. In his second chapter, he systematically lays out his distinctions (divisiones) and assumptions (suppositiones), then draws seven conclusions and 111  This is an example of the form of reasoning dubbed “Kilvington’s disputational meta-argument” in Kilvington, Sophismata, trans. Krtezmann, 31. See our Commentary on § 1.12.5. 112  See Bradwardine, Insolubilia, chaps. 3–5. 113  However, Marsilius of Inghen, whose solution Peter and Paul do not discuss, appears to have distinguished Albert’s solution from Buridan’s early view (see Spade, The Medieval Liar, 79). 114  See Strobino, “Truth and Paradox in Late XIVth-Century Logic,” 484. 115  See Spade, The Medieval Liar, 113–16.

Introduction33

notes seven corollaries. The basic idea is Swyneshed’s, to provide a solution which does not depend on postulating tacit, hidden, or consequential significates for insoluble propositions beyond what is clearly warranted—what they standardly suggest or indicate by the straightforward combination of their parts (in Heytesbury’s phrase, sicut termini communiter pretendunt). Instead, as Swyneshed had proposed, Paul tightens the criterion for truth, in order to exclude propositions that falsify themselves, while weakening the criterion for falsehood to admit those examples that do falsify themselves even if they are otherwise impeccable. Roger’s second and third notorious conclusions reappear as Paul’s fifth and second respectively. Paul will later describe the second conclusion as a fundamental principle, perhaps the fundamental principle (two other fundamental principles are mentioned in chapter 8, on merely apparent insolubles).116 Consequently, Paul defines an insoluble as a self-falsifying proposition, that is, “a proposition having reflection on itself wholly or partially implying its own falsity or that it is not itself true.”117 Paul’s adoption of Swyneshed’s solution to the insolubles and, in particular, his acceptance of Roger’s second conclusion as his fifth, overturns several claims Paul had made in the earlier chapter on consequence (De rationali) in the Logica magna, especially his third rule, according to which valid inference is always truth-preserving. As we saw when considering Roger’s solution in § 4, the simple Liar, deemed by both of them to be false, follows immediately from the (for them, true) statement that it is false. Both premise and conclusion state of the same thing (the conclusion) that it is false. What then is Roger’s account of validity? Spade claims that Roger believed that “signifying as it is” preserves validity.118 Paul, however, identifies validity with the opposite of the conclusion’s being incompatible with the premise: A valid inference which signifies in accordance with the composition of its elements may be defined as one in which the contradictory of its conclusion would be incompatible with the premise of that inference, given that these signify as they do; and by “as they do” I refer to what they customarily signify.119

 See Logica magna: Treatise on Insolubles, §ad 4.2.1.3, pp. 202–03 below in this volume.  Ibid., § 2.1.8, pp. 142–3. 118  See the introduction to Swyneshed, “Roger Swyneshed’s Theory of Insolubilia,” 105, 113 n. 35. Read, “The Rule of Contradictory Pairs, Insolubles and Validity,” (§ 4) argues that Spade is mistaken, though it remains unclear what Swyneshed’s account of validity is. 119  Logica magna: Capitula de conditionali et de rationali, ed. and trans. Hughes, 80: “Consequentia bona significans iuxta compositionem suarum partium dicitur esse illa cuius contradictorium consequentis potest repugnare antecedenti eiusdem, ipsis sic significantibus; et demonstro per ly sic ipsorum significata consueta.” He repeats this in the treatise on insolubles (§ 2.2.5, pp. 150–1 below). 116 117

34 Introduction

There is a further inconsistency within the Logica magna between the treatise on insolubles and the chapter on consequence in Paul’s treatment of what might be called the inferential Knower paradox:120 This is unknown to you, so this is unknown to you,

where both occurrences of “this” refer to the conclusion. In the Capitula de conditionali et de rationali, Paul presents this as a counterexample to his ninth rule, according to which knowledge is closed under known consequence.121 For the conclusion is an example of the Knower paradox and cannot be known, on pain of contradiction (knowledge entails truth and so if it were known it would be unknown). Therefore, the premise is not only true but known to be true (by the argument just given). Paul considers four possible responses (including restrictivism and cassationism as the second and third), suggesting in his preferred response, the fourth, that the premise is also unknown, like the conclusion. He promises that there will be “more about this when we come to deal with the insolubles.”122 When he comes to discuss this example in the fourth conclusion in the second chapter of the treatise on insolubles, Paul agrees that it is an insoluble, but claims that the premise is true: For it is evident that this inference is formally valid, because one cannot see how the opposite of the conclusion can be compatible with the premise. But the premise is known by you, because you know that the conclusion is not known, since it is an insoluble that asserts that it itself is unknown. And yet the conclusion is not known by you.123

This might appear to be a counterexample to the ninth rule of the chapter on consequence: knowledge is not closed under known consequence. However, he had included a caveat in his statement of the ninth rule: Suppose that a certain inference is valid, is known by you to be valid, is understood by you, and signifies primarily in accordance with the composition of its elements; suppose too that its premise is known by you, and that you know that what is false does not follow from anything that is true; then its conclusion is also known by you.124  The Knower paradox itself turns on the statement, “This proposition is unknown” or, “This proposition is unknown to you.” See, for example, Anderson, “The Paradox of the Knower.” 121  Logica magna: Capitula de conditionali et de rationali, ed. and trans. Hughes, 197. 122  Ibid., 200. 123  Logica magna: Treatise on Insolubles, § 2.3.4, pp. 164–5 below. 124  Logica magna: Capitula de conditionali et de rationali, ed. and trans. Hughes, 195, § 13: “Si aliqua est consequentia bona, scita a te esse bona et intellecta a te, significans primo iuxta compositionem suarum partium, et antecedens est scitum a te, sciendo quod ex nullo vero sequitur falsum, et consequens eiusdem est scitum a te.” 120

Introduction35

For in the “Insolubles” we do know (from the fifth conclusion) that the false does follow from the true, so that the conditions of the ninth rule are not satisfied. Paul also seems to be extending the notion of insoluble in § 2.3.4. In § 2.1.8, cited above, insolubles are defined as self-falsifying propositions—and indeed he remarks that it follows that many propositions called insolubles by others are not really insoluble insofar as they do not fit his definition.125 Swyneshed, in fact, had already extended the conception of insoluble to include propositions that are relevant to inferring themselves not to be known,126 and Paul seems to be following Swyneshed implicitly in § 2.3.4. However, in chapter 8 of the “Insolubles,” Paul includes “This is not known by you” among the merely apparent insolubles: “… although they are not insoluble since they do not have reflection of falsity on themselves.”127 In the rest of the treatise, Paul considers, in chapter 3, the objections which Heytesbury had directed at Swyneshed’s solution, many of which he had already addressed in discussing the conclusions and corollaries in chapter 2. In chapter 4, he shows at some length how his solution deals with the much-discussed example where Socrates says only, “Socrates says a falsehood.” In chapter 5, he extends the account to deal with other examples, such as the one where Socrates says that his sole business is to be hung on the gallows. These are not obviously insolubles until the background scenario is added. In chapter 6, he moves to examples—like “A falsehood exists”—which become insoluble in a suitable scenario, and in chapter 7 he addresses examples involving exclusive and exceptive propositions, such as “Only a false proposition is exclusive” (assuming it is the only exclusive proposition) and “No proposition except A is false” (where this is A and is the only exceptive proposition). Turning to the discussion of insolubles in the Sophismata aurea, we find this to be very derivative from that in the Logica magna. The work consists of a collection of fifty sophisms, many very familiar from other collections of sophisms, many turning on an equivocation between compounded and divided senses (what would nowadays be called a scope ambiguity), such as, “Every proposition or its contradictory is true” and “Everything false if it is impossible is not true.” The final sophism, no. 50, is “Socrates says a falsehood,” given that there is only one Socrates and that is all he says. The sophismatic arguments leading to contradiction are mostly drawn, essentially verbatim, from § 4.2 of the treatise on insolubles in the Logica magna (with a final argument drawn from § 3.1). Paul then presents four conclusions and associated corollaries in order to defuse the sophismatic arguments. The first of these conclusions is in effect a statement of the Swyneshed programme to solve the  See Logica magna: Treatise on Insolubles, § 2.1.8.3, pp. 144–5 below.  See Swyneshed, Insolubilia, ed. Spade, § 81. 127  Logica magna: Treatise on Insolubles, § 8.1, pp. 256–7 below. 125 126

36 Introduction

i­nsolubles without resort to any hidden signification; for Paul claims that “every subject-predicate insoluble signifies exactly according to the composition of its terms.”128 His first corollary consequently rejects all those solutions which turn on there being such a secondary signification, describing the standard opinion as specifying that second significate as “that it is true.”129 This is the modified Heytesbury solution. The subsequent conclusions repeat six of the seven conclusions of the second chapter of the insolubles treatise in the Logica magna. The work concludes with a discussion of the paradox of signification, “This proposition signifies other than it is,” drawn from Logica magna, § 3.2. 6. The Authorship of the Logica magna “This is the Logic of the most excellent friar Paul of Venice.”130 At the end of the Logica magna preserved in manuscript M, a fifteenth-century anonymous note of possession attributes the work to frater Paul of Venice. For more than five centuries, none of the biographers and scholars of Paul questioned this attribution. It was Perreiah who first cast doubt on Paul’s authorship, trying to bring conclusive evidence against it which he derived from analyses of (1) the external sources, (2) surviving manuscript material, and (3) the style and doctrine of the Logica magna in comparison with works undoubtedly penned by Paul.131 Eventually, he concluded that “the results of all of these investigations have been mainly negative: most indications lead to the conclusion that the Logica magna should not be included in Paul of Venice’s corpus operum.”132 This claim, however, has not been well received and we, too, are not persuaded by Perreiah’s arguments.133 In our view his interpretations are not strongly substantiated. Moreover, he often uses conjectural data and hypotheses to build arguments that he presents as conclusive. Nevertheless, his arguments have not been answered in print in any detail, which is the task to which we now turn.

 Appendix B, § 2.1, pp. 406–07 below.  Ibid., § 2.1.1.1, pp. 408–09 below. 130  M fol. 245rb: Ista est logica excellentissimi viri fratris Pauli de Venetiis et constitit michi ducatos Decem (et constitit michi ducatos Decem is written over earlier text that is partially erased and no longer legible). 131  See Perreiah, “A Concordance: Logica parva–Logica magna” and Perreiah, Paul of Venice, chaps. 4–5. 132  Perreiah, Paul of Venice, 93. 133  See, for example, Conti, Esistenza e verità, 2 n. 1: “Inaccettabile … la tesi che Paolo Veneto non sia l’autore della Logica magna, che va contro una tradizione secolare molto ben attestata e non viene suffragata da sufficienti dati filologici, storici ed esegetici.” 128 129

Introduction37

6.1  External Evidence 6.1.1  Perreiah’s External Evidence Reconsidered Perreiah focuses his attention on fifteenth- and early sixteenth-century sources which apparently testify to Paul’s authorship of the Logica magna. Particularly relevant to his denialist thesis are the references to Paul’s work found in the funeral oration for Paul delivered by Cristoforo Barzizza, and in Antonio Cittadini da Faenza’s commentary on Paul’s Logica parva. One of the earliest references to the Logica parva occurs in Barzizza’s funeral oration, delivered on 30 June 1429, where Barzizza lists the massive corpus of Paul’s works. Among the logical works, he first mentions a logica minor, which is indisputably identified with the Logica parva, secondly a summa maior in which Paul explains all the subtleties of British logicians, then the commentaries on Aristotle’s logical works and, finally, alia sophistica dubia.134 The summa maior has been associated in many minds with the Logica magna, but not in Perreiah’s. According to him, the summa maior should be identified with the Quadratura, as he claims fifteenth-century commentators like Cittadini did. In his Commentary on the Logica parva, Perreiah argues, Cittadini uses the “collective term” dialecticae maiores to refer to the Quadratura. And since for Perreiah the sophistica dubia mentioned by Barzizza allude to the Sophismata aurea, if summa maior does not refer to the Quadratura—as Perreiah suggests—then the latter “would be the only major work known to be by Paul of Venice … omitted by Barzizza’s list.”135 In our view, Perreiah misidentifies both texts: it is much more plausible that sophistica dubia refers to the Quadratura seu quatuor dubia and summa maior to the Logica magna. This hypothesis finds support in Cittadini’s commentary, particularly in its prologue. There Cittadini clearly distinguishes between the opusculum Logica parva, the dialecticae maiores, and the quaestiones quatuor sophismatum. Furthermore, he warns the reader about the inconsistencies among these three works, which might seem to imply a double authorship for these works. Thus, the Logica parva looks as though it could have been written by an author more intelligent and erudite than the simple-minded Paul, who could have authored the dialecticae maiores and the quatuor questiones: Although ⟨Paul⟩ remains consistent with himself in this work [i.e., the Logica parva], however, if a reader attentively examines the books of his dialecticae maiores and of the Quatuor questiones on sophisms, he will perhaps believe that Paul was not the author of this work [i.e., the Logica parva], since he is often at variance with himself and does not seem to be the same person at all. And, I admit, I cannot be led to think that this booklet, which is remarkably skillfully written, comes from Paul’s workshop, because its author  The list is cited in Perreiah, Paul of Venice, 76.  Ibid., 77.

134 135

38 Introduction

seems to have been much more clever and learned than Paul was or than it can be claimed and judged ⟨that Paul was⟩ on the basis of his books remaining to us. And that is mostly evident from the dialecticae maiores, in which you can find many things which either are completely wrong or totally inconsistent with what is said here [i.e., in the Logica parva].136

Perreiah completely misunderstood this text, thinking that Cittadini questions the authenticity of Paul’s other logical writings, specifically the Quadratura, rather than Paul’s authorship of the Logica parva. Moreover, misled by a mis-transcription of the text,137 Perreiah failed to distinguish between the dialecticae maiores and the quattuor questiones. Accordingly, he mistakenly explained that Cittadini doubted the authenticity of the Quadratura (alone) not because of its doctrinal discrepancies with the Logica parva, but because of the internal inconsistency in the Quadratura: “its author is inconsistent with himself and does not seem to be the same person.”138 Does Cittadini have in mind the Logica magna, rather than the Quadratura, in the above passage as well as in the many places of his commentary where he mentions the dialecticae maiores? It seems so, as emerges by comparison of Cittadini’s direct and indirect quotations with the passages of the dialecticae maiores to which Cittadini alludes. Consider, for example, Cittadini’s discussion of Paul’s theories of the proposition. In the Logica parva, a proposition is defined as “an indicative utterance signifying what is true or false (oratio verum vel falsum significans) … ‘indicative’ because only an indicative is a proposition and not an imperative or an optative or subjunctive.”139 Commenting on this passage, Cittadini points out the differences between these words and the definition of proposition found in the dialecticae maiores. There a proposition is defined as an utterance (enunciatio) which is capable of signifying something true or false (veri aut falsi significativa), but which does not necessarily actually signify something true or false—and as not necessarily “indicative”: It is surprising that in the dialecticae maiores Paul fell so much into error as to be the only one who grants that an optative and an imperative and other utterances are propositions, 136  Antonio Cittadini, In Dialectica minori Pauli Veneti, fol. 2r: “Licet autem ipsemet de hoc opere sibi sepe consentiat, si tamen dialecticorum suorum maiorum ac principalium questionum quae quattuor sunt sophysmatum quoque volumina diligenter examinentur, fortasse lector putabit illum huius operis auctorem non fuisse quod a se ipso plerumque dissideat nec idem omnino esse videatur. Neque ego ut verum de me fateor adduci possum ut putem opusculum hoc, quod miro quodam artificio compositum est, e Pauli officina emanasse: multo quippe ingeniosior eruditiorque auctor is fuisse videtur quam vel Paulus extiterit vel ex libris suis, qui nobis relicti sunt, perpendi ac vindicari possit. Id vero ex dialecticis maioribus potissimum liquet, in quibus plurima repperias, quae vel omnino errata sunt vel hiis que hic traduntur nullatenus consentanea.” 137  The text is cited in Perreiah, Paul of Venice, 81: erratum in, corrige tamen; err. maiorem, corr. maiorum; err. quorum cor. questionum. 138  Ibid., 82. 139  Paul of Venice, Logica parva, trans. Perreiah, 123.

Introduction39

which here [sc. in the Logica parva] he does not deem to be propositions and which everyone else does not consider to be truth- and falsity-bearers.140

Cittadini then conjectures that this diversity may be due to two factors: either Paul did not author the Logica parva, or he followed ideas fully divergent from those of the dialecticae maiores.141 Pace Perreiah, who takes in dialecticis maioribus to refer to the Quadratura (without however providing any precise reference),142 the description of propositions alluded to by Cittadini is found at the beginning of Part II of the Logica magna: A proposition is a perfect and well-formed utterance (enunciatio) capable of signifying something true or false … Not only a declarative utterance is said to be a proposition, but also an imperative, precative, optative, or subjunctive one; … although these types of utterances are not explicitly declarative, it suffices that they are implicitly ⟨declarative⟩, insofar as they are subordinated to mental declarative propositions.143

Another interesting disparity to which Cittadini draws attention between the two logical works concerns the treatment of the syntagm “e converso” in the discussion of subject-predicate propositions. Unlike the Logica parva, the dialecticae maiores consider the syntagm “e converso” as a full proposition and equate it to implicit subject-predicate propositions. Not surprisingly, Cittadini prefers the view of the  Cittadini, In Dialectica minori, fols. 16v–17r: “Mirum autem est quod Paulus in dialecticis maioribus adeo prolapsus sit ut optativam et imperativam caeterasque orationes, quas hic propositiones esse non putat quasque caeteri omnes veritatis falsitatisque expertes esse arbitrantur, solus propositiones esse concedat.” 141  See ibid., fol. 16r–v: “Quidam dialectici [sc. Menghus Bianchellus?] qui definitionem Pauli [sc. in logica parva] ab omni calumnia defendere volentes ita intelligendam esse dixerunt: Propositio est ratio [sic] indicativa verum vel falsum significans quiescenter … quasi Paulus etiam concedat infinitivas orationes, quae imperfectae sunt, verum significare vel falsum. Quod si Paulus in dialecticis maioribus definitionem ita ut hii dicunt intelligi debere existimat, nihil curandum est. Ut enim diximus, auctor is vel Paulus non fuit vel si Paulus fuit, nihil hoc in libro audiendus est: illic enim sententias quasdam secutus est, quae hiis quae hic dicuntur nullo pacto convenient” (“Some logicians [sc. Menghus Bianchellus?], wishing to defend Paul’s definition [sc. in the Logica parva] against every calumny, said that it should be understood in this way: A proposition is an indicative utterance signifying a truth or falsehood without further ado … as if even Paul admits that infinitive utterances, which are imperfect, signify a truth or a falsehood. And it does not matter if in the dialecticae maiores Paul thinks that the definition should be understood as they say. For as we said, the author—whether he was Paul or not— should not be taken into account in anything in this book, for there he followed some doctrines which in no way agree with what is said here.”) 142  Perreiah, Paul of Venice, 81. 143  Logica magna (E fol. 101ra–b, M fol. 128rb–vb): “Propositio est enuntiatio congrua et perfecta veri aut falsi significativa … Non solum indicativa enuntiatio propositio dicitur, verum etiam imperativa, deprecativa, optativa et subiectiva … huiusmodi enuntiationes … sunt indicative et si non explicite, sufficit quod implicite, quia subordinantur indicativis mentalibus.” 140

40 Introduction

Logica parva and rejects the other,144 which is exactly the view found in the Logica magna.145 6.1.2  Theories of Insolubles in Italy (from the End of the Fourteenth to the MidFifteenth Century) In his search for external evidence, Perreiah scrutinizes fifteenth- and early sixteenthcentury commentaries on the Logica parva since this is where one might most expect to find some testimony of Paul’s authorship of the Logica magna. Yet, he claims to find “only scant mention and almost no attention paid” to the Logica magna. One might arrive at a different picture, however, if one broadens the scope of this search for external evidence. Accordingly, we have considered the insolubilia literature produced in Italy between the end of the fourteenth and the mid-fifteenth century, and particularly some Recollectae. These lecture notes on insolubles, usually copied by students and at times written by the masters themselves, bear a close relationship with classroom teaching and disputations. They usually have the first chapter (on insolubles) of Heytesbury’s Regulae solvendi sophismata as their base text. This is due to fact that the Regulae were likely mandatory reading for undergraduates studying logic at northern Italian universities, such as Padua and Bologna, in the first half of the fifteenth century, just as they were at the end of the century, as codified by the 1496 statutes of the faculty of arts of the University of Padua. The various Recollectae on Heytesbury’s text that we have considered consist of series of objections and replies which lack a theoretical framework as comprehensive as that provided by fourteenth-century treatises on insolubles, including Paul of Venice’s. Usually, an author of Recollectae deals with the opinions presented by Heytesbury, either just Heytesbury’s own or also the other three views Heytesbury considered,146 and elaborates on Heytesbury’s definition of an insoluble proposition and  See Cittadini, In Dialectica minori, fol. 18v: “Existimat autem Paulus in dialecticis maioribus id quod dicimus e converso propositionem esse, quandoque quidem affirmativam, cum affirmativam sequitur propositionem, quandoque vero negativam, cum negationi comitatur, hoc modo: nullus homo est lapis et e converso … Ego vero non dixerim hanc dictionem … propositionem esse” (“In the Dialecticae maiores Paul admits that the expression ‘conversely’ is a proposition, affirmative when it follows an affirmative proposition, but negative when it follows a negation, in this way: no man is a stone and conversely… But I would not say that this expression [dictio] is a proposition.”) 145  See Logica magna (1499, fol. 102ra): “De propositione que non solum extremis caret immo et copula principali, sicut dicendo: omnis homo est animal et e converso, certum est quod ly e converso est propositio cathegorica” (“Concerning a proposition which lacks not only the extremes but also the main copula, as in saying ‘Every man is an animal and conversely,’ it is certain that ‘e converso’ is a subject-predicate proposition.”) 146  Cajetan (In regulas Gulielmi Hesburi recollecte, fol. 7va) identified the author of the first opinion with Roger Swyneshed, of the second with John Dumbleton, and of the third with Richard Kilvington. 144

Introduction41

its scenario (casus). Some authors also briefly sketch their own definitions of truth, falsity, and signification of propositions, before devoting a longer and more detailed discussion to formally deducing, attacking, and defending the five rules delineated by Heytesbury. Here is an example: I am writing ⟨this⟩ booklet as a consequence of the solutions to objections which are ordinarily proposed in all the schools and which are solved only childishly or through a more despicable evasion, by saying that Heytesbury did not understand them.147

The Recollectae usually have didactic purposes, their authors mainly aiming to provide the students who play the role of respondents in obligational disputations on insolubles with sets of stock arguments pro and con and with strategies and arguments for neutralizing tricky objections raised by the opponents. At the beginning of the 1390s, Peter of Mantua wrote an influential logical textbook, the Logica, whose last chapter is devoted to insolubles. Here, Peter delineates his view on insolubles, which departs from the traditional solutions of Heytesbury, Bradwardine, Albert of Saxony, and Peter of Ailly, and which Paul of Venice criticizes in the Logica magna.148 On the basis of his account of truth, which differentiates one criterion for the truth of self-referential and another for the truth of non-self-referential propositions, Peter of Mantua concludes that insoluble propositions are true in one sense and false in another.149 Peter’s students at Bologna probably included Angelo of Fossombrone, who became professor of logic and philosophy at Bologna (1395–99) and then of natural philosophy in Padua. An outcome of Angelo’s lessons, likely in Bologna, is his treatise on insolubles.150 In the opening lines of his work, Angelo informs us that in school The second and third opinions occupy the eighth and ninth positions in Paul of Venice’s list (cf. § 1.8–1.9, pp. 87–9 below). 147  Paul of Pergula, In regulas insolubilium Heytesbury, MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2139, fols. 67rb–80va, here fol. 68vb: “Opusculum condo propter solutiones instantiarum que communiter per omnes scholas adducuntur et non solvuntur nisi pueriliter vel cum fuga miseriora dicendo quod hisber non intellexit de hiis.” And similarly, at the end of his Insolubilia (MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2130, fol. 171va, and MS. Venice, Biblioteca Nazionale Marciana, Lat. 6.30 (2547), fol. 71va) Angelo of Fossombrone writes: “Et similiter est dicendum in similibus in quibus vel de quibus per hec dicta respondens cum aliquali solertia poterit habere certitudinem in respondendo” (“And one should reply in a like manner in similar cases in which or about which with diligence of some sort the respondent shall be certain in replying through what was said”). 148  Peter appears to have been influenced by Wyclif’s ideas. See Spade and Read, “Insolubles,” §§ 4.1 and 4.3. 149  On Peter of Mantua’s view, see Paul’s addition to his discussion of the thirteenth opinion, §§ 1.13.7–1.13.7.3, and our commentary. 150  The treatise is preserved in four manuscripts, listed by Spade, The Medieval Liar, 49 n. 25. We have used the MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2130 and MS. Venice, Biblioteca Nazionale Marciana, Lat. 6.30 (2547), where Angelo’s Insolubilia follow the chapter on future

42 Introduction

disputations of his time on insolubles, respondents commonly upheld a restrictivist view, which however he deems untenable and against which he proposes a series of objections.151 Except for this preliminary introduction, the content and inner arrangement of Angelo’s treatise are close to that of the Recollectae delineated above. Similarly to these, moreover, Angelo’s work exhibits a markedly didactic purpose, aiming to teach students how to respond correctly in dialectical exercises. This might account for Angelo’s decision to deal with Heytesbury’s view alone and to avoid any analysis or even mention of ancient or modern authors. The narrow educational scope of the work might also explain the prominent role assigned to obligations in his treatment of insolubles. Angelo’s responses to the objections raised against Heytesbury’s view depend not only on Heytesbury’s five rules but are grounded in obligational rules, as Angelo clearly says before replying to an argument against the third rule: In order to reply to arguments against the third rule, the respondent should keep the rules of obligation mentioned above close at hand, for example, which inference and which proposition should be granted, which inference should be granted as valid, and which proposition should be granted as true, and so on.152

Faithful to his base text, Angelo upholds Heytesbury’s view concerning insolubles, although not its original version but rather a developed form. He openly rejects the modified Heytesbury solution by denying that the respondent is required to specify the additional signification of insolubles: To the third argument against that rule [sc. the third] it is easily replied by saying that the respondent should not make known what that proposition signifies exactly, but he should respond only on the basis of what is proposed by the questioner.153 c­ ontingents extracted from the Logica magna. A summary of the content of the treatise is offered by Spade, The Medieval Liar, 49–52. 151  See Angelo of Fossombrone, Insolubilia (Vat. lat. 2130, fol. 61ra; Lat. 6.30 (2547), fol. 163ra): “Una communis oppinio in materia insolubilium ad quam communiter habent respondentes refugium dicit quod terminus que est pars propositionis aut secum convertibilis non potest supponere pro tota propositione cuius est pars neque pro eius pertinente … ut antecedente vel consequente ad illam aut repugnante eidem” (“A common opinion in the matter of insolubles, in which respondents usually take refuge, says that a term which is part of a proposition or one convertible with it cannot supposit for the whole proposition of which it is part nor for anything related to it … such as its premise or conclusion or a proposition inconsistent with it.”) 152  Ibid. (Vat. lat. 2130, fol. 68vb; Lat. 6.30 (2547), fol. 167rb): “Respondendo ad argumenta contra tertiam regulam, debet homo bene habere pre manibus illas regulas obligatorias superius recitatatas ut que sit consequentia concedenda et que sit propositio concedenda, que consequentia sit concedenda esse bona et que propositio sit concedenda esse vera et sic de aliis.” 153  Ibid. (Vat. lat. 2130, fol. 69rb; Lat. 6.30 (2547), fol. 167va: “Ad tertium argumentum contra illam regulam respondetur faciliter [facilius Lat. 6.30 (2547)] dicendo quod non tenetur respondens c­ ertificare quid illa adequate significet, sed secundum quod arguens proponit ita respondens debet respondere.”

Introduction43

Angelo’s treatise seems to have enjoyed some circulation in Northern Italy. It is the main source of the In regulas insolubilium Heytesbury written by Paul of Venice’s pupil, Paul of Pergula (d. 1455). In the In regulas, Paul of Pergula aligns himself with Heytesbury’s view and dismisses the modified Heytesbury solution, which he endorses in his Logica, an introductory textbook whose main source is Paul of Venice’s Logica parva.154 The In regulas has a merely didactical purpose and is constructed as a series of standard objections (communes obiectiones) to Heytesbury’s view, to which Paul often replies by referring explicitly to the argument elaborated by Angelo. In addition to references to Angelo, who is the most frequently cited recent author, Paul discards Peter of Mantua’s arguments and accepts instead those of Nicolò Fava, a Bolognaise professor who had an animated debate with Paul of Venice (Bologna, 1424) and who was Paul of Pergula’s promotor to the degree of arts (Bologna, 1425). Interestingly, all three sources (Peter, Angelo, and Nicolò) were prominent figures in the Bolognaise academic milieu. Paul of Pergula’s friend (and also a student of Paul of Venice’s and later his successor in the chair of natural philosophy) Cajetan of Thiene wrote the Recollectae super Regulas Hentisbery (also known as In regulas Gulielmi Hesburi recollecte) while teaching in Padua between 1422 and 1430. The Recollectae share some doctrinal and formal features with the works of Angelo and Paul of Pergula (whom Cajetan never mentions), such as the adoption of Heytesbury’s solution and the use of some of the same stock arguments. Yet, the Recollectae differentiate themselves from Angelo’s and Paul’s works in that they critically consider not only Heytesbury’s position, but also the other three opinions, which Angelo and Paul neglected. In view of our search for external evidence, it is particularly worth considering Cajetan’s treatment of those alternative views, and especially of the first and longest one. First, Cajetan correctly attributes this first opinion to Roger Swyneshed; then he introduces four preliminary assumptions to it, which he derives neither from Swyneshed nor from Heytesbury, but rather from the Logica magna: (1) on the truth and falsity, necessity and contingency of propositions;155 (2) on the convertibility of propositions: “all propositions having the same primary significate are convertible”;156

 See Paul of Pergula, In regulas (MS. Vat. lat. 2139, fol. 73vb); Paul of Pergula, Logica and Tractatus de sensu composito et diviso, 137. 155  Cf. Paul of Venice, Insolubles, §§ 2.2.1–2, and Sophismata, § 2.2.3.1. 156  Cajetan, In regulas, fol. 7va: “Omnes ille propositiones ad invicem convertuntur quarum est idem [corr., illud in textu] primarium significatum”; cf. Paul of Venice, Insolubles, § 2.2.3. 154

44 Introduction

(3) on types of formally valid inferences: from an inferior to its superior, from an expounded to its exponents or vice versa, and from a universal to the exclusive with the terms transposed and vice versa, etc.;157 (4) “A part of a proposition can supposit for the whole of which it is part and for anything related to it.”158

Subsequently, Cajetan lays down seven conclusions concerning Swyneshed’s theory and attacks them by referring to Heytesbury’s arguments, which are taken for granted and not quoted, and moreover by putting forward three further arguments, the second of which is also found in Paul of Venice.159 The first of these additional arguments is directed against Swyneshed’s claim that there are some self-referential propositions—for example, “This proposition does not signify as is the case”—that signify things to be in some way and yet neither signify principally as is the case nor as is not the case because they are relevant to inferring that they themselves do not signify principally as is the case.160 Interestingly, in his reply to this first argument, as well as in the replies to two of Heytesbury’s objections,161 Cajetan implicitly admits that Swyneshed and some of his followers have different viewpoints about propositions like “This proposition does not signify as is the case.” Whilst Swyneshed grants that there are such propositions, the alternative view does not admit propositions that neither signify as is the case nor as is not the case and considers them as insolubles. The latter position seems to have been Paul of Venice’s, who is not a faithful follower of Swyneshed on that point and does not list or mention such propositions in his division of propositions.162 Even more interesting are Cajetan’s responses to Heytesbury’s objections to Swyneshed’s opinion, since they very frequently correspond verbatim to Paul of Venice’s replies.163 Heytesbury’s first argument is directed against the common basic principle of the other three views, namely, that insolubles signify only as the words suggest, which he denies. In order to show the untenability of this tenet, Heytesbury argues that if the proposition “It is not as  Cf. Paul of Venice, Insolubles, §§ 2.2.4–2.2.5.  Cajetan, In regulas, fo. 7va: “Pars propositionis potest pro suo toto supponere aut illi pertinente”; cf. Paul of Venice, Insolubles, § 2.2.6. 159  See Cajetan, In regulas, fols. 7va–8va. Cf. conclusion 1 with Sophismata § 2.1.2, conclusion 2 with Insolubles § 2.3.5, conclusion 3 with Insolubles § 2.4.3, conclusion 4 with Insolubles § 2.3.1, conclusion 6 with Sophismata §§ 2.3.4–2.3.4.1, conclusion 7 with Sophismata §§ 2.2.3–2.2.4. And cf. argument 2 with Insolubles §§ 4.3.2.1 and ad 4.3.2.1. 160  See Swyneshed, Insolubilia, ed. Spade, 180, § 2 and 215, § 99. 161  See Paul of Venice, Insolubles, § 3.1.1 and § 3.2. 162  See ibid., §§ 2.1–2.1.8. 163  See Heytesbury, Insolubilia, ed. Pozzi, 216–26, with some passages taken almost verbatim from Paul of Venice, Insolubles, §§ ad 3.1.1, ad 3.2–3.2.1, ad 3.3, ad 3.4–3.6. 157 158

Introduction45

Socrates says it is” signifies only as its words suggest within a specific scenario, then a contradiction follows. For example, if it is granted that it is not as Socrates says it is, then one can argue: It is not as Socrates says it is, therefore it is true that it is not as Socrates says it is. … Then one argues like this: it is true that it is not as Socrates says it is, and Socrates says ⟨only⟩ that, therefore it is true that it is as Socrates says it is; and if so, then Socrates says it as it is, and then: therefore it is as Socrates says it is.164

Cajetan shows how this objection can be neutralized from Swyneshed’s viewpoint. He points out that the inference, “It is not as Socrates says it is, therefore it is true that it is not as Socrates says it is” is not valid if the term “true” is taken nominally as a term of first intention, namely, as one of the transcendentals, or as a term of second intention suppositing for a true proposition. It is valid only if “true” is taken adverbially as “truly,” as the magister understands it (secundum intellectum magistri). It is more than likely that the anonymous master referred to by Cajetan is his former master Paul of Venice, whose replies to Heytesbury’s arguments found in the Logica magna correspond verbatim to Cajetan’s. This clearly emerges, for example, in the answer just mentioned: Paul of Venice (§ ad 3.1.1) As for the confirmation: when one says, “it is not as Socrates says it is, therefore it is true that it is not as Socrates says it is,” I grant the inference and the conclusion. Then further: “it is true that it is not as Socrates says it is and Socrates says ⟨only⟩ that, therefore Socrates says what is truly so,” I deny the inference … But one should add in the premises that it is not incompatible that what Socrates said is true, which I deny.

Cajetan of Thiene As for the confirmation: when one says, “it is not as Socrates says it is, therefore it is true that it is not as Socrates says it is,” “true” is taken differently … adverbially as “truly,” according to the understanding of the master, and in that way the inference is good …. As to the first inference, when one argues: “it is true that it is not as Socrates says it is and Socrates says ⟨only⟩ that, therefore Socrates says what is truly so,” I deny the inference … But one should add in the premises that what Socrates said is not relevant to inferring that Socrates does not say what is truly so, which is false.165

164  Heytesbury, Insolubilia, ed. Pozzi, 216, § 2.0311: “Non est ita sicut Sortes dicit, ergo verum est quod non est ita sicut Sortes dicit … Arguitur ergo sic: verum est quod non est ita sicut Sortes dicit et Sortes dicit solo sic, ergo Sortes dicit sicut verum est ⟨esse⟩; et si sic, ergo Sortes dicit sicut est; et ultra: ergo est sicut Sortes dicit.” Cf. Paul of Venice, Insolubles, § 3.1.1. 165  Cajetan, In regulas, fols. 7vb–8ra: “Et ad confirmationem: cum dicitur: non est ita sicut Sor dicit ergo verum est quod non est ita sicut Sor dicit, distinguitur de li verum … adverbialiter secundum

46 Introduction

Moreover, it is from Paul, rather than from Heytesbury, that Cajetan borrows verbatim his three arguments against Dumbleton’s view.166 Thus Perreiah’s claim does not seem entirely correct, that “within 15 years of Paul’s death the text … was not on the minds of the best logicians of the day—John Anthony of Imola, Cajetan of Thiene or Paul of Pergula whose works contain no mention of the Logica magna.”167 If Cajetan offers us only implicit testimony of Paul’s authorship of the Logica magna, we can find explicit evidence in some anonymous and unfinished Recollectae preserved in the Vatican library.168 Similarly to Cajetan’s, this treatise devotes some space to the other three opinions and particularly to Swyneshed’s. Yet, contrary to Cajetan, the anonymous author proposes only Heytesbury’s objections and not Paul’s objections to Swyneshed. The first of Heytesbury’s arguments exclusively directed against Swyneshed aims to establish the contradiction that follows from accepting a scenario containing a self-referential proposition like “This proposition signifies other than it is,” call it A.169 The replies of the Recollectae’s anonymous author to it are a verbatim repetition of Paul’s response: In his Logic [sc. § ad 3.2–3.2.1], Master Paul replies to this argument by admitting the scenario and granting the insoluble and that it is not wholly as A signifies. And then when it is argued “It is not wholly ⟨as A signifies⟩ and A signifies that it is in some way, therefore A signifies other than it is,” he grants the inference and the conclusion. And when it is argued that thus B signifies consequentially as it is, he grants ⟨the inference⟩. And then he denies that A signifies only as it is, even though it signifies only as B ⟨signifies⟩, because it is inconsistent that it is true.170

When were these Recollectae composed? Were they written after the 1499 edition of the Logica magna, so that we can hypothesize that the anonymous author attributes intellectum magistri pro vere et illo modo consequentia est optima … sed prima consequentia cum arguitur sic: verum est est quod non est ita sicut Sor dicit et sor dicit sic, ergo Sor dicit sicut verum est esse, negatur consequentia, sed oporteret adere in antecedente quod dictum Sortis non est pertinens ad inferendo [sic] Sor non dicere sicut verum esse quod est falsum.” 166  Cf. ibid., fol. 8ra with Paul, Insolubilia, §§ 1.8.1–3. 167  Perreiah, Paul of Venice, 92. 168  The manuscript is MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2154, fols. 29va–34vb; see Spade, The Medieval Liar, 36 n. 13. 169  Heytesbury, Insolubilia, ed. Pozzi, 218, §§ 2.061–2; cf. Paul of Venice, Insolubilia, § 3.2–3.2.3. 170  Recollectae, Vat. lat. 2154, fol. 32va: “Ad hoc argumentum respondet Magister Paulus in logica sua admisso casu concedendo insolubile et concedendo quod non est ita totaliter sicut A significat et cum arguitur: non est ita totaliter etc et A aliqualiter esse significat, ergo A significat aliter quam est, concedit consequentiam et consequens; et cum arguitur quod tunc B significat assertive sicut est, concedit, et negat ulterius quod A significet precise sicut est, licet significet precise sicut B, quia repugnat esse verum.” Cf. Paul of Venice, Insolubilia, § ad 3.2–3.2.1.

Introduction47

the authorship of the Logica to Paul in the wake of the attribution in the edition? Unfortunately, we do not know either the dating of the anonymous Recollectae or their relative chronology as compared with the works of Paul and Cajetan. We can therefore only attempt to date them on the basis of internal and external conjectures. The work is preserved in a miscellaneous manuscript gathering tracts of varied origin composed at different times. We might notice that the Recollectae bear more similarity with Cajetan’s work than with Paul of Pergula’s In regulas. Both authors mention Peter of Mantua and Ferebrigge, but not other authors such as Angelo di Fossombrone. And, what is more relevant, both deal with the three other opinions, devoting significant space to Swyneshed’s. These similar features might hint at a close proximity in date and a common place of origin for both, namely, Padua, where Swyneshed’s view seems to have had followers among the most prominent university professors at the beginning of the fifteenth century. Paul of Venice advocated it in his Sophismata, and in the Logica magna—if we believe he was its author. But he was not its only advocate. The famous logician and philosopher Blaise of Parma (d. 1416), Paul’s predecessor in the chair of philosophy at Padua, upheld a view on insolubles derived from Swyneshed and similar to that found in the Logica magna and described by Cajetan. This emerges from reading his 68 Questiones super Tractatus Logice Magistri Petri Hispani, which represent the outcome of Blaise’s teaching activity.171 There, in fact, he subscribes to Swyneshed’s account of truth and falsehood172 and to his three conclusions, namely, that (1) a false proposition can signify only as it is: When you say that every false proposition signifies other than it is, I deny it because this proposition, “Socrates says a falsehood,” together with the scenario of insolubles, is false and yet it signifies only as it is;173

 The work is preserved in two manuscripts: MS. Venice, Biblioteca Nazionale Marciana, Lat. 6.63, and MS. Oxford, Bodleian Library, Can. misc. 471, which adds some sophisms at the end. The manuscripts probably represent two independent reportationes: the Venice manuscript seems to have resulted from Blaise’s classroom lectures at Bologna (in 1378–84), and the Oxford manuscript from his teaching at Padua in 1384–88. 172  See Blaise of Parma, Questiones super Tractatus Logice Magistri Petri Hispani, ed. Biard and Vescovini, 126–7: “Ideo dicunt sapientes et bene quod … hec propositio vel illa significat precise sicut est, nec sese falsificat, nec falsificatur per aliquam; igitur ista est vera” (“Therefore, wise persons correctly say that this or that proposition signifies only as it is and it does not falsify itself nor is it falsified by another proposition, therefore it is true”). 173  Ibid,. 378–9: “Cum dicis quod omnis propositio falsa significat aliter quam est, negatur istud, quoniam hec propositio: Sor dicit falsum, cum casu de insolubili, est falsa et tamen ipsa significat precise sicut est.” 171

48 Introduction

(2) in a formally valid inference the false can follow from the true, and (3) two mutually contradictory propositions can both be false: In the matter of insolubles many wise persons affirm this conclusion that the false follows from the true, and affirm the other ⟨conclusion⟩ that in the matter of insolubles contradictory propositions are both false at the same time. And I have also said this in the question in which I asked whether contradictory propositions are both true at the same time or false at the same time [sc. book 1, qu. 20] … In every inference in which no self-falsifying proposition occurs, the truth follows from the truth.174

Blaise subscribes to almost all the controversial main tenets of Swyneshed’s theory of paradox discussed, the only exception being the claim that there are some selfreferential propositions signifying things to be in some way which principally signify neither as is the case nor as is not the case—just as we find in Paul of Venice. 6.2  A Reassessment of Surviving Manuscript Sources 6.2.1 Parvus error in principio maximus est in fine Perreiah partly grounds his denial of Paul’s authorship of the Logica magna in palaeographical evidence. He had knowledge first-hand of the manuscripts preserving the Logica magna wholly or partially, namely: • MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2132 (M), ante 1443: the only witness preserving the Logica magna in its entirety; • MS. Venice, Biblioteca Nazionale Marciana, Lat. 6.30 (L), not before 1444: fols. 44ra–61rb contain the whole treatise on future contingents (Logica magna, Tractatus de necessitate et contingentia futurorum), as spelled out in the explicit: Explicit dubium de futuris producendis contingentibus extractum a logica magna excellentissimi viri Magistri Pauli de Veneciis ordinis heremitarum (fol. 61rb); • MS. Parma, Biblioteca Palatina, Parmense 1023 (K), not before 1459: half a column of the last folio, fol. 156vb, contains an abridgement of the beginning of the first treatise (on terms) of the Logica magna (Tractatus de terminis 2, 6, 70, 76).  Ibid. 375–8: “Multi sapientes dicunt in materia insolubilium hanc conclusionem quod ex veris sequitur falsum, et ponunt aliam que est quod in materia insolubilium contradictoria sunt simul falsa. Et hoc etiam dixi in una questione in qua querebam utrum contradictoria sint simul vera vel simul falsa [sc. I, qu. 20] … Cuiuslibet consequentie in qua non concurrit aliqua propositio se ipsa falsificans, ex vero sequitur verum.” Cf. also ibid., 156.

174

Introduction49

Perreiah conjectures that L, K and the 1499 edition (E) of the Logica magna were all copied directly from M, but he does not put forward any evidence supporting this supposition. On this hypothetical basis, he concludes that “Vaticanus Latinus 2132 is the only fifteenth century witness to the claim that Paul of Venice was the author of the Logica magna: there is no independent textual witness to that claim.”175 Such uniqueness, he adds, casts serious doubts on the authorship of the work. This conclusion, however, rests on a false premise since the genealogical relationship among the surviving witnesses of the Logica magna cannot be the one delineated by Perreiah. A relation of direct dependence of L, K, and E from M, such as Perreiah proposes, presupposes that each of the dependent copies (L, K, and E) contains all the significant errors present in M, plus at least one additional individual error for each of the copies, an error peculiar to each copy and not found in M. Perreiah’s hypothesis, however, does not meet these conditions, as we shall see. M and E. M cannot be the exemplar of E because M has individual errors not shared by E, which the editor of E could hardly have identified and corrected. In the treatise on obligations (Tractatus de obligationibus ), M has two non-homoeoteleutic 19-word omissions, one non-homoeoteleutic 14-word omission, plus other individual omissions and independently replicable errors not shared by E. Similar observations can be made about other edited treatises of the Logica magna. In the treatises “On Truth and Falsehood” (Tractatus de veritate et falsitate propositionis) and “On the Signification of the Proposition” (Tractatus de significato propositionis) M has one non-homoeoteleutic 15-word omission at 112, nos.16–17, plus at least four homoeoteleutic 15-word omissions (56, nos. 12–13; 70, nos. 18–19; 82, nos. 18–19; 196–8, n. 31), shorter omissions and significant individual errors not shared by E. In the treatise on insolubles, M has two sizeable omissions per homoeoteleuton which are not shared by E: (1) a 28-word omission at 140 (falsum est … est falsa); (2) a 23-word omission at 186 (et non … dicit et). Theoretically it is possible that the editors of E restored the missing lines, yet it is unlikely they actually did so, since they did not usually intervene with corrections and conjectures where these were needed in order to improve the inaccurate text we read in E.176 Moreover, M has 175  Perreiah, Paul of Venice, 93. It is worth noting that, among the many marginalia and notanda written by a second hand in M, in the right margin of fol. 21rb, in a comment on the view of the author of the Logica magna (Logica magna: Tractatus de suppositionibus, 16), we read “opinio .p.,” an abbreviation that might be expanded as p⟨auli⟩. Moreover, in the right margin of fol. 32va, we read, “In suo sophismate: omne co⟨lora⟩tum est, concedit iste magister propositionem huic similem esse disiunctivam,” which seems to refer to Paul’s Sophismata (Sophismata aurea, ed. 1483, sophism 3, fol. 9v). 176  Examples of this imprecision are errors such as 134: divisiones] dictiones E, 152: formetur erit vera] formaliter currit vere E; 172: ut quinta declarat conclusio] sexta conclusio declarat E, plus numerous single and multiple word omissions proper to E.

50 Introduction

some homoeoteleutic multiple-word omissions and single-word omissions that are not found in E as well as individual errors not shared with E. In light of what has been said about M’s errors, it seems highly improbable that M matches the text of the Logica magna as prepared by Paul himself. M, in fact, presents even more errors which it is unlikely Paul himself would have made. Here are some examples: Treatise on Insolubles, § 1.5: sorte intelligente] Sortem intelligere M E; § 4.1.2: est tibi dubium] ex tibi dubio M E; § 4.3.2.1: negatum] oppositum negati M E; § ad 4.3.2.1: creditum a Platone] creditum a Sorte M. Similarly, in the Tractatus de veritate et falsitate propositionis et tractatus de significato propositionis, 16, line 16 quarti] tertii M E, and in the Tractatus de obligationibus, 40, line 33: sequitur situaliter] situatur situaliter M. This conjecture seems to receive support also from two codicological data. First, at fols. 4vb–10rb (see above, § 2) the text sequence is wrong and this error in order can be explained as the result of an accident that occurred when the text was copied. Secondly, Paul’s own draft of the Logica magna seems to have consisted of sexternions (see § 6.2.2 below), while M consists of quinternions. M and L. In the Tractatus de necessitate et contingentia futurorum, at 41, lines 22–23 M omits 19 words, while L (fol. 48ra) and E have the entire text. This discrepancy is hardly explicable as an emendatio ope ingenii; moreover, at 64, lines 9–11 M has a non-homoeoteleutic15-word omission, while both L (fol. 50va) and E have the full text; then, at 91, lines 4–6 M has a non-homoeoteleutic 20-word omission, E omits only 6 words, and L has the most complete and meaningful text (generatus numquam E] generatus ex illo ligno ab A igne numquam L). Furthermore, M has other multiple-word omissions (between 11 and 15 words) per homoeoteleuton, where both L and E have the complete text, for instance at 10, lines 23–11,1 (L fol. 44vb); 20, lines 15–16 (L fol. 46ra); and 25, lines 19–22 (L fol. 46rb). At times L has a better text than M and E, for example at 70, line 1 (L fol. 51rb), 100. The explicit of L cited above (fol. 61rb) unquestionably ascribes the authorship of the Logica magna to Paul. It is very similar to what we read at the end of M: Ista est logica excellentisimi viri fratris Pauli de Veneciis. Unlike M, L specifies Paul’s membership in the Hermits and qualifies him as magister rather than as frater, which might be explained by the fact that Paul was (or had become) a renowned professor by the time L was copied. M and K (fol. 156rb, half column). The text in K is too short to conjecture any relationship with either M, L, both or neither, as can be seen from its transcription below. It consists of excerpts from the treatise on terms, specifically from sections on the definition of categorematic and syncategorematic terms, on syncategorematic terms like the adjectives, on the significative force of signs, and on the reversibility of the relation of signification between things and terms. The copyist has consistently

Introduction51

turned Paul’s direct speech (“I reply”) into indirect (“It is replied”) or reported speech (“he replies”) and, interestingly, on one occasion he has openly attributed the work to Paul (“Paul replies”). In the following transcription, references to the Tractatus de terminis appear in the margins. 2, 11–13

6, 24–7, 5

In

logica magna.

Terminus categorematicus est signum tam implicite quam explicite simplex de communi lege non extremorum aliqualiter unitivum sed alterius a se et suo consimili per se in notitiam deductivum. Terminus sincategorematicus est signum officii executivum, nullius a se et suo consimili sine nova impositione per se significativum.177 Utrum adiectiva178 possint179 esse simplici180 aut de facto181 et videtur quod non quia ly omnis et182 nullus et huiusmodi non sunt termini categorematici, ergo183 aliquod184 aliud adiectivum. Patet consequentia185 vel detur causa diversitatis. Respondet Paulus186 quod adiectiva sunt in multiplici differentia. Quedam enim non significant a se et suis similibus condistinctum187 ut omnis et nullus. Quedam significant, sed dependenter, per188 habitudinem ad alterum extremorum, ut albus,189 niger, alba et190 nigra in masculino vel in feminino genere et etiam191 in neutro genere ly192 album vel nigrum pure adiective sumptum. Tertio sunt aliqua que significant huiusmodi significata distincta193 per se et non per unum194 ad aliud ut sunt huiusmodi adiectiva et195 non tenta pure adiective sed in neutro genere

 significativum] signum E  aliqua add. M E 179  possint] possunt M E 180  simplici] termini categorematici M E 181  sint add. M E 182  et om. M E 183  ergo] igitur M E 184  aliquod] nec aliquid M E 185  Patet consequentia inv. M E 186  Respondet Paulus] Respondetur M E 187  similibus condistinctum] consimilibus distinctum M E. The syntagm “similibus condistinctum” is found in Logica magna: Tractatus de suppositionibus, 16 and 18. 188  per] propter E 189  vel add. M E 190  et] vel M E 191  etiam om. M E 192  ly] ut M E 193  distincta] sumpta M E 194  unum] respectum M E 195  et om. M E 177 178

52 Introduction

7, 23–71, 3

70, 16–17 76, 13–22

­substantive.196 Dicit197 ergo quod adiectiva primo modo sumpta non sunt termini categorematici sed sincategorematici, ut dictum est. Secundo modo sumpta sunt termini categorematici sed dependenter tales. Sicut enim significant dependenter, ita categorematici sunt dependenter. Dubium est tale198 numquid signum non significaret nisi imponeretur ad significandum. Respondetur199 quod quedam est impositio facta per signa que non sunt dictiones,200 quedam que est201 per dictiones. Primo modo conceditur202 dubium, quia tunc talis impositio fit per liberam doctrinam vel naturalem experientiam ut circulus vel folia ante tabernam sive203 latratus canum. Secundo modo204 negatur205 dubium, quia certum est quod ille206 terminus homo significat mihi hominem quando non imponitur ad significandum. Sed contra207 Philosophus208 decimo209 et primo Elengorum dicit210 quod oportet ante disputationem presupponere significationem terminorum.211 Sed hic212 oportet si tunc significaret213 sine impositione. Respondetur214 quod Philosophus intelligit215 de terminis ignotis, sic216 de A B buba217 vel pronominibus ubi nescitur quid demonstrator. 218 Utrum res possunt219 ita bene significare terminus sicut e contra, respondetur220 concedendo illud. Quia res apprehense aliquando ita faciunt venire intellectum in221 cognitionem

 substantive] substantivata M E  dicit] dico M E 198  dubium est tale] Primum M E 199  respondetur] pro primo dubio dicitur M E 200  et add. M E 201  que est om. M E 202  conceditur] concedo M E 203  sive] et E 204  modo om. M E 205  negatur] nego M E 206  ille] iste M E 207  istam responsionem arguitur sic per add. M E 208  Philosophus] Aristotelem M E 209  decimo (erratum)] quarto (correctum) E 210  dicit om. M E 211  significationem terminorum inv. M E 212  hic (dub.)] hoc non M E 213  significaret] significarent M E 214  respondetur] ad istud dicitur M E 215  intelligit] intendit E 216  sic] sicut M E 217  buba] ba bau M E 218  ad secundum dubium cum queritur add. M E 219  possunt] possint M E 220  respondetur] respondeo M E 221  in] ad E 196 197

Introduction53

et reducunt ita terminus ad memoriam, sicut e contra.222 Consequentia patet223 et antecedens probatur. Quia cum video Sortem vel Platonem quem scio significari per illum224 terminum Sortes vel Plato, ita bene intellectus meus refertur225 in ipsum terminum Sortes vel Plato, sicut quando audio nominari illum226 terminum227 ratione cuius intellectus refertur228 in cognitionem ipsius Sortis vel Platonis. Terminus prime intentionis est terminus mentalis significans rem que non est terminus dato quod esset idest posito quod illa res esset terminus non significaretur ab aliquo termino ut terminus, sed in eo quod res est ut res distingui habet contra terminum ex spiritu(?) [vel scripto?] p⟨etri?⟩ de rubeis. Deo Gracias.

It seems more consistent with the philological evidence that there are at least two independent textual witnesses (M and L) to the claim that Paul of Venice was the author of the Logica magna. To this one could add the ascription of its authorship in the short fragment in K. 6.2.2 Which Scribe Might Have Written Vat. lat. 2132? A Working Hypothesis Manuscript M itself does not offer any particular insight for solving this puzzle, since it displays only the writing of a single scribe. By a palaeographical comparison between M and more than 300 manuscripts preserving Paul of Venice’s works, Perreiah concluded that the hand of M very closely resembles that of Arnoldus Fabri of Sint-Truiden. On 29 May 1428 at Perugia, this scribe completed the latter part of MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2134, which contains the Quadratura and is in gothic script. In 1438 he wrote the MS. Vatican City, Biblioteca Apostolica Vaticana, Urb. lat. 1488 in humanistic script. For Perreiah, this fact testified to the modification of Arnoldus’s handwriting over ten years. Perreiah assumed that Arnoldus ran a scriptorium in Perugia. It would be surprising, however, that a professional scribe wrote only two manuscripts, or that we have only two manuscripts connected with Arnoldus’s workshop—and that M, being written in gothic script, lies much closer chronologically to 1428 than 1438.229  e contra] termini apprehensi reducunt res quas significant ad memoriam M E  patet] tenet M E 224  illum] istum M E 225  refertur] fertur M E. M and E have the correct reading. 226  illum] istum M E 227  Sortes vel Plato add. M E 228  refertur] fertur M E 229  See Perreiah, Paul of Venice, 91. We have used the MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2134 in part for establishing the text of the Quadratura in the Appendix. 222 223

54 Introduction

We shall attempt to present an alternative hypothesis as to the origin of M. The only possibly useful information we can gather from the manuscript itself is provided by a note of possession on the lower margin of fol. 247v: “On 29 July 1443, with great effort and difficulty, I got this book back from Master Matthew.” In our view, Perreiah correctly conjectured that the tenacious borrower of M was likely Mattheolus Perusinus. However, Perreiah erroneously depicted him as an almost unknown professor who was teaching in Perugia in 1443 and was a mere “specialist in the art of memory.” Instead, Matteolo Mattioli (d. c.1473) was a much renowned fifteenth-century physician. He obtained his degree at Padua (December 1432), and taught at Perugia, Bologna, Ferrara (1443–47), and Padua (1437–9, 1447–1451?, 1453–1473). In Padua, Matteolo was a colleague and friend of the famous physician and bibliophile Pietro Tommasi. He became acquainted with humanistic culture and such scholars of his age as Francesco Filelfo, who translated Hippocrates’s On the Passions and Errors of the Soul and On Breath. Sometime in the 1440s, Filelfo loaned his codices of Hippocrates to Matteolo, who failed to return them promptly, however. For in 1453 Filelfo was still sending requests for return of the borrowed manuscripts indirectly to Pietro Tommasi and directly to Matteolo himself, who did eventually return them to their owner. If Matteolo was the borrower of the Logica magna, it is plausible that the lender who was striving for its return in 1443 was Pietro Tommasi. Pietro spent his youth (1390s) between Venice and Padua, where he obtained his degrees in arts (1397) and medicine (1402) and where he then taught medicine. He formed close friendships with Pier Paolo Vergerio, Peter of Mantua, Anthony de Monte, Almericus of Serravalle,230 and the Hermits Paul Francis of Venice and Paul of Venice, of whom he also later became a colleague.231 Quite relevant for the history of the Logica magna and M is Pietro’s correspondence in the academic year 1396–7 (which Pietro spent in Pavia) with Paul Francis of Venice and Giovanni Ludovico Lambertazzi (d. 1401), who was Pietro’s father-in-law, a bibliophile, and professor of jurisprudence at Padua. From his epistolary, we know that Pietro was constantly informed about Paul of Venice’s latest works, and that he usually requested copies of them from Lambertazzi, who regularly let scribes make copies of philosophical works for him. In 1396, Pietro had already previewed an opuscule that Paul of Venice had completed earlier, the Logica parva, and  Anthony de Monte might be the homonymous copyist who in 1395 copied Segrave’s and Henry Anglicus’s treatises on insolubles and Heytesbury’s De sensu composito and diviso in MS. Oxford, Bodleian Library, Can. misc. 219. In that same manuscript, in 1391 Almericus of Serravalle had copied Strode’s consequences, suppositions, and his Tractatus secundus de principiis loycalibus. On this manuscript, see Maierù, “Le ms. Oxford Canonici misc. 219 et la ‘Logica’ de Strode.” 231  See Cessi, “La giovinezza di Pietro Tomasi erudito del secolo XV,” and Segarizzi, “La corrispondenza familiare di un medico erudito del quattrocento.” Paul Francis and Paul Nicoletti (our Paul) are different persons. 230

Introduction55

by the beginning of the following year he was aware that Paul was working on a new, massive work. On 17 January 1397, replying to a previous epistle of Tommasi’s, Lambertazzi informed Pietro that he had seen the first part of Paul’s new work and that Paul had given a copy to Lambertazzi so that he could pass it to Pietro: The Questions of Magister Messinus are finished, and now I am having a copy made of John of Casale’s Question and I will have both ⟨treatises⟩ bound in a single volume, so that John of Casale’s Question will be bound first and then Messinus’s Questions in the same volume, followed by two or three blank quinternions … I will send them to you either through Master Matthew, colleague of your Master John, who will soon come to Pavia, or through a servant of Master Marsilius of Santa Sophia … or through some other reliable servant. I have seen Brother Paul’s work, which will consist of thirty sexternions and, I think, fifty peciae and more. He has finished the first part, which consists of fifteen sexternions, and has begun the second part, which according to him will be similar in length to the first. The work is not corrected and he does not have time to correct it because he is busy teaching four classes per day. Eventually, persuaded by my requests, he supplied a copy of it on the proviso that you promise not to show or copy it for anyone before it is both finished and corrected. I think it will be expensive ⟨to copy it⟩, because it will be necessary to write in two columns and the text is huge, and one can find scribes only at high cost … I believe, however, the time and means to correct it will be granted, if you will plead with Brother Paul Francis in your letters.232

Lambertazzi’s description of a work of 200 or more folios subdivided into two parts having almost the same length, matches with the general features of the Logica magna. We might hypothesize that once Paul had amended the text, either ­Lambertazzi or Tommasi made a copy, and that this copy is the current MS. M.233 Sometime after 1432, when Matteolo Mattioli became magister, Tommasi might 232  Segarizzi, “La corrispondenza,” 10 (with a few corrections to the Latin): “Quaestiones magistri Mesini sunt complete, nunc vero scribi facio Questionem Iohannis de Casali, quas simul compaginari faciam in uno volumine hoc modo quia primo Questio Iohannis de Casali, postea Questiones magistri Mesini, dehinc duo vel tres quinterni vacui … in eodem volumine ligabuntur quas per magistrum Mathiam socium tui magistri Iohannis cito venturum Papiam vel per famulum magistri Marsilii … vel per alium nuncium fidelem tibi mittam. Opus fratris Pauli vidi, quod erit XXXta sexternorum et meo iudicio Lta peciarum et ultra. Complevit primam partem, que est XV sexternorum, inchoavit secundam, quam dicit futuram similem in quantitate scripture. Opus incorreptum est, nec habet tempus corrigendi propter quatuor lectiones, quibus singulis diebus intendit. Optulit tandem victus precibus meis copiam eius, ut tamen sponsio per te fieret quod nemini ostenderes, nemini copiam faceres, donec et perfectum et correctum esset. Ego quidem considero magnitudinem expense, quia oportebit binam facere scripturam et scriptura est magna, nec inveniuntur scriptores nisi caro pretio … Cogito tamen quod si instabis tuis litteris fratri Paulo Francisco, tempus et modus dabitur corrigendi.” 233  Perreiah, Paul of Venice, 116–18 rejects this hypothesis. Unlike him, we take the term corrigere in its general sense of “to amend.”

56 Introduction

have lent him the manuscript, which he could eventually have had returned on 29 July 1443 not without great efforts, similarly to Filelfo, who later struggled with having his Hippocrates returned from Matteolo. 6.3 Stylistic and Doctrinal Discrepancies between the Logica magna and Paul’s Authentic Works On the basis of a stylistic analysis, Perreiah pointed out that the Logica magna’s author often uses expressions (such as adverbial modifiers) and syntagms which one rarely encounters in Paul’s genuine works. This linguistic uniqueness of the Logica magna when compared with Paul’s authentic works led Perreiah to doubt its authenticity. But this claim is not conclusive because it rests on too small a sample: for Perreiah only provides the case for the construction “restare + verb or conjunction,” which is frequently found in the Logica magna but is almost absent in Paul’s other works.234 A further claimed peculiarity of the Logica magna is the failure to mention the author’s sources, particularly recent ones, which are often introduced anonymously (quidam, aliqui, etc.), while older authorities—such as Aristotle, Boethius, and Peter of Spain—are mentioned instead. “How can these glaring departures of the Logica magna from what is standard practice in Paul of Venice’s known works be explained?”235 An observation by Perreiah himself can help answer this question. He remarked, in fact, that Paul is silent on his sources in those texts that have a direct didactic purpose, namely, in logical works that were probably written at about the same time as the Logica magna: “with the exception of texts prepared ‘ad utilitatem scholarium’ such as the Logica parva, the Quadratura and the Sophismata aurea, all of Paul’s texts are replete with the names of his sources.”236 Therefore, Paul’s reluctance to name his modern sources in the Logica magna does not seem so extraordinary, just as his implicit references to Peter of Mantua, which play a pivotal role in Perreiah’s denialist thesis, are not particularly astonishing. Perreiah noticed a special antipathy by the author of the Logica magna to the ideas in Peter’s Logica. For Perreiah, in fact, the “concerted critique” against Peter of Mantua “is a dominant theme of the second [treatise, sc. on supposition] and it is resumed in the last [treatise, sc. on insolubles]. It exhibits a firm effort by the author of the Logica magna to define his own teachings on logic in relation to those of Peter of Mantua.”237 As we shall  See ibid., 85–7.  Ibid., 90. 236  Ibid. 237  Ibid., 102. 234 235

Introduction57

see, in the “Insolubles” the main polemical target is not Peter of Mantua but Heytesbury, to whom Paul devotes around 1000 words of attack in chapter 1 and then further space in the rest of the treatise, and Peter of Ailly, against whom Paul’s arguments go on for around 1500 words, compared with little more than 300 words on Peter of Mantua.238 Perreiah further observed that the style and tone of the criticism of Peter “suggest that Peter of Mantua and the author of the Logica magna have had a live exchange of views: ‘Ad istud argumentum respondet hic magister [Peter of Mantua] negando etc.’”239 This is a bold claim. In the Logica magna one can come across similar sentences in which, however, the author does not refer to Peter, but to other recent thinkers. In the treatise on obligations, for instance, while presenting various replies to an argument, Paul introduces one response in a way similar to Perreiah’s exemplary quotation: “Ad istud argumentum respondet unus magister dicens admittendo … Puto quod huius magistri intentio bona sit; tamen verba non recte procedunt.”240 The editor of the volume, E. J. Ashworth, proposes to identify the unus magister with either Richard Billingham or Albert of Saxony:241 whichever the unnamed master was, the author of the Logica magna (whose composition Perreiah dated to 1393–5) could hardly have had a live exchange of view with him. A strong argument Perreiah puts forward against Paul’s authorship of the Logica magna is the doctrinal discrepancy between the two Logicae. At times, indeed, the two treatises do not exactly cover the same material, and when they do, they do not always endorse the same views on specific topics. As examples one could cite the theories of proposition mentioned above, supposition, ampliation, proof of terms, consequences, and insolubles.242 Such disagreement did not pass unnoticed by fifteenth-century commentators like Cittadini nor obviously by Paul himself, who, however, does not seem over-concerned about inconsistencies among his works. This emerges from the initial warning to readers of the Sophismata: I have looked to introduce the minds of young ⟨students⟩ to this discipline [sc. logical sophisms] in the easiest and softest way. If in these things I determine something that seems to be inconsistent with my other works, this should not be ascribed to carelessness or inaccuracy or changeability of mind, but to the inclination of the crowd of students by which I am greatly overwhelmed. Thus some time ago I have decided that the common opinions, and not my own, should be expounded first, because they are easier and more  See Insolubles, chap. 1, esp. §§ 1.12 and 1.13, below.  Perreiah, Paul of Venice, 102. See Logica magna: Tractatus de suppositionibus, 64–5: “That master answers this argument by denying ….” 240  Logica magna: Tractatus de obligationibus, 320–1: “One master replies to this argument by admitting … I think that the intention of this master is correct; but his words do not do him justice.” 241  Ibid., p. 321, fn. 92. 242  See Perreiah, “A Concordance: Logica parva—Logica magna.” 238 239

58 Introduction

in use. And in this work I will not attempt to approve or reprove the sayings of earlier authors, just as I will not devote myself to defending someone in every part, but only the more useful and better doctrine.243

Readers of the Sophismata who also know the Logica parva and the Quadratura might be perplexed by reading, for example, two different solutions to insolubles in works by the same author. For they would find the widespread and dominant modified Heytesbury solution in the Logica parva and in the Quadratura, but that of Swyneshed in the Sophismata. The latter is also the view endorsed by the author of the Logica magna. At least for insolubles and obligations, therefore, the Logica magna is consistent with some teaching found in an authentic work of Paul’s. It is still possible to deny Paul’s authorship of the Logica magna and to suggest that Paul merely borrowed the material for the Sophismata from the Logica magna. But then the inconsistency problem would arise concerning the Sophismata themselves, since it would need to be explained why Paul chose to follow Swyneshed’s view rather than the modified Heytesbury solution, which he had advocated in the Logica parva and in the Quadratura. A possible way to account for the doctrinal divergences among Paul’s logical works is to appeal to the didactic context in which they were written and the different audiences they address. The Logica magna is a book of logic for intermediate or advanced students who have already been taught the rudiments of logic and thus are prepared to hear theories which might sound counterintuitive at first, to receive more in-depth explanations, and to deal with more difficult rules. However, the Logica parva is a textbook of logic for beginners which, while touching upon the typical topics treated in fourteenth-century Oxford-style summulae, does not abound in subtleties and avoids both complicated doctrines, which could confuse students, and metaphysical commitments or speculations. And, at least for insolubles, it does not seem to reflect Paul’s own view on the subject, as Paul himself admits at the end of the treatise on insolubles in the Logica parva: Notice that not everything I have said here, or in other treatises, I have said according to my own view, but partly according to the view of others, in order to enable young beginners to progress more easily.244  Paul of Venice, Sophismata aurea, ed. 1483, fol. 2ra: “Curavi iuniorum mentes ad hanc facilius, ad hanc dulcius deducere facultatem [sc. sophismata]. In quibus et si quedam determinata relinquam que aliis scripturis meis repugnare videantur non oblivioni nec inadvertentie nec animi volubilitati imponatur, sed inclinationi solum qua turba scolarium non parum afficior, cum non propria sed communia, tum quia faciliora tum quia magis in usu, in primis dudum manifestanda decrevi. Nec in hoc opere veterum dicta approbare vel reprobare conabor sicut nec per totum aliquem defensare studebo, sed solum utiliorem atque capaciorem doctrinam.” 244  Paul of Venice, Logica parva, ed. Perreiah, 150, checked against MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 5363, fol. 39rb: “Nota quod non quecumque fuit locutus hic, seu in ceteris 243

Introduction59

Paul and his students would not have been bothered by such doctrinal divergence.245 Certainly, his student Paul of Pergula would not have been surprised, for he adopted a similar strategy in his logical works. In his commentary on Strode’s consequences, which in 1496 became a curricular text in Padua, Paul admits altogether sixteen types of molecular proposition, eight affirmatives and eight negatives. He explicitly rejects the view he had adopted in his Logica, according to which there are only three main types of molecular proposition, namely, conditionals, copulatives and disjunctives. He justifies this doctrinal difference by remarking that the two works were directed to different audiences having different backgrounds, competences, and expectations: ⟨Earlier authors⟩ passed on to us only three types of molecular propositions not equivalent in signifying. And in the Summulae I followed this, in order not to create boredom in younger ⟨students⟩ because of the multiplicity, but now the truth of things has to be disclosed to intermediate and advanced students.246

6.4  An Evaluation of the Hypotheses on the Authorship of the Logica magna Having rejected Paul’s authorship of the Logica magna, Perreiah offers an alternative according to which the “original narration and first inscription” of the work occurred in Bologna between 1393 and 1395.247 On “that hypothesis Tommaso of Coderonco was very likely an important influence on, if not a principal source for, that work.”248 After 1397, he says, the second part of the Logica magna was completed and corrected, probably by Angelo of Fossombrone in Padua between 1399 and 1401.249 This hypothesis, however, presents disadvantages similar to those raised by Perreiah against Paul’s authorship. In the first chapter of the second part of the Logica magna, at fol. 129rb of MS. M, the author discusses some anonymous opinions on tractatibus, ego dixi secundum intentionem propriam, sed partim secundum intentionem aliorum, ut iuvenes incipientes proficere facilius introducantur.” We are grateful to Mark Thakkar for bringing this passage to our attention. 245  One of the present editors taught classical logic for thirty years, as part of the curriculum, and indeed published a short textbook on it (Read and Wright, Formel Logik), while his first research monograph was a plea for non-classical logic (Read, Relevant Logic). 246  Paul of Pergula, Dubia magistri Pauli Pergulensis, fol. 73va: “⟨Antiqui⟩ tradiderunt nobis tres ypotheticarum species tantum in significando non equivalentes. Et ego in summulis hoc fui assecutus ne Iunioribus fastidium afferatur ex multitudine sed hoc pro introductis et provectis iam rerum veritas aperienda erat.” Cf. Paul of Pergula, Logica and Tractatus de sensu composito et diviso, 17. 247  Perreiah, Paul of Venice, 102. 248  Ibid., 118. 249  See ibid., 102, 118–19.

60 Introduction

­ ropositions and discards the one according to which the dictum is a proposition: p “those people saying that ‘that God exists’ (deum esse) is a proposition assert a falsehood.” On the right margin of fol. 129rb, a different hand than the copyist’s—the same hand that elsewhere correctly identifies the polemical target as Peter of Mantua250—reveals that the author’s attack is directed “against Messinus” (contra Mesinum), that is to say, Professor Tommaso of Coderonco. Moreover, if Angelo of Fossombrone completed the second part of the Logica magna, and thus wrote or at least corrected the insolubles, then the issue of the doctrinal inconsistency of the author of the Logica magna would arise again. For, as we have seen, Angelo would have held two irreconcilable opinions on insolubles in his works, namely, Heytesbury’s in his treatise on insolubles and Swyneshed’s in the Logica magna. We are more inclined to accept the hypothesis that Paul is the author of the Logica magna, in favor of which there is considerably more intrinsic and extrinsic evidence than there is for any alternative. The author of the work refers to himself as an Augustinian and as “Paul.”251 Although there are some doctrinal inconsistencies between the Logica magna and Paul’s other works, nonetheless there are remarkable, even verbatim, similarities with others, as in the case of the Sophismata and the treatment of obligations in the Logica parva. Furthermore, Perreiah rules out Paul’s authorship because “the suggestion that Paul of Venice composed the Logica magna would be equivalent to the supposition today that a young American graduate student who was a teaching assistant could also write nine or ten complex books in his spare time.”252 This analogy, however, is less compelling if we substitute for the young American graduate student someone closer in time to Paul, like Walter Burley, who was an extremely prolific writer while he was bachelor (c.1297–1300/01) and teacher of arts in Oxford (1301–07), aged 23 to 32 years old, the same as Paul. Finally, all external evidence points to Paul’s authorship, such as ascriptions in colophons and acknowledgments by fifteenth-century authors, as detailed in § 6.2.1 above.

250  For identification of Peter of Mantua, see fol. 21vb top left margin: posicio petri de mantua (corresponding to Logica magna: Tractatus de suppositionibus, 52: “insurrexit quidam modernus … asserens in suis scriptis”); fol. 24va left margin: contra posicionem petri de mantua; fol. 25ra top margin: responsio suptilior ad posicionem petri de mantua. 251  In Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 95, Paul introduces the third opinion concerning the signification of propositions by saying that it “is common among the doctors of my order and is chiefly the opinion of Master Gregory of Rimini.” The relevant passages are quoted in Perreiah, Paul of Venice, 112. 252  Perreiah, Paul of Venice, 102.

Introduction61

7. Editorial Principles We have established the Latin text using the Vatican manuscript (M), which we have consulted both in situ and online,253 and the 1499 edition (E) held at the Bibliothèque nationale de France (Paris).254 Although M and E do not frequently display significant differences, when divergences arose we have generally followed the readings of M, except where the readings of E were clearly preferable. M generally presents fewer obvious errors; furthermore, when we could check the text against Paul’s sources, M agrees more often with them than does E. In a few places it was necessary to depart from both M and E. Thus, where M and E did not have a fully satisfactory variant and we could check the readings of M and E against Paul’s sources—especially in chapters 1–3—we have followed Paul’s sources when they had better readings. In a few cases, where both M and E had mistaken (or apparently mistaken) readings, we have emended the text and listed the readings of M and E in the critical apparatus. The critical apparatus records relevant variants, such as multiple- and single-word omissions, and also inversions of words or phrases. We have not noted the recurring cases in which M has ille and E iste or vice versa, or where M has igitur and E ergo and vice versa, nor differences in spelling between M and E or scribal corrections. We have preferred to adopt the medieval manuscript spellings, including e for ae, and correlarium, Sortes, and periurius for corollarium, Socrates, and periurus. On the other hand, we have adopted modern (English) punctuation as the meaning of the text requires. The section headings and division into paragraphs are ours. In translating the text, we have tried to stay as close as possible to the Latin text and to be as consistent as possible. In some cases, we have inserted words in ⟨angle brackets⟩ in order to make the translation more explicit and clearer. 8. Conspectus Siglorum, Signorum,

et

Abbreviationum

8.1  Conspectus Siglorum Sigla Codicum B: MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2133 (Quadratura) F: MS. Paris, Bibliothèque Nationale de France, lat. 6433A (Quadratura) M: MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2132 (Logica magna) P: MS. Padua, Biblioteca Universitaria, 925 (Sophismata aurea)  Available at https://digi.vatlib.it/view/MSS_Vat.lat.2132.  Available at https://gallica.bnf.fr/ark:/12148/bpt6k603439.image.

253 254

62 Introduction

V: MS. Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2134 (fols. 1ra–64vb = Sophismata aurea, fols. 65ra–161rb = Quadratura) Sigla Editionum E: Logica magna (Venice: Octavianus Scotus, by Albertinus Vercellensis, 1499) E1: Sophismata aurea (Pavia: Nicolaus Girardengus de Novis, 1483) E2: Sophismata aurea (Venice: Bonetus Locatellus, by Octavianus Scotus, 1493) Q: Quadratura (Venice: Bonetus Locatellus, by Octavianus Scotus, 1493) 8.2  Conspectus Signorum In textu latino et anglico ⟨ ⟩ = uncis acutis indicantur litterae vel verba ab editoribus addita [ ] = uncis angulatis indicantur verba ab editoribus deleta In apparatu critico … – … = a verbo vel verbis usque ad verbum vel verba (e.g.: dicit – falsum = a verbo dicit usque ad verbum falsum) 8.3  Conspectus Abbreviationum in Apparatu Critico a = prima columna a.c. = ante correctionem add. = addidit, –erunt b = secunda columna corr. = correximus dub. = dubitanter eds. = editiones fol., fols. = folium, folia inv. = invertit, –erunt (in) marg. = in margine mss. = codices om. = omisit, omiserunt om. (hom.) = omisit, omiserunt per homoeoteleuton p.c. = post correctionem r = facies folii vel paginae recta v = facies folii vel paginae versa

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Pauli Veneti Logica Magna Pars Secunda, tractatus 15

Paul of Venice, Logica Magna Part II, treatise 15

78

Latin Text

Insolubilia* Huic operi finem impositurus ultimum insinuandum sophismatum,1 quod insolubile nuncupatur, intimare propono, ut sophismatum apparentia disparere2 videatur. In principio ergo opiniones cum suis offensivis varias recitabo ut opinioni ultimo sustinende, licet non ultimo invente, maior3 adhibeatur fides. ⟨Capitulum Primum. Opiniones variae super insolubilia⟩ 1.1 Prima ergo

opinio ponit quod insolubile solvendum est per fallaciam figure dictionis, ut sorte dicente: sortes dicit falsum, dicit quod illa est falsa. Et si arguitur: sortes dicit hoc falsum, igitur sortes dicit falsum,4 negat consequentiam dicendo quod5 ibi est fallacia figure dictionis, quia in antecedente virtute demonstrationis6 supponit iste terminus falsum pro dicto a sorte,7 sed in consequente solum pro aliis. Ideo fit ibi mutatio ab hoc falso ad falsum aliud ab hoc. Quare ibi est fallacia figure dictionis. 1.1.1 Hec opinio deficit primo quia8 sorte dicente istam: sortes dicit falsum, stat quod dicat istam: homo est asinus, et sic illa non esset falsa sed vera. 1.1.2 Secundo, quia negat consequentiam qua arguitur ab inferiori ad suum superius.

 M 236ra, E 192rb] Insolubilia hic incipiunt add. M  sophismatum] sophisma E 2  disparere] disparare E, non add. M 3  maior] om. M 4  dicit hoc – falsum] om. M 5  quod] om. E 6  demonstrationis] dictionis E 7  a sorte] sortis E 8  quia] quod E *

1



English Translation

79

Insolubles To conclude this work I propose to explain the last of the sophisms to be introduced, which is called “insoluble,” so that the plausibility of sophisms is seen to disappear. So to begin with I will rehearse the various opinions with their drawbacks in order to give more credence to the opinion to be finally upheld, although it was not the last to be discovered.

⟨Chapter One: Various Opinions on the Insolubles⟩ 1.1 Then the

first opinion claims that insolubles should be solved by ⟨invoking⟩ the fallacy of form of expression, e.g., supposing Socrates says Socrates says a falsehood, ⟨the first opinion⟩ says that this proposition is false. And if one argues: Socrates says this falsehood, so Socrates says a falsehood, ⟨the first opinion⟩ denies the inference by saying that there is a fallacy of form of expression here because in the premise, by reason of the demonstrative, the term “falsehood” supposits for what was said by Socrates, but in the conclusion only for other ⟨falsehoods⟩. Hence there is a change here from this falsehood to a falsehood different from it. That is why there is a fallacy of form of expression here. 1.1.1 This opinion is defective, first, because if Socrates says Socrates says a falsehood, it could be that he ⟨also⟩ says A man is an ass, and thus it would not be false but true. 1.1.2 Secondly, ⟨it is defective⟩ because it denies an inference by which one argues from an inferior ⟨term⟩ to its superior.

80

Latin Text

1.1.3 Tertio,

quia solvit per fallaciam figure dictionis, unde dato adhuc quod ly falsum pro alio9 supponeret in antecedente et pro alio10 in consequente, adhuc non sequitur quod sit fallacia figure dictionis, sicut patet intuenti defectus consurgentes ex fallacia illa, quos recitare esset11 longum. 1.2 Secunda opinio solvit insolubilia per fallaciam secundum non causam ut causam, ita quod consequentia facta non valet quia assumitur non causa pro causa; antecedens12 (E192va) enim videtur esse causa consequentis et non est. 1.2.1 Ista opinio etiam non bene solvit per fallaciam istam, sicut habet videri intuendo defectus huius fallacie. Verumtamen,13 quia principale fundamentum huius et prime opinionis14 est quod pars non potest supponere pro toto cuius est pars, ideo posterius hoc improbando, harum15 patebit insufficientia. 1.3 Tertia opinio dicit quod sorte dicente: sortes dicit falsum, hoc verbum dicit, licet sit presentis temporis,16 debet tamen intelligi pro tempore sive17 instanti immediate precedente tempus prolationis. Ideo negat eam dicendo ipsam esse falsam. Et tunc ad argumentum: hec est falsa et sortes dicit istam, igitur sortes dicit falsum, dicunt isti quod hoc verbum dicit pro diversis temporibus verificatur in antecedente et in consequente, ideo nullum inconveniens sequitur. 1.3.1 Contra hanc opinionem arguitur sic: nichil dicit18 ante tempus prolationis sed precise in tempore prolationis, igitur illud verbum dicit non debet intelligi pro tempore vel instanti precedente tempus prolationis.  alio] aliquo E  alio] aliquo E 11  esset] essent M 12  antecedens] antecedentis M 13  Verumtamen] om. M 14  opinionis corr.] opinioni M, opinionum E 15  harum] horum E 16  temporis] om. M 17  sive] om. E 18  dicit] dicitur M 9

10

1.1.3 Thirdly,

English Translation

81

⟨it is defective⟩ because it solves ⟨the insoluble⟩ by ⟨invoking⟩ the fallacy of form of expression, for, even given that “falsehood” supposits for one thing in the premise and another in the conclusion, even so it does not follow that it is a fallacy of form of expression, as is clear to anyone who looks at the defects arising from this fallacy, which it would be tedious to rehearse. 1.2 The second opinion solves the insolubles by ⟨invoking⟩ the fallacy of false cause, so that the inference made ⟨in § 1.1⟩ is not valid, because a non-cause is taken for the cause, in that the premise seems to be the cause of the conclusion when it isn’t. 1.2.1 This opinion also fails to yield a solution by ⟨invoking⟩ this fallacy, as must be seen by reflecting on the defects identified in this fallacy. Nevertheless, because the main ground of both this and the first opinion is that the part cannot supposit for the whole of which it is a part, their deficiency will be shown by disproving this ⟨main ground⟩ later. 1.3 The third opinion says that supposing Socrates says Socrates says a falsehood, the word “says,” although it is in the present tense, must, however, be understood as referring to a period or instant immediately preceding the time of the utterance. So ⟨this opinion⟩ denies it, saying that it is false. And then in response to the argument: This is false and Socrates says it, so Socrates says a falsehood, they say that the word “says” is truly predicated for different times in the premise and in the conclusion and so nothing inconsistent follows. 1.3.1 I argue against this opinion like this: ⟨Socrates⟩ says nothing before the time of utterance but only at the time of utterance, so the word “says” should not be understood as referring to a time or instant preceding the time of utterance.

82

Latin Text

1.3.2 Item

dato quod hec solutio solveret insolubilia fundata in actu dicendi, non tamen19 solvit alia in aliis actibus fundata, ut posito20 quod sit scripta ista propositio: sortes legit falsum, quam sortes legat et nullus alius, aut quod intelligat hanc in mente sua et nullam aliam sortes, qui est omnis homo: sortes21 intelligit falsum. 1.4 Quarta opinio ponit quod nullus potest dicere se dicere falsum, nec intelligere se intelligere falsum, nec aliqua propositio potest esse ex qua insolubile (M 236rb) possit22 generari. 1.4.1 Hec opinio repugnat sensui et intellectui. Nam notum est cuilibet quod homo potest aperire os suum et formare istas voces: ego dico falsum, aut scribere23 consimiles et easdem legere, quorum quodlibet est insolubile, ut patet. 1.5 Quinta opinio ponit24 quod sorte dicente se25 ipsum dicere falsum nichil dicit, et26 sorte intelligente27 se intelligere falsum, nichil intelligit, et ita de aliis. 1.5.1 Similiter hec opinio falsa est, quia sortes sic dicens dicit litteras, sillabas, dictiones et orationes, ut alias ostendi.28 1.5.2 Preterea sortes auditur29 loqui, igitur dicit aliquid. 1.5.3 Item isti haberent dicere quod si scripta esset hec propositio30 et nulla alia: falsum est, nichil esset scriptum, quod est manifeste impossibile. 1.6 Sexta opinio ponit quod insolubile nec est verum nec falsum, sed medium indifferens ad utrumque.31

 non tamen] inv. E  posito] posita M 21  sortes] om. E 22  possit] posset E 23  scribere] sedere E 24  ponit] dicit E 25  se] in add. M 26  et] quod E 27  sorte intelligente corr.] sor intelligere E, sortem intelligere M 28  ostendi corr.] ostendidi M E 29  sortes auditur] inv. E 30  propositio] opinio M 31  utrumque] utrumlibet M 19 20

1.3.2 Again,

English Translation

83

although this solution would solve insolubles that are based on acts of saying, nevertheless it does not solve others based on other acts if, e.g., this proposition is written: Socrates reads a falsehood, which Socrates reads, and no-one else does; or if Socrates, who is the only man, understands in his mind: Socrates understands a falsehood, and no other proposition. 1.4 The fourth opinion claims that no one can say that he says a falsehood or understand that he understands a falsehood, nor can there be a proposition from which an insoluble could be generated. 1.4.1 This opinion is repugnant to the senses and the intellect: for everyone knows that a man can open his mouth and form the words: I am saying a falsehood, or write similar things and read them, any of which is an insoluble, as is clear. 1.5 The fifth opinion claims that, supposing Socrates says that he himself says a falsehood, he says nothing, and supposing Socrates understands that he understands a falsehood he understands nothing, and so on. 1.5.1 This opinion is similarly false because if Socrates says this he utters letters, syllables, words and sentences, as I have shown previously. 1.5.2 Moreover, Socrates is heard to speak, so he does say something. 1.5.3 Again, they would have to say that if this proposition: A falsehood exists, were written and no other, nothing would be written, which is clearly impossible. 1.6 The sixth opinion claims that an insoluble is neither true nor false but in the middle between the two.

84

Latin Text

1.6.1 Hii similiter

errant quia quelibet propositio est vera vel falsa, sed quodlibet insolubile est propositio, igitur quodlibet insolubile est verum vel falsum. 1.6.2 Item dicat sortes:32 ego non dico verum, et non aliud quod non sit hec vel pars eius, et queritur an sortes dicit verum vel non dicit33 verum. Si unum illorum, habeo intentum. Si neutrum, sequitur medium inter contradictoria, quod est impossibile. 1.7 Septima opinio ponit quod insolubile est solvendum per fallaciam equivocationis. Nam cum sortes34 dicit: sortes dicit falsum, distinguunt35 de ly dicere penes equivocationem. Potest enim significare dicere exercitum vel conceptum. Et vocant36 dicere exercitum quod est in exercitio, et37 est illius quod est in dici et non dictum complete;38 dicere vero conceptum dicunt cum homo prius39 dixit aliquid vel aliquale et statim post dicat se dicere illud vel tale. Verbi gratia, dicat sortes: deus est, et statim post: sortes dicit verum. Dicit ergo hec opinio quod incipiente sorte loqui, sortes dicit falsum, si dicere accipiatur pro dicere exercito, verum est;40 si pro dicere concepto, falsum est. 1.7.1 Et si arguitur: nullum falsum dicitur a sorte, et hoc41 dicitur a sorte, igitur ⟨hoc⟩ non est falsum, dicunt quod maior verificatur pro dicere concepto et minor pro dicere exercito, et ideo non concludit. 1.7.2 Sed hec solutio non valet, quia ponatur quod fiat locutio de dicere exercito et fiet communis deductio. 1.7.3 Preterea hec solutio non habet locum nisi in42 insolubilibus fundatis in actu dicendi et sic sequitur quod non est generalis solutio.  dicat sortes] inv. E  dicit] om. E 34  sortes] om. E 35  distinguunt] distinguit M 36  vocant] vocat E 37  exercitio et corr.] exercitio M, exercito et E 38  complete] completum M 39  homo prius] inv. E 40  est] om. E 41  hoc] hec E 42  in] om. M 32

33

1.6.1 They

English Translation

85

are similarly mistaken because every proposition is true or false, but every insoluble is a proposition, so every insoluble is true or false. 1.6.2 Again, suppose Socrates says I do not say a truth and nothing else which is not this or part of it, and one asks whether Socrates says a truth or does not say a truth. If one of them, I have what I wanted. If neither, it follows that there is a middle between contradictories, which is impossible. 1.7 The seventh opinion claims that insolubles should be solved by ⟨invoking⟩ the fallacy of equivocation. For when Socrates says Socrates says a falsehood, they disambiguate the word “say,” for it can mean saying as performed or as conceived. And they call a saying “as performed” when it is in the course of being performed, and it belongs to what is being said and ⟨is⟩ not ⟨yet⟩ said completely; they call a saying “as conceived,” however, when someone first said something or some sort of thing and immediately afterwards says that he says that thing or something of that sort. For example, suppose Socrates says God exists, and immediately afterwards Socrates says a truth. Then this opinion says that if Socrates begins to say Socrates says a falsehood, if “say” is taken as saying as performed, it is true, but if taken for saying as conceived, it is false. 1.7.1 And if one argues ⟨like this⟩: No falsehood is said by Socrates, and this is said by Socrates, hence ⟨this⟩ is not a falsehood, they say that the major premise is true of saying as conceived and the minor premise of saying as performed, and so ⟨the argument⟩ does not work. 1.7.2 But this solution does not work because we may assume that we are talking about saying as performed, and then the usual counterargument will still stand. 1.7.3 Moreover, this solution applies only to insolubles that are based on acts of saying, and thus it follows that it is not a general solution.

86

Latin Text

1.8 Octava

opinio ponit quod nullum insolubile est verum vel43 falsum, quia nullum tale est propositio. Quamvis enim44 quodlibet insolubile sit oratio indicativa significans sicut est aut sicut non est, eius significatio tamen ad hoc non sufficit ut ipsum45 propositio nominetur. 1.8.1 Contra istam opinionem arguitur, nam46 ex ipsa sequitur quod alique sunt due enunciationes quarum idem est significatum adequatum et tamen una est propositio et non alia. Patet assignatis istis: hoc est falsum, et: hoc est falsum, utrobique demonstrando secundam enunciationem. 1.8.2 Item sequitur quod aliqua enunciatio est non propositio et fiet propositio per solam mutationem in alio distante per mille milia.47 Probatur, nam posito quod non esset alia48 enunciatio complexa nisi ista: falsum est, et patet quod non est propositio secundum istam opinionem, quia est insolubile;49 et fiet propositio per novam (E 192vb) productionem illius: nullus deus est, nam50 per productionem illius51 illa desinit52 esse insolubilis. 1.8.3 Item sequitur quod hec non est copulativa: hoc est falsum et (M 236va) hoc non est falsum, demonstrata prima parte, nec ista est53 disiunctiva: hoc est falsum vel hoc non est falsum, eodem demonstrato. 1.9 Nona opinio ponit quod insolubile est verum vel falsum, sed non est verum neque54 falsum.

 vel] nec M  enim] omne vel add. E 45  ipsum] ipsa E 46  nam] quia E 47  milia] milearia E 48  alia] aliqua E 49  insolubile] impossibile E 50  nam] quia E 51  illius] om. E 52  desinit] desinat M 53  est] erit M 54  neque] nec E 43 44

1.8 The eighth

English Translation

87

opinion claims that no insoluble is true or false, because no insoluble is a proposition. For although every insoluble is an indicative utterance signifying as is the case or as is not the case, its signification, however, does not warrant its being called a proposition. 1.8.1 I argue against this opinion, for it follows from it that there are two utterances whose exact significate is the same and yet one is a proposition and not the other. This is shown by taking This is false and This is false, in both cases referring to the second utterance. 1.8.2 Again, it follows that some utterance is a non-proposition but becomes a proposition through a mere change in something else a thousand miles away. Proof: suppose that the only (propositionally) complex utterance was A falsehood exists. It is clear that it is not a proposition according to this opinion, since it is an insoluble, and it becomes a proposition by the fresh utterance of God does not exist, for by its utterance the first one ceases to be an insoluble. 1.8.3 Again, it follows that This is false and this is not false is not a conjunction, referring to the first part, nor is This is false or this is not false a disjunction, referring to the same thing. 1.9 The ninth opinion claims that an insoluble is true or false but is neither true nor false.

88

Latin Text

1.9.1 Contra

istam opinionem procedunt argumenta facta55 contra aliam. Arguitur tamen sic, reducendo oppositum opinionis: et sit a unum insolubile. Nam a est verum vel falsum, sed a non est verum, igitur a est falsum. Consequentia tenet a toto disiuncto sine limitatione precedente ad alteram56 partem. 1.10 Decima opinio solvit insolubilia per fallaciam secundum quid et simpliciter, dicens quod insolubile est difficilis paralogismus57 secundum quid et simpliciter ex reflexione alicuius actus supra se cum determinatione privativa vel negativa proveniens. Deinde in solvendo insolubilia58 dicit quod hec consequentia non valet: hoc falsum dicitur a Sorte, igitur falsum dicitur a Sorte, dato quod Sortes dicat consequens et non aliud quod non sit pars eius, quia arguitur a secundum quid ad simpliciter, quia antecedens solum cathegorice significat, consequens vero ypothetice quia significat se esse verum et se esse falsum. 1.10.1 Contra hanc opinionem arguitur primo contra diffinitionem. Nam insolubile non est syllogismus nec paralogismus,59 sed sola propositio cathegorica vel ypothetica non syllogistica. 1.10.2 Secundo, contra solutionem arguitur sic: nam universaliter sequitur: hic homo currit, igitur homo currit; hoc verum est propositio, igitur verum est propositio. Igitur per idem universaliter sequitur: hoc falsum dicitur a Sorte, igitur falsum dicitur a Sorte. Et confirmatur nam iste terminus hoc falsum est terminus discretus, igitur habet terminum communem illativum, sed non alium quam illum terminum60 falsum, igitur propositum. 1.10.3 Tertio, nam si illa consequentia non valet, stet oppositum consequentis cum antecedente et stabunt ista simul hoc falsum dicitur a Sorte, et tamen nullum falsum dicitur a Sorte, quod non videtur possibile.  facta] om. E  alteram] aliam E 57  paralogismus] persogismus M, propter add. E 58  insolubilia] om. E 59  paralogismus] persogismus M 60  terminum] tantum E 55 56

1.9.1 The arguments

English Translation

89

made against the other opinion work against this one. But I argue ⟨in addition⟩ by deducing the opinion’s opposite like this: let A be an insoluble. Then A is true or false, but A is not true, so A is false. The inference holds from the ⟨topic from the⟩ disjunctive whole, absent the prefixed determination, to the other. 1.10 The tenth opinion solves insolubles by ⟨invoking⟩ the fallacy of the restricted and unrestricted, saying that an insoluble is a difficult paralogism of the restricted and unrestricted resulting from the reflection of some act on itself with a privative ⟨or negative⟩ determination. Whence in solving insolubles it says that this inference is not valid: This falsehood is said by Socrates, so a falsehood is said by Socrates, given that Socrates says the conclusion and nothing else which is not part of it. For this is to argue from the restricted to the unrestricted, for the premise only signifies in a non-compound way but the conclusion in a compound way, since it signifies itself to be true and itself to be false. 1.10.1 Against this opinion, I argue first against the definition. For an insoluble is not a syllogism nor a paralogism but a single subject-predicate or compound non-syllogistic proposition. 1.10.2 Secondly, I argue against the solution like this: for these are valid universally: This man is running, so a man is running; This truth is a proposition, so a truth is a proposition. So by parity of reasoning this is valid universally: This falsehood is said by Socrates, so a falsehood is said by Socrates. This is confirmed since the term “this falsehood” is a discrete term, so there is a general term that allows an inference, but none other than this term “falsehood,” so ⟨we have⟩ what ⟨we⟩ wanted. 1.10.3 Thirdly, because if that inference is not valid the opposite of the conclusion would be consistent with the premise, and we could have at the same time that this falsehood is said by Socrates and yet no falsehood is said by Socrates, which does not seem to be possible.

90

Latin Text

1.11 Undecima

opinio favens immediate opinioni ponit quod omnis propositio insolubilis significat se esse veram et se esse falsam, intelligendo de adequato significato. Nam, ut dicit, omnis propositio cathegorica significat se esse veram, quia61 omnis propositio cathegorica significat idem esse vel non esse pro quo supponit subiectum et predicatum, et idem esse62 vel non esse est propositionem affirmativam vel negativam esse veram. Igitur omnis propositio cathegorica sive affirmativa sive negativa significat se ipsam esse veram, et omnis propositio insolubilis falsificat se, ideo omnis propositio insolubilis significat se esse veram et se63 esse falsam. 1.11.1 Contra istam opinionem argutum est in primo capitulo tractatus64 de veritate et falsitate. Verumtamen adhuc arguitur sic: nam data illa opinione, sequitur quod propositio possibilis est propositio impossibilis. Probatur: et65 capio istam propositionem: hoc non est verum, eodem demonstrato, et patet per opinionem quod ipsa est possibilis; et quod ipsa sit impossibilis66 sequitur ex suis dictis, quia significat hoc non esse verum et hoc esse67 verum, quod est impossibile. Et sic sequitur quod quelibet propositio insolubilis implicat contradictionem et est contradictionis illativa, quod non admittit hec positio. Et68 consequentia patet ex eadem ⟨ratione⟩. 1.11.2 Item sequitur quod hec consequentia est bona: hoc non est verum, igitur hoc est verum, eodem demonstrato. Probatur, nam iste propositiones convertuntur: hoc non est verum, et: hoc non est verum et hoc est verum, quia idem est significatum utriusque, sed ex illa copulativa sequitur quod hoc est verum, igitur similiter ex illa cathegorica: hoc non est verum.

 omnis propositio insolubilis – quia] om. (hom.) E  esse] est add. E 63  et se] vel E 64  capitulo tractatus] inv. M 65  et ] quia E 66  impossibilis] probatur quia add. E 67  esse] est E 68  et] om. E 61 62

1.11 The eleventh

English Translation

91

opinion, agreeing with the previous opinion, claims that every insoluble proposition signifies itself to be true and itself to be false, understood as referring to its exact significate. For, as it says, every subject-predicate proposition signifies itself to be true because every subjectpredicate proposition signifies that what its subject and predicate supposit for is or is not the same thing, and ⟨for them⟩ to be the same or not to be the same thing is ⟨what it is⟩ for an affirmative or negative proposition to be true. Therefore, every subject-predicate proposition, whether affirmative or negative, signifies itself to be true, and every insoluble proposition falsifies itself, so every insoluble proposition signifies itself to be true and itself to be false. 1.11.1 I argued against this opinion in the first chapter of the treatise On the Truth and Falsity ⟨of propositions⟩. However, I also argue ⟨against it⟩ like this: for given this opinion, it follows that a possible proposition is an impossible proposition. Proof: I take this proposition: This is not true, referring to itself, and it is clear according to this opinion that this proposition is possible; and that it is impossible follows from what it says because it signifies that this is not true and that this is true, which is impossible. And thus it follows that any insoluble proposition involves a contradiction and implies a contradiction, which this position does not allow. The inference is clear for the same ⟨reason⟩. 1.11.2 Again, it follows that this inference holds: This is not true, therefore this is true, referring to the same thing. Proof: for these propositions are convertible: This is not true and This is not true and this is true, because the significate of each is the same. But from the conjunctive one it follows that this is true, so similarly from the subject-predicate one: “This is not true,” ⟨it follows that this is true⟩.

92

Latin Text

1.11.3 Item

sequitur quod ista non sunt contradictoria: hoc est falsum, et: hoc non (M 236vb) est falsum, eodem demonstrato, quia prima significat hoc esse falsum et hoc esse verum et secunda significat hoc non esse falsum et hoc esse verum; sed ista non contradictoriantur:69 hoc esse falsum et hoc esse verum, et: hoc non esse falsum et hoc esse verum, nec in se nec in suis signis,70 igitur etc.71 1.11.4 Alia argumenta possent fieri que dimitto gratia brevitatis.

⟨Opinio Hentisberi⟩ 1.12.1 Duodecima

opinio, que iam communiter ab omnibus72 sustinetur, est ista: quod propositio insolubilis est propositio de qua fit mentio in aliquo casu que, si cum eodem casu precise significet sicut termini pretendunt, sequitur ipsam esse veram et ipsam esse falsam. Verbi gratia, si ponatur casus de insolubili et non ponatur qualiter illud insolubile debeat significare, respondendum est omnino sicut extra tempus, sicut si ponatur73 quod sortes dicat illam: sortes dicit falsum, nullo alio posito, proposita dubitanda est ista: sortes dicit falsum. 1.12.1.1 Si autem ponitur quod insolubile significet sicut termini pretendunt ⟨non tamen precise⟩,74 admittitur casus et conceditur insolubile, et dicitur ipsum esse falsum.

 contradictoriantur] contrariantur E  signis] significatis E 71  etc] etiam E 72  iam communiter ab omnibus] ab omnibus communiter M 73  ponatur] poneretur M 74  non tamen precise corr.] om. M E; we follow Heytesbury (Pironet, “William Heytesbury and the treatment of Insolubilia,” 285, 292) 69 70



English Translation

93

1.11.3 Again,

and

it follows that these are not contradictories: This is false

This is not false, referring to the same thing, because the first signifies that this is false and that this is true, and the second signifies that this is not false and that this is true. But That this is false and that this is true and That this is not false and that this is true are not contradictories, either in themselves or in their signs, so etc. 1.11.4 Other arguments could be given, which I omit for the sake of brevity. ⟨Heytesbury’s Opinion⟩ twelfth opinion, which is now generally maintained by everyone, is that an insoluble proposition is a proposition mentioned in some scenario such that, if in that scenario it signifies only as the terms suggest, it follows that it is true and that it is false. For instance, if a scenario concerning an insoluble is put forward and it is not stipulated how that insoluble should signify, one should respond in altogether the same way as outside the time ⟨of the obligation⟩: e.g., if it is stipulated that Socrates says Socrates says a falsehood, nothing else being stipulated, when Socrates says a falsehood is proposed it should be doubted. 1.12.1.1 But if it is stipulated that the insoluble should signify as the terms suggest ⟨but not necessarily only in that way⟩, the scenario ⟨should be⟩ admitted and the insoluble ⟨should be⟩ granted, and it ⟨should be⟩ said that it is false.

1.12.1 The

94

Latin Text

1.12.1.2 Et si

dicitur: hec est falsa: sortes dicit falsum, igitur significat sicut non est, sed significat (E 193ra) quod sortes dicit falsum, igitur non est ita quod sortes dicit falsum,75 negatur consequentia; sed oportet addere in minori quod significet precise sic; quod76 si ponitur negatur omnis talis casus. 1.12.2 Contra descriptionem datam insolubilis77 arguitur: nam sequitur quod nulla est propositio insolubilis sine casu; consequens est78 falsum, quia stat nullo79 casu posito quod ista: Sortes dicit falsum, dicatur a sorte significando sicut termini pretendunt et nulla alia que non sit pars illius. Sed tunc esset insolubilis propositio, igitur etc. 1.12.2.1 Item illa mentalis: omnis propositio est falsa, nullo casu posito, est insolubilis quia significat quod omnis propositio est80 falsa cum sit propositio naturaliter significans. Igitur etc. 1.12.3 Contra illud81 dictum, videlicet non admittendo casum de insolubili significante precise etc., quero quid intelligit per ly precise: aut notam exclusionis exclusive tentam, aut idem quod principaliter, primarie vel adequate. 1.12.3.1 Si dicitur82 primo modo, sequitur per idem quod nulla alia propositio significat precise taliter83 vel taliter esse, quia quelibet infinita84 significat, et per consequens non deberet85 negari ille casus solum in materia insolubilium, verum etiam86 in qualibet alia materia. Sed forte dicitur quod quelibet propositio insolubilis significat necessario duobus modis, propositio autem non insolubilis precise uno modo.  igitur non est – falsum] ergo etc. E  quod] et E 77  insolubilis] de insolubili E 78  est] om. E 79  nullo] uno E 80  est] sit E 81  illud] aliud E 82  dicitur] quod add. M 83  precise taliter] inv. E 84  infinita] indefinita E 85  deberet] debet E 86  etiam corr.] et M E 75 76

1.12.1.2 And

English Translation

95

if one says: ‘Socrates says a falsehood’ is false, so it signifies as is not the case, but it signifies that Socrates says a falsehood, so it is not the case that Socrates says a falsehood, the inference is denied; but one should add in the minor premise that it signifies only in that way. If that is stipulated every such scenario ⟨should be⟩ rejected. 1.12.2 I argue against the given definition of an insoluble: for it follows ⟨from it⟩ that there is no insoluble proposition without a scenario. The conclusion is false for it is possible that although no scenario is proposed, still the proposition Socrates says a falsehood is said by Socrates signifying as the terms suggest, and ⟨that he says⟩ no other proposition which is not part of it. But then there would be an insoluble proposition. Hence ⟨there is an insoluble proposition without any scenario⟩. 1.12.2.1 Again, the mental proposition Every proposition is false, with no scenario put forward, is an insoluble proposition because it signifies that every proposition is false since it is a proposition signifying naturally. Hence ⟨there is an insoluble proposition without any scenario⟩. 1.12.3 Against what he said, namely, “by not admitting a scenario where an insoluble signifies only etc.,” I ask what he understands by “only”: either a sign of exclusion taken exclusively, or the same as “principally,” “primarily” or “exactly.” 1.12.3.1 If it is meant in the first sense, it follows by parity of reasoning that no other proposition signifies only that it is in this or that way, because any proposition signifies indefinitely many things, and consequently the scenario should not be rejected in the matter of insolubles alone, but also in any other matter. But perhaps someone will say that any insoluble proposition necessarily signifies in two senses, but a non-insoluble proposition in only one sense.

96

Latin Text

1.12.3.1.1 Contra

istam responsionem arguitur sic: ex illa responsione sequitur quod aliqua propositio significat uno modo et significabit duobus modis per solam mutationem alterius propositionis. Patet consequentia dato quod non87 sint nisi iste due: falsum est, et: nullus deus est, et in fine huius hore negativa desinat88 esse. Quo posito patet quod illa: falsum est, non significat nisi uno modo, cum non sit insolubilis; sed in fine hore erit insolubilis, igitur habebit duo significata et hoc per solam89 mutationem alterius. 1.12.3.1.2 Secundo sic: aut igitur propositio insolubilis significat duobus modis iuxta viam undecime opinionis, aut per modum prioris et posterioris, primarii significati et secundarii. Non potest dici primo modo quia reprobatum est ibidem. Si dicitur secundo modo, capio illud insolubile: hoc est falsum, quod significet hoc esse falsum, et alio modo, quod sit a, et primum significatum b, et arguitur sic: significatum a est omnino ignotum tam opponenti quam respondenti, quia non est aliquis qui sciat ipsum intelligere vel nominare, igitur ratione ipsius non debet propositio dici90 vera vel falsa. 1.12.3.1.3 Item sic: aut91 a est principalius quam b vel e contra, vel sunt92 eque principalia. Non potest dici primum, quia b egreditur a significatis93 adequatis (M 237ra) partium, sicut ostensum est in tractatu de veritate et falsitate. Si secundum, igitur illud insolubile est verum, quia b est verum et94 principalius significatum, modo a principaliori debet denominari propositio talis vel talis. Si tertium, sequitur illud insolubile esse simul95 verum et falsum, quia non videtur ratio quare sit falsum ratione a et non verum ratione b, ex quo sunt eque principalia.

 dato quod non] non dato quod E  desinat] desinet E 89  per solam] solum per E 90  propositio dici] inv. E 91  aut] vel E 92  sunt] om. M 93  a significatis] assignatis E 94  et] est add. E 95  esse simul] inv. E 87 88

1.12.3.1.1 Against

English Translation

97

that response I argue like this: from that response it follows that some proposition signifies in one sense and will signify in two senses by a mere change in another proposition. The inference is clear given there are only these two ⟨propositions⟩: A falsehood exists and God does not exist, and at the end of one hour the negative ⟨one⟩ ceases to exist. On that assumption, it is clear that ⟨at first⟩ A falsehood exists signifies only in one sense since it is not an insoluble proposition, but that at the end of the hour it will be an insoluble proposition and so will have two significates and this by the mere change in the other proposition. 1.12.3.1.2 Secondly, ⟨I argue against it⟩ like this: therefore either an insoluble proposition signifies in two ways as in the eleventh opinion, or by means of a prior and a posterior, ⟨that is,⟩ a primary significate and a secondary. It can’t be said in the first way because that was disproven there. If it is said in the second way, I take the insoluble: This is a falsehood, which signifies this to be a falsehood, ⟨call this⟩ primary significate B, and in another sense, call it A, and I argue like this: significate A is altogether unknown both to the opponent and to the respondent, because there is no-one who knows how to understand or name it, so on account of ⟨A⟩ the proposition should not be called true or false. 1.12.3.1.3 Again, ⟨I argue against it⟩ like this: either A is more principal than B or vice versa, or they are equally principal. It cannot be said to be the first because B is generated from the exact significates of the parts, as was shown in the treatise On the Truth and Falsity ⟨of propositions⟩. If the second, then the insoluble is true because B is true and ⟨it is⟩ the more principal significate; yet the proposition should take its attributes from the more principal. If the third, it follows that the insoluble is both true and false, because there seems to be no reason why it is false by reason of A and not true by reason of B on the assumption that they are equally principal.

98

Latin Text

1.12.3.2 Sed96

forte dicitur in principio quod precise idem est quod adequate, principaliter 97 seu primarie, ita quod nulla propositio insolubilis significat adequate98 sicut termini pretendunt, cum hoc non sit possibile quia sequitur contradictio, sed significat adequate aliud99 significatum falsum quod non assignatur in arte insolubilium. 1.12.3.2.1 Contra istud secundum membrum arguitur sic. Si ista propositio: falsum est, foret in scripto et multe alie propositiones forent cum100 illa, possibile foret 101 quod aliquis homo videret ipsam; ponatur igitur quod multe alie propositiones102 sint cum illa: falsum est, et quod ipsa principaliter ex impositione significet quod falsum est; sit sortes unus homo qui continue per totam vitam suam fuerit103 instructus ad concipiendum per illam: falsum est, quod falsum est. Isto posito, pono quod sortes videat hanc propositionem in scripto et corrumpatur quelibet alia, quo posito sortes concipit per illam falsum esse,104 igitur illa adhuc sibi representat falsum esse, et per consequens non desinebat sic significare principaliter. 1.12.3.2.2 Secundo: talis propositio:105 deus est, principaliter significaret106 deum esse si nulla alia esset, et illa similiter: homo est asinus,107 hominem esse asinum si nulla alia esset, quare ergo illa: falsum est, non posset principaliter significare falsum esse si nulla alia esset?

 Sed] om. M  adequate principaliter] inv. M 98  adequate ] precise E 99  aliud] illud E 100  cum] illis add. M 101  foret] esset E 102  alie propositiones] inv. M 103  fuerit] fuit M 104  esse] est E 105  propositio] om. E 106  significaret] significat M 107  asinus] nullum add. E 96 97

1.12.3.2 But

English Translation

99

perhaps someone will say at the start ⟨in § 1.12.3⟩ that “only” is the same as “exactly,” “principally” or “primarily,” so that no insoluble proposition signifies exactly as the terms suggest, since this is not possible because a contradiction follows, but it signifies exactly another false significate which is not specified in the discipline of insolubles. 1.12.3.2.1 Against this second response I argue like this: if the proposition: A falsehood exists were written down, and many other propositions with it, it would be possible that some man should see it. Assume, therefore, that there are many other propositions with A falsehood exists and that it principally signifies by imposition that a falsehood exists. Let Socrates be one man who continuously for his whole life was taught to understand by A falsehood exists that a falsehood exists. Given this ⟨scenario⟩, assume that Socrates sees this proposition written down and every other is destroyed. Supposing this, Socrates conceives by it that a falsehood exists, so it still represents to him that a falsehood exists and consequently has not ceased to signify principally in this way. 1.12.3.2.2 Secondly, the proposition God exists would principally signify that God exists if there were no other proposition, and similarly A man is an ass that a man is an ass if there were no other, so why could A falsehood exists not signify principally that a falsehood exists if there were no other?

100

Latin Text

1.12.3.2.3 Tertio

arguitur sic: cuilibet propositioni extra tempus obligationis assignatur suum adequatum significatum oraculo vive vocis, ut patet in omnibus discurrendo, ergo sequitur quod illa propositio: hoc est falsum, se ipsa demonstrata, aut ista: omnis propositio est falsa, habet significatum explicite assignandum. Sed non videtur aliud posse explicite assignari nisi hoc, hoc esse108 falsum aut omnem propositionem esse falsam, igitur etc. 1.12.4 Ex ista opinione sequitur quod aliqua est propositio cathegorica109 vera que adequate aliqualiter esse significat110 et non potest habere contradictorium opposito111 modo adequate significans. Patet de ista pro-(E 193rb)-positione: aliqua propositio non est falsa, que vera est adequate significando aliquam propositionem non esse falsam. Et tamen hec que est sua contradictoria: omnis propositio est falsa, non adequate significat omnem propositionem esse falsam quia est propositio insolubilis. 1.12.5 Ulterius arguitur contra modum solvendi insolubilia, nam ponatur quod sortes dicat illam: falsum dicitur, et non proferatur ab aliquo aliqua propositio nisi illa, vel pars eius, que significet falsum dici et non precise, sicut ponit hec opinio. Tunc propono: falsum dicitur, et conceditur secundum viam illam,112 dicendo tamen113 ipsum esse falsum. Sed contra: sequitur: falsum dicitur114 principaliter significat deum esse, igitur falsum dicitur115 est propositio necessaria. Consequentia tenet, scita a te esse bona, antecedens est dubitandum a te, igitur116 consequens non est a te negandum.117 Antecedens patet quia cum toto casu est tibi dubium an illa: falsum dicitur118, significet119 principaliter deum esse.  esse] est E  cathegorica] om. E 110  adequate – significat] aliqualiter esse significat adequate E 111  opposito] aliquo E 112  viam illam] inv. E 113  tamen] tantum M 114  dicitur corr.] est M E; we follow Peter of Mantua, Insolubilia, O4va 115  dicitur corr.] est M E; ista propositio in Peter of Mantua, loc.cit. 116  igitur] et add. E 117  a te negandum] negandum a te E 118  dicitur corr.] est M E 119  significet] significat E 108 109

1.12.3.2.3 Thirdly,

English Translation

101

I argue like this: every proposition’s exact significate is assigned to it outside the time of the obligation by oral pronouncement, as is clear by running through all cases, so it follows that this proposition: This is false, referring to itself, or that the proposition: Every proposition is false, has a significate that has to be explicitly assigned. But it seems that the only one that can be explicitly assigned is that this is false, or that every proposition is false, hence etc. 1.12.4 From this opinion it follows that there is a true subject-predicate proposition which exactly signifies it to be in some way and which cannot have a contradictory signifying exactly in the opposite way. This is clear in the case of the proposition: Some proposition is not false, which is true exactly signifying that some proposition is not false. But its contradictory: Every proposition is false does not exactly signify that every proposition is false because it is an insoluble proposition. 1.12.5 I argue further against ⟨this⟩ way of solving insolubles, for assume that Socrates says A falsehood is said, and it or part of it is the only proposition said by anyone, and let it signify that a falsehood is said though not only ⟨that⟩, as this opinion claims. Then I propose A falsehood is said, and it is granted according to that view, by saying, however, that it is a falsehood. But on the contrary, this is valid: “A falsehood is said” principally signifies that God exists, therefore “A falsehood is said” is a necessary proposition. The inference holds, is known by you to be good, the premise should be doubted by you, so the conclusion should not be denied by you. The premise is clear because given the whole scenario it is in doubt for you whether A falsehood is said signifies principally that God exists.

102

Latin Text

1.12.5.1 Deinde

argumentum quod movet hec opinio ad probandum quod illa sit sequens: falsum dicitur,120 non valet, scilicet121 sortes dicit illam: falsum dicitur, que significat falsum dici et nulla propositio dicitur nisi illa vel pars eius, igitur falsum dicitur, sicut non sequitur: hec est vera: homo currit, que significat sortem122 currere, igitur (M 237rb) sortes currit. ⟨Opinio Petri de Alliaco⟩







1.13.0 Tertiadecima

opinio ponit plura puncta, aliqua per modum conclusionum, aliqua per modum suppositionum, aliqua per modum propositionum vel correlariorum. Sed omnia illa breviter ponam per modum conclusionum et correlariorum. 1.13.1 Prima conclusio est ista: nulla res creata potest distincte representare seipsam formaliter sed bene obiective. Patet quia nulla res creata potest esse propria et distincta cognitio formalis sui ipsius, quia si aliqua esset, et quelibet esset cum non esset maior ratio de una quam de alia. Verbi gratia, dicimus quod ymago regis significat regem non quidem formaliter sed obiective. Conceptus vero mentalis quem nos habemus de rege significat regem non quidem obiective sed formaliter quia est formalis cognitio regis. Si autem debet se representare distincte, hoc est123 obiective per aliam noticiam et non formaliter per seipsam.124 1.13.2 Secunda conclusio: nulla propositio mentalis proprie dicta potest significare seipsam esse veram vel125 seipsam esse falsam. Probatur quia aliter sequeretur quod aliqua cognitio ⟨creata⟩ esset propria et distincta cognitio126 formalis sui, quod est contra primam conclusionem.  dicitur corr.] est M E  scilicet] om. M 122  sortem] hominem E 123  est] erit E 124  formaliter per seipsam] per seipsam formaliter E 125  vel] nec E 126  aliqua cognitio –cognitio corr.] aliqua cognitio propria et distincta esset cognitio M E; cf. Peter of Ailly, Conceptus et insolubilia, c1vb 120 121



English Translation

103

1.12.5.1 From

here, the argument which this opinion gives to show that A falsehood is said follows is not valid, namely that Socrates says A falsehood is said, which signifies that a falsehood is said, and no proposition is said except that or part of it, therefore a falsehood is said, just as this is not valid: “A man is running,” which signifies that Socrates is running, is true, therefore Socrates is running. ⟨Peter of Ailly’s Opinion⟩

1.13.0 The thirteenth opinion makes many points, some by way of conclusions,

some by way of assumptions, some by way of propositions or corollaries. But I will make them all briefly by way of conclusions and corollaries. 1.13.1 The first conclusion is this: no created thing can distinctly represent itself formally, but rather as an object. This is shown because no created thing can be a proper and distinct formal cognition of itself, for if something were, then anything would be since there would be no more reason for one than for another. For example, we say that an image of the king signifies the king not in fact formally but as an object. However, a mental concept which we have of the king signifies the king not in fact as an object but formally because it is a formal cognition of the king. But if it must represent itself distinctly this is as an object by another conception and not formally through itself. 1.13.2 Second conclusion: no strictly mental proposition can signify itself to be true or itself to be false. Proof: because otherwise it would follow that some created cognition was a proper and distinct formal cognition of itself, which is contrary to the first conclusion.

104

Latin Text

1.13.2.1 Correlarium:127

Ex ista conclusione sequitur quod intellectus non potest formare propositionem universalem mentalem proprie dictam significantem omnem propositionem mentalem esse falsam, sicut istam mentalem: omnis propositio mentalis est falsa, intelligendo subiectum supponere pro seipsa, nec aliquam mentalem proprie dictam significantem aliquam aliam esse falsam que significat eandem128 esse falsam, sicut istam mentalem:129 hoc est falsum, demonstrando istam aliam: hoc est falsum, que demonstrat primam, nec etiam mentalem proprie dictam que significet suam contradictoriam esse veram, sicut istam mentalem: hoc est verum, demonstrando suam contradictoriam. 1.13.2.2 Etiam sequitur quod intellectus non potest formare aliquam propositionem mentalem proprie dictam130 ypotheticam cuius una pars significet ipsam131 ypotheticam esse falsam sicut talem:132 deus est et hec133 copulativa est falsa, demonstrando ipsam copulativam. 1.13.3 Tertia conclusio est ista: pars propositionis mentalis proprie dicte non potest supponere nec134 pro ipsa propositione cuius est pars nec pro contradictorio eiusdem propositionis, nec pars propositionis significantis ad placitum potest supponere pro propositione mentali proprie dicta sibi correspondente nec aliqua alia ex qua sequitur propositionem mentalem significare se esse veram aut se esse falsam. 1.13.3.1 Ex quo135 sequitur quod si illa mentalis formaretur et nulla alia: omnis propositio mentalis est universalis, ipsa esset falsa.

 Correlarium] marg. M, om. E  eandem] eadem M 129  istam mentalem] ista mentalis E 130  proprie dictam] om. E 131  ipsam] om. M 132  talem corr.] talis M E 133  hec ] ipsa E 134  nec] om. E 135  Correlarium] marg. M 127 128

1.13.2.1 Corollary:

English Translation

105

from this conclusion it follows that the intellect cannot formulate a universal strictly mental proposition signifying that every mental proposition is false, e.g., the mental proposition: Every mental proposition is false, understanding the subject to supposit for itself; nor a strictly mental proposition signifying that some other proposition is false that signifies it in turn to be false, e.g., the mental proposition: This is a falsehood referring to another ⟨mental⟩ proposition: This is a falsehood which refers to the first; nor even a strictly mental proposition which signifies its contradictory to be true, e.g., the mental proposition: This is a truth referring to its contradictory. 1.13.2.2 It also follows that the intellect cannot formulate a compound strictly mental proposition part of which signifies the compound proposition to be false, e.g., God exists and this conjunction is false, referring to that conjunction. 1.13.3 The third conclusion is this: part of a strictly mental proposition cannot supposit for the proposition of which it is part nor for the contradictory of the same proposition, nor can part of a proposition signifying at the pleasure of the impositor supposit for the strictly mental proposition corresponding to it, nor for any other from which it follows that a mental proposition signifies itself to be true or itself to be false. 1.13.3.1 From this it follows that if the mental proposition: Every mental proposition is universal were formed and no other, it would be false.

106

Latin Text

conclusio: propositio vocalis vel scripta vel etiam136 mentalis improprie dicta potest habere137 reflexionem supra se, quia138 quelibet talis significat ad placitum et non naturaliter, obiective et non formaliter; sed mentalis proprie dicta est signum naturaliter representans et formaliter nec est in potestate nostra ⟨per⟩ tale signum significare quod volumus sicut est de signo vocali vel139 scripto vel mentali140 improprie dicto. 1.13.4.1 Correlarium:141 Ex ista conclusione sequitur quod omnis propositio insolubilis est propositio vocalis, scripta vel mentalis improprie dicta et quod cuiuslibet talis pars potest supponere pro toto cuius est pars. 1.13.5 Quinta conclusio est ista:142 cuilibet143 propositioni insolubili correspondet aliqua propositio mentalis proprie dicta vera et aliqua alia144 proprie dicta falsa. Patet de ista: (E 193va) hoc est falsum, seipso demonstrato, que correspondet uni tali mentali: (M 237va) hoc est falsum, eodem demonstrato, que vera est. Et145 secunda pars probatur, nam illa vocalis est falsa, igitur significat aliquam mentalem esse falsam, sed non istam dictam, igitur aliam videlicet: hoc est falsum, demonstrando primam mentalem demonstrantem vocalem vel scriptam.146 1.13.5.1 Ex ista conclusione sequuntur aliqua correlaria: primum quod quelibet propositio insolubilis et similiter eius contradictoria est propositio plures quia sibi correspondent plures mentales inconiuncte. 1.13.5.2 Secundum correlarium: alique sunt propositiones omnino similes in voce et de terminis pro eodem147 supponentibus quarum una est propositio plures et non alia. Patet de istis: hoc est falsum, et: hoc est falsum, ubi per utrumque hoc demonstretur secunda propositio.

1.13.4 Quarta

 etiam] esset E  potest habere] que haberet E 138  quia] et M 139  vel] om. M 140  vel mentali] om. M 141  Correlarium] marg. M, om. E 142  ista] om. M 143  cuilibet] cuiuslibet E 144  alia] aliqua M 145  Et] om. M 146  scriptam] scriptum E 147  eodem] eisdem E 136 137

1.13.4 Fourth

English Translation

107

conclusion: a spoken or written proposition or even a loosely mental one can have reflection on itself since any of these signifies at the pleasure of the impositor and not naturally, as an object and not formally. But a strictly mental one is a sign representing naturally and formally, and it is not in our power to signify what we wish through such a sign, as it is for a spoken or written or loosely mental sign. 1.13.4.1 Corollary: it follows from this conclusion that every insoluble proposition is a spoken, written or loosely mental proposition, and that part of any such proposition can supposit for the whole of which it is a part. 1.13.5 The fifth conclusion is this: to any insoluble proposition there corresponds a true strictly mental proposition and another false strictly ⟨mental⟩ one. This is clear in the case of ⟨A⟩ This is a falsehood, referring to itself, which corresponds to the mental proposition: ⟨B⟩ This is a falsehood, referring to the same ⟨spoken proposition A⟩, which is true. And the second part is proven, for the spoken proposition ⟨A⟩ is false, so ⟨A⟩ signifies some mental proposition to be false but not the said ⟨B⟩, so another, namely, ⟨C⟩ This is a falsehood referring to the original mental one ⟨B⟩ that refers to the spoken or written one ⟨A⟩. 1.13.5.1 From this conclusion there follow some corollaries: the first, that any insoluble proposition and similarly its contradictory is a manifold proposition because there correspond to it many mental ones unconjoined. 1.13.5.2 Second corollary: there are some propositions altogether similar in speech and with terms suppositing for the same of which one is a manifold proposition and not the other. It is clear of these: This is a falsehood and This is a falsehood, where each ⟨occurrence of⟩ “this” refers to the second proposition.

108

Latin Text

1.13.5.3 Tertium

correlarium: quelibet propositio insolubilis est simul vera et falsa et similiter sua contradictoria, quia sibi correspondent due mentales quarum una est vera et reliqua est148 falsa, non tamen est simpliciter vera nec simpliciter falsa sed secundum quid. 1.13.5.4 Quartum correlarium: ad nullam propositionem insolubilem vel eius contradictoriam est danda unica responsio, scilicet ipsam simpliciter concedendo vel ipsam simpliciter negando. Patet ex predictis nam secundum Aristotelem secundo Elencorum “ad propositionem plures non est danda unica responsio,” sed quelibet talis est propositio plures, igitur etc. 1.13.6 Hic149 autem per longum processum locutus sum150 propter duo: primo quia iudicio meo adhuc non fuit aliquis qui clarius loqueretur151 in hac152 materia quam iste. Secundo propter quendam novum scriptorem circa insolubilia imitatorem huius, qui de toto processu solum tria inserit correlaria quinte conclusionis, videlicet primum tertium et quartum, que non sufficienter declarant intentionem153 istius nec154 materiam155 insolubilium iuxta opinionem meo156 modulo declaratam. Et licet via ista sit probabiliter157 sustinenda, tamen mihi non in toto vera videtur. Ad quod ostendendum arguo contra quedam dicta.

 est] om. E  Hic] hec M 150  sum] est M 151  loqueretur] loquetur E 152  hac] om. E 153  intentionem] materiam E 154  nec] om. E 155  materiam corr.] materie M E 156  meo] nostro E 157  sit probabiliter] inv. E 148 149

1.13.5.3 Third

English Translation

109

corollary: every insoluble proposition is at the same time true and false, and similarly its contradictory, because there correspond to it two mental propositions, one of which is true and the other false, but it is not unrestrictedly true nor unrestrictedly false but restrictedly. 1.13.5.4 Fourth corollary: no single answer should be given to any insoluble proposition or its contradictory, namely, unrestrictedly granting or unrestrictedly denying it. This is clear from what has been said, for according to Aristotle in Part II of De Sophisticis Elenchis “no single answer should be given to a manifold proposition,” but each of these is a manifold proposition, so ⟨no single answer should be given to any insoluble proposition or its contradictory⟩. 1.13.6 Now here I have spoken at length for two ⟨reasons⟩: first, because in my judgment there was not anyone who spoke more clearly on this matter than ⟨Peter of Ailly⟩ did. Secondly, on account of a recent writer on insolubles imitative of him, who from the whole of ⟨Peter of Ailly’s⟩ discussion draws only three corollaries from the fifth conclusion, namely, the first, third and fourth, which do not sufficiently explain ⟨Peter of Ailly’s⟩ meaning nor the matter of insolubles according to ⟨his⟩ opinion as well as ⟨I have⟩ explained it. And although ⟨Peter of Ailly’s⟩ view may probably be sustained, however, to me it does not seem to be wholly correct. To show this I argue against some of the claims.

110

Latin Text

1.13.6.1 Et

primo contra primam conclusionem. Nam alique intelligentie propter earum nobilitatem sunt noticie formales sui et aliorum cum non intelligant per accidentia superaddita, quia aliter genus accidentium esset infinitum, et non sunt noticie sui nisi158 distincte, quia in confuso non seipsas intelligant,159 igitur aliqua creatura160 potest esse proprie et distincte161 formalis noticia sui. 1.13.6.2 Secundo: intentiones mentales distincte intelliguntur et non per alias noticias ab ipsis distinctas,162 igitur formaliter cognoscuntur. Consequentia patet cum maiori et minorem probo: quia aliter esset processus in163 infinitum in noticiis. Ymmo due species accidentium essent in mente,164 quod est contra communem viam; et consequentia patet, quia semper ad cognitionem unius noticie requireretur165 alia et sic in166 infinitum. Similiter non videtur que species posset mediare inter intellectionem et intellectionem intellectionis. 1.13.6.2.1 Admissa adhuc ista conclusione alie non sequuntur, quia dato quod nihil sit formaliter cognitio sui distincte167 non tamen tollitur quin aliquid sit formaliter cognitio sui confuse. Nam ille terminus ens mentalis naturaliter significat omne ens et cum ipse sit ens, naturaliter significat se semper formaliter intelligendo; nam si non significaret se, sequeretur quod ille terminus ens in voce vel in scripto esset communior quam ille terminus ens in mente.

 nisi] om. E  intelligant] intelligunt dub. M 160  creatura] causa M 161  distincte] determinate E 162  distinctas corr.] distinctis M E 163  in] om. E 164  in mente] immediate E 165  requireretur] requiritur E 166  in] om. E 167  distincte] determinate E 158 159



English Translation

111

1.13.6.1 First, against the first conclusion. For some intelligences in virtue of their

superior nature are formal conceptions of themselves and of others since they do not comprehend them through added accidents, because otherwise the genus of accidents would be infinite. And they are conceptions of themselves only distinctly, because they do not comprehend themselves indistinctly, therefore some created thing can be a formal conception of itself properly and distinctly. 1.13.6.2 Secondly, mental contents are comprehended distinctly and not by other conceptions distinct from them, so they are cognized formally. The inference is clear, as is the major premise, and I prove the minor premise: for otherwise there would be a regress ad infinitum in conceptions. In fact, there would be two types of accidents in the mind which is contrary to the standard view; and the inference is shown by the fact that for the cognition of one conception another would always be required and so on ad infinitum. Similarly, it is not clear what species could mediate between a comprehension and a comprehension of a comprehension. 1.13.6.2.1 Even accepting this conclusion, the other ones do not follow, because given that nothing is formally a cognition of itself distinctly, that does not prevent the possibility that something is formally a cognition of itself indistinctly. For the mental term “being” naturally signifies every being and since it is itself a being it naturally signifies itself, always taking this formally. For if it did not signify itself, it would follow that the spoken or written term “being” would be broader than the term “being” in the mind.

112

Latin Text

1.13.6.2.2 Probo

quia quicquid significat ly ens in mente significat ly ens168 ad placitum et non e contra, quia ly ens ad placitum significat ly ens in mente et non ipse seipsum. Et sic sequitur quod ly ens ad placitum significans non subordinatur illi termino mentali ens nec omnem (M 237vb) suam significationem dependenter sumit ab illo, quod est contra hanc opinionem. 1.13.6.2.3 Concesso ergo quod ly ens significat omne ens, sequitur a pari quod ly propositio significat omnem propositionem et ly verum omne verum et ly falsum omne falsum. Cum ergo compositio vel divisio mentalis absque limitatione non tollat significationem termini, ymmo potius eam169 confirmet, sequitur quod170 quelibet illarum: omnis propositio est falsa, omnis propositio est vera, significat seipsam esse falsam vel seipsam171 esse veram. Prima enim significat omnem172 propositionem esse falsam et secunda173 omnem propositionem esse veram et cum ipsa sit propositio, significat174 seipsam esse veram aut seipsam esse falsam, quod est contra secundam175 conclusionem et176 eius primum correlarium. 1.13.6.3 Contra tertiam conclusionem procedit idem argumentum, tamen arguitur specialius sic: data illa conclusione sequitur quod aliquid significat indefinita propositio quod non significat sua universalis et aliqua est singularis indefinite177 que non est singularis universalis; probatur et capio istas duas mentales proprie dictas, idest naturaliter significantes (E 193vb) iuxta modum intelligendi istius doctoris, videlicet: omnis propositio est, et: aliqua propositio est, quarum prima sit a et secunda b, et patet178 quod tam a quam b est verum et a significat b et non seipsum et e contra, igitur etc. Similiter hec singularis: ista propositio est, est singularis b et non a, igitur intentum.  in mente significat ly ens] om. (hom.) M  eam] eque E 170  quod] om. E 171  seipsam] om. M 172  omnem] omne E 173  secunda] secundam M 174  significat] significans M 175  secundam] primam E 176  et] contra add. E 177  indefinite] indefinita E 178  patet] om. E 168 169

1.13.6.2.2 Proof:

English Translation

113

because whatever ⟨the term⟩ “being” in the mind signifies, ⟨the term⟩ “being” ⟨signifying⟩ by imposition signifies ⟨that same thing too⟩ and not conversely, for “being” ⟨signifying⟩ by imposition signifies “being” in the mind but ⟨”being” in the mind does⟩ not ⟨signify⟩ itself. Thus it follows that “being” signifying by imposition is not subordinated to the mental term “being” nor does it take all its signification dependently from it, which is contrary to ⟨Peter of Ailly’s⟩ opinion. 1.13.6.2.3 So granting that “being” signifies every being, it follows equally that “proposition” signifies every proposition and “truth” every truth and “falsehood” every falsehood. Then since mental composition and division without restriction does not remove the signification of a term (on the contrary, it rather confirms it), it follows that each of these, Every proposition is false, Every proposition is true signifies itself to be false or itself to be true. For the first signifies every proposition to be false and the second, every proposition to be true and since ⟨each⟩ is a proposition it signifies itself to be true or itself to be false, which is contrary to the second conclusion and to its first corollary. 1.13.6.3 The same argument works against the third conclusion, but I argue in particular like this: given that conclusion, it follows that some indefinite proposition signifies something that its universal does not signify, and there is some singular of an indefinite which is not a singular of the universal. Proof: take these two strictly mental propositions—that is, signifying naturally according to this doctor’s way of taking them— namely, Every proposition exists and A proposition exists, and let the first be A and the second B. Then it is clear that both A and B are true and A signifies B and not itself and conversely, so ⟨some indefinite proposition signifies something that its universal does not signify⟩. Similarly, the singular proposition: This proposition exists is a singular of B and not of A, so ⟨we have⟩ what was wanted.

114

Latin Text

1.13.6.4 Item

hec propositio vocalis vel scripta: omnis propositio est, non est insolubilis, quia vera secundum eum, igitur uni soli mentali debet subordinari, huic videlicet: omnis propositio est, sed nulla propositio ad placitum significans abundantius significat quam mentalis cui subordinatur secundum istam opinionem, quia quelibet talis, ut dicit, est extrinsece propositio et significativa per habitudinem ad propositionem mentalem. Sed illa in scripto significat simpliciter omnem propositionem esse, se et illam mentalem, igitur mentalis etiam significat se.179 Hec conclusio in reprobatione subsequentis opinionis fortius improbabitur. 1.13.6.5.1 Contra quintam conclusionem arguo180 sic: ista consequentia non valet: falsum est, igitur hoc est falsum, quocumque demonstrato, cum stet181 antecedens esse verum et consequens falsum, igitur hec:182 falsum est, non subordinatur alicui tali: hoc est falsum. Patet consequentia quia semper a subordinato ad183 subordinans valet consequentia, et per consequens hec opinio non verum dicit in hoc quod hec in scripto: falsum est, subordinetur duabus talibus: falsum est, que est vera, et huic: hoc est falsum, prima mentali demonstrata, que est falsa. 1.13.6.5.2 Secundo arguitur sic. Si non esset aliqua propositio in mundo nisi ista: falsum est, ipsa esset falsa, sed tunc non subordinaretur alicui184 mentali, igitur non est sufficiens ratio quare aliqua propositio insolubilis est falsa quia subordinatur uni mentali false.

 se] om. M  arguo] arguitur E 181  cum stet] quia stat E 182  hec corr.] hoc M E 183  ad] a M 184  alicui] tali add. E 179 180

1.13.6.4 Again,

English Translation

115

the spoken or written proposition: Every proposition exists is not an insoluble, because it is true according to him and so should be subordinate to a single mental proposition, namely, to Every proposition exists. But according to this opinion no proposition signifying at the pleasure of the impositor signifies more abundantly than the mental proposition to which it is subordinate, because each of them, as he says, is a proposition extrinsically and is significative in virtue of its relation to a mental proposition. But the written one signifies unrestrictedly that every proposition is, ⟨including⟩ itself and the mental one, so the mental one also signifies itself. This ⟨third⟩ conclusion will be disproven ⟨even⟩ more strongly in disproof of the next opinion. 1.13.6.5.1 I argue against the fifth conclusion like this: the inference: A falsehood exists, so this is a falsehood, whatever is referred to, is not valid, since it is possible for the premise be true and the conclusion false. So A falsehood exists is not subordinate to any ⟨occurrence of⟩ This is a falsehood. The inference is clear because an inference is always valid from a subordinate to the subordinating, and consequently this opinion does not affirm the truth in ⟨claiming⟩ that the written proposition: A falsehood exists is subordinate to these two: A falsehood exists, which is true, and to This is a falsehood referring to the first mental proposition, which is false. 1.13.6.5.2 Secondly, I argue like this: if the only proposition in the world were A falsehood exists, it would be false, but then it would not be subordinate to any mental one, so that it is subordinate to a mental one is not a sufficient reason why an insoluble proposition is false.

116

Latin Text

1.13.6.5.3 Tertio

arguitur sic. Non magis est propositio plures aliqua propositio insolubilis quam hec: canis currit, cum hec subordinatur tribus mentalibus et insolubilis solum duabus, sed hec: canis currit, univocatur per sequestrationem duarum mentalium tertia remanente significante adequate latrabilem canem currere, et significaret185 correspondenter186 solummodo hec: canis currit in voce vel in scripto, que sic sumpta conceditur187 esse vera. Igitur per idem hec: hoc est falsum, seipsa demonstrata, poterit univocari ita quod non subordinabitur nisi uni tali mentali: hoc est falsum, demonstranti priorem scriptam vel vocalem. Quo habito hec esset falsa: hoc est falsum, et tamen non haberet nisi unam veram mentalem nec posset habere dummodo sic univocaretur, sicut et illa univocari potest: canis currit, modo supradicto, igitur sequitur quod propositionem insolubilem non recte hic magister solvebat.

⟨Opinio Petri de Mantua⟩ 1.13.7 Alius

(M 238ra) magister ideo favens huic opinioni, sed non in modo subordinationis assignate, ponit quod huiusmodi termini verum et188 falsum sunt termini equivoci, deducta ipsorum transcendentia, sed solum ut de signis complexe significantibus dicuntur. Propositio ergo ait duobus modis189 potest esse vera.

 significaret corr.] sic M E  correspondenter] et add. E 187  conceditur] concederetur M 188  et] vel M 189  modis] media E 185 186



English Translation

117

1.13.6.5.3 Thirdly, I argue like this: an insoluble proposition is no more a manifold

proposition than is A dog is running, since the latter is subordinate to three mental ones and the insoluble to only two. But A dog is running is disambiguated by removal of two of the mental ones, the third one remaining signifying exactly that a dog that barks is running, and likewise correspondingly the spoken or written proposition: A dog is running would signify only in this way, which may be granted to be true. Therefore by parity of reasoning This is a falsehood, referring to itself, can be disambiguated so that it is subordinate to only one mental ⟨occurrence of⟩ This is a falsehood referring to the original written or spoken one. In this way, This is a falsehood would be false but would and could be related to only one true mental proposition so long as it were disambiguated in this way, just as A dog is running can be disambiguated in the above way. Therefore it follows that this master has not solved insoluble propositions correctly. ⟨Peter of Mantua’s Opinion⟩

1.13.7 Therefore,

another master agreeing with this opinion but not in the manner of subordination specified, claims that terms like “truth” and “falsehood” are equivocal terms, setting aside their transcendental sense, but only when they are said of signs signifying complexly. So he says a proposition can be true in two senses.

118

Latin Text

1.13.7.1 Uno

modo dicitur propositio190 vera quoniam ipsa verificatur non propter supposita suorum terminorum, quorum ipsa propositio est suppositum,191 scilicet quod propositio non redditur vera ex eo quod pars eius supponat pro ipsamet nec pro pertinente ad ipsam, sicut hec: deus est. Et illo modo propositiones que sunt de terminis prime intentionis vel impositionis sunt192 simpliciter et proprie vere vel false quoniam propositio ex ethimologia193 id est194 sermonis virtute idem significat quod pro alio positio.195 1.13.7.2 Alio modo dicitur propositio vera quoniam verificatur pro seipsa aut pro alio pertinente et illo modo hec propositio est vera: hoc est verum, seipsa demonstrata et hec196 ⟨similiter:⟩ hoc est falsum, seipsa demonstrata, non simpliciter sed secundum quid, sed est falsa primo modo quia ipsa non verificatur nisi pro supposito sue partis cuius ipsa est suppositum. 1.13.7.3 Concludit ergo quod ille propositiones sunt plures: hoc est falsum, non197 hoc est verum, et sic de aliis, quare cum proponitur aliqua illarum non est secundum unam responsionem respondendum,198 sed dicendum quod hoc est falsum secundum primum membrum divisionis199 posite et verum secundum aliud. Et ita dicendum est de suis contradictoriis: non hoc est falsum, et: hoc est verum, negando quod hoc sit falsum secundo modo et concedendo quod est falsum primo modo.

 propositio] esse add. E  propositio est suppositum] est propositio, supponit E 192  sunt] sequitur E 193  ethimologia] et hius? M 194  id est] om. E 195  positio] om. M 196  hec] om. M 197  non] om. E 198  respondendum] om. M 199  divisionis] dictionis E 190 191

1.13.7.1 In one

English Translation

119

sense a proposition is said to be true because it is made true ⟨but⟩ not on account of the supposita of its terms of which the proposition itself is a suppositum. That is, the proposition does not depend for its truth on part of it suppositing for ⟨the proposition⟩ itself nor for something logically related to it. An example is God exists; and in that sense propositions which are constructed from terms of first intention or imposition are unrestrictedly and properly true or false. Because “proposition,” from its etymology, i.e., from the literal meaning of the word, means the same as “putting in place for something else.” 1.13.7.2 In the other sense, a proposition is said to be true because it is made true of itself or something else logically related ⟨to it⟩, and in that sense the proposition: This is a truth, referring to itself, is true, not unrestrictedly but restrictedly, and ⟨likewise⟩ This is a falsehood, referring to itself. But it is false in the first sense because it is only made true by the suppositum of the part of it of which ⟨the proposition⟩ is the suppositum. 1.13.7.3 So he concludes that the propositions: This is a falsehood ⟨and⟩ Not: this is a truth are manifold, and similarly for others, whence when one of them is proposed one should not respond with a single response, but one should say that this is false according to the first sense in the distinction proposed and true in the other. And the same should be said of their contradictories, Not: this is a falsehood and This is a truth, denying that it is a falsehood in the second sense and granting that it is a falsehood in the first sense.

120

Latin Text

1.13.7.4 Sed

hec declaratio non solvit insolubilia sed potius se involvit. Nam capio verum et falsum secundo modo et probo quod sic sumendo hec est falsa: hoc est falsum, se demonstrato. Nam si ipsa sit vera et significet200 adequate hoc esse falsum, igitur verum est hoc esse falsum. Consequentia tenet apud eum (E 194ra); et ultra: verum est hoc esse falsum, igitur hoc est falsum, sic sumendo; et sic201 habeo quod idem est verum et falsum secundo modo dicto202, quod ipse203 negat. ⟨Opinio Gualteri de Segrave⟩

1.14 Quartadecima

opinio, que est fundamentum multarum204 precedentium et iam205 plurium sophistarum206 qui plus subterfugere quam respondere conantur,207 ponit insolubilia solvenda fore208 penes fallaciam accidentis secundum quam parologismi dupliciter fiunt aut ex variatione medii vel alterius209 extremorum. Ex variatione medii ut si medium210 pro alio supponat in maiori quam in minori vel211 e contra. Et ita dicatur ex variatione extremi.

 significet corr.] significat M E  sic] ita M; we have preferred the reading of E since it matches Mantua, Insolubilia, O6va 202  dicto corr.] dictis M E 203  ipse] ipsa M 204  multarum] multorum M 205  iam] ideo E 206  sophistarum] sophismatorum E 207  conantur] conatur E 208  fore] facta E 209  vel alterius corr.] aut alterius E, vel alterum M, we follow Segrave’s text 210  si medium] inv. M 211  vel] et E 200 201



English Translation

121

1.13.7.4 But

this explanation does not solve insolubles but rather increases the obscurity. For I take “truth” and “falsehood” in the second sense and prove that, taking ⟨them⟩ this way, the proposition: This is a falsehood is false, referring to itself. For if it is true and exactly signifies that this is a falsehood, then it is true that this is a falsehood. The inference holds according to him; and further: it is true that this is a falsehood, so this is a falsehood, taking ⟨the terms⟩ this way. Thus I can conclude that the same thing is a truth and a falsehood said in the second sense mentioned, which he denies.

⟨Walter Segrave’s Opinion⟩ 1.14 The fourteenth

opinion, which is the basis of many previous opinions and now of even more sophismatists, who try more to evade than to respond, claims that insolubles are to be solved by the fallacy of accident, according to which “paralogisms arise in two ways, either from variation of the middle or of one of the extremes. From variation of the middle, e.g., if the middle ⟨term⟩ supposits for something else in the major than in the minor ⟨premise⟩,” or vice versa—and the same sort of thing may be said of the variation of an extreme.

122

Latin Text

1.14.1.1 Dicit

ergo212 hec opinio213 quod sorte dicente:214 sortes dicit falsum, ipse non215 dicit falsum. Et tunc ad argumentum: sortes dicit hoc et hoc est falsum, igitur sortes dicit falsum, negat216 consequentiam dicendo quod est fallacia accidentis ex variatione extremi. Ille enim terminus falsum pro aliquo supponit in minori pro quo non supponit in conclusione. 1.14.1.2 Similiter si arguitur ex opposito dicti217 a sorte: nullum falsum dicitur a sorte, hoc est falsum, igitur hoc non dicitur a sorte, hec218 est fallacia accidentis ex variatione medii. Pro aliquo enim supponit ille terminus falsum in minori pro quo non supponit in maiori.219 1.14.2 Ad quod ostendendum presupponit quod in nulla propositione pars supponit pro toto cuius est pars nec convertibili220 cum toto nec opposito totius nec antecedenti221 ad totum. Ex quo patet quod illa propositio: sortes dicit falsum, significat quod sortes dicit falsum, non quidem falsum quod ipse dicit sed falsum distinctum ab illo; sed quia nihil dicit nisi illam propositionem, ideo ipsa est falsa. 1.14.3 Contra istam opinionem procedunt omnia222 argumenta facta contra secundam conclusionem et tertiam alterius opinionis. 1.14.3.1 Tamen arguo specialiter sic: et propono istam: hoc est falsum. Si queritur quid demonstras per ly hoc, dico quod223 eandem propositionem cuius est pars. Et si dicitur quod non possum demonstrare illam, contra: ly hoc est signum ad placitum, (M 238rb) igitur possum illam demonstrare.  ergo] om. M  hec opinio corr.] hoc opinio E, hec responsio M 214  dicente] falsum add. M 215  non corr.] om. M E; we follow Segrave’s text—see the Commentary 216  negat] negant E 217  dicti corr.] dicto M E; we follow Segrave’s text—see the Commentary 218  hec] hic M 219  in minori pro quo non supponit in maiori corr. ] in maiori pro quo non supponit in minori M E; we follow Segrave’s text 220  convertibili corr.] convertibile M E 221  antecedenti corr.] antecedente M E 222  omnia] viam et M 223  quod] om E 212 213

1.14.1.1 Hence

English Translation

123

this opinion says that “if Socrates says Socrates says a falsehood, he does not say a falsehood. And then ⟨in reply⟩ to the argument: Socrates says this, and this is a falsehood, so Socrates says a falsehood, it denies the inference, saying that it is a fallacy of accident from variation of the extreme. For the term ‘falsehood’ supposits for something in the minor ⟨premise⟩ for which it does not supposit in the conclusion.” 1.14.1.2 “Similarly, if one argues from the opposite of what was said by Socrates: No falsehood is said by Socrates, this is a falsehood, so this is not said by Socrates, this is a fallacy of accident from variation of the middle, for the term ‘falsehood’ supposits for something in the minor ⟨premise⟩ for which it does not supposit in the major.” 1.14.2 To support ⟨these claims, this opinion⟩ assumes that “in no proposition does the part supposit for the whole of which it is a part nor for anything convertible with the whole nor for the opposite of the whole nor for the antecedent to the whole.” From this it is clear that the proposition: Socrates says a falsehood signifies that Socrates says a falsehood, but definitely not the falsehood that he says but a falsehood distinct from it. But because he says only that proposition, it is false. 1.14.3 All the arguments made against the second and third conclusions of the ⟨thirteenth⟩ opinion work against this opinion. 1.14.3.1 But I argue in particular like this, and I propose: This is a falsehood. If you ask what I refer to by “this,” I say: the very proposition of which it is part. And if you say that I cannot refer to it, I reply: “this” is a sign ⟨signifying⟩ at the pleasure of the impositor, so I can refer to it ⟨if I so please⟩.

124

Latin Text

1.14.3.1.1 Item

ly hoc seipsum significat naturaliter, igitur potest totam illam propositionem ad placitum significare. Patet consequentia cum illa propositio ab ipso distinguatur. 1.14.3.1.2 Item per ly hoc possum demonstrare sortem vel unum alium terminum, igitur et illam propositionem. Consequentia patet quia eque bene a ly hoc distinguitur illa propositio sicut sortes vel plato. 1.14.3.1.3 Item quicquid possum intelligere et qualitercumque, sic possum224 demonstrare; sed possum intelligere ly hoc propositionem cuius est pars significare aut pro eadem supponere, igitur sic possum demonstrare. Si ergo dicitur in principio, sicut est dicendum, quod illam propositionem possum demonstrare, arguitur sic: quicquid ly hoc demonstrat potest significare225 et pro eodem supponere, quia non est extremum propositionis absque suppositione, sed demonstrat illam propositionem cuius est pars, igitur pro eadem supponit; quod si conceditur dicendo quod non est inconveniens in terminis positivis, sed bene in privativis, hoc est nimis ad placitum et sine ratione. Nam ille terminus falsum, cum sit terminus communis, citius poterit supponere, licet in confuso, pro illa propositione quam aliquis terminus discretus. 1.14.3.2 Secundo arguitur sic: sequeretur226 quod omnis propositio est falsa et hec est propositio et tamen hec non est falsa, dato quod non essent plures propositiones quam ille due: omnis propositio est falsa, et: nullus deus est. 1.14.3.3 Tertio arguitur sic: in istis propositionibus: ens est, aliquid est, propositio est, complexum est, pars supponit pro toto cuius est pars licet disiunctive. Igitur in talibus: omnis propositio est, omne complexum est, pars supponit pro toto cuius est pars copulative. Antecedens patet quia non potest dari causa negationis.

 sic possum] om. M  demonstrat potest significare] significat potest demonstrare E 226  sequeretur corr.] sequitur M E 224 225

1.14.3.1.1 Again,

English Translation

125

“this” naturally signifies itself, so it can ⟨only⟩ signify that whole proposition at the pleasure of the impositor. The inference is clear since that proposition is distinct from it. 1.14.3.1.2 Again, I can refer to “Socrates” or another term by “this,” so to that proposition too. The inference is clear because that proposition is just as distinct from “this” as “Socrates” or “Plato.” 1.14.3.1.3 Again, whatever and however I can understand, I can refer to in the same way, but I can understand “this” to signify a proposition of which it is part or to supposit for the same proposition, so I can refer ⟨to it⟩ in this way. Thus, if one says at the beginning, as one should say, that I can refer to that proposition, I argue like this: whatever “this” refers to it can signify, and supposit for it too, because there is no extreme of a proposition without supposition, but it refers to that proposition of which it is part, so it supposits for it too. And if it is granted, by saying that it is not impossible in the case of positive terms but ⟨is impossible⟩ in the case of privative ones, this is an imposition too far and without good reason, for the term “falsehood,” since it is a general term, could sooner supposit, although in an indistinct way, for that proposition than any discrete term. 1.14.3.2 Secondly, I argue like this: given that there were no more propositions than these two: Every proposition is false and God does not exist, it would follow that every proposition is false and ⟨Every proposition is false⟩ is a proposition, and yet it is not false. 1.14.3.3 Thirdly, I argue like this: in these propositions: A being exists, Something exists, A proposition exists, A propositional complex exists, the part supposits for the whole of which it is part, although disjunctively. Therefore, in these: Every proposition exists, Every propositional complex exists, the part supposits for the whole of which it is part conjunctively. The premise is clear because no reason can be given to deny it.

126

Latin Text

1.14.3.4 Quarto

arguitur sic: sorte dicente: omne falsum dicitur a sorte, audiat plato subiectum; tunc plato per illud intelligit omne falsum, igitur falsum dictum a sorte; igitur subiectum illius propositionis dicte a sorte sic significat et per consequens pro suo toto supponit. 1.14.3.4.1 Hic solet responderi quod non sequitur: ille terminus falsum significat totum cuius est pars, igitur supponit pro illo; restringitur enim per copulam vel per227 predicatum respectu cuius supponit, unde non est idem significare et supponere sicut patet. Nam ille terminus homo aliter supponit respectu verbi de presenti et aliter respectu verbi de preterito vel futuro et pro aliis et aliis supponit et tamen semper eosdem significat. 1.14.3.4.2 Hec responsio veritatem dicit sed non solvit argumentum. Concesso quod in illa propositione: omne falsum est, significat ly falsum omne falsum quod est et suum totum, arguitur sic: verbum228 de se est229 indeterminatum ad hoc suppositum vel ad illud (E 194rb), igitur communicat subiecto distributo suppositionem non plus pro uno significato de quo idem verbum verificatur quam pro alio. Cum igitur in ista propositione:230 omne falsum est, distribuatur ly falsum per signum et supponat ratione verbi de presenti, igitur supponit pro omni falso quod est; sed ipsa propositio est unum falsum, igitur supponit pro illa. 1.14.3.4.3 Confirmatur: nam qualis est habitudo illius termini homo ad sua significata, talis est habitudo illius termini falsum ad sua significata. Sed in tali propositione: omnis homo est, supponit ly homo231 pro omni homine qui est, igitur et in232 illa: omne falsum est, supponit ly falsum233 pro omni falso quod est.

 per] om. E  verbum] ubi E 229  est] om. M 230  in ista propositione] ista propositio E 231  supponit ly homo] ly homo supponit E 232  in] om. E 233  supponit ly falsum] ly falsum supponit E 227 228

1.14.3.4 Fourthly,

English Translation

127

I argue like this: if Socrates says Every falsehood is said by Socrates, and Plato hears the subject, then Plato understands by it every falsehood, therefore ⟨Plato understands⟩ the falsehood said by Socrates. Hence the subject of the proposition said by Socrates signifies accordingly, and consequently, it does supposit for the whole of which it is part. 1.14.3.4.1 Here it is usual to reply “that ⟨the inference⟩ The term ‘falsehood’ signifies the whole of which it is part, therefore it supposits for it is not valid, for ⟨the term⟩ is restricted by the copula or by the predicate relative to which it has supposition. Hence, to signify and to supposit are not the same, as is clear.” For the term “man” supposits one way with respect to a verb in the present tense and differently with respect to a verb in the past or future tense, and it supposits for different things, but it always signifies the same things. 1.14.3.4.2 This response is correct but does not refute the argument. Once it is granted that in the proposition: Every falsehood exists, “falsehood” signifies every falsehood which exists, including the whole of which it is part, I argue like this: the verb is indeterminate of itself as to this suppositum or that, so it gives supposition to a distributed subject no more for one significate of which that verb is truly predicated than for another. Since therefore in the proposition: Every falsehood exists, “falsehood” is distributed by the quantifier and supposits in virtue of a verb in the present tense, so it supposits for every falsehood which exists. But that proposition is one falsehood, so it supposits for it. 1.14.3.4.3 This is confirmed, since whatever the relation of the term “man” to its significates, the same relation holds between the term “falsehood” and its significates. But in the proposition: Every man exists, “man” supposits for every man who exists, hence also in Every falsehood exists, “falsehood” supposits for every falsehood that exists.

128

Latin Text

1.14.3.4.4 Item

confirmatur: si non esset aliquis homo in mundo preter sortem, stultitia foret asserere illum terminum homo in hac propositione: aliquis homo est, supponere pro illo qui non esset et non pro sorte. Sed si non esset alia propositio in mundo quam illa: falsum est, que esset a, non esset aliud falsum quam a, igitur stultitia similiter foret concedere ly falsum supponere pro alio ab a et non pro a. 1.14.3.5 Quinto arguitur ad principale sic: dicat sortes234 tantum illam: omne dictum a sorte est falsum, que sit a, tunc a est falsum, ut patet; et ibi est esse in toto et dici de omni, igitur nihil est (M 238va) subiecti de quo non dicatur predicatum. Consequentia tenet per diffinitionem dici de omni positam in primo Priorum; sed a est aliquid subiecti quia est unum dictum a sorte, igitur de a dicitur predicatum, et per consequens pro ipso supponit. 1.14.3.5.1 Confirmatur sic: in hac propositione: omnis homo est animal, non esset dici de omni si de235 aliquo verificaretur236 subiectum de quo non verificaretur237 predicatum aut si aliquis homo esset de quo non diceretur subiectum nec predicatum. Si ergo in illa propositione: omne dictum a sorte est falsum, nec subiectum nec predicatum verificatur pro a, quod de facto est dictum a sorte et falsum, sequitur quod in a non sit dici de omni. 1.14.3.5.2 Sed forte conceditur quod ibi non est dici de omni. Contra: aliquod dictum a sorte est falsum et non est dictum a sorte quin illud est falsum, igitur ibi est dici de omni. Consequentia tenet quia in aliis universaliter tenet argumentum.

 Dicat sortes] sortes dicit E  de] pro E 236  verificaretur] verificantur E 237  verificaretur] verificatur E 234 235

1.14.3.4.4 Again,

English Translation

129

this is confirmed: if there were no man in the world but Socrates, it would be stupid to claim that the term “man” in the proposition: Some man exists supposited for someone who did not exist and not for Socrates. But if there were no other proposition in the world than A falsehood exists, call it A, there would be no other falsehood than A, so it would similarly be stupid to grant that “falsehood” supposited for something other than A and not for A. 1.14.3.5 Fifthly, returning to the main point, I argue like this: “suppose Socrates says only Everything said by Socrates is a falsehood, call it A. Then A is a falsehood, as is clear; and there we have ‘to be in ⟨something as in⟩ a whole’ and the dici de omni, so nothing belongs to the subject of which the predicate is not said. The inference holds on the basis of the definition of the dici de omni stated in the first ⟨book⟩ of the Prior Analytics. But A is something belonging to the subject” because it is something said by Socrates, “so the predicate is said of A”, and consequently it supposits for it. 1.14.3.5.1 It is confirmed like this: in the proposition: Every man is an animal, we would not have the dici de omni if the subject were true of something of which the predicate was not true, or if there were a man of which neither subject nor predicate were ⟨truly⟩ said. Hence if in the proposition: Everything said by Socrates is a falsehood neither the subject nor the predicate is true of A, which in fact is said by Socrates, and a falsehood, it follows that the dici de omni does not hold of A. 1.14.3.5.2 But perhaps it is granted that we do not have the dici de omni here. On the contrary: something said by Socrates is a falsehood, and the only thing said by Socrates is a falsehood, so we do have the dici de omni here. The inference holds because the argument holds universally in other cases.

130

Latin Text

1.14.3.6 Sexto

arguitur sic: dicat sortes238 tantum illam vel239 partem eius tantum: falsum dicitur a sorte, que sit a, tunc a est falsum, sicut notum est; et illa exceptiva240 est vera241: nullum falsum preter a dicitur a sorte, quia utraque eius exponens est vera, videlicet: falsum a242 dicitur a sorte, et: nullum falsum non a dicitur a sorte, igitur subiectum huius: nullum falsum dicitur a sorte, supponit pro a quia aliter non fieret exceptio a suo toto et per consequens ⟨subiectum⟩ exceptive243 supponit pro a. 1.14.3.6.1 Confirmatur: omnis exceptiva propria repugnat sue preiacenti et non potest poni repugnantia nisi exceptiva244 significet quod a dicatur a sorte, igitur etc. 1.14.3.7 Septimo arguitur sic: sequitur ex illa opinione quod illa universalis: omnis propositio universalis est falsa, non est universalis illius indefinite: propositio universalis est falsa, nec illius245 singularis: ista propositio universalis est falsa, demonstrato insolubili, quia non supponunt pro eodem; et consequenter sequitur quod illa non sunt contradictoria: omnis propositio universalis est falsa, et: aliqua propositio universalis non est falsa, que tamen cuilibet sano capiti videntur246 contradicere. Et consequentia tenet quia non supponunt pro eodem.

 sortes] sortem M  vel] om. E Ma.c. 240  exceptiva corr.] exclusiva vel exceptiva M, exclusiva E; we follow Bradwardine, Insolubilia, ¶ 3.2.2 and Segrave’s text (see Commentary) 241  vera corr.] igitur M E; we follow Bradwardine, loc. cit., and Segrave (see Commentary) 242  a corr.] om. M E; we follow Bradwardine, loc. cit. 243  exceptive corr.] exceptiva dicta a sorte M; exclusiva E; we follow Bradwardine, loc. cit. 244  exceptiva corr.] exclusiva M E; we follow Bradwardine, loc. cit. 245  illius] illa E 246  videntur] videretur M 238 239



English Translation

1.14.3.6 Sixthly,

it:

131

I argue like this. Suppose Socrates says only this or only part of

A falsehood is said by Socrates, call it A. Then A is a falsehood, as we know, and the exceptive proposition: No falsehood except A is said by Socrates, is true because each of its exponents is true, namely, The falsehood A is said by Socrates and No falsehood ⟨which is⟩ not A is said by Socrates. Therefore, the subject of No falsehood is said by Socrates supposits for A because otherwise there would not be an exception from its whole ⟨viz in “No falsehood ⟨which is⟩ not A is said by Socrates”⟩; and consequently ⟨the subject of⟩ the exceptive supposits for A. 1.14.3.6.1 This is confirmed: every proper exceptive is inconsistent with its prejacent, and there cannot be an inconsistency ⟨between the exceptive and its prejacent⟩ unless the exceptive signifies that A is said by Socrates, therefore etc. 1.14.3.7 Seventhly, I argue like this. It follows from this opinion that the universal proposition: Every universal proposition is false is not the universal proposition corresponding to this indefinite proposition: A universal proposition is false, nor to this singular proposition: This universal proposition is false referring to the insoluble ⟨universal⟩, because they do not supposit for the same thing; and consequently, it follows that Every universal proposition is false and Some universal proposition is not false are not contradictories, which, however, seem to any sane head to contradict ⟨one another⟩. The inference holds on the grounds that they do not supposit for the same thing.

132

Latin Text

1.14.3.7.1 Etiam

sequitur quod aliqua propositio universalis non supponit pro eo pro quo sua singularis. Probatur et pono quod non sint plures propositiones totales quam ille:248 omnis propositio singularis est falsa, et: ista propositio singularis est falsa, seipsa demonstrata. Quod illa sit singularis illius universalis patet quia universaliter sequitur tamquam a superiori ad suum inferius: omnis propositio singularis est falsa, hec est propositio singularis, igitur hec est falsa, et sic patet conclusio.249 (M 238va) 247

 eo] eodem E  ille] ista E 249  et sic patet conclusio] om. E 247 248



English Translation

1.14.3.7.1 Also,

133

it follows that some universal proposition does not supposit for that for which its singular ⟨supposits⟩. Proof: assume that there are no more whole propositions than these: Every singular proposition is false and This singular proposition is false, referring to itself. That it is a singular of this universal is clear because this is universally valid, as from a superior to its inferior: Every singular proposition is false ⟨and⟩ this is a singular proposition, therefore this is false, and thus the conclusion is evident.

134

Latin Text

⟨Capitulum Secundum. Opinio Auctoris⟩ 2.1 Pro declaratione

quintedecime opinionis, quam scio valentium fuisse antiquorum tres articuli inferuntur: primus continet terminorum divisiones,1 secundus preambulas suppositiones, tertius intentum per conclusiones. ⟨Divisiones⟩

2.1.1 Quantum2

ad primum sit hec prima divisio: omne insolubile aut oritur ex actu nostro aut ex proprietate vocis. Actus nostri sunt duplices, quidam interiores et quidam exteriores. Interiores sunt hii qui fiunt ⟨ex parte mentis⟩, ut cogitare, ymaginari3 et huiusmodi. Exteriores sunt hii4 qui fiunt ex parte corporis, ut dicere, loqui et similia. Et ista sunt insolubilia orientia ex actu nostro: sortes dicit falsum, sortes intelligit falsum, et similia. Vocis 5 autem proprietates sunt ut subici, appellare, esse verum6 pro7 se, esse verum pro alio a se, non esse verum pro se et pro alio (E 194va) a se, et ita (M 238vb) de esse8 falsum et non esse falsum pro se vel pro alio. Unde huiusmodi insolubilia: falsum est, nullum verum est, propositio non verificatur pro se, et: aliqua propositio non appellatur a suo subiecto vel predicato, ex vocis proprietate9 nascuntur.

 divisiones] dictiones E  Quantum] Exemplum M 3  cogitare ymaginari] inv. E 4  hii] illi E 5  Vocis] Voces E 6  verum corr.] vel falsum add. M E 7  pro] per M 8  esse] se E 9  vocis proprietate] inv. E 1 2



English Translation

135

⟨Chapter Two: The Author’s Opinion⟩ 2.1 In order

to explain the fifteenth opinion, which I know to have been that of able forerunners, three sections are subjoined: the first contains the divisions of terms, the second preliminary assumptions, the third what is aimed at through ⟨some⟩ conclusions. ⟨Divisions⟩

2.1.1 Regarding

the first section, let this be the first division: every insoluble either arises from an act of ours or from a property of an expression. Acts of ours are of two sorts: some interior and some exterior. Interior ones are those which are performed ⟨by the mind⟩ like thinking, imagining and suchlike. Exterior ones are those which are performed by the body like saying, speaking and so on. Insolubles arising from an act of ours are: Socrates says a falsehood, Socrates understands a falsehood and so on. Now properties of an expression are such as being a subject, naming, being true of itself, being true of something other than itself, not being true of itself or of something other than itself, and likewise for being false and not being false of itself or of something other than itself. Whence insolubles of this sort: A falsehood exists, No truth exists, A proposition is not true of itself, and Some proposition is not named by its subject or predicate originate from a property of an expression.

136

Latin Text

2.1.2 Secunda divisio est ista: propositionum quedam habent reflexionem

supra se, quedam autem non. Propositio habens reflexionem supra se est illa cuius significatio refertur10 ad se ut:11 omne complexum est, aut: hoc est falsum,12 seipso demonstrato. Propositio non habens reflexionem supra se dicitur esse illa cuius significatio13 non refertur ad se, ut: deus est, et: homo est asinus. 2.1.3 Tertia divisio est ista: propositionum habentium reflexionem supra se quedam habent huiusmodi reflexionem immediate, quedam habent huiusmodi reflexionem mediate.14 Propositio habens immediate reflexionem15 supra se dicitur esse illa cuius significatio non prius refertur16 ad aliud quam ad se. Verbi gratia, hoc est verum, hoc est falsum, seipso demonstrato, omnis propositio est vera, omnis propositio est falsa. Propositio non immediate habens reflexionem supra se dicitur esse illa cuius significatio prius terminatur17 ad aliud quam ad se. Verbi gratia, hoc est verum, et: hoc est verum, demonstrando per primum hoc secundam propositionem et per secundum primam. Idem18 dico de istis: hoc est falsum, et: hoc est falsum, demonstrando ut prius.

 refertur] reflectitur M  ut] et ad M 12  est falsum] inv. E 13  significatio corr.] significatum M E 14  mediate] mediante E 15  immediate reflexionem] inv. E 16  refertur] referetur M 17  terminatur] refertur E 18  Idem] Ideo E 10 11

2.1.2 The second

English Translation

137

division is this: some propositions have reflection on themselves, but some do not. A proposition having reflection on itself is one whose signification relates to itself, such as: Every propositional complex exists or This is false, referring to itself. A proposition not having reflection on itself is said to be one whose signification does not relate to itself, e.g., God exists and A man is an ass. 2.1.3 The third division is this. Some propositions having reflection on themselves have such reflection immediately, some have such reflection mediately. A proposition having reflection on itself immediately is said to be one whose signification does not relate to something else before ⟨relating⟩ to itself. For example, This is true, This is false, referring to itself; Every proposition is true, Every proposition is false. A proposition having reflection on itself non-immediately is said to be one whose signification is directed to something else before ⟨relating⟩ to itself. For example, This is true and This is true, where the first “this” refers to the second proposition and the second ⟨“this”⟩ to the first proposition. I say the same of these: This is false and This is false, referring ⟨to each other⟩ as before.

138

Latin Text

2.1.4 Quarta

divisio: propositionum habentium reflexionem supra se, quedam sic se19 habent quod ipsarum significationes solum terminantur ad se, ut: hoc est verum, hoc est falsum, seipsis demonstratis. Quedam autem sic se habent quod ipsarum significationes terminantur ad se et ad alia a20 se, ut: omnis propositio est vera, omnis propositio est falsa. Non enim solum significant21 se esse veras vel se esse falsas, verum et alias propositiones distinctas a se. 2.1.5 Quinta divisio: propositionum habentium reflexionem supra se quedam seipsas ponunt, quedam autem se destruunt, modo loquendi Philosophi22 quarto Metaphysice ubi ponit quasdam orationes semetipsas destruere. Propositio seipsam ponens est enunciatio seipsam quiescenter verificans, sicut quelibet illarum: hoc est verum, se demonstrato, omnis propositio est vera. Propositionum autem se destruentium quedam se destruit affirmative, quedam negative. Propositio se destruens affirmative est propositio significans23 se aut asserens se esse falsam, ut hec propositio: hoc24 est falsum, seipsa demonstrata, aut hec: omnis propositio est falsa. Propositio autem destruens se25 negative est propositio significans se non esse veram assertive, ut: hoc non est verum, seipsa demonstrata, aut: nulla propositio est vera. Has omnes in processu vocabo propositiones seipsas falsificantes ut expedicius loquar et verius similiter, iuxta consonantiam lingue latine.

 se] om. E  a] ad E 21  significant] solum add. E 22  loquendi philosophi] loquendo philosophus E 23  significans] falsificans M 24  hoc] hec dub. M 25  destruens se] inv. E 19 20



English Translation

fourth division: some propositions having reflection on themselves are such that their significations are only directed to themselves, such as This is true, This is false referring to themselves. But some are such that their significations are directed to themselves as well as to something other than themselves, such as: Every proposition is true, Every proposition is false. For these propositions not only signify themselves to be true or to be false, but also other propositions distinct from themselves. 2.1.5 The fifth division: some propositions having reflection on themselves endorse themselves, but some undermine themselves, as the Philosopher put it in the fourth book of the Metaphysics ⟨chap. 8, 1012b14–15⟩, where he claims that some utterances undermine themselves. A proposition endorsing itself is an utterance which stably verifies itself, like either of these: This is true referring to itself, Every proposition is true. Some self-undermining propositions undermine themselves affirmatively, some negatively. An affirmatively self-undermining proposition is a proposition that signifies or implies that it itself is false, like the proposition: This is false referring to itself, or Every proposition is false. But a negatively self-undermining proposition is a proposition signifying ⟨overtly or⟩ consequentially that it itself is not true, e.g., This is not true referring to itself, or No proposition is true. To speak both more conveniently and more truly, in conformity with the Latin language, in the course of this work I will dub all of these “self-falsifying propositions.”

2.1.4 The

139

140

Latin Text

2.1.6 Sexta

divisio: propositionum significantium se esse falsas, quedam significant se esse falsas de per se, id est omni casu circumscripto, quedam vero sic significant de per accidens, hoc est26 ex positione27 alicuius casus. Exemplum primi: de qualibet illarum: hoc est falsum, hoc non est verum, seipsis demonstratis; omnis propositio est falsa, nulla propositio est vera. Exemplum secundi: sortes dicit falsum, quam nullus haberet28 asserere esse insolubilem sine casu. Non tamen dico quin possit esse insolubilis sine casu modo superius posito in redargutione duodecime opinionis. 2.1.7 Septima divisio: propositionum seipsas falsificantium, quedam sunt totaliter (M 239ra) illative suarum falsitatum, quedam autem partialiter29 solum. Propositio falsificans se totaliter illativa sue falsitatis30 est illa que sine medio infert se esse falsam aut se non esse veram. Exemplum patet de istis: hoc est falsum, [igitur]31 hoc non est verum, seipsis demonstratis. Sequitur enim sine medio: hoc est falsum, igitur hoc est falsum; hoc non est verum, igitur hoc non est verum, demonstrando per ly hoc consequentis antecedens suum, quod est insolubile.32 Propositio partialiter33 sue falsitatis illativa est illa que sine medio non infert seipsam esse falsam ⟨vel seipsam non esse veram⟩. Verbi gratia, omnis propositio est falsa, falsum est, sortes dicit falsum. Unde non sequitur ex aliqua illarum pro hoc quod sit falsum, quocumque demonstrato, sed bene cum medio, sic arguendo: omnis propositio est falsa,34 hec est propositio, illa universali  est] om. M  positione corr.] impositione M E; we follow Peter of Ailly, Conceptus et insolubilia, b8rb 28  haberet] habet M 29  partialiter] participaliter E 30  sue falsitatis] inv. E 31  igitur] add M E, delevimus 32  insolubile] impossibile E 33  partialiter] participaliter E 34  falsum est – omnis propositio est falsa] om. (hom.) M 26 27

2.1.6 The sixth

English Translation

141

division: some propositions signifying that they themselves are false signify that they themselves are false in themselves (de per se), that is, absent any scenario, while some propositions signify this co-incidentally (de per accidens), namely from putting forward some scenario. Examples of the first ⟨kind of self-falsifying propositions⟩: any of these: This is false, This is not true, referring to themselves; Every proposition is false, No proposition is true. An example of the second: Socrates says a falsehood, which no one would have to assert to be insoluble without a scenario. However, I am not denying that it could be insoluble without a scenario in the way set out above ⟨§ 1.12.2⟩ in the refutation of the twelfth opinion. 2.1.7 The seventh division: some self-falsifying propositions wholly imply their own falsity, but some only partially. A self-falsifying proposition wholly implying its own falsity is one that without an additional premise implies that it itself is false or that it itself is not true. These are clear examples: This is false, This is not true, referring to themselves. For these are valid without an additional premise: This is false, therefore this is false, This is not true, therefore this is not true, where “this” in the conclusion refers to the premise, which is an insoluble. A self-falsifying proposition partially implying its own falsity is one that without an additional premise does not imply that it itself is false ⟨or that it itself is not true⟩. For example, Every proposition is false, A falsehood exists, Socrates says a falsehood. Thus, it does not follow from any one of them that this, no matter what is referred to by “this,” is false, but it does ⟨follow⟩ with an additional premise, arguing like this: Every proposition is false, this is a proposition (referring to that universal proposition), therefore this proposition is false.

142

Latin Text

demonstrata, igitur hec est falsa. Etiam sequitur: falsum35 est, et hec est omnis propositio, igitur ipsa est falsa. Etiam sequitur: sortes dicit falsum et non dicit nisi illam, igitur illa est falsa. Eadem divisio potest esse de propositionibus se verificantibus. Nam quedam sunt illative totaliter suarum veritatum, quedam autem parti-(E 194vb)-aliter.36 Exemplum primi: hoc est verum, se demonstrato. Exemplum secundi: omnis propositio est vera. 2.1.8 Ultima divisio: propositionum habentium reflexionem supra se, quedam est insolubilis, quedam non. Propositio habens reflexionem supra se non insolubilis est illa que non est totaliter nec partialiter37 illativa sue falsitatis aut se non esse veram,38 sicut39 sunt huiusmodi: hoc est verum, hoc non est falsum, seipsis demonstratis; omnis propositio est vera, nulla propositio est falsa. Propositio insolubilis est propositio habens supra se reflexionem sue falsitatis aut se non esse veram totaliter vel partialiter illativa, ut: hoc est falsum, hoc non est verum, seipsis demonstratis; hoc est verum, demonstrando suum contradictorium, et sic de aliis multis que ex predictis possunt assignari. Quare et cetera. 2.1.8.1 Ex predictis sequitur quod insolubile non dicitur illud quod nullo modo possit solvi, quia sic nullum predictorum40 esset insolubile, quod est falsum. Nec etiam dicitur insolubile ex hoc quod difficiliter41 possit solvi, quia multa sunt difficilia ad solvendum que tamen non vocantur insolubilia. Sed dicitur insolubile modo priori.

 falsum] falsa E  partialiter] participaliter E 37  partialiter] participaliter E 38  veram corr.] falsam M E 39  sicut] ut E 40  predictorum esset] esset predictorum et M 41  difficiliter] difficultate E 35 36



English Translation

143

Also, this is valid: A falsehood exists and this is the only proposition, therefore this proposition is false. This is valid too: Socrates says a falsehood and he says only that proposition, therefore that proposition is false. The same division can be applied to self-verifying propositions. For some of them wholly imply their own truth, but others partially. An example of the first is: This is true, referring to itself. An example of the second is: Every proposition is true. 2.1.8 The last division: some propositions having reflection on themselves are insoluble, some are not. A proposition having reflection on itself which is not insoluble is one which does not imply, either wholly or partially, its own falsity or that it itself is not true, like these: This is true, This is not false, referring to themselves, and Every proposition is true, No proposition is false. An insoluble proposition is a proposition having reflection on itself wholly or partially implying its own falsity or that it is not itself true, such as: This is false, This is not true, referring to themselves, and This is true referring to its contradictory. And so on for many other examples which can be identified on the basis of what has been said. Therefore ⟨insolubles are self-falsifying propositions⟩. 2.1.8.1 From what has been said it follows that an insoluble is not said to be something which could not be solved in any way, because otherwise none of those mentioned above would be an insoluble, which is false. Nor is an insoluble so called because it could ⟨only⟩ be solved with difficulty, since there are many things which are difficult to solve and which, nevertheless, are not called “insolubles.” But it is called an “insoluble” in the above manner.

144

Latin Text

2.1.8.2 Secundo

sequitur quod nulla propositio habet supra se reflexionem nisi illa in qua ponitur terminus appropriate42 significans propositionem, sicut sunt tales termini:43 verum, falsum, universale, particulare, affirmativum, negativum, concedendum, negandum, dubitandum, et huiusmodi. Non tamen oportet quod omnis propositio44 in qua ponitur talis terminus habeat reflexionem supra se, sicut patet de istis: falsum est, que iam est vera, et de hac similiter: hoc est verum, demonstrando illam: deus est. Non enim habet reflexionem supra se, sed ipsius significatio solum ad id45 quod demonstrat46 dirigitur et cetera. 2.1.8.3 Tertio sequitur quod multe propositiones insolubilia nominantur que non sunt propositiones insolubiles quia non habent reflexionem supra se, ut patet de istis: sortes fingit se esse sophistam47, sortes scit se errare, sortes maledixit platoni, sortes optat48 malum ciceroni, sortes non pertransibit pontem, plato non habebit denarium, et huiusmodi, posito quod quilibet dicat talem sic adequate significantem. 2.1.8.4 Quarto sequitur quod non omnis propositio significat se esse veram, nec omnis propositio49 affirmativa cathegorica50 significat suum subiectum et predicatum supponere pro eodem, nec omnis negativa51 cathegorica significat subiectum et predicatum non supponere pro eodem, quorum opposita52 sustinet undecima opinio superius impugnata. Et consequentia patet quia non quelibet talis habet reflexionem supra se.

 appropriate] approprietate M  termini] om. M 44  propositio] illa M 45  id] illud E 46  demonstrat] demonstratur E 47  sophistam] sophismatam M 48  optat] optavit E 49  propositio ] om. M 50  affirmative cathegorica] inv. E 51  negativa] necessaria E 52  opposita] oppositum E 42 43

2.1.8.2 Secondly,

English Translation

145

it follows that no proposition has reflection on itself except one in which there is a term which specifically signifies a proposition, such as these terms: “truth,” “falsehood,” “a universal,” “a particular,” “an affirmative,” “a negative,” “thing to be granted,” “thing to be denied,” “thing to be doubted,” and suchlike. Yet this does not require that every proposition containing such a term has reflection on itself, as is clear in the case of A falsehood exists, which is indeed true, and similarly in the case of This is true referring to God exists. For it does not have reflection on itself, but its signification is directed only to what it refers to and so on. 2.1.8.3 Thirdly, it follows that many propositions are called “insolubles” which are not insoluble propositions, since they do not have reflection on themselves, as is clear from these: Socrates supposes himself to be a sophist, Socrates knows himself to be mistaken, Socrates bad-mouthed Plato, Socrates wishes a misfortune to happen to Cicero, Socrates will not cross the bridge ⟨see § 5.4⟩, Plato will not have a penny ⟨see § 5.3⟩, and so on, provided that someone says such a proposition as it exactly signifies. 2.1.8.4 Fourthly, it follows that not every proposition signifies that it itself is true, nor does every affirmative subject-predicate proposition signify that its subject and predicate supposit for the same thing, nor does every negative subject-predicate proposition signify that the subject and the predicate do not supposit for the same thing, the opposites of which are held by the eleventh opinion criticized above ⟨§ 1.11⟩. And the inference is clear, since not every such proposition has reflection on itself.

146

Latin Text

2.1.8.5 Quinto53

sequitur contra eandem opinionem quod non omnis cathegorica habet duas significationes, quarum una est materialis et alia formalis. Dicit enim quod hec: homo est animal, significatione materiali significat hominem ⟨ad extra⟩ esse animal, sed significatione formali ipsam esse veram, et quod subiectum et predicatum supponunt pro eodem; et ita proportionabiliter54 dicit de negativa; sed hec sunt55 (M 239rb) falsa, ut patet. Nam hec propositio: hoc est verum, se demonstrato, non habet huiusmodi duas56 significationes, sed solum illam que significat hoc esse verum.57 Dicere enim quod significat se esse verum58 et iterum idem esse verum, replicatio inutilis est et modicum favens materie insolubilium. ⟨Suppositiones⟩

ad secundum articulum alique preambule suppositiones sunt ponende. 2.2.1 Prima suppositio est ista: quod propositio vera est illa cuius adequatum significatum59 est verum et non repugnat ipsam esse veram. Patet ex dictis in de veritate et falsitate propositionum.

2.2 Quantum

 quinto] septimo E  proportionabiliter] proportionaliter E 55  sunt] est E 56  duas] om. M 57  verum] veram M 58  verum] veram M 59  adequatum significatum] inv. E 53 54



English Translation

147

2.1.8.5 Fifthly,

it follows contrary to the same opinion that not every subjectpredicate proposition has two significations, one of which is material and the other formal. For it says that A man is an animal signifies that a man ⟨outside the mind⟩ is an animal according to its material signification, while according to its formal signification it signifies that it itself is true, and that the subject and the predicate supposit for the same thing; and it says something analogous about negative propositions. But these are false, as is clear. For this proposition: This is true referring to itself does not have these two significations, but only one which signifies that it is true. For to say that it signifies that it itself is true and again that the same thing is true is useless repetition and scarcely valuable to the matter of insolubles. ⟨Assumptions⟩

2.2 In the second section, some preliminary 2.2.1 The first assumption is this:

assumptions are laid down.

A true proposition is one whose exact significate is true and ⟨for which⟩ it is not inconsistent that the proposition is true. This is clear from what has been said in the treatise On the Truth and Falsity of Propositions ⟨ch. 9, thesis 1⟩.

148

Latin Text

2.2.2 Secunda

suppositio: propositio falsa dicitur esse illa que falsificat se, aut cuius falsitas non consurgit ex terminis sed ex adequato significato falso. Ex quibus sequitur quod aliqua est propositio falsa cuius adequatum significatum est verum. Patet de illa: hoc est falsum, se demonstrato. Quod autem ipsa sit falsa patet. Nam ipsa asserit se esse falsam, igitur est falsa; et tamen eius adequatum significatum60 est verum, quia hoc esse falsum est verum. Etiam sequitur quod omnis propositio falsificans se est falsa, licet non omnis verificans se est61 vera. Hec enim: omnis propositio est vera, verificat62 se, et tamen non est vera, ut patet. 2.2.3 Tertia suppositio est ista: ille due propositiones ad invicem convertuntur quarum idem est adequatum significatum. Probatur, nam sint63 a et b due tales que habeant idem adequatum significatum. Et arguo sic: a et b sunt due64 propositiones de extremis omnibus similibus in voce vel in scripto aut in mente, et de consimili copula. Et non est demonstratio65 (E 195ra) in una pertinens66 ad propositionem quin sit in alia, igitur ille ad invicem convertuntur. 2.2.4 Quarta suppositio est ista: ab inferiori ad suum superius affirmative et sine termino faciente sophisma est consequentia formalis. Patet in consequentiis.

 adequatum significatum] inv. E  est] esse E 62  verificat] verificari E 63  sint corr.] sunt M E 64  due] om. E 65  demonstratio] denominatio M 66  pertinens] pertinenter M 60 61

2.2.2 The second

English Translation

149

assumption: A false proposition is one which either falsifies itself or whose falsity does not arise from its terms, but from its false exact significate. From these ⟨assumptions⟩ it follows that there is a false proposition whose exact significate is true. This is clear in the case of This is false referring to itself. Now it is obvious that this proposition is false. For it asserts that it itself is false, therefore it is false; and yet its exact significate is true, because it is true that this is false. It also follows that every selffalsifying proposition is false, although not every self-verifying proposition is true. For Every proposition is true is self-verifying, but it is not true, as is clear. 2.2.3 The third assumption is this: Two propositions having the same exact significate are convertible. Proof: let A and B be two propositions having the same exact significate. I argue like this: A and B are two spoken or written or mental propositions all of whose extremes are similar and whose copulas are alike. And there is no reference to a proposition in one ⟨of them⟩ without the same reference in the other, therefore, they are convertible. 2.2.4 The fourth assumption is this: An inference from an inferior to its superior in the affirmative, where there is no term that produces a sophism, is formally valid. This is clear ⟨from what has been said⟩ in the ⟨chapter⟩ “On inferences.”

150

Latin Text

2.2.5 Quinta

suppositio: ab universali ad quamlibet suarum exponentium aut ad suas singulares cum debito medio est bonum argumentum. Et ita ab universali ad exclusivam de terminis transpositis et e contra est bonum argumentum67 et formale. Et breviter suppono68 omnes regulas consequentiarum formalium quibus asseritur formaliter valere argumentum ab hoc ad illud. Hec suppositio patet quia aliter staret aliquam consequentiam non valere et oppositum consequentis formaliter repugnare69 antecedenti, cuius oppositum premaxime suppono. Volo namque concordanter cum omnibus quod si est aliqua consequentia significans ex compositione suarum partium, et oppositum consequentis repugnat formaliter antecedenti, quod illa consequentia70 sit bona et formalis. 2.2.6 Ultima suppositio est ista: pars propositionis potest supponere pro suo toto cuius est pars et pro quolibet pertinente vel impertinente71 ad ipsum, universaliter loquendo tam in mente quam in scripto quam in voce. Hec suppositio patet ex hiis quibus impugnantur tertiadecima et quartadecima opiniones.

 et e contra –argumentum] est bonum argumentum et e contra E  suppono (dub. M)] suppositio E 69  repugnare] repugnaret M 70  consequentia] om. E 71  vel impertinente] om. E 67 68

2.2.5 The fifth

English Translation

151

assumption: An argument from a universal proposition to any of its exponents or to any of its singulars with the required middle, is ⟨formally⟩ valid. And accordingly: An argument from a universal proposition to the exclusive with the terms transposed, and vice versa, is formally valid. And, briefly, I assume all the rules of formal inferences in which it is stated that an argument from this to that is formally valid. This assumption is clear since, otherwise, it would be possible both that some inference is not valid and that the opposite of the conclusion is formally incompatible with its premise, the opposite of which, however, I assume strongly. For, in agreement with everybody, I mean that if there is an inference that signifies by the composition of its parts and the opposite of the conclusion is formally incompatible with the premise, that inference is formally valid. 2.2.6 The last assumption is this: A part of a proposition can supposit for the whole of which it is a part, and for anything, whether or not logically related to the whole, speaking generally of mental, written and vocal ⟨expressions⟩. This assumption is clear from the criticisms of the thirteenth and fourteenth opinions.

152

Latin Text

2.2.6.1 Contra

istam suppositionem arguit72 tertiadecima opinio tenens quod in mente pars non potest supponere pro toto cuius est pars, sed bene in voce vel in scripto. Sumit73 enim hanc mentalem: omnis cognitio mea74 quam non intelligo alia75 intellectione, est in mente mea. Ista propositione formata in mente,76 si dicis quod subiectum illius supponat pro illa propositione, igitur per diffinitionem suppositionis subiectum ipsius verificatur de pronomine demonstrante illam propositionem mediante copula talis propositionis. Et per consequens ista singularis si formetur erit vera77, scilicet: hec est cognitio quam non intelligo alia intellectione,78 demonstrando illam propositionem. Hoc79 autem (M 239va) nullo modo potest dici quia ex eo quod demonstro illam, intelligo eam80 alia81 intellectione ab ea, puta per ipsum pronomen demonstrativum. Et per consequens demonstrando illam universalem, illa singularis includit contradictionem,82 quare sequitur quod subiectum illius universalis non supponit pro seipsa. 2.2.6.2 Secundo capit illam aliam mentalem: omnis propositio mentalis est universalis. Si dicis83 quod predicatum supponit pro ipsa, igitur per diffinitionem suppositionis huiusmodi predicatum verificabitur de pronomine demonstrante illam propositionem mediante copula ipsius. Et per consequens hec singularis si formaretur esset84 vera: hec est universalis, demonstrata illa prima, quod est manifeste falsum. Quare et cetera.  arguit] arguitur E  sumit] secundum E 74  mea] om. E 75  alia corr.] aliqua M E; we follow Peter of Ailly’s text (Conceptus et insolubilia, c3va). 76  mea2 – mente] om. (hom.) E 77  formetur erit vera] formaliter currit vere E 78  hec est – intellectione corr.] hec cognitio quam non intelligo alia (aliqua E) intellectione est M E 79  Hoc corr.] Hec M E; we follow Peter of Ailly (loc.cit.). 80  eam] om. M. 81  alia corr.] aliqua M, om. E; we follow Peter of Ailly (loc.cit.). 82  alia intellectione – contradictionem] om. E 83  dicis] in quantum add. E 84  formaretur esset] formatur esse E 72 73

2.2.6.1 The

English Translation

153

thirteenth opinion argues against this assumption, holding that in the mind a part cannot supposit for the whole of which it is part, although it can in speech or writing. For ⟨the author of this opinion⟩ takes the mental proposition: ⟨A⟩ Every cognition of mine that I do not comprehend by means of another comprehension is in my mind. When this proposition is formed in the mind, if you say that its subject supposits for that proposition ⟨itself⟩, then, by the definition of supposition, its subject is truly predicated, via the copula of the proposition, of a pronoun that refers to that proposition. And consequently, this singular proposition, if it is formed, will be true, namely, ⟨B⟩ This is a cognition that I do not comprehend by means of another comprehension, referring to that ⟨first⟩ proposition. But that cannot be maintained at all, because by referring to that proposition, I comprehend it by means of a comprehension different from it, namely, by the demonstrative pronoun. Consequently, by referring to that universal proposition, that singular proposition includes a contradiction. For that reason, it follows that the subject of that universal proposition does not supposit for itself. 2.2.6.2 Secondly, ⟨the author⟩ takes this other mental proposition: ⟨C⟩ Every mental proposition is universal. If you say that the predicate supposits for the proposition itself, then, by the definition of supposition, this kind of predicate will be truly predicated, via its copula, of a pronoun referring to that proposition. And consequently, this singular proposition: ⟨D⟩ This is universal, referring to the first proposition, if it were formed, would be true; which is clearly false. Therefore ⟨it is clear that the predicate does not supposit for that very proposition⟩.

154

Latin Text

2.2.6.3 Alia

argumenta facit quartadecima opinio, quorum solutiones ex articulo subsequente85 patebunt. ad 2.2.6.1 Respondetur ergo ad istas rationes. Et primo ad primam, que verbaliter difficilis est et non significative. ⟨1⟩ Primo nego illam propositionem assumptam tanquam impossibilem, videlicet: “omnis cognitio mea quam non intelligo alia86 intellectione,87 est in mente mea,” dicendo quod impossibile est me88 habere in mente mea89 aliquam cognitionem et non intelligere illam.90 Concedo91 consequenter quod eadem intellectio est formalis cognitio sui, quod apud illam opinionem videtur impossibile. ⟨2⟩ Secundo dicitur, concesso antecedente gratia disputationis, negando consequentiam quia non sequitur: talis terminus supponit pro illo, igitur verificatur de illo mediante pronomine demonstrativo. Nam in illa propositione: chymera92 est intelligibilis, subiectum supponit pro chymera et tamen non verificatur de pronomine demonstrante chymeram. Item dato quod non esset aliqua propositio93 in mundo nisi illa scripta vel vocalis: omnis propositio est, subiectum supponit pro ipsa propositione, ut94 concedit hec opinio,95 et tamen subiectum non verificatur de pronomine demonstrante illam ex quo ipsa est omnis propositio. Et cum dicitur: illa est diffinitio suppositionis, dicitur negando ut96 habet videri in97 tractatu de suppositionbus.

 subsequente] sequente E  alia] aliqua M E 87  intellectione] intentione E 88  me] vix M 89  mea] om. E 90  illam] eam E 91  Concedo] Concedendo M 92  chymera] alia E 93  aliqua propositio] inv. E 94  ut] patet vel ut add. E 95  opinio] positio M 96  ut] prout M 97  in] om. M 85 86

2.2.6.3 The

English Translation

155

fourteenth opinion proposes other arguments, whose solutions will become clear in the next section ⟨§ 2.3⟩. ad 2.2.6.1 I reply to these arguments. And firstly, to the first argument, which is complicated in its words but not in meaning. ⟨1⟩ First, I deny the proposition in question as impossible, namely, ⟨A⟩ Every cognition of mine that I do not comprehend by means of another comprehension is in my mind, by saying that it is impossible that I have some cognition in my mind and ⟨yet⟩ do not comprehend it. Accordingly, I grant that the same comprehension is a formal cognition of itself, which seems impossible according to that ⟨thirteenth⟩ opinion. ⟨2⟩ Secondly, granting the premise for the sake of argument, I reply by denying the inference, since this is not valid: This term supposits for ⟨A⟩, therefore it is truly predicated of ⟨A⟩ by means of a demonstrative pronoun. For in this proposition: A chimera is intelligible the subject supposits for a chimera and yet the subject is not truly predicated of a pronoun referring to a chimera. Again, given that there was no other proposition in the world except the written or spoken proposition: Every proposition exists, the subject supposits for the proposition itself, as this opinion grants, and yet the subject is not truly predicated of a pronoun referring to that proposition, since this proposition is the only proposition. And when one says “This is the definition of supposition,” I deny it, as should be seen in ⟨my⟩ treatise On Suppositions.

156

Latin Text

⟨3⟩ Tertio,

adhuc concesso consequente, non sequitur quod hec debeat esse vera si formetur: hec ⟨est⟩ cognitio quam non intelligo alia98 intellectione;99 sed sufficit quod hec sit vera per alium a te formata: hec cognitio quam non intelligis vel que non intelligitur a te, est in mente tua. Et tunc non oportet quod intelligas illud pronomen demonstrativum aut cognitionem illam per illud pronomen demonstrativum. ⟨4⟩ Quarto dicitur quod in predicta propositione assumpta stat quod subiectum supponat pro toto vel non pro eodem supponat. Verbi gratia, si dicerem: omnis vox quam non audio est, et in proferendo illam propriam vocem100 audirem, tunc subiectum non supponeret pro suo toto sed pro distincto101 ab eodem. Si autem non audirem propriam vocem, ita quod surdus illam enunciationem dictam (E 195rb) proferrem,102 supponeret subiectum pro suo toto. Ita dico in proposito, admisso quod quelibet103 propositio mentalis sit quedam cognitio et intellectio et quod illam propositionem significatam intelligam104 et aliqua alia sit cognitio vel intellectio quam non intelligam, dico quod subiectum in casu isto non supponit pro toto cuius est pars. Si autem nec illam intellectionem intelligerem, subiectum pro suo toto supponeret, ut dictum est.

 alia corr.] aliqua M E; we follow Peter of Ailly (Conceptus et insolubilia, c3va.).  intellectione] est in mente tua add. E 100  propriam vocem] inv. E 101  distincto] demonstrato E 102  proferrem] proferet E 103  quelibet] quedam E 104  intelligam] intelligo E 98 99







English Translation

157

⟨3⟩ Thirdly, even if the conclusion is granted, it does not follow that, if it is formed, this proposition: ⟨B⟩ This ⟨is⟩ a cognition that I do not comprehend by means of another comprehension, should be true; but it suffices that the proposition: This cognition that you do not comprehend, or that is not comprehended by you, is in your mind, formed by someone other than you, is true. And ⟨if so⟩, then it is not required that you comprehend the demonstrative pronoun or the cognition by means of the demonstrative pronoun. ⟨4⟩ Fourthly, I reply that in the aforesaid proposition in question, it is possible either that the subject supposits for the whole ⟨of which it is part⟩ or that it does not supposit for it. For example, if I said Every sound that I do not hear exists and while uttering that I heard my own voice, then the subject would not supposit for the whole ⟨of which it is part⟩, but for something else. If, however, I did not hear my own voice, so that I proffered the said utterance as ⟨one who is⟩ deaf, then the subject would supposit for the whole ⟨of which it is part⟩. Thus, I say as regards the example in question, given that any mental proposition is a certain cognition and comprehension, and ⟨given⟩ that I comprehend the signified proposition and ⟨that⟩ there is some other cognition or comprehension that I do not comprehend, then I say that in that case the subject does not supposit for the whole of which it is a part. But if I do not comprehend that comprehension, then the subject would supposit for the whole ⟨of which it is part⟩, as has been said.

158

Latin Text

⟨5⟩ Quinto



reduco105 argumentum suum contra eum sic: sumendo hanc vocalem: omnis propositio non demonstrata est, quam pono me debere proferre et nullam aliam, et patet iuxta hanc opinionem quod subiectum supponit pro suo toto, igitur per diffinitionem suppositionis subiectum ipsius verificatur de pronomine demonstrante illam propositionem mediante copula (M 239vb) talis propositionis. Et per consequens ista singularis, si formetur,106 erit vera: hec propositio non demonstrata est, demonstrando illam universalem. Sed hoc non est minus inconveniens quam illud quod107 contra me108 putat concludere. ad 2.2.6.2 Ex predictis patet solutio secundi argumenti. Concedendo109 ergo quod si illa esset omnis mentalis, subiectum ipsius pro toto supponeret et hec esset vera: hec propositio est universalis, ipsa mentali demonstrata, verum est quod ipsa singularis extra mentem vera esset110, et sua universalis mentalis. Sed si ambe essent in mente, singularis esset vera et universalis falsa. Patet ergo conclusionaliter111 ex predictis quod aliqua universalis est vera que esset falsa ex sola assistentia alicuius sue singularis. Patet de qualibet tali: omnis propositio est universalis, in mente, in voce112 vel in113 scripto, dato quod aliqua talis esset et nulla alia, semper de propositione totali intelligendo et non de illa que est alterius pars. Quare et cetera.

 reduco] deduco E  formetur] formatur E 107  illud quod] id quam M 108  me] hic add. M 109  Concedendo] concedo E 110  vera esset] est vera M 111  conclusionaliter] conclusio nam E 112  in mente, in voce] in voce mente M 113  in] om. M 105 106





English Translation

159

⟨5⟩ Fifthly, I turn his argument against ⟨the author of the thirteenth opinion⟩ like this, by taking the spoken proposition: Every proposition is not referred to. Suppose I have to utter this and no other proposition. Then it is clear that according to this opinion the subject supposits for the whole ⟨of which it is part⟩. Therefore, by the definition of supposition, the subject of this proposition is truly predicated of a pronoun referring to that proposition via the copula of this proposition. And consequently, if the following singular proposition is formed: This proposition is not referred to, referring to the universal proposition, it is true. But this is no less absurd than what he seeks to conclude in opposition to me. ad 2.2.6.2 From what has been said the solution to the second argument is clear. Granting therefore that if ⟨Every mental proposition is universal⟩ were the only mental proposition, its subject would supposit for the whole ⟨of which it is part⟩ and this would be true: This proposition is universal, referring to the mental proposition itself, then it is true that the same singular proposition outside the mind would be true, as well as the corresponding universal mental proposition. But if both were in the mind, then the singular would be true and the universal false. Thus, finally it is clear from what has been said that some universal is true that would be false merely through the presence of any of its singular ⟨instances⟩. This is clear concerning any proposition like this: Every proposition is universal, mental, spoken or written, given that some such proposition exists and no other, and always understanding it with regard to a whole proposition and not any proposition which is part of another. Therefore, etc.

160

Latin Text

⟨Conclusiones⟩ 2.3 Tertius

articulus quasdam declarat conclusiones. prima est ista: aliqua sunt duo contradictoria inter se contradicentia, quorum unum est impossibile et reliquum contingens. Probatur conclusio et capio duas tales: hoc est impossibile, et: hoc non est impossibile, semper demonstrando primam. Et patet quod ipsa sunt contradictoria: sunt enim due propositiones singulares de terminis discretis, de consimilibus extremis et oppositis qualitatibus,114 igitur sunt contradictoria. Quod autem primum sit impossibile probatur, nam hec propositio: hoc est impossibile, vel est possibilis vel115 impossibilis. Si secundum, habeo116 intentum. Si primum, igitur eius adequatum significatum est possibile, igitur possibile est illam esse impossibilem. Sed nunc non aliter significat quam tunc significaret, ut suppono, igitur nunc est impossibilis, quod erat intentum. Quod autem sua contradictoria sit contingens probatur, nam possibile est hoc esse impossibile, ut patet; et possibile est hoc non esse impossibile, quia possibile est hoc non esse; et illa propositio sic117 adequate significat et non est propositio insolubilis, igitur est contingens. Patet consequentia in omnibus aliis. Ex118 predicta conclusione sequitur quod non implicat contradictionem duo contradictoria inter se contradicentia esse simul falsa. Patet119 ex quo unum est impossibile et reliquum contingens.

2.3.1 Quarum

 qualitatibus] qualitatis E  vel] aut M 116  habeo] habetur M 117  sic] sit E 118  Correlarium] marg. M 119  Patet] om. M 114 115



English Translation

161

⟨Conclusions⟩ 2.3 The third section presents 2.3.1 The first conclusion is:

some conclusions.

There are two mutually contradictory contradictories, one of which is impossible and the other contingent. I prove the conclusion. Take these two propositions: ⟨A⟩ This is impossible and ⟨B⟩ This is not impossible, each referring to the first proposition. It is clear that they are contradictories, for they are two singular propositions with discrete ⟨subject⟩ terms, with similar extremes and with opposite qualities; therefore they are contradictories. Now I prove that the first is impossible. For this proposition: ⟨A⟩ This is impossible is either possible or impossible. If the latter, I have what I wanted. If the former, then its exact significate is possible, therefore it is possible that ⟨A⟩ is impossible. But ⟨A⟩ does not signify now differently than it would signify then, as I suppose, therefore ⟨A⟩ is now impossible, which is what I aimed to prove. And I prove that its contradictory ⟨B⟩ is contingent. For it is possible that ⟨ A⟩ is impossible, as is clear; and it is possible that ⟨A⟩ is not impossible, since it is possible that ⟨A⟩ does not exist. And proposition ⟨B⟩ signifies exactly in that way and is not an insoluble proposition, therefore it is contingent. The inference is clear in all other respects. From the aforementioned conclusion it follows that no contradiction is involved in two mutually contradictory contradictories being simultaneously false. It is clear since the one is impossible and the other contingent.

162

Latin Text

2.3.2 Secunda

conclusio est ista: aliqua contradictoria inter se contradicentia sunt simul falsa. Probatur, nam hec est falsa: hoc est falsum, se demonstrato, quia falsificat se. Et hec120 similiter: hoc non est falsum ⟨demonstrando primam⟩, quia asserit hoc non esse falsum, quod est falsum. Quare121 et cetera. Et quod illa sint contradictoria patet ex deductione priori. Et ita dico de aliis insolubilibus: sortes dicit falsum, sortes non122 dicit falsum. Non tamen dico quod sic semper sit123 in materia insolubilium quod duo contradictoria sint124 simul falsa, sicut patet de istis: omnis propositio est falsa, et: aliqua propositio non est falsa. 2.3.3 Tertia conclusio: aliqua duo subcontraria125 sunt simul falsa. Probatur: et sumo126 istas duas propositiones: aliqua propositio negativa est vera, et: aliqua propositio negativa non est vera, supponendo quod non sit aliqua alia propositio quam aliqua istarum vel pars eius. Isto posito, patet quod illa sunt subcontraria, quia omnia requisita ad subcontrarietatem127 ibidem reperiuntur. Sed quod ipsa sint simul128 falsa patet. Nam hec est falsa: aliqua propositio negativa non est vera, cum se falsificet ex quo non est129 alia negativa preter ipsam. Altera similiter est falsa, videlicet: aliqua propositio negativa est vera, quia asserit aliquam esse veram et nulla est vera, igitur et cetera. Et si ex predictis aliquis correlarie130 vellet dicere131 quod per idem est possibile duo contradictoria esse simul vera et similiter duo contraria, non valet argumentum. Unde (M 240ra) non est possibile extra materiam insolubilium ut patet, nec (E 195va) etiam in materia insolubilium, ut liquet singillatim querendo. Igitur132 et cetera.  hec] hoc M  falsum. Quare] consequens igitur E 122  sortes non] nullus sortes M 123  sic semper sit] semper sit sic E 124  sint] sunt E 125  subcontraria] sunt contraria E 126  sumo] summo E 127  subcontrarietatem] contrarietatem E 128  ipsa sint simul] ista sit E 129  cum – est] quia se falsificat cum non sit E 130  correlarie] om. E 131  dicere] contradicere E 132  igitur] om. E 120 121

2.3.2 The second

English Translation

163

conclusion is this: There are some mutually contradictory contradictories that are simultaneously false. Proof: for the proposition: This is false, referring to itself, is false since it falsifies itself. And similarly this proposition: This is not false ⟨referring to the first proposition, is false⟩ because it asserts that this ⟨first proposition⟩ is not false, which is false. Therefore, etc. And it is clear from the previous argument that these are contradictories. And I say the same about other insolubles, ⟨e.g.⟩: Socrates says a falsehood, Socrates does not say a falsehood. However, I do not say that in the case of insolubles two contradictories are always simultaneously false, as is clear from these: Every proposition is false and Some proposition is not false. 2.3.3 Third conclusion: There are some pairs of subcontraries that are simultaneously false. Proof: take these two propositions: Some negative proposition is true and Some negative proposition is not true, assuming that there is no other proposition than either of these or part of them. Supposing this, it is clear that they are subcontraries, because they meet all the requirements of subcontrariety. And it is clear that they are simultaneously false. For Some negative proposition is not true, is false, since it falsifies itself, given that there is no other negative proposition except it. Similarly, the other proposition, namely, Some negative proposition is true, is false, because it asserts that some ⟨negative proposition⟩ is true, yet there is no ⟨negative proposition⟩ that is true, therefore etc. And if from what has been said someone wanted to state as a corollary that, by parity of reasoning, it is possible that two contradictories are simultaneously true and, similarly, two contraries, the argument is not valid. For this is not possible outside the case of insolubles, as is clear, nor even in the case of insolubles, as is evident by studying ⟨them⟩ one by one, therefore etc.

164

Latin Text

2.3.4 Quarta

conclusio: aliqua consequentia est133 bona et formalis, scita a te esse talis, significans ⟨adequate⟩ ex compositione suarum partium, et antecedens est scitum a te et consequens non est scitum a te. Hanc conclusionem fere omnes permittunt transire tamquam possibilem faciendo hanc consequentiam: hoc est nescitum a te,134 igitur hoc est nescitum a te, demonstrando per utrumque hoc consequens eiusdem135 consequentie. Patet namque consequentiam illam esse bonam et formalem, quia non videtur quomodo stabit oppositum consequentis cum antecedente et antecedens136 est scitum a te, quia scis illud consequens non137 sciri, cum sit insolubile asserens se nesciri. Et tamen consequens138 non est scitum a te. Probatur, quia si est scitum a te, igitur est verum et significat adequate ipsum nesciri a te, igitur verum est ipsum nesciri a te,139 igitur tu non scis illud consequens. Et sic qualitercumque dicatur sequitur illud consequens non esse scitum a te. 2.3.5 Quinta conclusio: aliqua consequentia est bona et formalis significans adequate ex compositione suarum partium principalium et antecedens est verum et consequens falsum. Probatur, nam facio hanc consequentiam: hoc est falsum, igitur hoc est falsum, per utrumque hoc demonstrando consequens. Et patet quod est ⟨bona et⟩ formalis eo quod oppositum contradictorium consequentis repugnat antecedenti eiusdem. Quis enim diceret140 ista stare simul: hoc est falsum, et: hoc non est falsum, eodem demonstrato? Certe nullus141 maius volens vitare inconveniens. Sed quod antecedens illius consequentie sit verum patet, cum non sit propositio insolubilis et sic adequate significans verum, scilicet hoc esse142 falsum consequente demonstrato. Quod autem consequens sit falsum patet, cum sit propositio insolubilis seipsam falsificans.  consequentia est] inv. E  a te] hec E 135  eiusdem] illius E 136  antecedens] consequens E 137  non] est add. E 138  consequens] antecedens E 139  igitur – a te] om. (hom.) E 140  diceret] diceretur M 141  nullus] nullum M 142  esse] est E 133 134

2.3.4 Fourth

English Translation

165

conclusion: There is a formally valid inference, known by you to be so, signifying ⟨exactly⟩ by the composition of its parts, where the premise is known by you, yet the conclusion is not known by you. Almost everyone agrees to let this ⟨fourth⟩ conclusion go through as possible by forming this inference: This is unknown to you, therefore this is unknown to you, where each ⟨occurrence of⟩ “this” refers to the conclusion of the inference. For it is evident that this inference is formally valid, because one cannot see how the opposite of the conclusion can be compatible with the premise. But the premise is known by you, because you know that the conclusion is not known, since it is an insoluble that asserts that it itself is unknown. And yet the conclusion is not known by you. Proof: because if it is known by you, then it is true and it signifies exactly that it is unknown to you, therefore it is true that it is unknown to you, therefore you do not know the conclusion. And thus whatever one says, it follows that the conclusion is not known by you. 2.3.5 Fifth conclusion: there is a formally valid inference, signifying exactly by the composition of its principal parts, where the premise is true and the conclusion is false. Proof: for I form this inference: This is false, therefore this is false, where each ⟨occurrence of⟩ “this” refers to the conclusion. And it is clear that it is formally ⟨valid⟩, since the contradictory opposite of the conclusion is incompatible with its premise. For who would claim that these are compatible: This is false and This is not false, referring to the same thing? Surely, no-one who wishes to avoid a greater absurdity. But it is clear that the premise of this inference is true, since it is not an insoluble proposition and it signifies a truth exactly, namely that this is false, referring to the conclusion. And it is clear that the conclusion is false, since it is a self-falsifying insoluble proposition.

166

Latin Text

2.3.6 Sexta

conclusio: aliqua consequentia est bona et formalis significans adequate ex compositione suarum partium et antecedens est possibile et consequens impossibile. Patet de tali consequentia: hoc est impossibile, igitur hoc est impossibile, semper demonstrando consequens. Et patet quod est consequentia143 bona et formalis, quia contradictorium consequentis antecedenti repugnat. Et antecedens est possibile, ymmo verum cum non sit propositio insolubilis144 et ipsius adequatum significatum sit verum. Et quod consequens sit impossibile, superius est deductum. Et hic rogo legentem ut145 consideret circa quartam conclusionem, et non apparebunt sibi quinta et sexta inconvenientes. Cum enim opponitur: illa consequentia est bona ⟨et formalis et⟩ scita a te esse bona et etc.146 et antecedens est scitum a te, igitur et consequens, dicitur negando consequentiam, quia oportet addere in antecedente: et non repugnat consequens sciri. Ita dico in proposito quod non sequitur: hec147 consequentia est bona et formalis et antecedens est verum vel possibile, igitur et consequens est verum vel possibile; sed oportet addere in antecedente: et non repugnat consequens esse verum vel possibile, quod si additur negabitur antecedens et in propositis consequentiis pro hac parte, quare etc. 2.3.7 Septima conclusio: alique due propositiones ad invicem convertuntur et tamen una est vera vel possibilis et reliqua falsa vel impossibilis. Patet de talibus:148 hoc est falsum, et: hoc est falsum, secunda demonstrata; aut de istis: hoc est impossibile, et: hoc est impossibile, una illarum149 solummodo demonstrata150. Quod autem tales convertantur151 patet ex tertia suppositione, cum idem sit ­adequatum

 consequentia] om. M  non sit propositio insolubilis] sit non insolubilis propositio M 145  ut] quod E 146  et etc.] om. E 147  hec] ista E 148  talibus] ista E 149  una illarum] uno illorum M 150  demonstrata] demonstrato M 151  convertantur] convertuntur E 143 144

2.3.6 Sixth

English Translation

167

conclusion: there is a formally valid inference, signifying exactly by the composition of its parts, where the premise is possible and the conclusion is impossible. This is clear as regards this inference: This is impossible, therefore this is impossible, in both cases referring to the conclusion. And it is clear that it is a formally valid inference, since the contradictory of the conclusion is incompatible with the premise. And the premise is possible, certainly in as much as it is not an insoluble proposition and its exact significate is true. And that the conclusion is impossible has been proven above ⟨§ 2.3.1⟩. And here I ask the reader to reflect upon the fourth conclusion, and then the fifth and sixth will not seem absurd to him. For if one objects: That inference ⟨in 2.3.4⟩ is ⟨formally⟩ valid and known by you to be valid, ⟨and signifies by the composition of its parts⟩, and the premise is known by you, therefore so too ⟨is⟩ the conclusion, I reply by denying the inference, because one should add in the premise: and it is not incompatible that the conclusion is known. Similarly, I say in the case at hand that this is not valid: This inference ⟨“This is impossible, therefore this is impossible”⟩ is formally valid, and the premise is true or possible, therefore the conclusion is also true or possible, but one should add in the premise: and it is not incompatible that the conclusion is true or possible. If that is added, the premise will be denied, and the same is true in the proposed inferences, therefore etc. 2.3.7 Seventh conclusion: some pairs of propositions are mutually convertible, and yet one is true or possible and the other is false or impossible. It is clear as regards these propositions: This is false and This is false, ⟨both⟩ referring to the second; or of these: This is impossible and This is impossible,

168

Latin Text

significatum152 utriusque prius assignatarum153 et ita aliarum.154 Similiter nullius illarum155 contradictorium [consequentis]156 est alteri sic adequate significando compossibile, ut patet intuenti naturam incompossibilium,157 igitur ad invicem convertuntur. Secunda pars conclusionis ex dictis manifeste habetur.

⟨Correlaria⟩ 2.4.1 Ex predictis

sequitur158 universalem esse veram et suam subalternam esse falsam. Patet de tali: omnis propositio160 particularis affirmativa est falsa, que vera est; et hec est falsa: aliqua propositio particularis161 affirmativa est falsa, dato quod ille sunt omnes propositiones, saltim totales. 2.4.2 Secundo sequitur162 universalem esse falsam (M 240rb) et quamlibet suam singularem esse veram et cuilibet supposito subiecti ⟨ipsius universalis⟩ unam singularem correspondere. Patet de tali: omnis propositio universalis est falsa, dato quod sit omnis universalis, que falsa est quia seipsam falsificat; et hec163 cum casu illo esset vera: ista ⟨propositio⟩ universalis est falsa, eandem universalem demonstrando. Et patet quod esset quelibet singularis eius ⟨universalis⟩ cuilibet supposito ⟨subiecti⟩ illius ⟨universalis⟩ correspondens, cum tale suppositum unicum diceretur. 159



 significatum] om. M  assignatarum] significatarum vel signatarum M 154  aliarum] alium E 155  nullius illarum] nullum illorum E 156  consequentis] add. M E, delevimus 157  incompossibilium] incompossibilitatum M 158  Primum correlarium] marg. M 159  subalternam] subalternantem M 160  propositio] om. M 161  propositio particularis corr.] propositio M, particularis propositio E 162  Secundum] marg. M 163  hec] hoc M 152 153



English Translation

169

both referring to only one of them. And that they are convertible is clear from the third assumption, since the exact significates of the first pair as well as of the other pair are the same. Similarly, the contradictory of neither of them is compatible with the other, signifying exactly in that way, as is clear to anyone who reflects on the nature of incompatibles. Therefore ⟨these propositions⟩ are mutually convertible. The second part of the ⟨seventh⟩ conclusion is evident on the basis of what has been said. ⟨Corollaries⟩



2.4.1 From

what has been said it follows that there is a true universal proposition whose subalternate is false. It is clear regarding this proposition: Every particular affirmative proposition is false, which is true; while this is false: Some particular affirmative proposition is false, given that these are the only propositions, at least ⟨the only⟩ whole propositions. 2.4.2 Secondly, it follows that there is a false universal proposition each of whose singulars is true, and where to each suppositum of the subject ⟨of the universal proposition⟩ there corresponds one singular proposition. It is clear regarding this proposition: ⟨A⟩ Every universal proposition is false, given that it is the only universal proposition. ⟨A⟩ is false since it falsifies itself; and in that scenario ⟨where A is the only universal proposition⟩, this proposition would be true: ⟨B⟩ This universal proposition is false, referring to that same universal proposition ⟨A⟩. And it is clear that ⟨B⟩, corresponding to each suppositum ⟨of the subject⟩ of the ⟨universal⟩ ⟨A⟩, would be every singular ⟨of the universal⟩, since it would be said that there is only one such suppositum.

170

Latin Text

2.4.3 Tertio

sequitur164 quamlibet singularem esse falsam et universalem esse veram tali singulari correspondentem. Patet et suppono quod non sit aliqua universalis nisi illa: omnis propositio singularis est falsa, nec aliqua singularis nisi illa: hec propositio166 singularis est falsa, seipsa demonstrata. Et patet veritas cor-(E 195vb)relarii. 2.4.4 Quarto sequitur167 expositam vel exponendam esse veram et quamlibet suam168 exponentem esse169 falsam. Patet170 dato quod non essent plures propositiones quam ille tres vel partes ipsarum: omnis propositio particularis affirmativa et universalis negativa est falsa, aliqua propositio particularis affirmativa et universalis negativa est falsa, et: nulla est propositio particularis affirmativa et universalis negativa quin ipsa sit falsa. Quelibet enim exponentium se falsificat171 et exposita non, sed pro suis exponentibus verificatur. 2.4.5 Quinto sequitur172 e contra173 exponentes esse veras et expositam falsam. Patet dato quod hec174 esset omnis propositio175 universalis affirmativa,176 videlicet: omnis universalis affirmativa est falsa. 2.4.6 Sexto sequitur177 universalem esse veram et suam exclusivam falsam. Patet dato quod hec exclusiva: tantum falsum178 est exclusiva, esset omnis exclusiva, tunc ipsa esset falsa et hec esset vera: omnis exclusiva est falsa. 165

 Tertium correlarium] marg. M  esse] om. M 166  propositio] om. E 167  Quartum] marg. M 168  suam] suarum partium E 169  esse] om. M 170  patet] et add. E 171  se falsificat] inv. E 172  Quintum] marg. M 173  e contra] e converso E 174  hec] hoc M 175  propositio] om. E 176  affirmativa] affirmative E 177  Sextum] marg. M 178  falsum] falsa E 164 165

2.4.3 Thirdly,

English Translation

171

it follows that ⟨there are cases where⟩ each singular is false and the universal corresponding to those singulars is true. It is clear, assuming that there is no other universal proposition than: Every singular proposition is false, nor any other singular proposition than: This singular proposition is false, referring to itself. And the truth of the corollary is patent. 2.4.4 Fourthly, it follows that ⟨there are cases where⟩ an expounded proposition or one to be expounded is true and each of its exponents is false. This is clear given that there are no more propositions than these three or their parts: Every particular affirmative or universal negative proposition is false, There is a particular affirmative and a universal negative proposition which are false, and There is no particular affirmative or universal negative proposition that is not false. For each of the exponents falsifies itself and the expounded proposition does not, but is made true by its exponents. 2.4.5 Fifthly, it follows conversely that ⟨there are cases where⟩ the exponents are true and the expounded proposition false. This is clear given that the only universal affirmative proposition is Every universal affirmative is false. 2.4.6 Sixthly, it follows that ⟨there are cases where⟩ a universal proposition is true and its exclusive is false. This is clear given that this exclusive: Only a falsehood is an exclusive proposition were the only exclusive proposition. Then it would be false and this would be true: Every exclusive proposition is false.

172

Latin Text

2.4.7 Septimo

sequitur179 e contra180 universalem esse falsam et ⟨suam⟩ exclusivam veram. Patet dato quod hec esset quelibet universalis: omnis propositio universalis est falsa, ex quo falsificaret se falsa esset;181 et hec exclusiva esset vera: tantum falsum est propositio universalis. 2.5 Et licet concedam huiusmodi conclusiones, non tamen nego consequentiam aliquam nec dico esse falsam iuxta doctrinam quinte suppositionis, ita quod ab exclusiva ad suam universalem et e contra semper est argumentum bonum et182 formale, stat tamen antecedens esse verum sine consequente, ut quinta declarat conclusio.183 Et si contra predicta instaretur allegando dictum aliquod Aristotelis seu alterius authentici in hac parte, dicatur tale intelligi extra materiam insolubilium. Similiter si instetur contra me dicendo quod omnium istorum opposita184 affirmavi, dico quod non sum locutus oppositum alicuius istorum, nisi forte verbaliter, quia talia semper intellexi extra materiam insolubilium. Sunt enim ordinanda verba iuxta exigentiam materie tractande; esset enim tedium et nimia prolixitas totiens recitare idem quod plus ad multiplicationem verborum fecit185 quam utilitatis.

 Septimum] marg. M, om. E  e contra] e converso M 181  falsa esset] esse falsam E 182  bonum et] om. M 183  ut – conclusio] sexta conclusio declarat E 184  istorum opposita corr.] istarum opposita M, istorum opposito E 185  verborum fecit corr.] verborum facit M, fecit verborum E 179 180

2.4.7 Seventhly,

English Translation

173

it follows conversely that ⟨there are cases where⟩ a universal proposition is false and ⟨its⟩ exclusive is true. This is clear given that this were the only universal: Every universal proposition is false, which would be false since it falsifies itself; and this exclusive proposition would be true: Only a falsehood is a universal proposition. 2.5 And although I grant conclusions of this sort, I do not deny any inference, nor do I say that it is false according to the teaching of the fifth assumption, so that from an exclusive to its universal, and vice versa, is always a formally valid argument, even though the premise can be true and the conclusion not, as the fifth conclusion states. And if someone objects to what has been said by adducing some dictum of Aristotle’s or some other authority in this matter, the answer should be that this ⟨dictum⟩ has to be understood as excluding the case of insolubles. Similarly, if someone objects to me by saying that I have claimed the opposite of all of them, I reply that I did not say anything opposite to any of them, unless perhaps merely verbally, since I always understood them as excluding the case of insolubles. For words should be disposed according to what is required by the matter to be dealt with; for to rehearse the same thing so often would be tedious and too prolix, resulting more in an increase of words than of usefulness.

174

Latin Text

⟨Capitulum Tertium. Obiectiones et Responsiones⟩ 3.1 Contra

fundamentum huiusmodi opinionis1 arguitur probando nullum insolubile significare precise sicut termini pretendunt, reducendo sepe contradictionem tali casu admisso. Pono ergo primitus quod sortes dicat solummodo illam2 vel partem eius: non est ita sicut sortes dicit, que sic precise significat3 iuxta hanc opinionem. Isto posito, quero si est ita sicut sortes dicit vel non. 3.1.1 Si dicitur quod non est ita sicut sortes dicit, pono cum casu quod plato dicat istam solummodo4: non est ita sicut sortes dicit, et arguitur sic: non est ita sicut sortes dicit, et ita solummodo dicit plato; igitur ita est5 sicut plato dicit. Et ultra: ita est sicut plato dicit et sortes dicit totaliter sicut plato dicit et solum sic; igitur ita est sicut sortes dicit, quod est contradictorium (M 240va) prius concessi. Confirmatur, ⟨si⟩ non est ita sicut sortes dicit, igitur verum est quod non est ita sicut sortes dicit. Consequentia tenet quia sequitur universaliter: tu non curris, igitur verum est quod tu non curris. Tunc ⟨arguitur⟩ sic: verum est quod non est ita sicut sortes dicit, et sortes ⟨solummodo⟩ dicit sic, igitur sortes dicit sicut verum est esse; igitur sortes dicit sicut est, et per consequens ita est sicut sortes dicit.

 opinionis] positionis E  solummodo illam] inv. E 3  significat] significet E 4  istam solummodo] inv. E 5  ita est] inv. E 1 2



English Translation

175

⟨Chapter Three: Objections and Replies⟩ 3.1 Arguments

can be given against the basic idea of the ⟨present⟩ kind of opinion, by showing that no insoluble signifies only as the terms suggest, commonly by deducing a contradiction when the following scenario is admitted. First, therefore, assume that Socrates says only this or part of it: It is not as Socrates says it is, which according to this opinion signifies only in that way. Given this ⟨scenario⟩, I ask whether it is as Socrates says it is or not. 3.1.1 If one answers that it is not as Socrates says it is, I extend the scenario by assuming that Plato says only this: It is not as Socrates says it is, and I argue like this: it is not as Socrates says it is, and Plato says only that, therefore it is as Plato says it is. And further: it is as Plato says it is and Socrates says wholly what Plato says and only that; therefore it is as Socrates says it is, which is the contradictory of what was granted earlier. This is confirmed ⟨if⟩ it is not as Socrates says it is, then it is true that it is not as Socrates says it is. The inference holds, for this is valid universally: You are not running, therefore it is true that you are not running. Then ⟨I argue⟩ like this: it is true that it is not as Socrates says it is, and Socrates says ⟨only⟩ that, therefore Socrates says what is truly so; therefore Socrates says it as it is, and consequently, it is as Socrates says it is.

176

Latin Text

3.1.2 Si autem

in principio conceditur6 tamquam sequens quod ita est sicut sortes dicit, arguitur sic: ita est sicut sortes dicit, et plato solummodo dicit quod non est ita sicut sortes dicit, igitur non est ita sicut plato dicit. Tunc ⟨arguitur⟩ sic: non est ita sicut plato dicit, et plato solummodo dicit sicut sortes dicit et e contra, igitur non est ita sicut sortes dicit, quod est oppositum prius concessi. Confirmatur, nam sequitur: ita est sicut sortes dicit, igitur falsum est quod non est ita sicut sortes dicit. Consequentia patet7 quia sequitur universaliter: tu es, igitur falsum est te non esse. Tunc arguitur sic: falsum est quod non est ita sicut sortes dicit, et sortes sic dicit solummodo, igitur sortes dicit sicut falsum est esse. Et per consequens non est ita sicut sortes dicit. 3.1.3 Idem argumentum potest fieri ponendo quod quilibet homo dicat solummodo: non est ita sicut aliquis homo dicit, que sic precise significet. Quo posito queratur ut prius si ita est8 sicut aliquis homo dicit vel non et reducatur contradictio quacumque parte sumpta contradictionis. 3.2 Secundo arguitur sic: et pono quod a sit illa: hec propositio significat aliter quam est, eadem demonstrata,9 que precise sic significet. Quo posito, vel ita est totaliter sicut a propositio significat vel non.

 conceditur] concedatur E  patet] tenet E 8  est] sit E 9  demonstrata] demonstrato E 6 7

3.1.2 If, however,

English Translation

177

at the very beginning one grants as following ⟨from the scenario⟩ that it is as Socrates says it is, I argue in this way: it is as Socrates says it is, and Plato says only that it is not as Socrates says it is, therefore it is not as Plato says it is. Then ⟨I argue⟩ like this: it is not as Plato says it is, and Plato says only what Socrates says and conversely, therefore it is not as Socrates says it is, which is the opposite of what was granted earlier. This is confirmed, for this is valid: it is as Socrates says it is, therefore it is false that it is not as Socrates says it is. The inference is clear, for this is valid universally: You exist, therefore it is false that you do not exist. Then I argue like this: it is false that it is not as Socrates says it is, and that is all that Socrates says, therefore it is false that it is as Socrates says it is. And consequently, it is not as Socrates says it is. 3.1.3 The same argument can be made by assuming that every man says only It is not as any man says it is, which signifies only in that way. Given this ⟨scenario⟩, I ask as before if it is as any man says it is or not, and a contradiction may be deduced when either side of the contradiction is assumed. 3.2 Secondly, I argue like this. Let A be: This proposition signifies other than it is referring to itself, which signifies only in that way. Given this ⟨scenario⟩, either it is wholly as proposition A signifies or not.

178

Latin Text

3.2.1 Si non

est ita, contra sequitur: non est ita totaliter10 sicut a propositio11 significat, et a propositio aliqualiter ⟨esse⟩ significat, igitur a propositio significat aliter quam est. Quo concesso, ponatur quod b sit una propositio precise significans quod hec12 significat13 aliter quam est, demonstrata14 a, et sit15 eius contradictorium c, precise (E 196ra) significans quod hec non significat aliter quam est. Istis positis sequitur quod si a propositio significat aliter quam est et b propositio significat precise quod a propositio significat aliter quam est, igitur ita est totaliter sicut b propositio significat, et a propositio totaliter significat16 sicut b et e contra, igitur ita est totaliter sicut a significat, et si sic, igitur a non significat aliter quam est. 3.2.2 Si autem dicitur in principio quod ita est totaliter sicut a significat, contra: tunc c est verum, et sequitur: c est verum, et17 b et c sunt contradictoria, igitur b est falsum. Tunc ⟨arguitur⟩ sic: b est falsum et non falsificat se, igitur b significat aliter quam est et a totaliter significat sicut b significat, igitur a significat aliter quam est, et per consequens non ita est18 totaliter sicut a significat. 3.2.3 Et ita in casu isto sequitur contradictio. 3.3 Tertio arguitur sic: et pono quod a sit illa particularis: aliqua propositio significat aliter quam est, que sit omnis propositio et significet precise sicut termini pretendunt. Isto posito, quero si aliqua propositio significat aliter quam est vel nulla. 3.3.1 Si dicitur quod aliqua propositio significat aliter quam est et a est omnis propositio, igitur a significat aliter quam est. Tunc ⟨arguitur⟩ sic: a significat aliter quam est et a significat precise quod aliqua propositio significat aliter quam est, igitur non est ita quod aliqua propositio significat aliter quam est, et per consequens nulla propositio significat aliter quam est, quod est oppositum primi ⟨concessi⟩.  totaliter] om. E  propositio] totaliter add. E 12  hec] hoc E 13  significat] precise add. E 14  demonstrata] demonstrato M 15  sit] sic E 16  totaliter significat] inv. E 17  et] om. E 18  ita est] inv. E 10 11

3.2.1 If it is

English Translation

179

not ⟨wholly as A signifies⟩, on the contrary, this is valid: It is not wholly as proposition A signifies, and proposition A signifies ⟨that it is⟩ in some way, therefore proposition A signifies other than it is. Having granted that, let B be a proposition which signifies only that this proposition signifies other than it is, referring to A, and let C be the contradictory of B, signifying only that proposition A does not signify other than it is. Given these ⟨stipulations⟩, it follows that if proposition A signifies other than it is, and proposition B signifies only that proposition A signifies other than it is, then it is wholly as proposition B signifies, and proposition A signifies wholly as B does and vice versa, therefore, it is wholly as A signifies, and if so, then A does not signify other than it is. 3.2.2 If, however, at the beginning one answers that it is wholly as A signifies, on the contrary: then C is true; and this is valid: C is true and B and C are contradictories, therefore B is false. Then ⟨I argue⟩ in this way: B is false and it does not falsify itself, therefore, B signifies other than it is, and A signifies wholly as B signifies, therefore A signifies other than it is, and consequently it is not wholly as A signifies. 3.2.3 So a contradiction follows in this scenario. 3.3 Thirdly, I argue like this. Let A be this particular proposition: Some proposition signifies other than it is, and let it be the only proposition, and let it signify only as the terms suggest. Given this, I ask whether any proposition signifies other than it is or none does. 3.3.1 If one answers that some proposition signifies other than it is, and A is the only proposition, then A signifies other than it is. Then ⟨I argue⟩ like this: A signifies other than it is, and A signifies only that some proposition signifies other than it is, therefore, it is not the case that any proposition signifies other than it is, and consequently no proposition signifies other than it is, which is the opposite of what was first ⟨granted⟩.

180

Latin Text

3.3.2 Si autem

dicitur in principio quod nulla propositio significat aliter quam est, igitur a non significat aliter quam est, et a aliqualiter ⟨esse⟩ significat, igitur a significat precise sicut est; sed a significat precise quod aliqua propositio significat (M 240vb) aliter quam est, igitur ita est quod aliqua propositio significat aliter quam est. Et per consequens aliqua propositio significat aliter quam est, quod est oppositum principalis assumpti et concessi. 3.4 Quarto arguitur sic: data positione sequitur quod aliud quam necessarium contradicit impossibili. Probatur: et pono quod a sit illa propositio: hec propositio non est necessaria, demonstrata seipsa, que sic precise significet; et sit b eius contradictorium: hec ⟨propositio⟩ est necessaria, que sic precise significat, demonstrando a. Isto posito, ⟨sequitur quod⟩ b est impossibile quoniam ipsum implicat contradictionem. Asserit enim a esse necessarium, sed a esse necessarium implicat contradictionem, ex quo assertive significat se non esse necessarium. Arguitur ergo19 sic: a non est necessarium propter implicationem contradictionis et a contradicit b impossibili, igitur aliud quam necessarium contradicit impossibili, quod erat probandum. 3.5 Quinto arguitur sic: nam sequitur quod due sunt propositiones quarum utraque contradicit eidem propositioni impossibili et tamen una istarum est necessaria et alia contingens. Probatur: et pono quod c sit una talis propositio sicut est a illa videlicet: hec ⟨propositio⟩ non est necessaria, demonstrando a, que sic precise significet. Et sequitur, ut notum est, quod c est necessarium quia eius adequatum significatum est necessarium et ipsum c non est insolubile20, et a non est necessarium,21 ut deductum est, nec etiam impossibile quia ipsum est verum; igitur est contingens. Tunc arguitur sic: a est contingens et c necessarium, quorum quodlibet contradicit b impossibili, ⟨sequitur⟩ igitur conclusio ⟨proposita⟩.

 ergo] enim E  insolubile] impossibile M 21  necessarium corr.] necessariam M E 19 20

3.3.2 If, however,

English Translation

181

at the beginning one answers that no proposition signifies other than it is, then A does not signify other than it is, but A signifies ⟨that it is⟩ in some way, therefore A signifies only as it is; yet A signifies only that some proposition signifies other than it is, therefore it is the case that some proposition signifies other than it is. And, consequently, some proposition signifies other than it is, which is the opposite of what was first assumed and granted. 3.4 Fourthly, I argue like this: given ⟨this⟩ position, it follows that something other than the necessary contradicts the impossible. Proof: let A be this proposition: This proposition is not necessary, referring to itself, which signifies only in that way, and let B be its contradictory: This ⟨proposition⟩ is necessary, referring to A and signifying only in that way. Given this, ⟨it follows that⟩ B is impossible because it implies a contradiction. For it asserts that A is necessary, yet that A is necessary implies a contradiction since it signifies consequentially that it is not necessary. Therefore I argue like this: A is not necessary because ⟨its being necessary⟩ implies a contradiction, and A contradicts the impossibility B, therefore something other than the necessary contradicts the impossible, which is what was to be proven. 3.5 Fifthly, I argue like this: for it follows that there are two propositions, each of which contradicts the same impossible proposition, and yet one of them is necessary and the other contingent. Proof: assume that C is a proposition just like A, namely, This ⟨proposition⟩ is not necessary, referring to A and signifying only in that way. It follows, as is clear, that C is necessary, for its exact significate is necessary and C is not an insoluble. And A is not necessary, as was proven, nor ⟨is it⟩ impossible, for it is true; therefore it is contingent. Then I argue like this: A is contingent and C ⟨is⟩ necessary and each of them contradicts the impossibility B, therefore the ⟨proposed⟩ conclusion ⟨follows⟩.

182

Latin Text

3.6 Sexto sequitur



quod aliqua propositio est que simul est necessaria et contingens. Probatur de a, nam quod a sit contingens probatur, nam a est verum et non necessarium, igitur contingens. Et quod ipsum a sit necessarium probatur, nam adequatum significatum a est necessarium et a non est insolubile nec seipsum falsificans, igitur a est necessarium. Item c convertitur cum a quia sequitur: hoc non est necessarium, igitur hoc non est necessarium, et e contra,22 semper demonstrando a; sed c est necessarium, ut dictum est, igitur et a. 3.7 Alia contra hanc positionem contradicentes movent argumenta, quorum solutiones in altero capitulo conclusionaliter proponuntur. ad 3.1 Ideo ad hec respondendo23 motiva, assero ut prius insolubile significare precise sicut termini pretendunt vel saltem contradictionem non implicat, ut asserunt oppositum sustinentes. Unde hec sunt contraria: omnis propositio est falsa, nulla propositio est falsa; omnis propositio est vera, nulla propositio est vera, quorum unum non significat copulative nec duobus modis, igitur nec alterum. Consequentia patet quia qualitercumque asserit unum contrariorum, opposito modo contrarie asserit reliquum. Quod autem illa sint contraria24 patet quia universalis affirmativa et universalis negativa de eisdem subiectis et25 predicatis supponentibus26 debito modo sunt contrarie27 in figura. Idem dicitur de contradictoriis, ut: omnis propositio est falsa,28 aliqua propositio non est falsa;29 nulla propositio est (E 196rb) vera, aliqua propositio est vera. Quod ista sunt contradictoria patet30 per regulam contradictoriorum et nulla particularis significat copulative aut duobus modis, igitur nec universalis aliqua illarum, quarum tamen quelibet est insolubilis.31  e contra] e converso E  respondendo] responsio M 24  illa sint contraria] iste sunt contrarie E 25  et] om. M 26  supponentibus] suppositionibus M 27  contrarie] contraria E 28  falsa] vera E 29  falsa] vera E 30  patet] certum est M 31  insolubilis] insolubile E 22 23

3.6 Sixthly,

English Translation

183

it follows that there is some proposition which is necessary and contingent at the same time. I prove it about A ⟨“This proposition is not necessary”⟩, for I prove that A is contingent: for A is true and not necessary, therefore ⟨it is⟩ contingent. And I prove that A is necessary: for the exact significate of A is necessary and A is not an insoluble and does not falsify itself, therefore A is necessary. Again, C and A are convertible since this is valid: This is not necessary, therefore this is not necessary and vice versa, referring to A throughout; yet C is necessary, as has been said, therefore A is too. 3.7 The opponents raise other arguments against this position, whose solutions are presented as conclusions in chapter ⟨2⟩. ad 3.1 Replying to these arguments, therefore, I assert as above ⟨§ 3.1⟩ that an insoluble signifies only as the terms suggest or at any rate that it does not imply a contradiction, as the opponents claim it does. Hence, these are ⟨pairs of⟩ contraries: Every proposition is false, No proposition is false; Every proposition is true, No proposition is true, one of which does not signify conjunctively or in two ways, therefore neither does the other. The inference is clear because in whatever way one of ⟨two⟩ contraries asserts, the other asserts in the opposite and contrary way. Yet that these are contraries is clear because a universal affirmative and a universal negative with the same subjects and predicates suppositing in the way required are contraries in the square of opposition. The same is said of ⟨pairs of⟩ contradictories, such as: Every proposition is false, Some proposition is not false; No proposition is true, Some proposition is true. That these are contradictories is clear according to the rule of contradictories, and none of the particular propositions signifies conjunctively or in two ways, therefore neither does either universal corresponding to these particulars ⟨signify conjunctively or in two ways⟩, yet each of the universals is an insoluble.

184



Latin Text

Confirmatur: nam ille convertuntur: omnis universalis falsa est, et: aliqua universalis falsa est, quia universalis et particularis de secundo adiacente convertuntur, sed illa particularis non significat copulative nec duobus modis, igitur nec universalis, non obstante quod sit insolubilis, dato quod ipsa foret omnis propositio. Unde ista propositio: falsum est, non significat nisi uno modo. Non ergo video unde significaret copulative vel duobus modis, dato quod quelibet alia propositio32 corrumperetur,33 quia non a se haberet illam novam impositionem nec ab aliquo alio.34 ad 3.1.1 Ad primam rationem, admisso casu, concedo quod non est ita sicut sortes dicit, sicut concederem quod sortes non dicit35 verum, unde insolubile copulatum (M 241ra) seipso36 ⟨tamquam⟩ falsum solummodo est concedendum et quodlibet secum convertibile non tamquam verum, sed tamquam sequens. Hec enim: hoc est falsum, seipsa demonstrata, semper est concedenda,37 licet non concedatur ipsam esse veram. Et tunc ad argumentum: non est ita sicut sortes dicit et ita solummodo38 dicit plato, igitur ita est sicut plato dicit, concedo39 consequentiam et consequens. Et tunc ultra: ita est sicut plato dicit et sortes dicit totaliter sicut plato dicit, igitur ita est sicut sortes dicit, nego consequentiam. Unde posito quod sortes dicat illam: sortes dicit falsum, cum ceteris particulis, et plato illam: sortes dicit falsum, patet quod ita est sicut plato dicit, quoniam dicit verum cuius veri adequatum significatum40 est verum. Et dicit totaliter41 sicut sortes dicit, quia dicit unam propositionem secum convertibilem; et tamen non est ita sicut sortes dicit, quoniam ipse non dicit verum. Et ita in priori casu propositio dicta a sorte est falsa et dicta  propositio] om. E  corrumperetur] corrumpetur E 34  alio] etc. add. M 35  dicit] diceret E 36  seipso] seipsum E 37  est concedenda] concedendam est M 38  solummodo] solutio E 39  concedo] concedendo M 40  significatum] om. M 41  dicit totaliter] inv. E 32 33





English Translation

185

This is confirmed. For these are convertible: Every false universal proposition exists and Some false universal proposition exists, because a universal and a particular proposition of the second component are convertible, yet the particular does not signify conjunctively or in two ways, therefore neither ⟨does⟩ the universal, even though it is an insoluble proposition, given that it is the only proposition. Hence this proposition: A falsehood exists signifies in just one way. So I do not see how it could signify conjunctively or in two ways, given that every other proposition was destroyed, since it would not receive that new imposition through itself nor through anything else. ad 3.1.1 To the first argument: admitting the scenario, I grant that it is not as Socrates says it is, just as I would grant that Socrates does not say a truth, for the reason that an insoluble conjoined with itself should be granted only ⟨as⟩ false and anything convertible with it ⟨should be granted⟩ not as true but as following. For this proposition: This is false, referring to itself, should always be granted, although I do not grant that it is true. And then ⟨in reply⟩ to the argument: “It is not as Socrates says it is, and Plato says only that, therefore it is as Plato says it is,” I grant the inference and the conclusion. And then further ⟨to the argument⟩: “It is as Plato says it is and Socrates says wholly what Plato says, therefore it is as Socrates says it is,” I deny the inference. For, supposing that Socrates says this proposition: Socrates says a falsehood, with the other elements ⟨of the scenario⟩, and Plato ⟨says⟩ this proposition: Socrates says a falsehood,

186





Latin Text

a platone est vera,42 ideo ita est sicut plato dicit et non est ita sicut sortes dicit, non obstante quod convertibiliter dicant idem. Ideo debuit sic argui: Ita est sicut plato dicit et43 sortes omnino dicit sicut plato dicit nec repugnat dictum sortis esse verum, igitur ita est sicut sortes dicit. Et ad confirmationem: quando44 dicitur: non est ita sicut sortes dicit, igitur verum est quod non est ita sicut sortes dicit, concedo45 consequentiam et consequens.46 Tunc ultra: verum est quod non est ita sicut sortes dicit et sortes ⟨solummodo⟩ dicit sic, igitur sortes dicit sicut verum est esse, nego consequentiam sicut non sequitur: verum est quod sortes non dicit verum et sortes dicit sic solummodo, videlicet: sortes non dicit verum, igitur sortes dicit sicut verum est esse. Sed debet addi in antecedente quod non repugnat dictum sortis esse verum, quod negatur. ad 3.2 Ad secundam rationem dico, admisso casu, quod non est ita totaliter sicut a significat, et ita consequenter concedo quod a47 significat aliter quam est. ad 3.2.1 Et tunc ad argumentum: a propositio significat aliter quam est et b significat solummodo quod a significat aliter quam est, igitur ita est totaliter sicut b significat, concedo consequentiam et consequens; et tunc ad argumentum: ita est totaliter sicut b significat et a propositio totaliter significat48 sicut b et e contra, igitur ita est totaliter sicut a49 significat, nego consequentiam, sed deberet addi in antecedente quod non repugnat a esse verum et hoc negatur. Unde a falsificat se ex quo asserit se significare aliter50 quam est, quare repugnat a51 esse verum.  vera] et add. E  et non – dicit et] om. (hom.) M 44  quando] cum E 45  concedo] concedendo M 46  consequens] et add. E 47  a] om. E 48  totaliter significat] inv. E 49  a] propositio add. E 50  significare aliter] inv. E 51  a] est add. E 42 43







English Translation

187

it is clear that it is as Plato says it is, since he says a truth and the exact significate of that truth is true. And he says wholly what Socrates says, since he says a proposition convertible with it; nevertheless, it is not as Socrates says it is, since he does not say a truth. And thus in the earlier scenario ⟨in § 3.1.1⟩, the proposition said by Socrates is false and the proposition said by Plato is true, therefore it is as Plato says it is and it is not as Socrates says it is, even though they say the same thing convertibly. Therefore ⟨the opponent⟩ should have argued like this: “It is as Plato says it is, and Socrates says altogether what Plato says, and it is not incompatible that what Socrates said is true, therefore it is as Socrates says it is.” As for the confirmation: when one says: “it is not as Socrates says it is, therefore it is true that it is not as Socrates says it is,” I grant the inference and the conclusion. Then further: “it is true that it is not as Socrates says it is and Socrates says ⟨only⟩ that, therefore Socrates says what is truly so,” I deny the inference, just as this is not valid: “It is true that Socrates does not say a truth, and Socrates says only that, namely: ‘Socrates does not say a truth,’ therefore Socrates says what is truly so.” But one should add in the premises that it is not incompatible that what Socrates said is true, which I deny. ad 3.2 To the second argument: I say, admitting the scenario, that it is not wholly as A signifies, and so consequently, I grant that A signifies other than it is. ad 3.2.1 And then ⟨in reply⟩ to the argument: “A signifies other than it is, and B signifies only that A signifies other than it is, therefore it is wholly as B signifies,” I grant the inference and the conclusion; and then ⟨in reply⟩ to the argument: “It is wholly as B signifies and proposition A wholly signifies as B does and vice versa, therefore it is wholly as A signifies,” I deny the inference, but one would need to add in the premises that it is not incompatible that A is true, and this I deny. For A falsifies itself, because it asserts itself to signify other than it is, and this is why it is incompatible for A to be true.

188

Latin Text

tertiam rationem: concedo illam: aliqua propositio significat aliter quam est, et consequenter quod a significat aliter quam est. ad 3.3.1 Et tunc ad argumentum: a significat aliter quam est et a significat precise quod52 aliqua propositio significat aliter quam est, igitur non est ita quod aliqua propositio significat aliter quam est, negatur consequentia sed oportet addere in antecedente quod a non sit propositio insolubilis aut se ipsam falsificans. aliter ad 3.1-3.3  Potest53 tamen ad hec omnia aliter responderi negando semper, admisso casu, utrumque contradictoriorum,54 videlicet: ita est sicut sortes dicit, non est ita sicut sortes dicit; ita est sicut a significat, non est ita sicut a significat; aliqua propositio significat aliter quam est, nulla propositio significat aliter quam est55. Sicut enim non est inconveniens duo contradictoria esse simul falsa in materia insolubilium, ita non est inconveniens eadem simul negari in eadem materia, et precipue quando insolubilia habent principaliter reflexionem ad significationem propriam, ut in predictis motivis experientia docuit. ad 3.4 Ad quartam respondetur concedendo conclusionem56 quam asseruit alterius capituli57 prima conclusio. ad 3.5 Ad quintam respondetur similiter concedendo58 conclusionem quia sequitur ex priori. ad 3.6 Ad sextam: cum infertur quod aliqua est propositio necessaria et contingens, nego consequentiam. Et ad probationem dico quod a est contingens et non (M 241rb) necessarium. Et cum arguitur: adequatum significatum a est necessarium et a non59 est propositio insolubilis nec se falsificans, igitur a est (E 196va) necessarium, nego consequentiam, sed oportet addere in antecedente: et non repugnat a esse necessarium, quod negatur. Unde licet a non sit insolubile nec seipsum falsificans, tamen pertinet ad materiam insolubilium quia60 habet immediate reflexionem supra se.

ad 3.3 Ad

 quod] et E  Ponit (dub.) aliam responsionem (dub.)] marg. M 54  contradictoriorum] contradictorium E 55  nulla – est] om. M 56  concedendo conclusionem] concedo consequentiam E 57  capituli] casui E 58  concedendo] concedo E 59  non] om. Ma.c. 60  quia] quod E 52 53



English Translation

the third argument: I grant this proposition: Some proposition signifies other than it is, and consequently that A signifies other than it is. ad 3.3.1 And then ⟨in reply⟩ to the argument: “A signifies other than it is, and A signifies only that some proposition signifies other than it is, therefore, it is not the case that any proposition signifies other than it is,” I deny the inference, but it is necessary to add in the premises that A is not an insoluble proposition, or one falsifying itself. aliter ad 3.1-3.3 But one can respond to all these ⟨arguments⟩ in another way, always admitting the scenario, by denying each of the contradictories, namely, It is as Socrates says it is, It is not as Socrates says it is. It is as A signifies, It is not as A signifies. Some proposition signifies other than it is, No proposition signifies other than it is. For just as it is not impossible for two contradictories to be false at the same time in the case of insolubles, so it is not impossible for them to be denied at the same time in the same case, and especially when the insolubles principally have reflection on their own signification, as experience has taught in the foregoing arguments. ad 3.4 To the fourth argument: I respond by granting the conclusion which the first conclusion of chapter 2 ⟨§ 2.3.1⟩ asserted. ad 3.5 To the fifth argument: I respond similarly by granting the conclusion because it follows from the previous one. ad 3.6 To the sixth argument: when it is inferred that there is some proposition which is ⟨both⟩ necessary and contingent, I deny the inference. To its proof, I say that A is contingent and not necessary. When it is argued: “The exact significate of A is necessary and A is not an insoluble proposition and does not falsify itself, therefore A is necessary,” I deny the inference. But one needs to add in the premises: “and it is not inconsistent that A is necessary,” which I deny. For although A is not an insoluble and does not falsify itself, nevertheless it has to do with the matter of insolubles because it has reflection immediately on itself.

ad 3.3 To

189

190

Latin Text

⟨Capitulum Quartum: Insolubile Famosum⟩ predictis potest colligi,1 non dubito, omnium propositorum insolubilium responsio manifesta. Verumtamen, quia idem sepius exemplificatum intelligentie clarius innotescit, ideo famosum insolubile resumo disputandum, ponendo quod sortes dicat illam propositionem: sortes dicit falsum, que sit a, et nullam aliam, que sic precise seu adequate significet, quod in presenti responsione variari2 non debet. 4.1.1 Quo posito, propono a et quero utrum sit verum vel falsum. Si dicitur quod sit verum, contra: cum toto casu stat quod unus sortes sit omnis sortes; quo supposito, sequitur quod a est falsum. Si autem dicitur quod a est falsum, contra: cum toto3 casu stat quod sint duo sortes, quorum primus dicat a et secundus quod nullus deus est; quo supposito cum casu, sequitur quod a est verum. ad 4.1.1 ⟨1⟩ Respondetur admittendo casum. Et cum proponitur a, concedo quoniam ex casu sequitur sortem dicere falsum. Nam ex casu sequitur quod aliquis sortes dicit4 falsum vel nullus sortes dicit falsum. Si aliquis sortes dicit falsum, igitur sortes dicit falsum. Si nullus sortes dicit falsum et sortes dicit quod sortes dicit falsum, sic adequate significans, igitur sortes dicit falsum. Propterea cum proponitur a vel sibi consimile,5 concedatur. ⟨2⟩ Et cum queritur utrum a sit verum vel falsum, concedatur illa disiunctiva: a est verum vel a est falsum, et dubitetur quelibet eius pars, quoniam cum toto6 casu stat quod a est verum et cum eodem casu stat quod a est7 falsum, ut obiectio ostendit manifeste.

4.1 Ex

 potest colligi] colligi posse E  responsione variari corr.] responsionem variari M, responsione variare E 3  toto] om. E 4  dicit] dicat M 5  sibi consimile] consimilis E 6  toto] om. M 7  verum – est] om. (hom.) E

1 2



English Translation

191

⟨Chapter Four: The Familiar Insoluble⟩ 4.1 I do not

doubt that from what has been said one could gather a clear response to all the insolubles that have been presented. Since, however, the same thing is made known to the intellect more plainly when it is shown very often through examples, I return therefore to the familiar insoluble that needs to be analysed, assuming that Socrates utters the proposition: Socrates says a falsehood and no other proposition, call it A, signifying only or exactly in that way, which must not be changed in the present response. 4.1.1 Given this ⟨scenario⟩, I propose A and I ask whether it is true or false. If one answers that it is true, on the contrary: it is consistent with the whole scenario that there is only one Socrates; assuming this, it follows that A is false. If, however, one answers that A is false, on the contrary: it is consistent with the whole scenario that there are two Socrates, the first of whom says A and the second that God does not exist; supposing this and given the scenario, it follows that A is true. ad 4.1.1 ⟨1⟩ I respond by admitting the scenario. And when A is proposed, I grant ⟨it⟩ since it follows from the scenario that Socrates says a falsehood. For it follows from the scenario either that some Socrates says a falsehood or that no Socrates says a falsehood. If any Socrates says a falsehood, then Socrates says a falsehood. If no Socrates says a falsehood and Socrates says that Socrates says a falsehood, signifying exactly that, then Socrates says a falsehood. For this reason, when A is proposed or ⟨a proposition⟩ similar to it, it should be granted. ⟨2⟩ And when one asks whether A is true or false, the disjunction: A is true or A is false should be granted, and doubt should be expressed about each of its parts since it is consistent with the whole scenario that A is true and it is consistent with that same scenario that A is false, as the objection clearly shows.

192

Latin Text

4.1.2 Ad idem



secundo arguitur sic: et pono quod unus sortes sit omnis sortes et dicat istam propositionem: sortes dicit falsum, que sit a, significans precise vel adequate quod sortes dicit falsum. Isto posito, propono a et quero si est verum vel falsum. Si dicitur quod est8 verum, contra: cum casu stat quod sortes9 dicat a propositionem et nullam aliam, quo posito cum casu sequitur a esse falsum. Quia igitur dubium est in casu si sortes dicat aliam propositionem ab a, ideo est tibi dubium:10 a est verum. Et per consequens male conceditur a esse verum. Si vero dicitur a esse falsum, contra: cum casu stat sortem dicere illam: chymera est, quo supposito cum eodem ⟨casu⟩ stat a esse11 verum. Sed hoc est tibi dubium, igitur tu dubitas an a sit falsum. Et per consequens male respondetur concedendo12 a esse falsum. ad 4.1.2 Ideo respondetur ut prius, concedendo13 a cum proponitur, quia vel sortes dicit a solummodo vel aliquam aliam propositionem. Si dicit a solummodo, sequitur quod sortes dicit falsum. Si dicit aliam14 propositionem, sit illa b. Et quero si b est verum vel falsum. Si falsum, igitur sortes dicit falsum. Si verum et sortes non dicit plures propositiones quam a et b, ut suppono, igitur a est falsum, et sortes dicit a, igitur sortes dicit falsum. Cum vero queritur si a est verum vel falsum, concedatur illud disiunctum seu disiunctiva eidem equivalens, et dubitetur (M 241va) quelibet illarum: a est verum, a est falsum, quia cum casu stat quelibet illarum, ut argumentum satis notificavit.

 est] om. M  sortes] om. E 10  est tibi dubium corr.] ex tibi dubio M E 11  stat a esse] a est M 12  concedendo] concedo E 13  concedendo] concedo E 14  aliam] aliquam E 8 9



English Translation

4.1.2 On the



193

same ⟨point⟩, I argue secondly like this: assume that there is only one Socrates and that he utters the proposition: Socrates says a falsehood call it A, signifying only or exactly that Socrates says a falsehood. Given this ⟨scenario⟩, I propose A and I ask whether it is true or false. If one answers that it is true, on the contrary: it is consistent with the scenario that Socrates utters the proposition A and no other proposition; supposing this and given the scenario, it follows that A is false. Therefore, since in the scenario it is in doubt whether Socrates utters another proposition besides A, A is true is in doubt for you. And consequently you incorrectly grant that A is true. If, however, one answers that A is false, on the contrary: it is consistent with the scenario that Socrates says A chimera exists. Supposing this, it is consistent with the same ⟨scenario⟩ that A is true. But this is in doubt for you, therefore you should express doubt whether A is false. And consequently you respond incorrectly by granting that A is false. ad 4.1.2 Therefore I respond as before, by granting A when it is proposed, since either Socrates says only A or some other proposition ⟨as well⟩. If he says only A, it follows that Socrates says a falsehood. If he utters another proposition ⟨as well⟩, let this be B. And I ask whether B is true or false. If false, then Socrates says a falsehood. If true, and Socrates does not utter any proposition other than A and B, as I am assuming, then A is false and Socrates says A, therefore Socrates says a falsehood. But if one asks whether A is true or false, the disjunctive ⟨predicate⟩ or a disjunction equivalent to it should be granted, and doubt should be expressed about each of these: A is true, A is false, since each of them is consistent with the scenario, as the argument has made sufficiently clear.

194

Latin Text

4.1.3 Tertio



arguitur ad idem sic: et pono quod unus sortes sit omnis sortes et dicat illam et nullam aliam: sortes dicit falsum, que sit a. Isto posito, propono a querendo si est verum vel falsum. Si dicitur quod est verum, contra: cum casu stat quod a significet precise sortem dicere falsum; quo posito, a non est verum. Si autem dicitur quod a est falsum, contra: cum casu stat quod a significet precise deum esse; quo posito, a est verum. Arguitur ergo sic: ante casum dubitasses quamlibet illarum: a est falsum, a est verum. Sed iam15 quelibet illarum stat cum casu, igitur quelibet illarum est dubitanda. ad 4.1.3 Dicitur dubitando a, quia a est dubium et impertinens, cum16 non sequatur17 nec repugnet18. Et ita, cum proponitur: a est verum, a est falsum, dubitetur19 quelibet illarum quemadmodum et20 ante casum, ex quo quelibet earum21 stat cum casu. Eodem modo respondetur si ponetur22 cum toto casu a significare sicut termini pretendunt, non ponendo precise vel adequate; unde ex tali casu numquam sequitur sortem dicere falsum.

 iam] eam E  cum] tamen E 17  sequatur] sequitur E 18  repugnet] repugnat E 19  dubitetur] dubitatur E 20  et] om. M 21  earum] istarum E 22  ponetur] poneretur E 15 16





English Translation 4.1.3 Thirdly,

195

on the same ⟨point⟩ I argue like this: assume that there is only one Socrates and that he utters this proposition and no other proposition: Socrates says a falsehood, call it A. Given this ⟨scenario⟩, I propose A and I ask whether it is true or false. If one answers that it is true, on the contrary: it is consistent with the scenario that A signifies only that Socrates says a falsehood; given this ⟨scenario⟩, A is not true. But if one answers that A is false, on the contrary: it is consistent with the scenario that A signifies only that God exists; given this ⟨scenario⟩, A is true. Therefore, I argue like this: before the scenario was set up, you were in doubt about each of these: A is false, A is true. But now each of them is consistent with the scenario, therefore you should express doubt about each of them. ad 4.1.3 I answer by expressing doubt about A, since A is in doubt and irrelevant, since it does not follow nor is it inconsistent. And thus, when one proposes A is true, A is false doubt should be expressed about each of them just as ⟨each of them was in doubt⟩ before the scenario was set up, since each of them is consistent with the scenario. I respond in the same way if one assumes that in the whole scenario A signifies as the terms suggest, without adding “only or exactly ⟨in that way⟩”; so from such a scenario, it never follows that Socrates says a falsehood.

196

Latin Text

4.2 Ex quibus

sequitur quod a propositio insolubilis nominari non debet nisi ponatur in casu quelibet illarum particularum23: unus sortes est omnis sortes, et dicit a et nullam aliam, que sic precise seu adequate significat.24 Et ita correspondenter25 dicatur in aliis insolubilibus similibus26 non determinatis, originem habentibus ex actu nostro, sive interiori27 sive exteriori, ut: sortes audit falsum, plato non vi-(E 196vb)det verum, sortes cogitat28 falsum, plato intelligit impossibile. In insolubilibus autem demonstrativis29 seu particularibus vel indefinitis frustra ponuntur tot particule, ut: iste homo dicit falsum, aliqua propositio non est vera. Ad hoc30 namque quod prima sit insolubilis non requiritur ponere quod unus31 iste homo sit omnis iste32 homo nec pro secunda oportet exprimere aliquem dicere illam, sed sufficit quod ista sit omnis propositio et precise significet ut termini pretendunt. Notandum tamen propter cavillatores,33 cum ponitur sortem dicere illam: sortes dicit falsum, et nullam aliam, intelligitur sortem dicere illam solummodo vel partem eius. Et cum dicitur ponendo quod illa: aliqua propositio est falsa, sit omnis propositio, intelligitur non sicut verba sonant simpliciter, sed quod illa sit omnis propositio totalis. Quare etsi sua pars propositio34 dicatur, partialis solummodo et non totalis debet denominari.35 4.2.1 Pono igitur quod sortes existens omnis sortes dicat istam propositionem et nullam aliam: sortes dicit falsum, sic precise et adequate significantem, que sit a. Quo posito, sequitur ex dictis quod a est falsum.  particularum] particulariter E  significat] significet E 25  correspondenter] correspondetur E 26  similibus] singularibus M 27  sive interiori] om. M 28  cogitat] verum add. M 29  demonstrativis] de terminis E 30  hoc] hec M 31  unus] om. E 32  iste] om. E 33  cavillatores] cavillationes E 34  propositio] proprie M 35  denominari] nominari M, etc add. E 23 24



English Translation

4.2 From these

197

considerations it follows that A should only be called an insoluble proposition if in the scenario one adds all of these qualifications: there is only one Socrates, and he utters A and no other proposition, signifying only or exactly in that way. And the same should be said correspondingly in the ⟨case of⟩ other similar indeterminate insolubles arising from an act of ours, whether interior or exterior, such as: Socrates hears a falsehood, Plato does not see a truth, Socrates thinks a falsehood, Plato understands an impossibility. But it is pointless to add so many qualifications in the ⟨case of⟩ demonstrative or particular and indefinite insolubles such as: This man says a falsehood, Some proposition is not true. In fact, for the first ⟨of these⟩ to be an insoluble proposition it is not necessary to add that one this man is the only this man1 nor for the second is it necessary to state explicitly that someone says it, but it suffices that this is the only proposition and that it signifies only as the terms suggest. Because of quibblers, however, it should be noted that when I assume that Socrates says Socrates says a falsehood and no other proposition, I mean that Socrates says only that or part of it. And when I say: assume that Some proposition is false is the only proposition, I do not understand it as the words signify literally, but ⟨I mean⟩ that it is the only whole proposition. So even if its part may be called “proposition,” it should be understood only as “partial” and not “whole.”

 The awkwardness of the English reflects that of the Latin. See the commentary.

1

198

Latin Text

4.2.1.1 Arguitur

ergo sic: a est falsum,36 et sortes dicit a, igitur sortes dicit falsum. Ista consequentia est bona et37 antecedens est verum, igitur et consequens; sed consequens est a, igitur a est verum. 4.2.1.2 Secundo arguitur sic: falsum dicitur a sorte, igitur sortes dicit falsum. Consequentia tenet a passiva ad suam activam et38 antecedens est verum, igitur et consequens; sed consequens est a, igitur a est verum. Quod autem antecedens sit39 verum patet quia suum adequatum significatum est verum, et tamen repugnat illam esse veram. 4.2.1.3 Tertio arguitur sic: contradictorium a est falsum, igitur a est verum. Consequentia tenet et antecedens probatur, quia hoc est falsum: nullus sortes dicit falsum, et hoc est contradictorium a, igitur contradictorium a est falsum. Consequentia tenet cum minori, et maiorem probo: Nam a est falsum, et aliquis sortes dicit a, igitur aliquis sortes dicit falsum. Vel sic: nullus sortes dicit falsum, igitur nullus sortes dicit a falsum. Consequentia (M 241vb) tenet a superiori distributo negative ad suum inferius, et consequens est falsum, igitur et antecedens. ad 4.2.1 Ad insolubile respondetur concedendo40 ipsum quoniam sive41 a sit verum sive42 falsum, sequitur quod sortes dicit falsum. Numquam tamen dicatur a esse verum, quia ex illo sequitur a esse falsum, sic arguendo: a est verum, igitur significatum adequatum43 a est verum. Sed adequatum significatum a est sortem dicere falsum, igitur verum est sortem dicere falsum, igitur sortes dicit falsum; sed non dicit nisi a, igitur a est falsum. Ideo conceditur quod44 a est falsum.

 arguitur – falsum] om. (hom.) E  bona ] et add. E 38  et] sed E 39  sit] est E 40  concedendo corr.] concedo M E 41  quoniam sive] quod suum E 42  sive] et suum a sit E 43  significatum adequatum] inv. E 44  quod] om. M 36 37

4.2.1 Therefore,

English Translation

199

assume that Socrates, who is the only Socrates, says this proposition and no other: Socrates says a falsehood, signifying only and exactly in that way, call it A. Given this ⟨scenario⟩, it follows from what has been said that A is false. 4.2.1.1 Accordingly, one argues like this: A is false, and Socrates says A, therefore Socrates says a falsehood. This inference holds and the premise is true, therefore so too is the conclusion; but the conclusion is A, therefore A is true. 4.2.1.2 Secondly, one argues like this: A falsehood is said by Socrates, therefore Socrates says a falsehood. The inference holds from the passive to the corresponding active, and the premise is true, therefore so too is the conclusion; but the conclusion is A, therefore A is true. And that the premise is true is clear, since its exact significate is true, yet it is inconsistent that ⟨the active⟩ is true. 4.2.1.3 Thirdly, one argues like this: The contradictory of A is false, therefore A is true. The inference holds, and I prove the premise, for No Socrates says a falsehood is false, and this is the contradictory of A, therefore the contradictory of A is false. The inference holds, as does the minor premise, and I prove the major premise: for A is false, and some Socrates says A, therefore some Socrates says a falsehood. Or like this: No Socrates says a falsehood, therefore no Socrates says the falsehood A. The inference holds from the superior distributed negatively to its inferior, and the conclusion is false, therefore so too is the premise, ⟨that is, “No Socrates says a falsehood” is false⟩. ad 4.2.1 I respond to the insoluble by granting it. Since either A is true or false, it follows that Socrates says a falsehood. By no means, however, do I say that A is true, since from that it follows that A is false, arguing like this: A is true, therefore the exact significate of A is true. Yet the exact significate of A is that Socrates says a falsehood, therefore it is true that Socrates says a falsehood, therefore Socrates says a falsehood; but ⟨Socrates⟩ says only A, therefore A is false. So, I grant that A is false. And if one argues in a similar way, as before: A is false, therefore the exact significate of A is false,

200



Latin Text

Et si arguitur similiter, ut prius: a est falsum, igitur significatum adequatum45 a est falsum, nego consequentiam. Unde sequitur generaliter: a est verum vel necessarium, igitur adequatum significatum est huiusmodi, quia antecedens non se falsificat, nec asserit a esse falsum; sed illa: a est falsum, asserit a esse falsum, ideo ad concludendum significatum adequatum a esse falsum debuit addi in antecedente: et a non falsificat se, quod repugnat casui. ad 4.2.1.1 Ad primam ergo responsionem concedo consequentiam et consequens, et dico quod illa est vera: sortes dicit falsum. Non tamen est a, sed est similis a et46 secum convertibilis. Ubi tamen ponatur cum priori casu quod illa sit a, concedo consequentiam et consequens. Et cum dicitur: illa consequentia est bona et antecedens est verum, igitur et consequens, nego consequentiam. Ymmo extra materiam insolubilium non valet illa47 forma, ut ostensum est in materia consequentiarum, quia oportet48 addere in antecedente quod propositio significat precise ex compositione suarum partium. Ubi tamen alia quereretur49 difficultas, non oporteret50 illud addere, quia semper intelligitur quemadmodum est in proposito. Presupponit namque argumentum quod consequentia significet ex compositione suarum partium solummodo. Ideo presupposito isto dico quod extra materiam insolubilium foret talis forma concedenda, ad presens tamen eodem51 presupposito non valet consequentia. Unde52 superius concessum53 est quod aliqua consequentia est bona et formalis significans precise ex compositione suorum terminorum et54 antecedens est verum et consequens falsum.  significatum adequatum] inv. E  et] ac E 47  valet illa corr.] valeret illa M, valet consequentia vel ista E 48  oportet] oporteret M 49  quereretur corr.] queretur M E 50  oporteret] oportet E 51  eodem] eadem E 52  unde] ut E 53  concessum] concessa E 54  et] om. M 45 46



English Translation

201

I deny the inference. In fact, it generally follows: A is true or necessary, therefore so too is the exact significate, since the premise does not falsify itself nor does it assert that A is false; yet A is false asserts that A is false, so in order to conclude that the exact significate of A is false, one should have added in the premises and A does not falsify itself, which is inconsistent with the scenario. ad 4.2.1.1 Therefore, to the first response I grant the inference and the conclusion, and I say that Socrates says a falsehood is true. But ⟨this⟩ is not A, but it is similar to A and convertible with it. However, when in the earlier scenario one assumes that it is A, I grant the inference and the conclusion. And when it is said The inference holds and the premise is true, therefore so too is the conclusion, I deny the inference. In fact, outside the case of insolubles that form does not hold, as was shown in the chapter on inferences, since one should add in the premises that the proposition signifies only by the composition of its parts. However, when another difficulty might be being investigated, that ⟨clause⟩ need not be added, since it is always understood in the same manner as ⟨it is⟩ in that case. For the argument ⟨above⟩ presupposes that the inference signifies only by the composition of its parts. So, assuming this, I say that outside the case of insolubles this form should be granted, yet on the same assumption, the inference does not hold in the present ⟨case⟩. Hence I have granted above that some inference holds formally, signifying only by the composition of its terms, and the premise is true and the conclusion is false.

202

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ad 4.2.1.2 Ad

secundam rationem dicitur consimiliter concedendo55 consequentiam esse bonam et quod antecedens est verum et consequens est56 falsum, dato quod sit57 a. Ubi tamen foret convertibile solum, concederem ipsum consequens esse verum sicut antecedens. ad 4.2.1.3 Ad tertiam rationem nego consequentiam: contradictorium a est falsum, igitur a est verum, quoniam in materia insolubilium sustinetur58 pro fundamento duo contradictoria inter se contradicentia esse duo falsa. 4.2.2 Secundo principaliter arguitur sic: et pono quod duo sortes sint et non plures,59 quorum quilibet dicat unam talem: sortes dicit falsum, et nullam aliam, sic precise significantem, quarum una sit a et alia b. (E 197ra) Isto posito, quero si a est verum vel falsum. Si dicitur quod verum, igitur eius60 adequatum significatum est verum. Sed eius adequatum significatum est sortem dicere falsum, igitur verum est sortem dicere falsum. Sed non est61 maior ratio de uno quam de alio, igitur uterque illorum dicit falsum, et alter illorum dicit a, igitur a est falsum. Si autem dicitur in principio62 quod a est falsum et non falsificat se, quia non habet reflexionem supra se sed supra b, igitur eius adequatum significatum est falsum, sed eius adequatum significatum est sortem dicere falsum, igitur falsum est sortem dicere falsum, igitur nullus sortes dicit falsum; sed aliquis sortes dicit a, igitur a non est falsum, quod est oppositum primi. Eodem modo potest argui de b.

 concedendo] concedo E  est] om. E 57  sit] foret E 58  sustinetur] sustinentur E 59  et non plures] om. M 60  eius] om. M 61  est] om. E 62  in principio] om. E 55 56

ad 4.2.1.2 To

English Translation

203

the second argument, I answer in just the same way by granting that the inference holds and the premise is true and the conclusion is false, given that it is A. However, where it was only something convertible ⟨with A⟩, I would grant that the conclusion itself was true just like the premise. ad 4.2.1.3 To the third argument, I deny the inference: The contradictory of A is false, therefore A is true, since in the case of insolubles I uphold as a fundamental principle that two mutually contradictory propositions may both be false. 4.2.2 The second main objection is this: assume that there are only two Socrates, each of whom utters a proposition of the form: Socrates says a falsehood and no other ⟨whole⟩ proposition, signifying only in that way, one of these ⟨utterances⟩ being A and the other B. Given this ⟨scenario⟩, I ask whether A is true or false. If one answers that ⟨it is⟩ true, then its exact significate is true. But its exact significate is that Socrates says a falsehood, therefore it is true that Socrates says a falsehood. But there is no more reason why it is one ⟨of them⟩ rather than the other, therefore each of them says a falsehood, and one of them says A, therefore A is a falsehood. If, however, at the beginning one answers that A is false and does not falsify itself, since it does not have reflection on itself but on B, then the exact significate of A is false. But its exact significate is that Socrates says a falsehood, therefore it is false that Socrates says a falsehood, therefore no Socrates says a falsehood. But some Socrates says A, therefore A is not a falsehood, which is the opposite of what was first ⟨granted⟩. And one can argue in the same way about B.

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admisso casu concedendo63 tam a quam b. Et dico quod utrumque illorum est falsum. Et tunc ad argumentum: a est falsum et non falsificat se, igitur adequatum significatum eius64 est falsum, concedo consequentiam et nego (M 242ra) secundam partem antecedentis. Unde quandocumque sunt due propositiones convertibiles quarum quelibet supra reliquam habet reflexionem falsitatis65, quelibet illarum seipsam falsificat et quelibet earumdem habet supra se reflexionem; et si non immediate et a se, tamen mediate et a suo convertibili, sicut est in proposito. 4.2.3 Tertio principaliter arguitur sic: et pono quod sortes sit omnis sortes, qui dicat istam: plato dicit falsum, et nullam aliam, sic precise significantem, que sit a; et quod66 plato sit omnis plato dicens hanc solummodo: sortes dicit falsum, sic precise significantem, que sit b. Isto posito, quero utrum a sit verum vel falsum67. Si dicitur quod verum, igitur eius adequatum significatum est verum; sed ipsius adequatum significatum est platonem dicere falsum, igitur verum est platonem dicere falsum. Sed non videtur ratio quare plato dicat falsum68 quin etiam sortes, igitur sortes dicit falsum et non dicit nisi a, igitur a est falsum. Si autem in principio conceditur69 quod a est falsum et70 a non falsificat se sed b, igitur eius adequatum significatum est falsum; sed eius adequatum significatum est platonem dicere falsum, igitur plato non dicit falsum, igitur pariformiter nec sortes dicit falsum, sed sortes dicit a, igitur a non est falsum.

ad 4.2.2 Respondetur

 concedendo] concedo E  significatum eius] inv. E 65  falsitatis] falsificatis E 66  quod] om. E 67  verum vel falsum] inv. M 68  falsum] et non sortes vel add. E 69  conceditur] dicitur E 70  et] sed E 63 64



English Translation

respond by admitting the scenario and granting both A and B. And I say that each of them is false. And then ⟨in reply⟩ to the argument: A is false and does not falsify itself, therefore the exact significate of A is false, I grant the inference and I deny the second premise. Thus whenever there are two convertible propositions each of which has reflection of falsity on the other, each of them falsifies itself, and each of these does have reflection on itself; and if not immediately and by itself, at least mediately and by means of some ⟨proposition⟩ convertible with it, as in the present ⟨case⟩. 4.2.3 The third main objection is this: assume that there is only one Socrates, who utters this: Plato says a falsehood and no other proposition, signifying only in that way, call it A; and that there is only one Plato who utters only this: Socrates says a falsehood, signifying only in that way, call it B. Given this ⟨scenario⟩, I ask whether A is true or false. If one answers that ⟨it is⟩ true, then its exact significate is true; but its exact significate is that Plato says a falsehood, therefore it is true that Plato says a falsehood. But there is no more reason why Plato says a falsehood and not also Socrates, therefore Socrates says a falsehood and he says only A, therefore A is false. If, however, at the beginning one grants that A is false and A does not falsify itself but ⟨falsifies⟩ B, then its exact significate is false. But its exact significate is that Plato says a falsehood, therefore Plato does not say a falsehood. So equally neither does Socrates say a falsehood, but Socrates says A, therefore A is not false.

ad 4.2.2 I

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Latin Text

quod tam a quam b est falsum, quia72 quodlibet eorumdem73 falsificat se; et si non immediate, sufficit quod mediate. A enim falsificando b falsificat se, quia b habet reflexionem supra a. Similiter b falsificando a falsificat se, quia a habet reflexionem supra b. 4.2.4 Quarto principaliter arguitur sic: et pono quod sortes sit omnis sortes et dicat illam et nullam aliam: plato dicit falsum, que sic precise significet,74 que sit a. Plato vero sit omnis plato et dicat illam et nullam aliam: sortes non dicit falsum, sic precise significantem, que sit b. 4.2.4.1 Isto posito, quero utrum a sit verum vel falsum. Si dicitur quod a est verum, igitur eius adequatum significatum75 est verum,76 sed a adequate significat platonem dicere falsum, igitur ita est quod plato dicit falsum; sed plato non dicit nisi sortem non dicere falsum, igitur non est ita quod sortes non dicit falsum, et per consequens sortes dicit falsum et non dicit nisi a, igitur a est falsum. Si autem dicitur quod a est falsum et a77 non falsificat se sed b, igitur eius adequatum significatum est falsum; sed eius adequatum significatum est platonem dicere falsum, igitur non est ita quod plato dicit falsum; sed plato dicit sortem non dicere falsum, igitur plato dicit verum. Et plato dicit solummodo78 sortem non dicere falsum, igitur sortes non dicit falsum et sortes dicit a, igitur a non est falsum, quod est oppositum primi.

ad 4.2.3 Respondetur71

 Respondetur] responditur M  quia] quod M 73  eorumdem] correspondens E 74  significet] significat M 75  significatum] om. M 76  verum] nullum E 77  a] om. E 78  dicit solummodo] inv. E 71 72



English Translation

respond that both A and B are false, since each of them falsifies itself; and if not immediately, it suffices that ⟨they do so⟩ mediately. For A falsifies itself by falsifying B, since B has reflection on A. Similarly, B falsifies itself by falsifying A, since A has reflection on B. 4.2.4 The fourth main objection is this: assume that there is only one Socrates and that he utters this and no other proposition: Plato says a falsehood, which signifies only in that way, call it A. Moreover, let there be only one Plato and let him utter this and no other proposition: Socrates does not say a falsehood signifying only in that way, call it B. 4.2.4.1 Given this ⟨scenario⟩, I ask whether A is true or false. If one answers that A is true, then its exact significate is true; but A signifies exactly that Plato says a falsehood, therefore it is the case that Plato says a falsehood. But Plato says only that Socrates does not say a falsehood, therefore it is not the case that Socrates does not say a falsehood, and consequently Socrates says a falsehood, and he says only A, therefore A is false. If, however, one answers that A is false and A does not falsify itself but ⟨falsifies⟩ B, then its exact significate is false. But its exact significate is that Plato says a falsehood, therefore it is not the case that Plato says a falsehood. But Plato says that Socrates does not say a falsehood, therefore Plato says a truth. And Plato says only that Socrates does not say a falsehood, therefore Socrates does not say a falsehood, and Socrates says A, therefore A is not false, which is the opposite of what was first ⟨granted⟩.

ad 4.2.3 I

207

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Latin Text

4.2.4.2 Modo

consimili potest argui de b. Si enim dicitur quod b est verum et b adequate significare sortem non dicere falsum, igitur sortes non dicit falsum; sed sortes dicit solummodo platonem dicere falsum, igitur plato dicit falsum et non dicit nisi b, igitur b est falsum. Quod si conceditur, arguitur sic: b est falsum et non falsificat se nec aliquid aliud, igitur eius adequatum significatum est falsum; sed b adequate significat sortem dicere falsum, igitur non est ita quod sortes dicit falsum; sed sortes solum dicit platonem dicere falsum, igitur non est ita quod plato dicit falsum, et plato non dicit nisi b, igitur b non est falsum. ad 4.2.4.1 Ideo respondetur ut prius quod tam a quam b est falsum, quia79 quodlibet illorum falsificat se mediate. Nam a asserit b esse falsum et b asserit a non esse falsum, igitur a asserit falsum esse ipsum non esse falsum et ita se reflexe falsificat. ad 4.2.4.2 Similiter b seipsum falsificat (M 242rb) quoniam significat adequate sortem ⟨non⟩ dicere falsum et sortes non dicit nisi platonem dicere falsum, quare etc. 4.2.4.3 Consimilis est modus respondendi servandus in insolubilibus discretis originem trahentibus a proprietate vocis, ut: hoc est falsum, seipso demonstrato, aut: hoc non (E 197rb) est verum. Quodlibet namque istorum est falsum, licet sua convertibila sint80 vera. Similiter si a foret ista: hoc est falsum, et b illa: hoc est falsum, demonstrando b per a et a per b, concederetur81 utrumque et negaretur aliquod eorum esse verum82. Item si a foret illa: hoc est falsum, et b illa: hoc est verum, demonstrando a per b et b per a, dicendum foret tam a quam b esse falsum, quoniam utrumque se falsificaret; et si non immediate, sufficit quod mediate, circulariter deducendo secundum quod claruit ex responsione quarti; quare83 etc.

 quia] et M  convertibilia sint] convertibilis sit E 81  concederetur] conceditur E 82  verum] falsum E 83  quare] om. M 79 80

4.2.4.2 And

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209

one can argue in the same way about B. For if one answers that B is true and that B signifies exactly that Socrates does not say a falsehood, then Socrates does not say a falsehood. But Socrates says only that Plato says a falsehood, therefore Plato says a falsehood; and he says only B, therefore B is false. And if that is granted, one argues in this way: B is false and it does not falsify itself nor anything else, therefore its exact significate is false. But B signifies exactly that Socrates says a falsehood, therefore it is not the case that Socrates says a falsehood. But Socrates says only that Plato says a falsehood, therefore it is not the case that Plato says a falsehood and Plato says only B, therefore B is not false. ad 4.2.4.1 Therefore, I respond as before that both A and B are false, since each of them falsifies itself mediately. For A asserts that B is false and B asserts that A is not false, therefore A asserts that it is false that it itself is not false and so it falsifies itself by way of reflection. ad 4.2.4.2 Similarly, B falsifies itself since it signifies exactly that Socrates does not say a falsehood and Socrates says only that Plato says a falsehood, therefore ⟨B falsifies itself by way of reflection⟩. 4.2.4.3 One should follow a similar way of responding in the case of discrete insolubles arising from a property of an expression, such as: This is false, referring to itself, or This is not true. For each of them is false, although ⟨propositions⟩ convertible with them are true. Similarly, if A were This is false and B This is false, where A refers to B and B to A, each of them should be granted and one should deny that either of them is true. Again, if A were This is false and B This is true, where B refers to A and A to B, one should say that both A and B are false, since each of them falsifies itself; and if not immediately, it suffices that ⟨they do so⟩ mediately, by way of a circular deduction, as was made evident in the answer to the fourth ⟨objection⟩. So ⟨they falsify themselves by way of reflection⟩.

210

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4.3 Soluto

famoso insolubili, consequenter alia restat84 solvere que ex illo correlarie deducuntur. 4.3.1 Ponatur igitur quod unus sortes sit omnis sortes, qui credat illam: sortes decipitur, que sit a, et nullam aliam, et significet precise quod sortes decipitur. 4.3.1.1 Isto posito, propono tibi illam: sortes decipitur. Si concedis, arguitur contra sic: sortes credit precise sicut est, igitur sortes non decipitur. Consequentia tenet et antecedens probatur: sortes credit precise quod ipse decipitur, per casum; et ita est quod ipse decipitur, per te, igitur ipse credit precise sicut est. Si autem negatur illa: sortes decipitur, contra: sortes credit aliqualiter ⟨esse⟩ et non decipitur, per te, igitur sortes credit precise sicut est; sed ipse credit precise quod sortes decipitur, igitur ita est quod sortes decipitur; et per consequens sortes decipitur. 4.3.1.2 Item quero utrum a sit verum vel falsum. Si dicitur quod verum, igitur significat adequate verum; sed a significat adequate sortem decipi, igitur sortes decipitur. Tunc arguitur sic: sortes decipitur, igitur credit falsum; sed sortes non credit nisi a, igitur a est falsum, quod si conceditur in principio, arguitur sic: a est falsum, igitur significat adequate falsum; sed a significat adequate sortem decipi85, igitur sortes non decipitur. Tunc arguitur86 sic: sortes non decipitur, igitur non credit falsum; sed sortes non credit nisi a, igitur a non est falsum, quod est oppositum primi.

 restat] restant E  sortem decipi] sortes decipitur E 86  arguitur] om. E 84 85

4.3 Now that

English Translation

211

the familiar insoluble has been solved, the next task is to solve other ⟨insolubles⟩ derived from it as corollaries. 4.3.1 So assume that there is only one Socrates, who believes this: Socrates is deceived and no other proposition, call it A, which signifies only that Socrates is deceived. 4.3.1.1 Given this ⟨scenario⟩, I propose this to you: Socrates is deceived. If you grant it, I argue on the contrary like this: Socrates believes only what is the case, therefore Socrates is not deceived. The inference holds and I prove the premise: Socrates believes only that he is deceived, according to the scenario; and it is the case that, according to you, he is deceived, therefore he believes only what is the case. And if someone denies this: Socrates is deceived, on the contrary: Socrates believes that ⟨it is⟩ in some way, and according to you he is not deceived, therefore Socrates believes only what is the case; but he believes only that Socrates is deceived, therefore it is the case that Socrates is deceived; and consequently Socrates is deceived. 4.3.1.2 Again, I ask whether A is true or false. If one answers that ⟨it is⟩ true, then it exactly signifies a truth; but A signifies exactly that Socrates is deceived, therefore Socrates is deceived. Then I argue like this: Socrates is deceived, therefore he believes a falsehood; but Socrates only believes A, therefore A is a falsehood. If that is granted at the start, I argue like this: A is a falsehood, therefore it exactly signifies a falsehood; but A signifies exactly that Socrates is deceived, therefore Socrates is not deceived. Then I argue in this way: Socrates is not deceived, therefore he does not believe a falsehood; but Socrates only believes A, therefore A is not a falsehood, which is the opposite of what was ⟨granted⟩ at the start.

212

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ad 4.3.1.1 Respondetur

admisso casu concedendo illam: sortes decipitur, et nego quod ipse credat precise sicut est. Et tunc ad argumentum: nego iterum quod sortes credat87 precise quod ipse decipitur, quoniam hoc est impossibile, cum ex illo sequatur contradictio. Sequitur enim: sortes credit precise seipsum decipi, igitur sortes decipitur. Tunc sic: sortes decipitur, igitur credit aliter quam est; sed sortes credit precise quod ipse decipitur, igitur non est ita quod ipse decipitur; et per consequens ipse non decipitur, quod est oppositum primi consequentis. Nec propositio illa ponebatur in casu, quoniam non est positum quod sortes credat88 precise seipsum decipi, quia non admitteretur, sed quod sortes credat89 precise sortem decipi, et hoc est admissum. Et consequenter dico quod non credit seipsum decipi, sed sortem alium; et quia non est alius sortes ab eo, propterea90 decipitur. 4.3.1.3 Et si arguitur sic: sortes credit precise quod sortes decipitur, et ita est quod sortes decipitur, igitur sortes credit precise sicut est, et per consequens non decipitur. ad 4.3.1.3 Nego consequentiam primam, sicut non sequitur: tu credis precise sortem legere, et ita est quod sortes legit, igitur tu credis precise sicut est. Dato enim quod essent (M 242va) duo sortes91 quorum unus esset albus et alius niger, et albus legeret et niger non legeret, quem tamen crederes precise legere, isto posito, antecedens est verum et consequens falsum, sed debet addi in minori: et ita est quod ille quem precise credis92 legere, legit, quod negatur.

 credat] credit E  credat] credit E 89  credat] credit E 90  propterea] igitur E 91  sortes] om. E 92  precise credis] tamen crederes precise E 87 88

ad 4.3.1.1 I

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213

respond by admitting the scenario and granting Socrates is deceived, and I deny that he believes only what is the case. And then ⟨in reply⟩ to the argument: I deny again that Socrates believes only that he is deceived, because this is impossible, insofar as a contradiction would follow from it. For this is valid: Socrates believes only that he himself is deceived, therefore Socrates is deceived. Then ⟨I argue⟩ like this: Socrates is deceived, therefore he believes other than what is the case; but Socrates believes only that he is deceived, therefore it is not the case that he is deceived; and consequently he is not deceived, which is the opposite of what was first concluded. Nor was that proposition: ⟨Socrates believes only that he himself is deceived⟩ assumed in the scenario, because it was not supposed that Socrates believes only that he himself is deceived, since ⟨that⟩ would not be admitted; but ⟨it was assumed⟩ that Socrates believes only that Socrates is deceived, and this was admitted. And consequently, I say that Socrates does not believe that he himself is deceived, but ⟨that⟩ another Socrates ⟨is deceived⟩, and since there is no other Socrates than him, for this reason he is deceived. 4.3.1.3 And suppose one argues like this: Socrates believes only that Socrates is deceived, and it is the case that Socrates is deceived, therefore Socrates believes only what is the case, and consequently he is not deceived. ad 4.3.1.3 ⟨Here⟩, I deny the first inference, just as: You believe only that Socrates is reading, and it is the case that Socrates is reading, therefore you believe only what is the case is not valid. For given that there were two Socrates, one of whom was white and the other black, and that the white one was reading and the black one was not reading, and, however, you believed only that ⟨the black one⟩ was reading—given this ⟨scenario⟩, the premises are true and the conclusion is false. But one should add in the minor premise: and it is the case that he, of whom alone you believe that he is reading, is reading, which I deny. As regards the original example, this is valid: Socrates believes only that Socrates is deceived, and it is the case that he whom Socrates believes is deceived is deceived, therefore Socrates believes only what is the case.

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Latin Text

Et ita in principali proposito sequitur: sortes credit precise quod sortes decipitur, et ita est quod ille quem sortes credit decipi decipitur, igitur sortes credit precise sicut est. Consequentia est bona sed minor est neganda, quia credit alium sortem a se decipi,93 et nullus talis decipitur ex quo positum est quod ipse sit94 omnis sortes. ad 4.3.1.2 Ad aliud vero, cum queritur utrum a sit verum vel falsum, dicitur quod falsum. Et tunc ad argumentum: a est falsum, igitur significat adequate95 falsum, non valet argumentum. Dictum est sepissime quod insolubilia sunt falsa et tamen eorum adequata significata sunt vera. Sed debet sic argui: a est falsum et non falsificat se, igitur significat adequate falsum. Consequentia bona, sed negatur minor. 4.3.2 Secundo principaliter arguitur sic: et pono quod sortes sit omnis sortes et credat illam et nullam aliam: plato decipitur, sic precise significantem, que sit a. Plato vero sit omnis plato et credat illam precise: sortes non decipitur, sic adequate significantem, que sit b. 4.3.2.1 Isto posito, propono tibi: plato decipitur. Si concedis, arguitur sic: plato decipitur, igitur credit falsum, sed non credit nisi quod sortes non decipitur, igitur falsum est sortem non96 decipi, igitur sortes decipitur. Tunc sic: sortes decipitur, igitur credit aliter quam est, sed sortes credit precise quod plato97 (E 197va) decipitur, igitur plato non decipitur. Si autem in principio negatur illa: plato decipitur, tunc arguitur sic: plato non decipitur et aliqualiter ⟨esse⟩ credit, igitur credit precise sicut est; sed plato credit precise quod sortes non decipitur, igitur sortes non decipitur. Tunc arguitur sic: sortes non decipitur et aliqualiter ⟨esse⟩ credit, igitur credit precise sicut est; sed sortes credit precise quod plato decipitur, igitur plato decipitur, quod est negatum.98 Eodem modo potest argui proponendo illam: sortes non decipitur, semper inferendo contradictorium responsionis.  a se decipi] decipi a se E  sit] est M 95  significat adequate] inv. E 96  non] om. E 97  plato] sortes E 98  negatum corr.] oppositum negati M E 93 94



English Translation

215

The inference holds, but the minor should be denied since he believes that another Socrates than him is deceived, yet there is no such ⟨Socrates⟩ who is deceived, for it was supposed ⟨in the initial scenario⟩ that there is only one Socrates. ad 4.3.1.2 To the other point, then, when it is asked whether A is true or false, I answer that it is false. And then ⟨in reply⟩ to the argument: A is a falsehood, therefore it exactly signifies a falsehood, the argument is not valid. I have repeatedly said that insolubles are false and yet their exact significates are true. But it should be argued in this way: A is a falsehood and it does not falsify itself, therefore it exactly signifies a falsehood. The inference holds, but I deny the minor premise. 4.3.2 The second ⟨derivative insoluble⟩ is this: assume that there is only one Socrates and that he believes this and no other proposition: Plato is deceived, signifying only in that way, call it A. Let there be only one Plato and let him believe only this: Socrates is not deceived, signifying only in that way, call it B. 4.3.2.1 Given this ⟨scenario⟩, I propose to you: Plato is deceived. If you grant ⟨it⟩, I argue like this: Plato is deceived, therefore he believes a falsehood, but he only believes that Socrates is not deceived, therefore it is false that Socrates is not deceived, therefore Socrates is deceived. Then ⟨I argue⟩ like this: Socrates is deceived, therefore he believes other than what is the case, but Socrates believes only that Plato is deceived, therefore Plato is not deceived. But if at the start Plato is deceived is denied, then I argue like this: Plato is not deceived, and he believes that ⟨it is⟩ in some way, therefore he believes only what is the case, but Plato believes only that Socrates is not deceived, therefore Socrates is not deceived. Then I argue like this: Socrates is not deceived and he believes that ⟨it is⟩ in some way, therefore he believes only what is the case. But Socrates believes only that Plato is deceived, therefore Plato is deceived, which was denied. It can be argued in the same way by proposing this: Socrates is not deceived, each time inferring the contradictory of the response.

216

Latin Text

4.3.2.2 Item



quero si a est verum vel falsum. Si dicitur quod verum, igitur significat adequate verum; sed a significat adequate platonem decipi, igitur plato decipitur, igitur credit falsum; et non credit nisi b, igitur b est falsum; sed b significat precise sortem non decipi, igitur falsum est sortem non decipi, igitur99 sortes100 decipitur, igitur credit falsum; et non credit nisi a, igitur a est falsum. Quod si conceditur in principio, arguitur sic: a est falsum, igitur significat adequate aliter quam est,101 sed a significat adequate platonem decipi, igitur102 non est ita quod plato decipitur, igitur non credit falsum et non credit nisi b, igitur b non est falsum; sed b significat precise sortem non decipi, igitur sortes non decipitur, igitur non credit falsum et non credit nisi a, igitur a non est falsum, quod est oppositum primi. Eodem modo potest argui de b respectu a, semper inducendo contradictoria. ad 4.3.2.1 Dicendum ad primum quod plato decipitur et quod ipse credit falsum. Et tunc ad argumentum: plato (M 242vb) credit falsum et non credit nisi sortem non decipi, igitur falsum est quod sortes non decipitur, nego consequentiam, sicut non sequitur: sortes dicit falsum et non dicit nisi illam: sortes non103 dicit verum, igitur falsum est quod sortes non dicit verum; sed debet addi in antecedente quod dictum sortis104 non falsificat se. Et ita in alia105 debet addi: et creditum a platone106 non falsificat se, quod negatur. Simili modo conceditur illa: sortes non decipitur. Et si arguitur sic: sortes non decipitur et sortes aliqualiter ⟨esse⟩ credit, igitur credit precise sicut est; sed ipse credit precise quod plato decipitur, igitur ita est quod plato decipitur, conceditur consequentia et consequens.  falsum est sortem non decipi igitur] om. (hom.) E  sortes] non add. E 101  adequate –est] aliter quam est adequate E 102  igitur] om. M 103  non] om. E 104  dictum sortis] inv. M 105  alia] chymera E 106  creditum a platone corr.] creditum a sorte M, traditam sorti E 99

100

4.3.2.2 Again,

English Translation

217

I ask whether A is true or false. If one answers that ⟨it is⟩ true, then it exactly signifies a truth. But A exactly signifies that Plato is deceived, therefore Plato is deceived, therefore he believes a falsehood; and he only believes B, therefore B is false. But B signifies only that Socrates is not deceived, therefore it is false that Socrates is not deceived, so Socrates is deceived. Therefore he believes a falsehood, and he only believes A, therefore A is false. If that is granted at the start, I argue like this: A is false, therefore it exactly signifies other than is the case. But A exactly signifies that Plato is deceived, so it is not the case that Plato is deceived, therefore he does not believe a falsehood, and he only believes B, therefore B is not false; but B signifies only that Socrates is not deceived, therefore Socrates is not deceived, therefore he does not believe a falsehood, and he only believes A, therefore A is not false, which is the opposite of what was ⟨granted⟩ at the start. It can be argued in the same way about B with respect to A, each time deriving ⟨pairs of⟩ contradictories. ad 4.3.2.1 I reply to the first ⟨point by granting⟩ that Plato is deceived and that he believes a falsehood. And then ⟨in reply⟩ to the argument: Plato believes a falsehood and he only believes that Socrates is not deceived, therefore it is false that Socrates is not deceived, I deny the inference, just as: Socrates says a falsehood and he says only “Socrates does not say a truth,” therefore it is false that Socrates does not say a truth, is not valid; but one should add in the premise that what Socrates says does not falsify itself. And in the same way one should add in the ⟨premises of the⟩ other ⟨argument⟩: And what Plato believes does not falsify itself, which I deny. In a similar way, I grant this: Socrates is not deceived. And if one argues like this: Socrates is not deceived and Socrates believes that ⟨it is⟩ in some way, therefore he believes only what is the case, but he believes only that Plato is deceived, therefore it is the case that Plato is deceived, I grant the inference and the conclusion.

218

Latin Text

ad 4.3.2.2 Concesso

igitur quod plato decipitur et quod107 sortes non decipitur, oportet dicere quod a est verum et quod b est falsum, quia a dicitur a sorte non decepto et b a platone decepto. Et tunc ad argumentum: b est falsum et b significat precise sortem non decipi, igitur falsum est sortem non decipi, negatur consequentia. Sed oportet addere quod b non falsificat se. Quare etc. 4.3.3 Tertio principaliter arguitur sic: et pono quod sortes sit omnis sortes et dicat istam et nullam aliam: sortes mentitur, sic adequate significantem, que sit a. 4.3.3.1 Isto posito, propono tibi: sortes mentitur. Si concedis, arguo quod non, nam sequitur: sortes mentitur,108 igitur sortes dicit falsum. Tenet consequentia supponendo quod omne mentiens dicat falsum.109 Tunc arguitur sic: sortes dicit falsum, igitur sortes dicit sicut non est, sed sortes dicit solummodo quod sortes mentitur, igitur non est ita quod sortes mentitur et per consequens sortes non mentitur. Si autem negatur illa: sortes mentitur, arguo110 sic: sortes non mentitur et sortes dicit aliquam propositionem, igitur sortes dicit sicut est. Tunc sic: sortes dicit sicut est et non dicit nisi quod sortes mentitur, igitur ita est quod sortes mentitur et per consequens sortes mentitur. 4.3.3.2 Item quero utrum a sit verum vel falsum. Si verum, igitur significat adequate sicut est, sed a significat adequate quod sortes mentitur, igitur ita est quod sortes mentitur, igitur ipse dicit falsum et non dicit nisi a, igitur a est falsum. Quod si conceditur, arguitur sic: a est falsum, igitur significat adequate aliter quam est; sed a significat adequate quod sortes mentitur, igitur non est ita quod sortes mentitur, igitur non dicit falsum et non dicit nisi a, igitur a non est falsum, quod est contradictorium concessi.

 quod] om. E  si concedis – mentitur] om.(hom.) M 109  falsum] et add. E 110  arguo] arguitur E 107 108

ad 4.3.2.2 Thus,

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219

granted that Plato is deceived and that Socrates is not deceived, one must say that A is true and that B is false, since A is said by Socrates, who is not deceived, and B by Plato, who is deceived. And then ⟨in reply⟩ to the argument: B is false and B signifies only that Socrates is not deceived, therefore it is false that Socrates is not deceived, I deny the inference. But one should add that B does not falsify itself, ⟨which I deny⟩. Therefore ⟨A is true⟩. 4.3.3 The third ⟨derivative insoluble⟩ is this: assume that there is only one Socrates and that he says this and no other proposition: Socrates is lying, signifying exactly in that way, call it A. 4.3.3.1 Given this ⟨scenario⟩, I propose to you: Socrates is lying. If you grant it, I argue that not, for this is valid: Socrates is lying, therefore Socrates says a falsehood. The inference holds supposing that everyone who is lying says a falsehood. Then I argue like this: Socrates says a falsehood, therefore Socrates says what is not the case, but Socrates says only that Socrates is lying, therefore it is not the case that Socrates is lying, and consequently Socrates is not lying. If, however, you deny the proposition: Socrates is lying, I argue like this: Socrates is not lying and Socrates says some proposition, therefore Socrates says what is the case. Then ⟨I argue⟩ like this: Socrates says what is the case and he says only that Socrates is lying, therefore it is the case that Socrates is lying and consequently Socrates is lying. 4.3.3.2 Again, I ask whether A is true or false. If true, then it exactly signifies what is the case, but A exactly signifies that Socrates is lying, therefore it is the case that Socrates is lying, therefore he says a falsehood and he says only A, therefore A is false. If that is granted ⟨at the start⟩, I argue like this: A is false, therefore it exactly signifies other than is the case. But A exactly signifies that Socrates is lying, therefore it is not the case that Socrates is lying, therefore he does not say a falsehood and he says only A, therefore A is not false, which is the contradictory of what was granted.

220

Latin Text

ad 4.3.3.1 Ideo

concedendum est quod sortes mentitur et quod ipse dicit falsum. Et tunc ad argumentum: sortes dicit falsum, igitur dicit sicut non est, concedatur111 consequens. Tunc ad argumentum: sortes dicit sicut non est et non dicit nisi quod sortes mentitur, igitur non est ita quod sortes mentitur, nego consequentiam,112 quia oportet addere in antecedente quod dictum sortis non se falsificat, quod negatur. ad 4.3.3.2 Per hoc patet quod a est falsum et non valet illa consequentia: a est falsum, igitur significat adequate aliter quam est. Sed oportet addere in minori quod a non se113 falsificat, quod iterum negatur. 4.3.4 Quarto principaliter arguitur sic: et pono quod sortes sit omnis sortes et iuret istam et nullam aliam: sortes est periurius, sic precise114 significantem, que sit a, supposito115 quod omnis periurans dicat falsum. Unde si iurarem tibi dare crastina die denarium et crastina die116 nullum darem tibi denarium, ego dicerer periu-(E 197vb)-rius quoniam iurarem unum falsum. 4.3.4.1 Isto posito, propono tibi: sortes est periurius. Si concedis, igitur sortes dicit falsum, sed sortes dicit solum quod ipse est periurius, igitur non est ita quod ipse est periurius. Si negatur,117 contra: sortes non est periurius, igitur (M 243ra) sortes non dicit118 falsum, sed sortes solummodo iurat quod ipse est periurius, igitur verum est quod ipse est periurius. Et per consequens sortes est periurius. Nec etiam est dubitanda quia oppositum repugnat casui. Sequitur enim: sortes non est periurius et sortes iurat quod ipse est periurius, igitur iurat verum, et per consequens est periurius. Unde insolubile numquam est negandum nec dubitandum in casu, quia semper oppositum repugnat ⟨casui⟩.

 concedatur] conceditur E  nego consequentiam] negatur consequentia E 113  non se] inv. E 114  precise corr.] primo M E 115  supposito] supponendo E 116  crastina die] inv. M 117  negatur] negetur M 118  non dicit] om. M 111 112

ad 4.3.3.1 It

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221

should be granted, therefore, that Socrates is lying and that he says a falsehood. And then ⟨in reply⟩ to the argument: Socrates says a falsehood, therefore he says what is not the case, the conclusion should be granted. Then ⟨in reply⟩ to the argument: Socrates says what is not the case and he says only that Socrates is lying, therefore it is not the case that Socrates is lying, I deny the inference, since one should add in the premises that what Socrates says does not falsify itself, which I deny. ad 4.3.3.2 Through this it is clear that A is false and that the inference: A is false, therefore it exactly signifies other than is the case does not hold. But one should add in the minor premise that A does not falsify itself, which I again deny. 4.3.4 The fourth ⟨derivative insoluble⟩ is this: assume that there is only one Socrates and that he swears this and no other proposition: Socrates is an oath-breaker, signifying only in that way, call it A, and suppose that every oath-breaker says a falsehood. Then, if I swore that I would give you a penny tomorrow and tomorrow I didn’t give you a penny, I would be labelled an oath-breaker, since I swore something false. 4.3.4.1 Given this ⟨scenario⟩, I propose to you: Socrates is an oath-breaker. If you grant ⟨it⟩, then Socrates says a falsehood, but Socrates says only that he is an oath-breaker, therefore it is not the case that he is an oathbreaker. If you deny it, on the contrary: Socrates is not an oath-breaker, therefore Socrates does not say a falsehood, but Socrates swears only that he is an oath-breaker, therefore it is true that he is an oath-breaker. And consequently Socrates is an oath-breaker. Nor should it be doubted, since ⟨its⟩ opposite is inconsistent with the scenario. For this is valid: Socrates is not an oath-breaker and Socrates swears that he is an oathbreaker, therefore he swears a truth. And consequently, he is an oathbreaker. For an insoluble should never be denied nor doubted in a scenario, since ⟨its⟩ opposite is always inconsistent ⟨with the scenario⟩.

222

Latin Text

ad 4.3.4.1 Dicendum

ut prius quod sortes est periurius et quod ipse iurat et dicit falsum. Et tunc ad argumentum, quando concluditur quod non est ita quod ipse sit periurius, non valet argumentum, quia debuit119 addi in antecedente quod dictum sortis non120 se falsificaret121 etc. Dicitur etiam quod a est falsum, quia falsificat se, et suum contradictorium, quia non se falsificat et significat adequate nullum sortem esse periurium, quod est falsum. 4.4 Et non solum in hoc ultimo verificatur quod dictum est, ymmo et in prioribus et122 posterioribus, quia talis est natura insolubilium.123

 quia debuit] sed debet E  non] om. E 121  falsificaret] falsificat E 122  et] in add. E 123  insolubilium] etc. add. E 119 120

ad 4.3.4.1 I

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223

reply ⟨in the same way⟩ as before that Socrates is an oath-breaker and that he swears and says a falsehood. And then ⟨in reply⟩ to the argument, when one concludes that it is not the case that he is an oath-breaker, the argument is not valid, since it should have been added in the premises that what Socrates says should not have falsified itself, ⟨which I deny⟩. I also say that A is false, since it falsifies itself, as ⟨is⟩ its contradictory, since ⟨although⟩ it does not falsify itself, it exactly signifies that no Socrates is an oath-breaker, which is false. 4.4 And what has been said holds true not only for this last case but also for previous and subsequent cases, since that is the nature of insolubles.

224

Latin Text

⟨Capitulum Quintum: De propositionibus non apparentibus insolubilibus1⟩ consimili insolubilia solvuntur que prima facie insolubilia non apparent, ut: sortes est albus2, sortes est eger, plato male respondet, sortes non habebit denarium. ⟨Primo principaliter arguitur sic⟩: suppono igitur quod omnis homo eger dicat falsum et solum talis, et omnis homo sanus dicat verum et solum talis. Isto supposito,3 pono quod sortes sit4 omnis sortes et dicat illam solummodo: sortes est eger, sic precise significantem, que sit a. 5.1.1 Isto posito cum supposito simul, quero utrum sortes sit sanus vel eger. Si dicitur quod est sanus et omnis sanus dicit verum, igitur sortes dicit verum. Et non dicit nisi quod ipse5 est eger, igitur ita est quod ipse6 est eger. Et per consequens non est sanus. Si autem dicitur in principio quod sortes est eger et omnis eger dicit falsum, igitur sortes dicit falsum. Sed sortes dicit solum quod ipse est eger, igitur falsum est quod ipse est eger. Et per consequens non est eger, cuius oppositum est concessum. 5.1.2 Item queritur si a est verum vel falsum. Si verum7 et significat precise quod sortes est eger, igitur sortes est eger,8 igitur ⟨sortes⟩ dicit falsum, et non dicit nisi a, igitur a est falsum; quod si conceditur, arguitur sic: a est falsum et a9 significat precise quod sortes est eger, igitur non est ita quod sortes est eger.

5.1 Modo

 De propositionibus – insolubilibus] marg. M  sortes est albus] om. E 3  supposito] posito E 4  sit] fuit E 5  ipse] sortes E 6  ipse] sortes E 7  verum] eger E 8  igitur – eger] om.(hom.) M 9  a] om. E 1 2



English Translation

225

⟨Chapter Five: On Propositions which Appear not to be Insolubles⟩ the same way we solve insolubles that at first glance appear not to be insolubles, such as: Socrates is white, Socrates is sick, Plato responds incorrectly, Socrates will not receive a penny. ⟨The first insoluble to discuss is this⟩: suppose then that all and only sick people say a falsehood, and that all and only healthy people say a truth. Given this ⟨scenario⟩, I assume that there is only one Socrates and that he says only the proposition: Socrates is sick, signifying only in that way, call it A. 5.1.1 Given what was assumed together with the scenario, I ask whether Socrates is healthy or sick. If one answers that he is healthy, and every healthy person says a truth, then Socrates says a truth. And he says only that he is sick, therefore it is the case that he is sick. And consequently, he is not healthy. If, however, at the start one answers that Socrates is sick and every sick person says a falsehood, then Socrates says a falsehood. But Socrates says only that he is sick, therefore it is false that he is sick. And consequently, he is not sick, which is the opposite of what was granted. 5.1.2 Again, I ask whether A is true or false. If true, and it signifies only that Socrates is sick, then Socrates is sick, therefore ⟨Socrates⟩ says a falsehood, and he says only A, therefore A is false. If that is granted ⟨at the start⟩, I argue in this way: A is false, and A signifies only that Socrates is sick, therefore it is not the case that Socrates is sick.

5.1 In

226





Latin Text ad 5.1.1 Respondetur10

admittendo positum et suppositum et concedo quod sortes est eger et quod ipse dicit falsum et nego illam consequentiam: sortes dicit falsum et dicit solum quod ipse est eger, igitur falsum est quod ipse est eger, quia oportet addere in antecedente quod dictum sortis non se falsificat. ad 5.1.2 Et consequenter dico a esse falsum et suum contradictorium similiter, nec hoc in hac materia pro inconvenienti reputatur. 5.2 Secundo ⟨principaliter⟩ arguitur sic: suppono quod omne dicens falsum male respondeat et omne dicens verum bene respondeat et solum talis. Deinde pono quod plato sit omnis plato et dicat illam solummodo: plato male respondet, sic adequate significantem, que sit a. Isto posito, quero utrum plato bene vel male respondeat.11 Si bene respondet, igitur dicit verum et non dicit nisi quod ipse male respondet, igitur ita est quod male respondet. Et per consequens non bene respondet. Si autem dicitur quod male respondet, igitur dicit falsum et non dicit nisi quod ipse male respondet, igitur non est ita quod ipse male respondet. Et per consequens non male respondet. (M 243rb) ad 5.2 Ideo dicitur ut prius quod plato male respondet et quod ipse dicit falsum. Et ad improbationem negatur consequentia,12 quia oportet addere13 in antecedente quod dictum platonis non se falsificat. Et ita consequenter dicitur14 quod a est falsum, quia falsificat se, et suum contradictorium similiter, quia significat aliter quam est et hoc adequate.

 Respondetur] Responditur M  vel male respondeat corr.] vel male respondet M, respondeat vel male E 12  negatur consequentia] nego consequentiam E 13  addere] addi E 14  dicitur] dicatur E 10 11



English Translation

respond by admitting what was assumed and the scenario and I grant that Socrates is sick and that he says a falsehood, and I deny the inference: Socrates says a falsehood, and he says only that he is sick, therefore it is false that he is sick, since one should add in the premises that what Socrates says does not falsify itself, ⟨which I deny⟩. ad 5.1.2 And consequently, I say that A is false, and so too is its contradictory, which I do not consider impossible in the case ⟨of insolubles⟩. 5.2 The second ⟨insoluble⟩ is this: suppose that all and only people who say a falsehood respond incorrectly, and all and only people who say a truth respond correctly. Then I assume that there is only one Plato and that he says only this: Plato responds incorrectly, signifying exactly in that way, call it A. Given this ⟨scenario⟩, I ask whether Plato responds correctly or incorrectly. If he responds correctly, then he says a truth and he says only that he responds incorrectly, therefore it is the case that he responds incorrectly. And consequently, he does not respond correctly. If, however, one answers that he responds incorrectly, then he says a falsehood, and he says only that he responds incorrectly, therefore it is not the case that he responds incorrectly. And consequently, he does not respond incorrectly. ad 5.2 Accordingly, I answer as before that Plato responds incorrectly and that he says a falsehood. And against the counterargument, I deny the inference, since one should add in the premises that what Plato says does not falsify itself, ⟨which I deny⟩. And consequently, I say that A is false, since it falsifies itself, and so too is its contradictory, since it exactly signifies other than is the case.

ad 5.1.1 I

227

228

Latin Text

principaliter arguitur sic: et suppono quod omne dicens verum habebit denarium et solum tale, et omne dicens falsum non habebit denarium et solum tale. Isto supposito, pono quod sortes sit omnis sortes et dicat illam et nullam aliam: sortes non habebit denarium, sic adequate significantem, que sit a. Isto posito, propono: sortes non habebit denarium. Si conceditur, igitur sortes16 non dicit verum, sed sortes non dicit nisi quod ipse non habebit denarium, igitur non est verum quod ipse non17 habebit denarium. Et per consequens sortes habebit denarium; quod si conceditur, arguitur sic: sortes habebit denarium, igitur sortes18 dicit verum et non dicit nisi quod ipse non habebit denarium, igitur verum est quod ipse non habebit denarium. Et per consequens sortes non habebit de-(E 198ra)-narium. 5.3.1 Item potest queri ut prius si a est verum vel falsum et qualitercumque dicatur reducatur contradictio. ad 5.3 Dicendum quod sortes non habebit denarium, quia aut iste erit omnis sortes aut erunt plures. Si erit19 omnis sortes, sequitur cum casu quod ipse non habebit denarium. Si autem erunt plures, aut aliquis illorum dicit falsum aut nullus. Si aliquis eorum20 dicet falsum,21 sequitur quod ille sortes non habebit denarium et ille est vel erit sortes, igitur sortes non habebit denarium. Si autem dicitur quod nullus preter istum qui modo est dicet falsum, sequitur quod sortes non habebit denarium, quia iste de quo fit mentio in casu non habebit denarium. Cum ergo arguitur: sortes non habebit denarium—cum singulis particulis casus—igitur sortes dicit falsum, non valet consequentia. Unde cum toto casu stat quod cras erit unus alius sortes dicens nullum deum esse; quo dato ipse non foret habiturus denarium. Et consequenter sortes presens non diceret falsum sed verum.

5.3 Tertio15

 Tertio] quarto M  sortes] om. E 17  non] om. E 18  sortes] om. M 19  si erit] sicut E 20  eorum] istorum E 21  falsum] om. E 15 16

5.3 The third

English Translation

229

insoluble is this: suppose that all and only people who say a truth will receive a penny, and that all and only people who say a falsehood will not receive a penny. Given what was supposed, I assume that there is only one Socrates and that he says this and no other proposition: Socrates will not receive a penny, signifying exactly in that way, call it A. Given this scenario, I propose: Socrates will not receive a penny. If one grants it, then Socrates does not say a truth, but Socrates says only that he will not receive a penny, therefore it is not true that he will not receive a penny. And consequently, Socrates will receive a penny. If that is granted ⟨at the start⟩, I argue in this way: Socrates will receive a penny, therefore Socrates says a truth and he says only that he will not receive a penny, therefore it is true that he will not receive a penny. And consequently, Socrates will not receive a penny. 5.3.1 Again, one can ask as before whether A is true or false and a contradiction may be deduced whatever one says. ad 5.3 I answer that Socrates will not receive a penny, since either there will be only this Socrates or there will be several ⟨Socrates⟩. If there will be only this Socrates, it follows from the scenario that he will not receive a penny. If, however, there will be several ⟨Socrates⟩, either some of them will say a falsehood or none of them. If any of them will say a falsehood, it follows that Socrates will not receive a penny, and he is or will be a Socrates, therefore a Socrates will not receive a penny. If, however, one answers that no one except this ⟨Socrates⟩ who now exists will say a falsehood, it follows that Socrates will not receive a penny, since this ⟨Socrates⟩ mentioned in the scenario will not receive a penny. Therefore, when one argues, adding the specific qualifications of the scenario: Socrates will not receive a penny, therefore Socrates says a falsehood, the inference does not hold. For it is consistent with the whole scenario that tomorrow there will be another Socrates saying that God does not exist; given this, he will not receive a penny. And consequently, the present Socrates would not say a falsehood but a truth.

230

Latin Text

ergo proponitur a, concedatur.22 Et cum dicitur utrum sit verum vel falsum, dubitetur utrumque, quia cum casu stat a esse verum et cum eodem casu stat a esse falsum, dato quod numquam erit aliquis sortes preter istum. Ad hoc ergo quod a fiat insolubile non debet poni quod unus sortes sit omnis sortes, sed quod omnis sortes erit iste sortes, demonstrando presentem. Quo posito, iterum concedo quod sortes non habebit denarium et consequenter quod ipse dicit falsum. Et tunc ad argumentum negatur consequentia, quia oportet addere quod dictum sortis non falsificat se, quod negatur quia a falsificat se, ideo est falsum, et similiter suum contradictorium, quia significat adequate falsum. 5.3.2 Item si poneretur in casu quod sortes diceret illam solummodo: sortes habebit denarium, ego concederem quod sortes habebit denarium et quod a est verum, dato23 quod illa esset a, sicut concederem sortem dicere verum, dato quod non diceret nisi illam: sortes dicit verum, cum aliis particulis ponendis in casu. 5.4 Quarto principaliter arguitur sic:24 suppono quod omne dicens verum pertransibit pontem et solum tale, et omne dicens falsum non pertransibit pontem et solum tale. Isto supposito,25 pono quod sortes sit omnis sortes et erit omnis sortes, qui dicat istam solum: sortes non pertransibit pontem (M 243va), sic adequate significantem, que sit a. 5.4.1 Isto posito, quero utrum sortes pertransibit pontem vel non. Si pertransibit pontem, igitur dicit verum, et non dicit nisi quod sortes non pertransibit pontem, igitur sortes26 non pertransibit pontem; quod si conceditur arguitur sic: sortes non pertransibit pontem, igitur non dicit verum, et ipse non dicit nisi quod ipse non pertransibit pontem, igitur27 non est verum quod ipse non pertransibit pontem. Et per consequens sortes pertransibit pontem.

ad 5.3.1 Cum

 concedatur] conceditur E  dato] data M 24  et add. E 25  supposito] posito E 26  sortes] om. E 27  non dicit verum – pontem igitur] om. (hom.) E 22 23



English Translation

231

ad 5.3.1 So when A is proposed, it should be granted. And when one asks whether

⟨A⟩ is true or false, doubt should be expressed about each alternative, since it is consistent with the scenario that A is true and it is consistent with the same scenario that A is false, assuming that there will never be any Socrates except this one. In order for A to be an insoluble it should not be assumed that there is only one Socrates, but that there will be only this Socrates, referring to the present one. Given this ⟨scenario⟩, I again grant that Socrates will not receive a penny and consequently that he says a falsehood. And then ⟨in reply⟩ to the argument, I deny the inference, since one should add ⟨in the premises⟩ that what Socrates says does not falsify itself, which I deny since A falsifies itself. Therefore it is false, as too is its contradictory, since it exactly signifies a falsehood. 5.3.2 Again, if one assumed in the scenario that Socrates said only this: Socrates will receive a penny, provided that it was A, I would grant that Socrates will receive a penny and that A is true, just as I would grant that Socrates says a truth, given that he only said: Socrates says a truth, together with the other qualifications that were assumed in the scenario. 5.4 The fourth insoluble is this: suppose that all and only people who say a truth will cross the bridge, and all and only people who say a falsehood will not cross the bridge. Given this, I assume that there is and will be only one Socrates and let him say only this: Socrates will not cross the bridge, signifying exactly in that way, call it A. 5.4.1 Given this ⟨scenario⟩, I ask whether Socrates will cross the bridge or not. If he will cross the bridge, then he says a truth, and he says only that Socrates will not cross the bridge, therefore Socrates will not cross the bridge. If that is granted ⟨at the start⟩, I argue like this: Socrates will not cross the bridge, therefore he does not say a truth and he says only that he will not cross the bridge, therefore it is not true that he will not cross the bridge. And consequently, Socrates will cross the bridge.

232

Latin Text

5.4.2 Item quero

si a est verum vel falsum. Si dicitur quod verum, igitur significat adequate verum, sed a adequate28 significat quod sortes non pertransibit pontem, igitur sortes non pertransibit pontem, igitur non dicit verum, et non dicit nisi a, igitur a non est verum. Si autem dicitur quod a est falsum, contra: cum toto29 casu stat quod a est verum, igitur non debet aliter responderi quam ante casum; sed ante casum dubitasses illam: a est verum, igitur et nunc30 dubitanda est. Quod autem illa stet cum casu probatur31 ponendo cum casu quod in b instanti erit unus sortes, qui numquam erit iste presens; possibile est enim quod erunt duo sortes quorum quilibet erit32 omnis sortes non in eodem instanti sed in diversis et quod secundus sortes dicet33 illam: chymera est. Isto posito, patet quod a est verum. ad 5.4.1–2 Ideo concedo quod sortes non pertransibit pontem et dubito quamlibet illarum: a est verum, a est falsum. Si tamen poneretur quod omnis sortes erit iste sortes, concederetur a esse falsum quia cum isto casu non stat quod erunt duo sortes quorum quilibet erit omnis sortes. Etiam si cum priori casu non poneretur quod sortes diceret istam: sortes non pertransibit pontem, sed illam: ego non pertransibo pontem, que esset a, dicerem consequenter quod a est falsum quia se falsificaret. 5.4.3 Et si arguitur sic: sortes34 dicit falsum, igitur sortes35 non pertransibit pontem, consequentia est bona et antecedens est verum, igitur et consequens; sed consequens est a, igitur a est verum.

 a adequate] inv. E  toto] om. E 30  nunc] non E 31  probatur] probetur M 32  erit] erunt E 33  dicet] dicat M 34  sortes] a E 35  sortes] a E 28 29

5.4.2 Again,

English Translation

233

I ask whether A is true or false. If one answers that it is true, then it exactly signifies a truth. But A exactly signifies that Socrates will not cross the bridge, therefore Socrates will not cross the bridge. Therefore he does not say a truth and he says only A, therefore A is not true. If, however, one answers that A is false, on the contrary: it is consistent with the whole scenario that A is true, therefore one should only reply as ⟨one replied⟩ before the scenario was set up. But before the scenario was set up, you were in doubt about A is true, therefore now too you should express doubt about it. Proof that it is consistent with the scenario: assume that at instant t1 there will be a Socrates, who will never be the ⟨same as the Socrates⟩ existing now, for it is possible that there will be two Socrates, each of whom will be the only Socrates, not at the same instant but at different instants, and that the second Socrates will say A chimera exists. Given this ⟨scenario⟩, it is clear that A is true. ad 5.4.1-2 Therefore, I grant that Socrates will not cross the bridge and I express doubt about each of these: A is true, A is false. If, however, one assumed that there would be only this Socrates, I would grant that A is false, since it is inconsistent with this scenario that there would be two Socrates each of whom would be the only Socrates. Moreover, if with the given scenario one did not assume that Socrates said Socrates will not cross the bridge, but this: I will not cross the bridge, call it A, I would consequently answer that A is false, since it would falsify itself. 5.4.3 But if one argues like this: Socrates says a falsehood, therefore Socrates will not cross the bridge, the inference holds and the premise is true, therefore so too is the conclusion, but the conclusion is A, therefore A is true.

234

Latin Text

quod consequens non est a, sed convertibile cum eo. Ubi tamen ponatur quod sit a, dico quod consequentia est bona et antecedens est verum et consequens falsum. Nec ex hoc sequitur quod alicuius bone consequentie contradictorium consequentis stat36 cum antecedente quia,37 licet consequens sit falsum, non tamen38 sequitur quod contradictorium sit verum, ymmo potius oppo-(E 198rb)-situm in materia insolubilium. Quare etc. 5.5 Post declarationem insolubilium singularium restat consequenter alia solvere, quorum39 aliqua sunt quanta, quedam vero non quanta nominantur.

ad 5.4.3 Dicitur

 stat] stet E  quia ] om. E 38  non tamen] inv. E 39  quorum] quarum M 36 37



English Translation

reply that the conclusion is not A but a proposition convertible with it. However, assuming that it is A, I answer that the inference holds and the premise is true and the conclusion is false. And from this it does not follow that the contradictory of the conclusion of some valid inference is compatible with the premise, since although the conclusion is false, yet it does not follow that the contradictory is true, but it is rather the opposite in the case of insolubles, therefore ⟨A is false, as too is its contradictory⟩. 5.5 Now that singular insolubles have been explained, the next task is to solve others, some of which are quantified ⟨insolubles⟩, whilst some are called non-quantified ⟨insolubles⟩.

ad 5.4.3 I

235

236

Latin Text

⟨Capitulum Sextum: De quantis insolubilibus⟩ 6.1 Ideo a particulari

incipiens vel indefinita, pono istum casum quod hec propositio: falsum est, sit omnis propositio, que sit a, et significet adequate et precise sicut termini pretendunt. Isto posito, quero utrum a1 sit verum vel falsum. Si verum et significat adequate quod falsum est, igitur ita est quod falsum est. Et a est omnis propositio, igitur a est falsum. Quod si conceditur arguitur sic: a est falsum et2 a significat adequate quod falsum est, igitur non est ita quod falsum est. Et per consequens a non est falsum. 6.1.1 Item sic: hoc est, demonstrando a, et hoc est falsum, igitur falsum est. Consequentia est bona et antecedens est verum, igitur et consequens. Et3 consequens est a, igitur a est verum. ad 6.1 Respondetur concedendo4 illam: falsum est. Et cum queritur utrum a sit verum vel falsum, dico quod falsum. Et tunc ad argumentum: non valet consequentia, quia oportet addere in (M 243vb) antecedente quod a non falsificat se, quod negatur. A enim est falsum non propter suum significatum, quia illud est verum, sed quia se falsificat. Et contradictorium a est falsum, dato quod ambe forent omnes propositiones, quia eius adequatum5 significatum est falsum. ad 6.1.1 Ad aliud, cum dicitur quod illa consequentia est bona, negatur tamquam repugnans. Sequitur enim: a est omnis propositio, igitur nulla consequentia est, quia ex opposito consequentis sequitur oppositum antecedentis. Non enim est possibile aliquam consequentiam esse et non esse plures propositiones. Verumtamen non sequitur, ut superius dictum est, illa: consequentia est bona, et antecedens est verum, igitur et consequens. 6.1.2 Consimiliter diceretur6 ad istas: aliqua propositio particularis est falsa, propositio indefinita non est vera. Unde quelibet illarum est  a] om. E  et] igitur E 3  et] sed E 4  Respondetur concedendo] Responditur concedo M 5  adequatum] om. E 6  diceretur] dicetur M 1 2



English Translation

237

⟨Chapter Six: On Quantified Insolubles⟩ 6.1 Hence,







to begin with particular or indefinite ⟨propositions⟩, I put forward this scenario, that this proposition: A falsehood exists, call it A, is the only proposition and signifies exactly and only as the terms suggest. Given this ⟨scenario⟩, I ask whether A is true or false. If it is true and signifies exactly that a falsehood exists, then it is the case that a falsehood exists. And A is the only proposition, therefore A is false. If that is granted ⟨at the start⟩, I argue like this: A is false and A signifies exactly that a falsehood exists, therefore it is not the case that a falsehood exists. And consequently, A is not false. 6.1.1 Again, ⟨I argue⟩ like this: This exists—referring to A—and this is false, therefore a falsehood exists. The inference holds and the premise is true, therefore so too ⟨is⟩ the conclusion. And the conclusion is A, therefore A is true. ad 6.1 I respond by granting A falsehood exists. And when it is asked whether A is true or false, I answer that it is false. And then ⟨in reply⟩ to the argument: the inference is not valid, since one should add in the premises that A does not falsify itself, which I deny. For A is false not because of its significate, since that ⟨significate⟩ is true, but because it falsifies itself. And the contradictory of A is false, supposing that these two were the only propositions, since its exact significate is false. ad 6.1.1 To the other ⟨argument⟩, when it is said that that inference holds, I deny it as inconsistent ⟨with the scenario⟩, for this is valid: A is the only proposition, therefore no inference exists, since the opposite of the premise follows from the opposite of the conclusion. For it is impossible that some inference exists and yet there does not exist more than one proposition. Nevertheless, as has been said before, this is not valid: The inference holds and the premise is true, therefore so too is the conclusion.

238

Latin Text

falsa, dato quod ipsa sit quelibet propositio particularis vel indefinita. Et consequenter dico quod aliqua propositio particularis est falsa et sua indefinita vera et e converso. 6.2 In materia autem universalium fit iste casus, quod tantum sint a et b propositiones, a verum et b ista: omne verum est a, que significet adequate ut termini pretendunt. Isto posito, propono: omne verum est a. Si conceditur, arguitur sic: omne verum est a et7 b significat adequate quod omne verum est a, igitur b est verum; sed b non est a, igitur non omne verum est a. Si autem negatur vel dubitatur illa, contra: oppositum repugnat casui. Probatur, nam sequitur: non omne verum est a, et a et b8 sunt omnes propositiones, igitur b est verum; sed b significat adequate quod omne verum est a, igitur omne verum est a, quod fuit negatum. 6.2.1 Item quero si b est verum vel falsum. Si dicitur quod verum, et b significat adequate quod omne verum est a, igitur omne verum est a. Et per consequens b non est verum. Si dicitur quod b9 est falsum et b10 significat precise quod omne verum est a, igitur non omne verum est a. Tunc sic: non omne verum est a et a et b sunt omnes propositiones, igitur b est verum. ad 6.2 Dicendum quod b11 est concedendum quotienscumque proponitur et negatur illa consequentia: omne verum est a12 et b sic adequate significat,13 igitur b est verum, quia debet addi in minori:14 et b non falsificat se, quod negatur, ex quo sequitur quod b est falsum. ad 6.2.1 Et ad improbationem negatur consequentia propter eandem causam. 6.2.1.1 Et si arguitur sic: aliquod verum est a, et nullum est verum quin illud sit a, igitur omne verum est a, consequentia est bona et  et] om. E  a et b] b et a E 9  b corr.] a M E 10  b corr.] a M E 11  b corr.] a M E 12  a] et a add. M 13  significat] significant M 14  minori] antecedente E 7 8

6.1.2 I would

English Translation

239

answer in just the same way to these: Some particular proposition is false, An indefinite proposition is not true. Thus, each of them is false, given that each is the only particular or indefinite proposition. And consequently, I answer that some particular proposition is false and its indefinite is true and conversely. 6.2 In the case of universal ⟨propositions⟩, let the scenario be that the only propositions are A and B, where A is true and B is Every truth is A, signifying exactly as the terms suggest. Given this ⟨scenario⟩, I propose Every truth is A. If it is granted, I argue in this way: every truth is A, and B signifies exactly that every truth is A, therefore B is a truth; but B is not A, therefore not every truth is A. But if one denies or expresses doubt about it ⟨that is, Every truth is A⟩, on the contrary: the opposite ⟨“Not every truth is A”⟩, is inconsistent with the scenario. Proof: for this is valid: Not every truth is A, and the only propositions are A and B, therefore B is a truth; but B signifies exactly that every truth is A, therefore every truth is A, which was denied. 6.2.1 Again, I ask whether B is true or false. If one answers that ⟨it is⟩ true, and B signifies exactly that every truth is A, then every truth is A. And consequently B is not true. If, however, one answers that B is false and B signifies only that every truth is A, then not every truth is A. Then ⟨I argue⟩ like this: Not every truth is A and the only propositions are A and B, therefore B is true. ad 6.2 I answer that B should be granted whenever it is proposed; and the inference: Every truth is A and B signifies exactly in that way, therefore B is a truth should be denied since one should add in the minor premise and B does not falsify itself, which I deny, whence it follows that B is false. ad 6.2.1 And against the counterargument, I deny the inference for the same reason.

240

Latin Text

antecedens est verum, igitur et consequens, quod est b, dicitur quod consequentia est bona et consequens est falsum non obstante quod antecedens sit verum. Eodem modo dicitur ad hanc: hoc15 est a et hoc est omne verum, igitur omne verum est a, quod consequens est falsum et antecedens verum, licet consequentia sit bona. 6.2.2 Secundo arguitur sic: et pono quod a, b, c sint16 omnes propositiones, quarum a et b sint vere et c sit ista: quelibet17 propositio est dissimilis istis, demonstrando a et b, que sic precise et18 adequate significat.19 Isto posito, propono: quelibet propositio est dissimilis istis. Si conceditur, arguitur sic: quelibet propositio est dissimilis istis et c sic precise20 significat, igitur c est verum; sed a et21 b et22 c sunt omnes propositiones, igitur omnis propositio est vera. Et per consequens non quelibet propositio est dissimilis istis. Si autem negatur vel dubitatur illa: quelibet propositio est (E 198va) dissimilis istis, arguitur sic: non quelibet propositio est dissimilis istis, et a, (M 244ra) b, c sunt omnes propositiones, quarum a et23 b sunt vere, igitur c est verum. Tunc sic: c est verum et c adequate significat quod quelibet propositio est dissimilis istis, igitur quelibet propositio est dissimilis istis, cuius oppositum est concessum.

 hoc] hec E  sint] sunt E 17  quelibet corr.] omnis M E 18  et] om. E 19  significat] significent E 20  precise] om. E 21   et ] om. E 22  et ] om. E 23  et] om. E 15 16

6.2.1.1 And

English Translation

241

if one argues like this: this inference holds: Some truth is A, and there is no truth except A, therefore every truth is A, and the premises are true, therefore so too is the conclusion (call it B): I answer that the inference holds and the conclusion is false, despite the fact that the premises are true. And to this: This is A and this is the only truth, therefore the only truth is A, I answer in the same way that even though the inference holds, the premises are true and the conclusion is false. 6.2.2 Secondly, I argue like this: assume that the only propositions are A, B and C, where A and B are true and C is Every proposition is unlike those two ⟨in truth-value⟩, referring to A and B, signifying only and exactly in that way. Given this ⟨scenario⟩, I propose: Every proposition is unlike those two ⟨in truth-value⟩. If you grant it, I argue like this: every proposition is unlike those two ⟨in truth-value⟩ and C signifies only like that, therefore C is true; but the only propositions are A and B and C, therefore every proposition is true. And consequently, not every proposition is unlike those two ⟨in truth-value⟩. If, however, you deny or express doubt about the proposition: Every proposition is unlike those two ⟨in truth-value⟩, I argue like this: not every proposition is unlike those two ⟨in truthvalue⟩, and the only propositions are A, B ⟨and⟩ C, of which A and B are true, therefore C is true. Then ⟨I argue⟩ like this: C is true and C exactly signifies that every proposition is unlike those two ⟨in truthvalue⟩, therefore every proposition is unlike those two ⟨in truth-value⟩, which is the opposite of what you granted.

242

Latin Text

6.2.2.1 Item

quero si c est verum vel falsum. Si verum et c adequate24 significat quod quelibet propositio est dissimilis istis, igitur quelibet propositio est dissimilis istis25; sed non a nec b, igitur c est dissimilis istis. Tunc sic: c est dissimilis istis et a et b sunt vera, igitur c est falsum. Quod si conceditur, arguitur sic: c est falsum et c26 adequate significat quod quelibet propositio est dissimilis istis, igitur non est ita quod quelibet propositio est dissimilis istis; sed non est maior ratio de una quam de qualibet, igitur nulla est dissimilis istis. Sed a, b, c sunt ⟨omnes propositiones⟩ quarum a, b sunt vere, igitur c est verum. ad 6.2.2 Ideo dicitur admittendo positum et negando27 illam: quelibet propositio est dissimilis istis. Nam sequitur: quelibet propositio est dissimilis istis, sed a est propositio, igitur a est dissimilis istis. Consequens est28 negandum, sed29 non minor, igitur maior. 6.2.2.2 Et si dicitur: continue concessisti insolubile et negasti ipsum esse verum, quare nunc non consequenter respondendo ipsum30 concedis? ad 6.2.2.2 Dicitur quod insolubile cuius significatio terminatur totaliter in seipsum est concedendum, ut: hoc est falsum, seipso demonstrato; sortes non dicit verum, in casu communi. Sed insolubile cuius significatio partialiter31 terminatur in seipsum, potest negari, ut: omnis propositio est falsa, nulla propositio est vera. Horum namque insolubilium significationes non solum ad se terminantur32, sed ad alia, videlicet ad illam: deus est, vel: homo est rome, quare quodlibet illorum33 negatur. Ubi tamen aliquod illorum foret omnis propositio, concederetur illud, quia sua significatio ultimo in seipsum terminaretur.34 Et ita in proposito dico quod propositi insolubilis significatio non solum terminatur35 ad se36  adequate] om. E  igitur – istis] om. (hom.) E 26  C] om. E 27  negando] nego E 28  est] falsum et add. E 29  sed] et E 30  ipsum] om. E 31  partialiter ] particulariter E 32  terminantur] terminatur M 33  quare quodlibet illorum corr.] quarum quodlibet illorum M, quare quelibet illarum E 34  terminaretur] terminetur M 35  terminatur] terminetur M 36  se] c E 24 25

6.2.2.1 Again,

English Translation

243

I ask whether C is true or false. If ⟨it is⟩ true and C exactly signifies that every proposition is unlike those two ⟨in truth-value⟩, then every proposition is unlike those two ⟨in truth-value⟩; but neither A nor B, therefore C is unlike those two ⟨in truth-value⟩. Then ⟨I argue⟩ like this: C is unlike those two ⟨in truth-value⟩ and A and B are true, therefore C is false. If that is granted ⟨at the start⟩, I argue in this way: C is false and C exactly signifies that every proposition is unlike those two ⟨in truthvalue⟩, therefore it is not the case that every proposition is unlike those two ⟨in truth-value⟩. But there is no reason to single out one rather than any other, therefore no proposition is unlike those two ⟨in truth-value⟩. But ⟨the only propositions⟩ are A, B ⟨and⟩ C, of which A ⟨and⟩ B are true, therefore C is true. ad 6.2.2 I answer by admitting the scenario and by denying Every proposition is unlike those two ⟨in truth-value⟩. For this is valid: every proposition is unlike those two ⟨in truth-value⟩, but A is a proposition, therefore A is unlike those two ⟨in truth-value⟩. The conclusion should be denied, but the minor ⟨premise should⟩ not ⟨be denied⟩, therefore the major ⟨premise should be denied⟩. 6.2.2.2 And if someone says: you have always granted an insoluble and denied that it is true, why consequently do you not answer by granting it now? ad 6.2.2.2 I answer that an insoluble whose signification is wholly directed to itself—such as: This is false (referring to itself), Socrates does not say a truth (in the usual scenario), should be granted. But an insoluble whose signification is ⟨only⟩ partially directed to itself—such as: Every proposition is false, No proposition is true, can be denied. For the significations of these insolubles are directed not only to themselves, but also to other things, e.g., to God exists or A man is in Rome, for which reason each of these ⟨partially self-directed insolubles⟩ is denied. However, when any of these ⟨partially self-directed insolubles⟩ is the only proposition, it would be granted since its signification would be directed ultimately to itself. Thus, in the case given I say that the signification of the proposed insoluble is directed not only to itself, but also to A and B. That is why I deny it. But if A and B were indefinite propositions and C was

244

Latin Text

sed ad a et37 b similiter. Ideo negatur. Sed si a et b forent propositiones indefinite et c illa: omnis propositio universalis est dissimilis istis, concederem c, quia sua significatio ultimo terminaretur in ipsum c, et consequenter dicerem quod c38 est falsum. ad 6.2.2.1 Ad aliud: cum queritur utrum c39 sit verum vel falsum, dico quod falsum. Et tunc ad argumentum concedo consequens, quod non quelibet propositio est dissimilis istis, quoniam nec a nec b, cum quelibet illarum sit vera. Et cum dicitur quod non est maior ratio de una quam de alia, dicitur quod ymmo, quia a et b sunt vere et c falsum quia se falsificat.40 6.3 Tertio arguitur sic: et pono quod a et b sint omnes propositiones, quarum a sit illa: chymera est, sic precise significans, et b illa: omnis propositio est falsa, sic adequate significans. Isto posito, propono: omnis propositio est falsa. Si conceditur, et b sic precise et adequate significat, igitur b est verum. Et per consequens non omnis propositio est falsa. Si negatur vel dubitatur, contra: detur oppositum: non omnis propositio est falsa; et aliqua propositio est, igitur aliqua propositio est vera. Et non a, igitur b.41 Tunc arguitur sic: b est verum et b adequate significat quod omnis propositio est falsa, igitur omnis propositio est falsa42, quod est negatum vel dubitatum. Eodem modo potest argui querendo si a est verum vel falsum, continue inducendo oppositum responsionis.43 ad 6.3 Respondetur concedendo44 quod omnis propositio est falsa, quoniam tam a quam b est falsum. a namque quia suum adequatum significatum est falsum et b quia se falsificat. Et tunc ad argumentum: omnis propositio est falsa et b sic precise significat, igitur b est verum, nego consequentiam, quia oportet addere45 quod b non se falsificat, quod negatur.  a et] om. E  c] om. E 39  c corr.] a M E 40  se falsificat] inv. E 41  b] et add. E 42  igitur – falsa] om. (hom.) E 43  responsionis] conclusionis M 44  Respondetur concedendo corr.] Responditur concedendo M, respondetur concedo E 45  addere] om. E 37 38



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245

Every universal proposition is unlike those two ⟨in truth-value⟩, I would grant C, since its signification would be directed ultimately to C itself, and consequently I would answer that C is false. ad 6.2.2.1 To the other point, when it is asked whether C is true or false, I answer that it is false. And then ⟨in reply⟩ to the argument: I grant the conclusion that not every proposition is unlike those two ⟨in truth-value⟩, because neither A nor B ⟨is unlike A and B in truth-value⟩, since each of them is true. And when it is claimed that there is no more reason why one rather than another, I say that there is, because A and B are true and C ⟨is⟩ false because it falsifies itself. 6.3 Thirdly, I argue like this: assume that the only propositions are A and B, where A is A chimera exists, signifying only in that way, and B is Every proposition is false, signifying exactly in that way. Given this ⟨scenario⟩, I propose: Every proposition is false. If you grant it, and B signifies only and exactly in that way, then B is true. And consequently, not every proposition is false. If you deny or express doubt about it, on the contrary: assume the opposite: Not every proposition is false; and ⟨since⟩ some proposition exists, therefore some proposition is true. But not A, therefore B. Then I argue like this: B is true and B signifies exactly that every proposition is false, therefore every proposition is false, which you denied or about which you expressed doubt. It can be argued in the same way by asking whether A is true or false, again inferring the contradictory of the response. ad 6.3 I respond by granting that every proposition is false, since both A and B are false. For A ⟨is false⟩ because its exact significate is false and B because it falsifies itself. And then ⟨in reply⟩ to the argument: Every proposition is false and B signifies only in that way, therefore B is true, I deny the inference, since one should add that B does not falsify itself, which I deny.

246

Latin Text

6.3.1 Sed forte

dicitur: in alio insolubili negasti illam: omnis propositio est falsa, omnis propositio est dissimilis istis, quia ipsarum significationes non terminabantur ultimo et totaliter46 ad se, sed partim (M244rb) ad se et partim ad alia. Cum igitur significatio b non terminetur totaliter ad ipsum b sed partim ad ipsum a, igitur eadem ratione est b negandum sicut47 aliqua aliarum. ad 6.3.1 Dicendum quod non est similitudo, quoniam significatio illius: omnis propositio est falsa, existentibus propositionibus veris, terminabatur48 particulariter ad propositiones veras, ratione quarum illa est neganda. Sed si non sunt nisi propositiones false, debet illa49 concedi. Cum igitur in casu isto a et b sunt omnes propositiones, quarum quelibet est falsa, igitur b est concedendum. Et ita apparet ratio diversitatis. 6.4 Quarto principaliter arguitur50 contra universales et indefinitas simul, ponendo quod a, b, c, d51 sint omnes propositiones adequate52 significantes sicut termini pretendunt, quarum a sit illa: deus est, (E 198vb) b:53 homo est ⟨animal⟩, c:54 homo est asinus et55 d illa: quot sunt vera, tot sunt falsa. Isto posito, propono d, et quero si56 est verum vel falsum. Si dicitur quod verum, et d significat adequate quod57 quot sunt vera tot sunt falsa, igitur verum est quod quot sunt vera tot sunt falsa; sed duo sunt vera, igitur duo sunt falsa. Et per consequens c et58 d sunt falsa, ex quo a et b sunt vera. Si vero dicitur quod d est falsum, et d adequate significat quod quot sunt vera tot sunt falsa, igitur non est ita quod  ultimo et totaliter] om. M  sicut] et add. M 48  terminabatur] terminabantur M 49  debet illa] debent iste E 50  principaliter arguitur] inv. E 51  d] om. E 52  adequate] precise E 53  b] ista add. E 54  homo est c] om. E 55  et] om. E 56  si] utrum E 57  quod] om. E 58  et ] om. M 46

47

6.3.1 But perhaps

English Translation

247

someone will say: in the ⟨case of the⟩ other insoluble you denied Every proposition is false, Every proposition is unlike those two ⟨in truth-value⟩, since their significations were not ultimately and wholly directed to themselves, but partly to themselves and partly to other things. Since therefore the signification of B is not wholly directed to B itself, but partly to A, then for the same reason B should be denied just as are any of the others. ad 6.3.1 I answer that ⟨the cases⟩ are not similar since, given that true propositions exist, the signification of Every proposition is false is partially directed to the true propositions and it should be denied because of them. But if only false propositions exist, it should be granted. Therefore, since in this scenario the only propositions are A and B and each of them is false, then B should be granted. And thus the reason for the difference ⟨between the two cases⟩ is clear. 6.4 Fourthly, I mainly argue against universals and indefinites taken together: assume that the only propositions are A, B, C, ⟨and⟩ D signifying exactly as the terms suggest, where A is God exists, B is A man is ⟨an animal⟩, C is A man is an ass, and D is There are just as many falsehoods as truths. Given this ⟨scenario⟩, I propose D and I ask whether it is true or false. If you answer that it is true, and D signifies exactly that there are just as many falsehoods as truths, then it is true that there are just as many falsehoods as truths; but there are two truths, therefore there are two falsehoods. And since A and B are true, consequently C and D are false. If, however, you answer that D is false, and D exactly signifies that there are just as many falsehoods as truths, then it is not the case that there are just as many falsehoods as truths, therefore there are more falsehoods than truths or conversely. But you cannot take the first ⟨option⟩, as is clear. Therefore, it seems that there are some truths, yet not as many falsehoods. Therefore, since C is false, it seems that A, B ⟨and⟩ D are

248

Latin Text

quot sunt vera tot sunt falsa, igitur plura sunt falsa59 quam vera60 aut e converso. Non est dicendum primum ut patet. Igitur videtur quod aliqua sint61 vera et non tot sint falsa. Cum igitur c sit falsum, videtur quod a, b, d sint vera, quia aliter tot essent vera quot essent falsa. Et per consequens d est verum. ad 6.4 Dicendum concedendo62 d et dicitur quod d est falsum. Et tunc ad argumentum: non valet consequentia, quia63 oportet addere quod d non falsificat se, quod negatur. 6.4.1 Consimiliter esset respondendum dato quod quinque propositiones essent64 omnes propositiones significantes precise sicut termini pretendunt, quarum due essent vere et due65 false et quinta foret illa: plura sunt falsa quam vera, aut illa: pauciora sunt vera quam falsa, que esset a. Quo posito concederetur a et consequenter quod a esset falsum, quia se falsificaret veluti alia insolubilia. Quare etc.

 falsa] vera E  vera] falsa E 61  aliqua sint] aliquod sunt M 62  concedendo] concedo E 63  quia] sed E 64  essent] forent M 65  vere et due] om. M 59 60



English Translation

249

true, because otherwise there would be as many falsehoods as truths. And consequently, D is true. ad 6.4 I answer by granting D and I say that D is false. And then in response to the argument: the inference does not hold, since one should add that D does not falsify itself, which I deny. 6.4.1 One should respond in just the same way supposing that there are only five propositions signifying only as the terms suggest, of which two are true and two false and the fifth is There are more falsehoods than truths, or There are fewer truths than falsehoods, call it A. Given this ⟨scenario⟩, A should be granted and consequently ⟨it should be said⟩ that A is false, since it falsifies itself just like the other insolubles, therefore etc.

250

Latin Text

⟨Capitulum Septimum: De non quantis insolubilibus⟩ 7.1 De non

quantis insolubilibus, ut puta exclusivis et exceptivis,1 restat dicendum. 7.1.1 Pono igitur quod hec:2 tantum propositio falsa est3 exclusiva, sit omnis exclusiva, que sit a et significet precise sicut termini pretendunt. Isto posito, quero si a est4 verum vel falsum. Si verum et a significat adequate quod tantum propositio falsa est exclusiva, igitur tantum propositio falsa est exclusiva; et a est omnis exclusiva, igitur a est falsum. Quod si conceditur, arguitur sic: a est falsum et a significat precise quod tantum propositio falsa est exclusiva, igitur non est ita quod tantum propositio falsa est5 exclusiva; et a est omnis exclusiva, igitur a est verum. ad 7.1.1 Respondetur concedendo6 a et dicitur quod est falsum. Et ad impugnationem negatur consequentia, quia oportet addere in antecedente quod non falsificat se, quod negatur. 7.1.2 Secundo arguitur sic: et pono quod a et b sint omnes propositiones, quarum quelibet significet adequate sicut termini pretendunt, et quod a sit illa: deus est, et b illa: tantum a est verum. Isto posito, propono tibi: tantum a est verum. Si concedis, et b sic significat precise,7 igitur b est verum, igitur non tantum a est verum. Quod si conceditur, iterum8 arguitur sic: non tantum a est verum et a et b sunt omnes propositiones, quarum a est verum, igitur b est verum. Tunc sic: b est verum (M 244va) et b adequate significat quod tantum a est verum, igitur tantum a est verum, cuius oppositum est concessum.

 exceptivis] exclusivis E (etiam infra, sed ultra non notavimus)  hec] hoc E 3  propositio falsa est] falsum est propositio E 4  si A est] utrum A sit E 5  est] sit M 6  concedendo] concedo E 7  significat precise] inv. E 8  iterum] verum E 1 2



English Translation

251

⟨Chapter Seven: On Non-Quantified Insolubles⟩ 7.1 It remains to deal with non-quantified insolubles, namely, exclusives and

exceptives.

7.1.1 Thus assume

that the only exclusive is Only a false proposition is exclusive, call it A, and that it signifies only as the terms suggest. Given this ⟨scenario⟩, I ask whether A is true or false. If ⟨A is⟩ true and A exactly signifies that only a false proposition is exclusive, then only a false proposition is exclusive; and the only exclusive proposition is A, therefore A is false. If that is granted ⟨at the start⟩, I argue in this way: A is false and A only signifies that only a false proposition is exclusive, therefore it is not the case that only a false proposition is exclusive; and the only exclusive proposition is A, therefore A is true. ad 7.1.1 I reply by granting A and I say that A is false. And against the counterargument, I deny the inference since one should add in the premises that ⟨A⟩ does not falsify itself, which I deny. 7.1.2 Secondly, I argue in this way: assume that the only propositions are A and B and each of them exactly signifies as the terms suggest, and that A is God exists and B is Only A is true. Given this ⟨scenario⟩, I propose to you: Only A is true. If you grant it, and B signifies only in this way, then B is true, therefore not only A is true. If that is granted ⟨at the start⟩, again I argue in this way: not only A is true; and the only propositions are A and B, where A is true, therefore B is true. Then ⟨I argue⟩ like this: B is true and B exactly signifies that only A is true, therefore only A is true, which is the opposite of what you granted.

252

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concedendo quod tantum a est verum, quia b est falsum ex quo se falsificat. Et ad improbationem nego consequentiam,9 quia debet addi in minori: et b non se falsificat, quod negatur. 7.1.2.1 Eodem modo responderetur10 ad hanc: tantum plato dicit verum11, dato quod plato diceret a et sortes b et nullam aliam sic precise significando. Unde in casu isto b esset falsum, quia falsificaret se, et suum contradictorium12 similiter, quia significaret adequate falsum. 7.1.3 Item cum13 casu hec est insolubilis: tantum exclusiva est falsa. Non tamen conceditur, quia14 sua significatio non totaliter terminatur ad se, sed etiam ad alia falsa. Ubi tamen ipsa foret omnis propositio, concederetur illa15, quare etc. 7.2 De exceptivis vero sit tale16 sophisma: nulla propositio preter a est falsa, ponendo quod ipsa sit a et omnis propositio, et quod17 ipsa significet18 precise19 sicut termini pretendunt. Isto posito, quero si a est verum vel falsum. Si verum, et a significat precise quod nulla propositio preter a est falsa, igitur nulla propositio preter a est falsa20, igitur a est falsum. Quod si conceditur, arguitur sic: a est falsum, et a significat precise quod nulla propositio preter a est falsa, igitur non est ita quod nulla propositio preter a est falsa; et a est omnis propositio, igitur a non est21 falsum. ad 7.2 Ideo dicitur ut prius concedendo a et negando ipsum esse verum. Et ad improbationem negatur consequentia, quia debet addi in antecedente quod a non falsificat se, quod est negatum sepissime.

ad 7.1.2 Dicitur

 consequentiam] et add. E  responderetur corr.] respondetur M E 11  verum] falsum E 12  contradictorium] significatum E 13  cum corr.] sine M E 14  quia] quod E 15  illa] om. E 16  tale] hec E 17  quod] om. E 18  significet] significat E 19  precise] om. M 20  igitur – falsa] om. (hom.) E 21  non est] inv. E 9

10



English Translation

answer by granting that only A is true, since B is false because it falsifies itself. And against the counterargument, I deny the inference since one should add in the minor premise: and B does not falsify itself, which I deny. 7.1.2.1 I would reply in the same way to Only Plato says a truth, given that Plato said A and Socrates B and no other proposition, signifying only in this way. Thus in this scenario B would be false, since it would falsify itself, and similarly its contradictory, since it exactly signifies a falsehood. 7.1.3 Again, given a scenario, this proposition: Only an exclusive is false is an insoluble proposition. However, I do not grant it, since its signification is not wholly directed to itself, but also to other things that are false. If, however, it was the only proposition, I would grant it, therefore etc. 7.2 Turning to exceptives, take the sophism: No proposition except A is false, assuming that it is A and the only proposition, and that it signifies only as the terms suggest. Given this ⟨scenario⟩, I ask whether A is true or false. If true, and A signifies only that no proposition except A is false, then no proposition except A is false, therefore A is false. If that is granted ⟨at the start⟩, I argue in this way: A is false, and A signifies only that no proposition except A is false, therefore it is not the case that no proposition except A is false; and the only proposition is A, therefore A is not false. ad 7.2 Accordingly, I answer as before by granting A and denying that it is true. And against the counterargument I deny the inference, since one should add in the premises that A does not falsify itself, which I have repeatedly denied.

ad 7.1.2 I

253

254

Latin Text

7.2.1 Eodem

modo est22 respondendum ad illam exceptivam: omnis propositio preter a est verum, annectendo sibi casum eundem. 7.2.2 Secundo arguitur sic: pono quod a, b, c sint omnes propositiones quarum quelibet significet adequate sicut termini pretendunt, et23 quod a24 b sint vere, c autem25 sit illa exceptiva: omnis propositio preter exceptivam est vera. Isto posito, propono: omnis propositio preter exceptivam26 est vera. Si conceditur et c adequate sic27 significat, igitur c est verum. Sed c est omnis exceptiva28, ut suppono, igitur omnis exceptiva (E 199ra) est vera; et per consequens non omnis propositio preter exceptivam est vera. Quod si conceditur, arguitur sic: non omnis propositio preter exceptivam est vera et a, b, c sunt omnes propositiones, quarum a, b sunt vere et c est omnis exceptiva, igitur c similiter est verum. Sed c adequate significat quod omnis propositio preter exceptivam est vera, igitur omnis propositio preter exceptivam est vera, quod est negatum. ad 7.2.2 Dicitur concedendo quod omnis propositio preter exceptivam est vera et quod c est falsum. Et tunc ad argumentum negatur consequentia propter defectum in alio recitatum. 7.2.3 Eodem modo respondetur29 posito quod quilibet homo excepto sorte dicat: deus est, et sortes illam solummodo: quilibet homo preter me dicit verum. In casu isto30 conceditur exceptiva sortis31 et quelibet consimilis sibi.32 Tamen dicitur quod est falsa, licet alie multe essent vere cum illa convertibiles.33 Nec hoc est inconveniens propter materiam insolubilium.34  est] om. E  et] om. E 24  a ] et add. E 25  C autem] et quod C E 26  preter exceptivam] exclusiva E 27  adequate sic] inv. E 28  exceptiva] exclusiva E 29  respondetur] quod add. E 30  isto] om. E 31  sortis] om. E 32  consimilis sibi ] sibi similis E 33  convertibiles corr.] convertibilis M E 34  insolubilium] quare etc add. E 22 23

7.2.1 I reply

English Translation

255

in the same way to the exceptive: Every proposition except A is a truth, attaching the same scenario to it. 7.2.2 Secondly, I argue in this way: assume that the only propositions are A, B ⟨and⟩ C and each of them signifies exactly as the terms suggest, and that A ⟨and⟩ B are true, and that C is the exceptive: Every proposition except an exceptive is true. Given this ⟨scenario⟩, I propose: Every proposition except an exceptive is true. If you grant it, and C signifies exactly in that way, then C is true. But the only exceptive is C, let us suppose, therefore every exceptive is true. And consequently not every proposition except an exceptive is true. If that is granted ⟨at the start⟩, I argue in this way: not every proposition except an exceptive is true and the only propositions are A, B ⟨and⟩ C, where A ⟨and⟩ B are true, and the only exceptive is C, therefore C similarly is true. But C signifies exactly that every proposition except an exceptive is true, therefore every proposition except an exceptive is true, which was denied. ad 7.2.2 I answer by granting that every proposition except an exceptive is true and that C is false. And then ⟨in reply⟩ to the argument, I deny the inference because of the missing ⟨premise⟩ pointed out in the other case. 7.2.3 I reply in the same way supposing that every man except Socrates says God exists, and Socrates only ⟨says⟩ Every man except me says a truth. In this scenario, I grant the exceptive ⟨said by⟩ Socrates and anything similar to it. I answer, however, that it is false even though there are many other true propositions convertible with it. And this is not absurd because ⟨we are dealing with⟩ the topic of insolubles.

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Latin Text

⟨Capitulum Octavum: De apparentibus insolubilibus⟩ 8.1 Respondere

sequitur consequenter ad quasdam cathegoricas que insolubilia apparent, licet insolubilia non existant, quia supra se1 falsitatis reflexionem non habent, ut: hoc non est scitum a te, aliquod propositum tibi est a te negandum, et huiusmodi. 8.1.1 Propono igitur tibi illam: hoc non scitur a te, seipso demonstrato, (M 244vb) que sit a. Si concedis eam,2 arguitur sic: tu concedis a bene respondendo et non ratione alicuius obligationis facte, igitur a est scitum a te. Consequentia tenet quia non conceditur propositio nisi quia3 scitur illa vel propter obligationem aliquam factam. Sed non est hic facta obligatio aliqua4 ratione cuius debeat5 a concedi, igitur si concedis a bene respondendo, hoc est quia tu scis a.6 Tunc arguitur sic: a est scitum a te, et tu scis quod a primo significat hoc7 non sciri8 a te, igitur tu scis hoc non sciri a te. Et per consequens hoc non scitur a te, demonstrato9 a, cuius oppositum est concessum. Si autem negatur vel dubitatur a, contra: ex opposito a cum uno scito a te sequitur a, igitur a non est negandum nec dubitandum.10 Consequentia tenet et antecedens probatur, nam sequitur: hoc scitur a te, demonstrato a, et scis quod hoc significat hoc non sciri a te, igitur tu scis hoc non sciri a te. Et per consequens hoc non scitur a te, quod erat primo negatum.

 se] om. M  eam] om. E 3  quia] quod M 4  aliqua] om. M 5  debeat] deberet M 6  a] et add. E 7  hoc] quod E 8  sciri] scitur E 9  demonstrato] adequato E 10  dubitandum] igitur add. E 1 2



English Translation

257

⟨Chapter Eight: On Merely Apparent Insolubles⟩ 8.1 The next

task is to reply to some subject-predicate propositions which appear to be insolubles, although they are not insolubles since they do not have reflection of falsity on themselves, such as: This is not known by you, Something proposed to you should be denied by you, and suchlike. 8.1.1 I therefore propose to you: This is not known by you, referring to itself, call it A. If you grant it, I argue in this way: you grant A responding correctly and not by reason of some current obligation, therefore A is known by you. The inference holds since a proposition is granted only because it is known or because of some current obligation. But here there is no current obligation by reason of which A should be granted, therefore if you grant A responding correctly, this is because you know A. Then I argue like this: A is known by you, and you know that A primarily signifies that this is not known by you, therefore you know that this is not known by you. And consequently this is not known by you, referring to A, which is the opposite of what you granted. But if you deny or express doubt about A, on the contrary: A follows from the opposite of A together with something known by you, therefore you should not deny or express doubt about A. The inference holds and I prove the premise. For this is valid: this is known by you, referring to A, and you know that this signifies that this is not known by you, therefore you know that this is not known by you. And consequently this is not known by you, which you denied at the start.

258

Latin Text

8.1.2 Secundo

quero si a propositio est scita a te vel non. Si dicis quod est scita a te et scis quod a propositio significat adequate11 hoc non sciri12 a te, demonstrato a, igitur tu scis hoc non sciri13 a te. Et per consequens hoc non scitur a te, demonstrato a. Si autem dicitur quod a non est scitum a te, contra: tu scis hoc non sciri14 a te, demonstrato a, et scis quod a significat adequate hoc non sciri a te, igitur tu scis a. Et per consequens a propositio est scita a te. 8.1.3 Tertio quero utrum a sit verum vel falsum. Si verum, et quodlibet verum potest sciri, sic precise significando, igitur a potest sciri, sic precise significando. Consequens est falsum, quia implicat contradictionem a sciri, sic precise15 significando, ut patuit. Si autem dicitur quod a est falsum, et a significat adequate hoc non sciri, igitur non est ita quod hoc non scitur a te. Consequentia tenet, quia a non falsificat se. Tunc ultra: non est ita quod hoc non scitur a te, igitur hoc scitur a te; et hoc est a, igitur a scitur a te. Consequens est falsum, ut sepe est ostensum. ad 8.1.1 Respondetur quod licet a non falsificet16 se, tamen asserit se nesciri. Sicut ergo quelibet propositio se falsificans17 est falsa, ita quelibet asserens se nesciri18 non scitur. Et sicut19 quodlibet insolubile cuius significatio totalis ultimo20 terminatur in se conceditur, ita et propositio non insolubilis asserens se nesciri vel dubitari et huiusmodi. Ideo concedo a cum mihi proponitur. Et tunc ad argumentum: tu concedis a bene respondendo et non ratione alicuius obligationis facte, igitur a est scitum a te, nego consequentiam. Unde quotienscumque mihi proponitur illa: hoc est falsum, se demonstrata,21 concedo illam bene respondendo, non ratione  adequate] quod add. E  sciri] scitum E 13  sciri] scitum E 14  sciri] scitum E 15  precise] adequate E 16  falsificet] falsificat M 17  se falsificans] inv. E 18  nesciri] non scire E 19  sicut] sic E 20  ultimo] om. M 21  se demonstrata corr.] se demonstrato M, demonstrato se E 11 12

8.1.2 Secondly,

English Translation

259

I ask whether proposition A is known by you or not. If you answer that it is known by you, and ⟨since⟩ you know that proposition A signifies exactly that this is not known by you, referring to A, then you know that this is not known by you. And consequently, this (referring to A) is not known by you. If, however, you answer that A is not known by you, on the contrary: you know that this is not known by you, referring to A, and you know that A signifies exactly that this is not known by you, therefore you do know A. And consequently proposition A is known by you. 8.1.3 Thirdly, I ask whether A is true or false. If true, and every truth can be known (signifying only as the terms suggest), then A can be known (signifying only as the terms suggest). The conclusion is false, since that A is known (signifying only as the terms suggest) implies a contradiction, as was clear. If, however, you answer that A is false, and A signifies exactly that this is not known, then it is not the case that this is not known by you. The inference holds, since A does not falsify itself. Then further: it is not the case that this is not known by you, therefore this is known by you; and this is A, therefore A is known by you. The conclusion is false, as has often been shown. ad 8.1.1 I respond that although A does not falsify itself, nevertheless it does assert that it itself is unknown. Therefore, just as each self-falsifying proposition is false, so too each proposition asserting that it itself is unknown is not known. And just as I grant each insoluble whose whole signification is ultimately directed to itself, so too ⟨I grant⟩ a non-insoluble proposition asserting that it itself is unknown, or is doubted or suchlike. Hence, I grant A when it is proposed to me. And then ⟨in reply⟩ to the argument: You grant A responding correctly and not by reason of any current obligation, therefore A is known by you, I deny the inference. Thus, whenever this is proposed to me: This is false,

260





Latin Text

a­ licuius obligationis facte, sed ratione sui veri significati. Et tamen illa non scitur a me, quia non est vera. Ita in proposito conceditur a, non ratione obligationis facte, sed sui significati veri. Non tamen scis a, quia repugnat a sciri ⟨a te⟩ propter reflexionem sui qua22 asserit se non sciri a te. ad 8.1.2 Ad secundum dicitur quod a non est scitum a me. Et tunc23 ad argumentum: tu scis hoc non sciri a te et scis quod a significat adequate hoc non sciri a te, igitur tu scis a, nego consequentiam, sicut non sequitur: tu scis hoc esse fal-(E 199rb)-sum et24 scis quod illa:25 hoc est falsum, se demonstrato, significat adequate hoc esse falsum, igitur tu scis illam: hoc est falsum, quia oportuit addi in antecedente: et illa non se falsificat. Ita in proposito debet sumi cum26 antecedente quod non repugnat27 a sciri ⟨a te⟩; et tunc concederem consequentiam et negarem antecedens pro illa parte. ad 8.1.3 Ad tertium dicitur quod a est verum, quia suum adequatum (M 245ra) significatum est verum et a non se falsificat. Et tunc ad argumentum: concedo quod a potest sciri, etsi non a me, quia in a fit mentio de me solummodo, ⟨sed⟩ sufficit quod ab alio. Quicumque enim preter me sciret adequatum significatum a, et sciret a sic adequate significare, sciret a. Sed ego non possum scire, quia repugnat me scire a. Verumtamen illa consequentia non valet: a propositio est vera, igitur potest sciri sic adequate significando. Unde dato quod a esset una illarum: hoc non scitur, aut: hoc non potest sciri, se demonstrato, a esset verum, quia significaret adequate verum et non se falsificaret.28 Et tamen a non posset sciri sic adequate significando, quia repugnaret a sciri, ut patet intuenti.

 qua corr.] quo M, que E  tunc] om. E 24  et] tu add. E 25  illa] om. E 26  cum] in E 27  repugnat] repugnet E 28  falsificaret] falsificat M 22 23







English Translation

261

referring to itself, I grant it responding correctly, not by reason of any current obligation, but by reason of its true significate. And yet it is not known by me, since it is not true. So too, in the case given, you grant A not by reason of some current obligation, but because of its true significate. Yet you do not know A, since it is inconsistent that A be known ⟨by you⟩, because of its reflection by means of which it asserts that it itself is not known by you. ad 8.1.2 To the second, I reply that A is not known by me. And then ⟨in reply⟩ to the argument: You know that this is not known by you and you know that A signifies exactly that this is not known by you, therefore you do know A, I deny the inference, just as this is not valid: You know that this is false and you know that “This is false” (referring to itself) signifies exactly that this is false, therefore you know “This is false,” since it should have been added in the premises: and that proposition does not falsify itself. Thus, in the case given we should assume together with the premises that it is not inconsistent that A be known ⟨by you⟩. And then I would grant the inference and I would deny that premise. ad 8.1.3 To the third I reply that A is true, since its exact significate is true and A does not falsify itself. And then ⟨in reply⟩ to the argument: I grant that A can be known, although not by me, since I alone am mentioned in A, ⟨but⟩ it suffices that ⟨A can be known⟩ by someone else. For anyone except me who knew the exact significate of A and knew that A signifies exactly in that way, would know A. But I cannot know ⟨A⟩, since it is inconsistent ⟨with A⟩ that I know A. Nevertheless, the inference: Proposition A is true, therefore ⟨A⟩ (signifying exactly as the terms suggest) can be known is not valid. Thus, given that A was either This is not known or This cannot be known, referring to itself, A would be true, since it would exactly signify a truth and would not falsify itself. And yet A (signifying exactly as the terms suggest) could not be known, since it would be inconsistent that A be known, as is clear to anyone who considers it.

262

Latin Text

8.1.4.1 Consimiliter

respondetur ad illud sophisma: aliquod propositum nescitur a te, ponendo quod sortes dicat tibi illam propositionem et nullam aliam, que sit a, sic adequate significantem, cum hoc quod nihil aliud proponatur tibi.29 Isto posito, si proponitur a30 concedo ipsum. Et si arguitur sic: tu concedis a bene respondendo et non ratione obligationis, igitur a est scitum a te, nego consequentiam, quia oportet addere: et non repugnat a sciri a te, quod negatur. Unde31 a sciri a te implicat contradictionem, quia32 sequitur: a scitur a te et tu33 scis quod a significat adequate quod quodlibet propositum tibi nescitur a te, igitur quodlibet propositum tibi nescitur a te; et a est quodlibet propositum tibi, igitur a nescitur a te. 8.1.4.2 Item non sequitur: tu scis quod quodlibet propositum tibi nescitur a te34 et scis quod a adequate sic significat,35 igitur tu scis a. Sed oportet sumere: et non repugnat a sciri. 8.1.4.3 Etiam non sequitur: a est verum, igitur potest sciri, secundum quod est supra ostensum. 8.2 Secundo principaliter arguitur sic: et propono tibi illam: hoc est negandum a te, demonstrato seipso,36 que sit a.

 proponatur tibi] inv. E  A] om. E 31  Unde] tantum E 32  quia] om. M 33  tu] om. E 34  igitur quodlibet propositum – nescitur a te] om. (hom.) E 35  sic significat] inv. E 36  demonstrato seipso] inv. E 29

30

8.1.4.1 I answer

English Translation

263

in just the same way to the sophism: Something proposed is unknown to you. Assume that Socrates says this to you and no other proposition, call it A, signifying exactly as the terms suggest, and that nothing else is proposed to you. Given this ⟨scenario⟩, if A is proposed, I grant it. And if one argues like this: You grant A responding correctly and not by reason of some obligation, therefore A is known by you, I deny the inference since one should add: and it is not inconsistent that A is known by you, which I deny. Thus, that A is known by you implies a contradiction, since this is valid: A is known by you and you know that A signifies exactly that everything proposed to you is unknown to you, therefore everything proposed to you is unknown to you; and A is everything proposed to you, therefore A is unknown to you. 8.1.4.2 Again, this is not valid: You know that everything proposed to you is unknown to you and you know that A signifies exactly like that, therefore you know A. But one should add as a premise: and it is not inconsistent that A be known. 8.1.4.3 Furthermore, this is not valid: A is true, therefore it can be known, according to what was shown above. 8.2 The second ⟨merely apparent insoluble⟩ is this: I propose to you this proposition: This should be denied by you, referring to itself, call it A.

264

Latin Text

8.2.1 Si concedis,

arguitur sic: hoc est a te negandum et tu concedis illam, igitur tu37 male respondes. Si autem negas, argitur sic: tu negas a et bene respondes, igitur a est a te negandum. Et suum contradictorium est, igitur suum contradictorium est a te concedendum, videlicet: hoc non est a te negandum, quod est oppositum concessi. Si vero dubitatur a, arguitur sic: a est falsum et tu non es obligatus, igitur a non est a te dubitandum. Consequentia tenet cum minori et maiorem probo, nam hoc non est a te negandum, demonstrato a, quia dicis quod est dubitandum et a adequate significat hoc esse negandum, igitur a est falsum. Patet consequentia, quia a non falsificat se. 8.2.2 Secundo quero si a est concedendum a te, negandum a te vel dubitandum a te. Si dicitur quod a est concedendum a te et a significat adequate a esse negandum a te, igitur a esse negandum a te est concedendum a te. Et per consequens a38 est negandum a te39. Quod si conceditur, arguitur sic: a est negandum a te et a significat precise a esse negandum a te, igitur a esse negandum a te est negandum a te. Et per consequens a non est negandum a te. Si autem in principio dicitur quod a est dubitandum a te,40 arguitur sic: a est dubitandum a te, igitur a non est negandum a te. Consequentia tenet, quia oppositum consequentis repugnat antecedenti. Tunc sic: a non est negandum a te et a significat primo a esse negandum a te, igitur a esse negandum a te non est negandum a te. Et per consequens a non est dubitandum a te. Consequentia tenet, quia ex opposito sequitur oppositum. Sequitur enim:41 a est dubitandum a te, igitur a esse negandum a te est negandum a te. Patet consequentia, quia cum quis dubitat aliquam propositionem, negat illam esse negandam42 ab illo.  tu] om. E  a] non add. M 39  a te] om. M 40  te] et add. E 41  sequitur enim] om. E 42  negandam] dubitandam E 37 38

8.2.1 If you

English Translation

265

grant it, I argue in this way: this should be denied by you and you grant it, therefore you respond incorrectly. However, if you deny ⟨A⟩, I argue in this way: you deny A and you respond correctly, therefore A should be denied by you. And its contradictory exists, therefore its contradictory should be granted by you, namely, This should not be denied by you, which is the opposite of what you granted. If, however, you express doubt about A, I argue in this way: A is false and you are not obligated, therefore you should not express doubt about A. The inference holds, as does the minor premise, and I prove the major premise: for this should not be denied by you, referring to A, since you answer that doubt should be expressed about it and A signifies exactly that it should be denied, therefore A is false. The inference is clear, since A does not falsify itself. 8.2.2 Secondly, I ask whether you should grant, deny or express doubt about A. If you answer that A should be granted by you, and ⟨given that⟩ A signifies exactly that A should be denied by you, then it should be granted by you that A should be denied by you. And consequently A should be denied by you. If it is granted ⟨at the start that A should be denied by you⟩, I argue in this way: A should be denied by you, and A signifies only that A should be denied by you, therefore it should be denied by you that A should be denied by you. And consequently A should not be denied by you. If, however, at the start you answer that you should express doubt about A, I argue in this way: you should express doubt about A, therefore A should not be denied by you. The inference holds, since the opposite of the conclusion is inconsistent with the premise. Then ⟨I argue⟩ like this: A should not be denied by you and A signifies primarily that A should be denied by you, therefore it should not be denied by you that A should be denied by you. And consequently you should not express doubt about A. The inference holds, since the opposite ⟨of the premise⟩ follows from the opposite ⟨of the conclusion⟩. For this is valid: you should express doubt about A, therefore it should be denied by you that A should be denied by you. The inference is clear, since when someone expresses doubt about some proposition, he denies that it should be denied by him.

266

Latin Text

8.2.3 Tertio

quero (M 245rb) utrum a sit verum vel falsum. Si dicitur quod verum, arguitur sic: a est verum non repugnans in aliqua obligatione, et tibi proponitur, igitur a est a te concedendum. Consequentia tenet et consequens in secundo argumento improbatur. Si autem dicitur quod a est falsum, arguitur sic: a est falsum non sequens in aliqua obligatione et tibi proponitur, igitur a est a te negandum. Consequentia patet et consequens iterum ibidem43 est reprobatum. ad 8.2.1 Ad primum (E 199va) respondetur concedendo a et tunc ad44 argumentum: hoc est a te negandum et tu concedis illam, igitur tu male respondes, nego primam partem antecedentis. 8.2.1.1 Et si dicitur: tu negas primam partem antecedentis et ipsa est a, igitur tu negas a; et prius concessisti a, igitur male respondes. ad 8.2.1.1 Dicitur quod ipsa non est a, sed simile45 a vel convertibile cum ipso. Unde sicut in insolubilibus duo invicem convertuntur, quorum unum est verum vel46 necessarium et alterum falsum vel impossibile, ita in eadem materia una propositio est concedenda et alia neganda, non obstante quod invicem convertantur.47 Si tamen ponitur quod prima pars antecedentis sit a, negatur consequentia. Sed oportet addere in antecedente quod a non pertinet ad materiam48 insolubilium, quod negatur. ad 8.2.2 Ad secundum dicitur quod a est concedendum a te et tunc ad argumentum: a est concedendum a te et a49 significat adequate a esse negandum a te, igitur a esse negandum a te est concedendum a te, negatur consequentia quia oportet addere quod50 a non asserit se esse negandum, quod est falsum. ad 8.2.3 Ad tertium concedo quod a est verum, et quod est concedendum a te. Et nego quod ⟨consequens⟩ sit reprobatum, quia iam est ostensum,51 ut patet.  iterum ibidem] verumtamen idem E  ad] om. E 45  simile] similis E 46  vel] et E 47  convertantur] convertuntur E 48  ad materiam] in materia M 49  a] om. E 50  quod] et E 51  ostensum] responsum E 43 44

8.2.3 Thirdly,

English Translation

267

I ask whether A is true or false. If you answer that ⟨it is⟩ true, I argue in this way: A is true and it is not inconsistent in some obligation, and it is proposed to you, therefore A should be granted by you. The inference holds and the conclusion is disproven in the second argument. If, however, you answer that A is false, I argue in this way: A is false and it does not follow in some obligation and it is proposed to you, therefore A should be denied by you. The inference is clear and the conclusion was again disproven in the same place. ad 8.2.1 To the first ⟨argument⟩, I answer by granting A and then ⟨in reply⟩ to the argument: This should be denied by you and you grant it, therefore you respond incorrectly, I deny the first premise. 8.2.1.1 If someone says: you deny the first premise and this is A, therefore you deny A; yet you granted it at the beginning, therefore you respond incorrectly. ad 8.2.1.1 I answer that ⟨the first premise⟩ is not A, but is something similar to A or convertible with it. Thus, just as in ⟨the case of⟩ insolubles two ⟨propositions⟩ are convertible, one of which is a truth or a necessity and the other a falsehood or an impossibility, so too in this case even though the two propositions are convertible, one proposition should be granted and the other should be denied. If, however, you claim that the first premise is A, I deny the inference. But one should add in the premises that A has nothing to do with the case of insolubles, which I deny. ad 8.2.2 To the second ⟨argument⟩ I answer that A should be granted by you and then ⟨in reply⟩ to the argument: A should be granted by you and A signifies exactly that A should be denied by you, therefore it should be granted by you that A should be denied by you, I deny the inference since one should add ⟨in the premises⟩ that A does not assert that it should be denied, which is false. ad 8.2.3 To the third ⟨argument⟩, I grant that A is true and that it should be granted by you. And I deny that ⟨the conclusion⟩ is disproven, because that has already been shown, as is clear.

268

Latin Text

ad 8.2.1-3 Hunc

modum respondendi assigno dato quod pro fundamento sustineatur hanc consequentiam valere: tu concedis a et bene respondes, igitur a est concedendum a te; similiter: tu negas a, et bene respondes, igitur a est a te negandum; et ita de ly dubitandum eodem modo. Cum igitur concedantur insolubilia et cetere propositiones habentes reflexionem totaliter supra se, et hoc bene respondendo, videtur quod quelibet talis sit concedenda. Unde proposito52 a, si negares, arguerem sic, sumendo eius oppositum: hoc non est a te negandum, demonstrando a, et tu negas a, igitur tu male respondes. aliter ad 8.2 Alium modum respondendi possum notare, concedendo53 a quotienscumque proponitur, et negando a esse concedendum a me, sustinendo fundamentaliter quod sicut propositio se falsificans est falsa et propositio asserens se nesciri non scitur, ita propositio asserens se esse negandam est neganda, ita quod licet concedatur a, negetur tamen ipsum esse concedendum54 et dicatur ipsum esse negandum. Nec hoc est inconveniens cum demonstratum fuerit in arte obligatoria. aliter ad 8.2.1 Et tunc ad primam rationem nego illam consequentiam: hoc est a te negandum et tu concedis illud, igitur tu male respondes, sicut non sequitur: tu concedis a et bene respondes, igitur a est a te concedendum, sed debet addi in minori quod a non asserit se esse negandum aut quod non repugnat a esse concedendum, et hoc negatur secundum istam responsionem. aliter ad 8.2.2 Ad secundam dicitur ut prius quod a est negandum a te. Et tunc ad argumentum negatur consequentia, quia oportet addere in antecedente quod a non asserit se55 esse negandum.

 proposito] posito E  concedendo] concedo E 54  negetur – concedendum] tamen negatur ipsam esse concedendam E 55  se] a E 52 53



English Translation

269

ad 8.2.1-3 I choose this way of replying given that one upholds as a fundamental principle that this inference holds: You grant A and you respond correctly, therefore A should be granted by you, ⟨and⟩ similarly ⟨that this holds⟩: You deny A and you respond correctly, therefore A should be denied by you, and so too in the same way about “should express doubt.” Therefore since insolubles as well as other propositions having reflection wholly on themselves are granted, and this when responding correctly, it seems that also each such proposition should be granted. Thus, when A is proposed, if you denied it, I would argue in this way, taking its opposite This should not be denied by you (referring to A) and you deny A, therefore you respond incorrectly. aliter ad 8.2 I might mention another way of responding⟨, namely,⟩ to grant A whenever it is proposed and to deny that A should be granted by me, upholding as a fundamental principle that just as a self-falsifying proposition is false and a proposition asserting that it itself is unknown is not known, so too a proposition asserting that it itself should be denied should be denied. So that although I grant A, nevertheless I deny that it should be granted and I say that it should be denied. And this is not absurd, as was shown in the art of obligations. aliter ad 8.2.1 And then to the first argument, I deny the inference: This should be denied by you and you grant it, therefore you respond incorrectly, just as this is not valid: You grant A and you respond correctly, therefore A should be granted by you. But one should add to the minor premise that A does not assert that it itself should be denied or that it is not inconsistent that A should be granted. And this is denied according to this ⟨way of⟩ responding. aliter ad 8.2.2 To the second ⟨argument⟩, I reply once again that A should be denied by you. And then ⟨in reply⟩ to the argument, I deny the inference, since one should add in the premises that A does not assert that it itself should be denied.

270

Latin Text

aliter ad 8.2.3 Ad

tertiam rationem56 dicitur quod a est verum. Et tunc ad argumentum inferens a esse concedendum, negatur consequentia, quia debuit addi in antecedente propositio sepe recitata. Harum responsionum magis placens sustineatur, quia indifferens sum57 ad utramque. (M 245va) 8.2.4 Modo consimili est respondendum ad istas propositiones: hoc non est concedendum a te, se demonstrato; aliquod tibi propositum est a te negandum, nullum tibi propositum est a te concedendum, dato quod tibi nihil aliud58 proponatur quam (E 199vb) una illarum. 8.3.1 Tertio principaliter arguitur sic: et propono tibi illam: hoc est tibi dubium, demonstrato seipso, que sit a. Si concedis a et a est tibi dubium, igitur tu concedis tibi dubium. Tunc sic: tu concedis tibi dubium59 et non es obligatus, igitur male respondes. Si negas a, arguitur sic: tu negas a et bene respondes, igitur a est a te negandum. Consequens est falsum, quia a non est a te concedendum, igitur per idem nec negandum. Consequentia tenet vel detur ratio60 diversitatis. Si autem tu dubitas a, arguitur sic: tu dubitas a bene respondendo, igitur hoc non esse tibi dubium est a te negandum, demonstrando a; et per consequens a est a te concedendum. Consequentia tenet quia si unum contradictoriorum est negandum, alterum est concedendum. Sed ista sunt contradictoria: hoc est tibi dubium et hoc non est tibi dubium, quorum primum est a et secundum est a te negandum.

 rationem corr.] responsionem M E  sum] sive E 58  tibi nihil aliud] nihil aliud tibi E 59  tunc – dubium] om. (hom.) E 60  ratio] causa E 56 57



English Translation

271

aliter ad 8.2.3 To the third argument I reply that A is true. And then ⟨in reply⟩ to the argument which infers that A should be granted, I deny the inference, since one should have added the oft-repeated clause in the premise. You can uphold whichever of these two responses you prefer, since I am indifferent between them. 8.2.4 And one can respond in the same ways to the propositions: This should not be granted by you referring to itself, Something proposed to you should be denied by you, Nothing proposed to you should be granted by you, supposing that nothing is proposed to you other than one of them. 8.3.1 The third ⟨merely apparent insoluble⟩ is this: I propose to you: This is in doubt for you (referring to itself), call it A. If you grant A and A is in doubt for you, then you grant something in doubt for you. Then ⟨I argue⟩ like this: you grant something in doubt for you and you are not obligated, therefore you respond incorrectly. If you deny A, I argue in this way: you deny A and you respond correctly, therefore A should be denied by you. The conclusion is false, since A should not be granted by you, therefore, by parity of reasoning it should not be denied by you. The inference holds, otherwise you should provide a reason to distinguish ⟨between the two cases⟩. If, however, you express doubt about A, I argue in this way: you express doubt about A, responding correctly, therefore you should deny that this is not in doubt for you, referring to A. And consequently A should be granted by you. The inference holds since if one of contradictories should be denied, the other should be granted. But This is in doubt for you, and This is not in doubt for you are contradictories and the first of these is A and the second should be denied by you.

272

Latin Text

ut in prioribus61 concedendo62 a. Et cum dicitur: tu concedis a et hoc est tibi dubium, demonstrando a,63 igitur tu concedis tibi dubium, concedo consequentiam et consequens. Et tunc ad argumentum: tu concedis tibi dubium et64 non es obligatus, igitur male respondes, nego consequentiam. Unde sicut in materia insolubilium conceditur falsum vel impossibile bene respondendo, ita in eadem65 conceditur dubium. 8.3.2 Idem66 modus respondendi servandus est si ponitur quod nihil tibi proponatur nisi illa: dubium tibi proponitur, que sic precise significet et sit a.67 Tunc queritur qualiter respondes ad a. Si dicitur concedendo,68 arguitur sic: tu concedis a69 et bene respondes, igitur a est a te concedendum et non ratione alicuius obligationis facte, igitur a est scitum a te. Tunc arguitur70 sic: a est scitum a te et nihil est71 tibi propositum nisi72 a, igitur nullum dubium tibi proponitur. Ista consequentia est bona et antecedens est concedendum a te, igitur et consequens.73 Sed consequens est contradictorium a, igitur a non est a te concedendum. Si negas, contra: illa non est a te concedenda, igitur nec74 neganda. Consequentia tenet vel detur ratio75 diversitatis. Si dubitatur, igitur dubium tibi proponitur. Illa consequentia est bona et antecedens est concedendum a te,76 igitur et consequens. Sed consequens est a, igitur a77 est a te concedendum.78

ad 8.3.1 Dicendum

 prioribus] pluribus M  concedendo] concedo E 63  demonstrando a] om. E 64  et] tu add. E 65  eadem] eodem M 66  Idem corr.] item M E 67  a] et add E 68  concedendo] concedo E 69  a] om. M 70  arguitur] om. E 71  est] om. M 72  nisi] om. E 73  consequens] concedendum a te igitur et consequens add. E 74  nec] non E 75  ratio] causa E 76  a te] om. M 77  a] consequens E 78  a te concedendum] concedendum a te E 61 62



English Translation

answer as in the previous cases by granting A. And when one argues: You grant A and this is in doubt for you (referring to A), therefore you grant something in doubt for you, I grant the inference and the conclusion. And then ⟨in reply⟩ to the argument: You grant something in doubt for you and you are not obligated, therefore you respond incorrectly, I deny the inference. For just as in the case of insolubles a falsehood or something impossible is granted when responding correctly, so too in this ⟨case⟩ something in doubt may be granted. 8.3.2 One should retain the same way of responding assuming that the only thing proposed to you is this proposition: Something in doubt is proposed to you, which signifies only as the terms suggest, call it A. Then one asks how you respond to A. If you answer by granting it, I argue in this way: you grant A and you respond correctly, therefore A should be granted by you; and not by reason of any current obligation, therefore A is known by you. Then I argue like this: A is known by you and only A is proposed to you, therefore nothing in doubt is proposed to you. This inference holds and the premise should be granted by you, therefore the conclusion too. But the conclusion is the contradictory of A, therefore A should not be granted by you. If you deny ⟨A⟩, on the contrary: this should not be granted by you, therefore neither should it be denied. The inference holds, otherwise you should provide a reason to distinguish ⟨between the two cases⟩. If you express doubt, then something in doubt is proposed to you. This inference holds and the premise should be granted by you, therefore the conclusion ⟨should be granted⟩ too. But the conclusion is A, therefore A should be granted by you.

ad 8.3.1 I

273

274

Latin Text

dicitur ut prius quod a est concedendum a me, et cum concluditur quod a est scitum, nego consequentiam, quia oportet addere in antecedente: et a non asserit se non sciri a te, quod negatur, quia a asserit se esse dubium tibi, et per consequens a asserit se non sciri a te. 8.3.3 Eodem modo est respondendum ad istas: omne propositum est dubitandum a te, nihil tibi proponitur nisi dubium. Quare etc.79

ad 8.3.2 Ideo

 etc] Nunc et semper eterni dei filio deo trino et uno flexis genibus et manibus coniunctis humiles preces refero etc. Ista est logica excellentissimi viri fratris Pauli de Venetiis et constitit michi ducatos Decem add. M, Clarissimi Doctoris Pauli Veneti ordinis divi Augustini opus quod magna logica appellatur. Correptum per Magistrum Franciscum ⟨de⟩ Macerata et Fratrem Jacobum de Fossano philosophie bachalarium ordinis minorum impressum Venetiis per diligentissimum virum Albertinum Verceliensem expensis domini Octaviani Scoti ac eius fratrum. Feliciter explicit. Anno domini 1499. Die 24 Octobris Finis add. E 79



English Translation

I answer as before that A should be granted by me, and when one concludes that A is known, I deny the inference, since one should add in the premises: and A does not assert that it itself is not known by you, which I deny since A asserts that it itself is in doubt for you. And consequently A asserts that it itself is not known by you. 8.3.3 One should respond in the same way to these: Everything proposed should be doubted by you, Only something in doubt is proposed to you. Therefore etc.

ad 8.3.2 Accordingly,

275

Commentary on Paul of Venice’s Treatise on Insolubles Chapter 1: Various Opinions on the Insolubles § 1.1 Paul starts

his treatise by discussing in turn seven of the opinions, or theories, from the list Bradwardine gives in his Insolubilia, starting in § 1.1 with the second solution presented by Bradwardine, which is perhaps that of (pseudo-)William of Sherwood.1 Bradwardine writes (Insolubilia, § 5.2, 85–7): They assert that a term cannot supposit for the whole of which it is part, and so on. If Socrates says “Socrates says a falsehood,” they say that this is false, and when one argues: “Socrates says this falsehood, so Socrates says a falsehood,” they claim that this is a fallacy of [form of expression].

The Latin phrase figura dictionis is variously translated by Hamblin (Fallacies, 25) as “figure of speech,” by Copenhaver (Peter of Spain, Summaries of Logic) as “figure of expression,” and by Klima as “figure of words” (Buridan, Summulae de dialectica, § 7.3.9). We follow PickardCambridge in his translation of schêma tês lexeôs in Aristotle’s De sophisticis elenchis2 in rendering what Boethius translated as figura dictionis as “form of expression.”3 Three matters invite discussion: 1) Why is it a fallacy of form of expression (that is, figura dictionis)? 2) Who suggested it might be invalid to argue from “Socrates says this falsehood” (sortes dicit hoc falsum) to “Socrates says a falsehood” (sortes dicit falsum)? 3) How should we translate sortes dicit falsum (and sortes dicit hoc falsum and sortes dicit verum)?  See Bradwardine, Insolubilia, § 5.1 and 224 n. 23 (where for “296” read “251”). Sherwood’s treatise was edited in Roure, “La problématique des propositions insolubles,” 248–61. We preface the author’s name with “(pseudo-)” since there is some uncertainty as to whether the author is really Sherwood himself or someone else. See, for example, De Rijk, “Some Notes on the Medieval Tract De Insolubilibus,” 90–3. 2  Aristotle, De sophisticis elenchis, trans. Pickard-Cambridge, chap. 4, 166b10–21. 3  Aristotle, De sophisticis elenchis. Translatio Boethii, 10, lines 12–22. 1

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We start with the first question. This diagnosis of insolubles goes back at least to the treatise on insolubles (doubtfully) attributed to William of Sherwood, where the author presents the doctrine of restrictio: We next come to the opinion of the restrictivists, who say that “falsehood” cannot supposit for the expression of which it is part, nor similarly the part for the whole. Thus when one says “I say a falsehood,” the sense is “I say another falsehood,” and this proposition is unrestrictedly false, ⟨so⟩ one should reply to “I say a falsehood” that it is false. When it is argued [see § 0.02, cited below] “so it is false that I say a falsehood, so I do not say a falsehood,” one should grant it, and when it is said, “and I say something, so a truth,” they say that there is ⟨a fallacy of⟩ form of expression, in that “I say” earlier [i.e., in “I say something”] copulated not for the utterance performed [i.e., “I say something”] but for other things conceived ⟨by it⟩; ⟨but⟩ now [in “I say a falsehood”] it stands for the actual utterance, so there is a change of copulation in the premises.4

The author is referring back to the presentation of the insoluble in § 0.02: For example, if I say “I say a falsehood,” and then one asks of it, did I say a truth or did I say a falsehood … if a falsehood, then that I say a falsehood is false, so I did not say a falsehood, so a truth.5

Nuchelmans (“The Distinction actus exercitus/actus significatus in Medieval Semantics,” 76) gives a helpful explanation of the author’s reasoning. When one says, “It is false that I say a falsehood” (Falsum est me dicere falsum), “say” supposits for—or copulates6—what I say, but when  Roure, “La problématique des propositions insolubles,” § 3.01, 251: “Sequitur de oppinione restringentium, qui dicunt quod ly falsum non potest supponere pro hac oratione cuius est pars, nec similiter pars pro toto. Unde cum dicitur: ego dico falsum, sensus est: “ego dico falsum aliud,” et est hec simpliciter falsa, respondendum est: falsum est ad hanc: ego dico falsum. Et cum arguit: “ergo falsum est me dicere falsum, ergo non dico falsum,” concedunt, et cum dicit: “et dico aliquale, ergo verum,” dicunt quod ibi est figura dictionis, eo quod ly dico prius copulavit, non pro hac dictione exercita, sed pro aliis conceptis; hec nunc stat pro dictione exercita, unde mutatur eius copulatio in premissis.” 5  Ibid., § 0.02, 248: “Verbi gratia, si dicam sic: ego dico falsum, et deinde queratur de eodem: aut dico verum, aut dico falsum … si falsum, ergo me dicere falsum est falsum, ergo non dico falsum, ergo verum.” 6  The terminology here does suggest the treatise dates from the time of Sherwood, midthirteenth century, when copulation (as a property of predicates) was distinguished from supposition (then just a property of subjects). See Roure’s footnote, ibid., 251 and Read, “Medieval Theories: Properties of Terms,” § 3. 4



Commentary279

I say, “I say a falsehood” (Ego dico falsum), “say” supposits (or copulates) only for other things, since self-reference is banned by restrictivism. As Nuchelmans suggests, we could also describe the fallacy as a change of supposition of falsum (as Paul does) rather than dico. In the premise, falsum (“a falsehood”) supposits for what I say, but in the conclusion it can’t (because of the ban on self-reference). That is what many medievals often took a fallacy of form of expression to consist in, namely, a change of supposition from premise to conclusion.7 The origin of this fallacy, more generally, is when words having the same apparent form are used for two (or more) different things. As Aristotle says: Other [fallacies] come about owing to the form of expression used, when what is really different is expressed in the same form, e.g., a masculine thing by a feminine termination … or again when a quality is expressed by a termination proper to quantity or vice versa, or what is active by a passive word, or a state by an active word.8

However, although the pseudonymous author makes much play subsequently with a distinction between the supposition of hoc falsum, suppositing for ego dico falsum, and the supposition of falsum in ego dico falsum, which cannot supposit for the proposition itself according to the restrictivists, even mentioning the problematic inference (2), from “Socrates says this falsehood” (sortes dicit hoc falsum) to “Socrates says a falsehood” (sortes dicit falsum), in § 5.06, the first discussion of inference (2) as such comes a few pages later in Roure’s article, in Walter Burley’s treatise on insolubles. Burley considers an objection: Moreover, if Socrates says that he says a falsehood, “Socrates says this falsehood” is true, from which it follows that Socrates says a falsehood, because anyone saying this falsehood says a falsehood unrestrictedly. To this it is commonly said that: “he says this falsehood, so he says a falsehood” is not valid, because to say this falsehood is to say a falsehood only restrictedly,9  See, for example, Buridan, Summulae de dialectica, 537: “Thus many people say that one mode of this fallacy arises from the change of supposition of some word or other.” 8  Aristotle, De sophisticis elenchis, trans. Pickard-Cambridge, chap. 4, 166b10–14; cf. also chap. 22. See also Hamblin, Fallacies, 25–7. 9  Burley, Insolubilia, § 1.04, 264: “Preterea Sorte dicente se dicere falsum, hec est vera: Sortes dicit hoc falsum; ex quo sequitur quod Sortes dicit falsum, quia quilibet dicens hoc falsum dicit falsum simpliciter. Huic dicitur communiter quod non sequitur: dicit hoc falsum, ergo dicit falsum, quia dicere hoc falsum non est dicere falsum nisi secundum quid.” 7

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to which he replies: To the other ⟨objection⟩ one says that when Socrates speaks like this, he says this falsehood. But from this it does not follow that he says a falsehood, nor because of this does it follow that to say this falsehood is to say a falsehood restrictedly, since it is not a falsehood restrictedly, but to say this falsehood is to restrictedly say a falsehood, for the qualification “restrictedly” should qualify “say,” not “falsehood.”10

Nonetheless, this is not completely clear. We get a clearer discussion in Ockham, particularly in his Expositio super libros Elenchorum, discussing De sophisticis elenchis, chap. 25: For take a scenario in which someone begins thus to say, “I say a falsehood” … [Aristotle] solves the aforesaid paralogism by saying that it is difficult to know whether it should be conceded unrestrictedly that he says a truth or lies; however, nothing rules out what he says being unrestrictedly false, yet that he does not unrestrictedly lie, but restrictedly says the truth. He wants to say that he does not unrestrictedly lie nor restrictedly say the truth, but he says this falsehood, which is that he says a falsehood. But from this it does not follow that he unrestrictedly says a falsehood. And thus this inference: “Socrates says this falsehood, so Socrates says a falsehood” is not valid, but it is a fallacy of the restricted and the unrestricted, because in “Socrates says a falsehood,” “falsehood” cannot supposit for this falsehood ⟨itself⟩, which is that Socrates says a falsehood. And one should respond to all insolubles in this way, namely, by denying the inference from a term used with a demonstrative pronoun referring to something contained, to that term used without that pronoun. And the only reason is because a general term cannot supposit in that proposition for that inferior, although in other propositions it can supposit for it.11  Ibid., § 4.04, 275: “Ad aliud dicitur quod Sortes sic dicendo dicit hoc falsum. Sed ex hoc non sequitur quod dicat falsum, nec propter [ms., proprie ed.] hoc sequitur quod dicere hoc falsum est dicere falsum secundum quid, cum non sit falsum secundum quid, sed dicere hoc falsum est dicere secundum quid falsum, unde ista determinatio secundum quid debet determinare ly dicere et non ly falsum.” 11  Ockham, Expositio super libros Elenchorum, II.10, §§ 4–5, 267–8: “Unde ponatur iste casus, quod aliquis incipiat sic loqui: ego dico falsum … Solvit praedictum paralogismum dicens quod difficile est scire an sit concedendum simpliciter quod talis dicit verum vel mentitur; tamen nihil prohibet quod illud quod dicit sit simpliciter falsum, et tamen quod non simpliciter mentiatur, sed secundum quid dicat verum. Vult dicere quod non simpliciter mentitur nec simpliciter dicit verum; et tamen dicit hoc falsum, quod est ipsum dicere falsum, sed ex hoc non sequitur quod simpliciter dicat falsum. Et ita talis consequentia non valet ‘Sortes dicit hoc falsum; igitur Sortes dicit falsum,’ sed est f­allacia 10



Commentary281

He gives a similar diagnosis in his Summa Logicae III-3, chap. 46: [W]hen Socrates begins to say, “Socrates is saying a falsehood,” and it is asked, “Is Socrates saying a truth or a falsehood?” it is to be said that Socrates says neither a truth nor a falsehood, just as it is to be granted that he says neither a truth nor a falsehood other than this. And in that case it does not follow: “This is true: ‘Socrates is saying a falsehood’; and Socrates is saying this; therefore, Socrates is saying a falsehood,” just as it does not follow: “Socrates is saying this; and this is false; therefore, Socrates is saying a falsehood other than this.” This is because these two propositions are equivalent: “Socrates is saying a falsehood” and “Socrates is saying a falsehood other than this,” because of the fact that in “Socrates is saying a falsehood” the predicate cannot supposit for this proposition. And if it is said: Here the argument is from an inferior to a superior without negation and without distribution; therefore, the inference is a good one—it is to be said that the inference is not valid except when the superior in the consequent can supposit for the inferior. Thus if, in “A man is an animal,” “animal” could not supposit for a man, the inference “Socrates is a man; therefore, Socrates is an animal” would not be valid. But in “Socrates is saying a falsehood,” the predicate cannot supposit for the whole proposition; therefore, “Socrates is saying this falsehood; therefore, Socrates is saying a falsehood” does not follow. By means of what has been said [above], the studious person can reply to all insolubles, if in solving them he wants to pay diligent attention to and inquire [into] the nature of insolubles. I leave this to clever people, because I inserted these [materials] about obligations and insolubles only to fill out this Summulae and lest so great a part of logic be left completely untouched.12 secundum quid et simpliciter. Et hoc quia in ista ‘Sortes dicit falsum,’ li ‘falsum’ non potest supponere pro hoc falso, quod est Sortem dicere falsum. Et per istum modum respondendum est ad omnia insolubilia, negando scilicet consequentiam ab aliquo termino sumpto cum pronomine demonstrativo demonstrante aliquod contentum, ad ipsum terminum sumptum sine tali pronomine. Et non est alia ratio nisi quia terminus communis non potest supponere in illa propositione pro illo inferiori, quamvis in aliis propositionibus possit pro eo supponere.” 12  Ockham, Summa logicae, III.3, chap. 46, 746, trans. Spade (http://pvspade.com/ Logic/docs/OckhamInsolubilia.pdf): “[Q]uando Sortes incipit sic loqui ‘Sortes dicit falsum,’ et quaeritur ‘aut Sortes dicit verum aut falsum,’ dicendum est quod Sortes neque dicit verum neque falsum; sicut concedendum est quod neque dicit verum neque dicit falsum aliud ab isto. Et tunc non sequitur ‘haec est vera: Sortes dicit falsum; et Sortes dicit hanc; igitur Sortes dicit falsum,’ sicut non sequitur ‘Sortes dicit hoc, et hoc est falsum, igitur Sortes dicit aliud falsum ab isto.’ Et hoc, quia istae duae aequivalent ‘Sortes dicit falsum’ et ‘Sortes dicit aliud falsum ab isto,’ propter hoc quod in ista ‘Sortes dicit falsum’ praedicatum non potest supponere pro ista propositione.

282 Commentary

Argument (2) also surfaces in Bradwardine’s Insolubilia, in an objection to his own theory that he considers. The objector writes: [F]rom the response just given, it follows that the following inference is valid: Socrates utters this falsehood, namely, “Socrates utters a falsehood”, so Socrates utters a falsehood. Hence there is here no fallacy [of the restricted and the unrestricted], contrary to Aristotle’s Soph. Elench. 2, in the chapter on the solution of the fallacy [of the restricted and the unrestricted],13 where it is shown by him that insolubles are paralogisms [of the restricted and the unrestricted].14

Bradwardine’s reply is to affirm that the inference is valid, and should not be confused with the problem of revenge, which does commit the fallacy of the restricted and the unrestricted.15 Argument (2) also appears in Kilvington’s notorious 48th sophism (Kilvington, Sophismata, translation and commentary, S48(ee), 139–40): And this reply is evident, as regards the claim that each insoluble is true [restrictedly and not unrestrictedly], from the reply of those who say that this inference is not acceptable: “Socrates says this [falsehood]—i.e., ‘Socrates says [a falsehood]’; therefore, Socrates says [a falsehood].”16 Et si dicatur: hic arguitur ab inferiori ad superius sine negatione et sine distributione, igitur est consequentia bona, dicendum est quod consequentia non valet, nisi quando illud superius in illo consequente potest supponere pro illo inferiori. Unde si in ista ‘homo est animal’ li animal non posset supponere pro homine, haec consequentia non valeret ‘Sortes est homo, igitur Sortes est animal.’ In ista autem ‘Sortes dicit falsum’ praedicatum non potest supponere pro tota ista propositione, ideo non sequitur ‘Sortes dicit hoc falsum, ergo Sortes dicit falsum.’ Per praedicta potest studiosus respondere ad omnia insolubilia, si solvendo ea velit naturam insolubilium diligenter advertere et inquirere. Quod relinquo ingeniosis, quia ista de obligationibus et insolubilius non inserui nisi propter istius Summulae complementum et ne tanta pars logicae totaliter dimitteretur intacta.” 13  See Aristotle, De sophisticis elenchis, trans. Pickard-Cambridge, chap. 25, 180a22–30. 14  Bradwardine, Insolubilia, § 7.11, 118–19: “[E]x ista responsione sequitur istam consequentiam esse bonam: Sortes dicit hoc falsum quod est Sortes dicit falsum, ergo Sortes dicit falsum. Ergo ibi non est fallacia secundum quid et simpliciter, quod est contra Aristotelem 2o Elenchorum, capitulo de solutione fallacie secundum quid et simpliciter, ubi apparet per eum quod insolubilia sint paralogismi secundum quid et simpliciter.” 15  See Paul’s tenth opinion (§ 1.10 ff.), and Dutilh Novaes and Read, “Insolubilia and the Fallacy Secundum Quid et Simpliciter,” 189–90. The problem of revenge is explained in the introduction to Bradwardine, Insolubilia, 20. 16  Kilvington, Sophismata, text edition, S48(ee), 143: “Et ista responsio quantum ad hoc quod dicitur, quod quodlibet insolubile est verum secundum quid et non simpliciter, patet ex responsione eorum qui dicunt quod ista consequentia non valet: ‘Socrates dicit



Commentary283

Finally, we come to question (3), how to translate sortes dicit falsum and similar phrases. We need to take into account at least the following points: a. dicit connotes a spoken expression (Paul will use other words for written utterances); b. falsum is a noun, and should be translated by a noun (or nounphrase), if possible; c. sortes dicit falsum will, we have seen, be paired very often with sortes dicit hoc falsum, so whatever translation is chosen for falsum should admit a demonstrative “this” in front of it. Various suggested translations are: “Socrates utters a falsehood” (contrary to requirement a), “Socrates speaks falsely” (contrary to b and c), “Socrates says something false” (contrary to c, and perhaps also b), “Socrates says what is false” or “What Socrates says is false” (contrary to c, and perhaps also b). We have chosen “Socrates says a falsehood” since it passes the abc test (and in parallel, “Socrates says a truth”), although we share the doubts of those who think them ungrammatical, or at best clumsy. Most of § 1.1 is taken by Paul straight from Bradwardine’s text (§ 5.2), as is much of Paul’s discussion of the next six opinions (§§ 1.2–1.7). The first fourteen solutions examined by Paul are also partially translated in Bocheński, History of Formal Logic, §§ 35.27–56, 241–9. § 1.1.1 In the corresponding passage in Bradwardine, Insolubilia, Bradwardine simply says that this account is “wrong for the same reasons as the first group,” referring to chaps. 3–4, where he criticized Walter Burley’s restrictivist theory in detail. But no objection like that levelled by Paul in § 1.1 appears there, and the objection looks very weak, simply observing that Socrates might have already said a falsehood previously. This is standardly ruled out by describing the scenario as one where Socrates says “Socrates says a falsehood” and nothing else, or at least no (other) falsehood. The correct translation of homo is of course contentious. We follow Kretzmann’s practice in Paul of Venice, Logica magna: Tractatus de terminis, which he describes on 292, note c to 25: homo, like “man,” is in hoc falsum, quod est “Socrates dicit falsum,” igitur Socrates dicit falsum.’” Note that the Kretzmanns translate secundum quid et simpliciter as “in a certain respect and absolutely” and falsum as “what is false.” For discussion of Kilvington’s theory of insolubles, see the comments on § 1.9 below.

284 Commentary

general gender-neutral, but on occasion gender-specific. There are many examples of this ambiguity in the Logica magna: see, for example, Logica magna: Tractatus de suppositionibus, 88. § 1.1.2 One term is inferior to another (its superior) if it falls under it, as “this falsehood” falls under “a falsehood,” or “man” under “animal.” See also below, § 2.2.4. § 1.1.3 The defects identified in the fallacy of form of expression are set out in, for example, Peter of Spain, Summaries of Logic, §§ 83–100, and Buridan, Summulae de dialectica, §§ 7.3.9–10. § 1.2 Paul’s second solution is the third in Bradwardine’s list, whose proponents have not yet been identified (see Bradwardine, Insolubilia, § 5.3 and at 224 n. 24). § 1.2.1 Paul discusses the main grounds for the first and second opinions in considering the fourteenth opinion (that of Walter Segrave) in § 1.14 below. § 1.3 Paul’s third solution is the fourth in Bradwardine’s list, that of transcasus (see Bradwardine, Insolubilia, § 5.4). The term transcasus does not occur in either Paul’s or Bradwardine’s text. But this view seems to be the one which Burley describes as follows: There is another opinion, for there are some who respond to the insolubles by transcasus. There is transcasus when some proposition is changed from truth to falsity and vice versa.17

One of its proponents may be the anonymous author of the treatise on insolubles edited by Braakhuis (“The Second Tract on Insolubilia”) and dubbed by him as “Tractatus Sorbonnensis alter de Insolubilibus,” although again the term transcasus is not used, and Braakhuis (ibid., 127–8) expresses some doubts about this identification. The author makes a crucial distinction between utterances which are not numerically but only specifically the same. For the medievals in general, utterances are tokens, each numerically distinct.18 Thus when I say, “I say a falsehood,” if I have not said anything previously, what I say is false; when

 Burley, Insolubilia, § 2.07, 270: “Alia opinio est: sunt enim quidam qui respondent ad insolubilia per transcasum. Et est transcasus quando aliqua propositio mutatur a veritate in falsitatem vel e converso.” See De Rijk, “Some Notes on the Medieval Tract De Insolubilibus,” 87. 18  See Nuchelmans, “The Semantics of Propositions,” 207–09. 17



Commentary285

I repeat it, I utter a numerically different proposition, which is true, referring to the first: When I first say that I say a falsehood, “I say a falsehood” (call it A) is false, because it is not true of itself, unless there is some earlier utterance of which it is true. But “I say a falsehood” said a second time (call it B) will be true, for it is true of the first, since it is numerically distinct from it.19

Again, § 1.3 is almost verbatim from Bradwardine, Insolubilia, § 5.4. § 1.3.1 Paul’s use of the Latin term precise will often be translated as “explicitly” or “only” (literally it means “cut short” or “clipped”), in particular when used in conjunction with significare (“signify”). See commentary on §§ 1.8.1 and 1.12.1 below. § 1.4 Paul’s fourth solution is the fifth solution in Bradwardine’s list, that of the nullifiers (cassantes) of potency. For a discussion of cassatio, see Spade and Read, “Insolubles,” § 2.5. Although cassationism (cassatio) is often described and rejected in insolubilia-treatises (see, for instance, Burley Insolubilia, § 2.03, 269), only two treatises before Bradwardine’s are known in which cassatio is the preferred solution, notably, the Insolubilia Monacensia, written around 1200: see De Rijk, “Some Notes on the Medieval Tract De Insolubilibus,” § 4. The other text, dating from around 1225, is described in Spade (The Medieval Liar, item XXI, 44). Both correspond more precisely to the nullifiers of act described in § 1.5. Most of §§ 1.4–1.4.1 is taken verbatim from Bradwardine, Insolubilia, § 5.5. § 1.5 Paul’s fifth solution is the sixth solution in Bradwardine’s list, that of the nullifiers of act. This position seems to be that expressed in the Insolubilia Monacensia: [I]t should be said that you say nothing, since generally with regard to any utterance, everything by whose assumption [the utterance] becomes an insoluble, should be nullified.20

 Braakhuis, “The Second Tract on Insolubilia,” 144: “[C]um primo dico me dicere falsum, hec est falsa: ‘ego dico falsum’—que vocetur a—quia pro se non verificatur, nisi aliqua dicta sit prius pro qua verificetur. Hec autem: ‘ego dico falsum’ secundo loco dicta, erit vera—que vocetur b—verificatur enim pro prima, cum sit diversa ab ea secundum numerum.” 20  De Rijk, “Some Notes on the Medieval Tract De Insolubilibus,” 106: “[D]icendum: ‘nil dicis,’ cum generaliter omne illud sit cassandum circa unumquodque enuntiabile quo supposito accidit insolubile circa illud.” 19

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Indeed, the anonymous author suggests the act cannot be performed at all, as set out in Paul’s fourth solution (§ 1.4): It should generally be noted that when in any utterance the act of the verb is determined by this expression “falsehood,” that act can especially in no way be performed about that utterance, because if it were so performed, then an insoluble would result.21

Most of §§ 1.5–1.5.2 is taken verbatim from Bradwardine, Insolubilia, § 5.6. § 1.5.1 Where Bradwardine (ibid.) writes, “as is clear enough to sense and anyone present” (ut sensui et omnibus astantibus satis patet), Paul simply says, “as I have shown previously” (ut alias ostendi). Unless Paul is referring to a previous treatise in the Logica magna (or possibly elsewhere), he would seem to be referring back to § 1.4.1, where he used a similar phrase, “repugnant to the senses and the intellect” (repugnat sensui et intellectui). § 1.6 Paul’s sixth solution is the seventh solution in Bradwardine’s list, that of the mediantes (the middle way). It may be that of Richard Kilvington: see Bradwardine, Insolubilia, 225 n. 28. (But see the ninth opinion below, which Cajetan attributes to Kilvington.) Strobino (“Truth and Paradox in Late XIVth-Century Logic,” 479) attributes this opinion to Swyneshed. However, as Dutilh Novaes (“A Comparative Taxonomy of Medieval and Modern Approaches to Liar Sentences,” 239) recognises, the problem with attributing the sixth opinion to Swyneshed is that §§ 1.6–1.6.2 are a direct quotation from § 5.7 in Bradwardine’s treatise, which was written ten years before Swyneshed’s (according to the usually accepted dating). Even attributing it to Kilvington strains chronology a little, since Bradwardine’s treatise is from the early 1320s, whereas Kilvington’s is from the mid-1320s. But it is reasonable to believe that Bradwardine and Kilvington traded ideas, just as we will see Heytesbury and Dumbleton could (see § 1.8 below). Indeed, Appendix A to Bradwardine, Insolubilia, in which a Bradwardinian response is given to Kilvington’s Sophism 48, suggests just this, if Appendix A is actually a later addition by Bradwardine himself and not by someone else writing on his behalf. Swyneshed’s theory could, of course, be a repetition or update of an earlier theory to which 21  Ibid., 107: “Unde generaliter notandum quod quando in aliquo enuntiabili actus verbi determinatur per hanc dictionem ‘falsum,’ specialiter ille actus nullo modo potest exerceri circa illud enuntiabile, quia dato quod exerceretur, inde sequeretur insolubile.”



Commentary287

­ radwardine is referring. But Swyneshed’s theory has distinctive feaB tures which are not mentioned in the sixth theory. Moreover, Paul’s own theory (the fifteenth opinion: see chap. 2) is essentially (an update of) Swyneshed’s, which would explain why Swyneshed’s theory is not discussed by Paul in this first chapter. Most of §§ 1.6–1.6.2 is taken verbatim from Bradwardine, Insolubilia, § 5.7. § 1.7 Paul’s seventh solution is the eighth solution in Bradwardine’s list, which Spade (“Five Early Theories in the Medieval Insolubilia-Literature,” 36–7) attributes to Duns Scotus: see Bradwardine, Insolubilia, “Introduction,” 14. Nuchelmans (“The Distinction actus exercitus/actus significatus in Medieval Semantics,” 74–6) claims that the distinction was drawn in the anonymous Insolubilia Monacensia, “although the author does not yet explicitly use the terminology of the opposition significatus/ exercitus.” However, as noted in the comment on § 1.5, the author of that treatise is a cassationist, and in fact such terminology is widespread. It occurs also in pseudo-Sherwood (see § 1.1 above: “‘I say’ earlier copulated not for the utterance performed but for other things conceived ⟨by it⟩” (“ly dico prius copulavit, non pro hac dictione exercita, sed pro aliis conceptis”), but as we saw, pseudo-Sherwood holds the first opinion about the insolubles that Paul discusses. Ockham (Quodlibeta, VII.6) explains the distinction as follows: I say that a performed act (actus exercitus) is one which is conveyed by the verb “is” or some similar word, which not only signifies that something is predicated of something else but performs the predication, by predicating one of the other, as by saying, “A man is an animal,” “A man runs,” “A man disputes,” and so on. A designated act (actus signatus) is one which is conveyed by the verb “be predicated” or “be subject” or “be verified” or “agree with,” which signify the same. For example, here is a designated act, “Genus is predicated of species,” and similarly here, “Animal is predicated of man”; but in this proposition “animal” is not predicated of “man,” because in this proposition, “animal” is the subject and is not the predicate, and so it is a designated act, because it is not the same to say, “Animal is predicated of man” and “A man is an animal,” just as it is not the same to say, “Genus is predicated of species” and “A species is a genus,” because one is true and the other false.22 22  Ockham, Quodlibeta VII, qu. 6, 731–2: “[D]ico quod actus exercitus est ille qui importatur per hoc verbum ‘est’ vel aliquod consimile verbum, quod non tantum significat aliquid praedicari de alio sed exercet praedicationem, praedicando unum de alio, sic

288 Commentary

Although Paul’s terminology is slightly different (actus conceptus rather than actus signatus or actus significatus), Spade (“Five Early Theories in the Medieval Insolubilia-Literature,” 36–7) claims that the opinion Bradwardine is discussing draws the same distinction between what the speaker is doing and what he is talking about. Scotus wrote: To the question, it should be said that one starting to say, “I say a falsehood” is unrestrictedly false in saying it, but true restrictedly … there is falsity in the designated act and truth in the performed act. So it follows, “It is true that I performed an act of speaking about a falsehood, so that about which I perform is a falsehood.”23

Nonetheless, note that Bradwardine (in a final sentence that Paul does not repeat) dismisses the seventh (his eighth) opinion partly because it does not appeal to the Aristotelian fallacy of the restricted and the unrestricted (secundum quid et simpliciter): In addition, this view does not work by (a fallacy) secundum quid et simpliciter, which, however, it should,24

whereas Scotus explicitly did so. Most of §§ 1.7–1.7.3 is taken verbatim from Bradwardine, Insolubilia, § 5.8. The reading dictum complete (“complete utterance”) in E matches Bradwardine’s text, which has non est dictum complete (with the verb est). The scribe might easily (as in M) copy dictum complete as dictum completum (rather than vice versa). Scotus’s text (514–15) does not use the phrase dictum completum/e in either form. dicendo ‘homo est animal,’ ‘homo currit,’ ‘homo disputat,’ et sic de aliis. Actus signatus est ille qui importatur per hoc verbum ‘praedicari’ vel ‘subici’ vel ‘verificari’ vel ‘competere,’ quae idem significant. Verbi gratia hic est actus signatus ‘genus praedicatur de specie,’ et similiter hic ‘animal praedicatur de homine’; et tamen in ista propositione non praedicatur animal de homine, quia in ista propositione animal subicitur et non praedicatur, et ideo est actus signatus, quia non est idem dicere ‘animal praedicatur de homine’ et ‘homo est animal,’ sicut non est idem dicere ‘genus praedicatur de specie’ et ‘species est genus,’ quia una est vera et alia falsa.” Cf. Ockham, Summa logicae I, chap. 66, 202. 23  Scotus, Super libros Elenchorum, qu. 53, ed. Vivès, 76; ed. Andrews et al., 513–15: “Ad quaestionem dicendum, quod sic incipiens loqui, Ego dico falsum, est simpliciter falsus in dicendo, verus tamen secundum quid … in proposito falsitas est in actu signato, et veritas in actu exercito. Unde sequitur, Verum est me exercere actum dicendi, circa falsum; ergo illud circa quod exerceo, est falsum.” 24  Bradwardine, Insolubilia, 92–3: “Preterea hec positio non assignat secundum quid et simpliciter, et tamen deberet.”



Commentary289

§ 1.7.2 The

“usual counterargument” seems to be the one in § 1.7.1 rejecting the diagnosis by the fallacy of four terms: nothing stops us taking the expressions to have the same sense in both premises. § 1.8 Paul’s eighth solution is the second solution in Heytesbury’s list. (The first opinion listed by Heytesbury was Swyneshed’s, which as noted will be the source for Paul’s own, the fifteenth solution.) § 1.8 is taken almost verbatim from Heytesbury. Spade (Heytesbury, Insolubles, trans. Spade, 73 n. 28) notes that Cajetan attributes this opinion to John Dumbleton: The first of these positions is that of Swyneshed, the second was put forward by Dumbleton, the third is that of Richard Kilvington in his Sophismata.25

Spade questions this attribution on the grounds that Dumbleton himself refers to Heytesbury’s solution. But Heytesbury and Dumbleton were together in Oxford in the mid-1330s (as Kilvington and Bradwardine in the 1320s), so would presumably have been mutually aware of each other’s work.26 Dumbleton’s view is, in brief, that an insoluble is a proposition in name only, lacking one crucial component of signification, namely, uptake, necessary for it to be true or false (that is, for things to be as it signifies or not), and so is not really a proposition at all.27 § 1.8.1 We have chosen to translate significatum adequatum as “exact significate.” Spade (Swyneshed, “Roger Swyneshed’s Theory of Insolubilia,” 106) renders adequatum as “adequate” (as is done throughout Paul of Venice, Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis too), and equates it with “total” and “principal,” referring to Maierù, Terminologia logica della tarda scolastica, 490 n. 2, citing Strode’s Consequentiae from his Logica. The term is

 Cajetan, In regulas Gulielmi Hesburi recollecte, fol. 7va: “Prima harum positionum est suisset, secunda ponitur a dulmentone, tertia est ricardi climentonis in sophismatibus suis.” 26  Problems with Cajetan’s attributions are discussed, but set aside, by Hanke, “Cajetan of Thiene on the Logic of Paradox,” 74 n. 10. 27  For detailed discussion of Dumbleton’s solution, see Bartocci, “John Dumbleton on Signification and Semantic Paradoxes.”

25

290 Commentary

found in Burley,28 Gregory of Rimini,29 Richard Brinkley, and many others. Brinkley for one seems definitely to equate adequatum with totale and primum: [F]irst it should be known that that thing or things are called the significate of the proposition which are signified by the proposition. For example, the significate of the proposition “Socrates sees Plato” is the thing Socrates, the thing Plato, and the seeing by which Socrates sees. These things are all the significates of this proposition. So from their being duly arranged one with the others there results and is constituted the whole, exact and immediate significate of that proposition.30

As Cesalli (“Pseudo-Richard of Campsall and Richard Brinkley,” 106) observes, the exact significate is, for Brinkley, more than the sum of its parts, and in that sense, it is the whole, or total, significate. But not for Paul. Moreover, although adequatum can mean “adequate,” “exact” (or “core” or “minimal”) better captures Paul’s idea, as expressed in his explanation of his theory in treatise 11 of Part II of the Logica magna, chap. 9, thesis 3: [T]he term “man” signifies man, being, substance, animal, Socrates, Plato, and all present, past, and future or imaginable men, from which its total significate is composed. Nonetheless, it is not the exact significate of that  See Burley, Expositio super Praedicamenta Aristotelis, in Expositio super artem veterem Porphyrii et Aristotelis, cap. de priori, fol. 47va: “Et ex hoc patet quod per propositionem in voce et etiam in conceptu significatur aliqua res complexa quae non est proprie aliqua res praecise significata per subiectum nec res significata per praedicatum, sed aggregatum ex his. Et illa res, quae est ultimum et adaequatum significatum propositionis in voce et in conceptu, est quoddam ens copulatum. Et propter hoc potest dici propostio in re.” 29  See Nuchelmans, Theories of the Proposition, 211: “[S]ciendum est quod non solum enuntiationes ipse dicuntur vere vel false: sed earum enuntiabilia seu significata adequata.” (Gregory of Rimini, Super primum Sententiarum, prol. qu. 1, 10: “It should be known that not only statements themselves are true or false, but what they state, i.e., their exact significates.”) 30  See Brinkley, Richard Brinkley’s Theory of Sentential Reference, 34: “[P]rimo est sciendum quod significatum propositionis vocantur illud, vel illa, quod vel quae per propositionem significatur, ut significatum istius propositionis ‘Sortes videt Platonem’ est haec res Sortes, et haec res Plato, et visio qua Sortes videt. Et hae res sunt omnia significata per istam propositionem. Et ideo ex istis debite ad invicem ordinatis resultat et integratur totale, et adaequatum, et primum significatum illius propositionis” (Fitzgerald gives an alternative translation). Although Fitzgerald (ibid., 32) provisionally dates Brinkley’s Summa between 1356 and 1363, Cesalli (“Pseudo-Richard of Campsall and Richard Brinkley,” 84) and others now date it earlier, between 1345 and 1350. 28



Commentary291

term … From these remarks, it is clear that “man” does not immediately and exactly signify being, substance, or animal. For although it would signify any such things under its own concept and distinctly, its distinct signification does not stop at any of them, but descends to man, which is naturally posterior to being, substance, and animal. But it signifies nothing posterior to man—such as Socrates or Plato, etc.—distinctly under its own concept, but only confusedly. Therefore, the term “man” exactly signifies man and no other significate that is inferior or superior to man … Fifth conclusion: The exact significate of a proposition is things being in some way such that their being in that way arises implicitly or explicitly from the exact significates of the parts.31

Thus for Paul, adequatum is certainly not equivalent to totale (“total,” “whole”). It is a very exact part of the total signification, but only because, for Paul, the total signification is much more extensive than for some other authors. We see this in his discussion of primary and secondary (or consequential) signification in the first treatise in the Logica magna (Tractatus de terminis, 108–13). He writes: It was said [above] that each term secondarily signifies whatever an inferior or superior term primarily signifies. Thus the term “man” immediately or primarily signifies man but secondarily signifies animal, body, substance, this white man, and so on.32

Note that we have translated primo as “immediately” (and not as “primarily,” as do Adams and Fitzgerald) to distinguish it from primarie. A  Paul of Venice, Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 192–6: “Nam ille terminus ‘homo’ significat hominem, ens, substantiam, animal, Sortem, Platonem, et omnes homines praesentes, praeteritos, et futuros, aut imaginabiles; ex quibus fit totale significatum illius quod tamen non est significatum adaequatum eiusdem … Ex quibus patet quod ly homo non significat primo et adaequate ens, substantiam, aut animal, quia licet quodlibet tale significet sub propria ratione et distincte, tamen in nullo illorum sistit sua significatio distincta, sed descendit in hominem posteriorum naturaliter ente, substantia, et animali. Quia vero nihil posterius homine distincte significat sub propria ratione, ut Sortem vel Platonem et sic de aliis, sed solum in confuso, ideo significat hominem adaequate et non aliud significatum inferius aut superius ad hominem … Quinta conclusio est ista: significatum adaequatum propositionis est aliqualiter esse qualiter esse implicite vel explicite egreditur a significatis adaequatis partium” (for an alternative translation, see ibid., 193–7). 32  Logica magna: Tractatus de terminis, 108–09: “Dicitur enim ibidem quod quilibet terminus secundarie significat quicquid terminus inferior vel superior primarie significat, ita quod iste terminus ‘homo’ significat primo vel primarie hominem; secundarie vero significat animal, corpus, substantiam, istum hominem album, et sic ultra.” 31

292 Commentary

passage from Paul’s Quadratura (see Appendix B, § 2.2.3) is relevant here: It should be said that it is because it immediately signifies a truth that any proposition is true, and it is because it immediately signifies a falsehood that any proposition is false, where outside the case of insolubles “immediately” means the same as “exactly.” But in the case of insolubles it means the same as “principally.” Hence “A man is an animal” is true because it immediately signifies a truth, that is, it exactly ⟨signifies⟩ the truth that a man is an animal; but “This is false,” referring to itself, is false because it immediately signifies a falsehood, that is, it principally ⟨signifies⟩ a falsehood, namely, that this is false and that this is not false.33 § 1.8.2 The

two counterarguments in §§ 1.8.1–2 are similar to those in Heytesbury’s Insolubilia (Regulae solvendi sophismata, fol. 5vb; trans. Spade, 37–8; ed. Pozzi, 228). On the translation of Nullus deus est as “God does not exist,” see the comment on § 1.13.6.3. § 1.9 Paul’s ninth solution is the third in Heytesbury’s list. See Heytesbury, Regulae solvendi sophismata, fol. 4vb; trans. Spade, 19; ed. Pozzi, 212. Spade (Heytesbury, Insolubles, trans. Spade, 74 n. 29) notes that Cajetan, who studied with Paul of Venice in Padua (see Hanke, “Cajetan of Thiene on the Logic of Paradox,” 71 n. 2), attributes this opinion to Richard Kilvington (see comments on §§ 1.6 and 1.8 above), writing in Oxford in the mid-1320s. For a discussion of Kilvington’s somewhat obscure view, see Kretzmann’s discussion in Kilvington, Sophismata, 361–74, and Read, “The Calculators on the Insolubles,” § 3. Kretzmann summarizes Kilvington’s account as claiming that “each insoluble is true [restrictedly] and false [restrictedly] and neither true nor false [unrestrictedly]” (Kilvington, Sophismata, 362).34 Kilvington explains: Suppose, for example, that Socrates says this: “Socrates says what is false” and nothing else. In that case I say that the term “what is false” can be taken in 33  “Est ergo dicendum quod ex eo aliqua propositio est vera quia significat primo verum et ex eo aliqua propositio est falsa quia significat primo falsum, ita quod ly primo extra materiam insolubilium idem sonat quod adequate. Sed in materia insolubilium ⟨idem⟩ sonat quod principaliter. Hec ergo: homo est animal, est vera quia significat primo verum idest adequate verum quod est hominem esse animal, et hec: hoc est falsum, seipso demonstrato, est falsa, quia significat primo falsum idest principaliter falsum videlicet hoc esse falsum et hoc non esse falsum.” 34  See S48(cc). But Kilvington appears to qualify the statement in S48(ll): “I say, then, that no insoluble that is presently under discussion is [unrestrictedly] true or



Commentary293

one way for what is false [restrictedly] and in another way for what is false [unrestrictedly]. If for what is false [restrictedly], it is to be granted. If for what is [unrestrictedly] false, then it must be denied that Socrates says what is false.35 § 1.9.1 Presumably,

the reference to the “arguments against the other one” is to §§ 1.8.1–3 against the eighth opinion. Paul uses the verb reducere several times, apparently to mean “deduce.” See, for instance, §§ 3.1, 3.1.3, 5.3.1. The topic from the disjunctive whole (locus a toto disiuncto) is a subtopic of the topic from the whole to its part and is not present in Boethius’s list of Ciceronian and Themistian intrinsic topics (or loci). Two places where this topic has been discussed are Green-Pedersen, The Tradition of the Topics in the Middle Ages, and Kann, “Zur Behandlung der dialektischen Örter.” But it is remarkable how different the rules are that each medieval author includes under this heading. First, consider what Lambert of Auxerre writes in his Summa: A disjunctive whole occurs when two things are joined by a disjunctive conjunction. An argument holds destructively from this whole to its part, but one holds constructively from a part to its whole. An example of the first: “Socrates neither runs nor argues; therefore Socrates does not run.” An example of the second: “Socrates runs; therefore Socrates runs or he argues.” The topical maxim from a disjunctive whole: When the whole thing disjoined is destroyed, its part is also destroyed; and when a part is posited, its whole is also posited.36

[­unrestrictedly] false; instead each is true [restrictedly] and false [restrictedly]” (Kilvington, Sophismata, 142). See Kretzmann’s comments ibid., 368 and 374. 35  Kilvington, Sophismata, trans. Kretzmann, S48 (mm), 142. Latin text ibid., 146: “Ut, verbi gratia, posito quod Socrates dicat istam: ‘Socrates dicit falsum’ et nullam aliam. Tunc dico quod iste terminus ‘falsum’ sumi potest uno modo pro falso secundum quid et alio modo pro falso simpliciter. Si pro falso secundum quid, concedenda est. Si pro falso simpliciter, tunc est negandum quod Socrates dicit falsum.” 36  Lambert of Auxerre, Logica, or Summa Lamberti, trans. Maloney, § 776, 157–8; ed. Alessio, 129: “Totum disiunctum est quando aliqua duo coniunguntur per coniunctionem disiunctivam; ab isto toto ad suam partem tenet argumentum destructive; a parte vero ad suum totum tenet constructive. Exemplum de primo: Sortes non currit vel disputat ergo Sortes [non: add. Maloney] currit. Exemplum de secundo: Sortes currit ergo Sortes currit vel disputat. Locus a toto disiunctum [sic! Maloney corrects it to disiuncto] maxima: destructo toto disiuncto destruitur eius pars et posita parte ponitur totum.” See Green-Pedersen, The Tradition of the Topics, 244, 317.

294 Commentary

That is, from not-(p or q) to not-p from not-(p or q) to not-q

and from p to (p or q) from q to (p or q).

These last two (often dubbed “addition”) are a common thread running through the various authors. In his Summulae de dialectica, Buridan writes: [A]gain, there is the [locus] that goes from the part of a disjunctive whole to its whole and from any categorical [component] of a disjunctive proposition to the disjunctive proposition, for example, “Socrates runs; therefore, Socrates runs or Plato disputes” … The final part of this chapter is about the loci concomitant with substance that are not listed by the author [i.e., Peter of Spain]. And neither our author, nor even Themistius, listed these, for they are either not dialectical loci or they are easily reduced to the ones that have been listed.37

That is, Buridan lists only addition. Towards the close of his discussion of the topics, however, he returns to the question as to whether the topic from division is a dialectical topic: But some people say that the locus from division is not to be counted among the dialectical loci, given that an argument based on it is a formally valid consequence. But to maintain [the opinions of] Boethius and Themistius and our author, and generally everybody else who has listed this locus among the dialectical loci, I say that the following is an argument from division: “A is an animal and is not rational; therefore it is irrational”; and this [inference] is not formal[ly valid], but it would indeed be rendered formal[ly valid] by adding the proposition “Every animal is rational or irrational.”38  Buridan, Summulae de dialectica, trans. Klima, § 6.4.18, 456–7; Summulae de locis dialecticis, ed. Green-Pedersen, 83–4: “Item a parte totius disiuncti ad suum totum, id est a qualibet categorica alicuius disiunctivae ad disiunctivam, ut ‘Socrates currit; ergo Socrates currit vel Plato disputat’ … Ista ultima pars huius capituli est de locis a concomitantibus substantiam non enumeratis ab auctore. Et auctor et etiam Themistius non enumeraverunt istos locos, quia non sunt proprie dialectici, vel quia faciliter reducuntur ad enumeratos.” See Green-Pedersen, The Tradition of the Topics, 316. 38  Buridan, Summulae de dialectica, trans. Klima, § 6.6.4, 490–1; Summulae de locis dialecticis, ed. Green-Pedersen, 125: “Aliqui etiam dicunt locum a divisione non esse computandum inter locos dialecticos, quia argumentum per ipsum est consequentia formalis. Sed sustinendo Boethium et Themistium et communiter auctorem nostrum, 37



Commentary295

That is, from (p or q) and not-p infer q.

In his commentary on Buridan, Johannes Dorp agrees that the topic from division is a dialectical topic and that the expanded argument Buridan gives is formal. But he continues: But I say that that inference is not founded on the topic from division, but in the rule “from a disjunctive whole with the denial of one part to its other part is a valid inference,” which is a rule of the hypothetical syllogism; and that the inference holds by virtue of that rule.39

The form of Paul’s reasoning seems most similar to what is found in Agostino Nifo, Super libros Posteriorum Analyticorum Aristotelis commentaria (and before that in Buridan and Dorp), namely, modus tollendo ponens (also known as disjunctive syllogism), a form not mentioned (at least, not under the heading locus a toto disiuncto) by Lambert or Albert: For in the divisive method there are two lines of reasoning, one a syllogism, the other an enthymeme. For example, if I wish to show that every man is an animal … the second [line] is an enthymeme in this manner, “Every man is an animal or inanimate, but he is not inanimate, so he is an animal.” For this is an enthymeme, in which one reasons from a disjunctive whole with the denial of one part to the other, or it is some compound reasoning, in which one reasons from a disjunctive whole with the denial of one part to the other.40

qui omnes istum locum enumeraverunt inter locos dialecticos, ego dico quod hoc argumentum est a divisione ‘A est animal, et non est rationale; ergo est irrationale’; et non est ibi consequentia formalis, sed bene efficeretur per additionem istius propositionis ‘omne animal est rationale vel irrationale.’” 39  Dorp, Perutile compendium totius logice Joannis Buridani, r2vb: “Sed dico ultra quod illa consequentia non est fundata in loco a divisione, sed in illa regula: a tota disiunctiva cum destructione unius partis ad alteram eius partem, est bona consequentia, que quidem est una regula sillogismorum ypotheticorum et quod illa consequentia teneat virtute illius regule.” 40  Nifo, Super libros Posteriorum Analyticorum Aristotelis commentaria, lib. 2, fol. 46ra–b: “Nam in diuisiua methodo sunt duo processus, alter syllogismus, alter enthymema. Verbi causa, si voluero probare omnem hominem esse animal … Secundus vero est enthymema hoc pacto: Omnis homo est animal vel inanimal, sed non est inanimal, ergo est animal. Hoc enim enthymema est, in quo proceditur a toto disiuncto cum destructione alterius partis ad alteram, vel est processus quidam hypotheticus, in quo proceditur a toto disiuncto cum destructione alterius partis ad alteram.”

296 Commentary

This rule is given, not under the topical heading, but as a rule for compound propositions, in the Logica oxoniensis and in Paul’s Logica parva: The third rule: From an affirmative disjunction with the denial of one of its disjuncts to its other disjunct is a valid inference. “You run or you sit, but you do not sit; therefore, you run.”41

And again in the Logica magna: The inference, “Either you are yourself or a stick is standing in the corner, and you are not yourself; therefore a stick is standing in the corner,” is formally valid, since it argues from a disjunction, together with the denial of one disjunct, to the other disjunct.”42

That is, from (p or q) and not-p infer q

Nonetheless, Paul’s qualification, sine limitatione precedente, is puzzling. A similar phrase, per limitationem precedentem, is found in Paul’s Quadratura. The issue there is an ambiguity in universal propositions with a disjunctive subject, such as, “Every man or ass is an ass” (omnis homo vel asinus est asinus). The ambiguity does not come across easily in English. In Latin the proposition can mean, “Every man is an ass or an ass is an ass” (with narrow scope of “every” only over “man”) or, “Everything which is a man or an ass is an ass” (with wide scope of “every” over both “man” and “ass”). Just as we have done with “which is” in English, Paul employs the phrase quod est to mark the ambiguity, separating omnis from homo vel asinus to force the distribution of omnis to cover the whole subject homo vel asinus rather than just its first part.43  Paul of Venice, Logica parva, ed. Perreiah, 64. For the Logica Oxoniensis, see Heytesbury, Sophismata asinina, 563, or William of Osma, De consequentiis, 30. 42  Paul of Venice, Logica magna, Capitula de conditionali et de rationali, 96–7: “Tu es tu vel baculus stat in angulo, et tu non es tu; igitur baculus stat in angulo. Consequentia ista est formalis, a disiunctiva cum destructione unius partis ad aliam partem.” 43  See Paul of Venice, Quadratura, Part IV, chap. 9 (B, fol. 113ra; Q, fol. 67rb; V, fol. 148vb): “Nam in ista: omnis homo vel asinus est asinus, non distribuitur totum disiunctum, licet sit subiectum, sed solum prima pars. Secunda autem pars patet [corr., probatur B Q V] ex isto exemplo: omne quod est homo vel asinus est homo [vel asinus est homo add. V]. Totum enim disiunctum distribuitur ratione contractionis facte a relativo. Unde [om. Q] in hac propositione: omnis homo animal est, non distribuitur ly animal, et tamen dicendo: omnis homo qui est animal est [non – est om. B], distribuitur ly animal propter contractionem relativi. Et ita dicitur in proposito quod non distribuitur secunda pars extremi hypotetici nisi contrahatur per limitationem precedentem.” 41



Commentary297

“A is true or false” is similarly ambiguous. It can mean, “A is what is true or false” (with a disjunctive, compound predicate) or, “A is true or A is false” (with two separate predicates). Only construed the second way, as a propositional disjunction, is the inference “from the disjunctive whole” (de toto disiuncto) valid, that is, absent the restricting prefix “what is.” But presumably what the authors of the ninth opinion meant was that an insoluble is what is true or false but not true and not false. § 1.10 Paul’s tenth solution is arguably Bradwardine’s own (his ninth), though Spade (The Medieval Liar, 82) expresses uncertainty as to whether the tenth opinion is indeed Bradwardine’s. § 1.10 certainly cites Bradwardine’s definition of insolubles. See Bradwardine, Insolubilia, § 2.1. As mentioned in our introduction (§ 4), Bradwardine’s discussion of the insolubles, written in the early 1320s, produced a seismic change in subsequent treatments. Hitherto, the prevailing opinion had been that of the restrictivists. Subsequently, although not many adopted Bradwardine’s own theory, few (Walter Segrave, author of the fourteenth opinion discussed by Paul, was one) defended the restrictivist view while many other proposals were framed in response to Bradwardine’s account. His account turned on two major proposals: first, Bradwardine took a familiar account of truth, namely, that a proposition is true if and only things are wholly (or only) as it signifies, and observed that insolubles might signify more than at first appears; secondly, he held that signification is closed under implication, so that a proposition signifies anything that follows from anything it signifies (his second postulate). On the basis of these two ideas (plus a commitment to bivalence in the form, “Every proposition is either true or false and not both,” his first postulate), he was able to give a subtle argument to show that every proposition which signifies that it itself is false, or that it is not true, also signifies that it is true. See Bradwardine, Insolubilia, § 6.5.2. Consequently, all such propositions (and so, arguably, all insolubles) are false, since things cannot be wholly as they signify, that is, both true and false. Note Translation: “For in ‘Every man or ass is an ass,’ the whole disjunction is not distributed, even though it is the subject, but only the first part. But the second part is clear from this example: ‘Everything which is a man or ass is a man.’ For the whole disjunction is distributed by reason of the restriction made by the relative. Hence in this proposition, ‘Every man an animal is,’ ‘animal’ is not distributed, but in saying, ‘Every man who is an animal exists,’ ‘animal’ is distributed on account of the restriction of the relative. And thus it is said in the first case that the second part of the compound extreme is not distributed unless it is restricted by the prefixed determination, ⟨that is, ‘which is’⟩.”

298 Commentary

that Bradwardine agrees that the inference rejected in § 1.10 is not valid. See Bradwardine, Insolubilia, § 7.11. § 1.11 Spade (The Medieval Liar, 82) identifies Paul’s eleventh solution as that of Albert of Saxony, writing in Paris in the 1350s. Albert’s solution was similar to Bradwardine’s, but closer to, if not identical with, the solution that Buridan proposed in the 1330s but later rejected (or rather, revised). Following the early Buridan, Albert held that all propositions (not just insolubles, as Bradwardine had argued) signify their own truth. However, Buridan firmly rejected the idea of a propositional correlate of the proposition distinct from correlates (significates and supposita) of the terms.44 But Albert seems to have accepted some form of propositional significate in his Questions on the Posterior Analytics.45 Whether or not Albert himself used the term significatum adequatum for it, it sounds similar to Richard Billingham’s notion of the modus rei (state of a thing, or state of affairs), attacked by Paul in the first chapter of the treatise “On the Signification of a Proposition” in his Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis.46 If this is right, it confirms Spade’s attribution of the eleventh solution to Albert. Both Albert and Buridan give the rather simplistic argument in this paragraph for this claim, Albert in the chapter on insolubles in his Perutilis logica,47 Buridan some fifteen or more years earlier in his Questions on the Posterior Analytics.48 The argument turns on two premises: first, that every affirmative proposition signifies that what its subject and predicate supposit for are the same (and a negative proposition that they are different); secondly, that an affirmative proposition is true if what its subject and predicate supposit for are the same (otherwise false) while a negative is true if what its subject and predicate supposit for are different (otherwise false). The second premise, the nominalist account of truth, was famously proposed by Ockham in the second book of his Summa  See, for example, Klima, John Buridan, § 10.2, 219–21.  See Nuchelmans, Theories of the Proposition, § 14.2.3, 240–2. 46  For the attribution of the view to Billingham, see Bertagna, “Ferrybridge’s Logica,” 33 and Cesalli, “Propositions,” 261. 47  Translated in Albert of Saxony, “Insolubles,” 340–1. For the Latin text from the Perutilis logica, see Albert of Saxony, Logik, 1102–04. He repeats this position in his Questions on the Posterior Analytics, qu. 11, for which see Albert of Saxony, Albert of Saxony’s Twenty-Five Disputed Questions on Logic, Appendix II, 358. 48  Cited in the introduction (§ 4). See also Read, “The Liar Paradox from John Buridan Back to Thomas Bradwardine,” § 3. 44 45



Commentary299

logicae, an account taken up by Buridan (see, for example, Summulae de dialectica, Sophismata, chap. 2, fourteenth conclusion, 858–9; Summulae de practica sophismatum, 45–6). The first premise perhaps just turns on the simple idea that in a proposition of the form “A is B,” what is asserted is that A (that is, the referent of “A”) is the same as B (that is, the referent of “B”), and similarly for “A is not B.” The argument as a whole (and the idea how to solve insolubles) may have its origin in Girald Odo’s diagnosis of the liar paradox: [T]he proposition, “I say a falsehood” … has four things wrong with it … The fourth, where the difficulty is lurking, is that it signifies the predicate to be united and not united with the subject at the same time. This is clear because any affirmative signifies union of this sort, and this proposition is affirmative, so it signifies the predicate to be united with the subject. But the predicate “falsehood” signifies in an affirmative proposition that the predicate is not united with the subject to which it is attributed. Thus it is here because the proposition is affirmative and its subject supposits for the whole, and its predicate denotes that whole. So it signifies by the form of the expression that the extremes are united, and by the sense of the predicate that they are not united.49

However, Buridan will have been strongly opposed to many of Odo’s ideas, for Odo was as much a realist as Buridan was a nominalist about universals and worldly correlates of predicates and propositions. Much of § 1.11 is identical to the opening lines of Peter of Mantua, Insolubilia, O2rb–va. It is perhaps surprising that Paul does not discuss Buridan’s later account of insolubles. However, Buridan’s ideas were influential in the late fourteenth century mostly through his successors, such as Albert of Saxony and Peter of Ailly, as he gained recognition only at the turn of the century through texts such as Dorp, Perutile compendium totius logice Joannis Buridani (written in the 1390s, around the same time as Paul’s treatise).  Odo, Logica, 396–7: “[H]ec propositio ‘ego dico falsum’ … habet has quatuor malitias … Quarto vero, ubi latet lepus, notat predicatum uniri et non uniri subiecto simul et semel. Quod patet quia: quelibet affirmativa notat huiusmodi unionem; hec autem propositio est affirmativa; quare notat predicatum uniri cum subiecto; sed istud predicatum ‘falsum’ notat non uniri predicatum cum subiecto in propositione affirmativa, cui attribuitur. Sic autem est hic quoniam hec propositio est affirmativa et eius subiectum supponit pro tota ea, et eius predicatum denotat eam totam. Quare ipsa notat ex forma enuntiandi extrema uniri, et ex ratione predicati notat ea non uniri.” Odo was writing in Paris in the early 1320s. His own solution is that insolubles are implicitly contradictory, and consequently false.

49

300 Commentary

reference in the first sentence is to Logica Magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis (chap. 1, 4–15). Ex eadem in the last sentence has been expanded to ex eadem ratione (“for the same reason”). But it could be short for ex eadem opinione (“according to the same opinion”). § 1.11.2 We have translated consequentia est bona (literally, “the inference is good”) as, “the inference holds,” for to be valid is what it is for an inference to be a good one. It does not require that the premises are true, only that the conclusion follows validly from the premises. When Paul writes that propositions are convertible (convertuntur), he does not necessarily mean that they are equivalent (see, e.g., § 2.3.7), though arguably that is true here. Strictly, in Aristotelian logic, the conversion of a proposition requires that the subject and predicate are interchanged and the quality possibly changed. See Smith, “Aristotle’s Logic,” § 5.2. But Paul uses the terms convertere and convertibilis more broadly. For the argument to work, it is crucial that the premise is an insoluble, so eodem demonstrato (literally, “referring to the same”) must mean that both occurrences of hoc (“this”) refer to the premise, which must be the first, so that “This is not true” will be an insoluble. Then, according to Albert’s opinion, it will signify both that it is not true and that it is true. § 1.11.3 Again, eodem demonstrato must mean that both occurrences of hoc refer to the first proposition in order for that proposition to be an insoluble. The problem here is how to contradict an insoluble on Albert’s (or for that matter, Bradwardine’s) theory. On their theories, if the first proposition is an insoluble, it signifies both that it is false and that it is true, so simply saying, “This is not false” will not serve. Similarly, if Bertrand Russell was right about the analysis of “The King of France is bald,” “The King of France is not bald” will not contradict it (Russell, “On Denoting,” 485). Russell realized (ibid., 490), just as Bradwardine (Insolubilia, § ad 7.3) before him, that the scope of “not” is ambiguous, so that properly to express the contradictory of the insoluble, “This is false,” “not” must be given widest scope: “Not: this is false,” where the sub-clause “this is false” is expanded to its analysans: “Not: this is false and this is true,” that is, “This is not false or this is not true,” equivalently (given bivalence), “This is true or this is false.” The contradictory of the self-contradictory insoluble “This is false” is then a tautology. § 1.11.1 The



Commentary301

Heytesbury’s Opinion (The Medieval Liar, 82) identifies this as Heytesbury’s solution developed in the insolubles chapter of his Regulae solvendi sophismata (“Rules for Solving Sophisms”). However, some aspects of the account here correspond to a text whose anonymous author Spade dubs “pseudoHeytesbury” (The Medieval Liar, 35) and which was edited in Pironet, “William Heytesbury and the Treatment of Insolubilia.” Pironet writes (265): “the text is not by Heytesbury nor does it follow Heytesbury’s exact principles. The identity of the author is still a mystery.”50 The whole discussion is framed in terms of the theory of obligations.51 Heytesbury distinguishes the scenario (casus) of an insoluble from the insoluble proposition itself:

§ 1.12.1 Spade

[A] scenario of an insoluble is one in which mention is made of some proposition such that if in the same scenario it signifies only as its words commonly suggest, from its being true it follows that it is false and vice versa … [A]n insoluble proposition is one of which mention is made in some insoluble scenario such that if in the same scenario it signifies only as its words commonly suggest, from its being true it follows that it is false and vice versa.52

Pironet (“William Heytesbury and the Treatment of Insolubilia,” 256) suggests translating the Latin term precise (or praecise in classical orthography) as “only,” and we will follow her in this as better reflecting Heytesbury’s (and Paul’s) meaning. Paul includes precise in his list of exclusive terms in Part I (see E, fol. 34ra). He distinguishes five distinct

 In fact, it is the basis of the solution put forward in the Logica parva. See our introduction, § 5, and Logica parva, trans. Perreiah, 237–56, ed. Perreiah, 128–50. The significant difference from Heytesbury’s solution is that Heytesbury declines to specify what the additional signification of insolubles is, whereas pseudo-Heytesbury and the Logica parva specify that it also signifies that it is true (Pironet, “William Heytesbury and the Treatment of Insolubilia,” 276, 292; Logica parva, trans. Perreiah, 241, ed. Perreiah, 133). It is called “the modified Heytesbury solution” in Spade and Read (2021), § 3.5. 51  The penultimate treatise of the Logica magna, immediately preceding the current treatise on insolubles, is on obligations. It was edited and translated in Paul of Venice, Logica magna: Tractatus de obligationibus. On obligations, in addition to the introduction, § 3, see Dutilh Novaes and Uckelman, “Obligations,” and Yrjönsuuri, Obligationes. 52  Already cited in the introduction, § 4. For an alternative translation, see Heytesbury, Insolubles, trans. Spade, 46. Spade and others often transliterate casus as “casus” or “case.” 50

302 Commentary

uses of exclusive terms, depending on the scope of the exclusion the term has within the proposition. He writes of the fourth type: An exclusive proposition of the fourth type is one in which the exclusive term governs only part of the predicate or subject; e.g., “I see only Socrates” or “Proposition A signifies only as is the case.”53

In his Logica parva, Paul writes in explanation of the phrase “non ponendo praecise”: [O]ne ought to add in the antecedent an exclusive or exceptive expression, i.e., “A signifies praecise that no proposition is true” or “A does not signify save (nisi) that no proposition is true.”54

Both Pironet (“William Heytesbury and the Treatment of Insolubilia,” 256) and Spade (The Medieval Liar, 112, and Heytesbury, Insolubles, trans. Spade, 46) render pretendunt, literally “extend,” as “pretend,” but that does not capture its use in Heytesbury and his successors, which, we think, is “to indicate, signify or suggest.” Note that Paul omits communiter from his use of the phrase. There seem to be two textual traditions recording Heytesbury’s theory of insolubles. Spade (“The Manuscripts of William Heytesbury’s Regulae solvendi sophismata”) lists 33 manuscripts in total (and three incunabula) of Heytesbury’s Regulae, commenting on their sheer variety (ibid., 275): “it is not clear that any of the extant manuscripts or incunabula is good enough to be used as a ‘base’ text … [M]y collation … shows that … the textual tradition is hopelessly contaminated.” §§ 1.12.1 and 1.12.1.1 are also found in Peter of Mantua (Insolubilia, O3rb–va). However, since § 1.12.1.2 is not found in Peter’s text, it may be that they were working from a common source, rather than that Paul was copying Peter’s text. Paul, along with Peter of Mantua and pseudo-Heytesbury, states Heytesbury’s account of an insoluble as one in which “it follows that it is true and that it is false.” Pozzi, in his edition of Heytesbury (ed. Pozzi, 236) gives the reading found in Peter of Mantua and Paul: ­sequitur  Cited in Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 260: “Propositio exclusiva quarti ordinis est illa in qua dictio exclusiva determinat solum partem predicati aut subiecti ut ego video tantum sortem, a propositio significat precise sicut est.” 54  Logica parva, trans. Perreiah, 238; ed. Perreiah, 129: “[D]ebuit addi in antecedente dictio exclusiva vel exceptiva, videlicet ‘A significat praecise quod nulla propositio est vera’ vel ‘A non significaret nisi quod nulla propositio est vera.’” 53



Commentary303

eam esse veram et ⟨etiam⟩ eam esse falsam. But oddly, in his summary of Heytesbury’s position, Pozzi writes: The scenario of an insoluble is a situation in which a proposition is posited such that, if it signifies only as its terms commonly suggest, from the truth of the proposition together with the scenario the falsity of that proposition follows and vice versa in a circular deduction.55

Pozzi (Heytesbury, Insolubilia, ed. Pozzi, 212) is following only two of the same manuscripts (and the 1494 edition) as Spade, but neither mentions any variants at this point.56 One might think the difference insignificant, since the two readings could be seen as equivalent, on the grounds that “A is true if and only if A is false” is equivalent to “A is true and false.” But this holds only on certain assumptions, notably bivalence (that A is either true or false). Indeed, in the context of insolubles and paradox, those assumptions might themselves be in question, as bivalence, and indeed excluded middle, are for Swyneshed.57 Heytesbury presents his solution in the form of five rules. § 1.12.1 seems to conflate the first and second rules. Heytesbury writes: ⟨First rule:⟩ If someone constructs a scenario of an insoluble, either he specifies how that insoluble should signify or he does not. If not, then when the insoluble is proposed, one should respond to it exactly as one would respond when the scenario is not assumed … Secondly, notice that if a scenario of an insoluble is posited, and together with that it is assumed that the insoluble signifies only as its terms commonly suggest, the scenario may in no way be admitted.58  Heytesbury, Insolubilia, ed. Pozzi, 205: “Il caso di un insolubile è la situazione in cui si pone in essere una proposizione tale che, se significa precisamente come i suoi termini comunemente pretendono, dalla verità della proposizione con il caso segue la falsità della stessa proposizione e viceversa in una deduzione circolare.” 56  On the absence of “and vice versa” in Peter’s formulation, see also Strobino, “Truth and Paradox in Late XIVth-century logic,” § 3.1.2, 488 n. 24. 57  See Read, “Paradoxes of Signification,” 350. 58  Pironet, “William Heytesbury and the Treatment of Insolubilia,” 279–80: “Si fiat casus de insolubili, aut ponitur qualiter illud insolubile debeat significare aut non. Si non, proposito illo insolubili, respondendum est ad illud omnino sicut responderetur non supposito illo casu … Secundo est advertendum quod si ponatur casus de insolubili, et cum hoc supponatur quod illud insolubile praecise significet sicut termini illius communiter praetendunt, casus ille nullatenus admittatur.” For an alternative translation, see Heytesbury, Insolubles, trans. Spade, 47–8. 55

304 Commentary

Note that the first rule does not say that the proposition should be doubted; that is an addition made by pseudo-Heytesbury: [I]f a scenario of an insoluble is put forward and it is not specified how that insoluble should signify, one should respond to it as irrelevant. For example, supposing that Socrates says the proposition “Socrates says a falsehood” and no other, then when the proposition “Socrates says a falsehood” is proposed, one should respond by doubting it just as outside the scenario. The reason is that it is consistent for it to be true and likewise false; so it is consistent with the scenario that Socrates says a truth and a falsehood; and so since it is irrelevant to the scenario, one should not do otherwise than to respond to it as it was outside the scenario.59

The second rule reappears in Paul’s summary in § 1.12.2 below. Heytesbury’s solution is remarkably like Bradwardine’s. Like Bradwardine, he suggests that insolubles signify more than at first appears. In his third rule, Heytesbury says: Thirdly, if one constructs a scenario for an insoluble, and together with that one assumes that the insoluble signifies as its terms suggest, but not only in that way, then when this scenario is accepted, the insoluble has to be granted as following, whenever it is proposed, but one should deny that it is true, as incompatible.60

The phrase “as following” refers to the practice of obligations, in which propositions following from what has already been granted (and the opposite of what has been denied) should be granted, and “as ­incompatible”  Pironet, “William Heytesbury and the Treatment of Insolubilia,” 291: “[S]i ponatur casus de insolubili, et non ponatur qualiter illud insolubile debeat significare, supposito illo insolubili, respondendum est sicut ad impertinens. Verbi gratia, supposito quod Socrates dicat talem propositionem ‘Socrates dicit falsum’ et nullam aliam, tunc proposita ista propositione ‘Socrates dicit falsum,’ respondendum est ad istam dubitative sicut ante casum. Et causa est haec quia stat illam esse veram et similiter falsam; ideo cum casu stat quod Socrates dicit verum et falsum; et ideo cum illa sit impertinens casui, non est aliter respondendum ad illam quam fuerat ante casum.” Hunter (cited ibid., 303) and Holland ( John of Holland: Four Tracts on Logic, 128) also say that the insoluble should be doubted. 60  Heytesbury, Insolubilia, ed. Pozzi, 238: “Tertio si fiat casus de insolubili, et cum hoc supponatur quod illud insolubile significet sicut termini ipsius [praecise] praetendunt non tamen sic praecise, admisso illo casu, concedendum est illud insolubile quocumque loco proponatur tamquam sequens, et negandum est illud esse verum tamquam repugnans.” Pozzi, rightly in our view, brackets the first occurrence of praecise as a mistaken insertion. See also Pironet, “William Heytesbury and the Treatment of Insolubilia,” 285; for an alternative translation, see Heytesbury, Insolubles, trans. Spade, 49. 59



Commentary305

similarly refers to the obligation to deny what is incompatible with what has been granted and the opposite of what has been denied. However, unlike Bradwardine, Heytesbury declines to say what that additional signification is, resorting to the conventions of obligations theory to justify that reluctance: [S]upposing that Socrates is saying only the proposition “Socrates says a falsehood” and that it signifies that Socrates says a falsehood, but not only that, the proposition “Socrates says a falsehood” should be granted when it is proposed … But if someone asks in this scenario what the proposition that Socrates said signified other than that Socrates is saying a falsehood, I say to him that the respondent does not have to resolve or answer that question. For from the scenario it follows that the proposition signifies other than that Socrates is saying a falsehood, but the scenario does not specify what that is; hence the respondent does not have to answer that question any further.61

On pain of contradiction (paradox) the proposition cannot signify only what it appears to say, so it must signify something else. Then “Socrates says a falsehood” has to be granted, but that it is true has to be denied: [T]he insoluble has to be granted as following [from the positum etc.] whenever it is proposed, and that it is true has to be denied as inconsistent [with the positum etc.].62

Just because it is granted does not mean it is true. Falsehoods have to be granted in an obligation, when they follow from a false positum. § 1.12.1.1 Heytesbury adds non tamen praecise, that is, “but not only ⟨in that way⟩,” in his third rule, cited above, as does pseudo-Heytesbury (see Pironet, “William Heytesbury and the Treatment of Insolubilia,” 292). This is  Pironet, “William Heytesbury and the Treatment of Insolubilia,” 286: “Verbi gratia, supposito quod Sortes dicat solum hanc propositionem Sortes dicit falsum et quod illa sic significet non tamen praecise sic scilicet quod Sortes dicit falsum, concedenda est haec propositio Sortes dicit falsum, cum proponitur … Si autem quaeratur in illo casu quod significavit illa propositio dicta a Socrate aliter quam quod Socrates dicit falsum, huic dicitur quod respondens non habet istam quaestionem solvere sive determinare: quia ex casu sequitur quod illa propositio aliter significat quam quod Socrates dicit falsum, sed casus non certificat quid illud sit; ideo non habet respondens quaesitum illud ulterius determinare.” For an alternative translation, see Heytesbury, Insolubles, trans. Spade, 49–50. Heytesbury, Insolubilia, ed. Pozzi, 240, has a slightly different reading. 62  Pironet, “William Heytesbury and the Treatment of Insolubilia,” 285: [C]oncedendum est illud insolubile quocumque loco proponatur tamquam sequens, et negandum est illud esse verum tamquam repugnans.” 61

306 Commentary

crucial to Heytesbury’s view. Peter of Mantua, Insolubilia, O3va, does the same, contrasting the second with the third rule: Secondly, if it is specified that an insoluble signifies only as the terms suggest, the scenario should be rejected and not admitted. But if it is specified that the insoluble signifies as the terms suggest, without specifying only ⟨in that way⟩, the insoluble should be granted as following (tamquam sequens) and it should be denied that it is true, if it is proposed that it is true.63 § 1.12.1.2 This paragraph does not appear in Peter of Mantua, Insolubilia. § 1.12.2 Similar to a passage in Peter of Mantua, Insolubilia, O3vb. § 1.12.2.1 Again, similar to a passage in Peter of Mantua, ibid. § 1.12.3 This is Heytesbury’s second rule, cited in § 1.12.1.1. For the relation

between precise, principaliter, primo, and adequate, see the comment on § 1.8.1 above and our introduction, § 4. § 1.12.3.1.1 Paul made a similar objection to the eighth opinion in § 1.8.2. § 1.12.3.1.2 For the disproof of the first way, see § 1.11.2–3. In the disproof of the second way, note that B is the primary sense (i.e., that this is a falsehood) and A is the other, secondary sense. § 1.12.3.1.3 For the proof that the primary sense (that is, the exact significate) is generated from the exact significates of the parts, see Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, “On the Significate of the Proposition,” ninth way, thesis 5, 196–7. § 1.12.3.2 Here Paul refers back to § 1.12.3, where he proposed two possible senses of precise (“only”). Having explored the first (that it is simply taken exclusively), he turns to the second. He appears here to treat “principally,” “primarily,” and “exactly” as equivalent. Cf. the passages from the Quadratura cited in the comment on § 1.8.1 and in our introduction, § 4. § 1.12.3.2.3 Within an obligation, the scenario (casus) can specify, among other things, the signification of certain propositions and, in particular, their signification outside the time of the obligation, that is, when they are irrelevant (impertinens).  Cited in Strobino, “Truth and Paradox in Late XIVth‑Century Logic,” 489: “Si autem ponatur quod insolubile significet precise sicut termini pretendunt, casus est reiciendus et non admittendus. Sed si ponitur quod insolubile significet sicut termini pretendunt, non ponendo ‘precise,’ concedendum est insolubile tamquam sequens et negandum est quod sit verum, si proponitur esse verum.” 63



Commentary307

to Aristotle (De interpretatione, chap. 7, 17b16–25) the contradictory of “Every A is B” is “Not every A is B,” or equivalently “Some A is not B.” Aristotle writes:

§ 1.12.4 According

Let us call an affirmation and a negation which are opposite a contra­ diction. I speak of statements as opposite when they affirm and deny the same thing of the same thing (De interpretatione, tr. Ackrill, chap. 6, 17a32–34).

For example, Aristotle reads “Every A is B” as “B belongs to every A,” so its denial reads “B does not belong to every A,” that is, “Not every A is B.” Thus the contradictory is formed by a syntactic transformation, and is not defined semantically. Jones, “Truth and Contradiction in Aristotle’s De Interpretatione 6–9,” following Whitaker, Aristotle’s De Interpretatione, argues that in consequence, Aristotle denies the rule of contradictory pairs, according to which contradictories must take opposite truth-values. Consequently, it is possible for some authors, like Swyneshed and Paul, to claim that every proposition has a contradictory (denying of the subject what the other affirms of it) and that an insoluble and its contradictory are both false. See Swyneshed, Insolubilia, ed. Spade, 189; see further §§ 2.3.2, 2.4.5, and ad 3.1 below. § 1.12.5 The reference is to Heytesbury’s third rule, cited in discussion of § 1.12.1 above. The argument which runs, “The inference holds, is known by you to be good, the premise should be doubted by you, so the conclusion should not be denied by you” is an instance of what Kretzmann called a “disputational meta-argument”; see Kilvington, Sophismata, trans. Krtezmann, 31. The argument is presented more fully in Peter of Mantua, Insolubilia, O4va: Against the third rule of this opinion … one argues as follows: let it be put forward that Socrates is saying, “A falsehood is said” and that no other proposition except this one, or part of it, is uttered by anyone else, [and] that [the proposition] signifies that a falsehood is being said—however, not only that, as this view likes [to stipulate] … Next, “A falsehood is said” is proposed and it is granted according to this view. In addition to that, “This is false” is proposed, which is also granted according to this view. But to the contrary, ⟨consider this inference⟩: this proposition, “A falsehood is said” principally signifies that there is a God, hence this proposition is necessary. The inference is valid, known to be such and so on; and you ought to be in doubt about the premise; hence you ought not to deny its conclusion. The i­nference

308 Commentary

holds according to this view; and you ought to be in doubt about the premise according to the whole scenario; hence you ought not to deny the conclusion.64

We have chosen to follow Paul’s source, Peter of Mantua, in keeping the example proposition, “A falsehood is said,” constant throughout the paragraph. The point is that, in the context, the sense of “A falsehood is said” is in doubt, since it has not been specified. So one has to doubt (that is, express ignorance) whether it signifies that there is a God. But if it does, then it is a necessary proposition; the inference is valid. In that case, one cannot deny the conclusion, for if one denies the conclusion of a (single-premise) valid inference, one is committed to denying its premise, and we have agreed that we should doubt it. But Heytesbury’s third rule tells us to deny that it is true, and consequently that it is necessary. Contradiction. Yrjönsuuri (Obligationes, 141) claims that Kilvington’s meta-argument is incompatible with the obligational rules that Heytesbury employs. Indeed, in his Regulae, Heytesbury (“The Verbs ‘Know’ and ‘Doubt,’” 449) explicitly rejects Kilvington’s style of argument. However, this assumed incompatibility is not clear, since within the responsio antiqua, responses can be changed. Before the conclusion of some accepted inference is denied, its premise may have been doubted. But it is certainly true that by Burley’s rules, that doubt cannot be maintained once the conclusion has been denied, for it is now pertinens repugnans. § 1.12.5.1 Note that, as mentioned in the commentary on § 1.8.1, “man” does not only (precise) signify man, but also Socrates and Plato, albeit confusedly, so “A man is running” signifies, among many other things, that Socrates is running.

 Cited, with slightly different translation, in Strobino, “Truth and Paradox in Late XIVth‑Century Logic,” 497: “Contra autem tertiam regulam istius opinionis … sic arguitur: quia ponatur quod Sortes dicat istam ‘falsum dicitur,’ et non proferatur ab aliquo alia propositio nisi ista aut eius pars, que significet falsum dici non tamen precise, sicut illi positioni placet … Deinde proponitur ista ‘falsum dicitur’ et conceditur secundum istam positionem. Et ultra proponitur ista ‘hec est falsa,’ que etiam conceditur secundum istam positionem. Sed contra quia sequitur: hec propositio ‘falsum dicitur’ principaliter significat deum esse, igitur ista propositio est necessaria. Consequentia est bona scita esse talem etc.; et antecedens est a te dubitandum; igitur consequens eius non est a te negandum. Patet consequentia secundum istam positionem; et antecedens cum toto casu est dubitandum; igitur consequens non est a te negandum.”

64



Commentary309

Peter of Ailly’s Opinion (The Medieval Liar, 83) notes that this is Peter of Ailly’s solution. Peter’s treatise is one of a pair, on Concepts and Insolubles, written in Paris in 1372. Whereas Bradwardine and Albert of Saxony claimed that insolubles signify both that they are false and that they are true, Peter argued that they are ambiguous. Moreover, he claimed that only written and spoken propositions (or mental propositions that signify by imposition, not naturally) can be insoluble. He distinguishes strictly mental propositions (or, as Spade translates it, “properly so called”—but it is not that they are properly called propositions, or mental propositions, but that they are strictly, or properly, called mental), which signify naturally, from loosely mental propositions (which Spade renders as “mental propositions improperly so called”), which do signify by imposition—see comment on § 1.13.2. They can be insoluble too. In strictly mental propositions there cannot be any type of self-reference; in that respect, at the strictly mental level Peter’s solution is restrictivist (see § 2.2.6.1). For Peter, all self-referring propositions are false (see § 1.13.3.1). What insolubles signify are (strictly) mental propositions, but equivocally, either the mental proposition which says that the insoluble is false, or the (mental) proposition which says that the first mental proposition, saying that the insoluble is false, is itself false. Thus Peter says that insolubles are both true (in one sense) and false (in the other). Peter develops his view in opposition to a view he attributes to Gregory of Rimini, like his own, but claiming that spoken and written insolubles signify a conjunctive mental proposition composed of the two mental propositions between which, Peter says, the insolubles equivocate. See Spade and Read, “Insolubles,” § 4.2, and Spade’s introduction to his translation of Peter of Ailly, Concepts and Insolubles. § 1.13.1 Paul’s first conclusion combines Peter’s third and fourth assumptions, which had stated that no created thing (including angelic intelligences) but only the divine intellect can be a proper and distinct formal cognition of itself: see Peter of Ailly, Concepts and Insolubles, trans. Spade, 72 (§§ 275–6). The image of the king is taken from Peter’s discussion of his first assumption. The distinction between the formal and objective self-comprehension or knowledge seems to be a facet of the more general epistemological differentiation between immediate and mediate knowledge. For Peter of Ailly’s distinction between signifying formally and as an object (obiective, translated by Spade as “objectively”), see 72 (§ 274):

§ 1.13.0 Spade

310 Commentary

“to signify something objectively is nothing else than to be an object of some formal cognition … to signify formally is nothing else than to be a formal cognition of some object.” Spade (n. 733 on p. 147) likens Peter’s usage to Descartes’ in Meditations III: “‘formal reality’ is the reality a thing really is or has, while ‘objective reality’ is the reality it represents.” Note that “objectively” (obiective) does not have its modern sense here (which it acquired around 1800), but means (literally) “as an object (of thought, or of sight, etc.).” On this point, see Bos and Read, Concepts, § 4.3.1, 68–9. Concerning Paul’s argument that there is no more reason in favour of one option rather than its opposite, see Spade’s comment (Peter of Ailly, Concepts and Insolubles, trans. Spade, 117 n. 318) that a similar argument of Peter’s “is a good example of a common form of argument in late medieval authors … Since there is no reason to prefer one alternative to the other, the two must be treated alike; hence, one must either grant them both or grant neither.” § 1.13.2 Paul’s second conclusion conjoins Peter’s first and second conclusions: see ibid., 73, 78 (§§ 282, 302). Here we have amended Paul’s text in line with Peter’s. For the distinction between strictly and loosely mental propositions, see ibid., 36–7, §§ 94–96. Spade observes that Peter seems to take the distinction directly from Gregory, who himself traces it to Augustine and Anselm: Mental statements are of two kinds. For one kind are those mental statements which are images or likenesses of spoken statements derived from exterior sounds in the mind, or made up by the mind itself … and these ⟨mental words⟩ are not the same in all men. But some are in Greek, others in Latin, yet signifying the same … The other kind indeed are mental statements which are not likenesses of any sounds, and do not differ among men according to the difference of sounds, but these ⟨mental words [sc. concepts]⟩ are specifically the same for all, and naturally signify the same thing that spoken ⟨sounds⟩, which are subordinated to these ⟨mental words⟩ in signifying, signify by institutional imposition. They are those words which belong to no language, and precede spoken words sounding out loud and mental words of the first sort, as Augustine says.65  Gregory of Rimini, Super primum Sententiarum, prol., qu. 1, art. 3, 30: “Duplex est genus enuntiationum mentalium: quoddam enim est earum, quae sunt vocalium enuntiationum imagines vel similitudines ab exterioribus vocibus in animam derivatae vel per ipsam animam fictae … et istae non sunt eiusdem rationis in omnibus hominibus, sed aliae sunt in Graeco, aliae in Latino, etiam idem significantes … Quoddam vero est 65



Commentary311

The reference is to Augustine’s De Trinitate: Whoever, then, can understand the word, not only before it sounds, but even before the images of its sound are contemplated in thought—such a word belongs to no language, that is, to none of the so-called national languages, of which ours is Latin … [T]his word cannot be uttered in sound nor thought in the likeness of sound, such as must be done with the word of any language; it precedes all the signs by which it is signified. 66

Gregory also refers to a passage in Anselm’s Monologion: We can speak of a single object in three ways. For we may speak of it either (1) by perceptibly employing perceptible signs (i.e., signs which can be perceived by the bodily sense) or (2) by imperceptibly thinking to ourselves these same signs, which are perceptible outside us, or (3) neither by perceptibly nor by imperceptibly using these, but by inwardly and mentally speaking of the objects themselves by imagining them.67 § 1.13.2.1 Paul

here conjoins the first three of Peter’s four corollaries to the first (and second) conclusions: see Peter of Ailly, Concepts and Insolubles, 77–8 (§§ 297–300). § 1.13.2.2 Peter’s fourth corollary to the first (and second) conclusions: see ibid., 78 (§ 301). § 1.13.3 The first, second, and fourth of Peter’s four further conclusions: see ibid., 80–1 (§§ 313, 318, 322). We follow Ashworth (“Review of genus enuntiationum mentalium, quae nullarum sunt similitudines vocum, nec secundum illarum diversitatem in hominibus diversificantur, sed eadem sunt secundum speciem apud omnes, id ipsum naturaliter significantes quod vocales eis subordinatae ad significandum ad placitum et per institutionem significant. Et istae sunt illa verba, quae nullius linguae sunt. Et vocalia verba exterius sonantia et etiam mentalia primo modo dicta praecedunt secundum sententiam Augustini.” 66  Augustine, On the Trinity, trans. McKenna, 186, 188; Augustine, De Trinitate, ed. Mountain, XV.x–xi, 485, 488: “Quisquis igitur potest intellegere verbum non solum antequam sonet, verum etiam antequam sonorum eius imagines cogitatione volvantur (hoc est enim quod ad nullam pertinet linguam, earum scilicet quae linguae appellantur gentium quarum nostra latina est) … ad illud verbum hominis … quod neque prolativum est in sono neque cogitativum in similitudine soni quod alicuius linguae esse necesse sit, sed quod omnia quibus significatur signa praecedit.” 67  Anselm, Monologion, trans. Hoskins and Richardson, 18–19; ed. Schmitt, chap. 10, 24–5: “Quia rem unam tripliciter loqui possumus. Aut enim res loquimur signis sensibilibus, id est quae sensibus corporeis sentiri possunt sensibiliter utendo; aut eadem signa, quae foris sensibilia sunt, intra nos insensibiliter cogitando; aut nec sensibiliter nec insensibiliter his signis utendo sed res ipsas vel corporum imaginatione vel rationis intellectu pro rerum ipsarum diversitate intus in nostra mente dicendo.”

312 Commentary

Copenhaver’s Translation of Peter of Spain’s Summaries of Logic”) in translating ad placitum as “by imposition” and not, for example, as “by convention” or “arbitrarily.” § 1.13.3.1 See Peter of Ailly, Concepts and Insolubles, 82 (§ 325, penultimate sentence). § 1.13.4 Peter’s “main and last conclusion”: see ibid., 82 (§§ 327–8). The phrase “reflects on itself” (habet reflexionem supra se)—meaning, essentially, selfreference—will be at the heart of Paul’s own theory. See § 2.1.2 below. The phrase occurs in Bradwardine’s definition of insolubles: see the tenth opinion in § 1.10. § 1.13.4.1 Corollaries to the fourth conclusion: see Peter of Ailly, Concepts and Insolubles, 82 (§§ 329–30). § 1.13.5 Peter’s first and second conclusions from chap. 4 of his Insolubles: ibid., 85–6 (§§ 348 and 350). The illustration of the example “This is false” is found at 87, § 353. Peter’s theory says that the spoken proposition also signifies a false mental proposition which says that the true mental proposition to which the spoken proposition corresponds is false. Here is a diagram:

§ 1.13.5.1 The

four corollaries are found ibid., 88–9, 93 (§§ 357, 359, 361–2, 378). The first, that the corresponding mental propositions are not conjoined, is what distinguishes Peter’s theory from Gregory’s. See Spade’s introduction, ibid., 12. A manifold proposition (propositio plures) is one that has more than one meaning or signification or that is subordinated to more than one mental proposition. On the notion of a manifold



Commentary313

proposition, see Nuchelmans, Late-Scholastic and Humanist Theories of the Proposition, 119–24. § 1.13.5.4 Spade (Peter of Ailly, Concepts and Insolubles, 161 n. 980) refers to De sophisticis elenchis, chap. 30, 181a37–39. See also: chap. 17, 176a14–15, translatio Boethii, 38, ll. 9–10: “ergo non oportet ad duas interrogationes unam responsionem dare” (“so it is not necessary to give one response to two questions”). Cf. Hamesse, Auctoritates Aristotelis, 333 n. 20. § 1.13.6 For the “recent writer,” see § 1.13.7 below, discussing the views of Peter of Mantua, who wrote in the late 1380s and early 1390s shortly before Paul. The antipathy of the author of the Logica magna to Peter of Mantua’s ideas forms a strong part of Perreiah’s argument that the attribution of the work to Paul is mistaken. See our introduction, § 6. § 1.13.6.1 Paul’s counterargument against Peter of Ailly should be contextualized within the long-standing dispute about angelic epistemology and psychology, which involved many thirteenth- and fourteenth-century authors such as Thomas Aquinas, Henry of Ghent, Giles of Rome, John Duns Scotus, and William of Ockham.68 The medieval debate on angelic cognition focused on the means through which angelic intelligences comprehend God, themselves, and created things—material as well as immaterial. The intelligences endowed with a “superior nature,” here mentioned by Paul, are the angelic intelligences; Paul admits that these intelligences can be “formal cognitions of themselves and of others” (noticie formales sui et aliorum), namely, that they can have an immediate or direct self-cognition and cognition of other things. Unfortunately, he does not specify whether the noticie formales aliorum comprehend only created things or extend to God too. Paul’s claim is subsequently explained: angelic intelligences perform their cognitive acts immediately or directly (“they do not comprehend through added accidents”). That is, angels do not have a mediate (or objective) knowledge of their object through additional intelligible species—Paul’s “added accidents.”69 § 1.13.6.2 If this was the angels’ mode of cognition, argues Paul, then there would be an infinite regress since the intelligible species (s1) would necessarily be preceded by an additional intelligible species (s2) in order to be comprehended; but then, in its turn s2 would need to be preceded by an added intelligible species s3, and so on. Like authors such as Ockham, 68 69

 We are grateful to Robert Pasnau, who drew our attention to this discussion.  On intelligible species, see Conti, “Paul of Venice,” § 5.

314 Commentary

Paul admits that angelic intelligences have immediate ­comprehension of themselves and of “others” (known formally). Paul specifies that such immediate comprehension proper to angelic intelligences is a distinct and proper comprehension (“some created thing can be a formal conception of itself properly and distinctly”), that is, the object of angelic cognition is comprehended in a precise and well-defined way.70 § 1.13.6.2.2 Again, a reductio argument: the spoken term “being” takes its signification from the mental term “being,” and among its significates is the mental term “being,” even on Peter’s account. But if Peter’s theory prevents the mental term having itself as a significate, that is inconsistent. § 1.13.6.2.3 The sense in which mental composition and division confirm the signification of a term—in this case, the term “being”—is that they rely on that signification. § 1.13.6.3 Once again, this is an ad hominem argument: if the mental proposition A, “Every proposition exists” cannot signify itself (on Peter’s view), then the subordinate proposition B, “A proposition exists” would signify something that the superordinate proposition A did not, which is absurd. Moreover, “This proposition exists,” referring to A, would be a singular of B but not of A, which again is absurd. Propositions like omnis propositio est, falsum est, nullus deus est are tricky to translate idiomatically into English. In general, we translate est occuring without a predicate (secundum adiacens, as the medievals would say, that is, as second component—see comment on § ad 3.1) as “exists”: for instance, “A falsehood exists” for falsum est. This is to give consistency, for when the subject has a determiner, omnis or aliquis or ista, it is necessary to use “exist”: for example, “Every proposition exists” for omnis propositio est and “This proposition exists” for ista propositio est. § 1.13.6.4 Spoken and written propositions are propositions extrinsically, that is, as Paul says, only in virtue of their being subordinate to the corresponding mental proposition. Paul reaffirms the claim that the mental proposition signifies itself here, giving further disproof of Peter’s third conclusion (see § 1.13.3).  On angelic cognition, see Thomas Aquinas, Summa theologiae I, qu. 55, art. 1, in corp.; I, qu. 56, art. 1, in corp.; cf. also I, qu. 87, art. 1, in corp. and ad 2; I, qu. 87, art. 3, in corp.; Quaestiones disputatae de veritate, qu. 8; Henry of Ghent, Quodlibeta V, qu. 14; Giles of Rome, De cognitione angelorum; Ockham, Quaestiones in librum secundum Sententiarum, qu. 12–13. See also Perler, “Thought Experiments.” 70



Commentary315

§ 1.13.6.5.1 Paul’s

argument is ad hominem. On Peter’s theory, the written proposition, “A falsehood exists” (Falsum est) signifies, and so is subordinate to, two mental propositions, one saying that there is a falsehood, the other that the first mental proposition is false. If one proposition is subordinate to another, the first entails the second. But both the written and mental propositions “A falsehood exists” are true, whereas the mental proposition saying that the (first) mental proposition is false is false. Peter’s theory thus has the unacceptable consequence that there is a valid inference whose premise is true and conclusion false:

Paul’s own theory also has the implication that a valid argument can have true premises and a false conclusion—see § 2.3.5. However, Peter would find it unacceptable as an implication of his theory, which is why this is an ad hominem argument. § 1.13.6.5.3 The three mental “dogs” are the mammal, the star, and the fish. Peter of Mantua’s Opinion § 1.13.7 It is

possible that this section (§§ 1.13.7–1.13.7.4) is a later addition, since it is unnumbered, yet discusses another opinion, that of Peter of Mantua, which was arguably formulated only shortly before Paul of Venice was writing the Logica magna (see introduction, §§ 5–6). Note, however, that Paul’s discussion of the eleventh, twelfth, and fourteenth opinions follows closely the first three opinions discussed by Peter of Mantua before Paul presents his own. Whereas the first ten opinions Paul discusses have an English, largely Oxonian, origin, Peter of Man-

316 Commentary

tua’s text draws on Parisian logical ideas, including those of Marsilius of Inghen and William Buser, besides those actually discussed in opinions 11, 12 and 13. Wyclif, who strongly influenced Peter of Mantua, distinguishes three senses of “true” and “false,” including a transcendental sense in which every proposition is true, and truth is equivalent to being: see Spade and Read, “Insolubles,” § 4.1. Peter discusses these three senses, but, as Spade (The Medieval Liar, 83) notes, the idea that a proposition can be true in either the second or the third senses is deemed by Peter of Mantua sufficient to diagnose the insolubles. Our translation of duobus modis as “two senses” follows Strobino’s translation in Strobino, “Truth and Paradox in Late XIVth‑Century Logic,” 504. After the remarks about transcendentals, §§ 1.13.7–1.13.7.2 come almost verbatim from Peter of Mantua, Insolubilia, O5vb–O6ra (as cited in Strobino, “Truth and Paradox in Late XIVth‑Century Logic,” § 3.2, 505 n. 45). Although Paul gives Peter’s account of insolubles as an addendum to Ailly’s, it bears little resemblance to it except that although both Ailly and Mantua cast the solution within the framework of the fallacy of the restricted and unrestricted (secundum quid et simpliciter), Mantua holds that insolubles are true in one sense and false in another. But Mantua rejects the idea—found in Bradwardine, Heytesbury, Albert, Ailly, and others—that explains multiple senses by reference to different significates or mental propositions. Rather, his idea consists, in brief, in giving a different account of truth for self-referential propositions from those that are not self-referential. § 1.13.7.1 At the beginning of this paragraph Paul’s text offers a better reading than Mantua’s, for the latter adds suppositorum after quorum, which does not fit with the context non propter supposita suorum terminorum, quorum [suppositorum] supposita ipsa propositio … (Peter of Mantua, Insolubilia, O5vb). § 1.13.7.2 Strobino (“Truth and Paradox in Late XIVth Century Logic,” 505 n. 45) suggests that non has been omitted before est in hoc est verum in Mantua’s text, and suggests correcting it by changing hoc est verum (“This is true”) to hoc non est verum (“this is not true”). However, this change might not be necessary if one considers the fact that in the edition (Mantua, Insolubilia, O6ra) as well as in the text quoted by Strobino the line seipsa demonstrata et hec ⟨similiter:⟩ hoc est falsum is missing (perhaps it is an omission per homoeoteleuton). Paul’s text suggests that Mantua’s text is correct as it stands.



Commentary317

verbatim from Peter of Mantua, Insolubilia, O6va, quoted and translated in Strobino, “Truth and Paradox in Late XIVth Century Logic,” 506. Three differences between the two texts are worth mentioning: 1) While Paul has unam, Mantua has unicam in the sentence, “alia illarum non est secundum unam responsionem respondendum.” 2) In the last clause (“negando quod hoc sit falsum secundo modo”), Mantua’s text wrongly has verum, which Strobino chooses over falsum from the 1477 edition of Mantua’s Logica, against falsum in all six manuscripts. The sense supports what we read in Paul and in the manuscripts of Mantua’s treatise. 3) At the end of the last clause Peter has hoc before falsum, which Paul omits, so that Mantua’s text reads: “concedendo quod hoc est falsum primo modo.” § 1.13.7.4 See Spade’s refutation of Wyclif’s theory (to which Mantua’s is similar): Spade and Read, “Insolubles,” §§ 4.1 and 4.3, and Wyclif, Summa insolubilium, xxxiii.

§ 1.13.7.3 Almost

Walter Segrave’s Opinion § 1.14 Spade (The Medieval Liar, 83) correctly identifies this as Walter Segrave’s

(or Sexgrave’s) position.71 Walter’s opinion is a version of restrictivism, the first opinion criticized, and dismissed at length, in Bradwardine, Insolubilia, chaps. 3–4. That perhaps explains why Paul started his list of opinions with Bradwardine’s second, leaving the first till now. Walter was a fellow of Merton College, Oxford (like Bradwardine) from at least 1321 until 1338.72 Spade observes that the treatise must have been written before 1333, when one of the surviving manuscripts of the work was written (MS. Erfurt, Bibliotheca Amploniana, Quarto 276). Since Walter discusses and clearly rejects Bradwardine’s own solution (Paul’s tenth opinion), it must also have been written after Bradwardine’s treatise, which is dated to 1321–24. Unlike other advocates of restrictivism, who appealed to the fallacy of the restricted and the unrestricted to explain the fallacy in insolubles, Segrave attributes the error in insolubles to the fallacy of accident. This

71  The current editors are preparing an edition and English translation of Segrave’s treatise. For detailed discussion of Segrave’s solution, see Read, “Walter Segrave’s ‘Insolubles’: a Restrictivist Response to Bradwardine.” 72  See Emden, Biographical Register, 3:1664.

318 Commentary

fallacy is the first of Aristotle’s fallacies “outside language.” The reference to “accident” here should not be confused with accident as contrasted with substance. Rather, Peter of Spain (Summaries of Logic, 359) writes, “the accident in question is the same as what is not necessarily a consequence” (illud accidens est idem quod non necessarium in consequendo). Aristotle himself described it as follows: Accident occurs when in the same way anything whatever is treated as being in the thing as subject and in the accident.73

As Gambra (“Medieval Solutions to the Sophism of Accident,” 441) says, “one of the most influential and repeated doctrines among the Scholastics tries to show that the fallacy of accident is produced because the middle term varies in some way in the two premises.” In point of fact, Burley extends this variation to the extremes as well: [I]t should be realised that the fallacy of accident sometimes results from a variation of the middle term and sometimes from a variation of the major or minor extreme.74

In this section, Paul is quoting directly from Walter’s treatise (chap. 4: MSS. Erfurt, Bibliotheca Amploniana, Octavo 76, fol. 24vb, and Quarto 276, fol. 159vb; MS. Oxford, Bodleian, can. misc. 219, is acephalic). “Sophismatist” is chosen as a translation of sophista in this case, to speak of those who deal in sophisms. The term sophista is used in medieval treatises to refer to several types of person, from undergraduates (neophytes in logic) to sophisticated thinkers, including those writing advanced treatises on logic and philosophy with their constant use of sophisms to illustrate and advance their argument. § 1.14.1.1 Paul is again quoting verbatim from Walter’s treatise. Like Segrave, Paul omits stipulation of the familiar scenario that would seem to be needed, namely, that this is all Socrates says, which Paul mentions only in § 1.14.2. Both Erfurt manuscripts of Segrave’s treatise75 read, “if Socrates says: ‘Socrates says a falsehood,’ Socrates does not say a falsehood” (sorte dicente: Sortes dicit falsum, sortes non dicit falsum). The subsequent inference fails 73  Peter of Spain, Summaries of Logic, 357: “accidens fit quando similiter quidlibet fuerit assignatum rei subiecte et accidenti inesse,” quoting Aristotle, De sophisticis elenchis, chap. 5, 166b28–30. 74  Walter Burley, Tractatus super librum Elenchorum, cited in William Ockham, Expositio super libros Elenchorum (Part II, chap. 9, § 2), 23. 75  The Oxford manuscript is incomplete and does not include this passage.



Commentary319

because in the minor premise “falsehood” refers to “Socrates says a falsehood” (so the first premise is true by the scenario and the second premise is true because Segrave says Socrates says a falsehood, and “falsehood”—in the second premise—can supposit there for what Socrates says), but the conclusion is false (as Segrave has just said) because “falsehood” cannot supposit there for the whole of which it is part. Hence, if that is all Socrates says, then he does not say a falsehood. There is a variation of supposition of one of the extremes and so a fallacy of accident. § 1.14.1.2 Verbatim from Walter’s text. We have again corrected Paul’s text from comparison with Walter’s. § 1.14.2 Paul’s summary of the restrictivist position. § 1.14.3.3 As we remarked when commenting on § 1.13.6.3, propositions like ens est, aliquid est, deus est, falsum est are hard to render in English. Trickier still is omnis homo est (see § 1.14.3.4.3): “Every man exists.” The argument at § 1.14.3.3 seems to rely on the equivalence between particular (or indefinite) propositions and disjunctions (“Some A …” iff “This A … or that A … or …”) and universal propositions and conjunctions (“Every A …” iff “This A … and that A … and …”). A very similar argument is found in Bradwardine, Insolubilia, § 3.1.4. § 1.14.3.4 Paul’s argument is almost identical to Bradwardine’s ibid., § 3.1.5. § 1.14.3.4.1 In the first part, Paul is quoting verbatim from Walter’s text (Octavo 76, fol. 27ra, Quarto 276, fol. 160rb), while in the second he is summarizing it. § 1.14.3.5 Paul is quoting verbatim from Bradwardine, Insolubilia, § 3.1.7. See Prior Analytics, book 1, chap. 8, 30a2–3: “We should give the same account of ‘being in something as in a whole’ as of ‘to be said of all.’” For the dici de omni, see Prior Analytics, book 1, chap. 1, 24b28–29: “We speak of ‘being said of all’ when nothing can be found of the subject of which the other will not be said.” § 1.14.3.5.2 This really just repeats what was argued in 1.14.3.5.1. The dici de omni applies universally to cases where “nothing can be found of the subject of which the other will not be said,” and so it is with what was said by Socrates. § 1.14.3.6 Paul’s argument as given in M and E seems to fail to establish the inconsistency which is claimed to follow from Segrave’s restrictivist opinion. But it is sufficiently similar to an argument in Bradwardine’s Insolubles to which Segrave replies that we have emended Paul’s text here to accord with Bradwardine’s original argument (ibid., § 3.2.2) and Segrave’s summary of it and his response.

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Bradwardine’s argument claims that, given a specific scenario like the one introduced in § 1.14.3.6, the restrictivist ban on self-reference can lead to inconsistency in the case of exceptive propositions and their exponents. Segrave’s summary reads: Moreover, this exceptive is true: “No falsehood except A is said by Socrates,” where A is “A falsehood is said by Socrates.” It is then asked if the subject of the prejacent ⟨that is, “No Falsehood is said by Socrates”⟩ supposits for A or not. If so, we have what we claimed. If not, then here there is no excepting of a part from the whole, therefore it is not an exceptive.76

As given in M and E, Paul takes A to be the exclusive proposition, “Only a falsehood is said by Socrates” in place of “A falsehood is said by Socrates” (in both Bradwardine and Segrave). We conjecture that this confusion was the result of someone writing exclusiva instead of exceptiva (their contractions exva are similar), misconstruing the scope of tantum and consequently replacing vera with igitur. In the chapter on “Exceptive Expressions” in the third treatise of the Logica magna, Paul acknowledges two types of exceptive propositions: An exceptive proposition of the first type is one in which the term from which the exception is made and the part excepted occur in the same part of the utterance … An exceptive proposition of the second type is one in which the term from which the exception is made and the part excepted do not occur in the same part of the utterance, rather the main copula is placed between them.77

“No falsehood except A is said by Socrates” is an exceptive of the first type and is expounded through three exponents: 1) the prejacent resulting from the omission of the excepted part;78 2) an existential exponent,  Walter Segrave, Insolubilia, Octavo 76, fol. 26va, and Quarto 276, fol. 160rb: “Preterea: hec exceptiva est vera: Nullum falsum preter A dicitur a sorte (sit A hoc: Falsum dicitur a sorte). Queritur ergo utrum subiectum preiacentis supponat pro A vel non. Si sic, habetur propositum. Si non, ergo non est ibi extra captio partis a toto, ergo non exceptiva.” 77  E, fol. 38rb: “Propositio enim exceptiva primi ordinis est illa in qua terminus a quo fit exceptio et pars extra capta ponuntur in eadem parte orationis, ut omnis homo preter sorte currit, nemo preter sortem currit … Propositio exceptiva secundi ordinis dicitur esse illa in qua terminus a quo fit exceptio et pars extra capta non ponuntur ex eadem parte propositionis, immo inter ea mediat copula principalis ut omnis homo currit preter sorte, nemo scit omnia nisi sortes.” 78  E, fol. 38ra: “Preiacens exceptive dicitur esse illud quod remanet dempta dictione exceptiva cum parte extra capta” (“The prejacent of an exceptive is said to be what remains when the exceptive word and the excepted part are removed”). 76



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which seems to be Paul’s novel addition to the usual two exponents of exceptives (i.e., the first and third), yet which does not appear in the argument in § 1.14.3.6; 3) the prejacent to whose subject the part excepted preceded by non- is added. Paul writes: A negative exceptive proposition of the first type should also be expounded through three exponents, the first of which is a true affirmative in which that from which an exception was made is affirmed of the part excepted. The second is an affirmative proposition in which the opposite of the term excepted is affirmed of that from which the exception was made … The third exponent should be the prejacent of the exceptive adding the opposite of the term excepted to the term from which the exception was made. For example: No man except Socrates runs is expounded like this: Socrates runs and There is some man ⟨who is⟩ not Socrates and No man ⟨who is⟩ not Socrates runs.”79

Where Paul writes “No falsehood ⟨which is⟩ not A is said by Socrates” (Nullum falsum non a dicitur a sorte), Bradwardine wrote “No falsehood other than A is said by Socrates” (Nullum falsum aliud ab a dicitur a sorte). Paul’s preference for non- over alius/aliud ab appears to be that the former is gender-neutral, which the latter is not (see E, fol. 38va). § 1.14.3.6.1 Segrave’s text continues: Moreover: in every exceptive, the prejacent is incompatible with the exceptive, which is not true in the ⟨above⟩ case unless in the prejacent the subject supposits for A.80

 E, fol. 38vb (M fol. 49va): “Propositio exceptiva negativa primi ordinis est etiam per tres exponentes exponenda, quarum prima est vera affirmativa in qua illud respectu cuius fit exceptio affirmatur de parte extra capta. Secunda est una propositio affirmativa in qua oppositum termini extra capti affirmatur de illo a quo fit exceptio … Tertia exponens debet esse preiacens exceptive addendo termino a quo fit exceptio oppositum termini extra capti. Verbi gratia, hec: nullus homo preter sortem currit, sic exponitur: sortes currit, et: aliquis homo non sortes est, et: nullus homo non sortes [sortes om. E] currit.” 80  Segrave, Insolubilia, Octavo 76, fol. 26va, and Quarto 276, fol. 160rb: “Preterea: in omni exceptiva preiacens repugnat exceptive, quod non est verum in proposito nisi in preiacente subiectum supponat pro A.” 79

322 Commentary

In his response, he distinguishes proper from improper exceptives: To the next: I grant that this exceptive is true, “No falsehood except A is said by Socrates,” about which it should be noted that some exceptives are proper and others are improper. ⟨An exceptive is⟩ proper when a part is removed from a whole suppositing for that ⟨part⟩ and it is acknowledged that the exceptive is inconsistent with its prejacent. ⟨An exceptive is⟩ improper when something is removed which is signified by that from which the exception is made, although it does not supposit for it.81 And so it is in the present case. Sometimes ⟨a proposition⟩ is an exceptive when ⟨the exception⟩ is made in neither of these ways, and then it is most improper, like this: “No ass except a man runs.”82

That an exceptive is inconsistent with its prejacent is a standard rule concerning exceptives, stated by Paul as his second rule for exceptives: The second main rule is that every proper exceptive is inconsistent with its prejacent.83 1.14.3.7 Paul’s

dismissal of the consequence of the restrictivist solution that a universal affirmative and the corresponding negative particular might not be contradictories as insane is ironic in light of his own view, on which mutually contradictory contradictories may be simultaneously false (see § 2.3.2). 1.14.3.7.1 Equally ironic is his dismissal of these consequences of Segrave’s position in light of his own conclusions and corollaries in chap. 2 (e.g., §§ 2.4.2–3).

 See “Logica oxoniensis, De consequentiis,” in Heytesbury, Sophismata asinina, 544–5. A proper exceptive is one where the items excepted constitute a non-empty proper subset of that from which they are excepted; an improper exceptive is one for which they do not. Paul has a much fuller account of proper exceptives: E, fol. 38rb. 82  Segrave, Insolubilia, Octavo 76, fol. 27rb; Quarto 276, fol. 160rb; Oxford, Bodleian, can. misc. 219, fol. 1ra: “Ad aliud concedo quod hec exceptiva est vera: Nullum falsum preter A dicitur a sorte, pro quo sciendum est quod aliqua est exceptiva propria et aliqua impropria. Propria quando fit extra captio partis a suo toto supponente pro illa et intelligitur quod exceptiva repugnat sue preiacenti. Impropria quando fit extra captio alicuius quod significatur per illud a quo fit exceptio, licet pro illo non supponat, et sic est in proposito. Aliquando est exceptiva quando neutro modo fit ⟨extra captio⟩, et tunc est improprissima, ut hec: Nullus asinus preter hominem currit.” 83  E, fol. 39va: “Secunda regula principalis est ista quod omnis exceptiva propria repugnat sue preiacenti.” See also Walter Burley, On the Purity of the Art of Logic, 261; Heytesbury, “Logica Oxoniensis”; Buridan, Treatise on Consequences, 91; and Kretzmann, “Syncategoremata, Sophismata, Exponibilia,” 224. 81



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Chapter 2: Paul’s Own Opinion § 2.1 The “divisions”

are essentially definitions, in line with Aristotle’s theory of definition in the Posterior Analytics, book 2, chap. 13 (see also Metaphysics, book Z, chap. 12). The “assumptions” are perhaps more than mere assumptions: they are postulates to which Paul will appeal in establishing his Conclusions. § 2.1.1 The twofold division follows Bradwardine, Insolubilia, 93, § 6.1.1 and the Logica parva (ed. Perreiah, § 6.2, 128; trans. Perreiah, 237). Others, such as Swyneshed (Insolubilia, ed. Spade, 184, § 10), have a threefold division, including “a mixture of an act of ours with a property of speech.” Paul subsumes these hybrid cases under acts of ours. The phrase verum pro se could also be translated as “true in virtue of itself.” But in this context it seems to be a predicative sense of “of.” § 2.1.2 We have gone against M and E, which read significatum, to read significatio in the last sentence, following the other occurrences of significatio refertur earlier in § 2.1.2 and in 2.1.3. § 2.1.5 We will see that every proposition undermining itself is false, but not every proposition endorsing itself is true, since not every proposition verifying itself is true. See the example “Every proposition is true” in §§ 2.1.7 and 2.2.2 below. See also Swyneshed’s comments in his Insolubles (ed. Spade, 187, § 20). Note the change from Aristotle’s “self-undermining” (ἑαυτοὺς ἀναιρεῖν), which is rendered as destruere se in Moerbeke’s Latin translation of the Metaphysics,84 to Swyneshed’s “self-falsifying” (falsificare se). Paul is here following Swyneshed. This offers a better Latin rendition of the meaning of Aristotle’s words, since according to Swyneshed and Paul, at least, propositions do not annihilate themselves, but merely falsify themselves, although cassationists might claim that insolubles annihilate themselves. (See Bradwardine, Insolubilia, §§ 5.5–5.6, and the commentary on § 1.8 above.) The phrase significat assertive seems to have been a common way at the time of referring to a secondary or consequential signification, contrasted with the primary or exact signification. See De Rijk, “Semantics in Richard Billingham and Johannes Venator,” 175, and Nuchelmans, Late-Scholastic and Humanist Theories of the Proposition, 45.  Aristotle, Metaphysica, Translatio Guillelmi de Moerbeka, book 4, chap. 8 (vol. 1, 91, line 655).

84

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is taken almost verbatim from Peter of Ailly (Concepts and Insolubles, trans. Spade, § 255, 68), who in his turn seems to have drawn the distinction between per se and per accidens self-falsifying propositions from Marsilius of Inghen’s Insolubilia (see Spade’s comment ibid., 142 n. 684). On this refutation of the twelfth opinion, see § 1.12.2. For the scenario that needs to be set in order to make the proposition “Socrates says a falsehood” an insoluble proposition, see § 4.2. § 2.1.7 As noted in the comment on § 2.1.5 above, “Every proposition is true” is self-verifying, since “Every proposition is true, this is a proposition, so this is true” is valid, but the universal proposition is false, since not every proposition is true. § 2.1.8 This completes the justification, in §§ 2.1.6–2.1.8, for defining insolubles as self-falsifying propositions. See the definition at the end of § 2.1.5. Cf. Peter of Ailly, Concepts and Insolubles, § 253, 67. We have corrected se non esse falsam (as in M and E) to se non esse veram in the description of non-insolubles to maintain the symmetry with the ensuing account of insolubles, and in line with the examples which follow. § 2.1.8.1 § 2.1.8.1 is taken by Paul straight from Peter of Ailly’s text (ibid., § 242, 64–5). As Spade remarks, Peter’s words are again very close to Marsilius’s (ibid., 139 nn. 642, 644–5.) A popular definition of “insoluble” was something that is not impossible to solve, but can be solved only with difficulty. See Burley, Insolubilia, 268–9, §§ 2.01–02; Ockham, Summa logicae, 744; Albert of Saxony, Logik, § 6.1, 1100. § 2.1.8.2 Most of § 2.1.8.2 is taken verbatim from Peter of Ailly (Concepts and Insolubles, § 243, 65). § 2.1.8.3 Paul subsequently treats the last two on the list—“Socrates will not cross the bridge” and “Plato will not have a penny”—as “not-obvious insolubles,” namely propositions which are insolubles only within specific scenarios: see commentary on § 5.1. These covert insolubles are dealt with in chap. 5. Up to the sophism “Socrates wishes a misfortune to happen to Cicero,” the text is taken from Peter (Concepts and Insolubles, § 252, 67). Here are some references to discussions elsewhere of these examples by authors who take them to be genuine insolubles: for “Socrates supposes himself to be a sophist” and “Socrates knows himself to be mistaken,” see Albert of Saxony85 and Marsilius of Inghen.86 The sophism § 2.1.6 § 2.1.6

 See Albert of Saxony, Logik, 1162 and 1166 respectively.  Marsilius of Inghen, Insolubilia, MS. Vatican City, Biblioteca Apostolica Vaticana, Pal. lat. 995, fol. 82v.

85 86



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“Socrates badmouthed Plato” is found in Buridan’s Summulae as well as in Marsilius’s Insolubilia.87 The last of Buridan’s sophisms in his Summulae is “Socrates wishes Plato evil” (Buridan, Summulae de Dialectica, 996). Paul discusses “Socrates will not cross the bridge” in § 5.4; this sophism is found in many authors’ treatises on insolubles, such as Bradwardine, Insolubilia, § 8.8.1, ps.-Heytesbury (Pironet, “William Heytesbury and the Treatment of Insolubilia,” 294), in the Summa insolubilium attributed to John Wyclif,88 and in the chapter on conditionals in Wyclif’s Logicae continuatio (Wyclif, Tractatus de logica, vol. 2, 208). Paul discusses the insoluble “Socrates will not have the penny” in § 5.3; it is found in Bradwardine, Insolubilia, § 8.1, and in Johannes Venator (also known as John Hunter).89 Although the last two are considered insolubles, specifically not-obvious insolubles, in the Logica magna, they appear as merely apparent insolubles in the Logica parva (trans. Perreiah, VI.8, 253–4). § 2.1.8.4 Paul’s claim that “not every proposition signifies that it itself is true” is taken verbatim from Peter of Ailly’s second corollary to his descriptions (Concepts and Insolubles, § 244, 65). The opinion that every proposition signifies that it itself is true was maintained by many authors, including most famously Buridan (in his early Quaestiones in duos libros Aristotelis Analyticorum Posteriorum) and Albert of Saxony, and was rejected by Marsilius of Inghen. Albert of Saxony (“Insolubles,” 339–42) gives the same proof that every proposition signifies that it itself is true as that offered by Buridan, which we quoted in the introduction, § 4. The rest of § 2.1.8.4 is taken verbatim from Peter’s third corollary (Concepts and Insolubles, § 247, 66.) Here, Peter attacks “a certain master’s opinion,” namely, that of Marsilius of Inghen, who in his Insolubles wrote: Fourth assumption: Every affirmative subject-predicate proposition affirms that ⟨its⟩ subject and predicate supposit for the same thing … Fifth assumption: every negative subject-predicate proposition signifies that the subject does not supposit for the same thing for which the predicate ⟨supposits⟩.90  See Buridan, Summulae de dialectica, 995–6, and Marsilius, Insolubilia, fol. 82v.  See Wyclif, Summa insolubilium, 61–2. In a forthcoming paper, Mark Thakkar argues that this treatise is in fact by John Tarteys. 89  See Pironet, “William Heytesbury and the Treatment of Insolubilia,” 320. 90  See Marsilius, Insolubilia, fol. 67r: “Quarta suppositio: Omnis propositio categorica affirmativa affirmat subiectum et predicatum supponere pro eodem … Quinta suppositio: Omnis categorica negativa significat subiectum supponere non pro illo pro quo supponit predicatum.” (Spade gives a different translation in Peter of Ailly, Concepts and 87 88

326 Commentary

is taken almost entirely from Peter’s text (Concepts and Insolubles, § 249, 66), in which, however, it is not specified that according to its formal signification, “A man is an animal” signifies “that it itself is true.” Perhaps Paul added this extra clause in order to link the eleventh opinion with Peter’s text. In § 249, Peter’s polemical target is still Marsilius. Specifically, Peter criticises Marsilius’s distinction between material and formal signification of propositions as it is found in Marsilius’s Insolubilia:

§ 2.1.8.5 § 2.1.8.5

From this [the fourth assumption] it follows as a corollary that an affirmative subject-predicate proposition has two significations, namely, a material ⟨signification, concerning⟩ a thing ⟨existing⟩ outside of the mind—e.g. according to this signification, ⟨“A man is an animal” ⟩ signifies that a man ⟨existing⟩ outside of the mind is an animal. The other ⟨is the⟩ formal ⟨signification⟩ by which it signifies that it is the same thing for which “man” and “animal” supposit. Or, more clearly, it signifies that “man” and “animal” supposit for the same thing … It follows as a corollary that every ⟨negative subject-predicate proposition⟩ has two significations, one concerning a thing ⟨existing⟩ outside of the mind, namely, the material ⟨signification⟩, and another formal one, concerning ⟨the proposition⟩ itself and its terms.91

of Venice, Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, “On the Truth and Falsity of Propositions,” chap. 9, thesis 1, 62. “Exact significate” is Paul’s equivalent for Roger’s “principal signification.” See comment on § 1.12.3.2 above.

§ 2.2.1 See Paul

Insolubles, 141 n. 660.) It seems that Paul conflates Albert of Saxony’s view (which he might have known either directly or from an intermediate source and who might have been Paul’s polemical target in § 1.11) with Marsilius’s view, which is that of the master whom Peter criticizes. It is unclear if Paul correctly identified the quidam magister or if he thought it was Albert. As noted in the introduction, Albert’s view was one of those criticized by Marsilius. 91  See Marsilius, Insolubilia, fol. 67r–v: “Ex quo sequitur corollarie quod affirmativa categorica duas habet significationes videlicet materialem de re ad extra. Exemplum ut hec: homo est animal, hac significatione significat hominem ad extra esse animal. Et aliam formalem qua significat idem esse pro quo supponit ly homo et ly animal. Vel clarius significat quod ly homo et ly animal supponit pro eodem … Corollarie sequitur quod omnis talis [sc. categorica negativa] habet duas significationes, unam de re extra videlicet materialem, et aliam formalem et de se et de suis terminis.” Spade gives a different translation in Peter of Ailly, Concepts and Insolubles, 141 nn. 665–6.



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§ 2.2.2 See ibid.,

thesis 3, 62–4. The claim that “some false proposition signifies principally as it is” was Swyneshed’s first iconoclastic conclusion. See Swyneshed, Insolubilia, ed. Spade, 188, § 25. § 2.2.3 See § 2.3.7 for Paul’s use of this assumption. § 2.2.4 One term is inferior to another (and the other superior to the first) if every instance of the first is also one of the second, and not vice versa. For example, “man” is inferior to “animal,” for every man is an animal, and consequently, the inference from “A man is running” to “An animal is running” is formally valid, for the two propositions are in the affirmative mood. The inference from the negative, “A man is not running” to “An animal is not running” is clearly invalid. Paul here refers to a work on inferences (in consequentiis). However, there is no chapter or treatise in the Logica magna called “De consequentiis.” See Perreiah’s comment in Logica parva, trans. Perreiah, 331 and 335, noting another place in the Logica magna which refers to this treatise: All the rules concerning valid and invalid inferences which were given in the Tract on Inferences should be basically upheld here as well.92

Ashworth refers the reader to the treatise De hypotheticis, which was partially edited in Logica magna: Capitula de conditionali et de rationali, which contains a chapter on inferences (De rationali). Thesis 4 reads: An inference from an inferior term to one of its superiors, in the case of affirmative propositions and in the absence of any impediment, is both formally valid and valid because of its form.93

On the distinction between inferences which are formally valid and those valid because of their form (de forma), see Hughes’s discussion ibid., 258 n. 183. Note, however, that translating bona de forma as “valid because of its form” is tendentious. The inferences are valid because the contradictory of the conclusion is incompatible with the premises. Bona de forma simply signifies that they are valid in form: see ibid., 104: “An inference which is valid in form may be defined as one where every inference of the same form is valid” (consequentia bona de forma dicitur esse illa cui quaelibet consimilis in forma est bona).  Logica magna: Tractatus de obligationibus, 33: “[O]mnes regulae superius adsignatae in Tractatu Consequentiarum de consequentia bona vel non bona sunt hic fundamentaliter sustinendae.” 93  Ibid., 113: “[A]b inferiori ad suum superius affirmative sine impedimento est consequentia formalis et de forma.” 92

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and exposition, see Logica magna: Tractatus de terminis, 248–9, and Logica magna: Capitula de conditionali et de rationali, 232–3 n. 66, where Hughes writes:

§ 2.2.5 On exponents

In the form of analysis known as exposition, a proposition is displayed as deducible from certain other propositions (its exponents) and as equivalent to the conjunction of these.94

The exponents of a universal proposition of the form “Every S is P” are “Some S is P” and “Nothing which is an S is not P.” Its singulars are all propositions of the form “This is P,” and the required middle is “This is S.” For Paul’s account of formal validity, see Logica magna: Capitula de conditionali et de rationali, 89 and 122, the first general rule. One might wonder whether an inference such as “This is false, so this is false,” where both occurrences of “this” refer to the conclusion, runs counter to Paul’s claim in this paragraph that he accepts all the rules of formal inference. For the conclusion is false, since it falsifies itself, but the premise is true, for the same reason. But in fact, the opposite of the conclusion, “This is not false,” referring to the conclusion, is also false—indeed, they are both necessarily false—and so, like the conclusion, is also incompatible with the premise. See Paul’s fifth conclusion in § 2.3.5 below. § 2.2.6 This is Bradwardine’s third assumption (Insolubilia, 97, § 6.3), for which he had argued at length in chaps. 3–4. Swyneshed takes it for granted, without discussion, in his Insolubles. § 2.2.6.1 Almost verbatim from Peter of Ailly, Concepts and Insolubles, trans. Spade, §§ 313–14. There are two main differences between Paul’s and Peter’s texts: 1) Throughout § 2.2.6.1 Paul has aliqua intellectione, which does not fit with the context, while Peter has the better reading alia intellectione, which therefore we have followed. 2) Hec est cognitio quam non intelligo alia intellectione (see n. 78 in the apparatus): we have followed Peter of Ailly’s text in putting est before cognitio—although Peter has Hoc est cognitio, while Paul has Hec cognitio—since if we place est at the end of the sentence, as M and E do, it does not work. Although Paul is not strictly speaking plagiarizing Peter here, but quoting him in order to reject his arguments later, Spade (ibid., 153 n. 824) points out that Peter was here himself plagiarizing the English Franciscan Roger  See also Andrews, “Resoluble, Exponible, and Officiable Terms.”

94



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Roseth.95 The argument refers to the definition of supposition, which is not given here by either Paul or Peter. Nor does Paul appear to be referring to his own definition of supposition, which was given in Part I, treatise 2 (Logica magna: Tractatus de suppositionibus, 2–3). Rather, he picks up Peter’s reference to his own definition. Spade, in his comment on § 314 (Concepts and Insolubles, 153 n. 821) refers back to § 145, where Peter criticizes those who define truth in terms of supposition, having already defined supposition in terms of truth, a vicious circularity. Spade (see nn. 403–06 on 124) suggests Peter is criticizing Buridan’s theory, accepting Buridan’s account of supposition, but rejecting his account of truth. Buridan wrote: [S]upposition, as it is taken here, is the taking of a term in a proposition for some thing or things, so that when it or they are pointed out (demonstrato vel demonstratis) by the pronouns “this” or “these,” or equivalent ones, the term is truly affirmed of the pronoun by means of the copula of the proposition. For example, in the proposition “A horse runs” the term “horse” supposits for every horse that exists, for of whichever is pointed out it would be true to say: “This is a horse.” And in the proposition “A horse ran” the term “horse” supposits for every horse that exists or existed, because it would be true to say of every such, if it were pointed out, “This is or was a horse,” and the same goes for the future.96

However, Spade also notes that Roger Roseth was writing before 1337, in which case it would probably not be Buridan’s account of supposition that was originally referred to, but that of some earlier English writer— say, Ockham—whose definition reads much the same: [W]hen a term stands for something in a proposition in such a way that we use the term for the thing and the term or its nominative case (if it is in an

 Part of Roseth’s Sentences Commentary has now been published: see Roseth, Lectura super Sententias. 96  Buridan, Summulae de dialectica, trans. Klima, 866; Summulae de practica sophismatum, ed. Pironet, 53: “Est autem suppositio, prout hic accipitur, acceptio termini in propositione pro aliquo vel aliquibus, quo demonstrato vel quibus demonstratis per ista pronomina ‘hic,’ ‘hoc,’ ‘haec’ vel aequivalentia, ille terminus vere affirmaretur de isto pronomine mediante copula illius propositionis. Verbi gratia, in hac propositione ‘equus currit,’ iste terminus ‘equus’ supponit pro omni equo qui est, quia, quocumque demonstrato, esset verum dicere ‘hoc est equus.’ Et in ista propositione ‘equus cucurrit,’ ‘equus’ supponit pro omni equo qui est vel qui fuit, quia de omni tali, si demonstraretur, esset verum dicere ‘hoc est vel fuit equus,’ et similiter de futuro.” 95

330 Commentary

oblique case) is truly predicated of the thing (or the pronoun referring to the thing), the term supposits for that thing.97

Then the argument of Peter’s that Paul is considering runs as follows: take the proposition: (A) Every cognition of mine that I do not comprehend by means of another comprehension is in my mind

and suppose that its subject, “cognition of mine that I do not comprehend by means of another comprehension,” supposits for proposition (A). Then, by the Ockham/Buridan account of supposition, the following proposition: (B) This is a cognition of mine that I do not comprehend by means of another comprehension,

must be true, where the subject of (B), “this,” refers to (A). But this means that I have comprehended (A) by a different comprehension, namely, whatever concept allows the demonstrative pronoun to refer to it. So (B) is false, and consequently (A) is not self-referential and its subject cannot supposit for the whole proposition of which it is a part. Spade’s text (Concepts and Insolubles, § 314) has “singular” where Paul has “universal” in the last sentence Paul quotes from Peter. Paul omits Peter’s final sentence from § 314: “And by parity of reasoning the same thing can be said for other cases, therefore etc.” As Spade observes (ibid., 153 n. 824), not so. (A) is a “specially contrived case,” and the argument does not generalize to establish the universal claim given in § 313, as Paul shows in his response at § ad 2.2.6.1. § 2.2.6.2 Almost verbatim from Peter of Ailly, Concepts and Insolubles, § 315. The conclusion has been filled in from Peter of Ailly’s text. The argument is similar to that in § 2.2.6.1, applied this time to show that the predicate of a mental proposition cannot always supposit for the mental proposition itself. But it does not seem sound. Take the proposition: (C) Every mental proposition is universal

 Ockham’s Theory of Terms, trans. Loux, 189; Summa logicae I.63, 193: “[Q]uando terminus in propositione stat pro aliquo, ita quod utimur illo termino pro aliquo de quo, sive de pronomine demonstrante ipsum, ille terminus vel rectus illius termini si sit obliquus verificatur, supponit pro illo.” 97



Commentary331

and suppose that its predicate, “universal,” supposits for proposition (C). Then, by the Ockham/Buridan account of supposition, the following proposition: (D) This is universal

must be true, where the subject of (D) refers to proposition (C). And that is true—(C) is universal. Spade, in his comment on § 315 (see ibid., 153 n. 828) of Peter of Ailly’s text, notes that with the formation of (D), (C) will be false. But that misses the point and fails to establish the claim at the start of § 2.2.6.1, as Paul will observe at § ad 2.2.6.2. § ad 2.2.6.1 ⟨2⟩: on the controversy of whether “A chimera is intelligible” (or thinkable), see Priest and Read, “Intentionality: Meinongianism and the Medievals,” 426–7, referring to Paul’s discussion in Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 13, and Logica magna: Tractatus de scire et dubitare, 76. Paul’s definition of supposition in the second treatise of the Logica magna is: “Supposition is the signification of a term for some thing(s) but only while functioning in a proposition as an extreme [or part of an extreme].”98 See also Paul’s elaboration of the definition of personal supposition in Logica magna: Tractatus de suppositionibus, 83, but beware of Perreiah’s at best misleading translation. How supposition was related to signification was a matter of some dispute in the fourteenth century. See, for instance, Albert of Saxony, Quaestiones circa logicam, qu. 12 (113–17): “Does a term supposit for everything it signifies?” In general, supposition was defined either in terms of “standing for” (see the quotation from Ockham in the comment on § 2.2.6.1) or in terms of “taking” (acceptio), following Peter of Spain, Summaries of Logic, 240: “Supposition is taking a substantive term in place of something. Supposition and signification are different …” (Suppositio vero est acceptio termini substantivi pro aliquo. Differunt autem suppositio et significatio …). Segrave defined supposition in terms of “signifying,”99 as the author(s?) of the Logica cantabrigiensis did: see 98  Tractatus de suppositionibus, 2: “Suppositio est significatio termini extremaliter se habentis et non extremaliter unitivi pro aliquo vel aliquibus in propositione.” 99  Walter Segrave, Insolubilia, Octavo 76, fol. 24vb, and Quarto 276, fol. 159vb: “To supposit for something is to signify an extreme of the union signified by the copula… the extremes of a proposition take supposition from such a coupling. To supposit for its supposita is to signify them to be the extremes of that union in reality which the copula signifies” (“Supponere ergo pro aliquo est significare ipsum extremum unionis significate

332 Commentary

De Rijk (“Logica Cantabrigiensis,” 299), where we read: “And note that supposition is the signification of a categorematic term in an utterance according to how it is placed in that utterance.”100 Paul may be following the Logica cantabrigiensis in identifying supposition with a type of signification, or possibly Wyclif, who wrote in his Logica: “But it should be noted first that the definition of supposition is that supposition is the signification of a categorematic term which is one extreme of a proposition in relation to the other extreme. An extreme of a proposition is the subject or predicate.”101 ⟨3⟩: Paul pedantically points out that the singular proposition required by Peter’s preferred account of the supposition of a term in one proposition need not in fact be uttered by the same person uttering that proposition. Note the switch from the first person in (B) to the second person in the proposition uttered by a third person. ⟨5⟩: Paul supposes a scenario (perhaps in an obligation, supposing that there is only this one proposition) where you are required to utter it. Then Peter’s preferred account of supposition will require that the proposition, “This proposition is not referred to,” where the subject refers to that first proposition, should be true. But that is a contradiction in terms and clearly absurd. Thus Peter’s account of supposition is mistaken. § ad 2.2.6.2 Clearly, if both propositions were in the mind, the universal proposition would be false, for “This proposition is universal” would be a singular mental proposition. § 2.3.1 Following Swyneshed (and Aristotle), Paul takes pairs of contradictories, contraries, and subcontraries to be defined syntactically, not semantically (as would be more natural nowadays).102 In line with Aristotle’s account per copulam … Extrema igitur propositionis suppositionem capiunt a tali copulatione (copulativa mss). Supponere pro suis suppositis est significare illa esse extrema illius unionis ex parte rei quam significat copula”). 100  “Et nota … quod suppositio est significatio termini categorematici positi in oratione secundum quod ponitur in eadem oratione.” See also De Rijk, “Logica Oxoniensis,” 158, where De Rijk comments that he knows no corresponding anonymous Oxford treatise on supposition. 101  Wyclif, Tractatus de logica, vol. 1, 39: “Sed notandum est primo pro quidditate suppositionis quod suppositio est significatio termini kathegor⟨emat⟩ici qui est extremum propositionis in comparatione ad aliud extremum. Et est extremum in propositione subiectum vel predicatum.” 102  See Parsons, “The Traditional Square of Opposition,” § 2: on the semantic account, contradictories cannot both be true or both be false, contraries cannot both be true, and subcontraries cannot both be false.



Commentary333

of contradictories (see comment on § 1.12.4 above), “This is impossible” and “This is not impossible” affirm and deny the same thing (being impossible) of the same thing, namely, the first of them. There is some similarity here to Heytesbury’s argument against Swyneshed and others in his Insolubles (trans. Spade, 31, § 25), where he argues that on their view, given “This is necessary” and “This is not necessary,” each referring to the second, the second is impossible even though they are contradictories and the first is not necessary. In the first leg of the argument here, Paul appears to be trying to prove the S5-thesis that whatever is possibly impossible is simply impossible. Think in terms of possible worlds.103 If a proposition is possible (or possibly true), then it is true in some possible world. So if the exact significate of “This is impossible” is possible, there is some possible world where it is true, that is, where it is impossible. But if it is impossible (there) then it is false everywhere; and if it is false everywhere, it is (simply) impossible (here). So if it is possibly impossible, it is (simply) impossible. This is equivalent to the more standard form of the S5-thesis, that what is possibly necessary is (simply) necessary. (See also Paul’s Sophismata aurea, sophism 50, § 2.2.0.2 in Appendix B.) In the second leg of Paul’s argument, note that, although (A) is an insoluble, (B) is not, for it is not self-referential, as Paul points out. (B) is contingent, since it is possible that (A) is impossible (as we have shown in the first leg of the argument), and it is possible that (A) is not impossible, for it might not even exist, and on the medieval account of the truth and falsity of negative propositions like (B), they are true if what they purport to refer to does not exist. § 2.3.2 Paul’s point here is that “This is false” (referring to itself) and “This is not false” (referring to the first proposition, its contradictory) are both false. It was Swyneshed who famously claimed that two contradictory propositions can both be false. See our introduction, § 4. This was the third of three iconoclastic claims Swyneshed made: the other two were that some false proposition signifies as things are (see § 2.2.1), and that there is a formally valid inference with true premise and false conclusion (see § 2.3.5). As in § 2.3.1, “This is false” and “This is not false” are contradictories, since they affirm and deny the same thing (being false) of the same thing, namely, the first of them. For further discussion, see Read, “Swyneshed, Aristotle, and the Rule of Contradictory Pairs.”  See von Wright, “Interpretations of Modal Logic,” 166.

103

334 Commentary

Paul points out at the end of this paragraph that contradictories do not always have the same truth-value even in the case of insolubles. “Every proposition is false” falsifies itself, and so is an insoluble, even if there are other true propositions. In that case, “Some proposition is not false” is true, even though it contradicts an insoluble. § 2.3.3 The requirement that two particular propositions be subcontraries is that they have the form “Some A is B” and “Some A is not B,” or semantically that they can both be true but cannot both be false. Swyneshed’s third conclusion (see comment on § 2.3.2), and similarly this example, show that the semantic characterization is inadequate. Note that Aristotle himself does not introduce the notion of subcontrariety as such. Barnes et al. remark that “the words ‘subcontrary’ and ‘subaltern’ … we find as logical terms first in Alexander [of Aphrodisias].”104 The concept is found first in Apuleius (called by him “subpares”).105 Aristotle himself wrote: Verbally four kinds of opposition are possible, viz universal affirmative to universal negative, universal affirmative to particular negative, particular affirmative to universal negative, and particular affirmative to particular negative, but really there are only three: for the particular affirmative is only verbally opposed to the particular negative.106

That there is no corresponding pair of contradictories that are both true follows from the observation in the comment on § 2.1.7 above that a self-verifying proposition need not be true, even though a self-falsifying one is certainly false. § 2.3.4 Although in the argument of the fourth conclusion Paul labels “This is unknown to you” an insoluble, technically speaking it is for him only an apparent insoluble, that is, it is not an insoluble but a sophism, as he will say in the chapter dedicated to apparent or specious insolubles at the end of the treatise (chap. 8, § 8.1). The reason he gives there is that it does not in fact falsify itself. The sophism, “This proposition (or this conclusion) is unknown to you” is the famous Knower Paradox (see also Bradwardine, Insolubilia, “Introduction,” § 7, and Sorensen, “Epistemic Paradoxes,” § 5.1).

 Alexander of Aphrodisias, On Aristotle, Prior Analytics, trans. Barnes et al., 106 n. 29.  See Londey and Johanson, “Apuleius and the Square of Opposition,” 167. 106  Aristotle, Prior Analytics, ed. and trans. Tredennick, book 2, chap. 15, 63b23–26. 104

105



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We have added “exactly” (adequate) in the statement of the fourth conclusion since Paul adds it in the proof that the conclusion of the inference “This is unknown to you, therefore this is unknown to you” is unknown to you, and it seems necessary for it to work. In the statement of the corresponding conclusion in the Sophismata, Paul adds “only” (precise) at this point, in line with his usage in that work. On the equivalence between adequate and precise in Paul’s usage, see comment on § 1.12.3. § 2.3.5 This is Swyneshed’s second iconoclastic conclusion. One might wonder, then, what Paul’s (and Swyneshed’s) account of validity is, if a valid argument can have true premises and false conclusion. They cannot define a valid argument as one that is truth-preserving. Swyneshed’s statement that “since a proposition claiming that a proposition falsifying itself signifies as it is follows from some propositions each of which signifies principally as it is, it therefore follows that so it will be that a proposition following from them signifies as it is,”107 led Spade to infer that Swyneshed took preservation of “signifying as it is” as sufficient for validity. (See the discussion of Swyneshed’s account of validity in the introduction § 5.) For Paul, the requirement is the incompatibility of the opposite of the conclusion with the premise. See § 2.2.5.108 As noted in the introduction (§ 5), Paul’s fifth conclusion flatly contradicts a general rule he stated in the chapter on inference (De rationali) of the Logica magna. His third rule there stated: If the premise of a valid inference which signifies primarily in accordance with the composition of its parts is true, then the conclusion is also true. The rule can be proved as follows: If the premise of an inference which is valid etc. is true and its conclusion is false, and if the contradictory of its conclusion exists, then that contradictory is true; it is therefore consistent with the premise, since every independent truth is compatible with any other independent truth; but if that is so, the inference is not valid.109  Swyneshed, Insolubilia, ed. Spade, 191–2 § 35: “Cum igitur propositio ponens quod propositio falsificans se significat sicut est sequatur ex aliquibus propositionibus quarum quaelibet significat principaliter sicut est, sequitur igitur quod ita erit quod propositio sequens ex illis significat sicut est.” 108  For further discussion, see Read, “The rule of Contradictory Pairs, Insolubles, and Validity.” 109  Logica magna: Capitula de conditionali et de rationali, 140: “Tertia regula est ista: Si alicuius consequentiae bonae, significantis primo iuxta compositionem suarum partium, antecedens est verum, et consequens est verum. Probatur ista regula: Quia si alicuius 107

336 Commentary

The proof of this rule turns on the rule of contradictory pairs, denied in the second conclusion above (§ 2.3.2). It is surprising that Paul should have overlooked this inconsistency, particularly since a little later in that chapter on rules of inference, he recognises that his ninth rule needs qualification in order to accommodate the insoluble that proves the fourth conclusion above (§ 2.3.4). He writes: The ninth main rule is: suppose that a certain inference is valid, is known by you to be valid, is understood by you, and signifies primarily in acordance with the composition of its parts; suppose too that its premise is known by you, and that you know that what is false does not follow from anything that is true; then its conclusion is also known by you.110

The third argument against this rule presents the inference: This is unknown by you, therefore this is unknown by you.111

Paul’s offers four defences, the last of which runs: The fourth way of replying is to accept the inference “This is unknown by you, therefore this is unknown by you” in the given scenario, and to grant that all the clauses in the rule are satisfied, but to deny that the premise is known; the reason being that just as it is inconsistent for the conclusion to be known, so it is for the premise. But more about this when we come to deal with the insolubles.112

However, as we have seen, when he does come to ponder this insoluble in § 2.3.4 of the present treatise, he concedes that the premise is known, even though the inference is known to be valid and the conclusion is not known. See also his discussion of the sophism “This is not known by you” in chap. 8 (“On Merely Apparent Insolubles”) below. consequentiae bonae etc, antecedens est verum et consequens falsum, et contradictorium consequentis est, ipsum est verum; igitur stat cum antecedenti, quia omne verum independens cuilibet alteri vero independenti est compossibile; et si sic, illa consequenta non valet.” 110  Ibid., 195: “Nona regula principalis est ista: si aliqua est consequentia bona, scita a te esse bona et intellecta a te, significans primo iuxta compositionem suarum partium, et antecedens est scitum a te, sciendo quod ex nullo vero sequitur falsum, et consequens eiusdem est scitum a te.” 111  Ibid., 197: “Hoc est nescitum at te, igitur hoc est nescitum a te.” 112  Ibid., 200: “Quarto modo respondetur, concedendo consequentiam istam, ‘Hoc est nescitum a te, igitur hoc est nescitum a te’ in casu illo, cum omnibus illis particulis; sed negando quod antecedens est scitum. Sicut enim repugnat consequens sciri, ita et antecedens. Sed de hoc magis in materia insolubilium.”



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§ 2.3.6 On Paul’s

extension of Swyneshed’s second conclusion to modal and epistemic cases, see Hanke, “Paul of Venice and Realist Developments,” 302. Note that the sixth conclusion contradicts the fifth rule in his chapter on inference, which stated: “If the premise of a valid inference which signifies primarily in accordance with the composition of its premise and conclusion is possible, its conclusion is possible.”113 Paul says that the fifth and sixth conclusions will not seem so absurd when one realizes that one can proceed from knowledge of the premises to knowledge of the conclusion of these paradoxical arguments, which he admits are valid, only when one adds an extra condition to the effect that it is not incompatible (repugnans) with the premises that the conclusion is known, a proviso which he claims is false. The principle of the closure of knowledge under known entailment is contentious, and the subject of much contemporary discussion: see, for example, Dretske, “Epistemic Operators.” This is the first point at which Paul introduces what one might consider an ad hoc device which he will use repeatedly in chaps. 3–8, starting in § ad 3.1.1, to counter a series of objections to his solution. Paul’s proposal is that, given his fifth Conclusion, we need an extra premise to protect against such counterexamples. We need to know not only that the inference is valid (that the premises entail the conclusion) and that the premises are true, but also that knowledge of the conclusion (and consequently, the truth of the conclusion) is compatible with knowledge of, and hence the truth of the premises. § 2.3.7 In the final sentence, when Paul speaks of the second part of the conclusion, he means “one is true or possible and the other is false or impossible.” In the first part of the explanation, Paul states again the reasons for the convertibility of the two pairs of propostions given as examples (i.e., “This is false” and “This is false”; “This is impossible” and “This is impossible,” each time referring to the second member of each pair), which are respectively true, false, possible, and impossible. § 2.4.1 The second proposition, “Some particular affirmative proposition is false,” falsifies itself if by hypothesis it is the only particular proposition, and then the first proposition, “Every particular affirmative proposition is false,” is true.

 Ibid., 150: “Quinta regula est ista: si alicuius consequentiae bonae, significantis primo iuxta compositionem sui antecedentis et consequentis, antecedens est possibile, et consequens similiter est possibile.”

113

338 Commentary § 2.4.2 “Every

universal proposition is false” falsifies itself, so it is false. But if it is the only universal proposition, it has only one singular, namely, “This universal proposition is false,” referring to the (only) universal proposition, “Every universal proposition is false,” and accordingly the singular proposition is true. Note that the assumption is that there is only one singular for each instance of the universal, regardless of how that object is referred to in the singular, usually by a demonstrative attached to the term (subject or predicate) in question. See also Paul’s Sophismata aurea, sophism 50, § 2.4.4.1 in Appendix B. § 2.4.4 Let the three propositions be A, B, and C: A: “Every particular affirmative or universal negative proposition is false.” B: “There is a particular affirmative and a universal negative proposition which are false.” C: “There is no particular affirmative or universal negative proposition that is not false.”

Then by the normal medieval account, A is exponible, analysable into its exponents, B and C. See the comment on § 2.2.5 above. In general, “Every S is P” has exponents “Some S is P” and “There is no S which is not P.” However, the matter is complicated by the fact that proposition A has a compound subject. In the Latin, this is expressed by a conjunction: propositio particularis affirmativa et universalis negativa. But Paul does not mean every proposition which is both a particular affirmative and a universal negative is false. “Every P and Q is R” is arguably ambiguous, meaning either that everything which is both P and Q is R or that everything which is either P or Q is R. But that first interpretation won’t work here. What Paul must mean is that every proposition which is either a particular affirmative or a universal negative is false, for clearly no proposition is both a particular affirmative and a universal negative. So it is idiomatic in English to express it with a disjunction as “Every particular affirmative or universal negative proposition is false.” What that means is that every particular affirmative proposition is false and that every particular universal negative proposition is false. In logical symbols, (PvQ) → R is equivalent to (P→R) & (Q→R). Now the affirmative exponent of “Every particular affirmative or universal negative proposition is false” would appear to be “Some particular affirmative or universal negative proposition is false.” But this does not really express the first exponent of A. That is captured by something closer to the Latin, like “Some particular affirmative and universal



Commentary339

­ egative proposition is false,” provided this means both that some parn ticular affirmative proposition is false and that some universal negative proposition is false, since A, as we have seen, covers both types of proposition. A clearer expression is, perhaps, what we have given as B: “There is a particular affirmative and a universal negative proposition which are false.” The negative exponent of A is again universal, so like A it is best expressed by “or”: “There is no particular affirmative or universal negative proposition that is not false.” B is a particular affirmative and C is a universal negative, so A says that B and C are both false, but it does not falsify itself, for it is neither particular nor negative. However, B and C each falsify themselves, since B is, by hypothesis, the only particular affirmative and C is the only universal negative. So A is true, even though its exponents, B and C are both false, and we have Paul’s fourth corollary. § 2.4.5 By the same doctrine, “Every universal affirmative is false” is expounded, or analysed, as “Some universal affirmative is false and there is no universal affirmative that is not false.” Clearly, “Every universal affirmative is false” falsifies itself, but it makes both the exponents true, on the given hypothesis that it is the only universal affirmative. § 2.4.6 “Only a falsehood is an exclusive proposition” falsifies itself if it is the only exclusive, and so “Every exclusive proposition is false” is true. See Paul’s Sophismata aurea, sophism 50, § 2.4.1 in Appendix B. § 2.4.7 Paul has already, in the fifth conclusion (§ 2.3.5), endorsed Swyneshed’s claim that a valid argument can have true premises and a false conclusion. In the same way, Paul accepts the standard doctrine that an exclusive proposition of the form “Only B is A” is equivalent to a universal proposition of the form “Every A is B,” despite the fact that, in the special case of insolubles, one can be true and the other false. Chapter Three: Objections and Replies § 3.1 The arguments

in this paragraph and those following are drawn from Heytesbury, Insolubilia, ed. Pozzi, 216–18, §§ 2.031–2.04 (trans. Spade, 23–5). Heytesbury’s arguments are directed against the common view of Swyneshed, Dumbleton, and Kilvington that, as Heytesbury puts it, insolubles signify only as the terms suggest (see the comment on § 1.12.1 above). Paul will respond (in § ad 3.1) that in his view, insolubles do signify only as the terms suggest (as Heytesbury had put it).

340 Commentary § 3.1.1 This

is a familiar revenge scenario, applied here to a paradox of signification (or in this case, saying) rather than one of truth and falsehood. See Read, “Bradwardine’s Revenge.” Our additions to the Latin text follow Heytesbury (ed. Pozzi, § 2.0311). § 3.1.2 Paul’s phrase, sortes dicit sicut verum est esse (for example in the penultimate line of § 3.1.1) translates smoothly into English as “Socrates says what is truly so.” However, the parallel phrase to the Latin sortes dicit sicut falsum est esse, namely, “Socrates says what is falsely so,” is not idiomatic in English, nor is the more literal “Socrates says it is as it is false that it is.” We have tried to capture the sense more idiomatically by reversing the order, to render it as “it is false that it is as Socrates says it is.” § 3.1.3 On the translation of non est ita sicut aliquis homo dicit as “It is not as any man says it is,” see Swyneshed, Insolubilia, ed. Spade, 25 n. 37. Spade observes that the argument works only if we assume that no one else in the world is at that time saying what is the case. But that is built into Paul’s example, when Paul specifies that “every man says only ….” § 3.2 The arguments in this paragraph and those following are drawn from Heytesbury, Insolubilia, ed. Pozzi, 218, §§ 2.06–2.062 (trans. Spade, 26–7). Following Heytesbury, Paul sets out to show that if it is not wholly as A signifies, then it is wholly as A signifies, and conversely, if it is wholly as A signifies, then it is not wholly as A signifies. But it must either be wholly as A signifies or not. Either way, a contradiction follows. It is interesting that in expounding Heytesbury’s objection, Paul adds the qualification “wholly” (totaliter), which does not appear in Heytesbury’s text (as we have it—but see Spade, “The Manuscripts of William Heytesbury’s Regulae solvendi sophismata” on the large variations in the manuscript tradition of Heytesbury’s treatise, mentioned earlier). § 3.2.1 First, Paul assumes that it is not wholly as A signifies and shows that it follows that it is wholly as A signifies, and so we have a contradiction. Let B signify that A signifies other than it is. First, if it is not wholly as A signifies, and assuming A signifies something, then A signifies other than it is. It follows that it is wholly as B signifies. But B affirms of A just what A does, so it must be wholly as A signifies. Contradiction. § 3.2.2 Now let C be the contradictory of B, affirming of A what B denies of A, that is, C signifies that it is as A signifies. Assuming that it is wholly as A signifies, it follows that C is true, and so its contradictory, B, is false. Given that B does not falsify itself, B must signify other than it is, and since A and B signify the same thing, A must also signify other than it is, and so it is not wholly as A signifies. Contradiction again.

§ 3.3 The arguments

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in this paragraph and those following are drawn from and simplify those in Heytesbury, Insolubilia, ed. Pozzi, 220, §§ 2.07– 2.073 (trans. Spade, 27–8). § 3.4 The argument in this paragraph is drawn from Heytesbury, Insolubilia, ed. Pozzi, 222–24, §§ 2.10–2.101 (trans. Spade, 31). The puzzle arises because of the syntactic definition of “contradictory” that we noted above when commenting on § 1.12.4. Just as “This proposition is false” (referring to itself) and “This proposition is not false” (referring to the first proposition) are contradictories according to Aristotle’s definition of “contradiction” in De interpretatione, chap. 6, since “they affirm and deny the same thing of the same thing,” but are nonetheless (according to Swyneshed and Paul, among others) both false, so too A and B are contradictories by Aristotle’s purely syntactic definition: B affirms the necessity of A and A denies it. Nonetheless, A is not necessary because, as Spade comments (Heytesbury, Insolubles, trans. Spade, 31 n. 63), if A were necessary (as B says), it would be true and so not necessary, since that is what A says. So A is not necessary. But B is impossible, for as Paul shows, it implies a contradiction. That appears puzzling, for on the semantic account of contradiction, contradictories should have opposite truth-values, as the rule of contradictory pairs says. But as we have seen, Swyneshed—and Paul following him—think it is a necessary consequence, perhaps even a virtue. § 3.5 The argument in this paragraph is drawn from Heytesbury, Insolubilia, ed. Pozzi, 224, § 2.102 (trans. Spade, 32). A was shown not to be necessary in § 3.4, and so C is necessary, since it is necessary that A is not necessary: for the supposition that A is necessary implies a contradiction. Again, Paul accepts the conclusion here (see § ad 3.5 below). § 3.6 The argument in this paragraph bears some resemblance to that in Heytesbury, Insolubilia, ed. Pozzi, 224, § 2.111 (trans. Spade, 34). Whereas in § 3.5 the objector (Heytesbury) argued that A and C, while both being contradictories of B, were respectively necessary and contingent, here the objection is that the very same proposition A is both necessary and contingent. Recall that it was shown in § 3.4 that A (i.e., “A is not necessary”) is not necessary, and so is true and consequently contingent. The first proof that A is also necessary, which is not in Heytesbury, depends on the 5-principle (characteristic of the modal logic S5) that all modalities are necessary, that is, that not only is any proposition of the form “P is necessary” itself necessary (the characteristic principle of S4, the 4-principle), but any proposition of the form “P is

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possible” is also necessary. But A is of that form, since “A is not necessary” is equivalent to “It is possible that not-A.” Again, think in terms of possible worlds: then if P is possible, there is a world where P is true, so “Possibly, P” is true at every world. Consequently, “Necessarily, P is possible” is true. In particular, “A is not necessary,” being of the form “P is possible,” is necessary. But that is A itself, so A is necessary. In his response (in § ad 3.6), Paul will agree that A is not an insoluble, since it does not falsify itself, and that the exact significate of A (namely, that A is not necessary) is necessary, but he will deny that it follows that A itself is necessary. A has enough of the flavor of insolubles, he says, in that it has immediate reflection on itself, to reject such an inference. The second proof (starting “Again, C and A are convertible …,” due to Heytesbury), takes the inference “A is not necessary, therefore A is not necessary,” arguing first from A to C (each of which has the form “A is not necessary”), then from C to A. So A and C are convertible: in particular, C entails A. But it was argued in § 3.5 that C is necessary. So by the K-principle (that anything entailed by a necessary truth is itself necessary), A is necessary. For the K-, 4- and 5-principles, see Garson, “Modal Logic,” §§ 6–7. § 3.7 These conclusions were presented in §§ 2.3 and 2.4. In particular, Heytesbury complains (ed. Pozzi, 220, § 2.08) that the view entails that there is a formally valid inference with true premise and false conclusion. That was Paul’s fifth conclusion (see § 2.3.5 above), and Swyneshed’s second conclusion (Swyneshed, Insolubilia, ed. Spade, 189). § ad 3.1 Up till here, Paul has not committed himself to the claim in the first sentence, implicitly endorsed by Swyneshed (who does not use this terminology, which seems to be due to Heytesbury), that insolubles signify only as the terms commonly suggest (sicut termini communiter pretendunt), but he here accepts the hypothesis of his opponent in § 3.1 (to wit, Heytesbury) that the opponent aimed to show was untenable. His reluctance to endorse the claim (“or at any rate …”) is perhaps simply due to the fact that this is not terminology he or Swyneshed would use in their own person. However, Paul had used this terminology twenty times in his Logica parva (ed. Perreiah, 128–48), where—as noted in the introduction, § 5—he had followed Heytesbury in adopting the modified Heytesbury solution. Recall that the crux of the discussion in chap. 3 is whether to accept that insolubles signify only as their terms suggest, or whether they signify more than that, as Bradwardine, Heytesbury, pseudo-Heytesbury, Albert



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of Saxony, Peter of Ailly (see §§ 1.10, 1.11, 1.12, and 1.13), and many others had claimed. They all claimed that insolubles signified conjunctively. This Swyneshed and Paul deny. Note that, although for Paul, “Every proposition is false” and “Some proposition is not false” are contradictories by virtue of their signification (one denies what the other asserts), they may both be false, if, for example, one of them falsifies itself. See the comment on § 1.12.4 and 2.3.4 above. By describing a proposition as being “of the second component” (de secundo adiacente), Paul meant that it was compounded only of a subject and the copula, which was then considered as an existential predicate; while propositions “of the third component” (de tertio adiacente) were those containing subject, copula and predicate, where the copula simply links the subject to a substantival or adjectival predicate. See Del Punta and Adams’s comment in Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 266–7. Just as we noted, in the comment on § 2.4.5, that “Every A is B” is expounded as “Some A is B and nothing that is A is not B,” so too “Every A exists” is expounded as “Some A exists and nothing that is A does not exist.” But the second conjunct here is clearly redundant, so universals and particulars of the second adjacent are equivalent. § ad 3.1.1 The phrase copulatum seipso (“conjoined with itself”) in the first sentence here is puzzling. It could be a Moorean paradox: “What I am saying is false. What I just said is false, that is, I don’t believe it.” (On Moore’s paradox, see McGinn, “Wittgenstein and Moore’s Paradox.”) But the general idea that one can be forced to grant falsehoods in an obligational disputation as following is familiar. (Recall Burley’s rule, cited in the introduction, § 3:114 “When a false contingent proposition is posited, one can prove any false proposition that is compossible with it.”) The phrase tamquam sequens (that is, pertinens sequens—relevant and following) is explained in the introduction, § 3. Paul reminds us of two cases discussed earlier which are analogous to the situation with the example in § 3.1.1: “It is not as Socrates says it is.” This forms a problematic example without using the notion of truth, namely, “This is false,” referring to itself, and “Socrates says a falsehood,” where that is all Socrates says, and Plato goes on to observe that Socrates says a falsehood. He argued in § 2.2.2 that although “This is false,”  See Burley, “Obligations,” 391, § 3.61.

114

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r­eferring to itself, is indeed false, “This is false,” where “this” refers to the first proposition, is in contrast true. Similarly, he now observes that what Plato says is true, even though it converts with what Socrates says, which is false. For they have the same true exact significate, which makes Plato’s utterance true but fails to make Socrates’ utterance true since Socrates’ utterance falsifies itself. Paul’s diagnosis of the opponent’s (that is, Heytesbury’s) argument is that the final step fails, where the opponent reasoned: “it is as Plato says it is and Socrates says wholly what Plato says and only that [notice that the ‘and only that’ bit is lacking in Paul’s reply in § ad 3.1.1]; therefore it is as Socrates says it is.” Socrates and Plato may well have said wholly the same (their utterances have the same exact significate), but that does not prevent their utterances having different truth-values. An extra premise is needed, namely, that it is not incompatible that what Socrates said is true. This move may seem ad hoc, but it is the same move as he made in § 2.3.6, and will make again and again, with variants, in §§ ad 3.3.1, ad 3.6, ad 4.2.1.1, ad 4.3.2.2, ad 4.3.3.1, ad 4.3.3.2, ad 5.1.2, ad 5.2, ad 5.3.1, ad 6.1, ad 6.3, ad 6.4, ad 7.1.1, 8.1.4.1, ad 8.2.1.1, ad 8.2.2 (and aliter ad 8.2.1 and aliter ad 8.2.2), and ad 8.4. Nonetheless, one might argue that the cases are not analogous. The clause, “therefore it is true that it is as Socrates says it is,” which Heytesbury adds in the confirmatory argument, actually weakens it. Consider the difference between the paradoxical propositions “This proposition is false” and “This proposition is not true.” One can show that if the former is false it is true and if it is true it is false (and so both true and false) only on the assumption that it is either true or false. So one can avoid paradox by denying bivalence, and concluding that this proposition is neither true nor false. But the analoguous argument for the second example does not allow that solution: one can show that if it is not true then it is true and if it is true then it is not (and so it is both true and not true) only on the assumption that it is either true or not true. So to avoid contradiction one would have to deny excluded middle rather than bivalence, concluding that the second proposition is neither true nor not true. That is a more radical conclusion, which does not just revise the account of truth, but is arguably contradictory. What Paul should add in the premise is that it is not incompatible that it is as Socrates says it is. But one might argue that he cannot deny this, for that it is as Socrates says it is does seem to be incompatible with Paul’s claimed solution, namely, that it is not as Socrates says it is. See comment on § ad 3.1–3.3 below.



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Note that Paul does not need to reply to § 3.1.2–3 since he rejects the first leg of the argument in § 3.1.1. § ad 3.2.1 Paul’s response to Heytesbury’s argument is to fault the penultimate step in § 3.2.1. § ad 3.3.1 One might be alarmed that the extra premise required has now become “A is not an insoluble, or one falsifying itself.” If Paul really meant this, then the solution would simply be ad hoc, and no real explanation. What the extra premise really requires is, “it is not incompatible that A does not signify other than it is,” but of course (as in § ad 3.1.1 above) this is incompatible with Paul’s claim that A signifies other than it is. In fact, Paul vacillates over whether to claim that propositions such as A are insoluble or not. See comments on chap. 8 below, and also § ad 3.6, where Paul stops short of extending the notion of insoluble to include propositions which do not falsify themselves, but more generally have reflection on themselves. § aliter ad 3.1–3.3 Paul’s responses in § ad 3.1–§ ad 3.3 above are his own, saying that the various sophismata signify other than it is. The alternative response he gives here is Swyneshed’s, which is to deny that these sophismata either signify as it is or other than it is. See Swyneshed, Insolubilia, ed. Spade, 218, §§ 103–04. Thus Paul now wants to deny both that it is as Socrates says it is and that it is not as Socrates says it is (and mutatis mutandis for the other two cases). But it is unclear that this is possible without also asserting both that it is as Socrates says it is and that it is not as Socrates says it is. But he certainly does not want to do that, nor did Swyneshed. § ad 3.4 Paul accepts the fourth conclusion, thus outsmarting Heytesbury.115 He refers to the first conclusion of § 2.3, where it was already shown that “there is a pair of contradictory propositions that are mutually contradictory, one of which is impossible and the other is contingent.” § ad 3.6 See comment on § 3.6. See also § 2.1.8. Chapter Four: The Familiar Insoluble § 4.1 Here

Paul returns to the familiar insoluble, with which his treatise opened and which we noted occurs as the standard example of an insoluble in so many of his predecessors. See comment on § 1.1. (Parts of  See the spoof definition of “outsmarting” in Dennett et al., The Philosophical Lexicon: “to embrace the conclusion of one’s opponent’s reductio ad absurdum argument.”

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§§ 4.1–4.2.1.3 are translated in Bocheński, History of Formal Logic, §§ 35.11–12.) The point in the last sentence ruling out change of signification may be that Paul wants to avoid any future imposition which might change either the signification of A or whether it is exact or not. This is in line with his fundamental approach, following Swyneshed, of rejecting the proposal that insolubles have a further implicit signification. § 4.1.1 Paul’s aim in § 4.1.1 is to show the necessity of including all the rather pedantic caveats or qualifications: that there is only one Socrates, that he says only one thing, and that it signifies exactly one way. Note that Paul treats sortes as a general term, and if it is left open whether there is one or more than one Socrates, contradiction can be avoided. So for a genuine insoluble, it is necessary to add the condition that there is only one Socrates. See Hughes’s comment in Logica magna, Capitula de conditionali et de rationali, 232 (n. 64), referring to Logica magna, Tractatus de terminis, 129–33. See in particular Paul’s discussion ibid., 133–9. § ad 4.1.1 The scenario has not explicitly stated that there is only one Socrates. So, in response, Paul suggests that the first Socrates might refer to the second Socrates, who says something false (that God does not exist). So what the first Socrates says is unproblematically true, avoiding selfreference. ⟨1⟩ Given in the scenario that Socrates says something, either some Socrates says a falsehood or none does. The inference from “some Socrates says a falsehood” to “Socrates says a falsehood” is from a particular to an indefinite, given that Paul is treating “Socrates” (that is, sortes) as a general term, as in the inference from (for instance) “Some men are running” to “Men are running” ⟨2⟩ As we noted in the introduction, there is constant allusion, particularly in this chapter, to the practice and terminology of obligations. In such obligations, one is frequently obliged to deny a proposition one knows to be true (if it is inconsistent with the positum and other things that have been granted) or grant one that is known to be false. Similarly, one may be required to express doubt about a proposition about which one is not in any doubt, that is, that one knows to be true (or knows to be false). See Read, “Richard Kilvington and the Theory of Obligations,” 402 (reprint in History of Logic and Semantics, 256). § 4.1.2 Here the missing caveat is that Socrates says only one thing. He has added the qualification that there is only one Socrates.



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phrase illud disiunctum seu disiunctiva (which we have translated as “the disjunctive predicate or a disjunction”) would seem to correspond to the common distinction made at the time in supposition theory between a disjunct and a disjunctive descent (descendere disiunctim vel disiunctive). See Ashworth, “Descent and Ascent from Ockham to Domingo de Soto,” § 5. In this case, the disjunct predicate is “true or false,” and “A is true or false” is equivalent to the disjunctive proposition “A is true or A is false.” § 4.1.3 Now the missing qualification is that what Socrates says signifies exactly as the terms suggest. The upshot (§ 4.2) will be that all three qualifications or caveats are needed. § ad 4.1.3 The mode of reasoning in objection § 4.1.3 is not questioned in the response, on the grounds that the proposition in question is irrelevant, that is, it neither follows from the scenario nor is inconsistent with it. Heytesbury’s first rule is that in an insoluble scenario where it is not specified how the insoluble proposed signifies, the respondent should respond just as he would respond outside the scenario.116 PseudoHeytesbury (Pironet, “William Heytesbury and the Treatment of Insolubilia,” 291) disagrees, arguing that the insoluble should be doubted. This view is echoed by John Hunter (see introduction, § 3, and the comment on § 1.12.1 above): § ad 4.1.2 Paul’s

For example, suppose that this Socrates is the only Socrates, and that he utters this proposition and no other proposition, “Socrates says a falsehood,” and then “Socrates says a falsehood” is proposed. To this one should respond by expressing doubt, and the reason is that it is consistent with the scenario both that it is true and that it is false.117

The reasoning seems to be similar to that in § 1.12.5 above (drawn from Peter of Mantua): we are not told how the insoluble signifies, and it might signify that God exists, or that he does not, so might be either true or false—we do not know which.  See Heytesbury, Insolubilia, trans. Spade, 47; ed. Pozzi, 238: “respondendum est ad illud, proposito illo insolubili, omnino sicut respondetur ad illud non supposito illo casu.” 117  Pironet, “William Heytesbury and the Treatment of Insolubilia,” 303: “Verbi gratia, suppono quod iste Socrates sit omnis Socrates, et quod ipse dicat istam propositionem et nullam aliam ‘Socrates dicit falsum,’ tunc proponatur istam ‘Socrates dicit falsum.’ Ad illam respondendum est dubitative, et causa est quia stat cum casu quod illa sit vera et quod illa sit falsa.” 116

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The same response is invoked again in § 5.4.2 below. Sinkler (“Paul of Venice on Obligations,” 484–5) notes that Paul himself invokes a similar principle in the obligations treatise in the Logica magna (Tractatus de obligationibus, 72–3) when arguing against the responsio nova. Burley seems indeed to equate the scenario with what is in fact true: Suppose Socrates is black, and suppose it is posited that Socrates is white. The scenario, however, does not obligate but rather makes definite. And because I can be definite about the truth of one opposite and maintain the other opposite as true, the positio that posits that Socrates is white can be admitted even though the truth of the matter, as was already said, is that Socrates is black.118

However, Paul gives a fuller discussion in Logica magna (Tractatus de obligationibus, 190–3). § 4.2 For the insolubles arising from the twofold types of acts of ours—namely, interior and exterior—see the first division in chap. 2, § 2.1.1. When Paul describes the insolubles as “indeterminate,” perhaps what he means is that the above cases were indeterminate and so not necessarily insolubles, but needed to be made more precise, as he said at the end of the sixth division of chap. 2 (see § 2.1.6). For particular or indefinite insolubles, see § 6.1. When Paul writes that “one this man is the only this man” (unus iste homo est omnis iste homo) what he means is that only one man is referred to by the singular term “this man” (the discrete term, as Paul would have called iste homo). That is ensured by the fact that “this man” is a singular term, referring by definition to only one man. So the caveat is not needed. The quibblers mentioned in the final paragraph of this section point out, first, that when Socrates says, “Socrates says a falsehood” he does not only say that, but he also, of necessity, says, “Socrates”; secondly, that when Paul supposes that “Some proposition is false” is the only proposition, that is false, for in saying “Some proposition is false,” the speaker has also apparently uttered “Some proposition is.” So do not take me literally, Paul responds. Take me to be saying that some propo Burley, “Obligations,” § 3.03, 378–9; for the Latin text, see Green, An Introduction to the Logical Treatise De Obligationibus, vol. 2, 46: “Sit Socrates niger, et ponatur Socrates esse album. Sed casus non obligat sed certificat, et quia possum esse certus de veritate unius oppositorum et sustinere reliquum pro vero, potest admitti positio quae ponit Socratem esse album, prius dicto in rei veritate: ‘Socrates est niger.’” See also Yrjönsuuri, Obligationes, 309. 118



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sition is the only whole or complete proposition uttered. “Some proposition is” would be a whole proposition if uttered alone on another occasion, but here it is only part of a proposition. § 4.2.1.2 Bocheński (History of Formal Logic, § 35.12) translates much of chap. 4, including this paragraph. However, he has “the antecedent is A” where the text (in both manuscript and edition) reads sed consequens est a, that is, “the conclusion is A.” Indeed, the whole passage is confused in his translation. Note that the exact significate of both the premise and the conclusion is true, for it is the claim that A is false. § 4.2.1.3 Paul has already observed in § 4.2.1 that A is false, because it clearly falsifies itself. The opponent’s aim here is to show that A is true (instead or as well, it does not matter). The opponent takes it as obvious that if the contradictory of A is false, then A is true, by the rule of contradictory pairs (see comment on §§ 1.12.4 and 3.4). Paul will deny this in his response (§ ad 4.2.1.3). The bulk of the objection consists in showing that the contradictory of A (i.e., “No Socrates says a falsehood”) is false. The second argument (“Or like this …”) is another way to prove that “No Socrates says a falsehood” is false, relying on the observation already made that A is false. “No Socrates says the falsehood A” is false because the scenario specifies that Socrates says A. § ad 4.2.1 Although, according to Paul, if a proposition is true (insoluble or not), its exact significate is true, if it is false we cannot infer that its exact significate is false. For his account of truth requires that it does not falsify itself even if its exact significate is true, and so it may be false, if it falsifies itself, even if its exact significate is true. See §§ 2.2.1–2. § ad 4.2.1.1 Paul’s discussion of the failure of such inferences is in Logica magna: Capitula de conditionali et de rationali, 11–19. For example, Paul says that in “If you are a man you are an animal and conversely,” “conversely” is a conditional even though it lacks an antecedent, a consequent, and a conditional sign: “conversely” here signifies that if you are an animal you are a man, but not by the composition of its parts. (See Hughes’s note 35 on 218–19. See also the fourth, fifth, and sixth conclusions in §§ 2.3.4–6 above.) The qualification that “the inference signifies only by the composition of its parts” refers to the fifth conclusion of chapter 2, at § 2.3.5. Recall that the fifth conclusion stated, “some formally valid inferences signify exactly by the composition of their principal parts, where the premise is true and the conclusion is false.” So we cannot infer that the conclusion is true from the fact that the inference is valid and the premise is true.

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Paul’s utterance of “Socrates says a falsehood” is convertible with A but is not A. “Convertible” does not mean either identical or equivalent. As we have seen, two propositions may be convertible (namely, if they use the same words to say the same thing about the same thing—see § 2.3.7) but have different truth-values, as here. A is false, but Paul’s statement, which effectively says that A is false, is true. § ad 4.2.1.3 What Paul describes here as a “fundamental principle” is his second conclusion of chap. 2, at § 2.3.2. § 4.2.2 This is very similar to what is sometimes dubbed the “no-no paradox” (which will come at § 4.2.4.3), in which two propositions, A, which says that B is false, and B, which says that A is false, refer to one another. See Sorensen, Vagueness and Contradiction, 165. See also Swyneshed’s discussion in his Insolubilia, ed. Spade, §§ 76–7. On the “no more reason”form of argument, see the comment on § 1.13.1. Note that sortes dicit falsum (“Socrates says a falsehood”) is doubly indefinite, given that Paul treats sortes as a general term (see comment on § 4.1.1). We have inserted “whole” in the translation of et nullam aliam (“and no other proposition”), in light of Paul’s discussion at the end of § 4.2. § ad 4.2.2 Paul follows Swyneshed in holding that in the no-no paradox (and its variants) both propositions falsify each other. Swyneshed writes: A proposition falsifying itself indirectly is a proposition signifying principally as it is or other than it is and that so signifying falsifies another proposition falsifying it. An example: let A be a proposition signifying principally that B is false, and let B be a proposition signifying principally that A is false. And let there be only one A and only one B. Then on this assumption, it cannot be claimed that each is true, but it is necessary to claim that one is false. Nor can there be any better reason to say that A is false than that B is false, or vice versa. Therefore, in the scenario described, it is necessary to claim that each is false. And in this way from A it follows that B is false, and from B it follows that A is false. So from A it follows that A is false indirectly via B, and from B it follows that B is false indirectly via A. And thus A falsifies B directly and itself indirectly.119  Swyneshed, Insolubilia, ed. Spade, 182: “Propositio falsificans se mediate est propositio significans principaliter sicut est vel aliter quam est et ipsa sic significando falsificat propositionem aliam a se falsificantem se. Exemplum: Sit a una propositio significans principaliter quod b est falsum, et b una propositio significans principaliter quod a est falsum. Et sit tantum unum a et tantum unum b. Tunc hoc posito, non potest poni quod utrumque sit verum, sed oportet ponere quod alterum sit falsum. Nec potest ratio

119



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Sorensen (Vagueness and Contradiction, 166) notes that the no-no paradox seems to have a consistent, but puzzling, solution, forming a secondorder paradox: one of them is true and the other false, but there seems no way of discovering which is which. See Read (“Symmetry and Paradox,” § 3) for an argument, similar to Swyneshed’s, that this secondorder paradox gives reason to think that they are both false. § 4.2.3 Another variant of the no-no paradox. § ad 4.2.3 Again, as in § ad 4.2.2, Paul claims that A and B falsify themselves insofar as they falsify each other. § 4.2.4 One might call this the “yes-no paradox,” although it is more commonly known as the “postcard paradox.” See Haack, Philosophy of Logics, 135: a postcard has, “The sentence on the other side of this postcard is true” on one side, and, “The sentence on the other side of this postcard is false” on the other. In its no-no form, the postcard has, “The sentence on the other side of this postcard is false” on both sides. See also Swyneshed’s discussion in his Insolubilia (ed. Spade, § 78). § 4.2.4.3 For insolubles arising from the property of an expression, see § 2.1.1. By “discrete insoluble” (insolubile discretum) Paul presumably means any insoluble with a discrete term as subject (or more generally, by “discrete proposition” any proposition with a discrete term as subject). The expression does not seem to occur anywhere else. Paul gives a lengthy discussion in the third chapter of the first treatise (Logica Magna: Tractatus de terminis, 120–215) of the notion of a discrete term (terminus discretus vel singularis). Cf. § 2.3.1 and the comment on § 4.1.1 above. § 4.3 Although Paul describes the other paradoxes as derived from the familiar one as corollaries, it is really that their solutions are corollaries of the solution to the familiar one. § 4.3.1 Also discussed by Swyneshed (Insolubilia, ed. Spade, § 66). § 4.3.1.1 Ipse decipitur per te (literally, “he is deceived by you”): this is puzzling. We have translated the phrase as “he is deceived on the basis of what you have granted,” for the reasoning is based on the assumption that you grant “Socrates is deceived.” Sortes credit aliqualiter ⟨esse⟩ (“Socrates believes that it is in some way”): previously, we have encountered the phrase significat aliqualiter ⟨esse⟩ (“it assignari quare magis a est falsum quam b est falsum nec e converso. lgitur, illo casu posito, oportet ponere utrumque fore falsum. Et sic ex a sequitur b fore falsum, et ex b sequitur a fore falsum. Et ita ex a sequitur a fore falsum mediante b. Et ex b sequitur b fore falsum mediante a. Et ita a falsificat b immediate et se ipsam mediate.”

352 Commentary

signifies that it is in some way”), for example at § 3.3.2, to rule out the possibility that an expression does not signify anything at all. So too here, the addition ensures that there is something about which Socrates is deceived. On the translation of credit precise sicut est as “believes only what is the case,” see Spade’s comments in Heytesbury, Insolubles, trans. Spade, 60–4. § ad 4.3.1.1 Paul writes, “Nor was that proposition assumed in the scenario”: we speculate that the proposition in question is the premise of the argument which he claims leads to contradiction. Paul’s distinction here is reminiscent of recent discussions of belief (and other attitudes) de se: see, for instance, Lewis, “Attitudes de dicto and de se,” discussing a famous scenario proposed by John Perry about “lost Lingens,” lost in Stanford Library. Lingens has read a biography of Rudolph Lingens without, however, realizing it is a biography of himself. § ad 4.3.1.2 In § 2.2.2 Paul had clearly stated that insoluble propositions are false, insofar as they are self-falsifying, but their exact significates are true. This is the core of Paul’s diagnosis of insolubles. § 4.3.2 Secundo principaliter arguitur sic (literally, “Secondly, it is principally argued like this”): what Paul is doing in this section is to solve other insolubles closely related to the “familiar insoluble,” “Socrates says a falsehood” (see § 4.3). In §§ 4.3.1, 4.3.2, 4.3.3, and 4.3.4 he adapts his solution to a succession of such insolubles. Swyneshed discusses the current example in his Insolubilia, ed. Spade, § 69. § ad 4.3.2.1 Like Bradwardine (see comment on § 1.10 above), Paul rejects the inference from “A signifies that p” and “A is false” to “not-p” (downwards transmission of falsity), just as he rejects that from “A signifies that p” and “p” to “A is true” (upwards transmission of truth). The reason is different, of course, for Bradwardine attributes the failure of the inference to the possibility that A signifies something else and that it is that which fails, whereas Paul, following Swyneshed, attributes it to his strengthened account of truth (and correspondingly weakened account of falsity). § ad 4.3.2.2 The inference requires the extra premise that B does not falsity itself, which Paul of course will deny. § 4.3.3 This is the classic liar paradox. See Swyneshed’s discussion in his Insolubilia, ed. Spade, § 70. § 4.3.3.2 Here and in later occurrences of the formula Quod si conceditur (“if that is granted”) we have added the phrase “at the start” to make the structure of the argument clearer, on analogy with Paul’s own use at § 4.3.1.2 and § 4.3.2.2 of the phrase in principio.



Commentary353

§ 4.3.4 As noted in the introduction, § 4, the medievals seem to have taken their

inspiration for the insolubles from Aristotle’s example in De sophisticis elenchis, chap. 25, of the man “who swears that he will break his oath, and then breaks it, keeps this particular oath only” (De sophisticis elenchis, trans. Pickard-Cambridge, chap. 25, 180b1–2). See Spade and Read, “Insolubles,” § 1.3. Swyneshed’s brief discussion is in Insolubilia, ed. Spade, § 72. The twist added by the medievals is to identify the two oaths that Aristotle seems to be envisioning. § ad 4.3.4.1 Again, two contradictories are both false, as noted in Paul’s second conclusion, § 2.3.2. The insoluble is false since it falsifies itself, and its contradictory is false not because it is self-falsifying but because its exact significate is false. § 4.4 Paul concludes the chapter by observing that what he has just said about the oath-breaking insoluble applies quite generally to insolubles: they and their contradictories are both false: the insoluble because, although its exact significate is true, it falsifies itself, and its contradictory because its exact significate is false. Chapter Five: On Propositions which Appear not to be Insolubles § 5.1 Note that,

although these examples do not at first glance appear to be insolubles, and so are not obviously insolubles, what Paul shows is that, given certain scenarios, they are indeed insolubles—covert insolubles, one might say; see commentary on § 2.1.8.3. The examples are found in Swyneshed’s Insolubilia (ed. Spade, § 73), where they are examples of the third member of the division he makes there. Although Paul does not discuss the first example, “Socrates is white,” Swyneshed does so (ibid., § 87). § 5.1.1 Paul’s response in § ad 5.1.1 suggests that we should take him literally when he appears to be distinguishing the scenario from what was proposed (the positum), as is normal in obligations. Swyneshed discusses this example ibid., §§ 85–6. § ad 5.1.1 What Socrates says falsifies itself, given the scenario in which all and only sick people say a falsehood, for we can then infer from Socrates’ statement that he is sick that he says a falsehood. So Paul will deny the added minor premise stating that what Socrates says does not falsify itself.

354 Commentary § ad 5.1.2 Paul

is clearly alluding here to his second conclusion (at § 2.3.2), so when Paul says he does not consider it impossible, he surely means in any cases of insolubles, not just this one. Readers who are puzzled why Paul should add, “and so too is its contradictory” should note that § 5.1.2 and Paul’s reply (and much else in this chapter) are drawn from §§ 85–6 of Swyneshed’s treatise, where Swyneshed considers an objection: If “Socrates is sick” is granted, then like this: Socrates is sick, and every sick man says a falsehood, therefore, Socrates says a falsehood, and he only says “Socrates is sick,” therefore, “Socrates is sick” is false, and it has a contradictory, therefore, its contradictory, namely, “Socrates is not sick,” is true. Accordingly, it is as it principally signifies, and it signifies principally that Socrates is not sick, therefore, it is the case that Socrates is not sick, which is the opposite of what was granted. Solution: having admitted the scenario, it should be granted that Socrates is sick.120 And it should be granted that Socrates says a falsehood. And it should be granted that “Socrates is sick” is false because it falsifies itself. And it should be granted that it has a contradictory. And the inference, “therefore, its contradictory is true” should be denied, as is clear by the last thesis.121

That last thesis (of Swyneshed’s) is his third, corresponding to Paul’s second conclusion, rejecting the rule of contradictory pairs (RCP). Thus, in his reply to the argument in §§ 5.1.1–2, Paul implicitly replies to the full argument of Swyneshed’s by admitting that two contradictories can both be false. § 5.3 This is a classic insoluble, though not discussed by Swyneshed. See Bradwardine, Insolubilia, § 8.1, and the other authors listed in our commentary on § 2.1.8.3. This insoluble also appears in the Logica parva (trans. Perreiah, VI.8, 253), where, however, it is considered a merely apparent insoluble.  Pozzi, Il mentitore e il Medioevo, 194, § 6.011, amends the text to read concedendum est quod Sortes est aeger. 121  Swyneshed, Insolubilia, ed. Spade, 210: “Si conceditur, ‘Sortes est aeger,’ tunc sic: Sortes est aeger; et omnis homo aeger dicit falsum; ergo, Sortes dicit falsum; et solum dicit illam ‘Sortes est aeger’; ergo, haec est falsa ‘Sortes est aeger’; et habet contradictorium; ergo, suum contradictorium, scilicet, ‘Sortes non est aeger,’ est verum. Et ultra: igitur, ita est sicut illa principaliter significat; et illa significat principaliter quod Sortes non est aeger; ergo, ita est quod Sortes non est aeger, quod est oppositum concessi. Solutio: admisso casu, haec est concedenda ‘Sortes est aeger.’ Et concedendum est quod Sortes dicit falsum. Et concedendum est quod illa est falsa ‘Sortes est aeger’ quia se ipsam falsificat. Et concedendum est quod habet contradictorium. Et neganda est illa consequentia: igitur, suum contradictorium est verum, sicut patet per conclusionem ultimam.” 120



Socrates will not receive a penny and he is or will be a Socrates”: note that according to the medieval account of ampliation, “A will be B” is understood to be true if what is A will be B or what will be A will be B. See Read, “Richard Kilvington and the Theory of Obligations,” § 4. § 5.3.2 Given the scenario in § 5.3, Socrates’ utterance is equivalent to the Truth-teller. In its classic version, it turns on the proposition, “This proposition is true.” (See Read, “Symmetry and Paradox,” § 5.) It is consistent that it is true (for then things are as it signifies), and consistent that it is false (for then things are not as it signifies). Paul here implicitly claims that it is true, in claiming that what Socrates says when he says, “Socrates says a truth” (if that is all he says, and he is the only Socrates ever, and so on) is true, and consequently, that his utterance of “Socrates will receive a penny” (again, in the circumstances specified in the scenario) is also true. Neither utterance is an insoluble, by Paul’s lights, since neither falsifies itself. See his definition at § 2.1.8. § 5.4 Another classic insoluble, though again not discussed by Swyneshed. See Bradwardine, Insolubilia, § 8.8.1, and our commentary on § 2.1.8.3. In the Logica parva (trans. Perreiah, VI.8, 254) it is listed among merely apparent insolubles. § 5.4.2 In § 5.4 Paul specifies that there is now and will be only one Socrates. But perhaps they are two different Socrates. Then if that future Socrates says something false, then the present Socrates, saying (falsely) that Socrates will not cross the bridge, says something true. In his reply, Paul observes that we need the stronger condition that this, the present Socrates, is the only Socrates (ever). Paul’s opponent appeals to a standard feature of obligational disputations when he claims that if one is in doubt about a proposition before a scenario was set up, then one should express doubt about it afterwards too. The principle must presumably be restricted to irrelevant propositions, stating generally that irrelevant propositions should receive the same response within a scenario as outside it. A similar mode of reasoning was invoked in § 4.1.3: see the discussion of that paragraph and § ad 4.1.3 above. § ad 5.4.3 Again, Paul is alluding to his first and second conclusions, §§ 2.3.1–2. § 5.5 The insolubles which Paul calls quanta (“quantified” in our translation) are universal, particular, and indefinite subject-predicate propositions (propositiones categorice) of such forms as “Every A is B,” “Some A is B,” and “An A is B.” Those he calls non quanta (“non-quantified” in our translation) that he discusses are exclusives and exceptives, of the form “Only A is B” and “Every A except B is C.” In the Logica magna,

§ ad 5.3 “That

Commentary355

356 Commentary

­ ractatus de terminis, 18–19, he observed that exclusives have no quanT tity. Exclusives and exceptives are exponible, and so are implicitly molecular propositions (propositiones hypothetice), and no molecular propositions have a (simple) quantity. Chapter Six: On Quantified Insolubles § 6.1 Whether

a proposition is singular or quantified depends on its subject. Paul writes in the second chapter of the first treatise of Part II: A particular proposition is one whose subject is a non-pregnant term122 determined by only one particular quantifier, or by several ⟨signs⟩ equivalent to it or at least implying it,123 which is part (or are parts) of the particular proposition … An indefinite proposition is one whose subject is a non-pregnant general term not determined by any quantifier or by any signs ⟨equivalent to it⟩, which is part (or are parts) of the indefinite proposition … A singular proposition is one whose subject is a discrete term without an immediate quantifier or a general term determined by only one demonstrative pronoun, which is part of the singular proposition.124

The phrase which we have translated as “which is part (or are parts) of the … proposition” (existente vel existentibus aliquid ipsius) is designed to exclude the subjects of prejacents which are determined by a particular sign or quantifier in virtue of their being dependent upon the corresponding particular proposition, so that the quantifier is not part of the prejacent, but is part of the whole proposition, namely, the particular proposition. 122  Logica magna, Tractatus de terminis, 2, describes “nothing” (nihil) as a pregnant term since “it is not inwardly simple, being subordinated to the complex ‘no thing’ (nulla res), which is composed of a categorematic and a syncategorematic.” A non-pregnant term would have no such implicit negation or other logical constituent. 123  For example, non nullus = aliquis; accordingly, non nullus homo est asinus is equivalent to aliquis homo est asinus. 124  E, fol. 109ra–b (M, fol. 139rb–vb): “Particularis propositio est illa in qua subicitur terminus non pregnans solo signo particulari determinatus vel pluribus eidem equivalentibus vel saltim ipsum inferentibus existente vel existentibus aliquid ipsius … Propositio indefinita est illa in qua subicitur terminus communis non pregnans nullo signo nec signis determinatus existente vel existentibus aliquid ipsius … Propositio singularis est illa in qua subicitur terminus discretus sine signo immediate aut terminus communis [cathegorematicus E] solo pronomine demonstrativo determinatus existente aliquid ipsius.”



Commentary357

Here is Paul’s explanation of the corresponding clause in the definition of universal propositions: Fourthly, I said “which is part (or are parts) of it” referring by “it” to the proposition, because there are propositions in which the subject is a predicative term determined only by a universal quantifier but which are not universal. For in “Every man is an animal” its prejacent, which is part of it— namely, “man is an animal”—is an indefinite proposition, but the predicative term in it is determined by a universal sign because the same term is subject both of the universal and of its prejacent. So if the subject of the universal proposition is determined by a universal quantifier, so too is the subject of the prejacent but differently, for the subject of the universal proposition is determined by some part of that universal proposition, but the subject of the prejacent is not determined by some part of the prejacent, because the universal quantifier is part of the whole universal proposition and not part of the prejacent.125

Accordingly, in particular propositions: Proof that it is determined by a particular quantifier: the subject of a particular proposition and of the indefinite prejacent is the same, but the subject of the particular proposition is determined by the particular quantifier, and so too is the subject of the prejacent. And thus it is clear that the premise is true. And that the conclusion is false is clear because the prejacent of this particular is only indefinite. Hence to infer that conclusion, it is necessary to assume that the subject in the premise is determined only by the particular quantifier as part of it. But this is false of the prejacent of the particular in that the particular quantifier is not part of it but only of the whole particular proposition.126  E, fol. 107vb (M, fol. 138ra): “Quarto dicebatur existente vel existentibus aliquid ipsius referendo li ipsius ad propositionem quia aliqua est propositio in qua subicitur terminus cathegorematicus solo signo universali determinatus et tamen non est universalis, unde in ista: omnis homo est animal, est suum preiacens quod est pars ipsius, videlicet: homo est animal, que est indefinita et tamen in ea subicitur terminus cathegorematicus signo universali determinatus quia idem est subiectum universalis et sue praeiacentis. Si ergo subiectum universalis determinatur per signum universale, a pari et subiectum preiacentis, sed in hoc est diversitas, nam subiectum universalis determinatur per aliquid istius universalis sed subiectum praeiacentis non determinatur per aliquid ipsius praeiacentis, quia signum universale est pars totius propositionis universalis et non est pars preiacentis.” 126  E, fol. 109ra–b (M, fol. 139va–b): “et quod determinetur signo particulari probatur: idem est subiectum particularis et preiacentis indefinite, sed subiectum particularis determinatur signo particulari, igitur et subiectum preiacentis; et sic patet quod antecedens est verum. Et quod consequens sit falsum patet quia preiacens istius particularis est

125

358 Commentary

So one might assume that “Socrates says a falsehood” is singular, since its subject is a discrete term (even though its predicate is quantified). However, as noted above (see comment on § 4.1.1), Paul denies that “Socrates” is a singular term, claiming that it is a general term just like “human” (homo). Certainly, “A falsehood exists” is quantified (and indefinite) since its subject is an indefinite term. There is no perfect translation of particularis: strictly, it means “partial” (the Latin translation of Aristotle’s Greek). But to speak of a partial proposition would be misleading. The standard translation is “particular,” but that is unfortunately ambiguous, usually meaning a certain proposition, whereas here what is meant is any proposition of a certain type or form, namely, “Some A is B.” § ad 6.1.1 Note that Paul defines an inference as a molecular proposition in Logica magna, Capitula de conditionali et de rationali, 79: An entailment proposition (propositio rationalis) is a molecular statement that implicitly or explicitly joins two propositions together by an entailment sign, which is commonly expressed as “hence” or “therefore.”127

That is, an inference (consequentia) consists of at least two propositions. The final clause of § ad 6.1.1 denies the third rule in Part IV of the chapter on inferences in the Logica Magna, as noted in our comment on § 2.3.5 above. § 6.1.2 Note that “Some particular proposition is false” is a particular proposition, and “An indefinite proposition is not true” is an indefinite proposition. Then we have: (1) Aliqua propositio particularis est falsa. (2) Propositio particularis est falsa.

The particular proposition (1) is false since it is an insoluble, and accordingly the corresponding indefinite (2) is true. Conversely, (3) Propositio indefinita est falsa. (4) Aliqua propositio indefinita est falsa. indefinita solum. Ad concludendum ergo illud consequens oportet sumere in antecedente quod illud subiectum determinatur solo signo particulari existente aliquid ipsius. Modo hoc est falsum de praeiacente particularis eo quod signum particulare non est pars eius sed bene totius particularis.” 127  On the following page (80) he writes: “Since the word ‘entailment’ (rationalis) can be replaced by ‘inference’ (consequentia), I shall now stop using it and in this chapter I shall adopt this alternative terminology which other people use.”



Commentary359

The indefinite proposition (3) is false, and its corresponding particular (4) is true. § 6.2 This sophism is presented by Swyneshed, Insolubilia (ed. Spade, § 56). He gives essentially the same proof that B cannot be granted, but a more explicit proof that B cannot be denied or doubted: If it is denied or doubted, on the contrary: A is true, and no other proposition besides A is true, therefore, every truth is A. The major is clear by the scenario, and the minor follows ⟨from it⟩. This is proven like this: the opposite of the minor is a possible proposition, and if it is supposed with the scenario it happens to be the opposite of the scenario, therefore, the minor follows from the scenario. The claim is proven like this: assume the opposite of the minor with the scenario and argue: some assume is true other than A, and there is no proposition other than A besides B, therefore, B is true; and B principally signifies that every truth is A, therefore, so it is, and B is not A, nor is A B, therefore, B is not true, which is the opposite of one part of the scenario.128 § ad 6.2 This is Swyneshed’s reply in his Insolubilia § ad 6.2.1 The disproof in § 6.2.1 was of B’s being

(ed. Spade, § 57). false, and the inference he denies is: “If, however, one answers that B is false and B signifies only that every truth is A, then not every truth is A.” Again, one needs to add that B does not falsify itself, which of course Paul denies. § 6.2.1.1 See § 2.3.5. § 6.2.2 Swyneshed presents much the same example (ibid., § 58). However, besides specifying what true propositions A and B are—namely, “God exists” (Deus est) and “A man exists” (Homo est)—Swyneshed takes C to be “Every universal proposition is unlike them ⟨in truth-value⟩” (Quaelibet propositio universalis est dissimilis istis). This avoids the oddity that might be seen in Paul’s example, where clearly A and B are propositions but are not unlike themselves in truth-value. But see § 6.2.2.1, where this is acknowledged. Swyneshed says explicitly that istis refers to A and B (pono quod per ly istis solum demonstrentur primae duae).  Swyneshed, Insolubilia, ed. Spade, § 56, 200: “Si negatur vel dubitatur, contra: a est verum; et nulla alia propositio ab a est vera; igitur, omne verum est a. Major patet per casum. Et minor est sequens. Quod probatur sic: Oppositum minoris est propositio possibilis; et si illa ponatur cum casu accidit oppositum casus; ergo, minor sequitur ex casu. Et assumptum sic probatur. Et ponatur oppositum minoris cum casu et arguitur: Aliqua est propositio vera alia ab a; et nulla est alia propositio ab a nisi b; igitur, b est verum; et b principaliter significat quod omne verum est a; igitur, ita est; et nullum b est a, nec a est b; igitur, b non est verum, quod est oppositum unius particulae casus.” 128

360 Commentary § 6.2.2.1 On the “no more reason” form of argument, see the comment on § 1.13.1.

When Paul says there is no reason to single out one rather than any other, he is arguing from “Not every proposition is unlike A and B” to “No proposition is unlike A and B.” Since the only other proposition is C, this entails that C is not different in truth-value from A and B, and so C is true just as A and B are. The reasoning seems unnecessarily convoluted. § ad 6.2.2.2 The distinction between insolubles whose signification is totally or partially directed to themselves is explained in Paul’s third and fourth distinctions, §§ 2.1.3–4. § 6.3.1 Paul is referring to his discussion in § ad 6.2.2.2. § ad 6.3.1 B is false but should nonetheless be granted because its exact significate is true. Note that particulariter is equivalent to partialiter, meaning “partially.” Cf. the comment on particularis in § 6.1. § 6.4 We have amended B, given in both M and E as homo est (“A man exists”), to read homo est animal (“A man is an animal”), in line with Buridan’s tenth sophism in chap. 8 of his Sophismata (Summulae de dialectica, 976; Summulae de practica sophismatum, 163–4), and the same in Marsilius of Inghen’s Insolubilia (Pozzi, Il mentitore e il Medioevo, insoluble 265, 361). Chapter Seven: On Non-Quantified Insolubles translation of non quanta as “non-quantified,” see the comment on § 5.5. Paul claims that exclusives and exceptives have no quantity. See Logica magna: Tractatus de hypotheticis, 4: just as the temporal connective in a temporal proposition (for example, “when” in “When Socrates is running Plato is disputing”) “cancels quantification, ⟨so too⟩ do exclusive, exceptive, and reduplicative terms” (immo potius impedit sicut dictiones exclusivae exceptivae et reduplicativae). § 7.1.2.1 A is still “God exists,” while “Only Plato says a truth” is B, as in insoluble 261 in Pozzi (Il mentitore e il Medioevo, 361). In that case, B falsifies itself in the same way. § 7.1.3 As it stands in the manuscript and edition, this paragraph is puzzling. Without a scenario (sine casu, as we find in both M and E), there are false propositions which are not exclusives, in which case “Only an exclusive is false” is simply false. Given a scenario (cum casu) it is possible to ensure that “Only an exclusive is false” is an insoluble, for instance by stipulating that it is the only exclusive proposition. Sine casu § 7.1 For the



Commentary361

can plausibly be taken as a scribal slip for cum casu, which is the emendation we propose. Pozzi (ibid., 360) finds this particular insoluble only in John of Holland’s Insolubles (see John of Holland, John of Holland: Four Tracts on Logic, 135), and in a commentary on John’s treatise.It is also listed in a Compendium of John Dumbleton’s solution to insolubles: see Bartocci and Read, “John Dumbleton on insolubles.” John provides a scenario in which there are only three propositions: A (“God exists”), B (“A man is an animal”), and C (“Only an exclusive is false”). Recall that Paul defines an insoluble as “a proposition having reflection on itself wholly or partially implying its own falsity or itself not to be true” (§ 2.1.8). Only in some such scenario does “Only an exclusive is false” satisfy this definition and is a partially self-directed insoluble, which therefore should be denied, as Paul says in § ad 6.2.2.2. § 7.2 When Paul infers that A is false from there being no proposition except A that is false, he relies on the rule that “No B except A is C” entails that A is C. See the comment on § 1.14.3.6.1 above. By the rule stated there, “No B except A is C” is inconsistent with “No B is C” and so entails “A is C.” Equivalently, since A is a negative exceptive of the first type, where the exception and what is excepted are placed in the same part of the proposition, A must be expounded by three propositions, of which the first is “A is C.” For Paul’s account of the exposition of exceptive propositions, see commentary on § 1.14.3.6. § 7.2.1 Presumably, by the “same scenario” Paul means that “Every proposition except A is a truth” is A and is the only proposition, and that it signifies only as the terms suggest. § 7.2.2 Note that, though this is not stated in the scenario, Paul is assuming that C is the only exceptive, as he invokes this assumption later in the paragraph. § ad 7.2.2 The missing premise “pointed out in the other case” is the one in § ad 7.2. § 7.2.3 The point is that two propositions are convertible if they have the same exact significate, but one can be true and the other false if their exact significate is true but the latter falsifies itself. See comment on § ad 4.2.1.1. Chapter Eight: On Merely Apparent Insolubles similar categorization in his Insolubilia (ed. Spade), V § 100, 215–6. § 8.1.1 The sophism “This is not known by you” is called the “knower paradox” in contemporary literature. Paul briefly dealt with the similar ­proposition, § 8.1 See Swyneshed’s

362 Commentary

“This is unknown to you” (hoc est nescitum a te) in his fourth conclusion, where however he described it as an insoluble: see § 2.3.4 and the commentary on it. Swyneshed had classified “This is unknown to you” as an insoluble, on the grounds that it is relevant to inferring itself to be unknown (Insolubilia, §§ 80–1). Paul’s argument that A cannot be denied is clear enough: if “A is not known by you” were denied, it would follow that A was known by you, that is, you know that A is not known by you, from which it follows (by the factivity of knowledge) that A is not known by you—which you denied. But why can A not be doubted? Paul omits the argument, which presumably should run as follows: if you doubt A when not obligated, then you do not know A (that is the normal rule for irrelevant propositions which are unknown). But “You do not know A” is just A, so you have granted A after all, when you claimed to doubt it. The exact phrase obligatio facta (translated here as “current obligation”) occurs several times in the treatise on obligations (Logica magna: Tractatus de obligationibus, 276–8, 346, 354, 376), together with positio facta and impositio facta. In each case it refers to the current obligation, whether a case of positio or impositio. § 8.1.2 Note the move from “you answer that A is not known by you” to “you know that this is not known by you, referring to A,” that is, from knowledge de dicto to knowledge de re. § 8.1.3 There was a long debate in the fourteenth century about the object of scientific knowledge.129 Three different theories were prominent: the so-called res theory, that knowledge has things as its object, held, for example, by Walter Chatton; the complexum theory, that the object of knowledge is the mental proposition, held, for example, by Ockham; and Adam Wodeham’s and Gregory of Rimini’s theory of the complexe significabile, that the object of knowledge is some sort of propositional significate. Paul settles the issue by nominally accepting Gregory’s theory but taking literally Gregory’s protestations that the complexe significabile, or in his terms, the exact significate, is nothing more than things being in some way (see thesis 5 in the chapter “On the Significate of the Proposition” in the Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 197). Thus, for Paul, to 129  See Nuchelmans, Theories of the Proposition, chaps. 11–13; Nuchelmans, Late-­ Scholastic and Humanist Theories of the Proposition, § 7.2; and Tachau, Vision and ­Certitude in the Age of Ockham, Part III.



Commentary363

know a proposition is to know its primary or exact significate. See Logica magna: Tractatus de scire et dubitare, 162–3, where Paul says that the proposition “You know proposition A” is a describable proposition (propositio descriptibilis), that is, one in which a hyper-intensional term (terminus officiabilis) is applied to a proposition.130 Accordingly, Paul gives his account of “I know proposition A” in the chapter on hyperintensional terms: Hence this proposition, “I know proposition A” is described and proven like this: I know the primary significate of proposition A, which I know is primarily signified by A, therefore I know proposition A.131

Recall the passage from Paul’s Quadratura cited in the comment on § 1.8 above, which equates the primary or immediate significate (significatum primarium/primo) with its exact significate (significatum adequatum). To know A is to know its immediate, primary, or exact significate. The claim that every truth can be known is nowadays called the “principle of knowability.” It lies at the heart of Fitch’s paradox (see Fitch, “A Logical Analysis of Some Value Concepts”). Fitch’s paradox is paradoxical only in that it provides a counterexample to the principle of knowability. That a contradiction follows from the assumption that A is known was shown in § 8.1.2. That it is false that A is known by you was shown in § 2.3.4, and in the treatise “De rationali” (Logica magna: Capitula de conditionali et de rationali, 197 and 199–201). § ad 8.1.1 Paul accepts that A asserts that it is itself unknown, but unlike Swyneshed, does not extend the notion of insoluble to include such cases. Note that A can be known (see § ad 8.1.3), only not by you; hence we have added “by you” in the last sentence, and again in § ad 8.1.2. 130  See E, fol. 70va (M, fol. 89vb): “Propositio descriptibilis ergo est illa in qua aliquod predictorum verborum vel participiorum existens primum probabile in propositione incomplexum determinat” (“A describable proposition is one in which one of the said verbs or participles, occurring as the first provable ⟨term⟩ in the proposition, qualifies a simple term”). See also Paul of Pergula, Logica and Tractatus de sensu composito et diviso, 78: “Propositio descriptibilis est in qua terminus concernens actum mentis determinat incomplexum” (“A describable proposition is one in which a term involving a mental act qualifies a simple term”). 131  E, fol. 70va (M, fol. 89vb): “Unde hec propositio: ego scio a propositionem, sic describitur et probatur. Ego scio significatum primarium a propositionis quod scio primarie significari per a, ergo scio a (propositionem M).” For alternative translations, see Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 275 n. k, and Logica magna: Tractatus de scire et dubitare, xvii.

364 Commentary § ad 8.1.3 Paul

here rejects the principle of knowability for reasons similar to those offered by Fitch in his eponymous paradox (as discussed in Read, “Thomas Bradwardine and Epistemic Paradox”). For that reason, Paul rejects the very first inference in § 8.1.3, that if A is true, then A can be known (under its standard interpretation), since he rejects the major premise, that every truth can be known (under its standard interpretation). Note that he switches terminology slightly, from sic precise significando to sic adequate significando, which we have tried to reflect in the translation. § 8.1.4.3 As noted, this follows immediately from Paul’s analysis in § ad 8.1.3. Paul once again explicitly rejects the principle of knowability. § 8.2.1 The argument against granting A simply goes from what has been granted (namely, “This should be denied by you”) and the fact that it was granted to the conclusion that you responded incorrectly—by granting something that should have been denied. The argument against denying A shows first that A should be denied, and then that it should not be denied, which is the opposite of what was granted, namely, that A should be denied. § 8.2.3 The reference to the second argument is to § 8.2.2. Paul appears to be appealing here to the tenth rule from his “Obligations”: Every false proposition which does not follow, and is known not to, should be denied during the period of the obligatio; and every true proposition which is not inconsistent should be granted.132

In her note to this passage (Logica magna, Tractatus de obligationibus, 65 n. 10) Ashworth points out that the rule is found in Strode’s treatise, one of Paul’s sources, as we mentioned in the introduction. Although, as Ashworth observes, the rule is not standardly given explicitly, it follows immediately from the standard rules of the responsio antiqua. Indeed, it is invoked by the author of the treatise attributed to William Sherwood: For example, [suppose] it really is the case that Socrates does not exist. “A” names [the proposition] that Socrates is running or you are standing, “B,” that Socrates is moving or you are not standing. Let A and B be posited together. Then [propose this]: “Socrates is moving.” It seems as if this must  Logica magna, Tractatus de obligationibus, 64–5: “Omne falsum non sequens scitum esse tale infra tempus obligationis est negandum et omne verum non repugnans est concedendum.” 132



Commentary365

be denied since it is false and does not appear to follow [from the conjunction in the positio] … Similarly, “A” names [the proposition] that “you respond badly” and “you do not know that you respond badly” are alike [with respect to truth-value], and “B,” that you are awake. Let it be posited that A and B are alike [with respect to truth-value]. Then [let this be proposed]: “you are awake.” This is true and not incompatible [with anything previously granted]; therefore, it is to be granted.133

Although the inferences in accord with the rule (“This must be denied since it is false and does not appear to follow,” “This is true and not incompatible, therefore, it is to be granted”) are used to establish a sophism, it is not these inferences as such that are rejected by the author; rather, in the first case, the assumption, as he writes that “it does not appear to follow” (when in fact it does) and in the second, a mistaken move later in the sophistical argument. § ad 8.2.1.1 The convertibility between a true or necessary insoluble and a false or impossible insoluble was Paul’s seventh onclusion: see § 2.3.7. Paul might seem undecided as to whether this is an insoluble or not (see comment on § 8.1.1). He starts by simply likening the case to that of insolubles, but ends the paragraph by denying that A has nothing to do with insolubles. Should we take him at face value: that this is like an insoluble, but not one, and so is an example of a merely apparent insoluble; or as claiming that this is an insoluble after all? § ad 8.2.3 The disproof was rejected in § ad 8.2.2. § aliter ad 8.2 The alternative response seems to be an afterthought: see Paul’s comment at § aliter ad 8.2.3. The examples here are supposed to be merely apparent insolubles. The reference to where this was shown in his “Obligations” is Logica magna: Tractatus de obligationibus, 219. § aliter ad 8.2.2 Note that the argument he rejects here is not the one rejected in § ad 8.2.2 but the argument against denying A: A should be denied by you, and A signifies only that A should be denied by you, therefore it should be denied by you that A should be denied by you. 133  Sherwood, “William of Sherwood’s Treatise on Obligations,” 256–7. Cf. Green, An Introduction to the Logical Treatise De Obligationibus, §§ 1.40 (196) and 1.46 (199): “Verbi gratia, in rei veritate Socrates non est. Appellat A: Socratem currere vel te stare, B: Socratem moveri vel te non stare. Componantur A et B. Deinde: Socrates movetur. Haec videtur esse neganda, cum sit falsa et non videtur sequi … Item appellat A: te male respondere et te nescire te male respondere esse similia, B: te vigilare. Ponatur A et B esse similia. Deinde: tu vigilas. Hoc est verum non repugnans, ergo est concedendum.”

366 Commentary § 8.3.1 The

argument against denying A is puzzling. The idea seems to be this: we have shown that you should not grant A, since you should not grant something in doubt for you when you are not obligated. For the same reason, you should not deny A, since you should not deny something in doubt for you when you are not obligated, unless you can show some difference between the two cases. § ad 8.3.1 Something false or impossible may be granted in the case of insolubles, even when one is not obligated, on the grounds that its exact significate is true, and for the same reason in the present case something in doubt may be granted.

Appendices

368 Appendices

A. Pauli Veneti Quadratura Quatuor formabo dubia† … primo utrum eadem consequentia sit bona et mala;1 secundo utrum eadem propositio sit vera et falsa; tertio utrum de eodem sint verificabilia disparata;2 quarto utrum duo repugnantia possint esse simul vera vel simul falsa … Primum Dubium⟨: utrum eadem consequentia sit bona et mala ⟩ 1.15* Capitulum de insolubilibus.3 1.15.1 Quintodecimo principaliter4 ad questionem arguitur sic:5 ista consequentia est bona: a significabit precise quod quodlibet verum erit falsum, ergo a erit falsum: et hec eadem non valet, igitur questio vera. 1.15.1.1 Tenet consequentia et antecedens probatur. Et primo pro secunda parte, nam casu possibili posito antecedens est verum et consequens falsum, igitur consequentia non valet. Antecedens probatur: et pono quod quamdiu a erit, a significabit precise quod quodlibet verum erit6 falsum, et ita erit quod quodlibet verum erit7 falsum8 quamdiu a erit.9 Isto posito antecedens est verum per casum, et quod consequens sit falsum probatur, nam10 quamdiu a erit, a erit11 verum, igitur12 non erit falsum. Antecedens probatur, nam13  B 1ra, F 1ra, Q 2ra, V 65ra  mala] non bona F 2  verificabilia disparata] inv. F * B 10ra, F 7vb, Q 8rb, V 73rb 3  Capitulum de insolubilibus] om. B 4  principaliter] om. F V 5  sic] om. B Q 6  erit] est F 7  erit] est F 8  et ita – erit falsum] om. (hom.) V 9  et ita – a erit] om. B 10  nam] quoniam V 11  erit] om. F 12  igitur] a add. F 13  nam] quam F †

1



Appendices369

A. Paul of Venice, Quadratura I will formulate four doubtful questions … firstly, whether the same inference can be both valid and invalid; secondly, whether the same proposition can be both true and false; thirdly, whether disparate things are verifiable of the same thing; fourthly, whether two incompatibles can be both true or both false … First Doubtful Question ⟨i.e., whether the same inference can be both valid and invalid ⟩ 1.15 Chapter 15: On Insolubles1 1.15.1 Regarding the ⟨first⟩ doubtful question, I argue like this: This inference, “A will signify only that everything ⟨that is or will be⟩2 true will be false, so A will be false,” ⟨call it B,⟩ is valid, and that inference ⟨B⟩ is invalid, so the question ⟨is⟩ true. 1.15.1.1 The

argument is valid and I prove the premises, and first, the second premise: for there is a possible scenario in which the premise of inference ⟨B⟩ is true and its conclusion false, so inference ⟨B⟩ is not valid. I prove the premise: let us assume that as long as A will exist, A will signify only that everything true will be false, and ⟨that⟩ it will be the case that everything true will be false as long as A will exist. On this assumption, the premise ⟨of B⟩ is true according to the scenario; and I prove that the conclusion ⟨of B⟩ is false, because as long as A will exist, A will be true, so it will not be false. I prove the premise, because as long as A will exist, it will be the case that everything true will be false, and as long as A will exist, A will signify only that everything true will be false, therefore as long as A will exist, A will be true.

370 Appendices

quamdiu a erit, ita erit quod quodlibet verum14 erit falsum, et quamdiu a erit, a significabit precise quod quodlibet verum erit15 falsum, igitur quamdiu a erit, a erit verum.16 1.15.1.2 Sed iam probatur prima pars antecedentis, videlicet quod illa consequentia est bona, quoniam17 oppositum consequentis repugnat antecedenti. Hec enim repugnant: a significabit precise quod quodlibet verum erit falsum, et: a non erit falsum. Probatur, et facio istam consequentiam: quodlibet verum erit falsum, a non erit falsum,18 igitur a non erit verum.19 Ista consequentia tenet in quarto secunde figure, et quamdiu a erit,20 erit ita sicut significatur per antecedens, igitur quamdiu erit a, erit21 ita sicut significatur per consequens, et ita a erit et non erit verum nec falsum, quod erit impossibile. Confirmatur sic: a non erit falsum, et a erit, igitur a erit verum quamdiu erit; arguo ergo22 sic: omne verum erit falsum, a erit verum, igitur a erit falsum. Ista consequentia est bona, et quamdiu a erit,23 erit ita sicut significatur per antecedens, igitur quamdiu a erit,24 erit ita sicut significatur per consequens, quod repugnat secunde parti principalis copulative facte ex antecedente et opposito consequentis.

 quod quodlibet verum] verum quod quodlibet F  erit] est Q 16  igitur quamdiu – erit verum] om. Ba.c. F Q V 17  quoniam] quia V 18  falsum] verum B 19  verum] falsum B 20  a erit] est a F 21  ita sicut – a erit] om. (hom.) V 22  ergo] ego F 23  a erit] inv. F V 24  a erit] inv. F 14 15



Appendices371

1.15.1.2 But

now I prove the first premise ⟨of the argument⟩, namely, that inference ⟨B⟩ is valid, because the opposite of the conclusion is incompatible with the premise. For these are incompatible: A will signify only that everything true will be false and A will not be false. Proof: I form this syllogism: Everything true will be false, A will not be false, therefore A will not be true. This syllogism holds in Baroco, and as long as A will exist, it will be as the premises signify. Hence, as long as A will exist, it will be as the conclusion signifies. So A will exist and will be neither true nor false, which will be impossible. That is shown in this way: A will not be false, and A will exist, so A will be true as long as it will exist. Hence I argue like this: Everything true will be false, A will be true, therefore A will be false. This syllogism holds ⟨in Darii⟩, and as long as A will exist, it will be as the premises signify, therefore as long as A will exist, it will be as the conclusion signifies, which is incompatible with the second conjunct of the conjunction composed of the premise and the opposite of the conclusion ⟨of the original inference B⟩.

372 Appendices 1.15.2.1 Circa

materiam huius rationis pono quattuor conclusiones, quarum prima est ista: a significabit precise quod quodlibet album erit nigrum, et ita erit (V 73va) quod quodlibet album erit nigrum, et tamen a numquam erit verum. Probatur:25 et pono quod a sit ista propositio: quodlibet album erit nigrum, (F 8ra) que significat et significabit precise quamdiu erit quod quodlibet album erit nigrum et26 quamdiu a (B 10rb) erit27 non erit ita quod quodlibet album erit nigrum,28 sed post desinitionem29 a30 erit ita quod quodlibet album erit nigrum. Isto posito patet prima pars conclusionis. Sed secundam probo, nam quamdiu a erit a significabit precise quod quodlibet album erit nigrum, et quamdiu a erit, erit31 ita quod non32 quodlibet album erit nigrum per casum, igitur quamdiu a erit, a erit33 falsum, et per consequens numquam erit verum.

 Probatur] om. V  et] quod add. F 27  a erit] inv. F 28  et quamdiu – nigrum] om. (hom.) V 29  desinitionem] decisionem V, desitionem Q passim 30  a] eius V,] om. F 31  erit] a F 32  et quamdiu a erit, erit ita quod non] sed post desinitionem a erit ita quod V 33  erit a erit] est F 25 26

1.15.2.1 Concerning

is this:

Appendices373

this reasoning I present four conclusions, of which the first

A will signify only that everything white will be black, and it will be the case that everything white will be black, and yet A will never be true. Proof: assume that A is the proposition: Everything white will be black, which signifies and will signify only as long as it will exist that everything white will be black; and ⟨that⟩ as long as A will exist it will not be the case that everything white will be black, but after the destruction of A it will be the case that everything white will be black. On this assumption, the first part of the Conclusion is clear. But I prove the second part,3 for as long as A will exist, A will signify only that everything white will be black, and as long as A will exist, it will be the case that not everything white will be black, according to the scenario, therefore as long as A will exist, A will be false, and consequently it will never be true.

374 Appendices 1.15.2.1.1 Ex

ista conclusione sequitur correlarie34 quod ista consequentia non valet: a significabit precise quod quodlibet verum erit falsum, et ita erit35 quod quodlibet (Q 8va) verum erit falsum, igitur a erit verum. Posito enim36 quod a sit37 illa propositio: quodlibet verum erit falsum, sic38 precise significans quamdiu erit, et quod post desinitionem a39 erit ita quod quodlibet verum erit falsum, quod non erit ita40 ante desinitionem ipsius41 a, antecedens est42 verum, et consequens falsum,43 tamen bene sequitur: a significabit precise quod quodlibet verum erit44 falsum, et tunc ita erit quod quodlibet verum erit45 falsum, igitur a erit verum.46 Etiam sequitur: a significabit precise quod quodlibet album erit47 nigrum, et tunc ita erit quod quodlibet album erit nigrum,48 igitur a erit verum.

 correlarie] om. B V  erit] est F 36  enim] om. F 37  sit] est V 38  sic] sicud V 39  a] om. F 40  quod non erit ita] nam B, et non V 41  ipsius] om. B 42  antecedens est] inv. F V 43  falsum] et add. V 44  erit] est F 45  erit] est F 46  et tunc – verum] om. V 47  erit] est F 48  et tunc – nigrum] om. (hom.) F 34 35



Appendices375

1.15.2.1.1 From

this conclusion there follows as a corollary that this inference is not valid: A will signify only that everything true will be false, and it will be the case that everything true will be false, so A will be true. For suppose that A is the proposition: Everything true will be false, signifying only in that way as long as it will exist; and that after A’s destruction it will be the case that everything true will be false, which will not be the case before the destruction of A. The premises are true and the conclusion false. Nonetheless, this is valid: A will signify only that everything true will be false, and at that time it will be the case that everything true will be false, so A will be true. This is also valid: A will signify only that everything white will be black, and at that time it will be the case that everything white will be black, so A will be true.

376 Appendices 1.15.2.2 Secunda

conclusio est ista:49 aliqua est propositio significans solum cathegorice principaliter que aliquando significabit ypothetice principaliter,50 et tamen nulla in ea51 fiet mutatio, nec nova adveniet illi impositio. Ista conclusio patet de ista: omnis propositio est falsa,52 ponendo quod ipsa aliquando erit53 omnis propositio; ipsa enim solum significat54 cathegorice principaliter sicut et suum contrarium, scilicet:55 nulla propositio est falsa, et quando ipsa erit56 omnis propositio significabit principaliter quod omnis propositio est falsa et quod ipsa est vera, quemadmodum et alia insolubilia, quorum significationes reflectuntur ad se totaliter.57 1.15.2.2.1 Ex ista conclusione sequitur correlarie quod non est possibile58 omnem propositionem esse falsam, et hanc: omnis propositio est falsa esse omnem propositionem, et59 significare precise quod omnis propositio est falsa. Probatur, quia si est possibile, ponatur in esse,60 et sit61 a illa:62 omnis propositio est falsa; quero tunc an63 a est verum vel falsum. Si verum, et a significat precise quod omnis propositio est falsa, igitur omnis propositio est falsa, et a est propositio, igitur a est falsum, quod repugnat primo concesso. Si64 dicitur quod a est falsum, et a precise significat65 quod omnis propositio est falsa, igitur non omnis propositio est falsa, quod est oppositum casus.66  ista] quod add. B  ypothetice principaliter] inv. V, ypothetice F 51  nulla in ea] in ea nulla F 52  falsa] falsum F 53  erit] ipsa add. V 54  solum significat] inv. F V 55  scilicet] om. F V 56  erit] om. F 57  totaliter] om. F 58  possibile] quod add. V 59  esse omnem propositionem et] om. F 60  esse] ergo add. B V 61  sit] illa add. B 62  illa] propositio add. V 63  an] si B V 64  Si] Sed F 65  precise significat] inv. F V 66  casus] Tenet ista consequentia per hoc membrum quia si a est falsum tunc non est ita sicut per a precise significatur add. B marg. 49 50

1.15.2.2 The

Appendices377

second conclusion is this:

There is some proposition signifying principally purely predicatively which at some time will signify principally in a compound way. Nonetheless, there will be no change in it, nor will any new imposition be added to it. This conclusion clearly holds of this proposition: Every proposition is false, supposing that at some time it will be the only proposition, for it signifies principally purely predicatively just as does its contrary, namely, No proposition is false, and when it will be the only proposition it will signify principally that every proposition is false and that it is true, just like other insolubles, whose significations reflect wholly on themselves. 1.15.2.2.1 From this conclusion there follows as a corollary that it is not possible for every proposition to be false and for Every proposition is false to be the only proposition and to signify only that every proposition is false. Proof: because if it is possible, suppose it is the case, and let A be Every proposition is false, then I ask whether A is true or false. If true, and A signifies only that every proposition is false, then every proposition is false. But A is a proposition, therefore A is false, which is inconsistent with what was first granted. If it is said that A is false, and A signifies only that every proposition is false, then not every proposition is false, which is the opposite of what was supposed.

378 Appendices 1.15.2.2.2 Ex

eadem conclusione cum suo correlario sequitur67 quod non est possibile quod quamdiu a erit,68 erit69 ita quod quodlibet verum erit falsum, et quod70 quamdiu a erit, a71 significabit precise quod quodlibet verum erit falsum.72 Probatur, quia si est possibile,73 ponatur ergo74 in esse, et quero si a est verum vel falsum. Si verum, et quamdiu erit significabit precise quod quodlibet verum erit falsum, igitur quamdiu a erit, erit ita75 quod quodlibet verum erit falsum.76 Arguitur tunc77 sic: quodlibet verum erit falsum, a erit verum,78 igitur a erit falsum. Ista consequentia est bona, et quamdiu a erit ipsa erit79 bona significans precise iuxta compositionem (B 10va) suarum80 partium, et quamdiu a erit, erit antecedens81 verum, igitur quamdiu a erit consequens erit verum, et per consequens quamdiu a erit, ipsum erit falsum. Si82 dicitur quod a erit falsum, contra: quamdiu a erit, ita erit quod quodlibet verum erit falsum, et quamdiu a erit significabit precise quod quodlibet verum erit falsum83, igitur quamdiu a erit a84 erit verum, et ita sequitur contradictio.

 cum suo correlario sequitur] sequitur cum suo correlario F  erit] erat F 69  erit] est F (etiam infra, sed ultra non notavimus) 70  quod] om. V 71  a] om. V 72  quod quamdiu – falsum] om. (hom.) F 73  possibile] om. F 74  ergo] om. F 75  erit ita] inv. V 76  igitur – falsum] om. (hom.) F 77  Arguitur tunc] inv. F 78  verum erit falsum, a erit verum] a erit verum erit falsum V 79  erit] om. F 80  suarum] om. F V 81  erit antecedens] inv. F V 82  si] ideo add. V, vero add. F 83  et quamdiu – falsum] om. (hom.) F 84  a] ipsa V 67 68

1.15.2.2.2 From

Appendices379

the same conclusion together with its corollary, it follows that it is not possible that as long as A will exist it will be the case that everything true will be false, and that as long as A will exist, A will signify only that everything true will be false. Proof: because if it is possible, suppose that it is the case, and I ask whether A is true or false. If true, and as long as it will exist it will signify only that everything true will be false, then as long as A will exist, it will be the case that everything true will be false. Then I argue like this: everything ⟨that is or will be⟩ true will be false, A will be true, so A will be false. This inference is valid, and as long as A will exist it will be valid signifying only according to the composition of its parts, and as long as A will exist, the premises will be true, therefore as long as A will exist the conclusion will be true, and consequently, as long as A will exist, it will be false. If it is said that A will be false, on the contrary: as long as A will exist, it will be the case that everything true will be false, and as long as A will exist it will signify only that everything true will be false, therefore as long as A will exist it will be true, and so a contradiction follows. ⟨So it is not possible that as long as A will exist it will be the case that everything true will be false, and that as long as A will exist, A will signify only that everything true will be false.⟩

380 Appendices 1.15.2.3 Tertia

conclusio est ista: possibile est omnem propositionem esse falsam, et hanc: omnis propositio est falsa, significare adequate omnem (V 73vb) propositionem esse falsam. Probatur: possibile est quod omnis propositio sit falsa, et quod hec: omnis propositio est falsa, que sit a, sit propositio ut85 est manifestum. Quo admisso, a significat adequate aliquid vel aliqualiter, sed non videtur qualiter nisi (F 8rb) quod omnis propositio est falsa, igitur etc.86 Unde admisso quod omnis propositio est falsa,8788 et quod a adequate significat89 omnem propositionem esse falsam, dico quod a est falsum. Et si arguitur sic: a est falsum, et a90 significat adequate quod omnis propositio est falsa,91 igitur non omnis propositio est falsa, nego consequentiam—licet92 teneret93 extra materiam insolubilium—sed hec debet esse minor: a significat precise quod omnis proposito est falsa, aut ista: a significat principaliter quod omnis proposito est falsa, quorum utrumque negatur, quia dicitur in casu isto quod a significat omnem propositionem esse falsam et a esse94 verum. Et hoc significatum copulativum dicitur principale significatum a,95 licet non adequatum sed solum prima pars.

 ut] hoc B V  etc] om. B 87  est falsa] om. V 88  igitur – falsa] om. (hom.) F 89  adequate significat] inv. F V 90  a] om. V 91  falsa] nego consequentiam add. F 92  licet] sed F V 93  teneret] tenet F 94  esse] est V 95  a] om. F 85 86

1.15.2.3 The

Appendices381

third conclusion is this:

It is possible for every proposition to be false and for “Every proposition is false” to signify exactly that every proposition is false. Proof: it is possible that every proposition is false, and that Every proposition is false is a proposition, call it A, as is evident. In that case, A signifies something exactly or in some way, but the only way ⟨it signifies⟩ seems to be that every proposition is false, hence ⟨A signifies exactly that every proposition is false⟩. Thus, accepting that every proposition is false, and that A signifies exactly that every proposition is false, I say that A is false. And if one argues like this: A is false, and A signifies exactly that every proposition is false, therefore not every proposition is false, I deny the inference—although it would hold outside the context of insolubles—but the minor premise should be A signifies only that every proposition is false, or A signifies principally that every proposition is false, each of which I deny, because I claim in this scenario that A signifies that every proposition is false and ⟨that⟩ A is true. This conjunctive significate is called the principal significate of A, although ⟨it is⟩ not the exact ⟨significate⟩ but only the first part ⟨is⟩.4

382 Appendices 1.15.2.3.1 Ex

ista conclusione sequitur quod possibile est quod96 quamdiu a erit, erit97 ita quod quodlibet verum erit falsum, et quam-(Q 8vb)diu a erit98 sic99 adequate significabit.100 Nam posito hoc101 et admisso, nullum sequitur inconveniens, quia in casu isto dico quod a erit falsum quamdiu a102 erit, et quod non significabit103 precise quod quodlibet verum erit falsum, sed104 bene significabit illud adequate. Sed ultra hoc significabit quod a erit verum, quod negatur, ita quod adequatum significatum erit quod quodlibet verum erit falsum,105 penes quod non dicitur a verum nec falsum, et principale erit quod quodlibet verum erit falsum106 et a erit verum, penes quod dicetur107 a falsum ratione secunde partis ipsius, igitur etc.108 1.15.2.4 Quarta conclusio est ista: aliqua sunt duo contradictoria inter se contradicentia que fient non contradictoria per solam desinitionem cuiuslibet propositionis universalis alterius ab aliqua illarum.109 Probatur: et pono quod omnis propositio universalis preter istam: omnis propositio universalis est falsa, desinet esse in illa hora, ita quod in fine hore hec propositio, que sit a: omnis propositio universalis est falsa, erit omnis propositio universalis. Isto posito patet quod a et: aliqua propositio universalis non est falsa, que sit b, sunt duo contradictoria quia habent omnes110 conditiones requisitas ad contradictionem. Sed111 quod in fine hore a et b non erunt contra quod] om. F  erit] om. F 98  erit] a add. F V 99  sic] sicut F 100  significabit] significat V 101  hoc] om. F 102  a] om. F V 103  significabit] significat V 104  sed] licet V 105  falsum] et a erit verum add. F 106  penes – erit falsum] om. (hom.) V 107  dicetur] dicitur Q 108  igitur etc] om. B 109  illarum] om. F 110  omnes] om. B 111  Sed] Secundum V 96 97



Appendices383

1.15.2.3.1 From

this conclusion it follows that it is possible that as long as A will exist, it will be the case that everything true will be false, and as long as A will exist, it will signify exactly ⟨that everything true will be false⟩. For assuming and accepting this, nothing inconsistent follows, because in this scenario I claim that A will be false as long as A will exist, and that it will not signify only that everything true will be false, but rather it will signify that exactly. But moreover, it will signify that A will be true, which I deny, so that the exact significate will be that everything true will be false, according to which A is not said to be true or false, and the principal ⟨significate⟩ will be that everything true will be false and ⟨that⟩ A will be true, according to which A is said to be false by reason of its second conjunct, hence etc. 1.15.2.4 The fourth conclusion is this: There are two mutually contradictory contradictories which will not be contradictories by the sole destruction of every universal proposition other than one of them. Proof: suppose that every universal proposition except Every universal proposition is false ceases to exist in the next hour so that at the end of the hour this proposition, call it A: Every universal proposition is false will be the only universal proposition. In this ⟨scenario⟩ it is clear that A and Some universal proposition is not false,

384 Appendices

dictoria probatur, quia tam (B 10vb) a quam b tunc erit falsum, igitur illa112 non erunt contradictoria.113 Antecedens probatur, nam quod a erit falsum manifestum est, quia erit insolubile seipsum falsificans. Et quod b erit falsum probatur, nam b114 in fine hore erit una115 particularis negativa non insolubilis cuius quelibet116 singularis est117 falsa et cuilibet supposito subiecti correspondebit una singularis, igitur b tunc erit falsum. Quod autem ⟨fuerint non contradictoria⟩ per solam desinitionem cuiuslibet alterius universalis patet, quia remanentibus aliis universalibus que iam sunt, illa118 continue erunt contradictoria, igitur quod fiant non119 contradictoria,120 hoc est per desinitionem illarum. 1.15.2.4.1 Ex ista conclusione sequitur correlarie quod licet illa duo: quodlibet verum erit falsum, et: aliquod verum non erit falsum, sint contradictoria, tamen non essent121 contradictoria si universalis fieret insolubilis. Patet quia quelibet illarum esset falsa, sed contradictorium universalis122 daretur123 per negationem prepositam toti, et illud significaret disiunctive opposito modo ad universalem, videlicet quod aliquod verum non erit falsum vel quod a non erit verum. Et hoc est verum ratione secunde partis.

 igitur illa] in ipsa F  erunt contradictoria] inv. V 114  b] om. V 115  una] om. Q 116  cuius quelibet] cuilibet F 117  est] erit V 118  illa] et add. V 119  fiant non] inv. V 120  igitur quod fiant non contradictoria] om. (hom.) F 121  essent] erunt F 122  universalis] om. B 123  daretur] datur F 112 113



Appendices385

call it B, are two contradictories because they meet all the conditions required for a contradiction. But that at the end of the hour A and B will not be contradictories is proved because both A and B will then be false, so they will not be contradictories. The premise is proved for it is evident that A will be false, because it will be an insoluble falsifying itself. And that B will be false is proved for at the end of the hour B will be a non-insoluble particular negative each of whose singulars is false and to each suppositum of the subject will correspond one singular, so B will then be false. But that ⟨they will not be contradictories⟩ by the sole destruction of every other universal is clear, because were other universals which exist now to remain, ⟨A and B⟩ will continue to be contradictory, hence that they become non-contradictory is by the destruction ⟨of the other universals⟩. 1.15.2.4.1 From this conclusion it follows as a corollary that although these two: Everything true will be false and Something true will not be false are contradictories, nonetheless they would not be contradictories if the universal were insoluble. This is clear because each of them would be false, but the contradictory of a universal proposition would be given by preposing a negation to the whole, and that would signify disjunctively in the opposite way to the universal, namely, that some truth will not be false or that A will not be true. And this is true by reason of the second disjunct.

386 Appendices



ad 1.15.1 Per

hoc patet responsio ad argumentum principale124 concedendo illam consequentiam: a significabit precise quod quodlibet verum erit falsum, igitur125 a erit falsum, et ad improbationem non admitto casum, quia implicat contradictionem ut clare est ostensum. Quare etc.126 … Post expeditionem primi dubii accedo ad secundum quod querit utrum eadem propositio est vera et falsa127† … 2.2 Capitulum de antecedente ad concludendum propositionem veram vel falsam.128* 2.2.1 Secundo principaliter ad questionem arguitur sic: possibile est quod illa propositio: homo est animal sit vera et falsa, igitur questio vera. Tenet consequentia,129 quia illa propositio: homo est animal130 est necessaria, et antecedens131 arguitur sic:132 et signo illam propositionem scriptam: homo est animal, que sit a, ad quam concurrant133 eque primo sortes et plato, et quod primo134 (V 92va) sortes concipiat135 per eam quod136 homo est animal, plato vero quod homo est asinus. Isto posito arguitur sic: a propositio eque primo significat verum sicut falsum, et e converso, igitur significat primo verum, et significat primo falsum, et per consequens ipsa est vera et falsa. Consequentia137 patet, quia non videtur ratio quare sit magis138 vera quam falsa, vel e contra, cum

 principale] om. B F V  igitur] tamen F 126  Quare etc] om. B V 127 †   B 37ra, F 26ra, Q 25vb, V 92rb 128 *   B 37vb, F 26va, Q 26rb, V 92rb 129  Tenet consequentia] inv. V, tenet F 130  est animal] inv. Q 131  antecedens] aliquis F 132  arguitur sic] om. B, probatur V 133  concurrant] concurrat F, concurrent Q 134  sortes et plato, et quod primo] om. (hom.) F 135  sortes concipiat] inv. V 136  quod] om. B 137  Consequentia] om. F 138  sit magis] inv. V 124 125



Appendices387

ad.1.15.1 From this it is clear how to respond to the original argument, ⟨namely⟩ by granting inference ⟨B⟩: A will signify only that everything true will be false, therefore A will be false, and as for the counter-instance, I do not accept the scenario, because it implies a contradiction, as has been clearly seen. Hence etc. … … After dealing with the first doubtful question I turn to the second, which asks whether the same proposition ⟨can be⟩ both true and false … 2.2 Chapter 2: On the premises for concluding that a proposition is true or false. 2.2.1 Regarding the ⟨second⟩ doubtful question, I argue like this: It is possible that the proposition “A man is an animal” is both true and false, so the question ⟨is⟩ true. The inference holds, because the proposition: A man is an animal is necessary, and I argue in support of the premise like this: I indicate this written proposition: A man is an animal, call it A, which Socrates and Plato notice for the first time at the same time, and ⟨suppose⟩ Socrates immediately understands by it that a man is an animal, while Plato ⟨immediately understands by it⟩ that a man is an ass. Supposing this, I argue like this: proposition A just as immediately signifies a truth as a falsehood, and vice versa, therefore ⟨A⟩ immediately signifies a truth and immediately signifies a falsehood and consequently it is both true and false. The inference is clear, because there does not seem to be any reason why it is true rather than false, or vice versa, since it just as immediately signifies a truth as a falsehood and vice versa. For it immediately signifies a truth to Socrates, and it immediately signifies a falsehood to Plato, hence ⟨the proposition is both true and false⟩.

388 Appendices

eque primo significet139 verum sicut falsum, et e converso. Significat enim primo verum140 sorti, et platoni significat primo falsum,141 quare etc.142 2.2.2.1 Pro solutione huius argumenti pono quatuor conclusiones, quarum prima est ista: Non ex eo aliqua propositio affirmativa est vera, quia subiectum et predicatum supponunt pro eodem. Probatur, quia143 aliqua est propositio affirmativa que non habet predicatum, ut ista: homo est, et tamen ipsa144 est vera, ergo non oportet quod propositionis affirmative vere145 subiectum et predicatum supponant146 pro eodem. Idem147 ostendo de propositione habente predicatum et subiectum, nam hec est falsa: (Q 26va) sortes est sortes et plato, et tamen ipsa est affirmativa cuius subiectum et predicatum148 supponunt pro eodem, quia pro sorte. Similiter hec est vera: populus est populus, et tamen subiectum et predicatum non supponunt pro eodem, quia non supponunt pro aliquo, sed149 pro aliquibus. 2.2.2.1.1 Ex ista conclusione sequitur correlarie quod non ex eo aliqua propositio negativa est vera, quia subiectum et predicatum non supponunt pro eodem. Patet ex contradictoriis duarum propositionum150 in conclusione. Similiter tamen: homo non est animal.151 Ista est negativa falsa, et tamen subiectum et predicatum non supponunt pro eodem.

 significet] significaret B  primo verum] inv. F 141  falsum] verum V 142  quare etc] om. F Q 143  quia] om. V 144  ipsa] om. Q 145  vere] om. V 146  supponant] supponunt B Q 147  Idem] Illud F 148  subiectum et predicatum] predicatum et subiectum V 149  sed corr.] licet mss et ed., ponant add. V 150  duarum propositionum] inv. B, propositionum contradictorium V 151  animal corr.] asinus mss et ed. 139 140



Appendices389

2.2.2.1 For the

resolution of this argument I present four conclusions, of which the first is this: It is not because the subject and predicate supposit for the same thing that any affirmative proposition is true.5 Proof: because there is some affirmative proposition which does not have a predicate, for example, A man exists, and yet it is true, so it is not necessary that the subject and predicate of a true affirmative proposition supposit for the same thing. I show the same for a proposition having a predicate and subject, for Socrates is Socrates and Plato is false, and yet it is an affirmative proposition whose subject and predicate supposit for the same thing, namely, for Socrates. Similarly, A people is a people

is true, but the subject and predicate do not supposit for the same thing, because they do not supposit for some thing but for some things. 2.2.2.1.1 From this conclusion it follows as a corollary: It is not because the subject and predicate do not supposit for the same thing that any negative proposition is true. This is clear for the contradictories of the two propositions in the conclusion. But similarly, A man is not an animal6 is a false negative proposition but the subject and predicate do not supposit for the same thing.

390 Appendices 2.2.2.2 Secunda

conclusio est ista: Non ex eo aliqua propositio est vera, quia qualitercumque illa152 significat ita (F 26vb) est. Probatur,153 nam ista propositio est vera: homo est animal, et tamen non154 qualitercumque illa significat ita est. Probatur: ipsa155 significat hominem esse asinum et sic non est, ergo156 non qualitercumque ipsa157 significat ita est. Patet consequentia158 cum minori, et maiorem probo: subiectum significat hominem, et copula esse, et predicatum asinum, et tota propositio significat precise159 iuxta compositionem suorum terminorum, igitur tota propositio160 significat hominem esse asinum. Et quod161 predicatum significat162 asinum manifestum est, quia significat omne163 animal, et omnis asinus est animal, igitur etc. 2.2.2.2.1 Ex ista conclusione sequitur correlarie quod non ex eo aliqua164 propositio est falsa, quia aliter est quam per eam significatur. Patet ut prius, concedo ergo quod aliqua propositio vera significat aliter quam est, ut illa: homo est animal, et aliqua propositio falsa significat sicut est, ut illa: homo est asinus, que significat hominem esse.165

 illa] om. V  Probatur] patet V 154  et tamen non] non tamen V 155  et tamen – Probatur ipsa] om. (hom.) B F 156  ergo] qua add. B 157  ipsa] om. Q 158  consequentia] om. F 159  precise] om. F 160  tota propositio] om. F 161  quod] quia F 162  significat] significet V 163  omne] om. B 164  eo aliqua] aliquo V 165  esse] asinum add. V 152 153

2.2.2.2 The

Appendices391

second conclusion is this:

It is not because however it signifies so it is that any proposition is true.7 Proof: for this proposition is true: A man is an animal, but it is not that however it signifies so it is. Proof: it signifies that a man is an ass, and it is not so, so it is not that however it signifies so it is. The inference is clear, as is the minor premise, and I prove the major premise: the subject signifies man, and the copula being and the predicate ass, and the whole proposition signifies only according to the composition of its terms, so the whole proposition signifies that a man is an ass. And that the predicate signifies ass is evident, because it signifies every animal, and every ass is an animal, therefore etc. 2.2.2.2.1 From this conclusion it follows as a corollary: It is not because it is otherwise than is signified by it that any proposition is false. This is clear as before, therefore I grant that some true proposition signifies other than it is, for example, A man is an animal, and some false proposition signifies as it is, for example, A man is an ass, which signifies that a man exists.

392 Appendices 2.2.2.3 Tertia

conclusio est (B 38ra) ista: Non ex eo aliqua propositio est vera, quia ita est totaliter166 sicut per eam significatur. Probatur: aliqua167 propositio est vera cuius totale significatum est falsum, et aliqua propositio168 est falsa cuius totale significatum est verum, ergo conclusio vera. Consequentia169 tenet, et prima pars antecedentis probatur:170 hec propositio171 est vera: tu es homo, et eius totale significatum est falsum, igitur etc. Antecedens arguitur172 pro minori: quia te esse me est falsum, et te esse me est totale eius significatum,173 igitur etc. Minorem probo sic: te esse me est significatum174 illius, ut patet,175 quia subiectum significat176 te, et predicatum177 me, et non est178 significatum partiale eius,179 quia non significatur per aliquam eius partem, igitur est180 significatum totale,181 ex quo significatur per illam propositionem. Consimiliter probatur secunda pars antecedentis, nam hec est falsa: homo non est animal, et eius totale significatum est verum, videlicet hominem non esse asinum, significatur enim significatum (V 92vb) illud per totam et non per partem.

 totaliter] formaliter F  aliqua] antecedens V 168  propositio] per se F 169  consequentia] om. F 170  prima – probatur] probatur prima pars antecedentis V, sic add. F 171  propositio] om. F 172  arguitur] probatur V 173  eius significatum] inv. F V 174  Minorem – significatum] om. V 175  ut patet] om. B F 176  significat] om. F 177  predicatum] significat add. V 178  est] om. F V 179  partiale eius] inv. V 180  est] om. B F V 181  totale] om. F 166 167

2.2.2.3 The

Appendices393

third conclusion is this:

It is not because it is wholly as is signified by it that any proposition is true.8 Proof: some proposition is true whose whole significate is false, and some proposition is false whose whole significate is true, hence the conclusion is true. The inference holds, and the first premise is proved ⟨like this⟩: this proposition is true: You are a man and its whole significate is false, hence ⟨some true proposition has a false whole significate⟩. I argue for the minor premise: because your being me is false and your being me is its whole significate, therefore ⟨some false proposition has a true whole significate⟩. I prove the minor premise ⟨i.e. “your being me is its whole significate”⟩, like this: your being me is its significate, as is clear, because the subject signifies you, and the predicate me, and it’s not its partial significate, because it is not signified by some part of it, therefore it is its whole significate, since it is signified by the proposition. Similarly, the second premise ⟨“some proposition is false whose whole significate is true”⟩ is proven, for this is false: A man is not an animal, and its whole significate is true, namely, a man’s not being an ass, for that significate is signified by the whole proposition and not by a part.

394 Appendices 2.2.2.3.1 Ex

ista conclusione sequitur correlarie quod propositio vera habet multa182 totalia significata falsa et propositio falsa habet multa183 totalia significata vera. Prima pars patet de ista: deus est ens, que184 significat deum esse asinum, deum esse capram, deum esse lapidem, et sic ultra, quorum185 quodlibet186 est significatum falsum, et est significatum totale187 eius. Secunda pars patet de ista: homo est asinus, que significat hominem esse animal, hominem esse corpus, hominem esse substantiam, et ita de multis aliis,188 quorum quodlibet189 est verum190 et totale significatum eiusdem, igitur191 etc.192 2.2.2.4 Quarta conclusio est ista: Non ex eo aliqua propositio est vera, quia ipsa significat precise sicut est. Probatur: nulla propositio193 significat precise sicut est, igitur conclusio vera. Tenet consequentia,194 et195 probatum est antecedens196 in alia conclusione, quoniam quelibet197 propositio vera significat198 infinitas falsitates et quelibet propositio falsa infinitas veritates, quare199 sequitur immediate quod nulla proposito falsa200 significat precise201 aliter quam est, aut sicut non est. Opposita autem202 illorum aliquando admittuntur non quia vera, sed quia non implicant  habet multa] inv. V  habet multa] inv. V 184  que] quomodo V 185  quorum] quarum F 186  quodlibet] plus V 187  est significatum totale] totale significatum B V 188  aliis] om. B V 189  quodlibet] plus V 190  et est significatum totale – est verum] om. F 191  igitur] quare B 192  igitur etc.] quam non V, om. F 193  ipsa significat – propositio] om. F 194  Tenet consequentia] inv. B V, tenet F 195  et] aliquis add. F 196  antecedens] om. F 197  conclusione – quelibet] questione consequentia tenet V 198  significat] tenet V 199  quare] quia F 200  falsa] om. B F 201  precise] sicut est add. B 202  autem] tamen V, om. F 182 183

2.2.2.3.1 From

Appendices395

this conclusion it follows as a corollary:

A true proposition has many false whole significates and a false proposition has many true whole significates.9 The first part is clear from this: God is a being, which signifies that God is an ass, that God is a goat, that God is a stone, and so on, each of which is a false significate and is a whole significate of it. The second part is clear from this: A man is an ass, which signifies that a man is an animal, that a man is a body, that man is a substance, and so on for many others, each of which is true and a whole significate of it, therefore etc. 2.2.2.4 The fourth conclusion is this: It is not because it signifies only as it is that any proposition is true.10 Proof: no proposition signifies only as it is, therefore the conclusion is true. The inference holds, and the premise was proved in another conclusion, because each true proposition signifies infinitely many falsehoods and each false proposition infinitely many truths,11 from which it follows immediately that no false proposition signifies only other than it is, or as it is not. But the opposites of these are sometimes accepted not because they are true but because they do not imply a contradiction: when they are accepted, nothing impossible follows.12

396 Appendices

contradictionem,203 quibus admissis, nullum sequitur inconveniens. 2.2.2.4.1 Ex ista conclusione sequitur correlarie quod ista consequentia non valet: a propositio204 significat sicut est et non aliter quam sicut est, igitur a205 significat precise sicut est.206 Sit enim a207 ista: homo est asinus, et patet quod consequens est falsum, sed probatur antecedens sic: a propositio significat hominem esse,208 sed hominem esse est sicut est et non aliter quam sicut est, igitur a significat sicut est et non aliter quam sicut est,209 et ex isto consequente210 non sequitur quod a211 significat sicut est et non significat aliter quam sicut est, quia antecedens est de copulato extremo et consequens copulativa212 quibus est213, 214 admixta negatio quare etc.215 2.2.3 Finaliter est ergo216 dicendum quod ex eo aliqua propositio est vera quia significat primo217 verum et ex eo aliqua propositio est falsa quia significat primo falsum, ita quod ly primo extra materiam insolubilium idem sonat quod adequate; sed in materia insolubilium ⟨idem⟩ sonat quod218 principaliter219. Hec ergo: homo est animal, (B 38rb) est ve-(Q 26vb)-ra quia significat primo verum idest adequate verum220 quod est hominem esse animal, et hec:221  contradictionem] om. F  a propositio] propositio illa V 205  a] om. V 206  est] eum Q 207  a] om. F 208  esse] asinum add. F 209  igitur a – sicut est] om. (hom.) B F 210  consequente] antecedente F 211  a] aliqua V 212  copulativa] categorica B Q 213  quibus est] cuius est F 214  antecedens est – quibus est] quibus B, om. Q 215  quare etc] om. F 216  est ergo] inv. V 217  significat primo] inv. Q 218  quod adequate – quod] om. (hom.) B F 219  sonat – principaliter] principaliter sonat V 220  verum] om. F 221  hec] hoc V, om. F 203 204

2.2.2.4.1 From

Appendices397

this conclusion it follows as a corollary:

This inference is not valid: proposition A signifies as it is, and not other than it is, therefore A signifies only as it is. Let A be this proposition: A man is an ass, then it is clear that the conclusion is false ⟨by the fourth conclusion⟩, but the premise is proven like this: proposition A signifies that a man exists, but that a man exists is how it is and not other than it is, therefore A signifies as it is and not other than it is, and from this conclusion it does not follow that A signifies as it is and does not signify other than it is, because the premise has a conjoint extreme and the conclusion is a conjunctive proposition to which a negation is adjoined, hence etc. 2.2.3 Finally, then, it should be said: It is because it immediately signifies a truth that any proposition is true, and it is because it immediately signifies a falsehood that any proposition is false, where outside the case of insolubles “immediately” means the same as “exactly”; but in the case of insolubles it means the same as “principally.” Hence A man is an animal is true because it immediately signifies a truth, that is, it exactly ⟨signifies⟩ the truth that a man is an animal; but This is false, referring to itself, is false because it immediately signifies a falsehood, that is, it principally ⟨signifies⟩ a falsehood, namely, that this is false and that this is not false.

398 Appendices



hoc est falsum, seipso demonstrato, est falsa, quia significat primo falsum idest principaliter falsum videlicet222 hoc esse223 falsum et hoc non esse224 falsum. ad 2.2.1 Hiis suppositis respondetur ad rationem225 dicendo quod, admisso illo casu,226 illa: homo est animal, est vera et non falsa, et nego quod ista227 significet228 primo falsum, sed significat primo verum, scilicet hominem esse animal, quia229 hoc est suum adequatum significatum230 et nullum231 aliud. Et cum dicitur: illa significat primo platoni232 falsum, dico negando tenendo ly primo233 ut prius officiabiliter et non exponibiliter vel234 comparative. Licet enim plato per illam primo apprehendat235 falsum,236 tamen non237 significat illi primo falsum, ex quo ipsa238 non significat adequate falsum. Significat igitur dicta propositio239 pri-(F 27ra)-marie verum, ratione cuius dicitur vera, et secundarie falsum, ratione cuius non dicitur vera nec falsa, quare etc.240

 falsum videlicet] scilicet V  esse] est F 224  esse] est F 225  ad rationem] om. V 226  casu] copulativam V, quod add. Q 227  ista] non V 228  significet] significat Q 229  quia] et Q 230  adequatum significatum] inv. F V 231  nullum] non Q 232  platoni] plura V 233  tenendo ly primo] ly primo cedendo V 234  vel] aut B F 235  apprehendat] apprehendet Q 236  falsum] illa add. V 237  tamen non] inv. F 238  ipsa] om. F 239  propositio] prima F 240  quare etc] om. F 222 223



Appendices399

ad 2.2.1 Assuming

these ⟨conclusions and their corollaries⟩, I respond to the ⟨original⟩ argument by saying that, accepting the scenario, A man is an animal is true and not false, and I deny that it immediately signifies a falsehood; but it immediately signifies a truth, namely, that a man is an animal, because this is its exact significate and no other. And when it is said: It immediately signifies a falsehood to Plato, I deny it, holding as before that “immediately” is used officiably and not exponibly or comparatively.13 For although Plato immediately apprehends a falsehood by it, nonetheless it does not immediately signify a falsehood to him, since it does not signify a falsehood exactly. Therefore the said proposition primarily signifies a truth, and for that reason is said to be true, and it secondarily ⟨signifies⟩ a falsehood, but it is not for that reason said to be true or false, hence etc.

400 Appendices

B. Pauli Veneti Sophismata aurea, sophisma 50* 1.0 Ultimum

sophisma: sortes dicit verum si sortes dicit falsum. Pono quod sortes sit omnis sortes et dicat solummodo hanc: sortes dicit falsum, sic precise significantem, que sit a. 1.1 Isto posito quero241 utrum sortes dicat242 verum vel falsum. Si verum et non dicit nisi quod sortes dicit falsum, ergo verum est quod sortes dicit falsum, ergo sortes dicit falsum. Quod si conceditur, arguitur sic:243 sortes dicit falsum, et non dicit nisi quod sortes dicit falsum, ergo falsum est sortem dicere falsum. Ergo sortes non dicit falsum, et dicit aliquam propositionem, ergo sortes dicit verum. Et per consequens, a primo ad ultimum, si sortes dicit falsum, sortes dicit verum. Quod est sophisma. 1.1.1 Item quero utrum a sit verum vel falsum. Si verum et a significat precise sortem dicere falsum, ergo ita est quod sortes dicit falsum, et non dicit nisi a, ergo a est falsum. Quod si conceditur, arguitur sic:244 a est falsum, sed a significat precise sortem dicere falsum, ergo falsum (P 62rb) est sortem dicere falsum, ergo sortes non dicit falsum, sed sortes non dicit nisi a, ergo a non245 est falsum. 1.1.2 Confirmatur, nam falsum dicitur a sorte, ergo sortes dicit falsum. Tenet consequentia a passiva ad suam activam, antecedens est verum, ergo et consequens; sed consequens est a vel convertibile cum a, ergo a est verum. Quod antecedens sit verum246 arguitur: hoc dicitur a sorte, demonstrando a, et hoc est falsum, ergo falsum dicitur a sorte. Patet consequentia per resolutionem, antecedens est verum, ergo et consequens.

 P 62ra, V 63va, E1 p5ra, E2 53rb.  quero] om. P 242  dicat] dicit P V 243  sic] om. P E1 E2 244  sic] om. P E2 245  a non] om. E2 246  verum] falsum E1 *

241



Appendices401

B. Paul of Venice, Sophismata aurea, Sophism 50 1.0 The final

sophism: Socrates says a truth if Socrates says a falsehood. Suppose that there is only one Socrates, and he says only this proposition: Socrates says a falsehood,

signifying only in that way, call it A. ⟨scenario⟩, I ask whether Socrates says a truth or a falsehood. If a truth, and he says only that Socrates says a falsehood, then it is true that Socrates says a falsehood. So Socrates says a falsehood. If that is granted ⟨at the start⟩, I argue like this: Socrates says a falsehood, and he says only that Socrates says a falsehood, so it is false that Socrates says a falsehood. So Socrates does not say a falsehood, and he says some proposition, so Socrates says a truth. Consequently, from first to last, if Socrates says a falsehood, Socrates says a truth. That is the sophism. 1.1.1 Again, I ask whether A is true or false. If true, and A signifies only that Socrates says a falsehood, then it is the case that Socrates says a falsehood, and he says only A, so A is false. If that is granted ⟨at the start⟩, I argue like this: A is false, but A signifies only that Socrates says a falsehood, so it is false that Socrates says a falsehood. So Socrates does not say a falsehood. But Socrates says only A. So A is not false. 1.1.2 This is confirmed, for a falsehood is said by Socrates, therefore Socrates says a falsehood. The inference holds from the passive to the corresponding active, the premise is true, therefore so too is the conclusion; but the conclusion is A or something convertible with A, therefore A is true. I argue that the premise is true: this is said by Socrates, referring to A, and this is a falsehood, so a falsehood is said by Socrates. The inference is clear by resolution,1 the premise is true, so the conclusion is too.2 1.1 Given this

402 Appendices 1.1.3 Potest

etiam argui a esse verum ex eo, quia suum contradictorium est falsum, scilicet247 nullus sortes dicit falsum. Et hoc concesso248 quod a sit falsum, quia falsificat se. 1.2 Secundo. Sint duo249 sortes et non plures, quorum quilibet dicat unam talem: sortes dicit falsum, et nullam aliam, que sic precise significet. 1.2.1 Isto posito quero utrum sortes dicat250 verum vel falsum. Si verum et non dicit nisi quod sortes dicit falsum, ergo sortes dicit falsum, sed non est maior ratio de uno quam de alio, ergo quilibet sortes dicit falsum. Quod251 si conceditur, arguitur sic:252 quilibet sortes dicit falsum, sed nullus sortes dicit (E1 p5rb) nisi quod sortes dicit falsum, ergo falsum est quod sortes dicit falsum; ergo sortes non dicit falsum et sortes253 dicit propositionem, ergo sortes dicit verum. Quare254 etc. 1.2.2 Item pono quod sortes sit (E2 53va) omnis sortes et dicat istam: plato dicit falsum, et nullam aliam, que sic precise significet. Plato vero sit omnis plato et dicat istam: sortes dicit255 falsum, et nullam aliam, que sic precise (V 63vb) significet. Isto posito arguitur256 ut supra. Si enim sortes dicit verum257 et non dicit nisi quod plato dicit falsum, ergo verum est quod plato dicit falsum, sed non est maior ratio de platone quam de sorte,258 ergo sortes dicit falsum. Quod si conceditur, arguitur sic:259 sortes dicit falsum et non dicit nisi quod plato dicit falsum, ergo falsum est platonem dicere falsum. Tunc sic: plato non dicit falsum, ergo per idem nec sortes. Patet consequentia vel detur causa diversitatis.  scilicet] si E1 E2  concesso] concedendo E1 E2 249  Sint duo] om. P 250  dicat] dicit V 251  Quod] om. E1 E2 252  sic] om. P 253  sortes] om. E1 254  Quare] igitur E1 E2 255  dicit] om. V 256  arguitur] arguatur V 257  verum] et non dicit verum add. E1 E2 258  de platone quam de sorte] de uno quam de alio E1 E2 259  sic] om. P E1 E2 247

248



Appendices403

1.1.3 It can

also be argued that A is true from the fact that its contradictory is false, namely, No Socrates says a falsehood.3

And we have granted that A is false, because it falsifies itself. argument: Let there be only two Socrates, each of whom utters a proposition of the form:

1.2 Second

Socrates says a falsehood, and no other proposition, signifying only in that way. this ⟨scenario⟩, I ask whether Socrates says a truth or a falsehood. If ⟨Socrates says⟩ a truth and he says only that Socrates says a falsehood, then Socrates says a falsehood. But there is no more reason why it is one ⟨of them⟩ rather than the other, therefore each Socrates says a falsehood. If that is granted ⟨at the start⟩, I argue like this: each Socrates says a falsehood, but each Socrates says only that Socrates says a falsehood, so it is false that ⟨either⟩ Socrates says a falsehood. Hence Socrates does not say a falsehood. But Socrates says a proposition, so Socrates says a truth. Therefore ⟨Socrates says a truth if Socrates says a falsehood⟩.4 1.2.2 Again, suppose that there is only one Socrates, and that he utters 1.2.1 Given

Plato says a falsehood and no other proposition, signifying only in that way; and that there is only one Plato, and that he utters Socrates says a falsehood and no other proposition, signifying only in that way. Given this ⟨scenario⟩, I argue as above. For if Socrates says a truth and he says only that Plato says a falsehood, then it is true that Plato says a falsehood. But there is no more reason why Plato should rather than Socrates, so Socrates says a falsehood. If that is granted ⟨at the start⟩, I argue like this: Socrates says a falsehood and he says only that Plato says a falsehood, so it is false that Plato says a falsehood. Then like this: Plato does not say a falsehood, so for the same ⟨reason⟩ neither does Socrates. The inference is clear, otherwise one should provide a reason to distinguish ⟨between the two cases⟩.5

404 Appendices 1.3 Tertio.

Sit sortes omnis sortes et dicat hanc et nullam aliam: plato dicit falsum, que sic precise significet. Plato vero sit omnis plato et dicat hanc et nullam aliam: sortes non dicit falsum, que sic precise significet.260 1.3.1 Isto posito quero utrum sortes dicit verum vel falsum. Si verum et non dicit nisi quod plato dicit falsum, ergo verum est platonem dicere falsum. Tunc sic: plato dicit falsum et non dicit nisi quod sortes non dicit falsum, ergo falsum est sortem non dicere falsum, et per consequens sortes261 dicit falsum. Quod si conceditur, contra: sortes dicit falsum et non dicit nisi quod plato dicit falsum, ergo plato non dicit falsum; et plato dicit propositionem, ergo plato dicit verum. Sed plato solummodo262 dicit quod sortes non dicit falsum, ergo verum est263 sortem non dicere falsum; et sortes dicit propositionem,264 ergo sortes dicit verum, quod erat probandum. 1.4 Quarto. Sit sortes omnis sortes et dicat hanc et nullam aliam: non est ita sicut sortes dicit, que sic precise significet. Plato etiam dicat solum265 consimilem sic precise significantem. Isto posito quero utrum266 ita est sicut sortes dicit vel non. 1.4.1 Si ita est sicut sortes dicit et plato dicit solummodo quod non est ita sicut sortes dicit,267 ergo non est268 ita sicut plato dicit, sed sortes dicit omnino sicut plato dicit, ergo non est ita sicut sortes dicit. Quod si conceditur, arguitur sic:269 non est ita sicut sortes dicit et plato dicit solummodo quod non est ita sicut sortes dicit, ergo ita est sicut plato dicit, sed270 sortes dicit omnino sicut plato dicit,271 ergo ita est sicut sortes dicit.  significet] significatur E1  sortes] non add. E1 E2 262  solummodo] solum P V 263  est] dicere add. E2 264  propositionem] istam add. E1 E2 265  solum] om. E1 266  utrum] numquid E2 267  dicit] dicat V 268  non est] om. E1 269  sic] om. P 270  Sed] sortes V 271  dicit] om. V 260

261



Appendices405

1.3 Third argument:

Let there be only one Socrates and let him utter this and no other proposition: Plato says a falsehood, which signifies only in that way, Moreover, let there be only one Plato and let him utter this and no other proposition: Socrates does not say a falsehood,

signifying only in that way. this ⟨scenario⟩, I ask whether Socrates says a truth or a falsehood. If ⟨Socrates says⟩ a truth, and he says only that Plato says a falsehood, then it is true that Plato says a falsehood. Then ⟨I argue⟩ like this: Plato says a falsehood and he says only that Socrates does not say a falsehood, so it is false that Socrates does not say a falsehood. Consequently, Socrates says a falsehood. If that is granted ⟨at the start⟩, on the contrary: Socrates says a falsehood and he says only that Plato says a falsehood, so Plato does not say a falsehood. And Plato says a proposition, so Plato says a truth. But Plato says only that Socrates does not say a falsehood, so it is true that Socrates does not say a falsehood; and Socrates says a proposition, so Socrates says a truth ⟨if Socrates says a falsehood⟩, which is what was to be proven.6 1.4 Fourth argument: let there be only one Socrates and suppose he says 1.3.1 Given

It is not as Socrates says it is, and no other proposition, signifying only in that way, and suppose Plato also says only a proposition of the same form ⟨i.e., “It is not as Socrates says it is”⟩, signifying only in that way. Given this ⟨scenario⟩, I ask whether it is as Socrates says it is or not. 1.4.1 If it is as Socrates says it is and Plato says only that it is not as Socrates says it is, then it is not as Plato says it is. But Socrates says altogether what Plato says, so it is not as Socrates says it is. If that is granted ⟨at the start⟩, I argue like this: it is not as Socrates says it is and Plato says only that it is not as Socrates says it is, so it is as Plato says it is. But Socrates says altogether what Plato says, so it is as Socrates says it is.

406 Appendices 1.4.2 Et confirmatur:

si non est ita sicut sortes dicit, ergo verum est quod non est ita sicut sortes dicit. (E1 p5va) Tenet consequentia quia sequitur universaliter: tu non curris, ergo verum est quod tu non curris. Tunc ⟨arguitur⟩ sic: verum est quod non est ita sicut sortes dicit et sortes ⟨solummodo⟩ dicit sic, ergo sortes dicit sicut verum est esse, et per consequens ita est sicut sortes dicit. 1.5 Ad oppositum arguitur sic:272 sophisma est una condicionalis affirmativa273 denominata a ly si, cuius contradictorium consequentis stat cum antecedente, quia he stant simul: sortes dicit falsum, et: sortes non dicit verum. Ergo sophisma est274 falsum. ⟨2. Conclusiones⟩ 2.1 Circa predicta

sit hec prima conclusio: omne insolubile categoricum significat adequate (P 62va) iuxta compositionem suorum terminorum, ita quod ista: omnis propositio est falsa, significat adequate omnem propositionem esse falsam, et ipsa:275 hoc276 est falsum, se277 ipsa demonstrata, significat adequate hoc esse falsum. 2.1.0.1 Probatur, nam he contrariantur: omnis propositio est falsa, nulla propositio est falsa, sed secunda significat adequate nullam propositionem esse falsam, ergo prima significat adequate omnem propositionem esse falsam.278

 sic] om. P  affirmativa] om. E1 274  est] om. P 275  ipsa] ita P 276  hoc] hec E1 (etiam infra, sed ultra non notavimus) 277  se] seu E1 E2 278  omnem propositionem esse falsam] propositionem falsam P 272 273



Appendices407

1.4.2 This is

confirmed: if it is not as Socrates says it is, then it is true that it is not as Socrates says it is. The inference holds because this is valid universally: You are not running, so it is true that you are not running.

Then ⟨I argue⟩ like this: it is true that it is not as Socrates says it is, and Socrates says ⟨only⟩ that, so Socrates says what is truly so. Consequently, it is as Socrates says it is.7 1.5 But on the contrary, I argue like this: the sophism is an affirmative conditional, indicated by ⟨the expression⟩ “if,” the contradictory of whose consequent is consistent with the antecedent, for these can stand together: Socrates says a falsehood and Socrates does not say a truth. So the sophism is false. ⟨2. Conclusions⟩ 2.1 The first

conclusion about these matters is this: every subject-predicate insoluble signifies exactly according to the composition of its terms, so that Every proposition is false signifies exactly that every proposition is false, and This is false referring to itself, signifies exactly that this is false. for these propositions are contraries:

2.1.0.1 Proof:

Every proposition is false, No proposition is false, but the second signifies exactly that no proposition is false, so the first signifies exactly that every proposition is false.

408 Appendices 2.1.0.2 Item

he contradicunt: hoc est falsum, hoc non est falsum, ut colligitur de mente279 Aristotelis primo Perihermineias. Sed ista: hoc non est falsum, significat adequate hoc non esse falsum, ergo hec:280 hoc est falsum, significat adequate hoc esse falsum. Et ita281 dicatur de illis: sortes dicit falsum, sortes non dicit verum. 2.1.1 Ex ista conclusione sequitur primo quod insolubile categoricum non significat copulative nec duobus modis, scilicet adequate, quia secundum significatum aut est notum aut est282 ignotum. 2.1.1.1 (V 64ra) Si ignotum, ergo insolubile non est concedendum, negandum nec283 dubitandum. Si notum, sit ergo quod284 hec:285 hoc est falsum, significet286 adequate hoc esse falsum et hoc esse verum, ut communiter dicitur ratione reflexionis. Et sequitur quod insolubile istud287 est impossibile simpliciter, quia suum significatum adequatum est impossibile simpliciter. Sed tale insolubile est concedendum secundum communem opinionem, ergo impossibile simpliciter est concedendum, quod est falsum.288 2.1.1.2 Item hec mentalis naturaliter significans: ista289 propositio est falsa, non significat iam copulative nec duobus modis, ergo nec sic significaret si esset omnis propositio. Patet consequentia, quia propositio naturaliter significans non potest variare modum significandi, sicut nec potest absolvi a sua naturali290 significatione.

 mente] intentione V  hec] om. E1 E2 281  ita ] idem E2 282  aut est] aut E1 E2, vel V 283  nec] vel P2 284  ergo quod] ergo per P, ita quod E1 E2 285  hec] hoc P 286  significet] significant P E1 E2 287  insolubile istud] id insolubile P 288  quod est falsum] om. E1 289  ista] aliqua V, hec E1 E2 290  naturali] om. V E1 E2 279 280

2.1.0.2 Again,

Appendices409

these propositions are contradictories:

This is false, This is not false, as one can infer from Aristotle’s opinion in the first book of De interpretatione.8 But This is not false signifies exactly that this is not false, so This is false signifies exactly that this is false. And the same sort of things may be said of Socrates says a falsehood, Socrates does not say a truth. 2.1.1 The first

corollary9 following from this conclusion is that a subjectpredicate insoluble does not signify (exactly, that is) conjunctively or in two ways.10 For the second significate is either known or unknown.11 2.1.1.1 If unknown, then the insoluble should not be granted, denied or doubted.12 If ⟨it is⟩ known, then let This is false signify exactly that this is false and this is true, as is standardly said by reason of the self-reflection.13 It follows that this insoluble is unrestrictedly impossible, because its exact significate is unrestrictedly impossible. But this insoluble is to be granted by the standard opinion,14 so the unrestrictedly impossible is to be granted, which is false. 2.1.1.2 Again, the mental proposition: This proposition is false, which signifies naturally,15 does not currently signify conjunctively nor in two ways, so neither would it so signify if it were the only proposition. The inference is clear, because a proposition signifying naturally cannot vary its mode of signifying, just as it cannot be divorced from its natural signification.

410 Appendices 2.1.2 Secundo

sequitur quod non implicat contradictionem insolubile significare precise ut termini pretendunt291, quia non implicat contradictionem istam:292 hoc est verum,293 precise significare hoc esse verum, aut illam: hoc non294 est falsum, significare precise hoc non (E1 p5vb) esse falsum, seipsa demonstrata. Ergo per idem non implicat contradictionem istam: hoc est falsum, seipsa demonstrata, significare precise hoc esse falsum. 2.1.2.1 Deinde non implicat contradictionem has duas: falsum est, chymera est, esse omnes propositiones et significare precise sicut295 termini pretendunt. Ergo per idem296 non implicat contradictionem297 istam: falsum est, esse omnem propositionem et298 significare precise falsum esse, vel sequitur quod aliqua propositio mentalis299 per solam desitionem300 alterius incipit esse non propositio aut variat modum significandi et significatum adequatum, quod est dissonum.

 pretendunt] precedunt V  istam] ista P 293  verum] hoc add. P 294  non] om. P 295  sicut] ut E1 E2 296  idem] modo add. E1 E2 297  contradictionem] om. E1 298  et] om. E2 299  mentalis] b (dub.) P 300  desitionem] destructionem (dub.) mentalis P 291 292



Appendices411

2.1.2 The second

corollary is that it does not imply a contradiction that an insoluble signifies only as its terms suggest, because that This is true ⟨referring to itself,⟩ signifies only that this is true, or that This is not false, referring to itself, signifies only that this is not false do not imply a contradiction. So for the same reason, that This is false,

referring to itself, signifies only that this is false does not imply a contradiction. 2.1.2.1 Nor does it imply a contradiction that these two: A falsehood exists A chimera exists are the only propositions and that they signify only as the terms suggest. So for the same reason it does not imply a contradiction that A falsehood exists is the only proposition and that it signifies only that a falsehood exists. Otherwise, it ⟨would⟩ follow that some mental proposition begins to be a non-proposition by the mere destruction of another proposition or changes its mode of signifying and its exact significate, which sounds wrong.

412 Appendices 2.1.3 Tertio

sequitur quod aliqua propositio est falsa cuius adequatum (E2 53vb) significatum est verum. Patet,301 quia hec est falsa: hoc est falsum,302 ex quo falsificat se et suum adequatum significatum303 est verum, scilicet304 hoc esse305 falsum. Nec hoc306 est contra dicta alibi, quia ibi locutus sum de propositione cuius307 non est suum significatum et cuius veritas vel falsitas non dependet ex sola compositione terminorum, sed308 ex veritate vel309 falsitate sui significati, qualiter non est in proposito. 2.1.3.1 Propositio ergo est falsa si suum adequatum significatum est falsum aut si asserit se310 esse falsam; et propositio vera est si suum adequatum significatum est verum et311 non asserit se312 esse falsam aut si313 asserit immediate seipsam esse veram, sicut est ista: hoc est verum, seipso demonstrato. Et notanter dico: immediate, quia hec: omnis propositio est vera, asserit se esse veram et tamen non est vera. 2.1.4 Quarto sequitur quod duo contradictoria inter se contradicentia sunt duo falsa. Nam hec est falsa: hoc est falsum,314 seipsa315 demonstrata, quia falsificat se, et hec est falsa: hoc non est falsum, priori demonstrata, quia suum adequatum significatum est falsum.

 Patet] om. P V  falsum] quia add. P 303  significatum] om. P 304  scilicet] hoc add. E1 305  esse] est E2 306  hoc] om. E1 E2 307  cuius] que P V 308  sed] scilicet E2 309  vel] et P E1 310  se] seipsam E1 E2 311  et] aut si P 312  se] seipsam E1 E2 313  si] om. E1 E2 314  hoc est falsum] om. P 315  seipsa] se V E1 E2 301 302



Appendices413

2.1.3 The third

corollary is that there is some false proposition whose exact significate is true.16 This is clear because This is false

is false, since it falsifies itself, and its exact significate, namely, that this is false, is true. Nor is this contrary to what I said elsewhere, because there I spoke of a proposition of which it is not the significate and whose truth or falsity did not depend only on the composition of the terms, but on the truth or falsity of its significate, as is not the case here. 2.1.3.1 So a proposition is false if its exact significate is false or if it asserts that it itself is false; and a proposition is true if its exact significate is true and it does not assert that it itself is false or if it directly asserts that it itself is true; for example, This is true, referring to itself. Note that I say “directly,” because Every proposition is true ⟨indirectly⟩ asserts that it itself is true but is not true. corollary is that there are two mutually contradictory contradictories that are both false.17 For

2.1.4 The fourth

This is false, referring to itself, is false because it falsifies itself, and This is not false referring to the earlier proposition, is false because its exact significate is false.

414 Appendices 2.1.4.1 Consimiliter

dicitur quod aliqua duo subcontraria316 in casu isto317 sunt simul falsa, scilicet: aliqua propositio negativa est vera, aliqua propositio negativa non est vera, dato quod iste sint omnes propositiones. Et si arguitur per idem duo contraria vel duo318 contradictoria esse duo vera, negatur consequentia, cum hoc319 non sit reperibile in aliqua materia. (P 62vb) 2.2 Secunda conclusio. Aliqua inter se contradicunt quorum unum est impossibile et reliquum contingens. Patet de istis: hoc est impossibile, et:320 hoc non est impossibile, continue demonstrando primam, que sit a. 2.2.0.1 Nam321 quero utrum a sit possibile vel impossibile. Si impossibile, habeo intentum. 2.2.0.2 Si possibile, ergo suum adequatum significatum est possibile. Sed322 suum adequatum significatum est hoc esse impossibile, ergo possibile323 est hoc esse impossibile.324 Tunc sic: possibile est hoc esse (E1 p6ra) impossibile, sed non aliter iam325 significat326 a quam tunc significaret,327 quia suppono328 quod (V 64rb) a naturaliter significet,329 ergo a est propositio impossibilis.

 subcontraria] contradictoria P  isto] om. V E1 318  duo] om. E1 E2 319  hoc] om. P E1 E2 320  et] om. P E1 E2 321  Nam] om. P 322  Sed] quia E1 E2 323  possibile] impossibile P 324  impossibile] possibile E2 325  iam] om. P E1 E2 326  significat] ista add. P E1 E2 327  significaret] significat P 328  suppono] supposito P 329  significet] significat P E2 316 317



Appendices415

2.1.4.1 Similarly,

I say that there are some pairs of subcontraries that are simultaneously false in this scenario, namely, Some negative proposition is true, Some negative proposition is not true,

given that these are the only propositions.18 And if someone concludes, by parity of reasoning, that there are two contraries or two contradictories that are both true, I deny the inference, since this cannot be found in any context. 2.2 The second conclusion: there are some mutual contradictories one of which is impossible and the other contingent.19 This is clear concerning these ⟨contradictories⟩: ⟨A⟩ This is impossible and ⟨B⟩ This is not impossible, in both cases referring to the first, call it A. 2.2.0.1 For I ask whether A is possible or impossible. If ⟨A⟩ is impossible, I have what I wanted. 2.2.0.2 If ⟨A⟩ is possible, then its exact significate is possible. But its exact significate is that ⟨A⟩ is impossible, so it is possible that ⟨A⟩ is impossible. Then ⟨I argue⟩ like this: it is possible that ⟨A⟩ is impossible, but A does not signify now differently than it would signify then, because I assume that A signifies naturally, so A is an impossible proposition.20

416 Appendices 2.2.0.3 Quod

autem330 suum contradictorium sit contingens arguitur, nam possibile est hoc esse impossibile,331 ut patet; et possibile est hoc non esse impossibile,332 quia possibile est hoc non esse; et ista propositio sic adequate significat et non est propositio insolubilis, ergo est333 contingens. 2.2.1 Ex ista conclusione sequitur primo quod aliud quam necessarium contradicit impossibili. Patet in casu isto, sed aliter arguitur ponendo quod a sit ista: hoc non est334 necessarium, et b sit335 illa: hoc est necessarium, semper demonstrando a. 2.2.1.1 Isto posito ⟨sequitur quod⟩ b est impossibile propter implicationem contradictionis. Asserit enim a esse necessarium, sed a esse necessarium implicat contradictionem, ex quo asserit se non esse necessarium. Arguitur ergo sic: a non est necessarium propter implicationem contradictionis et a contradicit b impossibili, ergo aliud quam necessarium contradicit impossibili. 2.2.2 Secundo sequitur quod duarum propositionum quarum una est necessaria et alia contingens quelibet336 contradicit propositioni impossibili. 2.2.2.1 Pono quod c sit ista: hoc non est necessarium, demonstrato337 a, et sequitur quod c est necessarium, quia eius significatum adequatum est necessarium et ipsum c non est insolubile nec asserens se non esse necessarium, a vero non est necessarium, ut deductum est, nec etiam impossibile quia ipsum est verum; ergo est338 contingens. Tunc arguitur sic:339 a est [et]340 contingens et341 c necessarium, sed quodlibet342 istorum contradicit b impossibili, ergo etc.  autem] om. E1  impossibile] possibile P 332  impossibile] quia possibile est hoc non esse impossibile add. E1 E2 333  est] et P 334  est] om. P E2 335  sit] om. V E1 E2 336  quelibet] istarum add. P 337  demonstrato] demonstrata P, demonstrando E1 E2 338  est] et V 339  sic] om. P 340  et] add. mss. E1 E2, delevimus 341  et] om. E1 E2 342  quodlibet] quelibet P E1 E2 330

331



Appendices417

2.2.0.3 But I

argue that its contradictory ⟨B⟩ is contingent. For it is possible that ⟨A⟩ is impossible, as is clear; and it is possible that ⟨A⟩ is not impossible, because it is possible that ⟨A⟩ does not exist;21 and proposition ⟨B⟩ signifies exactly in that way and is not an insoluble proposition, therefore it is contingent. 2.2.1 The first corollary following from this conclusion is that something other than the necessary contradicts the impossible.22 This is clear in the given scenario, but I argue in a different way by supposing that A is This is not necessary, and B is This is necessary, where both refer to A. this, ⟨it follows that⟩ B is impossible because it implies a contradiction. For it asserts that A is necessary, yet that A is necessary implies a contradiction since it asserts ⟨consequentially⟩ that it itself is not necessary. Therefore I argue like this: A is not necessary because ⟨its being necessary⟩ implies a contradiction, and A contradicts the impossibility B, therefore something other than the necessary contradicts the impossible. 2.2.2 The second corollary is that each of two propositions, one of which is necessary and the other contingent, contradicts ⟨the same⟩ impossible proposition.23 2.2.2.1 Suppose that C is 2.2.1.1 Given

This is not necessary, referring to A. Then it follows that C is necessary, for its exact significate is necessary and C is not an insoluble nor does it assert ⟨consequentially⟩ that it itself is not necessary, while A is not necessary, as was proven, nor ⟨is it⟩ impossible, for it is true; therefore it is contingent. Then I argue like this: A is contingent and C ⟨is⟩ necessary, but each of them contradicts the impossibility B, so etc.

418 Appendices 2.2.3 Tertio

sequitur quod aliqua propositio est contingens cuius adequatum significatum est necessarium. 2.2.3.1 Patet de a. Tamen si alicuius propositionis adequatum significatum esset343 necessarium et ista propositio non assereret se non esse necessariam, ista esset necessaria. Ideo c est necessarium, non obstante quod a sit contingens cum quo convertitur. 2.2.4 Quarto sequitur quod aliqua propositio est necessaria cuius adequatum significatum est contingens. 2.2.4.1 Probatur. Hec est vera: hoc est verum, demonstrata seipsa, et ipsa344 est falsa: hoc est falsum, demonstrata seipsa, et ista est impossibilis: hoc est impossibile, demonstrata seipsa, ergo per idem hec est necessaria: hoc est necessarium, demonstrata seipsa. Tamen suum adequatum significatum est contingens, quia hoc est necessarium et possibile est quod hoc345 non sit necessarium, ex quo ista propostio potest non esse. 2.2.4.2 Nec talia sunt inconvenientia ubi propositio est vera vel falsa ex sola compositione terminorum. Et346 consequenter dicatur quod aliqua propositio est necessaria cuius contradictorium est contingens. Patet de propositione dicta cuius contradictorium est: hoc non est necessarium, cum347 ista sit falsa et possit348 esse vera sic significando, ex quo suum contradictorium potest non esse.

 esset] est E1  ipsa] ista E2 345  hoc] om. E1 E2 346  Et] om. E1 E2 347  cum] cuius E1 E2 348  possit] potuit E1 E2 343

344



Appendices419

2.2.3 The third

corollary is that there is a contingent proposition whose exact significate is necessary. 2.2.3.1 This is clear concerning A. But if the exact significate of some proposition were necessary and that proposition did not imply that it itself was not necessary, it would be necessary. Hence C is necessary, notwithstanding that A, with which it is convertible, is contingent. 2.2.4 The fourth corollary is that there is a necessary proposition whose exact significate is contingent. 2.2.4.1 Proof: This is true, referring to itself, is true, and This is false, referring to itself, is false, and This is impossible, referring to itself, is impossible, so for the same reason This is necessary, referring to itself, is necessary. But its exact significate is contingent, because it is necessary and it is possible that it is not necessary, since that proposition might not exist.24 2.2.4.2 There are no such puzzles when a proposition is true or false solely from the composition of its terms. Consequently, I say that there is a necessary proposition whose contradictory is contingent. This is clear regarding the aforementioned proposition whose contradictory is This is not necessary, since this is false and can be true signifying in that way, since its contradictory might not exist.

420 Appendices 2.3 Tertia conclusio.

Aliqua consequentia est bona et349 formalis, scita a te350 esse talis, significans precise351 iuxta compositionem suorum terminorum, cuius antecedens est scitum a te et consequens non est scitum a te. Patet de ista: hoc est nescitum352 a te, ergo hoc est nescitum a te, demonstrando consequens per utrumque hoc. Scis enim353 bene quod contradictorium consequentis repugnat formaliter antecedenti, ergo tu354 scis istam consequentiam esse formalem. Et quod355 antecedens sit scitum a te et consequens nescitum a te patet, quia bene scis quod repugnat consequens sciri a te, ex quo asserit se non sciri356 a te. 2.3.1 Ex ista conclusione sequitur primo quod aliqua est consequentia bona (E1 p6rb) et357 formalis significans precise iuxta compositionem suarum partium, cuius antecedens est verum et consequens falsum. Patet de illa: hoc est falsum, ergo hoc est falsum, per utrumque hoc358 demonstrando consequens. 2.3.2 Secundo sequitur quod aliqua est consequentia bona et formalis significans precise iuxta compositionem359 suarum partium, cuius antecedens est possibile (P 63ra) et (V 64va) consequens impossibile. Patet de ista: hoc est impossibile, ergo hoc est impossibile, demonstrando consequens per utrumque360 hoc. 2.3.3 Tertio sequitur quod aliqua est consequentia bona et formalis, significans precise iuxta compositionem suarum partium, cuius antecedens est necessarium et consequens contingens. Patet de ista: hoc non est necessarium; ergo hoc ⟨non⟩ est necessarium, continue demonstrando consequens per utrumque demonstrativum. (E2 54ra)  et] om. E1 E2  a te] precise E2, om. E1 351  precise] om. E2 352  est nescitum] non est scitum E1 E2 353  enim] om. P 354  tu] om. V 355  quod] om. P 356  non sciri] nesciri E1 E2 357  et] om. P 358  hoc] hec E1, om. P 359  iuxta compositionem] ex compositione V 360  utrumque] ly add. P 349

350



Appendices421

2.3 The third

conclusion: there is a formally valid inference, known by you to be so, signifying only according to the composition of its terms, whose premise is known by you and whose conclusion is not known by you. This is clear concerning this inference: This is unknown to you, therefore this is unknown to you,

where each ⟨occurrence of⟩ “this” refers to the conclusion. For you well know that the contradictory of the conclusion is formally inconsistent with the premise, so you know that the inference is formally valid. That the premise is known by you and the conclusion unknown to you is clear, because you well know that it is inconsistent for the conclusion to be known by you since it asserts that it itself is not known by you.25 2.3.1 The first corollary following from this conclusion is that there is a formally valid inference, signifying only according to the composition of its parts whose premise is true and conclusion false. This is clear concerning: This is false, so this is false, where each ⟨occurrence of⟩ “this” refers to the conclusion.26 2.3.2 The second corollary is that there is a formally valid inference, signifying only according to the composition of its parts, whose premise is possible and conclusion impossible. This is clear concerning this inference: This is impossible, therefore this is impossible, where each ⟨occurrence of⟩ “this” refers to the conclusion.27 2.3.3 The third corollary is that there is a formally valid inference, signifying only according to the composition of its parts, whose premise is necessary and conclusion contingent.28 This is clear regarding this inference: This is not necessary, so this is ⟨not⟩ necessary, where each demonstrative refers to the conclusion.29

422 Appendices 2.3.3.1 Idem

intelligo de illa consequentia: hoc est necessarium, ergo hoc est necessarium, demonstrando per utrumque hoc antecedens. 2.3.4 Quarto sequitur quod alique propositiones ⟨ad⟩ invicem convertuntur,361 quarum una est vera vel362 possibilis et altera363 falsa vel364 impossibilis. Patet de istis: hoc365 est falsum,366 ⟨et:⟩ hoc est falsum, demonstrando primam, et de illis: hoc est impossibile, et: hoc est impossibile, demonstrando secundam. 2.3.4.1 Et consequenter dicatur quod verum convertitur cum impossibili et necessarium cum contingenti. Patet,367 nam iste convertuntur: hoc est impossibile, et: hoc est impossibile,368 demonstrando secundam, quarum prima est vera et secunda impossibilis. Iste etiam convertuntur: hoc est necessarium, et: hoc est necessarium, demonstrando secundam, quarum prima est contingens et secunda necessaria.

 convertuntur] convertitur E1  vel] et P E1 E2 363  altera] alia P 364  vel] et P E1 E2 365  hoc] hec V 366  falsum] ergo add. P E1 E2 367  patet] consequentia add. P 368  impossibile] et hoc est impossibile add. P E1 361

362

2.3.3.1 I observe

Appendices423

that the same is true of this inference:

This is necessary, so this is necessary, where each ⟨occurrence of⟩ “this” refers to the premise.

2.3.4 The fourth corollary is that there are some propositions that are mutually

convertible, of which one is true or possible and the other false or impossible.30 This is clear of these: This is false,

⟨and⟩ This is false, referring to the first, and of these: This is impossible, and This is impossible, referring to the second. it may be said that truth converts with impossibility and necessity with contingency. This is clear since these propositions are convertible:

2.3.4.1 Consequently,

This is impossible and This is impossible, referring to the second, where the first is true and the second impossible. Also, these propositions are convertible: This is necessary and This is necessary, referring to the second, where the first is contingent and the second necessary.

424 Appendices 2.4 Quarta

conclusio. Aliqua propositio exponibilis est vera cuius quelibet exponentium est falsa. Patet dato quod iste tres essent omnes propositiones,1 semper intelligendo de propositionibus totalibus, videlicet: omnis propositio2 particularis affirmativa et universalis negativa est falsa, aliqua propositio particularis affirmativa et universalis negativa est falsa,3 nulla est propositio4 particularis affirmativa et5 universalis negativa quin ipsa sit falsa.6 Quelibet enim exponentium falsificat se et non exposita. 2.4.0.1 E contra est etiam concedendum exponibilem esse falsam7 et exponentes veras, dato quod hec8 esset omnis universalis affirmativa9: omnis universalis affirmativa est falsa. 2.4.1 Ex ista conclusione sequitur primo universalem esse veram et exclusivam esse10 falsam. Patet dato quod hec esset omnis exclusiva: tantum falsum est propositio exclusiva. Tunc ista exclusiva esset falsa et hec universalis vera: omnis propositio exclusiva est falsa. 2.4.1.1 E contra etiam est concedendum universalem esse falsam et suam exclusivam esse11 veram. Patet dato quod hec esset omnis propositio universalis affirmativa:12 omnis propositio universalis est falsa. In casu isto hec13 universalis esset falsa, quia falsificat14 se, et hec exclusiva esset vera: tantum falsum est propositio universalis.

 propositiones] om. E1 E2  propositio] om. P 3  falsa] et add. E1 E2 4  propositio] om. E1 5  et] om. E1 E2 6  falsa] igitur etc. add. E1 E2 7  exponibilem esse falsam] exponibilibus esse falsa P 8  hec] hoc V 9  omnis universalis affirmativa] om. V 10  esse] om. E1 E2 11  esse] om. V 12  affirmativa] om. E1 E2 13  hec] omnis P 14  falsificat] significat V 1

2



Appendices425

2.4 The fourth

conclusion: there is a true exponible proposition each of whose exponents is false.31 This is clear given that these three are the only propositions (always understanding that these are whole propositions): Every particular affirmative or universal negative proposition is false, There is a particular affirmative and a universal negative proposition which are false, There is no particular affirmative or universal negative proposition that is not false.

For each of the exponents falsifies itself and the expounded proposition does not. 2.4.0.1 Conversely, it must also be granted that the exponible is false and the exponents true, given that the only universal affirmative proposition is this:32 Every universal affirmative is false. 2.4.1 The first

corollary following from this conclusion is that there is a true universal proposition whose exclusive is false.33 This is clear given that the only exclusive proposition were this: Only a falsehood is an exclusive proposition. Then this exclusive would be false and this universal true: Every exclusive proposition is false.

2.4.1.1 Conversely,

it should also be granted that there is a false universal proposition whose exclusive is true34. This is clear given that the only universal affirmative proposition were this: Every universal proposition is false. In this scenario this universal proposition would be false, because it falsifies itself, and this exclusive would be true: Only a falsehood is a universal proposition.

426 Appendices 2.4.2 Secundo

sequitur quod aliqua exlusiva est falsa cuius sua exceptiva est vera. Patet dato quod iste essent15 omnes propositiones: tantum (E1 p6va) propositio falsa est exclusiva, nulla propositio preter propositionem falsam est exclusiva.16 2.4.2.1 E contra etiam est concedendum quod aliqua exclusiva est vera cuius sua17 exceptiva est falsa. Patet dato quod iste sunt omnes propositiones: tantum exceptiva est falsa, nulla propositio preter exceptivam est falsa.18 2.4.3 Tertio sequitur universalem esse veram et suam subalternam esse falsam. 2.4.3.1 Patet dato quod iste sint19 omnes propositiones affirmative: omnis propositio particularis affirmativa est falsa, aliqua propositio particularis affirmativa est falsa, aut ille omnes negative: nulla propositio particularis20 negativa21 est vera, aliqua propositio particularis negativa22 non est vera. 2.4.3.2 E contra etiam est concedendum quod aliqua indefinita23 de secundo adiacente cum verbo de presenti sine determinatione aliqua est vera et sua universalis est24 falsa. Patet dato quod iste sint omnes propositiones: propositio universalis falsa est, omnis propositio universalis falsa est. 2.4.4 Quarto sequitur universalem esse veram et quamlibet25 suam singularem esse falsam. Patet dato quod iste sint omnes propositiones: omnis propositio singularis26 est falsa, hec propositio singularis est falsa, seipsa demonstrata.

 essent] sint V, sic E2  exclusiva] et add. P E1 E2 17  sua] preiacens add. E1 E2 18  falsa] et add. P E1 E2 19  sint] etc. add. V 20  particularis] om. E1 21  negativa] non add. E2 22  negativa] om. V 23  indefinita] infinita E2 24  est] om. V 25  quamlibet] quemlibet V 26  singularis] particularis P 15

16



Appendices427

2.4.2 The second

corollary is that there is a false exclusive proposition whose exceptive is true. This is clear given that these were the only propositions: Only a false proposition is exclusive. No proposition besides a false proposition is exclusive.

2.4.2.1 Conversely,

it should also be granted that there is a true exclusive proposition whose exceptive is false. This is clear given that these are the only propositions: Only an exceptive is false. No proposition besides an exceptive is false.

2.4.3 The third

corollary is that there is a true universal proposition whose subalternate is false.35 2.4.3.1 This is clear given that these are the only affirmative propositions: Every particular affirmative proposition is false, Some particular affirmative proposition is false, or these the only negative propositions: No particular negative proposition is true, Some particular negative proposition is not true. 2.4.3.2 Conversely,

it should also be granted that some indefinite proposition with a present-tense verb without any determination as second component is true and the corresponding universal false. This is clear given that these are the only propositions: A false universal proposition exists. Every false universal proposition exists.

2.4.4 The fourth

corollary is that there is a true universal proposition each of whose singulars is false.36 This is clear given that these are the only propositions: Every singular proposition is false, This singular proposition is false, referring to itself.

428 Appendices 2.4.4.1 E contra

etiam est concedendum universalem esse falsam et quamlibet eius singularem esse veram et cuilibet supposito subiecti unam singularem correspondere. Patet dato quod iste essent28 omnes propositiones: omnis propositio universalis est falsa, (V 64vb) hec propositio universalis est falsa. 2.4.4.2 Et si contra predicta adducuntur29 auctoritates seu mea dicta30 alibi, dicatur quod illa31 intelliguntur extra materiam insolubilium. 27

⟨3. Responsiones ad argumenta⟩



ad 1.1 Ad

primum concedo quod sortes dicit falsum. Et tunc ad argumentum: sortes dicit falsum et non dicit nisi32 quod sortes dicit falsum, ergo falsum est sortem dicere (P 63rb) falsum, negatur consequentia, sed oportet addere in antecedente quod dictum33 sortis non falsificat se. ad 1.1.1 Consimiliter dico quod a est falsum. Et tunc ad argumentum: negatur consequentia, quia oportet addere in antecedente34 illud35 quod dictum est.

 quamlibet] quemlibet V  essent] sint P 29  adducuntur] inducerentur V 30  mea dicta] dictam P, materia dicta E1 E2 31  illa] iste P E1 E2 32  nisi] om. P 33  dictum] est add. P 34  in antecedente] om. E1 35  illud] id P E1 E2 27 28



Appendices429

2.4.4.1 Conversely,

it should also be granted that there is a false universal proposition each of whose singulars is true, and where to each suppositum of the subject ⟨of the universal proposition⟩ there corresponds one singular proposition.37 This is clear given that these were the only propositions: Every universal proposition is false, This universal proposition is false.

2.4.4.2 If someone

adduces ⟨some⟩ authorities or what I said elsewhere against what I have said ⟨here⟩, I reply that they are to be understood outside the context of insolubles.38 ⟨3.  Responses to the Original Arguments⟩



ad 1.1 In

response to the first argument I grant that Socrates says a falsehood. And then to the argument: Socrates says a falsehood, and he says only that Socrates says a falsehood, so it is false that Socrates says a falsehood,



I deny the inference, but one should add in the premises that what Socrates said does not falsify itself. ad 1.1.1 Similarly, I say that A ⟨that is, “Socrates says a falsehood”⟩ is false. Then to the argument, I deny the inference, because one should add in the premise what I said.

430 Appendices





ad 1.1.2 Item

cum dicitur: falsum dicitur a sorte, ergo sortes dicit falsum, dico quod consequens36 est verum sicut37 antecedens,38 sed non est a, licet sit convertibile39 cum a. Modo non habeo pro inconvenienti quod duorum convertibilium unum sit verum et reliquum falsum, sicut40 non habeo pro inconvenienti quod duo contradictoria inter se contradicentia sint41 simul falsa.42 ad 1.2 Ad secundum respondetur quod quocumque sorti significato stat quod iste43 dicat verum et stat quod iste dicat falsum. Unde si conceditur unum44 illorum dicere verum, concedatur45 alterum dicere falsum et e contra. Et46 si queritur ratio diversitatis, dicatur quod istud sequitur ex casu cum uno concesso. ad 1.2.2 Pariformiter est dicendum47 ad aliud, nam concesso quod sortes dicit verum, dicatur quod plato dicit falsum et e contra.48 Et cum dicitur: non est maior ratio,49 respondeatur50 quod sic, quia conceditur51 sortem dicere verum tamquam impertinens et platonem dicere falsum tamquam sequens ex posito cum uno concesso. Et52 si conceditur53 propositionem dictam a sorte esse veram, dicatur propositionem54 dictam a platone esse falsam, et e converso.

 consequens] antecedens P E1 E2  sicut] et add. E2 38  antecedens] consequens P E2 39  convertibile] convertibilis E2 40  non habeo – sicut] om. (hom.) V 41  sint] sunt P E1 E2 42  falsa] vera P 43  iste] illa V 44  unum] primum V E1 E2 45  concedatur] conceditur P E1 E2 46  Et] om. P E1 E2 47  dicendum] dictum E1 E2 48  e contra] e converso E1 E2 49  ratio] de uno quam de alio add. P V, de uno add. E1 50  respondeatur] respondatur V 51  conceditur] dicitur P, contingit E1 E2 52  Et] Etiam V 53  conceditur] concedetur E1 54  dictam a sorte esse veram, dicatur propositionem] om. P 36

37



ad 1.1.2 Again,

Appendices431

when ⟨in confirmation⟩ it is said:

A falsehood is said by Socrates, so Socrates says a falsehood,





I say that the conclusion is true just like the premise, but the former is not A, although it is convertible with A.39 But I do not find it inconsistent that one of two convertibles is true and the other false, just as I do not find it inconsistent that two mutually contradictory contradictories are both false.40 ad 1.2 To the second argument I respond that whichever Socrates is signified it is possible that he says a truth and it is possible that he says a falsehood. Whence, if it is granted that one of them says a truth, it should be granted that the other says a falsehood and conversely. And if a reason for the difference is sought, one should say that it follows from the scenario with what was granted. ad 1.2.2 One should reply in like manner to the other point, for once it is granted that Socrates says a truth, one should say that Plato says a falsehood and conversely. And when it was said: “there is no more reason,” one should respond that there is, because that Socrates says a truth is granted as ⟨true and⟩ irrelevant and that Plato says a falsehood ⟨is granted⟩ as following from the positum together with what was granted. And if it is granted that the proposition said by Socrates is true, I say that the proposition said by Plato is false, and conversely.

432 Appendices





ad 1.3 Ad

tertium concedo quod tam sortes quam plato dicit55 falsum. Et tunc ad argumentum: sortes dicit falsum et non dicit nisi quod plato dicit falsum, ergo falsum est platonem dicere falsum, negatur consequentia, quia oportet addere in minori quod dictum sortis non falsificet se. Modo tam56 propositio sortis quam57 propositio platonis falsificat se, licet non immediate, sed una mediante alia.58 ad 1.4 Ad quartum concedo quod non est ita59 sicut sortes dicit, sicut in (E1 p6vb) alio casu concederetur60 quod sortes non dicit verum. ad 1.4.1 Et tunc ad argumentum: non est ita sicut sortes dicit et plato dicit solummodo sicut sortes dicit, ergo ita est sicut plato dicit, concedo consequentiam et consequens. Deinde cum arguitur: ita est sicut plato dicit et sortes dicit solummodo61 sic,62 ergo ita est sicut sortes dicit, negatur consequentia. Posito enim quod tam sortes quam plato cum ceteris particulis63 dicat istam: sortes dicit falsum, patet quod ita est sicut plato dicit, quia dicit verum et tamen non64 ita est sicut sortes dicit, quia non dicit nisi falsum, non obstante quod sortes solummodo65 dicat66 sicut plato dicit. ad 1.4.2 Et ad confirmationem concedo primam consequentiam et nego secundam, sicut non sequitur: verum est quod sortes non dicit verum et67 sortes dicit sic solummodo, videlicet quod sortes non dicit verum, ergo sortes dicit sicut verum est esse, sed debet addi in antecedente quod non repugnat dictum sortis esse verum, quod negatur.

 dicit] dicunt P E1 E2  tam] om. E1 57  quam] et E1 58  mediante alia] aliud mediate P 59  ita] quod add. P 60  concederetur] conceditur P E1 E2 61  solummodo] solum E2 62  dicit solummodo sic] sic solummodo dicit V 63  particulis] paribus E1 E2 64  et tamen non (corr.)] et non tamen E2, et tamen P E1, non tamen V 65  solummodo] solum E2 66  dicat] dicit V 67  et] quia E1 E2 55

56



Appendices433

ad 1.3 To

the third argument, I grant that both Socrates and Plato say a falsehood. Then to the argument: Socrates says a falsehood and he says only that Plato says a falsehood, so it is false that Plato says a falsehood,41



I deny the inference, because one should add in the minor premise that what Socrates said does not falsify itself. But both Socrates’ proposition and Plato’s proposition falsify themselves, admittedly not immediately, but one through the other. ad 1.4 To the fourth argument, I grant that it is not as Socrates says it is, just as in another scenario I would grant that Socrates does not say a truth. ad 1.4.1 Then to the argument: It is not as Socrates says it is and Plato says only what Socrates says, so it is as Plato says it is,42 I grant the inference and the conclusion. Next, when it is argued: It is as Plato says it is and Socrates says only that, so it is as Socrates says it is,43 I deny the inference. For supposing that along with the other elements ⟨of the scenario⟩ both Socrates and Plato say Socrates says a falsehood,



it is clear that it is as Plato says it is, because he says a truth, but it is not as Socrates says it is, because he says only a falsehood, notwithstanding that Socrates says only what Plato says. ad 1.4.2 ⟨In response⟩ to the confirmation, I grant the first inference44 and deny the second, because this is not valid: It is true that Socrates does not say a truth and Socrates says only this, namely, that Socrates does not say a truth, so Socrates says what is truly so. But one should add in the premises that it is not inconsistent that what Socrates says is true, which I deny.45

434 Appendices

⟨4. Responsio ad unam communem difficultatem⟩ 4.1 Consimiliter

respondetur ad unam communem68 difficultatem,69 quando ponitur quod a sit hec propositio: hec propositio significat aliter quam est, et b70 una consimilis,71 c72 vero73 suum contradictorium semper demonstrando a, sic quod quelibet illarum propositionum74 significat precise sicut termini75 pretendunt. Isto posito queritur utrum ita est totaliter76 sicut a significat vel non. 4.1.1 Si sic, ergo c est verum. Sed c est contradictorium b, ergo b est falsum. Tunc sic: b est falsum et non falsificat77 se, ergo significat78 aliter quam est. Et a significat precise79 sicut b significat, ergo a significat aliter quam est, et per consequens non ita est totaliter80 sicut a significat. 4.1.2 Quod si conceditur, contra: non est ita totaliter81 sicut a significat, et a82 significat aliqualiter ⟨esse⟩, ergo a significat aliter quam est. Sed b significat precise quod a significat aliter quam est, ergo ita est totaliter sicut b significat. Tunc sic: ita est totaliter sicut b significat, sed a significat precise sicut significat83 b, ergo ita est totaliter sicut a significat, quod iterum est oppositum concessi. ad 4.1.1 Dicendum quod non est ita totaliter sicut a significat et quod a significat aliter quam est et consequenter84 quod ita est totaliter sicut b significat.  communem] consimilem P E1, om. E1  difficultatem] consimilem (dub.) add. P 70  b] c E1 71  consimilis] et add. P 72  c] b E1 73  vero] om. P 74  propositionum] om. P 75  termini] tibi add. E1 E2 76  est totaliter] sit E1 E2 77  falsificat] fallit E1 E2 78  significat] om. V 79  precise] om. V 80  ita est totaliter] est ita formaliter P, est ita precise et totaliter E1 E2 81  totaliter] formaliter P 82  significant et a] om. V 83  significat] om. E1 E2 84  consequenter] conceditur P E1 E2 68 69



Appendices435

⟨4.  Response to a Standard Argument⟩ 4.1 I respond

similarly to a standard difficulty, when it is supposed that A is this proposition: This proposition signifies other than it is,46

and B is a similar one, and in addition C is their contradictory, referring throughout to A, assuming that each of these propositions signifies only as the terms suggest.47 Given this ⟨scenario⟩, I ask whether it is wholly as A signifies or not. 4.1.1 If it is, then C is true. But C is the contradictory of B, so B is false. Then ⟨I argue⟩ like this: B is false and it does not falsify itself, so it signifies other than it is. And A signifies only as B signifies, so A signifies other than it is. Consequently, it is not wholly as A signifies.48 4.1.2 If that is granted, on the contrary: it is not wholly as A signifies, and A signifies that it is in some way, therefore A signifies other than it is. But B signifies only that A signifies other than it is, so it is wholly as B signifies. Then ⟨I argue⟩ like this: it is wholly as B signifies, but A signifies only as B signifies, so it is wholly as A signifies, which is again the opposite of what was granted.49 ad 4.1.1 My response is that it is not wholly as A signifies, and that A signifies other than it is, and consequently that it is wholly as B signifies.50

436 Appendices

ad 4.1.2 Et

tunc ad argumentum quo concluditur quod ita est totaliter85 sicut a significat, negatur consequentia, sed debet addi in antecedente quod non repugnat a esse verum aut quod a non asserit se esse falsum, quod negatur, ergo etc.

 totaliter] formaliter P

85



ad 4.1.2 Then

Appendices437

to the argument in which it is concluded that it is wholly as A signifies, I deny the inference, but one would need to add in the premises that it is not incompatible that A is true or that A does not assert that it itself is false, which I deny,51 hence etc.

Notes on the Appendices On Appendix A  See also chaps. 27 and 29 of the third Part.  The added text here (“that is or will be”) follows the standard medieval account of ampliation of future-tensed propositions. For Paul’s account, see Logica parva, trans. Perreiah, chap. II, § 8. It should be assumed to hold uniformly of all examples below, but will be added explicitly only when the structure of the argument is clearer by its addition. 3  That is, that A will never be true. 4  See § 2.2.3 below. 5  See Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, the First Way, 1–15. 6  Amending the incunable and manuscripts, which all read, “A man is not an ass.” 7  See Albert of Saxony, “Insolubles,” 338. 8  See Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, the Fourth Way, 36–43. 9  See Logica magna: Tractatus de terminis, 61, 97–121. 10  See Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, the Second Way, 15–25. 11  It was proven in the corollary to the third conclusion. 12  This appears to be an allusion to one of the first rules of obligations, that a positum should be accepted unless it is impossible or implies a contradiction. See Dutilh Novaes, “Ralph Strode’s Obligationes,” 343, and our introduction, § 3. 13  See Logica magna: Tractatus de veritate et falsitate propositionis et Tractatus de significato propositionis, 55 and 262 n. e. 1 2

On Appendix B  Resolution is closely related to exposition, amounting in effect to the expository syllogism. On exposition and resolution, see Logica magna: Tractatus de terminis, 225–9, and Andrews, “Resoluble, Exponible, and Officiable Terms,” 10: “An example from Billingham, ‘A man runs’ (‘homo currit’) is resolved into the syllogism: ‘This runs, and this is a man, therefore a man runs.’” 2  For the confirming argument see § 4.2.1.2 of the treatise on insolubles in the Logica magna. (Further references to the Logica magna will be to the treatise on insolubles unless otherwise specified.) 3  For an expanded version of this argument see § 4.2.1.3 of the Logica magna. On Paul’s view that “Socrates” is a general term see commentary on § 4.1.1 of the Logica magna, where it is clear that Paul treats “Socrates” as a general term. So “Socrates says a falsehood” is taken as indefinite or particular, that is, as “Some Socrates says a falsehood,” not as a singular proposition. 4  For the first part of the second argument, see § 4.2.2 of the Logica magna. 1

440

Notes on the Appendices

 For the second part of the second argument, see § 4.2.3 of the Logica Magna.  For the third argument, see §§ 4.2.4–4.2.4.1 of the Logica Magna. (Actually, what was to be proved was that if Socrates says a falsehood then he says a truth, so the first half of the paragraph, lifted from the Logica Magna, is irrelevant.) 7  For the fourth argument, see § 3.1–3.1.2 of the Logica magna. 8  See De interpretatione, chap. 6. 9  Paul draws four further consequences from each of this four main conclusions. We will refer to each of these further consequences as a corollary. 10  The alternatives rejected here appear to be all those solutions following Bradwardine which postulate a second, implicit, signification, such as Heytesbury, the modified Heytesbury solution, and many more, including Paul’s own solution in the Logica Parva and in the Quadratura. See introduction, § 4, and Spade and Read, “Insolubles,” §§ 3.4 and 4.2. 11  Perhaps notum (literally, “known”) here amounts to “specified” or “announced,” recalling Heytesbury’s claim that the respondent “is under no obligation to specify (certificat) what the secondary signification is,” whereas pseudo-Heytesbury specifies that it is “that the proposition is true” (and Gregory that the first conjunct of the mental proposition is false). See introduction, § 4, and Pironet, “William Heytesbury and the Treatment of Insolubilia,” 276. 12  The alternative would seem to be not to admit the insoluble. 13  See Pironet, “William Heytesbury,” 292). Thus Paul takes “the standard opinion” to be the modified Heytesbury solution, which he presented in the Logica parva. On self-reflection, see Logica magna, § 1.13.4 (discussing Peter of Ailly’s solution) and § 2.1.2 (developing his own solution). 14  See Pironet, “William Heytesbury,” 292. 15  As do all mental propositions: see Spade and Read, “Insolubles,” § 3.4. 16  This corollary was inferred from the second assumption (namely, that “a false proposition is one which either falsifies itself or whose falsity does not arise from its terms, but from its false exact significate”) in the Logica magna (§ 2.2.2). 17  This corollary, and its proof, are the same as the second conclusion and proof in the Logica magna (§ 2.3.2). 18  This observation and the caveat match the third conclusion in the Logica magna (§ 2.3.3). 19  This conclusion, and its proof, are the same as the first conclusion and proof in the Logica magna (§ 2.3.1). 20  In effect, Paul is arguing here for the S5-thesis, that what is possibly impossible is impossible. However, note that the modal expressions here are predicates, not operators. 21  Recall that for the medievals, negative propositions are true if the subject is empty. 22  See the fourth objection in the Logica magna (§ 3.4) and Paul’s reply in §ad 3.4, where he observes that this follows from his first conclusion (Logica magna, § 2.3.1), that “there are two mutually contradictory contradictories, one of which is impossible and the other contingent.” Nonetheless, this seems a strange observation, since what contradicts the impossible is the possible, not the necessary. 23  See the fifth objection in the Logica magna (§ 3.5) and Paul’s reply in §ad 3.5. 24  For something to be necessary is for it not to be possible that it is false, that is, if it exists, it is true. But it might not exist. One might conjecture that all exact significates which refer to a contingent being and can be true are contingent. 25  The third conclusion, and its proof, are the same as the fourth conclusion and proof in the Logica magna (§ 2.3.4). 26  The first corollary, and its proof, are identical with the fifth conclusion and proof in the Logica magna (§ 2.3.5). 27  The second corollary, and its proof, are the same as the sixth conclusion and proof in the Logica magna (§ 2.3.6). The conclusion was shown to be impossible in § 2.2.0.2 above. 5 6



Notes on the Appendices

441

28  The reasoning is a little different in the sixth objection in the Logica magna (§ 3.6) and Paul’s reply in § ad 3.6. 29  Although absent from all the witnesses (manuscripts and editions), sense seems to demand repeating non (“not”) in the conclusion. The conclusion was shown to be contingent in § 2.2.4.1. 30  This corollary, and its proof, are the same as the seventh conclusion and proof in the Logica magna (§ 2.3.7). 31  This conclusion, and its proof, are the same as the fourth corollary and proof in the Logica magna (§ 2.4.4). 32  This observation, and its proof, are the same as the fifth corollary and proof in the Logica magna (§ 2.4.5). 33  This corollary, and its proof, are identical with the sixth corollary and proof in the Logica magna (§ 2.4.6). 34  This observation, and its proof, are the same as the seventh corollary and proof in the Logica magna (§ 2.4.7). 35  This corollary, and its proof, are the same as the first corollary and proof in the Logica magna (§ 2.4.1). 36  This corollary, and its proof, are the same as the third corollary and proof in the Logica magna (§ 2.4.3). 37  This observation, and its proof, are the same as the second corollary and proof in the Logica magna (§ 2.4.2). 38  See the end of § 2.4.5 of the Logica Magna. 39  For the reply’s confirmative argument see ad § 4.2.1.2 of the Logica Magna. 40  See the Fourth Corollary to the First Conclusion and Logica Magna § ad 4.2.1.3. 41  Actually, this is a slight misquotation. The original argument ran: “Socrates says a falsehood and he says only that Plato says a falsehood, so Plato does not say a falsehood.” 42  This seems to be a more troubling misquotation. The relevant part of the argument seems to be: “it is not as Socrates says it is and Plato says only that it is not as Socrates says it is, so it is as Plato says it is.” 43  Again, a slight misquotation. The relevant passage seems to be: “it is as Plato says it is. But Socrates says in every way what Plato says. So it is as Socrates says it is.” 44  That is: “if it is not as Socrates says it is, then it is true that it is not as Socrates says it is.” 45  This reply is the same as Paul’s reply to the first objection in the Logica magna (§ ad 3.1.1). 46  Cf. Logica magna, § 3.2. 47  Cf. Logica magna, § 3.2.1. 48  Cf. Logica magna, § 3.2.2. 49  Cf. Logica magna, § 3.2.1. 50  Cf. Logica magna, § ad 3.2. 51  Cf. Logica magna, § ad 3.2.1.

Indices Names Albert of Saxony, 12, 57, 295, 299, 316, 325, 331, 439 n. 7 solution to insolubles, 26, 28 n. 100, 32, 41, 298, 300, 309, 324, 325 n. 90, 342 Almericus of Serravalle, 54 Angelo of Fossombrone, 41–3, 47, 59–60 Anthony de Monte, 54 Antonio Cittadini da Faenza, 1, 37, 38 n. 136 Arnoldus Fabri of Sint-Truiden, 53 Aristotle, ix, 37, 56, 173, 300, 332, 334, 358 De interpretatione, 11, 23, 26, 307, 341, 409, 440 n. 8 De sophisticis elenchis, 11, 15, 109, 277, 279–80, 282, 313, 318, 353 Metaphysics, ix, 3, 139, 323 Posterior Analytics, 323 Prior Analytics, 129, 319, 334 Topics, 11 Ashworth, Earline Jennifer, xi, 12, 14–15, 57, 311, 327, 364 Blaise of Parma, 3 n. 8, 47–8 Bocheński, Jósef Maria, 28 n. 100, 349 Boethius, Anicius Manlius Severinus. 11, 26, 56, 277, 293–4 Bottin, Francesco, 1 n. 3, 2 n. 6, 3 n. 7, 28 n. 100 Cajetan of Thiene, 40 n. 146, 43–7, 286, 289, 292 Conti, Alessandro, xi, 1 n. 3, 2 n. 6, 3 n. 7 Copenhaver, Brian, 277 Cristoforo Barzizza, 37 Francesco Filelfo, 54, 56 Giovanni Ludovico Lambertazzi, 54–5 Gregory of Rimini, 24, 26–8, 60 n. 251, 290, 309–12, 362, 440 n. 11

Henry Anglicus, 54 n. 230 Johannes Sharpe, 1 John Anthony of Imola, 46 John Buridan, 26, 279, 284, 294–5, 325, 329–31, 360 solution to insolubles, 24–6, 32, 298–9 John Dumbleton, 286 solution to insolubles, 17, 32, 40 n. 146, 46, 289, 339, 361 John Hunter (Venator), 19–21, 28, 304 n. 59, 325, 347 John of Holland, 20, 28, 304 n. 59, 361 John Tarteys, 1, 325 n. 88 John Wyclif, 1, 41, 316–17, 325, 332 Marsilius of Inghen 27, 32, 316, 324–6, 360 Mattheolus Mattioli Perusinus, 54 Menghus Bianchellus, 39 n. 141 Nicolò Fava, 43 Paul Francis of Venice, 54–5 Paul (Nicoletti) of Venice, passim fundamental principle, 33, 203, 269, 350 Logica magna, authenticity of, 2 n. 4, 12, 28, 36–60 Logica parva, 1–2, 4, 12, 21, 28–31, 37–40, 43, 54, 56, 58, 60, 296, 301 n. 50, 302, 323, 325, 327, 342, 354–5, 440 n. 10, n. 13 Order of the Hermits of St. Augustine, 1 Quadratura, 2, 28, 30–1, 37–9, 53, 56, 58, 292, 296, 306, 363, 368–99, 440 n. 10 Sophismata aurea, xi, 2, 28, 35, 37, 49 n. 175, 56–8, 333, 338–9, 400–37 Paul of Pergula, 41, 43, 46–7, 59, 363 n. 130 Perreiah, Alan, 1 n. 3, 2 n. 6, 3 n. 7, 4, 29, 30 n. 103, 36–40, 46, 48–9, 53–60, 313, 327, 331

444 Indices Peter Abelard, 11 Peter of Ailly, 140 n. 27, 152 n. 75, n. 79, n. 81, 156 n. 98, 299, 316 solution to insolubles, 26–7, 32, 41, 57, 103–04, 109, 113, 309–15, 324–5, 328, 330–1, 343, 440 n. 13 Peter of Candia, 12 Peter of Mantua, 12, 15, 31–2, 41, 43, 47, 54, 56–7, 60, 100 n. 114, n. 115, 116–21, 299, 302, 306–08, 313, 315–17, 347 Peter of Spain, 56, 284, 294, 318, 331 Pier Paolo Vergerio, 54 Pietro Tommasi, 54–5 Pironet, Fabienne, 24, 301–02 Ralph Strode, 12, 15, 54, 59, 289, 364 Richard Billingham, 57, 298, 439 n. 1 Richard Brinkley, 290 Richard Ferrybridge (Ferebrigge), 30 n. 105, 47 Richard Kilvington, 32, 40 n. 146, 282, 282 n. 16, 286, 289, 292, 339 disputational meta-argument, 32 n. 111, 307–08 Richard Swyneshed (Swineshead), 14 Robert Alyngton, 1 Robert Eland (Fland), 20 Roger Swyneshed, 14–15 solution to insolubles, 18, 21–4, 28, 31–3, 35, 40 n. 146, 43–8, 58, 60, 286–7, 289,

303, 307, 323, 327–8, 332–5, 337, 339–43, 345–6, 350–5, 359, 361–3 Roger Whelpdale, 1 Spade, Paul Vincent, 20, 27, 28 n. 100, 33, 285, 287–9, 292, 297–8, 301–03, 309–10, 312–13, 316–17, 324, 325 n. 90, 328–31, 335, 340–1, 352 Thakkar, Mark, 325 n. 88 Thomas Bradwardine, 130 nn. 240–244, 277, 283–9, 316–17, 319–21, 323, 325, 328, 342, 352, 354–5, 440 n. 10 solution to insolubles, 15–22, 24, 27, 32, 41, 282, 297–8, 300, 304–05, 309, 312 Tommaso of Coderonco, 59–60 Walter Burley, 12–14, 60, 308, 318, 343, 348 solution to insolubles, 16, 279, 283–5, 290 Walter Segrave, 16 n. 59, 32, 54 n. 230, 120–33, 284, 297, 317–22, 331 Whitaker, C. W. A., 23, 307 William Buser, 12, 316 William Heytesbury, 3, 40, 44–6, 54 n. 230, 92 n. 74, 289, 292, 316, 333, 339–42, 344–5 solution to insolubles, 18–22, 27–9, 31–3, 35, 41–3, 57, 60, 93–103, 286, 301–08, 340–2, 344–5, 347, 440 n. 10, n. 11

Subjects cassationism (cassatio), 16–17, 34, 285, 287, 323 cognition, 111, 153–7, 313–14, 330 formal, 103, 309–10 proper and distinct, 309 composition and division (mental), 113, 314 composition of terms, signifying by, 33–4, 36, 151, 165, 167, 201, 335–7, 349, 379, 391, 407, 413, 419, 421 comprehension, 111, 153–7, 314, 330 concept, 103 (conceptus), 291 (ratio) conception (noticia), 103, 111, 314 consequence, theories of, 11, 33–4

dici de omni, 129, 319 exponent, 151, 320–1, 338–9 exponible, 338, 356, 399, 425 exposition, 328, 361, 439 expository syllogism, 439 extreme, conjoint, 397 fallacy of accident, 32, 121–3, 317–9 of equivocation, 85 of false cause, 81 of form of expression, 79–81, 277–9, 284

of the restricted and unrestricted (secundum quid et simpliciter), 29, 89, 109, 119, 279–83, 288, 293, 316 Heytesbury solution, modified, 21, 24, 27–9, 31, 36, 42–43, 58, 301, 342, 440 imposition (first and second), 119 see also intention, first and second inference, theories of by parity of reasoning, 89, 95, 117, 163, 271, 330, 415 from inferior to superior, 149 from superior distributed negatively to its inferior, 199 see also consequence, theories of insolubles arising from an act of ours, 135, 197, 323 arising from a property of an expression, 135, 209, 351 definition, 23, 35, 40, 89, 95, 297, 312, 324, 355, 361 exceptive, 251–5, 355–6, 360–1, 427, see also proposition, exceptive exclusive, 171–3, 251–3, 339, 355–6, 360–1, 425–7, see also proposition, exclusive exterior, 135, 197, 348 interior, 135, 197, 348 merely apparent, 10, 33, 35, 257–75 non-quantified, 10, 251–5, 355, 360–1 quantified, 10, 237–49, 355–60 particular, 237–9, 337–9, 348, 358, 425–7, 439 indefinite, 237–9, 247, 348, 358–9, 439 universal, 239, 245, 338–9, 359, 383–5, 425–9 intelligences, 111, 309, 313–14 intention (first and second), 45, 119 obligations, 11–15, 18–19, 42, 49, 57, 269, 281, 301, 304 responsio antiqua, 12–14, 308, 364 responsio nova, 12, 14, 348 time of, 101, 306 officiably, 399 Oxford Calculators, 14

Indices445 paradox, ix, 15, 26, 36, 46, 303, 305, 337, 340, 344 epistemic, 11, 23–4, 34–5, 257–63, 334, 337, 361–4 Fitch’s, 363–4 inferential Knower, 34, 421 Knower, 24, 34, 257, 334, 361 liar, 15, 18, 22–3, 25, 27, 33, 299, 352 Moorean, 343 no-no, 203–09, 350–1 semantic, 11 yes-no (or postcard), 207, 351 precise (“only”), 19, 46 n. 170, 47 n. 172, n. 173, 80, 92–4, 98–100, 174–82, 188–96, 200–6, 210–20, 224, 236–40, 244–52, 258, 264, 272, 285, 301–02, 306, 308, 335, 352, 364, 368–82, 386, 390, 394–6, 400–04, 410, 420, 434 prejacent, 131, 320–2, 356–7 properties of terms, 11 proposition (propositio), 38–40, 119 convertible, 16–17, 42 n. 151, 43, 91, 123, 149, 167–9, 183–7, 201–05, 209, 235, 255, 267, 300, 342, 350, 361, 401, 419, 423, 431 endorsing itself, 139, 323 exceptive, 6, 35, 131, 302, 320–2 exclusive, 6, 35, 44, 151, 302, 320 falsifying itself, 22, 47 n. 172, 179, 183, 187–9, 201–09, 215–23, 227, 231–3, 237–9, 245, 249–53, 259–61, 265, 328, 334–5, 337–40, 342–5, 349–50, 353–5, 359–61, 385, 403, 413, 425, 429, 433–5, 440 n. 16, see also proposition, self-falsifying of the second component, 185, 314, 343, 427 having reflection on itself, 16, 33, 107, 137, 203–05, 342, 361 immediately 137, 189 non-immediately 137, 205 loosely mental, 107, 309–10 manifold (propositio plures), 107–09, 117–19, 312 self-falsifying, 33, 35, 48, 139–43, 165, 259, 269, 323–4, 334, 352–3, see also proposition, falsifying itself self-undermining, 139, 323 self-verifying, 143, 149, 324, 334

446 Indices strictly mental, 103–07, 113, 309 propositional complex (complexum), 87, 125, 137, 362 represent, 99 as an object, 103 formally and naturally, 107 resolution, 401, 439 n. 1 restrictivism (restrictio), 16–17, 32, 34, 42, 278–9, 283, 297, 309, 317, 319–20, 322 rule of contradictory pairs, 23, 307, 336, 341, 349, 354 scenario (casus), 12–13, 18–20, 28–31, 35, 41, 45–7, 93–5, 99–101, 141, 169, 175–9, 185–207, 211–15, 219–21, 225–33, 237–55, 263, 280, 283, 301–08, 318–20, 324, 332, 336, 340, 346–55, 359–61, 369, 373, 381–3, 387, 399–405, 415–17, 425, 431–5 self-comprehension, 309 sense, transcendental, 117, 316 sign of exclusion, 95 significate, 26, 29–31, 33, 36, 99, 127, 237, 261, 298, 314, 409 exact, 29, 31, 87, 91, 101, 147–9, 161, 167, 181–3, 187–9, 199–209, 237, 245, 261, 289–91, 306, 326, 333, 342, 344, 349,

353, 360–3, 366, 383, 399, 409–19, 440 n. 16, n. 24 primary, 43, 97, 363 principal, 30–1, 97, 381–3 total/whole, 290, 393–5 signify as the terms suggest (sicut termini pretendunt), 19, 28–9, 93–5, 99, 175, 179, 183, 195–7, 237–9, 247–55, 259–63, 273, 306, 339, 347, 361, 411, 435 consequentially, 46, 139, 181 exactly, 23, 31, 36, 42, 99–101, 117, 121, 145, 161, 165–9, 191–9, 207–11, 215–23, 227–33, 237–47, 251–55, 259–67, 291–2, 303, 306, 347, 349, 381–3, 397–9, 407–09, 417 primarily, 34, 95, 99, 257, 265, 291, 306, 335–7, 363 principally, 22, 24, 30–1, 44, 48, 95, 99–101, 189, 292, 306–07, 327, 335, 350, 354, 359, 377, 381, 397 term equivocal, 117 privative, 16, 125 topic, 293–6 from the disjunctive whole, 89, 293, 297

Manuscripts Erfurt, Bibliotheca Amploniana, Octavo 76, 318–22 Erfurt, Bibliotheca Amploniana, Quarto 276, 317–22 Oxford, Bodleian Library, Can. misc. 219, 54 n. 230, 318, 322 Oxford, Bodleian Library, Can. misc. 471, 47 n. 171 Padua, Biblioteca Universitaria, 925, 61 Paris, Bibliothèque nationale de France, lat. 6433A, 61 Parma, Biblioteca Palatina, Parmense 1023, 4, 5 n. 15, 48

Vatican City, Biblioteca Apostolica Vaticana, Pal. lat. 995, 27 n. 96, 324 n. 86 Vatican City, Biblioteca Apostolica Vaticana, Urb. lat. 1381, 2 n. 4, 38 n. 136, 39 n. 140, n. 141, 40 n. 144 Vatican City, Biblioteca Apostolica Vaticana, Urb. lat. 1488, 53 Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2130, 41 n. 147, n. 150, 42 nn. 151–3 Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2132, vi, 3, 48–9, 53, 61 Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2133, iv, 2 n. 6, 61 Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2134, 2 n. 6, 53, 62



Indices447

Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 2139, 41 n. 147, 43 n. 154 Vatican City, Biblioteca Apostolica Vaticana, Vat. lat. 5363, 58 n. 244

Venice, Biblioteca Nazionale Marciana, Lat. 6.30 (2547), 4, 7 n. 25, 41 n. 147, n. 150, 42 nn. 151–3, 48 Venice, Biblioteca Nazionale Marciana, Lat. 6.63, 47 n. 171

Insolubles1 Aliqua propositio est falsa, 196 Aliqua propositio negativa non est vera, 162, 196, 414 Aliqua propostio non appellatur a suo predicato, 134 Aliqua propositio non appellatur a suo subiecto, 134 Aliqua propositio particularis affirmativa est falsa, 168, 426 Aliqua propositio significat aliter quam est, 178, 188 Aliquod propositum nescitur a te, 262 Aliquod propositum tibi est a te negandum, 256 Deus est et hec copulativa est falsa, 104 Dubium tibi proponitur, 272 Ego dico falsum, 82 Ego non dico verum, 84 Falsum dicitur, 100, 102 Falsum dicitur a sorte, 88, 130 Falsum est, 82, 86, 96–8, 114, 134, 236 Hec propositio significat aliter quam est, 176, 434 Hec propositio singularis est falsa, 170, 426 Hoc est falsum, 86, 96, 100, 104, 106, 118, 122, 138, 142, 148, 164, 184, 208, 398, 408, 412 Hoc est negandum a te, 262–70 Hoc est nescitum a te, 164

Hoc est verum, 142 Hoc non est concedendum a te, 270–2 Hoc non est scitum a te/Hoc non scitur a te, 256–60 Hoc non est verum, 90, 138, 142, 208 Iste homo dicit falsum, 196 Nihil tibi proponitur nisi dubium, 274 Non est ita sicut aliquis homo dicit, 176 Non est ita sicut sortes dicit, 174, 184, 186–8, 404 Nulla propositio est vera, 138 Nulla propositio preter a est falsa, 252 Nullum verum est, 134 Omne verum est a, 238–40 Omnis propositio est falsa, 94, 100, 138, 244–6, 376, 380, 406 Omnis propositio preter a est verum, 254 Omnis propositio universalis est falsa, 168, 170, 172, 382, 424, 428 Omne propositum est dubitandum a te, 274 Omnis universalis affirmativa est falsa, 424 Plato decipitur, 214–18 Plato dicit falsum, 204–06, 402–04 Plato intelligit impossibile, 196 Plato male respondet, 224–26 Plato non videt verum, 196 Plura sunt vera quam falsa, 248 Propositio non verificatur pro se, 134

 The present list contains only the insolubles mentioned in Paul of Venice’s texts edited in this volume (Logica magna, Quadratura, Sophisma 50), not those mentioned in the introduction or commentary. 1

448 Indices Quelibet propositio est dissimilis istis, 240–4 Quilibet homo preter me dicit verum, 254 Quot sunt vera tot sunt falsa, 246–8 Sortes audit falsum, 196 Sortes cogitat falsum, 196 Sortes decipitur, 210–12 Sortes dicit falsum, 78, 80, 84, 92, 122, 134, 162, 184, 190–202, 400–02, 408 Sortes est albus, 224 Sortes est eger, 224–6 Sortes est periurius, 220–2

Sortes Sortes Sortes Sortes Sortes Sortes

intelligit falsum, 82, 134 legit falsum, 82 mentitur, 218–20 non dicit verum, 408 non habebit denarium, 224, 228–30 non pertransibit pontem, 230–4

Tantum a est verum, 250–2 Tantum exclusiva est falsa, 252 Tantum falsum est exclusiva, 170, 424 Tantum plato dicit verum, 252 Tantum propositio falsa est exclusiva, 250, 426

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