Treatise on Consequences 9780823257218

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Treatise on Consequences

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Medieval Philosophy Texts and Studies series editor

Gyula Klima Fordham University editorial board

Richard Cross Brian Davies Peter King Brian Leftow John Marenbon Robert Pasnau Giorgio Pini Richard Taylor Jack Zupko

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Treatise on Consequences John Buridan Translated and with an Introduction by Stephen Read Editorial Introduction by Hubert Hubien

Fordham University Press • New York • 2015

Copyright © 2015 Fordham University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means— electronic, mechanical, photocopy, recording, or any other—except for brief quotations in printed reviews, without the prior permission of the publisher. Fordham University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Fordham University Press also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Visit us online at www.fordhampress.com. Library of Congress Cataloging-in-Publication Data Buridan, Jean, 1300–1358. [Tractatus de consequentiis. English] Treatise on consequences / John Buridan ; translated and with an introduction by Stephen Read ; editorial introduction by Hubert Hubien. — First edition. pages cm. — (Medieval philosophy: texts and studies) Includes bibliographical references and index. ISBN 978-0-8232-5718-8 (cloth : alk. paper) 1. Logic—Early works to 1800. 2. Proposition (Logic) 3. Logic, Medieval. 4. Syllogism. I. Read, Stephen, 1947– translator. II. Title. BC60.B8713 2014 160—dc23 2014002167 Printed in the United States of America 17 16 15 5 4 3 2 1 First edition

contents Acknowledgments ix Introduction 1 Editorial Introduction by Hubert Hubien 53 Book I Consequences in General and Consequences Between Assertoric Propositions 63 Chapter 1: The Truth and Falsity of Propositions 63 Chapter 2: The Causes of the Truth and Falsity of Propositions 64 Chapter 3: The Definition of Consequence 66 Chapter 4: The Division of Consequences 68 Chapter 5: The Supposition of Terms 69 Chapter 6: The Ampliation of Terms 71 Chapter 7: The Matter and Form of Propositions 74 Chapter 8: Conclusions 75

Book II Consequences Between Modal Propositions 95 Chapter 1: Modal Propositions 95 Chapter 2: The Division of Modal Propositions into Composite and Divided 95 Chapter 3: The Division of Divided Modal Propositions into Affirmative and Negative 96 Chapter 4: The Ampliation of the Terms of Divided Modal Propositions 97 Chapter 5: Equivalences 98 Chapter 6: Conclusions about Divided Modals 99 Chapter 7: Conclusions about Composite Modals 105

Book III Syllogisms Between Assertoric Propositions Part I. Syllogisms between Propositions Containing Direct Terms Chapter 1: The Division of Consequences 113 Chapter 2: The Syllogism 115

viii contents Chapter 3: The Division of Terms into Finite and Infinite 116 Chapter 4: Conclusions 116

Part II. Syllogisms between Propositions Containing Oblique Terms Chapter 1: Propositions Containing Oblique Terms 128 Chapter 2: The Syllogistic Extremes and Middle 129 Chapter 3: Conclusions 130

Book IV Syllogisms Between Modal Propositions 140 Chapter 1: Syllogisms between Composite Modal Propositions 140 Chapter 2: Syllogisms between Divided Propositions of Necessity and Possibility 143 Chapter 3: Syllogisms between Divided Modal Propositions of Each-Way Contingency 155 Chapter 4: Syllogisms between Reduplicative Propositions 159

Notes 163 Glossary 177 Index 181

Ac know ledg ments First, I must express particular gratitude to members of the Medieval Logic Reading Group at the University of St Andrews, in particular Catarina Dutilh Novaes, participating mostly via the Internet, and Spencer Johnston, my PhD student. Those meetings, between 2008 and 2012, were invaluable, as we worked through the text and the translation line by line, even word by word. I am also grateful to the referees, in particular to Terence Parsons, for their often very detailed comments on the translation and the Introduction. Nonetheless, if any inaccuracies or confusions remain, I must take full responsibility for them. I must also acknowledge my debt to other colleagues at St Andrews and to the very supportive research environment in the Department of Philosophy and within the Arché Philosophical Research Centre. Much of the work was done during the Foundations of Logical Consequence project, funded by the Arts and Humanities Research Council of the United Kingdom. Among other colleagues, Sarah Broadie often gave me valuable advice. I am also grateful to the librarians at the University of Liège and at the Vatican Library (Biblioteca Apostolica Vaticana) for providing me with photocopies of the manuscripts containing Buridan’s Tractatus de Consequentiis. I am grateful to Professor Hubien’s daughter, Mme Sophie Hubien, for permission to include an English translation of his Editorial Introduction to his 1976 edition of the Latin text.

Stephen Read

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Treatise on Consequences

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Introduction Stephen Read

1. John Buridan John Buridan was born in Béthune in Picardy, in northern France, in the late 1290s. He spent his entire academic career in the arts faculty at the University of Paris, dying around 1360. In this regard, his career was unusual. Most other scholars either joined one of the religious orders (John Duns Scotus and William Ockham were Franciscans, Paul of Venice an Augustinian), or at least proceeded from their studies in arts to one of the higher faculties, usually theology (Scotus and Ockham did, as did Walter Burley, Thomas Bradwardine, William Heytesbury, and many others). Many of their works on logic and other arts matters were composed while studying in the higher faculty (often as a source of income from teaching), but eventually their interests moved onto theological matters. Buridan remained resolutely an arts man throughout his career. As he writes in his Tractatus de Consequentiis (Treatise on Consequences), the subject of the present study, “whether . . . syllogisms in divine terms are formally valid . . . I leave to the theologians, and . . . it is not for me, an Arts man, to decide regarding the foregoing beyond what was said” (Book III, Part I, Chapter 4, first conclusion). Our first firm evidence for Buridan’s career is when he was elected rector in 1327, in charge of all teaching at the University of Paris, a responsible post with a new incumbent elected every three months. He held it from mid-December 1327 until mid-March 1328. He must, therefore, have started his university studies some five or ten years earlier, for the rectorship was normally held by a young teaching master. He had not been a wealthy student, for we know that he held a benefice for needy students at the College of Lemoine. He continued to be supported by such benefices throughout his career, one of which, perhaps the main one, was that of the Collegiate 1

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Church of St Pol-sur-Ternoise at Arras. He seems to have held this benefice for life, which aids us in dating his death, since it was bestowed on his successor in 1361. There is evidence that he died on October 11, probably in 1360, and certainly no earlier than July 1358.1 The University of Paris was divided by the origins of its students into four nations, the French comprising those from France roughly south of Paris, Spain, and Italy; the English (later “English-German”) made up of those from Britain, the German and Slavic-speaking countries, and Scandinavia; the Norman, those from Normandy; and the Picardian, those from an area running from what is now Picardy northeast through present-day Belgium up to the Meuse. Béthune and Arras lie centrally in this area. Teaching, and even social life, seems to have been heavily segregated between the nations. Buridan himself was involved in disputes among the nations; we know he was still active in Paris in 1358 since he was party to a dispute with the English nation over whose right it was to examine a certain student. Each nation, for example, had its own teaching rooms, and only in exceptional cases would a student from one nation study under a master from another. Thus it is highly unlikely that Albert of Saxony was a student of Buridan’s (as has sometimes been supposed), since Albert belonged to the English nation.2 Albert shares certain doctrines with Buridan, but he has much more in common with English masters of the time such as Burley and Ockham and is highly critical of Buridan on several counts.

1.1. Buridan’s Works By the time Albert arrived in Paris in 1351, Buridan had been teaching and writing there for perhaps twenty-five years. He was a prolific author, though given his secular interests, his writings are restricted almost entirely to commentaries on Aristotle and two logical treatises, the Treatise on Consequences and the Summulae de Dialectica. The Summulae is a mammoth work, running to some one thousand pages in its recent English translation.3 It clearly went through many revisions over the course of at least twenty years, and its nine component treatises survive in versions that are apparently from different periods, some as early as 1340, others probably revised as late as the mid-1350s. It is written ostensibly as a commentary on the Summulae (or Tractatus—literally, Treatises) of Peter of Spain, composed in the early to mid-thirteenth century. Indeed, Buridan seems to have rescued this work from obscurity by his commentary. In at least one of the compo-

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nent treatises, he replaced Peter’s text with a text of his own on which to comment. Buridan’s Summulae consists of eight main treatises (on propositions, predicables, categories, supposition, syllogisms, loci, fallacies, and demonstration). In addition, some manuscripts include a ninth treatise on sophismata, which also seems to have had a separate existence. The treatises on propositions and on syllogisms will particularly concern us in relation to our study of the Treatise on Consequences. Hubien, in his introduction to his edition of the Latin text, dates the Treatise on Consequences to 1335. His argument is based on rather amusing internal evidence concerning a particular logical example Buridan uses. He wants to give an example of material consequence, one dependent on the meaning of the descriptive terms it contains. He writes: “A white cardinal has been elected Pope, so a master of theology has been elected Pope” (see Book I, Chapter 4), and from “I see him,” he infers, “I see a deceitful man.” Who was this white cardinal? So asks Hubien rhetorically, replying that it must be Jacques Fournier, famous as a scourge of the Cathars, a Cistercian (whose monastic cloak was consequently white), created cardinal in 1327 and elected Pope Benedict XII in December 1334. He had been a fierce opponent of Ockham in the late 1320s, when Ockham was summoned to the papal seat at Avignon to answer charges of heresy, and of the terminism that Ockham represented and Buridan admired. Hubien is surely right in his identification of the white cardinal. It would be a brave, indeed foolhardy, cleric, however, who made quite so obvious his distaste for the current pope. We certainly have a terminus a quo in the election of Benedict XII; but would Buridan include this example before Fournier’s death in 1342? On the other hand, the treatise also contains evidence of a relatively early composition in Buridan’s career in his discussion of the liar paradox. Buridan’s diagnosis of the liar paradox has been much discussed in the past sixty years, at least since Moody’s Truth and Consequence in Medieval Logic of 1953. It has become clear that his view evolved over the years. In his Sophismata he describes a possible solution to the liar paradox as one he used to hold, a position we find in the Treatise on Consequences. The liar paradox is an example of a type of proposition called an insoluble by the medievals. They were particularly difficult sophisms. For example, suppose I say, “The proposition which I am uttering is false” (see Book I, Chapter 5). According to Buridan’s account of truth (see Book I, Chapter 2),

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an affirmative subject-predicate proposition is true if subject and predicate supposit for the same. So, by an argument we find in his Quaestiones Elencorum,4 every proposition signifies, or as he says there, at least implies, its own truth. For an affirmative proposition signifies its subject and predicate to supposit for the same thing or things, and we will see later that this is the condition of its being true, so it signifies that it is true. (A similar argument, mutatis mutandis, applies to negative propositions.) Hence, “The proposition which I am uttering is false” signifies both that subject and predicate supposit for different things (that is what “false” means) and that they supposit for the same (since it is affirmative). So it signifies both that it is false and that it is true. Hence it is self-contradictory, and thus false. The important thing to note here regarding chronology is that in the present treatise, the Consequentiae, he speaks only of each proposition signifying that it itself is true, that is, the position he later described as an earlier view of his own. He later revised the view to say that each proposition implies (but does not signify) its own truth. The reason was his vehement opposition to any suggestion that there is anything in the world with propositional or propositionlike structure, an idea he rejected along with his nominalistic rejection of real universals. A particular articulation of such an idea, invoking “complexly signifiables” (complexe significabilia), was famously put forward in Paris in 1344 by Gregory of Rimini, drawing on ideas of Adam Wodeham’s, presented in Oxford in 1331.5 But it did not take thirteen years for Wodeham’s ideas to reach Paris from Oxford, as Tachau points out.6 It was almost certainly taken up by Nicholas of Autrecourt in his Sentences commentary, now lost, and partly led to the Condemnation at Paris in December 1340 of Autrecourt’s theses. That there is no mention of complexe significabilia, and Buridan’s rejection of them, in the Consequentiae would seem to support Hubien’s contention that Buridan’s treatise belongs to the mid-1330s. However, Zupko argues that Buridan would not have found Wodeham’s version of the doctrine objectionable.7 Rather, it was Rimini’s reification of the complexly signifiables to which Buridan vehemently objected. In fact, a later date is strongly suggested by an issue that arises in Book II over the correct analysis of modal propositions. Buridan appears there to be rejecting claims made by an anonymous author known nowadays as Pseudo-Scotus, since his works appear in the Opera Omnia of Duns Scotus, first edited by Wadding in 1639 and reprinted by Vivès in the 1890s. In fact, there are arguably two authors deserving the epithet, since neither the Questions

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on the Prior Analytics nor the Questions on the Posterior Analytics printed there is by Scotus, though they are demonstrably not by the same author. The Questions on the Posterior Analytics is attributed in one manuscript to a John of Cornwall and may indeed date from the time of Scotus himself. The Questions on the Prior Analytics, however, shows knowledge (from a Parisian perspective) of the doctrine of complexly signifiables. It also shows knowledge of Buridan’s Questions on the Posterior Analytics, as we will see. Thus, the Consequentiae are not only later than Pseudo-Scotus’s treatise, but later than the Questions on the Posterior Analytics, an early work of Buridan’s itself dating from whenever the doctrine of complexly signifiables reached Paris. This does not categorically rule out a date for the Consequentiae in the 1330s, but it does favor a date in the 1340s, before Rimini provoked Buridan to give voice to his opposition to the complexe significabilia. Buridan’s version of terminism, a more temperate and Aristotelian, and less Augustinian version than Ockham’s, had a huge influence in the century and a half following his death. Many of Buridan’s works were printed when the age of printing arrived in the late fifteenth century: the commentaries on Aristotle’s Physics, Metaphysics, Ethics, and Politics; the Sophismata, which was printed in 1496; and the Treatise on Consequences, printed at Paris in 1493, 1495, and 1499. It might seem that the whole of the compendious Summulae de Dialectica was published in 1499, from the title Perutile compendium totius logicae Joannis Buridani. But the title continues, “cum praeclarissime solertissimi viri Joannis Dorp expositione” (“with the most expert and brilliant commentary of master John Dorp”), and what we find is Dorp’s own commentary on the text (largely that of Peter of Spain) on which Buridan had commented, and hence almost nothing of Buridan’s survives in that work, apart from his influence on Dorp. The genuine Summulae has now been published in an outstanding English translation by Gyula Klima, but even today not all the Latin text has appeared in print. Moreover, there have been three English translations of the Sophismata, Klima’s, as part of the whole Summulae, another of just the Sophismata by Theodore Scott in 1966, and a third, of just Chapter 8, dealing with the semantic paradoxes, by George Hughes in 1982, all predating the publication in 2004 of Pironet’s edition of the Latin text.8 The text of the Treatise on Consequences was published in 1976, when Hubert Hubien produced a meticulous edition of the Latin text, based on

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the three surviving manuscripts and the incunabula, together with a brief introduction.9 An English translation was published in 1985, but is very unreliable;10 hence the occasion for this new translation. The English translation given here follows Hubien’s text with only a very small number of changes made after consulting the manuscripts. Hubien’s introduction, originally in French, is also translated here into English.

2. Truth There were two great spurs to medieval logical theory. One was the rediscovery of Aristotle’s logic (the logica nova) in the twelfth century. This was preceded by the development of logical theories by Peter Abelard in the early decades of that century. Abelard was at first aware only of the logica vetus, that is, Porphyry’s Isagoge, Aristotle’s Categories and De Interpretatione, and the logical works of Boethius. The upshot was that when medieval scholars came to read the rest of Aristotle’s Organon, they read them under the influence of Abelard, while Abelard’s thought was allowed to roam free, given the relative paucity of logical theory in the logica vetus and the very elementary and unsophisticated introductions of Boethius’s, in comparison to the power of Abelard’s originality. For example, while Boethius’s De Syllogismo Categorico summarizes Aristotle’s reduction of the theory of the assertoric syllogism to the perfect moods of the first figure, it omits not only Aristotle’s further theory of the modal syllogism entirely but also Aristotle’s careful proofs of the invalidity of the other putative (assertoric) moods. Once faced with Aristotle’s own text in the Prior and Posterior Analytics, and his account of fallacy in the Sophistici Elenchi, the medievals went on to develop an increasingly sophisticated theory of consequence, a general theory by which the particular theories of assertoric and modal consequence could be underpinned and explained. In many ways, what we find in Buridan’s Consequentiae, even more than his Summulae, is the pinnacle of sophistication and development in medieval times of a general theory of logical consequence.

2.1. The Theory of Signification In order to understand Buridan’s theory of consequence we need to look first at his theories of signification (meaning) and of truth. Along with other medievals, following Aristotle, Buridan recognizes three kinds of language:

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written, spoken, and mental. Written and spoken language is conventional by its conventional attachment to mental language, but mental language was taken to be natural, not conventional, because the constituents of mental language, concepts, are formed by a natural, or causal, process by similarity to external things. Hence, mental language or mental expressions (orationes) are primary and give meaning to spoken and written expressions by this conventional attachment. Any of these expressions can be simple or complex, and one type of complex expression is a proposition. It is important to realize, however, that all such expressions, including mental expressions, are for Buridan concrete contingent particulars. They exist only if they are uttered or inscribed (spoken or written down) or thought. In particular, things might be as some proposition signifies (e.g., Buridan might be running) but unless someone says or thinks that Buridan is running, the proposition that Buridan is running would not be true, since it would not exist. Absolutely everything is, for Buridan, a particular, and although some particulars exist of necessity (e.g., God), propositions, including mental propositions, do not, and depend for their existence on the contingency of someone uttering, inscribing or thinking them. The view of signification that Aristotle adumbrates at the opening of De Interpretatione, as understood by Buridan, made spoken and written expressions signs of mental expressions and consequently signs of things. But note that, whereas for Ockham, mental expressions are signs of things, that is not Buridan’s view. For Ockham, both spoken and mental expressions are signs of things (and indeed, spoken expressions are not signs of mental expressions, but, as he puts it, conventionally subordinated to them). In contrast, for Buridan, mental expressions are naturally similar to things outside the mind, but they are not signs of them. Rather, spoken expressions are signs of mental expressions and so derivatively signs of external things. But this is only true of simple expressions. Simple mental expressions— that is, concepts—are similar to things outside the mind, but, as mentioned before, Buridan does not believe there is propositional complexity in external things. Such complexity exists only in spoken, written or mental propositions themselves. Spoken and written terms get their meaning or signification by signifying mental terms (concepts), which are their immediate significates, and consequently their ultimate significates are the things conceived by and similar to those concepts. The spoken and written propositions get their

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meaning or signification by being compounded of meaningful parts, and so immediately signify complex compounds of concepts. But their ultimate significates are not themselves in that way complex, but are only the ultimate significates of the constituent terms. For example, the written proposition, “A man is white,” signifies the spoken proposition, which immediately signifies the concepts “man” and “white” and their combination, the mental proposition, but it ultimately signifies only men and white things. In particular, “A man is white” and “A man is not white,” and generally any proposition and its contradictory, have the same (ultimate) signification.

2.2. The Theory of Supposition This aspect of Buridan’s theory of signification has important consequences for his theory of truth. We see this in Book I, Chapter 1. For if a proposition and its contradictory signify the same, then things being as it signifies cannot be a sufficient condition for a proposition to be true, on pain of contradiction. Yet “things being as it signifies” (ita est sicut significat) was a very common account of the truth of propositions in Buridan’s day; we find it in Thomas Bradwardine, Albert of Saxony, and many others.11 But Buridan needs a different account of truth. It is not completely unrelated to his account of signification, but it needs to be mediated by his account of supposition. The theory of the supposition of terms was a peculiarly medieval notion, not matching any modern semantic one. The basic idea was to identify those among the things signified by a term that were in fact being spoken about in a particular proposition, that is, a particular utterance. For instance, in the above example, “A man is white,” when uttered, it is presently existing men and white things that are being spoken about—it is a present-tense proposition, so the subject supposits for the men there are at present and the predicate (for Buridan) for those things that are currently white. In “A fat man was running,” on the other hand, the subject is ampliated by the past tense of the verb, but at the same time restricted by the adjective “fat,” to supposit for those things that are or were fat men, while the predicate supposits for what was running at some past time. By Buridan’s account, such an affirmative subject-predicate proposition is true if subject and predicate supposit for the same thing or things—that is, for something in common. So if Socrates, for example, was at some time fat, but at another time thin and while thin went running one day, the proposition is true: Subject and predicate both supposit for Socrates, who was at

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some past time a fat man and at some, possibly other, past time was running. Thus we can see how the doctrines of signification, supposition, ampliation, and restriction of terms combine to give the truth-conditions of subjectpredicate propositions. Buridan spells them out at length in his Sophismata (Chapter 2, fourteenth conclusion). We have just stated the truth-conditions for affirmative particulars. There need also to be clauses for universal affirmatives (true when the predicate supposits for everything the subject supposits for), universal negatives (true when the predicate supposits for nothing the subject supposits for), and particular negatives (true when the corresponding universal affirmative, its contradictory, is false). Nonetheless, these clauses are sufficiently cumbersome that Buridan says (Book I, Chapter 1) that he will often say that a proposition is true if “things are altogether as it signifies,” provided this is understood as shorthand for the appropriate clause about the supposition of its terms and not literally about some supposed significate of the proposition.

2.3. Causes of Truth One distinctive facet of Buridan’s account of truth in Treatise on Consequences and in Summulae de Dialectica is his talk of “causes of truth.” For example, we noted in the penultimate paragraph that Socrates having run at some time and having been a fat man at another—or the same—would make the proposition, “A fat man was running,” true, so it would be one cause of the truth of that proposition. Buridan writes (Book I, Chapter 2): “I understand by “a cause of truth” of a proposition whatever is enough for the proposition to be true.” Hubien, and Klima following him in his translation of Summulae de Dialectica 1.6.5 (56 n.), insert “propositio” (i.e., “proposition”) here. Klima cites Book I, Chapter 2 of the Treatise on Consequences as saying: “And by the “causes of truth” of a given proposition I understand propositions such that any of them would suffice for the truth of the given proposition.” But construing the causes of truth of a proposition as themselves propositions is Hubien’s speculative addition. The word propositio does not occur here in the manuscripts. Indeed, it would be circular to claim that what makes propositions true are (the same or other) propositions—for what makes those further propositions true? There is a danger, of course, in suggesting that it is not propositions but something else that makes propositions true—things being thus and so, for example, Socrates having gone

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running. But that need not be construed as a fact or state of affairs or a truthmaker, a thing. Buridan has clearly set his face against any such entities. In David Lewis’s phrase,12 truth supervenes on how things are, not on what there is. How things are, are the causes of the truth of propositions, for Buridan. Buridan notes in Book I, Chapter 2 that one proposition can have more causes of truth than another, all those of the first plus more besides. When one proposition has the same causes of truth as and possibly more than another in this way, it is a consequence of that other. As Buridan says in the opening sentence of Book I, Chapter 1, he will “treat of consequence by recording their causes.” Surprisingly, he then omits talk of causes of truth when defining consequence in Book I, Chapter 3, but he returns to the notion in his conclusions about consequence in Chapter 8, particularly in the eighth conclusion, which we will discuss later.

3. Consequence In proceeding to expound and explain Buridan’s account of consequence in his Treatise on Consequences, I will not follow Buridan’s order of exposition completely. His treatise is in four books. The first is a general discussion of consequence and related notions, such as truth, supposition, and ampliation, and concluding with a succession of conclusions (theorems) about consequence in general. Book II discusses modal propositions, those containing terms like “necessary” and “possible,” again establishing a succession of conclusions about the logical relations between such propositions. Book III introduces the theory of the syllogism, inherited by the medievals from Aristotle, but given a substantially different basis by Buridan, using notions introduced in Book I. Finally, Book IV concludes with Buridan’s theory of the modal syllogism, again developed in a radically different way from Aristotle’s own account. What Buridan’s definition of consequence amounts to in Book I, Chapter 3 is necessary truth-preservation, that is, preservation of the “causes of truth.” However, as he says, we cannot take it literally like that, since if, for example, the consequent does not exist, it cannot be true, as we observed. We could say that consequence is preservation of signification, that whenever things are altogether as the antecedent signifies, they are as the consequent signifies, and indeed, Buridan is happy to settle for this, with the caveat that

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“signifies” must be understood as shorthand, as we also noted earlier, for the complex criterion in terms of supposition. Apart from rewriting the necessary truth-preservation criterion in his own idiosyncratic terms, this is a familiar account of consequence, shared by medievals and moderns. What is more distinctive of Buridan, and places him further from his contemporaries and closer to the moderns is his account of the division of consequence into formal and material. For example, “Socrates is a human, so Socrates is an animal” is, for Ockham, for example, a formal consequence:13 Animal is part of the form of human. Buridan, in contrast, relates form in formal consequence to the form of the propositions concerned, not the objects. A formal consequence is one that holds “in all terms,” that is, for all substitutions of terms retaining the same ( propositional) form. So “It is a human, so it is an animal” would not be formally valid for Buridan, for replacing “human” and “animal” by “donkey” and “stone” we obtain “It is a donkey, so it is a stone,” which is not a valid consequence.

3.1. Formal and Material Consequence In his De Puritate Artis Logicae, written in Paris in the 1320s, Walter Burley contrasted these two different conceptions of formal consequence:14 “A formal consequence is of two kinds: one kind holds by reason of the form of the whole structure. Conversion is such a [consequence], and syllogism, and so on for other consequences that hold by reason of the whole structure. Another kind is a formal consequence that holds by reason of the form of incomplex terms. For example, a consequence from an inferior to a superior affirmatively is formal, and nevertheless holds by reason of the terms.” The former conception, the structural conception of the formality of consequence, is thoroughly modern. Indeed, many contemporary logicians think that all logically valid consequence is formal in this way. The view has its most famous articulation in Tarski’s paper on “The Concept of Logical Consequence.” It depends crucially on the concept of form, and preservation or sameness of form, of a proposition and so on a clear distinction between logical and descriptive vocabulary. Two propositions share the same form if they differ only in descriptive vocabulary, replacing one or more descriptive terms by others, as do, for example, “Every man is white” and “Every donkey is running,” both of the form “Every S is P.” In Book I, Chapter 7, Buridan identifies the logical/descriptive distinction with the distinction between

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categorematic and syncategorematic terms. The latter do not signify anything themselves, but only in conjunction with categorematic expressions. The descriptive terms constitute the matter of a proposition, and its form, therefore, is what is common structure to propositions that differ only in the categorematic terms they contain. However, Buridan does not identify consequence with formality in the way Tarski did. Tarski attempted to reduce the necessity in the consequence relation to the uniformity of the relation between premises and conclusion: for Tarski, consequence holds only if truth is preserved in all terms.15 Buridan, however, takes necessity as primitive, and in addition to formal consequence, also admits material consequence, where things being how one proposition signifies necessitate things being how another signifies (with the usual caveat about “signifies”) but not in virtue of form, that is, not in all terms, so that other proposition pairs sharing their form with the first do not show such necessitation. We have seen an example above concerning the meaning of the descriptive terms “human” and “animal.” A further subdivision of material consequence, one that Buridan shares with his contemporaries, is between simple consequence (consequentia simplex, sometimes translated as “absolute consequence”) and as-of-now consequence (consequentia ut nunc). These distinctions run right through medieval logic, from Abelard and earlier.16 Any material consequence can be reduced to a formal consequence by the addition of an extra premise, and constitutes a simple consequence if the additional premise is necessarily true, an as-of-now consequence if it is merely true. For example, the material consequence “If Socrates is human, Socrates is an animal” is a simple material consequence for it is made formal by the addition of the necessary truth “All humans are animal”; the material consequence “If a white cardinal has been elected Pope, then a deceitful man has been elected Pope” is an as-of-now consequence, made formal by the addition of what Buridan implies is the present truth, “The white cardinal is a deceitful man.” Buridan recognizes in his first conclusion in Book I, Chapter 8 that the notion of as-of-now consequence trivializes the notion, for any proposition follows as-of-now from a false proposition and a true proposition follows as-of-now from any other. More interesting, perhaps, is the fact that any impossible proposition has as simple material consequence any other proposition, since it is impossible for things to be as an impossible proposition signifies, and so impossible that they so be and an arbitrary consequent false.

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In the same way, a necessary proposition follows simply and materially from any other. If the impossibility is a matter of form, say, in a formal contradiction, then the consequence is formal, as too if the necessary truth is formally necessary. See the seventh conclusion. These consequences hinge on the same point as is brought out in the sixth conclusion of Book I, Chapter 8, namely, that any necessary truth can be suppressed in a simple consequence, and any truth can be suppressed in an as-of-now consequence. For suppose C follows simply from A and B, where B is necessarily the case. Then it is impossible for things to be as A and B signify and not as C signifies. But if a conjunction is impossible one of whose conjuncts is necessary, the other conjunct must be impossible. So, given that B is necessary, it is impossible for things to be as A signifies and not as C signifies. So C follows simply from A alone. The necessity B can be suppressed.

3.2. The Modes of Supposition The eighth conclusion picks up the terminology of “causes of truth” from Chapter 2 and effectively equates consequence with having the same but possibly additional causes of truth. It has an important consequence in the tenth and eleventh conclusions concerning distributed terms. The notion of supposition, which we met earlier, concerned not only what a term supposited for but also how it supposited—the modes of supposition. When, in Buridan’s terminology, a term supposits for (some of) its ultimate significates, it was said to have personal supposition, or to supposit personally. If it supposits for itself or other occurrences of the same term, or for its immediate significate, it has material supposition. For example, in “Animal has three syllables,” or “Animal is a genus,” “animal” supposits not for animals but in the first case for a spoken or written term and in the latter for a mental term or concept, and so in both cases it has material, not personal, supposition.17 Other authors, such as Ockham, in fact distinguish these cases, restricting material supposition to supposition for a written or spoken term, and calling supposition for the mental term or concept, simple supposition. (More generally, in other authors, simple supposition is supposition for the universal, which nominalists such as Ockham and Buridan take to be at best the mental term or concept—the only universals are names.) Buridan does allude to simple supposition on one occasion in Book III, Part I, Chapter 3, but only in passing, and in general he subsumes it under material supposition.

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In “Every human is an animal” and “Every animal is running,” “animal” supposits for animals, so it has personal supposition, but it supposits differently in the two cases. From the second, we can infer “The donkey Brownie is running,” for the supposition of “animal” is, we might say, distributed over all animals; but we cannot infer from the first that every human is Brownie. “Animal” supposits differently and nondistributively in the first proposition. But it does not supposit for any determinate animal, either, for each human is a different animal. Buridan and other medievals said in a perceptive metaphor that the supposition of “animal” in both propositions is confused, but whereas in the second it is confused and distributive, in the first it is, they said, merely confused. A variety of glosses and rules concerning the modes of common personal supposition were given over time by different authors, but by the time of Ockham and Buridan the modes of supposition of common nouns or general terms suppositing personally were characterized in terms of descent to and ascent from singular propositions. Suppose a general term “S” occurs, perhaps along with a determiner or syncategorematic expression “σ” (e.g., “a,” “some,” “every,” “no”) in a proposition φ, forming φ(σS). If “S” were “animal,” the above examples would be φ1 (an S) and φ2 (every S). The singular instances are formed by replacing “σS” by a singular term such as “Brownie” or “this S.” Then the mode of supposition of “S” in φ(σS) is:18 1. determinate if one can ascend from any singular and descend disjunctively but not conjunctively 2. otherwise, confused and a. distributive if one can descend conjunctively to any singular b. otherwise, merely confused, when (in some cases) one can descend to a disjunct term.19 Thus, in “Some human is running,” both “human” and “running” have determinate supposition (both “This human is running” and “Some human is this runner” suffice for its truth and one can descend disjunctively but not conjunctively on each term); in “Every human is running,” “human” has confused and distributive supposition (one can descend to “This human is running and that human is running and so on”), while “running” has merely confused supposition (the only descent possible is to “Every human is this runner or that runner and so on,” though one could ascend from “Every human is this runner” if it were true).

Introduction

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Buridan’s tenth conclusion observes that one can formally infer a proposition with an undistributed term (that is, one with determinate or merely confused supposition) from the corresponding proposition where that term is distributed, but not vice versa. Looking at the above definition, we see that the conjunctive condition in confused and distributive supposition imposes a stronger constraint and so permits fewer causes of truth than the disjunctive condition in the case of determinate and merely confused supposition. Nonetheless, the conjunctive case does fall under the disjunctive condition, so the latter includes the same but additional causes of truth, and thus the consequence holds by the eighth conclusion. The converse clearly fails since the disjunctive condition does not warrant the conjunctive. The tenth conclusion of Book I will be central to Buridan’s demonstration of syllogistic consequence in Book III.

3.3. Nonnormal Propositions Buridan’s longest and most detailed discussion among the conclusions of Book I concerns the conversions described in the fourteenth conclusion. These conversions were central to Aristotle’s demonstration of syllogistic consequence, which we will consider later. Both here and in Summulae de Dialectica, Buridan introduces the notion of negative propositions in nonnormal form, namely, where the predicate precedes the negation. This is a particularly extreme example of an increasing regimentation of Latin by medieval thinkers. By the late Middle Ages, Latin was no longer a language of everyday speech, though it was used as a lingua franca in intellectual and political circles. As such, it evolved and took on aspects of the character of the new languages of everyday speech, French in particular. For example, the French definite article ly was coopted into the Latin of logic texts in place of iste terminus to indicate material supposition. Iste and ille were increasingly used as definite articles rather than demonstrative adjectives, and talis replaced them in the latter role. Moreover, the rather free word order of classical Latin, with a preference for subject-object-verb,20 was replaced by the subject-verb-object order of the vernacular languages.21 Such a fixed word order is important for the rules of consequence such as the rule that a universal sign confuses the term immediately following it distributively and any term mediately following it merely confusedly (as we noted, in “Every human is running,” “human” has confused and distributive supposition and “running” has merely confused supposition).22 This means that the order is crucial and no longer free. The standard way of negating a

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subject-copula-predicate proposition was to place a negation before the copula (in Latin): “Socrates is white” (Sortes est albus) becomes “Socrates is not white” (Sortes non est albus). “Not” (non) is then recognized as having the power to distribute the term immediately following it (if previously undistributed, removing the distribution if it was distributed already): “Socrates is not white” implies that Socrates is not this white thing, and not that white thing and so on conjunctively. If we now place the predicate before the verb (and its negation) in what to those of Buridan’s day would seem an unnatural way, we obtain what Buridan calls the nonnormal way of speaking (de modo loquendi inconsueto). For example, “donkey” is distributed in “Some animal is not a donkey,” but “donkey” is not distributed, he says, in “Some animal (some) donkey is not” (Quoddam animal asinus non est)—the latter is true if some animal is not some donkey (“some” in English has a similar power of overriding the distributive power of the negation), whereas the former is true only if some animal is not any donkey. If we compare what Buridan writes here with, e.g., Boethius’s De Syllogismo Categorico, we can see how Latin has changed and become regimented. For Boethius’s normal way of writing the O-proposition “Some animal is not a donkey” is “Quoddam animal asinus non est,” with the negation after the predicate (with the verb).23 All the subject-predicate propositions of Aristotle’s syllogistic are written in this normal SOV (subjectobject-verb) way. Boethius would not agree with Buridan that prefixing the predicate to the negation removes the distributing force of the negation. Nonetheless, the regimentation serves a useful purpose for Buridan in allowing him to convert O-propositions like “Some animal is not a donkey.” Aristotle’s assertoric syllogistic focused on just four forms of subjectpredicate propositions, displayed in the traditional Square of Opposition (implicit in Aristotle, first known to occur in Apuleius).24 A- and O-propositions, and E- and I-propositions, are contradictories (cannot both be true or both false)—O-propositions are often expressed by Aristotle as “Not every S is P”; A- and E-propositions are contraries (cannot both be true), and I- and O-propositions are subcontraries (cannot both be false); and A-propositions entail the corresponding I-proposition, and E-propositions the corresponding O-proposition as subalterns. See Table 1. Conversion consists in inverting subject and predicate. I- and E-propositions convert simply, that is, retaining the quantity (affirmative or negative): “Some S is

Introduction

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Table 1: The Square of Opposition

P” entails “Some P is S,” and “No S is P” entails “No P is S.” A- and O-propositions do not convert simply: “Every S is P” does not entail “Every P is S,” nor does “Some S is not P” entail “Some P is not S.” But “Every S is P” does entail (according to Aristotle) “Some P is S.” (We will discuss this later.) So Aristotle said that the A-form converts accidentally ( per accidens, that is, partially). Even this does not seem possible for the O-form. Buridan notes as an exception to the fourteenth conclusion that if nonnormal propositions are allowed, the O-form “Some S is not P” does convert, to “Some P (some) S is not.”25

3.4. Ampliation The fourteenth conclusion rehearses the standard doctrine of conversion, and it adds a further conversion valid for Aristotle but not included by him among the conversions: the accidental conversion of the E-form “No S is P” into “Some P is not S.”26 As Buridan notes, the proof, covering many cases, is long and difficult, but it illustrates a further regimentation, of the separation of the copula from the predicate. Often, the copula and predicate are combined in a single word—for example, “Socrates runs,” which can be spelled out as “Socrates is a runner.” Sometimes indeed, all three are combined in one word, for example, “Thunder,” which means, “There is thunder.” Much of Buridan’s discussion concerns the possible addition of the pronominal phrase “that which,” which plays a very large role in Buridan’s treatment

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of consequence. The reason is that it serves to override, or in some cases to make explicit, the effects of ampliation. We noted the phenomenon of ampliation earlier, but it deserves further comment. Ampliation is the power of certain verbs, of their tenses, and of certain predicates to enlarge the range of supposition of the subject of propositions in which they occur. As we saw, the past tense ampliates the subject to supposit for past significates as well as present; the future tense ampliates the subject to supposit for future significates as well as present; and modal auxiliaries ampliate the subject to supposit for possible significates—always in disjunction with present significates, Buridan observes. Among predicates, the most common ampliating predicate is “dead,” which ampliates the subject to supposit for past significates. See Book I, Chapter 6. It is a matter of some dispute whether Buridan believes there are things that do not exist—mere possibles ( possibilia). His language certainly suggests it. At Book II, Chapter 4, he writes: “A divided proposition of possibility has a subject ampliated by the mode following it to supposit not only for things that exist but also for what can exist even if they do not. Accordingly, it is true that air can be made from water, although this may not be true of any air that exists. So the proposition ‘B can be A’ is equivalent to ‘That which is or can be B can be A.’” By boiling water, we can bring into existence air that did not previously exist. George Hughes gives a further argument that Buridan was committed to there being merely possible beings, things that do not exist. At Book IV, Chapter 2, Buridan considers the proposition, “Every horse can sleep,” which according to his account of the ampliation of possibility-propositions has a subject ampliated to possible horses; but no actual thing that is not a horse can be a possible horse, so Buridan must admit pure possibilia.27 Lagerlund contests this conclusion in an argument we will consider later in this introduction. Buridan makes great use of the phrase “that which” to override the ampliating power of verbs and predicates. For example, “A human is dead” is true, since “dead” ampliates “human” to supposit for past humans, including, for instance, Socrates. But “That which is a human is dead” is false, for anything that is now a human is alive and not dead. When humans die, according to Aristotelian doctrine, they cease being human.28 This power to prevent ampliation is noted in the twelfth conclusion of Book I, Chapter 8. But in the fourteenth conclusion it serves to preserve conversions. For example, the I-proposition “A human will run” does not convert to “A run-

Introduction

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ner will be human,” but we must first spell out the ampliation explicitly, “That which is or will be human will run” and then convert it to “That which will run is or will be human.” The same use of the “that which” construction is sometimes needed when converting propositions that contain terms in an oblique case (usually the accusative and the genitive) and propositions containing exponible terms. Exponible terms create exponible propositions, ones that appear to be simple subject-predicate propositions but are really disguised conjunctions and need analysis (as, e.g., Russell claimed was true of propositions containing definite descriptions29). For example, “Only S is P” is analyzed as “S is P and nothing other than S is P,” a conjunction of an I-proposition and an E-proposition, both of which are simply convertible into “P is S and no P is other than S,” which in turn is equivalent to “Every P is S,” so the exponible proposition “Only S is P” does convert.

4. Syllogisms Before considering Buridan’s account of syllogistic, it will be useful first of all to review Aristotle’s own account of the syllogism in his Prior Analytics, and before that, to consider the four forms of proposition that make up the Square of Opposition, which for the most part are the propositions that Aristotle considers in his theory of the assertoric syllogism. (“Assertoric” translates “de inesse,” meaning “of (simple) inherence,” that is, (non-modal, simple) subject-predicate propositions.) Aristotle claimed in De Interpretatione, chapters 6, 7, and 10, that corresponding contrary, subcontrary, subaltern, and contradictory propositions were, in a certain sense, each “opposites” of a given proposition. In fact, this doctrine is somewhat puzzling, at least if we render the O-proposition as “Some S is not P.” For this seems false if there is no S. So the corresponding A-proposition, “Every S is P” should be true in that case, but then it would not entail its subaltern “Some S is P.” This certainly puzzled Brentano, and led Śleszyński and Łukasiewicz to assume that Aristotle meant his syllogisms to apply only to subjectpredicate propositions with non-empty terms. They claimed that only on that assumption did the relations of the Square of Opposition hold. Łukasiewicz’s interpretation was adopted by Bochenski and the Kneales in their influential histories of logic, and now seems to be the orthodox account of the syllogism.30

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Plate I: Buridan’s Square of Opposition (From MS Vatican Pal.lat. 994 f. 6r, reproduced with permission of the Biblioteca Apostolica Vaticana)

4.1. The Square of Opposition This was not the interpretation given by Buridan and his contemporaries.31 Starting with the I-proposition, “Some S is P,” they took it that this was false if there was no S, and so the corresponding A-proposition, “Every S is P” was similarly false in that case. Consequently, the O-proposition was true if there was no S, as was the E-proposition, “No S is P,” which entails it. Aristotle, we noted, often wrote the O-proposition as “Not every S is P” (indeed, literally as “P does not belong to every S”), and Buridan includes this form as equivalent to “Some S is not P” in his Square of Opposition at Summulae de Dialectica 1.4. (See Plate I.) The propositions at the O-corner read: “Some man is not running,” “Several men are not running,” “Not every man is running,” “The other of these is not running,” “Some part of man is not a man,” “Some man is not an animal.” The medieval interpretation of Aristotle was that affirmative propositions (A and I) have existential import and negative propositions (E and O) do

Introduction

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not. This means that (the negative proposition) “Some S is not P” (Quoddam S non est P) is different from (the affirmative proposition) “Some S is notP” (Quoddam S est non P), a distinction emphasized by Aristotle at Prior Analytics I, 46. The latter entails the former but not conversely, since the negative proposition has more causes of truth. Both Aristotle and the medievals rejected obversion (equating “Some S is not P” with “Some S is not-P”), which holds in traditional logic.32

4.2. The Syllogistic Figures Aristotle’s great idea in his doctrine of the syllogism was that all consequence could be reduced to the pairwise deduction of successive conclusions. Buridan says that he understands Aristotle to mean by a syllogism a collection (in the final analysis, a pair) of propositions from which a conclusion can be inferred (Book III, Part I, Chapter 4). Concentrating on the A, E, I, and O forms, there are forty-eight possible pairs sharing a middle term in common, sixteen in each of three figures: first, where the middle term is subject of one and predicate of the other; second, where it is predicate of both; and third, where it is subject of both. The other two terms are called the extremes. In the first figure, the conclusion can be direct (where the subject of the conclusion was subject in its premise) or indirect (where the subject of the conclusion was predicate in its premise). Buridan raises the possibility of counting the indirect moods of the first figure as a separate, fourth, figure (Book III, Part I, Chapter 2). But the cases differ only in the order of the premises and whether one counts the predicate or the subject of the conclusion as the major extreme. Buridan simply calls the first premise the major premise and its extreme (that is, the term other than the middle term) the major term. (Calling the predicate of the conclusion the major term and its premise the major premise began only during the Renaissance, harking back to late antiquity, and quickly led to the recognition of four figures as the orthodoxy.33) The task Aristotle set himself was to distinguish those pairs of premises that yield a syllogistic conclusion from those that do not.

4.3. Ecthesis Aristotle bases his demonstration of validity on the so-called dictum de omni et nullo, stated at the end of Prior Analytics I, 1. It is essentially a definition of what it is to predicate one thing of another: “We speak of ‘being predicated

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of all’ when nothing can be found of which the other will not be said, and the same account holds for ‘of none.’”34 The validity of the perfect syllogisms, namely, the direct syllogisms of the first figure, is based on this definition, and that of the remaining syllogisms is reduced to those by conversion and reductio per impossibile. Buridan mentions Aristotle’s approach in the ninth conclusion of Book III, but his general approach is very different, in a way that laid the foundation for the theory of the syllogism in the traditional logic of the eighteenth and nineteenth centuries. At the start of Book III, Part I, Chapter 4, he cites the principles: “ ‘Whatever are the same as one and the same are the same as each other,’ and ‘Two things are not the same as each other if one is the same as something and the other is not.’ ” Lagerlund and King identify these principles as Aristotle’s dictum de omni et nullo.35 But they are not. They are the medievals’ understanding of what Aristotle says in Prior Analytics I, 6, under the epithet ecthesis, which became for the medievals “the expository syllogism.” Bonaventure writes:36 “By the expository syllogism: for of necessity it follows, as is said in the Prior [Analytics] ‘this A is B, this A is C with the same demonstrated, therefore C is B’; and this syllogism is founded on the self-evident principle ‘whatever are the same as one and the same are the same as each other.’” Again, after noting that Aristotle validates the perfect syllogisms by the dictum de omni et nullo, Buridan discusses the general rules concerning syllogisms, and remarks (Summulae de Dialectica 5.1.8, 313): “We should look at the principles by virtue of which affirmative and negative syllogisms are valid. I declare, therefore, that every affirmative syllogism holds by virtue of the principle ‘whatever things are said to be numerically identical with one and the same thing, are also said to be identical between themselves,’” and later (315) that all negative syllogisms are valid by the corresponding negative expository principle. The medievals’ interpretation of Aristotle’s method of ecthesis (setting out, or exposition) was that the term Aristotle introduces was a singular term. Aristotle writes in Prior Analytics I, 6 (28a23–25): “The demonstration [of Darapti] can also be carried out . . . by setting out. For if both terms belong to all S, and one chooses one of the Ss, say N, then both P and R will belong to it, so that P will belong to some R.”37 It has been a constant puzzle since ancient times whether the term “N,” which Aristotle introduces here, is a singular or general term. But as Alexander of Aphrodisias and many others have urged,38 it cannot be a general term, since then ecthesis would simply be an instance of Darapti itself, so the attempted demonstration of Darapti

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would be circular. Aristotle must have intended it to be a singular term, as he says “one of the Ss,” so that one of the Ss, N, is P and the very same S, viz. N, is R. This connects the two premises. For the same reason, the middle term in a syllogism must be distributed, Buridan says in the sixth conclusion of Book III, Part I, Chapter 4, so that the premises can be joined together effectively. Provided one occurrence of the middle term is distributed, take an instance of the other, then the distribution will ensure that that instance is included and the premises relate to the same thing. Otherwise, Buridan writes: “[if] the middle is not distributed in either [premise] it is possible that its conjunction with the major extreme is true for one thing and its conjunction with the minor is true for another.” He spells this out in the twenty-third of his Questions on the First Book of the Prior Analytics:39 “It is asked about the rules that Aristotle proposes here, namely, in the first [book] of the Prior [Analytics], whether by them it can be known whether syllogisms are valid or not. The first rule is that if the middle was a general term and distributed in neither premise, no syllogism would be valid, that is, it would not be valid in virtue of its form. The reason for this rule is that an affirmative syllogism only holds in virtue of this principle ‘whatever are said to be the same as one and the same, they too are said to be the same as one another,’ ” and he also gives the corresponding negative principle for negative syllogisms as in Book III, Part I, Chapter 4.

4.4. Syllogistic Validity Thus the sixth conclusion of Book III gives a necessary condition for syllogistic validity. This is in marked contrast to Aristotle’s approach. To show premise pairs not to constitute syllogisms, Aristotle uses the method of counterinstances. That is, to show that a premise pair is not a syllogism, he gives a triad of terms to substitute for the extremes and the middle term such that, first, the premises and a universal affirmative coupling of the extremes are all true, and another triad such that the premises and a similar universal negative are all true. Hence, no particular negative can follow, in virtue of the first triad, and no particular affirmative in virtue of the second, and consequently no universal conclusion either, of which the particulars are subalterns. Aristotle does this systematically, but seriatim, for direct conclusions from every nonsyllogistic pair in each figure, that is, for thirtyfour premise pairs. (He does not complete the task for indirect conclusions in the first figure.) In contrast, Buridan now has a general principle that will

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show the invalidity of many of these cases of invalidity: twelve cases altogether, or sixteen including indirect first-figure conclusions. Buridan brings it all together in his seventh and eighth conclusions, with reference back to the second conclusion. Between them, they give necessary and sufficient conditions for inferring a conclusion from a pair of assertoric subject-predicate premises. As we noted, in each figure there are sixteen ways of linking premises of the four forms. The second conclusion shows that four moods in each figure are useless, namely, those with both premises negative. For the expository principles cannot adduce anything from premises both of which deny an identity. As Buridan points out, that Brownie is not A, not B, not C, and so on does not allow us to infer anything about A, B, or C, either affirmatively or negatively. Brownie does not provide a suitable middle term if all we know of him is negative. Together with the verdict of the sixth conclusion, this rules out the eight pairs ee, ia, oa, oe, eo, ii, oi, and oo in the first figure. The other eight pairs can produce a conclusion, the six identified by Aristotle and two more (ao and io) with a nonnormal conclusion. Buridan’s Figure I Figure I Premises aa ea ai ei ae ie ao io

Conclusion Direct a (Barbara) e (Celarent) i (Darii) o (Ferio) X X X X

Weakened i (Barbari) o (Celaront) X X X X X X

Nonnormal X X X o o oa o o

Indirect i (Baralipton) e (Celantes) i (Dabitis) X o (Fapesmo) o (Frisesomorum) X X

a. At SD 5.1.8, p. 318, Buridan says that we can conclude to a nonnormal E-proposition: Some M is P, No S is M, so Every S (some) P is not. “P” is undistributed in the conclusion, just as it is in the major premise. The same reasoning would support an inference to a nonnormal E-proposition from ae in the first figure, ie and oa in the second, and ae, ao, and ie in the third.

For example, the additional first-figure moods not recognized by Aristotle read:

Introduction

Every M is P Some S is not M So some S (some) P is not

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Some M is P Some S is not M So some S (some) P is not

In each case, the conclusion is an O-proposition of nonnormal form in which both S and P are undistributed, but M is distributed in the first premise of the first syllogism and in the second premise of both. So each satisfies the conditions of the second, seventh, and eighth conclusions. The conclusions also convert to the indirect nonnormal conclusion “Some P (some) S is not.” Moreover, whenever a normal O-conclusion (in, e.g., Ferio) can be inferred (in each figure), a nonnormal O-conclusion also follows, as we noted previously, by the tenth conclusion of Book I, that “from every proposition containing a distributed term there follows in a formal consequence a proposition with the same term not distributed, the rest remaining the same.” We can provide the same analysis of the second figure. Again, the second conclusion rules out the premise pairs ee, eo, oe, and oo, while the sixth conclusion rules out the premise pairs aa, ai, ia, and ii, where the middle term would not be distributed. The remaining eight pairs all produce at least one conclusion, though again two conclude only nonnormally. Buridan’s Figure II Figure II

Conclusion

Premises ae ea ei ao ie oa oi io

Direct e (Camestres) e (Cesare) o (Festino) o (Baroco) X X X X

Weakened o (Camestrop) o (Cesaro) X X X X X X

Nonnormal X X o o o o o o

Indirect e (Camestre) e (Cesares) X X o (Tifesno) o (Robaco) X X

The two moods Tifesno and Robaco (called Fitesmo and Boraco at Summulae de Dialectica 5.4.2–3) seem to be Buridan’s own invention, and it is questionable whether they really differ from Festino and Baroco, resulting simply from inverting the order of the premises and the order of the terms

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in the conclusion.40 The genuinely new non-Aristotelian moods are those with a nonnormal conclusion: Some P is not M Some S is M So some S (some) P is not

Some P is M Some S is not M So some S (some) P is not

In the third figure, the third conclusion rules out purely negative premise pairs and the sixth purely particular ones, namely, ee, eo, oe, oo, ii, io, and oi (oo is both purely negative and purely particular). That leaves nine useful premise pairs. There are no weakened moods in the third figure, and again one might suspect Lapfeton, Carbodo, and Rifeson (called Fapemton, Bacordo, and Fisemon at Summulae de Dialectica 5.5.2–3) of being an artificial fabrication. Buridan’s Figure III Figure III Premises aa ea ai ia oa ei ae ao ie

Conclusion Direct i (Darapti) o (Felapton) i (Datisi) i (Disamis) o (Bocardo) o (Ferison) X X X

Weakened X X X X X X X X X

Nonnormal X o X X o o o o o

Indirect i (Daraptis) X i (Datisis) i (Disami) X X o (Lapfeton) o (Carbodo) o (Rifeson)

4.5. Buridan’s Novelties Even the additional moods obtained by permitting the nonnormal O-form conclusions may seem artificial. But Buridan proceeds to two genuinely novel and worthwhile extensions of Aristotle’s theory by observing the effect of ampliation on syllogistic validity, and exploring the possibility of syllogisms involving propositions containing oblique terms, that is, terms in the accusative and, for the most part, the genitive. He notes that, unsurprisingly, syllogistic consequences are preserved when the “that which” construction is employed to prevent ampliation. The remaining conclusions

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of Part I of Book III concern the way ampliation affects distribution. In a sense, the way distribution affects truth is the converse of that due to causes of truth. Causes of truth are disjunctive—the move from fewer to more causes of truth is valid, just as that from A to “A or B.” Distribution has the opposite effect. If ampliation (or anything else, for that matter) extends the distribution of a term (as, e.g., the ampliation of “dead” or the past tense in the predicate extends the distribution of a distributed subject to both past and present significates), syllogistic validity is affected—for the effect of wider distribution is conjunctive, resulting in fewer causes of truth. This is the common feature running through conclusions 10–12. From broader to less broad distribution is valid, the converse invalid, whereas from broader to less broad undistributed is invalid, the converse valid. When Buridan turns to the topic of oblique terms in syllogisms in Part II of Book III, his remit is in principle very broad, but in practice he concentrates on two main cases. He points out in Summulae de Dialectica 5.8.1 that even Aristotle’s formulation of syllogistic propositions, e.g., “A belongs to every B,” contains terms in an oblique case: what is logically the subject, B, is in the dative case. But that is not the type of example Buridan is concerned with in the so-called oblique syllogism. The idea here is that (logical) subject or predicate may contain an oblique term that functions in itself as the extreme or the middle term, so that the extremes or middle are not simply the subject or predicate of the premise or the conclusion. One example concerns possessives, as in “Every bishop’s donkey is running.” Here the whole term “bishop’s donkey” may be distributed by “every,” or perhaps only “donkey,” or even, if “every” is in the (Latin) genitive, “bishop”—which we might better express in English as “A donkey of every bishop is running.” Then it may be that “bishop” is the middle term or the extreme, the other logical part (respectively, extreme or middle) being made up of the rest of the proposition, shared between subject and predicate.41 Once again, it is the expository principles that are applied, so careful analysis of the logical form of the premises is needed to check that the middle term does indeed connect the extremes in the way those principles require.

4.6. Intensional Verbs The same is true for the other case Buridan deals with here, concerning the objects of certain verbs, often called intensional verbs, “know,” “believe,” “comprehend,” “judge,” “promise,” which Buridan lists at the start of Book III, Part

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II, Chapter 3. In order to describe the functioning of these terms, Buridan gave new life to an old term, “appellation,” whose original role in the properties of terms had been largely subsumed by supposition.42 He said these terms had the power to make terms that they governed (regere) appellate the associated concept. For example, he spells out “Socrates does not know prime matter” as “Socrates no prime matter knows (or has knowledge of) according to the concept (or intension) according to which it is called ‘prime matter.’” The problem goes back to Aristotle’s De Sophisticis Elenchis (chapter 24), concerning the problem of the Hidden Man, or the Man in the Mask: if Coriscus, or your father, whom you know, is approaching (but hidden in some way, perhaps by a mask), it seems that you both know and do not know the person approaching—as your father, he is known, but as the one approaching, he is not. Aristotle proposed solving the puzzle by the fallacy of accident, since he thought it should share a solution with fallacies of the form, “This dog is a father, and yours, so this dog is your father”:43 in both cases, Aristotle says, “there is no necessity that all the same attributes [of something] should belong to all of the thing’s predicates and to their subject as well” (166b31). Just as “This is your father” is a counterexample to the inference “This is an S, this is a P, so this is an S P,” so too is “Your father is known to you, your father is approaching, so your father is known to you approaching.” Ockham tried to tease the sense out of this and attributed the Hidden Man fallacy to one kind of fallacy of accident.44 But Buridan breaks the connection with the other fallacies of accident, and attributes it specifically to the power of the term “know.” Buridan also distinguishes, in a way that should now seem familiar, the effect of terms like “know” on accusatives following them from a lesser effect on those preceding them (and so which they do not govern).45 The phrase “appellative name,” originally just a grammarian’s expression for a general term or common noun, came with Buridan to describe a term that connotes a property in addition to what it signifies and supposits for. For example, “white” signifies and supposits for white things, but it connotes whiteness as an individual property (a trope46 or moment), which those things have; “blind” signifies blind things but connotes sight as a property they lack. Cognitive verbs describing mental acts such as knowledge, belief, judgment, understanding, and promising have the power, Buridan said, to induce a further appellation, this time for the concept immediately signified by the subsequent accusative. We see here yet another example of the regi-

Introduction

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mentation of word order where this power is said to operate fully only on accusatives preceded by the cognitive verb. “The one approaching I know” is true, Buridan says, for I know him “under a disjunction” of appellations.47 So it is entailed by “I know the one approaching,” which requires that I know him under that particular appellation, but not vice versa.

4.7. Infinite Terms At the end of Book III, Buridan considers the effect of introducing so-called “infinite” terms, that is, negated terms such as “nonhuman,” “what is not a stone” and so on. He had noted in the second conclusion that such infinite terms create exceptions to the rule that no syllogism results from two negative premises. (In point of fact, so do nonnormal O-propositions.) The eighteenth conclusion observes that by use of infinite terms we can draw a negative conclusion from two affirmatives, and also from two negative premises. But this does not mean that obversion is valid, as we noted earlier, and Aristotle pointed out in Prior Analytics I, 46: “Some S is not P” (i.e., “Not every S is P”) is a negative proposition, true if nothing is S; “Some S is non-P” is affirmative, true only if something is S (and not P). So the latter entails the former, but not vice versa, and they are not equivalent. Thus, as Buridan notes, “from affirmatives there follows a negative” (since the latter has more causes of truth), and from negatives there also follows a negative; but an affirmative cannot follow from negatives. The connection of a finite term to its corresponding infinite term can also act as the bridge in a middle term where otherwise there would be a fallacy of four terms. Even so, an existence assumption about a negative premise is needed, to allow the corresponding affirmative to be inferred. For example, suppose we take “Every M is P” and “No S is non-M” in the first figure. To infer “Every S is P” we need to know that there are Ms (for “No S is nonM” is true if there are no non-Ms or Ss). We then infer “Every S is P” from “Every M is P” and “Every S is M” by Barbara. What Buridan says here is little more than a hint, inviting his readers to work out the multifarious cases for themselves.

5. Modality In sixteen chapters of Prior Analytics, which seem to have been added at a later date by Aristotle himself, Aristotle extended his account of syllogistic

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to cover modal propositions, that is, propositions in which the copula is modified by the expressions “necessarily” and “possibly.” He had already discussed these modifiers in the final chapters of the De Interpretatione (12–14). Concerning “possibly” he makes a crucial distinction at Prior Analytics I, 13 (32b4–18). “Possible” may be simply the contradictory of “impossible,” in which case, what is necessary is also possible; but “possible” may also be opposed to “necessary,” so that “possible” describes not simply what is not impossible but what is also not necessary. The first sense of “possible” is sometimes called “one-sided possibility,” the latter “two-sided possibility.”48 Buridan himself in general reserves “possible” for the first sense and “contingent,” or sometimes “contingent each way,” for the latter, for if a proposition is contingent then things can be as it signifies and can also fail to be as it signifies. What is contingent is not necessary, and so may possibly fail to obtain. Unfortunately, even though Aristotle has two words for “possible” (dunaton and endexesthai), he uses both in both senses, often but not always noting whether he means possible in the weaker sense (not impossible) or stronger (neither impossible nor necessary). This equivocation runs right through Aristotle’s discussion of the modal syllogism. As a matter of fact, he only takes “possibility” premises in the stronger sense (contingency) but often considers possibility conclusions in the weaker sense. The distinction seems to have crept up on him rather late, for in De Interpretatione 13 he first proposes a “square of opposition” for modal terms (22a22) where “possible” has the stronger sense, only to revise it at 22b10ff, by equating “possible” with “not necessarily not,” that is, the weaker sense.

5.1. Composite and Divided Modals Aristotle makes two further distinctions that are crucial for Buridan’s account of modal propositions, but that Aristotle himself fails to apply to them. The first is between the divided and composite (or compounded) sense of certain propositions, introduced by Aristotle at Sophistici Elenchi 4 (166a23– 38) and discussed there in Chapter 20. This is wider than but subsumes the more recent de re/de dicto distinction. Aristotle points out that in the divided sense, it is true that a man can walk while sitting, but false in the composite sense. Yet there is no sign of this distinction in the chapters on the modal syllogism. It was a distinction of which Peter Abelard made much, and his observations were highly influential on his fourteenth-century successors, including Buridan, though it seems largely absent from thirteenth-century

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treatments.49 The distinction between composite and divided modals runs right through Books II and IV of Treatise on Consequences and the comparable discussions in Summulae de Dialectica. Its absence from Aristotle’s account of the modal syllogism was thought by many to vitiate his treatment. For Aristotle follows his approach to the assertoric syllogism by basing his demonstrations of modal validity on the same principles, including conversions. Those conversions seem to hold only if the modal propositions have their composite sense. Yet in Prior Analytics 9, Aristotle seems to take modal propositions in the divided sense, for he declares that the four perfect syllogisms, Barbara, Celarent, Darii, and Ferio, yield a necessity conclusion if the major premise is also given a necessity modality while the minor premise remains assertoric, but not if the major premise is assertoric and the minor given the necessity modality. (Traditionally, the valid modal syllogisms are referred to as Barbara LXL and so on, the invalid moods as Barbara XLL and so on.50) For taken in the composite sense, Barbara LXL is open to a counterinstance, originally due to Aristotle’s successor, Theophrastus, as head of the Lyceum:51 animal belongs of necessity to every human; now suppose human belongs to all that moves, that is, only humans run. It is still false, he claimed, that animal belongs of necessity to all that moves. Theophrastus proposed, in opposition to Aristotle’s theory, the so-called ad peiorem rule, that the modality of a valid conclusion was never stronger than that of the weaker premise. Applying the distinction between composite and divided modals, one can see that the major premise and conclusion of Theophrastus’s example are both true in the divided sense, that is, as it is often put, de re, but though the premise is true in the composite sense (that is, de dicto), the conclusion is false taken de dicto. However, if Aristotle really did intend the de re interpretation of the necessity propositions in Prior Analytics 9, his method of proof, using conversion, is illicit. Thus, either Aristotle was confused, and his theory of the modal syllogism is unreliable, or he intended some other interpretation of modal propositions, neither de re nor de dicto, as others have urged.52

5.2. Modal Ampliation Buridan, we will see, rejects both Barbara LXL and XLL, and does not try to defend Aristotle’s verdicts. The reason he rejects Barbara LXL (which we have just noted seems to be valid de re) is that he takes the subjects of all divided

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modal propositions to be ampliated to the possible. This seems undeniable in the case of possibility propositions, and even of contingency propositions. Taken in the divided sense, “Some S is possibly P,” “Every S is possibly P,” “Some S is contingently P,” and so on all have among their causes of truth the possibility of things that are not S but could be S possibly and contingently being P. Buridan concludes from this that the subjects of divided propositions of necessity are also ampliated to the possible; see the second conclusion in Book II. Otherwise, he says, “Every S is necessarily P” would not contradict “Some S can fail to be P.” In fact, in the eighth chapter of the first treatise of Summulae de Dialectica he sets out a figure with eight vertices, his famous modal octagon,53 to clarify the various relations between distinct modal propositions. (See Plate II; we will represent and formalize the basic structure in Table 2.) “Every S is necessarily P” is the contradictory of “Some S is necessarily not P,” which is equivalent to “Not every S is possibly P,” whose subject is ampliated to the possible. Similarly, “Some S is possibly P,” again with S ampliated to the possible, contradicts “No S is possibly P,” in turn equivalent to “Every S is not possibly P,” that is, “Every S necessarily fails to be P.” Hence the subjects of necessity modals must also be ampliated to the possible. What then are the causes of truth of “S can be P”? There is disagreement over this, Buridan says at Book II, Chapter 4. Should we read “S can be P” as a disjunction, “That which is S can be P or that which can be S can be P,” or as “That which is or can be S can be P,” a single proposition with a disjunct term as subject?54 Now if “S” were a singular term, these would be equivalent. This was in many ways the point of the doctrine of terms underlying supposition theory. Take “Every S is P.” “P” has merely confused supposition, so we descend to “Every S is this P or that P and so on,” with a disjunct term, a single proposition not equivalent to a disjunction. However, if we now descend on “S,” which still has confused and distributive supposition, we obtain a conjunction of propositions of the form “This S is this P or that P and so on,” where each conjunct is in turn equivalent to a disjunctive proposition “This S is this P or this S is that P and so on.” Similarly, if we descended first on “S” in “Every S is P” to obtain conjuncts of the form “This S is P,” “P” no longer has merely confused supposition, but in “This S is P,” “P” has determinate supposition, since “This S is P” is equivalent to “This S is this P or this S is that P and so on.” Thus, an assertoric proposition containing a disjunct term resists expansion into a disjunction of propositions only if there is some other term,

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Plate II: Buridan’s Modal Octagon (From MS Vatican Pal.lat. 994 f. 11v, reproduced with permission of Biblioteca Apostolica Vaticana)

such as “every,” impeding it. That is not so in “That which is or can be S can be P,” where there is no determiner on “S.” But consider “Every S can be P.” As Buridan says (Book II, Chapter 4), there is here “just one subject and just one predicate and a single simply predicative proposition, and the subject is distributed all at once by a single distribution.” “Everything that is or can be S can be P” is not equivalent to “Everything that is S can be P or everything which can be S can be P.” The former is equivalent to “Everything that is S can be P and everything which can be S can be P,” as he says (Book II, Chapter 5, second conclusion): consequently, “it would

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be necessary to analyze particulars by disjunctions and universals by conjunctions.” That is not only awkward, but in fact even then it gives the wrong answer with universals of necessity, such as “Every S is necessarily P.” This is equivalent, he says, to “Everything that is or can be S is of necessity P,” and not to a conjunction “Everything that is S is of necessity P and everything that can be S is of necessity P.” For let “S” be “creating” and “P” be “God,” and suppose God is not now creating.55 Then “Everything that is creating is necessarily God” is false, since nothing is creating, and so the whole conjunction would be false. But “Everything that is or can be S is of necessity God” is made true by God, who can be creating and is necessarily God. Thus “Everything creating is of necessity God” is true and its contradictory “Something creating necessarily fails to be God” is false. Thus in general, “S can be P” should be analyzed using a disjunct term, “That which is or can be S can be P,” equivalent to a disjunction in the absence of a binding determiner, but in general not analyzed as a disjunction. Although necessity implies truth, which in turn implies possibility, as Buridan agrees, the ampliation of the subject in divided modals, unless it is restricted by the “that which” construction, means that necessity propositions (in the divided sense) do not imply the corresponding assertorics, nor do the assertorics imply the corresponding possibility propositions: see the third and fourth conclusions of Book II, Chapter 6. Nonetheless, necessity propositions do imply the corresponding possibility propositions, since the subject is ampliated to the possible in both. In the fifth and sixth conclusions of Book II, Chapter 6, Buridan points out which of Aristotle’s modal conversions are valid and which fail for divided modals. In brief, for divided modals, AaLB entails BiMA, and both AaMB and AiMB entail BiMA, and so by contraposition, AeLB entails BeLA and BoLA, but the rest all fail.56

5.3. Composite Modals Finally, consider composite modals. In each case, a mode is predicated of a dictum, which has material supposition for a proposition. (So, as Buridan notes in Book II, Chapter 7, an expression does not always have material supposition for itself, since a dictum is not a proposition. The same point underlies the twelfth conclusion.) We noted that Buridan’s nominalism means that the only propositions are individual, particular tokens. Accord-

Introduction

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ingly, although Buridan allows that composite modals can be universal, particular, indefinite or singular, the corresponding modals are all equivalent. As he says in the ninth conclusion of Book II, “Some ‘B is A’ is possible” entails “Every ‘B is A’ is possible,” since all (token) propositions of the form “B is A” signify the same, and so are true or false together. Note that it is crucial here to distinguish the singular proposition “‘Some B is A’ is possible” from the particular affirmative “Some ‘B is A’ is possible.” Actually, one might criticize Buridan ad hominem here. For Buridan’s response to the revenge problem for the liar paradox 57 is to say that, while, for instance, Socrates’s utterance of “Socrates speaks falsely” signifies both that Socrates’s utterance is false and that it is true (and so is false), Buridan’s utterance of “Socrates speaks falsely” signifies both that Socrates’s utterance is false and that Buridan’s utterance is true (and so is true). Thus two tokens of the same proposition may signify differently.58 Hence Buridan seems committed to counterexamples to the ninth conclusion of Book II by what he says at Book I, Chapter 5. The same point appears to undermine the fifteenth conclusion of Book II, where Buridan says that “it follows, ‘B is A, so it is true that B is A,’ ” for although Socrates speaks falsely, it is not true that Socrates speaks falsely, that is, Socrates’s utterance of “Socrates speaks falsely” is not true.

5.4. Pure Possibilia Lagerlund rejects the earlier claim that Buridan admits pure possibilia by claiming to derive the Barcan formula from Buridan’s assumptions.59 As a matter of fact, commitment to the Barcan formula does not actually oblige one to deny mere possibilia. The formula says (in one form) that (for any F) if it is possible that something is F, then there is something that can be F. That does not necessarily show that only things that exist can be F. Even so, Lagerlund’s argument is faulty. Ruth Barcan’s aim in proposing the formula named after her was to reduce de re modality (Quine’s third and unacceptable grade of modality) to de dicto modality (the arguably acceptable second grade).60 In the eighteenth conclusion of Book II, Buridan claims that a divided universal negative implies a composite universal with a negated dictum, that is, “Every B is necessarily not A” (i.e., “No B is possibly A”) entails “It is necessary that no B is A.” This follows by contraposition from the exception to the seventeenth conclusion, whereby “It is possible that some B is A” entails “Some B can be A,” which Buridan takes

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to be self-evident. Lagerlund claims that “Every B is necessarily not A” is equivalent to “Everything is such that necessarily, if it is B then it is not A,” in symbols: (*)

(∀x)(MBx → L¬Ax) ↔ (∀x)L(Bx → ¬Ax)

If so, then the eighteenth conclusion is equivalent to (∀x)L(Bx → ¬Ax) → L(∀x)(Bx → ¬Ax), which is an instance of the Barcan Formula. However, (*) is false, invalid in a one-element model whose sole element is B and not A, but might be A and not B, for it fails from right to left. So there is no reason to doubt our earlier conclusion that Buridan admitted pure possibilia.

6. Modal Syllogistic At the start of Book IV, Buridan distinguishes two uses Aristotle makes of the phrase “simply assertoric” (de inesse simpliciter). The literal sense of the phrase is that “simply” means “without addition or qualification.” But Aristotle also has a somewhat confusing technical sense of the term, when he contrasts “simply assertoric” with “assertoric as-of-now” (de inesse ut nunc), that is, without qualification with respect to time or with such qualification ( Prior Analytics 34b7ff). The syllogism Aristotle is discussing here is Barbara XQM, which has an obvious counterexample: suppose only men are moving, while it is contingent whether horses move. One cannot legitimately conclude that horses might be men. But if we suppose that the major premise is not just true now, but true “simply,” that is, at all times, then the counterexample fails. Buridan interprets Aristotle here as adding as a hypothesis that the assertoric premise is necessary, and points out that this is equivalent to replacing the assertoric premise by a composite proposition of necessity.

6.1. Composite Modal Syllogisms As Lagerlund points out (172–173), the notion of simply assertoric premises means that, although Buridan shows little explicit interest in modal syllogisms with composite modals, stating just three conclusions for them in

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Book IV, composite modals implicitly reappear in conclusions 11–14 and 19 there. The first conclusion turns on the basic modal principle that necessity and possibility (and contingency) are closed under necessary consequence.61 In symbols: L( p → q) ๵ Lp → Lq L( p → q) ๵ Mp → Mq L( p → q) ๵ Qp → Qq Taking the modality in the composite sense, we do not need to analyse the dictum, so these are theses of propositional modal logic. The same is true of Conclusion 2, which states the basic modal equivalences that Aristotle arrived at only with difficulty in De Interpretatione, mentioned earlier, by singling out the sense of “possible” that simply contradicts “impossible” and is compatible with “necessary” (see also Book II, Chapter 5): L¬p ๶๵ ¬Mp Taking truth as a further alethic modality, Buridan observes that truth commutes with negation (T¬p ๶๵ ¬Tp), equivalent to bivalence if one equates falsehood with truth of the negation.

6.2. Epistemic Modalities The third conclusion of Book II dismisses any syllogistic consequence employing epistemic modalities in the composite sense, in particular, denying that epistemic modalities might be closed under consequence. The objection turns on the simple point that the consequence itself might not be recognized. This leaves open the possibility, however, that there might be syllogisms in other modalities in the divided sense. Buridan does not consider these in Treatise on Consequences. But he had raised the issue in question 40 of his Questions on the Prior Analytics. He there repeats the claim that “from composite premises concerning ‘know,’ ‘believe,’ ‘suppose,’ and the like, there are no valid syllogisms like those of necessity or possibility.” He concedes that since knowledge entails truth, one can reason validly from composite premises about knowledge to a syllogistic conclusion like this: “‘I know that every B is A, I know that every C is B, so every C is A’ . . . but this is not the case with ‘suppose,’ ‘appears,’ or ‘believe.’ ”

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In the Questions on the Prior Analytics, he then turns to epistemic modals taken in the divided sense, proposing three positive conclusions: that in the first and third figures, a conclusion with the same mode as the major premise follows given an assertoric minor premise—for example, “Every B is supposed to be A, every C is B, so every C is supposed to be A” (matching Barbara) and “Every C I suppose to be A and every C is B, so B I suppose to be A” (matching Darapti). In the second figure, if the negative premise (there always is one in the valid moods in the second figure) has a negated mode and the affirmative premise an affirmed mode, an assertoric conclusion follows, for instance, “Every B I suppose to be A, no C do I suppose to be A, so no C is A” (matching Camestres). In Summulae de Dialectica 5.6.8, he dismisses even divided epistemic syllogisms in all figures except for the mode “know,” claiming the latter are valid in the first and third figures, invalid in the second. For example, “Every B I know to be A, every C I know to be B, so every C I know to be A” is said to be valid by the dictum de omni. As usual, the counterexamples concern things seen indirectly from afar as in the example of the Hooded Man. The Buridan of the Questions on the Prior Analytics would agree, since knowledge entails truth, so the minor premise entails that every C is B and so the example accords with the first conclusion given before. But in Treatise on Consequences, Buridan passes over these cases in silence, concentrating on the alethic modalities, necessity and possibility, as he says at Book II, Chapter 1.

6.3. Divided Modal Syllogisms Buridan has a basic system of modal syllogistic for divided modals, which he then adapts in two ways: first, by replacing any assertoric premise with a simply assertoric one—as we have seen, such a premise is equivalent to a composite modal, so this adapted system is essentially one with mixed divided and composite modals; second, by restricting the modal components with his popular “that which” phrase, canceling the ampliation of the subject. Buridan employs two methods for showing validity, reduction to the dictum de omni et nullo using those conversions that are still valid, on the one hand, and the expository syllogism, on the other. Thom has a useful notation for each of the various propositions, modal and assertoric, which encapsulates a semantic justification. For example, “Every A is necessarily B” becomes a† → b*, meaning that the possible (including actual) As are

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Table 2: Representation of Buridan’s modal octagon in Thom’s notation

included among the necessary Bs, and so on.62 See Table 2 for a representation of Buridan’s modal octagon in Thom’s notation, and Table 3 for a representation using first-order logic.63 Note the underline of a and d in the A- and I-propositions to indicate the existence of the subject (as we noted earlier), and the lack of underline in the O-proposition—the counterexample to the inclusion of a in b need not exist. Using this notation, we can form proofs of various valid modal syllogisms. For example, Darii MXM is valid: b† → a†, cೌb, so c†ೌa†. Suppose b† → a† and cೌb, i.e., ∃d, c ← d → b. Then, since b → b † and c → c †, c† ← c ← d → b → b† → a†, so ∃d, c† ← d → a†, that is, c†ೌa†.64 As Buridan succinctly puts it in the tenth conclusion: “There is a perfect syllogism by explicit subsumption under the distribution of the major,” that is, b is explicitly asserted in the major premise to be subsumed under a† (b → b† → a†).

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Table 3: Representation of Buridan’s modal octagon in first-order logic

Buridan also gives various proofs of invalidity for modal moods. For example, in the same conclusion, he shows by counterinstance that Barbara XMM is invalid, for even if only horses were running, every human could still run but none of them could be a horse, that is, b → a, c† → b† and c†|a† are compossible, so a†ೌc† and everything that entails it (including c† → a†) cannot follow. Hence, neither is Darii XMM valid, and similarly neither Celarent nor Ferio XMM, since the first cause cannot fail to be God even if God is not creating, while as first cause he can create. Tables 4, 5, and 6 summarize the valid conclusions drawn in the standard system, one for each figure. The mode of the major premise is given in the left-hand column, of the minor in the top row, and the mode of any permitted conclusion in their intersection. The justification for each conclusion is keyed to Treatise on Consequences where Buridan explicitly states it, to Summulae de Dialectica where the latter extends Treatise on Consequences consistently, and to Thom’s analysis where Thom demonstrates further valid conclusions not explicitly stated by Buridan.65 Recall that Buridan, following Aristotle, considers the validity of modal syllogisms only where the assertoric counterpart is valid; and that in Book III, Part I, Chapter 4, he has excluded from examination all weakened moods, and all indirect moods from the same premises. So the only indirect moods he admits as novel (apart from his own nonnormal conclusions) are Fapesmo and Frisesomorum. It is surprising, nonetheless, that in

Table 4: Valid conclusions for divided modals in the first figure Figure I

L

L

La

L′ M M′ X Q Q′

L′

M

M′

Lb, Mc, Celarent Xd Darii, Ferio L, L′j n

M

—k Mo

r

Mu, Celarent Xv Mab, Qac

M′ Xw

—l s

—x Mad, Qae

a. TC IV, conclusion 4, and SD 5.6.5. b. TC IV, conclusion 4, and SD 5.7.5, rule 1. c. Thom 9.2a. d. Thom 9.7a. e. TC IV, conclusion 16, and SD 5.7.3, rule 1. f. TC IV, conclusion 16. g. SD 5.7.7, rule 1, and TC IV, conclusion 21, which also appears at SD 5.7.1, rule 3. h. Thom 9.34. i. Thom 9.31. j. TC IV, conclusion 7, and SD 5.6.6. k. TC IV, conclusion 7. l. Ibid. m. SD 5.7.3, rule 1. n. TC IV, conclusion 4, and SD 5.7.5, rule 1. o. TC IV, conclusion 4, and SD 5.6.4. p. TC IV, conclusion 10, and SD 5.7.2, rule 5. q. TC IV, conclusion 21, and SD 5.7.6, rule 1. r. TC IV, conclusion 7.

M′ My

X

Q

Darii, Ferio Le, Barbara, Celarent Xf m L′ Darii, Ferio Mp M′t Darii, Ferio Mz Darii, Ferio Qaf Q′ah

Lg, Mh, Celarent Xi

Q′

Mq —aa Qag —ai

s. Thom 9.49a and SD 5.6.4. t. TC IV, conclusion 10, and SD 5.7.2, rule 4. u. SD 5.7.3, rule 2. v. TC IV, conclusion 15. Rejected at SD 5.7.3, rule 2; see Thom, 178. w. TC IV, conclusion 15. x. TC IV, conclusion 10, and SD 5.7.2 give a counterexample. y. TC IV, conclusion 10. z. Contrary to Thom 9.10a, which claims that all XXM moods are valid. aa. SD 5.7.4, rule 2. ab. Thom 9.37 and TC IV, conclusion 21. ac. TC IV, conclusion 23, and SD 5.7.7, rule 1. ad. SD 5.7.6, rule 1. ae. TC IV, conclusion 23, and SD 5.7.6, rule 2. af. TC IV, conclusions 21 and 24, and SD 5.7.4, rule 5. ag. SD 5.6.6 and TC IV, conclusions 21 and 23. ah. SD 5.7.4, rule 4. ai. SD 5.6.6.

Table 5: Valid conclusions for divided modals in the second figure Figure II

L

L

La

L′ M M′ X Q

L′

M

M′

X

Q

Festino Le, Camestres, Baroco Xf

Lg, Mh, Cesare, Camestres Xi

—

—q

—r

—x

Festino My

—z

—ac

—

—ad

Lb, Mc, Cesare, Camestres Xd X, Mj

m

n

M′k, Xl p

L , M , Cesare, Camestres Xo

Q′

L′s, M′t, Xu v

M , Cesare, Camestres Xw aa M , Cesare, Camestres Xab

Q′ a. TC IV, conclusion 5, and SD 5.6.5. b. TC IV, conclusion 5, and SD 5.7.5, rule 2. c. Thom 9.6a. d. Thom 9.1a. e. TC IV, conclusion 17. f. TC IV, conclusion 17, and SD 5.7.3, rule 4. g. SD 5.7.7, rule 2. h. Thom 9.35. i. Thom 9.32. j. TC IV, conclusion 8. k. TC IV, conclusion 7. l. TC IV, conclusion 8. m. TC IV, conclusion 5, and SD 5.7.5, rule 2. n. Thom 9.3b. o. Thom 9.8b. p. TC IV, conclusion 5, and SD 5.6.4 give a counterexample.

—ae q. TC IV, conclusion 12, gives a counterexample. r. SD 5.7.6, rule 3. s. Thom 9.49b. t. TC IV, conclusion 7. u. TC IV, conclusion 8. v. SD 5.7.3, rules 3 and 4. w. TC IV, conclusion 17. Thom 9.10 claims mistakenly that Festino and Baroco XLX are valid. x. TC IV, conclusion 12, gives a counterexample. y. Thom 9.11b. z. SD 5.7.4, rule 1. aa. Thom 9.38. ab. Thom 9.33. ac. SD 5.7.6, rule 3. ad. SD 5.6.6 gives a counterexample. ae. Ibid.

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his consideration of the modal syllogism, Buridan seems not to consider these indirect moods, at least, not explicitly. His only mention of indirect conclusions comes in Book IV, Chapter 4, with regard to reduplicative syllogisms. Take the fourth conclusion of Book IV, for example, that there is always a valid syllogism to a conclusion with the same modality as the major premise if all three propositions have modalities of possibility or necessity. This is true for the perfect moods, as Buridan notes. But it fails for the indirect moods Frisesomorum LML and MMM. Take the terms “runs,” “human,” and “animal,” where “runs” is the middle term. Then some runner can, and must, be human, and no animal necessarily runs. Yet it does not follow that some human is not necessarily, or even not possibly, an animal, for only humans can be human, so all humans are necessarily animal. In Table 5, we see, for example, that Cesare MLL is valid. Buridan’s proof reduces it per impossibile to Ferio MMM, which in turn is valid as follows: suppose b†|a* and c†ೌb†. Then ∃d, c† ← d → b†|a*, so c† ඃ a*. In the light of the objection at the end of the fifth conclusion of Book I, that while “Everything running is a horse; every human is running; so every human is a horse” is valid, its premises are possible but not its conclusion, the validity of MMM syllogisms may seem surprising. They are not all valid, as we have seen for Frisesomorum MMM, and no MMM syllogism is valid in the second figure at all. But the ampliation of the subject ensures that the direct MMM moods are all valid in the first and third figures. Tables 4–6 also include Buridan’s system with modals restricted by a “that which” phrase. We mark a restricted modal with a prime, thus L′, M′, Q′. The picture Buridan presents is rather incomplete. One may at first be puzzled by what Buridan writes in the seventeenth conclusion of Book IV. He notes, correctly, that a negative major premise of necessity, with an assertoric minor, yields a conclusion in the second figure only if that conclusion is particular. He then writes: “The first part of the conclusion claims that Cesare and Festino are valid to a conclusion of necessity, but not to a universal.” What he must mean, but expresses badly, is that Cesare LXL is invalid, since by conversion of the major premise we obtain Celarent LXL, which, he has noted in the sixteenth conclusion, is invalid; while Festino LXL is valid, since it converts in the same way to Ferio LXL, valid by the sixteenth conclusion. To see that Ferio LXL is valid, consider its premises: b†|a† and cೌb. We need to show c† ඃ a*. So suppose c† → a*. Then,

Table 6: Valid conclusions for divided modals in the third figure Figure III

L

L

La, Xb

L′ M M′ X Q

L′

M

M′

Lc, Md L′i

l

L′j m

M

M M, M′p s

Darapti, Disamis X Mw, Qx

Mq t

Darapti, Disamis M Qy

Q′ a. TC IV, conclusion 6, and SD 5.6.5. b. Thom 9.9b. c. TC IV, conclusion 6, and SD 5.7.5, rule 3. d. Thom 9.2b. e. Thom 9.10b. f. TC IV, conclusion 18, and SD 5.7.3, rule 5. g. SD 5.7.7, rule 3. h. Thom 9.36. i. TC IV, conclusion 9, and SD 5.6.6. j. TC IV, conclusion 9. k. Thom 9.48b. l. TC IV, conclusion 6, and SD 5.7.5 rule 3. m. TC IV, conclusion 6, and SD 5.6.4. n. TC IV, conclusion 13, and SD 5.7.2, rule 7. o. TC IV, conclusion 21, and SD 5.7.4 and 5.7.6, rule 1. p. TC IV, conclusion 9. q. SD 5.6.4. There is a slip in Klima’s translation at the foot of 342. Hubien’s Latin text reads: “Verum est tamen quod in tertia figura ualerent syllogismi

X

Q

Xe, Darapti, Felapton, Datisi, Ferison L f k L′ Darapti, Felapton, Datisi, Ferison Mn M′r Datisi Mu Disamis, Bocardo Mz, Datisi, Ferison Qaa

Lg, Mh

Q′

Mo

Darapti, Disamis Mv Qab

—ac ex talibus praemissis ad conclusionem sine ‘quod est,’ scilicet in qua non prohiberetur ampliatio” (“It is true, however, that in the third figure syllogisms from such premises to a conclusion without [the phrase] ‘that is’ would be valid, namely, in which ampliation is not prevented”). r. Thom 9.46b. s. SD 5.7.3, rule 6. t. TC IV, conclusion 13, and SD 5.7.2, rule 6. u. Thom, 179. v. TC IV, conclusion 21; SD 5.7.4, rule 8, and 5.7.6, rule 1. w. Thom 9.39. x. TC IV, conclusion 23, and SD 5.7.7, rule 3. y. TC IV, conclusion 23, and SD 5.7.6, rule 4. z. SD 5.7.4, rule 6, gives counterexamples. aa. TC IV, conclusion 25, and SD 5.7.4, rule 6. ab. SD 5.6.6; TC IV, conclusion 23. ac. SD 5.6.6.

Introduction

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since cೌb, ∃d, b† ← b ← d → c → c† →a* → a†, which contradicts the major premise. To see that Celarent LXL is invalid, recall that Buridan departs from Aristotle in rejecting Barbara LXL too, since he claims that the subject of the major premise is ampliated to the possible. So there is no middle term—we have possible Bs in the major premise and actual Bs in the minor. Buridan invites us to adapt the counterexample to Celarent XMM from the tenth conclusion: No moon is possibly the Sun (i.e., Every moon is necessarily not the Sun), every planet shining on our hemisphere is the moon (let us suppose), but it does not follow that no planet shining (that is, possibly shining) on our hemisphere can be the Sun. Thom claims that Baroco and Festino XLX are valid, and that consequently, so are Barbara and Celarent XXM.66 Certainly, the two pairs stand or fall together. However, his reasoning is mistaken. c → c†ೌd|a† ← a ← b does not warrant cೌd|b, for d may be only possibly c, not actually c. Buridan does not mention XXM moods, presumably because these would not be new syllogistic pairs, for as he says at the end of the fourth conclusion, he concentrates on the validity only of moods that are already valid for assertorics. King denies two of Buridan’s claims in the sixth conclusion, that is, the validity of Camestres MLL and Festino LML.67 However, Buridan is correct: they are both valid. For Camestres MLL, suppose a† → b† and c†|b†. Then a† → b†|c†, so a†|c† and so c†|a†. For Festino LML, suppose a†|b† and c†ೌb†. Then ∃d, c† ← d → b†|a†, so c† ඃ a†. King also disputes Buridan’s claim in the sixteenth conclusion that Celarent LXX is valid.68 But clearly, given b†|a† and c → b, we have c → b → b†|a† ← a, so c|a. King also seems to dispute the validity of Ferio LXL, which we established earlier. Furthermore, King seems to think that Buridan claims in his statement, “but there does follow an assertoric universal,” that we can draw a universal negative assertoric conclusion from the premises of Ferio with a major premise of necessity. But Buridan, as we noted, and following Aristotle’s lead, only considers applying modalities to the premises and conclusion of valid nonmodal syllogisms (else, as Hughes observes, he would have to survey some 86,016 cases). There is no reason to suppose that Buridan means his remark to apply to premises whose unmodalized form would not yield a universal conclusion.

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6.4. Syllogisms with a Simply Assertoric Premise Next, consider Buridan’s mixed system of modal syllogisms with a simply assertoric premise. Let us mark the simply assertoric premise as Xs. The full story is set out in Tables 7–9. Buridan’s procedure here merits some comment. The simply assertoric premise is not a modal proposition. (See Book IV, Chapter 1.) Rather, we have a syllogism with one assertoric and one modal premise, with the further condition (or as he calls it, “hypothesis, or supposition”), that the assertoric premise is, if true, necessarily true. Buridan’s example in the eleventh conclusion of Book IV is: Every B is A Every C can be B So every C can be A. He then formulates a further syllogism whose premises are the contradictory opposite of the original conclusion together with the original minor premise, and whose conclusion he shows is inconsistent with the supposition or hypothesis that the original major premise was necessary: Some C necessarily fails to be A Every C can be B So some B necessarily fails to be A. This conclusion can be true in three different ways, but Buridan shows that however it is true, it is always inconsistent with the necessity of “Every B is A.” Thus the original syllogism is valid on the assumption that its major premise is necessary.

6.5. Syllogisms of Contingency In Book IV, Chapter 3, Buridan proceeds to discuss syllogisms with premises and conclusions of contingency. Recall that “is contingent” means “is neither necessary nor impossible,” that is, “is possibly and possibly fails.” Thom represents the contingency of A by a‡, and it is immediate that a‡ → a†, as Buridan observes in the eighth conclusion of Book II, and that a‡|a*, as he says in Book IV, Chapter 3. Where Aristotle seems to have

Table 7: Valid conclusions with a simply assertoric premise in the first figure Figure I

L

M

Xs

Q

L M Xs Q

N/A N/A Lc N/A

N/A N/A Md N/A

La Mb N/A Qf, Mg

N/A N/A Me N/A

a. TC IV, conclusion 19. b. Thom 9.16a. c. TC IV, conclusion 14. d. TC IV, conclusion 11, and SD 5.7.2.

e. Thom 9.40. f. Thom 9.24. g. Thom 9.41.

Table 8: Valid conclusions with a simply assertoric premise in the second figure Figure II

L

M

Xs

Q

L M Xs Q

N/A N/A Lc N/A

N/A N/A Md N/A

La —b N/A —

N/A N/A Me N/A

a. TC IV, conclusion 19. b. TC IV, conclusion 12. c. Thom 9.14a.

d. Thom 9.13a. e. Thom 9.42.

Table 9: Valid conclusions with a simply assertoric premise in the third figure Figure III L M Xs Q

L N/A N/A Disamis, Bocardo XLLc N/A

M N/A N/A Datisi, Ferison Md N/A

a. TC IV, conclusion 20. b. TC IV, conclusion 14, but stated only for Disamis and Bocardo. c. Stated at SD 5.7.3 for Disamis. See Thom 9.16a. d. TC IV, conclusion 14.

Xs a

L Mb N/A Disamis, Bocardo Mf, Qg e. Thom 9.43. f. Thom 9.44. g. Thom 9.25.

Q N/A N/A Datisi, Ferison Me N/A

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proposed that the subject of propositions of contingency is ampliated to the contingent (see 32b25–30), Buridan realizes that it is more correctly ampliated to the possible, just like propositions of possibility and of necessity. Aristotle claimed at Prior Analytics 32a30–b1 that All possible [that is, contingent] premises convert to one another. I do not mean that affirmative ones convert to negatives, but that those that are affirmative in form convert with respect to opposites. So, for example, “[contingently] belonging” converts to “[contingently] not belonging,” “[contingently] belonging to all” converts to “[contingently] belonging to none” or “not to all,” and “[contingently] belonging to some” converts to “[contingently] not belonging to some.” What he seems to mean is that Qa propositions are equivalent to Qe and Qi to Qo, as Buridan observes in the seventh conclusion of Book II. Accordingly, Thom (171) represents divided A-, E-, I-, and O-propositions as shown in Table 10. However, this representation is not quite right, for “Every B is contingently A” (and equivalently, “Every B contingently fails to be A”) has existential import. For “Every B is contingently A” is equivalent to “Every B is possibly A and possibly fails to be A,” and so entails “Every B is possibly A,” which we know from Book II has existential import. So the correct representation is as shown in Table 11.

Table 10: Thom’s representation of contingency propositions Every B is contingently A Some B is contingently A

(Qa)

b† → a‡

(Qe)

(Qi)

b† ೌ a‡

(Qo)

Every B contingently fails to be A Some B contingently fails to be A

Table 11: Correct representation of contingency propositions in Thom’s notation Every B is contingently A Some B is contingently A

(Qa)

b† → a‡

(Qe)

(Qi)

b† ೌ a‡

(Qo)

Every B contingently fails to be A Some B contingently fails to be A

Introduction

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Table 12: Square of Opposition for Contingency Propositions

Recall that in Chapter 2, Buridan said that he would speak only of modal syllogisms with affirmed mode. However, this is possible only for the modes of necessity and possibility, which are duals, that is, “not necessarily” is equivalent to “possibly not” and so on. The case is different for contingency modals. We could introduce a new modality equivalent to “not contingently,” but in its absence, Buridan needs to treat directly of propositions of contingency with a negated mode. In fact, in line with his interpretation of “No B is necessarily A” as “Every B is not necessarily A” (that is, “Every B possibly fails to be A”), “No B is contingently A” is such a proposition, that is, “Every B is not contingently A.” Moreover, just as “Every B is contingently A” is equivalent to “Every B contingently fails to be A,” “Every B is not contingently A” (that is, “No B is contingently A”) is equivalent to “Every B does not contingently fail to be A.” If we represent propositions of contingency with negated mode as (Qa), (Qe), and so on, we then have a new Square of Opposition for modal propositions of contingency, shown in Table 12. Given that a‡ → a†, it is immediate that any M premise can be strengthened to the corresponding Q premise, by prefixing, that the consequent follows from whatever the antecedent follows from, as Buridan observes in the twenty-first conclusion. Looking at Table 4, for example, QLM and QMM are valid in the first figure since MLM and MMM are, and LQL, LQM and Celarent LQX follow from LML, LMM and Celarent LMX. The same pattern is repeated in Tables 5 and 6. Contingency conclusions follow in the first and third figures, as the twenty-third conclusion notes, but none in the second figure, as Buridan

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points out in the twenty-seventh conclusion. For example, consider QMQ in figure I. From b† → a‡ and c† → b† we obtain c† → a‡ by the dictum de omni et nullo, as Buridan says, validating both Barbara and Celarent QMQ, and from b† → a‡ and c† ೌ b†, c† ೌ a‡ follows, validating Darii and Ferio QMQ, from which QQQ follows in figure I by the observation of the twentyfirst conclusion. However, there is no contingency conclusion even in the first and third figures if the minor premise is of contingency unless the major premise is too, as he notes in the twenty-sixth conclusion. Moreover, there is no valid contingency conclusion in the second figure, since the extremes are the subjects in the premises, and so ampliated to the possible, not the contingent.

6.6. Reduplicative Syllogisms In the final chapter, Buridan elaborates on comments Aristotle makes in Prior Analytics I, 38 concerning reduplicative propositions. These propositions involve expressions such as “in so far as,” “qua” or “as such.” Bäck suggests that Aristotle calls them “reduplicative” because the “qua”-phrase in the major premise repeats or reduplicates the subject; for example, “The good is known qua good, all justice is good, so justice is known qua good.”69 Buridan identifies four components to a reduplicative proposition of the form, “A is B qua C” or “A qua C is B,” the three terms A, B, and C and the reduplicative expression “qua.” Aristotle claimed that reduplicative syllogisms (he seems to have had Barbara exclusively in mind) were valid only if the “qua”-phrase was joined to the major term (49a11): this led the medievals to construe “qua C” with the predicate B, rather than the subject A, as Buridan observes in IV 4. Aristotle’s interest in “qua”-propositions arises from their role in scientific knowledge, the object of the demonstrative syllogism in the Posterior Analytics. A is B qua C if C is an explanation or necessary accompaniment to B. For example, a triangle has internal angles equal to two right angles qua triangle, not qua isosceles or qua figure.70 According to Bäck,71 “A is B qua C” holds (immediately) if and only if A is B, A is C, every C is B, indeed, necessarily if anything is C it is B, and there is no middle term (different from C and B) for a demonstration that C is B. A is B qua C (exactly, or precisely, or convertibly) if and only if A is B, A is C, every C is B, necessarily if anything is C it is B, and every B is C. Buridan’s final conclusion, the twenty-eighth, confirms Aristotle’s statement at Prior Analytics I, 38 that the reduplicative term must be placed with the major

Introduction

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term to obtain a reduplicative syllogism, that is, to obtain a valid reduplicative conclusion. Take the first figure: if the reduplicative term goes with the major term, we have a valid syllogism immediately by the dictum de omni et nullo: Every/no A is B qua C, every/some D is A, so every/some D is/is not B qua C, effected by the middle term A. Similarly, in the third figure, the qualification of the major term carries over straightforwardly to the conclusion. It follows, Buridan notes, that there are no reduplicative syllogisms in the second figure, since the middle term is the predicate in both premises, so the reduplication would not carry through to the conclusion. For the same reason, a reduplicative minor term in the first and third figures would not carry over to a qualification of the predicate in the conclusion. The chapter on reduplicatives seems rather an afterthought, connected to a chapter on modal syllogisms only tenuously. Indeed, the final sections of Book IV are very terse, not to say cryptic. Buridan packs a great deal of information into each conclusion, and each provides a rich source of further research. Indeed, the whole treatise constitutes a fine resource for further contemplation. It provided this for medieval thinkers for the ensuing two centuries, and can still inspire thoughtful reflection in our own day.

7. Conclusion Buridan’s treatise on consequences is a highly original and influential study of the concept of logical consequence. It moves not only beyond Aristotle’s ideas, which had dominated medieval thought, but also beyond the already insightful developments of the logica modernorum. In particular, there are three crucial innovations made by Buridan. First, in Book I he establishes a clear notion of formal consequence, defined as necessary truth-preservation in all terms, that is, independent of subject matter. This removes any epistemic or semantic aspect of formal consequence that affected previous accounts, perhaps resulting from association with forms as concepts or essences, and concentrates purely on the logical aspects of the propositions involved independent of the descriptive terms they contain. Second, in Book II he sets aside Aristotle’s reduction of syllogistic validity to the perfect moods of the first figure, themselves validated by the dictum de omni et nullo. In its place, he identifies the basis of the syllogism as the relative distribution of the terms, articulated through the expository syl-

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logism, that is, ecthesis (or “setting out”), tentatively mooted by Aristotle himself in the Prior Analytics. Buridan sums up the marks of a valid syllogism (that the middle term must be distributed in at least one premise, that any term distributed in the conclusion must be distributed in its premises, and that the conclusion is negative if and only if just one premise is negative) in a series of conclusions. These marks became the “rules of the syllogism” in traditional logic. Buridan’s third innovation is his account of modal consequence and the modal syllogism in Books II and IV. He abandons attempts by his predecessors to remain faithful to Aristotle’s seemingly confused account and sets out his own theory of modal validity with the dual modes of necessity and possibility and the complex mode of contingency. All three innovations remained highly influential through the rest of the history of scholastic logic, into the sixteenth century, when interest in logic waned in the face of the renaissance and the reformation. Many of Buridan’s insights were regained in the resurgence of logic in the nineteenth and twentieth centuries, and extended much further. But there is still much to be learned from revisiting this remarkable treatise.

Editorial Introduction Hubert Hubien

Perhaps no science has known more vicissitudes than logic. One of the first to be created and assiduously cultivated by the Greeks, transmitted by the Syrians to Arabic-speaking peoples, it was able, despite being reduced to some elementary texts, to traverse the High Latin Middle Ages to blossom in the twelfth century and experience at the end of the thirteenth a flowering comparable to that which it has experienced in our own time. Then, toward the end of the fifteenth century, in one or two generations, it disappeared almost entirely. For more than three hundred years, one finds under the name of “logic” little more than a confused mixture of rhetoric, psychology, and epistemology (in proportions that varied with the taste of the author and of the times), where one uncovers with difficulty sole remnants of a great shipwreck, the theory of the categorical syllogism and of immediate inferences. And one must wait until the middle of the [nineteenth] century to see the efforts of some mathematicians restore vigor to this ancient discipline. This renaissance of logic did not have the effect of restoring the medieval tradition, confounded, out of pure ignorance, in the justified contempt that clothed later works. So much was this so that when, around 1930, some curious logicians, such as Heinrich Scholz, Jan Lukasiewicz, and the late Jan Salamucha, wished to study the history of their discipline, they found before them a veritable terra incognita.1 The oblivion was such that in 1956, the Rev. Father Bochenski could write: “we know this whole development so little that we are not even in a state to cite the names of the most important logicians.”2 And in fact, the greater part of this vast literature still lies unknown in the depths of libraries, either in manuscripts badly or incompletely described, sometimes even uncatalogued, or in renaissance editions whose text is often highly corrupt. 53

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Among the rare monuments of this literature, which we know a little, some notice has been taken of a treatise on consequences preserved in three incunable editions that attribute it to John Buridan. Father Bochenski used it in his excellent history of logic, as did E. A. Moody in the pioneering work that is Truth and Consequence in Medieval Logic.3 This treatise is, in fact, of great interest, and one can ask why it has not yet been the object of a modern edition. The reply is twofold: on one hand, because one might doubt the attribution to John Buridan, on the other hand, and above all, because the text furnished by the incunables is so corrupt that the boldest of editors would hesitate before the number of conjectural corrections that they require. In fact, the three incunables, as we will show later, copy one another, and Edmond Faral, in his thesis of 1946 on the manuscripts and editions of the works of John Buridan,4 had warned us that no manuscript of this treatise was known. Concerning the authenticity of this work, here is what the late Philotheus Boehner wrote in 1955 in a review of Moody’s work cited earlier: “Moody rightly makes good use of the Consequentiae of Buridan: it is medieval. Whether, however, it really is the work of Buridan is still open to question. Up to now I have not been able to discover a single manuscript of this work.”5 One must acknowledge that Boehner’s doubts were a little exaggerated. As he has himself very properly written elsewhere, “it is a good rule of sound historical research not to question the faithfulness of a historical document unless there is a grave reason for doubt.”6 Now our three editions expressly attribute the work to John Buridan, and one sees no reason to attribute any superiority to a manuscript document over a printed document. The first obstacle was then more apparent than real. It was quite otherwise with the second: the impossibility of restoring the text in a satisfactory manner without the aid of a new witness. Boehner was right on this point: one had to discover a manuscript of these Consequentiae. A hope could arise from reading the thesis of Mrs. Graziella Federici Vescovini, “Su alcuni manoscritti di Buridano.”7 The author signaled the existence, in a codex of the national library in Florence,8 of a work entitled Consequentie Iohannis Biridani, of which she gave the incipit and explicit: Incipit: “Incipiunt consequentiae magistri Iohannis Biridani. Consequentia est antecedens et consequens cum nota. . . .” Explicit: “Quidquid enim est asinus est homo uel asinus. Expliciunt consequentiae magistri Iohannis Biridani.”9 Alas, a mere reading of this incipit allows

Editorial Introduction

55

us to state that this text is not that of our incunables. Moreover, it seems certain that the work is not by John Buridan: The definition of consequentia with which he begins is not that of the Picardian master,10 but it definitely belongs to a member of the English school,11 which is confirmed by the fact that the Florentine codex otherwise contains only works by Richard Billingham, Peter of Candia, and William Heytesbury, apart from the Obligationes Iohannis Busti and an anonymous Obiectiones Consequentiarum.12 Must one despair of ever being able to edit the text of this remarkable work? Not so, for the manuscripts so much desired do exist. As early as 1875, Fiess and Grandjean mentioned one in their catalogue of the manuscripts of the library of the University of Liège13 and this reference was repeated, in 1938, by Dom H. Bascour, in his article, “Buridan” in the Dictionnaire d’Histoire et de Gëographie Ecclésiastiques.14 By what mystery this fact escaped Edmond Faral we will doubtless never know. Nevertheless, the credit that the reputation of its author attached to the thesis cited earlier always misled all those who dealt with the Consequentiae Buridani until a happy chance placed Dom Bascour’s article before our eyes. Moreover, we have had the good fortune to retrieve two other witnesses in the manuscripts of the Vatican Library. These manuscripts would have appeased Boehner’s doubts, had he known them. In fact, their colophons expressly attribute the work to John Buridan. But there is more. In the first conclusion of Book III, Chapter 4, the author considers the objection that certain syllogisms are not “formal consequences” (that is, valid whatever their terms) because one can find counterexamples formed with terms from the Holy Trinity. Our three incunables here read: “Utrum autem secundum alium modum locutionis syllogismi de forma ualent in his diuinis et quae sit illa forma relinquo theologis. Et est notandum et semper in memoria habendum quod, quia non pertinet ad istam de praedictis ultra determinare. . . .”15 This passage is enough to prove that the author of our Consequentiae was not a theologian, and it is known that John Buridan, in contrast with most of the great scholastics, never left the arts faculty. But the text of our incunables must be emended, and should read, with the manuscripts, not “quia non pertinet ad istam,” which does not mean anything, or “quia non pertinet ad istam artem,” which would be added if one had to make a conjecture, but “quia non pertinet ad me artistam.”16 Never has a reading (lectio difficilior!) asserted itself

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with better evidence. Now compare the text thus emended with another extracted from the tertia lectura of Buridan’s questions on the Physics of Aristotle: “Iam aliqui dominorum et magistrorum improperauerunt mihi de hoc quod aliquando in quaestionibus philosophicis intermisceo aliqua theologicalia, cum hoc non pertineat ad artistas. Sed ego cum humilitate respondeo quod ego bene uellem non esse ad hoc adstrictus, sed omnes magistri cum incipiunt in artibus iurant quod nullam quaestionem pure theologicam disputabunt, utpote de trinitate uel de incarnatione, et ultra iurant quod si contingat eos disputare uel determinare aliquam quaestionem quae tangat fidem et theologiam, eam pro fide determinabunt et rationes in oppositum dissoluent, prout eis uidebuntur dissoluendae.”17 Can one desire stronger proof of the authenticity than results from comparing these two texts and the explicit and parallel endorsement of the incunables and the manuscripts? Having attributed the text with certitude to John Buridan, it does not seem impossible to date it. This time, we rely on no evidence beyond the text. In Book I, Chapter 4, treating of the distinction between “consequentiae formales” and “materiales,” Buridan tells us that the latter are very commonly used: “Et istis consequentiis utuntur saepe uulgares,”18 and adds this example (ibid.): “ut si dicamus ‘cardinalis albus est electus in papam,’ concludemus ‘ergo magister in theologia est electus in papam.’”19 Now, if there is a constant feature of logical literature, it is that its examples are stereotypical. For nearly two millennia, as Callius, Dion, and Socrates have been walking and disputing with the same earnestness as their distant Stoic ancestors, Walter Burley or Paul of Venice inform us of the same incontestable truth: “if it is day, it is light.” The strange appearance of this white cardinal must make us suspect some allusion to a contemporary event. What is a white cardinal? For certain, a Cistercian, from the color of his habit (is John of Mirecourt not most often “the white monk”?). The fourteenth century knew four white cardinals, only one of whom became pope: Jacques Fournier, under the name of Benedict XII. That the election of this man was a disagreeable surprise to Buridan is not strange: he could expect nothing good of the declared adversary of Ockham,20 of a man whose honesty and incorruptibility compensated poorly for his intolerant fanaticism.21 Jacques Fournier was elected on December 20, 1334, and crowned on January 8, 1335. So it seems very plausible to suggest that our Consequentiae were written in 1335.

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The preceding considerations seem to us to establish with certainty that the present work is indeed from the pen of John Buridan and, with very strong probability, that it was written in 1335. It remains for us to justify the text that we publish and to give the reader an overview of what it contains. The text is attested by the Liège manuscript, the two Vatican manuscripts and the three incunables cited earlier, which we designate by the signs L, P, V, A, B, and C respectively. We believe it unnecessary to describe the three incunables, because they have already been in the Gesammtkatalog der Wiegendrucke, under the numbers 5749, 5750, and 5751; all three are quarto printings at Paris, by Félix Baligault about 1493, by Antoine Caillaut around 1495 and by Gui Marchant in 1499 respectively. The Liège manuscript is found in codex 114C of the University library; here is the description: Paper, end of the fourteenth century, 274 ff., 20.5 × 30.5 cm. 2ra–112ra. Iohannes Bvridanvs: Quaestiones in Physicam. Incipit: “Tabula quaestionum libri primi Physicorum magistri Iohannis Buridan in uico straminum Parisius anno domini MoIIIo66o pronuntiatarum.” Explicit: “Expliciunt quaestiones totius Physicorum magistri Iohannis Buridan de ultima lectura finitae in profesto Philippi et Iacobi apostolorum de mane Parisius.” 112va–172vb. Gerardvs de Kalkar: Quaestiones in librum De anima. “Summo fauente et disponente auxilio sine quo nullum rite fundatur exordium, praesentis est propositi secundum mei ingenii exigui pusillam facultatem . . . Circa librum de Anima quaeritur primo utrum scientiae  traditae in libro de Anima anima sit subiectum proprium et adaequatum.” 173ra–192vb. Anon.: Quaestiones in librum De anima. “Quaeritur utrum de anima possit esse scientia. Et arguitur quod non. Nullius insensibilis potest esse scientia.” 193ra–203va. Iohannes Bvridanvs: Tractatus de consequentiis. 204. blank 205ra–[211ra]. Anon.: Tractatus de consequentiis. “Circa scientiam consequentiarum ponenda sunt aliqua praeambula.”

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211ra–215vb. Anon.: Quaestiones de terminis potentiae actiuae et passiuae. “Diuersis difficultatibus circa terminos potentiae actiuae et passiuae quibus plures prouecti capiuntur coactus, quasdam quaestiunculas, et primo de termino potentiae actiuae in motu locali. . . .” 216ra–274rb. Anon.: Quaestiones in Physicam. “Circa primum librum Physicorum quaeritur utrum de naturaliter mobilibus potest esse scientia.” The two Vatican manuscripts whose description follows are known to us only by microfilms. Palatinus Latinus 994, end of the 14th century, 160 ff. 2ra–137vb. Iohannes Bvridanvs: Summa logica. “Sicut dux est saluator exercitus, sic ratio cum eruditione est dux uitae—Aristotiles in quadam epistola ad Alexandrum . . . .” 138ra. Notes on sophisms. 138rb–138vb. blank 139ra-vb. A table 140ra–152va. Iohannes Bvridanvs: Tractatus de consequentiis. 153ra–159va. Gvillelmvs Bvser: Tractatus de obligationibus. “Obrogatum quorundam discipularum et sociorum . . . .” 159va–160rb. Anon.: Poem. “En quanta desolatio Angliae praestatur. . . .” Vaticanus Latinus 3020, anno 1384, 155 ff. 1ra–104ra. Iohannes Bvridanvs: Summa logica. “S—Aristotiles in quadam epistola ad Alexandrum . . . .” This work is incomplete; the scribe has stopped in the middle of the sixth treatise (De locis). 105–121 missing. 122ra–154ra. Iohannes Bvridanvs: Tractatus de consequentiis. 155r. blank 155v. “Ista logica est mei Francisci de Nigris quam emi a Magistro Virgilio de Taruisio pretio (a word not deciphered) anno domini moccccovio de mense nouembris. In Bononia.”

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What value should we accord to these witnesses? One sees on first inspection that incunable B is a simple reproduction of incunable A. Some abbreviations have been expanded, some new errors have been introduced; apart from that, this second edition reproduces the first with almost photographic fidelity, including the punctuation. It is not the same with incunable C. The differences that it shows with the text of A are often very great. It is the only one to carry the name of an editor, Albert Fantin, O.F.M., of whom we know nothing else, who has printed at the head of the text the following dedicatory epistle: Albertus Fantinus Italus Bononensis, frater minor, Iustiniano Fantino, germano suo salutem plurimam dicit. Compulisti me assiduis uocibus, suauissime mi germane, ut tibi non minus ac germano cuius consequentiis te familiariter accommodare debeas significarem. Ego uero ipse, qui minime honestis tuis petitionibus reluctari putaui, perlegi consequentias magistri Iohannis Buridani, in hisque perlectis mirum ordinem, miram doctrinam, miram sententiam miramque breuitatem inueni, adeo ut mihi esset summae admirationi. Sed quia ipsae quibusdam erroribus atque uitiis (nescio quorum, sed, ut puto, impressorum) erant refertae, decreui eas ab huiuscemodi erroribus atque uitiis expiare expiatasque tibi, suauissime germane, praeter quem habui neminem, offerre, ut illas anima hilari accipias, acceptasque perlegas, perlectasque serues, seruatasque ames, amatasque pro uirili portione defendas. Vale et me ames ut soles.22 Albert, one sees, says nothing of the text on which he has labored. Nevertheless, it is clear that it must concern one of the first two incunables, because the errors are attributed to “printers.” Examination of the text of C confirms this inference. Indeed, many corrupt passages have been emended and many lacunae have been filled in, but, except when it concerns minor and obvious corrections, they never agree with the manuscripts. As we will see, Albert did not have access to any manuscript; his emendations are purely conjectural. One example suffices, drawn from the second  conclusion of Book II, Chapter 6.23 To make easy comparison, we reproduce in parallel the texts of incunable A, of Albert and of the Liège manuscript. The text of A is evidently corrupt: One cannot attribute to Buridan the enormous absurdity of declaring one proposition equivalent to another on the grounds that it is its contradictory. Albert has realized this and has introduced

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A Ergo haec est neganda “nullum creans potest esse deus,” quia contradicit praedictae; ergo aequipolleret huic Et similiter neganda est ista “omnem creantem necesse est non esse deum”; ergo contradictoria huic ultimae est concedenda.

C Quia haec est neganda “nullum creans potest esse deus”, quae contradicit praedictae; ergo aequipollentes huic similiter negandae sunt, scilicet istae “omnis creans impossibiliter est deus” et “omnem creantem necesse est non esse deum”; ergo contradictoria huic ultimae est concedenda

L Ergo haec est neganda “nullum creans potest esse deus”, quia contradicit praedictae; ergo aequipollens huic est simpliciter neganda, scilicet “omnem creantem necesse est non esse deum”. Et similiter neganda est ista “omnem creantem necesse est non esse deum”; ergo contradictoria huic ultimae est concedenda.

the conjecture that we have just seen. Having done this, in place of writing “omne creans impossibile est esse deum,” as Buridan would have done, he has followed Italian habits and has written “omnis creans impossibiliter est deus,” a phrase unknown to the Parisian master, but frequently found in such authors as Paul of Venice, Peter of Mantua, and Paul of Pergula. It is clear, then, that Albert has labored on one of the first two incunables. It is even possible to determine that it concerns the first and not the second. In fact, one finds in C some small typographical mistakes present in A but absent from B. The mutual independence of A, V, L, and P is sufficiently proved by the existence of lacunae proper to each of these witnesses. Let us cite the most important: for A, II.6.250–255 and IV.3.46–53; for V, II.6.244–7.41; for L, I.8.646–663; last, for P, which does not show any lacunae as extensive as those which have been cited, I.8.679–681, II.6.226–227, 272–273, III.1.206–208, IV.1.73–74, 2.294–295. The analysis of common mistakes reveals the existence of two archetypes, one peculiar to P and A, the other to V and L. Finally, the presence of mistakes common to these four witnesses lets us infer the existence of an archetype common to all four and distinct from the original. The parentage of our six witnesses can then be represented by the stemma shown in the accompanying illustration.

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O  α PP

   

β

Q  Q  Q

P

P

γ

Q  Q



A

Q  Q  Q

B

PP PP

V

Q

L

C

It remains to say a word about the content of the work. Contrary to what one might suppose from the title, it is not a simple monograph, but rather a systematic exposition of the whole of logic of the time, at least of what we could, in modern terms, call the syntax, the semantics being touched on only lightly. The four books treat in turn of the general theory of consequences, corresponding to our propositional calculus, and of immediate inferences between nonmodal categorical propositions; of immediate inferences between modal propositions; of nonmodal categorical syllogisms; and finally of modal syllogisms. If the title Treatise on Consequences is justified, it is that the whole matter is reorganized by means of the fundamental concept of consequence. The method is very modern: we have here what is without doubt the first attempt, albeit very imperfect, at an axiomatic exposition of logic on the basis of the propositional calculus.24 The succession of theorems, “conclusiones,” is preceded by the announcement of “suppositiones,” which are really the ancestors of modern axioms; “haec suppositio,” Buridan writes are the start of Book I, Chapter 2, “non indiget probatione.”25 What Buridan’s Consequentiae prefigures is not the Port-Royal Logic, but Frege’s Grundgesetze and the first part of the Principia Mathematica. A final word on the principles that have guided the preparation of this edition. Academic texts of the Middle Ages present extremely numerous variants, though most of them are without importance; thus our four witnesses furnish on average five variants for each two lines of printed text. One could not dream of giving a complete apparatus collating them. That is why all individual variants, as far as the above stemma allows, have been resolutely discarded with the exception of those that exhibit some interest in

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themselves, even of simple curiosity. On the other hand, it was necessary to note the most banal variants common to two witnesses in order to establish and control the stemma. Regarding the text itself, the spelling adopted is the modern orthography of classical Latin, which seems to us desirable for texts of this genre. The differences with medieval usage are superficial: the spelling of diphthongs, of some individual words like “sicud” for “sicut,” and of some Greek words: “cathegoricus” or “categoreuma.” Reproducing such spellings teaches nothing to readers familiar with manuscripts and could only hinder others. We have nonetheless retained in all the examples the spelling “Sortes.” It seems clear, in fact, that, at least in the fourteenth and fifteenth centuries, “Sortes” is not an abbreviation for “Socrates” but has become an independent word, as is shown by the absence of any sign of abbreviation in the manuscripts and above all the example “Sortes is a bisyllable” that one sometimes finds. Finally, for the convenience of the reader, titles have been given to the chapters. Consequentiae reuerendi magistri Iohannis Buridani incipiant feliciter.26 Signs and Abbreviations P codex Palatinus latinus 994 of the Bibliotheca Apostolica Vaticana V codex Vaticanus latinus 3020 of the same library L codex 114 C of Liège University Library A first Paris edition {} additions by the editor ? illegible passage add. incorrect additions in the manuscripts or the edition om. omitted suppl. added transp. words transposed. Translated by Stephen Read

Book I Consequences in General and Consequences between Assertoric Propositions Chapter 1: The Truth and Falsity of Propositions In this book I wish to treat of consequences by recording their causes, [p. 17]1 as far as I can, about which much has already been adequately demonstrated by others. But they have not in point of fact been reduced to the first causes by which [those consequences] are said to hold. It will be necessary first to set out some assumptions. In this first chapter I want in fact to make clear why a true proposition is said to be true, and a false one false, and a possible one possible, and an impossible one impossible, and a necessary one necessary. Some claim that every true proposition is true because things are altogether as it signifies they are, namely, in the thing or things signified in reality. But I believe this is not so literally speaking. Because if Colin’s horse, which cantered well, is dead, “Colin’s horse cantered well” is true, but things are not in reality as the proposition signifies, because the things have perished. Or we may suppose that they have been completely annihilated, so that there is nothing signified in reality, indeed there is nothing in reality in any way, so nothing one way or the other, but nonetheless this proposition is true because things were in reality as the proposition signifies they were. In the same way, “The Antichrist will preach” is true, not because things are in reality as the proposition signifies, but because things will be in reality as the proposition signifies they will be. Similarly, “Something that never will be can be” is true, not because things are as the proposition signifies, but because things can be as it signifies they can be. And so it is clear that it is necessary to assign causes of truth to different types of proposition in different ways, and attentive readers can now understand from what has been said how they should be [p. 18] assigned to affirmative propositions. The same points can explain how to assign the causes 63

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of their falsity. For since it is impossible for the same proposition to be both true and false, and if it is formed it is necessarily either true or false, one must assign the cause of truth and the cause of falsity to the one proposition in opposite ways. So the cause of falsity of an assertoric present-tense affirmative will be that things are not altogether as it signifies them to be, and of a past-tense one, that things were not altogether as it signifies them to have been, and of one of possibility that things can be other than how it signifies they can be, or could be [other than how it signifies] they could be, or could have been, and so on; and similarly for other types and situations. Regarding negatives, it should be understood that every negative has or can have an affirmative as its contradictory, and whatever is the cause of truth of the one proposition is the cause of falsity of its contradictory, and vice versa. Hence, it is impossible in any way that they are both true or both false. So every true negative assertoric present-tense proposition is true because things are not altogether as an affirmative contradictory to it would signify them to be if it were put forward. And so similarly of those of possibility or of necessity, in the past tense or the future. And every false negative present-tense assertoric is false because things are altogether as an affirmative contradictory to it would signify them to be if it were put forward. And similarly in their way for other tenses and moods. Thus, an affirmative assertoric present-tense proposition is said to be possible if things can be altogether as it signifies them to be, and necessary if things must be altogether as it signifies them to be. And if it is past-tense, then it is possible if it was possible for things to have been altogether as it signifies them to have been. And similarly in their way for others. I will not discuss this or state or detail it any further, but assume it. And since names signify by convention, I lay down, on grounds of brevity, that from now on I will use this form of expression that some have been accustomed to use, in expressing the cause of truth of any kind of proposition, namely, that things are altogether as it signifies, [p. 19] and my audience should take the sense not according to the strict meaning of the words but according to the tenor of what has been said. This [concludes] what is set down in the first chapter.

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Chapter 2: The Causes of the Truth and Falsity of Propositions Now, in the second chapter, once again, since the causes on account of which the same proposition is rendered true or false are manifold, I will set down some things about these kinds of causes’ being more or being fewer according to the different suppositions of the terms. Let me say first that every proposition with an undistributed general term, or one similar in form to it, has or can have more causes of truth than a proposition with the same general term distributed, other things being equal. I understand by “the causes of truth” of a proposition whichever of them2 is enough for the proposition to be true. And the [first] assumption does not require proof, but only the observation that a proposition with an undistributed term, or a formally similar one, has or can have on account of that term as many causes of truth as that term supposits for. For example, if I say, “A human is running,” this would be true if only Socrates were running, and if only Plato were, and so on, and no less if they all were. But if the term is distributed, it can only have one cause of truth on account of that term, namely, that it holds for all of them, not only for one or two. I say “or a formally similar one” since “A human is an ass” cannot have a cause of truth, since it cannot be true, but a formally similar one can, e.g., “A horse is an animal” or “Water is hot.” So the form of the proposition does not rule out that anything with an undistributed term can have more causes of truth than [it would] with the term distributed, but it is ruled out [p. 20] by the form of the proposition that with a distributed term it have more than one. This is what from now on I shall mean by a proposition having more causes of truth or less. Thus in the same way I say concerning undistributed terms that a proposition with a general term suppositing confusedly has more causes of truth than a proposition with the term suppositing determinately by whose distribution it is confused; that is, it is not itself ruled out on account of the form of the proposition that it have many, but the converse is. For example, let us see the difference between the two propositions, “Every B is A” and “A is every B.” Clearly, “A is every B” has in respect of “A” as many causes of truth, in the given sense, as there are As, because it would be true if this A were every B, and similarly if that other A were every B, and so on. But “Every B is A” has as many causes of its truth, because it also

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would be true if every B were this A, and similarly if every B were that A, and so on. But in addition it has another cause of truth, namely, if this B is this A and that other B is that other A and a third B is a third A and so on; for “Every human is an animal” is true in this way. This cause of truth is ruled out for determinate supposition. Thus moreover I say that both terms being undistributed but suppositing determinately, then there are more causes of truth than if one were distributed and the other confused without distribution. This is clear because every cause of truth enough to make “Every B is A” true is enough to make “B is A” true, but not vice versa. Therefore a proposition has most causes of truth with each term undistributed and fewer with one term distributed and the other confused without distribution, and fewer still with one distributed and the other used determinately without distribution, and fewest of all with both distributed. This [concludes] what is set down in the second chapter.

Chapter 3: The Definition of Consequence Next, in the third chapter, it must be seen what we should understand by “consequence,” what by “consequent” and what by “antecedent.” [p. 21] For in all disciplines it is necessary first to know what the terms [mean]. To this end, I say that propositions are divided into subject-predicate and compound propositions. Now a consequence is a compound proposition; for it is constituted from several propositions conjoined by the expression “if” or the expression “therefore” or something equivalent. For these expressions mean that of propositions conjoined by them one follows from the other; and they differ in that the expression “if” means that the proposition immediately following it is the antecedent and the other the consequent, but the expression “therefore” means the converse. Some say that every such compound proposition, namely, one conjoining several propositions by “if” or by “therefore,” is a consequence; and then consequence is divided because some are true and some false. Others say that if it is false it should not be called a consequence, but only if it is true. But this is not a matter for dispute, since names signify by convention; and whether it is or is not, in this treatise I shall mean by “consequence” a true consequence, and by “antecedent” and “consequent” I shall mean propositions of which one follows from the other in a true or good consequence.

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Now antecedent and consequent are so called relative to one another; so they should be defined relative to one another. Hence, many say that of two propositions one is antecedent to the other if it is impossible for the one to be true without the other being true, and one is consequent to the other if it is impossible for the one not to be true when the other is true, so that every proposition is antecedent to every other proposition for which it is impossible for it to be true without the other being true. But this definition is defective or incomplete, because “Every human is running, so some human is running” is a good consequence, but it is possible for the first to be true the second not being true, when the second does not exist at all. So some say the given definition should be supplemented like this: the one proposition is antecedent to the other proposition if it is impossible that it be true the other not being true when they are formed together. But I say that this definition is even now not good, because “No proposition is negative, so no ass is running” is not a good consequence, [p. 22] but according to the second definition given one must concede that it is good. I prove the main claim because the opposite of the antecedent does not follow from the opposite of the consequent, that is, this does not follow: some ass is running, so some proposition is negative. Now the second claim is obvious since it is impossible for the first, what is called the antecedent, to be true, so it is impossible for it to be true the other not being true. Therefore, some give a different definition, saying that one proposition is antecedent to another, which is such that it is impossible for things to be altogether as it signifies unless they are altogether as the other signifies when they are proposed together. But even this definition is not true, literally speaking, because it assumes that every true proposition is true because things are altogether as it signifies, which was earlier denied. But it was said that we may use this manner of speaking in the sense given earlier; so we grant that definition. Indeed, we will also often use the manner of speaking according to the first definition earlier clearly disproved, because it has counterinstances in few consequences. But whatever manner of speaking we use, we intend the sense given before. Now consequence can be defined like this: a consequence is a compound proposition composed of antecedent and consequent, meaning that the antecedent is antecedent and the consequent is consequent; the description follows from the expressions “if” or “therefore” or an equivalent, as was said earlier. This [concludes] what is set down in the third chapter.

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Chapter 4: The Division of Consequences Next, in the fourth chapter, a division of consequences is presented, namely, into material and formal. A consequence is called formal if it is valid in all terms retaining a similar form. Or if you want to put it explicitly, [p. 23] a formal consequence is one where every proposition similar in form that might be formed would be a good consequence, e.g., “That which is A is B, so that which is B is A.”3 A material consequence, however, is one where not every proposition similar in form would be a good consequence, or, as it is commonly put, which does not hold in all terms retaining the same form; e.g., “A human is running, so an animal is running,” because it is not valid with these terms: “A horse walks, so wood walks.” It seems to me that no material consequence is evident in inference except by its reduction to a formal one. Now it is reduced to a formal one by the addition of some necessary proposition or propositions whose addition to the given antecedent produces a formal consequence. E.g., if I say “A human is running, so an animal is running,” I will establish the consequence by adding that every human is an animal; for if every human is an animal and a human is running, it follows in a formal consequence that an animal is running. For everyone arguing enthymematically endeavors to prove his consequences in this way if they are not formal. Because I have started to speak of the distinction of consequences, I want to say further that some material consequences are called simple consequences because they are simply speaking good consequences, since it is not possible for the antecedent to be true the consequent being false. Others, which are not simply speaking good, are called as-of-now consequences because it is possible for the antecedent to be true without the consequent, but are good as-of-now, because, things being as a matter of fact as they are, it is impossible for the antecedent to be true without the consequent. Ordinary people often use these consequences, e.g., if we say, “A white cardinal has been elected Pope,” we infer “So a master of theology has been elected Pope,” and if I say “I see him,” you will infer “therefore you certainly see a deceitful man.” Now this consequence is reduced to a formal one by adding a true, but not necessary, proposition or propositions, e.g., in the examples given, because the white cardinal is master of theology and because this man is a deceitful [p. 24] man. In the same way we have

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here a good consequence, assuming that the only men are Socrates, Plato and Robert: “Socrates is running, Plato is running and Robert is running; so every man is running,” because the consequence can be made perfect by this truth: “Every man is Socrates or Plato or Robert.” It should be understood that promissive consequences are as-of-now consequences of this kind. For example, if Plato says to Socrates, “If you were to come to me I will give you a horse,” this proposition may by chance be a true consequence, but also by chance be a false proposition and not a consequence. For if the antecedent is impossible, namely, that Socrates cannot come to Plato, the consequence is simply true, because anything follows from the impossible, as will be affirmed later. But if the antecedent is false but not impossible, then the consequence is good as-of-now, because anything follows from a falsehood in an as-of-now consequence, as will be affirmed later,4 as long as we extend the name of consequences as-of-now to consequences as-of-then, whether of the past or of the future or of any other specific time. But if the antecedent is true, namely, that Socrates will come to Plato, then we may even then declare the consequence good, since it can be made formal by adding truths to it, namely like this: whatever Plato wants to do in the future and what he can do if his desire continues and all circumstances apply according to which he wants it, and if he is not prevented, he will do it when and how he wants and can do it (modifying this proposition so that it is true according to the ninth book of Aristotle’s Metaphysics); but Plato wants to give a horse to Socrates, who is going to come to him, when he comes to him; therefore Plato will give Socrates a horse. So if these propositions about Plato’s wishes and powers are true, Plato uttered a true consequence as-of-now to Socrates; but if they were not true, then Plato told Socrates a falsehood, and not a consequence, and if Plato believed the propositions adjoined or similar ones not to be true, he lied to Socrates. This [concludes] what is set down in the fourth chapter. [p. 25]

Chapter 5: The Supposition of Terms Now in the fifth chapter, I set down further that an affirmative proposition means that the terms supposit for what are the same, or were or will be or can be the same, depending on the kind of proposition. For if I say “A is B,” I mean that A and B are the same, and if I say “A was B,” I mean that A was the same as B, and so on.

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A negative proposition means the opposite, namely, that what the subject supposits for and what the predicate supposits for are not the same, or that they were not the same or will not be, and so on. We need to describe different kinds of proposition differently. For depending on the expressions used, we describe universal [propositions] one way and particulars another, for example, that nothing for which the subject supposits is the same as anything for which the predicate supposits, or for a particular, that something for which the subject supposits is not the same as something for which the predicate supposits. So the reason “A chimera is a chimera” is not true is because the subject supposits for nothing, and so for nothing the same [as the predicate supposits for]. Thus, I take it that no affirmative proposition is true in which some term does not supposit for anything present, past, future or at least possible. Hence, assuming that it is impossible for there to be a chimera, I consider “A chimera is thinkable,” “A chimera is a matter of opinion,” “A chimera is signified by the term ‘chimera’” are false. But I do not intend at present to answer the objections that can be adduced against this claim, because it would need a special treatise to itself. Even if these points are granted, it does not follow that an affirmative assertoric present-tense proposition whose terms supposit for the same thing is true, because in a proposition asserting itself to be false there can be terms suppositing for the same thing even though it is false; for example, if someone says “The proposition that I am uttering is false.” The reason is that, although from its form that proposition means that what the terms supposit for are the same, and they are, in addition, according to the signification of the predicate, it means that they are not the same. For to call any [p. 26] proposition false is to mean that what [the terms supposit] for are not the same. So that proposition means that they are the same and not the same, and so, although things are as it signifies, they are not altogether how it signifies, and so it is false.5 In the same way it is not necessary that every negative proposition whose terms do not supposit for the same is true, a counterinstance being one that denies itself to be true—for example, if I say “The proposition that I utter is not true.” The sophisms that are called “insoluble” are grounded on this point, but I will say no more about them at present. But I believe that every affirmative is true whose terms supposit in a way proportionate to its form, provided that it does not say of itself, formally or explicitly, or as a consequence or implicitly, that it is false. And the same holds mutatis mutandis for negatives.

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This [concludes] what is set down in the fifth chapter. If someone does not concede it, or what was set down earlier, or will be set down later, I do not care, for one must concede them or something similar, and once conceded, the same things follow about consequences, which will be inferred subsequently.

Chapter 6: The Ampliation of Terms Now, in the sixth chapter, let us take it that there is a difference between subject and predicate in propositions in that the predicate always follows the tense of the verb in suppositing for a time. For example, if I say, “A human will be white,” this would be false assuming that all present humans were white, provided that in the future none of them were white. And similarly of the past. It is the same if the time the verb [refers to] is determined [in some other way]. If one says “A human will be white on Sunday,” this would be false even if all humans are now white and will be in the future except on Sunday. But if no human is white and on Sunday a white human were born, “A human will be white on Sunday” would be true. And similarly mutatis mutandis for the past and the present. But the subject of a proposition is not restricted in this way to the tense of the verb, but always retains its supposition for present things, and moreover, if the verb is of some other tense, it is ampliated along with present things to those of the tense of the verb. Then it follows according to Aristotle that a past- or future-tense proposition has two causes of truth, e.g., “One laboring was healthy”; for this can be true either because someone who is laboring was healthy or because someone who was laboring was healthy. And “B will be A” can be true either because that which is B will be A or because that which will be B will be A. Thus I say that this verb “is” in the present tense as the copula in a proposition makes both the subject and the predicate strictly supposit for present things, unless the predicate works differently, about which I will speak later. But these verbs “will be” or “was,” and other similarly tensed verbs, determine the predicate to its time and ampliate the subject to its time along with the present time. This may be {what is meant} by [the adage] “The predicate appellates its form,” or something of that {sort}.6 Thus, because possibility is about the future and all that is possible, the verb “can be” similarly ampliates the supposition of the subject to everything

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that can be. So “B can be A” has two senses or two causes of truth, namely, because that which is B can be A or because that which can be but is not B can be A. The same holds for “cannot be,” because terms supposit for the same in contradictories. Consequently, one must say the same of “necessarily must be” and “necessarily must not be,” since they are equivalent to impossibilities, and similarly of “not necessarily” and “not necessarily not,” since they are equivalent to possibilities. But once again, most people, perhaps everyone, treats the following propositions as true: “Colin’s horse is dead,” “My grandfather, or the Pope, or the King, is dead”; similarly, “Many humans are to be begotten,” “Many are able to be begotten” and even if no rose now exists, “A rose is thinkable,” and “That which is not can be recalled,” and “Many [p. 28] are the possible things that never are, will be or have been.” Because this way of speaking is perfectly idiomatic, it seems to me that such future- and past-tense predicates ampliate their subjects like past- and future-tense verbs. Similarly, some predicates connote possibility, like the verb “can”; e.g., “Whatever can be is possible,” “Whatever can be begotten is begettable.” But subjects do not ampliate predicates; rather, they modify them according to the peculiarities of the words that follow. Hence, “Something dead is a human” is not true, although “Some human is dead” is true. Again, it should be noted that some verbs, whatever their tense, render indifferent to time the accusatives that follow them and that they govern, such as the verbs “know,” “think,” and “understand,” in that something can be understood without any reference to time. For example, the difference between the expressions “I strike a horse,” “I set fire to a house,” “I boil water,” and suchlike on one hand, and these on the other, “I think of a rose,” “I hope for health,” “I desire a good wine,” and suchlike, whereas uses of the first type of verb listed apply to things of the present without a mediating concept, uses of verbs of the second type listed apply to things that are not concepts by means of mediating concepts that may be indifferent to the present, the past and the future. For example, if I think of a human, it certainly follows that I have that concept present to me, but it does not follow that the human of whom I am thinking is present, because that concept can be of absent things, either past or future. Such verbs ampliate their accusatives to past and future, and indeed also to possibles; for God can consider things that neither are, were, nor will be, but are only possibles. Similarly, verbs designating an act of desire or of promising or of obligation by means

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of a cognition ampliate in this way, for example, “I promise or owe you a horse,” or “I remember the past,” “I hope for the future,” and so on. In a similar way, passive verbs ampliate subjects, such as, “A rose is thought [p. 29] of,” “Wine is desired,” “A horse is promised to you,” and so on. Participles or verbal nouns, when used as predicates, also ampliate subjects just like the verbs mentioned. So it is true to say “A rose is thinkable” even if there is no rose, “A horse that must be generated has been promised to you,” or “is promissable,” “Roses are promised to you,” or speaking simply, “That which does not yet exist is promissable.” From this it is clear that such propositions with ampliative verbs and predicates have more senses or more causes of truth, if the subject is used without distribution, than propositions with nonampliative verbs and predicates. For “B is A,” if the predicate is not ampliative, has only the cause of truth that that which is B is A; but “B will be A” has as causes of truth either that that which is B is A or that that which will be B will be A. “A human is dead” also has as causes of truth either that he who is a human is dead or that he who was a human is dead. Now, “A human is white” has as cause of truth only the first type described or something like it. But “A is thinkable” or “A is thought of” has as causes of truth that that which is A, or that which was A, or that which will be A or even that which can be A is thinkable. From which it is also clear that an ampliated term is related to one not ampliated as the more general to the less general; for in whatever way the ampliated one supposits, as determined by the form of the proposition, [it supposits] for more supposita than the nonampliated one. To remove any doubt, it should be noted that when I say, “A human is dead,” it is not necessary that the predicate supposit for present things, but rather for past things, in that it applies to past time; but it is necessary, when the verb is in the present tense, that it supposits for those which in respect of the present are past. What was said earlier should also be understood in this way, namely, that the verb determines the predicate to its time. For if the predicate is in the present tense or without tense, it supposits for things of the tense of the verb. Thus, if I say “Animals were living in Noah’s ark,” this term “living,” since it is in the present tense, supposits for things [p. 30] which were then present. But if I say, “At the time of Aristotle Averroes was yet to be born,” this term “yet to be born,” since it is in the future tense, supposits for those which at that time were future, although now they are past. Thus it is clear

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that future-tense participles can supposit for what are neither present nor future, but in fact past and perished, but relative to the tense of the verb they were future. And similarly the other way round, if I say, “The Antichrist will be dead.” This [concludes] what is set down in the sixth chapter.

Chapter 7: The Matter and Form of Propositions Now, because we have spoken of the form of a proposition and of the division of consequences into formal and material, it is necessary in the seventh chapter to set down what we say pertains to the form of a consequence and of a proposition and what to the matter. I say that when we speak of matter and form, by the matter of a proposition or consequence we mean the purely categorematic terms, namely, the subject and predicate, setting aside the syncategoremes attached to them by which they are conjoined or denied or distributed or given a certain kind of supposition; we say all the rest pertains to the form. So we say that the copulas of both simple subject-predicate and compound propositions pertain to the form, as do negations, [other] signs, the number of propositions and terms and the mutual relation of all these, and relations of anaphoric terms and modes of signifying pertaining to the quantity of the proposition, for example, whether discrete or general, and many other things that the attentive reader can recognize if they occur. For example, in what has been said, mutually diverse forms result from the copulas of modals distinguishing them both between themselves and from assertorics; affirmatives are of different forms from negatives on account of the negations and [other] signs and universals from particulars; a singular proposition is of a different form from an indefinite on account of the terms’ being discrete rather than general [p. 31]; and “A human is a human” and “A human is an ass” are of different forms on account of the number of terms, as too in these consequences, or compound propositions, “Every human is running, so some human is running” and “Every human is running so some ass is walking.” Similarly, “Every human is an animal” and “An animal is every human” are of different form on account of the order [of the terms], and similarly, these consequences, “Every B is A, so some B is A” and “Every B is A, so some A is B.” Again, the conjunctive “A human is running and a human is not running” is of a different form from

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“A human is running and he or she is not running” on account of the anaphor, on account of which the second, by its very form, is impossible but the first is not. The attentive reader will recognize when other matters occur that pertain to the form, such as whether the terms are oblique or not; for “A human’s is an ass” and “A human is an ass” are not of the same form, and similarly in other cases. This [concludes] what is set down in the seventh chapter.

Chapter 8: Conclusions Now that these things have been set down, I add these principles: in every contradiction one of the contradictories must be true and the other false and it is impossible for them both to be true or false at the same time; again, every proposition is true or false and it is impossible for the same proposition to be true and false at the same time. From these postulates we can infer the following: First Conclusion: From every impossible proposition any other follows and every necessary proposition follows from any other. [p. 32] This conclusion is immediate from the meaning of “antecedent” and “consequent.” For it is impossible for an impossible proposition to be true, or for things to be altogether as it signifies. So it is impossible for things to be altogether as it signifies without them being altogether as another signifies. Similarly, it is impossible for things not to be altogether as a necessary {proposition} signifies. So it is impossible for things not to be altogether as it signifies when they are altogether as another signifies. It should be noted that this conclusion has to be adapted to the case of as-of-now consequences, so that from every false proposition any other follows in an as-of-now consequence, and every truth follows from any other [proposition] in an as-of-now consequence. For it is impossible, things being as they are as of now, that a proposition that is true not be true. So it is not possible for it not to be true when any other is true. If the expression is in the past or future tense, then it may be called a consequence as-of-then, or however you wish to call it. For example, this follows in an as-of-now, or as-of-then, or as-of-now-for-then consequence: “If the Antichrist will not be begotten, Aristotle never was.” Because, although it is simply true that the Antichrist can fail to be going to be, it is impossible things being as they are going to be that they will be such that

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he will not be; for he will be and it is impossible that he will be and will not be. Second Conclusion: From every proposition any other follows whose contradictory cannot obtain at the same time as it, and from no proposition follows any other whose contradictory can obtain at the same time as it. I understand by a proposition’s obtaining the same as that it be true or the same as that things be altogether as it signifies, and by obtaining at the same time I understand the same as that things be at the same time altogether as they signify. Therefore let us suppose it is impossible that proposition A obtain at the same time as proposition B. Then I say that the contradictory of B follows from A, [p. 33] that is, not-B. Proof: because either A cannot obtain, so it is impossible, and then anything follows from it. Or A can obtain, and then it is necessary that if A obtains either B or not-B obtains, because one or other part of a contradiction must always obtain. But it is impossible that if A obtains B obtains, by hypothesis. So it is necessary that if A obtains, not-B obtains, and consequently, it is impossible that if A obtains not-B does not obtain. So from A not-B follows. Now to prove the second part of the conclusion stated. For if A and not-B obtained at the same time, we would have A obtaining at the same time as not-B obtains. But B and not-B cannot obtain at the same time. So it is possible that A obtains when B does not obtain. So B does not follow from A. Third Conclusion: In every good consequence, the contradictory of the antecedent must follow from the contradictory of the consequent, and every proposition formed as a consequence is a good consequence if the contradictory of the antecedent of the said consequence follows from the contradictory of the consequent of the said consequence. The second part of this conclusion is usually put in this way: every consequence is good in which the opposite of the antecedent follows from the opposite of the consequent. But I have not put it this way because it would literally speaking be a petitio principii. For that formulation already says that there is a consequence, a consequent and an antecedent, and so a good consequence. The first part is proved by supposing that B follows from A. Then we say that not-A follows from not-B. Because either it does, or else it is possible for A to obtain at the same time as not-B, by what went before. But it is

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necessary that if A obtains so does B. So B and not-B would obtain together, which is impossible. The second part of the conclusion is clear for the same reason. Fourth Conclusion: In any good consequence whatever follows from the consequent follows from the antecedent, and the consequent follows from whatever the antecedent [p. 34] follows from, and similarly, put in the negative, whatever does not follow from the antecedent does not follow from the consequent, and the antecedent does not follow from whatever the consequent does not follow from. This conclusion has four parts. The first part is clear. Because if B follows from A and C from B, then it is impossible for A to obtain without B. And necessarily if B obtains, it obtains together with C. So it is impossible for A to obtain without C, and this means it follows. The second part of the conclusion is similarly clear. Because if B follows from A, and is antecedent to C, I say that it is necessary for C to follow from A. This holds by the first part of the conclusion, namely, that B follows from A and C from B; so C from A. The third part of the conclusion is also obvious. Because let us suppose that B follows from A and C does not follow from A. Then I say that C does not follow from B. Because not-C can obtain along with A, by the second conclusion; so also with B, because if A obtains B obtains of necessity. So it follows, by the second conclusion, that C did not follow from B. The fourth part is also clear. Supposing that B is consequent on A and that B does not follow from C, then I say that A does not follow from C. Because not-B obtains along with C, by the second conclusion, and not-A follows from not-B, by the third conclusion. So not-A obtains along with C. So A does not follow from C. Fifth Conclusion: It is impossible for what is false to follow from truths or what is impossible from the possible or what is not necessary from what is necessary. Proof: because if both parts of a consequence, or either of them, that is, the antecedent or the consequent, were not formed, there would not be a consequence. But if both are formed as a consequence it is impossible for things to be as the antecedent signifies without their being as the consequent signifies; this is clear by the definition. Therefore, it is then impossible for the antecedent to be true without the consequent being true. So it is impossible for the one to be true with the other false.

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The same is clear concerning possibility. Because if the antecedent is possible then [p. 35] it is possible for things to be as it signifies, and on that assumption it is necessary for things to be as the consequent signifies. So the consequent is possible, not impossible. The same is clear concerning necessity. Because it is impossible for the antecedent to obtain without the consequent. So if it is always necessary for the antecedent to obtain it will always be necessary for the consequent to obtain, and this means the consequent is necessary. From this conclusion, by an equivalence, we can infer that when the antecedent and consequent are formed together, if the antecedent is true the consequent is true, and if possible then possible, and if necessary then necessary. Conversely, if the consequent is false the antecedent is false, if the consequent is impossible the antecedent is impossible, and if the consequent it not necessary the antecedent is not necessary. But this notwithstanding, it can happen that the antecedent can be true while the consequent cannot be true, for example, “Every proposition is affirmative” can be true but not “No proposition is negative,” even though the second follows from the first. So also as a corollary it follows that it is not at all the same for a proposition to be possible and for a proposition to be able to be true, or also for a proposition to be impossible and for a proposition not to be able to be true. Whereas “No proposition is negative” is possible, since it follows from the possible, nevertheless it cannot be true. Moreover, this notwithstanding, it is possible for the true and necessary to follow from the false or impossible, for it was said earlier that anything follows from the impossible. But there may be a sophistical objection to this fifth conclusion. For this is a good syllogistic consequence: “Everything running is a horse; every human is running; so every human is a horse”; but each of the premises is possible while the conclusion is impossible. Solution: neither of those premises constitutes the whole antecedent to the conclusion as stated. Rather, the antecedent is a conjunction composed from the two premises, that is, “Everything running is a horse and every human is running”; and this conjunction is impossible, just like the conclusion. Sixth Conclusion: [If] from any proposition with some necessity or necessities adjoined to it there follows some conclusion, the same conclusion follows from the same proposition alone without the adjoining of that necessity or those necessities.

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Proof: because suppose A is a proposition from which with many necessities adjoined to it the conclusion B follows; I say that B follows from A. Because if B follows from those necessities without A, then it is necessary, so it follows from anything. But if B does not follow from those necessities without A, then either A is impossible, and then anything follow from it, or A is possible. Then either it is impossible if A obtains for B not to obtain, and then B follows from A, or it is possible if A obtains for B not to obtain. If an opponent suppose this, then, since A cannot obtain without all the necessities obtaining at the same time, it follows that it is possible if A obtains with all the necessities for B not to obtain. Then B does not follow from A along with those necessities adjoined to it, which is contrary to the hypothesis. In the same way I would say that [if] from any proposition with many truths or some truth adjoined there follows some conclusion, the same conclusion follows from the same proposition alone in an as-of-now consequence, but not in a simple consequence. This is proved in a similar way to the main conclusion. Seventh Conclusion: From every conjunctive proposition consisting of two mutual contradictories any other [proposition] follows in a formal consequence. From this it immediately follows that anything follows from such [a proposition], indeed also from anything implying a contradiction, because any such [proposition] is impossible. But it is necessary to see in what way this is a formal consequence. [p. 37] So I say that from this conjunction, “Every B is A and some B is not A,” anything follows and in the same way in whatever terms this consequence is formed. Proof: for example, this follows: every B is A and some B is not A, so a stick stands in the corner. For from “Every B is A and some B is not A” it follows that every B is A, since from a conjunction each of its conjuncts follows. Then from “Every B is A” it follows that either every B is A or a stick stands in the corner, since anything implies itself in disjunction with anything else. Then from this and the second part of the original antecedent let me argue like this: every B is A or a stick stands in the corner; and some B is not A; so a stick stands in the corner. This is a case of disjunctive syllogism7—that from a disjunction, if either [disjunct] is denied, the other may be inferred. So, from first to last, from the original antecedent the stated conclusion follows, because whatever follows from the consequent follows from the antecedent.

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In accordance with this deduction, some have believed that this is a formal consequence: “Only a father exists, so a stick stands in the corner.” But this is not true, because its form is not valid in other terms, e.g., “Only God exists, so an ass stands in a stable.” Nor is what they said true. For they said that these are formal consequences: “Only a father exists, so a son exists,” and again, “A son exists, so not only a father exists.” Neither of these is formal, because there are counterinstances in other terms: for it does not follow, “Only God exists, so an ass exists,” nor does this follow, “A being exists, so not only one exists.” Again, from this conclusion we can see how every consequence from an impossible antecedent may be reduced to a formal consequence by the addition of some necessity. For if the antecedent is impossible, its contradictory is necessary, and by adding it we obtain anything by a formal consequence, as was said. So we see clearly what some marvel at, namely, how this is a consequence: [p. 38] “There is nothing, so an ass is running.” I explain it {like this}: “There is something” is necessary, and adding it to the premise the conclusion follows. Similarly with the consequence, “An ass is running, so an ass is moving.” For how does it follow? Since “Everything running is moving,” or at least, “If something is running, everything running is moving,” is necessary, by adding it to the antecedent the consequence is made explicit and formal.8 This also shows how every as-of-now consequence from a false antecedent can be reduced to a formal consequence by the addition of a truth, namely, the contradictory of the antecedent. But in this regard, some will be unclear how every consequence whose consequent is necessary may be reduced to a formal one—for example, “An ass is running, so God is just.” I maintain that this consequence may be reduced to a consequence with an impossible antecedent by the third conclusion. For this follows, “No God is just, so no ass is running,” and so it follows [that the first consequence is formal too]. Eighth Conclusion: All propositions having an equal number and the same causes of truth are equivalent, and all having more follow from those having fewer and the same as some of that greater number, but not conversely. This is often stated by saying that it is not a good consequence from more causes of truth to one, but it is a good consequence from one to more. This conclusion, according to what was said earlier, should be understood not only about how things actually are but also about what is possible or

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not inconsistent. If this proceeds from the form of the proposition, the consequences will be formal; if not, but from the signification of the terms, as in, “A human is running, so one capable of laughter is running,” then they will not be formal consequences. Hence properly speaking, the first part of the conclusion [p. 39] should be put like this: “Any two propositions of which neither can have any cause of truth that is not a cause of truth of the other,” indeed clarifying the subsequent part: “Any two propositions of which neither can have any cause of things’ being altogether as it signifies that is not {a cause of things’ being altogether as the other signifies},” in the sense explained earlier. For it was said that whatever manner of speaking we used we meant the sense explained earlier, and we will not repeat this again. This conclusion is proved by means of the first conclusion. For it is selfevident that for any two propositions, it is impossible for things to be altogether as the one signifies without their being altogether as the other signifies if the cause of things’ being as the one signifies is all the same as and none other than how the other signifies, that is, there is no cause of things’ being as the one signifies {that is not a cause of their being} as the other signifies. Similarly, the second part of the conclusion is manifest, because if there can be no cause of things’ being as A signifies that is not a cause of things’ being as B signifies, although the converse may be possible, it is clear that A cannot be true without B, or things be as A signifies unless they are as B signifies. The third part is also clear, for if there is a cause of the truth of A that is not a cause of the truth of B, then A can be true without B; and so {B} does not follow {from A}. From this conclusion all the [subsequent] equivalences and conversions follow. But it is necessary to list them, and they are listed in the following conclusions. Ninth Conclusion: All propositions having the same quality and the same terms suppositing in the same way and with the same copula are equivalent. The reason is that their causes of truth are equal in number and the same, since there can only be a difference in the cause of truth of one proposition [p. 40] relative to another if they differ in their terms, or the terms supposit differently or their copulas are different, or the quality of the propositions is different, which in turn can be reduced to the copula if one says that an external negation is a matter of the copula.

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This is further confirmed because if this conclusion did not hold, it would seem to be by a transposition [of the terms], which cannot be maintained. For an affirmative signifies this to be the same as that, either for some, if it is a particular proposition, or for any suppositum if it is a universal proposition. Now it is necessary that if this is the same as that, then that is the same as this. So conversion is not prevented. Similarly, a negative signifies that this is not to be the same as that, and it is necessary that if this is not the same as that, then that is not the same as this. All formal equivalences of simple subject-predicate assertoric propositions are included under this conclusion, and also all equivalences of those of necessity to others of necessity, of those of possibility to others of possibility, and also of other modalities except equivalences of those of contingency by conversion into the opposite quality. For according to the said conclusion, “Socrates is not running” and “It is not that Socrates is running” follow from each other as equivalents, taking the negations externally; similarly, “Every human is not running” and “No human is running,” by the first rule of equivalents;9 also, “It is not that no human is running” and “Some human is running,” “Not every human is running” and “Some human is not running,” and “It is not that some human is running” and “No human is running,” by the second rule of equivalents; also “Not every human is not running” and “Some human is running,” and “It is not that no human is not running” and “Some human is not running,” by the third rule of equivalents.10 Similarly, “Of any human no ass is running” and “Of no human is an ass running” are equivalent; and similarly, “Of any contradiction one part is true” and “Of no contradiction is neither part true.” The same may be maintained for [p. 41] all others. For example, concerning modals, “Every B is necessarily A” and “No B is necessarily not A” are equivalent; similarly, “No B is possibly A” and “Every B is not possibly A,” and so on for the others. Again, simple conversions are included under this conclusion. But I will present [separate] conclusions about them to make their veracity more explicit. Tenth Conclusion: From every proposition containing a distributed term there follows in a formal consequence a proposition with the same term not distributed, the rest remaining the same, but never conversely. The reason derives from the eighth conclusion and from what was set down {in} the second {chapter}. For a proposition with an undistributed

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term has more causes of truth than {that} with the term distributed, other things being equal. Consequences by subalternation and conversions of universals into particulars hold by reason of this conclusion; so too it follows: “Of no human is an ass running, so of every human some ass is not running,” and similarly for others. Eleventh Conclusion: From no proposition with a term suppositing merely confusedly on account of the distribution of a preceding term does there follow a proposition with that term suppositing determinately with the distribution not removed, but it does follow with the distribution removed. This conclusion is demonstrated by the same [reasoning] as the previous one. So it does not follow, “Every B is A, so A is every B,” but it does follow, “so A is B,” for the conditions are then satisfied. Similarly, it does not follow, “Any human’s ass is running, so an ass of any human is running,” but it does follow, “so an ass of some human is running.” Twelfth Conclusion: None of the [following] consequences is formal: “B was A, so that which is B was A,” or “B will be A, so that which is B will be A,” or even “B is A, so that which is B is A,” but the converses are formal consequences; yet if the propositions were universal affirmatives, none of them would be formal consequences; again, none of these consequences is formal: {nothing that is B was A, so no B was A,” nor “Nothing that is B will be A, so no B will be A,” nor} “Nothing that is B is A, so no B is A,” but the converses are formal; but if the propositions were particular negatives, none would be formal consequences. This conclusion has six parts. The first is clear because this construction, “B is A,” permits the ampliation of the subject if the predicate is ampliative, for example, “A human is dead.” But the [other] construction, “That which is B is A,” does not permit the ampliation of the subject, namely, of “B”; for [B] is contracted and restricted to the present by the verb “is” in the present tense, which precedes it. Thus if the predicate is ampliative, “B is A” has more causes of truth than “That which is B is A,” and from many to fewer is not a good consequence. The second part is clear for the same [reason]. Note first that, in it, and in the fifth part, by “converse” we do not mean that the terms are transposed, namely, the subject and predicate, but transposition of [one] construction [into the other], “B is A” into “That which is B is A.” Then the second part

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is proved, because there are no causes of the truth of “That which is B is A” that are not causes of that of “B is A,” although there are counterinstances in divine terms. For although “That which is the Father is the Son” is granted concerning God, “The Father is the Son” is not granted; but this is unique to the Holy Trinity, in that it is at the same time a Trinity and an Indivisible Unity, which pagan philosophers did not generally consider. My own position is that, since names are a matter of [p. 43] convention, I will call consequences “formal” that have counterinstances only in these terms, whereas properly speaking they should not be called “formal.”11 The third part of the conclusion is clear from this, that each [of those] propositions has a cause of truth that the others do not have. Even if the term [A] is ampliative, “Everything which is B is A” is true if it holds only for everything {which exists}, whereas “Every B is A” would not have this cause of truth. For example, as soon as Noah left the Ark, it was true to say “Everyone who is a human was in Noah’s Ark,” but it was not true to say “Every human was in Noah’s Ark,” because it would follow that everyone who was a human was in Noah’s Ark. So more is distributed in “Every A is B,” if the predicate is ampliative, than in “Everything that is A is B,” and from the distribution of fewer the distribution of more does not follow, for example, “Every human . . . so every animal . . .” Similarly and conversely, “Every B is A” can be true even if there is no B, for example, “Every rose is thinkable” even if there is no rose, and in that circumstance “Everything that is B is A” would not be true. The other three parts of the present conclusion are proved by combining the first three parts of the conclusion with the third conclusion. Note that a proposition with the subject ampliated by the predicate should be analyzed by a disjunctive subject combining the present tense with the tense or tenses appropriate to the ampliation, for example, “B will be A” as “That which is or will be B will be A,” and “A human is dead” as “The one who is or was a human is dead,” and “The Antichrist can be a man” as “He who is or can be the Antichrist can be a man,” and “A rose is thought of” as “That which is or was or will be or can be a rose is thought of.” It is also clear that for the same reasons there is no formal consequence from “is” used predicatively to “is” used existentially. For this does not follow, “The Pope is dead, so the Pope is,” or “The Pope is creating, so the Pope is,” or “A rose is thinkable, so a rose is”; but it does follow, “The Pope is dead, so the Pope is or was,” and for others in their own way. [p. 44]

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Thirteenth Conclusion: For any propositions without an ampliative predicate, these consequences are good, for instance, “B is A, so that which is B is A,” and conversely; similarly, “Every B is A, so everything that is B is A,” and conversely; and similarly for negatives. For no term supposits for something different or in a different way in one than in the other and there is the same copula. Similarly, in these cases there is a valid consequence from “is” used predicatively to “is” used existentially. Fourteenth Conclusion: From every universal or particular assertoric affirmative there follows a particular affirmative by conversion of the terms; and from every universal negative there follow a universal and a particular negative; and from no particular negative does any other follow in virtue of form. This conclusion should be understood in respect of negatives only for negatives in normal form. Now I call it normal form where an external negation precedes the predicate, for example, “No B is A,” “Some B is not A.” But I call the form nonnormal where the predicate precedes the negation and so is not distributed by it, for example, “Every B A is not,” “Some B A is not.” For the present conclusion does not hold of nonnormal form. For from “Every sun a planet is not,” which is true, no universal follows by conversion, since the predicate is not distributed in the antecedent, while in the consequent, where it would become the subject, it would be distributed; whence “Every planet the sun is not” is false. But from a particular negative in this nonnormal form there always follows another particular negative, since it follows, “If B A is not then A B is not,” by the ninth conclusion, since the terms remain the same and the supposition of the terms is the same. Therefore, the conclusion is stated for propositions [p. 45] in normal form, where by “conversion of terms” we understand the transposition of terms, namely, that the subject becomes the predicate and the predicate the subject. On this assumption, the proof is still long and difficult, on account of the manifold diversity of forms of assertoric propositions, whose distinct forms we will now describe. Hence, I say that a simple subject-predicate proposition should have a subject, a predicate and a copula. All three are sometimes implicit in one word, for example, if I say “Thunder,” “Rain,” “Hail,” or also if I say, with an impersonal verb, “[There] is running,” “[There] is reading,” “[There] is speaking.” Now sometimes the subject is explicit but the predicate and copula are implicit in the same word, for example, “A human runs,” “A

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human grows,” “A human exists”; also sometimes the predicate is explicit but the subject and copula are together implicit, for example, “Happen a human run,” that is, “It happens that a human is running,” “Properly you study,” that is, “It is proper that you study,” “Humans sees Socrates,”12 that is, “Humans seeing is Socrates.” But sometimes the three are explicit—for example, “A human is white.” In the last type, conversion is straightforward; so we will begin with them. We say that the copula is either in the present tense, or in the past tense, or in the future. If it is in the present tense, then either both terms are ampliative to another time, or neither is ampliative, or the predicate alone is ampliative, or only the subject. If neither is ampliative, then there is clearly conversion of a universal negative and particular affirmative into ones similar to themselves, because both have the same terms, suppositing in the same way and with the same copula, and so follow from each other by the ninth conclusion. If from a particular affirmative a particular affirmative follows, it also follows from the universal, because whatever follows from the consequent follows from the antecedent. But from a universal affirmative there does not follow a universal because the predicate of the first is not distributed in the first but it is distributed in the second. Similarly, it is clear that a particular negative is not converted, because its subject is not distributed, but in the other, when it becomes the predicate, it is distributed. Now if the copula is in the present tense and either the predicate or the subject is ampliative, [p. 46] or both, then either the proposition to be converted is formed in this way, “B is A,” “B is not A,” and similarly for universals, or in this way, “That which is B is A,” “That which is B is not A,” and similarly for universals. Then if the term both in the one to be converted and in the result is formed with “that which is,” then it works the same way as if the terms were not ampliative, for in this construction any ampliation of the subject by the predicate is prevented. So it follows that these conversions with “that which is” are formal; for they hold both in ampliative terms and in non-ampliatives. Now if the propositions to be converted are formed without “that which is” and the predicate is ampliative, it is necessary to convert affirmatives by disjoining the subject of the converted proposition in the present tense with the tense or tenses appropriate to the ampliation, for example, “A human is dead, so the dead one is or was a human,” “A rose is thinkable, so a thinkable

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is, will be or was or can be a rose,” and so on. Otherwise there would be a process from more causes of truth to fewer, or the first proposition would have some cause of truth that the second did not have, and so it would not be a {good} consequence. But a universal negative, if the predicate is ampliative, does not require this kind of disjunction, provided that the subject is not ampliative. For there is no change in the supposition of the predicate of the first but only a change in the supposition of the subject of the first from ampliated to nonampliated, and in this way, negatively and distributively, it is a good consequence. For one denying what is in more denies what is in less, and one denying a disjunction denies each part; for example, if nothing that is or was B is A it follows that nothing that is B is A, and also that nothing that was B is A. Now if the propositions in question are formed without “that which is” and the subject is ampliative and the predicate not, then affirmatives are converted without a disjunction and without “that which is.” For one can proceed from what is not ampliated to what is ampliated and not distributed; so from fewer causes of truth to more is a good consequence. But a universal negative must be converted by adding “that is” to the subject of the result to prevent the ampliation, [p. 47] for instance, “Nothing dead is a human; so nothing that is a human is dead.” Otherwise there would be a process from less great distribution to greater distribution, where there is not a good consequence. But if each term is ampliative, for example, “Destroying is begetting,” it is necessary, as before, to convert affirmatives by a disjunction and negatives by “that which is.” Now if the copula is in the past or future tense, then the proposition is either formed with “that which is” or without “that which is.” If with “that which is,” then the conversion of all is made by retaining the same expressions, for instance, “That which is white was black, so that which was black is white”; similarly, “That which was white will be black, so that which will be black was white”; similarly, “Nothing that was white is black, so nothing that is black was white”; and so on. For the terms continue to supposit for the same things in the same way. The conversions can be proved by expository syllogisms. But if the propositions in question do not use “that which is,” for example, “A white thing was black” or “Nothing white was black,” then it is best to add this expression “that which” together with the verb before the subject of the result of conversion and to place the copula “is” and

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a disjunction of the appropriate sort in the predicate of the result, for example, “A white thing was black, so that which was black is or was white,” “Nothing white was black, so nothing that was black is or was white.” For in this way the terms continue to supposit for the same things in the same way. Now if the subject is explicit and the predicate and copula are implicit in the same word, then if possible the predicate should be separated from the copula, and then conversion is made as before—for example, “A human is running,” that is, “A human is a runner”; “A human will run,” that is, “A human will be a runner.” But sometimes, perhaps because of lack of words in grammatical usage or perhaps from matters of meaning, it is not possible to make this separation while keeping the same meaning, for instance, “A human comes to be,” “A human is loved.”13 For because with these verbs there is no present-tense participle, it is not possible to make a suitable analysis. For if I say “A human is having-been-made,” “A human is having-beenloved,”14 it is doubtful that these are equivalent to the originals, because those giving counterinstances would say that in the first moment of love a human [p. 48] is loved but has not yet been loved, and that which comes to be is not, as Aristotle observed,15 so is not having-come-to-be. So then the proposition should be converted by adding to the subject the {pro}noun “that which” with that word and the copula “is,” and if the predicate is ampliative one should also put a disjunction in the predicate of the resulting proposition, as was said before, for example, “A human is loved, so that which is loved is a human,” “A rose is thought of, so that which is thought of is, was or will be or can be a rose.” And because these verbs “comes to be,” “is created,” “begins” mean that a thing is when it was not earlier, or will be when it is not now, and these verbs “ceases,” “is destroyed” and suchlike mean the converse, namely, that a thing is and later it will not be, or was when it is not now, therefore for these verbs the conversions are made like this: “A comes to be, so that which comes to be is or will be A,” and “A is destroyed, so that which is destroyed is or was A.” If there is a predicate after the verb “comes to be,” for example, “A white thing comes to be black” or “B comes to be A,” it is safest to convert [like this], “so that which comes to be black is or was white,” and “that which comes to be A is or was B,” and similarly for “begins” and “ceases.” But when the verb “is” occurs existentially, my view is that we should analyze “is” into subject-predicate form using the participle “being” in the singular disjoined with “beings” in the plural, if the consequence is to be formal. For God is, and an army is, and God is a being and not beings, and

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an army is beings and not a being. So this does not follow in a formal consequence, “A is, so A is a being,” but it does follow, “so it is a being or beings.” So it is converted [like this], “A is, so a being or beings is or are A”; {and} separating each from the other it is converted into this compound, “so a being is A or beings are A.” But if, as a matter of meaning, it appears that the subject supposits for some single being, it can be converted, “A is, so a being is A”; and if it appears that the subject does not supposit for a single thing but only collectively for many that are not one single thing, it should be converted [like this] “A is, so beings are A,” for example, “A people is, so beings are a people.” As a corollary, I infer that it does not follow in a formal consequence, “A is, so A is a being,” but it does follow [p. 49] that it is a being or beings. Similarly, it does not follow, “A and B are, so A and B are beings.” For if the names are synonymous and supposit for the same simple thing— for example, both names are imposed synonymously to signify Peter’s soul— then A and B are, but they are not beings but a being, just as the Creator and God are, but they are not some things but some thing. Because we have started to talk about collective nouns, it must be realized that it is very problematic how “An army is round” or “An army is square” are to be converted. For it is false to say that a square is an army and that squares are an army. Moreover, grammar, not allowing an adjective to be subject unless it made into a substantive in the neuter gender, does not permit us to say, “A round is an army.” Therefore, I say that since this proposition, “An army is round,” is true only in the sense that an army is a round army, not because it is something round or some round things, therefore in order to convert it, it is necessary to add to the predicate of what is to be converted the substantive “army”; and then the conversion is clearly like this: “so a round army is an army.” If, however, the predicate is explicit and the subject and copula are implicit in the verb, then it is possible to make explicit what the verb has as its subject; for example, if it was said, “Properly you act well,” that is, “It is proper that you act well,” “Happen a human run,” that is, “It happens that a human is running,” “Human sees Socrates,” that is, “Human seeing is Socrates.”16 If there is no participle available or in use, a noun may be invented, and what is needed should be added, and what needs analysis should be analyzed, until we have subject, predicate and copula explicitly: for example, “Concerning the king to do well,” that is, “It is of concern to the king to do well,” “It is disgusted my soul with my life,” that is, “Disgusting to my soul is my life”

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or “something about my life,” “It grieves me to have been a human,” that is, “That which grieves me is to have been a human.” For the present we are not concerned whether propositions of this sort are true properly speaking or not, but only in what way they should be analyzed so that subject, predicate, and copula occur explicitly. Concerning propositions in which the whole thing is implicit in one word I say, as before, that they should be analyzed in such a way that the whole thing is made [p. 50] explicit; e.g., “Rain,” that is, “Rain falls from the sky,” “Snow,” that is, “Snow falls from the sky,” “Thunder,” that is, “There is thunder” or “A sound is made in the sky,” or however you wish to analyze these words; similarly, “[There] is running,” that is, “Someone is running,” “[There] is reading,” that is, “Someone reads” or perhaps “Some man or woman reads or some men or women read.” Also, it is necessary in some conversions to ensure that no grammatical error is made. For example, it does not follow, “No phoenix is [he] a female, so no female is a phoenix,” because in the first proposition “phoenix” is restricted to male phoenixes and not in the second. Similarly, it does not follow, “No human is [he] a woman, so no woman is human.” Instead, so that the term supposits for the same in the antecedent and the consequent, some restriction of some kind to the gender in which it is used in the antecedent should be added to the consequent, by saying, “so no female is a male phoenix” and “no woman is a male human.” Again, it is necessary to be attentive to propositions with oblique terms, about which there are different views. For if only an oblique term precedes the copula, many say that it alone is the subject of the proposition, for example, “Him sees Socrates.”17 Even if a nominative accompanies and precedes the oblique {term} that itself precedes the verb, they say that only the oblique {term} is the subject and all the rest is the predicate. They say resolution should be made so that the oblique {term} is made into a nominative and the copula “is” added, and a relative {pronoun} “that which” added to the predicate in the same case as the oblique {term}; for example, “A horse riding is a man,” that is, “A horse is that riding which is a man,” “Any man’s ass is running,” that is, “Any man is that of whom an ass is running,” and so on, and then it is clear how the conversion should be made. But I say that without doubt in such an analysis the consequence is good, provided, however, that there is no ampliation of the subject, because then it would be necessary to add a disjunction to the analysis. For it does not

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follow, “Him saw Noah,18 so he is [he] whom saw Noah,” but [p. 51] this does follow, “so he is or was [he] whom saw Noah”; and it does not follow, “The Antichrist’s preachings are in the future, so the Antichrist is,” but this does follow, “so he is or will be.” But I do not agree that the oblique {term} is the subject of the proposition; rather, I believe that the nominative, whether explicit or understood, should always be taken as the subject, either on its own or with the determiners occurring in the proposition. So I hold that in “Any human’s ass is running,” the whole [phrase] “human’s ass” is the subject. It is converted to “so one running is a human’s ass”; for it is necessary to remove the distribution so that the supposition is changed from confused to determinate. “Of no contradiction is each part true” is converted into “No truth is each part of some contradiction.” “A human a horse is seeing” is converted into “Someone a horse seeing is a human”; for I hold that in the first proposition this expression “horse” belongs to the predicate, even though it precedes the copula; so too if I say “Every horse a human is seeing,” I believe that “horse” belongs to the predicate in as much as it determines it; so it is converted into “so someone seeing a horse is a human.” But whether this opinion is right or the earlier one is not a matter for the theory of consequences to resolve. For it suffices that in each construction there are good consequences, which has been shown. Finally, we should be aware that if certain special syncategorematic expressions occur in propositions so that to express their sense the propositions need several exponents—for example, exceptive, exclusive, and reduplicative propositions—then they should better be called compound than subjectpredicate and should first be analyzed into their exponents and only then, in accord with their exponents, should they be converted. For example, “Every human besides Socrates is running” is analyzed as “Socrates is not running and every {human} other than Socrates is running,” then converted [p. 52] into “so someone running is not Socrates and someone running is a human other than Socrates.” But only one exponent of “Every animal some human excepted is running” can be converted, since a particular negative is not convertible. So too, “Only an animal is a human” is analyzed as “An animal is a human and nothing other than an animal is a human,” then converted into “so a human is an animal and no human is other than an animal”; and because this conjunction is equivalent to this universal affirmative “Every human is an animal,” it is said that therefore the first proposition is also converted into a universal affirmative. Perhaps “For a shield to be white

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whiteness is required” should be analyzed as [saying] that a shield can be white and it cannot be white without whiteness. And so on. This [concludes] what is said about conversions of propositions formed from general terms. Fifteenth Conclusion: From every singular proposition there follows another proposition by conversion of the terms, for example, “Socrates is running, so someone running is Socrates,” “Plato is not running, so no one running is Plato.” For the supposition of the terms remains the same. But we should be aware of ampliations, as was said. For it does not follow, “The Antichrist is to be begotten, so that which is to be begotten is the Antichrist,” but it does follow, “that which is to be begotten is or will be the Antichrist.” If the singular [term] consists of a general term with a demonstrative adjective added to it, then it can be converted simply, namely into another singular qualified in the same way, for instance, “This human is running, so this runner is a human”; similarly, “This human is not running, so this runner is not a human.” [p. 53] Note incidentally that from every such singular [proposition] there follows a particular or an indefinite except in the case where in a negative there would not be anything demonstrated for which the general term might supposit. For if [a piece of] wood is pointed to, it does not follow, “This human is not running, so a human is not running.” Sixteenth Conclusion: Conversion by contraposition is not a formal consequence but is valid on the assumption that all the terms supposit for something. Conversion by contraposition is not strictly conversion, since the terms do not remain the same but are changed from finite to infinite. A conversion of this sort of a universal affirmative fails where a finite predicate supposits for every being, because then the infinite term opposite to it would supposit for nothing, so it would not be possible to form a true affirmative with it. For example, it does not follow, “Every human is a being, so every nonbeing is a nonhuman,” because the first is true and the second false. Similarly, conversion of a particular negative fails where a finite subject supposits for nothing, because then the infinite term opposite to it supposits for everything, so it cannot be truly denied of any term that supposits for something. For example, it does not follow, “A chimera is not a human; so a nonhuman is not a nonchimera,” because the first is true and second false;

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for the contradictory of the second is true, namely, “Every nonhuman is a nonchimera.” This proves the first part of the conclusion. The second part is shown by supposing that [we are dealing with] a negation acting infinitely, that is, that it prevents a term suppositing for what [the term] used finitely would supposit for and makes it supposit for everything else if there is anything else. So it is equivalent to say, “A nonhuman {is a nonhuman” or “A nonanimal is a nonanimal” and “A nonhuman} is other than a human” or “A nonanimal is other than an animal,” though one [p. 54] could prevent the equivalence just stated on account of a failure in the supposition of the predicate, in that in Metaphysics X19 it is said that “every being is either the same as or diverse from every being,” while “being is not the same as or diverse from nonbeing, but nonidentical;” so a human, while a nonchimera, is not other than a chimera. Then on these assumptions it will be clear that the opposite of the antecedent follows from the opposite of the consequent. We start with the consequence, “Every B is A, so every non-A is non-B,” that is, “so everything other than A is other than B.” The opposite of the consequent is “Something other than A is not other than B,” from which it follows that that which is other than A is the same as B, in that “every being is the same as or other than every being,” and we are assuming that both A and non-A are beings and also both B and non-B. Then it follows, “Another than A is the same as B, so a B is other than A,” whence, “so a B is not A,” as will be seen in the following conclusion, and this contradicts the original antecedent. The conversion of a particular negative may be proved similarly. For we start with the consequence, “Some B is not A, so something other than A is not other than B.” Then the contradictory of the consequent is “Everything other than A is other than B,” whence, “so nothing other than A is B,” from which, “so no B is other than A,” whence, “so every B is the same as A,” in that “every being is the same as or other than every being;” and so it follows that every B is A, which contradicts the original [proposition]. Seventeenth Conclusion: From every affirmative there follows a negative by changing the predicate from finite to infinite, other things remaining the same, but from a negative to an affirmative is not a formal consequence, but it is a consequence on the assumption that all the terms supposit for something. The first part of the conclusion is verified because a finite and an infinite term do not supposit for the same; so of whatever either of them is truly affirmed it is necessary that the other is denied. [p. 55] The consequence is

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clearly proved since the opposite of the antecedent follows from the opposite of the consequent. For example, take the consequence, “Every B is A, so no B is non-A,” or “so no B is other than A.” The opposite of the consequent is “Some B is other than A,” from which it follows that that B is not the same as A, since nothing is both the same as and different from the same thing; so some B is not A, and this contradicts the original [proposition]. The consequence, “Every B is non-A, so no B is A,” is proved in the same way. For the opposite of the consequent is “Some B is A”; so some B is not other than A, and this contradicts the original [proposition], which said, “Every B is other than A.” The second part of the conclusion is clear, since the consequence fails when the subject does not supposit for anything. For it does not follow, “A chimera is not a human, so a chimera is a nonhuman.” The third part is clear. Because if B exists it is necessary that it is either A or other than A; therefore if it is not A, it is other than A, and if it is not other than A, it is A. This is the end of Book I, which was about consequences in general and more particularly about formal consequences between one assertoric subjectpredicate [proposition] and another.

Book II Consequences between Modal Propositions Chapter 1: Modal Propositions In this second book, [p. 56] we will treat a particular kind of consequences, namely, between modal propositions. We set down some assumptions about them. For first we take it that although many [different] modes occur in modal propositions, the main ones of which Aristotle and other masters have treated are modals of possibility and impossibility, of necessity and contingency, and of truth and falsity. So I intend only to treat of these. But it should be noted that propositions are not said to be of necessity or of possibility in that they are possible or necessary, but from the fact that the modes “possible” or “necessary” occur in them, and the same for other modes. So a proposition can indeed be necessary when it is not one of necessity and possible when it is not one of possibility, and conversely. For example, “A human is an animal” is necessary according to Aristotle, but it is purely assertoric, and “A human of necessity runs” is [a proposition] of necessity but it is not necessary, but false and impossible, and “It is possible that a human is an ass” is one of possibility, but it is impossible.

Chapter 2: The Division of Modal Propositions into Composite and Divided Now, in the second chapter, we must acknowledge that modal propositions of this sort are commonly of two types. For some are called “composite” and others “divided.” [p. 57] They are called composite when a mode is the subject and a dictum is the predicate, or vice versa. I call the terms “possible,” “necessary,” “contingent” and the like “modes.” I call a “dictum” that whole occurring in the proposition in addition to the mode and copula and negations and signs or other 95

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determinations of the mode or the copula. For example, I call the following composite: “That a human runs is possible,” “It is necessary that a human is an animal.”1 The subject of the first of these is “that a human runs” and the predicate is “possible”; the subject of the second is “necessary,” and the predicate is “that a human is an animal.” They are called “divided” when part of the dictum is the subject and the other part the predicate. The mode attaches to the copula as a determination of it. For example, “A human can run” or “A human is possibly running”; similarly, “A human is of necessity a runner” or “A human is necessarily running,” and the like. In these, the subject is “human” and the predicate is “runner.” The copula is the whole phrase “can be” or “is possibly” or “is of necessity” or “is necessarily.” For this proposition “A human is possibly running” should be analyzed into “A human is possibly a runner,” just as “A human is running” is [analyzed] into “A human is a runner.”

Chapter 3: The Division of Divided Modal Propositions into Affirmative and Negative Now, in the third chapter, we should note that some divided [modal] propositions are simply affirmative, when no negation occurs in them, for example, “A human is possibly white” and “God is necessarily just.” Others are negative, and they are of two sorts. In some the negation occurs in the mode, in that it precedes it, for example, “A human is not possibly an ass” and “No human is possibly an ass.” In others the negation does not occur in the mode but follows it, for example, “A human is possibly not white” and “God [p. 58] is necessarily not wicked.” Some are in doubt whether these last should properly speaking be called affirmative or negative. But whatever they say, I believe they should be called negative, both because the proposition “B is possibly not A” is equivalent to “B is not necessarily A,” which is clearly negative, and because an affirmative proposition is not true if any term supposits for nothing, but “A chimera is necessarily not an ass” is true, and consequently so is “A chimera is possibly not an ass.” There are others in which the negation occurs twice, once [attached] to the mode, the other to the predicate, for example, “B is not possibly not A” or “B is not necessarily not A.” I hold that these are really affirmatives, since they are equivalent to ones that are clearly affirmative. For “B cannot not be A” is equivalent to “B is necessarily A,” which is plainly affirmative, and

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“No B is necessarily not A” is equivalent to “Every B can be A,” which is plainly affirmative. And thus it is seen that some divided modal propositions, both affirmative and negative, have a negated mode, and others a mode not negated.

Chapter 4: The Ampliation of the Terms of Divided Modal Propositions Now, in the fourth chapter, it should be realized that a divided proposition of possibility has a subject ampliated by the mode following it to supposit not only for things that exist but also for what can exist even if they do not. Accordingly, it is true that air can be made from water, although this may not be true of any air that exists. So the proposition “B can be A” is equivalent to “That which is or can be B can be A.” Some2 say that it is equivalent to a compound disjunction, namely, to “That which is B can be A or that which can be B can [p. 59] be A.” But I do not accept this, because saying this and saying what I said earlier are very different. For on the former reading, “A creating God can fail3 to be God” is false, since its contradictory is true in any situation. For nothing that is or can be a creating God can fail to be God, since all and only God is or can be a creating God and he cannot fail to be God. But if one takes the second reading, one must grant “A creating God can fail to be God” on the hypothesis that God is not now creating, because it would be equivalent to this disjunction, “He who is a creating God can fail to be God or he who can be a creating God can fail to be God.” This disjunction must be granted because its first disjunct is true under the hypothesis, since a creating God is nothing and that which is nothing can fail to be God, that is, a creating God is not God and that which is not God can fail to be God. Note that these two interpretations agree in that on each interpretation the proposition “B can fail to be A” can have three causes of truth. The first is that something is B and it can fail to be A. The second is that something can be B, although it is not, and it can fail to be A. The third cause on the second interpretation is that nothing is B that can be A, although it can be [B]; and this is not sufficient on the first interpretation, but it is required that nothing is B nor can be that can be A. Similarly, I shall say the same of those of the past or of the future. For the proposition “B will be A” is equivalent to “That which is or will be B

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will be A” and not equivalent to the disjunction “That which is B will be A or that which will be B will be A.” The interpretations differ in that if God is not now creating and will create tomorrow, “A creator will fail to be God” is false, but it would be true on the other interpretation. But the reason why I maintain the first interpretation and not the second is because if I say “Every B can be A,” there is just one subject and just one predicate and a single simply predicative proposition, and the subject is distributed all at once by a single distribution. So it seems better to analyze it by a single predicative proposition with a single subject [p. 60] and a single predicate, even though the ampliation of the subject makes the subject of the analysans a disjunction of the verb “can be” with the verb “is.” Accordingly, the way of analyzing universals and particulars will be the same, namely, by always disjoining the verb “can be” to the verb “is.” But on the other interpretation it would be necessary to analyze particulars by disjunctions and universals by conjunctions. I maintain that “Everything creating was God” is true although nothing is now creating, and this would be false on the other interpretation. However, I do not think that either interpretation can be disproved, for the reason that names and utterances are conventionally significative expressions. But I will follow the first interpretation in what follows.

Chapter 5: Equivalences Now, in the fifth chapter, I take it as Aristotle did and others do too, namely, that “necessarily” and “impossibly not” are equivalent, and “necessarily not” and “impossibly” are also equivalent. For in itself it seems clear that of everything that necessarily is, it is impossible that it not be, and conversely, of everything that necessarily is not, it is impossible that it be. I also take “impossibly” and “not possibly” to be equivalent, because a negation is implicit in the term “impossibly.” So “B is not possibly A” and “B is impossibly A” are equivalent, and similarly, “B cannot be A” and “B is not possibly A,” because “can be” and “is possibly” mean the same. Similarly, “Every B is impossibly A” and “Every B is not possibly A” are equivalent, and “No B is possibly A” and “No B can be A.” Too, I take it that a universal affirmative contradicts a particular negative, and a universal negative a particular affirmative in the same way, so that in the negative the negation governs the mode. For example, [p. 61] “Every B

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is possibly A” contradicts “Some B is not possibly A”; similarly, “No B can be A” [contradicts] “Some B can be A.” And similarly for other modes.

Chapter 6: Conclusions about Divided Modals Further, in the sixth chapter, it will be supposed that modal propositions can be expressed without any restriction of the subject by “that which is” or “that which was,” and the like, for example, “Every B can be A,” “Every C is necessarily D,” and so on. But sometimes they can also be expressed by adding “that which is” or “that which was” or “that which can be” and suchlike to the subject. For example, “That which is B can be A,” and then “B” is restricted to suppositing only for those which are B. Similarly, “That which was B is necessarily A,” and then B is restricted to suppositing only for those which were B, and similarly for the future. Similarly, we can say “That which can be B is necessarily A,” or “That which is necessarily B is contingently A,” and so on. And these are self-explanatory. Now on these assumptions, we will first speak of divided modals and later of composite ones, and first of those in which the subject is taken without the addition of “that which is” or “that which can be,” and so on. First Conclusion: From any proposition of possibility, there follows as an equivalent another of necessity and from any of necessity another of possibility, such that if a negation was attached either to the mode or to the dictum or to both in the one it is not attached to it in the other and if it was not attached in the one it is attached in the other, other things remaining the same. For example, “B is necessarily A” is equivalent to “B is not [p. 62] possibly not A.” This is clear from what we agreed in the fifth chapter, since “not possibly” and “impossibly” come to the same. And for the same reason “Every B is necessarily A” is equivalent to “Every B is not possibly not A,” and consequently also to “No B is possibly not A.” Then it follows that their contradictories are mutually equivalent, namely, “No B is necessarily A” and “Every B is possibly not A”; and similarly, “Some B is not necessarily A” and “Some B is possibly not A.” Similarly, “B is necessarily not A” and “B is not possibly A” are equivalent, since as I said, “not possibly” and “impossibly” come to the same. For the same reason, “Every B is necessarily not A” is equivalent to “Every B is not

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possibly A,” and also to “No B is possibly A.” So similarly their contradictories are equivalent, namely, “No B is necessarily not A” and “Every B is possibly A”; and similarly, “Some B is not necessarily not A” and “Some B is possibly A.” From what has been said it should be inferred as a corollary that if something was established for [propositions] of possibility having an affirmed mode it will be established for those of necessity having a negated mode, since these are equivalent other things remaining the same, as was shown. Similarly, if something was established for those of necessity having an affirmed mode it will be established for those of possibility having a negated mode, since these are also equivalent, as was said. So we will only establish [conclusions] for those of possibility and of necessity that have an affirmed mode; when we speak of [propositions] of possibility or of necessity, we will always understand [those] having an affirmed mode. This should always be borne in mind in the following conclusions. Again, we will also establish nothing for those of impossibility, since they are equivalent to those of possibility that have a negated mode; for [p. 63] “not possibly” and “impossibly” come to the same, and so those of impossibility are equivalent to those of necessity. Second Conclusion: In every divided proposition of necessity the subject is ampliated to supposit for those that can be. This conclusion seems clear. For otherwise those of necessity would not be equivalent to those of possibility having a negated mode, since in those of possibility the subject is clearly granted to be so ampliated. So the proposition “B is necessarily A” is analyzed as “That which is or can be B is necessarily A” and “Every B is necessarily A” is analyzed as “Everything that is or can be B is necessarily A,” and similarly for negatives. This is clearly shown if this kind of ampliation is granted for those of possibility. For “Some B can fail to be A” contradicts “No B can fail to be A,” which is equivalent to “Every B is necessarily A.” This is also true of the results of analysis, since “That which is or can be B can fail to be A” contradicts “Nothing that is or can be B can fail to be A,” which is equivalent to “Everything that is or can be B is necessarily A.” From this one can infer as a corollary that “Something creating is of necessity God” and “Something creating is necessarily God” are true, even on the hypothesis that God is not now creating. Similarly, the universal [claim], namely, that everything creating is of necessity God, is true, because

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it is equivalent to “Everything that is or can be creating is of necessity God,” and this is simply true and necessary. But if such propositions with an ampliated subject were analyzed by compound disjunctive or conjunctive propositions, namely, particulars by disjunctions and universals by conjunctions, as some suggest,4 “Everything creating is necessarily God” would be false and “Something creating necessarily fails to be God” would be true on the said hypothesis. But one may object that that which is not is not necessarily God, [p. 64] and there is not anything creating, by hypothesis; so something creating is not necessarily God. Again, everything is which is of necessity God; but there is not anything creating, so something creating is not of necessity God. These objections are not sound. For “Something creating can be God” is granted on the hypothesis given, even if it were analyzed by a compound disjunction. So “Nothing creating can be God” should be denied, since it contradicts it. Then what is equivalent to it should be simply denied, namely, “Everything creating necessarily fails to be God.” A similar [argument shows that] “Everything creating is possibly not God” should be denied. Therefore, the contradictory of this last should be granted, namely, “Something creating is of necessity God” or “Something creating is necessarily God”; for I claim these are essentially the same proposition. Thus the objections are solved by saying that everything that is necessarily God, or indeed anything that is possibly God, is and always was and always will be. So since that which is creating is necessarily God, that which is creating always necessarily is, and he always was, is and will be, but was not always creating nor always will be creating nor is now creating on the given hypothesis. So the premises of the said objections are granted and the syllogistic form denied, which will be clear in what follows. Third Conclusion: From no proposition of necessity does there follow an assertoric or vice versa, except that from a universal negative of necessity a universal negative assertoric follows. I speak always of those with an affirmed mode. Also I mean if the subject is not restricted by “that which is”; and I say this because “That which is B is necessarily A, so B is A” does follow. First, it is clear that from an assertoric there does not follow one of necessity, since an assertoric can describe a contingency of which that of necessity is not true. For example, “Every, or some, human necessarily runs” does not

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follow [p. 65] from “Every human runs”; similarly, “Every, or some, human necessarily fails to run” does not follow from “No human runs.” Nor does it follow: “Everything creating is necessarily God, so the one creating is God,” since the first is true in a situation in which God is not now creating and the second is false. The cause of falsity of the consequent lies in inferring one part from a disjunct term. Similarly, it does not follow: “Some planet shining on our hemisphere is necessarily not the sun, so some planet shining on our hemisphere is not the sun,” because in a situation where of the planets only the sun is now shining on our hemisphere, the first proposition would be true, as is clear by its analysis, and the second would be false. The reason is that the subject in the [proposition] of necessity supposits for other things besides the sun, because of the ampliation, whereas in the assertoric it does not supposit for them. But it is clear that “Every B is necessarily not A, so no B is A” is a good consequence. For if B in both the first and the second supposits for something, then in the first, B is distributed for everything for which it is distributed in the second, and perhaps also for more besides; so if the first is true of everything, the second will also be true of everything. But if B does not supposit for anything in either, then the cause of truth will be the same for both; so neither will be true without the other. But if B supposits for something in the first but for nothing in the second, then the second will be true, whether the first is true or false. So no situation can be given in which the first is true while the second is false. Fourth Conclusion: From no proposition of possibility does there follow an assertoric or vice versa, except that from every affirmative assertoric proposition there follows an affirmative particular of possibility. I speak always of affirmed modes. The conclusion is proved by the fact that affirmative particulars of possibility contradict universal negatives of necessity, and universal affirmatives particular negatives, and similarly universal negatives [p. 66] of possibility affirmative particulars of necessity and particular negatives universal affirmatives. But between assertorics and those of necessity, except universal negatives of necessity, there are no consequences; so neither will there be between their contradictories, which are all the assertorics and all those of possibility, except a particular affirmative of possibility. The consequence holds by the third conclusion of Book I. But it is clear that from every assertoric affirmative there will follow a particular affirmative of possibility, since in the previous conclusion the

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contradictory of the antecedent followed from the contradictory of the consequent. This fourth conclusion can also be shown separately, because it clearly follows if some B is A that it can be A. It is also clear that “Every white thing can be black, so a white thing is black” does not follow, nor does “Everything running can fail to be running, so everything running is not running” follow. Nor does “Something creating is not God, so something creating can fail to be God” follow, because if God is not now creating, the first is true and the second false. What went before can be confirmed by this conclusion, since they mutually convert by the third conclusion of Book I. It should also be realized that from every proposition of necessity there follows a proposition of possibility, and not conversely, because what is necessarily some way is possibly that way, and what is necessarily not some way is possibly not that way. But since this is self-evident, I have not called it a conclusion. Fifth Conclusion: From every affirmative of possibility there follows by conversion of the terms a particular affirmative of possibility, but not a universal, and from no negative of possibility does there follow by conversion of the terms another of possibility. The first part is clear by an expository syllogism. For if B can be A, then designate such a B as C. Then this C is or can be B and the same thing can be A; so that which can be A is or can be B. From this it follows that it can be B, since from being B it follows [p. 67] that it can be B, by the previous conclusion. Then “That which can be A can be B, so A can be B” follows. This last consequence is clear because A supposits as broadly in “A can be B” as in “That which can be A can be B”; so they are equivalent. Then it is immediately clear that from a universal affirmative a particular affirmative also follows by conversion. For a particular follows from the universal, by subalternation, and from a particular there follows a particular, by conversion; so it follows from the universal, because what follows from the consequent follows from the antecedent. But that a universal does not follow from a universal is proved as in the case of assertorics. The second part of the conclusion is proved because “Every God can fail to be creating, so something creating can fail to be God” does not follow, because the first is true and the second false. The reason is some sort of restriction that the verb “to be” imposes on the predicate that it does not impose on the subject. So it affects the predicate, not only the verb, to say “B can be A” and “B can have been, or be about to be A.”

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Sixth Conclusion: From no proposition of necessity does there follow by conversion of the terms another of necessity, except that from a universal negative there follows a universal negative. The first part is clear, because “Everything creating is necessarily God, so God is necessarily creating” does not follow, since the first is true and the second false. Similarly, in the case of a particular negative, “Some animal necessarily fails to be an ass, so some ass necessarily fails to be an animal” does not follow. But the second part of the conclusion is proved, because the contradictory of the antecedent follows from the contradictory of the consequent. For “Some A can be B, so some B can be A” follows by the previous conclusion. But it should be realized that every affirmative of necessity can be converted by use of “that which” in the result of the conversion. For example, “Something creating is necessarily God, so that which is necessarily God is or can be creating” does follow. [p. 68] It should also be realized that affirmatives of necessity can be converted into an affirmative of possibility. For example, “B is necessarily A, so A can be B” follows. The reason is that from a [proposition] of necessity that of possibility follows. For example, “B is necessarily A, so B can be A” follows; from there, by conversion, “so A can be B”; so, from first to last, and so on. Seventh Conclusion: Every proposition of each-way contingency having an affirmed mode is converted into [one of] the opposite quality with an affirmed mode, but none is converted if the result of conversion or what was converted had a negated mode. The first part is clear by definition, for it is called contingent when it can either be or not be. So “Every B is contingently A, so every B contingently fails to be A” follows, and conversely. Similarly, “Some B is contingently A, so some B contingently fails to be A” follows, and conversely. But the second part is also clear because “It is contingently, so it is not contingently” does not follow, indeed, they are opposed. For “Every B is contingently A” and “No B is contingently A” are contraries. So it is clear that from every proposition of contingency having an affirmed mode there follow both an affirmative and a negative proposition of possibility. So it is well said that contingency excludes necessity and impossibility. Whence from every proposition of necessity having an affirmed mode there follows a proposition of contingency having a negated mode.

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Eighth Conclusion: No proposition of contingency can convert in terms into another of contingency, but any having an affirmed mode can be converted into another of possibility. [p. 69] The first part can be seen because “God is contingently creating, so something creating is contingently God” does not follow, for the first is true and the second false, since everything creating is necessarily God. If the affirmative is not converted, then also the negative is not converted, because they are equivalent or follow from one another. Similarly, with a negated mode, “Nothing creating is contingently God, so no God is contingently creating” does not follow, because the first is true and the second false. But the second part of the conclusion is evident from the fact that from any [proposition] of contingency having an affirmed mode there follows an affirmative of possibility, which is converted into another of possibility. So from first to last, and so [the conclusion follows].

Chapter 7: Conclusions about Composite Modals Having clarified these matters, we now turn to composite modals. About these, it must first be said, that, as some do say, they can be universal or particular, or indefinite or singular. For example, “Every possibility is that B is A” is universal and “It is a possibility that B is A” is indefinite, and similarly one can speak of “some possibility” or “this possibility.” Here “possibility” is taken not for what can be but for a possible proposition, which is said to be possible in so far as things can be altogether as it signifies. So in the examples above, saying “Every possibility is that B is A” is the same as to say “Every possible proposition is that B is A.” If this noun “possibility” were used in any other way, there would be an equivocation, and such a proposition would not be called one of possibility. The same must be said of the other modes. It should also be noted that in the proposition “Every possibility is that B is A,” the predicate “that B is A” supposits materially for the proposition “B is A,” and does not supposit for itself, since the phrase “that B is A” is not a proposition.5 [p. 70] Now if the mode is the predicate and the dictum is the subject, the proposition can still be universal, particular, and so on. So “Everything that is that B is A is a possibility” is universal, and “Something that is that B is A is a possibility” is particular, and “{That which is that B} is A is a possibility”

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is indefinite, and “This that B is A is a possibility” is singular. Or also just as we can say “This proposition ‘B is A’ is possible,” so we can say indefinitely, “A proposition ‘B is A’ is possible,” universally, “Every proposition ‘B is A’ is possible,” and particularly, “Some proposition ‘B is A’ is possible.” But if it is asked what the quantity of the proposition “Every B being A is a possibility” is,6 I would say that if the whole [phrase] “every B being A” is the subject suppositing materially for the proposition “Every B is A,” then it is indefinite. But if we said that the subject was “B being A” suppositing for the proposition “B is A” and the sign “every” is taken significatively, then the proposition would be universal, and it would mean “Every proposition ‘B is A’ is possible.” Similarly, if I say “No B being A is a possibility,” if the whole [phrase] “no B being A” is the subject, then the proposition is indefinite and affirmative; but if the subject is only “B being A” and the expression “no” is used significatively, then it is a universal negative. Then there follow some conclusions. Ninth Conclusion: In all composite modals in which the dictum is subject, from a particular there follows a universal, the rest being unchanged. For example, this follows: “Some proposition ‘B is A’ is possible, so every proposition ‘B is A’ is possible,” and similarly for truth and falsity, contingency and necessity. The reason is that among all the propositions “B is A,” each signifies whatever the others signify and altogether as the others signify. So if things are as one signifies, they are as any other signifies, and if not, not; so if [p. 71] one is true the other is true, and if false, false.7 Similarly for possibility, necessity and other modes. What I say concerning particular to universal, so I say concerning singular to universal. For this proposition, namely, “The proposition ‘B is A’ is possible” cannot be true unless “Every proposition ‘B is A’ is possible” is true. But this also holds for negatives, because if some proposition “B is A” is not true there follows “therefore no proposition ‘B is A’ is true.” For the causes of truth of the proposition “Some proposition ‘B is A’ is not true” are either that there is no proposition “B is A,” and then the subject supposits for nothing, and so the negative proposition is true; or that there is a proposition “B is A” but it is not true. But from the first of these causes it follows not only that some such is not true, but also that no such is true, and we have what was declared. Similarly, from the second cause of truth it follows

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that no such is true, since every cause of the truth of the one is a cause of the truth of any, if there are many; so what was claimed always follows. Tenth Conclusion: Any composite modal in which the dictum is subject is simply converted in [its] terms except a universal affirmative, which is only converted accidentally.8 First, a particular affirmative is converted into a particular affirmative, for example, “Some proposition ‘B is A’ is possible, so something possible, that is, some possible proposition, is the proposition ‘B is A.’ ” This is proved just as in the assertoric case, and can be proved by an expository syllogism. Having proved this, it is clear that a universal is also converted into a particular, for whatever follows from the consequent follows from the antecedent. But the universal is not converted simply, for it does not follow, if every proposition “B is A” is possible, that every possible proposition is the proposition “B is A.” Also a {universal} negative is converted simply, for [p. 72] the same reason as assertorics. For there is the same supposition of terms in both and there is no ampliation. A particular negative is converted into a particular negative. For the first is false only if the subject supposits for something, as is the second; and if the subject supposits for something it is equivalent to a universal, which itself is converted; so the first cannot be true without the second. Singulars are treated just like assertorics. Eleventh Conclusion: Every composite modal in which the mode is subject is converted simply, except a particular negative, which is not converted. This conclusion is evident for all cases as it was for assertorics, except that the case of a universal affirmative is converted simply because evidently it is converted into a particular and from that particular the universal follows. Twelfth Conclusion: Every affirmative composite modal of truth, of possibility, and of necessity is converted as regards the dictum in whatever way the dictum would itself convert. I call “is converted as regards the dictum” the mutual transposition of the terms of the dictum and not the transposition of the overall subject and the overall predicate. For example, “It is possible that some human is running, so it is possible that some runner is a human”; similarly, “It is necessary

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that every God is just, so it is necessary that some just person is God”; similarly, “That no human is an ass is true, so that no ass is a human is true.” These conversions hold by the fifth conclusion of Book I, namely, that if the antecedent is true, the consequent is true, and if [p. 73] possible, possible, and if necessary, necessary, supposing that they are formed together. Then it should be noted that these conversions are simply good consequences only if an existence postulate about the terms9 is added. For example, it is possible that the proposition “B is A” is true and that “A is B” is not true because it is not formed. Nor is the objection of some people valid, namely, that [since] a consequence cannot be formed unless each [of its parts] is formed, so if the consequence were formed the existence of the terms [would follow] without being added as a postulate. First, I say that a consequence [may] be formed without the formation of those propositions for which the terms supposit, as in saying “It is possible for B to be A, so it is possible for A to be B”; for “B to be A” is not a proposition, although it supposits for a proposition. Second, I say that it does not suffice for a simply good consequence that it is impossible for things to be as the antecedent signifies unless they are as is signified by the consequent when the consequence is formed. Rather, it is simply required that it is impossible for things to be as is signified by the antecedent unless they are as is signified by the consequent. So it does not follow, “Some proposition is affirmative, so some proposition is particular,” which is clear because from the contradictory of the consequent the contradictory of the antecedent does not follow. Thirteenth Conclusion: Every particular dictum is converted into a universal in an affirmative composite modal proposition of falsity or of impossibility, but a universal is not converted into a particular; and a universal negative dictum and a particular affirmative convert simply, but a universal affirmative is not converted. All these hold because if the consequent is false or impossible it is necessary that the antecedent is false or impossible, and the converse is not necessary because the true can easily follow from the false and the possible from the impossible. Add to [the thirteenth] conclusion, as before, the existence postulate about the terms, namely, that the propositions were formed.10 [p. 74] Fourteenth Conclusion: Every dictum in an affirmative composite proposition of contingency is converted according to the opposite quality into the contradictory dictum, not into the contrary. For the sense of the proposition “It is contingent (I mean, each way) that B is A” is that “B is A” can be true and can be false. So it is also necessary

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for its contradictory to be of the same kind, since contradictories are not at the same time true or at the same time false. But this is not so for contraries. For although “Every intelligent being is God” can be true or false, nonetheless, “No intelligent being is God” cannot be true. So it does not follow, “It is contingent that every intelligent being is God, so it is contingent that no intelligent being is God.” Fifteenth Conclusion: From every affirmative composite proposition of truth its dictum follows and conversely; and from every such [proposition] of necessity its dictum follows and propositions of truth and of possibility follow, but not conversely; and from every such [proposition] of truth that of possibility follows but not conversely; and from every proposition a proposition of possibility of which it is the dictum follows, but not conversely; and from every such [proposition] of contingency one of possibility follows but not conversely. I have put all of these forward together because they are evident, but perhaps they should really be put forward as postulates rather than conclusions. For every necessary proposition is possible and true, and not conversely, and also every true or contingent [proposition] is possible and not conversely. It indeed follows, “B is A, so it is possible that B is A,” and not conversely; also it follows, “B is A, so it is true that B is A,” and conversely, at least given an existence postulate about the terms.11 This exhibition of examples suffices for me for this conclusion. [p. 75] Sixteenth Conclusion: If an affirmative composite proposition of necessity is true, whatever follows from its dictum is necessary and similarly whatever follows from [the dictum of one] of possibility is possible, and whatever from one of truth is true; but this is not the case for divided [propositions] except those of truth. The first part of this conclusion follows by the rule, “If the antecedent is true, the consequent is true, if possible then possible, and if necessary then necessary.” The second part follows because although “Everything creating is necessarily God” is true, nonetheless, it need not be necessary that it follows from “Something creating is God,” for it might be false. Similarly, although “A white thing can be black” is true, nonetheless, from “A white thing is black” there follows an impossibility. Note that the common rule “Supposing an assertoric to be possible nothing impossible follows” must be understood in accordance with this conclusion. Now there is a doubt whether it is permissible that every divided

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proposition of possibility warrants an assertoric, that is to say that the doubt is whether and in what way there corresponds to every true divided proposition of possibility a possible assertoric proposition. Then I say first that it is not necessary that if a divided universal of possibility is true, the [corresponding] universal assertoric is possible. For whereas every star existing in the zodiac is possibly shining on our hemisphere, nonetheless, “Every star existing in the zodiac is shining on our hemisphere” is by the laws of nature impossible. So if the universal should be formed as an assertoric, it should be formed as an assertoric divisively for its singulars. I also say, second, that it is not necessary that if a divided particular or indefinite [proposition] of possibility is true, an assertoric is possible retaining the same terms. For whereas a white thing can be black, nonetheless, “A white thing is black” is not possible. So I say that if such a one, namely, [p. 76] where the subject is a connotative term, should warrant an assertoric it should warrant [one] with a demonstrative pronoun demonstrating that for which the subject supposited; for example, if a white thing can be black, then it follows that “This is black” is possible, demonstrating that for which “white thing” supposited. Even so, this is not universally true unless sometimes a change is made in the predicate, namely, when the predicate implies opposition to an assertoric present-tense proposition. For example, this human can be white at a future time; it does not follow that “This human is white at a future time” is possible. Similarly, although the matter of water can take the form of air that does not exist, nonetheless, “This matter takes the form of air that does not exist” is not possible. Therefore, it would be necessary to remove from the predicate the phrases “at a future time” and “that does not exist.” Seventeenth Conclusion: From no affirmative composite of possibility does there follow a divided one of possibility with the mode affirmed, or conversely, except that from an affirmative composite with an affirmed dictum there follows a divided particular affirmative. First, the exception is evident, because if “[Some] B is A” is possible, “some B can be A” evidently follows. But although “Everything running is a horse” is possible, it does not follow universally that everything running can be a horse, because an ass may be running, but it cannot be a horse. So only a particular follows. Now it is clear that the converse never follows. For although everyone asleep can be awake, nonetheless “Someone asleep is awake” is not possible;

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similarly, in the negative, although every star in the zodiac shining on our hemisphere can fail to shine on our hemisphere, nonetheless, “A star in the zodiac shining on our hemisphere does not shine on our hemisphere” is not possible. [p. 77] Similarly, conversely, although “Nothing creating is God” is possible, nonetheless, “Something creating is possibly not God” is not true, because everything creating is necessarily God. Eighteenth Conclusion: From no composite affirmative of necessity does there follow a divided one of necessity with an affirmed mode, nor conversely, except that from a divided universal negative there does follow a composite universal with a negated dictum. First, the exception is proved. For from “[Some] B is A” being possible it follows that [some] B can be A, by the previous conclusion. So, by the third [conclusion] of Book I, it follows “No B can be A, so ‘[Some] B is A’ is not possible.” So, by an equivalence, it follows “Every B is necessarily not A, so ‘[Some] B is A’ is not possible,” from which it follows that its contradictory is necessary, for every impossibility has a necessary contradictory. Then it follows that “No B is A” is necessary, and this is what was said. Now the first part of the conclusion is clear. For although according to Aristotle, “Every horse is an animal” is necessary, nonetheless, no horse is necessarily an animal, in that every horse could fail to exist, and in consequence, not be an animal. Similarly and conversely, everything creating is necessarily God, but “Something creating is God” is not necessary. And so also, although it is necessary that no one sleeping is awake, nonetheless, no one sleeping is necessarily awake. Concerning the particular negative, it is also clear that from “Something running necessarily fails to be a horse” it does not follow that [p. 78] “Something running is not a horse” is necessary, because supposing that only a human is running, the first is true and the second false. Nineteenth Conclusion: From no proposition, [whether] assertoric, of possibility or of necessity does there follow one of contingency with both modes affirmed; similarly, from none of contingency does there follow an assertoric or one of necessity, but there does follow one of possibility. The first part is clear from the fact that from no affirmative, [whether] assertoric, of possibility or of necessity does there follow a negative in the same terms, nor from a negative an affirmative, and any [proposition] of contingency contains in itself, formally or implicitly, an affirmative and a negative.

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The second part is clear from the fact that contingency excludes necessity and impossibility. Because the contingent can be and fail to be, it does not follow that it is or follow that it is not. The third part is also clear because the contingent is neither a necessity nor an impossibility. So it is possible and also possibly not, so that it follows “It is contingent, so it is possible that it is,” and “so it is possible that it is not,” and similarly, “It is contingent that it is not, so it is possible that it is” and “so it is possible that it is not.” These are true both for composite and for divided [modal propositions], as the conclusion says. Let everyone take care that they do not take divided modals for composites or vice versa, and do not take divided [modals] with a negated mode for divided [modals] with a negated dictum, since they are very different, as has been shown. This is the end of Book II, which concerned consequences between one simple predicative modal [proposition] and another.

Book III Syllogisms between Assertoric Propositions Part I: Syllogisms between Propositions Containing Direct Terms Chapter 1: The Division of Consequences Next, in this third book [p. 79], we will treat syllogistic consequences, about which we must set down a number of assumptions. First, that there are many kinds of consequences. For some hold only because of their matter, so that they are not formal: for example, enthymemes, inductions, and examples,1 and perhaps many others—for instance, if from an impossibility you infer whatever you please. None of these deserves to be called syllogisms. Now among formal consequences some are from one simple subjectpredicate to one simple subject-predicate [proposition], and then it is necessary for them to contain both terms besides the syncategoremes. Such are equivalences, subalternations and conversions, and consequences by interchange of some term between finite and infinite. These are not inductions nor examples nor syllogisms, as everyone agrees, because it is necessary for a syllogism to have several premises. Neither properly speaking are they enthymemes, since an enthymeme is an imperfect or truncated syllogism. For an enthymeme should be from one premise to a conclusion, such that the conclusion is then understood to follow syllogistically from that premise with another added to it. But, broadly speaking, they can be called enthymemes, if we wish to call all consequences from one simple subject-predicate [proposition] to another enthymemes. But enough has been said about these consequences. Other consequences are formal by virtue of a conjunct [following] from a conjunction or a disjunction [p. 80] from one of the disjuncts. For from every conjunction each of the conjuncts follows and from any proposition there follows every disjunction disjoining it with another. But these consequences are also not syllogistic, because in a syllogism something else should 113

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follow from the premises such that the conclusion does not contain each term occurring in the premise nor vice versa. Other consequences are formal by virtue of a condition, [that is,] by adjoining a proposition to a conditional proposition. One way can be by adjoining its antecedent to a conditional and inferring its consequent, another way by adjoining the contradictory of its consequent to the conditional and inferring the contradictory of the antecedent, and a third way by adjoining a conditional to a conditional such that the antecedent of the one is the consequent of the other and inferring another conditional in virtue of the principle: whatever follows from the consequent follows from the antecedent. An example of the first is that if it is A it is B, and it is A, so it is B; an example of the second is that if it is A it is B, but it is not B, so it is not A; an example of the third is, if it is A it is B, and if it is B it is C, so if it is A it is C. But I do not count these consequences as properly syllogistical, at least among those that I intend principally to treat, because I want to treat of simple syllogisms, which are simple consequences that do not contain a consequence or consequences in their premises. Again, other consequences are also formal on account of the formal impossibility of the premise or the formal necessity of the conclusion. For since from an impossibility anything follows and what is necessary follows from anything, if a proposition is impossible on account of its form there will be a formal consequence from it to anything, and if it is necessary on account of its form there will be a formal consequence from any other [proposition] to it. Now a conjunction made up of two contradictories or contraries is impossible on account of its form, as is [a proposition] in which some finite term is affirmed of its infinite counterpart or vice versa, and a disjunction made up of contradictories or of subcontraries is formally [p. 81] necessary, as is [a proposition] in which an infinite term is denied of its finite counterpart or vice versa. But I do not count these consequences to be syllogistical, because a syllogism should be specifically to some conclusion or conclusions, not just any, and from a specific premise, not just any indifferently. Again, there are also consequences [that are] formal by analyzing syncategoremes, from analysandum to analysans or vice versa, which I do not intend [to treat] because analysis of this sort is only an explanation of the meaning of the syncategoremes. There are also consequences [that are] formal by division, which Aristotle calls “weak syllogisms,”2 as when one member of an exhaustive division is denied and the other [member] is inferred: for example, “Every A is B or every

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A is C, but [some] A is not B, so every A is C,” or “Every A is B or C, but no A is B, so every A is C,” or “Every A is B or C, and some A is not B, so some A is C.” But again, whether such consequences should be called syllogistic or not, I do not intend to speak of them principally, in that in them the conclusion and one of the premises contain the same terms. But I wanted to list these consequences in this chapter lest they appear to have been overlooked. Yet others could be given, but these seem to me to be the more principal ones besides the syllogistic cases that we intend to speak of principally in this book. So we want to understand by “syllogism” in what follows only a formal consequence to a single subject-predicate conclusion by a middle [term] different from each of the extremes in the conclusion. This [concludes what] is set down in the first chapter. [p. 82]

Chapter 2: The Syllogism Now in the second chapter, we take it that every syllogism links the middle term in the premises with each extreme from the conclusion, so that on account of that linking the linking of the extremes is inferred, either affirmatively or negatively. Then it is clear that every syllogism, as we here intend syllogism, is made up from only three terms, namely, from two extremes that are the terms of the conclusion, and from a middle term with which those extremes are linked in the premises. It is also clear that there will be two premises in a syllogism, such that in one the middle is linked with one extreme and in the other with the other; and thus it is clear that the middle occurs in each premise and not in the conclusion. Now the first premise is called the major premise and the second the minor, and the extreme occurring in the major premise is called the major extreme and the one occurring in the minor is called the minor extreme. Moreover, it follows from this that there are only four figures of this kind of syllogisms. For the relation of the middle to the extremes in the premises as subject and predicate is called the syllogistic figure. This can happen in only four combinations. The first is when the middle is subject in the major premise and predicate in the minor; the second is when the middle is predicate in both; the third is when the middle is subject in both; and the fourth is the converse of the first, namely, when the middle is predicate in the major and subject in the minor. But it should be noted that the fourth figure differs from the first only in the transposition of the premises, and that transposition does not permit

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inferring another conclusion or prevent that inference, but affects whether the conclusion inferred is direct only when in the first figure and indirect in the fourth and vice versa. Now I call the conclusion direct when the major extreme is predicated of the minor, and I call it indirect when the minor extreme is predicated of the major. From this it is clear that once the first figure has been explained [p. 83] it will be superfluous to explain the fourth; so Aristotle does not mention it. This [concludes what] is set down [in] the second [chapter] in Book III.

Chapter 3: The Division of Terms into Finite and Infinite Now in the third chapter, we must lay down that some of the terms from which a syllogism is formed are finite, some infinite. The way of syllogizing with them is the same, provided, however, that the term in the two propositions in which it occurs is not interchanged between finite and infinite. For example, just as it follows, “Every B is A, every C is B, so every C is A,” it also follows, “Every B is non-A, every non-C is B, so every non-C is non-A,” and it also follows, “Every non-B is A and every C is non-B, so every C is A.” But when some term in the two propositions in which it occurs is interchanged between finite and infinite, then there will be other kinds of syllogisms, which will be explained later; for example, if we say “No non-B is A, no C is B, so no C is A.” Similarly, syllogisms are sometimes formed from direct terms and sometimes from oblique. First, those that are formed from direct will be explained, and first also those that are formed from assertoric, then from modal, premises. Moreover, in some syllogisms the middle is a general term, and in some the middle is a discrete term, and the latter should be called expository syllogisms, which we will explain first. It should also be realized that sometimes the subject in the propositions of a syllogism is ampliated by the predicate, and sometimes it is not ampliated, and this makes a great difference to syllogisms. This [concludes what] is set down in the third chapter. [p. 84]

Chapter 4: Conclusions Now in the fourth chapter, we lay down that affirmative syllogisms hold in virtue of the principle, “Whatever are the same as one and the same are the

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same as each other.”3 Hence, from the fact that the extremes are said in the premises to be the same as the one middle it is concluded in the conclusion that they are the same as each other. However, negative syllogisms hold by another principle: “Two things are not the same as each other if one is the same as something and the other is not.”4 Thereby an affirmative conclusion must be concluded from two affirmatives and a negative from an affirmative and a negative, since an affirmative proposition indicates identity and a negative nonidentity. These rules must be explained lest they misunderstood. Concerning the first rule, I say first that if these expressions “whatever” and “the same,” in the plural, are used collectively, the proposition, “Whatever are the same as one and the same are the same as each other,” meaning that this is the same as that, is not true, for matter and form [collectively] are the same as one and the same composite but the one is not the other. Second, I say that no things are divisively the same as one and the same, namely, such that each is the same as it (unless there is a divine counterexample, which I shall talk about later), because there are no things that are not many and mutually diverse, and consequently, no one thing is numerically the same as any of them. But then in what way is the principle valid? I reply that several names are separately said to be the same as one and the same discrete term. I do not say “are the same” but “are said to be the same”—that is, that they are affirmatively predicated of the same discrete term with the addition of the expression “same” and in this way they are truly the subject of it; as when we correctly say, “Socrates is the same as animal and Socrates is the same as human,” or conversely, “Animal is the same as Socrates and human is the same as Socrates,” from which it can be inferred that [some] animal is the same as a human, or that [some] human is an animal. So the rule [p. 85] should be put like this, literally speaking: Whatever terms are truly said separately to be the same as one discrete term, they are truly said to be the same as one another. As I observed, I do not say, “are the same,” but “are said to be the same.” Now if there is no discrete term, then it is necessary that the rule be expressed like this: Whatever terms are truly said separately to be the same as one general term by reason of the same thing for which that general term supposits, they are truly said to be the same as one another. For example, whereas “Socrates” and “Plato” are said separately to be the same as the

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general term “human,” it cannot be concluded that Socrates is the same as Plato, because Socrates and Plato are not said to be the same as human by reason of the same human; rather, Socrates is the same as one human and Plato as another. But if Socrates were the same as [some] human and Plato were the same as the same human, then it would be concluded that Socrates was Plato. In the same way regarding the rule for negatives, I say that one term is truly said not to be the same as another if one is truly said to be the same as some discrete term and the other is not truly said to be the same as it;5 so one can infer one from the other negatively. If the middle is a general term, it is necessary to add that it is for the same thing that one of the extremes is said to be the same and the other not the same. But it should be carefully noted that these rules do not hold in the case of divine terms, which supposit for one most simple thing that is also three. So although the Father is the same as simple God and the Son the same as the same God, nonetheless, the Son is not the Father; and although the same Father is God and not the Son, nonetheless it is false that the Son is not the same as God. But in other cases in which it is impossible that the same simple thing is both three and one, the rules are valid. Aristotle believed the rules to be valid in all cases because a counterexample cannot be given to them by human reason but only by faith.6 Then from these assumptions there follow the conclusions. [p. 86] First Conclusion: No syllogisms are formal in the common and customary way of speaking. I call “the common and customary way of speaking” the way of speaking without the addition of “that which is” or suchlike, as when I say, “Socrates is a human,” “Every human is an animal,” not by saying, “He who is Socrates is a human” or “Everything that is a human is an animal,” or the like. The reason for this conclusion is that terms may be found for which the form is not valid, for the rules given earlier are not valid for them. For example, this is not valid: “This God is the Father and this same God is the Son, so the Father is the Son”; similarly, this is not valid: “This Father is not the Son and this same Father is God, so God is not the Son”; similarly, this is not valid: “Every God is the Divine Father, every Divine Son is God, so every Divine Son is the Divine Father”; similarly, this is not valid: “No Son is the Divine Father, every God is the Son, so no God is the Divine Father.” There can be similar counterexamples to other moods.

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Now, whether according to another way of speaking syllogisms in divine terms are formally valid and what that form is, I leave to the theologians. And it should be noted and always kept in mind that, because it is not for me, an arts man,7 to decide regarding the foregoing beyond what was said, I will for the rest call syllogisms formal to the form of which one cannot give counterexamples other than in divine terms. I do not understand by this that they are simply formal, but only that they do not contain terms relating to the distinctness of the divine persons. Second Conclusion: No syllogism can be validly drawn from two negatives (as I said, I am speaking here of syllogisms in which no terms are interchanged between finite and infinite). The reason for this conclusion is that in such a case neither of the rules by [p. 87] which it was said that syllogisms hold is satisfied. For terms that are said to be the same as each other as much as terms that are said to be different from each other can be truly denied of the same thing. For Brownie is not a stone, Brownie is not a human, Brownie is not capable of laughter, and nonetheless it cannot be affirmatively concluded that a stone is a human, or negatively that a human is not capable of laughter. Third Conclusion: In every good syllogism, from each premise with the contradictory of the conclusion there follows the contradictory of the other premise; and also every syllogism is good where the contradictory of one premise follows from the other premise with the contradictory of the conclusion.8 This conclusion holds by the third conclusion of Book I. To see this, it should be realized that in a syllogism, neither premise [alone] deserves to be called the antecedent, because the conclusion can be false along with the truth of either premise, namely, if the other is false. Rather, the conjunction consisting of the two premises is the whole antecedent. So from the contradictory of the conclusion there follows the contradictory of that conjunction, and that contradictory is a disjunction of the contradictories of the premises. The premise that is taken with the contradictory of the conclusion denies one part of that disjunction. So the other part must be inferred, since a disjunction is not true unless one of its parts is true. Thus is the first part of the conclusion proved. The second part is proved similarly. For if from the contradictory of the conclusion along with one of the premises there follows the opposite of the other premise, then the premises, which are the antecedent, cannot obtain

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along with the contradictory of the conclusion. So the opposite of the antecedent is inferred from the contradictory of the consequent. [p. 88] Fourth Conclusion: Whatever conclusion follows from the premises of a given syllogism, it follows from one of them together with anything antecedent to the other; and whatever conclusion does not follow from those premises does not follow from one of them together with anything consequent to the other. It is from this conclusion that the rule is drawn that if there is no syllogism from two universal [premises] there will not be a syllogism from one universal and a particular, other things remaining the same, and whatever conclusion follows from a universal and a particular also follows from the two universals. The present conclusion follows by the fourth conclusion of Book I. For every conjunction is consequent to a conjunction made up from one of its parts and anything antecedent to the other {part}, and it is antecedent to a conjunction made up from one of its parts and anything consequent to the other part. Fifth Conclusion: In every figure, there is a valid expository syllogism from two affirmatives to an affirmative conclusion, and from one affirmative and a negative to a negative conclusion, whichever premise was affirmative. Each claim holds by the rules stated in the assumptions. These syllogisms are most evident in the third figure—for example, “This C is an A and the same C is a B, so [some] B is A.” For since the C and the A are the same, if a B is the same as the C and not [the same] as the A, then the B will be the same and not the same as the same thing, which is impossible. Similarly, negatively, like this: “This C is not an A and the same C is a B, so [some] B is not A”; for otherwise the same impossibility would follow as before. If these syllogisms in the third figure are granted, it follows that they are valid in the second, because they would turn into the third figure immediately by conversion of the premises.9 Similarly, it follows that they are valid in the first, for they would turn into the third figure by conversion of the minor [premise].10 [p. 89] But it should be noted that in negative syllogisms, if the major extreme is not distributed, a direct conclusion cannot be inferred according to the normal way of speaking, in which the negation precedes the predicate, for the major would be distributed in the conclusion when it was not distributed

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in the premises.11 Similarly, if the minor is not distributed an indirect conclusion cannot be inferred according to the customary way of speaking, for the same reason. But when the major was distributed a direct conclusion can be inferred, and if the minor was distributed an indirect one can be inferred. If neither is distributed, then it is necessary to infer without distribution of the predicate according to a nonnormal way of speaking, for example, “[Some] animal is not Socrates, [some] human is Socrates, so [some] human [some] animal is not.” From this it may be inferred that to conclude negatively and directly and according to the customary way of speaking by an expository syllogism it is necessary that in the first and third figure the major [premise] is negative, and in the second figure it is necessary that it is negative and universal. From now on, everything I say should be understood for syllogisms formed from general terms. And this is how I propose the following conclusions. Sixth Conclusion: No syllogism is valid in which the middle is distributed in neither premise, unless the middle is used in the minor with a relative of identity. For the rules by which syllogisms hold require that if the middle is a general [term] the extremes are linked by reason of the same thing for which this general term supposits, as explained earlier. Since the middle is not distributed in either [premise] it is possible that its conjunction with the major extreme is true for one thing and its conjunction with the minor is true for another; and from this no conjunction of the extremes with one another can be inferred unless the middle is brought together by a relative of identity to hold for the same thing in the minor premise as that for which it was verified in the major. But then the syllogism is valid, and clearly holds by the rules given above, and it is [p. 90] effectively an expository syllogism; for example, “[Some] B is an A and [some] C is the same B, so [some] C is A.” So such syllogisms hold in all moods {in} which expository syllogisms hold. This is one way in which it is possible to argue from two particulars. But I will speak about the other [way] when there is a syllogism without such a relative. Seventh Conclusion: In every figure, if the middle was distributed in one of the premises, there is always a valid syllogism by concluding to a conclusion of one extreme with the other extreme, unless ampliation prevents it and [the premises] are not both negative.

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The reason is that in such an arrangement one of the rules presented above by which syllogisms hold will always be satisfied. Because if the middle was distributed in one of the premises, then it is necessary if that premise is true that the term is [distributed] for all those for which it supposits. So the other premise cannot be true unless it is true for something the same as that for which the other was true; and so the extremes are shown in the middle to be linked by reason of something the same as that for which the middle supposits. If the middle was distributed in both premises, then the extremes are shown to be linked with it on account of everything for which it supposited. Now in what way and why an exception should be made by reason of ampliation will be seen in what follows. But it is necessary to attend carefully to what should properly be concluded. So to direct us to what should properly be concluded, there follows another conclusion. Eighth Conclusion: If the minor extreme was distributed in the premises a direct universal conclusion can be inferred, and if not, not; if the major extreme was distributed in the premises an indirect universal conclusion can [p. 91] be inferred, and if not, not; if the predicate of a negative conclusion was distributed in the premises the conclusion should be formed in the customary way of speaking; and if it was not distributed, then the conclusion should be formed by placing the negation after the predicate. These all follow from the tenth conclusion of Book I. For a term should never be distributed in the conclusion that was not distributed in the premises; but it can be distributed if it was distributed, because the extreme can be inferred to be linked to the other extreme for those things for which it is linked with the middle and not for others. It should be noted that by these three conclusions, that is, the sixth, seventh, and eighth, and by the second, the number of all the useful ways of syllogizing in any of the three figures both direct and indirect is made manifest. For in each figure there are sixteen ways of linking by a combination of universal and particular and affirmative and negative in the two premises. Because either both are universal, or both are particular, or the major is universal and the minor particular or vice versa. And each of these four ways is divided into four, for either both are affirmative, or both negative, or the major affirmative and the minor negative, or vice versa. Now of these ways of linking in each figure there are four useless moods, namely, those that arise from two negative [premises]. Moreover, in every

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figure there is a fifth useless mood, namely, from two particular affirmatives, since the middle is [then] distributed in neither premise. In the first figure, if the major was particular, whether affirmative or negative, and the minor a universal affirmative, then there is no valid syllogism, for the middle is not distributed. So there are seven useless moods. An eighth mood is also useless, if the major is a particular negative and the minor a particular affirmative, for the middle is not distributed. But the other eight moods are useful. For first, if the minor is negative and the major affirmative, there are four useful moods, since the middle is distributed in the minor premise. But in these four moods, if the minor was [p. 92] particular, then neither extreme is distributed in the premises; so a conclusion cannot be inferred either directly or indirectly in the customary way of speaking, but it is necessary to place the predicate in the conclusion before the negation. Now if the minor was universal, then, since the minor extreme is distributed and the major not, a conclusion can be inferred indirectly according to the customary way of speaking, but not directly. Thus, there are the two moods Fapesmo and Frisesomorum.12 But if each is affirmative and the major universal, of whatever quantity the minor is, a syllogism will be valid, since the middle is distributed in the major. Thus, there are two moods in each of which both a direct and an indirect conclusion can be inferred. The moods concluding directly are Barbara and Darii, while those concluding indirectly are Baralipton and Dabitis. Now if the major is a universal negative and the minor was affirmative, whether universal or particular, then there are two valid moods concluding directly, namely, Celarent and Ferio. In these two moods, if the minor was universal an indirect conclusion can also be drawn, and it will be Celantes, but if the minor was particular, an indirect conclusion can only be drawn in the nonnormal way of speaking, since the minor extreme is not distributed. Thus, we have eight useful moods.13 But six of them are only valid in the customary way of speaking, namely, Barbara, Celarent and Darii, which conclude both directly and indirectly, and Ferio, only concluding directly, and Fapesmo and Frisesomorum, only concluding indirectly.14 It seems to me that Aristotle takes a syllogism not to be composed of premises and conclusion, but composed only of premises from which a conclusion can be inferred; so he postulated one power of a syllogism [to be] that from the same syllogism15 many things can be concluded. So in the first

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figure in addition to the four moods concluding directly and according to the customary way of speaking, Aristotle describes only two other moods that, also according to the customary way of speaking, conclude indirectly, namely, Fapesmo and Frisesomorum, and rejects those that conclude only according to the nonnormal way of speaking. Nor did he list [p. 93] Baralipton, Celantes, and Dabitis in addition to Barbara, Celarent, and Darii, since according to the definition they do not differ from them. Nothing follows in the second figure from pure negatives or from pure affirmatives, because the middle would not be distributed. So there are eight useless moods. The other eight moods are useful, namely, Cesare and Camestres, concluding both directly and indirectly; Festino and Baroco, concluding directly only in the customary way of forming a conclusion. Moreover, {if} the premises of Festino and Baroco are transposed, there will be two more moods concluding only indirectly, which can be called “Tifesno” and “Robaco,” which are proved by reduction to Festino and Baroco simply by transposing the premises. The other two moods can only be concluded in nonnormal form, namely, if both premises are particular, one affirmative and the other negative.16 Nothing follows from pure negatives or pure particulars in the third figure, because the middle would not be distributed. So there are seven useless moods. The other nine moods are useful for making a conclusion, and according to the customary way of speaking. For Darapti, Disamis, and Datisi are valid for concluding both directly and indirectly; but Felapton, Bocardo, and Ferison are only valid for concluding directly. But conversion of these three, namely, Lapfeton, Carbodo, and Rifeson, are valid for concluding only indirectly, and are reduced to direct ones by transposition of the premises.17 That said, I want to note otherwise that in the following conclusions, I will speak only of moods in which one can infer a conclusion in the customary way of speaking. Ninth Conclusion: Ampliative predicates do not affect the above syllogistic moods if the subject of the premise and conclusion [p. 94] is used with the addition of “that is,” for example, “Everything that is B is A, everything that is C is B, so everything that is C is A,” and the same for other figures and moods. The reason is that in this way of speaking the ampliation is prevented. It is also manifest that according to this way of speaking, the first four of Aristotle’s moods are perfect, evidently holding explicitly by the dictum

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de omni et nullo. The other moods of Aristotle’s that conclude in the customary way of speaking reduce to these four moods, just as Aristotle teaches that they reduce, because in this way of speaking the conversions that support them are formal. For although it does not follow, “Aristotle is dead, so a dead thing is Aristotle,” nonetheless it does follow, “He who is Aristotle is dead, so that which is dead is or was Aristotle.” Now all other moods, except Baroco and Bocardo, reduce by conversions to the four perfect moods of the first figure. But then Baroco and all other moods of the second figure can be reduced and proved “per impossibile,” namely, because the contradictory of the minor [premise] can be inferred in the first figure from the major [premise] and the contradictory of the conclusion; so, by the third conclusion, it follows that they were good syllogisms. Bocardo and all other moods of the third figure are proved in the same way, since the contradictory of the major [premise] can be inferred from the minor [premise] and the contradictory of the conclusion; so they were all good. Tenth Conclusion: An ampliative major extreme prevents a direct universal conclusion but does not prevent a particular or an indirect universal. This conclusion assumes that there is no other ampliative term besides the major extreme. Then the explanation for the conclusion is that the minor extreme is ampliated in the conclusion when it was not ampliated in the premises. So if the major extreme is ampliative there will be a process from the less broad to the broader. Now such a process is not valid if what is concluded [p. 95] is broader and distributed, but it is valid if what is concluded is not distributed. For “Every human . . . so every animal . . .” is not valid, but “[Some] human . . . so [some] animal . . .” is valid. Now the reason why an indirect universal conclusion is not prevented is that the minor extreme in such a conclusion is not ampliated, so there is process only from the nonampliated to the nonampliated. To make these things more evident to beginners, I show by counterexamples that universal conclusions cannot be inferred in Barbara or in Celarent. For a counterexample to Barbara, take the hypothesis that everything that is or that will be my horse is going to head for Rome, and that at present every horse is mine but tomorrow many horses will be born that will not be mine or head for Rome. Then take the syllogism, “Every horse of mine is going to head for Rome, and every horse is my horse, so every horse is

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going to head for Rome.” In the case posited, the conclusion is false and the premises true; so the syllogism is not valid. For a similar counterexample to Celarent, suppose that no horse of mine will head for Rome and that every horse is at present mine and tomorrow there will be others that will head for Rome but will not be mine. Then take the syllogism, “No horse of mine is going to head for Rome; every horse is my horse; so no horse is going to head for Rome.” It should also be noted that even if the predicate is not ampliative but the copula in the major premise was ampliative—for example, if it was of the past or future tense, the same holds as in this conclusion, for the same reason in both cases. Eleventh Conclusion: An ampliative middle does not prevent [a conclusion] in the third figure. For [in the premises of the third figure] the middle is the subject and the extremes are the predicate, and the predicate is not ampliated by the subject. So the extremes are not said to be broader or less broad in the premises than in the conclusion. In this conclusion it must be understood that only the middle term is ampliative. [p. 96] Twelfth Conclusion: An ampliative middle does not prevent universal negative moods in the first or second figure but it does prevent all others. The reason why it does not prevent universal negative [conclusions] is that the minor extreme is ampliated in the premises and is not ampliated in the conclusion. But a good consequence holds negatively from a broader distribution to one less broad. For it follows, “No animal is running, so no human is running.” But that sort of consequence does not hold affirmatively, for it can be prevented by the fact that the less broad term does not supposit for anything; for example, if nothing is a horse, it does not follow, “Every animal is alive, so a horse is alive.” There can also be counterexamples in Barbara. Although it is difficult to find a case, suppose that God brings it about by his absolute power, sometime towards the end of the world, that in one day all those who will then be old die and the youths remain, and that subsequently no one living is produced, and that the youths live until they are old and then die, and then the world ends. Therefore, on that day when it was supposed that the old died, it is argued like this: “Everyone going to die is a youth, everyone old is going to die, so everyone old is a youth.” The conclusion is false, but the premises

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are true according to the hypothesis; for the minor [premise] will clearly be seen to be true if it is analyzed.18 It is seen similarly why syllogisms [with] particular [conclusions] in the first and second figures are not valid. It is because in syllogisms with, as I mean here, direct particular conclusions, the minor premise is particular (for it is necessary in these two figures for the major [premise] to be universal), and so the minor extreme is not distributed. But the process from the broader undistributed to the less broad is not valid. From what has been said, it is also sufficiently evident that an ampliative middle does not prevent indirect syllogisms in these two figures in which the minor [premise] is a universal negative, for the same reason for which it does not prevent direct universal negative [conclusions]. It is also seen [p. 97] that an affirmative universal syllogism is conditionally good, that is, given the hypothesis of an existence assumption (constantia) about the subject of the conclusion, that is, that the subject in the conclusion supposits for something. Also, from the last three conclusions stated, the attentive reader can see in what way syllogisms are prevented or not prevented if the major extreme and the middle are both ampliative terms. Because if they make the same kind of ampliation—for example, both to the past or both to the future—a syllogistic [conclusion] is not prevented, for then there will not be a process from the broader to the less broad or vice versa, but from the equally broad to the equally broad. But if the major extreme ampliates to other things than does the middle, then what was said in the tenth conclusion to be prevented by the major extreme will be prevented. While if the middle ampliates to other things than does the major extreme, what was said to be prevented in the twelfth conclusion will be prevented.19 And if each ampliates to other things than does the other, both will be prevented, so that there will not be a formal consequence, although perhaps to some [people] a counterexample may not be immediately apparent. It should also be noted that the same must be said if the copula was ampliative in the minor premise in the first or second figure—for example, if it was in the past or future tense, just as if the middle was ampliative, for the same reason in each case.20 Enough has now been said in these conclusions about syllogisms with premises in the past or future tense. Now we must discuss syllogisms with oblique terms. [p. 98]

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Part II: Syllogisms between Propositions Containing Oblique Terms Chapter 1: Propositions Containing Oblique Terms Now we should discuss syllogisms containing oblique terms. Accordingly, it will first be explained what an oblique term is when it is used with a direct term that is governed by it as a determination of that direct term, just as an adjective is a determination of a substantive [term]. For just as when saying “A white horse is running,” the expression “white” determines the expression “horse” to supposit only for white ones, so if I say “Socrates’s horse is running,” the expression “Socrates’s” restricts the expression “horse” to supposit only for those that are Socrates’s. In this regard, I believe that just as an adjective, unless it is in the neuter gender of substantives, cannot alone replace the verb nor be the whole subject of a simple subject-predicate proposition, neither can an oblique term. However, I do not say this categorically, since it is not necessary for me to assume it in what follows. However, on the basis of what was said, it is clear that sometimes a combination of a direct and an oblique [term] is distributed by a single distribution, not each individually. For example, if I say, “Every human’s ass is running,” “ass” is not distributed, nor is “human’s” distributed, but the whole combination “human’s ass.” So a syllogism cannot be made by subsuming [something] under “ass” or “human’s.” Hence, it does not follow, “Every bishop’s ass is running, Brownie is an ass, so Brownie is running;” neither does it follow, [p. 99] “Every bishop’s ass is running, Socrates is a bishop, so Socrates’s ass is running,” because if Socrates does not have an ass, the conclusion is false, although the premises might be true. But if we said in the minor premise that Brownie is a [bishop’s] ass,21 then we could conclude, “so Brownie is running.” Nonetheless, sometimes the oblique term alone, without the direct term, is distributed, for example, if the distributive sign is added to the oblique one in the same case as the oblique and not in the same case as the direct. For example, in saying “Any human whatever’s ass is running,” “human” is distributed and “ass” is not distributed, nor [is] the whole “human’s ass.” It would be the same if I said “An ass of any human whatever is running.” For here “ass” supposits determinately, because no cause of confused [supposition] precedes it. Nonetheless, “ass” does not supposit alone but restricted

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by the distributed oblique [term] following it; so the said proposition would not be true if no asses were running or no ass running were an ass of any human whatever. Moreover, sometimes the direct [term] is distributed and not the oblique nor the combination of the direct and the oblique, but the oblique retains determinate supposition, namely, if it precedes the distributive sign, for example, in saying, “Of [some] human any ass whatever is running.” For it does not follow, “Of [some] human any ass whatever is running, so any human whatever’s ass is running.” But in the said proposition the term “ass” is not distributed straightforwardly, but restricted by the term “human” suppositing determinately. So it does not follow, “Of [some] human any ass is running, Brownie is an ass, so Brownie is running.” Similarly, because the whole combination is not distributed, it does not follow, “Of [some] human any ass is running, the bishop’s ass is [some] human’s ass, so the bishop’s ass is running,” because the bishop’s ass is perhaps lying in the stable and Socrates is the human whose every ass is running. So in this case, if we wish to include in a syllogism what is distributed in the proposition in question, we must form a minor premise in which a relative of identity is adjoined to the term “human” so that the middle is understood to hold of the same thing in the premises. For example, it does follow, “Of [some] human any ass is running, the bishop’s ass is an ass of that human, so the bishop’s ass is running.” For in this way the rule is preserved by which affirmative syllogisms were said to hold. This [concludes] what is set down in the first chapter. [p. 100]

Chapter 2: The Syllogistic Extremes and Middle Now in the second chapter, we take it that sometimes in syllogizing with oblique [terms] it is not necessary that the syllogistic extreme or the syllogistic middle is the extreme of either premise. For I call the extremes of the syllogism what are inferred in the conclusion to be linked by the fact that each is linked with the middle in the premises, and I call the extremes of the premises their subjects and predicates. So I say that the syllogistic middle is sometimes neither the subject nor the predicate of the major premise but part of the subject or predicate; and similarly, [sometimes] neither of the extremes is the subject or predicate either in the premises or in the conclusion. For example, here is a good

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syllogism: “Some human [it is] every horse [he or she] is seeing; Brownie is a horse, so some human [it is] Brownie [he or she] is seeing.”22 Now in this syllogism, the term “horse” is the middle, but it is neither the subject nor the predicate of the major premise. Indeed, the major extreme is these two terms “human” and “seeing”; for they are linked with the middle “horse” in the major premise. It is clear that one of these two terms is placed in the premise in the subject and the other in the predicate, so they are neither the subject nor the predicate. Indeed, the term “Brownie” is the minor extreme, and in the minor premise it is linked with what was said to be the middle. So it is clear that neither the major extreme nor the minor is the subject or the predicate of the conclusion, but the major extreme is linked with the minor in the conclusion in the same way in which it is linked with the middle in the major premise. This [concludes] what is set down in the second chapter.

Chapter 3: Conclusions Furthermore, in the third chapter, we take it that the predicate is restricted by a verb preceding [it] to supposit for the supposita at the time of the verb; for example, if I say, “Socrates yesterday saw a white human” or “He struck [p. 101] a running human,” the proposition is not true unless the human whom he saw was white when he saw him or that he whom he struck was running when he struck him. So if the predicate or term placed in the predicate is distributed then it is only validly used for the time for which the distribution was made. So it does not follow, “Yesterday I did not see a human running, Socrates is a human running, so yesterday I did not see Socrates”; similarly, it does not follow, “Never did I strike an old human, Socrates is an old human, so never did I strike Socrates.” But it would be necessary to argue like this: “Yesterday I did not see a human {running}, but whenever yesterday I saw a human Socrates was a human {running},” and then it follows, “so yesterday I did not see Socrates”; similarly, it would be necessary to argue like this: “Never have I struck an old human, but whenever I struck a human, Socrates was an old human, so never did I strike Socrates.” Again, as has been mentioned elsewhere, there are some verbs that render the accusatives following them and that they govern such that the act described by those verbs does not straightforwardly apply to the things for which those accusatives supposit, but apply to them by means of certain

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concepts of them described by those terms suppositing for them.23 Those verbs are “to know,” “to understand,” “to comprehend,” “to believe,” “to appear,” “to judge,” and also “to see,” “to hear,” “to imagine,” and so on, and consequently, “to seek,” “to want,” “to love,” “to desire,” “to hate,” and so on; and also “to promise,” “to sell,” “to buy,” “to owe,” “to give,” “to condemn,” and so on. Participles of these verbs and nouns derived from them have a similar power, when they occur, over the terms following them that they govern. Hence, such verbs or participles or nouns derived from them restrict the terms following them which they govern to supposit for that for which they supposit not absolutely but with the appellation of an intension or concept by means of which those terms signify what they signify. So, in syllogizing, one cannot interchange one term with another with a different intension, by descent under a distribution or by predicating that term of it by another term or terms. Here the mode of supposition [p. 102] is like simple or material supposition, in which it is not permissible to descend under a distributed term.24 For example, if it is necessary that every human is an animal and the one running is a human, it does not follow, “therefore it is necessary that the one running is an animal”;25 similarly, it does not follow, “Every God is and always was a triune God, and Averroes had knowledge of God; so he had knowledge of a triune God”; nor does it follow, “Socrates is ignorant of prime matter,” or “does not have knowledge of prime matter, and all prime matter is nature, so Socrates is ignorant of nature,” or “Socrates does not have knowledge of nature.” If it is as many suppose, namely, that the light of the sun has magnitude and shape, then I could see the light of the sun even though I did not see the magnitude or shape of the sun or the light, as when a rainbow is seen by me or when I see daylight before dawn. But if the said accusatives or terms that are governed by these sorts of verbs or participles precede those verbs or participles, then they are not restricted to appellate those intensions or concepts.26 Next, some conclusions are propounded. Thirteenth Conclusion: From any given proposition in which some term is distributed, whether direct or oblique ( provided its supposition is not material), a good syllogism is made by taking another term under that term in the minor [premise]. It seems to me that this conclusion holds by the nature and definition of distribution, and is evident to anyone who knows what is understood by the

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term “distribution.” These syllogisms are as perfect as syllogisms made in the first figure from simple and direct terms. So they can also be reduced to the first figure by their similarity, because they hold straightforwardly and immediately by the nature and definition of distribution and subsumption under a distributed term. The exception made about material supposition is not an exception to the definition and nature of distribution, because it is not necessary for a distributive sign to work in the same way when it occurs in a proposition materially as it might if it were used significatively. Similarly, the exception with the appellation of an intension or a concept and so on is also not inconsistent with the definition and [p. 103] nature of distribution, because a term is not simply distributed but [only] with some addition or determination understood or indicated by the definitions of those verbs. For example, if I say, “Socrates does not have knowledge of prime matter,” this term “prime matter” is not simply distributed; so it does not follow, “therefore no prime matter does Socrates have knowledge of.” But make the sense of the proposition explicit like this: “Socrates does not have knowledge of prime matter,” that is, “Socrates no prime matter has knowledge of according to the intension according to which it is called prime matter”; for this was the sense of that proposition. So you may subsume under the distribution that nature is prime matter and you may conclude, “so Socrates nature does not have knowledge of according to the intension according to which it is called prime matter.” Similarly, if I say, “Averroes does not have knowledge of a triune God, and every God always is and was a triune God,” it does not follow, “so Averroes does not have knowledge of God,” but it only follows, “so no God does Averroes have knowledge of according to the intension according to which he is called a triune God.” It was also said earlier27 that the exception about the appellation of a certain time does not remove the power and definition of distribution, because if the subsumption under the distributed term is made for the time for which it was distributed, the syllogism is valid in the earlier manner. So what was said earlier does not contradict the thirteenth conclusion, but we safely subsume the minor [term] under the major and safely infer the conclusion. So I say that it clearly follows, “[Some] B is every A and C is A, so [some] B is C,’ or also ‘so [some] C is B”; similarly, it follows “[Some] human [it is] every horse [he or she] sees, Brownie is a horse, so [some] human [it is] Brownie [he or she] sees”; similarly, “Of every human an ass is running,

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Socrates is a human, so of Socrates an ass is running”; similarly, “The earth is encompassed by every celestial sphere, the orbit of the moon is a celestial sphere, so the earth is encompassed by the orbit of the moon”; and similarly, in accordance with what was said, “Of the king every horse is running, Brownie is a horse of that king, so Brownie is running.” Fourteenth Conclusion: From every given affirmative premise with direct terms, and anything connected affirmatively to the subject directly or [p. 104] used obliquely, the same can be inferred to be connected to the predicate in the same way provided no difference of tense or appellation of an intension presents an obstacle. The reason for this conclusion is that if this is the same as that, everything that has some attribution to this has the same to that. For example, it follows, “Every human is running, [it is some] human you see, so [it is] someone running you see”; similarly, it follows, “Every horse is black, you have a horse in the stable, so you have something black in the stable”; similarly, “Every ass is running, [it is some] human’s [thing that] is an ass, so [it is some] human’s [thing that] is running.” For Aristotle syllogized in this way in Prior [Analytics, Book] I: “All wisdom is knowledge, [it is] of the good [that] wisdom is, so [it is] of the good [that] knowledge is.”28 However, on account of the difference in tense, it does not follow, “Every horse is running, [it is] a horse you had in the stable, so [it is] something running you had in the stable,” because the minor [premise] could be true not for some horse for which the term “horse” was distributed in the major [premise] but for another. But if the term were understood by using a relative of identity to stand for the same thing in the minor as in the major, then it would be a good syllogism—for example, “Every horse is running and you have one of those horses, so you have something running.” Similarly, an appellation of an intension can prevent a syllogism, for it does not follow, “Coriscus is the one approaching, you know Coriscus, so you know the one approaching,” but it does follow, “so the one approaching you know.” Fifteenth Conclusion: In any proposition in which an oblique term is placed first, if we use it as subject and the rest as predicate, syllogisms are valid in any figure in which they are valid with direct terms. The reason is that many take it to be true that the oblique term is the subject of the proposition and the rest is the predicate. Even if it is not really as they suppose, nonetheless every such proposition is equivalent to another

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in which a direct [term] is used in lieu of the oblique one as subject and the rest as predicate according the analysis given elsewhere.29 For example, the proposition “A human’s ass is running” is equivalent to “Some human is [one] whose ass is running”; so it is possible to syllogize with the one as is done with the other. So in the first figure, or something like it, there is a syllogism [p. 105] like this: “Any human whatever’s ass is running, Socrates is a human, so Socrates’s ass is running”; and in the second figure like this: “Any courtier whatever’s horse is running, of no priest is a horse running, so no priest is a courtier”; similarly, in the third figure like this: “Any human whatever’s ass is running, any human whatever’s horse is walking, so someone whose horse is walking is [one] whose ass is running.” Sixteenth Conclusion: If a compound or nearly compound proposition is formed with a categorematic distributive term in the first simple subjectpredicate part and its relative in the second simple subject-predicate part, from every term subsumed under that distributive [term] in respect of the same thing that was attributed to it, there can be inferred what was attributed to the relative, by forming propositions like “Whatever you bought, that you consumed,” or “However much Socrates is, such is Plato,” or “Whatever it was I bought, that I consumed,” or “However Socrates is related to Plato, so too is John to Robert,” or “Wherever Socrates is holding forth, there Plato is listening,” or “Whenever the king is angry, then his servants tremble.” I call {such} a proposition “compound” or “quasi-compound” because some say they are compound (and I believe that is true) while others say they are simple subject-predicate [propositions] although they have a closeness to compound ones. I do not care, in treating of consequences, what they are called; but I want by convention to call them “compound relatives.” It is already generally conceded that those of place and time are compound, and the reason is the same for these. Thus, just as sometimes I express in such propositions an interrogative or distributive in the category of quantity or quality or substance with its relative, so sometimes I express them both in the category of where or when.30 For example, just as I say, “Of whatever sort Socrates is, so too is Plato,” and “However much Socrates is, so much is Plato,” and “That which Socrates is, that is Plato,” so I say or can say, “Wherever Socrates is, [p. 106] there is Plato” and “When this one, then that one.” Similarly, just as sometimes I express what is sought of quantity or quality or substance and understand by it the relative, so often I do with the category where or when. For example, just as I say, “Socrates is what

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Plato is” and “Socrates is as much as, or how, Plato is,” so I say, “Socrates is when, or where, Plato is.” So there is equal reason that they all be called compound or all simple subject-predicate [propositions]. It should be noted that just as in our language we have interrogatives with their relatives pertaining to the categories of substance, quality, quantity, where and when, so perhaps the Greeks had them in other categories (and we could invent them), and perhaps they used them in a general interrogative “in what way, or how, is it related?” I do not really care about this very much, but only that if there were these interrogatives and corresponding relatives pertaining to them, the conclusion stated would be true of them. It seems to me that these syllogisms hold by the meaning and definition of distribution and of relation. For according to distribution, because a distributive pertaining to a category distributes the category across all terms supposited for by it, so any of them can be subsumed under it, and the relative of identity corresponding to it means that what is attributed to it can be attributed to what is subsumed under it. So it seems to me that such syllogisms should be said to be perfect by definition. It should be noted that these syllogisms are valid for any sort of simple subject-predicate [propositions] from which a compound with its relative is constructed. For it follows, “Of whatever sort Socrates is so too is Plato, Socrates is white, so Plato is white”; similarly, “However much you do not see, that much I do not see, two cubits [away] you do not see, so two cubits [away] I do not see”; similarly, it follows, “Whenever Socrates was, then Plato was not, yesterday Socrates was, so yesterday Plato was not”; similarly, it also follows, “Wherever Socrates is, there Plato is not, in church is [where] Socrates [is], so in church is not [where] Plato [is].” Nor in syllogisms of this sort need the same hypothesis be preserved in both simple subjectpredicate [propositions] of the first compound [premise]; indeed, it is permissible [p. 107] to argue like this: “Whomever you see, that one is white, John [is whom] you see, so John is white”; similarly, “However long the ruler, that much you can measure, two cubits [long] is the ruler, so two cubits [long] you can measure.” For Aristotle syllogized in this way in the Prior [Analytics, Book] I: “Of whatever there is a science, of that there is a genus, of the good there is a science, so of the good there is a genus.”31 It should also be noted that in this way of syllogizing or taking care of syllogisms there are ampliations, namely, that from a broader undistributed [term] we may not infer one less broad, nor from one less broad may we

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infer one more broadly distributed. For it does not follow, “Whatever is dead it is not alive, a human is dead, so a human is not alive,”32 neither does it follow, “Whatever is alive it is not dead, every human is alive, so no human is dead.” Care is also needed with ampliation of tense, of forms or of intensions on the part of the predicate, because it does not follow, “Whatever sort of thing I bought such a thing I consumed, I bought something uncooked, so I ate something uncooked”; neither does it follow, “Whatever you have knowledge of, that thing I have knowledge of, you have knowledge of matter, so I have knowledge of matter,” because it may be possible that [what is in fact] matter I have knowledge of but I do not have knowledge of matter [as matter]. Seventeenth Conclusion: In the aforesaid mode of arguing, namely, through [terms] distributive over the categories and their relatives, it is not permissible to subsume under a distributive in one category a term of another category, except that one can subsume terms of other categories under [a term] in the subject distributive of substance, unless ampliation prevents it. For it does not follow, “However many are running as many are disputing, Socrates is running, so Socrates is disputing,” or “a white thing is running, so a white thing is disputing.” But it does follow, “Whatever you see I see, a white thing you see, so a white thing I see,” although it does not follow, “so I see a white thing,” neither does it follow like this: “Whatever thing you saw is what you see, a white thing you saw, so a white thing {you} see”; for ampliation prevents this consequence.33 [p. 108] But the reason why terms of other categories can be subsumed under a distributive [term] of substance and not conversely is that a relative of substance means identity simply while relatives of other categories do not mean identity simply but only similarity or equality or sameness of place and so on, and an affirmative proposition by which [something] is subsumed under a distributive [term] also means identity simply of that for which the subject supposits to that for which the predicate supposits, of whatever category the terms are. Many very specific things might be said about this mode of syllogizing, but the rest I leave to the scrupulous investigation of diligent students. Eighteenth Conclusion: In every figure, from two affirmatives, and also from two negatives, there follows a negative conclusion interchanging the middle between finite and infinite, unless ampliation prevents it.

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In this conclusion it must be ensured that the middle is distributed in at least one premise and that the conclusion is universal or particular, direct or indirect, and according to the normal or nonnormal way of speaking, as was explained in the sixth, seventh, and eighth conclusions. The ground for the conclusion is that whatever a finite term supposits for, the infinite [term] does not supposit for, and whatever a finite term does not supposit for, the infinite does supposit for, and vice versa. So it is necessary that those to which a finite and its infinite term are linked are said not to be the same, so that from affirmatives there follows a negative. So also from negatives there follows a negative, because either each of the terms to which finite and infinite terms are negatively linked supposits for something or one of them supposits for nothing. If one of them supposits for nothing then it is clear that the negative conclusion will be true. But if each of them supposits for something then it is impossible that a finite and an infinite term are [correctly] denied of the same, so it is necessary that they supposit for different things and accordingly the negative conclusion will be true. [p. 109] Now to make it more evident, this conclusion will be demonstrated separately for the three figures. First, I say that this is a good syllogism: “Every B is A and every C is non-B, so every C [some] A is not,” and we should not conclude “so no C is A” because A was not distributed in the premises. The syllogism is proved because the opposite of the minor [premise] follows from the opposite of the conclusion together with the major [premise]. For the contradictory of the conclusion is “Some C every A is” from which it follows that every A is C, and from there “Every A is C, every B is A, so every B is C,” and so by conversion, “Some C is B,” and this contradicts the minor [premise] in the original [syllogism]. I also say that in that syllogism an indirect conclusion could be concluded, namely, “so some A is not C.” Because it is immediately evident that the opposite of the minor [premise] follows from the opposite of the conclusion and the major [premise], because C was distributed in the minor premise. Similarly, in the second figure, there is a good syllogism like this: “Every B is A, every C is non-A, so no C is B.” It is proved because from the minor, “Every C is non-A,” there follows “No C is A,” and then it is a good syllogism without interchanging the middle between finite and infinite. Similarly, in the third figure, it follows: “Every C is A, every non-C is B, so some B [some] A is not.” It is proved because converting the minor [premise] produces the first figure.34 It is also proved per impossibile, because

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the contradictory of the conclusion is “Every B every A is,” from which it follows that every A is every B, from which, with “Every C is A,” it follows that every C is every B, and this contradicts the minor [premise]. It can also be shown that a negative follows from negatives. First, I say that this is a good syllogism: “No B is A, no C is non-B, so no C is A.” For the opposite of the minor [premise] follows from the opposite of the conclusion together with the major [premise] like this: “No B is A, some C is A, so some C is not B.” But this conclusion contradicts the minor [premise] of the first [syllogism], supposing that there is a C, for it follows, “There is a C and it is not B, so it is non-B.” But “[Some] C is non-B” clearly contradicts the minor [premise] of the first [syllogism]. Now that assumption, namely, that there is a C, is guaranteed by the statement of the contradictory of the original conclusion, provided there is no ampliation; for it follows, if there is no ampliation, “Some C is A, so there is some C.” From this it is immediately clear that in the second figure there is a good syllogism [p. 110] like this: “No B is A, no C is non-A, so no C is B,” because converting the major [premise] produces the first figure. Similarly, in the third figure, there is a good syllogism like this: “No C is A, no non-C is B, so no B is A,” because converting the minor [premise] again produces the first figure, and these universal negatives are convertible simply unless ampliation prevents it. It should be realized that such a mode of arguing is often used, though we do not realize its [actual] form. For we often argue like this: “What is not an animal is not a human, a stone is not an animal, so a stone is not a human”; and this mode of arguing is not valid as a matter of form unless we understand the major [premise] to be read universally, and if it is read universally then it is equivalent to “No non-animal is a human,” whence clearly it is a syllogism in the first figure interchanging the middle [term] between finite and infinite. Nineteenth Conclusion: From an affirmative major and negative minor [premise] in the first figure where the middle [term] changes between the premises from finite to infinite there follows an affirmative conclusion given an existence assumption about the negative terms, and similarly in the third figure from an affirmative and a negative, whichever [premise] is affirmative; but in the second figure nothing follows in this way. The reason it is valid in the first figure is that from the minor [premise], which is negative, there follows an affirmative by interchanging the predicate

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between finite and infinite, given an existence assumption about the terms, and then it will be a syllogism in the first figure with the middle unchanged.35 It is [valid] for the same reason in the third figure, namely because from a negative with an existence assumption about the terms there follows {an affirmative by interchanging the predicate between finite and infinite}. But that it is not valid in the second figure is seen by the fact that every human is an animal and no ass is a non-animal, while it does not follow that some ass is a human. But it is valid in the second figure if the major extreme is interchanged between finite and infinite. This is the end of Book III concerning syllogisms containing assertoric propositions.

Book IV Syllogisms between Modal Propositions Chapter 1: Syllogisms between Composite Modal Propositions Finally, in Book IV, [p. 111] we will consider syllogisms between modal propositions. First, it should be remembered that some modals are composite and some divided. The subject in all divided [propositions] of necessity or of possibility is ampliated to supposit for what can be unless the ampliation is prevented by the extra phrase “that which is” attached to the subject. So those in which the extra phrase is not added are called “of possibility” or “of necessity” simply, while those in which the extra phrase is added are called “of possibility” or “of necessity for those which are.” It should also be noted concerning mixed [syllogisms] that Aristotle was accustomed to distinguish [different types of] assertoric propositions, calling some “simply assertoric” and others “assertoric as-of-now.” But “simply” is not used here as it is described in Topics [Book] II,1 where he says “what I say without addition I say simply”; for then “Socrates is running” would be simply assertoric, because it is true without addition to say that it is assertoric. But Aristotle intends by “simply assertoric proposition” an assertoric proposition that is necessary and by “assertoric as-of-now” he intends an assertoric that is contingent. Then it should be realized that since a proposition [that is] assertoric asof-now is in the literal sense simply and absolutely assertoric, a syllogism [p. 112] with an assertoric premise does not hold in virtue of form if it is possible to find a counterinstance with a proposition [that is] assertoric asof-now. But although speaking simply it is not formal, it is possible that it will be good and formal on the hypothesis, that is, the assumption, that the assertoric [premise] is simply assertoric, that is, that it is necessary. It should be noted that adding the said hypothesis to a syllogism makes it valid just as adding a composite [proposition] of necessity in place of the assertoric 140

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one does. For example, it is effectively the same in syllogizing and in saying what is true or false to say that a human is an animal on the assumption that the proposition is simply assertoric and to say explicitly that “a human is an animal” is necessary. It should also be noted that although Aristotle in his examples seems to take such propositions as “Every human is of necessity an animal” and “Every white thing is necessarily not black” as true, nonetheless verification of such examples is not required since those propositions are simply false. For everything that can fail to be can fail to be an animal; but a human, such as Socrates or Plato, can fail to be; so a human can fail to be an animal, and this contradicts the claim that every human is of necessity an animal. Similarly for a white thing, since a white thing can become black, so it can be black. As to whether the proposition “A horse is an animal” is necessary, I believe it is not, speaking simply of a necessary proposition, since God can annihilate all horses all at once, and then there would be no horse; so no horse would be an animal, and so “A horse is an animal” would be false, and so it would not be necessary. But such [propositions] can be allowed to be necessary, taking conditional or temporal necessity, analyzing them as saying that every human is of necessity an animal if he or she exists, and that every human is of necessity an animal when he or she exists. It should also be said that Aristotle believed such [propositions] to be simply necessary because he thought that the eternity of the world and universal nature could not allow that at some time nothing was a horse or a dog. And it is true that it is not possible by natural means, although it is by a supernatural miracle, that at some time nothing [p. 113] is a horse, nothing the earth, nothing fire. So speaking only naturally such [propositions] as “A horse is an animal,” “Fire is hot,” should be taken as necessary, in the sense that it is not possible by nature, without a miracle, for them to be false; in what follows we will take such [propositions] to be necessary. This is what is initially set down. There follow some conclusions. First Conclusion: In every figure, nothing follows from composite [premises] both of possibility or of contingency, but from composite [premises] both of necessity or of truth there follows a composite conclusion of the same sort. This conclusion is appreciated by attending to the figure and moods of the contained propositions (the dicta), not to the premises as a whole, for

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the premises as a whole may be affirmative with an affirmed modal, though perhaps the dictum may be negated. The basis of this conclusion is that, as was said earlier,2 neither of the premises is the whole antecedent in a syllogism, but the conjunction composed of the two premises is the whole antecedent. Now every conjunction composed from truths is true and any composed of necessities is necessary, but one composed from possibilities might be impossible. For example, “Every human is running and some human is not running” is composed of possibilities, but it is impossible. So if both premises are necessary, the whole antecedent is necessary; hence the conclusion is necessary, if it is formed. And the same for truth. But if both premises are possible, it does not follow that the whole antecedent is possible; so it does not follow that the conclusion is possible. For example, it does not follow: “Everything running is a horse” is possible (or contingent), and the same for “Every human is running,” so “Every human is a horse” is possible (or contingent), since the premises were true but the conclusion false. [p. 114] Second Conclusion: In every syllogism containing composite propositions of necessity or of truth one may replace a [proposition] of necessity with one of impossibility with the contradictory dictum and replace one of truth with one of falsity also with the contradictory dictum. The reason is that they are equivalent and follow from each other by the law of contradictories. For it is necessary that if one [of a pair] of contradictories is necessary then the other is impossible and conversely, and if one is true then the other is false and conversely. So this is a good syllogism: “It is impossible that some B is not A and impossible that some C is not B, so it is impossible that some C is not A.”3 Similarly, since from [a composite proposition] of truth its dictum follows and from one of falsity the contradictory of its dictum, so this is a good syllogism: “It is false that some B is not A and false that some C is not B, so it is false that some C is not A,” that is, it may be concluded, “every C is A.” Third Conclusion: From composites of “know,” “think,” “doubt,” and similar modes, there is no valid syllogism. For it does not follow: “That every B is A is known by Socrates and that every C is B is known by Socrates, so that every C is A is known by Socrates.” For even if the two [premises] are known by him, nonetheless he may not construct a syllogism from them or realize that the third follows from them.

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Chapter 2: Syllogisms between Divided Propositions of Necessity and Possibility I will now stop speaking of composite modals and speak of divided ones. Whenever I speak of those of possibility or of necessity, I [p. 115] mean only those having an affirmed mode though they may have a negation falling on the dictum. It was seen earlier that, on account of the equivalences, if it is explained for them it will be explained for those having a negated mode.4 I will also speak only of direct conclusions and those in the normal way of speaking, because having explained them attentive readers can readily deal with the others. Fourth Conclusion: In the first figure with both [premises] of necessity or of possibility or one of necessity and the other of possibility there is always a valid syllogism to a conclusion of the same kind as the major [premise]. These are evident by the dictum de omni et nullo. They are all perfect syllogisms or nearly perfect. For if the major [extreme] is explicitly expressed by a disjunction of the verb “is” with the verb “can,” then if the minor is of possibility it will be clearly subsumed under the distribution of the major [extreme]; while if the minor is of necessity, the same is true, since that of possibility follows from that of necessity. When in the preceding conclusion I said or when I will say later, “is valid,” I always understand by “valid” not that it is valid in all combinations, but that it is valid in all moods that were claimed to be valid for assertorics. Fifth Conclusion: In the second figure, with both [premises] of necessity or one of necessity and the other of possibility there is always a valid syllogism to a conclusion of necessity, but no valid syllogism with both [premises] of possibility. That it is not valid from both [premises] of possibility is evident, because with the terms “create,” “God,” “first cause,” a negative conclusion would be false, and with the terms “run,” “human,” “horse,” an affirmative conclusion would be false, where the middle [term] is “create” or “run.” So an affirmative conclusion cannot follow, nor can a negative one follow in virtue of form. [p. 116] But that a conclusion of necessity follows when one or both [premises] are of necessity is proved by the fact that the opposite of the minor [premise] always follows from the major [premise] and the opposite of the conclusion, which is clear by the preceding conclusion if the syllogisms are formed.

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Sixth Conclusion: In the third figure, a conclusion of possibility always follows from two premises of possibility, and a conclusion of necessity from two [premises] of necessity, and a conclusion of the same kind as the major [premise] from one [premise] of necessity and the other of possibility. It is all shown by expository syllogisms. For in affirmative moods, if both [premises] are of possibility the conclusion, say, “that which can be B can be A,” evidently follows: let C be the middle [term]. But it follows further, “that which can be B can be A, so B can be A.”5 In negative moods, the conclusion “that which can be B can fail to be A” follows, from which “so B can fail to be A” follows. But if both [premises] are of necessity, then in affirmative moods, “that which is of necessity B is of necessity A” follows; so B is of necessity A.6 In negative moods, “that which is of necessity B of necessity fails to be A” follows; so B of necessity fails to be A. If, however, the major [premise] is of necessity and the minor of possibility, “that which can be B is of necessity A” follows in affirmative moods; so B is of necessity A. In negative moods, “that which can be B of necessity fails to be A” follows; so B of necessity fails to be A. But if the major [premise] is of possibility and the minor of necessity, then “that which is of necessity B can be A” evidently follows in affirmative moods; so B can be A. In negative moods, “that which is of necessity B can fail to be A” follows; so B can fail to be A.7 Again, it [can also] all be proved per impossibile, for the opposite of the major [premise] is inferred from the opposite of the conclusion and the minor [premise] in the first figure. Because many8 say that the subject in [propositions] of possibility or of necessity [p. 117] is always restricted to supposit only for those that are, I present the following conclusions. Seventh Conclusion: By restricting propositions of possibility or of necessity by “that which is,” there is a valid syllogism in the first figure with a major [premise] of necessity or of possibility if the minor [premise] is of necessity, but not if it is of possibility. I call it “restriction by ‘that which is’” when the proposition is formed as “that which is B is necessarily A” or “that which is B can be A,” and this is [what is meant by] the claim that in propositions of necessity or of possibility the subjects are said to supposit only for things that are.

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Then if the minor [premise] is of necessity it will evidently be subsumed under the distribution of the major [extreme]. For example, if I say, “Everything that is B is of necessity A and every C is of necessity B,” I subsume under “B” only things that are B. For everything that is of necessity B is in fact B and always will be B, and [this] ensures that “B” was distributed in the major premise for all the things that are B. From this, it is clear enough that although the major is restricted, if the minor is not restricted, the conclusion can be inferred without restriction. But if the minor is restricted, the conclusion should be restricted if a universal is concluded, lest there be process from the less to the more ampliated distribution. Now if the minor is of possibility it will not then be subsumed under the distribution of the major restricted by “that which is.” So a syllogism will not be valid. For it does not follow, “Everything that is shining is necessarily other than the moon and every moon can be shining; so {every moon can be other than the moon}.” Similarly, it does not follow, “Everything that is running can laugh, every horse can be running, so every horse can laugh.” Eighth Conclusion: In the second figure, with the premises restricted by “that which is,” there follows from [premises] both of necessity or one of necessity and the other of possibility a conclusion of possibility or assertoric restricted by “that which is,” but a conclusion of necessity does not follow. [p. 118] That a conclusion of necessity does not follow is shown by [this]: everything that is a planet shining on our hemisphere is necessarily the sun, and everything that is a planet shining beneath our hemisphere is necessarily not the sun; but it does not follow, “so everything that is a planet shining beneath our hemisphere is necessarily not a planet shining on our hemisphere,” since the premises can in fact be true [together], but the conclusion is false. Nonetheless, it would follow, “so nothing that is a planet shining beneath our hemisphere is a planet shining on our hemisphere.” And indeed from this conclusion there follows a conclusion of possibility, because it follows: “nothing that is B is A, so everything that is B can fail to be A.” For everything that is not A can fail to be A, so just as the said assertoric conclusion would follow, so the [conclusion] of possibility follows. That the said assertoric conclusion followed can be proved per impossibile, because from the major [premise] and the opposite of the conclusion the opposite of the

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minor [premise] follows in a mixture of necessary and assertoric, as will be seen later. Ninth Conclusion: In the third figure, with the premises restricted by “that which is,” if each is of necessity or of possibility, there always follows a conclusion of the same kind as the major [premise]. The whole thing can be shown by expository syllogisms and per impossibile. But it should be recognized that if the minor [premise] is of possibility the conclusion should not be restricted by “that which is.” For the minor extreme was not restricted. But if the minor is of necessity, then the conclusion can be inferred whether restricted or not restricted. For, as was said, everything that is of necessity B is in fact B.9 Hence, although it does not follow: “The one creating is necessarily God, so the one creating is God,” nonetheless it does follow: “That which {is creating is necessarily God}, so he is God.” Tenth Conclusion: In the first figure, there is no valid syllogism in virtue of form from an assertoric major and a minor of possibility, but there is a valid syllogism from a major of possibility [p. 119] and an assertoric minor to a particular conclusion of possibility, but not to a universal. The first part of the conclusion is shown by the fact that if everything running is a horse and every human can run it does not follow that a human can be a horse. Similarly, if nothing creating is God and every first cause can create, it does not follow that a first cause can fail to be God. Now it is evident that from a major of possibility and an assertoric minor there may follow a particular of possibility, if [the ampliation of] the major is stated explicitly. For there will be a perfect syllogism by evident subsumption under the distribution of the major—for example, “Everything that is or can be B can be A and [some] C is B, so [some] C can be A.” It works the same way if the major is a negative of possibility. And if the major is restricted by “that which is” the syllogism will be no less evident. But that a universal conclusion does not follow is shown because from the fact that every moon can be the lowest planet and every planet shining on our hemisphere is the moon (let us suppose), it does not follow “so every planet shining on our hemisphere can be the lowest planet.” For the conclusion is false, because if it is expounded by a disjunction of the verb “is” and the verb “can” there will be a clear counterinstance of the sun, which can be a planet shining on our hemisphere and cannot be the lowest planet.

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There is a similar counterinstance to Celarent by replacing the previous affirmative major with the negative [premise] “every moon can fail to be the sun.” And the reason why a universal conclusion does not follow is that the minor extreme is more ampliated in the conclusion than in the minor premise, and from less broadly ampliated in any way to more broadly ampliated is not a valid consequence. But it should be known that if the minor premise and the conclusion are formed with “that which is,” so that ampliation is prevented in both, then a universal conclusion can be inferred in Barbara and in Celarent, and there is a perfect syllogism. But if the minor is not formed with “that which is,” then an ampliative middle can prevent every conclusion restricted by “that which is” in all affirmative moods. For it does not follow, “Everything going to die can be alive; the Antichrist is going to die; so he who is the Antichrist can be alive.” But in negative moods such a restricted conclusion is not prevented, because if the subject of the conclusion supposited for nothing presently existing, being negative [the conclusion] would be no less true, just as the affirmative would not be true. [p. 120] Eleventh Conclusion: In the first figure there will always be a valid syllogism from an assertoric major and a minor of possibility to a conclusion of possibility assuming {that the major is simply assertoric}, that is, that the major is necessary. As I said earlier,10 when I say that some assertoric is necessary that means the same as when I say, as Aristotle was wont to do, that it is simply assertoric. The conclusion is proven by the fact that the opposite of the hypothesis, that is, of the supposition [that the major premise is necessary], follows from the opposite of the conclusion and the minor [premise].11 For taking the syllogism: “‘Every B is A’ is necessary and every C can be B, so every C can be A,” then we obtain this syllogism from the opposite of the conclusion and the minor [premise], “Some C necessarily fails to be A, every C can be B, so some B necessarily fails to be A.” This is a good syllogism, by the sixth conclusion, but the conclusion cannot obtain together with the necessity of the major [premise] of the first [syllogism]. This is shown like this: the proposition “Some B necessarily fails to be A” can have three causes of truth. The first is that “A” or “B” supposits for nothing; then the first major premise, saying that every B is A, was false, [and] so was not necessary. The second cause of truth is that something is B and the same thing necessarily fails to

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be A, and in that case “Every B is A” would [again] be false. The third cause of truth is that something can be B, although nothing is actually B, and that same thing necessarily fails to be A. Then supposing that it becomes B, since that is possible, then it will not be A. So then “Every B is A” will be false, so it was not necessary, although it might perhaps be true. Similarly, I say that, taking a negative case, this is a good syllogism: “‘No B is A’ is necessary, and every C can be B, so every C can fail to be A.” For from the opposite of the conclusion and the minor [premise] we may draw a conclusion that cannot hold together with the necessity of the major [premise], as was said of the affirmative moods. For this conclusion, “Some B is necessarily A,” cannot hold together with the necessity of “No B is A.” [p. 121] Twelfth Conclusion: In the second figure there is no valid syllogism in virtue of form from one [premise] of possibility and the other assertoric; but on the hypothesis that the assertoric [premise] is necessary, there is always a valid syllogism to a conclusion of possibility if the major [premise] is assertoric but not if the major is of possibility. To show that nothing is valid in virtue of form, whether the major [premise] is assertoric and the minor of possibility or conversely, we can find a counterinstance to all moods in these terms: “shine,” “moon,” “lowest planet”; always taking “shine” as the middle [term], “moon” as major extreme and “lowest planet” as minor. But that it is always valid on the said hypothesis that the major is [simply] assertoric is proved first for Cesare and Festino, for by conversion of the major they turn into the first figure. Then all moods are proved at once per impossibile, for in Cesare and Festino, from the minor [premise] and the opposite of the conclusion there follows not only the conclusion “Some B can be A,” but also the conclusion “Something that is of necessity B can be A,” which is clear enough by an expository syllogism. For example, take the first syllogism in Cesare, “No B is A (supposing this is necessary), and every C can be A, so every C can fail to be B.” Then there will be a syllogism [leading] to impossibility like this: “Every C can be A and some C is necessarily B”; designate a particular C, and say “This C can be A and this same C is necessarily B,” then it clearly follows, “So something that necessarily is B can be A,” and this conclusion cannot hold together with the necessity of “No B is A.” Similarly in Camestres and in Baroco, from the minor [premise] and the opposite of the

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conclusion there follows the conclusion, “Something that is of necessity B can fail to be A,” which cannot hold together with the necessity of “Every B is A.” Form the syllogisms if you wish. That these syllogisms are not valid if the minor is assertoric, even if it is necessary, is proved by counterinstances, which seem difficult to find. But suppose that we live in an upright sphere, so that every star sometimes has an elevation relative to us and sometimes a declination. Then there is a counterinstance to Cesare and Festino, for every star possibly [p. 122] fails to be above our hemisphere and every star elevated above our horizon is above our hemisphere, but it does not follow, “so some star elevated above our horizon can fail to be a star.” Similarly, there is a counterinstance to Camestres and Baroco, for every star possibly is above our hemisphere, and no star that is below our hemisphere is above our hemisphere, but it does not follow, “so some star that is below our hemisphere can fail to be a star.” Thirteenth Conclusion: In affirmative moods in the third figure from a combination of an assertoric [premise] and one of possibility there is a valid [syllogism] in virtue of form to a conclusion of possibility if the premise of possibility is universal, but otherwise not, and in negative moods it is valid if the major [premise] is universal and of possibility, otherwise not. This conclusion is proved first as regards the useful moods. First, if the major [premise] is universal and of possibility, as in Darapti and in Felapton, Datisi and Ferison, it is immediately evident that conversion of the minor results in the first figure. Then also, in affirmative moods, if the minor [premise] is universal and of possibility, as in Darapti and in Disamis, conversion of the major [premise] and of the conclusion and interchanging the premises results in the first figure having a major [premise] of possibility. So it is clear that in this way all these moods are valid. Next, the conclusion stated is proved regarding the useless moods. First, if the [premise] of possibility is particular, in affirmative moods there is a counterinstance in these terms, “sun,” “moon,” “planet shining on our hemisphere”; let “sun” be the major extreme, “moon” the minor extreme, and “planet shining on our hemisphere” the middle [term]. For in every combination of premises a situation can be set up so that the premises will be true and the conclusion false. Similarly, in negative moods, if the particular [premise] is of possibility, there will be a counterinstance in these terms: “sun,” “brightest planet” and “planet shining on our hemisphere”; let the middle [term] be “planet shining on our hemisphere.”

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Finally, we can also show that negative moods are not valid if the minor [premise] is of possibility, even if it is universal. For nothing creating is God [p. 123] (let us suppose) and everything creating can be the first cause, but it does not follow “so the first cause can fail to be God.” Fourteenth Conclusion: In the third figure, there is a valid [syllogism] from a combination of an assertoric [premise] and one of possibility to a conclusion of possibility on the condition that the assertoric [premise] is necessary. This conclusion is evident as regards those moods that were claimed in the previous conclusion to be valid without the condition. For adding the condition does not prevent the consequence. But Felapton, Datisi, and Ferison, if they have a minor [premise] of possibility, are reduced to the first figure by conversion of the minor [premise], and in the first figure a conclusion of possibility certainly follows on the stated condition. Similarly, Disamis, if it has a major [premise] of possibility, is reduced to the first figure by conversion of the major [premise] and of the conclusion and by interchanging the premises. But Bocardo is proved per impossibile, first, if the major [premise] is of possibility, because from the opposite of the conclusion with the major [premise] there follows the conclusion, “Some C necessarily fails to be B,” by the fifth conclusion,12 and this conclusion does not hold together with the necessity of the original minor [premise], which said “Every C is B.” While, if the minor [premise] is of possibility, then from the opposite of the conclusion and the minor [premise] there follows the conclusion, “Every C is necessarily A,” by the fourth conclusion,13 and the conclusion does not hold together with the necessity of the original major [premise], which said “Some C is not A.” Fifteenth Conclusion: In the first figure from an assertoric major [premise] and a minor [premise] of necessity a conclusion of necessity does not follow, nor does an assertoric conclusion follow except in Celarent. That a conclusion of necessity does not follow is evident, for every God is creating (let us suppose) and every first cause is of necessity God, but it does not follow that a first cause is necessarily creating. Similarly, no God is creating (let us suppose), every first cause is of necessity God, but it does not follow that a first cause necessarily fails to create. [p. 124] Now that an assertoric conclusion does not follow in Barbara and in Darii is proved, because supposing that God at present is not creating, it does not

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follow, “Every God is just and everything creating is necessarily God, so someone creating is just,” since the premises are true and the conclusion false in that situation. Also, that an assertoric conclusion does not follow in Ferio is clear, for suppose that at present the moon is not shining, then it is argued like this, “No lowest planet is other than the moon, something shining is of necessity the lowest planet,” but it does not follow, “so something shining is not other than the moon,” since the conclusion is false and the premises are true. For although the moon is not shining, nonetheless, this is true, “Something shining is of necessity the moon,” since something that can be shining is of necessity the moon or the lowest planet. But that an assertoric conclusion follows in Celarent is proved by taking this syllogism, “No B is A, every C is of necessity B, so no C is A.” Everyone knows that from “No B is A” there follows “Nothing that is B is A,” by the twelfth conclusion of Book I. Then we express the minor [premise] explicitly as “Everything that is or can be C is of necessity B.” From this it follows that everything that is or can be C is B, because that which is of necessity B is in fact B. Then the syllogism becomes, “Nothing that is B is A, everything that is or that can be C is B,” [from which] it clearly follows by the dictum de nullo, “so nothing that is or can be C is A.” From that, moreover, there follows “so no C is A,” so the conclusion follows from the original [premises]. It should be realized, however, that if the minor [premise], which is of necessity, is restricted by “that which is,” then in all moods of the first figure there follows an assertoric conclusion. For from a minor [premise] of necessity so restricted, there follows an assertoric minor also restricted. For it follows, “That which is B is necessarily A, so that which is B is A.” And then both premises are assertoric. Sixteenth Conclusion: From a major [premise] of necessity and an assertoric minor, there is always a valid syllogism in the first figure to a particular conclusion of necessity, but not to a universal.14 [p. 125] This conclusion is proved and expounded or modified in an altogether similar way, mutatis mutandis, to the way the second part of the tenth conclusion was proved and expounded or modified. But it must be said that there does follow an assertoric universal. Seventeenth Conclusion: From a negative major [premise] of necessity and an assertoric minor there is always a valid {syllogism} in the second figure to a particular conclusion of necessity, but not to a universal; but if

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the major is affirmative of necessity or assertoric, there is no valid syllogism to a conclusion of necessity, but it is valid to an assertoric conclusion. The first part of the conclusion claims that Cesare and Festino are valid to a conclusion of necessity, but not to a universal.15 The reason is that by conversion of the major [premise] Cesare immediately becomes Celarent and by conversion of the major [premise] Festino becomes Ferio. So the conclusion follows, just as it does in Celarent and in Ferio in the previous conclusion. But it is proved that no mood is valid if the major [premise] is assertoric, for in Cesare and in Festino there is a counterinstance like this: “Nothing creating is God (supposing this) and every first cause is of necessity God,” but it does not follow that some first cause necessarily fails to be creating. Similarly, there is a counterinstance against Camestres and Baroco, for every planet shining on our hemisphere is the sun (let us suppose), and every moon necessarily fails to be the sun, but it does not follow, “so the moon necessarily fails to be a planet shining on our hemisphere.” Similarly, it is clear that it is not valid if the major [premise] is affirmative, even if it is of necessity, for instance, in Camestres and Baroco, for every first cause is of necessity God and nothing creating is God (let us suppose), but it does not follow, “so the one creating necessarily fails to be a first cause.” But that the said moods are valid to an assertoric conclusion is clear,16 for from a [premise] of necessity there always follows an assertoric except in the case where it is true only for those which can be. But this case does not prevent the truth of a negative assertoric conclusion. [p. 126] Eighteenth Conclusion: In the third figure, a syllogism is always valid to a conclusion of necessity from a universal major of necessity and an assertoric minor, but from an assertoric major it is not valid to a direct conclusion of necessity, nor from a major of necessity if it is particular. The first part of the conclusion would be evident in all moods by expository syllogisms, namely, in Darapti, in Felapton, Datisi, and Ferison. It can also be proved per impossibile, since from the opposite of the conclusion with the minor the opposite of the major can be inferred, by the tenth conclusion. But that it is not valid to a conclusion of necessity when the major is assertoric is shown first for the affirmative moods. For every God is creating (let us suppose) and every God is necessarily the first cause, but it does not follow that the first cause is necessarily creating. Nonetheless, an indirect

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conclusion of necessity can be concluded. For interchanging the premises, the major would be of necessity and the conclusion direct. It is shown similarly for negative moods. For no God is creating (let us suppose) and every God is of necessity the first cause, but it does not follow that the first cause necessarily fails to create. Similarly, it is evident that it is not valid to a conclusion of necessity if the major is particular, even if it is of necessity. First, because supposing that the moon is not now shining, there is a counterinstance to Disamis like this: “Something shining is necessarily the moon, and everything shining is other than the moon,” but it does not follow, “so {something} other than the moon is necessarily the moon”; indeed, it is shown in this example that there does not follow an assertoric conclusion nor one of possibility. But if the major were restricted by “that which is,” there would follow a conclusion of necessity, which can be seen by an expository syllogism. There is also a counterinstance to Bocardo. Supposing that nothing is now shining except a star, nonetheless fire can be started that tomorrow will shine and that is necessarily not a celestial body. Then like this, “Something shining necessarily fails to be a celestial body and everything shining is a star,” it does not follow, “so a star necessarily fails to be a celestial body.” However, if the major were restricted by “that which is,” there would follow a conclusion of necessity. [p. 127] Nineteenth Conclusion: In the first figure and in the second figure, from one {premise} of necessity and the other assertoric there always follows a conclusion of necessity on the assumption, namely, that the assertoric is necessary. The conclusion is proved first in the first figure. For if the major is of necessity and the minor assertoric there follows without the assumption a conclusion of necessity, by the sixteenth conclusion, and given the assumption the consequence stated still holds, indeed is even more useful. For it is useful in universal moods, since it makes a universal conclusion of necessity follow, which without the assumption only follows if it is particular. For example, “Every B is of necessity A and every C is B ([given that] this is necessary),” it follows, “so every C is of necessity A.” For from the major and the opposite of the conclusion the opposite of the assumption follows like this, “Every B is of necessity A, some C can fail to be A,” it follows, by the fifth conclusion, “so some C necessarily fails to be B,” where the conclusion does not hold together with the necessity of the original minor

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{premise} saying that every C is B. Similarly, in Celarent, “Every B necessarily fails to be A and every C is B ([given that] this is necessary),” it follows, “so every C necessarily fails to be A.” For from the major and the opposite of the conclusion there follows, as before, “so some C necessarily fails to be B.” Now if the major is assertoric and the minor of necessity, then there is a valid syllogism like this, “Every B is A ([given that] this is necessary) and every C is of necessity B, so every C is of necessity A.” For from the opposite of the conclusion and the minor there follows not only the conclusion, “Some B can fail to be A,” but there also follows the conclusion, “Something that is of necessity B can fail to be A,” as can also be seen by an expository syllogism, and the conclusion does not hold together with the necessity of “Every B is A.” It is shown similarly in Celarent. For there is a valid syllogism like this, “No B is A ([given that] this is necessary) and every C is of necessity B, so every C necessarily fails to be A.” For from the opposite of the conclusion and the major there follows, “so something that is of necessity B can be A,” and the conclusion does not hold together with the necessity of the major saying “No B is A.” It is proved similarly in Darii and Ferio. Next, the conclusion is proved for the second figure, and first if the major is of necessity and the minor assertoric. For Cesare and Festino [p. 128] are reduced to the first figure by conversion of the major [premise]. For a universal negative is of necessity converted simply. But all four moods are proved at once per impossibile. For from the major [premise] and the opposite of the conclusion there always follows a conclusion that does not hold together with the necessity of the minor [premise]. Similarly, if the major is assertoric and the minor of necessity, then Cesare and Festino are as before reduced to the first figure by conversion of the major [premise]. Camestres is also reduced by conversion of the minor [premise] and of the conclusion and by interchanging the premises. But Baroco, and also all other moods mentioned, are proved per impossibile. For from the minor [premise] and the opposite of the conclusion there follows in the third figure a conclusion that cannot hold together with the necessity of the major [premise]. Twentieth Conclusion: In the third figure from a major [premise] of necessity and an assertoric minor a syllogism is always valid to a conclusion of necessity on the assumption, namely, that the assertoric is necessary, but it is not valid if the major was assertoric.

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The first part of the conclusion is evident if the major [premise] is universal, for without the assumption a conclusion of necessity follows, as is seen by the eighteenth conclusion, and the assumption does not impede that consequence. But if the major [premise] is particular, as in Disamis and Bocardo, the syllogism is proved per impossibile. For from the opposite of the conclusion and the major [premise] a conclusion follows that cannot hold together with the necessity of the minor. But the second part of the conclusion is proved by counterinstances. First, a counterinstance in affirmative moods is like this: “Every degree of the zodiac elevated above our horizon is elevated above our hemisphere ([given that] that is necessary), and every degree of the zodiac elevated above our hemisphere is of necessity a degree of the zodiac,” but it does not follow, “so some degree of the zodiac is of necessity elevated above our hemisphere.” Similarly for a counterinstance to negative moods. For no degree of the zodiac that is below our hemisphere is above our hemisphere and every degree of the zodiac that is below our hemisphere is necessarily a degree of the zodiac, but it does not follow, “so some degree of the zodiac necessarily fails to be above our hemisphere.”

Chapter 3: Syllogisms between Divided Modal Propositions of Each-Way Contingency If next we wish to deal with syllogisms between propositions of each waycontingency, the seventh conclusion of Book II should be recalled, namely, that every proposition of each-way contingency is converted into the opposite quality of dictum, but never into the opposite quality of mode, so that an affirmative is equivalent to a negative and a negative to an affirmative. Next, it should also be recalled that some propositions of contingency are composite and others divided, just as those of possibility and of necessity. Recall, moreover, that I do not intend to say any more about composites, but everything I say in what follows should be understood to concern divided17 [contingent modals]. Next, it should also be recalled that some divided [modals] of contingency have an affirmed mode and some a negated mode, just as was said of other modals. Now, in order that the explanation of those of contingency is reduced to the explanation of those of possibility and those of necessity, we should set

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out how those of contingency are related to those of possibility or of necessity. About this, it should be noted that from every proposition of contingency with an affirmed mode there follow both affirmative and negative propositions of possibility also with an affirmed mode. For it follows, “B is contingently A,” or “B contingently fails to be A,” “so B is possibly A,” and “so B possibly fails to be A.” Whence “B is contingently A” is equivalent to “B can be A and can fail to be A,” and so too is “B contingently fails to be A,” because affirmatives and negatives of contingency are equivalent. But it should be noted that a particular or indefinite of contingency is not equivalent to a conjunction made up of an affirmative and a negative of [p. 130] possibility unless the second [conjunct] of possibility is taken with a relative of identity. For this conjunction is true, “Some planet can be the moon and some planet can fail to be the moon,” but this is false, “Some planet is contingently the moon.” Next, from every proposition of necessity with an affirmed mode, whether affirmative or negative, there follows a proposition of contingency with a negated mode. For it follows, “B is necessarily A,” or also “B necessarily fails to be A,” “so B is not contingently A,” and “so B does not contingently fail to be A.” So “No B is contingently A” is equivalent to “Every B is necessarily A or necessarily fails to be A.” But we should not accept that a universal of contingency with a negated mode is equivalent to a disjunction made up of an affirmative and a negative of necessity with an affirmed mode. For this is true, “No planet is contingently the moon,” but this is false, “Every planet is necessarily the moon or every planet necessarily fails to be the moon.” In this connection, moreover, it should be realized that in a proposition of contingency the subject is ampliated to supposit for those that are and for those which can be, and it is not required that the subject supposit for those that are contingently. For God is contingently creating, but nothing that is contingently God [is] contingently creating, because nothing is contingently God, indeed, everything is necessarily God or necessarily fails to be God. On these assumptions we can present some conclusions. Twenty-First Conclusion: If in any given syllogism to a given conclusion having a premise of possibility with an affirmed mode that premise is replaced by one of contingency with an affirmed mode, whether affirmative or negative, there follows the same conclusion, and from whatever premises

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from which no conclusion of possibility with an affirmed mode follows a conclusion of contingency with an affirmed mode does not follow. The whole [conclusion] is proved by the fourth conclusion of Book I. For [a proposition] of possibility follows from one of contingency, as was said.18 Then [p. 131] as a corollary we conclude that from two negative [premises] of contingency there is a valid syllogism whenever it is valid from affirmatives of possibility. Twenty-Second Conclusion: From whatever premises there follows a conclusion of necessity with an affirmed mode, there [also] follows a conclusion of contingency with a negated mode. This conclusion is proved by the fourth conclusion of Book I, just as the previous one, for [propositions] of necessity imply [propositions] of contingency [with negated mode]. Twenty-Third Conclusion: In the first and third figures from a major [premise] of contingency, whether with an affirmed mode or a negated mode, there follows a similar conclusion of contingency if the minor [premise] is of necessity or of possibility or of contingency. This conclusion, in the case of the first figure, is shown by the dictum de omni et nullo, just as the fourth conclusion of this book was shown. But as regards the third figure, it can be shown by expository syllogisms and per impossibile, just as the sixth conclusion of this book was shown. Twenty-Fourth Conclusion: From a major [premise] of contingency and an assertoric minor, the first figure is valid to a particular conclusion of contingency, but not a universal. This conclusion is demonstrated just as the second part of the tenth conclusion of this book was demonstrated. For that a universal conclusion does not follow is seen because every human contingently laughs and everything running is a human (assuming it is so), [but] then the universal conclusion would be false. And if the major [premise] has a negated mode, there is a counterinstance, because no horse contingently laughs; everything running is a horse (assuming it is so), [but] the universal conclusion would also be false. [p. 132] Twenty-Fifth Conclusion: From a universal major [premise] of contingency in the third figure and an assertoric minor, there follows a conclusion also of contingency, but if the major is particular a conclusion of contingency does not follow. The first part of the conclusion is proved because in all moods of the third figure having a universal major [premise], if the minor, which is taken to be

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assertoric, is converted, the first figure results, which was said to be valid in the previous conclusion. But the second part is clear because while someone running is contingently laughing and everything running is a horse, nonetheless no horse is contingently laughing. It is the same if the major [premise] is taken to be negative, because it is equivalent to an affirmative. But if we speak of a negated mode, then there is a counterinstance, for someone thinking is not contingently creating and everyone thinking is God (let us suppose), but it does not follow, “so God is not contingently creating.” Twenty-Sixth Conclusion: In the first and third figures, there is no valid syllogism to a conclusion of contingency with an affirmed mode unless the major [premise] is of contingency with an affirmed mode; for there is nothing useful in a minor [premise] of contingency. First, it is clear in the first figure that even if everything creating is God and can be God, and is of necessity God, and that every first cause is contingently creating, nonetheless it is not true that the first cause is contingently God. Similarly, there will be a counterinstance in negative moods, for nothing running is a stone and every horse is contingently running, but it is not true that a horse may contingently fail to be a stone, for [then] it would contingently be a stone. Similarly, in the third figure, although every nonluminous planet is the moon and every nonluminous planet is contingently beneath our hemisphere, nonetheless it is not true that something beneath our [p. 133] hemisphere is contingently the moon. Similarly, if the major [premise] is of necessity or of possibility, for then {the premises} would be true and the conclusion false. It is the same in negative moods, for no nonluminous planet is the sun, but taking the minor [premise] as before, the conclusion will be false if it is one of contingency with an affirmed mode. Twenty-Seventh Conclusion: In the second figure, no conclusion of contingency with an affirmed mode follows. For if each premise is of necessity, there will be a counterinstance in the terms “planet,” “moon,” “stone,” where “planet” is the middle [term]. With these terms, the minor [premise] will be no less true if it is assertoric or of possibility, and the conclusion will always be false if it is contingent with an affirmed mode. And if both premises are of contingency, there will be a counterinstance in the terms “running,” “horse,” “human,” where the middle [term] is “running.”

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Chapter 4: Syllogisms between Reduplicative Propositions Twenty-Eighth (and Last) Conclusion: The first and third figures are valid to a reduplicative conclusion if the major [premise] is reduplicative, otherwise not, and the second figure is never valid to a reduplicative conclusion. It should be noted that there are four principal components to a reduplicative proposition: first, the principal subject, second, the principal predicate, third, the reduplicative term, and fourth, the reduplication, for example, “Humans are sensible qua animal”; for “humans” is the principal subject, “sensible” is the principal predicate, “animal” is the reduplicative term, and the expression “qua” is the reduplication. It should be realized that reduplication in an affirmative proposition signifies [p. 134] an immediacy or exactness of the principal predicate to the reduplicative term. So the principal predicate and the reduplicative term should not be the same, for it is pointless to signify an exactness or immediacy of the same thing to itself. So it is not a proper expression to say that a human is an animal qua animal or that an animal is sensible qua sensible. But sometimes the same term is the subject and also the reduplicative term, namely, if the principal predicate exactly fits the principal subject by a reduplication signifying the exactness or immediacy by a reduplication signifying immediacy. For thus we say that humans qua human are capable of laughter, but Socrates is not capable of laughter qua Socrates but qua human. It should be realized that although the reduplicative proposition is sometimes granted on account of immediacy alone—for example, “A human is an animal qua human”—and is sometimes granted on account of exactness alone—for example, “A triangle qua triangle has three angles equal to two right angles”—nonetheless, most properly a reduplicative is one in which exactness and immediacy run together. Now sometimes reduplicative expressions are used improperly, namely, specificatively and not reduplicatively, and then they specify a reason or sense in which the proposition should be understood, which without that specification would not be literally true, for instance, if we say that man is a species in that the term “man” supposits materially, and a form in that it is in pure potency first matter, and Homer is the archetypal poet. Propositions of this sort are not proper reduplicatives, so we do not intend [to deal] with them.

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But having noted this, one may doubt concerning reduplicatives whether the reduplicative term belongs to the subject, so that it is part of the whole subject, or belongs to the predicate, so that it is part of the whole predicate. One may also wonder in what way they are converted. To this it must be said, following Aristotle, that the reduplicative term should be taken with the major extreme, and in the first figure that is predicated both of the middle [term] and of the minor extreme; so [p. 135] the reduplicative term is part of the whole predicate. This is also shown by [the fact] that if it were part of the subject in the major premise, then it would be part of the syllogistic middle, and then it would not occur in the conclusion. So in the natural order one should say “Humans are capable of laughter qua human”; and if we sometimes say that humans qua human are capable of laughter, this should, nonetheless, be understood as if the terms were construed differently. So we should say that “human” is the subject and “capable of laughter qua human” is the predicate, notwithstanding [the fact] that in propounding it we prefer to put “qua human” before the copula. So the proposition “Humans qua human are capable of laughter” also is not converted into “One capable of laughter is human qua human,” for it would be improper, as was said, or it would be false and unintelligible, as Aristotle says in the Prior Analytics,19 but it should be converted into “That which is capable of laughter qua human is human.” Similarly, “Socrates is sensible qua animal” is converted [to] “so something that is sensible qua animal is Socrates.” Having noted this, I [can] establish the conclusion. First, it is evident by the dictum de omni et nullo that the first figure is valid with a reduplicative major [premise], for it is clearly directly subsumed under a distributed middle in the minor premise. The same is true whether the minor is used with reduplication or without reduplication, for in each case it is sufficiently subsumed under the middle. That the third figure is also valid when the major [premise] is reduplicative can be proved by expository syllogisms and per impossibile. But that the first figure is not valid with a reduplicative conclusion when the major [premise] is not reduplicative is seen because reduplication should not be added to the conclusion if it was not added to one of the premises, and if it were added to the minor premise, in which the middle [term] is the predicate, then it would be part of the middle [term], which does not occur in the conclusion, so what is concluded could never for this reason include the reduplication.

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The same is seen in the third figure, for if the conclusion is reduplicative, the reduplication must be in the predicate, as was said, and the predicate was the major extreme if the conclusion was direct, so the major [premise] must have been reduplicative. But if we want to conclude indirectly, then the minor [premise] should have been reduplicative. For example, [from] “Every human is an animal and every human is capable of laughter qua human,” it follows, “so some animal is capable of laughter qua human,” for interchanging [p. 136] the premises produces a direct mood. It must also be conceded that from a reduplicative minor [premise] it can be concluded like this, “Every human is an animal, every human is capable of laughter qua human, so something that is capable of laughter qua human is an animal”; but this conclusion is not reduplicative, although it does have a reduplicative subject. Now it is evident that the second figure is not valid with a reduplicative conclusion, first, because it does not conclude affirmatively, and second, because whichever premise included a reduplication, the reduplication would be part of the middle [term], in that the middle [term] is the predicate in each premise, so reduplication could never for this reason be added to the conclusion. However, it should be realized that every figure and mood inferring a negative assertoric without reduplication can be inferred from the same negative premises with reduplication. For a negative with reduplication follows from a negative assertoric without reduplication, for it follows, “B is not A, so B is not A qua C,” and “No stone is capable of laughter, so no stone is capable of laughter qua human.”

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Notes Introduction Stephen Read 1. See B. Michael, Johannes Buridan: Studien zu seinem Leben, seinen Werken und zur Rezeption seiner Theorien im Europa des späten Mittelalters (Berlin: Freie Universität Berlin, 1995), 1:79–238. See also G. Klima, John Buridan (Oxford: Oxford University Press, 2009), chapter 1, and J. Zupko, John Buridan (Notre Dame, Ind.: Notre Dame University Press, 2003), Introduction. 2. See, e.g., M. Fitzgerald, Albert of Saxony’s Twenty-Five Disputed Questions on Logic (Leiden: Brill, 2002), 1. 3. J. Buridan, Summulae de Dialectica: An Annotated Translation, with a Philosophical Introduction, trans. G. Klima (New Haven: Yale University Press, 2001). Hereafter SD. 4. J. Buridan, Questiones Elencorum, ed. R. van der Lecq and H. A. G. Braakhuis (Leiden: Ingenium, 1994), 92. 5. See G. Gál, “Adam Wodeham’s Question on the complexe significabile,” Franciscan Studies 37 (1977): 66–102. 6. K. Tachau, Vision and Certitude in the Age of Ockham (Leiden: Brill, 1988), chapter 12. 7. Zupko, John Buridan, 126. 8. J. Buridan, Sophisms on Meaning and Truth, trans. T. K. Scott (New York: Appleton-Century Crofts, 1966). Cf. T. K. Scott, Johannes Buridanus: Sophismata, (Amsterdam: Frommann-Holzboog, 1977); G. E. Hughes, John Buridan on Self-Reference: Chapter Eight of Buridan’s Sophismata (New York: Cambridge University Press, 1982); Buridan, Summulae de Practica Sophismatum, ed. F. Pironet (Antwerp: Brepols, 2004). 9. H. Hubien, Iohannis Bvridani: Tractatvs de Conseqventiis (Louvain: Publications Universitaires, 1976). 10. P. King, Jean Buridan’s Logic: The Treatise on Supposition; The Treatise on Consequences. Translated with a Philosophical Introduction (Amsterdam: Reidel, 1985). See S. Read, “Review,” Vivarium 25 (1987): 154–157. 11. Strictly speaking, they write “things being wholly (or only) as it signifies (ita est totaliter sicut significat).” See, e.g., Thomas Bradwardine, Insolubilia, trans. Stephen Read (Amsterdam: Peeters, 2010), Introduction §5.

163

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12. See, e.g., D. Lewis, “Critical Notice of D.M. Armstrong, A Combinatorial Theory of Possibility,” Australasian Journal of Philosophy 70 (1992): 211–224, esp. 218–219. 13. See G. de Occam, Summa Logicae, ed. G. Gál et al. (Allegany, N.Y.: Franciscan Institute Publications, 1974), III–3, chapter 1. 14. The Latin text is found in Walter Burleigh, De Puritate Artis Logicae, Tractatus Longior, ed. P. Boehner (Allegany, N.Y.: Franciscan Institute Publications, 1955), 86. There is an alternative English translation in W. Burley, On the Purity of the Art of Logic, trans. P. V. Spade (New Haven: Yale University Press, 2000), 173. 15. See, e.g., Carnap’s comments referred to by W. D. Hart in his review of Etchemendy’s The Concept of Logical Consequence in The Philosophical Quarterly 41 (1991): 491–492. 16. In fact, the distinction between simple and ut nunc truth goes back to Aristotle, Prior Analytics 34b7ff. See §5. 17. See SD 4.3.2. A tricky case would be, e.g., the word “word,” which might supposit for itself in either material or personal supposition. 18. See, e.g., SD 4.3.5–6. If σ is “no,” think of “No A is B” as “Every A is not B.” 19. At SD 4.3.6, Buridan writes: “Merely confused supposition is that in accordance with which none of the singulars follows separately while retaining the other parts of the proposition, and neither do the singulars follow disjunctively, in terms of a disjunctive proposition, although perhaps they do follow by a proposition with a disjunct term.” The qualification “perhaps” (i.e., in some cases) is explained at SD 4.3.8.4, when he discusses the confusion effected by cognitive terms like ‘know’: “The following [inference] is valid: ‘Every man is an animal, therefore every man is this animal or that animal.’ . . But in the present mode of confusion [i.e. of terms like “know”] it is not possible to descend to the supposita in terms of either a disjunctive proposition or a categorical with a [disjunct] term.” See §4.6. 20. See, e.g., Matthew Dryer, “Order of Subject, Object, and Verb,” World Atlas of Language Structures Online, at http://wals.info. 21. Another influence of the vernacular was the increasing use of ad and the accusative to replace the dative and ablative. See R. E. Latham and D. R. Howlett, eds., Dictionary of Medieval Latin from British Sources (Oxford: Oxford University Press for the British Academy, 1975), under the entry for ad: “as in CL [classical Latin], but used more extensively, like Fr. à, esp. in place of dat. or abl.” We see this in the use by Buridan and many of his contemporaries in the expression of logical laws such as “ad omnem propositionem impossibilem omnem aliam sequi et omnem propositionem necessariam ad omnem aliam sequi” (Treatise on Consequences [hereafter TC] Book I, Chapter 8, first conclusion): “from every impossible proposition, every other follows; and every necessary proposition follows from every other.”

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22. See, e.g., Burley, On the Purity of the Art of Logic; Tractatus Longior, chapter 4; Buridan, SD 4.3.7.1. 23. See, e.g., C. T. Thörnqvist, Anicii Manlii Severini Boethii De Syllogismo Categorico (Gothenburg: Gothenburg University Press, 2008), 20 ff. 24. See D. Londey and C. Johanson, The Logic of Apuleius (Leiden: Brill, 1987), 108–112. 25. Arguably, adding propositions in nonnormal form is equivalent to allowing quantification of the predicate: e.g., “No S (some) P is not” expresses “Every S is every P,” “Every S (some) P is not” expresses “No S is every P,” and so on. 26. “No S is P” converts simply to “No P is S,” which has subaltern “Not every P is S,” or “Some P is not S.” 27. See G. Hughes, “The Modal Logic of John Buridan,” in Atti del Convegno Internazionale di Storia della Logica. Le Teorie della Modalità, ed. G. Corsi et al. (Bologna: CLUEB, 1989), 93–111, at 97. See also C. Normore, “Buridan’s Ontology,” in How Things Are, ed. J. Bogen and J.E. McGuire, 189–203 (Antwerp: Reidel, 1985). 28. See, e.g., Aristotle, Parts of Animals I 1, 641a1: “a dead body has exactly the same configuration as a living one; but for all that is not a man,” and Meteorology IV 12, 389b32: “a dead man is a man only in name.” 29. See, e.g., Bertrand Russell, “On Denoting,” Mind 14 (1905): 479–493. 30. See R. Smith, “Aristotle’s Logic,” Stanford Encyclopedia of Philosophy (Summer 2011 Edition), ed. Edward N. Zalta, at http://plato.stanford.edu/ archives/sum2011/entries/aristotle-logic/, §5.2. On Śleszyński, see W. Bednarowski, “Hamilton’s Quantification of the Predicate,” Proceedings of the Aristotelian Society 56 (1955–56): 217–240, esp. 225. See also J. Łukasiewicz, Aristotle’s Syllogistic (Oxford: Oxford University Press, 1957), 72; I. M. Bochenski, A History of Formal Logic, trans. I. Thomas (New York: Chelsea, 1961), §13; W. Kneale and M. Kneale, The Development of Logic (Oxford: Clarendon Press, 1962), §6. 31. It was arguably not true to Aristotle, either; see Read, “Aristotle and Łukasiewicz on Existential Import.” 32. See, e.g., J. N. Keynes, Formal Logic (London: Macmillan, 1884), chapter 4; C. Read, Logic: Deductive and Inductive (London: De La More Press, 1914), chapter VIII §5. 33. See, e.g., E.J. Ashworth, Language and Logic in the Post-Medieval Period (Antwerp: Reidel, 1974), 224. 34. Aristotle, Prior Analytics Book I, trans. G. Striker (Oxford: Clarendon Press, 2009), 24b28–30. All citations from the first book Aristotle’s Prior Analytics are taken from this translation. 35. P. King, Jean Buridan’s Logic, 71; H. Lagerlund, “Medieval Theories of the Syllogism,” Stanford Encyclopedia of Philosophy (Spring 2010 Edition), at http://plato.stanford.edu/archives/spr2010/entries/medieval-syllogism, §7.

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36. My translation from Bonaventure, Commentaria in Quatuor Libros Sententiarum, Book I Distinction 33 Article I Question 3, at www.franciscan-archive .org/bonaventura/I-Sent.html. The formula actually occurs in Aristotle at De Sophisticis Elenchis, chapter 6 (168b31). 37. “Darapti” is a medieval designation for the third-figure mood aai, drawing a particular affirmative from two universal affirmatives. The mnemonic includes all moods recognized by Aristotle, extended by others added by Theophrastus. It reads: Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum; Cesare Camestres Festino Baroco; Darapti Felapton Disamis Datisi Bocardo Ferison.

38. 39. 40. 41.

42.

43. 44. 45. 46.

The first four terms refer to the direct conclusion in Figure I; the next five to indirect conclusions in that figure; the next four to Figure II; and the last six to Figure III. The first three vowels in each case denote the quality and quantity of premises and conclusions. The initial consonant indicates to which direct mood of the first figure a mood reduces. The role of the consonants c, m, p, and s is explained in SD 5.2.4, 321–324, citing Peter of Spain’s Tractatus, ed. L. M. De Rijk (Assen: van Gorcum, 1972), 52. However, since they describe Aristotle’s method of reduction, which Buridan eschews, they need not concern us here. See Alexander of Aphrodisias, On Aristotle Prior Analytics 1.1–7, trans. J. Barnes et al. (London: Duckworth, 1991), 99:27–30 (173). My own translation from the Latin text at http://la.wikisource.org/wiki/ Quaestiones_in_Analytica_priora. See Lagerlund, “Medieval Theories of the Syllogism,” §7. The same is true of Camestre and Cesares. Buridan himself concedes as much at SD 5.2.1, 320. For Buridan’s octagon of opposition for oblique terms, see S. Read, “John Buridan’s Theory of Consequence and His Octagons of Opposition,” in Around and Beyond the Square of Opposition, ed. Jean-Yves Beziau and Dale Jacquette (Berlin: Birkhäuser/Springer, 2012), 93–110. See S. Read, “Medieval Theories: Properties of Terms,” Stanford Encyclopedia of Philosophy (Spring 2011 Edition), at http://plato.stanford.edu/archives/ spr2011/entries/medieval-terms, §5. Aristotle is here responding to a sophism in Plato’s Euthydemus 298e. See G. Priest and S. Read, “Intentionality—Meinongianism and the Medievals,” Australasian Journal of Philosophy (2004): 416–435. Note again the need for a fixed and novel SVO order for propositions, with SOV a nonnormal alternative, or even OSV. See D.C. Williams, “On the Elements of Being: I,” Review of Metaphysics 7 (1953): 3–18; D. M. Armstrong, Universals (Boulder, Colo.: Westview Press, 1989), chapter 6.

notes to pages 29–35 47. 48. 49. 50.

51. 52.

53.

54.

55.

56.

57.

167

SD 9.4. Cf. G. Klima, John Buridan, 196. See, e.g., C. J. Shields, Aristotle (New York: Taylor & Francis, 2007), 415. Lagerlund, Modal Syllogistics in the Middle Ages (Leiden: Brill, 2000), 63. L marks a necessity proposition, M a possibility proposition, Q a contingency one, and X an assertoric. Note that calling a proposition a “necessity proposition” and so on, does not mean it is necessary, only that it states a necessity, as Buridan remarks in II 1. See Alexander of Aphrodisias, On Aristotle Prior Analytics, 124:8–25. One plausible alternative is Rescher’s. See his “Aristotle’s Theory of Modal Syllogisms and Its Interpretation,” in The Critical Approach to Science and Philosophy, ed. M. Bunge (New York: Free Press, 1964), 152–177. See, e.g., Lagerlund, Modal Syllogistics in the Middle Ages, 13–14. Another notable interpretation is given in M. Malink, Aristotle’s Modal Syllogistic (Cambridge, Mass.: Harvard University Press, 2013). The modal octagon also appears in Buridan’s Questiones longe super Perihermeneias, ed. R. van der Lecq (Nijmegen, 1983), 87. There is a transcription in G. Hughes, “The Modal Logic of John Buridan,” 109. The issue is raised by Pseudo-Scotus in his Questions on the Prior Analytics, Book I q. 26, in John Duns Scotus, Opera Omnia, ed. Vivès ( Paris 1891–95), 2:143 ff. Buridan argues in favor of the analysis with a disjunct term, whereas Pseudo-Scotus prefers the reading with a disjunctive proposition. See, e.g, Lagerlund, Modal Syllogistics in the Middle Ages, 187–189. See also G. Priest and S. Read, “Ockham’s Rejection of Ampliation,” Mind 90 (1981): 274–279, where Ockham’s arguments in favor of an ambiguity are considered. Buridan uses counterexamples concerning God and the first cause repeatedly. What we need to know is that on the medieval conception, only God can be the first cause, and so if anything is creating, it is necessarily God, but as a matter of fact, God is not now creating. Another example he uses again and again for counterexamples to modal syllogisms concerns the planets. One needs to know that for Buridan and his contemporaries, there were seven planets, starting from the lowest, the Moon, followed by Venus, Mercury, the Sun, Mars, Jupiter, and Saturn, any of which may or may not be shining on our hemisphere at any one time. To distinguish formally between the composite and divided senses, it is helpful to indicate the modality in divided modals by subscripting its sign to the copula, in line with Buridan’s own remark at TC II 2: “The mode attaches to the copula as a determination of it.” The revenge problem is this: suppose one claims that a paradoxical proposition is φ (e.g., is false, lacks truth-value, is meaningless, etc.). Now consider the claim that the claim that the proposition is φ. See R.M. Martin, ed., Recent Essays on Truth and the Liar Paradox (Oxford: Oxford University Press, 1984), 4. Buridan claims that the liar proposition (“This proposition is false”) is false. What should we say of Buridan’s own claim?

168

notes to pages 35–54

58. See Hughes, John Buridan on Self-Reference, 25. 59. Lagerlund, Modal Syllogistics in the Middle Ages, 160 n. 56. 60. See, e.g., Ruth Barcan Marcus, “A Backward Look at Quine’s Animadversions on Modalities,” in Perspectives on Quine, ed. R. B. Barrett and R.F. Gibson (Oxford: Blackwell, 1989), 230–243. 61. See Aristotle, Prior Analytics I 15, 34a22–24; Lewis and Langford, Symbolic Logic, 164; Hughes and Cresswell, A New Introduction to Modal Logic (London: Macmillan 1996), 20, 48; Buridan, TC I, 8, fifth conclusion. 62. See Thom, Medieval Modal Systems, 17–18. Arguably, a further axiom is needed to govern the existential underline, namely, if a → b then a → b, else we cannot prove, e.g., Celarent XXX. 63. The identification of the eight modal syllogistic propositions can also be seen in Hughes, “The Modal Logic of John Buridan,” 98. 64. Note that ∃ must be read noncommittally in Thom’s system, that is, as “for some ( possibly empty) term.” 65. Thom, Medieval Modal Systems, chapter 9. 66. Theorem 9.10, 177–178. 67. See King’s introduction to Jean Buridan’s Logic, 82. 68. Ibid., 362 n. 25. 69. Allan Bäck, “Syllogisms with Reduplication in Aristotle,” Notre Dame Journal of Formal Logic 23 (1982): 453–458. 70. See Posterior Analytics I, 4–6. 71. Allan Bäck, On Reduplication (Leiden: Brill, 1996), 394–396. See also Burleigh (Burley), De Puritate Artis Logicae, Tractatus Longior, 267–283.

Editorial Introduction Hubert Hubien 1. All the same, three of the four volumes of the work of Carl Prantl, Geschichte der Logik im Abendlande (Leipzig 1855–1870) are devoted to the Middle Ages. This enormous work is perhaps unique in the history of erudition: The author has spent twenty years of his life studying works and authors for whom he expressed a limitless contempt, in a discipline of which he understood nothing. The result is as one might imagine: save perhaps as a bibliographic catalogue (limited only to printed texts) and as a compilation of texts (the citations are plentiful), the value of Prantl’s work is, sadly, zero. 2. I. Bochenski, Formale Logik (Freiburg: Karl Alber, 1956), 170. 3. E. A. Moody, Truth and Consequence in Medieval Logic (Amsterdam: North-Holland, 1952). 4. E. Faral, “Jean Buridan. Note sur les manuscrits, les éditions et le contenu de ses ouvrages,” Archives d’histoire doctrinale et littéraire du moyen âge 15 (1946): 1–53. 5. Journal of Symbolic Logic 20 (1955): 44–45. 6. Introduction to his edition of Walter Burley, De Puritate Artis Logicae (New York, 1955), viii.

notes to pages 54–56

169

7. Rivista critica de Storia della filosofia 15 (1960): 413–427. 8. Cod. Magl. cl. V. 43 (Strozz. 120). 9. [“It begins: Here begin the Consequences of master John Buridan. A consequence is an antecedent and a consequent with a sign . . . It ends: For any ass is a man or an ass. Here end the Consequences of master John Buridan.”—trans.] 10. See Book I, Chapter 3, at the end. 11. A treatise on consequences beginning “Consequentia est antecedens et consequens . . .” has been noted by Father O. Kristeller (Iter Italicum vol. 2, p. 75) in codex 61 (ff. 83v 87v) of the capitulary library of Pistoia, which, like the Florentine codex, contains, with the exception of the De suppositionibus of Marsilius of Inghen, only works of the English school, of Ralph Strode, Richard Billingham, William Heytesbury, Walter Burley, and Peter of Candia. Moreover, according to an anonymous commentary on William Sutton’s treatise on consequences, Richard Swineshead defined consequence in this way: “consequentia est totum aggregatum ex antecedente et consequente cum nota consequentiae” (cited by James A. Weisheipl, “Roger Swyneshed, O.S.B.,” in Oxford Studies Presented to Daniel Callus, [Oxford, 1964], 245). Finally, Pits attributed a treatise beginning “Consequentia est antecedens et consequens . . .” to Richard Lavingham (cf. Gaudens E. Mohan, “Incipits of Logical Writings of the XIIIth–XVth Centuries,” Franciscan Studies 12 [1952]: 349–489, at 381). 12. These Obligationes Iohannis Busti are doubtless those which other manuscripts (e.g., Pal.lat. 994; cf. p. XX) attribute to a William Buser, of whom we know nothing. [But see C. H. Kneepkens, “The Mysterious Buser Again: William Buser of Heusden and the Obligationes tract Ob rogatum,” in English Logic in Italy in the 14th and 15th Centuries, ed. A. Maierù, 147–166 (Bibliopolis, 1982). —Trans.] 13. Bibliothèque de l’Université de Liège. Catalogue des manuscrits (without the name of the author), Liége 1875. 14. Vol. 10, col. 1370–1375, Paris 1938. 15. [“Now whether according to another way of speaking syllogisms in these divine terms are formally valid and what that form is I leave to the theologians. And it should be noted and always kept in mind that, because it is not for this to decide the foregoing. . . .” See also Book I, Chapter 8, conclusion 12; and O. Hallamaa, “Defending Common Rationality: Roger Roseth on Trinitarian Paralogisms,” Vivarium 41 (2003): 84–119. —Trans.] 16. [“Because it is not for me, an arts man.” —Trans.] 17. [“Now some of the lords and masters reproach me for sometimes mixing theological issues among philosophical questions, since it is not for arts men. But I respond with humility that I do not wish to be constricted in this way, but all masters, when they begin in arts judge that they may not dispute any purely theological question, either about the Trinity or the Incarnation, and judge further that if they happen to dispute or determine any question that

170

18. 19. 20. 21.

22.

23. 24.

25. 26.

notes to pages 56–63

touches on faith and theology, they will determine in favor of faith and reject the reasons against it, according as they seem to be dissolved.” —Trans.] Quaestiones in Physicam IV 8, cited by Anneliese Maier, Metaphysische Hintergründe der spätscholastischen Naturphilosophie (Rome, 1955), 3, from the Parisian edition of 1509 and the codex Vat.lat. 2163. See Book I, Chapter 4. [“For example, if we say, ‘A white cardinal has been elected pope,’ we infer ‘So a master of theology has been elected pope.’ ”— Trans.] See, inter alia, L. Jadin, “Benoit XII,” in Dictionnaire d’Histoire et d Géographie Ecclésiastiques, vol. 8, col. 116–135. On the opinion that his contemporaries had of this Pope, see the lives published by Baluze in Vitae Paparum Auinionensium, ed. G. Mollat, vol. I ( Paris, 1916). [“Albert Fantin, Italian from Bologna, Franciscan friar, to Justinian Fantin, his brother, greetings. You have supported me with assiduous words, my most sweet brother, so that I would signify to you no less as a brother whose needs you are bound to meet as a kinsman. Indeed, I myself, who thought to struggle least against your noble requests, have surveyed the Consequences of Master John Buridan, and in these perusals found marvelous regularity, marvelous doctrine, marvelous meaning, and marvelous concision, so much so that I have the greatest admiration. But because they were filled with certain errors and mistakes (I don’t know whose, but I suppose the printers’), I have decided to cleanse them from these sorts of errors and mistakes and to make amends, to offer them to you, most sweet brother, beyond whom I have had no one, so that you may accept them to be cheered in mind, and survey what you accept, and preserve what you survey, and love what you preserve, and defend what you love to the best of your power. Farewell, and may you love me as is your custom.” —Trans.] See Book II, Chapter 6, second conclusion. The late Rev. Father Boehner claimed this title for the De Puritate artis logicae of Walter Burley (Medieval Logic [Chicago, 1952], 89), written between 1325 and 1328, according to him (viii). Nevertheless, the deductive treatment, which in Buridan’s treatise spans the whole work, extends in Burley only for some pages (60–66 in the edition cited). See Book I, Chapter 2 [“The assumption does not require proof”]. [Let the Consequences of the Reverend Master John Buridan now begin. —Trans.]

Book I: Consequences in General and of Consequences between Assertoric Propositions 1. [] mark insertions by the translator; {} insertions by the editor. Page references rendered thus [p. 17] give the corresponding place in Hubien’s Latin text. All chapter headings were supplied by Hubien.

notes to pages 65–96

171

2. Not following the editor’s suggestion of inserting “propositiones” before “quarum quaelibet.” See the Introduction, §2.1. 3. This may seem fallacious. But see conclusion 12 of Chapter 8, where Buridan claims that “B is A” and “That which is B is A” are equivalent, though not formally so. So we can argue: “That which is A is B, so A is B, so B is A, so that which is B is A.” 4. See conclusion 1, Chapter 8. 5. This is Buridan’s early account of insolubles. See the Introduction, §1.1. 6. See, e.g., S. Read, “Medieval Theories: Properties of Terms,” Stanford Encyclopedia of Philosophy (Spring 2011 Edition), §5. 7. Literally, “per locum a divisione,” the topic from division. See, e.g., Boethius, De Topicis Differentiis, trans. E. Stump (Ithaca, N.Y.: Cornell University Press, 1978), 58–60. 8. Note that “Everything running is moving” is false if nothing is running. 9. See, e.g., J. Buridan, Summulae de Dialectica, trans. G. Klima (New Haven: Yale University Press, 2001), §1.5.2. 10. Ibid., §1.5.3. 11. Cf. Book III, Part I, Chapter 4. 12. Literally, “A human sees Socrates,” that is, “Socrates sees a human” (“Hominem videt Sortes”). 13. “Homo fit,” “Homo amatur.” 14. “Homo est factus,” “Homo est amatus.” 15. Possibly a reference to Aristotle, Physics VIII, 263b26–27, trans. D.W. Graham (Oxford: Clarendon Press, 1999): “If whatever exists without having existed previously must come to be, and if when it is coming to be, it does not exist, time cannot be divided into indivisible times.” 16. See note 13. 17. Literally, “A horse sees Socrates,” that is, “Socrates sees a horse” (“Equum videt Sortes”). 18. Literally, “A human saw Noah,” that is, “Noah saw a human” (“Hominem vidit Noah”). 19. A and L (the Paris edition and the Liège MS) both give “decimo.” See Aristotle, Metaphysics X c.3, 1054 b15ff: “Omne ens omni enti comparatum, aut est idem aut diversum”: cf. Aquinas, In Metaphysicae Aristotelis commentaria (Venice 1548), f.137E; Scotus, In Met. Q12 n.4, ed. Vivès ( Paris, 1891–95), vol. 7, 239a. Hubien follows MS V (Vat.lat. 3020) in referring to Metaphysics IV, and cites 1005 b19 ff.

Book II: Consequences between Modal Propositions 1. Buridan usually expresses the dictum using an accusative + infinitive phrase, which is much more idiomatic in Latin than in English. So for the most part, to remove clumsiness, I have rendered the dictum as a “that” clause. Occasionally, when it really matters, I use an accusative + infinitive phrase.

172

notes to pages 97–117

2. See Pseudo-Scotus, Questions on the Prior Analytics (included in John Duns Scotus, Opera Omnia, ed. Vivès ( Paris, 1891–95), vol. 2, Book I q. 26. See the Introduction, p. 000. 3. Literally, “A creating God can not be God.” However, it is too easy to confuse “can not” with “cannot.” It is important to realize, however, that “He who is a creating God can fail to be God” is negative, and so, as observed at the end of this paragraph, true if God is not now creating. I appeal to Buridan’s own observation (in Chapter 3) to show that such a construction forms a negative proposition, since “can fail to be” is equivalent to “is possibly not,” which is equivalent to “is not necessarily,” “which is clearly negative.” 4. Again, Pseudo-Scotus. See note 2. 5. Note that the Latin phrase translated here as “that B is A” could equally be rendered as “B to be A.” 6. Once again, the Latin phrase translated as “B being A” is the accusative + infinitive construction “B to be A.” 7. This seems to rob Buridan of a possible response to the revenge problem for insolubles. See, e.g., G. Hughes, John Buridan on Self-Reference (Cambridge: Cambridge University Press, 1982), 25. 8. Accidental conversion (i.e., per accidens) has the form “Every B is A, so some A is B.” 9. That is, the terms of the (composite) modal proposition, the dictum and the mode, not the terms of the dictum. 10. See note 9. 11. See note 9.

Book III: Syllogisms between Assertoric Propositions 1. See, e.g., Lambert, Logica, ed. F. Alessio (Florence, 1971), V: De Syllogismo, p. 107: “There are four types of argument, which are: syllogism, enthymeme, example and induction”; p. 111: “Enthymeme is an imperfect or truncated syllogism. . . . Induction is a progression from singulars to a universal. . . . Example infers from one similar to another similar, e.g., for controlling a ship you shouldn’t choose the sailor by lot but by skill, so for controlling the schools you should choose a master not by lot but by skill.” Cf. Peter of Spain, Tractatus, ed. L.M. De Rijk (Assen, 1972), V 3, pp. 56 ff. 2. Correcting Hubien’s “syllogismos infinitos” to “syllogismos infirmos,” which is in fact the reading in all three MSS. See Prior Analytics I 31 (46a31 ff): “It is easy to see that division by genera is only a small part of the method described above. For division is something like a weak syllogism,” trans. Striker (“Quoniam autem divisio per genera parva quaedam particula est dictae methodi, facile videre; est enim divisio velut infirmus syllogismus,” Aristoteles Latinus, trans. Boethius). 3. Buridan here departs from Aristotle in basing the validity of syllogisms on the principle of the expository syllogism. See the Introduction, §4.3.

notes to pages 117–124

173

4. Literally, “Any two of which one is the same as something as which the other is not the same, are not the same as each other.” 5. Literally, “of any two terms one of which is truly said to be the same as some discrete term and the other is not truly said to be the same as it, the one is truly said not to be the same as the other.” 6. See also the first conclusion here and the twelfth conclusion of Book I. 7. On this reading, see Hubien’s introduction. 8. This is Aristotle’s method of proof “per impossibile.” See the ninth conclusion here and, e.g., Aristotle, Prior Analytics Book I, trans. Striker, 70. 9. E.g., the first expository syllogism in the third figure in the previous paragraph is converted from the second-figure expository syllogism: “An A is this C and a B is the same C, so [some] B is A.” 10. E.g., the first expository syllogism in the third figure in the previous paragraph is converted from the first-figure expository syllogism: “This C is an A and a B is the same C, so [some] B is A.” 11. The conclusion of a negative syllogism is always negative, and the major extreme being the predicate of the conclusion is always distributed, whether the conclusion is universal or particular. 12. Recall the mnemonic: Barbara Celarent Darii Ferio Baralipton Celantes Dabitis Fapesmo Frisesomorum; Cesare Camestres Festino Baroco; Darapti Felapton Disamis Datisi Bocardo Ferison. See the Introduction, §4.3. 13. Here, by “mood,” Buridan simply means a way of combining the premises that are useful in producing a conclusion. 14. Thus in Figure I, we have: Barbara/ Baralipton

Celarent

Darii/ Dabitis

MaP SaM SaP/PiS

MeP SaM SeP/PeS

MaP SiM SiP/ PiS

Ferio

Fapesmo

Frisesomorum

(new)

(new)

MeP SiM PoS/PSo

MaP SeM PoS

MiP SeM PoS

MaP SoM SPo

MiP SoM SPo

Here, e.g., SoP denotes an O-proposition in the customary way of speaking, with subject S and predicate P, and SPo denotes the corresponding O-proposition in the nonnormal way of speaking, where the predicate precedes the negated copula. There are also the three weakened moods, Barbari, Celaront, and Celantop. 15. That is, pair of premises. 16. Thus in Figure II, we have:

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notes to pages 124–132

Camestres/ Camestre

Cesare/ Cesares

Festino

Baroco

Tifesno

Robaco

(new)

(new)

PaM SeM SeP/PeS

PeM SaM SeP/PeS

PeM SiM SoP

PaM SoM SoP

PiM SeM PoS

PoM SaM PoS

PoM SiM SPo

PiM SoM SPo

There are two weakened moods, Cesaro and Camestrop. 17. Thus in Figure III, we have: Darapti/ Datisi/ Daraptis Felapton Datisis MaP MaS SiP/PiS

MeP MaS SoP

Disamis Bocardo Ferison Lapfeton Carbodo Rifeson

MaP MiP MoP MiS MaS MaS SiP/PiS SiP/PiS SoP

MeP MiS SoP

MaP MeS PoS

MaP MoS PoS

MiP MeS PoS

There are no weakened moods in Figure III. 18. The minor premise, “Everyone old is going to die,” is analyzed as “Everyone who is or will be old is going to die.” 19. Hubien’s edition follows MS L in reading “undecima” (eleventh) here. But nothing was said to be prevented in that conclusion. I have followed MS V in reading “duodecima” (twelfth). 20. For example, if there is ampliation in the major premise to the future and in the minor premise to the past, Barbara will be invalid, that is, a direct universal affirmative conclusion will be prevented: e.g., “All living things will die, all dead things were living,” but it does not follow that “all dead things will die.” 21. Correcting the text, which reads “Brownie is a human’s ass.” All three MSS have asinus hominis (human’s ass). 22. As a general principle, if the English is awkward, so too is Buridan’s Latin. (My apologies to Buridan where this is not true.) 23. See the Introduction, §4.6. 24. It is interesting that Buridan includes simple supposition here, which he famously rejected in later writings. However, simple supposition is not mentioned again, as it might be, e.g., in the thirteenth conclusion. Note also the similarity to earlier theories, such as Burley’s, in which “horse” in, e.g., “I promise you a horse” has simple supposition (suppositing for the universal). 25. Note that this example is not of appellation of a concept, but of the analogue, namely, of material supposition (that is, the supposition of “human” in “every human is an animal”). 26. Note that later, in his Sophismata, he says that the terms preceding the appellating term also appellate a concept, but only every concept under a disjunction. See Sophismata, Chapter 4. Cf. G. Klima, John Buridan, §8.6. 27. At the beginning of Chapter 3.

notes to pages 133–152

175

28. 29. 30. 31. 32.

Aristotle, Prior Analytics I, 36, 48b12. In the fourteenth conclusion of Book I. Cf. Buridan, Summulae de Dialectica, trans. Klima, §3.7.1, 202. Aristotle, Prior Analytics I 36, 48b22. So “dead” ampliates, but “not alive” does not, any more than “alive” does. The conclusion is contradictory, for “human” can supposit there only for living humans. 33. That is, it was white when you saw it, but is not necessarily white any longer. 34. The converted minor premise is “Some B is not C,” and then the syllogism has the form of the first of Buridan’s novel syllogisms in the first figure: “Every C is A, some B is not C, so some B (some) A is not.” 35. If the premises are MaP and either SeṀ or SoṀ (where M and Ṁ are one finite, the other infinite), the minor premise is equivalent to SaM or SiM (given that there is an S), and so it becomes a syllogism in Barbara or Darii.

Book IV: Syllogisms between Modal Propositions 1. 2. 3. 4. 5.

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

15.

Topics II 11, 115b29. See the third conclusion of Book III. This is effectively (composite) Barbara LLL. See Book II, Chapter 5. Hubien’s text in this paragraph seems problematic, and I have amended it in line with the MSS. Where Hubien has “Modo sequitur: ‘C potest esse A, C potest esse B,’ uel sic: ‘quod potest esse C potest esse A, quod potest esse C potest esse B’; {‘ergo quod potest esse B potest esse A’}. Sequitur ultra ‘quod potest esse B potest esse A; ergo B potest esse A,’” I have chosen to follow MS P, Pal.lat. 994: “Modo sequitur ultra: quod potest esse B potest esse A, igitur B potest esse A.” This inference fails for universals but is valid for particulars, and all conclusions are particular in the third figure. Once again, this is valid for particulars, not for universals. For example, Abelard and Ockham. See Thom, Medieval Modal Systems, 53, 143. See the seventh conclusion herein. Book IV, Chapter 1. See the Introduction, §6.4, for a clarification of the reasoning in this paragraph. By Baroco LML. By Barbara LML. Note that here Buridan, correctly in light of his interpretation of universal modal statements, rejects Aristotle’s claim that Barbara and Celarent LXL are valid. See the Introduction, §5.2. The context should make clear that what Buridan means here is that Cesare LXL is invalid (having a universal conclusion), while Festino LXL is valid. See the Introduction, §6.3.

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notes to pages 152–160

16. That is, Camestres and Baroco LXX. 17. Hubien titles the chapter “Syllogisms between Composite Modal Propositions,” but the chapter is explicitly about divided modal propositions of contingency. 18. See the eighth conclusion of Book II. 19. Prior Analytics I, 38.

Glossary antecedent (antecedens) [see also “consequent”]: Buridan treats consequences (inferences—consequentiae) and conditionals interchangeably. See, for example, Book I, Chapter 3: “A consequence is a compound proposition; for it is constituted from several propositions conjoined by the expression “if” or the expression “therefore” or something equivalent.” For him, one proposition is antecedent to another if the other follows from it: “of two propositions one is antecedent to the other if it is impossible for the one to be true without the other being true.” as a matter of fact / as of now (ut nunc): Material consequences are divided by Buridan into those that are “simple” or hold unqualifiedly, and those that “are good as-of-now, because, things being as a matter of fact as they are, it is impossible for the antecedent to be true without the consequent” (Book I, Chapter 4). Barbara: A medieval mnemonic for the first mood of the first syllogistic figure. See Introduction, §4.3. can fail to be (potest non esse): Literally, “can not be,” but because this is easily confused with “cannot,” I have translated it as “can fail to be.” composite (composita) [see also “divided”]: Throughout medieval logic, a distinction was made between composite (or compounded) and divided senses of propositions, what we might call a “scope distinction.” In De Sophisticis Elenchis 20, Aristotle describes a fallacy of composition and division, giving as one example: “I saw a man being beaten with my eyes.” In the composite (implausible) sense, “beaten with my eyes” is taken together; in the divided sense, “beaten” and “with my eyes” are divided from one another, so the sentence means “I saw with my eyes a man being beaten.” The main application of the distinction in Buridan’s treatise is to modal propositions: in the composite sense, the mode is predicated of the whole proposition taken 177

178

Glossary

together; in the divided sense, the mode is taken with the predicate, which is divided from the subject. See Book II, Chapter 2. compound (hypothetica) [see also “subject-predicate”]: Propositions consisting of more than one proposition conjoined with “and,” “or,” “if,” “when,” “where,” “because,” and so on, are compound, as are others that are implicitly compound, such as “Only As are B” (equivalent to “As are B” and “Nothing other than As are B”), propositions of contingency (see Book II, seventh conclusion) and reduplicative propositions (see Book IV, Chapter 4). consequence (consequentia): An inference in which one or more propositions are antecedent to another, the consequent. Buridan distinguishes valid (or good—bona) consequences from others, but he often means “valid consequence” by “consequence.” See Book I, Chapter 3. consequent (consequens) [see also “antecedent”]: One proposition is consequent to others if it follows from them in a valid consequence. dictum: “That whole occurring in the proposition in addition to the mode and copula and negations and signs or other determinations of the mode or the copula” (Book II, Chapter 2). For example, in “That a human flies is not possible,” “that a human flies” is the dictum and “possible” is the mode. divided (divisa) [see also “composite”]: Modal propositions in the divided sense (or divided modal propositions) are where “the mode attaches to the copula as a determination of it,” and so the predicate is divided from the subject. It applies to modal propositions a more general distinction where one part of a proposition is divided from the rest (as opposed to being “compounded” with it). human (homo): The Latin term is usually taken to include both male and female, so although sometimes clumsy, “human” captures the sense better in English than “man.” infinite (infinita): An infinite term (subject or predicate) is one that is negated, for example, “nonhuman” or “unjust.” There is thus a crucial distinction between a negative proposition such as “Some chimera is not human” (true, since there are no chimeras) and the corresponding affirmative proposition with infinite predicate “Some chimera is nonhuman” (false for the same reason).

Glossary

179

mode/mood (modus): The Latin term is ambiguous, sometimes referring to the mode (e.g., possible, necessary) and sometimes to the mood (of a syllogism, e.g., Barbara). negation, external (negatio negans): A negation may negate only a term, forming an infinite term, or it may negate the copula (of a subjectpredicate proposition and so the whole proposition), when it is an external negation. normal form/nonnormal (modum consuetum/inconsuetum): Literally, “in the usual way (of speaking).” Buridan takes the normal word order to be “subject-copula-predicate,” or “subject-verb-object” (SVO), so that inverting copula and predicate is “nonnormal” or unusual, giving a special interpretation to the proposition. now (autem): Often translated as “but”; however, there is rarely the contrast that “but” would indicate, so “now” in its nontemporal sense as a sentence connector (“hesitation word”) is preferable. premise ( propositio): I usually translate propositio as “proposition,” but occasionally it is meant in the more precise sense of “proposition used as a premise of an inference.” set down (supponantur): At the start of each chapter, Buridan sets down certain principles that seem more than just assumptions. simple (simplex): Sometimes translated as “absolute,” meaning “without qualification.” simply assertoric (de inesse simpliciter): A special sense of “simple,” meaning “without qualification as to time,” that is, “necessary.” subject-predicate (categorica) [see also “compound”]: Often translated as “categorical,” which is rather misleading, since it does not have the usual meaning of “categorical” in English but derives from the Greek term for “predicate” and applies to any simple predicative proposition such as “Water is hot,” “No animals are running,” or “Socrates lies.” supposit (supponere): Supposition was one of the “properties of terms” that the medievals introduced into their theoretical vocabulary to describe the semantics of propositions.

180

Glossary

think of, thinkable (intelligibilis, intelligitur): A standard example for Buridan of an intensional verb, so that what is “thinkable” or “can be thought of” need not exist (though for Buridan, it must be possible, so that, e.g., “A chimera is thinkable” is false, since chimeras are impossible. See Book I, Chapter 5).

Index Abelard, Peter, 6, 12, 30, 175n8 Albert of Saxony, 2, 8 Alexander of Aphrodisias, 22 ampliation, 8–9, 10, 17–19, 71–74, 83–88, 90, 92, 116, 174n20, 175n32 effect on distribution, 26–27, 121–122, 124–27, 135–136, 138, 145, 146–147 modal, 31–34, 38, 43, 44, 45, 48, 50, 97–98, 100–102, 107, 140, 156 appellation, 28–29, 45, 71, 131–33, 174nn25,26 Aristotle, 6, 7, 28–31, 34, 50, 51–52, 69, 71, 88, 111, 160, 164n16, 165n31, 166n43, 177 commentaries on, 2, 5, 56 on modality, 36–37, 46, 48, 95, 98, 147 on the syllogism, 10, 15–17, 19–31 passim, 40, 45, 114, 116, 118, 123–125, 133, 135, 140–141, 166n37, 172n3, 173n8, 175n14 Averroes, 131–132 Bäck, Allan, 50 Barcan formula, 35–36 Billingham, Richard, 55, 169n11 Bochenski, I.M., Father, 53–54 Boehner, Philotheus, 54, 170n24 Boethius, Anicius Manlius Severinus, 6, 16 Bonaventure, Saint, 22 Bradwardine, Thomas, 1, 8

Buridan, John, 1–52 passim, 54–62, 169n9, 171n5, 172n3, 172n7, 174nn22,24, 175nn14,15, 179, 180 life, 1–2 Questiones Elencorum, 4 Questions on the Prior Analytics, 5, 23, 37–38 Sophismata, 3, 5, 9, 174n26 Summulae de Dialectica, 2, 5, 9, 15, 20, 22, 25–26, 27, 31, 32, 38, 40 Treatise on Consequences, 1, 2–3, 5, 9, 10, 31, 37, 38, 40, 59, 61, 170n22 Burley (Burleigh), Walter, 1, 2, 11, 56, 169n11, 170n24, 174n24 Buser, William, 55, 58 complexe significabilia, 4–5 composition and division, 30–31, 32, 95–96, 99, 140, 155, 167n56, 177–178 composition, 34–35, 36–37, 38, 105–112, 140–142, 172n9, 175n3 division, 18, 31–32, 34, 35, 37–38, 38–45, 48, 96–98, 99–105, 109–112, 140, 143–58 passim, 176n17, 178 concepts, 7–8, 13, 28, 51, 72, 131, 132, 174nn25,26 consequence, 6, 15, 17–18, 21, 37, 51, 61, 63–94 passim, 171n3 as-of-now (ut nunc), 12–13, 68–69, 75, 79, 80, 177. See also proposition: assertoric as-of-now as-of-then, 69, 75

181

182

index

consequence (continued) definition of, 10–11, 66–67, 178 division of, 11–13, 68–69 formal, 11–12, 25, 51, 55, 56, 68, 74–75, 79–84, 86, 88–89, 92, 93, 94, 113–115, 118–119, 125, 127, 140, 169n15 example, 113, 172n1 induction, 113, 172n1 material, 3, 11–13, 56, 68, 74–75, 177 modal, 51, 95–112 passim, 140–158 passim promissive, 69 simple (absolute), 12–13, 68, 79, 114, 164n16, 177, 179 See also syllogism contingency, 7, 37, 52, 101, 104, 106, 112, 140. See also proposition: (of) contingency convention, names signify by, 7, 64, 66, 84, 98, 134 conversion, 11, 15–19, 34, 48, 81–83, 85–93, 103–105, 107–109, 113, 160, 172n8 in proof of syllogistic validity, 22, 25, 31, 38, 43, 120, 124–125, 137–138, 148–150, 152, 154, 158, 173nn9,10 copula (“is”), 17, 74, 81, 85–91, 95–96, 167n56, 173n14; ampliative, 71, 126–127 used existentially, 85, 88 used predicatively, 85 dictum, 34, 35, 95–6, 105, 106, 107–111, 141–142, 155, 171n1, 172n9, 178 dictum de omni et nullo, 21–22, 38, 50, 51, 151, 157, 160 disjunctive syllogism, 79 distribution, 13–15, 16, 27, 33, 39, 51, 65–66, 82–83, 85–86, 91,

98, 120–123, 128–132, 143, 173n11 ampliated, 84, 87, 102, 125–127, 135–136, 145 of the middle term, 23–25, 52, 121–122, 124, 137, 160 See also supposition, theory of: modes of: confused and distributive division, 171n7, 172n2 Duns Scotus, John, 1, 5 ecthesis, 21–23, 52. See also syllogism: expository enthymeme, 68, 113, 172n1 equivalents, rules of, 82 example, 113, 172n1 existence postulate (constantia), 108, 109, 127, 139 existential import, 20–21 extreme, 21, 23, 27, 115–130 passim, 145–149, 160–161, 173n11 Fantin, Albert, 59–60, 170n22 Faral, Edmond, 54, 55 Federici Vescovini, Graziella, 54 figure, syllogistic, 21–26, 38, 40–51 passim, 115, 120–127, 132–139, 141–161 passim, 166n37 first, 6, 24, 173n14, 175n34, 177 fourth, 21, 115 second, 25, 173n16 third, 26, 173n9, 174n17, 175n6 form, 11–13, 51, 65, 68, 70–71, 136, 159 and matter, 74–75, 117 See also consequence: formal; proposition: nonnormal; syllogism: form Fournier, Jacques (Pope Benedict XII), 3, 56, 170n21 Frege, Gottlob, 61 Gregory of Rimini, 4

index Heytesbury, William, 1, 55, 169n11 Hubien, Hubert, ix, 3, 4, 5–6, 9, 44, 53–62, 171n19, 172n2, 174n19, 175n5, 176n17 Hughes, George, 5, 18, 45 induction, 113, 172n1 insoluble, 3–4, 70, 171n5, 172n7. See also liar paradox intension, 131–133, 136 intensional verb, 27–29, 180 John of Mirecourt, 56 King, Peter, 22, 45 Klima, Gyula, 5, 9, 44 Kneale, William and Martha, 19 Kristeller, Oscar, Father, 169n11 Lagerlund, Henrik, 18, 22, 35–36 Lavenham (Lavingham), Richard, 169n11 liar paradox, 3–4 revenge paradox, 35, 167n57, 172n7 See also insoluble logic, 19, 37, 39–40, 52, 53–54, 61 medieval, 12, 177 traditional, 21–22, 52 logica modernorum, 51 Marsilius of Inghen, 169n11 mental language, 7–8, 13 modality, 29–50 passim, 167n56 mode, 18, 34, 38, 40, 49, 52, 95–112 passim, 142–143, 155–158, 167n56, 172n9, 177–179 negated, 38, 49, 100, 104, 105, 107, 112, 143, 155–158 of signifying, 74 See also supposition: modes of mood, syllogistic, 6, 24–26, 38, 40, 118–156 passim, 166n37, 173n13, 177, 179

183

Barbara, 177 indirect, 21, 40, 43 invalid, 6, 31, 40 modal, 40–41, 43, 45, 141–161 passim perfect, 6, 43, 51, 64, 124–125 useless, 122–124, 149 weakened, 26, 40, 173nn14,16, 174n17 Moody, Ernest, 3, 54 nations (at University of Paris), 2 necessity, 7, 12–13, 37, 46, 77–78, 80, 141, 147–150. See also proposition: (of) necessity negation, 37, 74, 81–82, 85, 93, 95–96, 98–99, 120, 143, 178–179 predicate precedes, 15–16, 85, 122–123 See also terms: infinite Nicholas of Autrecourt, 4 nominalism, 4, 13, 34 nonnormal way of speaking. See proposition: nonnormal Ockham, William, 1–14 passim, 28, 56, 167n54, 175n8 Paris, University of, 1–2, 4–5, 11 Paul of Pergula, 60 Paul of Venice, 1, 56, 60 Peter of Candia, 55, 169n11 Peter of Mantua, 60 Peter of Spain, 2, 5 petitio principii, 76 possibilia, 18, 35–36 possibility, 28, 37, 71–72 one-sided, 30 two-sided (contingency), 30, 37 See also proposition: (of) possibility Prantl, Carl, 168n1 premise, 179; major, 21, 115; minor, 21, 115

184

index

proposition, 63–66, 74–75, 179 assertoric, 19, 63–94 passim, 113–139 passim assertoric as-of-now, 140 compound, 66, 67, 74, 91, 97, 101, 134, 135, 177–178 (of) contingency, 30, 32, 36–37, 46–50, 104–105, 108–109, 111–112, 141–142, 155–158, 167n50, 176n17, 178 exceptive, 91 exclusive, 91 exponible, 19, 91 modal, 10, 29–35, 37–38, 49, 52, 64, 95–112 passim, 140–158 passim (of) necessity, 31–32, 34, 43, 45, 48, 82, 140, 167n50 nonnormal, 15–17, 24–26, 29, 40, 85, 121, 123–124, 137, 165n25, 173n14, 179 oblique. See terms: oblique (of) possibility, 18, 32, 34, 64, 82, 167n50 reduplicative, 91, 159–160, 176. See also syllogism: reduplicative simply assertoric, 36, 38, 46–47, 140–141, 147, 148, 179 Pseudo-Scotus, 4–5, 167n54, 172n4 reductio per impossibile, 22, 43, 125, 137, 144, 145–146, 148, 150, 152, 154–155, 157, 160, 173n8 relative, 67, 90, 134–136 of identity, 121, 129, 133, 135, 156 restriction, 8–9, 71, 83, 90, 99, 103, 128–130, 144, 147, 151, 153 by appellation of a concept, 131 by “that which,” 34, 38, 43, 101, 144–147, 151, 153 signification, 6–8, 9, 10, 70, 81 sophisms (sophismata), 3, 58, 70, 166n43

square of opposition, 16–17, 19, 20, 30, 49 Strode, Ralph, 169n11 subalternation, 16, 19, 23, 83, 103, 113, 165n26 supposition, theory of, 3, 4, 8–9, 10–11, 92, 164nn17,19 modes of, 13–15, 131: confused, 65, 91, 128; confused and distributive, 14–15, 32; determinate, 14–15, 32, 65–66, 83, 91, 128–129; material, 13, 15, 34, 105–106, 131–132, 159, 164n17, 174n25; merely confused, 14–15, 32, 66, 83, 164n19; simple, 131, 174n24 Sutton, William, 169n11 Swineshead, Richard, 169n11 syllogism, 3, 6, 10–11, 19–29, 53, 61, 113–161 passim, 172n1, 173n11 expository, 38, 51–52, 87, 103, 107, 116, 121, 144, 146, 148, 152, 154, 157, 160, 173nn9,10 in divine terms, 1, 55, 117, 118–119, 169n15 form, 101 indirect conclusion, 21, 23–24, 25, 26, 40, 43, 116, 121, 122, 123, 124, 125, 127, 137, 152–153, 161, 166n37 infinite terms create exceptions, 29, 92, 119 modal, 6, 10, 30–31, 36–51, 52, 141–158 passim oblique, 26, 27, 116, 127–134 reduplicative, 43, 50–51, 159–161 rules of, 22, 23, 42, 52, 117–119, 120, 121, 122 validity and invalidity, 6, 21, 22, 23–24, 26, 27, 38, 40, 43, 45, 51, 52, 172n3 weak, 114, 172n2 See also mood, syllogistic

index terminism, 3, 5 terms categorematic, 11–12, 74, 134 disjunct, 14, 32, 34, 102, 164n19, 167n54 divine, 84, 118–119. See also syllogism: in divine terms exponible, 19 infinite, 29, 92–94, 113, 114, 116, 119, 136–139, 175n35, 178–179 major, 21, 39, 50–51 middle, 21, 23–29, 43, 51–52, 113–161 passim minor, 21, 51 oblique,19, 26, 75, 90–91, 166n41. See also syllogism: oblique reduplicative. See proposition: reduplicative

185

syncategorematic, 11–12, 14, 91, 114 Theophrastus, 31, 166n37 Thom, Paul, 38–40, 42, 45, 46, 48, 168n64 truth, 3–4, 6, 8, 10, 27, 34, 37–38, 63–66, 75, 79–80, 91, 95, 109, 141–142, 164n16 causes of, 9–10, 13, 15, 21, 27, 29, 32, 63–66, 71–73, 80–81, 83–84, 87, 97, 102, 106–107, 147–148 necessary truth, 12, 13 way of speaking customary, 118, 121–125, 173n14, 179 nonnormal. See proposition: nonnormal Wodeham, Adam, 4

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Medieval Philosophy Texts and Studies The Vatican Mythographers Ronald E. Pepin The Logic of the Trinity: Augustine to Ockham Paul Thom Later Medieval Metaphysics: Ontology, Language, and Logic Charles Bolyard and Rondo Keele, eds. Ens rationis from Suárez to Caramuel: A Study in Scholasticism of the Baroque Era Daniel D. Novotný Treatise on Consequences Translated by John Buridan, with an introduction by Stephen Read and an editorial introduction by Hubert Hubien Intentionality, Cognition, and Mental Representation in Medieval Philosophy Gyula Klima, ed.