Alexander of Aphrodisias

On Aristotle Prior Analytics 1.14-22

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Alexanderof Aphrodisias

On AristotlePrior Analytics 1.14-22

Translated by Ian Mueller withJosiah Gould

Introduction, Notes andAppendices by

Ian Mueller

LON DON • NEW DELHI • NEW YORK • SY DN EY

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The present translations have been made possible by generous and imaginative funding from the following sources: the National Endowment for the Humanities, Division of Research Programs, an independent federal agency of the USA; the Leverhulme Trust; the British Academy; the Jowett Copyright Trustees; the Royal Society (UK); Centro Internazionale A. Beltrame di Storia dello Spazio e del Tempo (Padua); Mario Mignucci; Liverpool University; the Leventis Foundation; the Humanities Research Board of the British Academy; the Esmée Fairbairn Charitable rust; the Henry Brown Trust; Mr and Mrs N. Egon; The Netherlands Organisation for Scientifi c Research (NWO/GW). The editor wishes to thank Jonathan Barnes, Tad Brennan, Kevin Flannery, Pamela Huby, Michael B. Papazian, Richard Patterson and Donald Russell for their comments on the volume, and Han Baltussen for preparing the volume for press.

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Contents

Editor’s Note viiPreface viii

Introduction

I. Assertoric syllogistic 4II. Modal syllogistic without contingency 9

II.A. Conversion of necessary propositions 9II.B. NN-combinations 11II.C. N+U combinations 13

III. Modal syllogistic with contingent propositions 19III.A. Strict contingency and its transformation rules 19III.B. Alexander and the temporal interpretation ofmodality 23III.C. Conversion of necessary propositions 25III.D. Conversion of contingent propositions 28 III.D.1. Conversion of affirmative contingent propositions 28 III.D.2. Non-convertibility of negative contingent propositions 31III.E. Syllogistic and non-syllogistic combinations 35 III.E.1. CC premiss combinations 35 III.E.2. U+C premiss combinations 37 III.E.3. N+C premiss combinations 44

IV. Theophrastus and modal logic 52Notes 54Summary (overview of symbols and rules) 59

Translation: The commentary

Textual Emendations 731.14-16 The first figure 75

1.14 Combinations with two contingent premisses 751.15 Combinations with an unqualified and a contingent premiss 83

1.16 Combinations with a necessary and a contingent premiss 118

1.17-19 The second figure 1341.17 Combinations with two contingent premisses 1341.18 Combinations with an unqualified and a contingent premiss 1491.19 Combinations with a necessary and a contingent premiss 155

1.20-2 The third figure 1661.20 Combinations with two contingent premisses 1661.21 Combinations with an unqualified and a contingent premiss 1701.22 Combinations with a necessary and a contingent premiss 175

Notes 185

Appendix 1. The expression ‘by necessity’ (ex anankês) 229Appendix 2. Affirmation and negation 231Appendix 3. Conditional necessity 232Appendix 4. On Interpretation, chapters 12 and 13 237Appendix 5. Weak two-sided Theophrastean contingency 239Appendix 6. Textual notes on Aristotle’s text 243

Bibliography 246English-Greek Glossary 248Greek-English Index 251Subject Index 266Index Locorum 269

vi Contents

Editor’s Note

This text, translated in two volumes, is a very important one becauseAlexander’s is the main commentary on the chapters in which Aristotleinvented modal logic, i.e. the logic of necessary and contingent (possible)propositions. Because it is more technical than the other texts in thisseries, Ian Mueller explains the modal logic in his masterly introduc-tion, which takes an exceptional form, being couched in logical symbolspartly of his own devising. All symbols are explained on first occurrence.Symbols are entirely excluded from the translation itself, and this canbe consulted freely by those who do not wish to master the entire modalsystem.

Aristotle also invented the theory of the syllogism, and in this volumehe extends this theory to include syllogisms containing contingentpropositions, the contingent being what may or may not happen.

December 1998 R.R.K.S.

Preface

This translation has been literally decades (two) in the making. JosiahGould, acting on a suggestion of Ian Mueller, prepared a first draft ofthe translation. Mueller produced a second draft and, then, in consult-ation with Gould, a third and final version with introduction, notes,appendices, and indices. We are certain that errors remain, but knowthat there would have been many more without the advice of TadBrennan, Glenn Most, Richard Patterson, Robin Smith, and severalanonymous readers whose friendly but stern admonitions turned usfrom some paths. We take full responsibility for remaining on otherpaths despite their counsel.

December 1998 I.M. J.G.

Introduction

We offer here a translation of Alexander of Aphrodisias’ commentary onchapters 14-22 of Aristotle’s Prior Analytics, in which Aristotle presentswhat we call his modal logic as applied to contingent as well as necessaryand what we call unqualified propositions. In a separate volume wehave translated Alexander’s commentary on chapters 8-12, in whichAristotle treats arguments not involving contingent propositions, andalso chapter 13, in which Aristotle discusses contingency in general, andpart of chapter 17 in which he treats the conversion of contingentpropositions. Chapters 1, 2 and 4-7 of the Prior Analytics constitute aself-contained presentation of what we will call non-modal or assertoricsyllogistic. Alexander’s commentary on this material (and on chapter 3)has been admirably translated and discussed by Barnes et al. We referthe reader to their introduction for information about Alexander,ancient commentaries, and the general character of Alexander’s com-mentary on the whole of Prior Analytics.

In making choices for how to deal with our task, we have alwaysbegun by consulting Barnes et al. for guidance, and in many cases havefollowed their practices. But the greater difficulty of the material wehave to present here has led us to diverge from them in some significantways. First of all, in our notes and discussions we have relied on aquasi-formal symbolism. We hope that the symbolism is enlightening;we are confident that a full exposition of our text not using someformalism would run to much greater length. To put this another way,if one used a formal symbolism one could encapsulate the full contentof Alexander’s commentary in many fewer pages than Alexander hasused.

We have, however, not thought it a good idea to introduce formalsymbolism into the translation itself. Our major departure from Alex-ander’s text is that we have used considerably more variables thanAlexander uses; we have sometimes done the same thing in our trans-lations of Aristotle. To give one example, at 121,4-6, where Alexanderwrites,

In both cases the conclusion proved is a particular negative necessaryproposition of which the opposite is ‘It is contingent of all’ (endekhetaipanti).

we have translated,

In both cases the conclusion proved is a particular negative necessaryproposition of which the opposite is ‘It is contingent that X holds of all Y’.

There is in general no way for a reader to tell whether, e.g., thetranslation ‘A holds of all B’ is literal or corresponds to something like‘holds of all’ or ‘universal affirmative’ without consulting the Greektext.1

The modern literature on Aristotle’s modal logic is substantial anditself difficult. The interpretations offered have been quite diverse, anda number of them have connected the modal logic with Aristotelianmetaphysics. We did not see any way to enter into these variousinterpretations, and we have thought it best to focus on what we wouldcall logical content, which seems to us also to be the focus of Alexander’scommentary. In fact, it seems to us that Alexander’s frequently ex-pressed perplexities about what Aristotle says are a more accuratereflection of Aristotle’s presentation of modal logic than is the work ofmany subsequent interpreters who have attempted to turn the modallogic into a coherent system. In our notes and discussions we haveprimarily tried to extract the logical content of Alexander’s torturedprose. Just as we have not devoted much attention to modern interpre-tations of Aristotle’s logic, so we have not devoted much to parallelpassages in other ancient texts. In both cases limitations of time, ofknowledge, and of space have constrained us.

We have included a complete translation of the text of Aristotle asread by Alexander (insofar as we can infer that) in the lemmas; we haveinserted texts as lemmas in places where there is no lemma in theedition of Wallies, on which our translation is based; and we havesometimes produced a stretch of Aristotelian text more than once. Thereader can identify the exact extent of the lemmas in Wallies’ text,because any material added by us is put between square brackets. Ourown judgment is that the lemmas are matters of convenience; they tellus more about the practice of scribes and later teachers than about thepractices of ancient commentators.

In our translation we have adopted the unusual practice of placingnote references at the beginning of paragraphs which we judge to beespecially difficult to follow. We believe that some readers will find ituseful to have an account of what Alexander is going to say before tryingto follow his own words. Those who prefer to make their own waythrough the text can simply ignore those notes initially and recur tothem as they deem necessary.

In section I of this introduction we give a brief and schematicpresentation of non-modal syllogistic to familiarize the reader withterminology and with some of our apparatus for representing Alexan-

2 Introduction

der’s discussions. In section II we try to give at least a partial overviewof chapters 8-12 and the part of chapter 3 relevant to it. But we shallpostpone the discussion of some of that material because it presupposesthe discussion of contingency, which we take up in section III, where wealso deal with chapters 13-22. Finally in a brief fourth section wediscuss the treatment of modality by Theophrastus, the associate ofAristotle. On occasion the reader may find it useful to refer to the formalSummary which follows this introduction.

Here is an outline of the contents of the first 22 chapters of the PriorAnalytics:

1.1-3 Introductory material 1.1 Preliminary definitions 1.2 Conversion of unqualified propositions 1.3 Conversion of necessary and contingent propositions

1.4-7 Combinations with only unqualified premisses 1.4-7 Combinations with two unqualified premisses 1.4 First figure 1.5 Second figure 1.6 Third figure 1.7 Further remarks

1.8-12 Combinations with at least one necessary but no contingent premiss 1.8 Combinations with two necessary premisses 1.9-11 Combinations with one necessary and one unqualified premiss 1.9 First Figure 1.10 Second Figure 1.11 Third Figure 1.12 Summarizing remarks on necessity

1.13-22 Combinations with a contingent premiss 1.13 Discussion of contingency 1.14-16 The first figure 1.14 Both premisses contingent 1.15 One premiss unqualified 1.16 One premiss necessary 1.17-19 The second figure 1.17 Both premisses contingent 1.18 One premiss unqualified 1.19 One premiss necessary 1.20-2 The third figure 1.20 Both premisses contingent 1.21 One premiss unqualified 1.22 One premiss necessary

Introduction 3

I. Assertoric syllogistic (1.1, 2, 4-7)

For the most part it suffices for understanding Aristotelian modalsyllogistic to have only a schematic understanding of the non-modal orassertoric syllogistic as it is developed in the first six chapters of thePrior Analytics. We present such a schematic representation here inquasi-formal terms which we will also rely on in our commentary.Qualifications of this schematic representation will be introduced onlyas they are needed.

A. Terms are capital letters from the beginning of the alphabet: A, B, C,D, ..., (standing for general terms such as human or animal).

B. Propositions. There are four types of propositions:

XaY (read X holds of all Y or All Y are X)XeY (read X holds of no Y or No Y are X)XiY (read X holds of some Y or Some Y are X)XoY (read X does not hold of some Y or Some Y are not X)

where X and Y are terms (called respectively the predicate and thesubject of the proposition).2 These propositions are sometimes referredto as a-propositions, e-propositions, etc. Propositions of the first twokinds are called universal, those of the last two particular; a- andi-propositions are called affirmative, e- and o- negative. Universalityand particularity are called quantity, affirmativeness and negativenessquality. When we wish to represent a proposition in abstraction fromits quantity and quality we write, e.g., XY.

C. Pairs of propositions with one common term are called combinations,and assigned to one of three figures:

First figure XZ ZYSecond figure ZX ZYThird figure XZ YZ

As members of combinations propositions are called premisses. In theschemata given, Z is called the middle term of the combination, and Xand Y are called extremes or extreme terms. In addition X is called themajor term, Y the minor term; and the premiss containing the majorterm is called the major premiss, the one containing the minor term theminor premiss.

D. The major problem for Aristotle is to determine which combina-tions are syllogistic, that is imply a proposition (called the conclusion)with the major term as predicate and the minor as subject.3 Aristotle

4 Introduction

restricts himself to considering the strongest conclusion implied by asyllogistic combination. In the first figure he recognizes the followingsyllogistic combinations:4

1. AaB BaC AaC (1.4, 25b37-40)2. AeB BaC AeC (1.4, 25b40-26a2)3. AaB BiC AiC (1.4, 26a23-5)4. AeB BiC AoC (1.4, 26a25-7)

As a preliminary notation for these syllogisms (which will be compli-cated when we take up modal syllogistic) we introduce

1. AAA12. EAE13. AII14. EIO1

where the letters give the quality and quantity of the propositionsinvolved and the subscripted number gives the figure. When we wish torepresent just a pair of premisses we write such things as

EE_1

to represent the pair

AeB BeC

We will, in fact, use something like this notation for pairs of pre-misses, but after some hesitation, we have decided also to use themedieval names for the categorical syllogisms in the belief that mostpeople who work on syllogistic will find them easier to read than themore abstract symbolism. Those unfamiliar with the names need onlyremember that the sequence of vowels in the medieval names repro-duces the sequence of letters in the symbolism we have introduced; forfurther clarity we will add to the names numerical subscripts indicatingthe figure.5 Thus we will refer to the four first-figure syllogisms as

1. Barbara12. Celarent13. Darii14. Ferio1.

Aristotle calls these four syllogisms complete (teleios, rendered ‘perfect’by Barnes et al). Aristotle says that a syllogism is complete if it ‘needsnothing apart from the assumptions in order for the necessity

conclusion> to be evident’ (1.1, 24b22-4). Modern scholars have disputedwhat Aristotle means here,6 but Alexander clearly thinks that thecomplete syllogisms receive a kind of justification from the so-calleddictum de omni et nullo, which he takes to give an account of therelations expressed by ‘a’ and ‘e’:7

For one thing to be in another as in a whole and for the other to bepredicated of all of the one are the same thing. We say that one thing ispredicated of all of another when it is not possible to take any of it ofwhich the other is not said. And similarly for of none. (1.1, 24b26-30)

Alexander understands this passage to be saying something like:

XaY if and only if it is not possible to take any Y which is not an X; XeY if and only if it is not possible to take any Y which is an X.

Alexander’s treatment of Barbara1 and Ferio1 show how he invokes thedictum in the treatment of complete syllogisms:

Let A be the major extreme, B the middle term, and C the minor extreme.If C is in B as in a whole, B is said of every C. ... Therefore, it is not possibleto take any C of which B is not said. Again, if B is in A as in a whole, A issaid of every B. Hence it is not possible to take any of B of which A is notsaid. Now, if nothing of B can be taken of which A is not said, and C issomething of B, then by necessity A will be said of C too. (54,12-18)

If something of C is in B as in a whole,8 and B is in no A, then A will nothold of some C. For something of C is under B; but nothing of B can betaken of which A is said. Hence A will not be said of that item of C whichis something of B. (60,27- 61,1)

Whether one thinks that for Aristotle complete assertoric syllogisms aresimply self-evident or – in agreement with Alexander – that theirvalidity depends on the dictum de omni et nullo affects one’s under-standing of Aristotle’s conception of logic, but it does not affect one’sunderstanding of which assertoric combinations are syllogistic. In thecase of modal syllogistic the situation changes. At least in antiquity thedictum played a role in disputes about whether certain combinationsare syllogistic. We will say more about the issue in section II.C.

E. At this point it is convenient to describe the principal procedure bywhich Aristotle shows that a combination is non-syllogistic. We wouldnormally show that a given first-figure combination XY, YZ does notyield a specific conclusion XZ by specifying concrete terms which, whensubstituted for X, Y, and Z, make XY and YZ true and XZ false. Thus toshow that AaB and BeC do not imply AeC we can point out that although‘All humans are animals’ and ‘No cows are humans’ are true, ‘No cows

6 Introduction

are animals’ is false. Aristotle shows that a first-figure combination ofspecific premisses XY and YZ yield no conclusion (of the relevant kindwith X as predicate and Z as subject) by giving two interpretations, onewhich makes XY, YZ, and XaZ true, the other of which makes XY, YZ,and XeY true. This procedure works because of the following relationsamong propositions:

a. XaY and XoY are contradictories, i.e., XaY if and only if (XoY) (so that also XoY if and only if (XaY));

b. XeY and XiY are contradictories, i.e., XeY if and only if (XiY) (so that also XiY if and only if (XeY));

c. XaY and XeY are contraries, i.e., they cannot be true together (although they might both be false).9

Given these relationships, an interpretation making XaZ true rules outany negative conclusion XZ and an interpretation making XeZ true rulesout any affirmative conclusion.

On pp. 12-14 Barnes et al. discuss Alexander’s understanding of thismethod of rejecting non-syllogistic pairs and say, ‘He always misunder-stands it.’ In a footnote they add, ‘He may seem to get it right at in An.Pr. 101,14-16 and 328,10-20; but in these passages it seems reasonableto think that he has succeeded by mistake.’ We agree with this assess-ment of Alexander. He consistently treats the method of rejection as amatter of showing that both XaY and XeY (or their analogues in modalsyllogistic, ‘X holds of all Y by necessity’ and ‘X holds of no Y bynecessity’) follow from a pair of premisses. We have signalled Alexan-der’s misapprehension in cases where – if we have understood himcorrectly – it has led him to express a false opinion or made hisdiscussion less cogent than it might be, and sometimes we have done soin the many more numerous passages where Alexander’s misdescrip-tion of what is going on is harmless. But we have frequently left it tothe reader to realize that in a given passage Alexander speaks about,e.g., P3 following from P1 and P2 when he should be speaking about allthree propositions being true.10

F. Aristotle shows that second- and third-figure assertoric combinationsare syllogistic by completing them or reducing them to first-figuresyllogisms. Reductions are either direct or indirect. Direct reductionsmake use of the following rules of conversion enunciated and discussedby Aristotle in the second chapter of the Prior Analytics:

EE-conversion: XeY YeX (25a14-17)AI-conversion: XaY YiX (25a17-19)II-conversion: XiY YiX (25a20-2)

Introduction 7

Aristotle uses terms to reject the possibility of any kind of O_-conversionat 25a22-6.

The second-figure syllogisms are:

1. Cesare2 AeB AaC BeC (1.5, 27a5-9)2. Camestres2 AaB AeC BeC (1.5, 27a9-15)3. Festino2 AeB AiC BoC (1.5, 27a32-6)4. Baroco2 AaB AoC BoC (1.5, 27a36-b3)

The first three of these are completed directly. We indicate the way inwhich we will describe their reductions or proofs (deixeis), as Alexandermost frequently calls them, in the Summary. Baroco2 is justified indi-rectly by reductio ad absurdum: from the contradictory of the conclusionand one of the premisses, one uses a first-figure syllogism to infer thecontradictory of the other premiss. For our representation of the argu-ment see the Summary, which gives similar representations for thethird figure. These derivations for modally unqualified propositions areworth learning since in general Aristotle tries to adapt them to modallyqualified propositions. In the directly derivable cases he faces fewproblems so that many of the main issues for them arise already inconnection with the first figure. However, the addition of the modaloperators causes special problems in the indirect cases.

G. Aristotle seems to assume the completeness of his reduction proce-dures, that is, he assumes that any combination can either be refutedby a counterinterpretation or reduced to a first-figure syllogism. He alsoassumes that the system is consistent in the sense that one cannot giveboth a counterinterpretation and a reduction for a given syllogism.These assumptions are correct for assertoric syllogistic, and they makepossible another method of showing a combination non-syllogistic: showthat the rules do not allow the combination to be reduced to a first-figuresyllogism.11 Aristotle does not use this method in assertoric syllogistic,but he does apply it in modal syllogistic (e.g. at 1.17, 37a32-6), andAlexander does it even more frequently. The applications of this methodare not up to the standards of modern proof theory, but they aregenerally corrrect.

A more important point is that the modal syllogistic is not consistent,so that a derivation does not suffice to show that a counterinterpreta-tion is impossible, and a counterinterpretation does not suffice to showthat a derivation is impossible. Alexander is aware of some of the casesin which this is true,12 but – as is frequently the case in the commentary– he does not seem to be aware of either the depth of the problem createdby this situation or its devastating effect on Aristotle’s modal syllogistic.

8 Introduction

II. Modal syllogistic without contingency(1.3, 25a27-36 and 8-12)

As a first approximation modal syllogistic can be understood as anextension of assertoric syllogistic brought about by adding for everyproposition P of assertoric syllogistic the propositions ‘It is necessarythat P’ and ‘It is contingent that P’. The issues which arise in connectionwith the notion of contingency are considerably more complex thanthose which arise in connection with necessity. Unfortunately some ofthe issues which arise in connection with necessity are inextricablybound up with contingency. We are going to try to abstract from thoseissues here, and return to them after we have discussed contingency.We shall adopt the abbreviation NEC(P) for various Greek expressionswhich we take to have the sense of ‘It is necessary that P’. Ultimatelywe will use abbreviation CON(P) for ‘It is contingent (usually endekh-etai) that P’, using the word ‘possible’ informally (and in the translationof such expressions as dunaton, dunatai, enkhôrei, hoion, estai). Weshall call a proposition NEC(P) a necessary proposition, CON(P) acontingent proposition; if NEC(P) (CON(P)) is true we will say that P isnecessary (contingent). To be explicit we shall call a proposition ofassertoric syllogistic an unqualified proposition.13 We will define variousformal notions in the same way as before, but we will extend ourrepresentation of syllogisms and combinations. Assertoric Barbara1now becomes:

Barbara1(UUU)

and the assertoric combination AE_1 becomes:

AE_1(UU_)

The following examples should make the notation to be employed clear:

Barbara1(NUN) NEC(AaB) BaC NEC(AaC)Bocardo3(NCU) NEC(AoC) CON(BaC) AoBEA_2(CU_) CON(AeB) AaC

II.A. Conversion of necessary propositions (1.3, 25a27-36)

Aristotle accepts the same conversion laws for necessary propositionsas for unqualified ones, that is, he accepts:

EE-conversionn: NEC(XeY) NEC(YeX) (25a29-31)AI-conversionn: NEC(XaY) NEC(YiX) (25a32-4)II-conversionn: NEC(XiY) NEC(YiX) (25a32-4)

Introduction 9

To justify EE-conversionn Aristotle writes,

If it is necessary that A holds of no B, it is necessary that B holds of no A;for if it is contingent that B holds of some, it will be contingent that Aholds of some B. (25a29-32)

Aristotle here appears to reduce EE-conversionn to:

II-conversionc: CON(XiY) CON(YiX)

a law which he does not take up until 25a40-b3, and which he appearsto justify by citing EE-conversionn. For AI-conversionn and II-conver-sionn Aristotle writes,

If A holds of all or some B by necessity, it is necessary that B holds of someA. For if it is not necessary, A will not hold of some B by necessity.(25a32-4)

apparently taking for granted that

NEC(BiA) NEC(AiB)

which, if it is not just another formulation of II-conversionn itself, wouldseem to involve some such reasoning as the following. AssumeNEC(AiB) and NEC(BiA). Then since:

(i) NEC(P) CON( P) ( N C )

CON(BeA). But:

(ii) CON(XeY) CON(YeX) (EE-conversionc)

So CON(AeB), and since:

(iii) CON(P) NEC( P) (C N )

NEC(AiB), contradicting NEC(AiB)

The problem with this reconstruction is not simply that Aristotle relieson laws concerning contingency which he has not yet discussed, but (i)and (ii) are laws which Aristotle rejects at 1.17, 36b35-37a31. In thecourse of doing so he denies that an indirect argument works by denyingan instance of:

CON(P) NEC( P) ( C N )

10 Introduction

which is equivalent to (i). Aristotle is, however, committed to C Nand its equivalent:

NEC(P) CON( P) (N C ).

Since we cannot hope to clarify this situation without looking at Aris-totle’s treatment of contingency and Alexander’s understanding of it, weshall for now simply take for granted the conversion laws for necessarypropositions and turn to Aristotle’s application of them. However, beforedoing so we mention one other law assumed by Aristotle:

P NEC( P) (U N )

that is, if a proposition holds, its contradictory is not necessary.

II.B. NN-combinations (1.8)

The perfect parallelism between the conversion laws for unqualified andnecessary propositions greatly simplifies the treatment of NN combina-tions in chapter 8, and Aristotle’s discussion is very succinct. Theprincipal value of Alexander’s commentary on chapter 8 is its scholas-ticism, the concrete filling out of what Aristotle describes in outline. Wehere follow Alexander’s account. Aristotle assumes that an NN combi-nation is syllogistic if the corresponding UU combination is, and thatthe former will yield the conclusion NEC(P) if the latter yields theconclusion P. The argument that the converses of these assumptionsholds has three steps. The first two are stated briefly in the followingpassage:

For, if the terms are posited in the same way in the case of holding and inthat of holding by necessity – or in the case of not holding – there eitherwill or there won’t be a syllogism , except that they willdiffer by the addition of holding or not holding by necessity to the terms.For the privative converts in the same way, and we will give the sameaccount of ‘being in as a whole’ and ‘said of all’. (29b37-30a3)

Alexander points out that Aristotle means to include all conversion rulesin this remark (120,20-5), and he applies the reference to the dictum deomni et nullo to the first figure (120,13-15), a sure sign that he takesAristotle to be treating the first-figure NNN syllogisms as complete.Thus the argument is that the parallel first-figure combinations aresyllogistic of parallel conclusions and that conversion will generate theparallel directly verified syllogisms in the second and third figures. Theonly remaining problem concerns:

Introduction 11

Baroco2(NNN) NEC(AaB) NEC(AoC) NEC(BoC)Bocardo3(NNN) NEC(AoC) NEC(BaC) NEC(AoB)

the UUU analogues of which were established indirectly. This whole way of looking at modal syllogistic is basic to Aristotle.

Roughly, one can say that for Aristotle the fundamental question is todecide which modal analogues of the complete first-figure assertoricsyllogisms are syllogistic14 and then to ask whether the second- andthird-figure analogues of syllogisms can be derived in ways analogousto those in which the first-figure ones were. Only when a derivationcannot be provided does Aristotle look for counterinterpretations. Inother words, Aristotle does not appear to first raise the questionwhether a second- or third-figure combination is syllogistic, but firstasks what, if any conclusion can be derived from the combination by aderivation of the type used with the analogous assertoric combination.If that analogous derivation fails he looks for a counterinterpretation.If he can’t find one and decides there isn’t one, he looks for an alterna-tive derivation.

If we try to copy the indirect derivations of Baroco2(UUU) andBocardo3(UUU) for the corresponding NNN cases we run into the samekind of problems we encountered with Aristotle’s indirect argumentsfor the conversion laws for necessary propositions. We here give theindirect arguments which we would seem to need, first for:

Baroco2(NNN) NEC(AaB) NEC(AoC) NEC(BoC)

Assume NEC(AaB), NEC(AoC), and NEC(BoC). Then ( N C )CON(BaC). Now, if we had Barbara1(NCC), we could infer CON(AaC),which implies (C N ) NEC(AoC), contradicting NEC(AoC).The argument for

Bocardo3(NNN) NEC(AoC) NEC(BaC) NEC(AoB)

is quite analogous. Assume NEC(AoC), NEC(BaC), and NEC(AoB).Then ( N C ) CON(AaB). So, if we had Barbara1(CNC), we couldinfer CON(AaC), which implies (C N ) NEC(AoC), contradict-ing NEC(AoC).

One obvious difficulty with these arguments is the use of N C, which, as we have said, Aristotle rejects. However, it is also truethat Aristotle sometimes uses the equivalent of this rule, namely C N. Indeed, he uses it without acknowledgement in arguing thatBarbara1(NC_) yields a contingent conclusion.15 Alexander is quite clearthat because of the use of C N the conclusion is of the form NEC (AaC), and that this is not equivalent to CON(AaC); it involveswhat we will call Theophrastean contingency because it was the notion

12 Introduction

of contingency highlighted by Theophrastus.16 The situation is suffi-ciently fluid that we might choose to allow Aristotle the use of N C in arguing for Baroco2(NNN) and Bocardo3(NNN). By itself thiswould take care of Bocardo3(NNN), since Aristotle takes Bar-bara1(CNC) to be complete at 1.16, 36a2-7. However, Barbara1(NC‘C’)17is not complete for Aristotle and requires an argument which invokesthe notion of contingency.

In any case it is quite clear that Baroco2(NNN) and Bocardo3(NNN)are valid. Aristotle chooses to verify them with what he calls an ekthesis.The ekthesis works on the premiss NEC(AoC), and involves taking apart D of C of which A does not hold by necessity. SubstitutingNEC(AeD) for NEC(AoC), we have in the case of Baroco2(NNN) aninstance of Camestres2(NNN) with the conclusion NEC(BeD); but D ispart of C, so NEC(BoC). For Bocardo3(NNN), one changes the secondpremiss to NEC(BaD) to get an instance of Felapton3(NNN). (In bothcases Alexander carries out the reduction to the first figure.) Alexanderdiscusses the character of the ekthetic arguments starting at 123,3-24,drawing a contrast between them and the ekthesis arguments of asser-toric syllogistic. At 123,18-24 he provides the important historicalinformation that Theophrastus preferred to postpone the treatment ofBaroco2(NNN) and Bocardo3(NNN) until he could establish them indi-rectly, that is, use some version of the argument we have just sketched.We discuss the question of how Theophrastus might have done this insection IV of the introduction.

II.C. N+U combinations (1.9-11)

In chapters 9-11 Aristotle takes up the N+U cases, devoting a chapterto each of the three figures. In 9 he takes as complete all the NUN andUNU analogues of the complete UUU first-figure syllogisms. Giventhese syllogisms, the direct derivations for the second- and third-figureN+U combinations are straightforward. The indirect cases are againproblematic. Aristotle decides that each of the four N+U cases of Baroco2and Bocardo3 yields only an unqualified conclusion. He gives no positiveargument for any of the four, but only uses terms to show that none ofthe four combinations yield a necessary conclusion. We shall discuss hisuse of terms to show that certain N+U combinations yield an unqualifiedconclusion in a moment. For now we simply remark that all four casesaccepted by Aristotle have simple indirect derivations. Alexander pointsout at 144,23-145,20 and 151,22-30 that the kind of ekthesis argumentwhich Aristotle used to establish Baroco2(NNN) could be used forBaroco2(UNN) and Bocardo3(NUN). Unfortunately, Alexander’s discus-sion of the implications of this situation in which a proof and acounterinterpretation conflict (145,4-20 and 151,22-30) is very indeci-sive, to say the least.

Introduction 13

We shall approach Aristotle’s treatment of the complete combina-tions in terms of the two cases of Barbara1. For Barbara1(UNU)Aristotle takes for granted that Barbara1(UN_) yields either an un-qualified or a necessary conclusion and offers two kinds of argumentsto show that the conclusion cannot be necessary. One is a specificationof terms, which, indeed, work if one assumes the truth of the followingpropositions:

(a) All animals are in motion;(b) It is necessary that all humans are animals;(c) It is not necessary that all humans are in motion.18

Unfortunately the use of these terms seems to cast doubt on Bar-bara1(NUN) since – to use an example of Theophrastus mentioned byAlexander at 124,24-5 – it would seem to be just as much true that:

(b) It is necessary that all humans are animals;(a’) Everything in motion is a human;(c’) It is not necessary that everything in motion is an animal.

‘All humans are animals’ is, of course, a standard example of anecessary truth. (a) and (a’) are typical problematic examples of anunqualified truth: they are not, in fact, true, but they are taken to betrue for the sake of making an argument, in Alexander’s terminology,they are ‘hypotheses’.19 Unfortunately, this way of interpreting unquali-fied statements makes it very difficult to see that there is any differencebetween unqualified and contingent propositions. Alexander raises thisissue in connection with Aristotle’s remarks at 1.15, 34b7-18 in thecontext of an apparent counterinterpretation to Barbara1(UC‘C’), andso we postpone considering it until our discussion of the U+C first-figure cases in section III.E.2.a.

A modern way of making a distinction between (c) and (c’) invokesthe distinction between what are called de re and de dicto necessity. Tosay that NEC(XaY) is true de dicto is to say that there is some lawlikeconnection between the notion of being a Y and the notion of being anX, so that just knowing that something is a Y is enough to know it is anX. Both (c) and (c’) are true de dicto because there is no such connectionbetween being an animal and being in motion or between the latter andbeing a human; knowing that something is an animal does not sufficeto tell us that it is in motion and knowing that something is in motiondoes not suffice to tell us it is a human. We find the notion of de renecessity hard to grasp, but perhaps the following will do. We mustimagine that individuals have necessary properties, that, for example,Socrates is necessarily a human being and an animal. Socrates hasthose properties no matter how he is described, e.g., as the anathema

14 Introduction

of the politicians. Now we say that NEC(XaY) is true de re if each of theY’s has the property of being necessarily X. If (a’) is true, then each ofthe things in motion is necessarily an animal, even though there is nolawlike connection between being in motion and being an animal. Thus,if (a’) is true, (c’) is in fact false on the de re interpretation. On the otherhand, (c) is true de re because no individual human being is necessarilyin motion.

The issues surrounding the de re/de dicto distinction and the inter-pretation of Aristotle’s modal syllogistic have received a great deal ofdiscussion, which we cannot recapitulate here.20 We shall occasionallyinvoke the distinction in our notes, but on the whole we shall leave itout of account since it does not come to the surface in Alexander’sremarks. In the Appendix on conditional necessity we discuss anotherdistinction which he does sometimes invoke, namely the distinctionbetween what is necessary without qualification and what is necessaryon a condition.

Aristotle’s brief remarks about the validity of Barbara1(NUN) havebeen taken as an expression of the notion of de re necessity. He says:

if A has been taken to hold ... of B by necessity and B just to hold of C ...,A will hold ... of C by necessity. For since A is assumed21 to hold ... of allB by necessity and C is some of the B’s, it is evident that [A will hold] ofC by necessity. (30a17-23)

Alexander’s paraphrase of this passage shows that he takes it to involvean application of the dictum de omni et nullo and hence to be anargument for completeness:

For since A is said of all B by necessity, and C is under B and is some ofB, A is also said of C by necessity. For what is said of all B by necessitywill also be predicated of what is under B by necessity – at least if beingsaid of all is ‘when nothing of the subject can be taken of which thepredicate will not be said’.22 But C is some of the B’s. For being said of allby necessity is taken in the same way ,as he said before in the case of necessary things: ‘For the privativeconverts in the same way, and we will give the same account of “to be inas a whole” and “said of all” ’ . (126,1-8)

For Alexander, then, the validity of Barbara1(NUN) depends on inter-preting NEC(AaB) as saying that no B can be taken of which A does nothold by necessity (to which we might add, ‘no matter how the B isdescribed’). Alexander explicitly refrains from committing himself onthe correctness of Aristotle’s position, but it is clear that he is quiteimpressed by the arguments of Theophrastus and Eudemus,23 who, asAlexander tells us, rejected Barbara1(NUN) in favour of Bar-bara1(NUU), and adopted what Bochenski (1947, p. 79) called ‘la règledu peiorem’ and we will call the peiorem rule, according to which the

Introduction 15

conclusion of a combination can be no stronger than its strongestpremiss.24 Throughout the commentary Alexander signals when a moveof Aristotle’s depends or appears to depend on his acceptance of first-figure NUN syllogisms, a clear indication that he thinks the move isproblematic.25 It may be that his ultimate position is that the notion ofnecessity is ambiguous. Commenting on a passage (1.13, 32b25-32) inwhich Aristotle says that contingency can be taken in two ways, Alex-ander writes:

But if ‘It is contingent that A holds of that of which B is said’ has twomeanings, so will ‘By necessity A holds of that of which B is said’ have twomeanings; for it will mean either ‘A holds by necessity of all of that ofwhich B is said unqualifiedly’ or ‘A holds by necessity of all of that ofwhich B is said by necessity’. But if this is true, it will not be the case that‘A is said of all B by necessity’ is equivalent to ‘A is said by necessity of allof that of which B is said’, as is said by some of those who show that it is true that the conclusion of a necessarymajor and an unqualified minor is necessary.(166,19-25)

Before he gives terms for rejecting Barbara1(UNN), Aristotle offers thefollowing argument against it:

But if the proposition AB is not necessary, but BC is necessary, theconclusion will not be necessary. For, if it is, it will result that A holds ofsome B by necessity – through the first and through the third figure. Butthis is false. But it is possible that B is such that A can hold of none of it.(30a23-8)

After giving terms Aristotle says that the proof that Celarent1(UNN)fails will be the same. Later, having affirmed Darii1(NUN) andFerio1(NUN), Aristotle rejects Darii1(UNN) and Ferio1(UNN):

But if the particular premiss is necessary, the conclusion will not benecessary; for nothing impossible results, just as in the universal syllo-gisms. Similarly in the case of privatives. Terms: motion, animal, white.(30b2-6)

It seems reasonably clear that Alexander is right to interpret Aristotle’sfirst rejection of Barbara1(UNN) as something like the following correctargument:

Assume that AaB and NEC(BaC) yield NEC(AaC). But NEC(AaC) andNEC(BaC) yield (Darapti3(NNN)) NEC(AiB). However, we ought to beable to make AaB true while making NEC(AiB) false. Hence, the assump-tion that Barbara1(UNN) holds is wrong.

We prefer the following paraphrase of this argument:

16 Introduction

Assume, as is possible, that AaB, NEC(AiB), NEC(BaC), and assumethat Barbara1(UNN) is valid. Then NEC(AaC), which with NEC(BaC)implies (Darapti3(NNN)) NEC(AiB), contradicting NEC(AiB). HenceBarbara1(UNN) is not valid.

We shall call such an argument against a rule of inference an incom-patibility rejection argument, meaning an argument which shows thatacceptance of a proposed rule of inference would allow one to derive aninconsistency from a set of compatible premisses, and we shall call anargument against the possibility of an incompatibility rejection argu-ment an incompatibility acceptance argument. In his remarks onDarii1(UN_) and Ferio1(UN_) Aristotle claims that he has incompatibil-ity acceptance arguments for all four first-figure UNU cases as well asincompatibility rejection arguments for the UNN cases. The formerclaim is incorrect in the case of Barbara1(UNU), since – once thecomplete Darii1(NUN) (or Darapti3(UNN)) is available – the argumentwe have given above could be formulated as a rejection of Bar-bara1(UNU).26 On the other hand, the claim is correct for the other threecases. We do the arguments. For:

Celarent3(UNU) AeB NEC(BaC) AeC

the two negative propositions entail nothing, and AeC and NEC(BaC)entail (Felapton3(UNU)) AoB which is certainly not incompatible withAeB.27 For:

Darii1(UNU) AaB NEC(BiC) AiC

and

Ferio1(UNU) AeB NEC(BiC) AoC

the conclusion and either premiss entail nothing. However, in the caseof these two the situation is exactly the same if the conclusion is takento be NEC(AiC) or NEC(AoC), as Alexander points out at 134,32-135,6and 135,12-19. Hence Aristotle cannot give incompatibility rejectionarguments for either Darii1(UNN) or Ferio1(UNN).

At 129,9-22 Alexander more or less shows that there is no incompati-bility rejection argument for Barbara1(NUN). The same is true for theother first-figure NUN cases.28 In commenting on the rejection of Bar-bara1(UNN) (128,3-129,7) and Celarent1(UNN) (130,27-131,4)Alexander contents himself with showing that incompatibility argu-ments work for rejecting these. However, as we have seen, when he getsto Aristotle’s specification of terms, he points out (129,23-130,24) thatvery similar terms would suffice for the rejection of Barbara1(NUN),

Introduction 17

and offers essentially Theophrastean considerations against Aristotle’sposition. He subsequently (131,8-21) tries to explain the differencebetween incompatibility rejection arguments and reductios, and thensays that Aristotle doesn’t seem to be entirely confident about theserejection arguments. This remark might seem out of place, given whatAlexander has said up to this point, but it is not if we realize thecomplications which we have already outlined. Alexander goes on togive his own method (132,5-7), which involves the attempt to produce areductio on the denial of a purported conclusion; if one is produced thepurported conclusion follows, if it isn’t, the purported conclusion doesnot. Application of the method requires Alexander to look ahead notonly to third-figure N+U (and UU) combinations, which is all right sincethese combinations reduce to first-figure ones, but – because the denialof a necessary proposition is a ‘contingent’ one – also to N+C (and U+C)combinations. The method appears to work for accepting Bar-bara1(UNU) and rejecting Barbara1(UNN), but it would commitAristotle to acceptance of Celarent1(UNN).29

Alexander is obviously in difficulty when he gets to Aristotle’s rejec-tion of Darii1(UNN) and Ferio1(UNN), since what Aristotle says orclearly implies is false: we cannot give incompatibility rejection argu-ments for these cases. Essentially Alexander considers variousalternatives without clearly espousing any one of them. We describe thetext, since it offers some difficulty. Alexander considers three alterna-tive interpretations. He first suggests (133,20-9) that Aristotle isintending to apply his method of incompatibility argumentation toDarii1(UNN) and Darii1(NUN). But now he claims that the methodwould not generate a contradiction if applied to Barbara1(NUN). Thisclaim is, of course, false, and in trying to defend it Alexander usesDarapti3(UNU) rather than the stronger Darapti3(UNN) which is ac-cepted by Aristotle.30 In any case, as we have seen, he subsequently(134,32-135,6 and 135,12-19) asserts correctly that Aristotle’s incom-patibility arguments will not work to reject either Darii1(UNN) orFerio1(UNN).

Alexander’s second alternative interpretation of Aristotle’s words(133,29-134,20) is his own method. He shows – more or less – that it willsuffice to confirm Darii1(NUU) but not Darii1(UNN). He does not pointout that it also confirms Darii1(NUN). Nor does he say anything aboutFerio1. In fact his method confirms both Ferio1(UNN) and Ferio1(NUN),hardly a satisfactory result from Aristotle’s point of view.31

Alexander’s third alternative is that Aristotle has in mind concretecounterinterpretations. This has the benefit of putting Aristotle onlogically sound ground, but it is hard to believe that this is what the textmeans.

18 Introduction

III. Modal syllogistic with contingentpropositions (1.13-22)

We have seen that full treatment of Aristotle’s discussion of the conver-sion of necessary propositions requires reference to his treatment ofconversion for contingent propositions. In III.A we say something aboutAlexander’s understanding of the notion of contingency and the rulesfor converting contingent propositions. In III.B we go into more detailon Alexander’s interpretation of the three modal notions, and in III.Cand D we look in more detail at his treatment of conversion for necessityand contingency. Finally, in III.E.1-3 we consider the various combina-tions involving contingent propositions.

III.A. Strict contingency and its transformationrules (1.13, 32a18-32b1)

At the beginning of his discussion of the transformation rules forcontingency in 1.3 Aristotle says that ‘to be contingent is said in manyways, since we say that the necessary and the non-necessary and thepossible are contingent’ (25a37-9). Commenting on this remark, Alex-ander writes:

He (sc. Aristotle) showed us the homonymy of ‘contingent’ in On Interpre-tation too. For we apply ‘It is contingent’ to what is neces-sary when we say that it is contingent that animal holds of every human;and to what holds if we say of what holds of something that it is contin-gent that it holds. Here he indicates what holds with the words ‘the non-necessary’; for what holds differs in this way from what is necessary whilesharing with it the fact of holding at the present time. (Note the expres-sion: what holds contingently is the same as what is signified by anunqualified proposition.) ‘Contingent’ is also applied to what is possible.He will explain what this means a little later on when he says ‘Those which are said to be contingent inasmuch as they hold for the mostpart and by nature – this is the way in which we determine the contin-gent ...’. (37,28-38,10)

We discuss the reference to On Interpretation in Appendix 4 (On Inter-pretation, chapters 12 and 13). At this point what is important is thatAlexander understands Aristotle to hold that we use ‘It is contingentthat’ in three different senses when we apply it to a proposition express-ing a necessary truth, a proposition expressing something which holdsbut is not necessary, and a proposition which expresses a mere possibil-ity. For Alexander it is only the third sense which gives the strictmeaning of contingency, the one which is central to Aristotle’s syllogis-tic. We also wish to signal the curious sentence in parenthesis callingattention to the notion of holding contingently (endekhomenôs). In the

Introduction 19

next sections we shall emphasize occurrences of this word by includingthe transliterated Greek.

In his comment on 25b14, Alexander says:

He set down only this sort of contingency – what holds for the most partand is by nature (for what is by nature is for the most part), since onlythis sort is useful in the employment of syllogisms. The possible alsocovers what holds in equal part and what holds infrequently, but syllo-gisms with material terms of this kind are of no use. (39,19-23)

In other words, holding for the most part is not the defining feature ofcontingency. Aristotle specifies the defining feature toward the begin-ning of chapter 13 when he announces what Alexander calls (on thebasis of 1.14, 33b21-3, 1.15, 33b25-31, and 1.15, 34b27-9) the diorismosof contingency:

I call P contingent or say it is contingent that P if P is not necessary andif, when P is posited to hold, nothing impossible will be because of it. Forwe call what is necessary contingent homonymously. (32a18-21)

It seems reasonably clear that Aristotle intends a biconditional here:

CON(P) iff (i) NEC(P) and (ii) no impossibility follows from P

The only clear and explicit use Aristotle makes of clause (ii) is in hisspecious justifications of certain first-figure UC and NC syllogisms,notably Barbara1(UC_) and Celarent1(UC_).32 Commenting on thediorismos Alexander argues that for Aristotle CON(P) rules out P aswell as NEC(P):

Since he is going to discuss syllogisms from contingent premisses, he firstdefines the contingent. He does not define it in its homonymous use sinceit is not possible to define something as it is used homonymously. Ratherhe isolates contingency as said of the necessary and the unqualified fromthe contingent. For he showed that the contingent is also predicated ofthese things.

By saying ‘when P is posited to hold’ he indicates that, in addition tonot being necessary, the contingent is not unqualified either. For what iscontingent according to the third adjunct33 is of this kind and it differsfrom what is necessary and what is unqualified because if P is said to bepossible (dunasthai), P is not yet (mêdepô) the case. So, P would becontingent in the strict sense if P is not the case and if when P is positedto be the case it has nothing impossible as a consequent. And he wouldhave spoken more strictly about the contingent if he said ‘P is not the caseand when P is posited to hold’. For although what is not the case is notnecessary, what is not necessary is not ipso facto not the case. (156,11-22)34

20 Introduction

Thus we may state ‘Alexander’s diorismos’ as:

CON(P) iff (i) P, and (ii) no impossibility follows from P35

Alexander frequently refers to this strict sense of contingency as contin-gency in the way specified (kata ton diorismon).

At 32a29-35 Aristotle announces rules of transformation for contin-gent propositions:

It results that all contingent propositions convert with one another. I donot mean that the affirmative converts with the negative, but rather thatwhatever has an affirmative form converts with respect to its antithesis,e.g., that ‘It is contingent that X holds’ converts with ‘It is contingent thatX does not hold’, and ‘It is contingent that A holds of all B’ converts with‘It is contingent that A holds of no B’ and with ‘It is contingent that A doesnot hold of all B’, and ‘It is contingent that A holds of some B’ convertswith ‘It is contingent that A does not hold of some B’, and the same wayin the other cases.

If one understands ‘ “It is contingent that X holds” converts with “It iscontingent that X does not hold” ’ to mean that CON(P) is equivalent toCON( P) and applies that understanding to modal syllogistic, theresult, taken in conjunction with other equivalences accepted by Aris-totle, is to make all contingent statements involving two terms A and Bequivalent and so to render syllogistic with contingency more or lessbankrupt. It seems certain that Aristotle does not intend this, and thethought that he might doesn’t even enter Alexander’s head.36 He takesAristotle’s point to apply only to so-called indeterminate propositions,that is, propositions which are ambiguous with respect to quantity.37This means that the relevant transformations for syllogistic are simply:

AE-transformationc:38 CON(AaB) CON(AeB)EA-transformationc: CON(AeB) CON(AaB) IO-transformationc: CON(AiB) CON(AoB)OI-transformationc: CON(AoB) CON(AiB)

Unfortunately, Aristotle does not offer any argument for any of theserules, but simply says,

For since the contingent is not necessary, and what is not necessary may(enkhôrei) not hold, it is evident that, if it is contingent that A holds of B,it is also contingent that it does not hold of B, and if it is contingent thatit holds of all, it is also contingent that it does not hold of all. And similarlyin the case of particular affirmations. (32a36-40)

Alexander does not choose to expand significantly on these remarks,

Introduction 21

telling us only that this position is ‘reasonable’ (eikotôs) given thediorismos of contingency.

When we add to these transformation rules the conversion rulesannounced at 1.3, 25a37-b3:

AI-conversionc: CON(AaB) CON(BiA)II-conversionc: CON(AiB) CON(BiA)

the result is still the equivalence of:

(ia) CON(AaB)(ib) CON(AeB)

and of all of:

(iia) CON(AiB)(iib) CON(AoB)(iic) CON(BiA)(iid) CON(BoA)

as well as the implication of any of (iia)-(iid) by either of (ia) or (ib). Onthe other hand, as we have already mentioned, Aristotle denies EE-con-versionc at 1.17, 36b35-37a31.

The equivalences Aristotle does accept have the effect of generatingwhat we will call waste cases of syllogistic validity. For example, sinceAristotle accepts:

Barbara1(CCC) CON(AaB) CON(BaC) CON(AaC)

the equivalence of (ia) and (ib) would also commit him to

EAA1(CCC) CON(AeB) CON(BaC) CON(AaC)AEA1(CCC) CON(AaB) CON(BeC) CON(AaC)EEA1(CCC) CON(AeB) CON(BeC) CON(AaC)

to give only examples with an a-conclusion. Aristotle’s handling of thewaste cases is not always perspicuous. He mentions some and notothers, and, for example, he chooses to endorse Celarent1(CCC) withoutmentioning EAA1(CCC). For the most part the waste cases are of nointerest, and we shall not worry about them. But in some places,particularly after Aristotle loses sight of – or perhaps interest in – thevarious notions of contingency which he has brought into play, Alexan-der addresses difficulties implicit in determining exactly what wastecase Aristotle is espousing.

22 Introduction

The diorismos of contingency appears to commit Aristotle to thefollowing instances of CON(P) NEC(P) (C N):

(i) CON(AaB) NEC(AaB)(ii) CON(AeB) NEC(AeB)(iii) CON(AiB) NEC(AiB)(iv) CON(AoB) NEC(AoB)

The first two of these propositions are clearly Aristotelian, but the lasttwo cause some difficulty. One can see in a rough way that if sense couldbe made of a de dicto reading of particular propositions these two wouldbe true de dicto, but false de re, since, for example, there might be someanimals, e.g., humans, for which it is contingent that they are white andother animals, e.g., swans for which it is necessary that they are white.We are not confident about Aristotle’s view of (iii) and (iv), but we notethat at 1.14, 33b3-8 (cf. 1.15, 35a20-4) he takes CON(Animal i White)and CON(Animal o White) to be true, whereas at 1.16, 36b3-7 (cf. 1.9,30b5-6) he takes NEC(Animal i White) and NEC(Animal o White) to betrue. The last pair seems reasonable enough on a de re reading, but thefirst pair seems to be false on such a reading.

Whatever Aristotle may have thought about (iii) and (iv), Alexanderis uneasy with violations of them. Thus, when Aristotle takes CON(Animal i White) and CON(Animal o White) as true, Alexander says(171,30-172,5) that a ‘truer’ choice of terms would involve takingCON(White i Walking) and CON(White o Walking) to be true. Thischoice is equally problematic on the intuitive de re reading which liesbehind Alexander’s acceptance of NEC(Animal i White) and NEC(Animal o White), but it allows him to preserve (iii) and (iv).

III.B. Alexander and the temporal interpretation ofmodality: preliminary remarks

At the beginning of chapter 2 Aristotle announces that ‘every proposi-tion says either that something holds or that it holds by necessity orthat it is contingent that it holds’ (25a1- 2). Alexander’s comment on thispassage helps to fill out our understanding of his conception of the threemodalities:

It is necessary to understand the word ‘categorical’ added to the words‘every proposition’, since he is now talking about such propositions andsyllogisms . ... Now in every categoricalproposition one term is predicated of another either affirmatively ornegatively, i.e., as holding or not holding of the subject; and if X holds ofY, it either holds always or holds at some time and doesn’t hold atanother. If what is said to hold holds always and is taken to hold always,the proposition saying this is necessary true affirmative; but a necessary

Introduction 23

negative true proposition is one which takes what by nature never holdsof something as never holding of it. But if X does not always hold of Y, ifit holds at the present moment, the proposition which indicates this is anunqualified true affirmative; and similarly a proposition which says thatwhat does not now hold does not now hold is an unqualified true negative.But if X does not hold of Y at the present time but can (dunamenon) holdof it and is taken in this way – i.e., as being able to hold – the propositionis a true contingent (endekhomenon) affirmative; and a proposition whichsays of what holds or does not hold but can (hoion) both hold and not holdthat it is contingent that it does not hold is a true contingent negative.(25,26-26,14)39

In this passage, as in many others, it is not entirely clear whetherAlexander is speaking about (in our formulations) the assertion that‘Animal a Human’ is a (true) necessary proposition, the assertion thatNEC(Human a Animal) or just the expression ‘NEC(Human a Animal)’.Let us begin by talking about the simple categorical propositions, AaB,AeB, AiB, AoB, which we represent by P. In this paragraph Alexandercommits himself to at least a partial temporal interpretation of neces-sity, contingency, and unqualified holding. Part of the difficulty inconstruing what Alexander has in mind here arises from his attempt todistinguish between affirmative propositions, which we shall temporar-ily represent as XaffY, and negative ones, which we shall represent asXnegY. We can construe Alexander’s account of the modalities asfollows:

XaffY is necessary iff X holds of Y always;XaffY is unqualified iff X holds of Y now but not always;XaffY is contingent iff X does not hold of Y now but can hold of Y.XnegY is necessary iff X never holds of Y;XnegY is unqualified iff X does not hold of Y now (but does hold at

some time);XnegY is contingent iff X can hold of Y and can not hold of Y.

One problem here is the obvious asymmetry between the definitions ofcontingency for affirmative and negative statements. We can see Alex-ander’s difficulty by considering the two possible ways of making thedefinitions symmetrical:

(i) XaffY is contingent iff X does not hold of Y now but can hold of Y; XnegY is contingent iff X holds of Y now, but can not hold of Y. (ii) XaffY is contingent iff X can not hold of Y and can hold of Y. XnegY is contingent iff X can hold of Y and can not hold of Y;

Of these two alternatives (ii) might seem to be preferable since Aristotleis committed to AE-, EA-, IO- and OI-transformationc. However, it is

24 Introduction

relatively certain that Alexander thinks of (ii) as something like afeature of contingency, whereas (i) is closer to a genuine analysis of it.For we have seen that for him the primary account of contingency isgiven by the diorismos, which he takes to imply that what is contingentdoes not hold. For this reason we take (i) instead of (ii) as the relevantaccount of contingency. We can then drop the distinction between affand neg, and write the three accounts as

(Nt) P is necessary iff P is always true;(Ut) P is unqualified iff P is true now and not always true;(C*) P is contingent iff P is not true now, but P can be true.

The assertion that ‘P can be true’ is ultimately of no more help inunpacking the notion of contingency than the assertion that nothingimpossible follows from the assumption that P. In both cases we areusing the notion of possibility to explain the notion of possibility.Unfortunately, Alexander does not seem to have any non-circular wayof explaining what ‘can be true’ means. However, it is useful to have inmind a strictly temporal version of (C*), since Alexander sometimesseems to flirt with the following idea:40

(Ct) P is contingent iff P is not true now, but P will be true at some time.41

It is clear that Nt allows one to give simple justifications of the conver-sion laws for necessary propositions and that Ct allows one to do thesame for not only AI-conversionc and II-conversionc, but also EE-conver-sionc. In order to indicate Alexander’s apparent flirtation with Ct weshall look at his account of Aristotle’s justification of the conversion lawsfor necessary propositions, which as we explained in section II.A, seemto rely on claims about contingency which Aristotle hasn’t proved or –worse yet – ultimately decides are false. However, before doing so, weshould mention that, insofar as Alexander equates contingency withpossibility, he explicitly assigns C* rather than Ct to Aristotle at184,9-11.

III.C. Conversion of necessary propositions (1.3, 25a27-36)

The laws in question are:

EE-conversionn: NEC(AeB) NEC(BeA)AI-conversionn: NEC(AaB) NEC(BiA)II-conversionn: NEC(AiB) NEC(BiA)

We recall Aristotle’s justification of EE-conversionn:

Introduction 25

If it is necessary that A holds of no B, it is necessary that B holds of no A;for if it is contingent that B holds of some, it will be contingent that Aholds of some B. (25a29-32)

Here is Alexander’s comment:

Here again he seems to have used the conversion of particular contingentaffirmative propositions in his proof for necessary universal negativeones, even though he has not yet discussed conversions of contingentpropositions. Or should we rather say this? He holds it to be agreed that(i) particular affirmative contingent propositions are opposite to univer-sal necessary negative ones since they are contradictories, and thereforeassumes this. Having assumed it, then, (ii) since if B holds of some A butnot by necessity, it is said that it is contingent that B holds of some A andthat it holds contingently (endekhomenôs) of some A, and (iii) since he hasproved that particular unqualified propositions convertwith themselves, he makes use of propositions of this kind. Thus he doesaway with the necessity by saying that it is contingent that A holds ofsome B because (iv) what holds of some – when it holds – converts.42 (v)But if it is contingent that B holds of some A, then either it already holdsof A or it is possible (hoion) that it will hold of it at some time. (vi) In thisway what holds of no B by necessity will at some time hold of some of it,which is impossible. (vii) He says a little later when he distinguisheskinds of contingency that what holds but not by necessity is said to becontingent. (viii) For he says that contingency signifies both what isnecessary and also what is not necessary but holds – and he now uses itin application to the latter case. (ix) And what holds contingently (endek-homenôs) of some or will hold of some is the opposite of what holds of noneby necessity. (36,7-25)

We propose the following interpretation of Alexander’s argument:

Aristotle takes for granted that NEC(XeY) is equivalent to ‘It is contin-gent that XiY’ (i). Hence (ii) he assumes NEC(BeA) and infers ‘It iscontingent that (BiA)’ and so (v) either BiA or it is contingent that B willhold of A at some time. But (iv) at the time BiA holds, AiB holds byII-conversionu. But this conflicts with the assumption NEC(AeB) (vi andix). Hence we see that Aristotle uses only II-conversionu. When in hisargument he seems to invoke II-conversionc he is using ‘contingent’ in thesense in which applies to what holds; he could just as well have written‘if B holds of some A, A holds of some B.’ (iii; vii-viii)

Alexander underlines this last point in a subsequent reference back tothis argument:

It is clear from this that in the previous proof too he used ‘It is contingentthat B holds of some A’ in connection with something unqualified; forthere ‘for if it is contingent that B holds of some’ should be understood tomean ‘For if B holds contingently (endekhomenôs) of some A’. (37,17-21;cf. 149,5-7)

26 Introduction

Clearly (vi) and (ix) presuppose Nt, but Alexander’s vocabulary showsthe same wavering between (C*) and (Ct) to which we have alreadycalled attention. There is a perhaps more serious problem raised by (i).Alexander offers no justification for how Aristotle can take this forgranted when he himself holds that CON(XiY) does not follow from NEC(XeY), since NEC(XeY) is compatible with NEC(XaY), whichis incompatible with CON(XiY). Perhaps when Alexander says thatAristotle takes (i) to be something agreed, he means that Aristotle istaking (i) as an endoxon, albeit one which he does not accept.

Alexander’s discussion of AI- and II-conversionn, to which we nowturn, throws some further light on his treatment of EE-conversionn.Alexander’s summary of the argument involves another (to us approxi-mate) use of temporal considerations and the same assertion of theequivalence of NEC(P) and ‘It is contingent that P’.

He proves that particular affirmative necessary propositions convertfrom both universal affirmative necessary and particular affirmativenecessary ones in the same way as he did in the case of privative universalones. For if A holds of all or some B by necessity, but B does not hold ofsome A by necessity, it will be contingent that B hold of no A at some time;for the negation of ‘It is necessary that B holds of some A’ is ‘It is notnecessary that B holds of some A’, which is equivalent to ‘It is contingentthat B holds of no A’, since ‘It is not necessary that B holds of some A’ and‘It is contingent43 that B holds of no A’ are the same. But when B holds ofno A, A will hold of no B (for this has been proved). Hence, A will not holdof all or some B by necessity. (37,3-13)

Insofar as there is anything new in Alexander’s discussion of AI- andII-conversionn, it comes when he tries to defend Aristotle against thecharge of using EE-conversionc:

It is clear that he has not conducted the proof with contingent negativepropositions; for he thinks that they do not convert. Rather he reduces to an unqualified one, subtractingnecessity from it.44 He makes this clear by no longer using the word‘contingent’ but simply saying ‘For if it is not necessary’. For he isassuming that unqualified propositions convert. (37,14-17)

Here Alexander lights on the fact that in the justification of AI- andII-conversionn Aristotle does not say something like ‘if NEC(BiA),then it is contingent that B holds of no A, and so it is contingent that Aholds of no B and so NEC(AiB)’, but simply ‘if NEC(BiA) then NEC(AiB)’.

Introduction 27

III.D. Conversion of contingent propositions

III.D.1 Conversion of affirmative contingent propositions (1.3,25a37-b3); more on Alexander and the temporal interpretation ofmodality

Aristotle argues for AI- and II-conversionc simultaneously. We wish toconsider what he says as an alternative to a simple argumentwhich one might have expected him to use. Suppose CON(YiX).Then ( C N) NEC(YeX). But (EE-conversionn) NEC(XeY), con-tradicting CON(XaY) or CON(XiY). Aristotle avoids such an argumentbecause of the use of C N; he later (1.17, 37a9-31) rejects theanalogous argument for EE- conversionc: assume CON(YeX); then( C ) NEC(YiX), so that (II-conversionn) NEC(XiY), contradict-ing CON(XeY). But Aristotle’s own argument for AI- and II-conversioncis very problematic:

Since to be contingent is said in many ways (since we say that thenecessary and the non-necessary and the possible are contingent) in thecase of contingent propositions, the situation with respect to conversionwill be the same in all cases of affirmative propositions. For if it iscontingent that A holds of all or of some B, then it will be contingent thatB holds of some A. For if of none, then A of no B; this has been provedearlier. (25a37-b3)

Alexander takes for granted that Aristotle’s argument must turn on thethree ways in which contingency is said, and that it will proceedindirectly by moving from:

(i) ‘It is not contingent that B holds of some A’

to:

(ii) a universal negative statement in which B is the predicate andA is the subject

and then to:

(iii) a universal negative statement in which A is the predicate andB is the subject

which contradicts:

(iv) ‘It is contingent that A holds of some B’

28 Introduction

To try to work out an interpretation satisfying these conditions Alexan-der takes it that there are three cases of (i):

(ia) possibility: CON(BiA)(ib) holding: (BiA)(ic) necessity: NEC(BiA)

and three corresponding antecedents of the conditional from which tofind an inconsistency:

(iva) CON(AiB)(ivb) AiB(ivc) NEC(AiB)

Case (b) is easy since (BiA), i.e., BeA, yields (EE-conversionu) AeB,contradicting AiB. Similarly, given C N , which Alexander pre-sumably again takes as ‘agreed’, case (c) reduces to EE-conversionn. Forcase (a) Alexander takes for granted N and gives his moststraightforward temporal argument: if NEC(BiA) then ( N C )CON(BeA), so that (Ct) at some time BeA, so that at that time AeB, soCON(AeB), contradicting NEC(AiB). He does not seem to notice that ifthis argument were correct it would establish EE-conversionc.

Alexander preserves for us something like such an argument ofTheophrastus and Eudemus for a version of EE-conversion for contin-gent propositions, although it too shows an unclear handling oftemporal considerations:

If it is contingent that A holds of no B, it is also contingent that B holdsof no A. For since it is contingent that A holds of no B, when it iscontingent that it holds of none, it is then contingent that A is disjoinedfrom all the things of B. But if this is so, B will then also have beendisjoined from A, and, if this is so, it is also contingent that B holds of noA. (220,12-16)

Alexander defends Aristotle against this argument:

It seems that Aristotle expresses a better view than they do when he saysthat a universal negative which is contingent in the way specified doesnot convert with itself. For if X is disjoined from Y it is not therebycontingently (endekhomenôs) disjoined from it. Consequently it is notsufficient to show that when it is contingent that A is disjoined from B,then B is also disjoined from A; in addition that B iscontingently disjoined from A. But if this is not shown, then it has notbeen shown that a contingent proposition converts, since what is sepa-rated from something by necessity is also disjoined from it, but notcontingently. (220,16-23; cf. 221,1-2)

Introduction 29

Alexander here seems to concede that if CON(AeB), then at some timeAeB and therefore BeA. But he insists that one cannot infer CON(BeA)because one doesn’t know that BeA holds contingently if we haveinferred BeA from AeB, where AeB holds contingently.45 It seems clearthat Alexander is invoking a distinction between the ways in whichthings hold. We cannot infer CON(P) from P unless we know that Pholds contingently.

Alexander uses the words ‘necessarily’ (anankaiôs) and ‘unquali-fiedly’ (huparkhontôs) as well as ‘contingently’ in the commentary.46Although Aristotle never uses any of these words in a logical context,they are also found in the other commentaries on his logical works. Forthe most part they are simply variants of expressions such as ‘It iscontingent that’, but we are convinced that Alexander wishes to putspecial weight on the ideas of holding contingently and of holding butnot holding necessarily. By insisting on the latter notion Alexander isable to maintain the position that unqualified propositions for Aristotledo not signify holding necessarily or eternally. But he has much moredifficulty with what the difference is between a contingent and anunqualified proposition. Indeed, his assertion at 38,5-7 that holdingcontingently correlates with ‘what is signified by an unqualified propo-sition’ is probably intended to justify the application of II-conversionuwhich Alexander detects in Aristotle’s justification of AI-conversionc.Similarly in his account of the justification of EE-conversionn Alexanderwants to stress that NEC(BeA) implies that BiA holds contingentlyto justify the alleged application of the same rule. If Alexander werewilling to use the temporal reading of the modal operators straightfor-wardly, he would have no difficulty, but, as we have seen, he insteadmixes the temporal reading with the idea of something holding contin-gently. But using that idea depends on blurring the distinction betweenwhat holds now and what holds at some time. To put this point anotherway, for Alexander’s reasoning to work, one has to assume that Aris-totle proves II-conversionu not just for propositions which hold now, butfor propositions which hold at some time. But, on the temporal readingof the modalities, that is to say that II-conversionu is or includesII-conversionc.

Although Alexander makes no such claim, it seems to us that hishandling of the modal conversion rules more or less commits him tosome such idea. Moreover the lumping together of unqualified andcontingent propositions is quite in keeping with Aristotle’s use of falsebut possible truths, e.g., ‘All animals are moving’ to interpret un-qualified propositions, and with his willingness to use the sameterms to verify corresponding contingent and unqualified proposi-tions.47 As Alexander explains in connection with the proposition ‘Nohorse is white’:

30 Introduction

For if someone requires that we take as universal what holds always butnot what holds at some time, he will be requiring nothing else than thatthe unqualified be necessary, since the necessary does always hold.Furthermore, he himself, when he is considering an unqualified proposi-tion with respect to terms does not ever consider it with respect to termsof this kind. (232,32-6; cf. 130,23-4)

If Alexander adhered to a strict temporal interpretation of contingencywhat he says here would implicitly commit him to the identification ofunqualified truths with propositions true at some time, that is, withcontingent propositions. He, of course, never makes this identification.If he had, he might have seen problems which face any interpreter tryingto understand why Aristotle accepts certain U+C combinations whilerejecting their CC analogues. There are many reasons why Alexandernever offers a strict temporal interpretation. Perhaps the most impor-tant is that for him the meaning of contingency is determined by thediorismos, not by any temporal account.

III.D.2. Non-convertibility of negative contingent propositions(1.3, 25b3-19; 1.17, 36b35-37a32)

Aristotle’s denial of EE-conversionc is controversial. Alexander’s discus-sion of it is dense, but is largely a scholastic defence of Aristotle’sposition. We will mention a few points in it, but we will mainly contentourselves with describing Aristotle’s text.

In the case of negative propositions, it is not the same. With those whichare said to be contingent inasmuch as they do not hold by necessity or theyhold but not by necessity, the case is similar, e.g., if someone were to saythat it is contingent that what is human is not a horse or that white holdsof no cloak. For of these examples the former does not hold by necessity,and it is not necessary that the latter hold – and the proposition convertsin the same way; for, if it is contingent that horse holds of no human, itwill be possible (enkhôrei) that human holds of no horse, and if it ispossible that white holds of no cloak, it is possible that cloak holds ofnothing white – for if it is necessary that it holds of some, then white willalso hold of some cloak by necessity (for this was proved earlier). Andsimilarly in the case of particular negatives. (25b3-14)48

Alexander understands Aristotle to be dealing here with the situationin which an unqualified or necessary proposition is said to be contingent,and to be conceding that EE-conversion does hold in those cases.According to Alexander, Aristotle illustrates necessity with the propo-sition ‘It is contingent that horse holds of no human’ and unqualifiedholding with ‘It is possible that white holds of no cloak’. Aristotle’sargument that the latter converts seems to be a straightforward indirectargument moving from ‘It is not possible that cloak holds of nothing

Introduction 31

white’ to ( C ) NEC(Cloak i White) to (II- conversionn)NEC(White i Cloak). Alexander insists on reparsing what Aristotle saysto make it fit the case of contingency as holding:

He says ‘for if it is necessary that it holds of some, then white will alsohold of some cloak by necessity’ since a particular affirmative necessaryproposition must be the opposite of a contingent universal negative one,and the unqualified proposition was assumed as contingent in its verbalformulation. And the verbal opposite will contain necessity, althoughwhat is signified by it will be particular affirmative unqualified. For thisis the opposite of a universal negative unqualified proposition. (39,4-11,our italics)

That is to say, according to Alexander, Aristotle uses the vocabulary ofnecessity although he expects us to understand that he is talking aboutunqualified propositions. Aristotle has little to say about the third case.He remarks that EE-conversionc fails and OO-conversionc works, butdefers discussion until chapter 17:

But those things which are said to be contingent inasmuch as they are forthe most part and by nature – and this is the way we specify contingency– will not be similar in the case of negative conversions. Rather auniversal negative proposition does not convert, and the particular doesconvert. This will be evident when we discuss contingency. (25b14-19)49

Aristotle’s actual argument for rejecting EE-conversionc is confusing fora number of reasons, one of which is his tacit reliance on the equivalenceof CON(XaY) and CON(XeY). He begins the rejection, which is what wehave called an incompatibility rejection argument, as follows:

It should first be shown that a privative contingent proposition does notconvert; that is, if it is contingent that A holds of no B, it is not necessarythat it is also contingent that B holds of no A. For let this be assumed andlet it be contingent that B holds of no A. Then, since contingent affirma-tions convert with negations – both contraries and opposites – and it iscontingent that B holds of no A, it is evident that it will also be contingentthat B holds of all A. But this is false. For it is not the case that if it iscontingent that X holds of all Y, it is necessary that it be contingent thatY holds of all X. So the privative does not convert. (36b35-37a3)

Here Aristotle takes for granted the equivalence of CON(XeY) andCON(XaY) and the compatibility of CON(XaY) (or equivalentlyCON(XeY)) and CON(YaX). We may represent his argument asfollows. Assume that EE-conversionc holds and that CON(AeB) (orequivalently CON(AaB)) and, what is possible, CON(BaA). Then(EE-conversionc) CON(BeA) and (EA-transformationc) CON(BaA), con-tradicting CON(BaA). Therefore EE-conversionc cannot be correct.50

Aristotle goes on to give terms for rejecting EE-conversionc:51

32 Introduction

Furthermore nothing prevents it being contingent that A holds of no B,although B does not hold of some A by necessity. For example, it iscontingent that white does not hold of any human being – for it is alsocontingent that it holds of every human being –, but it is not true to saythat it is contingent that human holds of nothing white. For it does nothold of many white things by necessity, but what is necessary is notcontingent. (37a4-9)

Using our symbols we represent what Aristotle says as follows: further-more, in some cases, CON(AeB) and NEC(BoA) (i.e., NEC (BaA)). Forexample, CON(White e Human), since CON(White a Human), but CON(Human e White), since NEC(Human o White) (since, e.g.,swans are not human by necessity) and nothing necessary is contingent.

We turn now to perhaps the most difficult part of Aristotle’s rejectionof EE-conversionc, his rejection of the following indirect argument for it:

Suppose CON(AeB) and CON(BeA). Then NEC (BeA), i.e.,NEC(BiA). But then (II-conversionn) NEC(AiB), contradictingCON(AeB).

Aristotle rejects the transition from CON(BeA) to NEC (BeA) or,equivalently, NEC(BiA). Underlying his rejection is the idea that, evenif NEC(BiA), one might have CON(BeA) because NEC(BoA). Thatis, although it is true that:

(NCe) NEC(BiA) v NEC(BoA) CON(BeA)

it is not true that:

* CON(BeA) NEC(BiA)

since one might have NEC(BoA) and NEC(BiA). For this discussionit is also useful to have the analogue of (NCe) for a-propositions:

(NCa) NEC(BiA) v NEC(BoA) CON(BaA)

What does not emerge clearly from Aristotle’s text is whether or not heaccepts the converses of (NCe) and (NCa), that is

( CeN) CON(BeA) NEC(BiA) v NEC(BoA)( CaN) CON(BaA) NEC(BiA) v NEC(BoA)

We discuss Alexander’s view of these two propositions in Appendix 5 onweak two-sided Theophrastean contingency. We now look at Aristotle’srejection of *. It begins at 37a14:

Introduction 33

It is not the case that if it is not contingent that B holds of no A, it isnecessary that B holds of some A. For ‘It is not contingent that B holds ofno A’ is said in two ways; it is said if B holds of some A by necessity andif it does not hold of some by necessity. (37a14-17)

We take Aristotle to here be asserting (NCe) and not ( CeN). He goeson to assert a consequence of (NCe) and its analogue for (NCa):

For if B does not hold of some A by necessity, it is not true to say that itis contingent that it does not hold of all, just as if B does hold of some Aby necessity, it is not true to say that it is contingent that it holds of all.(37a17-20)

That is,

(NCe’) NEC(BoA) CON(BeA)(NCa’) NEC(BiA) CON(BaA)

He now goes on to deny the analogue of * for a- and o- propositions, andinsist that we might have CON(BaA), NEC(BoA) and NEC(BiA):

So, if someone were to maintain that, since it is not contingent that Bholds of all A,52 it does not hold of some by necessity, he would take thingsfalsely. For it holds of all,53 but we say that it is not contingent that itholds of all because it holds of certain of them by necessity. (37a20-4)

Aristotle now says:

Consequently both ‘X holds of some Y by necessity’ and ‘X does not holdof some Y by necessity’ are opposite to ‘It is contingent that X holds of allY’. And similarly in the case of ‘It is contingent that X holds of no Y’.(37a24-6)

Clearly Aristotle is asserting the same thing about CON(BaA) andCON(BeA). What is not clear is whether he is simply asserting (NCa)and (NCe) or also ( CaN) and ( CeN). That he intends to make thestronger assertion is suggested by what he goes on to say about thealleged indirect proof of EE-conversionc:

It is clear then that with respect to things which are contingent and notcontingent in the way which we have specified initially it is necessary t